THE EFFECT OF A MATHEMATICAL TASK FOCUSED INTERVENTION ON PROSPECTIVE ELEMENTARY SCHOOL T EACHERS BELIEFS ABOUT MATHEMATICS INSTRUCTION By KRISTEN APRAIZ 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 2014
Â© 2014 Kristen Apraiz
To Briana and Gabe
4 ACKNOWLEDGMENTS My pursuit of a doctoral degree could not have been possible without the support of my professors, family, and friends. First, I would like to thank my adviser , Dr. Tim Jacobbe , and my committee members, Dr. Kent Crippen, Dr. David Miller, and Dr. Thomasenia Adams, for their time , feedback, and expertise . Second, I would like to thank my mom, Ronna Appleby. I am forever grateful for how you engrained in me the value of a strong work ethic , the importance of education, and the ability to persevere through life . I would like to th ank my husband, Gabe, for his endless support, understanding, and patience throughout these last four years. Finally , I would like to thank my friends and fellow graduate students for their encouragement and support. In particular, I would like to acknowl edge Dr. Julie Brown, Dr. Rich Busi, Dr. Cheryl Mclaughlin, and Rhonda Williams , for always listening and providing the opportunities to bounce ideas back and forth .
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 ! LIST OF TABLES ................................ ................................ ................................ ............ 8 ! LIST OF ABBREVIATIONS ................................ ................................ ............................. 9 ! ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 ! Working Definitions ................................ ................................ ................................ . 17 ! Theor etical Perspective ................................ ................................ .......................... 17 ! Mathematical Tasks and Cognitive Demand ................................ ........................... 23 ! Cognitive Dem and ................................ ................................ ............................ 23 ! Statement of the Problem ................................ ................................ ................. 30 ! Structure of the Dissertation ................................ ................................ ................... 37 ! 2 REVIEW OF LITERATURE ................................ ................................ .................... 38 ! Vision for Mathematics Instruction ................................ ................................ .......... 38 ! Mathematical Tasks ................................ ................................ ................................ 40 ! The Earlier Years of Mathematical Tasks ................................ ......................... 41 ! Teacher Education and Mathematical Tasks ................................ .................... 46 ! ................................ ..... 62 ! Tea cher Beliefs and Teacher Change ................................ ................................ .... 65 ! 3 METHODS ................................ ................................ ................................ .............. 72 ! Overview ................................ ................................ ................................ ................. 72 ! Methodology ................................ ................................ ................................ ........... 75 ! Procedures ................................ ................................ ................................ ............. 77 ! Participants ................................ ................................ ................................ ....... 77 ! Identification of participants ................................ ................................ ........ 77 ! Description of participants ................................ ................................ .......... 79 ! Setting ................................ ................................ ................................ ........ 79 ! Treatment ................................ ................................ ................................ ......... 80 ! Treatment group ................................ ................................ ........................ 80 ! Control group ................................ ................................ ............................. 82 ! Data Sources ................................ ................................ ................................ .... 83 ! Data Collection ................................ ................................ ................................ . 83 ! Beliefs instrument ................................ ................................ ...................... 83 ! Interviews ................................ ................................ ................................ ... 86 !
6 Mathematical Task Sort ................................ ................................ .................... 87 ! Elementary Mathematical Task Fo rm ................................ ............................... 88 ! Data Analysis ................................ ................................ ................................ ... 89 ! Limitations of the Study ................................ ................................ ........................... 91 ! Conclusion ................................ ................................ ................................ .............. 92 ! 4 RESULTS ................................ ................................ ................................ ............... 94 ! Mathematics and Mathematics Instruction ................................ ................................ ................................ ............ 94 ! Pre and Post Belief Survey ................................ ................................ ............. 94 ! The Beliefs of Heidi and Nora ................................ ................................ ........... 99 ! Past experiences and present beliefs about mathematics instruction ...... 103 ! Mathematics instruction should include authentic and inquiry based learning ................................ ................................ ................................ . 105 ! The role of the teacher in making ins tructional decisions ......................... 107 ! Cognitive aspects of student understanding ................................ ............ 107 ! Social and emotional aspects of teaching ................................ ................ 112 ! Preservice Te Tasks ................................ ................................ ................................ ................. 115 ! Pre and Post Intervention Mathematical Task Sort ................................ ...... 116 ! Comparing the Treatment Group to the Control Group ................................ .. 117 ! .................. 118 ! 5 DISCUSSION ................................ ................................ ................................ ....... 122 ! Importance of this study ................................ ................................ ........................ 122 ! Explanations for Results ................................ ................................ ....................... 126 ! The Intervention Provided PSTs with the Opportunity to Do Mathematics ..... 126 ! Belief Changes ................................ ................................ ............................... 134 ! Increased awareness of level of cogniti ve demand for mathematical tasks ................................ ................................ ................................ ..... 140 ! Effectiveness of the intervention ................................ .............................. 142 ! Contributions of this Investigation ................................ ................................ ......... 145 ! Conclusions, Limitations, and Future Research ................................ .................... 147 ! APPENDIX A ELEMENTARY MATHEMATICAL TASK ANALYSIS FORM ................................ 153 ! B TASKS FROM THE QUASAR ELEMENTARY SCHOOL TASK SORTING ACTIVITY ................................ ................................ ................................ .............. 154 ! C EXAMPLE OF QUESTIONS FOR GENERAL INTERVIEW GUIDE APPROACH 155 ! D COURSE OVERVIEW FOR 12 WEEKS OF THE STUDY ................................ .... 156 ! E OVERVIEW OF PEN PAL LETTER WRITING ................................ ..................... 158 !
7 F EXAMPLE LESSON PLAN ................................ ................................ ................... 159 ! G SAMPLE OF BELIEFS INSTRUMENT ................................ ................................ . 161 ! H CROSSTABULATION FOR BELIEF SURVEY DATA ................................ .......... 162 ! I CODING FOR THEMATIC ANALYSIS ................................ ................................ . 164 ! J ICAL TASKS ................................ ................... 165 ! K INFORMED CONSENT ................................ ................................ ........................ 169 ! LIST OF REFERENCES ................................ ................................ ............................. 171 ! BIOGRAPHICAL SKETCH ................................ ................................ .......................... 180 !
8 LIST OF TABLES Table page 3 1 Table of Participants ................................ ................................ ........................... 78 ! 4 1 Beliefs pretest significance values. ................................ ................................ ..... 96 ! 4 2 Actual count for belief change score for treatment and control groups ............... 96 ! 4 3 Belief change score significance values. ................................ ............................ 97 ! 4 4 Belief 5 cross tabulation values. ................................ ................................ ......... 99 ! 4 5 Belief 7 cross tabulation values. ................................ ................................ ......... 99 ! 4 6 Percentage of students in each group whose scores on Belief 5 increased 1, 2, 3, or 4 levels from presurvey to postsurvey ................................ .................... 99 ! 4 7 Percentage of students in each group whose scores on Belief 7 increased 1 or 2 levels from presurvey to postsurvey ................................ ............................ 99 ! 4 8 and Post Belief Scores ................................ ............... 100 ! 4 9 Descriptive Statistics on Mathematical Task Sort Scores ................................ . 117 ! 4 10 Comparison of P re Mathematical Task Sort Scores of Treatment and Control Groups ................................ ................................ ................................ .............. 118 ! 4 11 Analysis of the Task Sort Responses by Level of Cognitive Demand (n = 32 preservice teachers) ................................ ................................ ......................... 121 ! H 1 Belief 1 crosstabulation values. ................................ ................................ ........ 162 ! H 2 Belief 2 crosstab ulation values. ................................ ................................ ........ 162 ! H 3 Belief 3 crosstabulation values. ................................ ................................ ........ 162 ! H 4 Belief 4 crosstabulation values. ................................ ................................ ........ 162 ! H 5 Belief 5 crosstabulation values. ................................ ................................ ........ 162 ! H 6 Belief 6 crosstabulation values. ................................ ................................ ........ 16 3 ! H 7 Belief 7 crosstabulation values. ................................ ................................ ........ 163 !
9 LIST OF ABBREVIATIONS C BMS Conference Board of the Mathematical Sciences CCSS Common Core State Standards NCTM National Council of Teachers of Mathematics NMAP National Mathematics Advisory Panel NRC National Research Council PST Prospective Elementary School Teacher
10 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 THE EFFECT OF A MATHEMATICAL TASK FO CUSED INTERVENTION ON PROSPECTIVE ELEMENTARY SCHOOL T EACHERS BELIEFS ABOUT MATHEMATICS INSTRUCTION By Kristen Apraiz August 2014 Chair: Tim Jacobbe Major: Curriculum and Instruction The purpose of this study was to determine the extent to which element ary pr ospectiv e about mathematics instruction change as a result of participating in a 12 week intervention focused on learning about the level of cognitive demand of mathematical tasks and writing letters to third grade elementary students. A necessary component of implementing mathematical tasks effectively with students is to have a foundational understanding of mathematical task knowledge (Chapman, 2013). During the 12 week interventio n the treatment group learned about featur es of mathematical tasks that elicit low level and high level cognitive demand from students . The pr ospectiv e teachers wrote their own ma thematical tasks that they include d in letters to th ird grade elementary students. The situative learning perspe ctive informed the design of the intervention. Specifically, the intervention was designed to position participants to examine, do, and discuss mathematical tasks and participate in a letter writing exchange with third grade students. The study utilized mixed methods to investigate the pr ospective about mathematics instruction and knowledge of cognitive demand for mathematical
11 tasks. The I ntegrated M athematics and P edagogy (IMAP) Belief Survey (Philip et al., 2007) was used to co llect data on beliefs. Additionally , participants of the study complete d an elementary mathematical task sort on two occasions. In addition, two interviews were conducted for participants who exhibit ed changes in several beliefs and a change in one belief for the seven beliefs. Results from the study indicate that the treatment group experienced significant belief changes on two of the seven beliefs. These two beliefs most closely align with the intervention. Results from the mathematical task sort indica te that participants in the treatment group were more able to accurately identify mathematical tasks as either eliciting low or high level cognitive demand. The outcome of the study could provide guiding principles for designing the elementary mathemat ics methods course where pr ospective teachers focus on the level of cognitive demand of mathematical tasks and have opportunities to interact with elementary school students through authentic experiences.
12 CHAPTER 1 INTRODUCTION Perhaps the major challenge facing those who wish to improve the mathematics learning of U.S. students is taking seriously the fact that teaching must change . (Hiebert, 2009, Foreword, p. ix) Over a decade ago, the National Council of Teachers of Mathematics (NCTM) (2000 Additionally, students need to learn in a way that creates understanding by having opportunit ies to discuss mathematics and engage in complex tasks while drawing upon a variety of mathematical to pics (National Research Council, 2001). Students can have access to the aforementioned learning opportunities by having a teacher who has been trained to implement mathematics instruction using well t hought out mathematical tasks. J ust like students, prospective and inservice teachers need opportunities to explore mathematics, collaborate with peers, and learn pedagogical strategies to help reach all studen ts during mathematics instruction (Conference Boa rd of the Mathematical Sciences, 2012; S tein, Smith, Hennigsen & Silver, 2009). help facilitate a learning environment where students are conjecturing, refining mathematical ideas, and applying mathematical knowledge to relevant, real world tasks (National Governors Association Center for Best Practices (NGA) & Council of Chie f State School Officers (CCSSO), 2010). Establishing a classroom learning environment that incorporat es the desired student abilities takes practice, s upport, and time to accomplish. Prospective teachers learn about this type of teaching by reading course textbooks, watching videos, and
13 observing classrooms (Ball, 1990). Rarely do prospective teachers hav e opportunities to experiment w ith this type of teaching (Ball, 1990; Ebby, 2000; Hiebert, Morris & Glass, 2003; Kosko, Norton, C onn, & San Pedro, 2010). In order for students to learn mathematics at the desired level, teachers must be prepared to use math ematical tasks that will allow students to go deeply into the mat hematical content (Stein et al., 2009). Currently, with the widespread adoption of the Common Core Stat e Standards (CCSS) (NGA & CCSSO, 2010) for mathematics across the United States, teacher s have opportunities to reconsider the structure of lessons for students to learn mathematics through rigorous, cognitively demanding mathematical tasks, which emphasize justification, synthesis, and analysis of mathemat ics (Grossman, Reyna, & Shipton, 201 1). The mathematical tasks teachers choose to implement in their classroom provide opportunities to engage stu dents in the learning of mathematics (Chapman, 2013 ; National Research Council, 2001). Mathematical tasks are the problem s teachers present to students. For example, a mathematical task could ask the student for one answer or an extended response that requ i res an explanation. Well constructed mathematical tasks allow students to view mathematics as a connected discipline where mathematical reasoning, problem solving, and justifying the mathematics are needed to accomplish the mathematical task (National Cou ncil of Teachers of Mathematics [NCTM] , 2000; National Research Council, 2 001; Stein, Grover, & Henningsen, 1996) . In order to use mathematical tasks effectively in mathematics instruction, both in service and prospective teachers need
14 time to consider the possible outcomes mathematical tasks can elicit (National Research Council, 2001). Prior research has revealed that teachers who are provided with opportunities to learn and focus on the cognitive demands of mathematical tasks have experienced increases i n student learning and engagement during mathematics instruction (Boston & Smith, 201 1; Henningsen & Stein, 1997). Teachers who par ticipated in these studies had the opportunities to identify features of mathematical tasks that elicit low and high levels o f cognitive demand, learn how to maintain the level of cognitive demand from the initial phase of choosing the task through the implementation phase in the classroom , and examine student work related to the mathematical task (Henningsen & Stein, 1997). The process of selecting a mathematical task for mathematics instruction, implementing the task with students, and discussing student results takes time for teachers to learn (Henningsen & Stein , 1997). Therefo re, it is crucial for prospective teachers to hav e opportunities during their teacher preparation to develop and implement mathematical tasks through authentic activities, such as participating in a letter writing exchange (Kosko, et al., 2010). The aforementioned opportunities are encompassed in classro om environments where students construct their own knowledge of mathematics through probl em solving and experimentation. In order for students to experience mathematics as a gen uine discipline, both prospective and practicing teachers need to experience le arning mathematics the way policy documents recommend mathematics be taught (NCTM, 2000; Nati onal Mathematics Advisory Panel, 2008). The Conference Board of the Mathematical Sciences (CBMS) (2012) supports this view by recommending that
15 ofessional development for teachers must provide opportunities to Prospective teacher education needs to include experiences where teacher ns about the nature of Developing as a Teacher of Mathemat For instance, prospective elementary school teachers (PSTs) enter teacher educati on programs with about 13 years of passively watching teachers teach mathematics . These years of watching contribute to the pre conceived notions PSTs have about how math ematics should be taught (Ball, 1990; Ebby, 2000 , Ernest, 1989 ). More often than not, t he mathemati cs instruction they believe is the way to teach is the traditional approach to teaching where the teacher stands up in front of a room of students and lectures, while the students take notes and work on problems from textbooks (Bahr, Monroe & Shaha 2013) . During teacher education courses, PSTs experience a conflict between their own learning of mathematics and the new methods of teaching mathematics (Ebb y, 2000; Hiebert, et al., 2003). To further complicate the PSTs understanding of mathemat ics teaching, they are sometimes placed in educational field experiences where the teacher is not teaching students in the same manner as preservice teachers are learning in their teacher education courses ( Philip p , et al. 2007). With the implementation of the CCSS for mathematics, PSTs need opportunities where they can experiment with posing mathematical tasks to elementary students and assessing the cognitive demand of mathematical tasks in print form (CBMS, 2012) .
16 The CCSS for mathematics place an empha sis on preparing students for future careers and colleges. The standards strive to increase the rigor and relevance of mathematics in order to ensure that students are prepared with the knowledge and skills that will lead them to having success in colleges and careers (Grossman, et al., 2011). Traditionally, individual states have set the criteria for mathematics instruction. The CCSS hold all students across the nation to the same expectations for mathematical learning. The implementation of the CCSS for m athematics will require ongoing professional development for in service teachers and more exposure to mathematics for prospective teachers ( Association of Mathematics Teacher Educators, 2011). Not only will teachers need a deep level of mathematical conten t knowledge, they will also need to know how to structure learning experiences for students in order to implement the Standards for Mathematical Practice (NGA & CCSSO 2010). The purpose of this study is to determine the extent of change that can be ascert ained by an intervention focused on learning about the cognitive demand of mathematical tasks. The mathematical tasks would be posed through a letter writing exchange and a review would be made of the resulting PST beliefs about mathematics instruction. Th e hypothesis guiding the study is that providing PSTs with opportunities to consider the cognitive demand of mathematical tasks, solve those tasks and discuss the solutions, while posing mathematical tasks through a letter writing exchange will produce sig remainder of this chapter will provide justification for learning about the cognitive demand of mathematical tasks and posing mathematical tasks through a letter writing exchange.
17 Workin g Definitions Mathematics education tends to use common terms throughout the literature that may be defined different ly in other areas of education. In order to provide clarification for the reader, certain terms are defined. 1. M ATHEMATICAL TASK as a classroom activity, the purpose of which is to 1996, p.460). 2. P RACTICING OR INSERVI CE TEACHER a current teacher who is working in a school. 3. P ROSPECTIVE ELEMENTAR Y SCHOOL TE ACHER current postsecondary student who is enrolled in a teacher education program at a higher learning institution. 4. S TUDENTS MATHEMATICAL THINKIN G how students process mathematics in order to make mathematical decisions. 5. C OGNITIVE DEMAND the type o r level of thinking produced by the student to solve a mathematical task (Stein et at., 1996). 6. R EFORM MATHEMATICS IN STRUCTION refers to mathematics instruction that goes beyond teaching procedures and havi ng students recall facts; students are expected to apply knowledge, explore mathematics, and produce conjectures about how a mathematics concept is applied (NCTM, 2000) . Theoretical Perspective According to Chapman (2013), there are several factors that i nfluence the implementation of mathematical tasks: of learners, goal for task, instructional orientation, and beliefs about mathematics instruction. task knowledge is the determining factor for how tasks ar e implemented in the classroom. Mathematical task t heir development of mathematical thinking, and capture their interest and curiosity and (b) optimize the learning potential of su More
18 specifically, mathematical mathema Purpose fully selected mathemati cal tasks are a key component for effective mathematics instruct ion (National Research Council, 2001). Mathematical tasks are viewed as channels that provide opportunities for students to engage in problem solving and to see mathematics as a discipline with practical purposes, ( i.e., one which makes sense to learn ) (N ational Research Council, to elicit a high level of cognitive demand; however it is up to the teacher and student whether the cognitive mathematics instruction (i.e., how students learn) influence the impl ementation of the mathematical task (National Research Council, 2001). For instance, a teacher selects a mathematical task that exhibits a high level of cognitive demand because the task asks students to apply knowledge of a concept to a real world applic ation. From the time the mathematical task is first introduced through the duration of time the student attends to the task, the teacher has the ability to affect the level of cognitive demand. If the teacher believes that students should not struggle with mathematics, they may intervene in the process of making sense of the task and lower the level of cognitive demand by suggesting a possible alternative way to produce an answer. Th e aforementioned example is provided to show how a teacher can affect
19 the implementation of a mathematical task. Mathematics instruction leaves lasting engaged determine not only what substance they learn but also how they come to think about, de Preservice teachers can benefit from experimenting with posing mathematical tasks in order to learn how to use mathematical tasks effectively during instruction. Exposure to mathemati cal tasks during preservice teacher education could provide opportunities for PSTs to acqui re mathematical task knowledge. Researchers, Crespo (2003) and Norton and Kastberg (2012) used the authentic activity of letter writing with PSTs. The letter writing experience provided PSTs with the opportunity to mathematical task, and contempla te how to respond to students. Norton and Kastberg (2012) incorporated the level s of cognitive d emand into their research study, which involved secondary preservice teachers posing mathematical tasks to high school students through letter writing, in order to demonstrate the potential cognitive demand a task could elicit from a student . Crespo (2003) had PSTs focus on the type of mathematical tasks they asked students to complete. Both studies revealed that PST s progressed over time in their ability to pose cognitively demanding math ematical tasks. Chapman (2013) describes mathematical task knowledge dimensional and thus likely to be challenging for a teacher to construct without meaningful intervention to build on her or his initial sense ). This dissertation focu ses on building PST task know ledge, which is necessary to effectively implement mathematical tasks . Further, preservice teachers
20 develop ed select components of this knowledge by ta king part in learning about the cognitive demands of mathematical tasks and posing such tasks to elementa ry school students. Another component of the study is the mathematical instructional beliefs held by the PSTs. Preservice teachers hold beliefs about mathematics and mathematics instruction that date back to their yea rs as students (Ernest, 1989). Accordin g to Philipp et al. (2007) PST perpetuating belief that If I, a college student, do not know something, then children would not be expected to know it, and if I do know something, I If it is true that PSTs hold such self perpetuating beliefs, then challenging such beliefs is a demanding task for teacher educators to undertake. Philipp et al. (2007) posits that in order for PST s to care about learning mathematics for teaching, teacher educator s need to recognize that PST s care about c hildren first. In the study by Philipp et al. (2007), researchers used a guiding hypothesis that if PST s focused on und erstanding how children think about mathemat ics, then their beliefs about mathematics instructi on would change. The theory behind the belief change is for the PSTs to connect the mathematical content knowledge with knowledge of how to teach children to le arn and understand mathematics. Philipp et al. (2007) used the Circles of Caring model , which is an expansion of Nel Noddings (1984) theory about caring, to describe the process by which PSTs learn to care about learning mathematics for teaching. The premise behi nd the Circles of Caring model is to tap into the feelings of caring that PSTs have for children. Next, PSTs can see how children succeed in solving a mathematical task, and they are then more inclined to learn how to incorporate more
21 of these experiences for children. The PSTs will be able to see how mathematical thinking plays a role in solving mathematical tasks for children. Lastly, PSTs will want to know more about mathematics to help children. The desired outcome from the aforementioned study by Phili pp et al. (2007) was for the PSTs to change their beliefs pertaining to mathematics instruction from a formulaic, procedural process to a more conceptual based form of mathematics instruction. According to Philipp et al. (2007), the change in beliefs that the researchers were hoping to see in PSTs arrived through the act of positioning them in an environment where they engage in designed mathematical experiences (i.e. , watch videos of children solving problems, read case studies, examine student work) to ei ther act or to consider how to act with children. Due to the structure of the field experiences for the PSTs, Philipp et al. (2007) used the situative learning perspective to guide their research. The situative learning perspective is the lens through whic h this dissertation view s the Circles of Caring theory and the acquisition of mathematical task knowledge for PSTs . The situative learning perspective is a part of two other perspectives sociocultural and distrib uted cognition (Sawyer, 2006). The sociocu lturalist envisions the learner as an individual, who is part of a larger community (Lave, 1991). The activities that take place within the community help the individual acquire kn owledge (Cobb & Bowers, 1999). The activity is the unit of analysis in the s ituative learning pe rspective (Rogoff, 2008). Reform documents focused on improving teacher education (e.g., NMAP , NCATE ), have made recommendations for preservice teachers to have opportunities which allow them to be p art of the teaching com munity. For example,
22 one opportunity is a field experience where preservice teachers may observe, interact, and discuss classroom activity with peers and a teacher educator. Interactions with others in the environment help shape learning for t he individua l (Vygotsky, 1978). In order to promote an optimally effective learning community, the following elements were identified by researchers as critical to facilitate learning: the role of the teacher educator, the establishment of classroom norms, and the use of tools to help preservice teachers learn about teaching using mathematical tasks (Cady, Meier & Lubinski, 2006; Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz, 1991; Stein & Henningsen, 1997; Szydlik, Szydlik & Benson, 2003 ). For instanc e, in the study conducted by Szydlik et al (2003), the teacher educators facilitated a classroom learning environment, which focused on establishing social norms. The establishment of agreed upon social norms assisted the preservice teachers in working tog ether on problem solving tasks. During the course of the semester, the researchers were interested in how this type of learning environment affected the beliefs of preservice teachers about mathematics instruction. Several preservice teachers expressed a n ew understanding about mathematics as a connected dis cipline (Szydlik et al., 2003). For example, the preservice teachers were able to experience how several mathematical topics were needed to solve a mathematical task. Researchers using this theory of lea rning place an emphasis on the setting and the activities that happen within this setting (Putnam & Borko, 2000) . The tools that preservice teachers will acquire in order to learn to use mathematical tasks effectively develop through their experiences, whi ch are situated within authentic activities . Brown, Collins and Duiguid (1989) describe such activities by stating
23 are most simply defined as the ordinary pra According to Putnam and Borko (2000), solving skills that are important in out of school settings, whether or not the activities themselves mirror what practi Teacher preparation strives to prepare PSTs to take on leadership roles in the classroom that involve making instructional decisions on a daily basis. The experiences provided to PSTs by teacher educators during their teacher preparation should situate learning in a setting where the activity allows PSTs to examine their thinking by constructing meaning with their peers through social interaction (Lav e, 1991; Putnam & Borko, 2000). The authentic activities in this dissertation provide the context for mathematics learning to be viewed as a social process (Co bb & Bowers, 1999; Gee, 2008). The intervention for the study is designed to provide the PSTs the opportunities to work together on mathematical tasks and participate in an authentic letter writing exchange. The letter writing exchange served as an a ctivity, which follows the Circles of Caring theory , by allowing the PST to focus on a student and learn about how the student was thinking about mathematics. During the intervention, the cognitive demand of mathematical tasks was used as a focal point for PSTs to consider the opportunities to learn mathematics provided by mathematical tasks. Mathematical Tasks and Cognitive Demand Cognitive Demand For this study, t he task analysis guide from Stein, Smith, Henningsen & Silver (2000) will pr ovide a framework for analyzing the cognitive demand of mathematical tasks. The researchers
24 required of students in order to successfully engage wi Stein et al. (2000) use a task analysis guide to assess the level of cognitive demand a task is capab le of producing for a student. The task analysis guide consists of two categories: Lower Level Demands and Higher Level Demands. Within each category, there are two sub categ ories. For instance, the Lower Level Demand category incorporates memorization tasks and procedures without connections tasks (e.g., Solve. 25 Ã–5) . The Higher Level Demand category includes procedures with connections tasks and completion of mathematics t asks (e.g., t hink of a real life situation that describes the following problem: 124 Ã–12; write the problem and then solve it) . Descriptions for each of the sub categories are provided, so the teacher can analyze the mathematical task with the description t o determine the level of cognitive demand a mathematical task has the potential of producing . For example, a teacher can use the task analysis guide to categorize questions on a standardized test (e.g., The Florida Comprehensive Assessment Test) in order t the student. Stein et al. (2000) stress the importance of matching the task with the learning goal for the student. The task analysis guide was developed with the intention prior knowledge and allowing the teacher the opportunity to scaffold student thinking , with the ultimate goal of eliciting high level thinking from the students (Stein et al., 2000). With this in mind, the task analysis guide is a tool teachers can use to plan mathematics instruction. that are focused on the implementation of cognitively demanding tasks (Boston & Smith,
25 2011; Stein et al., 1996). S everal studies have focused on the cognitive demand of mathematical tasks in a professional development setting with teachers , and include the introduction of the task analysis guide to preservice teachers (e.g., Arbaugh & Brown, 2005; Boston & Smith, 2011; Kosko et al., 2010; Leung & S ilver, 1997; Norton & Kastberg, 2011; van den Kieboom & Magiera, 2010 ). Each of these studies focused on task knowledge by choosing mathematical tasks and determining the cognitive demand. It is important for teachers to exp erience mathematics, just as they want their students to experience the mathema tics (CBMS, 2012). One way to do this is through working in a supportive environment where teachers feel comfortable asking questions about mathematics and explaining their th in king (Boston & Smith, 2011). Stein and Smith (1996) investigated the use of cognitively demanding tasks with middle school teachers through their work on the Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) project. During the QUASAR project, middle school teachers had the opportunity to learn about the cognitive demand of mathematical tasks. As a result of this project, Stein and Smith (1996) were able to develop a Mathematical Task F ramework (MTF ) . The f ramework is designed to show the stages a teacher progresses through when using tasks in mathematics instruction. For this dissertation, the treatment group focus ed on the beginning stage of the MTF . The beginning stage for the MTF is how a m athematical task i s represented in curricular/instructional materials. Researchers Arbaugh and Brown (2005) conducted a study which focused on the MTF . A group of geometry teachers set forth on a journey to examine the
26 mathematical tasks in their curriculum materials while learning about t he levels of cognitive demand. They found that the time spent on analyzing the level of cognitive demand for mathematical tasks proved to be a catalyst for change in the majority of atical instructional practices. Acco rding to Arbaugh and Brown (2005), threatening way to start teachers thinking more deeply about thei Many aspects of the study by Arbaugh and Brown will be addressed with PSTs in the researc h design of this study. Boston and Smith (2011) used a task centric approach to professional development for secondary mathematics teachers who participated in a NSF funded project called Enhancing Secondary Mathematic s Teacher Preparation. The research ers found that the teachers who participated in the project were able to set up and enact high level tasks significantly better than a control group o f teachers. One in the everyday activities of teaching, thereby providing teachers with the opportunity to learn about practice from a close examination of 967). For instance, Darling Hammond (2013) asserts that when teachers have the oppo rtunity to collaborate with other teachers, they benefit from learning and sharing with each other and can directly apply their knowledge to the cla ssroom. Another key takeaway from the study was the recognition that teachers will enter professional develo pment with varying degrees of experience using cognitively demanding tasks. It is important to provide support for teachers as they progress, at their own pace, in recognizing and implementing these cognitively challenging instructional tasks.
27 The afore mentioned studies that used the MTF (Stein, Grover, & Henningen 1996) focused on professional development with secondary teachers. Researchers , Kosko et al. (2010), Osana, LaCroix, Tucker, and Desrosiers (2006), Norton and Kastberg (2011), and van de Kieb oom and Magiera (2010) used the framework and task analysis guide while working with PSTs on developing mathematics and mathematics teaching. In all the studies, PSTs demonstrated an understanding of how to assess mathematical tasks for cognitive demand. O ne aspect that the studies conducted by Kosko et al. (2010), Norton and Kastberg (2011), and van de Kieboom and Magiera (2010) encompassed was the opportunity for PSTs to implement l thinking. For instance, the PSTs who were part of the van de Kieboom and Magiera study were provided the opportunity to explore mathematical tasks in the ir mathematics content course. The exploration consisted of learning multiple ways to represent a sol ution to a problem, to focus on student misconceptions, and to focus on important aspects of the mathematical tasks that PSTs felt were esse ntial to student understanding. The mathematics educators selected the mathematical tasks for the PSTs . The educator s wanted the PSTs to focus on the mathematics and implementing task s with students in order to understand how students were thinking about mathematics. A component of the mathematics content course included field experiences where PSTs were able to impleme nt the mathematical tasks with students and reflect upon the process. Kosko et al. (2010) and Norton and Kastberg (2011) worked with preservice secondary teachers on learning how to pose mathematical tasks through letter writing.
28 The idea of letter writ ing came from the work of Crespo (2003), who originally studied how preservice teachers posed problems to their elementary students through a letter teachers incorporate the NCT M (2000) Process Standards (problem solving, reasoning and proof, communication, connections, representation) and use these to build toward higher levels of cognitive demand when writing le tters to high school students. Prior to sending the letter, the pre service teachers were asked to predict what processes and levels of cognitive demand the mathematical task would produce from the student. The reflective papers written by t he preservice teachers throughout the duration of the letter writing exchange. improvements over the semester in their ability to engage students in different mathematical processes at a h igh level Interestingly, the preservice teachers struggled when trying to distinguish between mathematical tasks that could be classified by the two high level categories which were: (1) procedures with connections and (2) doi ng mathematics. Researchers Osana , et al. (2006) had a similar result when PSTs were asked to classify mathematical tasks by cognitive demand. Osana et al. (2006) were interested in finding out if there was a relationship between the mathematical content k nowledge of PSTs and their ability to classify mathematical tasks using the task analysis g uide. The PSTs received a 45 minute lecture on cognitive demand levels of mathematical tasks during one session of th eir mathematics methods course. Afterwards, th e preservice teachers participate d in a mathematical task sort. The researchers found that preservice teachers who obtained
29 higher scores on the mathematical content knowledge test had a higher score for correctly sorting the mathematical tasks. Addition ally, researchers found that the PSTs lacked knowledge in understanding . This result led Osana and g may hinder their ability to classify accurately mathematical tasks, particular those that are specifically designed to stimulate genuine mathematical PSTs need opportunities to see how children think and respond to mathemati cal tasks (O sana et al., 2006). Furthermore, situating the learning for PSTs within an authentic activity such as a field experience or providing an in class activity (i.e., viewing videos of students working on mathematical tasks) would further their understanding ab out the cognitive demand a particular task is capable of eliciting in students (Norton & Kastberg, 2011; Osana et al., 2006). Learning about the potential cognitive demand a mathematical task can possess is one aspect of learning how to implement mathema tical tasks. Bringing awareness of the potential cognitive demand of a mathematical task allows practicing and preservice teachers the opportunity to consider the implications for classroom instruction (Arbaugh & Brown, 2005; Norton & Kastb erg, 2011; Osan a et al., 2006). Each of the studies t ask knowledge (Chapman, 2013). PSTs need opportunities to apply their knowledge, ks, and reflect upon their experience.
30 Statement of the Problem p.335). The National Mathem atics Advisory Panel (NMAP) (2008) stresses the need for teachers to have extensive mathematical content knowledge and knowledge about how mathematical concepts connect together so mathematics is viewed as a coherent discipline . Providing students with the opportunity to learn mathematics through worthwhile mathematical tasks has been shown to increase student achievement (Boston & Smith, 2011; National Research Counc il, 2001; Stein et al., 1996). Considering the desired outcome of student achievement, pres ervice teachers need learn about learning paths and tasks designed to help students progress along learni ng More specifically, preservice teach ers need to experience mathematics that allows them to struggle and persevere in solving a problem. This experience needs to take place in a supportive environment where preservice teachers can learn together and discuss the mathematics (Chapman, 2007). Es sentially, if preservice teachers experience the process of working through a challenging problem and the feeling of accomplishment afterwards, then possibly they will encourage their students to do the same. Teacher preparation and professional developme nt are the necessary components for teachers to strengthen their content knowledge and pedagogical knowledge (Na tional Research Council, 2001). Learning to teach differently from the way one was taught is a change that does not happen instantly (H iebert, M orris, & Glass, 2003). Teacher education courses are an ideal environment for preservice
31 teachers to learn how to develop learning, how to pose mathematical tasks, analyze the nking, while receiving support from their peers and teacher educators. Many aspects of planning mathematics lessons incorporate the implementation of challenging mathematical tasks d uring mathematics instruction. Stein, Grover, and Henningsen (1996) crea ted the conceptual framework of mathematical task s to describe how they occur during classroom instruction. The framework consists of three phases: 1) mathematical tasks represented in curricular materials, 2) mathematical tasks as set up by the teacher in the classroom, and 3) mathematical tasks as implemented by students in the classroom (p. 459). There is a further dimension that features such factors the pedagogical knowledge of teachers , which contribute s to the overall cognitive demand of the task. unced by the An example of task set up is expl aining directions to students. The researchers d lly work It is important to distinguish between these two phases in the conceptual framework because this is where the secondary level factors i nfluence the mathematical task. At both the task set up and task implementation phases, the mathematical tasks are examined for cognitive demand and the processes students engage in while working on the task. For example, in the set up phase , does the task ask students to apply a m emorized procedure? In the implementation phase , what are students doing to apply the memorized procedure? The framework takes into account
32 the dynamic nature of the classroom where interactions between teacher and students and between students shape how t he mathematical task is completed. Stein et al. (1996) provide insight about how a mathematical task can appear to exhibit high cognitive demand in the beginning phase of the framework; but factors may influence the outcome at the end, which sometimes lead s to a decline in cognitive demand. The conceptual framework provides a concrete way of understanding the process of implementing a mathematical task and the different factors that can e ffect implementation. The factors in the framework that e ffect task set up and task implementation are beneficial for professional developers and teacher educators when assist ing preservice and practicing teachers in developing skills to effectively implement mathematical tasks into instruction. One aspect of teaching that the mathematical task framework does not account s and mathematics instruction. Raymond (1997) found a strong connection between mathematics teaching practices and about the nature of mathematics have the potential to perpetuate mathematics teaching that is more traditional, even when teachers hold nontraditional beliefs about Researchers Arbau gh and Brown (2005) were puzzled by a teacher who exhibited an increase in understanding about the levels of cognitive demand of mathematical tasks through a task sorting activity, but did not change his instructi onal practice in the classroom. The researc hers were only able to speculate about the development and the lack of progress in changing his instructional practices. The
33 researchers inferred the possibility that about mathematics instruction did not allow him to change his instructional practices. Did the beliefs must be recognized and challenged in order to help them change th eir instructional practice (Pajares, 1992). Other research studies (e.g., Cobb et al., 1991; Ernest, 1989) have found that le in mathematics instruction. For example, Cobb et al. (1991) wanted to initiate change in t he instructional practices of second grade rrent practice, which they identified as problematic. The goal of the study was to reorganize the knowledge and beliefs that teachers hold about learning and teaching. The rese archers found that the teachers who participated in the project held beliefs that were more comp atible to socioconstructivism. Socioconstructivism is a learning theory, which in this study describes how the teachers viewed mathematics instruction. As a res ult, the project teachers provided their students with mathematics instruction focused on allowing students opportunities to construct their own mathematical knowledge through engagement with mathematical tasks. The teachers had opportunities to do the mat hematical tasks during a summer professional development institute. assessed prior to the professional development. The researchers could not say whether the professional development had an effect that worthwhile mathematical task The Common Core Standards for Mathematical Practice (2010) want students to learn math ematics through
34 the process of doing mathematics , with an end result of developing an understanding of mathematics. NCTM (1991) outlines the need for mathematical tasks to help students develop and understand mathematics, allow them to think on their own a bout how to arrive at a solution, and use different strategies. In order to know which type of mathematical task to use during instruction, PSTs need to consider three areas recommended by NCTM: mathematical content, the make up of the students, and ways in which students learn mathematics. When PSTs choose a mathematical task, they need to cons ider the mathematics involved. One way to assess the mathematics is do the mathematical tasks with other preservice teachers and discuss the results (Boston & Smith , 2011 ). Researchers Leung and Silver (1997) noticed that preservice teachers had a difficult time grasping the idea that students could have many different solutions for a mathematical task. PSTs need to think about the task they are asking students to do . For instance, is the task simply asking a student to recall procedures or does the task position students to reason about ma thematics and apply that knowledge? The mathematical content involved in the tasks establishes its academic value (NCTM, 1991). PSTs need to consider the students as learners of mathematics when selecting a mathematical task. For instance, prior knowledge students bring can provide a foundation for building to advanced mathematical concept s. disequilibrium i s important to consider in respect to how students learn (Carter, 2008). Disequilibrium creates conflict for students when learning is taking place and creates a healthy struggle in order for students to reach a desi red academic goal. Preservice teachers c an choose tasks that will challenge students on different academic levels.
35 The act of knowing about how students learn mathematics is essential knowledge for selecting and im plementing mathematical tasks. chosen tasks allow The proposed study provide d PSTs with the opportunity to develop an understanding of the mathematical task knowledge needed for teaching mathemat ics , as well as experience with types of mathematical tasks that vary in cogniti ve demand (Chapman, 2013; Smith & Stein, 1998) . Mathematical task knowledge is the knowledge teachers need to implement mathematical tasks effectively during mathematics instru ction (Chapman, 2013). PSTs need opportunities to experience the same type of learning they want their students to experience (Ball & Cohen, 1996). Through the course of the study, preservice teachers were exposed to learning about the task analysis guide, analyzing the cognitive demand of mathematical tasks, and created tasks that were posed to elementary students through a letter writing exchange . PSTs had the opportunity to construct a mathematical task based upon a Common Core State Standard (2010) appr opriate for the content taught in the elementary classroom. The PSTs predicted the cognitive demand of the mathematical task prior to sending the letter. Upon receiving a letter back from the student pen pal, PSTs then had the opportunity to see how studen ts respond ed to their ma thematical tasks and re evaluate the level of cognitive demand for that task. Another aspec t of the study was to determine which of the seven beliefs measured by the IMAP Belief Instrument (Philipp et al., 2007) showed significant c hange as a result of learning about the levels of cognitive demand for mathematical
36 beliefs about mathematics have been directly linked to their instructional practic es (Ernest, 1989; Raymond, 1997). Preservice teachers hold beliefs about mathematics and mathematics instruction that need to be challenged (Ernest, 1998; Pajares , 1992; Philipp et al., 2007). through which teachers view mathematics and mathematic s instruction (Pajares, 1992). Teacher educators would benefit from knowing how to influence a change in PSTs beliefs by structuring experiences in a mathematics education course that could challenge beliefs. Ther e fore the study was interested in discovering whether learning about the levels of cognitive demand for mathematical tasks, participating in discussions about the complexity of mathematical tasks, and creating tasks that are embedded in letters to elementar y school students affects preservice teache mathematics instruction. Research Questions This dissertation sought to explore how a 12 beliefs about mathematics instruction . The PSTs completed two mathematic al task sorts in order to demonstrate their ability to classify mathematical tasks as having low level or high level cognitive demand. Additionally, PSTs construct ed their own mathematical t asks implement ed with third grade elementary students through a le tter writing exchange . PSTs were asked to select a level of co gnitive demand before the task wa s implemented and then to re evaluate the task for the level of cognitive demand ba sed The study also provide d PSTs with a safe enviro nment to engag e in solving mathematical tasks and participate in discussions about mathematics ,
37 in order to strengthen mathematical task knowledge. The followi ng research questions guided the dissertation study: 1. To wh at extent did the intervention of learn ing about the levels of cognitive demand for mathematical tasks and implementation of mathematical tasks through letter writing wi th third grade students impact elementary preservice teachers beliefs about mathematics and mathematics instruction? 2. How do e lementary preservice teachers' beliefs about mathematics instruction influence their implementation of mathematical tasks? 3. level or low level cognitive demands and does this cha nge after a 12 week intervention specifically focused on learning about the levels of cognitive demand and implementation of mathematical tasks in letter writing with third grade students? Structure of the Dissertation The dissertation will consist of five chapte rs and appendices. Chapter 1 consists of an introduction, theoretical and conceptual frameworks, and an introduction to the lite rature that frames this study. Chapter 2 consists of an in depth review of literature relevant to the study. Chapter 3 de scribes the formal methods employed to conduct the research. Chapter 4 consists of the research results. And finally, Chapter 5 will consist of a conclusion, implications, and possible d irections for future research. Appendices include a copy of the task r eflection form , an example of the mathematical tasks used in the task sort, quantitative instrument samples, a schedule for the intervention, interview questions, example lesson plan, PST mathem atical tasks from the letter writing, and the IRB proposal .
38 CHAPTER 2 REVIEW OF LITERATURE Vision for Mathematics Instruction The National Council of Teachers of Mathematics has a vision for school mathematics. One interpretation of this vision could resemble the following example: Imagine a classroom where a teacher and students are engaged in learning through conversations centered upon the mathematical topic of equivalent fractions. The teacher is using his/her prior knowledge engage students in learning through the implemen tation of a mathematical task. The students work collaboratively to form conjectures about why two g iven fractions are equivalent. Students us e prior knowledge and m athematical tools (e.g. , fraction circles) to deepen their understa nding of mathematical content. A teacher pos es cognitively demanding questions to students in order to elicit higher cognitive responses such as asking students to explain how they are thin king about the mathematical task. NCTM describes this type of mathematical learning in detail in documents about reform mathematics instruction (NCTM, 2000). When some people recall a past mathematics class they may think of a situation where they are s itting in a seat and staring at a chalkboard where the t eacher is writing mathematics. The teacher may interact with students by posing questions such as, hat the answer is seventy two. The students are viewed as passive learners in this type of classr oom. Over the decades, the vision of mathematics learning that NCTM (2000) proposes has been mentioned in reform documents (e.g., Curriculum and Evaluation Standards for School Mathematics, Principles and Standards for School Mathematics). Unfortunately,
39 this type of instruction has not become the reality of mathematics teaching and learning in the United States . A video analysis of middle school mathematics lessons from the United States rev ealed that teachers have not changed the way they implement mathematics instruction (Hiebert et al., 2005). The video analysis conducted by Hi e bert and others (2005) revealed that teachers in the United States repeatedly taught low level mathematical skill s , as well as procedures , without connections. When Hiebert et al. (2005) compared the instructional practices of U.S. teachers to those of teachers in other countries , using video analysis of the Trends in International Mathematics and Science Study (TIMS S) 1999, they found there was a lack of opportunity for students to discuss mathema tics. The lessons from the United States revealed that teachers reinforced low level mathematics skills such as applying pro cedures without justification. Analyzing the vide os of the United States separately from the other countries would not have r evealed the differences. However, w hen compared to the other countries, it was evident that the instructional practices of U.S. teachers do not challenge students to apply mathemat ical knowledge or make mathematical conne ctions (Hiebert et al., 2005). The mathematical tasks used by the U.S. teachers had the potential to elicit a high level of cognitive demand. However, during the period of time in which the students were presented with the task and toward the end of the lesson, the teacher would lower the level of cognitive demand by providing answers or telling the students how to think about the mathematics. The researchers recommended continual support and professional developmen t for teachers that focused o Perhaps a suggestion for improving the instruction al methods of teachers would be to
40 focus . Additional research could concentrat e on learning about the potential level of cognitive demand of mathematical tasks . This chapter of the dissertation will examine the relevant literature on hematical thinking. Mathematical T asks The research studies of Doyle (1988), Hiebert and Wearne (1993), Cobb et al. (1991), Silver and Stein (1996), Stein and Henningsen (1997), and Stein et al. (2008) , lay the foundation for learning about the cognitive d emand of a mathematical task and how the components associated with using mathematical tasks in instruction a ffect student learning. Doyle (1988) categorized academi c tasks as familiar and novel. Cobb et al. (1991) provided evidence that a certain type of instruction with mathematical tasks supports problem based learning. Hiebert and Wearne (1993) examined the impact mathematical tasks and discourse had on student learning when teachers used reform based curriculum. The work of Silver and Stein (1996) and Stein and Henningsen (1997) led to a conceptual framework for mathematical tasks. Stein et al. (2008) built upon the conceptual framework for mathematical tasks by developing five practices teachers can use to facilitate learnin g by using mathematical task s. The contributions of these researchers have encouraged other researchers to delve deeper into the factors that contribute to the application of mathematical tasks. Currently, with the implementation of the Common Core State Standards (NGA & CCSSO, 2010) , mathematics education is undergoing a change . This is a challenging period in
41 mathematics education and it is important to study how the use of mathematical tasks can lead to changes for improving teacher effectiveness and student outcomes. The E arlier Y ears of M athematical T asks The first serious discussions about how the implementation of mathematical tasks affected the learning of students started around the time NCTM was working on developing a vision for how mathematics should be taught (Ball, 1991 ). Early research studies focused on how mathematical tasks were used in the classroom and the environmental factors that e ffected the implementation of these tasks (Doyle, 1988; Marx & Walsh, 1988 ). An example of one of the studies is from the researcher Doyle (1988), who examined 450 junior high academic tasks across the academic areas of Engl ish, science, and mathematics. Doyle argued that knowledge is constructed based on the academic tasks presented t o students during instruction. Doyle found that tea chers play a critical role in th e success of an academic task. Teachers affect how the students learn from the task and the work students produce from the task. l state or end product to be achieved, (b) a problem space or set of conditions and resources available to accomplish the task, (c) the operations involved in assembling and using resources to reach the goal state or generate the product, and (d) the impor tance of the task in the overall work The observational data from this study revealed that tasks appear differently at diff erent stages in the classroom. For instance, the task presented by the teacher may appear to elicit a h igh cognitive demand from the students; however, the students may interpret the task differently from the way the teacher intended and the task only elicits low cognitive demand because the student is using recall of a memorized procedure .
42 Based on Doyle two categories: fami liar and novel. Familiar tasks focused on the memoriz ation of facts and procedures. When students are engaged in familiar academic work, they may need to know complex know ledge; however, the outcome of the academic work is predictable with little opportunity for stud ents to extend their thinking. Novel tasks elicited a higher level of cognitive demand, in which students have opportunities to explore their thinking. There is not a set procedure or rule that will l ead the student to the answer. The type of task presented to students affects the l evel of work in the classroom. For instance, Doyle found that familiar tasks were implemented seamlessly with little disruption and distraction because students were applying procedures or recalling informati on. Novel tasks , on the other hand, tended to take more time and left room for unpredictability in student performance , which means teachers need to be able to address the stud thinking about the task and even model this thinking for students. Doyle found that math teachers tend to use familiar tasks that focus on computation and memorization of procedures. Additionally, there was no need for students to think deeply about mathematics when this occurred. Doyle hypothesized that curriculum materials might be a contributing factor explaining why familiar tasks were more pre valent in mathematics classes. Closer analysis of the mathematics curriculum revealed that the focus was on skills rather than concepts and problem is relevant to how students acquire mathematical knowledge. If teachers choose to implement tasks that focus on lower level cognitive skills , then students experience the
43 The results from the analysis of the 450 academic tasks indicate that the implementation of tasks is e ffected by curriculum, social order, c lassroom management, instructio n, and learning (Doyle, 1988). Teachers use familiar work to make the class manageable and productive. Familiar tasks can lead to student misconceptions because of the lack of opportunity for them to explore, an alyze, and dis cuss mathematics. Students working on a familiar task may just follow the procedure, but lack an understanding about how the mathematics is ac tually applied to the problem. Hence, Doyle (1988) is claiming that the tasks used during instruction relate direc tly to what students learn . If learning is designed to improve student outcomes, then further research needs to be done on effectively implementing mathematical tasks to support the student learning desired by reform policy documents. While Doyle worked on identifying mathematical tasks, Marx and Walsh (1988) were interested in the supporting classroom structure that assisted students in higher order thinking through the implementation of academic tasks. According to Marx and Walsh, there are three elemen are set, the cognitive plans students use to accomplish tasks, and the products students create as a result of their task The teacher was the key to students successfully com pleting an academic task th at matched the intended goals. For instance, when the teacher set clear goals for the academic tasks, the students were able to engage in the task a t the intended academic level. On the other hand, when clear goals were not made explicit to the students , negative effects on classrooms in addition to the confusion they create in the mind of
44 Marx and Walsh (1988) stress the importance of having teachers focus on stude memory, procedure, comprehension, or opinion needed to complete an academic task. In essence, the cognitive plans are tasks int o four categories (e.g., memorization tasks, procedures without connections tasks, procedures with connections tasks, and doing mathematics tasks) based upon their cognitive demand. The awareness of the level of cognitive demand an academic task has is an important piece of effective in struction (Marx & Walsh, 1988). In order for teachers to have this awareness, they need to have time to work on the academic task to know the steps the student wil l take in completing the task. Marx and Walsh argue that kind of cognitive engagement students utilize is of utmost importance rather than instructional time or time on t This statement further illustrates the need for teacher education that is focused on doing the mathematics and learning a bout where students may struggle with mathematics. Next, the study conducted by Hiebert and Wearne (1993) models the principles that researchers Doyle (1988) and Marx and Walsh (1988) argue need to occur in order for mathematical tasks to have an effect on student achievement. Researchers Hiebert and Wearne (1993) were interested in how student learning was supported through the implementation of mathematical tasks and classroom discourse. The researchers interpreted the interactions between mathematical ta sks and classroom discourse in second grade classrooms by observing the classrooms through the social cognitive and social constructivist perspectives. Hiebert and Wearne
45 d iscourse encourage product For instance, classrooms utilizing mathematical problems that required students to spend more time on the problem and had more than one way to represent the answer produced higher student performance than classrooms using mathematical problems that were focused on applying a memorized procedur e. Additionally, classrooms where the teachers asked more thought provoking questions to students , as opposed to teachers who asked questions that focused on recall of facts, created more opportunities for student achievement. In order to test the effect different instructional approaches had on students, the researchers observed two classrooms that implemented a different way of instruction that contrasted with the traditional textbook approach. The classrooms were chosen specifically by the researchers to demonstrate the e ffect teacher questioning had on student outcomes. The researchers compared observations of the two classrooms with four other classrooms that focused on proced ures and recall of mathematics. Overall, for all six classrooms the kind of questions that occurred the most were ones that focused on recall of f acts (Hiebert & Wearne, 1993). This kind of questi oning which focuses on recall of facts is prevalent during mathematics instruction across the United States (Hiebert et al., 2005). In conclusion, Hiebert and Wearne (1993) argue there are more factors that influence the learning that takes place when impl ementing mathematical tasks and classroom discour se. The findings from the ir research study were not able to pinpoint the factors or reasons why the classrooms with higher achieving students were more influenced by the type of task and exe mplified producti ve discourse. The researchers
46 were able to conclude that instructional tasks and discourse are related to the practice of teaching and le arning. Based upon student performance on the assessments, the classes that focused more on procedural questions were o ut performed by classes that spent more time on problems and used multiple representations. The conclusions made by Hiebert and Wearne indicate a need for further investigation into the factors that influence the implementation of academic tasks, the effec t curriculum materials have on teaching, and the questioning practices of teachers. Teacher Education and Mathematical Tasks strengthening teacher preparation, early career mentori ng and support, and ongoing professional development for teachers of mathematics at every level, with special Thus, in November of 2010 the National Council for Accreditation of Teacher Education Partnerships for Improved Student L A panel consisting of state officials, P 12 and higher education leaders, teachers, teacher educators, union r epresentatives, (NCATE, 2010, p. ii). The panel made it clear that a main goal was to strengthen teacher preparation by the needs of schools (NCATE, 2010, p. ii). The NCATE (2010) panel recommended that teacher candidates be prepared to ensure that all children mast er rigorous course content, be able to apply what they and
47 workforce Standards (NGA & CCSSO, 2010) f or mathematics initiative. For example, Achieve and the U.S. Education Delivery Institute (2012) released recommendations for state and district leaders to prepare for full implementation of the CCSS . In contrast to individual state standards, the CCSS wil l bring an increase in the cognitive demand required by students (Achieve & U.S. Education Delivery Institute, 2012) . Furthermore, the CCSS assessments (i.e., Smarter Balanced and Partnership for Assessment of Readiness for College and Careers), which are still in the developmental stages, have released assessment items showing the rigorous cognitive demands students will need to apply to complex mathematical tasks. Preservice teachers need to be prepared to meet these same challenges that their future stud ents will face on assessments. Specifically, NCATE (2010) recommends that preservice teacher educa tion focus on student learning. For instance, having preservice teachers focus on student learning will prepare them to incorporate instructional strategies knowledge to foster the growth of mathe matical knowledge in students. Developing are innovators and problem solvers, working with colleagues constantly seeking new and different ways of teaching students 2010, p.5). Next, several studies are discussed that describe how researchers contribute to teacher learning by focusing on mathematical tasks. Arbaug h and Brown (2005) sought to embed the professional learning of a group of geometry teachers into their everyday practice by focusing on the mathematica l tasks presented to students. They used a study group as a form of professional development
48 for the tea chers. The professional development was focused on learning about the cognitive demand of mathematical tasks. Arbaugh and Brown (2005) used a task sorting interview to measure the growth of levels of cognitive demand for teachers of mathematical tasks. Pri or to the start and at the end of the professional development, teachers were asked to sort a group of high school mathemati cal tasks at two different times. The categories created by the teachers were then compared using qualitative methods to draw conclu sions about their understanding of t he levels of cognitive demand. The results from this study reveal how teachers learned to use the mathematical task framework and became more cognizant of the tasks they chose to i mplement in their instruction. Additiona determine whether the level of cognitive demand for mathematical tasks increased because of the professional development. Overall, the levels of cognitive demand of the mathematical tasks used in instruction did n ot change when considering the teach ers as a group. Individually, some of the teachers showed an increase in the level of cognitive demand of the mathematic al tasks in their lesson plans. Researchers Arbaugh and Brown (2005) argue that the professional dev elopment focused on mathematical task implementation content knowledge by influencing the type of mathematical tasks the teachers chose to use during mathematics instruction. Another aspect of t his study that proved successful was providing teachers with the opportunity to collaborate with peers and discuss features of mathematical tasks , which produced low level and high level cognitive demand .
49 Teachers need professional oppor tunities to work with other teachers and discuss their work (Ba ll, 1993). When teachers have the opportunity to do mathematical tasks and then discuss the mathematical tasks with peers in a nonthreatening professional development setting, research has indi cated that their content and pedagogical knowledge are positively impacted (Chamberlin, 2005; Cobb et al., 1991; Kazemi & Franke, 2004; Prestage & Parks, 2007; Remillard & Bryans, 2004; Steele, 2005). Teachers need to understand where the students may stru ggle and where misconceptions can develop during the implementation of the mathematical task (CB MS, 2012; Stein et al., 2008). In order to understand how students may struggle with mathematics, teachers need to experience the mathematics by actually doing the mathematics (Ball & Cohen, 1999; CBMS, 2012). The purpose of having teachers work on a challenging mathematical task and experience the enjoyment of completing the task as a group is a feeling that teacher educators hope that teachers want their own s tudents to feel (CBMS, 2012). Working with other teachers in an educational setting is an ideal environment for teachers to benefit from seeing and hearing how others solve mathematical tasks (Chamberlin, 2005; Kazemi & Franke, 2004). The aforementioned op portunities rarely occur in a teacher preparation program (Ball, Thames & Phelps, 2008). In fact, most of the preparation for teachers to teach mathematics occurs in content courses (Ball et al., 2008). Ball et al. (2008) would argue that teachers need m ore opportunities to develop mathematical knowledge for teaching. Recently, researchers Ball et al. (2008) theorized about the type of knowledge teachers need in order to teach mathematics. Ball et al.
50 gical content knowledge by including the domains of knowledge of content and teaching (KCT) and knowledge of content and students (KCS) within the pedagogical content knowledge needed for teachers of mathematics. The knowledge of content and teaching is kn owing how to design effective mathematical instruction and the ability to facilitate learning by deciding (Ball et al., 2008). Knowledge of content and students encompasses the knowledge needed by the teacher to know where students will struggle in the process of learning mathematics, how to motivate students to learn mathematics by choosing interesting ways to present topics, and the ability to know if a mathematical task w ill be easy or hard (Ball et al., 2008). Both domains affect the instructional decisions teachers make when selecting mathematical tasks to use during classroom instruction. According to Ball et al. (2008) teachers need to know mathematics in ways usef ul for, among other things, making mathematical sense of student work and choosing powerful ways of representing the recommended knowledge needed for mathematics instruction , teacher preparation programs need to look beyond stand alone mathematical content courses and focus on experiences where preservice teachers have opportunities to develop knowledge of content and students and knowledge of content and teaching. The Mat hematical Education of Teachers (MET) II (2012) document calls for elementary teachers to experience learning similar to that by which students need to learn mathematics, (i.e., through the 8 Commo n Core Mathematical Practices). Many
51 preservice elementary teachers have the belief that mathematics is taught by following a set of rules and procedures (Ball, 1990; Kirtman, 2008; Ma, Millman, and Wells, 2008; Phillip p et al., 2007). These beliefs come from their own experiences in mathematics courses. The role of the teacher educator is to guide and facilitate the learning of preservice teachers (Ball & Cohen, 1999). According to NCTM (1991), preservice and revise their assumpti ons about the nature of mathematics, how it should be taught, activities . For example, researchers Cobb et al. (1991) used mathematical problem solving with elementary teach ers during a 1 week summer institute focused on student learning that was problematic. In the yearlong study conducted by Cobb et al. (1991), the socioconstructivist theory of knowledge was used to inform the instruction of second grade teachers, who were focused on implemen ting problem centered learning. This study incorporated reform based mathematics instruction into experimental classrooms and compared student achievement results with the controlled classrooms. The teachers who implemented reform based instruction received opportunities in the summer institute to work on mathematics problems that they implemented in their clas srooms during the school year. A benefit of having the opportunity to work on the mathematics problems with colleagues was that t hey were able to talk about possible trouble areas students may experi ence when solving the problem. The resea rchers argued that professional development contributed to the success of students who had teachers who implemented the mathematica l problems duri ng instruction. This is significant because it
52 shows that if teachers have the opportunity to experience mathematics as learners in an environment that promotes problem solving, discussion, and experimentation, then they may be more apt to foster the same type of environment with their own students. Additionally, findings from this study support the use of teacher and student discourse throughout instructio n. The interactions between teacher and students and peer student interactions supported learning ma thematics conceptually rather than procedurally, as m easured by a mathematics test. For example, the researchers explained teacher and student discourse as teacher questioning and student explanations that concent rated on mathematical problems. Another key to success for the teachers who participated in the summer institute was continual suppo rt from project staff. The staff met with the teachers throughout the year, and visits consisted of solving methods centered on topics that were relevant to 2 nd grade . This type of support is recommended by the MET II (2012) for quality professional development that is directly related to the instructional practices of teachers. A reoccurring theme for the reform of teacher educati on is the need for teachers p.3). An example of a professional development project that focused on the aforementioned opportunities is the Quantitative Understanding: Amplifying Student Achievement and Reasoning (Q UASAR) project (Silver & Stein, 1996). The premise of the project was for a mathematics educator to provide ongoing support to teachers as they made changes in their mathematics instruction by posing cognitively demanding
53 m athematical tasks to students. Students had opportunities to take an active role in their learning by engaging in mathematics through mathematics instructio n focused on cognitively demanding tasks. For instance, the researchers provided a snapshot of a three part lesson that them how to convert fractions to decimals. In this snapshot , the researchers emphasized classroom norms established by both the teachers and students, the questioning teacher s use d while students work ed collaboratively, and the explanations students share d with the class to show how they though t about the mathemat ical task. Silver and Stein (1996) reported that success was made possible for students when teachers had the instruction (p.513). Mathematical tasks from the QUASAR project we re analyzed for the cognitive demand elicited throughout m athematics instruction. In the set up phase of the mathematical task , cognitive demand was high. During the implementation phase, the cognitive demand fell. Some of the reasons for the decline in co gnitive demand were attributed to the teacher telling the students what to do instead of asking questions to the students in order to engage them in thinking deeply about mathematics, not ime to complete the task (Stein et al., 1996). On the other hand, the data collected and analyzed from the QUASAR project led the researchers to discover how to maintain student engagement throughout the implemen tation of a mathematical task. The top fiv e ways to maintain student
54 amount of time allotted for the mathematical task, the modeling of high level performance, and sustained encouragement from the teacher for ex planations and meaning from the students (Henningsen & Stein, 1997). These methods can help teacher educators model how to maintain student engagement within preservice teacher education courses. Next, Boston and Smith (2011) used a task centric approac h to professional development. Such professional development was part of a project titled Enhancing Second ary Teacher Preparation (ESP). Two years after the project ended, the researchers followed up with the participants to investigate whether or not the teachers were still using cognitively challenging tasks d uring mathematics instruction. Data sources for this study consisted of observations, instructional tasks submitted by teachers, and artifacts from the professional development. An important componen t to this professional development was having the participants complete a pre and post middle school mathematical task sort. Results from the task sort indicated that participants were able to recognize features of low level and high level mathematical tasks after participating in the professional development. The researchers concluded implement cognitively challenging instructional tasks, and a subset of these teachers were shown to have sustained the improvements more than a year after the project (Boston & Smith, 2011, p.970). Results from this study provide rationale for including more professional experiences for teachers that involve learning about the
55 cognit ive demand of mathematical tasks and examining such tasks used in their everyday instruction . Researchers Norton and Kastberg (2011) wanted to provide preservice teachers with opportunities to learn about the cognitive demand of mathematical tasks and t o practice implementing mathematical tasks with students. One of the challenges of preservice teacher education, noted by the researchers, is the fact that preservice teachers do not have access to their own group of students. Teacher educators must seek o ut opportunities that are meaningful and impactful to the learning and beliefs of preservice teachers. Researchers have used the mathematical task framework and task analysis guide with preservice teachers, but they focused on individual par ts instead of t he whole piece. For instance, Norton and Kastberg (2011) and Kosko et al. (2010) introduced preservice secondary teachers to mathematical tasks through letter writi ng. The letter writing exchange was used as an authentic activity to engage preservice teach ers in focusing on the level of cognitive demand of mathematical tasks and st As a result of this letter writing exchange , the engage student s in different mathematical processes at a high level of cognitive demand opportunity to learn how to construct mathematical tasks for students and the time to analyze how a stu dent thinks about mathematics (Crespo, 2003; Norton & Kastberg, 2011). Letter writing, focused on mathematical learning, was first used in a research study conducted by Crespo in 2003. Crespo (2003) found that preservice teachers, who
56 took part in a letter writing exchange with elementary school students, were provided with the opportunity to be the posers of problems rather than the problem solver. This opportunity allowed preservice teachers time to reflect upon the problems they asked of their elementary school pen pals . Later, w hen the responses from the pen pals came , ir letters and construct questions that would elicit more insight into their mathematical thinking. Initially, the preservice teachers were assigned a fourth grade elementary student. for preservice teachers to experiment with posing mathematical problems to students and for preservice teachers to learn which problems provided students with opportunities to justify their mathematical thinking. Thirteen preservice teachers volunteered to participate in the study. The data collected in the study consisted of the letters sent between preservice teachers and elementary students, weekly journals from the preservice teachers, and the final case report turned in at the end of the semester. The preservice teachers used se lf generated, class and course textbook problems for their letters. Crespo (2003) identified three approaches used by the preservice teachers when they began writing: 1) make problems easy to solve; 2) pose familiar problems; 3) pose problems blindly. F urther, Crespo (2003) found that the initial problems posed to the elementary students were comprised of arithmetic operations and only allowed for one answer, which is an example of low cognitive demand. The preservice teachers wanted to simplify the prob lems for the students by underlining and bolding key words or by
57 providing hints. Crespo speculated that the choice of problems included in the letters might have been made based upon the preservice teachers own mathematical ability or lack thereof. For in stance, when the problems were chosen blindly, students were more challenged by them because the problems lacked the hints seen in those problems where preservice teachers already knew the answer. Problems chosen blindly also resulted in preservice teacher s realizing that elementary students had the mathematical ability to work on challenging tasks. Over time, the preservice teachers changed the types of problems posed to elementary students. They began to ask more open ended problems that allowed for more exploration and justification. Crespo found three themes in the changes of the types of problems: 1) trying unfamiliar problems; 2) posing problems that challenged the p osing problems undertaken by the preservice teachers did not happen in isolation. In fact, preservice teachers reported that the teacher educator had an instrumental role in the changes associated with problem posing. The problems the preservice teachers e ngaged in during their teacher education class helped to expand their thinking about different types of mathematical problems. In essence, the experience of letter writing to elementary students while enrolled in the elementary mathematics methods course l ed to changes in the beliefs preservice teachers held about mathematical problems (Crespo, 2003). that k nowing mathematics for oneself may not be a reliable predictor of good problem
58 Ball et al. (2008) would agree with Crespo and suggest that preservice teachers need more experiences, which allow them to develop pedagogical content knowledge needed to become problem posers. Furthermore, Crespo (2003) claims that a critical mathematical thinking and ways of solving problems. In conclusion, Crespo suggest s the need for further research on how preservice teachers learn to pose and develop problems for students. Preservice teachers need authentic experiences where they can learn how to pose problems and reflect upon the experience to further their own knowl edge of teaching mathematics. Building upon the work of Crespo (2003), researchers Norton and Kastberg (2011) used case studies to describe the experiences of two preservice teachers, who took part i n a letter writing exchange. The case studies revealed ho w the preservice teachers struggled with how much assistance to provide the students with when trying to encourage them to res pond to the mathematical task. One preservice teacher focused on the solution he knew would give the correct answer and did not re spond to the st Instead, the preservice teacher tried to redirect the s tudent to follow his thinking. This incident reveals that preservice teachers may need more time and experience with solving mathematical tasks and learnin g there is more than one method to solve tasks. I n turn, van den Kieboom and Magiera (2010) had elementary preservice teachers work on mathematical tasks and share solutions in their methods course before implementing the mathematical tasks with elementa ry students. In the study, preservice teachers were enrolled in a mathematics content course, which first
59 introduced mathematical content and pedagogy for selecte d mathematical tasks involving fractions. The in course activities consisted of preservice tea chers learning from each other by showing multiple ways to represent the solution to a mathematical task, focusing on where students might develop misconceptions, and highlighting important math ematical features of the task. Next, preservice teachers had t he opportunity to implement the same mathematical tas ks with students. An important component to the study was the reflections the preservice teachers wrote after they evaluated their Ma giera (2010), facilitated a learning environment where the focus was on learning about mathematical tasks. Furthermore, van den Kieboom and Magiera (2010) describe the process of developing content knowledge for teaching as a cycle where the content course serves as a place for preservice teachers to learn together by doing the mathematics and reflecting on their experience implementing the same mat hematical tasks with students . The research study focuses on the first step in the Mathematical Task Framework , which is to identify mathematical tasks and work out solutions to the se tasks (Stein et al., 2000). The research study conducted by van den Kieboom and Magiera (2010) furthe r illustrate the beneficial practice of having preservice teachers learn how to implement mathematical tasks in a supportive environment where they can ask questions and examine and reflect upon their practice. First , Osana et al. (2006) investigated how e mathematical knowledge affected their ability to classify mathemati cal tasks by cognitive
60 demand. A teacher educator, who was familiar with the task analysis guide, instructed 26 preservice teachers on how to identify the cog nitive demand of mathematical tasks. The researchers using the criteria set forth in the task analysis guide created the mathematical tasks. Additionally, the researchers were interested in knowing if the length of a task e ffected the classification of th e task into a low level or high level cogni tive demand category. Results from the task sorting activity indicate that preservice teachers were able to classify L evel 1 tasks (memorization) with 74% accuracy and L evel 2 (procedures without connections) with 56% accuracy (Osana et al., 2006). As the level of cognitive demand for mathematical tasks increased, the ability to classify a task accurately as L evel 3 (procedures with connections) and L evel 4 (doing mathematics) decreased. This result directed Osan accurately mathematical tasks, particularly those that are specifically designed to stimulate genuine ma thema The researchers suggest that the next direction for future research should provide preservice teachers with the opportunity to work with child ren and mathematical tasks. The proposed opportunity would allow preservice teachers to learn about the level of cognitive demand required by the student to effectively work on the mathematical task. Furthermore, situating the preservice teachers in a field experience where they can learn about the cognitive demand a particular mathematical t ask is capable of eliciting in students is an authentic activity for developing knowledge for teaching mathematics (Borko & Putnam, 2000).
61 In essence, preservice teachers need opportunities where they can learn about mathematical content knowledge and pe dagogical content knowledge at the same time (Lampert & Ball, 1998) , which contributes to mathematical task knowledge ( Chapman , 2013) and mathematical knowledge for teaching (Ball et al., 2008). Examining mathematical tasks provides an opportunity to learn about the potential cognitive demand a task can hold f or potential student learning. Learning about the potential cognitive demand a mathematical task can possess is one aspect of learning how to implement such mathematical tasks. The act of bringing awar eness to the potential cognitive demand of a mathematical task allows practicing and preservice teachers the opportunity to consider the possible implications for classroom instruction (Arbaugh & Brown, 2005; Boston & Smith, 2011; Kosko et al., 2010; Norto n & Kastberg, 2011; Osana et al., 2006). Reform mathematics instruction posits the need for stude nts to have opportunities to extend their thinking and apply their knowledge; meanwhile they are constructing their own understanding of mathematics (NGA & CCS SO , 2010; NRC, 2001). These learning opportunities for students only come from well thought out mathematical tasks from the teacher which extend beyond applying low level mathematical procedures. In summary, the research studies presented in this section and the research known about effective professional development and implementing mathematical tasks with preservice teachers are the guiding force behind the design of the elementary mathematics methods course for the dissertation study. Learning about mat hematical tasks is related to the Standards for Mathematical Practice (NGA & CCSSO, 2010) . For instance , the Standards for Mathematical Practices describe how students should learn
62 mathematics and also relates how teachers need to experience mathematics in order to create instructional opportunities for their students to utilize the standards to learn mathem atics (CBMS, 2012). The elementary mathematics methods course was designed to provide PSTs with an opportunity to engage in solving mathematical tasks t ogether as a group, discuss solutions, consider the potential level of cognitive demand, and provide an opportunity to create their own mathematical tasks for the letter writing exchange. Using K nowledge of S M athematical T hinking When teachers are selecting mathematical tasks to use during mathematical instruction, it is important to consider the learning needs of the students (NC TM, 1991). student in learning (Brans ford et al., 1999). A mathematical task alone will not the Engle, Smith, and Hughes (2008) r ecommend five practices that teachers could use to orchestrate discuss ions about mathematical tasks. The practices are: 1) anticipating responses to the tasks during the explo r ation phase, 3) selecting particular students to present their mathematical responses during the discussion and summariz ing phase, 4) purposefully sequencing the student responses that would be displayed, and 5) helping the class make mathematical connect nses and the key ideas (p.12). The knowledge the teacher a crucial piece to the aforementioned five practices.
63 According to the Mathe matical Task Framework (Stein et al., 1996) one of the knowledge students will have about the ta thinking to engage them in the task, and where they might struggle when working the task. , they are able to use that knowledge to form worthwhile mathematical task s (Chamberlin, 2005). For example, Franke et al. (2009) studied questions asked by elementary teachers to students during an algebraic reasoning lesson to determine whether the questions asked allowed students to elaborate on their mathematical thinking. The findings from this study suggest that there is no guarantee that by asking a question a student will reve al their mathematical thinking. Some types of questions produced elaborations from the students about their mathematical thinking, but the re sult was not always consistent. further develop a mathematical concept. More research is needed to learn how to help teachers decide which questions to ask in order to further understand students mathematical thinking and which type of questions influence student participation and learning (Franke et al., 2009). Even & Tirosh (2008) stress the importance of providing teachers with the thi nking in teacher education. The Cognitive Guided Instruction (CGI) study (Carpenter, Fennema & Franke, 1996) specific research
64 2008, p.215). The CGI model emphasizes what students can do, not what they cannot do. One of the major facets of the study was positioning teachers to recognize the informal knowledge mathematics students already possess and how to use this knowledge in the development of formal mathemat ics (Carptenter et al., 1996). A in identifying problems their students can solve and the strategies that they use to solve them t han non Based on the significant results from the CGI study, the same principles have been used in both preservice and inservice teacher education. Cady, Meier, and Lubinski (2006) used the same principles as the CGI study (1996) along with NCTM (1991) practices in a study involving pract icing and preservice teachers. The study followed the participants over multiple years and gathered data at pre determined points to assess the beliefs of the participants, who wer e initially preservice teachers and then became pra cticing teachers. Interestingly, this data indicated that the experiences the preservice teachers had as participants in a project ical views l., 2006, p.296). When these preservice teachers transitioned to novice teachers, researchers were disappointed to report that d with al., 2006, p.296). In fact, the results the researchers were hoping to see initially were not seen until six years later, when the preservice teach ers became practicing teachers. Results from this study ind icate that changes in instructional practices of teachers takes time to evolve into the desired
65 reform based instructional practices. Once the novice teachers became comfortable in their role as teacher , they were able to slowly implement problem solving t asks in their rk as reformers imagine, but such knowledge does not offer clear guidance, for teaching of In essence, PSTs need to see how students th ink and mathematics and mathematics instruction (Chapman, 2013). There are several pieces of a hypot hetical puzzle, which need to assimilate in order for teachers to teach current reform mathematics. Teacher Beliefs and Teacher Change attempting to understand mathematics t eac Beliefs about teaching and learning are important to determine how a teacher can implement reform mathematics instruction ( Sztajn, 2003). For instance, a teacher may only decide to implement the mathematical content in reform m athematics standards and not use the recommended practices. Raymond ( mathematics formulated from experiences in mathematics, including beliefs about the nature of mathematics, learning mathematics, and tea the way these beliefs wer e formed (Pajares, 1992).
66 Preservice teachers have long formed their beliefs held unde rstandings, premises, or propositions about the world that are thought to be about mathematics instruction, even before they decided to become teachers (Ball, 1990; CBMS, 2012; Chapman, 2007; Ebby, 2000; Raymond , 1997; Szydli k et al., 2003). through which one looks when int Through the preservice 12 schooling experiences, they have formed their beliefs about mathematic s instruction. During that time, they experienced mathematics for the most part differently than the method by which their preservice teacher education courses and policy documents (e.g., CCSS, NCTM Principles and Standards) envi sioned mathematics would be taught. In fact, Ebby (2000) argues that preservice teacher education is a weak intervention for trying to change Preservice teachers may experience a slight belief change during their teacher preparation, but once they becom e teachers , they tend to adapt to the culture at the school (Cady et al., 2006; Manouchehri, 1997). How do the deeply held beliefs of elementary preservice teachers change? ore experience mathematics instruction differently from the way they were taught and to provide an opportunity to try mathematics instruction with students (Crespo, 2003). Sowde r (2007) suggests having teachers attend to students thinking as a driving force for elementary teachers to want to change their practice and learn more mathematics.
67 Several studies are discussed to show how researchers structured opportunities to challeng e the PSTs beliefs about mathematics and mathematics instruction. First, Raymond (1997) examined the beliefs of elementary teachers through case study methodology e ffected their instructional practice. She found that belie fs and practice had a reciprocal relationship. Additionally, nd practice . the actual instruction Raym ond observed in the classroom. Raymond explain ed that the cause of the mismatch between beliefs and practice is caused by the constraints from the classroom environment. Teacher educators planning a professional development should bring aw areness to the relationship between beliefs and practice through reflective discussions with the teachers (Borko, 2004). When teachers have the opportunit y to view an instructional practice different than their own, they then need to reflect upon the experience. The teacher educator in how t hey interpret the instruction. Philipp (2007) suggests that teachers need to change their instructional practices . T hen when they see results in their students, their beliefs will change. Although this may be true, Pajares (1992) claims changes in struction are influenced by results, which indicate improv ements in student achievement. A more complex description of belief change is advanced incorporated into existing beliefs in the ecology; accommodation takes place when new information is such that it cannot be assimilated and existing beliefs mu st be replaced or
68 reorganized. Both result in belief change, but accommodation requires a more radical e, teacher educators need to structure opportunities that For inst ance, researchers Vacc and Bright (1999), Szydlik, Szydlik, and Benson (2003) , and Philipp et al. ( 2007) studied the belief changes among elementary preservice teachers who participated in specific interventions in thei r teacher preparation courses. four occasions through out their teacher p reparation. The researchers found the greatest change in beliefs occu r r ed during the elementary mathematics methods course. In the course, the teacher educator implemented CGI instruction for five weeks with preservice teachers . According to Vacc and Brigh and solution strategies as a guide for planning instruction, listening to how children preservice teachers in this study with the reinforcement needed to s (p.108). Carefully structured learning opportunities can provide occasions for preservice teachers to explore their pre existing beliefs (Ernest, 1989). Subsequently, researchers Szydlik and colleagues (2003) focused on the classroom environment where preservice teachers negotiated social norms while learning mathem atics through problem solving. The researchers situated the learning of preservice teachers within a community of learners that focu sed on di scussing mathematics. During the mathematics content course, the teacher educator served as a facilitator for learning and did not provide assistance to the preservice teachers as they worked together t o solve mathematical problems. The instrument used to collect data
69 , as researchers found conflicting reflective journal responses. Therefore, the researchers were left to draw conclusions ab out the st udy from the qualitative data. The preservice teachers expressed a feeling of mathematics confidence as a result of the experience of working together to sol ve problems. Additionally, several preservice teachers expressed the opinion that they we re able to see mathematics as a sense making discipline . An important aspect of this study was the supportive environment that was formed by the teacher educa tor, which allow ed preservice teachers to participate in doing mathematics through an authentic ex perience. Researchers Philipp et al. (2007) were also in search of an authentic experience instruction. The researchers argued that preservice teachers hold the belief I, as a college student, do not know something, then children would not be expected to know p et al., 2007, p.439). In order to challenge the aforementioned belief, teacher ed ucators need belief. For this study, preservice teachers were randomly assigned to either one of the four treatment groups or the control group. There were two treatm ent groups focused on students participating in problem solving. The one difference between the groups was that one group had the opportunity to engage children in proble m solving. The two other treatment groups observed and reflected on their visits to elementary schools. For this
70 treatment, one group of preservice teachers was assigned to teachers who were identified as reform oriented. The other group of preservice teac hers was assigned to teachers who were at schools that were convenient to the university. Philipp et al. (2007) were interested in developing a belief instrument that would and mathematics instruction. Traditional belief inventories that utilized a Likert scale could ). In order to capture the beliefs of preservice teachers, the belief instrument was con structed to collect authentic responses to the following tasks in various domains whole number, two [ are ] in the domain of fractions, and one [ is ] a general teaching (Philipp et al., 2007, p.451). The belief instrument measures sev en beliefs about mathematics and mathematics instruction. Rubrics were developed to analyze the responses from the preservice teachers. Results from the belief instrument data developed more sophisticated beliefs about mathematics and mathematics were assigned to conveniently located classrooms had the highest percentage of part icipants with no increase in beliefs out of all the group. These results emphasize the importance of structuring a field experience that intentionally focuses on reform t teacher educators need to find experiences for them that will challenge their beliefs. The
71 act of placing preservice teachers haphazardly in convenient classrooms was shown t o dramatically e ffect their beliefs about mathematics and mathematics instruction. According to the above literature there is a need to understand the beliefs preservice teachers bring with them to their mathematics education courses. These beliefs serve as a lens through which they view mathematics instruc tion (Philipp, 2007). Teacher educators need to provide preservice teachers with opportuniti es to challenge their beliefs. This study explored how learning about the level of cognitive demand and develop ing mathematical tasks for elementary stud ents had an impact on preservice teachers beliefs about mathematics and mathematics instruction. Preservice teachers need an environment where they can practice constructing mathematical tasks and evaluate the eff ectiveness of such tasks . In addition , preservice teachers need experiences interacting with children to learn how students think about mathematics and respond to mathematical tasks. Changing preservice teachers beliefs about mathematics is a challenging task , but providing them with experiences that might impact their belief s has proved to be successful. This study builds upon research about cognitively demanding tasks and changing the beliefs of preservice teachers.
72 CHAPTER 3 METHODS Overview The purpo se of this study was to determine the extent to which elementary a result of participating in a 12 week intervention focused on learning about the level of cognitive demand of mathematical tasks and writing letters to third grade elementary students. A necessary component of implementing mathematical tasks effectively with students is to have a foundational understanding of mathematical task knowledge (Chapman, 2013). Mathematical task chers need in order to (a) select and develop tasks to of mathematical thinking, and capture their interest and curiosity and (b) optimize the learning potential of such t apman, 2013, p.1). During the 12 week interventio n the treatment group learned about features of mathematical tasks that elicit low level and high level cognitive demand from students . The preservice teachers construct ed their own ma thematical tas ks that they include d in letters to third grade elementary students. to affect the mathematical tasks that teachers use for mathematics instruction (Pajares, 1992; Philipp, 2007 ; Raymond, 1997). Therefore, an importa nt component of this study was to determine whether or not the 12 week intervention had an effect on the beliefs about mathematics and mathematics instruction for the treatment group. The I ntegrated M athematics and P e dagogy (IMAP) Belief Survey (Philip et al., 2007) was administered at the beginning of the elementary mathematics methods course for both the treatment and control group s, and then the survey was administered again during W eek 11 of the
73 study. Additionally , participants of the study complete d an elementary mathematical task sort on W eek 2 and W eek 12 of the study. The purpose of the elementary mathematical tas k sort was to see if participants could identify a mathematical task as having low level or high le vel cognitive demand. In addition, two interviews were conducted for participants who exhibit ed a high belief change and low belief change for the seven beliefs . The interview s provide d insight about how the intervention e ffe cted the participant s experience of designing and posing mathematical tasks for third grade elementary students. The following questions guide d the research study: 1. To wh at extent did the intervention of learning about the levels of cognitive demand for ma thematical tasks and implementation of mathematical tasks through letter writing wi th third grade students impact elementary preservice teachers beliefs about mathematics and mathematics instruction? 2. How do elementary preservice teachers' beliefs about ma thematics instruction influence their implementation of mathematical tasks? 3. level or low level cognitive demands and does this change after a 12 week intervention specifically f ocused on learning about the levels of cognitive demand and implementation of mathematical tasks in letter writing with third grade students? A quasi experiment al research design wa s used for this study (Plano Clark & Creswell, 2010). The research design w as chosen because the sample for the study is comprised of two groups of preservice teachers that were not randomly assigned. The study was conducted through the lens of the situated learning perspective (Putnam & Borko, 2000). The goal of the research wa s to provide a better understanding about how 12 week intervention that focus ed on the level of cognitive demand of mathematical tasks and writing letters to third grade studen ts. The outcome of the study could provide guiding
74 principles for designing an elementary mathematics methods course where preservice teachers focus on the level of cognitive demand of mathematical tasks and have opportunities to interact w ith elementary school students through authentic experiences. Participants were recruited to take part in the research through their elementary mathematics methods course. There were two sections of the same course offered in the Fall 201 3 semester. One section was the control group and the other was the treatment group. P reservice teachers were not randomly assigned to the groups. However, the instructors were randomly assigned to the groups of preservice teachers. The researcher for this study was the i nstructor for the treatment group. A third year graduate student in the mathemati cs education doctoral program was the instructor for the control group. Both instructors for the cont rol and treatment groups use d the same textbooks (i.e., Van d e Walle & Lovin, 2006; Sowder, Sowder & Nickerson, 2010) and cover ed the same mathematical content. The differences between the two groups were the opportunities that the treatment group had to learn about the levels of cognitive demand for mathematical ta sks and to participate in a letter writing exchange with third grade elementary students. Data was collected through the use of several instruments: 1) the IMAP beliefs survey (Philipp et al., 2007) ; 2 ) general interview guide approach with two p articipa nts (Turner, 2010); and 3) elementary mathematical task sort (Arbaugh & B rown, 2005; Osana et al., 2006). The data from the beliefs survey was analyzed using a Chi S qu are for one way designs ( Q uestion 1). A thematic analysis method described by Braun and Clarke (2006) (Question 2) was used for the two participant interviews. A Wil coxon
75 ranked signed test and Whitney Mann U test were used to analyze the elementary mathematical task sort (Question 3 ). Methodology This dissertation wa s a mixed methods stud y that aimed to examine whether the 12 week intervention had an effect mathematics and mathematics instruction and how they identify mathematical tasks as low level or high level cognitive demand. The resear ch was viewed through a situated learning lens (Putnam & Borko, 2000). The situative perspective focuses on how individuals participate in a group or in a classroom practice (Borko, 2004). Knowledge is situated in the activity in which it occurs (Putnam & Borko, 2000). Gee (2008) states the relationship between an individual with both a mind and body and an environment in which the individual Bas ed upon the theoretical principles of the situative perspective, preservice teachers need to participate in a setting which allows them to learn from each other while acquiring the tools necessary to identify the cognitive demand of mathematical tasks. Br learner. For example, knowledge of the level of cognitive demand of mathematical tasks could be a tool preservice teachers use to analyze or construct mathematical tasks. A preservice teacher could interact with this tool in a class setting, but not be able to fully apply it to a real life teaching situat ion. Therefore, this study provided PSTs with the opportunity to design mathematical tasks and pose the tasks with elemen tary students through a letter writing exchange on three occasions.
76 The intervention focus ed on learning about the level of cognitive demand for mathematical tasks. Learning about the conceptual mathematical task framework and task analysis guide has bee n shown to provide opportunities that allow preservice and inservice teachers time to explore mathematics and mathematics instruction in a supportive and safe learning environment (e.g. , Arbaugh & Brown, 2005; Boston & Smith; Kosko et al., 2010; Osana et a l., 2006; Silver & Stein, 1996). PSTs work ed in pairs to create a mathematical task for their elementary pen pal. The letter writing exchange paired an elementary student with one or two PSTs. There were more PSTs than elementary students, therefore some P STs had to share an elementary student. The PSTs were matched up randomly with the elementary students. The pen pals exchanged letters on three occasions during the study. Prior to sending out the mathematical ta sk, the PSTs determine d the cognitive demand of the ir own mathematical task and provide d a rational e for the selection of cognitive demand. Afterwards , when the PSTs receive d their mathematical task b ack from their pen pal, they reevaluate d the cognitive demand for the ir own mathematical task and p rovide d rationale for their rating . The PSTs record ed their observations on the elementary task analysis form modified from Kosko et al. (2010) (Appendix A ) . This mixed methodology was selected, as it was necessary to answer the questions posed for thi s study. Questions 1 and 3 allowed for quantitative measurement , mathematics instruction , and scoring of the mathematical task sort . I n addition , Question 2 utilized qualitative methods for the participant interviews . Both sets of data were analyzed separat ely and then the results were compare d. Finally, an interpretation of the results was made and then discussed
77 (Plano & Clark, 2010). The Chi S quare test was used to compare the pre and posttest data colle cted from the groups on the beliefs survey . For the mathematical task sort, a pre and post task sort score was determined for each participant. A Wilcoxon Signed Ranked test and Mann Whitney U test for two independent groups were used to analyze the pr e and post task sort data. The format for the mathematical task sort activity was derived from researchers Arbaugh and Brown (2005 ) who implemented a pre and post task sort with practicing geometry teachers. The same elementary mathematical tasks were used for each task sort activity (see Appendix B ). At the conclusion of the study tw o participant interviews were conducted. The interviewer asked questions about the intervention, specifically about the experiences the preservice teachers had during the course and the letter writing exchange. A thematic analysis was conducted on data collected from the participant interviews (Braun & Clarke, 2006 ). Procedures Participants Identification of participants The participants fo r this study were elementary pres ervice teachers enrolled in an undergraduate elementary mathematics methods course during the Fall 2013 semester, at a large research university in the southeastern United States. The beliefs data were collected from PSTs in the contro l (n = 33) and treatm ent (n = 30 ) groups. The interviews were conducted with two PSTs from the treatment group. The task sort data was collected from PST s in the control group (n = 34) and treatment group (n = 32). Since the data collection for the beliefs instrument and math ematical task sort happened on different days, the numbers of participants who completed the pre and post
78 assessments differ because some PSTs were not present on that day. Table 3 1 displays the participants for each method of data collection. The parti cipants were not randomly assigned to these groups; however instructors were randomly assigned to sections of the course. A graduate student teaching instructor taught the control group and the researcher, who is also a graduate student teaching instructor , taught the treatment group. Table 3 1. Table of Participants Control Group Treatment Group Beliefs Survey n = 33 n = 30 Interview n = 0 n = 2 Mathematical task Sort n = 34 n = 32 Th e general interview data was collected from two participants in th e treatment group (Turner, 2010). The researcher chose two participants based on changes in their belief scores. These participants were chosen through purposive sampling because of the intensity in their belief change. (Flick, 2009). One participant had s ignificant changes in beliefs for four out of the seven beliefs and the other participant had a belief change for one of the seven beliefs. Raymond (1997) created two of the questions and the researcher wrote the other eleven. The purpose of the questio ns was to learn more about ho e ffected their implementation of mathematical tasks (see Appendix C ). According to Turner (2010), this type of interview works well for researchers who have already established a rapport with pa rticipan ts, and the interview guide provide s structure to the conversations . All PSTs enrolled in the elementary ma thematics methods course were invited to participate in the study. All participants were asked to fill out a consent form. The participants a sked to participate i n the general interviews were provided with an
79 additional consent form. For the treatment group, three PSTs chose not to participate, and for the control group, four PSTs chose not to participate. Descrip tion of participants All elig ible participants (4 male, 73 females) were undergraduate students who were enrolled in their third semester of a teacher preparation program for elementary school teachers. This program offers both a Bachelor of Arts in Education and Master of Education. At the end of the program, preservice teachers can seek teacher c ertification for grades K 6. Th is program requires 124 undergraduate credit hours and 36 graduate credit hours. Prior to entering this study, each participant completed the first elementary mathematics course, as well as a mathematics course at or ab ove the college algebra level. The course used for this study is the second out of four required mathematics education courses at the university for the elementary education major. Setting The in tervention took place during scheduled cours e meetings throughout the first twelve weeks of the Fall 2013 semester. A twelve week timeline was based on timelines of similar studies involving mathematical tasks (e.g., Kosko et al., 2010 [12 weeks]; Norton & Kastberg, 2011 [10 weeks]; Osana et al., 2006 [1 day]; Vacc & Bright, 1999 [5 weeks]). The course met once per week for a three hour session. The tr eatment and control groups met on different days. The classroom for both the control and treatment groups w as set up to provide preservice teachers with the opportunity to work together in small groups. An overview of the topics for the intervention is in Appendix D . D ata fro m the beliefs instrument were collected during the first class meeting of the seme ster (pretest) and on the eleventh class meeting (posttest) . The mathematical task sort activity took place during the second c lass meeting (pretest) and twelfth class
80 meeting (posttest). Additionally, participants in the treatment group created a mathematical task for their elementary student pen pal on three occasions (sixth, ninth, and twelfth week). The eleme ntary task analysis form was completed prior to the elementary students receiving the tas k, and then afterwards when the PSTs receive d the responses fr om the elementar y students. This form also serve d as a reflect ion tool for PSTs as they looked back upon their letter writing exchange. The interview s took place during the fourteenth week of class to allow for the data from the beliefs instrument and mat hematical task sort to be analyzed. The interviews took place in a conference room adjacent to the room where the course was taught in the college of education during a convenient time of day for both the researcher and the participants. Treatment Treatm ent group The PSTs who were part of the treatment group for the study were introduced to the levels of cognitive demand for mathematical tasks, Mathematical Task Framework (Stein et al., 1996), and task analysi s guide (Stein & Smith, 1998). The intention f or the selected activities was to have PSTs examine, perform , and then discuss mathematical tasks that were in the curriculum and in other resources and to d etermine the cognitive demand. The PSTs also consider ed how students would approach a mathematical task and discuss ed possi ble challenges students might have with the task. During class sessions, PSTs discussed ways to help students persevere through challenging aspects of mathematical tasks without lowering the cognitive demand of the mathematical task s . Each of the classes was structured so PSTs had an opportunity to watch videos, view student work, and engage in discussions .
81 During Week 3, pairs of PSTs were randomly assigned to a third grade elementary student pen pal from a school in Texas. The ele mentary third grade teacher made the letter writing exchange possible. She contacted the mathematics education department at the University of Florida at the time that the researcher for this study was seeking a school to exchange letters with. Each classr oom at the elementary school was assigned a university to study for the whole year. The premise behind having each class study a university was to motivate the students to try to succeed academically and to go to college. The teacher described the school a s a rural Kindergarten through fifth grade elementary school. The majority of the students spoke a language other than English and most students were from low socioeconomic homes. After hurricane Katrina in 2005, many displaced families moved to the town w here the elementary school is located. The third grade class consisted of twenty two students (11 boys and 11 girls). The third grade students initiated the first correspondence of the letter writing exchange. The third grade students sent a pre made sur vey consisting of favorite things (e.g., color, candy, movie, etc.) to the PSTs. In pairs, the PSTs wrote a letter introducing themselves, completed the same pre made survey, and included a mathematical task based upon a pre selected Common Core State Stan dard (2010). During the study, PSTs were able to correspond with their pen pals on three occasions. An overview of the timeline for the pen pal letters with the corresponding standard is located in Appendix E . ced through the activities and tools participants interact wi week intervention, PSTs were exposed to a variety of mathematical tasks. An example lesso n plan is included in Appendix F . T he PSTs had the opportunity to lear n from each other as they solve d the
82 mathematical tasks presented in class . In addition, they had the opportunity to create mathematical tasks for their pen pals. The PSTs used features described in the task analysis guide (Stein et al., 2000) to create th eir tasks for their pen pal. Control group The c ontrol group for the study follow ed the typical format for the elementary mathematics met hods course at the university. This typical format consists of discussing previously assigned homework problems, int roduction to the new content through a student centered activity, followed by a discussion, and then a recap of the content and how it relates to teaching mathematics. The class activities were typically from the course textbook (e.g., Van de Walle & Lovin , 2006). Participants i n both the control and treatment groups were taught the same mathematical content on fractions. The control group did not par ticipate in assignments that were focused on the cognitive demand of mathemati cal tasks and the letter writi ng exchange with elementary school students. However, the control and treatment shared a common practicum experience. Each PST in both groups was assigned to an elementary school classroom in the same school district for the whole semester. The PSTs would go to their assigned classroom once a week. Throughout the course of the semester, the PSTs were asked to complete assignments in their practicum classrooms. The assignments consisted of creating and implementing reading and mathematics lessons, completin g a case study on one English for speakers of other languages student, and conducting a technology inquiry. The PSTs experiences in their practicum classrooms varied. For instance, some of the PSTs were able to create their own lessons, while other PSTs we re given pre made lessons to teach. The teaching experiences ranged from whole group to small group.
83 The practicum experience was a required component of the elementary education program for all third semester students. Data Sources Th e data sources for th is study consist ed of: the beliefs survey (Philipp et al., 2007) , two audio taped preservice teacher interviews , and mathematical task sort ( Smith et al., 2004 ) . There were also written artifacts from the preservice teacher interviews, written field note s, and video taped class sessions. The written field notes a nd videotaped class sessions provide d a resource for monitoring the implementation of the treatment. Specifically, the researcher viewed the video taped class sessions to ensure the content covere d in the control group was consistent with the treatment group . The instructors did not share teaching resources during the study. The video taped class sessions for the treatment group provided a resource for fidelity o f implementation of the treatment fo r the study. The beliefs survey was taken in the form of pencil and paper responses. The two partici pant interviews were audio recorded and transcribed. For the mathematical task sort, participants used a paper with pre cognit Data Collection Beliefs i nstrument A pretest and posttest were used to capture any effects the treatment had on beliefs. The beliefs instrument was created through funding provid ed by the Interagency Educational Research Initiative, National Science Foundation grant to researchers at San Diego State University. The researchers were interested in creating an instrument that could measure change in beliefs about mathematics an d mathematics instructions. They hypothesized that Likert type surveys were not capable
84 of measuring such chang e and were determined to develop a beliefs instrument that could provide a better understanding about PSTs beliefs. It took two years to develop the instrument (Ambrose, Phili pp, Chauvot, & Clement, 2003). The goal of the researchers account for the critical role they play in teaching a et a l., 2007, p. 450). Only one version of the instrument was p roduced in a W eb based format. A researcher modified the original W eb based format to a paper and pencil format. The original layout, design, directions, and questions remain the same. Original ly each question appeared on one screen , and so each question would appear on a separate sheet of paper. There was no time limit for the survey. A selection from this inst rument is provided in Appendix G . The beliefs instrument contain ed sixteen questio ns that include d complex situations and scenarios that require d respondents to make teaching dec isions (Philipp et al., 2007). The instrument was intended to capture beliefs that focus on implementing reform based teaching (Philipp et al., 2007). Originall y, the researchers created eight such beliefs but could only provide reliability and validity measure for seven beliefs. The seven beliefs included in the instrument are: Beliefs About Mathematics : 1. Mathematics is a web of interrelated concepts and procedur es (and school mathematics should be too). Beliefs About Learning or Knowing Mathematics, or Both : 2. correspond with understanding of the underlying concepts.
85 3. Understanding mathemat ical concepts is more powerful and more generative than remembering mathematical procedures. 4. If students learn mathematical concepts before they learn procedures, they are more likely to understand the procedures when they learn them. If they learn the pr ocedures first, they are less likely to learn the concepts. Beliefs About Ways Children (Students) Learn and Perform Mathematics : 5. Children can solve problems in novel ways before being taught how to solve such problems. Children in primary grades generall y understand more mathematics and have more flexible solution strategies than adults expect. 6. The ways children think about mathematics are generally different from the ways adults would expect them to think about mathematics. For example, real world conte , whereas symbols do not. 7. During interactions related to the learning of mathematics, the teacher should allow the children to do as much of the thinking as possible. The researchers engaged in a rigorous proces s to d evelop the instrument. Pilot work with in service teachers provided positive results for captur ing a range of belief scores. Due to the fact that this survey was not in a typical Likert Scale format, the usual tests used to determine validity and reliabili ty were not appropriate (Ambrose et al., 2003). Therefore, to ensure consistency of the survey, it was administered to 18 PSTs, which included follow up inte rviews with the PSTs (Ambrose et al., 2003). Additionally, five mathematics education researchers, with experience in teaching and researchin g up discussions (Philips et al., 2007). The result of dispensing the survey to the aforementioned parties was confirmation of the reliability and validity of the survey. Hence, the survey measured instruction) and these results were found each time the survey was administered. Reliability information for the sixteen questions on the beliefs instrument was never published.
86 The majority of the construction time spent on the instrument was with the rubrics. Seventeen rubrics for the instrument were constructed and each one took 72 person hours. The coders for a ll 17 rubrics calculated an 84% inter rater reliability. For this study, the instrument was administered to the participants at the beginning of the first scheduled class meeting for the Fall 2013 semester . The researcher administer ed the instrument by f ollowing the directions in the manual created by the authors (with the exception of the paper and pencil modifi cation). The participants record ed their answers directly on the paper, either by selecting an answer to a multiple choice response or filling in the s hort answer responses. There was a video component to the instrument, which was shown to the whole group at the s ame time. The participants were only identified by their study identif ication number, which they record ed on the instrument. Interviews Two weeks after the study, two participant interviews were conducted using the Genera l Interview Guide Approach (Turner, 2010). At the conclusion of the study, the researcher chose two participants from the treatment group . The participants were chosen b ased upon their belief change scores. Specifically, the range for the change scores were analyzed and the two participants were chosen for their growth or lack thereof out of the seven beliefs. One participant had belief changes on f ive of the seven be liefs and the other participant only had a belief change on one of the seven beliefs. Both participants were in third grade practicum classrooms at the same school. This characteristic is significant to the study because third grade students were also the pen pals for the letter writing exchange.
87 The intent of the interviews was to gain insight into how the preservice teachers interpreted their experience in the intervention. Pajares (1992) recommended that multiple data sources be used to understand the b eliefs of teacher s. The interviews took place in a conference room adjacent to the classroom where the course was held at the C ollege of E ducation. There was no time limit on the interv iew and each participant was asked the same qu estions. Each interview w as audio recorded and transcribed verbatim by the re searcher. The participants already had an established rapport with the researcher, which help ed the interview flow similar to a conversation (Turner, 2010). Mathematical Task Sort The design, use and an alysis of a mathematical task sort was informed by research established by Arbaugh and Brown (2005), Boston and Smith (2011), Osana et al. (2006), and Smith et al. (2004). The mathematical task sort provided an assessment of pre and post know ledge of the cognitive demands of mathematical tasks . In order to complete the task sort, participants were provided with sixteen elementary mathematical tasks on sheets of 8 .5 x 11 paper. These sixteen tasks were created through the QUASAR research stu dy (Stein et al., 2000). Each task was assigned a letter (e.g., A through P). The participants were provided with a table that consisted of rows labeled A through P and columns labeled as memorization, procedures without connections, procedures with connec tions, doing mathematics, and a rationale for classification of the mathematical task. The participa nts were asked to sort the mathematical tasks into four categories (e.g., memorization, procedures without connections, procedures with connections, doing m a thematics). The instructions were as follows: The papers in the plastic sleeves contain 16 elementary school level mathematics tasks.
88 Please read them through, and then sort them into one of the four categories provided in the table. Please provide rati onale for your category selection in the last column. The mathematical task sort was designed as a tool to be used with teachers in professional development settings (Smith et al., 2004). The purpose of the mathematical task sort was for participants to kn ow the potential a mathematical task has to meet the student learning goals and bring awareness to features of mathematical tasks (Stein et al., 2004). All participants could engage with the mathematical task sort at some level. The mathematical task sor t was scored using criteria established by Boston (2006), who created a scoring matrix for the task sort. A score of either a 1 or a zero was assigned to a participant identifying each individual task as either high or low cognitive demand. There was also a rationale column provided for the decision of the cognitive demand. Responses in this column were also assigned a score of 1 or zero. However, only a few participants in each group completed the rationale column. Therefore, the researcher did not include this information in the pre and post task sort. Elementary Mathematical Task Form Preservice teach ers in the treatment group focus ed on constructing mathem atical tasks for the pen pal letters. After the mathematical task was constructed, the PSTs chose a level of cognitive demand for the mathematical task and provided a rationale for this decision. They chose the level of cognitive demand based upon the features of the task explained in the Task Analysis Guide (Stein et al., 2000). Afte r the elementary st udent responded to the mathematical tas k, the PST re evaluate d the level of cognitive demand of the task and provide d a reason for the choice in level of that cognitive demand. Kosko et al. (2010) found that the forms provided evidence of the
89 abiliti es to predict levels of cognitive demand and indicated improvement over the period of the study. This form served as a tool for PSTs to consider the cognitive demand for the task they created for the pen pal letter. The PSTs were able to reflect upon the c ognitive demand of the task after the elementary student had the opportunity to try the task. Data Analysis In order to answer Research Question 1, t he data collected fro m the beliefs instrument was analyzed using a chi square test for one way designs (Sha velson, 1996). The beliefs instrument collects data for an independent variabl e that has two or more levels. Each of the seven beliefs that the instrument measured was treated independent ly. The chi square test was performed using the Statistics Package fo r th e Social Sciences (SPSS) 18.0. The five assumptions f or this test were considered. The assumptions are as follows (Shavelson, 1996, p.558): 1. Each observation must fall in one and only one category. 2. The observations in the sample are independent of one another. 3. The observations in the sample are measured by frequencies. 4. The expected frequency for each category is less than 5 for df than 10 for df = 1. 5. The observed values of with 1 degree of freedom must be corrected for continuity in order to use the table of values of critical . To address Research Question 2, t he transcribed interview data from the tw o participant interviews were analyzed using thematic analysis ( Braun & Clarke, 2006 ). Braun and Clarke (2006) describe the process of completing a thematic analysis in six phases. It is important to note that these phases do not represent a linear process; they denote a process that can take the form needed by the researcher to create themes.
90 Hence, the researcher can return to any phase at any point in the process of the thematic analysis (Braun & Clarke, 2006). These phases are as follows: Phase 1: fami liarizing yourself with the data Phase 2: generating initial codes Phase 3: searching for themes Phase 4: reviewing themes Phase 5: defining and naming themes Phase 6: producing the report The identified themes provid ed insight into the participant s beliefs about mathematics and mathematics instructi on. Throughout the analysis the researcher made note of a ny theme that was not consistent with the beliefs about mathematics instruction. The purpose of the interviews was to provide follow up information about the preservice Additionally, the researcher sought assistance from a fellow graduate student researcher to read the transcripts from the interviews and code the data in order to valida themes from the data created by the first researcher. Both researchers had a conversation about the themes that were identified in the data and found that the majority of the themes coincided with one another. In order to address research Question 3 , the mathematical task sort s were coded cognitive demand of mathematical tasks . The treatmen t group scores on the pre and post task sort were compared using one tailed Wilcoxon Signed Rank test for nonparametric data. This test was chosen to determine whether knowledge of the levels of cognitive demand increased after the 1 2 week intervention. The pre and post task sort data for both the treatment and control group was compared
91 using a Mann Whitney U test for two independent groups (Shavelson, 1996). This test was chosen to examine if there were differences in the distri butions of two groups or differences in the medians of two groups (Lund & Lund, 2013). The assumptions for the Mann Whitney U Test were as follows ( Shavelson, 1996 ): 1. You have one dependent variable that is measured at the continuous or ordinal level. 2. You have one independent variable that consists of two categorical, independent groups (i.e. a dichotomous variable). 3. You should have independence of observations, which means that there is no relationship between the observations in each group of the indepen dent variable or between the groups themselves. 4. You must determine whether the distribution of scores for both groups of your independent variable have the same shape or a different shape. This will determine how you interpret the results of the Mann Whi tney U Test. The analyses for these three research questions were utilized to provide an understanding of the changes that occurred during the course of the study. In the next section the potential limitations are identified by the researcher. Limitations of the Study There were sev eral limitations to the study. First, the sample population for the study was not randomly selected into groups. The groups of the participants were alre ady intact prior to the study. Upon acceptance into the teacher education p rogram , participants ha ve been assigned a cohort with whom they take almost every course together. The participants are in close proximity t o each other on a daily basis. Additionally, both groups participate d in a practicum experience in the same school d istrict one day a week. The participants were assigned to one classroom in grades Kindergarten through fifth grade. During the practicum experience the participants
92 observed mathematics instruction and implemented mathematics lessons. This could pose a pot ential threat to the internal validity of the study. The sample for the study is a convenience sample of preservice teachers enrolled in an elementary mathematics methods course as part of a required co urse in their program of study. Due to this fact, this may limit the generalizations that can be made to other populations of preservice teachers. The beliefs instrument also provides threats to i nternal validity of the study. For instance, there was only one copy of the instrument and it was administered twice in twelve weeks (i.e., befo re and after the treatment). Furthermore, the original instrument was va lidated as a web based survey. Another researcher transcribed the instrument to a paper and pencil format, keeping the original formatting and followin g the instructions of the belie fs instrument manual. There is no validation information for the new paper and pencil format of the instrument. The mixed methods research design was chosen to provide validity to the quantitative and qualitative data collec ted for the study. The design provide d the opportunity for the researcher to analyze the data separately and then compare the results. Finally , t he researcher posed a threat to the validity of the study because the researcher taught the treatment group . The participants could have become aware of the researcher s intentions for the treatment when they read the consen t form. The researcher tried to not influence the participants in any way during the study. Conclusion Th is dissertation study builds upon the research conducted by Arbaugh and Brown (2005), Boston and Smith (2011), Crespo (2003), Kosko et al. (2010), Norton and
93 Kastberg (2011), and Osana et al. (2006) by using the Mathematical Task Framework with elementary teachers and incorporating letter writing into the elementary mathematics course. The goal of the study wa s to build upon prior research studies which focused on providing PSTs with opportunit ies to learn about the level of cognitive demand of mathematical tasks and provide a better understanding about how beliefs change as a result of participating in a 12 week intervention that focus ed on the level of cognitive demand of mathematical ta sks and writing le tters to third grade students. Furthe rmore, the researcher was interested in whether or not the treatment had an impact on the beliefs. The PSTs had an opportunity to learn about the cognitive demand of mathematical tasks, as well as, analyze curriculum materials with the intention of identifying the cognitive demand of potential mat hematical tasks for elementary grades , participate in doing mathematics during the course , discuss mathematics and mathematics instruction as a group, and correspond with a third grade pen pal. Overa ll, the dissertation study provide d insight into the beliefs preservice elementary teachers hold about mathematics and mathematics and instruction. The study wa s intended to add to the body of literature on m athematical tasks instruction, and authentic experiences for PSTs in teacher education .
94 CHAPTER 4 RESULTS The results of the data analysis are presented in this chapter, organized into three sections that correspond to the three original research questions presented in Chapter 1. The first section describes the beliefs of the elementary preservice teachers, including the results of the pre and post belief survey. The second section provides closer analysis of how the beliefs of two participants were impacted by the intervention. This includes the qualitative analysis of two participant interviews. The third section explains the knowledge of preservice teachers about the cognitive demands of mathematical t asks . This in cludes the pre and post mathematical task analysis for the treatment group, a comparison of mathematical task sort scores for the treatment and control group, and an analysis of the mathematical tasks classified by the treatment group. Elementary Preservi Instruction Research Question 1: To wh at extent did the intervention of learning about the levels of cognitive demand for mathematical tasks and implementation of mathematical tasks through letter writ ing wi th third grade students impact elementary preservice teachers beliefs about mathematics and mathematics instruction? In order to answer this research question, a Chi Square analysis was used to analyze the data collected by the beliefs survey for bo th the treatment and control groups (n=63). The results of the comparisons between the treatment and control group are presented in the remainder of this section. Pre and Post Belief Survey The pre and post belief survey change scores serve as an indicat or of the
95 after 12 weeks. Originally, the pre belief survey was administered to seventy three PSTs. A total of sixty three PSTs completed both the pre and post belief surv ey. Due to attrition from the study, the scores of ten PSTs were not included in the data analysis. There were three PSTs in the treatment group and four PSTs in the control group who chose not to be part of the study. Additionally, one PST from the trea tment group withdrew from the course halfway through the intervention and two PSTs did not complete both the pre and post belief surveys. The researcher administered the pre and post belief surveys to both groups using the same directions on all four o ccasions. The researcher assigned a numerical score to each pre and post belief survey following the rubric for each of the seven beliefs. To ensure reliability when assigning a score to each of the surveys, the researcher calibrated by using the practice sets in the training manual, prior to scoring the surveys. After all of the pre and post surveys were assigned raw belief scores, the researcher assigned a change score based upon the following criteria: no increase or a negative value in belief score received a 0, increased one level received a 1, and increased two or more levels received a 2. First, a Chi Square analysis was conducted to determine if there was a difference b etween the pretest belief scores for the treatment and control groups (see Table 4 1 ). These results show that the pretest scores were not significantly different between the treatment and control groups for all seven beliefs. More specifically, the pretes t scores for both groups were very similar and hence did not indicate one group was higher than the other at a certain belief change score.
96 Next, the a ctual counts for each change score level for all s even beliefs for both gro ups is represented in Table 4 2. The descriptive statistics revealed a significant difference between the treatment (n=30) and control group (n=33) on two out of the seven beliefs (see Table 4 3 ) . The two beliefs are: Belief 5 (children can solve problem s in novel ways before being taught how to solve such problems) 2 (2) and Belief 7 (the teacher should allow the children to do as much of the thinking as possible) 2 (2) The Chi Square test revealed a change in beliefs for more participants in the treatment group than in the control group for Beliefs 5 and 7. The control group participants did experience changes on beliefs. However, these changes are not significant when comparing the actual counts versus the expected count for the two groups. Table 4 1 . Beliefs pretest significance values. Belief Pearson Chi Square Value Degrees of Freedom p value (2 sided) 1 4.016 3 0.260 2 3.654 3 0.301 3 2.073 3 0.557 4 1.451 3 0.694 5 1.037 3 0.792 6 3.590 4 0.464 7 3.841 2 0.147 "#$%&!' () !! *+,-#%!+.-/,!0.1!$&%2&0!+3#/4&!5+.1& ! 0.1!,1,6&/ ,!#/7!+./,1.%!41.-85 Change Score Belief 0 1 2 T C T C T C 1 20 20 7 7 3 6 2 20 26 5 3 10 7 3 14 17 10 7 6 9 4 17 19 6 9 7 5 5 7 19 11 9 12 5 6 19 16 9 8 5 6 7 14 27 10 6 6 0 * T represents treatment group and C represents control group
97 Tab le 4 3 . Belief change score significance values. Belief Pearson Chi Square Value Degrees of Freedom p value (2 sided) 1 0.859 2 0.651 2 1.254 2 0.534 3 1.280 2 0.527 4 0.904 2 0.636 5 8.497 2 0.014 6 0.692 2 0.708 7 11.004 2 0.004 The cross tab ulation in Table 4 4 and 4 5 below provide a breakdown of results for Belief 5 and Belief 7. For Belief 5, the treatment group had 23 participants experience at least one level of change in beliefs compared to the control group that only at least one lev el of belief change for 14 participants. More specifically, the actual count reveals that 12 participants in the treatment group had at least two levels of changes in beliefs, while the control group had 19 participants with no change in beliefs for Belief 5. Belief 5 involves presenting problems to children and allowing them to solve the problems before being taught how to solve them. Additionally, Table 4 6 and Table 4 7 provide the percentages for the each category of belief change by group. The letter writing exchange may have facilitated the change in this belief. The participants in the treatment group created problems for their pen pals and were only able to provide directions. The participants were not able to provide instruction and thus had to wa it to see how their pen pal responded to their problem. There is evidence for this conclusion from one of the interview participants who first analyzed the pen pal response and thought it was incorrect. Upon further inspection of the solution and the stude student was actually correct in her thinking and the solution was correct. The participant was anticipating that the pen pal student would solve the problem like she had intended
98 it to be solved, and was surprised when the pen pal student came up with an alternative solution to her preconceived one. The control group did not have the opportunity to interact with students in this manner and this may be the reason why 19 of the participants experienced no change in level in their beliefs for Belief 5. Belief 7 is focused on allowing students to do as much thinking about mathematics as possible without teacher interference. Belief 7 for the treatment group had actual counts o f 16 participants with a belief change of at least one level or higher. The control group had actual counts of 6 participants with one level change, while none had two levels of belief change. There were 14 participants with no belief change in the treatm ent group compared to 27 in the control group. An explanation for the slightly higher belief change for the treatment group as opposed to the control group could be due to the way the course was structured for the treatment participants. Beliefs 5, 6, and 7 are encompassed in the category, beliefs about ways children (students) learn and perform mathematics. Belief 6 was not significant in this study and this could have been due to a missing component in the intervention which would have allowed PSTs the o pportunity to see that children think differently than how adults would expect them to think about mathematics. However, t he treatment group experienced mathematics as a problem solving discipline. There were several occasions when participants were asked to work together on a task and share their solution process. The participants were also asked to consider the cognitive demand of the mathematical tasks and how to differentiate the task to accommodate different levels of cognitive demand. The remaining cross tabulations for the five beliefs are included in the Appendix H .
99 Table 4 4 . Belief 5 cross tabulation values. Group Change score 0, expected cou nt Change score 0, actual count Change score 1, expected count Change score 1, actual count Change score 2, expected count Change score 2, actual count Treatment 12.4 7 9.5 11 8.1 12 Control 13.6 19 10.5 9 8.9 5 Table 4 5. Belief 7 cross tabulati on values. Group Change score 0, expected count Change score 0, actual count Change score 1, expected count Change score 1, actual count Change score 2, expected count Change score 2, actual count Treatment 19.5 14 7.6 10 2.9 6 Control 21.5 27 8.4 6 3.1 0 Table 4 6 . Percentage of students in each group whose scores on Belief 5 i ncreased 1, 2, 3 , or 4 levels from presurvey to postsurvey 1 level increase 2 level increase 3 level increase 4 level increase In creased 1 level Treatment (n = 30) 37% 27% 10% 3% 77% Control (n = 33) 27% 9% 3% 0% 39% Table 4 7 . Percentage of students in each group whose scores on Belief 7 increased 1 or 2 levels from presurvey to postsurvey 1 level increase 2 level incre ase Increased 1 level Treatment (n = 30) 33 % 20 % 53 % Control (n = 33) 18 % 0% 18% The Beliefs of Heidi and Nora Research Question 2: How do elementary preservice teachers' beliefs about mathematics instruction influence their implementation of mathem atical tasks? In order to answer this question the researcher chose two preservice teachers from the treatment group that exhibited either a significant belief change or a minimal belief change across all seven beliefs measured by the survey. The partici pants were chosen from the treatment group through purposive sampling (Flick, 2009). One participant was chosen because she had few belief changes
100 and one participant was chosen because she had many changes. For clarification purposes the preservice teache r with a significant belief change on five out of seven beliefs will be referred to as Heidi and the preservice teacher with one belief change will be referred to as Nora. presented in Table 4 8 . Heidi had significant increases on five out of the seven beliefs and Nora had a belief change on one of the seven beliefs. Heidi had the highest score change for all seven beliefs. For example, on Belief 6, Heidi went from a pre survey score of a 0 to a post survey score of a 4. Nora had the lowest score change for all seven beliefs. For example, on Belief 7, Nora had a pre survey score of 2 and a post survey score of 0. Both participants were in their third semester in the elementary education progr am, placed at the same school in different third grade classes for their third semester practicum, and have taken the same courses together since entering the elementary education program. Table 4 8 . and Post Belief Scores Belief s Pre Belief Score Post Belief Score Heidi Belief 1 0 0 Belief 2 1 3 Belief 3 2 3 Belief 4 0 3 Belief 5 1 3 Belief 6 0 4 Belief 7 1 1 Nora Belief 1 1 0 Belief 2 1 0 Belief 3 0 1 Belief 4 2 1 Belief 5 1 1 Belief 6 1 0 Belief 7 2 0
101 Back ground information about the experience the PSTs had in their third semester is provided to understand the premise of the interview questions. During the third semester the PSTs were assigned to a practicum placement at an elementary school for one day a w eek. The PSTs were placed in pairs in elementary classrooms. The practicum experience was designed for PSTs to have an opportunity to work with English Language Learners and teach two lessons for mathematics and three lessons for reading. In the beginni ng of the 12 week intervention, the preservice teachers were asked to read two articles about mathematical tasks outside of the course. One article focused on making instructional decisions and planning mathematical tasks that would elicit high cognitive d emand (e.g., Breyfogle & Williams, 2008). The other article focused on implementing high level mathematical tasks (e.g., Smith, Bill & Hughes, 2008). For the practicum experience, the preservice teachers were asked to develop two mathematical lessons. The lessons were supposed to have a mathematical task that would maintain a level of cognitive demand. After the lesson, preservice teachers were asked to evaluate their success or lack of success implementing the mathematical task and maintaining the level of cognitive demand through a lesson reflection form. The interview questions began by asking about how the preservice teachers planned their lessons for the practicum, followed by their beliefs about mathematics instruction, and finally about the pen pal ex perience. The interviews were analyzed through thematic analysis (Braun & Clarke, 2006). Both interviews were completed on the same day, after the last class of the semester. The researcher had pre selected thirteen questions for the interviews. Two of t he
102 questions were from an interview conducted by researcher Raymond (1997), the other eleven questions were created by the researcher for this study. Each participant was asked the same thirteen questions. Throughout the interview, follow up questions were asked to clarify responses. Each participant interview was audio recorded and transcribed immediately following the interview. Additionally, the researcher took observation notes throughout both interviews. After the interviews were transcribed, the resea rcher became familiar with the data by reading the data through one time without taking notes. Then the data was reread and initial codes were generated. Next, the researcher completed another reading of the transcripts and sorted the data into potential t hemes (see Appendix I ). Once the initial themes were established, another reading of the data took place and themes were named and defined (Braun & Clarke, 2006). The goal of the interviews was to better understand the mathematical beliefs of two purpose fully selected preservice teachers. The following themes were identified in the data: 1. Past experiences and present beliefs about mathematics 2. Authentic and inquiry based learning 3. The role of the teacher in making instructional decisions a. Cognitive aspects of student understanding b. Social and emotional aspects of teaching In these sections each theme is defined and evidence from the participants is development and implemen tation of lessons and mathematical tasks.
103 Past experiences and present beliefs about mathematics instruction experiences. The participants try to present mathematics in the o pposite way than the way they were taught mathematics. Both participants were asked to explain the best way to learn mathematics. In each of their explanations, they included an example of their prior learning experience with mathematics. Heidi and Nora ha d different experiences experience. A comparison of the experiences is done through the support of the data. Nora had a negative experience with learning mathematics. This n egative experience made an impact on how she viewed mathematics and how she taught mathematics to students in practicum. When she taught her first lesson on place value, she was very nervous at the beginning and later relieved when the students did not ask any questions. These feelings reveal she was not comfortable with teaching mathematics during the first lesson. As the semester progressed Nora contributed experiences in the course, such as demonstrating how to solve mathematical tasks together as a grou p, thereby shaping how she envisioned mathematics should be taught. Nora wanted her students to experience success with mathematics and continually described how she used easy mathematics problems with the students. In one of her lessons she had students create mathematics problems for her and she went through the classroom answering the problems. She wanted students to see that she could answer the problems quickly and this was something that she wanted her students to aspire toward when learning mathema tics. Additionally, Nora wanted to make sure her students knew she believed in them and that she was learning along with them.
104 worksheets. I got frustrated with them and, like I wish it would have been more hands on Throughout the 12 week intervention, the course consisted of mathematical tasks that incorporated the use of manipulatives. Nora valued the use of manipulatives because at Nora was trying to change her belief about mathematics instruction, but there was a constant struggle for her to reclaim her own confidence in mathematics. Heidi had a contrasting experience with learning mathematics. She did not struggle to learn ma thematics. Heidi remembers learning procedures for easy topics such as mean, median, and mode. Her teachers used hands on activities when students had difficulties understanding the mathematics. I n order to find out how Heidi felt about this type of learni ng, the researcher asked her if she felt the activities should come first when learning a new mathematical topic. Heidi thought activities should indeed come first to help students understand the mathematical concept. She describes an experience in her pra cticum where she saw how the students did not understand the concept of multiplication and the procedure was taught to the students first. She felt the concept should have been taught before hand so students understood why to multiply. This example is the opposite from how Heidi was taught mathematics. She was trying to change her prior experience with mathematics instruction for her students.
105 hands f this characteristic is exhibited in the design of her lesson for creating bar graphs. Heidi felt strongly that students needed to collect their own data and create bar graphs based upon this data, instead of the alternative which was completing worksheet s where the data was already collected or made up. She felt students need to be motivated by fun mathematics lessons which allow students to figure out problems on their own, provide an explanation to demonstrate their understanding of the concept, and not become overwhelmed with the information. Mathematics instruction should include authentic and inquiry based learning Both participants want students to have an authentic learning experience. Their interpretation of an authentic learning experience means having students do worthwhile mathematics activities where the students are applying knowledge and see the relevance of mathematics. Nora felt it was important to connect mathematics to real life. Heidi feels the same way as Nora. However she approaches l earning through inquiry based instruction, where she wants students to figure out the mathematical tasks on their own. Both participants felt that instruction should be guided and include scaffolds for student learning. This section explains how each parti cipant describes what they value for mathematics instruction and how it relates to their beliefs about mathematics instruction. to show my kids how it really does relate to real life. Like multiplication is something that we use a lot. Division is something that we t o read a newspaper or to read anything they need to know, like, what does that mean. You know. A lot of pe
106 they should learn it. I think they should learn it based on real life and figuring out how it is g to real life; however when she describes the lessons she taught in practicum, she focuses more on the student outcomes than she does the connections between real life a nd mathematics. Later in the interview she comments about the lack of connections between mathematical topics and how she felt she was never taught how mathematics applied to real life. Heidi views mathematics instruction through inquiry based learning. She values the importance of having students find out the solutions to mathematical tasks on their own and for the students to explain their answer. One example to show how Heidi encouraged students to explain their work and present their solutions in sma ll groups is shown here: I could tell by how they explained by the words they used, did they know really explain. Because I noticed that some of the students would be just copying t heir neighbor. And so when those students tried to talk they words. The other students were doing re ally great. They could explain exactly what they were doing step by step. Heidi values student explanations. On another occasion, Heidi describes a situation when she assumed her pen pal got the incorrect answer based upon the with the student work can be seen in the appendix (see Appendix J ). When she read the correctly. Addition ally, Heidi made it a point to include directions for her pen pal to
107 explain and show her work. This suggests that Heidi wanted to know how the student was thinking about the mathematics and whether her pen pal understood the mathematical task. liefs about mathematics instruction align with how she implements mathematical tasks. Nora struggles with what she believes should happen with learning mathematics and how she implements mathematics instruction. Both participants believe learning mathemati cs should be fun for the students. They both feel guided learning and scaffolding instruction through hands on activities and manipulatives are a means to make mathematics instruction fun and engaging for students. This is in contrast to how they were taug ht mathematics in elementary school. The role of the teacher in making instructional decisions This theme was derived from the data through two sources: the cognitive aspects of student understanding and the social and emotional aspects of teaching. Both participants mentioned student understanding consistently throughout the interview; however they referred to it in different ways. Hence, the researcher felt it necessary to separate the theme into two sub themes to explain how the participants were descri bing the role of the teacher. First, the cognitive aspects of student understanding are discussed, followed by the social and emotional aspects of teaching. Cognitive aspects of student u nderstanding The participa nts focused on different aspects of mathematics instruction related to cognitive demand in both their lessons and pen pal letters. These aspects illustrate a picture of the way in which preservice teachers are learning to teach mathematics. Heidi focused o n cognitive aspects of student learning, while Nora focused on the emotional
108 aspects of student learning. A conclusion provides a description of similarities found between the approaches the participants used to teach mathematics. Heidi focused on the s implementing her lessons and writing her pen pal letters. One example is in her response to an interview question, which asked her if she focused on the cognitive demand of mathematical tasks or mathematical questions when developing her lessons. She stated: I think I definitely did but like I said according to student ability. So I figured not just all around the problem is high cognitive demand I feel while I was teaching it I feel like I was manipulating t hings on the spot. Like make it higher for some students. lesson and then, in the moment of teaching, she would change the mathematical problem. For instance, if student s were struggling with the problem, she would change the problem to an easier one, or, if they were able to do the original problem, provide an differentiating instruction for her s tudents. mathematical task was in the letter writing exchange. Heidi explains: kind of gave her an easier one and saw how she did with that and then based the other problems. She did very well with the first task. So then, like me and Krista made the next one more challenging. The pen pal letter writing exchange began with the elementary students send ing a filled out questionnaire about their interests (e.g., favorite color, favorite candy) to the mathematical ability. It was up to the preservice teacher to determine how to write a
109 mathematical task that was based upon a Common Core State Standard and elicit a preselected level of cognitive demand. Heidi explained that she started with an easy mathematical task, and then, when the student completed the task correctly she began to pose more challenging mathematical tasks. This is an example of how Heidi used scaffolding techniques to write mathematical tasks for her pen pal. Heidi felt it was important to start with easy problems and work toward more challenging ones. Another example comes from a worksheet she created for students on the topic of multiplication. Heidi used pre made groups for students to place objects in to illustrate the concept of grouping for multiplication. She chose this representation for the top ic because she felt it illustrated the concept of multiplication. As the worksheet progressed the pre made groups disappeared, allowing the students to make their own groups that illustrated the multiplication problem. Heidi was also concerned that studen ts understood mathematics conceptually. In the pen pal letters Heidi sent, she always asked her pen pal to provide a different way to solve the problem. Prior to the intervention Heidi describes her instruction as: Like just give them the problem and not ask them to explain it. Not ask give them the problem and they would just do the procedure. Rather now I know that to understand how they are thinking and why they answered that way is probably the most important part about solving math problems. Her response in dicates she is open minded to having students share their solution strategies, and mathematics should be taught in a non procedural manner. This a problem, more than ju st seeing a numerical response.
110 On the contrary, Nora was concerned with student outcomes. She taught her lessons with the goal of having the students know what they needed for the test. When asked if she focuses on the cognitive demand for mathematical t asks or mathematical reveals that the cognitive demand was not considered for her les sons or the pen pal letters. Nora made problems simple for her students and pen pal so they would get them correct. She wanted her students to feel successful with mathematics, and this meant answering mathematics problems correctly. Nora based instruction al decisions on making the mathematics easy for the students. Another example comes from her pen pal letters. She first wrote her mathematical tasks for her pen pal by only using words. Her pen pal would not respond and she assumed the problem was too ha rd for him. Instead of using words she drew pictures to illustrate a mathematics problem that she described as easy and was able to get a response from her pen pal. Later in the interview Nora describes how she knows when students understand the mathemat ics. Nora states: I would throw things in and I knew that they really understood it when they that means that they have a deeper understanding of just what I am saying or just what I expl ained. Nora thought students demonstrated their understanding of mathematics when they could quickly respond to a question. She made up questions on the spot to keep demonst rating understanding of the mathematical material.
111 Both participants were concerned with challenging the students academically. Heidi interpreted challenging as progressively creating mathematical tasks that increased cognitive demand. Nora interpreted challenging students as creating questions in the moment and students were able to answer them right away. For example, Nora states: Because they were answering the questions that I already planned to ask. Like they have already got the concept that I p lanned to teach them and I wanted to extend it a little more so I would ask, like, oh what if I did this? What if I did that? And they were able to answer those questions too. So I was like they really get this. They are not just understanding this at a s The example also reveals that Nora believes her students have a deeper level of understanding and that she is successful with her teaching methods. Heidi also created questions in the moment but she considered the in who would ask questions to the whole group of students she was teaching. Interestingly, neither participant planned in advance to challenge students during their lessons. They had to think quickly about how to engage t he students by posing questions, which they felt challenged the students. The pen pal letter writing exchange provided Heidi with the opportunity to think, plan, and consider how to pose a challenging mathematical task to her pen pal. In conclusion, both participants indicated they want students to experience success with easier mathematical tasks before they move toward more cognitively learning by building upon easy ski lls through less cognitively demanding mathematical tasks.
112 Social and emotional aspects of teaching On several occasions the participants mentioned the need to know students and motivate them to learn mathematics. The pen pal letter writing exchange was an opportunity for the participants to know their pen pal without physically meeting them. This opportunity is just one instance of both participants taking into account student interests when designing mathematical tasks. Nora and Heidi both indicated that they cared about students as learners. In this section, the social and emotional aspects of teaching with a focus on student understanding are described to illustrate how the N ora felt students need to have a teacher who supports and believes in them. students because students through her teaching and in her letters to her pen pal that she believes in them and cares, in order not to perpetuate the feelings she had when learning mathematics. For instance, sh e feels the students need to know the teacher is there to support them as they solve problems. Nora stresses the need to work as a team in the classroom and to solve problems together. In essence, Nora is trying to change the way she experienced mathematic s instruction. Nora also wanted her pen pal to feel like she was learning alongside him. For example, in this section, she describes the letter writing exchange where she and her pen pal are exchanging mathematics problems: I loved it. I was glad that he was feeling like he was almost like to give her some. Like he was feeling like it was equal. I would give him one and he would give me one. Which is the way I wanted him to feel, like
113 you to solve because I am a teacher but you can give me some, too, because I still need the practice too. It was very important to Nora that the students felt she was part of the community of learning. She wanted them to feel comfortable with her and with mathematics. This is another example of how she wanted to provide students with a different learning experience than the way she experienced mathematics. Heidi believed she motivated her pen pal to respond to the mathematical tasks by including the stude letters. Heidi describes how she made a connection with her pen pal in the following response: W e tried to base on her interests too so that would make her excited to do it. She was ver y excited and then she would get the problems. After she would get the problem she would write underneath it and color so it seemed like she was excited because we would, like, use her name in it. I think making them interested is like definitely a big th ing for us. In essence, Heidi and her classmates were excited and motivated to write the letters when the pen pals included something personal in the letters such as drawings or sentences about their life. The motivation for the letter writing exchange was evident in ways to motivate her pen pal and she valued this aspect of teaching. In another example from the interview, Heidi describes how a student from her practicu m classroom took pride and ownership of her work. On this occasion, Heidi explained how she knew she had a successful mathematics lesson. Following the lesson Heidi collected the student work for a class assignment. All of the students wanted their work back to show their parents. Heidi
114 was proud of her work so I feel like that is when you know when you have a successful different from Nora. Heidi interprets successful mathematics teaching as when students are excited and proud of their work. Nora interprets success as students arriving at the correct answer. Heidi and Nora both agreed that motivating students and makin g learning fun through hands on activities and manipulatives is important for teaching mathematics. Heidi considers students when planning lessons. For example, Heidi describes how she structures a lesson: Y ou are presenting them with a lot of information have to kind of teach the new, like, concepts. So I made sure to make that very explicit and like, not just ramble on because I know they would get overwhelmed. And definitely like modeling and think a louds. I think that was, li ke, the most important part of the structure because that really shows them, like, you are solving the problem and they get to see how they need to think about when they are solving it. This excerpt indicates that Heidi thinks it is important to model the mathematical behavior she expects of her students. Manipulative use was recognized by both participants as a way to have students engaged and motivated to learn mathematics. Nora wishes she were taught mathematics by using manipulatives and seeks out oppo rtunities to use them with her students in her practicum classroom. For instance, Nora felt the students had to know how to use the manipulatives in order to show the concept of place value. Nora and Heidi implemented their beliefs into mathematical tasks through hands on activities and using manipulatives in lessons. In conclusion, both participants took the time to create problems they felt their pen pals would be motivated to respond to. For example, Heidi would incorporate her pen terests into the mathematical problem. Nora wanted her pen pal to feel he could send her mathematics problems too. Heidi and Nora wanted students to be
115 excited about mathematics and therefore cared about the way the students learn mathematics. All in all, the past experiences with mathematics instruction that Heidi and Nora experienced has led them to want to change how they teach mathematics. Both participants want students to feel successful with solving mathematics problems. They did not want to have st udents struggle with mathematics. However, they both went about relieving the struggle in different ways. Both participants believed students should be taught mathematics through authentic and hands on activities. The themes illustrate how deeply intertwin ed mathematical beliefs are when the participants present mathematics to students. Heidi experienced significant belief changes in four of the seven beliefs. Evidence of her belief changes has been is seen in her interactions with her pen pal and designi ng motivating students to see mathematics differently. For instance, she wants students to life. Overall, experiences in the intervention and practicum classroom have impacted how Heidi and Nora interrupt mathematics and mathematics instruction. Tasks Research Question #3 : mathematical tasks as having high level or low level cognitive demands, and does this change after a 12 week intervention specifically focused on learning about the levels of cognitive demand and implementatio n of mathematical tasks in letter writing with third grade students? To answer this question, comparisons w ere made between the
116 treatment and post mathematical task sort scores and between the mathematical task sort scores of the treatment gro up and control group. The results of these comparisons are presented in the remainder of this section. Pre and Post Intervention Mathematical Task Sort The pre and post intervention mathematical task sort scores provide evidence of knowledge of cognitive demands prior to and follo wing their participation in a 12 week intervention. Thirty two preservice teachers participated in the pre and post interv ention mathematical task sort. For each of the 16 tasks in the mathematical task sor t, preservice teachers received 1 point for correctly classifying the task as high level or low level according to the Task Analysis Guide (Stein et al., 2000). The mathematical tasks classified as high The mathematical tasks classified as low level include The highest possible score on the mathematical task sort is 16 points. Scores on the pre mathematical task sort ranged from 6 to 1 4, with a mean score of 10.22. Post mathematical task sort scores ranged from 7 to 14, with a mean score of 11. 66. Results of the Wilcoxon Signed Rank tests for non parametric, paired data indicate that the increase of 1.44 between the means of the pre a nd post mathematical task sorts was significant (z = 2.694; p = 0.007 [two tailed]). These results suggest that for the treatment group of preservice teachers , their knowledge of cognitive demands of mathematical tasks increased following the 12 week inte rvention (see Table 4 9) . The effect of the increase will be described later in this section, and the analysis of the intervention will identify events that might have provided opportunities for this learning to occur.
117 Table 4 9 . Descriptive Statistics on Mathematical Task Sort Scores Pre Mathematical Task Sort Post Mathematical Task Sort n Mean (SD) Mean (SD) Treatment Group 32 10.22 (1.896) 11.66 (1.977) Comparing the T reatment G roup to the C ontrol G roup and post mathem atical task sort scores were compared to the task sor t scores of the control group. The results indicate whether the treatment group had a greater knowledge of the cognitive demands of mathematical tasks at the end of the study than the control group who d id not participate in the intervention. The pre mathematical task sort scores from the 34 control group preservice teachers ranged from 5 to 1 5 points, with a mean of 9.71. The post mathematical task sort scores from the 34 control group preservice teac hers ranged from 5 to 14, with a mean of 10.22. Results of the Mann Whitney U test comparing the mathematical task sort scores of the treatment group and the co ntrol group are listed in Table 4 10 . The Mann Whitney U test was conducted to determine if the re were differences in pre mathematical task sort scores, post mathematical task sort scores, and change scores between the treatment and control groups. Distributions of the pre mathematical task sort scores, post mathematical task sort scores, and change scores for the treatment and control group were similar, as assessed by visual inspection. The median pre mathematical task sort score s were not statistically significantly different between the treatment (Mdn = 10) and control (Mdn = 10), U = 485.5, z = .760, p = . 447. The median change scores for the mathematical task sorts were not statistically significantly different between the treatment (Mdn = 0) and control (Mdn =2),
118 U = 450.5, z = 1.207 , p = 0.228. The median post mathematical task sort score s were statistically significantly different between the treatment (Mdn = 12) and control (Mdn = 11), U = 353, z = 2.481, p = .013. The results indicate that both groups were relatively equivalent on the pre mathematical task sort, indicating both groups had prior knowledge of the cognitive demand of mathematical tasks. Additionally, when change scores were used as the dependent variable, there was no significant difference between the groups. However, the results indicate that the post mathematical task sort scores for the treatment group were significantly higher than the control group. The intervention, which focused on learning about the cognitive demand of mathematical tasks, appears to have provided preservice teachers with knowledge of mathematical tasks. Table 4 10 . Comparison of Pre Mathematical Task Sort Scores of Treatment and Control Groups Mean (SD) Mean Difference vs. Control Group Pre Mathematical Task Sort: Treatment Group (n = 32) 10.22 (1.896) 0.51 Control Group (n = 34) 9.71 (2 .223) NA Post Mathematical Task Sort: Treatment Group (n = 32) 11.66 (1.977) 1.22* Control Group (n = 34) 10.44 (2.205) *Results are significant at p < .05 [two tailed] A closer analysi s is provided to understand the significance in the pre and post mathematical task sort s cores for the treatment group. Specifically, information is provided on whether gains in pre and post mathematical task sort scores could be attributed to an improve level and low level
119 mathem atical tasks. Table 4 1 1 provides data to illustrate the nature of changes in Participants struggled with ident ification of high level tasks. The category with the percentage of incorrect classifications for both the pre and post mathematical task sort. Of the 160 instances in which DM tasks were classified on the task sort (i.e., 5 DM per participant times 32 participants), 118 incorrect classifications (74%) occurred on both the pre an d post mathematical task sort. As shown in Table 4 1 1 , all of the participants classified at least one DM task incorrectly. htly better than the DM tasks. There were five PWC tasks on the ta sk sort creating 160 instances. On the pre mathematical task sort, participants incorrectly classif ied 98 times out of the 160 instances (61%) and 84 times out of the 160 instances (53%) for th e post mathematical task sort. This result indicates that participants held prior knowledge of PWC tasks. It is possible that the intervention helped the particip ants r ecognize features of PWC tasks. Similar to the DM task, all of the participants classified at l east one PWC task incorrectly. Overall, the higher level tasks were more difficult for participants to classify. Participants incorrectly classified 63% of the higher level tasks for the post level task sort scores. Participants had d ifficulty correctly classifying DM tasks and only showed a slight improvement with classifying PWC tasks.
120 Table 4 1 1 illustrates that the treatment group participants had better succes s classifying low level tasks. On the post mathematical task sort, onl y 48% of the low level 4 tasks times 32 participants). On the pre mathematical task sort PWOC tasks were incorrectly classified in 101 of the 128 occurrences (79%). On the post mathematical task sort, 83 tasks (65%) were incorrectly tasks. Table 4 1 1 shows that only one teacher was correctly able to classify all PWOC on the post mathemat ical task sort. sort creating 64 instances (i.e., 2 tasks times 32 participants) wh ere MEM tasks were classified. Participants incorrectly classified 22 instances out of 64 (34%) on the pre mathematical task sort and 10 times out of 64 (16%) on the post mathematical task sort. Twelve participants incorrectly classified at least one MEM task on the pre mathematical task sort, and eight participants did so on t he post math ematical task sort. These results suggest that the participants exhibited a slightly improved ability to classify low level tasks on the post mathematical task sort. The descriptive data provide support to validate the increase in the treatment st mathematical task sort scores were not e ffected by repeated measures (i.e., the scores did not improve simply because the participants were completing the mathematical task sort for the second time) and the participants were not learning the task sort in the intervention. The treatment group improved in their ability to correctly classify mathematical tasks with low level cognitive demand.
121 In summary, participants in the treatment group had a sign ificant change in Beliefs 5 and 7, compared to the control group. The control group did not have a significant change in any of the seven beliefs, compared to the treatment group. However, all participants in the study had a belief change for at least one of the seven beliefs. The interview data from Heidi and Nora revealed that they want to teach mathematics differently from how they were taught. There are similarities and differences between the instructional approaches utilized by Heidi and Nora. The ana lyses of the mathematical task sort suggest that PSTs struggle with identifying high level cognitively demanding tasks. Chapter 5 will discuss the conclusions that can be made from the results of the statistical analyses and interviews presented in this chapter . Table 4 11. Analysis of the Task Sort Responses by Level of Cogni tive Demand (n = 32 preservice teachers) Level of Cognitive Demand # of Tasks Total # of classifications a # of incorrect classifications # of teachers incorrectly classifying a task at that level Pre Post Pre Post High Level: Doing Mathematics 5 160 118 118 32 32 Procedures with Connections 5 160 98 84 32 32 Low Level: Procedures without Connections 4 128 101 83 32 31 Memorization 2 64 22 10 12 8 a Tot al number of classifications is determined by multiplying the number of tasks at that level by 32 (the number of preservice teachers)
122 CHAPTER 5 DISCUSSION Importance of this study The results from this study are important to PST education, specifically the elementary mathematics methods course, because in the treatment group there appears to be a chang e in beliefs that are aligned with reform based mathematics instruction and an increase of knowledge about features of mathematical tasks. The treatment group had significant changes on two beliefs that are included in the broad out Specifically, these beliefs include Belief 5: Children can solve problems in novel ways before being taught how to solve such problems. Children in primary grades generally understand more mathema tics and have more flexible solution strategies than adults expect. They also include Belief 7: During interactions related to the learning of mathematics, the teacher should allow the children to do as much of the thinking as possible. Additionally, the r esults for the study indicate that the treatment group PSTs could accurately identify low level mathematical tasks after the 12 week intervention and had a significant increase in overall ability to identify mathematical tasks by low and high level cognit ive demand. This study contributes to research on elementary preservice teacher mathematics education by finding evidence through a belief survey, mathematical task sort, and two interviews, which provide indication of change in mathematics instruction be liefs and knowledge of the cognitive demand of mathematical tasks following participation in a 12 week intervention. The researcher designed and facilitated a series of learning experiences that allowed PSTs to have the opportunity to do mathematics,
123 analy ze mathematical tasks, create mathematical tasks, and pose mathematical tasks instruction and develop aspects of mathematical task knowledge (Crespo, 2003). In Chapter 1, the argument was presented that in order for students to learn mathematics at the desired level, teachers must be prepared to use mathematical tasks that will allow students to go deeply into the mat hematical content (Stein et al., 2009). Research has shown that when teachers are provided with opportunities to learn and focus on the cognitive demands of mathematical tasks, they experienced increases in student learning and engagement during mathematics instruction (Boston & Smith, 201 1; Henningsen & Ste in, 1997). Therefore, PST education serves as the stage where PSTs have opportunities to develop and experiment with posing mathematical tasks through authentic activities with students (Crespo, 2003). The purpose of this study was to determine the extent of such development, and included a 12 week intervention which focused on learning about the cognitive demand of mathematical tasks and posing mathematical tasks through a letter writing exchange. The study evaluated the effects of the levels of cognitive demand for mathematical phenomena, the study utilized separate pre and post assessments for the knowledge of levels of cognitive demand for mathematical tasks and the beliefs about mathematics instruction, along with two participant interviews from the treatment group. The results from this study provide evidence that following the 12 week intervention the treatment group had significant changes in 2 of the 7 beliefs. Additionally, the treatment group showed growth in their knowledge about levels of
124 cognitive demand of mathematical tasks. At the end of the intervention, the treatment group significantly increased: 1) their knowledge of the cognitive demands of mathemat ical tasks (i.e., their task sort scores); 2) their knowledge of Belief 5: Children can solve problems in novel ways before being taught how to solve such problems. Children in primary grades generally understand more mathematics and have more flexible sol ution strategies than adults expect (i.e., the change score for the belief); and 3) their knowledge about Belief 7: During interactions related to the learning of mathematics, the teacher should allow the children to do as much of the thinking as possible (i.e., the change score for the belief). The change scores for Beliefs 1, 2, 3, 4, and 6 did not increase significantly when However, when comparing the individual belief chan ges for each group, the treatment group had more participants who experienced belief changes for at least three of the seven beliefs than the control group. The task sort scores for the treatment group were significant when comparing pre and post scores. A comparison of the control and treatment group revealed a significant difference in the post mathematical task sort beliefs changed significantly compared to the control grou p, due to the participation in the 12 week intervention focused on mathematical tasks. According to the National Research Council (2001), approaches to teaching have little effect on student learning. Instead, successful interaction among three elemen ts:
125 are the keys to successful student learning (National Research Council, 2001, p.111 ). mathematical tasks is the initial step in the MTF. This growth is an important first step in recognizing the potential a mathematical task has in eliciting thinking about mathematics from a student. Research studies that have focused on engaging both preservice and inservice teachers in learning about the level of cognitive demand have ind icated growth in the type of mathematical tasks posed by participants (Arbaugh & Brown, 2005; Boston & Smith, 2011; Kosko et al., 2010; Silver & Stein, 1996). Researchers have also found that there are cases where a teacher elects to use a mathematical tas k with features of high cognitive demand and in the course of implementing the mathematical task, pedagogical decisions are made by the teacher which lower the overall cognitive demand of the mathematical task (Arbaugh & Brown, 2005; Silver & Stein, 1996). The instructional decisions teachers make e ffect the about mathematics instruction play an important role in the presentation of mathematics (Remillard & Bryans, 2004). A mathematics instruction depend upon the successful implementation of reform based mathematics instruction. Therefore, based on evidence from prior research about the positive effects learning about the cog nitive demand of mathematical tasks has on mathematics instruction, the results from this study show the potential to help create opportunities for learning in elementary preservice teacher mathematics education courses.
126 The intervention appears to have pr ovided PSTs with the opportunity to participate in a learning environment where they solved and discussed mathematical tasks together and acquired the tools necessary to identify the cognitive demand of mathematical tasks (Brown et al., 1989, Gee, 2008). T he following section will provide changes and knowledge of the level of cognitive demand of mathematical tasks as identified by the results of this investigation. Expl anations for Results This section offers explanations for the results obtained in the study. How did the selected for interviews use mathematical tasks? What types of ma thematical tasks did the PSTs have a difficult time identifying? Potential explanations for these questions are presented in the remainder of this section. The Intervention Provided PSTs with the Opportunity to Do Mathematics The intervention for the stud y was designed to provide the PSTs with opportunities to work together on mathematical tasks and participate in an authentic letter writing exchange. During the intervention, the cognitive demand of mathematical tasks was used as a focal point for PSTs to consider the prospects mathematical tasks provide students to learn mathematics. Throughout the 12 were presented with a variety of opportunities to engage with mathematical tasks. Aspects of the intervention provided PSTs with a c hance to develop mathematical task knowledge (Chapman, 2013) and challenge their beliefs about mathematics instruction. The intervention was designed and implemented in ways consistent with the situative learning theory (Brown et al., 1989; Cobb & Bowers, 1999; Gee, 2008; Lave,
127 1991; Putnam & Borko, 2000; Rogoff, 2008). For instance, the situative learning theory places an emphasis on the activities that take place within the community to help the individual acquire knowledge (Cobb & Bowers, 1999). Based up on the situative learning theory, in order for learning to occur the following components need to be included in the design of the learning environment: establish classroom norms, identify the role of the teacher educator as the facilitator of the group, p rovide opportunities to socially construct the meaning of mathematics with peers, and engage in activities which positioned PSTs to acquire the tools to learn about the cognitive demand of mathematical tasks. These four components were included in the inte rvention. First, to begin building a classroom learning environment the instructor and participants created course expectations. The initial activity in Session 1 of the intervention had the participants work together in groups of four to create expectati ons for the course, each other, and the instructor. Each group had the opportunity to share their expectations with the class. The instructor also shared the expectations she had for the participants. During Session 2, the instructor facilitated a discussi on about what it means to teach student centered mathematics. Many participants shared their own also find myself giving up halfway through the problem and waiting for the teacher to tell Following the discussion, the participants were asked how to convert an improper fraction to a mixed number. In small groups, the participants were asked to share their methods for converting. The objective for this activity was to find out how the participants were converting the improper fraction to a mixed number. Were they
128 recalling a procedure or did they represent an improper fraction with pictu res? The instructor wanted the participants to consider how they were taught mathematics. Next, the participants watched a video clip of a child struggling to remember the procedure for converting a mixed number to an improper fraction. The clip was intend ed to challenge or confirm their beliefs about mathematics teaching and learning. In this same session, the participants were introduced to the Standards for Mathematical Practice. This introduction included background knowledge about the Standards for Mat hematical Practice and a video where students used the Standards for Mathematical Practice in their learning of mathematics. This sequence of activities assisted in the creation of classroom norms. From the start of the intervention, the instructor made i t clear to the participants that her role was to facilitate learning and the role of the participants was to engage in learning about mathematics through mathematical tasks (Szydlik et al., 2003). Previous studies (e.g., Szydlik et al., 2003; van den Kiebo om & Magiera, 2010) made the role of the instructor explicit in the establishment of creating a learning environment where preservice teachers engaged in problem solving and generating mathematical explanations. The opportunities to participate in solving mathematical tasks in groups and discussing solution strategies has been recommended as an effective practice to help preservice and inservice teachers become comfortable using mathematical tasks during instruction (Remilard & Bryans, 2004). The mathematic al tasks used in the intervention were chosen based upon mathematical content and certain features of cognitive demand.
129 An explanation about the mathematical tasks and the ways in which the tasks were used in the intervention is provided to form an unders tanding of what events may have contributed to the results in the study. In Session 3, participants were first introduced to mathematical tasks that involve fractions. The participants were asked to work in small groups with four fraction mathematical task s that were taken from a National Assessment of Education Progress 2009 fourth grade test. After the participants worked on the tasks, they were asked to consider three questions: 1) What mathematical practices did you use to solve these tasks? 2) How are these two tasks similar? How are they different?; 3) What prior knowledge did students need to know in order to answer the questions? The participants were given the opportunity to discuss the results of each task and address the questions. Following this activity the participants were asked to work on a Brownie Problem (Flores & Klein, 2005), which can be classified as a fair sharing task. The participants solved the task as a group and then shared their results with the group. This activity provided the p articipants with the opportunity to see how their peers represented the solution to the problem in different ways. For Session 4 and 9, participants were asked to work on mathematical tasks for the Inside Mathematics (Noyce Foundation, 2012). The tasks se lected included a fifth grade fraction task that asked students to consider where to place fractions on a number line and compare fractions using fraction benchmarks, and a decimal task. Both tasks were given to participants to solve and discuss on respect ive occasions. Following the discussion, participants were given student work that showed how students interpreted the mathematical task. First the participants were asked to use a rubric to
130 score the students work. Second, the participants were asked ques tions that positioned them to analyze the student work responses and group them in categories. For ents would respond to the tasks and what decisions teachers need to make to help the student understand the mathematics needed to answer the task. In Session 6 a multiplication fraction task was presented to PSTs. This task posed a dilemma for some partic ipants because they were only familiar with the standard algorithm to multiply fractions. The task asked the participants to use pattern blocks to represent the multiplication and provide a reason for why the standard algorithm for fraction multiplication works. For Session 10, the participants were placed in groups and asked to work on a measurement task that involved finding measurements of Barbie and then were assigned to construct a real life model of Barbie. This task allowed the participants to decid e how to act in order to develop a real life model of Barbie. Several design features for the intervention focused on developing mathematical task knowledge through doing the mathematic tasks, discussing results, analyzing student work from the mathematica l task, and discussing features for the mathematical task that would elicit cognitive demand. The participants were formally introduced to the levels of cognitive demand during the second part of Session 4. The first activity was to complete two fraction tasks, selected by the instructor, which had features of low and high level cognitive demand. The participants were asked to identify the strategies they could use to solve
131 the two tasks, identify the mathematics con cepts, and make a list of similarities and differences between the tasks. The instructor provided sheets of poster paper for the participant responses for strategies, mathematics concepts, and similarities and differences. The poster papers were all combin ed into one list. Afterwards, the participants were asked to classify the two tasks as low or high cognitive demand and experience in the intervention with low and high l evel cognitive demand. In preparation for Session 5, participants were asked to review the third grade sample questions for the Partnership for Assessment of Readiness for College and Careers assessment items (e.g., Flower Gardens, Fractions on the Number evaluating the Mathematical Practices used to solve the task and the level of cognitive demand for the task. During class, the participants engaged in a discussion based upon their findings for mathematical tasks. A lot of questions arose from the participants about how to adequately prepare students for mathematical tasks. Many of the participants felt that if they were not able to answer the mathematical tasks on their own, their students would be unable to answer them as well. Following this discussion the preservice teachers completed a middle school mathematical task sort. The participants were asked to sort the middle school tasks into categories of their choice and to describe the categories on their paper. Then the p articipants discussed with their group the categories they created. Next, the instructor passed out a table with the four categories that represented the level of cognitive demand. The participants were asked to sort the tasks into one of the four categori es. Afterwards, participants were asked to share one task from each of the categories, identify features that allowed for
132 classification to that category, and describe what adaptations could be made to raise the cognitive demand (if the task was classified as low cognitive demand). The conclusion for the middle school task sort activity ended with a discussion about the features of mathematical tasks and the classification of cognitive demand for the tasks. For instance, will a task with a real life applic ation always be classified as high level cognitive demand? The goal for the discussion was for participants to realize that the cognitive demand in written form had the potential to complete the mathematical task. When the mathematical task was used in con text, the potential cognitive demand of the mathematical task may or may not be maintained based upon factors in the classroom. A limitation to the mathematical task sort is the lack of an opportunity to see how students interact with the mathematical task . Preservice teacher education faces the challenge provided to PSTs when they teach their own students (Norton & Kastberg, 2012). Therefore, this study sought to provide PSTs with an authentic opportunity, which was participating in a letter writing exchan ge. The letter writing exchange facilitated the development of mathematical tasks and allowed the PSTs to pose the tasks in the pen pal letters. The letter writing three important aspects of mathematics teaching practice: posing tasks, analyzing On three occasions, the participants had the opportunity to write letters to third grade students. These opportunities served as a time for participants to experiment with posing mathema tical tasks. The participants had the opportunity to write about anything they wanted to in the letters. For instance, some of the participants chose to share their experience in college, such as attending sporting events; others included information
133 about where they grew up. The only requirement was to include a mathematical task based upon a pre determined Common Core State Standard. The instructor selected the standard to reflect the content that the students were learning at their school. The participa nts had the opportunity to write their first pen pal letter during Session 6. For the letter they were asked to create a mathematical task using a second grade Common Core State Standard for partitioning circles and rectangles. The participants had the fre edom to create the mathematical task. The instructor suggested the use of the Task Analysis Guide to identify features of the mathematical tasks that exhibited the cognitive demand they desired from the task. Participants had the opportunity to write their second pen pal letter during Session 9. The mathematical task focused on a second grade standard that dealt with place value. For instance, if you have the number 897, explain what the 8, 9, and 7 represent. The third pen pal letter was written in Sessio n 11 and focused on a second grade standard for the foundations of multiplication. One of the intentions of the letter writing exchange was to make a connection between the university coursework and the actual practice of teaching (Kagan, 1992). Prior rese arch studies recommend structured mathematics (Kagan, 1992; Philipp et al., 2002; Philipp et al., 2007). All in all, the intervention provided learning experiences that focused on creating classroom norms, establishing roles of the instructor and participants, structuring activities where mathematical task knowledge was developed through social interactions with peers, and acquiring the tools to learn about the cognitive demand of mathematical tasks. The activities, which occurred during the intervention, are the unit
134 of analysis for this study (Rogoff, 2008). Situating the PSTs in a classroom environment, which focused on doing mathematical tasks, considering the cognitive demand a particular task elicited in students, and posing mathematical tasks to the student, culminates in an authentic activity which develops knowledge for teaching mathematics the cognitive demand of mathematical tasks and changes in mathematics instruction beliefs suggest that participants benefited from the opportunities presented to them in the intervention. Exposure to mathematical tasks in the intervention also contributed to Belief Changes need to be recognized in order for mathematical tasks to be properly implemented. With this in mind, the intervention was focused on providing opportunities, which challenged PSTs beliefs about mathematics instruction by using the situative perspective to inspire the design for the intervention and provide an authentic letter writing ex change experience. The treatment group had experiences in the intervention that positioned them to solve mathematical tasks together, discuss mathematics, and share solutions ( Szydlik et al., 2003). These experiences allowed participants to experience math ematics as a sense making discipline (Szydlik et al., 2003; van den Kieboom & Magiera, 2010). The quantitative data for beliefs indicate that more PSTs in the treatment group increased their belief scores on at least 3 beliefs, compared to the control grou p, who experienced less change in beliefs. One possible explanation for the score increase can be attributed to the intervention and specifically the letter writing exchange. PSTs in the
135 treatment group had the opportunity to experiment with posing mathema tical tasks through a letter writing exchange. This opportunity could have ignited a change in Belief 5 because PSTs had the chance to build a relationship with a pen pal, write et al., 2013). The letter writing exchange served as an important tool in promoting change in worked exclusively for 3 sessions with a third grade student on mathematical ta sks related to place value. All of the participants had the opportunity to work with students during the study, due to the practicum placement, which was a component of the third semester in the elementary teacher preparation program. The control and treat ment group both shared a common field experience during the 12 week study. Both groups were assigned to a K 5 classroom at the same school for one day a week. All participants had the same assignment of creating and implementing two mathematics lessons to a group of students. The difference between the two groups who contributed to a significant belief change for B eliefs 5 and 7 was the letter writing exchange that the treatment group took part in. The letter writing exchange contributed to a significant be lief change for B elief 5 due to the nature of the authentic activity. R esearchers Philip et al. (2007), reported similar findings for the Mathematical Thinking Experiences Live ( CMTE L ) group of PSTs. The researchers intentionally focused the CMTE understood mathematics they have not preselected mathematical tasks to implement with an individual child (p. 446). The letter
136 writing exchange incorporated the co nstruction of a mathematical task and provided PSTs with the opportunity to see how the student responded to the task. The PSTs in the treatment group did not have an opportunity to instruct students, hematical ability. The letter writing exchange was an opportunity for PSTs to experiment with posing mathematical tasks (Crespo, 2003). In essence, the letter writing exchange supported the change in B elief 5. Philip et al. (2007) found that the PSTs who w ere in the CMTE treatments experienced a larger increase in change of beliefs for both Belief 5 and 7 than the PSTs treatments. The MORE treatments had PSTs observe teachers who w ere either reform oriented or located in a classroom convenient to the college campus. Results from both studies indicate that experiences where PSTs implement mathematical tasks with nking are more likely to experience changes in Beliefs 5 and 7 than PSTs who are only observing mathematics instruction. Similar results for Beliefs 5 and 7 were recorded by Bahr et al. (2013). In their study, PSTs who were simultaneously enrolled in an el ementary mathematics methods course and attended a field experience at a local public school experienced significant changes in Beliefs 5 and 7 due to the direct application of knowledge they were taught in their university course. Consequently, both studi es, Philipp et al. (2007) and Bahr et al. (2013), experienced similar results for PSTs who had opportunities to work with children and engage children in learning mathematics. These results align with findings from this study in which PSTs had an opportuni ty to see how students learn.
137 The qualitative data provides evidence of the sophistication of beliefs held by Heidi and Nora. For instance, Heidi describes her desire to make mathematics meaningful to students through real life mathematical tasks. She was able to teach a group of students during her practicum experience and then use this same knowledge about teaching and apply it to constructing a mathematical task for her pen pal. According to Ernest (1989), knowledge of organization and management of teac hing mathematics is acquired through experimentation. Heidi had several opportunities to experiment with teaching and posing mathematical tasks, which contributed to her development of pedagogical content knowledge. PSTs experimented with writing and posi ng mathematical tasks. They wanted to get a response from their pen pal and the task set up did not always provide the response they were expecting. Therefore, they had to structure the task differently the next time they wrote the letter. The tasks from H eidi and Nora are shown in Appendix J . Learning how to present tasks to students was a challenge because the PSTs did not have the opportunity to respond to students in the moment and did not have prior the other hand, this challenge provided time for PSTs to consider how to respond to their pen pal, which is not always possible in a live classroom setting (Crespo, 2003). Additionally, PSTs had to experiment with methods to learn more about their pen pal practicum and letter writing exchange experiences provided different contexts for PSTs to experience how students learn mathematics. In the interviews there was conflicting evidence about what it meant for a student to be chall enged during mathematics instruction. For instance, on one hand, when Nora
138 described how she structured a lesson, she stated that students needed to be challenged and she was coming up with questions to do this. On the other hand, she wanted students to ex perience success and therefore she made it easy for them by giving them something she knew they would get so the students would be set up for success. A possible explanation for this occurrence can be explained by Philipp et al. have little mathematical experience with children, they are initially able to project only their own, too often negative, mathematical experiences onto those of children, with the result that they avoid placing children in challenging situations (e.g., nev er asking children to solve a problem before they have been shown students. However, her past experiences with learning mathematics strongly e ffect the experiences she pro vides her students with, and she does not want her students to feel the way she did when learning mathematics (e.g., frustrated, upset). Heidi provided similar explanations with regards to challenging students and having them feel successful. One differen ce between the way Nora and Heidi presented mathematics is that Heidi used scaffolding in her mathematics instruction by first using easy mathematical mathematical ability. S he also did the same thing when constructing and posing the mathematical tasks in the letter writing exchange. Crespo (2003) found that the types of problems created and posed by PSTs in a letter writing exchange and related questions for the problem chang ed over time. The participants in her study contributed these changes to their exposure to mathematical tasks in their university course. Similarity,
139 Heidi felt the intervention contributed to her learning more about mathematics concepts because of all the hands on activities. Heidi had significant changes on 4 of 7 beliefs measured in this study. These reported changes for Heidi are significant as we consider instruction ( Crespo, 2003; Philipp et al., 2007; Vacc & Bright, 1999). A finding from the interviews that was underlying the way both participants interacted with their students (both practicum and pen pal) was an element of caring. This study recognized the fact that PSTs care about students (Philipp et al., 2007). The notion of caring about students was present in the mathematics instruction presented by mathematics and did not want them to be embarrassed if they did not know the answer. She also wanted the students to see that mathematics was fun. Nora was interested in the emotional outcome of learning mathematics and did not focus on the mathematical content. In contrast, Heidi was concer ned with not overwhelming students with new information and therefore implemented mathematical tasks in a learning progression. Heidi felt the gradual increase in cognitive demand was the best way to reach all students. The Circles of Caring model (Philipp et al., 2007) was used to hook PSTs into he argument is that such conceptions have a powerful impact on teaching through such processes as the
140 Nora would probably benefit from another semester of a similar mathe matics course paired with a practicum experience, in order for her to continue to teach differently than the way she herself was taught (Nicol, 1999). Hence, this intervention might have more of an impact if it was introduced earlier in the elementary educ ation program. mathematical thinking during the letter writing exchange supported PSTs views of mathematics instruction. hat both PSTs were trying to teach mathematics in a way that was opposite from how they experienced mathematics instruction. This fact goes against the literature that argues teachers are more likely to teach mathematics the way they have been taught, and perpetuates the belief that mathematics is a set of rules and procedures with no connections (Ball, 1988; Cady et al., 2006; Kagan, 1992). Although it is clear that their beliefs about mathematics instruction influence the experiences they provide to their practicum students and pen pal, there is hope that these two PSTs are developing a vision for reform based mathematics instruction (Nicol, 1999). Explanations for the Increase d awareness of level of cognitive demand for mathematical tasks scores is their experience with analyzing features for mathematical tasks presented during the course of the intervention. Additionally, the treatment group had the opportunity to write, pose, and receive a response from a student for mathematical and post mathematical task sort, and were si gnificantly higher following their participation in the intervention than the scores of the control group who did not participate in the
141 intervention. Data from pre and post mathematical task sort scores by categories of cognitive demand reveal that parti cipants in the treatment group had the highest accuracy on tasks, which had the potential for eliciting low levels of cognitive demand. Osana et al. (2006) studied a similar population of elementary preservice teachers and had the same result for the class ification of low level cognitive demand mathematical tasks. Another significant finding was , the accuracy to classify a task as eliciting high levels (e.g., procedures with connections, doing mathe matics) of cognitive demand decreased for participants in the treatment group. A similar result was found in the literature (Osana et al., 2006; Kosko et al., 2010). Osana and colleagues (2006) theorized that the result of inaccurate high level classificat ions was due to the participants not having opportunities to work with children because they lacked the participants experienced the same result. Perhaps an explanation for the inaccuracy of classification for high level tasks is due to lack of exposure to high level tasks in the PSTs K 12 academic schooling (Crespo, 2003). The letter writing exchange was intended to support the PSTs in developing their understanding of the levels of cognitive demand. For instance, each PST completed a table that allowed them to hypothesize about the level of cognitive demand for the mathematical task they created and provide evidence for this decision prior to sending the pen pal letter (Norton & Kastberg, 2011). After the task came back, they were asked to re evaluate the level of cognitive demand. An analysis of whether or not the mathematical task created by the PST was correctly identified by cognitive demand was
142 not completed since t his form served as a reflective tool. However, the researcher did look over the tables after every letter writing occasion to see how the PSTs classified the mathematical tasks. The two participants who participated in interviews, both stated that they pos ed challenging tasks to their pen pals. The tasks from all three occasions for each participant can be seen in Appendix J . An extension of this study would be to analyze the mathematical tasks for features representative of the levels of cognitive demand and compare the classifications made by PSTs. PSTs in the treatment group were exposed to a variety of mathematical tasks during the 12 week intervention. The comparison of task sort scores for the two groups reveals that the treatment group scores were significant after the 12 week intervention. Hence, exposing PSTs to mathematical tasks, having them discuss the mathematical task, do the mathematical task, and create their own mathematical tasks led to increases in knowledge for identifying mathematical tasks by low and high levels of cognitive demand. Effectiveness of the intervention The effectiveness of the intervention in producing changes in PSTs knowledge of cognitive demand for mathematical tasks and changes in Belief 5 and 7 is notable given t he short amount of time in which the intervention occurred. The focus of mathematical tasks and the letter writing exchange in the intervention can be credited with contributing to those changes. The intervention provided exposure to the level of cognitive demand of mathematical tasks (Arbaugh & Brown, 2005; Boston & Smith, 2011; Silver & Stein, 1996), group participation in performing mathematics tasks (Crespo, 2003; Nicol, 1999; Szydlik et al., 2003) , and experimentation with writing and posing mathematic al tasks (Crespo, 2003; Norton & Kastberg, 2012). Crespo (2003)
143 suggests a lot of attention has been given to the ability of PSTs to solve mathematical tasks; however little attention has been given to their ability to construct and pose mathematical tasks to students. Based upon the research about mathematical tasks, the researcher felt it was important for the PSTs to consider the different features of mathematical tasks by examining mathematical tasks in curriculum materials and ones that serve as assess ment items, and then discussing the aspects of mathematical tasks. This study demonstrates that PSTs developed aspects of mathematical task knowledge as described by Chapman (2013). This knowledge is related to the mathematical knowledge needed for teachin g as defined by Hill et al. (2008). Hill et al. (2008) who used an egg metaphor to represent the knowledge a teacher needs to teach mathematics. Within the egg there exist two sides, which are divided into sections. The side that relates to this study is P edagogical Content Knowledge. Within Pedagogical Content Knowledge, the section that closely relates to the knowledge participants acquired through the intervention is the Knowledge of Content and Students (KCS). According to Hill et al. (2008), KCS is def content knowledge intertwined with On a small scale, the letter writing exchange positioned PSTs to create a mathematical task based upon a mathematical stan dard, to consider how the student would think about the mathematical task, and afterwards to examine how the student responded to the task. There were several tools which PSTs utilized to learn about mathematical tasks.
144 The Mathematical Tasks Framework and Task Analysis Guide were the tools, utilized by PSTs while designing mathematical tasks for the pen pal letters. Additionally, the Task Analysis Guide was used when analyzing and discussing mathematical tasks during intervention sessions. The Mathematical Tasks Framework was used to show the progression of how a mathematical task is enacted within mathematics instruction. Through consistent focus on the cognitive demand of mat hematical tasks, the tools provided to the treatment PSTs were useful in constructing mathematical tasks for their pen pal letters and identifying features of mathematical tasks. The design of the intervention contributed to the significance for 2 of 7 be liefs about mathematics instruction measured by the beliefs survey. For the treatment group, mathematics instruction and supported overall PST learning (Putnam & Borko, 2000). Nic ol (1999) recommended that PSTs have opportunities to work with students in order to produce a change in their beliefs about mathematics instruction. The letter writing experience was an authentic activity that presented opportunities to PSTs that might no t normally occur in an elementary classroom. The letter writing exchange helped maintain a sense of community because the PSTs cared about their pen pal students and spent time crafting mathematical tasks for their letters. In preparation for the future, the intervention could be improved in ways that would further influence the beliefs held by PSTs and challenge their ability to create cognitively challenging tasks. For instance, PSTs would benefit from more interactions with students through letter writi ng. Many of the participants provided verbal feedback
145 for the pen pal activity, and the most frequent response was that they wished they had more opportunities to correspond with their pen pals. During the 12 week intervention, the participants received on ly three letters. Additionally, PST could have benefitted from opportunities to collaborate on analyzing the mathematical tasks prior to sending them to the pen pal students and after the letters came back to the PSTs. These opportunities would have allowe d for PSTs to discuss how the pen pal students were approaching and solving the problems. These interactions could have led to a change in Belief 6, which was not significant in this study. Further, there are a couple of questions that were ignited from th is study and worth investigating in the future: Could more frequent interactions between PSTs and pen pals contribute to increased belief changes? ,and Would the mathematical tasks posed by the PSTs increase in cognitive demand? These are examples of quest ions that would be worthy of investigation in future studies. The following section will describe how the intervention and methodology utilized in this study builds on prior PST research and can inform future elementary preservice teacher mathematics cours es. Contributions of this Investigation This study contributes to the growing body of research on elementary mathematics teacher preparation. The results of this study provide evidence of the effectiveness for applying the situative learning perspective t o PST learning, as utilized in several other preservice teacher education studies (i.e., Crespo, 2003; Nicol, 1999; Philipp et al., 2008; Szydlik et al., 2003 ) . In this study, a situative learning approach to the design and facilitation of the intervention appeared to be successful in supporting development of knowledge and beliefs about mathematics instruction for PSTs. The PSTs had opportunities to engage in mathematics by actually doing mathematical tasks
146 (Crespo, 2003, Nicol, 1999; Szydlik et al., 2003; van den Kieboom & Magera, 2010), and contributed to learning within the intervention. Specifically, as suggested by prior studies (i.e., Crespo, 2003; Norton & Kastberg, 2012; Szydlik et al., 2003), engaging PSTs in solving mathematical tasks provided the m with an opportunity to construct their own tasks, and experiments with posing the task was a valuable tool for learning about the level of cognitive demand and influencing their beliefs about mathematics instruction. The PST interviews provide evidence t o support this claim. In order to offer meaningful experiences to PSTs, it is highly recommended by researchers (i.e., Bahr et al., 2013; Nicol, 1999; Norton & Kastberg, 2012; Philipp et al., 2007) to include authentic learning opportunities that focus on how students think and learn about mathematics. Teacher educators are faced with a dilemma about the best ways to prepare PSTs (MET, 2012). These dilemmas arise when teacher educators are met with resistance from the PSTs to learn mathematics. Many PSTs h old a preconceived notion that their mathematical knowledge is sufficient for teaching elementary school students and therefore do not see the value of learning more mathematics (MET, 2012; Philipp et al., 2007). Teacher educators face the test of structur ing experiences for PSTs that challenge their beliefs about mathematics and provide opportunities to engage students in learning mathematics. This study also contributes to research on effective field experiences for elementary preservice teachers (Bahr et al., 2013; Philipp et al., 2007). The current study focused on providing PSTs with opportunities to learn about the levels of cognitive demand and to participate in a letter writing exchange. This study built on the success of teacher educators who used the Task Analysis Guide and mathematical task sort
147 with practicing teachers in professional development settings (i.e., Arbaugh & Brown, 2005; Boston & Smith, 2011, Silver & Stein, 1996). In professional development studies, the participants had access to students and were able to use mathematical tasks in their mathematics instruction. The participants in this study did not have their own students. Therefore the letter writing exchange provided the authentic experience of writing and posing mathematical t asks to students (Crespo, 2003; Norton & Kastberg, 2012). In the following section, the conclusions, limitations, and suggestions for future research are presented. Conclusions, Limitations, and Future Research The results of this study are important for several reasons. First, the intervention contributed to a significant increase in Belief 5 and 7. Both of these beliefs represent points of the study. However, a limitation of the study was the fact that Belief 6 was not significant. In the future the intervention can include the component of having the PSTs share their letters with one another and analyze the mathematical tasks prior to sending the letters by solving the mat hematical task and discussing how the pen pal student would solve the task. Upon the letters arrival of the letters, the PSTs can analyze and discuss to discuss how the pen pal student solved the mathematical task. This initial component to intervention co uld potentially provide the opportunity for Belief 6 to change significantly. The significant increase in 2 of the 7 beliefs is important because it has been documented in the literature that trying to change PSTs beliefs is difficult due to years of prior experience as students themselves. Based upon the interview responses from the two participants who had either a high or low change in beliefs, it is difficult to determine
14 8 which aspects of the intervention had a significant overall effect on PSTs beliefs . Taking this into account, the intervention e ffected the PSTs in different ways depending on their university course) is not so focused on math. It is more our d iscovery of how to do elementary mathematics course because it did not feel like the mathematics instruction she experienced in this past. Taking this into consideratio n, the intervention did have an effect perspective about the intervention in this statement, I feel like that everything we talked about all year about the conceptual understandin g that definitely because beforehand if you would have told me to write a problem I would have done it how I learned. Like just give them the problem and not ask them to explain it, not ask them to critique g. Just give them the problem and they would just do the procedure. Rather now I know that to understand how they are thinking and why they answered that way is Heidi and Nora were selected bas ed on their contrasting results from the belief survey. It would have been helpful if a participant were chosen that had average belief changes to gain their perspective about the intervention. Although a conclusion from two participant responses is hard t o draw, it can be stated that the intervention did e ffect PSTs beliefs about mathematics instruction in different positive ways. One question that about mathemati cs and mathematics instruction once they make the transition to a practicing teacher? Second, exposure to different mathematical tasks, doing mathematics as a community, writing mathematical tasks, and posing mathematical tasks through letter
149 writing con and categorize the tasks by high and low level cognitive demand. A more in depth look at the results from the mathematical task sort reveals need for PSTs to have experience with high level cognitive demand tasks. The participants were most successful identifying the low level cognitive demand tasks. A suggestion to improve the knowledge of high level tasks could be to take low level tasks and change certain features of the t ask to potentially elicit a higher level of cognitive demand. Also, would the results have been different if the intervention was started earlier in the elementary education program as opposed to waiting until the third semester in the program. The researc her found that several PSTs in the treatment group struggled with their own mathematical content knowledge of fractions. Due to the lack of knowledge, they struggled with the fraction tasks and sometimes became frustrated when other PSTs were able to expla in the task and they did not understand. The researcher assumed that the PSTs were already prepared to take the course with the required prerequisite content knowledge. The PSTs lack of knowledge limited some of their interactions with the mathematical tas ks. An additional construct to measure for future studies involving mathematical tasks and beliefs is mathematical content knowledge. Third, this study utilized both quantitative and qualitative methods to analyze PSTs beliefs and mathematical task knowl edge. Reflection forms for mathematical tasks, pen pal letters, and videotaped class sessions supported the identified changes in the study. Future research studies would seek a larger sample size, a different instrument to collect data for mathematical ta sk knowledge, a longer time frame for data collection, and an additional instrument to measure mathematical content knowledge.
150 The sample for this study was a convenience sample and the groups were already intact. Prior to the beginning of this study, the majority of the participants had been in their group for one year. The participants were familiar with one another and took all of their courses together every semester. The two groups could have interacted with one another and discussed what was occurrin g in the different classes resulting in a diffusion threat to internal validity. Another threat to internal validity that this study faced was attrition. There were participants in both groups who chose not to participate in the study. Additionally, both groups had participants who did not complete the pre and post assessments for either beliefs or mathematical task sort. The data for these participants (e.g., the ones who chose not to participate and the ones with incomplete assessments) were removed fro m the data analysis, which poses another threat to internal validity. The beliefs instrument caused a threat to internal validity. First, there was only one form for the survey that was used for both the pre and post test. The participants could have bec ome familiar with the survey questions and therefore rushed through the post test by filling in the same information. Second, the original version of the survey was designed and implemented for use on the web. Therefore, it was only validated as a web base d survey. A researcher took careful time and precision to produce an identical paper and pencil form of the beliefs survey. The paper and pencil form of the beliefs survey was not validated and could pose a threat to the study. Next, the mathematical task sort was designed for use in professional development settings with practicing teachers. The mathematical task sort does not have validity or reliability information because it is a tool used to bring awareness to
151 features of mathematical tasks based upon the cognitive demand. Other researchers, tool. The researcher asked the participants in both groups to give reasons why they chose the classification category for the mathem atical task. Less than half of the and post mathematical task sort. Arbaugh and Brown (2005) and Boston and Smith (2011) used the reasons as qualitative data and incorporated it into the o verall score for the mathematical task sort. Unfortunately, this data was not available for this study due to lack of compliance from the participants. Finally, there exists the possibility of researcher bias. The researcher was the instructor for the tre atment group. Participants in the treatment group were aware that the researcher was completing this study for part of the degree requirements and may have felt obligated either to participate or give perceived desirable responses on the beliefs survey. Th e researcher made a conscious effort to stay true to the design structure of the intervention and communicate openly with participants about the study. Notes were taken after each class session for the treatment group, and both the treatment and control gr oup were videotaped. The video recordings were used to monitor the study. However, the implementation of the intervention could still have posed a threat to the internal validity. In summary, the researcher attempted to conduct a study following rigorous guidelines for education research. The results from this study are multifaceted and serve to inform the design of future elementary mathematics education courses. It would be beneficial to continue to use mathematical tasks with preservice teachers by
152 pro viding them with a classroom environment where they can discuss features and solutions of the mathematical tasks. Additionally , PSTs can extend their pedagogical knowledge by learning how to pose questions that engage students to further learn about th e mathematical task. Aspects of learning how to ask questions arose from the data in both PST interviews. Another avenue worth exploring is the connection between the mathematical content included in the mathematical task for the letter writing and the Com mon Core State Standards. Are PSTs correctly interpreting the Common Core State Standards? Lastly, it would be interesting to measure the PSTs beliefs at the beginning of the first semester of the elementary education program and then again at the end of t he third semester, after the intervention is implemented in the first semester. Vacc and Bright (1999) measured belief s at the beginning and ending of each semester during the elementary education program to determine which events contributed to a si gnificant belief change. Following a similar timeline, it would be beneficial to measure PSTs beliefs the first semester in the program and again in their third semester to determine whether the intervention in the third semester has a significant impact o n PSTs provide more data about the effects of the intervention. Future research studies can learn how to support PSTs as they experiment with writing and posing mathematical tasks for students.
153 APPENDIX A ELEMENTARY MATHEMATICAL TASK ANALYSIS FORM Name: _______________________ Date: ________________________ Level of Cognitive Demand Predicted Evidence for Prediction, or Reasons Observed Evidence for Observation, or Reasons Why Expected Level of Cognitive Demand Were not Observed Memorization Procedures without connections Procedures with connections Doing mathematics Use the student response to your task to reflect on and analyze ways in which you can improve future task design. **Form adapted from Kosko et al. (2010)
154 APPENDIX B TASKS FROM THE QUASAR ELEMENTARY SCHOOL TASK SORTING ACTIVITY
155 APPENDIX C EXAMPLE OF QUESTIONS FOR GENERAL INTERVIEW GUIDE APPROACH How did you develop your math lessons for your practicum experience? When you developed your lessons did you focus on the cognitive demand of the mathematical task(s) or mathematical questions? Describe how you implemented mathematical tasks in your teaching of mathematics. How did you know when students understood the mathemati cal tasks? What do you think is the best way for students to learn math? What are the three most important characteristics of good mathematics teaching? (Raymond, 1997) (Raymond, 1997) For t he pen pal assignment, what did you do to make a mathematical task for your student? Did your student provide you with insight about how they were doing the mathematical task? Describe your overall thoughts about the pen pal assignment. Did any experience in the course help prepare you to construct mathematical tasks?
156 APPENDIX D COURSE OVERVIEW FOR 12 WEEKS OF THE STUDY
157 Session 1: August 21 st , 2013 Session 2: August 28 th , 2013 Session 3: September 4 th , 2013 Session 4: September 11 th , 2013 Session 5 : September 18 th , 2013 Session 6: September 25 th , 2013 Introductions & Data Collection for Beliefs Instrument Data Collection for pre mathematical task sort Introduction to the Task Analysis Guide Middle School Mathematical Task Sort and discussion M ultiplication and Division Task Assignment: Complete four fraction tasks Compare features of fraction tasks Partition Task Fractions Task on ordering fractions with student work Adding and Subtracting Fraction Task First pen pal letter with mathematical task focused on partitioning shapes Part to Whole Tasks Assignment: Read article Designing and Implementing Worthwhile Tasks Assignment: Analyze 3 rd grade fraction tasks for the PARCC with the Task Analysis Guide Assignment: Read article Thinking through a lesson: Successfully implementing high level tasks Mathematical Practices Gallery Walk Session 7: October 2 nd , 2013 Session 8: October 9 th , 2013 Session 9: October 16 th , 2013 Session 10: October 23 rd , 2013 Session 11: October 30 th , 2013 Session 12: November 6 th , 2013 Dividing fraction task Decimal Task Fraction, Decimal, and Percent Task Measurement Task Beliefs Posttest Post Mathematical Task Sort Second Pen Pal Letter with mathematical task focused on place value Third Pen Pal letter with mathematical task focused on algebraic thinking
158 APPENDIX E OVERVIEW OF PEN PAL LETTER WRITING Session Date Common Core State Standard 6 September 25 th , 2013 CCSS.M ath.Content .2.G.A.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the w hole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. 9 October 16 th , 2013 CCSS.Math.Content.2.NBT.A.1 Understan d that the three digits of a three digit number represent amounts of hundreds, tens, and ones; e.g. 706 equals 7 hundreds, 0 tens, and 6 ones. 12 November 6 th , 2013 CCSS.Math.Content.2.OA .C.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
159 APPENDIX F EXAMPLE LESSON PLAN Session 5 Goals: Participants will learn about features of mathematical tasks. Participants will recognize that mathematical tasks vary in level of cognitive demand. Participants will learn about the Task Analysis Guide. Participants will discuss characteristics of mathematical tasks based upon the level of cognitive demand. Activity 1 : Comparing Tasks (15 minutes): Participants will work on two tasks called Necklace Task and the Musical Fractions Task Necklace Task Karen is stringing a necklace with b eads. S he puts green beads on 1/2 of the string and purple beads on 3/10 of the string. How much of the string does Karen cover with beads? Musical Fractions Task The fraction assigned to a musical note represents its value. The following are common values: 1/ 16, 1/8, ! , " How many combinations of these notes can you find that equal the value of a whole note? Participants will compare the features of the two tasks in dyads. They will make a list of similarities and differences. Next, they will discuss which t ask would provide the most information about Afterwards, results from the discussions will be shared. A list of qualities for both tasks will be made and compared. The participants will develop criteria for the two tasks. Activity 2: Mathematical Task Sort (60 minutes) : Participants will sort a set of mid dle school mathematical tasks. A handout will be provided where they can place the task into a category (memorization, procedures without connections, procedures with conne ctions, doing mathematics). Participa nts will complete the middle school mathematics task sort, first individually and then with a partner.
160 The results for the categories will be shared with the whole group and each task will be discussed. The participants will develop a set of criteria for each category. Next, the teacher educator will introduce the participants to the Task Analysis Guide. The participants will compare their categories to the Task Analysis Guide. A discussion will be held about the feature s of each task. Overview of Levels of Cognitive Demand (15 minutes): A discussion about the levels of cognitive demand and how they relate to student achievement in mathematics. Activity 3: Adding and Subtracting Task (45 minutes): In groups, participant s will complete the adding and subtracting task using pattern blocks. First, the participants are asked to complete the task on their own. Next, they are asked to share the results with the group. Lastly, participants are asked to create their own task usi ng the pattern blocks. Closure for the activity consists of a whole group discussion about the mathematical task. The teacher educator asks the participants to discuss the mathematical knowledge needed to complete the task, where students might struggle, a nd the potential to be a cognitively demanding task. Wrap up for the session (35 minutes): Discussion about homework problems from the previous class session. Discussion about the course reading from the previous class session and how it relates to this
161 APPENDIX G SAMPLE OF BELIEFS INSTRUMENT
162 APPENDIX H CROSSTABULATION FOR BELIEF SURVEY DATA Table H 1. Belief 1 crosstabulation values. Group Change score 0, expected count Change score 0, actual count Change score 1, expected count Change score 1, actual count Change score 2, expected count Change score 2, actual count Treatment 19 20 6.7 7 4.3 3 Control 21 20 7.3 7 4.7 6 Table H 2. Belief 2 crosstabulation values. Group Change score 0, expected count Change sc ore 0, actual count Change score 1, expected count Change score 1, actual count Change score 2, expected count Change score 2, actual count Treatment 21.9 20 3.8 5 4.3 5 Control 24.1 26 4.2 3 4.7 4 Table H 3. Belief 3 crosstabulation values. Group Chan ge score 0, expected count Change score 0, actual count Change score 1, expected count Change score 1, actual count Change score 2, expected count Change score 2, actual count Treatment 14.8 14 8.1 10 7.1 6 Control 16.2 17 8.9 7 7.9 9 Table H 4. Belief 4 crosstabulation values. Group Change score 0, expected count Change score 0, actual count Change score 1, expected count Change score 1, actual count Change score 2, expected count Change score 2, actual count Treatment 17.1 17 7.1 6 5.7 7 Control 18. 9 19 7.9 9 6.3 5 Table H 5. Belief 5 crosstabulation values. Group Change score 0, expected count Change score 0, actual count Change score 1, expected count Change score 1, actual count Change score 2, expected count Change score 2, actual count Treatm ent 12.4 7 9.5 11 8.1 12 Control 13.6 19 10.5 9 8.9 5
163 Table H 6. Belief 6 crosstabulation values. Group Change score 0, expected count Change score 0, actual count Change score 1, expected count Change score 1, actual count Change score 2, expected cou nt Change score 2, actual count Treatment 17.6 16 7.1 8 5.2 6 Control 19.4 21 7.9 7 5.8 5 Table H 7. Belief 7 crosstabulation values. Group Change score 0, expected count Change score 0, actual count Change score 1, expected count Change score 1, actua l count Change score 2, expected count Change score 2, actual count Treatment 19.5 14 7.6 10 2.9 6 Control 21.5 27 8.4 6 3.1 0
164 APPENDIX I CODING FOR THEMATIC ANALYSIS
165 APPENDIX J Tasks Task 1: Task 2:
166 Task 3:
167 Nora: Task 1:
168 Task 2: Task 3:
169 APPENDIX K INFORMED CONSENT
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180 BIOGRAPHICAL SKETCH Kristen Apraiz graduated from Florida State University in December 2003 with a Bachelors of Science in secondary mathematics education. After graduation, Kristen exte nded her studies by pursuing a Master of Science in mathemati cs education from Florida State University and graduated in August 2004. Immediately following graduation, Kristen started her first teaching position at New Smyrna Beach High School in New Smyr na Beach, Florida. She taught high school mathematics, coached the boys and girls swim teams, and was the adviser for the student government association. Kristen was also the teacher leader for the Algebra professional learning community. In 2008, Kristen accepted a faculty position at Daytona State College in the College of Adult Education. At Daytona State College, she taught high school mathematics courses and college preparatory mathematics courses. In 2010, Kristen enrolled at the University of Florida to begin working on a Doctor in Philosophy in curriculum and instruction with an emphasis on mathematics education. During her third year in the doctoral program she accepted a te aching position at Ivy Hawn Charter School for the Arts in Lake Helen, Florida. She served as the middle school mathematics teacher and elementary mathematics coach. Kristen graduated in August 2014 and is a clinical assistant professor in the School of Te aching and Learning in the College of Education at the University of Florida. She continues to research elementary preservice teacher education with a focus on the cognitive demand of mathematical tasks.