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Factors Influencing Success in Online High School Algebra

Permanent Link: http://ufdc.ufl.edu/UFE0041943/00001

Material Information

Title: Factors Influencing Success in Online High School Algebra
Physical Description: 1 online resource (195 p.)
Language: english
Creator: Liu, Feng
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Teaching and Learning -- Dissertations, Academic -- UF
Genre: Curriculum and Instruction (ISC) thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: At present, an increasing number of students at the K-12 level in the U.S. are taking courses online via virtual schools, which have been in existence since the end of the 20th century. Virtual schooling is becoming a mainstream option alongside traditional face-to-face learning environments. Even with its increasing popularity, very few empirical studies have been conducted to provide practical guidance for teaching, learning, research, and policy making in K-12 virtual schooling. Some leading virtual school organizations, such as the Southern Regional Educational Board and the International Association for K-12 Online Learning, have produced standards in these fields. However, many of the standards lack empirical support based on research conducted in virtual learning environments. Math has been identified as a very important force to push a society forward since it is considered a foundational subject. Many countries emphasize the improvement of math knowledge and they develop policies to attract more people to the field. The examination of success factors in the math field in general and Algebra in specific in virtual learning environments can provide better implementation strategies in virtual schools to improve student math and science achievement and increase the Science, Technology, Engineering, and Mathematics (STEM) workforce in U.S. The purpose of this study is to examine the factors including LMS utilization, teacher comment/feedback and student demographic information that can influence the success of Algebra courses in K-12 virtual learning environments. Students who completed Algebra and took the end-of-course (EOC) test and students who took one state standardized mathematics test at grade 6-8 level in a state virtual school in the Midwestern U.S region during 2008-2009 participated in this study. Student scores on these tests were collected by this virtual school. Hierarchical linear modeling (HLM) technique was used for data analysis to account for the influence of school characteristics on student scores. The results show these factors have different influences on student performance on the state standardized mathematics test and the Algebra EOC test. These findings have implications for quality online teaching, instructional design, and the policy-making process in virtual learning environments. Further research is proposed based on the results and limitations of this study.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Feng Liu.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Cavanaugh, Catherine S.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041943:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041943/00001

Material Information

Title: Factors Influencing Success in Online High School Algebra
Physical Description: 1 online resource (195 p.)
Language: english
Creator: Liu, Feng
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Teaching and Learning -- Dissertations, Academic -- UF
Genre: Curriculum and Instruction (ISC) thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: At present, an increasing number of students at the K-12 level in the U.S. are taking courses online via virtual schools, which have been in existence since the end of the 20th century. Virtual schooling is becoming a mainstream option alongside traditional face-to-face learning environments. Even with its increasing popularity, very few empirical studies have been conducted to provide practical guidance for teaching, learning, research, and policy making in K-12 virtual schooling. Some leading virtual school organizations, such as the Southern Regional Educational Board and the International Association for K-12 Online Learning, have produced standards in these fields. However, many of the standards lack empirical support based on research conducted in virtual learning environments. Math has been identified as a very important force to push a society forward since it is considered a foundational subject. Many countries emphasize the improvement of math knowledge and they develop policies to attract more people to the field. The examination of success factors in the math field in general and Algebra in specific in virtual learning environments can provide better implementation strategies in virtual schools to improve student math and science achievement and increase the Science, Technology, Engineering, and Mathematics (STEM) workforce in U.S. The purpose of this study is to examine the factors including LMS utilization, teacher comment/feedback and student demographic information that can influence the success of Algebra courses in K-12 virtual learning environments. Students who completed Algebra and took the end-of-course (EOC) test and students who took one state standardized mathematics test at grade 6-8 level in a state virtual school in the Midwestern U.S region during 2008-2009 participated in this study. Student scores on these tests were collected by this virtual school. Hierarchical linear modeling (HLM) technique was used for data analysis to account for the influence of school characteristics on student scores. The results show these factors have different influences on student performance on the state standardized mathematics test and the Algebra EOC test. These findings have implications for quality online teaching, instructional design, and the policy-making process in virtual learning environments. Further research is proposed based on the results and limitations of this study.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Feng Liu.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Cavanaugh, Catherine S.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041943:00001


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FACTORS INFLUENCING SUCCESS IN ONLINE HIGH SCHOOL ALGEBRA


By

FENG LIU















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010

































2010 Feng Liu
































To my family









ACKNOWLEDGMENTS

I offer a special thank you to my family my parents, sister, and brother their

unconditional love has brought me to where I am now. I would not have finished this

long journey without their support. I want to give a special thank you to my mom, who

has always been there to encourage me to move forward especially when I was feeling

down. My whole family provides the momentum for me to conquer all the difficulties that

I have been through during this long, and sometimes lonely, journey.

I thank Dr. Cathy Cavanaugh, my major advisor and committee chair. As my

primary mentor, she inspired me with her dedication to the research and practice in

virtual schooling and her motivating work spirit. As a leading researcher in K-12 virtual

schooling, she gave me confidence and provided me invaluable guidance to position

myself in this field. It is such an honor to be a mentee under her direction.

I thank Dr. Kara Dawson, Dr. James Algina, and Dr. Tim Jacobbe, my committee

member, for their help and support during this journey. I thank Dr. Dawson for

introducing me to the educational technology program, the ever-changing and exciting

field. She is my role model to balance life between family and the professional world. I

thank Dr. Algina, for his guidance on my minor: research, evaluation, and methodology.

His strictness in educational research truly helped me build the foundation to conduct

rigorous research in the field of education. I thank Dr. Jacobbe for his patience and

insightful questions that inspire me to advance my knowledge in math education.

In addition to my committee, I would like to thank my cohort and support group of

doctoral students who helped me go through this long process. I thank them for their

encouragement and support and all the inspiring stories they shared with me. All these

people have made my life much easier during the process.









Finally, I thank all the faculty and staff in the College of Education at University of

Florida who have shaped me in many ways and helped me reach this significant

milestone. I thank the University of Florida Graduate Alumni Fellowship Program, which

provided me with generous funding for four years. I would not have finished this journey

without its assistance.









TABLE OF CONTENTS

page

A C K N O W LE D G M E N T S ................................................................................. .. .... 4

LIST O F TA B LE S ...................................................................................... 9

LIST O F FIG URES........................................... ............... 10

A B S T R A C T ...................................................... 1 1

CHAPTER

1 IN T R O D U C T IO N ......................................................................................... .. ........ 13

B ackgro und .............................. .............. ...... 13
Problem Statem ent .................................... ......................... 14
Purpose Statem ent .................................... ......................... 16
R research Q questions ...................................................................... 16
S significance of the S tudy ....................................................................................... 17
D e lim stations ................................................................................................ ........ 18
Definition of the Terms.................................. ............... 19
O organization of the Study ......................................................... .. ....... 20

2 REVIEW O F LITERATURE ............................................................................. ..... 23

Intro d uctio n ...................................... ...................... ... ... .... .. .......... ................. 2 3
Research in Online/Distance Education and Significance of This Study ............. 25
Review of Literature .............................................. ...... ......... 27
Effectiveness of Online/Distance Education .................................... 27
Algebra/Mathematics Education ................................................................. 35
Learner characteristics variables ............................................................... 35
Learning environment variables ...... ............................................. 39
Algebra teaching and learning ................ ............................ 41
Success Factors in Online Learning ......................................................... 44
Teacher comments/teacher-student interaction ....................................... 46
Participation in online academy ic activities ................................................ 49
Race/ethnicity ..................... ............................................... 49
Participation in school free lunch/fam ily SES ............................................. 50
Learning ability/presence of individual educational plan .......................... 51
S c h o o l ty p e .............................................................. ............... 5 4
Conclusion............... ............................................................... 56

3 M ETH O D O LO G Y .............................................................. .................... 59

In tro d u ctio n ..................................... ................................................ .. 5 9
R e s e a rc h D e s ig n ............................................................................... ..... ........ 5 9


6









Participants and Data Collection.................. .......... ... .................... 60
Instrum ent..................... ................ ..... .............. 61
Algebra EOC Test ...... ...................... ................. 62
State Standardized Test ....... ............................... .... ............... 63
Data Analysis ........... .................................... 64
Lim stations ....................................... ................................. ............................ 66

4 R E S U LT S .......................... ................................ ............................ 68

Introduction ................... ........................... 68
S a m p le ........................................................................................ ........... 6 8
EOC Tests Taker .............................................................................. ........ ......... 68
State Standardized Test Taker........................................... .................... 69
RA Model ......... ......... .... .................. ............... 70
Coefficients for the Variables ............... .... ...................... 71
EOC Test............... ......... .......... .......... 72
State Standardized Mathematics Test................................... .... ........... 72
Descriptive Statistics, Standardized Coefficient, and Reduction of Variance.......... 72
Research Question 1 ........... ............................. 73
EOC Test ............... ... ................ ......... ............... 74
State Standardized Test ..................... ............................ 75
R e s e a rc h Q u e stio n 2 .............................................................................................. 7 5
EO C Test ......................................... ........................................ 77
State Standardized Test ..................... ............................ 77
Research Question 3 ............... ......... .......... ........... 77
EOC Test ............... ... ................ ......... ............... 79
State Standardized Test ..... .............................................................. ...... 80
Summary of Findings ................. ......... ......... .............................. 82

5 DISCUSSION AND IMPLICATIONS................................. 96

Introduction ................... ........................... 96
Summary of Study ................ ......... ......... ......... 96
Overview of the Problem .................... ........................ ....... ........ 96
Purpose Statement and Research Questions .......... ........ ....... .......... 97
Review of the Methodology .................. .............. .......... 98
F ind ings .. ............................................ .. ....... ..... ......................... 99
Research Question 1 ....... .................. .................. 100
R research Q question 2 .......... ........ ................... ............... .............. 104
Research Question 3 ......................... ... .. .. ........ ................ 107
Broad Implications for Online Course Design and Online Teaching .................... 111
C o n c lu s io n s .............. ..... ............ ............................... ........................................ 1 1 3

APPENDIX

A ALGEBRA I MULTIPLE CHOICE RELEASED SAMPLES................................. 121









B ALGEBRA I PERFORMANCE EVENT RELEASED SAMPLES ........................ 142

C STATE ALGEBRA STANDARDS ................................................ 145

D NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS MATHEMATICS
STANDARDS FOR GRADES 6-8................................................. 147

E MAP GRADE 6 RELEASED ITEMS SPRING 06........................................... 153

F STATE STANDARDS FOR MATHEMATICS AT GRADE LEVEL 6..................... 167

LIST OF REFERENCES ........... ..... ............ ........................... ............... 170

BIOGRAPHICAL SKETCH ........................... ............................ 195









LIST OF TABLES


Table page

3-1 Coding of the independent variables .............. ..... ... .......... ...... ..... 67

4-1 EOC test takers demographics....................... ..... ........................... 85

4-2 Standardized test takers demographics .............. ..... ................... .... 86

4-3 Overview of RA model for different datasets ...... ..... ..................................... 87

4-4 Least-squares estimates of fixed effects (with robust standard errors)............ 88

4-5 Least-squares estimates of fixed effects (with robust standard errors)............ 89

4-6 Ordinary Least-squares estimates of fixed effects .................... ........ ........... 90

4-7 Descriptive statistics for EO C test takers................................. ..................... 91

4-8 Descriptive statistics for standardized test takers ..................... ....... ............ 92

4-9 Standardized coefficients for EOC test takers...... ..... .................................... 93

4-10 Standardized coefficients for standardized test takers ............... ........... 94

4-11 A adjusted R -squares .......... ......... .......... .............. ............... .............. 95

5-1 Significance and Direction of the Effect of Factors.................................. 117

5-2 Alignment with National Standards in Quality Online Course........................ 118

A-1 Number and Operations Standard for Grades 6-8 Expectations ..... ........... 147

A-2 Geometry Standard for Grades 6-8 Expectations................................... 149

A-3 Measurement Standard for Grades 6-8 Expectations............................. 150

A-4 Data Analysis and Probability Standard for Grades 6-8 Expectations......... 151

A-5 Problem Solving Standard for Grades 6-8 ...... ..... ...................................... 152

A-6 Reasoning and Proof Standard for Grades 6-8 ........................................... 152

A-7 Communication Standard for Grades 6-8 ............................................... 152

A-8 Connections Standard for Grades 6-8 .................................................. 152

A-9 Representation Standard for Grades 6-8 ............ ................................... 152









LIST OF FIGURES


Figure page

1-1 Online Students at K-12 Level ............... ................................. 22









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

FACTORS INFLUENCING SUCCESS IN ONLINE HIGH SCHOOL ALGEBRA


By

Feng Liu

August 2010

Chair: Cathy Cavanaugh
Major: Curriculum and Instruction

At present, an increasing number of students at the K-12 level in the U.S. are

taking courses online via virtual schools, which have been in existence since the end of

the 20th century. Virtual schooling is becoming a mainstream option alongside traditional

face-to-face learning environments. Even with its increasing popularity, very few

empirical studies have been conducted to provide practical guidance for teaching,

learning, research, and policy making in K-12 virtual schooling. Some leading virtual

school organizations, such as the Southern Regional Educational Board and the

International Association for K-12 Online Learning, have produced standards in these

fields. However, many of the standards lack empirical support based on research

conducted in virtual learning environments.

Math has been identified as a very important force to push a society forward

since it is considered a foundational subject. Many countries emphasize the

improvement of math knowledge and they develop policies to attract more people to the

field. The examination of success factors in the math field in general and Algebra in

specific in virtual learning environments can provide better implementation strategies in









virtual schools to improve student math and science achievement and increase the

Science, Technology, Engineering, and Mathematics (STEM) workforce in U.S.

The purpose of this study is to examine the factors including LMS utilization,

teacher comment/feedback and student demographic information that can influence the

success of Algebra courses in K-12 virtual learning environments. Students who

completed Algebra and took the end-of-course (EOC) test and students who took one

state standardized mathematics test at grade 6-8 level in a state virtual school in the

Midwestern U.S region during 2008-2009 participated in this study. Student scores on

these tests were collected by this virtual school. Hierarchical linear modeling (HLM)

technique was used for data analysis to account for the influence of school

characteristics on student scores. The results show these factors have different

influences on student performance on the state standardized mathematics test and the

Algebra EOC test. These findings have implications for quality online teaching,

instructional design, and the policy-making process in virtual learning environments.

Further research is proposed based on the results and limitations of this study.









CHAPTER 1
INTRODUCTION

Background

The United States has experienced an extraordinary growth in online education at

the K-12 level since its emergence in the late 1990s: from single online course offerings

to large virtual schools today. Thousands of students were attracted to online education

because of the advantages it brings such as flexible and longer school time, more

educational opportunities, and increased access to resources (Cavanaugh, et al., 2004).

Several surveys have showed that at least one third of high school students had online

learning experience (Allen & Seaman, 2006; Setzer & Lewis, 2005). Figure 1-1 shows

the dramatic increase of K-12 online enrollment between 2001 and 2008 (Clark 2001;

Glass, 2009; Newman, Stein, & Trask, 2003; Peak Group 2002; Picciano & Seaman,

2009; Picciano & Seaman, 2007; Setzer & Lewis, 2005; Tucker 2007; Zandberg, Lewis,

& Greene, 2008). By 2016, this number is anticipated to reach 5-6 million and will keep

growing in the future (Picciano & Seaman, 2009). Only public school students were

included in this figure; the number will be higher if all other students are included, such

as those in private schools and home-schools (Picciano & Seaman, 2009).

The virtual school movement in the US is the outgrowth of independent study high

schools in many ways (Clark 2003). The first two virtual schools in the US, the Virtual

High School (VHS) and Florida Virtual School (FLVS), were both created in 1997

(Barbour & Reeves, 2009). By 2001, about 14 states had established state-wide virtual

schools and 40,000 to 50,000 students enrolled in courses offered by these schools

(Clark, 2001). In July 2005, 21 states offered statewide online programs (Watson &

Kalmon, 2005). By September 2006, 38 states have either state-led online learning









programs or online education policies, and 24 states have state-led online learning

programs (Watson & Ryan, 2006). As of September 2007, 42 states offered significant

hybrid learning programs (students in physical schools taking some courses online),

pure online programs (students in physical schools or at home taking most or all of

course online) or both (Watson & Ryan 2007). By fall, 2008, 44 states have online

offering for students and 34 states offer state-led online programs or initiatives (Watson,

Gemin, & Ryan, 2008). Response to a survey administered during 2007-08 showed

about 75% of public K-12 school districts were offering full or partial online courses

(Glass, 2009; Picciano & Seaman, 2009), approximately 10% increase since 2005-06

academic year (Picciano & Seaman, 2009). Additionally, another 15% were planning to

have online offerings within the next three years (Picciano & Seaman, 2009). Currently

all states offer online courses at school or district level (Cavanaugh 2007). Online

education is not seen as separate entity any more but one kind of educational approach

which can strengthen the public education and benefit the society at large (Watson &

Ryan, 2006). There is a need for deeper understanding of the success in virtual learning

environments for the better utilization of this education format to help improve student

academic achievement.

Problem Statement

Along with the extraordinary growth of online education in the US., some research

has been conducted to examine success factors in online learning environments.

According to Roblyer et al., (2008), there are two lines of research emerged to address

success factors in online learning: studies focusing on learner characteristics and

studies focusing on learning environment characteristics. Learner characteristics include

student cognitive factors such as locus of control and learning styles; prior technology









skills and attitudes; and experience and prior knowledge about course content. Learning

environment characteristics include technology support, course content area, and

accessibility to Internet. At present, no clear set of characteristics have been identified

to predict success in virtual learning environments, and no conclusive model has been

created to apply in online learning practice (Roblyer & Davis, 2008; Tallent-Runnels et

al., 2006). There is a gap regarding the establishment of one online success model to

help improve student academic achievement considering the quick development of

virtual schooling in the US. Learner characteristic variables including personal

effort/participation in academic activities, student learning ability/whether has individual

educational plan, race/ethnicity, and family background/participation in free or reduced

lunch programs, and learning environment variables including teacher comment

/instructor-student interaction and school type (private or public school) have been

proved in some studies to correlate to student academic achievement. However, these

variables have not been investigated systematically in one model regarding their effects

on student success in K-12 virtual learning environments.

Math knowledge is important for a citizen to fully participate in society. Math is the

most widely used subject among all the fields and almost every career uses math at

different levels (Saint Paul Public Schools, 2007). During the May 2003 commencement

address, the president of Society for Industrial and Applied Mathematics (SIAM),

Professor Doug Arnold mentioned math is the foundation to understand the world

around us and math knowledge can influence other sciences as well such as

economics, business, and sociology. He predicted that math will have huge impacts in

the 21st century, the digitalized and data-enriched century.









Math has been considered a very important force to push a society forward. Many

countries emphasize the improvement of math knowledge and they develop policies to

attract more people into this field. The underachievement of students in math subjects

at the K-12 level could lead to the lack of preparation for students to pursue advanced

degrees in these fields. This will cause a shortage in the workforce in math and other

sciences fields, which, in turn, could weaken the momentum for a country to move

forward in many aspects. The National Association of Manufacturers (NAM) believed

the shortage of workforce in Science, Technology, Engineering, and Mathematics

(STEM) fields can weaken manufacturers' abilities to ensure quality, productivity

efficiency, and customers' satisfaction (D'Amico, 2008). The quality of Algebra courses

is essential in building the number of students who are ready for advanced degrees in

STEM and career success in these fields. Growing number of students take math

courses online at K-12 level so there is a need to examine the quality of online math

courses and build one online success model to help improve student academic

achievement in general and Algebra/mathematics in specific. The present study was

designed to fill this gap.

Purpose Statement

Based on the lack of models for predicting success in high school Algebra courses

and the clear need for increased Algebra achievement, this study examines the problem

of identifying factors that influence online high school Algebra performance.

Research Questions

The research questions in this study are:

* Does the level of LMS utilization influence Algebra/mathematics performance in
online education?









* Does teacher comment or feedback influence Algebra/mathematics performance in
online education?

* Do student demographic information such as race/ethnicity, grade level, status in
virtual school, whether have individual educational plan (IEP), and participation in
free/reduced lunch programs influence Algebra/mathematics performance in online
education?

Significance of the Study

Even after more than 10 years of extraordinary growth in K-12 online learning, little

research has been done as compared to post-secondary education (Cavanaugh 2007;

Cooze & Barbour, 2005; Means et al., 2009; Picciano & Seaman, 2007; Picciano &

Seaman, 2009; Ronsisvalle & Watkins, 2005). The amount of evidence-based research

or empirical study applicable to guide educators' instruction and policy makers' decision

relevancies is slight (O'Dwyer, Carey, & Kleiman 2007). The dearth of studies on

academic achievement in K-12 virtual learning environments in comparison with that in

traditional learning environments (Cooze & Barbour, 2005; Means et al., 2009; Picciano

& Seaman, 2007; Picciano & Seaman, 2009; Smith, Clark, & Blomeyer, 2005; Watson,

2007) form the rationale for this study. This study can help discover certain

characteristics and good practices in online learning and incorporate them into the

instructional model of the K-12 virtual learning environment. This study could add to the

knowledge of effectiveness of online/distance education in helping improve student

academic achievement in the K-12 virtual learning environment. This will provide

valuable guidance for the better implementation and practice of K-12 virtual schooling.

Given the dearth of research on the factors of academic success in K-12 virtual

learning environments, this study could be beneficial to educators, course designers,

researchers, online program leaders, policy makers, and society at large. The

investigation of success factors in this study will provide a deeper understanding of









success in online learning in general and in the K-12 virtual learning environment in

specific. It has implications for the decision making process for virtual schools with

respect to the development of more efficient online courses in general and online

mathematics courses in specific. It also could help identify the success factors that

should be considered in the virtual learning process and guide management of virtual

schools to maximize their effectiveness to provide better assistance and supplement to

the traditional learning environment.

Delimitations

The study was conducted from Oct 2009 to June 2010. One state virtual school in

the Midwestern US region was chosen as the location in which the data were collected.

This virtual school has a big student population and large Algebra course enrollment

from which the researcher can draw the sample. This virtual school also has a

comprehensive data system enabling the use of advanced statistic model during data

analysis. This virtual school can represent the virtual school as a whole in US in many

respects such as the design of courses according to state and national standards, the

utilization of one single LMS to deliver the course materials, and the flexible timeline for

students to finish courses. However, this virtual also has its own characteristics that

may not be common in other virtual schools such as it recruits both full-time and part-

time students and it moves paced courses all along. Students statewide from bricks-

and-mortar public and private schools as well as home school students were eligible to

enroll in this virtual school. Students in this virtual school who completed Algebra

courses during 2008-09 and took the EOC test and students who took one state

standardized mathematics test at grade level 6-8 during 2008-2009 participated in this

study.









Definition of the Terms

Distance education has been practiced in various forms since its emergence in the

early 1900s, evolving from correspondence to broadcasting including radio and

television, to online education today (Moore & Kearsley, 1996). It has experienced an

extraordinary development in the 20th century and its practice will continue to grow in

the 21st century. Many distance education related terms have appeared: cybershool,

distance education, distance learning, e-learning, online education, online learning,

virtual school, and web-based learning. There are also multiple definitions for each of

these terms. In this study, the authors are using the definitions that have been broadly

cited though by no means are they the most accurate ones.

Distance education, defined by Keegan (1996), has four main components: (1)

quasi-permanent separation of teacher and learner, (2) the use of technical support to

bridge the distance, (3) two-way communication during the process, and (4) possible

non presence of learning groups. It is probably the most cited definition of DE in the

literature. Another very comprehensive definition of distance education is in a published

monograph by The Association for Education Communications and Technology

(Schlosser & Simonson, 2002): "Institution-based, formal education where the learning

group is separated, and where interactive telecommunications systems are used to

connect learners, resources, and instructors" (p. 1). Distance learning, defined by Allen

et al. (2004), is a course where the students and instructor will not be physically in the

same location during the teaching/learning process. Distance learning can be

conducted asynchronously using communication techniques such as e-mail,

audio/video recording, mail correspondence, and synchronously using techniques such

as television, radio, internet chat room, and telephone (Allen et al, 2004).









Allen and Seaman (2006) defined three types of online courses. Online is the

course where most or all of the content is delivered online. At least 80% of the

traditional face to face (f2f) classroom meeting time is replaced by online activity.

Blended/Hybrid is the course that combines online and traditional f2f delivery methods.

A considerable proportion (30 to 79%) of the content is delivered online. Web-facilitated

is the course where web-based technologies are used to facilitate learning. A proportion

(1 to 29%) of the content is delivered online.

Virtual school, defined by Clark (2000), is "a state approved and/or regionally

accredited school that offers secondary credit courses through distance learning

methods that include Internet-based delivery" (p. i). Russell (2004) defined virtual school

as "a form of schooling that uses online computers to provide some or all of a student's

education" (p. 2). Greenway and Vanourek (2006) described virtual schools as "a hybrid

of public, charter, and home schooling, with ample dashes of tutoring and independent

study thrown in, all turbocharged by Internet technology" (p. 36). A more recent study

conducted by Barbour and Reeves (2009), defined virtual school as "an entity, which

has been approved or accredited by a state or governing body within the state, that

offers secondary-level courses through distance delivery most commonly using the

Internet." (p. 412). This study examined the practice of virtual school following Clark's

and Barbour and Reeves' definition.

Organization of the Study

The remainder of the study is organized into five chapters and appendices

including some released test items and national and state Algebra and mathematics

standards. Chapter two is the review of the related literature regarding mathematics

courses specifically Algebra success factors and online learning success factors.









Chapter three presents the research design and methodology of the study. The

population and sampling technique, instruments that were used for data collection, and

the procedure of data analysis are described. Chapter four presents the results of the

data analysis and the findings based on the analysis. Chapter five contains the

summary, discussions and implications of the results, recommendations based on the

results, and conclusions. The study concludes with a bibliography and appendices.












1.2


1


0.8
0.7

S 0.6 Unit: Million
0. 0.5
LU
0.4
0.3


0.2 0.18

0.04
0
2000-01 2001-02 2002-03 2004-05 2005-06 2007-08
Academic Year


Figure 1-1. Online Students at K-12 Level









CHAPTER 2
REVIEW OF LITERATURE

Introduction

Online/distance education has greatly contributed to the American education

system with its broad advantages covering a variety of aspects including economic,

political, demographic, and pedagogical (Dede, 1990). Distance learning, especially

online learning, can decrease instructor and student travel costs and possibly increase

instructors' productivity (Bartley & Golek, 2004; Cavanaugh, 2001; Cornford & Pollock,

2003; Evans & Haase, 2001; Gallagher & McCormick, 1999; Glenn, 2001; Paulsen et

al., 1998). The communities benefit from online/distance learning because it can

increase educational opportunities that otherwise might be restricted by geographic

barriers or resource restriction and provide the flexibility for students at different levels

(Bogden, 2003; Helphinstine, 1995; Kerka, 1996; Parsad & Lewis, 2008; Patrick, 2004;

Shachar & Neumann, 2003). The online delivery method via learning management

systems can help the decision making process in regard to instruction and

administration issues to improve teaching and learning effectiveness through providing

data-enriched environments (NACOL & Partnership for 21st Century Skills, 2006). Well

designed online instruction can promote collaboration among peers and between

learners and instructors and co-construct their knowledge structure (Bartley & Golek,

2004; Blomeyer, 2002; Hassell & Terrell, 2004; Hill, 1997; Summers, Waigandt, &

Whittaker, 2005; Webster & Hackey, 1997), leading to the enhancement of higher-order

thinking skills and cognitive abilities (Blomeyer, 2002; Garrison, 2003) as well as

motivation for students with different learning styles (Butz, 2004; Hassell & Terrell,

2004). Online courses have worked well with a variety of learners including at-risk









students, students with some disabilities, and students with limited English proficiency

(Keeler et al., 2007; NACOL, 2009; Watson, Gemin, & Ryan, 2008). The advocates of

online learning believe it will transform teaching, learning, and schooling as a whole

(Cox, 2005).

As a comparatively new form of online education, virtual schooling has

experienced significant development since its emergence in the late 1990s and has

been accepted by more and more educators and students because of its great benefits.

With its help, small schools and rural schools especially can offer a wide range of high

quality courses that otherwise they can't offer (Donlevy, 2003). Virtual schooling gives

students more options for obtaining education (Butz, 2004; Clark & Berge, 2005;

Newman, Stein, & Trask, 2003; NACOL & the Partnership for 21st Century Skills, 2006).

Virtual school could help solve the inequality of educational opportunities caused by a

variety of reasons such as family income, geographical location, and school resources

(Blaylock & Newman, 2005; Cavanaugh, 2001; Clark & Berge, 2005; Hernandez, 2005;

Kellogg & Politoski, 2002; Newman, Stein, & Trask, 2003; Roblyer et al., 2007; Rose &

Blomeyer, 2007; Setzer & Lewis, 2005; Watson & Ryan, 2007). Virtual school can help

improve student learning outcomes and skills (Clark & Berge, 2005) through offering

individual instruction and flexibility to meet the specific needs of students (Keeler et al.,

2007; Kellogg & Politoski, 2002; Newman, Stein, & Trask, 2003).

With the support of advanced technologies and rigorous curriculum, virtual

schooling can help students master 21st century skills including global awareness, self-

directed learning, information and communications technology literacy, problem solving,

and time management and responsibility (NACOL & the Partnership for 21st Century









Skills, 2006; Watson & Ryan, 2006). Virtual school gives students who failed a course in

a traditional classroom the chance for remediation (Barker & Wendel, 2001; Freedman

et al., 2002; Glass, 2009; Newman, Stein, & Trask, 2003) and students who want

advanced courses an enriched curriculum such as advanced placement courses in

different fields including mathematics and science (Barker & Wendel, 2001; Butz, 2004;

Newman, Stein, & Trask, 2003; Watson, Gemin, & Ryan, 2008). Virtual school also can

benefit home school students through offering more educational opportunities that they

otherwise wouldn't have due to reasons such as their parents' lack of knowledge or

family resource limitation (Butz, 2004; Watson, Gemin, & Ryan, 2008).

Research in Online/Distance Education and Significance of This Study

Along with the growth of online education in the US, considerable research has

been conducted on online/distance education effectiveness with respect to improved

student academic performance and most of the studies have confirmed its

effectiveness. However, little research has been done to examine success factors in K-

12 online learning environments. In recent years, two lines of research emerged to

address online success factors: one focuses on learner characteristics and another one

focuses on learning environment characteristics (Roblyer et al., 2008). However, no

clear set of characteristics have been identified as online success factors and no

conclusive model has been created to apply in online learning practice (Roblyer &

Davis, 2008; Tallent-Runnels et al., 2006). Learner characteristics including participation

in academic activities, whether have IEP and learning environment characteristics

including teacher comment and school type (private or public school) have been proved

to correlate to student academic achievement. However, these variables have not been









investigated systematically in one model regarding their effects on student success in K-

12 virtual learning environments.

In a document about nation wide college student transcripts, Adelman (1995)

reported that math courses detained the top 7 places in the percentage of grades that

were withdrawals, incomplete, or no credit repeats. The first six were pre-college math

courses and the seventh was college Algebra. Clearly, math is a difficult subject for

many students including secondary level and higher education level. School Algebra is

therefore a key subject during the school reform discussion (Chazon & Yerushalmy,

2003). It has been critical for filtering the educational opportunities for high school

students to further study in college (Moses, 1994; Moses et al., 1989).

Algebra/mathematics is also a very important momentum to push a society to

move forward. Many career options are only open to students with advanced

mathematics skills in the job market (House, 1993). Stanic and Hart (1995) believe

mastering mathematics knowledge and being able to apply mathematics ideas are

critical for each member in a society to participate in the democratic processes and

have more career opportunities. The possession of more mathematical literacy for

everyone in the society is also the need for full participation in military service and shifts

in US and the worlds' economic systems (Secada, 1992).

The purpose of this study is to examine the factors including LMS utilization,

teacher comment/feedback and student demographic information that can influence the

success of Algebra courses in K-12 virtual learning environments. This study can help

discover certain characteristics and good practices in online learning and help

incorporate them into the instructional model of the K-12 virtual learning environment. It









also will add to the knowledge of the effectiveness of online/distance education in

helping improve student academic achievement in K-12 virtual schooling and provide

valuable guidance for better implementation and practice. The investigation of success

factors will provide a deeper understanding of success in the K-12 virtual learning

environment specifically in online Algebra/math courses to guide management of virtual

schools for maximizing their effectiveness to provide better assistance and

supplementation to the traditional learning environment.

Review of Literature

The review of literature in this chapter covers the effectiveness of online/distance

education, Algebra/mathematics education, and online success factors. The review of

literature on effectiveness of online/distance education presents the evidence for the

conduct of increasing research in online/distance education. It also provides the

rationale for the investigation of online success factors in this study. The review of

Algebra/mathematics education can grant the support for the selection of specific

courses in which the present study is conducted and demonstrates the relationships

between a traditional teaching format and online education. The review of online

success factors grounds the present work in the related studies and provides the

support for the selection of factors in the present study.

Effectiveness of Online/Distance Education

Well designed distance education courses/programs can provide effective learning

with innovative pedagogy, rich experience, and deep understanding of knowledge

(Cavanaugh, 2001). Many studies have been done to examine the effectiveness of

distance education. Research on distance education effectiveness has mainly focused

on several aspects: student learning outcomes, student-instructor interaction during the









learning process, and student and faculty attitude and satisfaction with the learning

experience (Gallagher & McCormick, 1999).

Cavanaugh (2001) conducted one meta-analysis study to examine the

effectiveness of interactive distance education in the K-12 learning environment. She

reviewed 19 experimental and quasi-experimental studies selected with strict criteria

including the focus of study, publication date, research design, and calculated effect

sizes to assess the effects of some technologies including video-conferencing and

online telecommunications on student achievement and to investigate the success

factors for effective distance education. All these studies covered a wide range of

subject areas and grade levels. The overall effect size for the 19 studies, 0.147,

indicated the small positive effect of distance education over traditional education. No

significant differences were found in grade levels, ability levels, content areas,

technology use, and achievement measure. The author concluded distance education

can be at least as effective as traditional education to help students achieve academic

goals and that offering distance courses at the secondary level could enrich the course

curriculum and students' knowledge structure.

Sherry, Jess, and Billig (2002) conducted one action research study to evaluate

the effectiveness of online learning in improving student media literacy and multimedia

techniques. They collected data quantitatively using surveys and qualitatively using

interview and focus groups from students and instructor. The results showed

technologies that are integrated into online learning can help students acquire a variety

of skills such as creating multimedia projects, editing digital artifacts, designing web

pages, and promoting student learning motivation. They concluded that the technology-









enriched learning environment in general and online education in specific has a positive

influence on student achievement.

Allen et al. (2002) reviewed 25 empirical studies comparing student satisfaction

between distance education and traditional classrooms. The criteria used to select the

studies included the presence of comparison group and sufficient statistical information

that effect sizes can be calculated. The results showed overall there was no significant

difference in satisfaction level though students showed a slight preference toward

traditional education over distance education. The researchers also examined the

effects of communication and interaction on student satisfaction level and found that

there was virtually no effect on communication methods (video, audio, written). They

supported the implementation of distance education by providing the evidence that

distance education will not reduce student satisfaction with the learning experience.

Aragon, Johnson, and Shaik (2002) examined the impact of learning style

preference on student academic success between online and traditional learning

environments to investigate the effectiveness of online learning. Thirty-eight students

taking a graduate-level instructional design course, with nineteen in a traditional

classroom and nineteen in an online course, taught by the same instructor in a

Midwestern university participated in this study. The two groups of students were

equivalent with respect to their demographic information such as age, undergraduate

GPA, and year of baccalaureate graduation. The researchers found that there were

differences in learning style between these two groups, though these differences were

not significant when student success was controlled. It indicated online education can

be as effective as traditional f2f education in helping students succeed academically









even though they have different learning style preferences. The researchers advocated

for the development of online education based on the results.

Swan (2003) looked beyond the no significant differences phenomena and

reviewed the literature on the effectiveness of online learning focusing on the three

types of interaction: student-instructor, student-student, and student-content. She

believed online education is effective as compared to traditional education, and some

unique characteristics of the online technology can be further utilized to improve the

online learning effectiveness. Based on the literature review, she gave some

suggestions for the improvement of online learning environments such as providing

timely and constructive feedback to students, integrating activities to establish online

community, encouraging students to share experiences and thoughts during their

learning process, and ensuring the clarity and consistency of the course materials.

Zhao et al. (2004) employed a meta-analytical approach to investigate the

effectiveness of distance education. The researchers found many individual studies

reporting significant differences between distance education and traditional education;

some found distance education more effective while others found traditional more

effective. They selected 51 out of thousands of articles for review with some criterion

such as that they needed to be journal articles, they must possess empirical data, and

they needed enough statistical information to calculate effect size. The researchers

analyzed several variables in this meta-analysis including study related variables such

as design of study, measurement employed, instruction related variables such as

instructor (status, involvement level), learner (status, background), curriculum (content

area, degree), and milieu (interaction, media, setting). Effect size was calculated for the









estimation of difference between distance education and traditional education. The

results showed overall there is no significant difference between distance education and

traditional education. However, the wide range of effect size (-1.43, 1.48) indicated

distance education was much more effective in some studies while was much less

effective in other studies. Interestingly, the researchers found publication time was a

factor for the effectiveness: studies published before 1998 are more likely to find no

significant difference while studies published after 1998 are more likely to find significant

effectiveness favoring distance education. The researchers believed that distance

education is getting better with the advance of technologies and design principles. The

instructor involvement was found to influence the effectiveness: traditional education

was more effective when involvement was low and distance education was more

effective when it was high. This confirmed the importance of instructor involvement in

the form of teacher feedback, and student-teacher discussion for successful distance

learning. The researchers also found the content area can predict the difference

between distance education and traditional education: distance education was more

effective in fields such as Business, Computer Science, and Medical Science; no

significant difference was detected in Social Science and Science fields; distance

education was slightly effective over traditional education in Military, Mathematics and

Specific Skills. Though the researchers did not examine the learner variables such as

gender, or learning styles, they believed groups with certain characteristics are more

likely to succeed in distance learning. This study also found a blended learning

environment mixing distance education and a certain amount of f2f meeting was most

effective and called for more comprehensive studies in the distance education field.









Cavanaugh et al. (2004) reviewed 14 studies to examine the effectiveness of K-12

distance education. The studies selected were related to distance education published

between 1999 and 2004 under the following criteria: type of publication, K-12 focus,

quantitative experimental or quasi-experimental studies, and enough statistical

information for the calculation of effect size. They specifically looked at the effects of

distance education on student academic achievement and the effects of different

features of distance education including content area, duration and frequency of

distance education, student grade level, school type, interaction, and instructor role on

academic achievement. The overall effect size, zero, showed distance education is as

effective as traditional education. The wide range of effect size (-1.158, 0.597) indicated

some distance education courses/programs were much better than traditional education

while others were much worse. Publication and methodological variables such as year

and type of publication, measurement employed in the study and statistical power, and

distance education experience variables such as duration and frequency of distance

education, instructor role, and type of interactions had no significant influence on effect

sizes. However, instructional and program variables such as student grade level, school

type, and content area did influence effect sizes significantly. The researchers

concluded with the promotion of implementation of K-12 distance education with close

collaboration among different stakeholders including teachers, researchers,

policymakers, developers, and parents, and more rigorous research in this field to guide

the practice and implementation of K-12 distance education.

Stewart et al. (2005) evaluated the effectiveness of one online case-based

continuing education program for family physicians in improving their knowledge and









skills. They randomly assigned the participants into experimental groups with the

implementation of intervention: online learning and control group without the

intervention. They analyzed the knowledge and skill growth measured by two

knowledge questionnaires and charts quantitatively and the posts and emails

qualitatively. The results showed the intervention had positive effects on knowledge

growth and the quality of practice for these physicians. The researchers confirmed the

promise of the broad implementation of online education in general.

Williams (2006) reviewed 25 comparative studies from 1990 to 2003 on distance

education in allied health science education to examine the learning effectiveness on

student achievement and the instructional design (ID) components contributing to the

effectiveness. The overall effect size, 0.15, with confidence interval from 0.07 to 0.23,

showed distance education was slightly more effective than traditional education with

respect to improved student achievement. The results also showed the integration of ID

components in distance courses had a positive effect on achievement. The researcher

suggested the effective distance education courses should incorporate various ID

components. The study was concluded with the promotion of distance education

courses/programs and a call for more research on the effect of different aspects such as

educational level on the effectiveness of distance education.

In 2009 US Department of Education (DOE) released a report about a meta-

analysis of empirical studies from 1996 to 2008 to evaluate the effectiveness of online

learning practice. The studies included in this meta-analysis were selected based on the

criteria: rigorous research design including random assignment or controlled quasi-

experimental design to contrast online to traditional education, objective learning









outcome was measured and enough information to calculate effect size. The

researchers found overall online students outperformed traditional students with respect

to their learning outcomes with effect size of 0.24 favoring online students. Hybrid

learning students had a larger gain over traditional students relative to purely online

students over traditional students with effect size of 0.35 favoring hybrid learning

students when comparing hybrid learning and traditional learning and effect size of 0.14

favoring online students when comparing purely online learning and traditional learning.

Some methodological variables including sample size, content knowledge, research

design, and equivalence of instructional approach were evaluated whether they

accounted for the results and equivalence of instructional approach was found to be a

significant moderator variable. This report confirmed that the combination of a variety of

elements in online learning such as instructional strategies, integrated technologies, and

students' effort rather than the delivery medium per se is what results in better learning

outcomes. The researchers also found that very few rigorous research studies of K-12

online learning effectiveness have been published. Only 7 out of 99 studies included in

this meta-analysis focused on K-12 level, and the effect size comparing online learning

and traditional learning at K-12 levels was not significant. Therefore, the researchers

caution the application of the results to K-12 virtual learning environments and more

rigorous research is needed to guide the practice and implementation of K-12 online

education.

The proved effectiveness of online/distance education in the literature provides the

support for the present study. Many effectiveness studies have focused on the student

outcome. This lends the relevance to the selection of student academic achievement as









the dependent variable in the present study. Another focus in many effective studies,

student-teacher interaction, presents the rationale for the investigation of teacher

comments which could be the indicator of student-teacher interaction in the present

study.

Algebra/Mathematics Education

Considerable educational and psychological research has been conducted to

identify the success factors in mathematics fields (Grouws, 1992). In the literature on

mathematics education, many researchers have focused on a variety of factors that

associated with mathematics learning including student attitude and background, family

background-socioeconomic, peer environment, instructor factors, and curriculum and

instruction (Beaton & Dwyer, 2002; Kifer, 2002; Wilkins, Zembylas, & Travers, 2002).

These factors have been categorized into three major topics by Schiefele and

Csikszentmihalyi (1995): learner characteristics such as learning styles, learning

strategies, and locus of control, home environment such as socioeconomic status (SES)

and family size, and school environment such as instructor experience, instruction

quality, technique support and resources. However, in other studies, these factors have

been grouped into two categories: learner characteristics and learning environment

characteristics which include home learning environment and school learning

environment (Catsambis, 1995; Ercikan, McCreith, & Lapointe, 2005; Ho et al., 2000).

To be consistent with Roblyer et al., (2008)'s study of K-12 distance education, we used

two groups of factors in the present study.

Learner characteristics variables

Research has shown learner' personal characteristics variables such as prior

knowledge and background, motivation or self-concept are strongly associated with









student mathematics achievement (Catsambis, 1995; Ercikan, McCreith, & Lapointe,

2005; Ho et al., 2000). Some affective variables such as self-concepts, attitudes toward

mathematics, self-confidence mathematics learning ability, motivation, locus of control,

and perceptions of the usefulness of mathematics have been found to relate to student

academic achievement in mathematics (Bassarear, 1991; Duranczyk, 1997; House,

1995; House, 1993; Marsh & Yeung, 1997; Reyes, 1984). Increased study time and the

taking of advanced coursework also can positively affect students' mathematics

achievement (Secada, 1992). Student English language proficiency is another factor for

student mathematics achievement in U.S. (Jacobson, 2000; Secada, 1992). Bilingual

students whose native language is not English will be likely to achieve higher

performance in mathematics if they receive the instruction in their native language

(Secada, 1992).

Ercikan et al. (2005) conducted one exploratory research study examining factors

that might affect students' achievement in mathematics and their participation in

advanced mathematics courses in three countries: Canada, Norway, and the US. They

found students' personal and home environment variables strongly affect their

achievement in mathematics and participation in advanced mathematics courses in

these three countries. These researchers specifically confirmed the relationship

between attitude toward mathematics and participation in advanced mathematics

courses in these countries and the relationship between SES related variables and

achievement in mathematics in the US.

Higbee and Tomas (1999) conducted one research study to examine the

relationship between non-cognitive variables including math anxiety, perceived









usefulness of mathematics, self control ability, and self confidence in one's ability to

learn mathematics and academic achievement in mathematics. The participants were

23 college freshmen and student score in their math courses was collected as the

indicator of academic achievement. The researchers found student attitudes toward

mathematics, motivation, self-management skills and self-confidence are related to

academic achievement. Jacobson (2000) examined success factors in high school

mathematics using a variety of statistical techniques: multiple regressions, ANOVA, and

path analysis with a sample size of 1205 high school students. She found student

beliefs/confidence in their mathematics learning ability and family background/SES have

strong effect on academic achievement in mathematics courses. Interestingly, she also

found student's primary language and writing ability are significant success factors in

mathematics learning.

Based on the review of literature on mathematics education, Reyes (1984)

asserted the affective variables including students' confidence in their mathematics

learning ability, attitude toward mathematics, and mathematics anxiety are related to

mathematics learning. Reyes believed students' confidence in mathematics learning or

self-concept about mathematics learning has a positive relationship with mathematics

achievement. Mathematics anxiety can negatively affect student mathematics

achievement. Reyes believed students' attitude toward mathematics and their perceived

usefulness of mathematics can affect their decision about the mathematics course they

will take. Students who valued mathematics more tended to take more mathematics

courses, which, in turn, could contribute to their higher achievement.









Edge and Friedberg (1984) conducted one study to evaluate the effect of student

ACT scores, high school prior knowledge in calculus, gender, family size, and high

school size on student academic achievement in the first college calculus course. They

found the long-term perseverance/self-control ability and student pre

experience/knowledge in Algebra can significantly affect student achievement in the first

semester of calculus for freshman. Schiefele and Csikzentmihalyi (1995) conducted one

research study using 108 high school freshmen and sophomores to examine the

relationships between interest, learning motivation, prior mathematics

knowledge/mathematic ability, student mathematics learning experience, and academic

achievement in mathematics. The researchers found mathematics ability is a significant

predictor of academic achievement, and the predictability of interest for achievement is

different for students at different grade levels. At 9th or 10th grade level, interest is a

good predictor of achievement.

Belcheir (2002) reported a research study on the exploration of variables that can

predict success in math courses. The sample of participants was 734 college students

enrolled in one intermediate Math course. This study included learner variables such as

student math learning attitude and dispositions, study skills, and student commitments.

This study did not find time on task as a good predictor of course success as expected.

However, the researcher further explained that some valuable information could be

missing in this study because the researcher did not collect information about how

students spent time studying and whether students felt the amount of time they could

use on the course was sufficient. Student motivation and commitment were found to be

the most significant predictors of success for the Algebra courses. The researcher also









emphasized that the instructors should let students know early on how they are

performing in order for them to succeed in the courses.

Learning environment variables

Research has shown learning environment characteristics variables including

family background and support, school or classroom resources, teacher/classroom are

strongly associated with student mathematics achievement (Catsambis, 1995; Ercikan,

McCreith, & Lapointe, 2005; Ho et al., 2000). High parent expectations (Cohen, 1987;

Marjoribanks, 1988; Scott-Jones, 1984; Seginer, 1986; Thompson, Alexander, &

Entwisle, 1988) can positively affect students' mathematics achievement. Instructional

strategies including the implementation of computer Algebra software (Elington, 2003;

Lawson, 1995; Mayes, 1995; Stephens & Konvalina, 2001), the use of other

technologies such as calculators in general and graphing calculators in specific

(Elington, 2003), and collaborative problem solving strategy and visual technique

support (Higbee & Thomas, 1999) can positively affect student mathematics

achievement. Teacher variables such as teaching behaviors are another type of factor

that related to student academic achievement (Schoen et al., 2003).

House and Telese (2008) investigated the relationships between instructional

strategies and self confidence in mathematics learning ability and Algebra academic

achievement in the US and Japan using 2003 TIMSS assessment data. The sample

includes 4244 students from Japan and 7862 students from the US. They found the

teaching and learning strategies and students' self confidence/beliefs in mathematics

learning ability are significantly associated with student Algebra academic achievement

in these two countries. Interestingly, they found the instructional strategy of cooperative

learning activities (working in small groups) negatively affected student achievement









and students who worked on problems on their own (active learning strategy) more

often tended to achieve higher performance. Students who can associate their

mathematics knowledge with their daily lives tended to achieve lower test score. More

research is needed on the effect of learning strategies (group work or independent

study) and the triangulation of a variety of academic measurements during the study of

Algebra learning factors.

Elington (2003) investigated the effects of calculators including basic, scientific,

and graphing on students' achievement and attitude levels through the examination of

54 studies, 26 of which targeted high school students. She found the use of calculators

in the testing system and instruction can increase students' strategic skills,

computational and conceptual skills, and problem-solving skills and promote students'

positive attitude toward learning mathematics. Based on the empirical studies, Hollar

and Norwood (1999) and Shoaf-Grubbs (1993) found the use of graphing calculators

can increase students' overall mathematics ability including the understanding of

function, the ability for modeling, interpreting, and translating. In additional to students'

increased ability in mathematics problem solving, the integration of technology during

the mathematics learning process can also promote collaboration among students

during group interactions and class discussions (Goos et al., 2003).

Wheland et al. (2003) examined two types of factors that affect student academic

performance in an intermediate Algebra course: instructor characteristics- English

speaking status (non-native English speaker), teaching assistant or adjunct faculty, and

student characteristics: attendance. The effect size was also calculated in the study.

The researchers found the instructor characteristics variables: English speaking status,









teaching assistant or adjunct faculties do not have significant effect on student

performance while student attendance can significantly affect academic performance.

Schoen et al. (2003) also analyzed teacher variables related to student achievement

within one reform-based project, the Core-Plus Mathematics Project involving 40

teachers and their 1466 students in 2 schools. They found teaching behaviors such as

following the guidance and recommendations of standards and aligning the instruction

with the high mathematics expectations are related to higher academic achievement.

Algebra teaching and learning

A significant amount of research has been conducted on Algebra teaching and

learning considering its importance as the momentum to push society to move forward.

The focus of Algebra teaching and learning research has been shifted from students'

understanding of Algebra activities to the way students construct meaning of Algebra

procedures and objects (Kieran, 2007). Based on these studies, recommendations and

suggestions have been provided to help improve Algebra teaching and learning quality.

Smith, diSessa, and Roschelle (1993) believe school Algebra instruction should build

upon the strengths and the resources within the perceptions students have based on

their own experience in relation to Algebra concepts. Students should grasp the ability

to solve ill-defined tasks that are more closely connected with the questions they may

have in the real life rather than the well-defined ones within the school settings

(Resnick, 1987). Based on the review of literature on Algebra teaching and learning for

students with different backgrounds, Secada (1992) recommended some instructional

strategies including increased school time, more mathematics course taking, use of

students' native language for instruction, direct instruction for structured curriculum and









basic mathematics skills, and divide of whole class activities into group and individual

work.

Considerable research has been conducted on the approaches to Algebra

learning. Bednarz, Kieran, and Lee (1996) reported the four approaches that have been

focused on at an international colloquium on Algebra in early 1990, including

generalization of numerical and geometric patterns and laws regarding Algebraic

relationships, functional situations, modeling of mathematical phenomena, and problem

solving. Drijvers (2003) described the similar approaches for Algebra learning in more

detail:

* Problem-solving approach: view Algebra as a way to solve problems that can be
expressed in equations.

* Functional approach: view Algebra as a way to investigate the functions and
relations among different variables.

* Generalization approach: mainly focus on the examination of patterns or models and
configurations, and focus on the generalization of relations among different
variables.

* Language approach: view Algebra as a way to convey mathematics ideas in which
Algebra is merely a representation structure composed of symbols without specific
context attached meaning.

Drijvers also identified some aspects that make learning Algebra difficult. These

include:

1. The difficulty for students to relate the formal algorithmic procedures with informal
while meaningful methods

2. The abstract characters of Algebra problem solving approaches that students can't
connect them with the concrete situations

3. The Algebraic language includes particular symbols and rules that are difficult for
students to grasp









Drijvers pointed out the importance of reification of expressions and formulas

during the Algebra learning process. Students need to possess the ability to

comprehend the structure and meaning of formulas and expressions, which as he called

"symbol sense" (p.49). Furthermore, Drijvers explained some key elements of one

theory for mathematics education: Realistic Mathematics Education (RME) and their

meaning and relations to Algebra learning:

* Guided reinvention and progressive mathematization: with the guidance from the
teacher, students are given the opportunities to develop formalized mathematics
knowledge by employing the informal strategies and apply the knowledge in the
concrete life situations.

* Didactical phenomenology: design activities that encourage students to develop their
own mathematics learning strategies

* Horizontal and vertical mathematization:

* Horizontal: by employing empirical methods for example observation and
experimentation, students can structure and solve the problem with mathematics
formulas or conventions.

* Vertical: based on these problem-solving experiences and beyond, students can
develop mathematical framework in regarding with the relations among symbol.

The review of Algebra/mathematics education demonstrates the relationships

between f2f learning environments and online education with respect to the factors of

academic achievement. It lends the support for the selection of success factors in the

present study. For example, many Algebra/mathematics education studies have

focused on a variety of factors that associated with mathematics learning outcome.

These factors include learner characteristics such as learning styles and learning

strategies, and learning environment characteristics such as family background and

school learning resources. This supports the categorization of factors in the present

study.









Success Factors in Online Learning

Given the fact of extraordinary development of online education in the last two

decades, little research has been conducted to examine success factors in the online

learning environment as compared to the traditional learning environment. With the high

early dropout and failure rates in the online learning environment (Carr, 2000a; Roblyer

& Elbaum, 2000), there is an urgent need for more research on success factors to

prevent students from dropping out of virtual or physical school and ensure their

success in this learning environment (Barbour & Reeves, 2009; Bernard et al., 2004a;

Butz, 2004; Dickson, 2005; McLeod et al., 2005). At the K-12 level, there is great

concern about the readiness for students to take online courses and succeed

academically because they are not socially and emotionally mature as compared to

students in higher education (Picciano & Seaman, 2007). The review of studies

examining success factors in online learning environments as well as traditional learning

environments can better guide the practice of K-12 online education.

Schrum and Hong (2001) administered a survey with 70 institutions and found

several factors can influence student success in e-learning environment: learning styles,

prior technology experience, personal disposition, study habits, and tools accessibility.

Brown and Liedholm (2002) conducted a comparative study between online education

and traditional education and found student's personal effort on learning tasks could

make a difference in academic performance. Swan (2002) investigated the correlation

between 22 course design factors and student academic achievement and satisfaction

with learning experience. She found three factors: transparent interface/clarity and

consistency in course design, instructor feedback/instructor-student interaction, and

dynamic online discussion are associated with the success of online learning. These









three factors could be three necessary steps for the establishment of an online learning

community (Swan et al., 2000), which can affect students' learning outcomes in online

education.

Roblyer and Marshall (2003) evaluated one instrument, the Educational Success

Prediction Instrument (ESPRI) which was created to predict success in Virtual High

School (VHS) courses. The constructs measured by this instrument related to success

in VHS courses included time management, achievement, motivation, self-

responsibility, prior technology skills, self-regulation, and self-confidence. They

evaluated ESPRI with 135 students in 13 virtual high schools and found ESPRI is a

reliable instrument (Cronbach a=0.92) to predict student success in the online learning

environment, and certain personality characteristics and attitude were associated with

online learning success. Roblyer et al. (2008) created the instrument: ESPRI-V2 based

on ESPRI and reevaluated it with a bigger sample size: 4110 students from VHS. They

found the four factors measured by ESPRI-V2: technology use/self-efficacy,

achievement beliefs, instructional risk-taking, and organization strategies can predict

student success in VHS courses though it's harder to predict failure, and the Cronbach

a, 0.92, indicated ESPRI-V2 is a reliable instrument. The researchers concluded the

combination of prior knowledge, cognitive characteristics such as self-efficacy and

achievement beliefs, and environment variables such as Internet accessibility and

technical support can predict student online success.

According to Roblyer et al. (2008), two lines of research emerged to address

success factors in online learning: studies focusing on learner characteristics and

studies focusing on learning environment characteristics. Learner characteristics include









student cognitive factors such as locus of control and learning styles; prior technology

skills and attitudes; and experience and prior knowledge about course content while

learning environment characteristics include technology support, course content area,

and accessibility to the Internet. At present, no clear set of characteristics have been

identified to predict the success of the virtual learning environment, and no conclusive

model has been created to apply in online learning practice (Roblyer & Davis, 2008;

Tallent-Runnels et al., 2006). Other learner characteristics including personal

effort/participation in academic activities, whether has individual educational plan,

race/ethnicity, and family background/participation in free/reduced lunch programs, and

learning environment characteristics including teacher comment/feedback/instructor-

student interaction and school type (private or public school) also have been proved in

some studies to correlate to student academic achievement. However, these factors'

influences have not been investigated systematically. The review of these factors in light

of the relationship with student academic achievement in other studies can provide

deeper understanding of success in online learning in general and the K-12 virtual

school environment in specific and shed light on the establishment of a model to predict

online learning success in general and online Algebra/mathematics learning in specific.

Teacher comments/teacher-student interaction

Teacher comments and student-teacher interaction is a critical component in

academic learning (Boaz, 1999; Laurillard, 1997; Parker, 1999; Schaffer & Hannafin,

1993; Summer, 1991; Swan, 2003; Williams, 2006). It can affect learning in traditional

f2f learning environments (Christophel, 1990; Kelly & Gorham, 1988; Rodriguez, Plax &

Kearney, 1996) and online learning environments (Blomeyer, 2002; Jiang & Ting, 2000;

Johnson et al., 2000; Swan, et. al., 2000; Swan, 2002; Swan, 2003; Tallent-Runnels et









al., 2006). Interaction, well described by Cavanaugh, is the "core of teaching" (2001, p.

3) and "at the heart of online learning" (2007, p. 6). The presence of interactivity is vital

for the quality education in distance learning (Blomeyer, 2002; Flottemesch, 2000;

NACOL, 2006; Parker, 1999; Zhao et al., 2004). It can help students evaluate their

learning progress and adjust the instructional strategies if necessary to improve the

learning outcome which will lead to a deeper understanding of knowledge (Hiebert &

Grouws, 2007; Parker, 1999; Schoenfeld-Tacher, McConnell, & Graham, 2001).

Student-instructor interaction can provide the social support for students during the

learning process, which is conducive to higher academic achievement and the

development of social skills (Parker, 1999). It also has a positive relationship with

students' satisfaction with their learning experience (Liaw & Huang, 2000; Swan, 2002;

Usun, 2004).

The educators' active facilitation in the form of teacher comment and feedback in

online learning is an important factor that influences students' academic performance

(Cavanaugh et al., 2005; Dickson, 2005; Ferdig, Papanastasiou, & DiPietro, 2005;

Hughes et al., 2005; Karp & Woods, 2003; Lin 2001; Peters 1999; Phipps & Merisotis,

2000; Smouse, 2005; Zucker, 2005). Jiang and Ting (2000) conducted one study to

examine instructor activity in online learning and found the student's perceived learning

is correlated with the number of feedback comments per student that the instructor

made. This relationship also has been confirmed by Swan et al. (2000) and Swan

(2002). Anderson and Kuskis (2007) argued many of the pedagogical benefits brought

by instructor feedback/student-teacher interaction such as those related to motivation

are relevant to distance education as well as the conventional classroom education.









Hughes et al. (2005) believed that teachers' individual feedback can increase

communication opportunities for students who are shy and may not participate in

academic activities, and these opportunities are helpful to develop closer relationships

between the instructor and students. Constructive and timely feedback from instructor is

one of the success factors for the practice of an effective virtual learning

course/program (Cavanaugh, 2004).

Frequent and open communication between students and instructor is identified as

an important component to build a virtual community during online learning (Lin 2001;

Murphy, Mahoney & Harvell, 2000). The development of a learning community in an

online K-12 course is considered an important factor for students' better academic

performance (Lin 2001; Oren, et al., 2002; Ronsisvalle & Watkins 2005; Wang & Newlin,

2000). Wang and Newlin (2000) argued the social support provided by the learning

community could improve students' academic achievement as well as their involvement

and interest in online academic activities. Oren, et al. (2002) believed teachers should

act as a moderator to facilitate and scaffold students learning and encourage various

interactions especially peer-to-peer interaction to let students learn from each other.

They emphasized teachers' supportive feedback to encourage student-to-student social

interactions for the formation of the social groups during the leaning process and

beyond. O'Dwyer et al. (2007) conducted a quasi-experimental study to examine the

impact of one Algebra I online initiative on students' learning outcomes and found that

online students themselves also highly value the student-instructor interaction during

the learning process.









Participation in online academic activities

The number of times students logged into the LMS and how long they stayed in

the LMS could be considered as the indicators of participation in online academic

activities. The time spent in academic activities has been identified as a very important

factor that has a strong effect on success in online education (Cavanaugh, 2007), face-

to-face instruction (Rocha, 2007), and blended programs (Cavanaugh, 2009). The

activities students engaged in during online study is a predictor of final scores, with

students who participate in academic activities at high level performing better than those

who do not in online learning (Wang & Newlin, 2000). Dietz (2002) believed one of the

most significant predictors of success is attendance which could be reflected by the

number of times students log into an LMS. These findings were confirmed by Dickson

(2005) that participation in online academic activities, which is measured by clicks in the

LMS, is a strong predictor of final scores in online learning.

Race/ethnicity

Considerable studies have been conducted on the relationship between

race/ethnicity and academic achievement in traditional learning environment. Racial

gaps in student test score are undeniable facts (Bali & Alvarez, 2004; Hall et al., 2000).

The student body in online K-12 schools often represents the community that is served

by the traditional school system (Ronsisvalle & Watkins, 2005). The findings in the

literature of the relationship between race/ethnicity and student academic achievement

in traditional learning environments could shed light on success factors studied in K-12

online learning environments.

Through a meta-analysis of 16 studies of race differences in mathematics

performance from grades 4 to 8, Lockhead et al. (1985) found Asian Americans usually









perform at the highest level in math, followed by Caucasian students, and then

Hispanic. All the three groups perform better than African American students. Hall et al.

(2000) also found significant differences in student math achievement among different

ethnicities in a study on gender and racial differences in mathematics performance

among 5th and 8th grade students in the United States. These differences continue at

the higher level. The math skills of most African American in 12th grade, as Barth (2001)

described, are only equivalent to the skills of Caucasian students in the eighth grade.

U.S. DOE released a report in 2004 about the gaps in academic achievement in

different content areas such as reading, math, and science among different racial and

ethnic groups based on the data collected since the mid-1980s. At 4th grade level, 41%

of whites and 38% of Asians were proficient readers while the number for African

Americans, Hispanics, and Native Americans was 13%, 15%, and 16%, respectively. In

mathematics, 48% of Asians and 43% of Whites achieved at proficient level while only

10% of African Americans, 16% of Hispanics, and 17% of Native Americans achieved at

this level.

Participation in school free lunch/family SES

Participation in school free or reduced programs has a correlation with student

academic achievement, and the magnitude of correlation is weaker as grade level rises

(McLoyd, 1998). Klein et al. (2000) conducted a research study on the relationship

between students' participation in free or reduced lunch programs and school test score

using the data about 2000 Texas 5th graders in reading and math. They found the

percentage of students participating in the free or reduced lunch program in a school

can affect the school's mean test score. The researchers believed participation in these

programs could be considered a sign of the level of poverty which has a strong









relationship with student academic performance at the school level. Participation in a

school lunch program was also frequently used as the measurement of student's family

Social Economics Status (SES) in the literature on student academic achievement

(Sirin, 2005). The level of the family support including the resources provided for

students and education values can influence student academic achievement (Hiebert &

Grouws, 2007). Higher SES families provide students more resources at home and

social capital, both of which can improve chances for their academic success (Coleman,

1988). A considerable body of research has been done on the relationship between

SES and student academic performance. The magnitude of this relationship was found

to be strong in two meta-analytic studies conducted more than twenty years apart from

each other: 0.343 in White's (1982) meta-analysis and 0.299 in Sirin's (2005) meta-

analysis. In K-12 online learning environment, participation in school reduced lunch

program/family SES could also be associated with student academic success.

Learning ability/presence of individual educational plan

Student learning ability is a factor that can influence student academic success

during the learning process (Keeler & Homey, 2007). The virtual school student body is

a diverse population including students with different learning disabilities (Dickson,

2005; Ferdig, Papanastasiou, & DiPietro, 2005). Virtual school offers individual

education plans for these students during the learning process. Therefore, whether or

not a student has an individual education plan could be a sign of the level of learning

abilities. The review of studies on the relationship between learning ability and

academic achievement could shed light on the decision making process to provide more

opportunities for students with special needs to succeed in the K-12 online learning

environment.









According to Keeler et al. (2007), students may bring some characteristics that

could physically or psychologically inhibit their access to the information or tools on the

Internet, preventing their success in the online learning environment. The virtual school

learning environment has the potential to bridge gaps between disabled students and

other students without learning disabilities with respect to the success opportunities in

online learning. For example, for students with different levels of learning disabilities,

technologies such as computer, internet, audio, video, animation, gaming and

simulation could help reduce their disadvantages as compared to students without

disabilities (Coombs & Banks, 2000; Richardson et al., 2004). Some instructional design

strategies have been recommended to ensure online courses meet the special needs of

students with disabilities (Keeler et al., 2007; Rose & Blomeyer, 2007). These include

assurance of accessibility to the information for students with disabilities and the

support in course materials and learning activities for these students. Virtual courses

need to be designed with accommodations specifically for students with disabilities to

access course materials and should benefit all learners under the framework of

Universal Design for Learning principals (Rose & Blomeyer, 2007). Since the

emergence of universal design technology and the requirement for the development of

Learning Management System (LMS) to integrate components to meet the special

needs of disabled students to align with the American with Disabilities Act (ADA)

(Watson, 2007), the opportunities to achieve higher performance for disabled online

students has been greatly increased. The proposition of early adoption of technology-

infused education for disabled online students (O'Connor, 2000) also will benefit their









online learning. Over a decade virtual school programs have successfully provided

quality education to students with special needs (Rose & Blomeyer, 2007).

However, even with the bridging gaps with regard to online success opportunities

for disabled online students, they are still underrepresented in online education (Kinash

& Crichton, 2007). For example, even though different learning management systems

such as WebCT and Blackboard are generally accessible to disabled students, there

are still inherent problems limiting them from fully utilizing the functions in their courses

therefore limiting their chances of success in online learning (Asuncion et al., 2006).

Many online courses still have barriers preventing students with disabilities from fully

accessing materials as other students do (Edmonds, 2004; Keeler & Homey, 2007),

which will affect their success in online learning.

Keeler and Homey (2007) conducted one study to evaluate the elements of online

course design that address students' special needs. They found some problems still

existed in the five categories of design elements: accessibility, website design,

technologies used, instructional methodologies, and support systems, which can

prevent students' special needs being fully addressed. According to National Center for

Education Statistics (NCES, 2000), though computer and communication technologies

may be especially beneficial for disabled students (Johnson, 1986), providing them the

access to these technologies could be more expensive than regular students because

they may need special equipment to use these technologies. The lack of

accommodations for students with special needs may exclude them from fully

participating in online learning (Keeler & Homey, 2007). Disabled students lack the

opportunities to use these communication technologies for a variety of reasons









including insufficiently trained special education teachers and inadequate support

services for them to use these technologies (NCES, 2000). This could lead to the

academic gap between students with disabilities and students without these disabilities

in online learning.

Students' learning ability could affect other academic performance in addition to

achievement such as academic engagement. Kersting (1997) interviewed 10 deaf

students to examine their learning experience and found these students had lower

academic engagement in learning activities unless they got sufficient support during

their learning process. Richardson, Long, and Foster (2004) compared deaf students

and their peers without hearing loss regarding academic engagement in distance

learning. There were 267 students with a hearing loss and 178 students without this

disability in an open university who participated in this study. The results showed

students with hearing loss could not perform well on communication and some other

tasks in comparison to students without this disability, which could affect the academic

achievement negatively for these disabled students.

School type

The gap in student achievement between private and public schools has been

documented in many studies (Chubb & Moe, 1990; Coleman & Hoffer, 1987; Coleman,

Hoffer, & Kilgore, 1982). In 2006, the US Department of Education's National Center for

Education Statistics (NCES) released a report about the academic achievement

differences in reading and math at grade levels 4 and 8 between private and public

schools (Braun, Jenkins, & Grigg, 2006). This report showed on average the private

school mean score was higher than the public school's. Students in private schools

achieved at higher levels academically than those in public schools. However, many









studies did not control other important variables such as student SES, or grade level

when examining the difference between public and private schools and many of them

have been done more than two decades ago, so the data could be already outdated.

Therefore, more studies with new data and new methodology are encouraged to be

conducted. A study (Lubienski & Lubienski, 2005) supported by the National

Assessment of Educational Progress (NAEP) compared student achievement in

mathematics between public and private schools using a student sample of 4th and 8th

graders. There were over 13000 4th grade students from 607 schools, 385 of which

were public schools and 222 were private schools, and over 15000 8th grade students

from 740 schools, 383 of which were public schools and 357 were private schools, who

participated in this study. The results showed overall students in private schools

outperformed their counterparts in public schools; however, after controlling for

student's SES, public schools outperformed private schools. The larger proportion of

high-SES students in private schools accounted for their overall outperformance. The

researchers called for more research on the examination of effectiveness of public and

private schools.

The review of success factors grounds the present work in the related literature. It

helps the establishment of the model in the present study with respect to the selection

of independent variables. For example, the family SES has been proved to relate to

student academic performance in many studies, which provides the support for the

inclusion of the participation in school reduced or free lunch programs which could be

the indicator of family SES in the model. Many studies in this literature are quantitative









studies. This can shed lights on the quantitative research method utilized in the present

study.

Conclusion

The review of literature on effectiveness of online/distance education

demonstrates well designed online/distance course can be as effective as its traditional

counterpart with respect to helping improve student academic achievement. This

presents the evidence for the increasing research in online/distance education and

provides the rationale for the present study. The review of literature on

Algebra/mathematics education illustrates that several issues such as learner prior

knowledge and learning ability, study time, and instructional strategies need to be

addressed during the process of Algebra/mathematics teaching and learning. It also

indicates a variety of approaches such as problem-solving, generalization of geometric

patterns and Algebraic relationships, and functional situations should be utilized to

improve learning efficiency. This section builds the connections between traditional

education and online education. Both of these two education formats share success

factors though the effect could be different in these two environments. The review of

success factors confirmed the relationship between student demographic information,

participation level in academic activities, and teacher comments and student academic

achievement. These are also the variables of interest in the present study.

Even after more than 10 years of extraordinary growth in K-12 online learning, little

research has been done as compared to post-secondary education (Cavanaugh, 2007;

Cooze & Barbour, 2005; Means et al., 2009; Picciano & Seaman, 2007; Picciano &

Seaman, 2009; Ronsisvalle & Watkins, 2005). The amount of evidence-based research

or empirical study applicable to guide educators' instruction and policy makers' decision









relevancies is slight (O'Dwyer, Carey, & Kleiman, 2007). Many states have no data on

the current practice of K-12 online education such as the number of students taking

courses online, the number of online programs existing, and how these programs are

managed (Watson, 2007). After review of 99 comparative studies regarding online

education versus traditional education published between 1996 and 2008, the U.S.

Department of Education found that only 7 of them involved K-12 learners (Means et al.,

2009). The development of K-12 online education is advancing differently from the

development of postsecondary online education (Picciano & Seaman, 2009).

Therefore, the practice of online education in higher education may not be applied to the

K-12 environment. The dearth of studies on academic achievement in K-12 virtual

learning environment in comparison with that in traditional learning environments form

the rationale for more quantitative research in this field to guide the implementation and

practice of online learning at this level (Cooze & Barbour, 2005; Means et al., 2009;

Picciano & Seaman, 2007; Picciano & Seaman, 2009; Smith, Clark, & Blomeyer 2005;

Watson, 2007). Quantitative data collection is the required research methodology to

support the understanding of the efficacy in K-12 virtual school (Smith, Clark, &

Blomeyer, 2005).

Currently the lack of new data regarding K-12 online learning is attributed to a

variety of reasons including lack of requirement in many states for data collection on

online students and the significant growth of K-12 online education practice causing the

difficulty of data collection (Picciano & Seaman, 2007). Research conducted in virtual

schools is rare because of its comparatively new practice, and currently available data

can't provide enough information for accurate estimation of its practice (Glass, 2009).









More research has been called for that focuses on students' academic performance,

particularly on the factors influencing the success of students in K-12 virtual learning

environments (Smith, Clark, & Blomeyer, 2005). The question of whether the factors

that affect students' achievement in the traditional school learning environment play the

same role in the virtual school learning environment remain to be answered. Academic

performance is considered as the single greatest indicator of school completion (Battin-

Pearson, Newcomb, & Abbott, 2000), and lowering the school dropout rate is a national

priority. The investigation of the factors that influence student academic performance in

virtual schools is of critical interest to educators, researchers, virtual school program

administrators, and policy makers.









CHAPTER 3
METHODOLOGY

Introduction

There are five sections in the methodology chapter: research design, population

and sample, instrumentation, data collection and analysis, and limitations. What type of

design employed in this study is explained in research design section. Then the

population that this study is targeting, sample that has been selected and sampling

techniques employed are described. Instruments utilized in this study are detailed in the

section of instrumentation. The process of data collection and analysis are then

explained. And limitations if any are pointed out at the end.

The purpose of this study is to examine the factors including LMS utilization,

teacher comment/feedback and student demographic information that can influence the

success of Algebra courses in K-12 virtual learning environments. The research

questions in this study are:

* Does the level of LMS utilization influence Algebra/mathematics performance in
online education?

* Does teacher comment or feedback influence Algebra/mathematics performance in
online education?

* Do student demographic information such as race/ethnicity, grade level, status in
virtual school, whether have individual educational plan (IEP), and participation in
free/reduced lunch programs influence Algebra/mathematics performance in online
education?

Research Design

The present study is descriptive in nature. The researcher described some factors'

predictability of Algebra learning outcome without intervening. The researcher collected

the data at the end of 2008-09 academic year. These data include student demographic









information, their participation level in online academic activities, teacher comments,

student EOC test score, and the score on one state standardized mathematics test.

The variables of interest in this study include: teacher comments

(TEACHERCOM), student participation level of online academic activities the number

of times students logged into the LMS (TOTALLOG), the time they stayed in the LMS

(TOTALMIN), and student demographic information whether students have IEP,

students' grade levels in their physical schools (GRADE), race/ethnicity (RACE),

students' status in the virtual school (full time or part time students, PT/FT), and the

participation in free or reduced lunch (FRL) programs. They are independent variables

in the study. Student EOC test score and the score on one state standardized

mathematics test are dependent variables in this study.

Participants and Data Collection

The data were collected during the 2008-09 academic year from one state virtual

school in the Midwestern US region. This virtual school was implemented in 2007. A

similar pilot project was conducted in spring, 2009, based on its first year (2007-08)

data. This dissertation builds on the results of the pilot project. However, the present

study is different from the pilot project in many respects. For example, the present study

investigated the success factors not only for Algebra EOC test but also for one state

standardized mathematics test which was missed in the pilot project. The present study

was conducted with different sample from the pilot project. Students statewide from

bricks-and-mortar public and private schools as well as home school students were

eligible to enroll in this virtual school. Enrolled students resided in most of the state's

school districts. The school hired content area teachers who met state certification and









other requirements. A single LMS was utilized by this virtual school to manage course

content and deliver instruction at the secondary level.

The students needed to take the EOC test at the end of each semester during

2008-09 academic year after they completed the course. Some students also took one

state standardized mathematics test at the end of academic year. Students who

completed the four Algebra courses: Algebra I (1st half), Algebra I (2nd half), Algebra II

(1st half), and Algebra II (2nd half) and took the EOC tests and the students who took the

state standardized mathematics test grade 6, 7, or 8 participated in this study (Due to

lack of information about students who took the standardized test grade 3, 4, or 5, they

were dropped from the study). The number of students who took the four Algebra EOC

tests was: 101, 75, 26, and 36 respectively. Due to the insufficient power for data

analysis caused by the small sample size of Algebra II (1st half) and Algebra II (2nd half),

26 and 36, these two groups were dropped from the study. Within the two Algebra I

groups, 64 out of 101 students in Algebra I (1st half) (63.4%) and 59 out of 75 students

in Algebra I (2nd half) (78.7%) were second year students in this virtual school. Students

who took the two Algebra EOC tests were from grades 8 to 12. Students who took the

standardized mathematics test grade 6, 7, or 8 were from grade 6 to 10 and the number

was: 74 (grade 6), 73 (grade 7), and 107 (grade 8). All of these participants were first

year students in this virtual school.

Instrument

Success in an online course can be measured by academic achievement including

the grades students earn and their performance on advanced placement exams

(Ronsisvalle & Watkins, 2005; Tallent-Runnels et al, 2006). Dickson (2005) used

student final score as the measure of student performance in online courses during the









data analysis in a study conducted to investigate the variability of student performance

in online courses. Full-time online schools assessed student achievement in the same

way as all public schools (Watson, Gemin & Ryan, 2008). Student achievement in many

supplemental online programs is also assessed by course grade or EOC test score

(Watson, Gemin, & Ryan, 2008). Indeed, some virtual schools and their teachers are

paid on the basis of successful students, defined as those passing their courses.

Algebra EOC Test

Students who took the Algebra I EOC test in the virtual school participated in this

study. The Algebra EOC tests were tests administered at the end of each semester in

this virtual school. According to the virtual school administration, they have high

correspondence to the state's Algebra and mathematics content standards. The

purpose of the EOC test, as described by the state's department of education (2009), is

to:

* Measuring student achievement and progress toward postsecondary readiness

* Identifying students' strengths and weaknesses

* Communicating expectations for all students

* Meeting state and national accountability requirements

* Evaluating programs


The Algebra I EOC test includes one session of multiple choice items and one

session of performance events (Missouri Department of Elementary and Secondary

Education (MDESE), 2009a). The items in the multiple choice session are developed

specifically for students in this state (see Appendix A for some released samples). The

items in the performance events session are longer, and focusing on more challenging









tasks that require students to work through different problems, arguments, or require

extended writing (see Appendix B for some released samples). These EOC tests are

intended to measure students' skills in number and operations, Algebraic relationships,

and data and probability. This state has its standards for Algebra (see Appendix C for

the state Algebra standards). The Appendices A-E provide the evidence of alignment

between the Algebra I EOC test and state Algebra standards.

State Standardized Test

Students who took one state standardized mathematics test grade 6, 7, or 8 after

they finished one year of study in this virtual school during 2008-09 academic year

participated in this study. This standardized mathematics test is aligned with the state

Show-Me Standards which are the educational standards of this state. For

mathematics, the Show-Me standards require students in state public schools to obtain

knowledge of (MDESE, 2008)

1. Addition, subtraction, multiplication and division; other number sense, including
numeration and estimation; and the application of these operations and concepts
in the workplace and other situations

2. Geometric and spatial sense involving measurement (including length, area,
volume), trigonometry, and similarity and transformations of shapes

3. Data analysis, probability and statistics

4. Patterns and relationships within and among functions and Algebraic, geometric
and trigonometric concepts

5. Mathematical systems (including real numbers, whole numbers, integers,
fractions), geometry, and number theory (including primes, factors, multiples)

6. Discrete mathematics (such as graph theory, counting techniques, matrices)


This grade level state standardized mathematics test is a standards-based test

designed to measure the skills for each grade of students in the state where this virtual









school is located (MDESE, c). It also has a national norm-referenced test that can be

used to compare students in this state with students across the country. This

component helps align the state standardized test with the National Council of Teachers

of Mathematics (NCTM) standards. See Appendix D for NCTM mathematics standards

for grades 6-8.

There are three types of questions in this grade level standardized mathematics

test: 1. multiple choice items that are developed specifically for students in this state or

the questions in the national norm-referenced survey; 2. constructed response items

that require students to provide the response rather than selecting the options among

different choices; 3. performance events items as described above in EOC test that are

longer, and focusing on more challenging tasks that require students to work through

different problems, arguments, or require extended writing (MDESE, c). See Appendix E

for released items of this standardized test (spring 2006) at grade level 6. This state has

standards for mathematics at different grade levels. See Appendix F for the state

standards for mathematics at grade level 6 (due to the space limit, the author did not

attach released items of this standardized test (spring 2006) at grade 7 and 8 and state

standards for mathematics at grade level 7 and 8). The Appendices D-F provide the

evidence of alignment between this state standardized mathematics test and NCTM

mathematics standards and state mathematics standards.

Data Analysis

Due to the very small sample size of minority groups including Asian American,

Hispanic American, Indian American, and African American in this study, these four

groups were combined as one category during data analysis in this study: Minority.

There are only two categories in the categorical variable: Racial/Ethnicity, Caucasian









American Students and Minority Students. Other categorical variables were coded

accordingly during data analysis. Table 3-1 shows the coding information.

Students who took the two Algebra EOC tests were from grades 8 to 12. Students

who took the standardized mathematics test grade 6, 7, or 8 were from grade 6 to 10.

These two sets of groups were overlap to some degree. Therefore, the analysis was

conducted for these two sets of groups separately. Some of the participants in state

standardized test groups will take Algebra course in this virtual school. The analysis of

these groups can add to the knowledge of success factors in Algebra.

This virtual school student body included students statewide from bricks-and-

mortar public and private schools as well as home school students. The physical

schools as well as the home schools that students attend could affect student academic

performance through school culture, technical support, and resources available for

students. Student test scores within the same physical school are not independent of

one another. Therefore, any evaluation of the influence of student level factors such as

grade level, race, and teacher comment on these scores must account for the influence

of school characteristics. To investigate the Algebra/mathematics success factors in the

K-12 online learning environment, Hierarchical Linear Modeling (HLM) technique was

used to account for the clustering of student score within one school caused by school

characteristics. HLM was carried out by the software program HLM 6.06 for data

analysis in this study. The fully unconditional or Random ANOVA (RA) model was

estimated at the beginning in order to partition the variance into within-school (Sigma

Square) and between-school (Tau) components. After that, all independent variables









were added into the model. Generalized estimating equation was applied for the

estimation of correlation coefficients.

Limitations

Limitations of this study include:

1. The small sample size could affect the power for statistic claims.

2. The coding strategy for race/ethnicity variable could mask influential information.

3. Many home school students and the very small number of students from many
different physical schools (some only had one student) may cause data analysis
difficulty.









Table 3-1. Coding of the independent variables
Variable Categories
Student status (part-time or full-time) 0: part time
1: full time
IEP (individual educational plan) 0: without individual educational plan
1: with individual educational plan
FRL (free or reduced lunch) 0: not in free or reduced lunch programs
1: in free or reduced lunch program
RACE 0: Caucasian American student
1: minority student









CHAPTER 4
RESULTS

Introduction

As stated in Chapter 1, the study reported here examined the success factors

including student demographic information, teacher comments, and student

participation level in online academic activities in K-12 virtual learning environments.

This chapter is organized based on the three research questions posed in Chapter 1. It

first describes the sample of this study. It then reports the effects of LMS utilization,

teacher comment, and student demographic information such as race/ethnicity and

whether have IEP on Algebra/mathematic achievement in virtual learning environments.

Sample

The data were collected during the 2008-09 academic year by a consulting

company that works with one state-led virtual school in the Midwestern US region. This

consulting company collected student demographic information and their performance

on two types of tests: EOC test and state standardized test. The researcher obtained

the data from this consulting company. The criteria for the participation in this study

were: (1). students who completed Algebra online courses during the 2008-09 year and

took the EOC test at the end of each semester; or (2)students in this virtual school who

took one state standardized mathematics test at the end of the 2008-09 academic year.

EOC Tests Taker

This virtual school offered four Algebra courses: Algebra I (1st half), Algebra I (2nd

half), Algebra II (1st half), and Algebra II (2nd half) during the 2008-09 academic year.

The number of students who completed these four Algebra courses and took the EOC

tests at the end of each semester during that academic year was: 101, 75, 26, and 36









respectively. These four groups participated in this study. However, due to the

insufficient power for data analysis caused by the small sample size of Algebra II (1st

half) and Algebra II (2nd half), 26 and 36, these two groups were dropped from the

study. Within the two Algebra I groups, 64 out of 101 students in Algebra I (1st half)

(63.4.2%) and 59 out of 75 students in Algebra I (2nd half) (78.7%) were second year

students in this virtual school. See table 4-1 for student demographic information. The

sample can be described as primarily white, not participating in school free or reduced

lunch programs, and part time virtual school students without individual educational

plans.

State Standardized Test Taker

There were 487 students in this virtual school during 2008-09 academic year who

took a state standardized mathematics test at the end of academic year. This

standardized mathematics test has different grade levels from 3 to 8. Due to lack of

information about students who took the state standardized mathematics test grade

level 3, 4 and 5, only students who took the state standardized mathematics test grade

level 6, 7, and 8 participated in this study. The number of students in these three groups

is 74, 73, and 107 respectively. All of these participants were first year students in this

virtual school. See table 4-2 for student demographic information. The sample can be

described as primarily white, participating in school free or reduced lunch programs, and

part time virtual school students with individual educational plans.

As stated in Chapter 3, the physical schools students attended could affect student

academic performance via the resources the school provided for students, technical

support, and school culture. Students' test scores within one school are not independent

of one another. Any evaluation of the variables at student level such as teacher









comments, grade level, and race on student test score must account for the influence of

school characteristics on this dependent variable. The Hierarchical linear modeling

(HLM) technique was carried out by the software program HLM 6.06 for data analysis to

account for the clustering of students' scores within one school caused by the school

characteristics. The fully unconditional or Random ANOVA (RA) model was utilized to

partition the total variance of student test score into within-school (Sigma Square) and

between-school (Tau) components at the beginning during the analysis. After that, all

the independent variables were added into the model. Generalized estimating equation

was then applied for the estimation of coefficients of the different variables.

RA Model

The RA model was estimated for each dataset to partition the variance of student

test score into within-school (Sigma Square) and between-school (Tau) components.

Level-1 Model

Y = BO + R

Level-2 Model

BO = GOO + UO

The Intra-Class Correlation (ICC) was calculated according to the formula: Tau/

(Tau+ Sigma Square) for each dataset. Results for the RA model are presented in Table

4-1.

Table 4-1 demonstrates the ICC for all the five datasets is equal to or above 0.7.

This finding shows the between-school variance was large in comparison with the

within-school variance for the five groups of students. This tells us the students within

the same school are similar with respect to their academic achievement in mathematics

and in Algebra particularly in comparison with the students from different schools.









Coefficients for the Variables

After estimating the RA model, all the independent variables were added into the

model at the level 1 (student level). With the exception of the data on the standardized

test in grade 6, the generalized estimating equation (GEE) procedure was then used to

estimate the coefficients of these variables. For the data on the standardized test in

grade 6, ordinary least square was used. The results are presented in Tables 4-2 to 4-4.

Summarization of the results is presented following Table 4-4.

Level-1 Model

Y = Y = BO + B1*(GRADE) + B2*(RACE) + B3*(FRL) + B4*(IEP) + B5*(PT/FT) +

B6*(TEACHERCOM) + B7*(TOTALLOG) + B8*(TOTALMIN) + R

Level-2 Model

BO = GOO + UO

B1 = G10

B2 = G20

B3 = G30

B4 = G40

B5 = G50

B6 = G60

B7 = G70

B8 = G80

The results of the GEE are presented in Tables.









EOC Test

Table 4-2 shows the estimate of the variable effect coefficients for the two Algebra

EOC tests: Algebra I (1) and Algebra I (2). The variables that have significant effects on

student final score on these two tests are highlighted in grey.

State Standardized Mathematics Test

Table 4-3 shows the estimate of the variable effect coefficients for the two state

standardized mathematics tests grade 7 and grade 8. The variables that have

significant effects on student final score on these two tests are highlighted in grey.

Ordinary Least Square (OLS) was applied for the dataset: mathematics

standardize test grade 6 for the estimate of effect coefficient. There are 74 students in

this group. Five of them are from the same school and all other 69 students are from

different schools. There is almost no clustering for student scores due to the small

sample size at student level (74) and comparatively large sample size at school level

(70). Thus, least-squares estimates with robust standard errors can't be applied to

correct the errors associated with the clustering of student scores within one same

school. Instead, OLS was applied for the estimates of coefficients of the independent

variables. Table 4-6 shows the results.

Descriptive Statistics, Standardized Coefficient, and Reduction of Variance

Descriptive statistics analysis was conducted for each group to demonstrate the

mean and variance of the factors (independent variables) and student score (dependent

variable). See table 4-7 and 4-8 for this information for these two sets of participants.

To compare among different factors with respect to the importance in determining

student score, standardized coefficient (3) was calculated according to the formula: 3k

=bk Sxk Sy (bk is the unstandardized coefficient, Sxk is the standard deviation of the









corresponding independent variable, and Sy is the standard deviation of the dependent

variable). See table 4-9 and 4-10 for standardized coefficients for these two sets of

participants. Standardized coefficient demonstrates that how increases in the

independent variables affect relative position within the group. For example, the

standardized coefficient of TEACHERCOM was 0.56 for Algebra I (2). It means with 1

standard deviation increase in TEACHERCOM, student test score increased 0.56

standard deviation.

The adjusted R-square (Rc2) was calculated according to the formula: Rc2= 1- VAR

e/VAR t (VAR e is the least squares estimates of the model with all the predicators, VAR

t is the least squares estimates of the RA model) to show the reduction of test score

variance from the RA model with the addition of the factors. See table 4-11 for adjusted

R-square for the five groups. The adjusted R-square also shows the variance that is

accounted for by these factors. For example, Rc2 is 0.15 for Algebra I (2). It means the

eight factors accounted for 15 percent of test score variance.

Research Question 1

Does the level of student participation in academic activities predict

Algebra/mathematics performance in online education?

The participation in online academic activities can be reflected through the number

of times students logged into the Learning Management System (LMS) and how long

they stayed in the LMS. This study used these two variables as the indicators of the

level of student participation. Other indicators of participation not collected by the virtual

school's data system, such as the time students spent online on academic tasks and the

time they spent on non academic tasks, will not be part of this study. Time on task has

been identified as a critical factor in relation to the improvement of understanding level









of subject matter (Bransford et al., 1999). As Bransford et al. mentioned, students need

to take time to make meaning of the concepts in the subject areas and build the

connections to their preexisting knowledge. Based on one study on the effect of one

Algebra I online learning model on students' academic outcome, O'Dwyer et al. (2007)

concluded online students spent more time on interacting with one another on academic

topics than their counterparts in traditional classroom. Peer-to-peer interaction could

help improve online students' learning outcome. The time spent in academic activities

has a strong effect on success in online education (Cavanaugh, 2007), face-to-face

instruction (Rocha, 2007), and blended programs (Cavanaugh, 2009). It can predict

student final grade in online learning (Wang & Newlin, 2000). To investigate the effects

of student participation in online academic activities on student achievement in

mathematics and Algebra in particular in online learning environments, the number of

times students logged into the LMS and how long they stayed in the LMS were

analyzed using HLM along with other factors in one single equation. Other student time

on task outside the LMS was not measured in the school data system.

EOC Test

Table 4-2 shows TOTALLOG (number of times student log into the LMS) had a

non significant effect (-0.03, p=0.230) on student final score for Algebra I (1) and the

direction shows students who logged into the LMS less tending to perform better than

those students who logged into the LMS more. The effect of TOTALLOG is significant

for Algebra I (2) (-0.02, p=0.006), with students who logged into the LMS less achieved

higher scores. There is a weak and non significant effect of TOTALMIN (total minutes

students stay in LMS) on final score for Algebra I (1) (0.0005, p=0.304), with students

who stayed in the LMS longer tending to achieve higher scores. The effect of









TOTALMIN is significant for Algebra I (2) (0.0004, p=0.008). The direction of the effect

tells us students who stayed in the LMS longer performed better.

State Standardized Test

Table 4-3 shows TOTALLOG has no significant effect (0.02, p=0.725) on student

score in the grade 7 mathematics standardized test. The direction shows students who

logged into the LMS more tending to perform better than those students who logged into

the LMS less. The effect of TOTALLOG is also not significant (-0.07, p=0.117) for the

grade 8 mathematics standardized test, with students who logged into the LMS less

achieving higher scores. Table 4-4 shows TOTALLOG also has no significant effect (-

0.04, p=0.414) for the grade 6 mathematics standardized test, with the same direction

as it in grade 8. Table 4-3 shows there is a weak and non significant effect of

TOTALMIN (0.0004, p=0.680) on student score in the grade 7 mathematics

standardized test, with students who stayed in the LMS longer tending to achieve higher

scores. The effect of TOTALMIN is nearly significant (0.001, p=0.057) for grade 8. The

direction of the effect tells us students who stayed in the LMS longer tended to perform

better. Table 4-4 shows there is a weak and non significant effect (0.0005, p=0.544) of

TOTALMIN for the grade 6 mathematics standardized test, with students who spent

more time in the LMS tending to achieve higher scores.

Research Question 2

Does teacher comment or feedback predict Algebra/mathematics performance in

online education?

Bransford et al. (1999) emphasized the importance of frequent feedback from the

instructors for students to monitor their learning process and evaluate their

understanding levels and the learning strategies during the learning process. Based on









the feedback, students could revise their thinking and enrich their knowledge structure

as they move along. Teacher feedback or teacher comment on student assignments,

papers, and projects has been identified as a critical factor that can influence student

academic performance in online education (Cavanaugh et al., 2005; Dickson, 2005;

Ferdig, Papanastasiou, & DiPietro, 2005; Hughes et al., 2005; Peters, 1999; Zucker,

2005). Phipps and Merisotis (2000) believed that student-teacher interaction and the

timely and constructive teachers' feedback to students' assignments and questions are

critical characteristics of the teaching/learning benchmarks for the quality of online

learning. Watson and Ryan (2006) showed there are big differences in students'

experiences between virtual classrooms with minimal teacher involvement and those

with greater student-teacher interactions via different means such as e-mail, online

message, online discussion forum, phone, etc. Based on a quasi-experimental study on

the impact of one state-wide Algebra I online initiative on students' learning outcomes,

O'Dwyer et al. (2007) found that online students highly value the student-instructor

interaction during the learning process.

The critical value of teacher feedback and teacher comment for success in online

learning is also applicable to students with special needs. Based on a study of students

with learning disabilities (SLD) and students with attention deficit hyperactivity disorder

(ADHD), Smouse (2005) found communication with and feedback from instructors was

the most valuable aspect of online courses. To investigate the effect of teacher

comment or teacher feedback on student achievement in mathematics and Algebra in

particular in online learning environments, the number of teacher comments was

analyzed using HLM along with other factors in one single equation.









EOC Test

Table 4-2 demonstrates the non significant effect of TEACHERCOM (teacher

comments) for Algebra I (1) (-0.04, p=0.450). Interestingly, the direction shows students

with fewer teacher comments tended to achieve higher scores than those with more

teacher comments. This could be due to the lower need for corrective feedback from

teachers for students with better performance during the learning process. The effect of

TEACHERCOM for Algebra I (2) is also not significant (0.01, p= 0.912). Its direction is

different from the one in Algebra I (1). In Algebra I (2), students who received more

teacher comments tended to perform better than those students who received less

teacher comments.

State Standardized Test

Table 4-3 demonstrates the significant effect of TEACHERCOM (0.46, p=0.041)

for the grade 7 mathematics standardized test. The direction shows students with more

teacher comments performed better in this test. The effect of TEACHERCOM for grade

8 is not significant (0.002, p=0.990), with the same direction as it is in the grade 7

mathematics standardized test. Table 4-4 shows there is no significant effect of

TEACHERCOM (-0.02, p=0.949) on student score in the grade 6 mathematics

standardized test. Interestingly, the direction shows students with more teacher

comments tended to achieve lower scores in this test.

Research Question 3

Do student demographic information, such as race/ethnicity, grade level, status in

virtual school, whether have IEP, and participation in free/reduced lunch programs,

predict Algebra/mathematics performance in online education?









The virtual school student body is a diverse population of learners that includes

students with different learning disabilities (Dickson, 2005; Ferdig, Papanastasiou, &

DiPietro, 2005). In the present study, the virtual school follows individual education

plans for students with special needs during the learning process. Therefore, whether or

not a student has an individual education plan could be considered as a sign of student

learning ability which can affect student academic achievement during the learning

process (Keeler & Homey, 2007). Currently, many popular LMS, such as WebCT and

Blackboard, still have different problems that can prevent students with disabilities from

fully utilizing their functions even though they are generally accessible to these disabled

online students (Asuncion et al., 2006). Students' learning ability may be related to their

learning outcome through some other factors such as academic engagement

(Richardson et al., 2003). Based on a study comparing online students with a hearing

loss and those without this disability with respect to the relationship between students'

academic engagement and their perceptions of the academic quality of the courses,

Richardson et al. (2003) found the correlation between hearing status and students'

academic engagement and their perceived academic quality of the courses. Students

with hearing disability can't perform well on communication and some other tasks during

online learning processes as compared to students without this disability. This, in turn,

may negatively impact these disabled students' academic achievement.

Research shows that eligibility for school free or reduced lunch programs

correlates with academic achievements, with students not participating in these

programs achieving better performance (McLoyd, 1998). Participation in these

programs could be considered as the indicator of the family poverty level, which has a









strong relationship with student academic achievement at school level (Klein et al.,

2000). Considerable research has also found student academic achievement difference

among different racial groups, with Hispanic and African American students lagging

behind Caucasian and Asian American students (Bali, 2004; Barth, 2001; Hall et al.,

2000; Lockhead et al., 1985). Though students' race/ethnicity and their participation in

school free or reduced lunch programs have been proved to correlate with student

academic achievement in traditional face-to-face education, their effects have not been

examined systematically for virtual learning environments. In this study, student

demographic information including race/ethnicity, participation in free/reduced lunch

programs, learning ability, grade level, and status in virtual school were investigated

with other factors in one single equation.

EOC Test

Table 4-2 shows the participation in free or reduced lunch programs has no

significant effect (-0.04, p=0.992) on student EOC test score in Algebra I (1), with

students who did not participate in these programs tending to achieve higher scores.

The non significant effect (-4.36, p=0.172) of the participation in these programs was

also observed for Algebra I (2), with the same direction as it for Algebra I (1). There is

no significant difference (0.93, p=0.826) in student EOC test score between Caucasian

American students and the minority students for Algebra I (1) (see Table 4-2). The

direction shows minority students tend to perform better than Caucasian American

students. No significant difference (-2.53, p=0.455) was also found for Algebra I (2).

However, the direction is different from Algebra I (1), with Caucasian American students

tending to perform better.









Student grade level has a significant effect (-3.89, p=0.030) on student EOC test

score for Algebra I (1), with students in lower grade levels achieving better scores than

their counterparts in higher grade levels. The significant effect (-3.36, p=0.005) of grade

level has also been found for Algebra I (2), with the same direction as for Algebra I (1).

Table 4-2 demonstrates the non significant effect (-1.37, p=0.825) of student learning

ability on EOC test score for Algebra I (1). The direction shows the students who do not

have individual educational plans tended to perform better. Similarly, the non significant

effect (-2.26, p=0.557) of student learning ability is observed for Algebra I (2), with the

same direction as for Algebra I (1). Table 4-2 shows student status in the virtual school

(full-time or part-time) has no significant effect (1.78, p=0.733) on the EOC test score for

Algebra I (1), with full-time students tending to achieve better performance than part-

time students. The non significant effect of student status (6.49, p=0.096) has also been

found for Algebra I (2), with the same direction as for Algebra I (1).

State Standardized Test

Table 4-3 shows the participation in free or reduced lunch programs has no

significant effect (-0.89, p=0.936) on student score in the grade 7 mathematics

standardized test, with students who participated in these programs tending to perform

better. However, the strong and significant effect of participation in these programs (-

61.40, p=0.000) has been found for the grade 8 standardized test, with the same

direction as it is for the grade 7 standardized test. A significant difference (-21.28,

p=0.046) in student score in the grade 7 mathematics standardized test between

Caucasian American students and minority students is observed (see Table 4-3). The

direction shows the Caucasian American students achieved higher scores than the

minority students in this test. Table 4-3 demonstrates there is no significant difference (-









9.31, p=0.254) between these two ethnicity groups for the grade 8 standardized test,

with the same direction as it for the grade 7 standardized test.

Student's grade level in his/her physical school has no significant effect (-0.17, p=

0.083) on score in the grade 7 mathematics standardized test, with students in lower

grade levels tending to achieve higher scores (see Table4-3). The non significant effect

of student grade level (1.71, p=0.623) has also been found for the grade 8 mathematics

standardized test (see Table 4-3). However, the direction is different for the grade 7

standardized test. Table 4-3 shows there is a strong and significant effect of student

learning ability (-41.90, p=0.001) on student test score in the grade 7 standardized test,

with students without individual educational plans performing better than those students

with educational plans in the virtual school. The significant effect of student learning

ability (21.92, p=0.022) was also observed for the grade 8 mathematics standardized

test. Interestingly, the direction tells us students with individual educational plans

achieved better scores. Table 4-3 demonstrates the non significant effect of student

status in virtual school on student score for the grade 7 standardized test (4.97,

p=0.614), grade 8 standardized test (-11.98, p=0.146). However, the two directions are

different, with the direction for the grade 7 test showing full-time students tending to

perform better than part-time students and the direction for grade 8 test showing part-

time students tending to perform better.

Shown in table 4-4, the participation in free or reduced lunch programs has no

significant effect (-26.54, p=0.496) on student score in the grade 6 mathematics

standardized test, with students who did not participate in these programs tending to

achieve better scores than their counterparts who participated in these programs. There









is a nearly significant difference (-21.48, p=0.068) in student score between Caucasian

American students and the minority students for the grade 6 standardized test. The

direction shows Caucasian American students tended to perform better than the

minority students. Student grade level has no significant effect (-8.13, p=0.106) on

student score in the grade 6 standardized test, with students who are in the lower grade

levels tending to achieve better performance. A non significant while strong effect was

observed for student learning ability (-40.10, p=0.348) in the grade 6 mathematics

standardized test (see Table 4-4). The direction tells us students without individual

educational plans tended to perform better than their counterparts who had educational

plans. Table 4-4 demonstrates the non significant effect (11.62, p=0.344) of student

status in the virtual school for the grade 6 standardized test, with full-time students

tending to achieve better scores than part-time students.

Summary of Findings

The purpose of this study is to examine the factors including LMS utilization,

teacher comment/feedback and student demographic information that can influence the

success of Algebra courses in K-12 virtual learning environments. The three research

questions formulated sought to (1) discover the influence of student participation in

online academic activities on student mathematics achievement in virtual learning

environments; (2) explore whether teacher comment or feedback can predict student

academic achievement in online mathematics courses; and (3) investigate the

differences in online mathematics achievement among students with different

demographic information.

The results of question one showed the influence of participation in online

academic activities on achievement could be different based on mathematics levels.









The indicators of student participation in online academic achievements in this study

include the number of times student logged into the LMS (TOTALLOG) and how long

they stayed in the LMS (TOTALMIN). TOTALLOG has a significant influence on student

performance in Algebra I (2) EOC test (-0.002, p=0.006) while not in Algebra I (1) EOC

test. The direction of the significant influence showed students who logged into the LMS

less performed better. Similarly, TOTALMIN also has a significant influence for Algebra I

(2) EOC test (0.0004, p=0.008) while not for Algebra I (1) EOC test. The direction of the

significant influence indicated students who spent more time in the LMS achieved better

performance. For the state standardized mathematics test, TOTALLOG has no

significant influence on student performance at all the three levels: grade 6 to 8.

Similarly, TOTALMIN also has no significant influence at the three levels.

The results of question two provided the evidence that teacher comment can affect

student mathematics performance at different levels depending on the type of

mathematics tests. In this study, teacher comment has no significant effect on student

achievement in the two Algebra courses: Algebra I (1) and Algebra I (2). For the state

standardized mathematics test, teacher comment has a significant effect on student

achievement at the grade 7 level (0.46, p=0.041) while not at grade 6 and grade 8

levels. The significant effect at the grade 7 level showed students with more teacher

comments performed better in the test.

The results of question three showed some demographic information was

predictive of student online mathematics achievement while others may not be and the

predictability also depended on the type and the level of the mathematics test. The

participation in free or reduced lunch programs, race/ethnicity (Caucasian American or









minority), student learning ability, and student status in the virtual school (full-time or

part-time) were not predictive of student performance in Algebra I (1) and Algebra I (2)

EOC tests. However, student grade level is predictive of student performance in Algebra

I (1) (-3.89, p=0.030) and Algebra I (2) (-3.36, p=0.005) EOC tests, with students in

lower grade levels achieved higher scores. For the state standardized mathematics test,

the participation in free or reduced lunch programs, student grade level and status in the

virtual school were not predictive of student performance at all three levels: grade 6 to

8. The participation in free or reduced lunch programs was a significant predictor only at

the grade 8 level (-61.40, p=0.000). The direction showed students not participating in

these programs performed better. Race/ethnicity was a significant predictor only for the

grade 7 level test (-21.28, p=0.046), with Caucasian American students performing

better than the minority students. Student learning ability was a significant predictor for

the grade 7 level (-41.90, p=0.001) and the grade 8 level (21.92, p=0.022) tests. They

have different directions. For the grade 7 level test, students without individual

educational plans performed better than those with individual educational plans while for

the grade 8 level test, students with individual educational plans performed better.









Table 4-1: EOC test takers demographics
GRADE 8: 4(4.0%), 9: 35(34.7%), 10: 37(36.6%), 11: 15(14.9%), 12: 10(9.9%)
Algebra I (1st RACE White: 82(81.2%), Other Minority: 19(18.8%)
half) FRL 0: 68(67.3%), 1: 33(32.7%)
IEP 0: 94(93.1%), 1:7(6.9%)
PT/FT 0: 87(86.1%), 1: 14(13.9%)
GRADE 7: 2(2.7%), 8: 13(17.3%), 9: 24(32.0%), 10: 24(32.0%), 11: 9(12.0%),
12: 3(4.0%)
Algebra I (2nd RACE White: 62(82.7%), Other Minority: 13(17.3%)
half) FRL 0: 53(70.7%), 1: 22(29.3%)
IEP 0: 70(93.3%), 1: 5(6.7%)
PT/FT 0: 62(82.7%), 1: 13(17.3%)










Table 4-2: Standardized test takers demographics
GRADE 6: 35(47.3%), 7: 28(37.8%), 8: 6(8.1%), 9: 4(5.4%), 10: 1(1.4%)
RACE White: 60(81.1%), Other Minority: 14(18.9%)
Standardized test Grade
6 FRL 0: 6(8.1%), 1: 68(91.9%)
IEP 0: 5(6.8%), 1:69(93.2%)
PT/FT 0: 61(82.4%), 1: 13(17.6%)
GRADE 7: 29(39.7%), 8: 32(43.8%), 9: 9(12.3%), 10: 2(2.7%), missing:
1 (1.4%)
Standardized test Grade RACE White: 59(80.8%), Other Minority: 14(19.2%)
7 FRL 0: 16(21.9%), 1: 57(78.1%)
IEP 0: 19(26.0%), 1: 54(74.0%)
PT/FT 0: 68(93.2%), 1: 5(6.8%)
GRADE 8: 63(58.9%), 9: 39(36.4%), 10: 4(3.7%), missing: 1(0.9%)
RACE White: 83(77.6%), Other Minority: 24(22.4%)
Standardized test Grade
8 FRL 0: 20(18.7%), 1: 87(81.3%)
IEP 0: 20(18.7%), 1: 87(81.3%)
PT/FT 0: 89(83.2%), 1: 18(16.8%)









Table 4-3: Overview of RA model for different datasets
Test Variables df Sigma Square Tau ICC
Algebra I (1) None 79 41.79 225.26 0.84
Algebra II (2) None 56 63.25 146.04 0.70
Standardized test Grade 6 None 69 43.33 1797.90 0.98
Standardized test Grade 7 None 63 468.66 1189.96 0.72
Standardized test Grade 8 None 93 401.44 1052.68 0.72











Test


Fixed Effect Coefficient
GRADE -3.89
RACE 0.93
FRL -0.04
IEP -1.37
PT/FT 1.78
TEACHERCOM -0.04
TOTALLOG -0.03
TOTALMIN 0.0005
GRADE -3.36
RACE -2.53
FRL -4.36
IEP -2.26
PTFT 6.49
TEACHERCOM 0.01
TOTALLOG -0.02
TOTALMIN 0.0004


Table 4-4: Least-squares estimates of fixed effects (with robust standard errors)


Standard Error
1.76
4.22
3.93
6.19
5.20
0.06
0.03
0.00
1.14
3.37
3.15
3.83
3.85
0.09
0.01
0.00


Algebra I (1)







Algebra I (2)


T-ratio
-2.21
0.22
-0.01
-0.22
0.34
-0.76
-1.21
1.04
-2.96
-0.75
-1.38
-0.59
1.69
0.11
-2.90
2.74


d.f.
92
92
92
92
92
92
92
92
66
66
66
66
66
66
66
66


P-value
0.030
0.826
0.992
0.825
0.733
0.450
0.230
0.304
0.005
0.455
0.172
0.557
0.096
0.912
0.006
0.008










Table 4-5: Least-squares estimates of fixed
Test Fixed Effect Coefficient


Standardized test
Grade 7






Standardized test
Grade 8


GRADE -0.17
RACE -21.28
FRL -0.89
IEP -41.90
PT/FT 4.97
TEACHERCOM 0.46
TOTALLOG 0.02
TOTALMIN 0.0004
GRADE 1.71
RACE -9.31
FRL -61.40
IEP 21.92
PT/FT -11.98
TEACHERCOM 0.002
TOTALLOG -0.07
TOTALMIN 0.001


effects (with robust standard errors)


Standard Error
0.10
10.50
11.03
11.93
9.81
0.22
0.06
0.00
3.47
8.11
9.57
9.39
8.17
0.16
0.04
0.00


T-ratio
-1.76
-2.03
-0.08
-3.51
0.51
2.09
0.35
0.41
0.49
-1.15
-6.42
2.33
-1.47
0.01
-1.58
1.92


d.f.
64
64
64
64
64
64
64
64
98
98
98
98
98
98
98
98


P-value
0.083
0.046
0.936
0.001
0.614
0.041
0.725
0.680
0.623
0.254
0.000
0.022
0.146
0.990
0.117
0.057










Table 4-6 Ordinary Least-squares estimates of fixed effects
Test Fixed Effect Coefficient Standard Error T-ratio d.f. P-value
GRADE -8.13 4.97 -1.64 65 0.106
RACE -21.48 11.59 -1.85 65 0.068
FRL -26.54 38.75 -0.69 65 0.496
Standardized test IEP -40.10 42.42 -0.95 65 0.348
Grade 6 PT/FT 11.62 12.18 0.95 65 0.344
TEACHERCOM -0.02 0.26 -0.06 65 0.949
TOTALLOG -0.04 0.05 -0.82 65 0.414
TOTALMIN 0.0005 0.00 0.61 65 0.544









Table 4-7 Descriptive statistics for EOC test takers


Test


Algebra I (1)


Algebra I (2)


Variables
FINAL GRADE
GRADE
RACE
FRL
IEP
PT/FT
TEACHERCOM
TOTALLOG
TOTALMIN
FINAL GRADE
GRADE
RACE
FRL
IEP
PT/FT
TEACHERCOM
TOTALLOG
TOTALMIN


N
101
101
101
101
101
101
101
101
101
75
75
75
75
75
75
75
75
75


Mean
70.82
9.92
.19
.33
.07
.14
20.78
215.36
10783.93
79.00
9.45
.17
.29
.07
.17
15.31
461.04
27425.76


Std. Deviation
15.598
1.026
.393
.471
.255
.347
21.390
125.476
6418.637
13.650
1.119
.381
.458
.251
.381
21.526
288.174
17097.182










Table 4-8 Descriptive statistics for standardized test takers
Test Variables N
MAP SCALE SCORE 74
GRADE 74
RACE 74
MAP FRL 74
Standardized test Grade IEP STUDENT 74
6TFT 74
PTFT 74


Standardized test Grade
7


TEACHERCOM
TOTALLOG
TOTALMIN
MAP SCALE SCORE
GRADE
RACE
MAPFRL
IEPSTUDENT
PTFT
TEACHERCOM
TOTALLOG
TOTALMIN


MAP SCALE SCORE 107 696.23 38.020
GRADE 107 8.36 .994
RACE 107 .22 .419
MAP FRL 107 .81 .392
Standardized test Grade 107 81 392
IEP STUDENT 107 .81 .392
8
PTFT 107 .17 .376
TEACHERCOM 107 11.28 18.569
TOTALLOG 107 293.93 202.395
TOTALMIN 107 18067.10 14548.311


Mean
664.07
6.76
.19
.92
.93
.18
11.32
354.65
23006.26
676.18
9.03
.19
.78
.74
.07
11.15
253.25
16720.51


Std. Deviation
41.827
.919
.394
.275
.253
.383
18.144
181.272
13370.676
41.913
10.704
.396
.417
.442
.254
19.715
141.193
9324.034









Table 4-9 Standardized coefficients for EOC test takers
Tests Factors Standardized Coefficients
GRADE -0.26
RACE 0.36
Algebra I (1) FRL -0.05
IEP -0.74
PT/FT 2.42
TEACHERCOM -2.47
TOTALLOG -0.18
TOTALMIN 0.03
GRADE -0.28
RACE -0.86
FRL -5.24
IEP -1.24
Algebra I (2) PTT 9.
PTIFT 9.85
TEACHERCOM 0.56
TOTALLOG -0.27
TOTALMIN 0.02









Table 4-10 Standardized coefficients for standardized test takers
Factors Standardized Coefficients
GRADE -0.18
RACE -9.21
MAPFRL -18.52
Standardized test IEP_STUDENT -36.89
Grade 6 PTFT 17.59
TEACHERCOM -0.95
TOTALLOG -0.40
TOTALMIN 0.04
GRADE -0.04
RACE -0.79
MAPFRL -0.93
IEP STUDENT -44.41
Standardized test
PTFT 2.86
Grade 7
TEACHERCOM 35.70
TOTALLOG 0.143
TOTALMIN 0.03
GRADE 0.04
RACE -3.92
MAPFRL -57.44
Standardized test IEP_STUDENT 21.92
Grade 8 PTFT -11.49
TEACHERCOM 0.10
TOTALLOG -0.76
TOTALMIN 0.07









Table 4-11 Adjusted R-squares
Tests VARe VAR t Rc
Algebra 1(1) 236.69 243.29 0.03
Algebra I (2) 159.04 186.32 0.15
Standardized test Grade 6 1412.24 1749.46 0.19
Standardized test Grade 7 1440.78 1756.73 0.18
Standardized test Grade 8 1146.03 1445.5 0.21









CHAPTER 5
DISCUSSION AND IMPLICATIONS

Introduction

This chapter summarizes the findings of the present study and presents the

important conclusions drawn from the data shown in Chapter 4. It also addresses the

implications for teaching, research, and policy making processes in the discussion of

the findings. This chapter presents the implications based on the three research

questions in the present study.

Summary of Study

To explore different success factors for online mathematics in general and online

Algebra in specific, the present study investigated the effect of a variety of variables on

student achievement on Algebra EOC tests and state standard mathematics tests. The

present study used the secondary data collected from a state led virtual school in the

Midwestern US region. The variables include learner characteristic variables such as

student demographic information and participation level in online academic activities

and learning environment characteristics variables such as teacher comment in the

present study.

Overview of the Problem

The U.S has experienced an astonishing growth in online education at K-12 level

during the past decade. The enrollment of K-12 virtual school students has increased

from 40,000 in 2000-01 academic year to 1 million in 2007-08 academic year (Clark

2001; Glass, 2009; Newman, Stein & Trask, 2003; Peak Group, 2002; Picciano &

Seaman, 2009; Picciano & Seaman, 2007; Setzer & Lewis, 2005; Tucker, 2007;

Zandberg, Lewis, & Greene, 2008). With the large population of online learners, it's









possible to evaluate the effectiveness of online courses. However, currently, there is no

one single model being created to predict online success and no clear set of

characteristics that have been identified in this regard (Roblyer & Davis, 2008; Tallent-

Runnels et al., 2006).

Math has been considered a very important force to push a society forward. Many

countries emphasize the improvement of math knowledge and they develop policies to

attract more people into this field. Having good academic performance in math subjects

at the K-12 level is important for students to pursue advanced degrees in this field. It will

help prepare more students to have careers in Science, Technology, Engineering, and

Mathematics (STEM) and increase workforce for U.S. in these fields, which could

provide strong momentum for this country to move forward in many aspects. The quality

of Algebra courses is essential in building the number of U.S. students who are ready

for advanced degrees in STEM and career success in these fields.

Purpose Statement and Research Questions

The purpose of this study is to examine the factors including LMS utilization,

teacher comment/feedback and student demographic information that can influence the

success of Algebra courses in K-12 virtual learning environments. The research

questions in this study are

* Does the level of LMS utilization influence Algebra/mathematics performance in
online education?

* Does teacher comment or feedback influence Algebra/mathematics performance in
online education?

* Do student demographic information such as race/ethnicity, grade level, status in
virtual school, whether have individual educational plan (IEP), and participation in
free/reduced lunch programs influence Algebra/mathematics performance in online
education?









Review of the Methodology

The present study is descriptive in nature. The researcher described the

relationship between some factors and student learning outcome without intervening.

The researcher received the secondary data from one state led virtual school in the

Midwestern US region that collected student demographic information, participation in

online academic activities, teacher comments, and academic achievement on Algebra

EOC tests and state standardized mathematics tests in the 2008-09 academic year.

The present study builds on the results of Liu and Cavanaugh (2010)'s study. This

virtual school was launched in 2007. Liu and Cavanaugh's study used the first year

(2007-08) data collected by this virtual school and the present study used the second

year data. The present study investigated success factors for both Algebra EOC tests

and one state standardized mathematics test while Liu and Cavanaugh's study only

investigated these factors for Algebra EOC tests. The data regarding student

demographic information, participation in online academic activities and teacher

comments were collected by the LMS utilized by this virtual school for course content

delivery. Student academic achievement on (1) the Algebra I, II EOC tests administered

at the end of semester designed based on the state Algebra standards and (2) one

state standard mathematics test, designed based on the state mathematics standards,

was collected. HLM was carried out by the software program HLM 6.06 for data analysis

in this study to account for the clustering of academic scores of students recruited from

the same physical school. The rest of chapter 5 will discuss the outcomes and review

the implications associated with the three research questions designed to examine the

impact of different success factors on student mathematics achievement. This chapter

will close with the conclusions drawn from the findings shown in chapter 4.









Findings

In the present study, RA model was analyzed at the beginning to partition the total

variance of student score into within-school and between-school components. The intra-

class correlation coefficient was calculated for the five groups and it was Algebra I (1) -

.84, Algebra II (2) -.70, Standardized test Grade 6 -.98, Standardized test Grade 7 -

.72, and Standardized test Grade 8 .72 respectively. This shows the between-school

variance was big in comparison with the within-school variance for all these five groups

especially for the Standardized test Grade 6 group. Partially, it could be attributed to the

small number of students per school. The big ICC indicated students from different

schools are different from each other with respect to their academic achievement. This

finding confirmed the gap between private and public schools in student academic

achievement found in other studies (Braun, Jenkins, Grigg, Tirre, Spellings, Whitehurst,

& Schneider, 2006; Demircioglu & Norman, 1999; Lubienski & Lubienski, 2005). It also

could indicate that it will take time for the standardized testing criterion to be well

implemented in the school system of this state.

In the present study, standardized coefficient (3) was calculated according to the

formula: Pk =bk Sxk Sy (bk is the unstandardized coefficient, Sxk is the standard

deviation of the corresponding independent variable, and Sy is the standard deviation of

the dependent variable). It can be used to compare among different factors with respect

to the importance in determining test score. Table 4-9 shows that for Algebra I (1)

group, student status (3 = 2.42) and teacher comment (3 = -2.47) were most important

factors, and for Algebra I (2) group, participation in school free or reduced lunch

programs (3 = -5.24) and student status (3 = 9.85) were most important factors. Table 4-

10 shows that for standardized test grade 6, participation in school free or reduced









lunch programs (3 = -18.52), whether have IEP (3 = -36.89) and student status (3 =

17.59) were most important factors. For standardized test grade 7, whether have IEP (3

= -44.41) and teacher comment (3 = 35.70) were most important factors while for

standardized test grade 8, participation in school free or reduced lunch programs (3 = -

57.44), whether have IEP (3 = 21.92) and student status (3 = -11.49) were most

important factors. These findings show that the same factors can influence student test

score differently for different online Algebra courses. The adjusted R-square (Rc2) was

also calculated according to the formula: Rc2= 1- VAR eNAR t to show the reduction of

test score variance from the RA model with the addition of the factors. Table 4-10

shows that the same set of factors accounted for student score variance at different

degree for different tests. All these findings demonstrated the complexity of the

investigation of factors influencing success in online Algebra.

The following Table 5-1 shows the summary of the significance and direction of

the factor effect on student academic performance in the five tests. The "+" sign

indicates the positive direction of the factor effect and the "-" sign indicates the negative

direction of the factor effect. The "X" sign indicates the factor had a significant effect on

student academic performance in the corresponding test.

Research Question 1

Does the level of LMS utilization influence Algebra/mathematics performance in

online education?

The time spent in academic activities has been identified as a very important factor

that has strong effect on success in online education (Cavanaugh, 2007), face-to-face

instruction (Rocha, 2007), and blended programs (Cavanaugh, 2009). Based on a study

for the investigation of the cognitive-motivational and demographic characteristics of


100









online students and the predictors for their success, Wang and Newlin (2000) found out

students who participate in online activities at a high level tend to perform well in the

online course. They concluded the total online course activity is a predictor of students'

final grades. Compared to the students in traditional classrooms, online students spend

more time in the virtual learning environments on interacting with one another on

academic topics (O'Dwyer et al., 2007). The peer-to-peer interaction, in turn, could help

improve online students' learning outcomes (Cavanaugh, 2007). In the present study,

the numbers of times students logged into the LMS and how long they stayed in the

LMS were considered the indication of student participation level on online academic

activities. The number of times students logged into the LMS also has been identified

as a strong predictor of student academic performance in online learning (Dietz, 2002;

Dickson, 2005).

Compared to traditional classroom instructors, online instructors lack of the regular

set of cues about students' confusion or frustration during the learning process such as

their facial expression and body positions. The measure of time students spent in the

online academic activities can provide online instructors the information about students'

understanding of content materials. A lower level of involvement in online activities in

the course at the beginning of the semester could be an early warning sign of failure

later during the learning process. Therefore, online instructors should closely monitor

these behaviors via LMS login data to prevent students who show warning signs at the

beginning from failure.

The influence of time students spent in the LMS was found to be positive for the

five groups and significant for Algebra I (2). These findings are aligned with the


101









statement Wang and Newlin (2000) made in their study mentioned above that students

participating in online academic activities at a higher level achieve better performance in

online learning. They echo the call for sustained time on task for cognitive learning

(Gallagher, 2009) and provide support for the emphasis of expanded learning time,

including with online courses, to improve academic achievement (Cavanaugh, 2009).

These findings could be explained by the call for changes in instructional practices in

mathematics education by many educational reformers such as the implementation of

standards for mathematics instruction from the National Council of Teachers of

Mathematics (NCTM, 1989, 1991) and the active involvement in academic activities is

one of their arguments (Forman, 1996). This also confirmed the value of increased

participation in learning activities in mathematics education emphasized in Forman's

article. However, it's surprising to the researcher that this factor only had a significant

effect for Algebra I (2) in this study considering many other studies already showed the

importance of time on task for the improvement of student achievement. Many of the

students taking Algebra I (2) course are from higher grade levels for credit recovery or

to make up failing grades in their physical schools. The increased engaged time on task

could be more effective with respect to the improvement of academic achievement than

the other four groups. Nevertheless, the significant effect of time spent in the LMS for 1

out of 5 groups calls for more studies on activities that engage students during their stay

in the LMS as an explanation for the findings.

The effect of the number of times students logged into the LMS on student

academic achievement is negative for the 4 out 5 groups and negative and significant

for Algebra I (2). To some degree, this is contradictory to the belief that the number of


102









times students login to the LMS is a strong and positive predictor of success in online

learning (Dietz, 2002; Dickson, 2005). It is possible that if students are logging into the

course environment more often, they are staying and working for shorter time periods,

negatively impacting their concentration on their studies. This also calls for more

research on the investigation of LMS utilization with bigger sample size and diversified

mathematics tests.

Implications Related to Research Question 1

Several implications for research, policy and practice can be drawn from the

outcomes associated with research question one even though the results found in this

study are mixed. These implications provide guidance for future studies to investigate

the effect of time on task and the form of activities students engaged in when they stay

in the LMS on academic achievement in virtual learning environments. The positive and

significant effect of the time spent in LMS for Algebra I (2) shows students who spent

more time in the LMS performed better than students spending less time in the LMS. It

is plausible that each log in session of high-performing students was longer than the

session length for lower-performing students, showing that high-performing students

may benefit from sustained time on task rather than more frequent but shorter time on

task. This explanation would support flexible online courses that allow students to stay

in the course for extended periods of time while working on complex and abstract

content. The positive influence of time spent in the LMS provides the support for the

improvement of many LMSs to make them more user-friendly with attractive interfaces

that motivate students to spend more time in the system engaging in academic activities

delivered during the learning process, as well as teaching practices that foster


103









connectedness among teachers and students and time management strategies for

students who do not have high self-regulation abilities.

As mentioned above, the significant effect of time spent in the LMS on only 1 out

of 5 groups and the contradictory directions of the effect of number of times student

logged into the LMS calls for more research in this field. Future researchers should

investigate the activities students engaged in each time they logged into the LMS and

the distribution of the logged in times throughout the semester for deeper understanding

of the effect of time on task on academic achievement in virtual learning environments.

This information could be used to help teachers have better knowledge about the

activities in which students are more interested and their engagement level in academic

activities during the learning process. Online instructors and course designers could

design and develop better online activities, specifically activities that are more

individualized, diverse, and authentic to increase engaged learning time.

Research Question 2

Does teacher comment or feedback influence Algebra/mathematics performance

in online education?

Educators' active facilitation of online learning and teachers' feedback are

important factors that influence students' academic performance during the learning

process (Dickson, 2005; Cavanaugh, Gillan, Bosnick, Hess, & Scott, 2005; Hughes,

McLeod, Brown, Maeda, & Choi, 2005; Ferdig, Papanastasiou, & DiPietro, 2005;

Zucker, 2005). Many of the pedagogical benefits brought by the student-teacher

interaction such as those related to motivation and feedback are relevant to distance

education as well as the conventional classroom education (Anderson & Kuskis, 2007).

Teacher individual feedback and comments are especially helpful for students who are


104









shy and may not participate in academic activities to increase their communication

opportunities (Hughes et al., 2005), which could help improve learning outcomes. Based

on the results of a study of students with learning disabilities (SLD) and students with

attention deficit hyperactivity disorder (ADHD), Smouse (2005) found communication

with and feedback from instructors was the most valuable aspect of online courses.

Timely and constructive teachers' feedback to students' assignments and questions

have been identified as the critical characteristics of the teaching/learning benchmarks

for the quality of online learning (Phipps & Merisotis, 2000).

The influence of teacher comments on student academic achievement is positive

and significant for standard test grade 7. This provides the evidence of the importance

of teacher comments and teacher-student interaction for the improvement of learning

outcomes in online learning environments (Cavanaugh et al., 2005; Peters, 1999;

Williams, 2006; Zucker, 2005). It confirmed the relationship between social interaction

and mathematics learning (Cobb, Yackel & Wood, 1992; Vogit, 1996). This finding also

align with the belief that in mathematics education, the interactions between adults

(instructor) and children (students) have impacts on the quality of learning outcome via

the psychological benefits brought to students such as critical thinking and self-

reflection (Oers, 1996) and on students' cognitive development (Voigt, 1994). In the

process of interaction or negotiation during mathematics learning, students can build the

connections between materials and mathematics terms (Voigt, 1994), which can be

proved, to some degree, by these this finding. However, it's surprising to the researcher

that the influence of this factor is found to be positive and significant for 1 out of 5

groups considering many studies have shown the positive effect of this factor on


105









academic achievement. Variables other than teacher comments especially the number

of teacher comments investigated in the present study such as teaching styles, quality

of teacher comments, etc., could have more influence on student achievement. Due to

the small sample size for each group in the present study, readers should be cautious

about the interpretation of the findings.

Implications Related to Research Question 2

The positive and significant effect of teacher comment or feedback for standard

test grade 7 provides the support for the statement about the importance of this factor in

other studies. This finding could shed light on the development of online courses that

integrate teacher feedback and teacher-student interaction as critical components

during the course design. It also indicates the importance of timely and constructive

feedback from online instructors for the success in online learning. However, the

significant effect of this factor for only 1 out 5 groups in the present study showed more

research is needed with a bigger sample size and on the form and content of teacher

feedback for insightful explanation. It also supports the call for more study about the

most effective interaction type, tools, and frequency for the participants in online

learning (Cavanaugh, 2007) for online instructors to better facilitate the learners to

achieve success. The integration of qualitative dimensions of interaction in future study

also will assist in the triangulation of quantitative dimensions as shown in the present

study to better understand the importance of teacher-student interaction in online

learning (Weiner, 2003).


106









Research Question 3

Do student demographic information such as race/ethnicity, grade level, status in

virtual school, whether have IEP, and participation in free/reduced lunch programs

influence Algebra/mathematics performance in online education?

Students' demographic information such as participation in free or reduced lunch

programs, race/ethnicity and whether have IEP has been proved to relate to students

academic achievement in other studies. McLoyd (1998)'s study showed there is a

correlation between student participation in school free or reduced lunch programs and

student academic achievement: the magnitude is weaker as grade level rises. Student

participation in these programs can be considered as the indicator of his/her family

Social Economics Status (SES) (Sirin, 2005), which has already been proved to affect

student academic performance in many studies (Coleman, 1988; Sirin, 2005; White,

1982). In the present study, the influence of participation in these programs on

achievement is negative for all the five groups and significant for standard test grade 8.

For standard test grade 8 group, students who did not participated in these lunch

programs achieved higher scores than students who participate in these programs. This

provides support for the findings regarding the correlation between this factor and

achievement in other studies mentioned above. The negative influence of this factor

found in the present study can add to the body of knowledge about the correlation

between SES and academic achievement. Other studies about the relationship between

participation in free/reduced lunch programs or SES and academic achievement are all

conducted in traditional learning environments. The result in the present study

demonstrates the possibility for the generalization of study findings between traditional

learning environments and virtual learning environments. On the other hand, the


107









influence of participation in free or reduced lunch programs is only significant for 1 out of

5 groups in this study is surprising to the researcher considering the body of research

demonstrating the correlation between this factor and student academic achievement.

The virtual school student body is a diverse population including students with

different learning disabilities (Dickson, 2005; Ferdig, Papanastasiou & DiPietro, 2005).

Virtual schools offer or support individual educational plans (IEP) for these students

during the learning process. Therefore, whether a student has an individual educational

plan could be a sign of the level of learning abilities. Many technologies utilized in virtual

school learning environments could help bridge gaps between students with disabilities

and students without these disabilities with respect to the success opportunities in

online learning (Coombs & Banks, 2000). However, students with disabilities are still

underrepresented in online education (Kinash & Crichton, 2007). The present study

provides some evidence for this claim. For example, the influence of IEP on student

achievement is negative and significant for standardized test grade 7 with students who

did not have an IEP (usually students without learning disabilities) performed better than

students who had an IEP (usually students with learning disabilities). However, for

standard test grade 8, the influence of IEP is positive and significant favoring students

with learning disabilities. This finding could indicate that the virtual school may be able

to help improve academic achievement for students at risk for failure in their physical

schools. It also could be a sign of bridging gaps between students with learning

disabilities and others without these disabilities with respect to their academic

performance possibly due to the academic support provided through the IEP.


108









Racial gaps in student test score have been proved in many other studies

conducted in traditional learning environments (Bali & Alvarez, 2004; Barth, 2001; Hall

et al., 2000; Lockhead et al., 1985). The student body in online K-12 schools represents

the community that is served by traditional school system (Ronsisvalle & Watkins,

2005). The findings about the racial gaps in student achievement in other studies could

apply to the present study as well. The significant racial difference for standardized test

grade 7 and nearly significant difference for standardized test grade 6 provides the

evidence for the findings in other studies. The directions of the difference in the two

groups show white American students perform better than other minority groups as a

whole. However, the finding that the significant racial difference was only found for 1 out

5 groups could be due to the coding system that combined different minority groups into

one category potentially masking important information regarding the differences in

student academic achievement among different racial groups. Future study could be

conducted to investigate these differences with bigger sample size. The effect of

student grade level in physical school was found to be negative and significant for two

Algebra I groups, with students from lower grade levels performing better than those

from higher grade levels. Students taking the standardized tests are from lower levels

(grade 6-8) compared with the students who took the Algebra I courses and Algebra I

EOC tests (most of them from grade 9-12). Algebra I is a required course for high

school graduation. Many students in higher grades such as grade 11, 12 take Algebra I

courses in this virtual school as credit recovery or remediation to make up failing grades

in their physical schools to meet the graduation requirement. It could be the explanation

for the negative and significant effect of this factor for the two Algebra I groups. The


109









effect of student status in the virtual school (part-time or full-time online students) was

not significant for all the five groups.

Implications Related to Research Question 3

In the present study, the influence of participation in free or reduced lunch

programs is negative for all the five groups and significant for standard test grade 8. For

standard test grade 8 group, students who participated in these lunch programs

achieved lower scores than students who did not participate in these programs. This

echoes the belief that family SES could affect student academic achievement via its

influence on parental involvement in virtual learning environments (Black, 2009). This

finding can guide the decision making process in the virtual schools by encouraging

them to be sensitive to the needs of students with low family SES background and to

take measures to bridge the gap in access to resources that could influence student

academic achievement. The effect of IEP on student achievement is negative for 4 out

of 5 groups and significant for standard test grade 7 in the present study. For standard

test grade 7 group, students with IEP (usually students with learning disabilities)

achieved lower performance than others without IEP. This could provide support for the

integration of instructional strategies such as hiring academic coaches or tutors or

advanced technologies during the online learning process to help students with

disabilities succeed.

The negative and significant effect of grade level for the two Algebra I groups

could be explained by the situation that many students in higher grade levels take

Algebra I courses in this virtual school to make up the credits lost in their physical

schools to meet the graduation requirement. This has the implication for the virtual


110









school during online course design for the implementation of certain strategies such as

peer support and online tutoring, or flexible timelines and multiple paths to help higher

grade students in Algebra I courses to achieve better performance. Online teachers also

should provide individual assistance based on the needs of different students.

Broad Implications for Online Course Design and Online Teaching

In September 2007, International Association for K-12 Online Learning (iNACOL)

endorsed the National Standards of Quality for Online Courses based on the Southern

Regional Education Board (SREB) Standards for Quality Online Courses. In February

2008, iNACOL released National Standards for Quality Online Teaching based on

SREB's Standards for Quality Online Teaching and Online Teaching Evaluation for

State Virtual Schools. The SREB's two sets of standards have been widely used by the

16 states in the southern United States. iNACOL' National Standards of Quality for

Online Courses standards were designed to "provide states, districts, online programs,

and other organizations with a set of quality guidelines for online course content,

instructional design, technology, student assessment, and course management."

(iNACOL, 2006, p.1). There are 6 categories in iNACOL standards:

1. Content
2. Instructional Design
3. Student Assessment
4. Technology
5. Course Evaluation and Management
6. 21st Century Skills.

Under each category there are a set of standards. National Standards for Quality

Online Teaching is designed to "provide states, districts, online programs, and other

organizations with a set of quality guidelines for online teaching and instructional

design." (iNACOL, 2008, p.1). There are 13 categories in these standards:


111









A. The teacher meets the professional teaching standards established by a state-
licensing agency or the teacher has academic credentials in the field in which he
or she is teaching.

B. The teacher has the prerequisite technology skills to teach online.

C. The teacher plans, designs and incorporates strategies to encourage active
learning, interaction, participation and collaboration in the online environment.

D. The teacher provides online leadership in a manner that promotes student
success through regular feedback, prompt response and clear expectations.

E. The teacher models, guides and encourages legal, ethnical, safe and healthy
behavior related to technology use.

F. The teacher has experienced online learning from the perspective of a student.

G. The teacher understands and is responsive to students with special needs in the
online classroom.

H. The teacher demonstrates competencies in creating and implementing
assessments in online learning environments in ways that assure validity and
reliability of instruments and procedures.

I. The teacher develops and delivers assessments, projects, and assignments that
meet standards-based learning goals and assesses learning progress by
measuring student achievement of learning goals.

J. The teacher demonstrates competencies in using data and findings from
assessments and other data sources to modify instructional methods and content
and to guide student learning.

K. The teacher demonstrates frequent and effective strategies that enable both
teacher and students to complete self- and pre- assessments.

L. The teacher collaborates with colleagues.

M. The teacher arranges media and content to help students and teachers transfer
knowledge most effectively in the online environment. (Instructional Design)

Under each category there are a set of standards. Many of the findings, derivative

outcomes, or implications in the present study align with the two sets of standards. The

following two tables show these alignments.


112









Conclusions

This dissertation examined the impact of some variables including students'

demographic information, teacher comments, and student utilization of the LMS on

academic performance in Algebra EOC tests and state standard mathematics tests

using a sample of students from a state led virtual school in the Midwestern U.S region.

The results show different variables affect student Algebra/mathematics achievement in

different ways. No single factor investigated in the present study has been found to be

significant for all five groups. It could be due to the limitations mentioned in Chapter 3:

Methodology. It also indicated that some other factors such as instructional strategies

utilized, teacher experience and student prior subject knowledge could have been

missed in the present study. They should be investigated in the future studies on

success factors in the virtual schooling. Outcomes of this study have some specific

implications for researchers, practitioners, and policy makers.

The results show the time student spent in the LMS has positive influence on

student academic achievement. This provides the support for the online instructional

designers or LMS developers to utilize more advanced technologies such as some

educational games and refine the course delivery system to motivate students learn the

content and spend more time engaging the academic activities. It also can lend

relevance to online instructors for the implementation of instructional strategies to

encourage students to focus on the learning tasks during their stay in the course

delivery systems. The results of data analysis in this study show the influences of many

factors are mixed. Some are positive, and others are negative. Even for the same

factor, the influence could be in different directions for different tests. This indicates that

the investigation of success factors of online learning is a complex process in which


113









quantitative methodology independently may not be able to effectively measure the

influence of the factors on academic achievement. Therefore, future research seeking to

investigate the success factors in online learning should utilize mixed methodology

incorporating quantitative and qualitative methods.

This dissertation has implications for policy-making processes at state and

national level regarding quality virtual schooling and research support. At state levels in

which the virtual school is implemented, effective and well-designed LMS should be

utilized for course delivery and management. The LMS interface should be user-friendly

that can attract students' attentions in longer periods during the learning process.

Components such as online forum, incorporation of social networking software, online

synchronous audio/video conferencing should be integrated in the LMS to encourage

more and diversified teacher-student and student-student interaction. To increase

success opportunities for all students, virtual schools should take some measures to

increase access of students from lower SES households the learning resources such as

additional lab time, one on one computer/laptop, or extra instructional time. Virtual

school should provide individualized assistance based on students' different needs such

as

* for students with learning disabilities it could be individual education plans

* for students taking the online courses for credit recovery it could be peer-to-peer
support or group projects

At national level, more support should be provided to help build better designed

state led virtual schools to increase access to more effective learning resource for all

students. More national standards regarding quality virtual schooling should be created

to guide the practice and implementation of state level virtual schools. Both at state and


114









national level, policy makers should grant more resources to support more empirical

study collecting quantitative and qualitative data to provide evidence for policies making

process. More research is needed on student academic achievement, online success

model, and longitudinal study on virtual school retention. One data system regarding

virtual school practice should be built both at state and national level from which the

researchers can draw the information they need to conduct the secondary research

similar to the present study. These secondary research studies can supplement the first

hand studies though they may have limitations such as missing information like the

present study that lacks of qualitative data for some factors.

Since the establishment of the first virtual school at the end of 20th century, it has

experienced an extraordinary development during the last one decade. However, with

its short history, K-12 virtual schools are still a relatively new concept for many

researchers and educators. Compared to online education at post-secondary level, little

research has been done in K-12 virtual learning environments (Cavanaugh 2007; Cooze

& Barbour, 2005; Means et al., 2009; Picciano & Seaman, 2007; Picciano & Seaman,

2009; Ronsisvalle & Watkins, 2005). The present study is the first research on success

factors in K-12 virtual learning environments. At present, no clear set of characteristics

have been identified to predict success in virtual learning environments, and no

conclusive model has been created to apply in online learning practice (Roblyer &

Davis, 2008; Tallent-Runnels et al., 2006). However, to help improve the practice and

implementation of virtual schooling, Smith et al. (2005) emphasized the empirical

studies on student academic achievement. Given the dearth of research on success

factors in K-12 online learning environments, this dissertation should serve as the


115









starting point for more studies utilizing both qualitative and quantitative methods to help

the development of one success model to improve student academic achievement in

virtual schooling.


116









Table 5-1: Significance and Direction of the Effect of Factors
Grade Race Free or IEP Student Teacher Number Time
Level Reduced Status Comment of Spent
ourse\Factor Lunch Times in the
Course\Factor
Logged LMS
into the
LMS
Algebra 1st half -X + + +
I 2nd half -X + + -X +X
Grade 6 + +
MAP Grade 7 -X -X + + X + +
Grade 8 + -X + X + +
TOTAL of 5 2 1 1 2 0 1 1 1










Table 5-2: Alignment with National Standards in Quality Online Course
Findings, derivative outcomes, or implications Aligned standards in iNACOL National Standards of
of the present study Quality for Online Courses
Tests used in the present study align with state Course tasks and assessments align with the
or national standards. required local, state, and national assessments that
are associated with the course. (A)
This virtual school hires the authorized course The course provider is authorized to operate in the
provider to implement the Learning state in which the course is offered. (E)
Management System (LMS) and content area
teachers who met state certification and other The teacher meets the professional teaching
requirements as online instructors, standard established by a state licensing agency or
the teacher has academic credentials in the field in
which he or she is teaching and has been trained to
teach online and to use the course. (E)
Flexible online courses that allow students to The course instruction includes activities that
stay in the course for extended periods of time engage students in active learning. (B)
while working on complex and abstract
content. The course provides opportunities for students to
engage in higher-order thinking, critical-reasoning
activities and thinking in increasingly complex ways.
(B)
Improvement of many LMSs to integrate The course design provides opportunities for
teaching practices that foster connectedness appropriate instructor-student interaction, including
among teachers and students. timely and frequent feedback about student
progress. (B)
Development of online courses that integrate
teacher feedback and teacher-student The course provides opportunities for appropriate
interaction as critical components during the instructor-student and student-student interaction to
course design. foster mastery and application of the material and a
plan for monitoring that interaction. (B)

Integration of instructional strategies or The course meets universal design principles,
advanced technologies during the online Section 508 standards and W3C guidelines to
learning process to help students with ensure access for all students. (D)
disabilities to succeed.


118









Table 5-3: Alignment with National Standards in Quality Online Teaching


Findings, derivative outcomes, or
implications of the present study
The importance of timely and
constructive feedback from online
instructors for the success in online
learning.


The finding that students who
participated in free or reduced lunch
programs achieved lower scores than
students who did not participate in
these programs for state standardized
test grade 8 group could guide the
decision making process in the virtual
schools by encouraging them to be
sensitive to the needs of students with
low family SES background and to take
measures to bridge the gap in access
to resources that could influence
student academic achievement.


Aligned standards in iNACOL National
Standards for Quality Online Teaching
Encourages interaction and cooperation
among students, encourages active
learning, provides prompt feedback,
communicates high expectations, and
respects diverse talents and learning
styles. (D The teacher provides online
leadership in a manner that promotes
student success through regular feedback,
prompt response and clear expectations.)
Establishes and maintains ongoing and
frequent teacher-student interaction,
student-student interaction and teacher-
parent interaction. (D)
Provides timely, constructive feedback to
students about assignments and questions.
(D)
Personalizes feedback (support, growth
and encouragement). (D)

Creates a warm and inviting atmosphere
that promotes the development of a sense
of community among participants. (C The
teacher plans, designs and incorporates
strategies to encourage active learning,
interaction, participation and collaboration
in the online environment)
Provides activities, modified as necessary,
that are relevant to the needs of all
students. (G The teacher understands and
is responsive to students with special
needs in the online classroom.)


119









Table 5-3. Continued
Findings, derivative outcomes, or
implications of the present study
Online teachers should provide
individual assistance based on the
needs of different students.


Aligned standards in iNACOL National
Standards for Quality Online Teaching
Provides activities, modified as necessary,
that are relevant to the needs of all
students. (G)

Personalizes feedback (support, growth
and encouragement). (D)

Provides evidence of effective learning
strategies that worked for the individual
student and details specific changes in
future instruction based upon assessment
results and research study (data-driven
and research- based). (J The teacher
demonstrates competencies in using data
and findings from assessments and other
data sources to modify instructional
methods and content and to guide student
learning.)


120










APPENDIX A
ALGEBRA I MULTIPLE CHOICE RELEASED SAMPLES



Algebra I


Directions to the Student


Today you will be taking Session I of the Missouri Algebra I Test. This
is a test of how well you understand the course level expectations for
Algebra I.


There are several important things to remember:
1 Read each question carefully and think about the answer. Then
choose the one answer that you think is best.
2 Make sure you completely fill in the bubble for the answer on your
answer sheet with a number 2 pencil.
3 If you do not know the answer to a question, skip it and go on. You
may return to it later if you have time.
4 If you finish the test early, you may check overyour work.
5 Do NOT wnte in your test booklet. Mark your answers directly on your
answer sheet with a number 2 pencil.


Copyright 0 208 by the Missouri State Department of Ecmcntary and Secondary Education


121














Algebra I


1. If the first Now = -9, which equation represents this sequence?

-9, -4, 1, 6, 11,...


A. Next

B. Next

C. Next

D. Next


Now 5

Now + 5

5 Now 1

5 Now I 1


2. Which inequality statement is true?

A. 8 < 78 <9

B. 38 < 1/78 < 40

C. 77 < 78 < 79

D. 6,083 < V78 < 6,085


Copyright 0 2008 by the Missouri State Department of Elementary and Secondary Education.


122














Algebra I


3. Daniel made a box-and-whisker plot of the ages of his cousins.





12 14 16 18 20 22 24 26 28 30


What is the median age of his cousins?

A. 24

B. 25

C. 27

D. 28



4. Given y = x2, how would the graph of y = x2 2 differ?

A. It shifts 2 units up.

B. It shifts 2 units down.

C. It shifts 2 units left.

D. It shifts 2 units nght.


Go On >


Copyright 2008 by the Missouri State Dcpartwieat of Elementary and Secondary Educaton.


123














Algebra I


5. Given the following fractions:
3 18 24 3 12
4' 29' 39' 5' 18
Which group below has the fractions in order from least to greatest?

A. 3 24 18 12 3
5' 39' 29' 18' 4
3 3 18 24 12
4' 5' 29' 39' 18
S3 12 24 3 18
5' 18' 39' 29
3D. 3 12 18 24
4' 5' 18' 29' 39


6. The automobile repair shop uses the following chart to determine labor
costs for each job.


Automobile
Repair Shop Costs

Hours Labor Cost

1 $25

2 $40

3 $55

4 $70



Which function should the automobile repair shop use to determine the
labor cost, C, for a job that takes h hours?

A. C- 15h

B. C- 15h I 10

C. C 25 1 15h

D. C-25h+15h




Copyright t) 2008 by the Missouri State Department of Elementary and Secondary Education.


124














Algebra I


7. A survey was administered to 500 high school students to determine
the type of music they prefer. The survey indicated that 22%
prefer rock, 26% prefer hip hop, 29% prefer pop, and 23% selected
"other." Which representation best illustrates the number of students
preferring each type of music?


Preferred Music


Type

Rock

Hip Hop
Pop
Other


Percent of
Students
22

26
29
23


Preferred Music


Preferred Music


Pop Other


Preferred Music


I I X
0 Rock Hip Pop Other
Hop


Go On 0


Copyright 0 2008 by the Missouri State Dcpartlrut of Ecmentary and Soondary Education










125


Rock


1 1














Algebra I


8. What is the value of the numerical expression below?

V/16 + 24- 23
3

A. 4

8. 6

C. 8

D. 10



9. Aaron listed the ages of all of his family members as shown below.

10,10, 10,10,10, 12, 14,14, 15,16,50,50,51, 53,80


What is the mean age of his family members?

A. 10

B. 14

C. 27

D. 70





















Copyright t) 2008 by the Missouri State Department of Elementary and Secondary Education.


126














Algebra I


10. What is the product of the following expression?

2x(x2 + x 5)


A. 2x I x 5

B. 2x+ + 2x 10

C. 2xd + 2x? 5x

D. 2x3 + 2x2 1x


Go On >


Copyright V 2008 by the Missouri State Dcpartlmet of Elementary and Secondary Education.


127













Algebra I


11. Beth and Jacob are graphing two equations on a coordinate grid. Beth has
graphed the equation y = x2 + 1.


xK- 1


If Jacob graphs y = x2 + 3, where will his graph be in relation to the graph
Beth made?

A. 2 units up

B. 3 units up

C. 2 units to the left

D. 3 units to the right











Copyright C 208 by the Missuri State Departmcnt of laemcntary and Secondary Education


128

















Algebra I



12. A survey was taken asking participants their age and the number of
minutes they exercise per week. The results of the survey are shown in the
scatterplot below.



Minutes of Exercise per Week


140


120
3
C

100

^-N
s so

( 60
a

2 40
LU
w .


* *


9


a


a a
.
,... .. ,- .
'. .. @@
i*


U 10 3

0 10 20 30


40 50 60


Age, in years



The data for people who are 30 to 39 years of age are not displayed. Based
on the scatterplot, how many minutes would a 30- to 39-year-old person be
expected to exercise?

A. 40-60 minutes

B. 60-80 minutes

C. 80-100 minutes

D. 100-120 minutes


Go On >


Copyright 2008 by the Missouri State Dcparntroet of Elemntary and Secondary Educaton.


129


I














Algebra I


13. Ben bought 8 notebooks for $24.50. Some of the notebooks were $2.50
each, and the others were $3.25 each. If x represents the number of least
expensive notebooks, which equation can be used to find the number of
least expensive notebooks purchased?

A. $5.75(8 x) $24.50


B. $2.50(x 8) 1 $3.25x

C. $2.50x + $3.25(8 x)

D. $2.50x + $3.25(x 8)


$24.50

$24.50

$24.50


14. The number 18 is 24% of which number?

A. 4.32

B. 75

C. 1331
3

D. 432

























Copyright 0 2008 by the Missouri State Department of Elementary and Secondary Education.


130













Algebra I


15. The graph of y = 2x 4 is shown below.


2x 4


If the slope of the line is doubled, the new equation is y = 4x 4.
Which of these is a correct comparison of the two lines?

A. The x-intercept and y-intercept change.

B. The x-intercept and y-intercept stay the same.

C. The x-intercept changes, and the y-intercept is the same.

D. The x-intercept is the same, and the y-intercept changes.


Go O(n


Copyright V 2008 by the Missouri State Dcptarliet of Elementary and Secondary Education.


131













Algebra I


16. The following line graph shows the test scores for 10 students on a unit
exam.


Test Results


x x x
x x x x x

50 60 70 80 90 100
Test Score


Which shape most accurately describes these data?


A. The data are skewed to the left.
B. The data are skewed to the right.
C. a bimodal or "U"-shaped curve
D. a normal or "bell"-shaped curve


17.

rn


Mary would like to plant grass in her backyard. Her backyard is a
rectangle that measures 10 yd by 8 yd. In the middle of her backyard is
a circular swimming pool that has a diameter of 5 yd. What is the area to
be planted with grass, to the nearest tenth of a square yard?

A. 1.5yd2


B. 19.6 yd
C. 60.4 yd
D. 80 yd'










Copyright t) 2008 by the Misouri State Department of Elementary and Secondary Education.


132














Algebra I


18. Which expression represents the output of the nth term?


Input 1 2 3 4 5 n
Output 1 3 5 7 9


A. n+2

B. n+11

C. 2n+1

D. 2n 1



19. What is the solution to the equation?

-12= 6 +


A. -27

B. -24

C. -12

D. -9


Go On >


Copyright V 2008 by the Missouri State Deparntroet of Elemntary and Secondary Eduation.


133














Algebra I


20. What is the mode of the data set displayed below?



Stem Leaf
1 00344444
2 2249
3 1123
4 678888899
5 001257
19 8

Key
1 13 = 13


A. 14

8. 48

C. 4and8

D. 14 and 48



21. Which number line below shows the set of numbers graphed correctly?

j3, -7 1 -2, -1
2'2'


A. 4 3 -2 -1 0 1 2 3 4


B. -4 -3 -2 -1 0 1 2 3 4



C. -4 -3 -2 -1 0 1 2 3 4


D. -4 -3 -2 -1 0 1 2 3 4




Copyright t 2008 by the Misouri State Department of Elementary and Secondary Education.










134













Algebra I


22. What is true about the slope and y-intercept of the two equations below?

4x + 3y = 12
-8x + 6y = 6


A. same slope, same y-intercept

B. same slope, different y-intercept

C. different slope, same y-intercept

D. different slope, different y-intercept



23. The diagram shows the outcomes of flipping a coin and rolling a die.


H T

1 2 3 4 5 6 1 2 3 4 5 6


H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6


Which statement regarding the diagram isfalse?

A. The probability of obtaining "H6" is 2 out of 12.

B. There are 12 possible outcomes in the sample space.

C. The chance of flipping "heads" and rolling a "5" is 1 in 12.

D. Flipping 'tails" and rolling a "2" represents about 8% of the possible outcomes
of the sample space.









Go Oan

Copyright 2008 by the Missouri State Dcparntlet of Elementary and Socondary Eduation.


135















Algebra I


24. The population of a type of bacteria triples every minute. The chart below
represents the population of bacteria after t minutes.


t

0
1
2
3
4
5


Bacteria
Population
1
3
9
27
81
243


Which type of function represents the data?

A. linear

B. quadratic

C. exponential

D. absolute value



25. What are the slope, m, and the y-intercept, b, of a line that passes through
the points (-3, 1) and (7, -5)?
3 -4
A. m= and b =

-5
B. m = and b = -4
3

C. m 5 and b -


5


Copyright t) 2008 by the Misouri State Department of Elementary and Secondary Education.










136












Algebra I


26. Given the following set of numbers:

-V4, -22, -2.3, -, -2.7


Which set is in order from least to gr,.''lervit?
A. I-2.7, -2.3, -22 -V

3' 2'
B. 1-2-7. -22-- -2.3,-V4i

C. V4',-I- 23 2.3, 2.7

D. V, -2.3, 5 -22, -2.7



27. Which of these shows the following expression factored completely?
6x2 + 15x 36

A. (2x 3)(x 1 4)
B. (6x I 9)(x 4)
C. 3(2x 3)(x + 4)
D. 3(2x + 3)(x 4)


Go On >


Copyright 0 2008 by the Missouri State Department of Elementary and Secondary Eduation.


137















Algebra I


28. A scatterplot is shown on the graph below.


30
300f


100 |


0
. *
*


. *0
* -o
*


2 4 6 8 10 12
2 4 6 8 10 12


Which of these could be a line of best fit?

A. y x + 100

B. y- x- 100

C. x 100

D. y 100

















Copyright C 208 by the Missouri State Departmecnt of MEezntary and S-condary Eduation


138













Algebra I


29. What is the equation of the function represented by this table of values?


x 2 1 0 1 2
y 3 3 15 75
Y 25 5


A. y 5x 1 3

B. y- 12x 3

C. y-3-5"

D. y-5-3x



30. The enrollment at High School R has been increasing by 20 students per
year. Currently High School R has 200 students attending. High School
T currently has 400 students, but its enrollment is decreasing in size by an
average of 30 students per year. If the two schools continue their current
enrollment trends over the next few years, how many years will it take the
schools to have the same enrollment?

A. 4 years

B. 5 years

C. 10 years

D. 20 years


Go O(n


Copyright 2008 by the Missouri State Dcptarltet of Elementary and Secondary Eduation.


139















Algebra I


31. What is the solution to the following inequality?

1(6 x) > -2


x 0

x-O

x -12

x 12


32. Which is a true statement about the data shown in the tables?


Table 1

x y
1 4
2 5
3 6
9 12
10 13


Table 2

x y
0 0


2
3
4


4
9
16


A. Both tables represent a linear relation.

B. Only Table 1 represents a linear relation.

C. Only Table 2 represents a linear relation.

D. Neither table represents a linear relation.













Copyright 0 2008 by the Missouri State Department of Elementary and Secondary Education.










140














Algebra I


33.
1rn


The length of a rectangle is 4 times its width. If the length of the rectangle
is cut in half, the new perimeter is which percent of the original perimeter?


A. 25%

B. 50%


C. 60%

D. 100%


34. What is the simplified form of the expression?
4x3y3


A. Y

B. 2Y
B.
x2

C. 2x'y

D. 2x"y



35. What is the solution for the system of equations?


y = 2x- 3
4x 3y = 31


A. ( 11, 25)

B. (-11, -19)

C. (11, 19)

D. (14, 25)


STOP


Copyright V 2008 by the Missouri State Deparntret of Elementary and Secondary Eduation.


141










APPENDIX B
ALGEBRA I PERFORMANCE EVENT RELEASED SAMPLES




Algebra I


Directions to the Student


Today you will be taking Session II of the Missouri Algebra I Test. This
is a test of how well you understand the course level expectations for
Algebra I.


There are several important things to remember:
1 Read the performance event carefully and think about how to answer
the questions.
2 Show all of the work that you did to answer the question with a
number 2 pencil. If a box is provided, make sure all of your work is
in the box. If a line is provided to write your answer on, be sure your
answer is on the line.
3 If you do not know the answer to a question, skip it and go on. You
may return to it later if you have time.
4 If you finish the test early, you may check overyour work.
5 There is not an answer sheet for this session of the test. Wnte or mark
your answers directly in your test book with a number 2 pencil.


Copyright 0 2W0 by the Missouri State Departrnet of Elemintary ad Smondary Education


142














Algebra I


1 .A line passes through the points (3, 5) and (-2, 7).

On the line below, write an equation of the line and graph it on the
coordinate grid.

Equation:


Copyright j 208 by the Missouri State DpTsutmncn of Elmantary and Sccondary Eduation


143


-x













Algebra I


* What is the y-intercept of the line?

* What is the slope of the line?

* Write a new equation with the same slope and a different y-intercept on
the line below.

Equation:

* On the lines below, explain how the graph of this line relates to the
original line.











* Write a new equation with the original y-intercept and a different slope on
the line below.

Equation:

* On the lines below, explain how the graph of this line relates to the
original line.


STOP


Copyright V 2008 by the Missouri State Dcpartleat of Elementary and Secondary Eduation.


144









APPENDIX C
STATE ALGEBRA STANDARDS

Number and Operations

N.1.A.AI compare and order rational and irrational numbers, including finding their

approximate locations on a number line

N.1.B.AI use real numbers and various models, drawing, etc. to solve problems

N.1.C.AI use a variety of representations to demonstrate an understanding of very large

and very small numbers

N.2.B.AI describe the effects operations, such as multiplication, division, and computing

powers and roots on the magnitude of quantities

N.2.D.AI apply operations to real numbers, using mental computation or paper-and-

pencil calculations for simple cases and technology for more complicated cases

N.3.D.AI judge the reasonableness of numerical computations and their results

N.3.E.AI solve problems involving proportions



Algebraic Relationships

A.1 .B.AI generalize patterns using explicitly or recursively defined functions

A. 1 .C.A compare and contrast various forms of representations of patterns

A.1.D.AI understand and compare the properties of linear and nonlinear functions

A. 1.E.AI describe the effects of parameter changes on linear, exponential growth/decay

and quadratic functions including intercepts

A.2.A.AI use symbolic Algebra to represent and solve problems that involve linear and

quadratic relationships including equations and inequalities

A.2.B.AI describe and use Algebraic manipulations, including factoring and rules of


145









integer exponents and apply properties of exponents (including order of operations) to

simplify expressions

A.2.C.AI use and solve equivalent forms of equations (linear, absolute value, and

quadratic)

A.2.D.AI use and solve systems of linear equations or inequalities with 2 variables

A.3.A.AI identify quantitative relationships and determine the type(s) of functions that

might model the situation to solve the problem

A.4.A.AI analyze linear and quadratic functions by investigating rates of change,

intercepts and zeros



Data and Probability

D. 1.A.AI formulate questions and collect data about a characteristic which include

sample spaces and distributions

D.1 .C.A select and use appropriate graphical representation of data and given one-

variable quantitative data, display the distribution and describe its shape

D.2.A.AI apply statistical measures of center to solve problems

D.2.C.AI given a scatter plot, determine an equation for a line of best fit

D.3.A.AI make conjectures about possible relationships between 2 characteristics of a

sample on the basis of scatter plots of the data


146









APPENDIX D
NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS MATHEMATICS
STANDARDS FOR GRADES 6-8

Table A-1. Number and Operations Standard for Grades 6-8 Expectations


Instructional programs from prekindergarten
through grade 12 should enable all students
to-
Understand numbers, ways of representing
numbers, relationships among numbers, and
number systems


Understand meanings of operations and how
they relate to one another


In grades 6-8 all students should-


* work flexibly with fractions, decimals, and
percent to solve problems;
* compare and order fractions, decimals, and
percent efficiently and find their
approximate locations on a number line;
* develop meaning for percent greater than
100 and less than 1;
* understand and use ratios and proportions
to represent quantitative relationships;
* develop an understanding of large numbers
and recognize and appropriately use
exponential, scientific, and calculator
notation;
* use factors, multiples, prime factorization,
and relatively prime numbers to solve
problems;
* develop meaning for integers and represent
and compare quantities with them.
* understand the meaning and effects of
arithmetic operations with fractions,
decimals, and integers;
* use the associative and commutative
properties of addition and multiplication and
the distributive property of multiplication
over addition to simplify computations with
integers, fractions, and decimals;
* understand and use the inverse
relationships of addition and subtraction,
multiplication and division, and squaring
and finding square roots to simplify
computations and solve problems.


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Table A-1. Continued
Instructional programs from
prekindergarten through grade 12 should
enable all students to-
Compute fluently and make reasonable
estimates


In grades 6-8 all students should-


* select appropriate methods and tools for
computing with fractions and decimals
from among mental computation,
estimation, calculators or computers,
and paper and pencil, depending on the
situation, and apply the selected
methods;
* develop and analyze algorithms for
computing with fractions, decimals, and
integers and develop fluency in their
use;
* develop and use strategies to estimate
the results of rational-number
computations and judge the
reasonableness of the results;
* develop, analyze, and explain methods
for solving problems involving
proportions, such as scaling and finding
equivalent ratios.


148









Table A-2. Geometry Standard for Grades 6-8 Expectations


Instructional programs from
prekindergarten through grade 12 should
enable all students to-
Analyze characteristics and properties of
two- and three-dimensional geometric
shapes and develop mathematical
arguments about geometric relationships


In grades 6-8 all students should-


* precisely describe, classify, and
understand relationships among types
of two- and three-dimensional objects
using their defining properties;
* understand relationships among the
angles, side lengths, perimeters, areas,
and volumes of similar objects;
* create and critique inductive and
deductive arguments concerning
geometric ideas and relationships, such
as congruence, similarity, and the
Pythagorean relationship.


Specify locations and describe spatial use coordinate geometry to represent
relationships using coordinate geometry and examine the properties of geometric
and other representational systems shapes;
use coordinate geometry to examine
special geometric shapes, such as
regular polygons or those with pairs of
parallel or perpendicular sides.
Apply transformations and use symmetry describe sizes, positions, and
to analyze mathematical situations orientations of shapes under informal
transformations such as flips, turns,
slides, and scaling;
examine the congruence, similarity, and
line or rotational symmetry of objects
using transformations.
Use visualization, spatial reasoning, and draw geometric objects with specified
geometric modeling to solve problems properties, such as side lengths or angle
measures;
use two-dimensional representations of
three-dimensional objects to visualize
and solve problems such as those
involving surface area and volume;
use visual tools such as networks to
represent and solve problems;
use geometric models to represent and
explain numerical and Algebraic
relationships;
recognize and apply geometric ideas
and relationships in areas outside the
mathematics classroom, such as art,
science, and everyday life.


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Table A-3. Measurement Standard for Grades 6-8 Expectations
Instructional programs from In grades 6-8 all students should-
prekindergarten through grade 12 should
enable all students to-
Understand measurable attributes of understand both metric and customary
objects and the units, systems, and systems of measurement;
processes of measurement understand relationships among units
and convert from one unit to another
within the same system;
understand, select, and use units of
appropriate size and type to measure
angles, perimeter, area, surface area,
and volume.
Apply appropriate techniques, tools, and use common benchmarks to select
formulas to determine measurements appropriate methods for estimating
measurements;
select and apply techniques and tools to
accurately find length, area, volume,
and angle measures to appropriate
levels of precision;
develop and use formulas to determine
the circumference of circles and the
area of triangles, parallelograms,
trapezoids, and circles and develop
strategies to find the area of more-
complex shapes;
develop strategies to determine the
surface area and volume of selected
prisms, pyramids, and cylinders;
solve problems involving scale factors,
using ratio and proportion;
solve simple problems involving rates
and derived measurements for such
attributes as velocity and density.


150









Table A-4. Data Analysis and Probability Standard for Grades 6-8 Expectations
Instructional programs from In grades 6-8 all students should-
prekindergarten through grade 12 should
enable all students to-
Formulate questions that can be formulate questions, design studies, and
addressed with data and collect, collect data about a characteristic
organize, and display relevant data to shared by two populations or different
answer them characteristics within one population;
select, create, and use appropriate
graphical representations of data,
including histograms, box plots, and
scatterplots.
Select and use appropriate statistical find, use, and interpret measures of
methods to analyze data center and spread, including mean and
interquartile range;
discuss and understand the
correspondence between data sets and
their graphical representations,
especially histograms, stem-and-leaf
plots, box plots, and scatterplots.
Develop and evaluate inferences and use observations about differences
predictions that are based on data between two or more samples to make
conjectures about the populations from
which the samples were taken;
make conjectures about possible
relationships between two
characteristics of a sample on the basis
of scatterplots of the data and
approximate lines of fit;
use conjectures to formulate new
questions and plan new studies to
answer them.
Understand and apply basic concepts of understand and use appropriate
probability terminology to describe complementary
and mutually exclusive events;
use proportionality and a basic
understanding of probability to make
and test conjectures about the results of
experiments and simulations;
compute probabilities for simple
compound events, using such methods
as organized lists, tree diagrams, and
area models.


151









Table A-5. Problem Solving Standard for Grades 6-8
Instructional programs from prekindergarten through grade 12 should enable all
students to-
build new mathematical knowledge through problem solving;
solve problems that arise in mathematics and in other contexts;
apply and adapt a variety of appropriate strategies to solve problems;
monitor and reflect on the process of mathematical problem solving.

Table A-6. Reasoning and Proof Standard for Grades 6-8
Instructional programs from prekindergarten through grade 12 should enable all
students to-
recognize reasoning and proof as fundamental aspects of mathematics;
make and investigate mathematical conjectures;
develop and evaluate mathematical arguments and proofs;
select and use various types of reasoning and methods of proof.

Table A-7. Communication Standard for Grades 6-8
Instructional programs from prekindergarten through grade 12 should enable all
students to-
organize and consolidate their mathematical thinking through communication;
communicate their mathematical thinking coherently and clearly to peers,
teachers, and others;
analyze and evaluate the mathematical thinking and strategies of others;
use the language of mathematics to express mathematical ideas precisely.

Table A-8. Connections Standard for Grades 6-8
Instructional programs from prekindergarten through grade 12 should enable all
students to-
recognize and use connections among mathematical ideas;
understand how mathematical ideas interconnect and build on one another to
produce a coherent whole;
recognize and apply mathematics in contexts outside of mathematics.

Table A-9. Representation Standard for Grades 6-8
Instructional programs from prekindergarten through grade 12 should enable all
students to-
create and use representations to organize, record, and communicate
mathematical ideas;
select, apply, and translate among mathematical representations to solve
problems;
use representations to model and interpret physical, social, and mathematical
phenomena.


152










APPENDIX E
MAP GRADE 6 RELEASED ITEMS SPRING 06


1 The circle graph below shows sales of different kinds of bikes at Bill's Bikes.

BILL'S BIKES

Other Hybrid Road
SKind Bike Bike
g jChild's
Bike
Mountain
Bike



Which kind of bike had approximately twice the sales as hybrid bikes?

s O road bike
O child's bike
O mountain bike
O other kind of bike




2 Ms. Williams is making a design using tiles, as shown below.



2iI1II1


Which statement describes the pattern?

O 1 large tile above and 2 small tiles below
O 1 large tile above and 3 small tiles below
O 2 large tiles above and 3 small tiles below
O 2 large tiles above and 4 small tiles below


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153













Use your protractor to help you solve this problem.
The picture below shows walking paths in a park.


Which angle is an acute angle?

0 angle L
0 anale M


0 angle N
0 angle 0


4 A rectangular field measures 40 feet wide by 20 feet long. Which of these shows how to find
the area of the field?


O 40 feet + 20 feet
O 40 feet X 2
O 2(40 feet + 20 feet)
O 40 feet x 20 feet


Go On >

[I .lge j


Copyright 2006 by the Missouri State Department of Elreentary and Secondary Education


154













5 Study the grid below.


0 2 3 4 5 6 7 8 9 10

Which two pairs of coordinates can be used so that the figure on the grid, when
completed, is a hexagon?
0 (2, 4),(8, 4)
0 (4, 4), (6, 4)
0 (4, 6), (6, 6)
0 (3, 6), (7, 6)


6 Each of the 3 sixth -rade classes at Jefferson School has 25 students. On picture day, a few
students were absent. Which expression shows how many sixth-grade students had their
pictures taken?

p =number of sixth-grade students who were absent on picture day


0 (25 p)x 3
0 (25 -p) 3
0 (25 3) p
0 (25 3)- p

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155


/7














7 Study Figure A and Figure B below.


Figure A


--Figure
Figure B


In the table below, fill in the correct number of faces, vertices, and edges for Figure A
and Figure B.


I F


Number
of Edges


Number Number
of Faces of Vertices


Figure A

Figure B


8 Karen baked cookies for 2 hours and 25 minutes. She finished baking cookies at 4:15 P.M.
What time did she begin baking cookies?
O 1:50 P.M.


O 2:40 P.M.

O 2:50 P.M.

O 6:40 P.M.


!
-I


Go On 1

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'. the Misouri State Department of Elementary and Secondary Education


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12 Kyle is 3 years older than twice his sister's age. His sister is 5 years old. How old is Kyle?

0 7
O

S 0 10

O 13






8



if




















Page 8
If
SI






4









-,'I,, .' _' :''_1,- Ir, .,, .r L'- I... r of lem entry and Secondary Education
0


157














14 Study the figures labeled A and B.









i *
i



Figure B shows Figure A after 1 transformation. Which transformation was used-a flip, a
slide, or a turn? Write your answer on the line.


In the figures above, draw the flip line, slide arrow, or turn point on the figures for the
transformation you chose.
























Go On

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i












8









1 7 Jodi wants to rent sports equipment to people at the park. She needs to know the
equipment that will rent most often. Which is the best method to gather the information?

S 0 observe the equipment that most people buy at a sports store

O survey adults at the library on Saturday afternoon
O survey elementary school students on the playground
O observe !h I quiplmernt that most people rent at different rental stands



8



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0


159















18 Study the triangle below-







6 cm 6.5 cm






--3 cm-


Area for a triangle = X base x height


Which of these expressions can be used to find the area of the triangle?

O 6+6.5+3

0 (3 x 6)

0 3 X 6 5

0 (3 x 6.5)


0
0

S

a







B
0
I









!
I

GoOn> o

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'. the Misouri State Depaitment of Elementary and Secondary Education


160















19 The table below shows the coordinates for points A, B, C, and D.



Point A B C D

Coordinates (1,5) (1,3) (4,3) (4,5)



Locate and label the points A, B, C, and D on the grid below.


0 123456


Connect the points to create a polygon. On the line below, write the name of the polygon.


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161















20 Which expression shows the same value as 3(10 + n)?

0 (3 i 40) (3 n)
S(3 x 40) + (3 n)
0 (3 + 40)x (3 + n)
0 (3 40) (3 n)




21 Study the table below. The table shows the information about 4 accounts that were opened
at the same time.

SAVINGS ACCOUNTS INFORMATION
SAVINGS ACCOUNTS IN FORMATION t


Amount When Monthly
ame Opened Deposit

Linda $25 $15

Michael $40 $10

Nelson $30 $14

Olivia $20 $20


Who will be the first to have $100 in his or her savings account?

0 Linda

0 Michael
0 Nelson
0 Olivia









:. the Misouri State Department of Elementary and Secondary Educatton
0


a







Sz
0
0
0
S
a




-I




!
I


Go On .

Page 13


162














22 Which 3-dimensional figure has 7 faces, 15 edges, and 10 vertices?


23 Study the table below.

MEASUREMENT
CONVERSIONS

1 foot= 12 inches
1 yard = 3 feet

1 mile = 1,760 yards

What is the best estimate for the number of inches in a mile?

O 40,000 inches
O 60,000 inches
O 90,000 inches
O 1: 71o0 inches


Page 14
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i. the Missour State Depaitment of Elementary and Secondary Education


163


O















25 Sela uses one type of bread and one type of filling to make each sandwich.


Filling

ham
tuna
cheese
peanut butter


Which diagram shows all the possible types of sandwiches Sela can make?


Bread Filling

ham
wheat ha
tuna
cheese
-white -" peanut butter

O



Bread Filling

wheat peanut butter

ham-cheese
white tuna









O


Bread Filling

wheat ham- cheese

white tuna

wheat peanut butter


O


Bread


Filling


ham
tuna
wheat t
cheese
peanut butter

ham
S. tuna
white tuna
cheese
peanut butter


0


Page 16
: '. i i. the Misouri Stte Depaitment of Elementary and Secondary Education


164


Bread

wheat
white
















26 Each of two snack stands at the soccer field collected data to compare sales. Which of these
does not show the number of hot dogs sold at each stand?


SNACK STANDS
25
20





Hot Dogs Soda Water Popcorn
Snacks

KEY
O Stand 1 I Stand 2


0 SNACK STANDS




Hot longs Soda


E 1 Stand 1
Popcorn Stand 2
Popcorn Water


SNACK STANDS

5





0*
Hot Dogs Soda Water Popcorn
Snacks

KEY
-0- Stand 1 "-- Stand 2


SNACK STANDS

Snacks Stand 1 Stand 2

Hot Dogs 1I 19
Soda IS 21
Watel I 5 I,
Pop,.,,r r, 19


t
U
a








!
a




Go On I)

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27 Study the figures on the grid below.








Figure I Figure 2




Which two transformations could be used to change Figure 1 to Figure 2 ?

0 a flip and a 'Jd-
O a slide and a flip
0 a counterclockwise 90 turn and a slide
0 a clockwise 90 turn and a *-lJe






















Page b
Copynph 2X.C by thle Missoun State Deparitmenr of Elementary and Secondary iLducLtion


166


STOP









APPENDIX F
STATE STANDARDS FOR MATHEMATICS AT GRADE LEVEL 6

Number and Operations

N.1.A.06 compare and order integers, positive rationals and percent, including finding

their approximate location on a number line

N.1.B.06 recognize and generate equivalent forms of fractions, decimals and percent

N.1.C.06 recognize equivalent representations for the same number and generate them

by decomposing and composing numbers, including expanded notation

N.1.D.06 use factors and multiples to describe relationships between and among

numbers, including whole number common factors and common multiples

N.2.B.06 describe the effects of addition and subtraction on fractions and decimals

N.3.C.06 add and subtract positive rational numbers

N.3.D.06 estimate and justify the results of addition and subtraction of positive rational

numbers

N.3.E.06 solve problems using equivalent ratios



Algebraic Relationships

A. 1.B.06 represent and describe patterns with tables, graphs, pictures, symbolic rules or

words

A.1 .C.06 compare various forms of representations to identify a pattern

A.1.D.06 identify functions as linear or nonlinear from a table or graph

A.2.A.06 use variables to represent unknown quantities in expressions

A.2.B.06 recognize equivalent forms for simple Algebraic expressions including

associative and distributive properties


167









A.3.A.06 model and solve problems, using multiple representations such as graphs,

tables, expressions and equations

A.4.A.06 compare situations with constant or varying rates of change



Geometric and Spatial Relationships

G.1 .A.06 identify the properties of one-, two- and three-dimensional shapes using the

appropriate geometric vocabulary

G.1.B.06 describe relationships between the corresponding angles and the length of

corresponding sides of similar triangles (whole number scale factors)

G.2.A.06 use coordinate geometry to construct geometric shapes

G.3.A.06 describe the transformation from a given pre-image to its image using the

terms reflection/ flip, rotation/ turn and translation/ slide

G.3.C.06 create polygons and designs with rotational symmetry

G.4.A.06 use spatial visualization to identify isometric representations of mat plans

G.4.B.06 draw or use visual models to represent and solve problems



Measurement

M. 1.A.06 identify and justify an angle as acute, obtuse, straight or right

M.1.C.06 solve problems involving elapsed time (hours and minutes)

M.2.A.06 estimate a measurement using either standard or non-standard unit of

measurement

M.2.B.06 select and use benchmarks to estimate measurements of 0-, 45-, 90-, 180-,

360- degree angles


168









M.2.C.06 describe how to solve problems involving the area or perimeter of polygons

M.2.E.06 convert from one unit to another within a system of measurement (mass and

weight)



Data and Probability

D.1 .A.06 formulate questions, design studies and collect data about a characteristic

D.1.C.06 interpret circle graphs; create and interpret stem-and-leaf plots

D.2.A.06 find the range and measures of center, including median, mode and mean

D.2.B.06 compare different representations of the same data and evaluate how well

each representation shows important aspects of the data

D.3.A.06 use observations about differences between 2 samples to make conjectures

about the populations from which the samples were taken

D.4.A.06 use a model (diagrams, list, sample space, or area model) to illustrate the

possible outcomes of an event


169









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BIOGRAPHICAL SKETCH

Feng Liu was born in 1973 in Liaocheng City, Shandong Province, China. As the

second child of one three-child family, he attended No.1 High School, Zhongyuan Oil

Field in Puyang City, Henan Province, China. Feng Liu graduated from Nanjing Normal

University's Computer Science Department with a Bachelor of Science in computer

science education in 1995. He has taught computer science courses at postsecondary

level including Nanjing Material Polytechnic School and Nanjing University of Finance &

Economics for eight and half years.

Feng Liu came to United States at January 2004 to further his education at

Georgia College & State University where he earned a Master of Education in

educational technology in May 2006. In August of 2006, Feng Liu enrolled as a doctoral

fellow in the Educational Technology program in School of Teaching and Learning at

the University of Florida (UF).

During his study at UF, Feng Liu has focused on research in the learning

technologies. His research interests include the investigation of online learning success

and the effectiveness of virtual schooling, the employment of advanced research

methods and statistical approaches in educational research, and the use of e-

game/simulation for knowledge gain, attitude change and motivation in areas such as

science and second language acquisition. He has several publications in these areas.


195





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1 FACTORS INFLUENCING SUCCESS IN ONLINE HIGH SCHOOL ALGEBRA By FENG LIU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

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2 2010 Feng Liu

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3 To my family

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4 ACKNOWLEDGMENTS I offer a special thank you to my family my parents, sister, and brother their unconditional love has brought me to where I am now. I would not have fini shed this long journey without their support. I want to give a special thank you to my mom, who has always been there to encourage me to move forward especially when I was feeling down. My whole family provides the momentum for me to conquer all the diffic ulties that I have been through during this long, and sometimes lonely, journey. I thank Dr. Cathy Cavanaugh, my major advisor and committee chair. As my primary mentor, she inspired me with her dedication to the research and practice in virtual schooling and her motivating work spirit. As a leading researcher in K 12 virtual schooling, she gave me confidence and provided me invaluable guidance to position myself in this field. It is such an honor to be a mentee under her direction. I thank Dr. Kara Dawson, Dr. James Algina, and Dr. Tim Jacobbe, my committee member, for their help and support during this journey. I thank Dr. Dawson for introducing me to the educational technology program, the ever changing and exciting field. She is my role model to balance life between family and the professional world. I thank Dr. Algina, for his guidance on my minor: research, evaluation, and methodology. His strictness in educational research truly helped me build the foundation to conduct rigorous research in the field of education. I thank Dr. Jacobbe for his patience and insightful questions that inspire me to advance my knowledge in math education. In addition to my committee, I would like to thank my cohort and support group of doctoral students who helped me go through this long process. I thank them for their encouragement and support and all the inspiring stories they shared with me. All these people have made my life much easier during the process.

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5 Finally, I thank all the faculty and staff in the College of Educa tion at University of Florida who have shaped me in many ways and helped me reach this significant milestone. I thank the University of Florida Graduate Alumni Fellowship Program, which provided me with generous funding for four years. I would not have finished this journey without its assistance.

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6 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 9 LIST OF FIGURES ........................................................................................................ 10 ABSTRACT ................................................................................................................... 11 CHAPTER 1 INTRODUCTION .................................................................................................... 13 Background ............................................................................................................. 13 Problem Statement ................................................................................................. 14 Purpose Statement ................................................................................................. 16 Resear ch Questions ............................................................................................... 16 Significance of the Study ........................................................................................ 17 Delimitations ........................................................................................................... 18 De finition of the Terms ............................................................................................ 19 Organization of the Study ....................................................................................... 20 2 REVIEW OF LITERATURE .................................................................................... 23 Introduction ............................................................................................................. 23 Research in Online/Distance Education and Significance of This Study ................ 25 Review of Literature ................................................................................................ 27 Effectiveness of Online/Distance Education ..................................................... 27 Algebra/Mathematics Education ....................................................................... 35 Learner characteristics variables ............................................................... 35 Learning environment variables ................................................................. 39 Algebra teaching and learning ................................................................... 41 Success Factors in Online Learning ................................................................. 44 Teacher comments/teacher student interaction ......................................... 46 Participation in online academic activities .................................................. 49 Race/ethnicity ............................................................................................ 49 Participation in school free lunch/family SES ............................................. 50 Learning ability/presence of i ndividual educational plan ............................ 51 School type ................................................................................................ 54 Conclusion .............................................................................................................. 56 3 METHODOLOGY ................................................................................................... 59 Introduction ............................................................................................................. 59 Researc h Design .................................................................................................... 59

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7 Participants and Data Collection ............................................................................. 60 Instrument ............................................................................................................... 61 Al gebra EOC Test ............................................................................................ 62 State Standardized Test ................................................................................... 63 Data Analysis .......................................................................................................... 64 Li mitations ............................................................................................................... 66 4 RESULTS ............................................................................................................... 68 Introduction ............................................................................................................. 68 Sample .................................................................................................................... 68 EOC Tests Taker .............................................................................................. 68 State Standardized Test Taker ......................................................................... 69 RA Model ................................................................................................................ 70 Coefficients for the Variables .................................................................................. 71 EOC Test .......................................................................................................... 72 State Standardized Mathematics Test .............................................................. 72 Descriptive Statistics, Standardized Coefficient, and Reduction of Variance .......... 72 Research Question 1 .............................................................................................. 73 EOC Test .......................................................................................................... 74 State Standardized Test ................................................................................... 75 Research Question 2 .............................................................................................. 75 EOC Test .......................................................................................................... 77 State Standardized Test ................................................................................... 77 Research Question 3 .............................................................................................. 77 EOC Test .......................................................................................................... 79 State Standardized Test ................................................................................... 80 Summary of Findings .............................................................................................. 82 5 DISCUSSION AND IMPLICATIONS ....................................................................... 96 Introduction ............................................................................................................. 96 Summary of Study .................................................................................................. 96 Overview of the Problem .................................................................................. 96 Purpose Statement and Research Questions .................................................. 97 Review of t he Methodology .............................................................................. 98 Findings .................................................................................................................. 99 Research Question 1 ............................................................................................ 100 Research Q uestion 2 ............................................................................................ 104 Research Question 3 ............................................................................................ 107 Broad Implications for Online Course Design and Online Teaching ..................... 111 Conclusions .......................................................................................................... 113 APPENDIX A ALGEBRA I MULTIPLE CHOICE RELEASED SAMPLES .................................... 121

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8 B ALGEBRA I PERFORMANCE EVENT RELEASED SAMPLES ........................... 142 C STATE ALGEBRA STANDARDS ......................................................................... 145 D NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS MATHEMATICS S TANDARDS FOR GRADES 6 8 ......................................................................... 147 E MAP GRADE 6 RELEASED ITEMS SPRING 06 .................................................. 153 F STATE STANDARDS FOR MATHEMATICS AT GRADE LEVEL 6 ..................... 167 LIST OF REFERENCES ............................................................................................. 170 BIOGRAPHICAL SKETCH .......................................................................................... 195

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9 LIST OF TABLES Table page 3 1 Coding of the independent variables .................................................................. 67 4 1 EOC test takers demographics ........................................................................... 85 4 2 Standardized test takers demographics ............................................................. 86 4 3 Overview of RA model for different datasets ...................................................... 87 4 4 Least squares estimates of f ixed effects (with robust standard errors) ............... 88 4 5 Least squares estimates of fixed effects (with robust standard errors) ............... 89 4 6 Ordinary Least squares estimates of fixed effects .............................................. 90 4 7 Descriptive statistics for EOC test takers ............................................................ 91 4 8 Descriptive statistic s for standardized test takers ............................................... 92 4 9 Standardized coefficients for EOC test takers .................................................... 93 4 10 Standardized coefficients for standardized test takers ....................................... 94 4 11 Adjusted R squares ............................................................................................ 95 5 1 Significance and Direction of the Effect of Factors ........................................... 117 5 2 Alignment with National Standards in Quality Online Course ........................... 118 A 1 Number and Operations Standard for Grades 6 8 Expectations ...................... 147 A 2 Geometry Standard for Grades 6 8 Expectations ............................................ 149 A 3 Measurement Standard for Grades 6 8 Expectations ...................................... 150 A 4 Data Analysis and Probability Standard for Grades 6 8 Expectations ............. 151 A 5 Problem Solving Standard for Grades 6 8 ....................................................... 152 A 6 Reasoning and Proof Standard for Grades 6 8 ................................................ 152 A 7 Communication Standard for Grades 6 8 ........................................................ 152 A 8 Connections Standard for Grades 6 8 ............................................................. 152 A 9 Representation Standard for Grades 6 8 ......................................................... 152

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10 LIST OF FIGURE S Figure page 1 1 Online Students at K 12 Level ............................................................................ 22

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11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FACTORS INFLUENCING SUCCESS IN ONLINE HIGH SCHOOL ALGEBRA By Feng Liu August 2010 Chair: Cathy Cavanaugh Major: Curriculum and Instruction At present, an increasing number of students at the K 12 level in the U.S. are taking courses online via virtual schools, which have been in existence since the end of the 20th century. Virtual schooling is becoming a mainstream option alongside traditional face to face learning environments. Even with its increasing popularity, ver y few empirical studies have been conducted to provide practical guidance for teaching, learning, research, and policy making in K 12 virtual schooling. Some leading virtual school organizations, such as the Southern Regional Educational Board and the Inte rnational Association for K 12 Online Learning, have produced standards in these fields. However, many of the standards lack empirical support based on research conducted in virtual learning environments. Math has been identified as a very important force to push a society forward since it is considered a foundational subject. Many countries emphasize the improvement of math knowledge and they develop policies to attract more people to the field. The examination of success factors in the math field in general and Algebra in specific in virtual learning environments can provide better implementation strategies in

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12 virtual schools to improve student math and science achievement and increase the Science, Technology, Engineering, and Mathematics (STEM) workforce in U.S The purpose of this study is to examine the factors including LMS utilization, teacher comment /feedback and s tudent demographic information that can influence the success of Algebra courses in K 12 virtual learning environments. Students who compl eted Algebra and took the endof course (EOC) test and students who took one state standardized mathematics test at grade 68 level in a state virtual school in the Midwestern U.S region during 20082009 participated in this study. Student scores on these tests were collected by this virtual school. Hierarchical linear modeling (HLM) technique was used for data analysis to account for the influence of school characteristics on student scores. The results show these factors have different influences on stude nt performance on the state standardized mathematics test and the Algebra EOC test. These findings have implications for quality online teaching, instructional design, and the policy making process in virtual learning environments. Further research is pro posed based on the results and limitations of this study.

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13 CHAPTER 1 INTRODUCTION Background The United States has experienced an extraordinary growth in online education at the K 12 level since its emergence in the late 1990s: from single online course of ferings to large virtual schools today. Thousands of students were attracted to online education because of the advantages it brings such as flexible and longer school time, more educational opportunities, and increased access to resources (Cavanaugh, et al., 2004). Several surveys have showed that at least one third of high school students had online learning experience (Allen & Seaman, 2006; Setzer & Lewis, 2005). Figure 1 1 shows the dramatic increase of K 12 online enrollment between 2001 and 2008 (Clark 2001; Glass, 2009; Newman, Stein, & Trask, 2003; Peak Group 2002; Picciano & Seaman, 2009; Picciano & Seaman, 2007; Setzer & Lewis, 2005; Tucker 2007; Zandberg, Lewis, & Greene, 2008). By 2016, this number is anticipated to reach 56 million and will ke ep growing in the future (Picciano & Seaman, 2009). Only public school students were included in this figure; the number will be higher if all other students are included, such as those in private schools and homeschools (Picciano & Seaman, 2009). The vi rtual school movement in the US is the outgrowth of independent study high schools in many ways (Clark 2003). The first two virtual schools in the US, the Virtual High School (VHS) and Florida Virtual School (FLVS), were both created in 1997 (Barbour & Reeves, 2009). By 2001, about 14 states had established statewide virtual schools and 40,000 to 50,000 students enrolled in courses offered by these schools (Clark, 2001). In July 2005, 21 states offered statewide online programs ( Watson & Kalmon, 2005) By September 2006, 38 states have either stateled online learning

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14 programs or online education policies, and 24 states have stateled online learning programs (Watson & Ryan, 2006). As of September 2007, 42 states offered significant hybrid learning programs (students in physical schools taking some courses online), pure online programs (students in physical schools or at home taking most or all of course online) or both ( Watson & Ryan 2007) By fall, 2008, 44 states have online offering for students and 34 s tates offer state led online programs or initiatives (Watson, Gemin, & Ryan, 2008). Response to a survey administered during 200708 showed about 75% of public K 12 school districts were offering full or partial online courses (Glass, 2009; Picciano & Seam an, 2009), approximately 10% increase since 200506 academic year (Picciano & Seaman, 2009). Additionally, another 15% were planning to have online offerings within the next three years (Picciano & Seaman, 2009). Currently all states offer online courses a t school or district level (Cavanaugh 2007). Online education is not seen as separate entity any more but one kind of educational approach which can strengthen the public education and benefit the society at large (Watson & Ryan, 2006). There is a need for deeper understanding of the success in virtual learning environments for the better utilization of this education format to help improve student academic achievement. Problem Statement Along with the extraordinary growth of online education in the US., s ome research has been conducted to examine success factors in online learning environments. According to Roblyer et al., (2008), there are two lines of research emerged to address success factors in online learning: studies focusing on learner characterist ics and studies focusing on learning environment characteristics. Learner characteristics include student cognitive factors such as locus of control and learning styles; prior technology

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15 skills and attitudes; and experience and prior knowledge about course content Learning environment characteristics include technology support, course content area, and accessibility to Internet. At present, no clear set of characteristics have been identified to predict success in virtual learning environments, and no conc lusive model has been created to apply in online learning practice (Roblyer & Davis, 2008; Tallent Runnels et al., 2006). There is a gap regarding the establishment of one online success model to help improve student academic achievement considering the q uick development of virtual schooling in the US. Learner characteristic variables including personal effort/participation in academic activities, student learning ability/whether has individual educational plan, race/ethnicity, and family background/partic ipation in free or reduced lunch programs, and learning environment variables including teacher comment /instructor student interaction and school type (private or public school) have been proved in some studies to correlate to student academic achievement However, these variables have not been investigated systematically in one model regarding their effects on student success in K 12 virtual learning environments. Math knowledge is important for a citizen to fully participate in society. Math is the most widely used subject among all the fields and almost every career uses math at different levels (Saint Paul Public Schools, 2007) During the May 2003 commencement address, the president of Society for Industrial and Applied Mathematics (SIAM), Professor D oug Arnold mentioned math is the foundation to understand the world around us and math knowledge can influence other sciences as well such as economics, business, and sociology. He predicted that math will have huge impacts in the 21st century, the digital ized and dataenriched century.

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16 Math has been considered a very important force to push a society forward. Many countries emphasize the improvement of math knowledge and they develop policies to attract more people into this field. The underachievement of students in math subjects at the K 12 level could lead to the lack of preparation for students to pursue advanced degrees in these fields. This will cause a shortage in the workforce in math and other sciences fields, which, in turn, could weaken the mome ntum for a country to move forward in many aspects. The National Association of Manufacturers (NAM) believed the shortage of workforce in Science, Technology, Engineering, and Mathematics (STEM) fields can weaken manufacturers abilities to ensure quality, productivity efficiency, and customers' satisfaction (DAmico, 2008) T he quality of Algebra courses is essential in building the number of students who are ready for advanced degrees in STEM and career success in these fields. Growing number of students take math courses online at K 12 level so there is a need to examine the quality of online math courses and build one online success model to help improve student academic achievement in general and Algebra/mathematics in specific The present study was d esigned to fill this gap. Purpose Statement Based on the lack of models for predicting success in high school Algebra courses and the clear need for increased Algebra achievement, this study examines the problem of identifying factors that influence onlin e high school Algebra performance. Research Questions The research questions in this study are: Does the level of LMS utilization influence Algebra/mathematics performance in online education?

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17 Does teacher comment or feedback influence Algebra/mathemati cs performance in online education? Do student demographic information such as race/ethnicity, grade level, status in virtual school, whether have individual educational plan (IEP) and participation in free/reduced lunch programs influence Algebra/mathem atics performance in online education? Significance of t he S tudy Even after more than 10 years of extraordinary growth in K 12 online learning, little research has been done as compared to post secondary education (Cavanaugh 2007; Cooze & Barbour, 2005; M eans et al., 2009; Picciano & Seaman, 2007; Picciano & Seaman, 2009; Ronsisvalle & Watkins, 2005). The amount of evidencebased research or empirical study applicable to guide educators instruction and policy makers decision relevancies is slight ( ODwy er, Carey, & Kleiman 2007) The dearth of studies on academic achievement in K 12 virtual learning environments in comparison with that in traditional learning environments ( Cooze & Barbour, 2005; Means et al., 2009; Picciano & Seaman, 2007; Picciano & Sea man, 2009; Smith, Clark, & Blomeyer, 2005; Watson, 2007) form the rationale for this study. T his study can help discover certain characteristics and good practices in online learning and incorporate them into the instructional model of the K 12 virtual learning environment. This study could add to the knowledge of effectiveness of online/distance education in helping improve student academic achievement in the K 12 virtual learning environment. This will provide valuable guidance for the better implementati on and practice of K 12 virtual schooling Given the dearth of research on the factors of academic success in K 12 virtual learning environments, this study could be beneficial to educators, course designers, researchers, online program leaders, policy m akers, and society at large. The investigation of success factors in this study will provide a deeper understanding of

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18 success in online learning in general and in the K 12 virtual learning environment in specific. It has implications for the decision maki ng process for virtual schools with respect to the development of more efficient online courses in general and online mathematics courses in specific. It also could help identify the success factors that should be considered in the virtual learning process and guide management of virtual schools to maximize their effectiveness to provide better assistance and supplement to the traditional learning environment. Delimitations The study was conducted from Oct 2009 to June 2010. One state virtual school in the Midwestern US region was chosen as the location in which the data w ere collected. This virtual school has a big student population and large Algebra course enrollment from which the researcher can draw the sample. This virtual school also has a comprehensive data system enabling the use of advanced statistic model during data analysis. This virtual school can represent the virtual school as a whole in US in many respects such as the design of courses according to state and national standards the utiliz ation of one single LMS to deliver the course materials, and the flexible timeline for students to finish courses. However, this virtual also has its own characteristics that may not be common in other virtual schools such as it recruits both full time and part time students and it moves paced courses all along. Students statewide from bricks andmortar public and private schools as well as home school students were eligible to enroll in this virtual school. Students in this virtual school who completed Alg ebra courses during 200809 and took the EOC test and students who took one state standardized mathematics test at grade level 68 during 20082009 participated in this study

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19 Definition of the T erms Distance education has been practiced in various forms since its emergence in the early 1900s, evolving from correspondence to broadcasting including radio and television, to online education today (Moore & Kearsley, 1996). It has experienced an extraordinary development in the 20th century and its practice wi ll continue to grow in the 21st century. Many distance education related terms have appeared: cybershool, distance education, distance learning, elearning, online education, online learning, virtual school and web based learning. There are also multiple definitions for each of these terms. In this study, the authors are using the definitions that have been broadly cited though by no means are they the most accurate ones. Distance education, defined by Keegan (1996), has four main components: (1) quasi permanent separation of teacher and learner, (2) the use of technical support to bridge the distance, (3) twoway communication during the process, and (4) possible non presence of learning groups. I t i s probably the most cited definition of DE in the litera ture. Another very comprehensive definition of distance education is in a published monograph by The Association for Education Communications and Technology (Schlosser & Simonson, 2002): Institutionbased, formal education where the learning group is separated, and where interactive telecommunications systems are used to connect learners, resources, and instructors (p. 1). Distance learning, defined by Allen et al. (2004), is a course where the students and instructor will not be physically in the same lo cation during the teaching/learning process. Distance learning can be conducted asynchronously using communication techniques such as email, audio/video recording, mail correspondence, and synchronously using techniques such as television, radio, internet chat room, and telephone (Allen et al, 2004).

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20 Allen and Seaman (2006) defined three types of online courses. Online is the course where most or all of the content is delivered online. At least 80% of the traditional face to face (f2f) classroom meeting t ime is replaced by online activity. Blended/Hybrid is the course that combines online and traditional f2f delivery methods. A considerable proportion (30 to 79%) of the content is delivered online. Webfacilitated is the course where webbased technologies are used to facilitate learning. A proportion (1 to 29%) of the content is delivered online. Virtual school, defined by Clark (2000), is a state approved and/or regionally accredited school that offers secondary credit courses through distance learning methods that include Internet based delivery (p. i). Russell (2004) defined virtual school as a form of schooling that uses online computers to provide some or all of a students education (p. 2). Greenway and Vanourek (2006) described virtual schools as a hybrid of public, charter, and home schooling, with ample dashes of tutoring and independent study thrown in, all turbocharged by Internet technology (p. 36). A more recent study conducted by Barbour and Reeves (2009), defined virtual school as an entity, which has been approved or accredited by a state or governing body within the state, that offers secondary level courses through distance delivery most commonly using the Internet. (p. 412). This study examined the practice of virtual school following Clarks and Barbour and Reeves definition. Organization of the S tudy The remainder of the study is organized into five chapters and appendices including some released test items and national and state Algebra and mathematics standards Chapter t wo is the review of the related literature regarding mathematics courses specifically Algebra success factors and online learning success factors.

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21 Chapter three presents the research design and methodology of the study. The population and sampling technique, instruments that were used for data collection, and the procedure of data analysis are described. Chapter four presents the results of the data analysis and the findings based on the analysis. Chapter five contains the summary, discussions and implicati ons of the results, recommendations based on the results, and conclusions. The study concludes with a bibliography and appendices.

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22 0.04 0.18 0.3 0.5 0.7 1 0 0.2 0.4 0.6 0.8 1 1.2 2000-01 2001-02 2002-03 2004-05 2005-06 2007-08 Academic Year Enrollment Unit: Million Figure 1 1 Online Students at K 12 Level

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23 CHAPTER 2 REVIEW OF LITERATURE Introduction Online/distance education has greatly contributed to the American education system with its broad advantages covering a variety of aspects including economic, political, demographic, and pedagogical (Dede, 1990). Distance learning, especially online learning, can decrease instructor and student travel costs and possibly increase instructors productivity (Bartley & Golek, 2004; Cavanaugh, 2001; Cornford & Pollock, 2003; Evans & Haase, 2001; Gallagher & McCormick, 1999; Glenn, 2001; Paulsen et al., 1998). The communities benefit from online/distance learning because it can increase educational opportunities that otherwise might be restricted by geographic barriers or resource restriction and provide the flexibility for students at different levels (Bogden, 2003; Helphinstine, 1995; Kerka, 1996; Parsad & Lewis, 2008; Patrick, 2004; Shachar & Neumann, 2003). The online delivery method via learning management systems can help the decision making process in regard to instruction and administration issues to improve teaching and learning ef fectiveness through providing dataenriched environments (NACOL & Partnership for 21st Century Skills, 2006). Well designed online instruction can promote collaboration among peers and between learners and instructors and coconstruct their knowledge struc ture (Bartley & Golek, 2004; Blomeyer, 2002; Hassell & Terrell, 2004; Hill, 1997; Summers, Waigandt, & Whittaker, 2005; Webster & Hackey, 1997), leading to the enhancement of higher order thinking skills and cognitive abilities (Blomeyer, 2002; Garrison, 2003) as well as motivation for students with different learning styles (Butz, 2004; Hassell & Terrell, 2004). Online courses have worked well with a variety of learners including at risk

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24 students, students with some disabilities, and students with limited English proficiency (Keeler et al., 2007; NACOL, 2009; Wat s on, Gemin, & Ryan, 2008). The advocates of online learning believe it will transform teaching, learning, and schooling as a whole (Cox, 2005). As a comparatively new form of online education, virtual schooling has experienced significant development since its emergence in the late 1990s and has been accepted by more and more educators and students because of its great benefits. With its help, small schools and rural schools especially can offer a wi de range of high quality courses that otherwise they can't offer (Donlevy, 2003). Virtual schooling gives students more options for obtaining education (Butz, 2004; Clark & Berge, 2005; Newman, Stein, & Trask, 2003; NACOL & the Partnership for 21st Century Skills, 2006). Virtual school could help solve the inequality of educational opportunities caused by a variety of reasons such as family income, geographical location, and school resources (Blaylock & Newman, 2005; Cavanaugh, 2001; Clark & Berge, 2005; Hernandez, 2005; Kellogg & Politoski, 2002; Newman, Stein, & Trask, 2003; Roblyer et al., 2007; Rose & Blomeyer, 2007; Setzer & Lewis, 2005; Watson & Ryan, 2007). Virtual school can help improve student learning outcomes and skills (Clark & Berge, 2005) through offering individual instruction and flexibility to meet the specific needs of students (Keeler et al., 2007; Kellogg & Politoski, 2002; Newman, Stein, & Trask, 2003). With the support of advanced technologies and rigorous curriculum, virtual schooling can help students master 21st century skills including global awareness, self directed learning, information and communications technology literacy, problem solving, and time management and responsibility (NACOL & the Partnership for 21st Century

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25 Skills, 2006; Watson & Ryan, 2006). Virtual school gives students who failed a course in a traditional classroom the chance for remediation (Barker & Wendel, 2001; Freedman et al., 2002; Glass, 2009; Newman, Stein, & Trask, 2003) and students who want advanced courses an enriched curriculum such as advanced placement courses in different fields including mathematics and science (Barker & Wendel, 2001; Butz, 2004; Newman, Stein, & Trask, 2003; Watson, Gemin, & Ryan, 2008). Virtual school also can benefit home school students through offering more educational opportunities that they otherwise wouldnt have due to reasons such as their parents lack of knowledge or family resource limitation (Butz, 2004; Watson, Gemin, & Ryan, 2008). Research in O nline/D istance E duc ation and S ignificance of T his S tudy Along with the growth of online education in the US, considerable research has been conducted on online/distance education effectiveness with respect to improved student academic performance and most of the studies have confirmed its effectiveness. However, little research has been done to examine success factors in K 12 online learning environments. In recent years, two lines of research emerged to address online success factors: one focuses on learner characteristics and another one focuses on learning environment characteristics (Roblyer et al., 2008). However, no clear set of characteristics have been identified as online success factors and no conclusive model has been created to apply in online learning practice (Roblyer & Davis, 2008; Tallent Runnels et al., 2006). Learner characteristics including participation in academic activities, whether have IEP and learning environment characteristics including teacher comment and school type (private or public school) have been proved to correlate to student academic achievement. However, these variables have not been

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26 investigated systematically in one model regarding their effects on student success in K 12 virtual learning environments. In a document about nation wide coll ege student transcripts, Adelman (1995) reported that math courses detained the top 7 places in the percentage of grades that were withdrawals, incompletes, or no credit repeats. The first six were precollege math courses and the seventh was college Algeb ra Clearly, math is a difficult subject for many students including secondary level and higher education level. School Algebra is therefore a key subject during the school reform discussion (Chazon & Yerushalmy, 2003). It has been critical for filtering the educational opportunities for high school students to further study in college (Moses, 1994; Moses et al., 1989). Algebra/mathematics is also a very important momentum to push a society to move forward. Many career options are only open to students wit h advanced mathematics skills in the job market (House, 1993). Stanic and Hart (1995) believe mastering mathematics knowledge and being able to apply mathematics ideas are critical for each member in a society to participate in the democratic processes and have more career opportunities. The possession of more mathematical literacy for everyone in the society is also the need for full participation in military service and shifts in US and the worlds economic systems (Secada, 1992). The purpose of this stu dy is to examine the factors including LMS utilization, teacher comment /feedback and s tudent demographic information that can influence the success of Algebra courses in K 12 virtual learning environments. This study can help discover certain characteristi cs and good practices in online learning and help incorporate them into the instructional model of the K 12 virtual learning environment. It

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27 also will add to the knowledge of the effectiveness of online/distance education in helping improve student academi c achievement in K 12 virtual schooling and provide valuable guidance for better implementation and practice. The investigation of success factors will provide a deeper understanding of success in the K 12 virtual learning environment specifically in online Algebra/math courses to guide management of virtual schools for maximizing their effectiveness to provide better assistance and supplementation to the traditional learning environment. Review of L iterature The review of literature in this chapter covers the effectiveness of online/distance education, Algebra/mathematics education, and online success factors. The review of literature on effectiveness of online/distance education presents the evidence for the conduct of increasing research in online/distanc e education. It also provides the rationale for the investigation of online success factors in this study. The review of Algebra/mathematics education can grant the support for the selection of specific courses in which the present study is conducted and demonstrates the relationships between a traditional teaching format and online education. The review of online success factors grounds the present work in the related studies and provides the support for the selection of factors in the present study. Effectiveness of O nline/ D istance E ducation Well designed distance education courses/programs can provide effective learning with innovative pedagogy, rich experience, and deep understanding of knowledge (Cavanaugh, 2001). Many studies have been done t o examine the effectiveness of distance education. Research on distance education effectiveness has mainly focused on several aspects: student learning outcomes, student instructor interaction during the

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28 learning process, and student and faculty attitude and satisfaction with the learning experience (Gallagher & McCormick, 1999). Cavanaugh (2001) conducted one metaanalysis study to examine the effectiveness of interactive distance education in the K 12 learning environment. She reviewed 19 experimental and quasi experimental studies selected with strict criteria including the focus of study, publication date, research design, and calculated effect sizes to assess the effects of some technologies including videoconferencing and online telecommunications on student achievement and to investigate the success factors for effective distance education. All these studies covered a wide range of subject areas and grade levels. The overall effect size for the 19 studies, 0.147, indicated the small positive effect o f distance education over traditional education. No significant differences were found in grade levels, ability levels, content areas, technology use, and achievement measure. The author concluded distance education can be at least as effective as traditional education to help students achieve academic goals and that offering distance courses at the secondary level could enrich the course curriculum and students knowledge structure. Sherry, Jess, and Billig (2002) conducted one action research stu dy to evaluate the effectiveness of online learning in improving student media literacy and multimedia techniques. They collected data quantitatively using surveys and qualitatively using interview and focus groups from students and instructor. The results showed technologies that are integrated into online learning can help students acquire a variety of skills such as creating multimedia projects, editing digital artifacts, designing web pages, and promoting student learning motivation. They concluded that the technology -

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29 enriched learning environment in general and online education in specific has a positive influence on student achievement. Allen et al. (2002) reviewed 25 empirical studies comparing student satisfaction between distance education and tradi tional classrooms. The criteri a used to select the studies included the presence of comparison group and sufficient statistical information that effect sizes can be calculated. The results showed overall there was no significant difference in satisfaction level though students showed a slight preference toward traditional education over distance education. The researchers also examined the effects of communication and interaction on student satisfaction level and found that there was virtually no effect on communication methods (video, audio, written). They supported the implementation of distance education by providing the evidence that distance education will not reduce student satisfaction with the learning experience. Aragon, Johnson, and Shaik (2002) ex amined the impact of learning style preference on student academic success between online and traditional learning environments to investigate the effectiveness of online learning. Thirty eight students taking a graduatelevel instructional design course, with nineteen in a traditional classroom and nineteen in an online course, taught by the same instructor in a Midwestern university participated in this study. The two groups of students were equivalent with respect to their demographic information such as age, undergraduate GPA, and year of baccalaureate graduation. The researchers found that there were differences in learning style between these two groups, though these differences were not significant when student success was controlled. It indicated onl ine education can be as effective as traditional f2f education in helping students succeed academically

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30 even though they have different learning style preferences. The researchers advocated for the development of online education based on the results. Swan (2003) looked beyond the no significant differences phenomena and reviewed the literature on the effectiveness of online learning focusing on the three types of interaction: student instructor, student student, and student content. She believed online education is effective as compared to traditional education, and some unique characteristics of the online technology can be further utilized to improve the online learning effectiveness. Based on the literature review, she gave some suggestions for the improvement of online learning environments such as providing timely and constructive feedback to students, integrating activities to establish online community, encouraging students to share experiences and thoughts during their learning process, and ensuring the clarity and consistency of the course materials. Zhao et al. (2004) employed a metaanalytical approach to investigate the effectiveness of distance education. The researchers found many individual studies reporting significant differences between distance education and traditional education; some found distance education more effective while others found traditional more effective. They selected 51 out of thousands of articles for review with some criterion such as that they needed to be journal articles, they must possess empirical data, and they needed enough statistical information to calculate effect size. The researchers analyzed several variables in this metaanalysis including study related variables such as design of study, measurement employed, instruction related variables such as instructor (status, involvement level), learner (status, background), curriculum (content area, degree), and milieu (interaction, media, setting). Effect size was calculated for the

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31 estimation of differenc e between distance education and traditional education. The results showed overall there is no significant difference between distance education and traditional education. However, the wide range of effect size ( 1.43, 1.48) indicated distance education was much more effective in some studies while was much less effective in other studies. Interestingly, the researchers found publication time was a factor for the effectiveness: studies published before 1998 are more likely to find no significant difference while studies published after 1998 are more likely to find significant effectiveness favoring distance education. The researchers believed that distance education is getting better with the advance of technologies and design principles. The instructor involvement was found to influence the effectiveness: traditional education was more effective when involvement was low and distance education was more effective when it was high. This confirmed the importance of instructor involvement in the form of teacher f eedback, and student teacher discussion for successful distance learning. The researchers also found the content area can predict the difference between distance education and traditional education: distance education was more effective in fields such as B usiness, Computer Science, and Medical Science; no significant difference was detected in Social Science and Science fields; distance education was slightly effective over traditional education in Military, Mathematics and Specific Skills. Though the researchers did not examine the learner variables such as gender, or learning styles, they believed groups with certain characteristics are more likely to succeed in distance learning. This study also found a blended learning environment mixing distance educati on and a certain amount of f2f meeting was most effective and called for more comprehensive studies in the distance education field.

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32 Cavanaugh et al. (2004) reviewed 14 studies to examine the effectiveness of K 12 distance educat ion. The studies selected were related to distance education published between 1999 and 2004 under the following criteri a : type of publication, K 12 focus, quantitative experimental or quasi experimental studies, and enough statistical information for the calculation of effect size. They specifically looked at the effects of distance education on student academic achievement and the effects of different features of distance education including content area, duration and frequency of distance education, student grade level, school type, interaction, and instructor role on academic achievement. The overall effect size, zero, showed distance education is as effective as traditional education. The wide range of effect size ( 1.158, 0.597) indicated some distance education courses/programs were much better than traditional education while others were much worse. Publication and methodological variables such as year and type of publication, measurement employed in the study and statistical power, and distance education experience variables such as duration and frequency of distance education, instructor role, and type of interactions had no significant influence on effect sizes. However, instructional and program variables such as student grade level, school type, a nd content area did influence effect sizes significantly. The researchers concluded with the promotion of implementation of K 12 distance education with close collaboration among different stakeholders including teachers, researchers, policymakers, develop ers, and parents, and more rigorous research in this field to guide the practice and implementation of K 12 distance education. Stewart et al. (2005) evaluated the effectiveness of one online case based continuing education program for family physicians i n improving their knowledge and

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33 skills. They randomly assigned the participants into experimental groups with the implementation of intervention: online learning and control group without the intervention. They analyzed the knowledge and skill growth measured by two knowledge questionnaires and charts quantitatively and the posts and emails qualitatively. The results showed the intervention had positive effects on knowledge growth and the quality of practice for these physicians. The researchers confirmed t he promis e of the broad implementation of online education in general. Williams (2006) reviewed 25 comparative studies from 1990 to 2003 on distance education in allied health science education to examine the learning effectiveness on student achievement and the instructional design (ID) components contributing to the effectiveness. The overall effect size, 0.15, with confidence interval from 0.07 to 0.23, showed distance education was slightly more effective than traditional education with respect to impr oved student achievement. The results also showed the integration of ID components in distance courses had a positive effect on achievement. The researcher suggested the effective distance education courses should incorporate various ID components. The study was concluded with the promotion of distance education courses/programs and a call for more research on the effect of different aspects such as educational level on the effectiveness of distance education. In 2009 US Department of Education ( DOE ) releas ed a report about a metaanalysis of empirical studies from 1996 to 2008 to evaluate the effectiveness of online learning practice. The studies included in this metaanalysis were selected based on the criteri a : rigorous research design including random as signment or controlled quasi experimental design to contrast online to traditional education, objective learning

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34 outcome was measured and enough information to calculate effect size. The researchers found overall online students outperformed traditional st udents with respect to their learning outcomes with effect size of 0.24 favoring online students. Hybrid learning students had a larger gain over traditional students relative to purely online students over traditional students with effect size of 0.35 fav oring hybrid learning students when comparing hybrid learning and traditional learning and effect size of 0.14 favoring online students when comparing purely online learning and traditional learning. Some methodological variables including sample size, content knowledge, research design, and equivalence of instructional approach were evaluated whether they accounted for the results and equivalence of instructional approach was found to be a significant moderator variable. This report confirmed that the combination of a variety of elements in online learning such as instructional strategies, integrated technologies, and students effort rather than the delivery medium per se is what results in better learning outcomes. The researchers also found that very few rigorous research studies of K 12 online learning effectiveness have been published. Only 7 out of 99 studies included in this meta analysis focused on K 12 level, and the effect size comparing online learning and traditional learning at K 12 levels was not significant. Therefore, the researchers caution the application of the results to K 12 virtual learning environments and more rigorous research is needed to guide the practice and implementation of K 12 online education. The proved effectiveness of onl ine/distance education in the literature provides the support for the present study. M any effectiveness studies have focused on the student outcome This lends the relevance to the selection of student academic achievement as

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35 the dependent variable in the present study. Another focus in many effective studies, student teacher interaction, presents the rationale for the investigation of teacher comments which could be the indicator of student teacher interaction in the present study Algebra / M athematics E d ucation C onsiderable educational and psychological research has been conducted to identify the success factors in mathematics fields (Grouws, 1992). In the literature on mathematics education, many researchers have focused on a variety of factors that asso ciated with mathematics learning including student attitude and background, family backgroundsocioeconomic, peer environment, instructor factors, and curriculum and instruction (Beaton & Dwyer, 2002; Kifer, 2002; Wilkins, Zembylas, & Travers, 2002). These factors have been categorized into three major topics by Schiefele and Csikszentmihalyi (1995): learner characteristics such as learning styles, learning strategies, and locus of control, home environment such as socioeconomic status (SES) and family size, and school environment such as instructor experience, instruction quality, technique support and resources. However, in other studies, these factors have been grouped into two categories: learner characteristics and learning environment characteristics w hich include home learning environment and school learning environment (Catsambis, 1995; Ercikan, McCreith, & Lapointe, 2005; Ho et al., 2000). To be consistent with Roblyer et al., (2008)s study of K 12 distance education we used two groups of factors i n the present study. Learner characteristics variables Research has shown learner personal characteristics variables such as prior knowledge and background, motivation or self concept are strongly associated with

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36 student mathematics achievement (Cats ambis, 1995; Ercikan, McCreith, & Lapointe, 2005; Ho et al., 2000). Some affective variables such as self concepts, attitudes toward mathematics, self confidence mathematics learning ability, motivation, locus of control, and perceptions of the usefulness of mathematics have been found to relate to student academic achievement in mathematics (Bassarear, 1991; Duranczyk, 1997; House, 1995; House, 1993; Marsh & Yeung, 1997; Reyes, 1984). Increased study time and the taking of advanced coursework also can posi tively affect students' mathematics achievement (Secada, 1992). Student English language proficiency is another factor for student mathematics achievement in U.S. (Jacobson, 2000; Secada, 1992). Bilingual students whose native language is not English will be likely to achieve higher performance in mathematics if they receive the instruction in their native language (Secada, 1992). Ercikan et al. (2005) conducted one exploratory research study examining factors that might affect students achievement in mathematics and their participation in advanced mathematics courses in three countries: Canada, Norway, and the US. They found students personal and home environment variables strongly affect their achievement in mathematics and participation in advanced math ematics courses in these three countries. These researchers specifically confirmed the relationship between attitude toward mathematics and participation in advanced mathematics courses in these countries and the relationship between SES related variables and achievement in mathematics in the US. Higbee and Tomas (1999) conducted one research study to examine the relationship between noncognitive variables including math anxiety, perceived

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37 usefulness of mathematics, self control ability, and self confidence in ones ability to learn mathematics and academic achievement in mathematics. The participants were 23 college freshmen and student score in their math courses was collected as the indicator of academic achievement. The researchers found student attitu des toward mathematics, motivation, self management skills and self confidence are related to academic achievement. Jacobson (2000) examined success factors in high school mathematics using a variety of statistical techniques: multiple regressions, ANOVA, and path analysis with a sample size of 1205 high school students. She found student beliefs/confidence in their mathematics learning ability and family background/ SES have strong effect on academic achievement in mathematics courses. Interestingly, she al so found student's primary language and writing ability are significant success factors in mathematics learning. Based on the review of literature on mathematics education, Reyes (1984) asserted the affective variables including students confidence in their mathematics learning ability, attitude toward mathematics, and mathematics anxiety are related to mathematics learning. Reyes believed students confidence in mathematics learning or self concept about mathematics learning has a positive relationship wi th mathematics achievement. Mathematics anxiety can negatively affect student mathematics achievement. Reyes believed students attitude toward mathematics and their perceived usefulness of mathematics can affect their decision about the mathematics course they will take. Students who valued mathematics more tended to take more mathematics courses, which, in turn, could contribute to their higher achievement.

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38 Edge and Friedberg (1984) conducted one study to evaluate the effect of student ACT scores, high s chool prior knowledge in calculus, gender, family size, and high school size on student academic achievement in the first college calculus course. They found the long term perseverance/self control ability and student pre experience/knowledge in Algebra ca n significantly affect student achievement in the first semester of calculus for freshman. Schiefele and Csikzentmihalyi (1995) conducted one research study using 108 high school freshmen and sophomores to examine the relationships between interest, learni ng motivation, prior mathematics knowledge/mathematic ability, student mathematics learning experience, and academic achievement in mathematics. The researchers found mathematics ability is a significant predictor of academic achievement, and the predictability of interest for achievement is different for students at different grade levels. At 9th or 10th grade level, interest is a good predictor of achievement. Belcheir ( 2002) reported a research study on the exploration of variables that can predict suc cess in math courses. The sample of participants wa s 734 college students enrolled in one intermediate Math course. This study include d learner variables such as student math learning attitude and dispositions, study skills, and student commitments. This s tudy did not find time on task as a good predictor of course success as expected. However, the researcher further explained that some valuable information could be missing in this study because the researcher did not collect information about how students spent time studying and whether students felt the amount of time they could use on the course was sufficient. Student motivation and commitment were found to be the most significant predictors of success for the Algebra courses. The researcher also

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39 emphasi zed that the instructors should let students know early on how they are performing in order for them to succeed in the courses. Learning environment variables Research has shown learning environment characteristics variables including family background and support, school or classroom resources, teacher/classroom are strongly associated with student mathematics achievement (Catsambis, 1995; Ercikan, McCreith, & Lapointe, 2005; Ho et al., 2000). High parent expectations (Cohen, 1987; Marjoribanks, 1988; Scot t Jones, 1984; Seginer, 1986; Thompson, Alexander, & Entwisle, 1988) can positively affect students' mathematics achievement. Instructional strategies including the implementation of computer Algebra software (Elington, 2003; Lawson, 1995; Mayes, 1995; Stephens & Konvalina, 2001), the use of other technologies such as calculators in general and graphing calculators in specific (Elington, 2003), and collaborative problem solving strategy and visual technique support (Higbee & Thomas, 1999) can positively aff ect student mathematics achievement. Teacher variables such as teaching behaviors are another type of factor that related to student academic achievement (Schoen et al., 2003). House and Telese (2008) investigated the relationships between instructional strategies and self confidence in mathematics learning ability and Algebra academic achievement in the US and Japan using 2003 TIMSS assessment data. The sample includes 4244 students from Japan and 7862 students from the US. They found the teaching and learning strategies and students' self confidence/beliefs in mathematics learning ability are significantly associated with student Algebra academic achievement in these two countries. Interestingly, they found the instructional strategy of cooperative lea rning activities (work ing in small groups) negatively affect ed student achievement

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40 and students who worked on problems on their own (active learning strategy) more often tended to achieve higher performance. Students who can associate their mathematics knowledge with their daily lives tend ed to achieve lower test score. More research is needed on the effect of learning strategies (group work or independent study) and the triangulation of a variety of academic measurements during the study of Algebra lea r nin g factors. Elington (2003) investigated the effects of calculators including basic, scientific, and graphing on students' achievement and attitude levels through the examination of 54 studies, 26 of which targeted high school students. She found the use of calculators in the testing system and instruction can increase students strategic skills, computational and conceptual skills, and problem solving skills and promote students positive attitude toward learning mathematics. Based on the empiric al studies, Hollar and Norwood (1999) and Shoaf Grubbs (1993) found the use of graphing calculators can increase students overall mathematics ability including the understanding of function, the ability for modeling, interpreting, and translating. In addi tional to students increased ability in mathematics problem solving, the integration of technology during the mathematics learning process can also promote collaboration among students during group interactions and class discussions (Goos et al., 2003). Wheland et al. (2003) examined two types of factors that affect student academic performance in an intermediate Algebra course: instructor characteristics English speaking status (nonnative English speaker), teaching assistant or adjunct faculty, and st udent characteristics: attendance. The effect size was also calculated in the study. The researchers found the instructor characteristics variables: English speaking status,

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41 teaching assistant or adjunct faculties do not have significant effect on student performance while student attendance can significantly affect academic performance. Schoen et al. (2003) also analyzed teacher variables related to student achievement within one reform based project, the Core Plus Mathematics Project involving 40 teachers and their 1466 students in 2 schools. They found teaching behaviors such as following the guidance and recommendations of standards and aligning the instruction with the high mathematics expectations are related to higher academic achievement. Algebra teaching and learning A significant amount of research has been conducted on Algebra teaching and learning considering its importance as the momentum to push society to move forward. The focus of Algebra teaching and learning research has been shifted from students understanding of Algebra activities to the way students construct meaning of Algebra procedures and objects (Kieran, 2007). Based on these studies recommendations and suggestions have been provided to help improve Algebra teaching and learning quality. Smith, diSessa, and Roschelle (1993) believe school Algebra instruction should build upon the strengths and the resources within the perceptions students have based on their own experience in relation to Algebra concepts. Students should grasp the ability to solve illdefined tasks that are more closely connected with the questions they may have in the real life rather than the well defined ones within the school settings (Resnick, 1987). Based on the review of literature on Algebra t eaching and learning for students with different backgrounds, Secada (1992) recommended some instructional strategies including increased school time, more mathematics course taking, use of students' native language for instruction, direct instruction for structured curriculum and

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42 basic mathematics skills, and divide of whole class activities into group and individual work. Considerable research has been conducted on the approaches to Algebra learning. Bednarz, Kieran, and Lee (1996) reported the four approaches that have been focused on at an international colloquium on Algebra in early 1990, including generalization of numerical and geometric patterns and laws regarding Algebraic relationships, functional situations, modeling of mathematical phenomena, and problem solving. Drijvers (2003) described the similar approaches for Algebra learning in more detail: Problem solving approach: view Algebra as a way to solve problems that can be expressed in equations. Functional approach: view Algebra as a way to investigate the functions and relations among different variables. Generalization approach: mainly focus on the examination of patterns or models and configurations, and focus on the generalization of relations among different variables. Language approach: view Algebra as a way to convey mathematics ideas in which Algebra is merely a representation structure composed of symbols without specific context attached meaning. Dri j vers also identified some aspects that make learning Algebra difficult. These inclu de: 1 The difficulty for students to relate the formal algorithmic procedures with informal while meaningful methods 2 The abstract characters of Algebra problem solving approaches that students cant connect them with the concrete situations 3 The Algebraic lan guage includes particular symbols and rules that are difficult for students to grasp

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43 Drijvers pointed out the importance of reification of expressions and formulas during the Algebra learning process. Students need to possess the ability to comprehend the structure and meaning of formulas and expressions, which as he called symbol sense (p.49). Furthermore, Drijvers explained some key elements of one theory for mathematics education: Realistic Mathematics Education (RME) and their meaning and relations t o Algebra learning: Guided reinvention and progressive mathematization: with the guidance from the teacher, students are given the opportunities to develop formalized mathematics knowledge by employing the informal strategies and apply the knowledge in the concrete life situations. Didactical phenomenology: design activities that encourage students to develop their own mathematics learning strategies Horizontal and vertical mathematization: Horizontal: by employing empirical methods for example observation and experimentation, students can structure and solve the problem with mathematics formulas or conventions. Vertical: based on these problem solving experiences and beyond, students can develop mathematical framework in regarding with the relations among symbol. The review of Algebra/mathematics education demonstrates the relationships between f2f learning environments and online education with respect to the factors of academic achievement It lends the support for the selection of success factors in th e present study. For example, many Algebra/mathematics education studies have focused on a variety of factors that associated with mathematics learning outcome. These factors include learner characteristics such as learning styles and learning strategies, and learning environment characteristics such as family background and school learning resources. This supports the categorization of factors in the present study.

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44 Success F actors in Online Learning Given the fact of extraordinary development of online e ducation in the last two decades, little research has been conducted to examine success factors in the online learning environment as compared to the traditional learning environment. With the high early dropout and failure rates in the online learning env ironment (Carr, 2000a; Roblyer & Elbaum, 2000), there is an urgent need for more research on success factors to prevent students from dropping out of virtual or physical school and ensure their success in this learning environment (Barbour & Reeves, 2009; Bernard et al., 2004a; Butz, 2004; Dickson, 2005; McLeod et al., 2005). At the K 12 level, there is great concern about the readiness for students to take online courses and succeed academically because they are not socially and emotionally mature as compared to students in higher education (Picciano & Seaman, 2007). The review of studies examining success factors in online learning environments as well as traditional learning environments can better guide the practice of K 12 online education. Schrum and Hong (2001) administered a survey with 70 institutions and found several factors can influence student success in elearning environment: learning styles, prior technology experience, personal disposition, study habits, and tools accessibility. Brown and Liedholm (2002) conducted a comparative study between online education and traditional education and found students personal effort on learning tasks could make a difference in academic performance. Swan (2002) investigated the correlation between 22 cours e design factors and student academic achievement and satisfaction with learning experience. She found three factors: transparent interface/clarity and consistency in course design, instructor feedback/instructor student interaction, and dynamic online dis cussion are associated with the success of online learning. These

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45 three factors could be three necessary steps for the establishment of an online learning community (Swan et al., 2000), which can affect students learning outcomes in online education. Ro blyer and Marshall (2003) evaluated one instrument, the Educational Success Prediction Instrument (ESPRI) which was created to predict success in Virtual High School ( VHS ) courses. The constructs measured by this instrument related to success in VHS course s included time management, achievement, motivation, self responsibility, prior technology skills, self regulation, and self confidence. They evaluated ESPRI with 135 students in 13 virtual high schools and found ESPRI is a 0.92) to predict student success in the online learning environment, and certain personality characteristics and attitude were associated with online learning success. Roblyer et al. (2008) created the instrument: ESPRI V2 based on ESPRI and reevaluated it with a bigger sample size: 4110 students from VHS. They found the four factors measured by ESPRI V2: technology use/self efficacy, achievement beliefs, instructional risk taking, and organization strategies can predict student success in VHS courses though its harder to predict failure, and the Cronbach V2 is a reliable instrument. The researchers concluded the combination of prior knowledge, cognitive characteristics such as self efficacy and achievement beliefs, and environment variables such as Internet accessibility and technical support can predict student online success. According to Roblyer et al. (2008), two lines of research emerged to address success factors in online learning: studies focusing on learner characteristics and studies focusing on learning environment characteristics. Learner characteristics include

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46 student cognitive factors such as locus of control and learning styles; prior technology skills and attitudes; and experience and prior knowledge about course content while learning environment characteristics include technology support, course content area, and accessibility to the Internet. At present, no clear set of characteristics have been identified to predict the success of the virtual learning environment and no conclusive model has been created to apply in online learning practice (Roblyer & Davis, 2008; Tallent Runnels et al., 2006). Other learner characteristics including personal effort/participation in academic activities, whether has individual educ ational plan, race/ethnicity, and family background/participation in free/reduced lunch programs, and learning environment characteristics including teacher comment/feedback/instructor student interaction and school type (private or public school) also hav e been proved in some studies to correlate to student academic achievement. However, these factors influences have not been investigated systematically. The review of these factors in light of the relationship with student academic achievement in other st udies can provide deeper understanding of success in online learning in general and the K 12 virtual school environment in specific and shed light on the establishment of a model to predict online learning success in general and online Algebra/mathematics learning in specific Teacher comments/ t eacher student interaction Teacher comments and s tudent teacher interaction is a critical component in academic learning (Boaz, 1999; Laurillard, 1997; Parker, 1999; Schaffer & Hannafin, 1993; Summer, 1991; S wan, 2003; Williams, 2006). It can affect learning in traditional f2f learning environments (Christophel, 1990; Kelly & Gorham, 1988; Rodriguez, Plax & Kearney, 1996) and online learning environments (Blomeyer, 2002; Jiang & Ting, 2000; Johnson et al., 2000; Swan, et. al., 2000; Swan, 2002; Swan, 2003; Tallent Runnels et

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47 al., 2006). Interaction, well described by Cavanaugh, is the "core of teaching" (2001, p. 3) and at the heart of online learning (2007, p. 6). T he presence of interactivity is vital for t he quality education in distance learning (Blomeyer, 2002; Flottemesch, 2000; NACOL, 2006; Parker, 1999; Zhao et al., 2004). It can help students evaluate their learning progress and adjust the instructional strategies if necessary to improve the learning outcome which will lead to a deeper understanding of knowledge (Hiebert & Grouws, 2007; Parker, 1999; SchoenfeldTacher, McConnell, & Graham, 2001). Student instructor interaction can provide the social support for students during the learning process, whi ch is conducive to higher academic achievement and the development of social skills (Parker, 1999). It also has a positive relationship with students satisfaction with their learning experience (Liaw & Huang, 2000; Swan, 2002; Usun, 2004). The educators active facilitation in the form of teacher comment and feedback in online learning is an important factor that influences students academic performance ( Cavanaugh et al., 2005; Dickson, 2005; Ferdig, Papanastasiou, & DiPietro, 2005; Hughes et al., 2005; Karp & Woods, 2003; Lin 2001; Peters 1999; Phipps & Merisotis, 2000; Smouse, 2005; Zucker, 2005). Jiang and Ting (2000) conducted one study to examine instructor activity in online learning and found the students perceived learning is correlated with the number of feedback comments per student that the instructor made. This relationship also has been confirmed by Swan et al. (2000) and Swan (2002). Anderson and Kuskis (2007) argued many of the pedagogical benefits brought by instructor feedback/student tea cher interaction such as those related to motivation are relevant to distance education as well as the conventional classroom education.

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48 Hughes et al. (2005) believed that teachers individual feedback can increase communication opportunities for students who are shy and may not participate in academic activities, and these opportunities are helpful to develop closer relationships between the instructor and students. Constructive and timely feedback from instructor is one of the success factors for the prac tice of an effective virtual learning course/program (Cavanaugh, 2004). Frequent and open communication between students and instructor is identified as an important component to build a virtual community during online learning ( Lin 2001; Murphy, Mahoney & Harvell, 2000) The development of a learning community in an online K 12 course is considered an important factor for students better academic performance ( Lin 2001; Oren, et al., 2002; Ronsisvalle & Watkins 2005; Wang & Newlin, 2000) Wang and Newlin ( 2000) argued the social support provided by the learning community could improve students' academic achievement as well as their involvement and interest in online academic activities. Oren, et al. ( 2002) believed t eachers should act as a moderator to faci litate and scaffold students learning and encourage various interactions especially peer to peer interaction to let students learn from each other. They emphasized teachers' supportive feedback to encourage student to student social interactions for the formation of the social groups during the leaning process and beyond. ODwyer et al. (2007) conducted a quasi experimental study to examine the impact of one Algebra I online initiative on students learning outcomes and found that online students themselves also highly value the student instructor interaction during the learning process.

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49 Participation in online academic activities T he number of times students logged into the LMS and how long they stayed in the LMS could be considered as the indicators of participation in online academic activities. The time spent in academic activities has been identified as a very important factor that has a strong effect on success in online education (Cavanaugh, 2007), faceto face instruction (Rocha, 2007), and blended programs (Cavanaugh, 2009). The activities students engaged in during online study is a predictor of final scores, with students who participate in academic activities at high level performing better than those who do not in online learning (Wang & Newlin, 2000). Dietz (2002) believed one of the most significant predictors of success is attendance which could be reflected by the number of times students log into an LMS. These findings were confirmed by Dickson (2005) that participation in online academic act ivities, which is measured by clicks in the LMS, is a strong predictor of final scores in online learning. Race/ethnicity Considerable studies have been conducted on the relationship between race/ethnicity and academic achievement in traditional learning environment. Racial gaps in student test score are undeniable facts (Bali & Alvarez, 2004; Hall et al., 2000). The student body in online K 12 schools often represents the community that is served by the traditional school system ( Ronsisvalle & Watkins, 2005) The findings in the literature of the relationship between race/ethnicity and student academic achievement in traditional learning environments could shed light on success factors studied in K 12 online learning environments. Through a metaanalysis of 16 studies of race differences in mathematics performance from grades 4 to 8, Lockhead et al. (1985) found Asian Americans usually

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50 perform at the highest level in math, followed by Caucasian students, and then Hispanic. All the three groups perform bett er than African American students. Hall et al. (2000) also found significant differences in student math achievement among different ethnicities in a study on gender and racial differences in mathematics performance among 5th and 8th grade students in the United States. These differences continue at the higher level. The math skills of most African American in 12th grade, as Barth (2001) described, are only equivalent to the skills of Caucasian students in the eighth grade. U.S. DOE released a report in 2004 about the gaps in academic achievement in different content areas such as reading, math, and science among different racial and ethnic groups based on the data collected since the mid1980s. At 4th grade level, 41% of whites and 38% of Asians were profic ient readers while the number for African Americans, Hispanics, and Native Americans was 13%, 15%, and 16%, respectively. In mathematics, 48% of Asians and 43% of Whites achieved at proficient level while only 10% of African Americans, 16% of Hispanics, and 17% of Native Americans achieved at this level. Participation in school free lunch/family SES Participation in school free or reduced programs has a correlation with student academic achievement, and the magnitude of correlation is weaker as grade level rises (McLoyd, 1998). Klein et al. (2000) conducted a research study on the relationship between students participation in free or reduced lunch programs and school test score using the data about 2000 Texas 5th graders in reading and math. They found the percentage of students participating in the free or reduced lunch program in a school can affect the schools mean test score. The researchers believed participation in these programs could be considered a sign of the level of poverty which has a strong

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51 relationship with student academic performance at the school level. Participation in a school lunch program was also frequently used as the measurement of students family Social Economics Status (SES) in the literature on student academic achievement (Sirin, 2005). The level of the family support including the resources provided for students and education values can influence student academic achievement (Hiebert & Grouws, 2007). Higher SES families provide students more resources at home and social capital, both of which can improve chances for their academic success (Coleman, 1988). A considerable body of research has been done on the relationship between SES and student academic performance. The magnitude of this relationship was found to be strong in tw o metaanalytic studies conducted more than twenty years apart from each other: 0.343 in Whites (1982) metaanalysis and 0.299 in Sirins (2005) metaanalysis. In K 12 online learning environment participation in school reduced lunch program/family SES c ould also be associated with student academic success. Learning ability/ p resence of i ndividual educational plan Student learning ability is a factor that can influence student academic success during the learning process (Keeler & Horney, 2007). The vir tual school student body is a diverse population including students with different learning disabilities (Dickson, 2005; Ferdig, Papanastasiou, & DiPietro, 2005). Virtual school offers individual education plans for these students during the learning process. Therefore, whether or not a student has an individual education plan could be a sign of the level of learning abilities. The review of studies on the relationship between learning ability and academic achievement could shed light on the decision making process to provide more opportunities for students with special needs to succeed in the K 12 online learning environment.

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52 According to Keeler et al. (2007), students may bring some characteristics that could physically or psychologically inhibit their acc ess to the information or tools on the Internet, preventing their success in the online learning environment. The virtual school learning environment has the potential to bridge gaps between disabled students and other students without learning disabilities with respect to the success opportunities in online learning. For example, for students with different levels of learning disabilities, technologies such as computer, internet, audio, video, animation, gaming and simulation could help reduce their disadv antages as compared to students without disabilities (Coombs & Banks, 2000; Richardson et al., 2004). Some instructional design strategies have been recommended to ensure online courses meet the special needs of students with disabilities (Keeler et al., 2007; Rose & Blomeyer, 2007). These include assurance of accessibility to the information for students with disabilities and the support in course materials and learning activities for these students. Virtual courses need to be designed with accommodations specifically for students with disabilities to access course materials and should benefit all learners under the framework of Universal Design for Learning principals (Rose & Blomeyer, 2007). Since the emergence of universal design technology and the requi rement for the development of Learning Management System ( LMS ) to integrate components to meet the special needs of disabled students to align with the American with Disabilities Act (ADA) (Watson, 2007), the opportunities to achieve higher performance for disabled online students has been greatly increased. The proposition of early adoption of technology infused education for disabled online students (OConnor, 2000) also will benefit their

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53 online learning. Over a decade virtual school programs have succes sfully provided quality education to students with special needs (Rose & Blomeyer, 2007). However, even with the bridging gaps with regard to online success opportunities for disabled online students, they are still underrepresented in online education (K inash & Crichton, 2007). For example, even though different learning management systems such as WebCT and Blackboard are generally accessible to disabled students, there are still inherent problems limiting them from fully utilizing the functions in their courses therefore limiting their chances of success in online learning (Asuncion et al., 2006). Many online courses still have barriers preventing students with disabilities from fully accessing materials as other students do (Edmonds, 2004; Keeler & Horney, 2007), which will affect their success in online learning. Keeler and Horney (2007) conducted one study to evaluate the elements of online course design that address students special needs. They found some problems still existed in the five categories of design elements: accessibility, website design, technologies used, instructional methodologies, and support systems, which can prevent students special needs being fully addressed. According to National Center for Education Statistics (NCES, 2000), t hough computer and communication technologies may be especially beneficial for disabled students (Johnson, 1986), providing them the access to these technologies could be more expensive than regular students because they may need special equipment to use t hese technologies. The lack of accommodations for students with special needs may exclude them from fully participating in online learning (Keeler & Horney, 2007). Disabled students lack the opportunities to use these communication technologies for a variety of reasons

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54 including insufficient ly trained special education teachers and inadequate support services for them to use these technologies (NCES, 2000). This could lead to the academic gap between students with disabilities and students without these dis abilities in online learning. Students learning ability could affect other academic performance in addition to achievement such as academic engagement. Kersting (1997) interviewed 10 deaf students to examine their learning experience and found these students had lower academic engagement in learning activities unless they got sufficient support during their learning process. Richardson, Long, and Foster (2004) compared deaf students and their peers without hearing loss regarding academic engagement in dis tance learning. There were 267 students with a hearing loss and 178 students without this disability in an open university who participated in this study. The results showed students with hearing loss could not perform well on communication and some other tasks in comparison to students without this disability, which could affect the academic achievement negatively for these disabled students. School type The gap in student achievement between private and public schools has been documented in many studies (Chubb & Moe, 1990; Coleman & Hoffer, 1987; Coleman, Hoffer, & Kilgore, 1982). In 2006, the U S Department of Educations National Center for Education Statistics (NCES) released a report about the academic achievement differences in reading and math at grade levels 4 and 8 between private and public schools (Braun, Jenkins, & Grigg, 2006). This report showed on average the private school mean score was higher than the public schools. Students in private schools achieved at higher levels academically than t hose in public schools. However, many

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55 studies did not control other important variables such as student SES, or grade level when examining the difference between public and private schools and many of them have been done more than two decades ago, so the data could be already outdated. Therefore, more studies with new data and new methodology are encouraged to be conducted. A study (Lubienski & Lubienski, 2005) supported by the National Assessment of Educational Progress (NAEP) compared student achievement in mathematics between public and private schools using a student sample of 4th and 8th graders. There were over 13000 4th grade students from 607 schools, 385 of which were public schools and 222 were private schools, and over 15000 8th grade students fro m 740 schools, 383 of which were public schools and 357 were private schools who participated in this study. The results showed overall students in private schools outperformed their counterparts in public schools; however, after controlling for students SES, public schools outperformed private schools. The larger proportion of high SES students in private schools accounted for their overall outperformance. The researchers called for more research on the examination of effectiveness of public and private schools. The review of success factors grounds the present work in the related literature. It helps the establishment of the model in the present study with respect to the selection of independent variables For example, the family SES has been proved to relate to student academic performance in many studies, which provides the support for the inclusion of the participation in school reduced or free lunch programs which could be the indicator of family SES in the model. Many studies in this literature are quantitative

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56 studies. This can shed lights on the quantitative research method utilized in the present study. Conclusion The review of literature on effectiveness of online/distance education demonstrates well designed online/distance course can be as effective as its traditional counterpart with respect to helping improve student academic achievement. This presents the evidence for the increasing research in online/distance education and provides the rationale for the present study. The review of literature on Algebra/mathematics education illustrates that several issues such as learner prior knowledge and learning ability, study time, and instructional strategies need to be addressed during the process of Algebra/mathematics teaching and learning. It also indicates a variety of approaches such as problem solving, generalization of geometric patterns and Algebraic relationships, and functional situations should be utilized to improve learning efficiency. This section builds the connections between tradi tional education and online education. Both of these two education formats share success factors though the effect could be different in these two environments. The review of success factors confirmed the relationship between student demographic information, participation level in academic activities, and teacher comments and student academic achievement. These are also the variables of interest in the present study. Even after more than 10 years of extraordinary growth in K 12 online learning, litt le research has been done as compared to post secondary education (Cavanaugh, 2007; Cooze & Barbour, 2005; Means et al., 2009; Picciano & Seaman, 2007; Picciano & Seaman, 2009; Ronsisvalle & Watkins, 2005). The amount of evidencebased research or empiric al study applicable to guide educators instruction and policy makers decision

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57 relevancies is slight ( ODwyer, Carey, & Kleiman, 2007) Many states have no data on the current practice of K 12 online education such as the number of students taking courses online, the number of online programs existing, and how these programs are managed (Watson, 2007). After review of 99 comparative studies regarding online education versus traditional education published between 1996 and 2008, the U.S. Department of Education found that only 7 of them involved K 12 learners ( Means et al., 2009). The development of K 12 online education is advancing different ly from the development of postsecondary online education (Picciano & Seaman, 2009). Therefore, the practice of onli ne education in higher education may not be applied to the K 12 environment. The dearth of studies on academic achievement in K 12 virtual learning environment in comparison with that in traditional learning environments form the rationale for more quantit ative research in this field to guide the implementation and practice of online learning at this level ( Cooze & Barbour, 2005; Means et al., 2009; Picciano & Seaman, 2007; Picciano & Seaman, 2009; Smith, Clark, & Blomeyer 2005; Watson, 2007). Quantitative data collection is the required research methodology to support the understanding of the efficacy in K 12 virtual school (Smith, Clark, & Blomeyer 2005). Currently the lack of new data regarding K 12 online learning is attributed to a variety of reasons including lack of requirement in many states for data collection on online students and the significant growth of K 12 online education practice causing the difficulty of data collection (Picciano & Seaman, 2007). Researc h conducted in virtual schools is rare because of its comparatively new practice, and currently available data cant provide enough information for accurate estimation of its practice (Glass, 2009).

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58 More research has been called for that focuses on students academic performance, particularly on the factors influencing the success of students in K 12 virtual learning environments (Smith, Clark, & Blomeyer 2005). The question of whether the factors that affect students achievement in the traditional school learning environment play the sam e role in the virtual school learning environment remain to be answered. A cademic performance is considered as the single greatest indicator of school completion ( Battin Pearson, Newcomb, & Abbott 2000), and lowering the school dropout rate is a national priority. The investigation of the factors that influence student academic performance in virtual schools is of critical interest to educators, researchers, virtual school program administrators, and policy makers.

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59 CHAPTER 3 METHODOLOGY Introduction Ther e are five sections in the methodology chapter: research design, population and sample, instrumentation, data collection and analysis, and limitations. What type of design employed in this study is explained in research design section. Then the population that this study is targeting, sample that has been selected and sampling techniques employed are described. Instruments utilized in this study are detailed in the section of instrumentation. The process of data collection and analysis are then explained. A nd limitations if any are pointed out at the end. The purpose of this study is to examine the factors including LMS utilization, teacher comment /feedback and s tudent demographic information that can influence the success of Algebra courses in K 12 virtual learning environments. The research questions in this study are: Does the level of LMS utilization influence Algebra/mathematics performance in online education? Does teacher comment or feedback influence Algebra/mathematics performance in online educati on? Do student demographic information such as race/ethnicity, grade level, status in virtual school, whether have individual educational plan (IEP), and participation in free/reduced lunch programs influence Algebra/mathematics performance in online educ ation? Research Design The present study is descriptive in nature. The researcher described some factors predictability of Algebra learning outcome with out intervening. The researcher collected the data at the end of 200809 academic year. These data incl ude student demographic

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60 information, their participation level in online academic activities, teacher comments, student EOC test score, and the score on one state standardized mathematics test The variables of interest in this study include: teacher com ments (TEACHERCOM), student participation level of online academic activities the number of times students log ged into the LMS (TOTALLOG), the time they stayed in the LMS (TOTALMIN), and student demographic information whether students have IEP students grade level s in their physical schools (GRADE), rac e/ethnicity (RACE), students status in the virtual school ( full time or part time students PT/FT), and the participation in free or reduced lunch (FRL) programs. They are independent variables in the study. Student EOC test score and the score on one state standardized mathematics test are dependent variables in this study. Participants and Data C ollection The data were collected during the 200809 academic year from one state virtual school in the Midwestern US region. This virtual school was implemented in 2007. A similar pilot project was conducted in spring, 2009, based on its first year (200708) data This dissertation builds on the results of the pilot project However, the present study is di fferent from the pilot project in many respects. For example, the present study investigated the success factors not only for Algebra EOC test but also for one state standardized mathematics test which was missed in the pilot project. The present study was conducted with different sample from the pilot project. Students statewide from bricksandmortar public and private schools as well as home school students were eligible to enroll in this virtual school. Enrolled students resided in most of the states school districts. The school hired content area teachers who met state certification and

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61 other requirements. A single LMS was utilized by this virtual school to manage course content and deliver instruction at the secondary level. Th e students needed to take the EOC test at the end of each semester during 200809 academic year after they completed the course. Some students also took one state standardized mathematics test at the end of academic year. Students who completed the four Algebra courses: Algebra I (1st half), Algebra I (2nd half), Algebra II (1st half), and Algebra II (2nd half) and took the EOC tests and the students who took the state standardized mathematics test grade 6, 7, or 8 participated in this study (Due to lack of information about stud ents who took the standardized test grade 3, 4, or 5, they were dropped from the study). The number of students who took the four Algebra EOC tests was : 1 01 75, 26, and 36 respectively. D ue to the insufficient power for data analysis caused by the small s ample size of Algebra II (1st half) and Algebra II (2nd half), 26 and 36, these two groups were dropped from the study. Within the two Algebra I groups, 64 out of 101 students in Algebra I (1st half) ( 63.4 %) and 59 out of 75 students in Algebra I (2nd hal f) ( 78.7%) were second year students in this virtual school. Students who took the two Algebra EOC tests were from grades 8 to 12. Students who took the standardized mathematics test grade 6, 7, or 8 were from grade 6 to 10 and the number was: 74 (grade 6) 73 (grade 7), and 107 (grade 8). All of these participants were first year students in this virtual school. Instrument Success in an online course can be measured by academic achievement including the grades students earn and their performance on advanc ed placement exams ( Ronsisvalle & Watkins, 2005; Tallent Runnels et al, 2006) Dickson (2005) used student final score as the measure of student performance in online courses during the

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62 data analysis in a study conducted to investigate the variability of s tudent performance in online courses. Full time online schools assessed student achievement in the same way as all public schools (Watson, Gemin & Ryan, 2008). Student achievement in many supplemental online programs is also assessed by course grade or EOC test score (Watson, Gemin, & Ryan, 2008). Indeed, some virtual schools and their teachers are paid on the basis of successful students, defined as those passing their courses. Algebra EOC T est Students who took the Algebra I EOC test in the virtual school participated in this study. The Algebra EOC tests were tests administered at the end of each semester in this virtual school. According to the virtual school administration, they have high correspondence to the states Algebra and mathematics content stan dards. The purpose of the EOC test, as described by the states department of education ( 2009), is to : Measuring student achievement and progress toward postsecondary readiness Identifying students strengths and weaknesses Communicating expectations for all students Meeting state and national accountability requirements Evaluating programs The Algebra I EOC test includes one session of multiple choice items and one session of performance events ( Missouri Department of Elementary and Secondary Educat ion (MDESE), 2009a). The items in the multiple choice session are developed specifically for students in this state (see Appendix A for some released samples). The items in the performance events session are longer, and focusing on more challenging

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63 tasks t hat require students to work through different problems, arguments, or require extended writing (see Appendix B for some released samples). These EOC tests are intended to measure students skills in number and operations, Algebra ic relationships, and data and probability. This state has its standards for Algebra (see Appendix C for the state Algebra standards). The Appendices A E provide the evidence of alignment between the Algebra I EOC test and state Algebra standards. State Standard ized Test Students who took one state standardized mathematics test grade 6, 7, or 8 after they finished one year of study in this virtual school during 200809 academic year participated in this study This standardized mathematics test is aligned with the state Show Me Standards which are the educational standards of this state For m athematics, the Show Me standards require students in state public schools to obtain knowledge of (MDESE, 2008) 1 A ddition, subtraction, multiplication and division; other number sense, includi ng numeration and estimation; and the application of these operations and concepts in the workplace and other situations 2 G eometric and spatial sense involving measurement (including length, area, volume), trigonometry, and similarity and transformations of shapes 3 D ata analysis, probability and statistics 4 P atterns and relationships within and among functions and Algebraic geometric and trigonometric concepts 5 M athematical systems (including real numbers, whole numbers, integers, fractions), geometry, and num ber theory (including primes, factors, multiples) 6 D iscrete mathematics (such as graph theory, counting techniques, matrices) Th is grade level state standardized mathematics test is a standards based test designed to measure the skills for each grade of st udents in the state where this virtual

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64 school is located (MDESE, c). It also has a national norm referenced test that can be used to compare students in this state with students across the country This component helps align the state standardized test with the National Council of Teachers of Mathematics (NCTM) standards See Appendix D for NCTM mathematics standards for grades 68. There are three types of questions in this grade level standardized mathematics test: 1. multiple choice items that are developed specifically for students in this state or the questions in the national norm referenced survey; 2. constructed response items that require students to provide the response rather than selecting the options among different choices; 3. performance events items as described above in EOC test that are longer, and focusing on more challenging tasks that require students to work through different problems, arguments, or require extended writing (MDESE, c ). See Appendix E for released items of this standardized test (spring 2006) at grade level 6. Th is state has standards for mathematics at different grade levels See Appendix F for the state standards for mathematics at grade level 6 (due to the space limit, the author did not attach released items of this standardized test (spring 2006) at grade 7 and 8 and state standards for mathematics at grade level 7 and 8). The Appendices D F provide the evidence of alignment between this state standardized mathematics test and NCTM mathematics standards and state mat hematics standards. Data A nalysis Due to the very small sample size of minority groups including Asian American, Hispanic American, Indian American, and African American in this study, these four groups were combined as one category during data analysis in this study: Minority. There are only two categories in the categorical variable: Racial/Ethnicity, Caucasian

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65 American Students and M inority Students. Other categorical variables were coded accordingly during data analysis. Table 3 1 shows the coding inf ormation. Students who took the two Algebra EOC tests were from grades 8 to 12. Students who took the standardized mathematics test grade 6, 7, or 8 were from grade 6 to 10. These two sets of groups were overlap to some degree. Therefore, the analysis was conducted for these two sets of groups separately. Some of the participants in state standardized test groups will take Algebra course in this virtual school. The analysis of these groups can add to the knowledge of success factors in Algebra. This virtu al school student body included students s tatewide from bricks andmortar public and private schools as well as home school students. The physical schools as well as the home schools that students attend could affect student academic performance through sc hool culture, technical support, and resources available for students. Student test score s within the same physical school are not independent of one another. Therefore, any evaluation of the influence of student level factors such as grade level, r ace, and teacher comment on these score s must account for the influence of school characteristics. T o investigate the Algebra/ mathematics success factors in the K 12 online learning environment, H ierarchical L inear M odeling ( HLM ) technique was used to account for the clustering of student score within one school caused by school characteristics. HLM was carried out by the software program HLM 6.06 for data analysis in this study. The fully unconditio nal or Random ANOVA (RA) model was estimated at the beginning in order to partition the variance into withinschool ( Sigma Square) and betweenschool (Tau) components. After that, all independent variables

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66 were added into the model. Generalized estimating equation was applied for the estimation of correlation coefficien ts. Limitations Limitations of this study include: 1 The small sample size could affect the power for statistic claims. 2 The coding strategy for race/ethnicity variable could mask influential information. 3 Many home school students and the very small number of students from many different physical schools (some only had one student) may cause data analysis difficulty.

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67 Table 31. Coding of the independent variables Variable Categories Student status (part time or full time) 0: part time 1: full time IEP (individual educational plan) 0: without individual educational plan 1: with individual educational plan FRL (free or reduced lunch) 0: not in free or reduced lunch programs 1: in free or reduced lunch program RACE 0: Caucasi an American student 1: minority student

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68 CHAPTER 4 RESULTS Introduction As stated in Chapter 1, the study reported here examined the success factors including student demographic information, teacher comments, and student participation level in online ac ademic activities in K 12 virtual learning environments. This chapter is organized based on the three research questions posed in Chapter 1. It first describes the sample of this study. It then reports the effects of LMS utilization teacher comment, and s tudent demographic information such as race/ethnicity and whether have IEP on Algebra/ mathematic achievement in virtual learning environments Sample The data were collected during the 200809 academic year by a consulting company that works with one stateled virtual school in the Midwestern US region. This consulting company collected student demographic information and their performance on two types of test s : EOC test and state standardized test. The researcher obtained the data from this consulti ng company. The criteri a for the participation in this study were : (1). students who completed Algebra online courses during the 200809 year and took the EOC test at the end of each semester; or (2)students in this virtual school who took one state standardized mathematics test at the end of the 2008 09 academic year. EOC T ests T aker This virtual school offer ed four Algebra courses: Algebra I (1st half), Algebra I (2nd half), Algebra II (1st half), and Algebra II (2nd half) during the 200809 academic year. The number of students who completed these four Algebra courses and took the EOC tests at the end of each semester during that academic year was : 1 01 75, 26, and 36

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69 respectively. These four groups participated in this study. However, due to the insu fficient power for data analysis caused by the small sample size of Algebra II (1st half) and Algebra II (2nd half), 26 and 36, these two groups were dropped from the study. Within the two Algebra I groups, 64 out of 101 students in Algebra I (1st half) ( 63.4.2%) and 59 out of 75 students in Algebra I (2nd half) ( 78.7%) were second year students in this virtual school. See table 41 for student demographic information The sample can be described as primarily white, not participating in school free or reduced lunch programs, and part time virtual school students without individual educational plans. State S tandardized T est T aker There were 487 students in this virtual school during 200809 academic year who took a state standardized mathematics test at the end of academic year. This standardized mathematics test has different grade levels from 3 to 8. Due to lack of information about students who took the state standardized mathematics test grade level 3, 4 and 5, only students who took the state standar dized mathematics test grade level 6, 7, and 8 participated in this study. The number of students in these three groups is 74, 73, and 107 respectively. All of these participants were first year students in this virtual school. See table 42 for student demographic information. The sample can be described as primarily white, participating in school free or reduced lunch programs, and part time virtual school students with individual educational plans. As stated in Chapter 3, the physical schools students attend ed could affect student academic performance via the resources the school provided for students, technical support, and school culture. Students test scores within one school are not independent of one another. Any evaluation of the variables at student level such as teacher

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70 comments, grade level, and race on student test score must account for the influence of school characteristics on this dependent variable. The Hierarchical linear modeling ( HLM) technique was carried out by the software program H LM 6.06 for data analysis to account for the clustering of students scores within one school caused by the school characteristics. The fully unconditional or Random ANOVA (RA) model was utilized to partition the total variance of student test score into w ithin school (Sigma Square) and betweenschool (Tau) components at the beginning during the analysis. After that, all the independent variables were added into the model. Generalized estimating equation was then applied for the estimation of coefficients o f the different variables. RA M odel The RA model was estimated for each dataset to partition the variance of student test score into within school (Sigma Square) and betweenschool (Tau) components. Level 1 Model Y = B0 + R Level 2 Model B0 = G00 + U0 The Intra Class Correlation (ICC) was calculated according to the formula: Tau/ (Tau+ Sigma Square) for each dataset. Results for the RA model are presented in Table 4 1 Table 41 demonstrates the ICC for all the five datasets is equal to or above 0.7. Th is finding shows the betweenschool variance was large in comparison with the within school variance for the five groups of students. This tells us the students within the same school are similar with respect to their academic achievement in mathematics and in Algebra particularly in comparison with the students from different schools.

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71 Coefficients for the V ariables After estimating the RA model, all the independent variables were added into the model at the level 1 (student level). With the exception o f the data on the standardized test in grade 6, t h e generalized estimating equation (GEE) procedure was then used to estimat e the coefficients of these variables. For the data on the standardized test in grade 6, ordinary least square was used. The results are presented in Tables 42 to 44. Summarization of the results is presented following Table 44. Level 1 Model Y = Y = B0 + B1*(GRADE) + B2*(RACE) + B3*(FRL) + B4*(IEP) + B5*(PT/FT) + B6*(TEACHERCOM) + B7*(TOTALLOG) + B8*(TOTALMIN) + R Level 2 Model B0 = G00 + U0 B1 = G10 B2 = G20 B3 = G30 B4 = G40 B5 = G50 B6 = G60 B7 = G70 B8 = G80 The results of the GEE are presented in Tables.

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72 EOC T est Table 42 shows the estimate of the variable effect coefficients for the two Algebra EOC tests: Algebra I (1) and Algebra I (2). The variables that have significant effects on student final score on these two tests are highlighted in grey State S tandardized M athematics T est Table 43 shows the estimate of the variable effect coefficients for the tw o state standardized mathematics tests grade 7 and grade 8. The variables that have significant effects on student final score on these two tests are highlighted in grey Ordinary Least Square (OLS) was applied for the dataset: mathematics standardize test grade 6 for the estimate of effect coefficient. There are 74 students in this group. Five of them are from the same school and all other 69 students are from different schools. There is almost no clustering for student scores due to the small sample siz e at student level (74) and comparatively large sample size at school level (70). Thus, least squares estimates with robust standard errors cant be applied to correct the errors associated with the clustering of student scores within one same school. Inst ead, OLS was applied for the estimates of coefficients of the independent variables. Table 46 shows the results. Descriptive S tatistics Standardized Coefficient and Reduction of Variance Descriptive statistics analysis was conducted for each group to de monstrate the mean and variance of the factors (independent variables) and student score (dependent variable) See table 47 and 48 for this information for these two sets of participants To compare among different factors with respect to the importance in determining student score, standardized coefficient ( ) was calculated according to the formula: k =bk Sxk Sy (bk is the unstandardized coefficient, Sxk is the standard deviation of the

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73 corresponding independent variable, and Sy is the standard deviation of the dependent variable). See table 49 and 410 for standardized coefficients for these two sets of participants. Standardized coefficient demonstrates that how increases in the independent variables affect relative position within the group. Fo r example, the standardized coefficient of TEACHERCOM was 0.56 for Algebra I (2). It means with 1 standard deviation increase in TEACHERCOM, student test score increased 0.56 standard deviation. The adjusted R square (Rc 2) wa s calculated according to t he formula: Rc 2= 1 VAR e/VAR t ( VAR e is the least squares estimates of the model with all the predicators, VAR t is the least squares estimates of the RA model) to show the reduction of test score variance from the RA model with the addition of the fact ors. See table 4 11 for adjusted R square for the five groups. The adjusted R square also shows the variance that is accounted for by these factors. For example, Rc 2 is 0.15 for Algebra I (2). It means the eight factors accounted for 15 percent of test score variance. Research Q uestion 1 Does the level of student participation in academic activities predict Algebra/mathematics performance in online education? The participation in online academic activities can be reflected through the number of times stud ents logged into the Learning Management System (LMS) and how long they stayed in the LMS. This study used these two variables as the indicators of the level of student participation. Other indicators of participation not collected by the virtual school's data system, such as the time students spent online on academic tasks and the time they spent on non academic tasks, will not be part of this study. Time on task has been identified as a critical factor in relation to the improvement of understanding level

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74 of subject matter (Bransford et al., 1999). As Bransford et al. mentioned, students need to take time to make meaning of the concepts in the subject areas and build the connections to their preexisting knowledge. Based on one study on the effect of one Al gebra I online learning model on students academic outcome, ODwyer et al. (2007) concluded online students spent more time on interacting with one another on academic topics than their counterparts in traditional classroom. Peer to peer interaction could help improve online students learning outcome. The time spent in academic activities has a strong effect on success in online education (Cavanaugh, 2007), faceto face instruction (Rocha, 2007), and blended programs (Cavanaugh, 2009). It can predict stu dent final grade in online learning (Wang & Newlin, 2000). To investigate the effects of student participation in online academic activities on student achievement in mathematics and Algebra in particular in online learning environments, the number of time s students logged into the LMS and how long they stayed in the LMS were analyzed using HLM along with other factors in one single equation. Other student time on task outside the LMS was not measured in the school data system. EOC T est Table 42 shows TO TALLOG (number of times student log into the LMS) had a non significant effect ( 0.03, p=0.230) on student final score for Algebra I (1) and the direction shows students who logged into the LMS less tending to perform better than those students who logged in to the LMS more. The effect of TOTALLOG is significant for Algebra I (2) ( 0.02, p=0.006), with students who logged into the LMS less achieved higher scores. There is a weak and non significant effect of TOTALMIN (total minutes students stay in LMS) on final score for Algebra I (1) (0.0005, p=0.304), with students who stay ed in the LMS longer tending to achieve higher scores. The effect of

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75 TOTALMIN is significant for Algebra I (2) (0.0004, p=0.008). The direction of the effect tells us students who stay e d in the LMS longer perform ed better. State S tandardized T est Table 43 shows TOTALLOG has no significant effect ( 0.02, p=0.725) on student score in the grade 7 mathematics standardized test. The direction shows students who logged into the LMS more tending to perform better than those students who logged into the LMS less. The effect of TOTALLOG is also not significant ( 0.07, p=0.117) for the grade 8 mathematics standardized test, with students who logged into the LMS less achieving higher scores. Table 44 shows TOTALLOG also has no significant effect ( 0.04, p=0.414) for the grade 6 mathematics standardized test, with the same direction as it in grade 8. Table 43 shows there is a weak and non significant effect of TOTALMIN (0.0004, p=0.680) on student s core in the grade 7 mathematics standardized test, with students who stay ed in the LMS longer tending to achieve higher scores. The effect of TOTALMIN is nearly significant (0.001, p=0.057) for grade 8. The direction of the effect tells us students who stay ed in the LMS longer tend ed to perform better. Table 44 shows there is a weak and non significant effect (0.0005, p=0.544) of TOTALMIN for the grade 6 mathematics standardized test, with students who spent more time in the LMS tending to achieve higher s core s. Research Q uestion 2 Does teacher comment or feedback predict Algebra/mathematics performance in online education? Bransford et al. (1999) emphasized the importance of frequent feedback from the instructors for students to monitor their learning pr ocess and evaluate their understanding levels and the learning strategies during the learning process. Based on

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76 the feedback, students could revise their thinking and enrich their knowledge structure as they move along. Teacher feedback or teacher comment on student assignments, papers, and projects has been identified as a critical factor that can influence student academic performance in online education ( Cavanaugh et al., 2005; Dickson, 2005; Ferdig, Papanastasiou, & DiPietro, 2005; Hughes et al., 2005; Peters, 1999; Zucker, 2005). Phipps and Merisotis (2000) believed that student teacher interaction and the timely and constructive teachers feedback to students assignments and questions are critical characteristics of the teaching/learning benchmarks for the quality of online learning. Watson and Ryan (2006) showed there are big difference s in students experiences between virtual classrooms with minimal teacher involvement and those with greater student teacher interactions via different means such as email, online message, online discussion forum, phone, etc. Based on a quasi experimental study on the impact of one state wide Algebra I online initiative on students learning outcomes, ODwyer et al. (2007) found that online students highly value the st udent instructor interaction during the learning process. The critical value of teacher feedback and teacher comment for success in online learning is also applicable to students with special needs. Based on a study of students with learning disabilities (SLD) and students with attention deficit hyperactivity disorder (ADHD), Smouse (2005) found communication with and feedback from instructors was the most valuable aspect of online courses. To investigate the effect of teacher comment or teacher feedback on student achievement in mathematics and Algebra in particular in online learning environments, the number of teacher comments was analyzed using HLM along with other factors in one single equation.

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77 EOC T est Table 42 demonstrates t he non significant effect of TEACHERCOM (teacher comments) for Algebra I (1) (0.04, p=0.450). Interestingly, the direction shows students with fewer teacher comments tended to achieve higher scores than those with more teacher comments. This could be due to the lower need for corrective feedback from teachers for students with better performance during the learning process. The effect of TEACHERCOM for Algebra I (2) is also not significant (0.01, p= 0.912). Its direction is different from the one in Algebra I (1). In Algebra I (2), students who received more teacher comments tended to perform better than those students who received less teacher comments. State S tandardized T est Table 43 demonstrates the significant effect of TEACHERCOM ( 0.46, p=0.041) for the grade 7 mathemat ics standardized test. The direction shows students with more teacher comments perform ed better in this test. The effect of TEACHERCOM for grade 8 is not significant (0.002, p=0.990), with the same direction as it is in the grade 7 mathematics standardized test. Table 44 shows there is no significant effect of TEACHERCOM ( 0.02, p=0.949) on student score in the grade 6 mathematics standardized test. Interestingly, the direction shows students with more teacher comments tended to achieve lower scores in t his test. Research Q uestion 3 Do student demographic information, such as race/ethnicity, grade level, status in virtual school, whether have IEP and participation in free/reduced lunch programs, predict Algebra /mathematics performance in online education?

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78 The v irtual school student body is a diverse population of learners that includes students with different learning disabilities (Dickson, 2005; Ferdig, Papanastasiou, & DiPietro, 2005). In the present study, the v irtual school follows individual educati on plans for students with special needs during the learning process. Therefore, whether or not a student has an individual education plan could be considered as a sign of student learning ability which can affect student academic achievement during the le arning process (Keeler & Horney, 2007). Currently, many popular LMS, such as WebCT and Blackboard, still have different problems that can prevent students with disabilities from fully utilizing their functions even though they are generally accessible to t hese disabled online students (Asunci on et al., 2006). Students learning ability may be related to their learning outcome through some other factors such as academic engagement (Richardson et al., 2003). Based on a study comparing online students with a h earing loss and those without this disability with respect to the relationship between students academic engagement and their perceptions of the academic quality of the courses, Richardson et al. (2003) found the correlation between hearing status and students academic engagement and their perceived academic quality of the courses. Students with hearing disability cant perform well on communication and some other tasks during online learning process es as compared to students without this disability. This in turn, may negatively impact these disabled students academic achievement. Research shows that eligibility for school free or reduced lunch programs correlates with academic achievements, with students not participating in these programs achiev ing better performance (McLoyd, 1998). Participation in these programs could be considered as the indicator of the family poverty level, which has a

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79 strong relationship with student academic achievement at school level (Klein et al. 2000). C onsiderable research has also found student academic achievement difference among different racial groups, with Hispanic and African American students lagging behind Caucasian and Asian American students (Bali, 2004; Barth, 2001; Hall et al 2000; Lockhead et al., 1985). Tho ugh students race/ethnicity and their participation in school free or reduced lunch programs have been proved to correlate with student academic achievement in traditional faceto face education, their effects have not been examined systematically for vir tual learning environments. In this study, student demographic information including race/ethnicity, participation in free/reduced lunch programs, learning ability, grade level, and status in virtual school were investigated with other factors in one singl e equation. EOC T est Table 42 shows the participation in free or reduced lunch programs has no significant effect ( 0.04, p=0.992) on student EOC test score in Algebra I (1), with students who did not participate in these programs tending to achieve hig her scores. The non significant effect ( 4.36, p=0.172) of the participation in these programs w as also observed for Algebra I (2), with the same direction as it for Algebra I (1). There is no significant difference (0.93, p=0.826) in student EOC test scor e between Caucasian American students and the minority students for Algebra I (1) (see Table 42). The direction shows minority students tend to perform better than Caucasian American students. No significant difference ( 2.53, p=0.455) w as also found for Algebra I (2). However, the direction is different from Algebra I (1), with Caucasian American students tending to perform better.

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80 Student grade level has a significant effect ( 3.89, p=0.030) on student EOC test score for Algebra I (1), with students in lower grade levels achiev ing better scores than their counterparts in higher grade levels. The significant effect ( 3.36, p=0.005) of grade level has also been found for Algebra I (2), with the same direction as for Algebra I (1). Table 42 demonstrates t he non significant effect ( 1.37, p=0.825) of student learning ability on EOC test score for Algebra I (1). The direction shows the students who do not have individual educational plans tended to perform better. Similarly, the non significant effect ( 2.26 p=0.557) of student learning ability is observed for Algebra I (2), with the same direction as for Algebra I (1). Table 42 shows student status in the virtual school (full time or part time) has no significant effect (1.78, p=0.733) on the EOC test scor e for Algebra I (1), with full time students tending to achieve better performance than part time students. The non significant effect of student status (6.49, p=0.096) has also been found for Algebra I (2), with the same direction as for Algebra I (1). State S tandardized T est Table 43 shows the participation in free or reduced lunch programs has no significant effect ( 0.89, p=0.936) on student score in the grade 7 mathematics standardized test, with students who participated in these programs tendi ng to perform better. However, the strong and significant effect of participation in these programs ( 61.40, p=0.000) has been found for the grade 8 standardized test, with the same direction as it is for the grade 7 standardized test. A significant differ ence ( 21.28, p=0.046) in student score in the grade 7 mathematics standardized test between Caucasian American students and minority students is observed (see Table 43). The direction shows the Caucasian American students achieved higher scores than the minority students in this test. Table 43 demonstrates there is no significant difference ( -

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81 9.31, p=0.254) between these two ethnicity groups for the grade 8 standardized test, with the same direction as it for the grade 7 standardized test. Students g rade level in his/her physical school has no significant effect ( 0.17, p= 0.083) on score in the grade 7 mathematics standardized test, with students in lower grade level s tending to achieve higher scores (see Table43). The non significant effect of stud ent grade level (1.71, p=0.623) has also been found for the grade 8 mathematics standardized test (see Table 43). However, the direction is different for the grade 7 standardized test. Table 43 shows there is a strong and significant effect of student le arning ability ( 41.90, p=0.001) on student test score in the grade 7 standardized test, with students without individual educational plans perform ing better than those students with educational plans in the virtual school. The significant effect of student learning ability (21.92, p=0.022) was also observed for the grade 8 mathematics standardized test. Interestingly, the direction tells us students with individual educational plans achieved better scores Table 43 demonstrates the non significant effect of student status in virtual school on student score for the grade 7 standardized test (4.97, p=0.614), grade 8 standardized test ( 11.98, p=0.146). However, the two directions are different, with the direction for the grade 7 test showing full time studen ts tending to perform better than part time students and the direction for grade 8 test showing part time students tending to perform better. Shown in t able 44 the participation in free or reduced lunch programs has no significant effect ( 26.54, p=0.496) on student score in the grade 6 mathematics standardized test, with s tudents who did not participate in these programs tending to achieve better scores than their counterparts who participate d in these programs. There

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82 is a nearly significant difference ( 21.48, p=0.068) in student score between Caucasian American students and the minority students for the grade 6 standardized test. The direction shows Caucasian American students tended to perform better than the minority students. Student grade level has no significant effect ( 8.13, p=0.106) on student score in the grade 6 standardized test, with students who are in the lower grade levels tending to achieve better performance. A n on significant while strong effect was observed for student learning abili ty ( 40.10, p=0.348) in the grade 6 mathematics standardized test (see Table 44). The direction tells us students without individual educational plans tend ed to perform better than their counterparts who had educational plans. Table 44 demonstrates the non significant effect (11.62, p=0.344) of student status in the virtual school for the grade 6 standardized test, with full time students tending to achieve better scores than part time students. Summary of Findings The purpose of this study is to examine the factors including LMS utilization, teacher comment /feedback and s tudent demographic information that can influence the success of Algebra courses in K 12 virtual learning environments. The three research questions formulated sought to (1) discover the influence of student participation in online academic activities on student mathematics achievement in virtual learning environments; (2) explore whether teacher comment or feedback can predict student academic achievement in online mathematics courses; and (3) investigate the differences in online mathematics achievement among students with different demographic information. The results of question one showed the influence of participation in online academic activities on achievement could be different bas ed on mathematics levels.

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83 The indicators of student participation in online academic achievements in this study include the number of times student logged into the LMS (TOTALLOG) and how long they stayed in the LMS ( TOTALMIN ). TOTALLOG has a significant influence on student performance in Algebra I (2) EOC test ( 0.002, p=0.006) while not in Algebra I (1) EOC test. The direction of the significant influence showed students who logged into the LMS less performed better. Similarly, TOTALMIN also has a signifi cant influence for Algebra I (2) EOC test ( 0.0004, p=0.008) while not for Algebra I (1) EOC test. The direction of the significant influence indicated students who spent more time in the LMS achieved better performance. For the state standardized mathemati cs test, TOTALLOG has no significant influence on student performance at all the three levels: grade 6 to 8. Similarly, TOTALMIN also has no significant influence at the three levels. The results of question two provided the evidence that teacher comment c an affect student mathematics performance at different levels depending on the type of mathematics tests. In this study, teacher comment has no significant effect on student achievement in the two Algebra courses: Algebra I (1) and Algebra I (2). For the s tate standardized mathematics test, teacher comment has a significant effect on student achievement at the grade 7 level (0.46, p=0.041) while not at grade 6 and grade 8 levels. The significant effect at the grade 7 level showed students with more teacher comments perform ed better in th e test. The results of question three showed some demographic information was predictive of student online mathematics achievement while others may not be and the predictability also depended on the type and the level of t he mathematics test. The participation in free or reduced lunch programs, race/ethnicity (Caucasian American or

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84 minority), student learning ability, and student status in the virtual school (full time or part time) were not predictive of student performanc e in Algebra I (1) and Algebra I (2) EOC tests. However, student grade level is predictive of student performance in Algebra I (1) ( 3.89, p=0.030) and Algebra I (2) ( 3.36, p=0.005) EOC tests, with students in lower grade levels achieved higher scores. Fo r the state standardized mathematics test, the participation in free or reduced lunch programs, student grade level and status in the virtual school were not predictive of student performance at all three levels: grade 6 to 8. The participation in free or reduced lunch programs was a significant predictor only at the grade 8 level ( 61.40, p=0.000) The direction showed students not participating in these programs performed better. Race/ethnicity was a significant predictor only for the grade 7 level test ( 21.28, p=0.046) with Caucasian American students performing better than the minority students Student learning ability was a significant predictor for the grade 7 level ( 41.90, p=0.001) and the grade 8 level (21.92, p=0.022) tests. They have different directions. For the grade 7 level test, students without individual educational plans performed better than those with individual educational plans while for the grade 8 level test, students with individual educational plans performed better.

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85 Ta ble 41: EOC test takers demographics Algebra I (1st half) GRADE 8: 4(4.0%), 9: 35(34.7%), 10: 37(36.6%), 11: 15(14.9%), 12: 10(9.9%) RACE W hite: 82(81.2%), Other Minority: 19(18.8%) FRL 0: 68(67.3%), 1: 33(32.7%) IEP 0: 94(93.1%), 1: 7(6.9%) PT/FT 0: 87(86.1%), 1: 14(13.9%) Algebra I (2nd half) GRADE 7 : 2(2.7%), 8: 13(17.3%), 9: 24(32.0%), 10: 24(32.0%), 11: 9(12.0%), 12: 3(4.0%) RACE W hite: 62(82.7%), Other Minority: 13(17.3%) FRL 0: 53(70.7%), 1: 22(29.3%) IEP 0: 70(93.3%), 1: 5(6.7%) PT/ FT 0: 62(82.7%), 1: 13(17.3%)

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86 Table 42: Standardized test takers demographics Standardized test Grade 6 GRADE 6 : 35(47.3%), 7 : 28(37.8%), 8 : 6(8.1%), 9 : 4(5.4%), 1 0 : 1(1.4%) RACE W hite: 60(81.1%), Other Minority: 14(18.9%) FRL 0: 6 (8.1%), 1: 68(91.9%) IEP 0: 5(6.8%), 1: 69(93.2%) PT/FT 0: 61(82.4%), 1: 13(17.6%) Standardized test Grade 7 GRADE 7 : 29(39.7%), 8: 32(43.8%), 9: 9(12.3%), 10: 2(2.7%), missing: 1(1.4%) RACE W hite: 59(80.8%), Other Minority: 14(19.2%) FRL 0: 16(21. 9%), 1: 57(78.1%) IEP 0: 19(26.0%), 1: 54(74.0%) PT/FT 0: 68(93.2%), 1: 5(6.8%) Standardized test Grade 8 GRADE 8: 63(58.9%), 9: 39(36.4%), 10: 4(3.7%), missing: 1(0.9%) RACE W hite: 83(77.6%), Other Minority: 24(22.4%) FRL 0: 20(18.7%), 1: 87(81.3 %) IEP 0: 20(18.7%), 1: 87(81.3%) PT/FT 0: 89(83.2%), 1: 18(16.8%)

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87 Table 43 : Overview of RA model for different datasets Test Variables df Sigma Square Tau ICC Algebra I (1) None 79 41.79 225.26 0.84 Algebra II (2) None 56 63.25 146.04 0.70 Standardized test Grade 6 None 69 43.33 1797.90 0.98 Standardized test Grade 7 None 63 468.66 1189.96 0.72 Standardized test Grade 8 None 93 401.44 1052.68 0.72

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88 Table 44 : Least squares estimates of fixed effects (with robust standard errors) Test Fixed Effect Coefficient Standard Error T ratio d.f. P value Algebra I (1) GRADE 3.89 1.76 2.21 92 0.030 RACE 0.93 4.22 0.22 92 0.826 FRL 0.04 3.93 0.01 92 0.992 IEP 1.37 6.19 0.22 92 0.825 PT/FT 1.78 5.20 0.34 92 0.733 TEACHERCOM 0.04 0.06 0.76 92 0.450 TOTALLOG 0.03 0.03 1.21 92 0.230 TOTALMIN 0.0005 0.00 1.04 92 0.304 Algebra I (2) GRADE 3.36 1.14 2.96 66 0.005 RACE 2.53 3.37 0.75 66 0.455 FRL 4.36 3.15 1.38 66 0.172 IEP 2.26 3.83 0.59 66 0. 557 PTFT 6.49 3.85 1.69 66 0.096 TEACHERCOM 0.01 0.09 0.11 66 0.912 TOTALLOG 0.02 0.01 2.90 66 0.006 TOTALMIN 0.0004 0.00 2.74 66 0.008

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89 Table 45 : Least squares estimates of fixed effects (with robust standard errors) Test Fixed Effect Coefficient Standard Err or T ratio d.f. P value Standardized test Grade 7 GRADE 0.17 0.10 1.76 64 0.083 RACE 21.28 10.50 2.03 64 0.046 FRL 0.89 11.03 0.08 64 0.936 IEP 41.90 11.93 3.51 64 0.001 PT/FT 4.97 9.81 0.51 64 0.614 TEACHERCOM 0.46 0.22 2.09 64 0.041 TOTALLOG 0.02 0.06 0.35 64 0.725 TOTALMIN 0.0004 0.00 0.41 64 0.680 Standardized test Grade 8 GRADE 1.71 3.47 0.49 98 0.623 RACE 9.31 8.11 1.15 98 0.254 FRL 61.40 9.57 6.42 98 0.000 IEP 21.92 9.39 2.33 98 0.022 PT/FT 11.98 8.17 1.47 98 0.146 TEACHERCOM 0.002 0.16 0.01 98 0.990 TOTALLOG 0.07 0.04 1.58 98 0.117 TOTALMIN 0.001 0.00 1.92 98 0.057

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90 Table 46 Ordinary Least squares estimates of fixed effects Test Fixed Effect Coefficient Standard Error T ratio d.f. P value Standardized test Grade 6 GRADE 8.13 4.97 1.64 65 0.106 RACE 21.48 11.59 1.85 65 0.068 FRL 26.54 38.75 0.69 65 0.496 IEP 40.10 42.42 0.95 65 0.348 PT/FT 11.62 12.18 0.95 65 0.344 TEACHERCOM 0.02 0.26 0.06 65 0.949 TOTALLOG 0.04 0.05 0.82 65 0.414 TOTALMIN 0.0005 0.00 0.61 65 0.544

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91 Table 47 Descriptive statistics for EO C test takers Test Variables N Mean Std. Deviation FINAL GRADE 101 70.82 15.598 GRADE 101 9.92 1.026 RACE 101 .19 .393 Algebra I (1) FRL 101 .33 .471 IEP 101 .07 .255 PT/FT 101 .14 .347 TEACHERCOM 101 20.78 21.390 TOTALLOG 101 215.36 125.47 6 TOTALMIN 101 10783.93 6418.637 FINAL GRADE 75 79.00 13.650 Algebra I (2) GRADE 75 9.45 1.119 RACE 75 .17 .381 FRL 75 .29 .458 IEP 75 .07 .251 PT/FT 75 .17 .381 TEACHERCOM 75 15.31 21.526 TOTALLOG 75 461.04 288.174 TOTALMIN 75 27425.76 17097.182

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92 Table 48 Descriptive statistics for standardized test takers Test Variables N Mean Std. Deviation Standardized test Grade 6 MAP_SCALE_SCORE 74 664.07 41.827 GRADE 74 6.76 .919 RACE 74 .19 .394 MAP_FRL 74 .92 .275 IEP_STU DENT 74 .93 .253 PTFT 74 .18 .383 TEACHERCOM 74 11.32 18.144 TOTALLOG 74 354.65 181.272 TOTALMIN 74 23006.26 13370.676 Standardized test Grade 7 MAP_SCALE_SCORE 73 676.18 41.913 GRADE 73 9.03 10.704 RACE 73 .19 .396 MAP_FRL 73 .78 .417 IEP _STUDENT 73 .74 .442 PTFT 73 .07 .254 TEACHERCOM 73 11.15 19.715 TOTALLOG 73 253.25 141.193 TOTALMIN 73 16720.51 9324.034 Standardized test Grade 8 MAP_SCALE_SCORE 107 696.23 38.020 GRADE 107 8.36 .994 RACE 107 .22 .419 MAP_FRL 107 .81 .392 IEP_STUDENT 107 .81 .392 PTFT 107 .17 .376 TEACHERCOM 107 11.28 18.569 TOTALLOG 107 293.93 202.395 TOTALMIN 107 18067.10 14548.311

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93 Table 49 Standardized coefficients for EOC test takers Tests Factors Standardized Coefficients Alge bra I (1) GRADE 0.2 6 RACE 0.3 6 FRL 0.0 5 IEP 0.74 PT/FT 2.42 TEACHERCOM 2.4 7 TOTALLOG 0.1 8 TOTALMIN 0.0 3 Algebra I (2) GRADE 0.2 8 RACE 0.86 FRL 5.24 IEP 1.2 4 PT/FT 9.85 TEACHERCOM 0.56 TOTALLOG 0.2 7 TOTALMIN 0.0 2

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94 Table 410 Standardized coefficients for standardized test takers Factors Standardized Coefficients Standardized test Grade 6 GRADE 0.1 8 RACE 9.2 1 MAP_FRL 18.52 IEP_STUDENT 36.89 PTFT 17.59 TEACHERCOM 0.9 5 TOTALLOG 0. 40 TOTALMIN 0.0 4 Standardized test Grade 7 GRADE 0.04 RACE 0.7 9 MAP_FRL 0.93 IEP_STUDENT 44.41 PTFT 2.8 6 TEACHERCOM 35.70 TOTALLOG 0.143 TOTALMIN 0.0 3 Standardized test Grade 8 GRADE 0.04 RACE 3.92 MAP_FRL 57.44 I EP_STUDENT 21.92 PTFT 11.49 TEACHERCOM 0. 10 TOTALLOG 0.76 TOTALMIN 0.07

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95 Table 411 Adjusted R squares Tests VAR e VAR t R c 2 Algebra I (1) 236.69 243.29 0.0 3 Algebra I (2) 159.04 186.32 0.1 5 Standardized test Grade 6 1412.24 1749.46 0 .19 Standardized test Grade 7 1440.78 1756.73 0.1 8 Standardized test Grade 8 1146.03 1445.5 0.2 1

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96 CHAPTER 5 DISCUSSION AND IMPLI CATIONS Introduction This chapter summarizes the findings of the present study and presents the important conclusions drawn from the data shown in Chapter 4. It also addresses the implications for teaching, research, and policy making process es in the discussion of the findings. This chapter presents the implications based on the three research questions in the present study. Summary of Study To explore different success factors for online mathematics in general and online Algebra in specific, the present study investigated the effect of a variety of variables on student achievement on Algebra EOC test s and state standard mathematics tests. The present study used the secondary data collected from a state led virtual school in the Midwestern US region. The variables include learner characteristic variables such as student demographic information and participation level in online academic activities and learning environment characteristics variables such as teacher comment in the present study. Overview of the P roblem The U.S has experienced an astonishing growth in online education at K 12 level during the past decade. The enroll ment of K 12 virtual school students has increased from 40,000 in 200001 academic year to 1 million in 200708 academic year (Clark 2001; Glass, 2009; Newman, Stein & Trask, 2003; Peak Group, 2002; Picciano & Seaman, 2009; Picciano & Seaman, 2007; Setzer & Lewis, 2005; Tucker, 2007; Zandberg, Lewis, & Greene, 2008). With the large population of online learners, its

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97 possible to evaluate the effectiveness of online courses. However, currently, there is no one single model being created to predict online suc cess and no clear set of characteristics that have been identified in this regard (Roblyer & Davis, 2008; Tallent Runnels et al., 2006). Math has been considered a very important force to push a society forward. Many countries emphasize the improvement o f math knowledge and they develop policies to attract more people into this field. Having good academic performance in math subjects at the K 12 level is important for students to pursue advanced degrees in this field. It will help prepare more students to have careers in Science, Technology, Engineering, and Mathematics ( STEM ) and increase workforce for U.S. in these fields which could provide strong momentum for this country to move forward in many aspects The quality of Algebra courses is essential in building the number of U.S. students who are ready for advanced degrees in STEM and career success in these fields. Purpose S tatement and R esearch Q uestions The purpose of this study is to examine the factors including LMS utilization, teacher comment /feed back and s tudent demographic information that can influence the success of Algebra courses in K 12 virtual learning environments. The research questions in this study are Does the level of LMS utilization influence Algebra/mathematics performance in onlin e education? Does teacher comment or feedback influence Algebra/mathematics performance in online education? Do student demographic information such as race/ethnicity, grade level, status in virtual school, whether have individual educational plan (IEP) and participation in free/reduced lunch programs influence Algebra/mathematics performance in online education?

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98 Review of the M ethodology The present study is descriptive in nature. The researcher described the relationship between some factors and student learning outcome without intervening. The researcher received the secondary data from one state led virtual school in the Midwestern US region that collected student demographic information, participation in online academic activities, teacher comments, and academic achievement on Algebra EOC tests and state standardized mathematics test s in the 200809 academic year. The present study builds on the results of Liu and Cavanaugh (2010) s study. This virtual school was launched in 2007. Liu and Cavanaughs study used the first year (200708) data collected by this virtual school and the present study used the second year data. The present study investigated success factors for both Algebra EOC tests and one state standardized mathematics test while Liu and C avanaughs study only investigated these factors for Algebra EOC tests. The data regarding student demographic information, participation in online academic activities and teacher comments were collected by the LMS utilized by this virtual school for cours e content delivery. Student academic achievement on (1) the Algebra I, II EOC tests administered at the end of semester designed based on the state Algebra standards and (2) one state standard mathematics test, designed based on the state mathematics standards was collected. HLM was carried out by the software program HLM 6.06 for data analysis in this study to account for the clustering of academic score s of students recruited from the same physical school. The rest of chapter 5 will discuss the outcomes and review the implications associated with the three research questions designed to examine the impact of different success factors on student mathematics achievement. This chapter will close with the conclusions drawn from the findings shown in chapter 4.

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99 Findings In the present study, RA model was analyzed at the beginning to partition the total variance of student score into withinschool and betweenschool components. The intraclass correlation coefficient was calculated for the five groups and it w as Algebra I (1) .84, Algebra II (2) .70, Standardized test Grade 6 .98, Standardized test Grade 7 .72, and Standardized test Grade 8 .72 respectively. This shows the betweenschool variance was big in comparison with the withinschool variance f or all these five groups especially for the Standardized test Grade 6 group. Partially, it could be attributed to the small number of students per school. The big ICC indicated s tudents from different schools are different from each other with respect to t heir academic achievement. This finding confirmed the gap between private and public schools in student academic achievement found in other studies (Braun, Jenkins, Grigg, Tirre, Spellings, Whitehurst, & Schneider, 2006; Demircioglu & Norman, 1999; Lubiens ki & Lubienski, 2005). It also could indicate that it will take time for the standardized testing criterion to be well implemented in the school system of this state. In the present study, standardized coefficient ( was calculated according to the formula: k =bk Sxk Sy (bk is the unstandardized coefficient, Sxk is the standard deviation of the corresponding independent variable, and Sy is the standard deviation of the dependent variable). It can be used to com pare among different factors with respect to the importance in determining test score. Table 49 shows that for Algebra I (1) = 2.42) and teacher comment = 2.47) were most important factors, and for Algebra I (2) group, partici pation in school free or reduced lunch = = 9.85) were most important factors. Table 410 shows that for standardized test grade 6, participation in school free or reduced

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100 = 18.52), whether have IE = = = = 35.70) were most important factors while for standardized test grade 8, participation in school = = 11.49) were most important factors. These findings show that the same factors can influence student test score differently for different online Algebra cour ses. The adjusted R square (Rc 2) was also calculated according to the formula: Rc 2= 1 VAR e/VAR t to show the reduction of test score variance from the RA model with the addition of the factors. Table 410 shows that the same set of factors accounted for student score variance at different degree for different tests. All these findings demonstrated the complexity of the investigation of factors influencing success in online Algebra. The following Table 51 shows the summary of the significance and direction of the factor effect on student academic performance in the five tests The + sign indicates the positive direction of the factor effect and the sign indicates the negative direction of the factor effect. The X sign indicates the factor had a sig nificant effect on student academic performance in the corresponding test Research Question 1 Does the level of LMS utilization influence Algebra/mathematics performance in online education? The time spent in academic activities has been identified as a very important factor that has strong effect on success in online education (Cavanaugh, 2007), faceto face instruction (Rocha, 2007), and blended programs (Cavanaugh, 2009). Based on a study for the investigation of the cognitivemotivational and demographic characteristics of

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101 online students and the predictors for their success, Wang and Newlin (2000) found out students who participate in online activities at a high level tend to perform well in the online course. They concluded the total online course activity is a predictor of students final grades. Compared to the students in traditional classrooms, online students spend more time in the virtual learning environments on interacting with one another on academic topics (ODwyer et al., 2007). The peer t o peer interaction, in turn, could help improve online students learning outcomes ( Cavanaugh, 2007) In the present study, the numbers of times students logged into the LMS and how long they stayed in the LMS were considered the indication of student part icipation level on online academic activities. The number of times students logged into the LMS also has been identified as a strong predictor of student academic performance in online learning (Dietz, 2002; Dickson, 2005). Compared to traditional classr oom instructors, online instructors lack of the regular set of cues about students' confusion or frustration during the learning process such as their facial expression and body positions. The measure of time students spent in the online academic activities can provide online instructors the information about students understanding of content materials. A lower level of involvement in online activities in the course at the beginning of the semester could be an early warning sign of failure later during the learning process. Therefore, online instructors should closely monitor these behaviors via LMS login data to prevent students who show warning signs at the beginning from failure. The influence of time students spent in the LMS was found to be positive f or the five groups and significant for Algebra I (2) These findings are aligned with the

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102 statement Wang and Newlin (2000) made in their study mentioned above that students participating in online academic activities at a higher level achieve better perfor mance in online learning T hey echo the call for sustained time on task for cognitive learning (Gallagher, 2009) and provide support for the emphasis of expanded learning time, including with online courses, to improve academic achievement (Cavanaugh, 2009). These findings could be explained by the call for changes in instructional practices in mathematics education by many educational reformers such as the implementation of standards for mathematics instruction from the National Council of Teachers of Math ematics (NCTM, 1989, 1991) and the active involvement in academic activities is one of their arguments (Forman, 1996). This also confirmed the value of increased participation in learning activities in mathematics education emphasized in Formans article. However, its surprising to the researcher that this factor only had a significant effect for Algebra I (2) in this study considering many other studies already showed the importance of time on task for the improvement of student achievement. Many of the students taking Algebra I (2) course are from higher grade levels for credit recovery or to make up failing grades in their physical schools. The increased engaged time on task could be more effective with respect to the improvement of academic achievement than the other four groups. Nevertheless, the significant effect of time spent in the LMS for 1 out of 5 groups calls for m ore studies on activities that engage student s during their stay in the LMS as an explanation for the findings The effect of the n umber of times students logged into the LMS on student academic achievement is negative for the 4 out 5 groups and negative and significant for Algebra I (2) To some degree, t his is contradict ory to the belief that the number of

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103 times students login to the LMS is a strong and positive predictor of success in online learning ( Dietz, 2002; Dickson, 2005) It is possible that if students are logging into the course environment more often, they are staying and working for shorter time periods, negatively impac ting their concentration on their studies. This also call s for more research on the investigation of LMS utilization with bigger sample size and diversified mathematics tests. Implications Related to Research Question 1 Several implications for research, policy and practice can be drawn from the outcomes associated with research question one even though the results found in this study are mixed. These implications provide guidance for future studies to investigat e the effect of time on task and the form of activities students engag ed in when they stay in the LMS on academic achievement in virtual learning environments The positive and significant effect of the time spent in LMS for Algebra I (2) shows students who spent more time in the LMS perform ed better than students spending less time in the LMS I t is plausible that each log in session of highperforming students was longer than the session length for lower performing students, showing that highperforming students may benefit from sustained time on t ask rather than more frequent but shorter time on task. This explanation would support flexible online courses that allow students to stay in the course for extended periods of time while working on complex and abstract content. The positive influence of t ime spent in the LMS provides the support for the improvement of many LMSs to make them more user friendly with attractive interfaces that motivate students to spend more time in the system engaging in academic activities delivered during the learning proc ess, as well as teaching practices that foster

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104 connectedness among teachers and students and time management strategies for students who do not have high self regulation abilities As mentioned above, the significant effect of time spent in the LMS on onl y 1 out of 5 groups and the contradictory directions of the effect of number of times student logged into the LMS call s for more research in this field. Future researchers should investigate the activities students engaged in each time they logged into the LMS and the distribution of the logged in times throughout the semester for deeper understanding of the effect of time on task on academic achievement in virtual learning environments. This information could be used to help teachers have better knowledge about the activities in which students are more interested and their engagement level in academic activities during the learning process. Online instructors and course designers could design and d evelop better online activities, specifically activities that are more individualized, diverse, and authentic to increase engaged learning time. Research Question 2 Does teacher comment or feedback influence Algebra/mathematics performance in online education? Educators active facilitation of online learning and teachers feedback are important factors that influence students academic performance during the learning process (Dickson, 2005; Cavanaugh, Gillan, Bosnick, Hess, & Scott, 2005; Hughes, McLeod, Brown, Maeda, & Choi, 2005; Ferdig, Papanastasiou, & DiPietro, 2005; Zucker, 2005). Many of the pedagogical benefits brought by the student teacher interaction such as those related to motivation and feedback are relevant to distance education as well as the conventional classroom education (Anderson & Kuskis, 20 07). Teacher individual feedback and comments are especially helpful for students who are

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105 shy and may not participate in academic activities to increase their communication opportunities (Hughes et al 2005), which could help improve learning outcomes. Based on the results of a study of students with learning disabilities (SLD) and students with attention deficit hyperactivity disorder (ADHD), Smouse (2005) found communication with and feedback from instructors was the most valuable aspect of online courses. Timely and constructive teachers feedback to students assignments and questions have been identified as the critical characteristics of the teaching/learning benchmarks for the quality of online learning (Phipps & Merisotis, 2000). The influence of teacher comments on student academic achievement is positive and significant for standard test grade 7. This provides the evidence of the importance of teacher comments and teacher student interaction for the improvement of learning outcomes in online learni ng environments (Cavanaugh et al., 2005; Peters, 1999; Williams, 2006; Zucker, 2005). It confirmed the relationship between social interaction and mathematics learning (Cobb, Yackel & Wood, 1992; Vogit, 1996). This finding also align with the belief that i n mathematics education, the interactions between adults (instructor) and children (students) have impacts on the quality of learning outcome via the psychological benefits brought to students such as critical thinking and self reflection (Oers, 1996) and on students cognitive development (Voigt, 1994). In the process of interaction or negotiation during mathematics learning, students can build the connections between materials and mathematics terms (Voigt, 1994), which can be proved, to some degree, by these this finding. However, its surprising to the researcher that the influence of this factor is found to be positive and significant for 1 out of 5 groups considering many studies have shown the positive effect of this factor on

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106 academic achievement. Var iables other than teacher comments especially the number of teacher comments investigated in the present study such as teaching styles, quality of teacher comments, etc., could have more influence on student achievement. D ue to the small sample size for each group in the present study, readers should be cautious about the interpretation of the findings Implications Related to Research Question 2 The positive and significant effect of teacher comment or feedback for standard test grade 7 provides the support for the statement about the importance of this factor in other studies. This finding could shed light on the development of online courses that integrate teacher feedback and teacher student interaction as critical components during the course design. It also indicates the importance of timely and constructive feedback from online instructors for the success in online learning. However, the significant effect of this factor for only 1 out 5 groups in the present study showed more research is needed wit h a bigger sample size and on the form and content of teacher feedback for insightful explanation. It also supports the call for more study about the most effective interaction type, tools, and frequency for the participants in online learning ( Cavanaugh, 2007) for online instructors to better facilitate the learners to achieve success. The integration of qualitative dimensions of interaction in future study also will assist in the triangulation of quantitative dimensions as shown in the present study to better understand the importance of teacher student interaction in online learning (Weiner, 2003).

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107 Research Question 3 Do student demographic information such as race/ethnicity, grade level, status in virtual school, whether have IEP and participation in free/reduced lunch programs influence Algebra/mathematics performance in online education? Students demographic information such as participation in free or reduced lunch programs, race/ethnicity and whether have IEP has been proved to relate to students academic achievement in other studies. McLoyd (1998)s study show ed there is a correlation between student participation in school free or reduced lunch program s and student academic achievement : the magnitude is weaker as grade level rises. Student parti cipation in these programs can be considered as the indicator of his/her family Social Economics Status (SES) (Sirin, 2005), which has already been proved to affect student academic performance in many studies (Coleman, 1988; Sirin, 2005; White, 1982). In the present study, the influence of participation in these programs on achievement is negative for all the five groups and significant for standard test grade 8. For standard test grade 8 group, s tudents who did not participated in these lunch programs achieve d higher scores than students who participate in these programs. This provides support for the findings regarding the correlation between this factor and achievement in other studies mentioned above. The negative influence of this factor found in the present study can add to the body of knowledge about the correlation between SES and academic achievement. Other studies about the relationship between participation in free/reduced lunch programs or SES and academic achievement are all conducted in traditi onal learning environments. The result in the present study demonstrates the possibility for the generalization of study findings between traditional learning environment s and virtual learning environment s. On the other hand, the

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108 influence of participation in free or reduced lunch programs is only significant for 1 out of 5 groups in this study is surprising to the researcher considering the body of research demonstrating the correlation between this factor and student academic achievement. The virtual school student body is a diverse population including students with different learning disabilities (Dickson, 2005; Ferdig, Papanastasiou & DiPietro, 2005). Virtual schools offer or support individual educational plans (IEP) for these students during the lear ning process. Therefore, whether a student has an individual educational plan could be a sign of the level of learning abilities. Many technologies utilized in virtual school learning environments could help bridge gaps between students with disabilities a nd students without these disabilities with respect to the success opportunities in online learning (Coombs & Banks, 2000) However, students with disabilities are still underrepresented in online education (Kinash & Crichton, 2007). The present study prov ides some evidence for this claim. For example, the influence of IEP on student achievement is negative and significant for standardized test grade 7 with s tudents who d id not have an IEP (usually students without learning disabilities) perform ed better th an students who ha d an IEP (usually students with learning disabilities). However, for standard test grade 8, the influence of IEP is positive and significant favoring students with learning disabilities. This finding could indicate that the virtual school may be able to help improve academic achievement for students at risk for failure in their physical schools. It also could be a sign of bridging gaps between students with learning disabilities and others without these disabilities with respec t to their academic performance possibly due to the academic support provided through the IEP.

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109 Racial gaps in student test score have been proved in many other studies conducted in traditional learning environments (Bali & Alvarez, 2004; Barth, 2001; Hall et al., 2000; Lockhead et al ., 1985). The student body in online K 12 schools represents the community that is served by traditional school system ( Ronsisvalle & Watkins, 2005) The findings about the racial gaps in student achievement in other studies could apply to the present study as well. The significant racial difference for standardized test grade 7 and nearly significant difference for standardized test grade 6 provides the evidence for the findings in other studies. The directions of the difference in the two groups show white American students perform better than other minority groups as a whole. However, t he finding that the significant racial difference was only found for 1 out 5 groups could be due to the coding system that combined different minority groups into one category potentially masking important information regarding the differences in student academic achievement among different racial groups. Future study could be conducted to investigate these differences with bigger sample size. The effect of student grade level in physical school was found to be negative and significant for two Algebra I groups, with students from lower grade levels perform ing better than those from higher grade levels. Students taking the standardized tests are from lower lev els (grade 68) compared with the students who took the Algebra I courses and Algebra I EOC tests (most of them from grade 9 12). Algebra I is a required course for high school graduation. Many students in higher grades such as grade 11, 12 take Algebra I courses in this virtual school as credit recovery or remediation to make up failing grades in their physical schools to meet the graduation requirement. It could be the explanation for the negative and significant effect of this factor for the two Algebra I groups. The

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110 effect of s tudent status in the virtual school (part time or full time online students) was not significant for all the five groups Implications Related to Research Question 3 In the present study, the influence of participation in free or reduced lunch programs is negative for all the five groups and significant for standard test grade 8. For standard test grade 8 group, s tudents who participated in these lunch programs achieved lower scores than students who did not participate in these pr ograms. This echoes the belief that family SES could affect student academic achievement via its influence on parental involvement in virtual learning environments (Black, 2009). This finding can guide the decision making process in the virtual schools by encouraging them to be sensitive to the needs of students with low family SES background and to take measures to bridge the gap in access to resources that could influence student academic achievement. The effect of IEP on student achievement is negative f or 4 out of 5 groups and significant for standard test grade 7 in the present study. For standard test grade 7 group, students with IEP ( usually students with learning disabilities ) achieved lower performance than others without IEP This could provide support for the integration of instructional strategies such as hiring academic coaches or tutors or advanced technologies during the online learning process to help students with disabilities succeed. The negative and significant effect of grade level for t he two Algebra I groups could be explained by the situation that many students in higher grade levels take Algebra I courses in this virtual school to make up the credits lost in their physical schools to meet the graduation requirement This has the impli cation for the virtual

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111 school during online course design for the implementation of certain strategies such as peer support and online tutoring, or flexible timelines and multiple paths to help higher grade students in Algebra I courses to achieve better p erformance. Online teachers also should provide individual assistance based on the needs of different student s. Broad I mplications for O nline C ourse D esign and O nline T eaching In September 2007, International Association for K 12 Online Learning (iNACOL) endorsed the National Standards of Quality for Online Courses based on the Southern Regional Education Board (SREB) Standards for Quality Online Courses In February 2008, iNACOL released National Standards for Quality Online Teaching based on SREBs Stan dards for Quality Online Teaching and Online Teaching Evaluation for State Virtual Schools The SREBs two sets of standards have been widely used by the 16 states in the southern United States. iNACOL National Standards of Quality for Online Courses stan dards were designed to provide states, districts, online programs, and other organizations with a set of quality guidelines for online course content, instructional design, technology, student assessment, and course management. (iNACOL, 2006, p.1). There are 6 categories in iNACOL standards: 1 Content 2 Instructional Design 3 Student Assessment 4 Technology 5 Course Evaluation and Management 6 21st Century Skills. Under each category there are a set of standards. National Standards for Quality Online Teaching is designed to provide states, districts, online programs, and other organizations with a set of quality guidelines for online teaching and instructional design. (iNACOL, 2008, p.1). There are 13 categories in these standards:

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112 A The teacher meets the profess ional teaching standards established by a statelicensing agency or the teacher has academic credentials in the field in which he or she is teaching. B The teacher has the prerequisite technology skills to teach online. C The teacher plans, designs and incorporates strategies to encourage active learning, interaction, participation and collaboration in the online environment. D The teacher provides online leadership in a manner that promotes student success through regular feedback, prompt response and clear expe ctations. E The teacher models, guides and encourages legal, ethnical, safe and healthy behavior related to technology use. F The teacher has experienced online learning from the perspective of a student. G The teacher understands and is responsive to students with special needs in the online classroom. H The teacher demonstrates competencies in creating and implementing assessments in online learning environments in ways that assure validity and reliability of instruments and procedures. I The teacher develops and delivers assessments, projects, and assignments that meet standards based learning goals and assesses learning progress by measuring student achievement of learning goals. J. The teacher demonstrates competencies in using data and findings from assessments a nd other data sources to modify instructional methods and content and to guide student learning. K The teacher demonstrates frequent and effective strategies that enable both teacher and students to complete self and preassessments. L The teacher collabor ates with colleagues. M The teacher arranges media and content to help students and teachers transfer knowledge most effectively in the online environment. (Instructional Design) Under each category there are a set of standards. Many of the findings, derivat ive outcomes, or implications in the present study align with the two sets of standards. The following two tables show these alignments.

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113 Conclusions This dissertation examined the impact of some variables including students demographic information, teacher comments, and student utilization of the LMS on academic performance in Algebra EOC tests and state standard mathematics tests using a sample of students from a state led virtual school in the Midwestern U.S region. The results show different variables a ffect student Algebra/mathematics achievement in different ways. No single factor investigated in the present study has been found to be significant for all five groups. It could be due to the limitations mentioned in Chapter 3: Methodology. It also indicated that some other factors such as instructional strategies utilized, teacher experience and student prior subject knowledge could have been missed in the present study They should be investigated in the future studies on success factors in the virtual s chooling. Outcomes of this study have some specific implications for researchers, practitioners, and policy makers. The results show the time student spent in the LMS has positive influence on student academic achievement. This provides the support for th e online instructional designers or LMS developers to utilize more advanced technologies such as some educational games and refine the course delivery system to motivate students learn the content and spend more time engaging the academic activities. It al so can lend relevance to online instructors for the implementation of instructional strategies to encourage students to focus on the learning tasks during their stay in the course delivery systems. The results of data analysis in this study show the influences of many factors are mixed. Some are positive, and others are negative. Even for the same factor, the influence could be in different directions for different tests. This indicates that the investigation of success factors of online learning is a compl ex process in which

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114 quantitative methodology independently may not be able to effectively measure the influence of the factors on academic achievement. Therefore, future research seeking to investigate the success factors in online learning should utilize mixed methodology incorporating quantitative and qualitative methods. This dissertation has implications for policy making process es at state and national level regarding quality virtual schooling and research support. At state level s in which the virtual school is implement ed effective and well designed LMS should be utilized for course delivery and management. The LMS interface should be user friendly that can attract students attentions in longer periods during the learning process. Components such as online forum, incorporation of social networking software, online synchronous audio/video conferencing should be integrated in the LMS to encourage more and diversified teacher student and student student interact ion To increase success opportunities for all students, virtual schools should take some measures to increase access of students from lower SES households the learning resources such as additional lab time, one on one computer/laptop, or extra instructi onal time. Virtual school should provide in dividualized assistance based on students different needs such a s for students with learning disabilities it could be individual education plans for students taking the online courses for credit recovery it could be peer to peer support or group project s At national level, more support should be provided to help build better designed state led virtual schools to increase access to more effective learning resource for all students. More national standards regarding quality virtual schooling should be created to guide the practice and implementation of state level virtual school s. Both at state and

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115 national level, policy makers should grant more resources to support more empirical study collecting quantitative and qualitative data to provide evidence for policies making process. More research is needed on student academic achievement, online success model, and longitudinal study on virtual school retention. One data system regarding virtual school practice should be built both at state and national level fr om which the researchers can draw the information they need to conduct the secondary research similar to the present study. These secondary research studies can supplement the first hand studies though they may have l imitations such as missing information like the present study that lacks of qualitative data for some factors Since the establishment of the first virtual school at the end of 20th century, it has experienced an extraordinary development during the last one decade. However, with its short h istory, K 12 virtual schools are still a relatively new concept for many researchers and educators. Compared to online education at post secondary level, little research has been done in K 12 virtual learning environments (Cavanaugh 2007; Cooze & Barbour, 2005; Means et al., 2009; Picciano & Seaman, 2007; Picciano & Seaman, 2009; Ronsisvalle & Watkins, 2005). The present study is the first research on success factors in K 12 virtual learning environments. At present, no clear set of characteristics have been identified to predict success in virtual learning environments, and no conclusive model has been created to apply in online learning practice (Roblyer & Davis, 2008; Tallent Runnels et al., 2006). However, to help improve the practice and implementation of virtual schooling, Smith et al. (2005) emphasized the empirical studies on student academic achievement. Given the dearth of research on success factors in K 12 online learning environments, t his dissertation should serve as the

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116 starting point for mor e studies utilizing both qualitative and quantitative methods to help the development of one success model to improve student academic achievement in virtual schooling.

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117 Table 51 : S ignificance and Direction of the E ffect of F a ctors Course\ Factor Gr ade Level Race Free or Reduced Lunch IEP Student Status Teacher Comment Number of Times Logged into the LMS Time Spent in the LMS Algebra I 1 st half X + + + 2 nd half X + + X + X MAP Grade 6 + + Grade 7 X X + + X + + Grade 8 + X + X + + TOTAL of 5 2 1 1 2 0 1 1 1

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118 Table 52: Alignment with National Standards in Quality Online Course Findings, derivative outcomes, or implications of the present study Aligned standards in iNACOL National Standards of Quality for Online Courses Tests used in the present study align with state or national standards. Course tasks and assessments align with the required local, state, and national assessments that are associated with the course. (A) This virtual schoo l hires the authorized course provider to implement the Learning Management System (LMS) and content area teachers who met state certification and other requirements as online instructors. The course provider is authorized to operate in the state in which the course is offered. (E) The teacher meets the professional teaching standard established by a state licensing agency or the teacher has academic credentials in the field in which he or she is teaching and has been trained to teach online and to use the course. (E) Flexible online courses that allow students to stay in the course for extended periods of time while working on complex and abstract content. The course instruction includes activities that engage students in active learning. (B) The course provides opportunities for students to engage in higher order thinking, critical reasoning activities and thinking in increasingly complex ways. (B) Improvement of many LMSs to integrate teaching practices that foster connectedness among teachers and students. Development of online courses that integrate teacher feedback and teacher student interaction as critical components during the course design. The course design provides opportunities for appropriate instructor student interaction, including timely and frequent feedback about student progress. (B) The course provides opportunities for appropriate instructor student and student student interaction to foster mastery and application of the material and a plan for monitoring that interaction. (B) Int egration of instructional strategies or advanced technologies during the online learning process to help students with disabilities to succeed. The course meets universal design principles, Section 508 standards and W3C guidelines to ensure access for all students. (D)

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119 Table 53: Alignment with National Standards in Quality Online Teaching Findings, derivative outcomes, or implications of the present study Aligned standards in iNACOL National Standards for Quality Online Teaching The importance of timel y and constructive feedback from online instructors for the success in online learning. Encourages interaction and cooperation among students, encourages active learning, provides prompt feedback, communicates high expectations, and respects diverse talent s and learning styles. (D The teacher provides online leadership in a manner that promotes student success through regular feedback, prompt response and clear expectations.) Establishes and maintains ongoing and frequent teacher student interaction, student student interaction and teacher parent interaction. (D) Provides timely, constructive feedback to students about assignments and questions. (D) Personalizes feedback (support, growth and encouragement). (D) Creates a warm and inviting atmosphere that pr omotes the development of a sense of community among participants. (C The teacher plans, designs and incorporates strategies to encourage active learning, interaction, participation and collaboration in the online environment) The finding that students wh o participated in free or reduced lunch programs achieved lower scores than students who did not participate in these programs for state standardized test grade 8 group could guide the decision making process in the virtual schools by encouraging them to be sensitive to the needs of students with low family SES background and to take measures to bridge the gap in access to resources that could influence student academic achievement. Provides activities, modified as necessary, that are relevant to the needs of all students. (G The teacher understands and is responsive to students with special needs in the online classroom.)

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120 Table 53. Continued Findings, derivative outcomes, or implications of the present study Aligned standards in iNACOL National Standards for Quality Online Teaching Online teachers should provide individual assistance based on the needs of different student s. Provides activities, modified as necessary, that are relevant to the needs of all students. (G) Personalizes feedback (support, growth and encouragement). ( D) Provides evidence of effective learning strategies that worked for the individual student and details specific changes in future instruction based upon assessment results and research study (datadriven and researchbased). (J The teacher demonstrates competencies in using data and findings from assessments and other data sources to modify instructional methods and content and to guide student learning.)

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121 APPENDIX A ALGEBRA I MULTIPLE CHOICE RELEAS ED SAMPLES

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142 APPENDIX B ALGEBRA I PERFORMANCE EVENT RELEASED SAMPLES

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145 APPENDIX C STATE ALGEBRA STANDARDS Number and Operations N.1.A.AI compare and order rational and irrational numbers, including finding their approximate locations on a number line N.1.B.AI use real numbers and various models, drawing, etc. to solve problems N.1.C.AI use a variety of representations to demonstrate an understanding of very large and very small numbers N.2.B.AI describe the effects operations such as multiplication, division, and computing powers and roots on the magnitude of quantities N.2.D.AI apply operations to real numbers, using mental computation or paper andpencil calculations for simple cases and technology for more complicated cas es N.3.D.AI judge the reasonableness of numerical computations and their results N.3.E.AI solve problems involving proportions Algebraic Relationships A.1.B.AI generalize patterns using explicitly or recursively defined functions A.1.C.AI compare and contrast various forms of representations of patterns A.1.D.AI understand and compare the properties of linear and nonlinear functions A.1.E.AI describe the effects of parameter changes on linear, exponential growth/decay and quadratic functions including intercepts A.2.A.AI use symbolic Algebra to represent and solve problems that involve linear and quadratic relationships including equations and inequalities A.2.B.AI describe and use Algebraic manipulations, including factoring and rules of

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146 integer exponents and apply properties of exponents (including order of operations) to simplify expressions A.2.C.AI use and solve equivalent forms of equations (linear, absolute value, and quadratic) A.2.D.AI use and solve systems of linear equations or inequaliti es with 2 variables A.3.A.AI identify quantitative relationships and determine the type(s) of functions that might model the situation to solve the problem A.4.A.AI analyze linear and quadratic functions by investigating rates of change, intercepts and z eros Data and Probability D.1.A.AI formulate questions and collect data about a characteristic which include sample spaces and distributions D.1.C.AI select and use appropriate graphical representation of data and given onevariable quantitative data, di splay the distribution and describe its shape D.2.A.AI apply statistical measures of center to solve problems D.2.C.AI given a scatter plot, determine an equation for a line of best fit D.3.A.AI make conjectures about possible relationships between 2 characteristics of a sample on the basis of scatter plots of the data

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147 APPENDIX D NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS MATHEMATICS STANDARDS FOR GRADES 6 8 Table A 1. Number and Operations Standard for Grades 6 8 Expectations Instructional programs from prekindergarten through grade 12 should enable all students to In grades 6 8 all students should Understand numbers, ways of representing numbers, relationships among numbers, and number systems work flexibly with fractions, decimals, and percent s to solve problems; compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line; develop meaning for percents greater than 100 and less than 1; understand and use ratios and proportions to represent quantitative relationships; develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation; use factors, multiples, prime factorization, and relatively prime numbers to s olve problems; develop meaning for integers and represent and compare quantities with them. Understand meanings of operations and how they relate to one another understand the meaning and effects of arithmetic operations with fractions, decimals, and integers; use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals; understand and use the inver se relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

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148 Table A 1. Continued Instructional programs from prekindergarten through grade 12 should enab le all students to In grades 6 8 all students should Compute fluently and make reasonable estimates select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods; develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use; develop and use strategies to estimate the results of rational number computations and judge the reasonableness of the results; develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

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149 Table A 2. Geometry Standard for Grades 6 8 Expectations Instructional programs from prekindergarten through grade 12 should enable all students to In grades 6 8 all students should Analyze characteristics and properties of two and threedimensional geometric shapes and develop mathematical arguments about geometric relationships precisely describe, classify, and understand relationships among types of two and threedimensional objects using their defining properties; understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects; create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship. Specify locations and describe spatial rel ationships using coordinate geometry and other representational systems use coordinate geometry to represent and examine the properties of geometric shapes; use coordinate geometry to examine special geometric shapes, such as regular polygons or thos e with pairs of parallel or perpendicular sides. Apply transformations and use symmetry to analyze mathematical situations describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling; examine the congruence, similarity, and line or rotational symmetry of objects using transformations. Use visualization, spatial reasoning, and geometric modeling to solve problems draw geometric objects with specified properties, such as side l engths or angle measures; use two dimensional representations of threedimensional objects to visualize and solve problems such as those involving surface area and volume; use visual tools such as networks to represent and solve problems; use g eometric models to represent and explain numerical and Algebraic relationships; recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

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150 Table A 3. Measurement Standard for Grades 6 8 Expectations Instructional programs from prekindergarten through grade 12 should enable all students to In grades 6 8 all students should Understand measurable attributes of objects and the units, systems, and processes of measurement understand both metric and customary systems of measurement; understand relationships among units and convert from one unit to another within the same system; understand, select, and use units of appropriate size and type to measure angles, perim eter, area, surface area, and volume. Apply appropriate techniques, tools, and formulas to determine measurements use common benchmarks to select appropriate methods for estimating measurements; select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision; develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of morecomplex shapes; develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders; solve problems involving scale factors, using ratio and proportion; solve simple problems involving rates and derived measurements for such attributes as velocity and density.

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151 Table A 4. Data Analysis and Probability Standard for Grades 6 8 Expectations Instructional programs from prekindergarten through grade 12 should enable all students to In grades 6 8 all students should Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population; select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots. Select and use appropriate statistical methods to analyze data find, use, and interpr et measures of center and spread, including mean and interquartile range; discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem and leaf plots, box plots, and scatterplots. Deve lop and evaluate inferences and predictions that are based on data use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken; make conjectures about possible relationshi ps between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit; use conjectures to formulate new questions and plan new studies to answer them. Understand and apply basic concepts of probability u nderstand and use appropriate terminology to describe complementary and mutually exclusive events; use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations; compute p robabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

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152 Table A 5. Problem Solving Standard for Grades 6 8 Instructional programs from prekindergarten through grade 12 should enable all students to build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical pr oblem solving. Table A 6. Reasoning and Proof Standard for Grades 6 8 Instructional programs from prekindergarten through grade 12 should enable all students to recognize reasoning and proof as fundamental aspects of mathematics; make and investigate mathematical conjectures; develop and evaluate mathematical arguments and proofs; select and use various types of reasoning and methods of proof. Table A 7. Communication Standard for Grades 6 8 Instructional programs from prekindergarten through grade 12 should enable all students to organize and consolidate their mathematical thinking through communication; communicate their mathematical thinking coherently and clearly to peers, teachers, and others; analyze and evaluate the mathematical thinking and strategies of others; use the language of mathematics to express mathematical ideas precisely. Table A 8. Connections Standard for Grades 6 8 Instructional programs from prekindergarten through grade 12 should enable all students to recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and apply mathematics in contexts outside of mathematics. Table A 9. Representation Standard for G rades 6 8 Instructional programs from prekindergarten through grade 12 should enable all students to create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; use representations to model and interpret physical, social, and mathematical phenomena.

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153 APPENDIX E MAP GRADE 6 RELEASED ITEMS SPRING 06

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167 APPENDIX F STATE S TANDARDS FOR M ATHEMATICS AT GRADE LEVEL 6 Number and Oper ations N.1.A.06 compare and order integers, positive rationals and percents, including finding their approximate location on a number line N.1.B.06 recognize and generate equivalent forms of fractions, decimals and percents N.1.C.06 recognize equivalent representations for the same number and generate them by decomposing and composing numbers, including expanded notation N.1.D.06 use factors and multiples to describe relationships between and among numbers, including whole number common factors and co mmon multiples N.2.B.06 describe the effects of addition and subtraction on fractions and decimals N.3.C.06 add and subtract positive rational numbers N.3.D.06 estimate and justify the results of addition and subtraction of positive rational numbers N .3.E.06 solve problems using equivalent ratios Algebraic Relationships A.1.B.06 represent and describe patterns with tables, graphs, pictures, symbolic rules or words A.1.C.06 compare various forms of representations to identify a pattern A.1.D.06 ide ntify functions as linear or nonlinear from a table or graph A.2.A.06 use variables to represent unknown quantities in expressions A.2.B.06 recognize equivalent forms for simple Algebraic expressions including associative and distributive properties

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168 A. 3.A.06 model and solve problems, using multiple representations such as graphs, tables, expressions and equations A.4.A.06 compare situations with constant or varying rates of change Geometric and Spatial Relationships G.1.A.06 identify the properties o f one, two and threedimensional shapes using the appropriate geometric vocabulary G.1.B.06 describe relationships between the corresponding angles and the length of corresponding sides of similar triangles (whole number scale factors) G.2.A.06 use c oordinate geometry to construct geometric shapes G.3.A.06 describe the transformation from a given preimage to its image using the terms reflection/ flip, rotation/ turn and translation/ slide G.3.C.06 create polygons and designs with rotational symmetr y G.4.A.06 use spatial visualization to identify isometric representations of mat plans G.4.B.06 draw or use visual models to represent and solve problems Measurement M.1.A.06 identify and justify an angle as acute, obtuse, straight or right M.1.C.06 solve problems involving elapsed time (hours and minutes) M.2.A.06 estimate a measurement using either standard or nonstandard unit of measurement M.2.B.06 select and use benchmarks to estimate measurements of 0 45 90 180 360degree angles

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169 M. 2.C.06 describe how to solve problems involving the area or perimeter of polygons M.2.E.06 convert from one unit to another within a system of measurement (mass and weight) Data and Probability D.1.A.06 formulate questions, design studies and collect data about a characteristic D.1.C.06 interpret circle graphs; create and interpret stem andleaf plots D.2.A.06 find the range and measures of center, including median, mode and mean D.2.B.06 compare different representations of the same data and evaluate how well each representation shows important aspects of the data D.3.A.06 use observations about differences between 2 samples to make conjectures about the populations from which the samples were taken D.4.A.06 use a model (diagrams, list, sample space, or area model) to illustrate the possible outcomes of an event

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195 BIOGRAPHICAL SKETCH Feng Liu was born in 1973 in Liaocheng City, Shandong Province, China. As th e second child of one threechild family, he attended No.1 High School, Zhongyuan Oil Field in Puyang City, Henan Province, China. Feng Liu graduated from Nanjing Normal Universitys Computer Science Department with a B achelor of S cience in computer scienc e e ducation in 1995. He has taught computer science courses at postsecondary level including Nanjing Material Polytechnic School and Nanjing University of Finance & Economics for ei ght and half years. Feng Liu came to United States at January 2004 to further his education at Georgia College & State University where he earned a Master of E ducation in e ducational t echnology in May 2006. In August of 2006, Feng Liu enrolled as a doctoral fellow in the Educational Technology program in School of Teaching and Learning at the University of Florida (UF). During his study at UF, Feng Liu has focused on research in the learning technologies. His research interests include the investigation of online learning success and the effectiveness of virtual schooling, the employment of advanced research methods and statistical approaches in educational research, and the use of egame/simulation for knowledge gain, attitude change and motivation in areas such as science and second language acquisition. He has several publicat ions in these areas.