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The efficacy of an acquisition strategies model for middle school students with learning problems

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The efficacy of an acquisition strategies model for middle school students with learning problems
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by Penny R. Cox.

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THE EFFICACY OF AN ACQUISITION STRATEGIES MODEL
FOR MIDDLE SCHOOL STUDENTS WITH
LEARNING PROBLEMS














By

PENNY R. COX


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


2001




























Copyright 2001

by

Penny R. Cox













ACKNOWLEDGMENTS

The efforts of many people have contributed to the success of this

research. I would like to express my appreciation to all who had a part in making this project possible.

I want to thank the students who participated in this study and to thank their parents for permission to work with their children. Without them, none of this could have happened.

I extend special thanks go to Lynn Jamison, Tarcha Rentz, Sari Heipp, Christina Zamora, Jody Joseph, and Darby Desmond, the teachers who so graciously allowed me to come into their classrooms and who offered their assistance and support in collecting data. I thank them for their interest in this project and their help in making it happen. I greatly appreciate their kindness in putting up with my interruptions and numerous requests for "just one more" probe sheet or piece of information.

I would also like to thank the members of my committee for their support in the preparation, planning, and implementation of this study. Dr. Mary Kay Dykes, Dr. Cecil Mercer, Dr. Maureen Conroy, and Dr. David Miller have provided valuable assistance and advice.

Dr. Mary Kay Dykes has proved to be a source of great encouragement throughout my doctoral program. She has been my committee chair, mentor, advisor, and friend. I thank her for believing in me.








I give special thanks to my husband, Dewey, without whose loving

encouragement and support this journey would never have begun. He has put up with an absent wife while continuing to bolster me throughout this process. I thank him for being with me and helping me through.

Finally, my parents, Jim and Terry Ritch, deserve more thanks than can be adequately expressed. They gave me the gift of valuing education. In 1966, my father left his long-time and secure position with the railroad to enter the ministry-a step of faith that meant giving up a financially secure future. He did so with the prayer that the Lord would see that his children were educated. His prayer has been answered.













TABLE OF CONTENTS


page


ACKNOW LEDGMENTS ................................

A BSTRA C T .........................................


. . . . . . . . . . . iii

. . . . . . . . . . v ii


CHAPTERS


1 INTRODUCTION ............................

Standards for Mathematics Learning .............
Statement of the Problem ......................
D efinitions ..................................
D elim itations ................................
Lim itations .................................
S um m ary ..................................


.. 1

.. 3


10
* 10 .10


2 REVIEW OF RELEVANT LITERATURE ..................


....... 12


Information Processing Theories ............................... 13
Theoretical Basis for Concrete, Representational, and Abstract
Instruction .............................................. 19
Math Achievement for Students with Learning Problems ............. 21
Criteria for Selecting Concrete-to-Representational-to-Abstract
Studies for Review ....................................... 25
Review of Relevant Concrete-to-Representational-to-Abstract
Lite rature ............................................... 26
S um m ary ................................................. 35

3 RESEARCH METHODOLOGY ................................ 37


Instructional Procedures ........
S etting .....................
Subjects ....................
M aterials ....................
Statistical Analysis ............

4 RESULTS ...................

Analyses of Data .............
Sum m ary ...................


. .........
..........
..........


. . . .. . . . . .. � � .� . . . . . . . . . . .
. .. � �.. . .� � . . . . . . ..� � . . . . . . . . .
. . . . . . . . . . . . � � �.. . . . . . . .
. . . . . . . . . . . . . .� , � . . �.. . . . . .








5 D ISC USSIO N ...................................


Summary of Research Questions and Findings .................... 56
Relevance of Findings to the Acquisition Strategies Model ........... 57
Lim itations ................................................ 58
Im plications ..................................... ........... 59
S um m ary ................................................. 60

APPENDICES

A SAMPLE PROBE SHEET .................................... 63

B INFORMED CONSENT LETTER ............................... 64

C ASSENT SCRIPT READ TO PARTICIPANTS ..................... 66

D PRETEST, POSTTEST, AND FOLLOW-UP SCORES FOR PARTIC IPANTS ......................................... 67

R EFER ENC ES ................................................. 71

BIOGRAPHICAL SKETCH ........................................ 76


........... 55













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 EFFICACY OF AN ACQUISITION STRATEGIES MODEL
FOR MIDDLE SCHOOL STUDENTS WITH LEARNING PROBLEMS

By

Penny R. Cox

August 2001

Chair: Mary Kay Dykes
Major Department: Special Education

This study tested the efficacy of an acquisition strategies model for facilitating the acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems. The effectiveness and efficiency of concrete-to-representational-to-abstract, concrete-to-abstract, representational-to-abstract, and abstract-only instructional sequences were compared. Ninety-six participants were randomly assigned to four treatment groups within their schools. Each group received instruction in the form of one of the sequences being examined. Concrete and representation-level instruction was implemented in small groups. Abstract level lessons were administered as whole-class activities.

Participants were enrolled in middle schools in one north central Florida school district. Instruction took place within those schools and was implemented








by special education teachers. Dependent variables were posttest and follow-up test scores.

Statistical analyses of the data revealed that posttest and follow-up test scores were significantly higher than pretest scores. No differences were found between posttest and follow-up measures. There were no differences between groups when pretest differences were controlled, indicating that all treatments were equally effective.

The various acquisition strategies of the model are supported by the

results of this investigation as effective and efficient instructional interventions. More involved and longer teaching interventions based on the model produced basically the same results but were time and personnel intensive. Abstract-only instruction produced results similar to those of the longer CRA, CA, and RA strategies but required less class time and minimal effort on the part of the teacher. Therefore, daily abstract-only level practice of multiplication facts was effective in improving the retention and retrieval of multiplication facts for this population.













CHAPTER 1
INTRODUCTION

Demands for more stringent academic standards for students are increasing. More states are requiring students to demonstrate grade level performance before being promoted to the next grade. The public is also more concerned about how American students compare to their peers in the world community.

Students with learning problems are expected to meet the same academic requirements as their normally achieving peers. Such demands can be especially arduous for this group since it is well documented that students with learning problems experience academic difficulties. The difficulties they encounter result in poor reading, writing, math, and social/emotional skills. Such difficulties persist into adulthood placing students with learning problems at risk for further problems on their jobs, in their communities, and in their homes and relationships (Cronin & Patton, 1993).

Efforts to alleviate such problems for students with learning problems must begin as early as possible in the school experience. Teachers need to know and use effective and efficient teaching methods to help students with learning problems to not only learn more but also to acquire skills after as little instructional time as possible. Such instructional demands make it especially important that the nature of effective instruction be understood. Knowing how to








teach students to maximize acquisition and retention of content is necessary for students to achieve academic success.

Instruction should be based on information processing models that explain how individuals incorporate information into their knowledge base. Generally, information that is received undergoes various types of processing at different levels of understanding. The use of effective processes transforms the information to create new understanding. Individuals then store the new knowledge in memory to be used during processing of more information in the future. Existing understanding combines with new information and processing strategies to build the individual's knowledge base.

The acquisition strategies model proposed by Baumeister and Kellas (as cited in Mercer & Snell, 1977) provides a format for instruction by which occurrence of memory and retrieval of data are facilitated by using strategies to enhance memory, retention and retrieval. According to this model, learners select strategies to code information that is received. After coding the information, it is practiced then recoded into a form for output. For the purpose of this research, concrete and representational coding strategies were used. Concrete strategies require subjects to manipulate three-dimensional objects to solve math problems. Representational strategies use drawings in place of objects. Concrete and representational strategies can be used within the same instructional sequence or independently of each other. After concrete and/or representational coding strategies are implemented, practice takes place. In this investigation, practice was done at the abstract level. Abstract strategies involve solving problems without the aid of manipulatives of drawings. During the









recoding phase, the learner uses the information and strategies to solve a math problem. Finally, the solution is made known by means of a motor output (e.g., speaking, writing).

It is important to identify coding strategies that are effective for helping students with learning problems increase the rate at which they acquire, retain, and retrieve data. Research to validate the effectiveness of concrete and representational coding strategies and abstract level practice is needed to help determine whether or not the use of such strategies constitutes effective, efficient instruction for students with learning problems.


Standards for Mathematics Learning

The National Council of Teachers of Mathematics (NCTM), the

professional organization of math teachers, addressed the issues of determining math curriculum and identifying effective teaching strategies for math skills. The NCTM, in collaboration with professionals in areas requiring math skill (e.g., engineers, information science specialists, scientists) developed and published a set of standards outlining math skills that students need at each level of instruction. The standards include descriptions of how the skills should be taught. The NCTM recommends that all students be guided to construct mathematical concepts through activities designed to facilitate discovery through problem solving.

The NCTM standards are supported by professionals within mathematics. The fact that the standards were developed by professional groups with expertise in mathematics and related areas is viewed as a strength (Giordano,






4

1993). The instructional methods recommended in the standards are supported as best practices because they are in the math education literature. However, such practices have not been validated for use with students with learning problems.

Support for NCTM standards outside the field of mathematics, specifically within the field of special education, has not come so readily. Objections to the NCTM standards are many. First, professional collaboration is not a sufficient basis for setting national standards (Rivera, 1993). Instead, research-based data are necessary to determine what skills and instructional practices should be recommended not just inclusion of ideas as best practice. Second, the absence of references to students with disabilities and other diverse backgrounds is a stated concern of special educators and researchers (Hofmeister, 1993; Hutchinson, 1993; Mercer, Harris, & Miller, 1993; Rivera, 1993). A related objection was expressed by Hutchinson (1993) who stated that it is more important to focus on the quality of teaching and learning for students with disabilities rather than on access to the same curriculum as their nondisabled peers. Hutchinson stated reservations about the NCTM's recommended instructional methods since there are no data to show that students develop concepts and acquire skills because they are exposed to mathematical experiences.

Special educators and researchers suggest that different teaching

approaches (especially for lower level skills) need to be used with students with learning problems. Research needs to be conducted to identify and validate models of curriculum and instructional practices that give rise to effective






5

instructional strategies for students with learning problems. Presently, no such model has been validated.


Mathematics Deficits in Students with Learning problems

Students with learning problems achieve below the level of their typically achieving peers in math. In studies conducted during the 1980s and 1990s, students with learning problems performed poorly on measures of minimum skills (Algozzine, O'Shea, Crews, & Stoddard, 1987 and achieved only half the expected growth for each year of school (Cawley, Parmar, Yan, & Miller, 1996). Causes of Mathematics Disabilities

Mathematics deficits in students with learning problems have been reported to be due to factors that can be grouped into two categories. One category is student factors. These factors are directly related to how students go about working math problems and how well they use the strategies and skills they have learned. For example, students with learning problems often choose incorrect or inefficient strategies to solve problems. In addition, students with learning problems work slowly and inaccurately.

The second category of factors responsible for deficits in math

performance is related to instruction. The first instructional factor contributing to poor math achievement concerns the lack of class time spent teaching students math concepts. Second, teachers often move from one concept to another before students have achieved mastery. These problems are exacerbated by the fact that no model has been determined as the basis for effective instructional practices.








Statement of the Problem

The effects of math deficits are cumulative. Students who have difficulty learning math facts are at risk for developing more and greater learning problems as they progress through school. Students who cannot develop strong basic skills have no foundation upon which to build more complex skills (Fleischner, Garnett, & Shepherd, 1982). It is essential that instruction be effective in order to maximize student learning. The development of a validated model of instruction which has demonstrated a positive, effective, and predictable impact on learning of math facts is a priority in order to provide efficient teaching practices. No such model is currently found in the literature.

Given the low level of math performance of students with learning

problems, it is important to identify instructional strategies that result in efficient and predictable learning. The acquisition strategies model proposed by Baumeister and Kellas (as cited in Mercer & Snell, 1977) provides a framework for instruction and test-specific coding strategies. The model provides flexibility in testing strategies in isolation or in combined with other strategies. The effectiveness of each coding strategy or combination of strategies can be determined by comparing the output of learners using various strategies.

Concrete and representational coding strategies and rote repetition in the form of abstract level practice are strategies that have been used in math instruction with students with learning problems. The concrete-torepresentational-to-abstract (CRA) instructional sequence has been effective for teaching math skills to students with learning problems (Mercer & Miller, 1992b?). However, research is needed to pinpoint more effective and more








efficient ways to teach students with learning problems. Toward that end, this investigation examined manipulations of the CRA instructional sequence to determine if both coding strategies and the abstract level practice are all necessary for efficient student learning. The experimental questions were as follows:

1. Is the CRA model effective in facilitating acquisition, retention, and

retrieval of multiplication facts for middle school students with learning problems?

2. Is the CA model effective in facilitating acquisition, retention, and

retrieval of multiplication facts for middle school students with learning problems?

3. Is the RA model effective in facilitating acquisition, retention, and

retrieval of multiplication facts for middle school students with learning problems?

4. Is an A-only model effective in facilitating acquisition, retention, and

retrieval of multiplication facts for middle school students with learning problems?

5. Are CRA, CA, RA, and A models equally effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems?

Comparisons of individual and group performance levels will help

determine if middle school students with learning problems acquire, retain, and retrieve math facts efficiently when their instruction does not include all three elements of the CRA sequence.

Definitions

Definitions for terms used in this research are provided below.

Abstract level of instruction "involves the use of numerals" (Mercer, 1997, p. 581). During abstract instruction, mathematical problems are solved without








the aid of objects or drawings. While this is the most time-efficient strategy, the effectiveness of isolated abstract level practice is not known.

In this research, accuracy refers to the number of correct responses on untimed practice activities at the concrete level and on timed probe sheets used during abstract instruction.

Arithmetic involves (a) real numbers in terms of their characteristics and relationships and (b) computations (primarily addition, subtraction, multiplication, and division) using real numbers (Gove, 1986).

Basic facts are addition facts with single digit addends, subtraction facts with minuends to 18 and subtrahends to nine, multiplication facts with factors to nine, and division facts representing inverse operations for basic multiplication facts.

The concrete level of instruction "involves the manipulation of objects (Mercer, 1997, p. 581). Students receiving concrete instruction solve mathematical problems using three-dimensional objects. Concrete coding strategies are basic processes for all initial tasks and instruction in math in the acquisition-strategies model proposed by Baumeister and Kellas (in Mercer & Snell, 1977).

A criterion is a predetermined level of performance for a task.

Students with low academic achievement who are not served in special education programs are at risk for dropping out of schools. The academic focus of drop out prevention programs is remediation of deficits in basic skills.








Students with learning problems are those students served in special education and drop out prevention programs whose academic achievement is below expected levels.

A probe samples a target behavior. In this investigation, probes of basic math facts consist of a single page of 60 multiplication facts. Students had one minute to complete as many multiplication facts as possible.

The number of times a behavior occurs during a specified period is rate. For this research, rate refers to the number of math facts correctly completed in one minute.

The phrase representational level of instruction is used synonymously with the terms semiconcrete and semiabstract. This level of instruction "involves working with illustrations of items in performing math tasks (Mercer, 1997, p. 581). Such illustrations can include pictures, dots, or tallies. Representational activities typically follow concrete strategies in math instruction. They are included in the acquisition strategies model by Baumeister and Kellas (as cited in Mercer & Snell, 1977) as coding strategies.

Special education consists of services for students with disabilities if such disabilities result in a need for specially designed instruction or related services (Heward, 2000).

Specific learning disabilities disorders "in one or more the basic

psychological processes involved in understanding or in using language, spoken or written, which may manifest itself in an imperfect ability to listen, think, speak, read, write, spell, or to do mathematical calculations" (U.S. Office of Education, 1977, p. 65083).








Delimitations

This investigation has the following delimitations. First the study was

limited to students served in one north central Florida school district. Second, it includes only middle school students in public schools. Third, only students receiving math instruction in special education and dropout-prevention classes were included. Fourth, multiplication facts were the only operation taught.


Limitations

Caution should be exercised in generalizing results of this research.

Since only students in some middle schools in one Florida school district were represented, results should not be extended to students in other areas. Further, generalizing findings to students in other types of educational settings or with other types of learning problems should be done carefully. Finally, multiplication facts were the focus of this investigation, so results should not be generalized to other skills.


Summary

Deficits in math performance for students with learning problems are well documented. Efforts to increase math skills of students with learning problems should include identification of effective instructional strategies. Additionally, examination of such strategies is needed to determine their most efficient uses.

This research examined the effectiveness of acquisition-strategies model proposed by Baumeister and Kellas (in Mercer & Snell, 1977). Manipulations of the CRA sequence were tested. Specifically, CA, RA, and A sequences were examined and compared to each other and to the CRA sequence to determine






11

their potential as effective and efficient interventions for increasing retention and retrieval of multiplication facts. Comparisons of student achievement determined how well students with learning problems acquired math facts with the shortened instructional sequences. Such comparisons were useful in deciding if the CA, RA, and/or A strategies were more efficient ways of helping students with learning problems acquire math facts.

A review of literature relevant to this research is presented in Chapter 2.

The methodology used for the investigation is described in Chapter 3. Results of the study are reported in Chapter 4 and discussed in Chapter 5.













CHAPTER 2
REVIEW OF RELEVANT LITERATURE As discussed in the previous chapter, students with learning problems do not achieve at the same levels in math as nondisabled students (Algozzine, O'Shea, Crews, & Stoddard, 1987). To alleviate such problems for this population, it is necessary to identify instructional strategies that are effective and efficient for enhancing retention and retrieval of information for students with learning problems. This study was designed to identify such strategies.

The purpose of this chapter is to discuss the theoretical basis for the investigation and to present a review of related professional literature. Specifically, literature regarding use of the concrete-to-representational-toabstract (CRA) instructional sequence to enhance retention and retrieval of information with students with learning problems will be reviewed.

This chapter includes five sections. First, the information processing

theories providing the foundation for this research are discussed. Second, the theoretical basis of concrete, representational, and abstract retention and retrieval strategies are discussed. Third, math achievement for students with learning problems is considered. Next, the criteria for selecting studies to be reviewed are listed. Finally, studies in which math instruction included elements of the concrete-to-representational-to-abstract instructional sequence are reviewed.









Information Processing Theories

The proposed research was designed to investigate the effects of the concrete-to-representational-to-abstract instructional sequence on the acquisition, retention, and retrieval of multiplication facts by students with learning problems. The study was based on information processing models that propose explanations for how information is assimilated by learners. Such models have been produced in attempts to understand what information individuals learn and how they acquire the information (Mercer, 1992). The implications for the use of such models in the field of education are obvious. A better and more complete understanding of how students learn is required to design effective and efficient instruction that capitalizes on the natural processes of assimilation of new knowledge. According to Siegler (1988) understanding such processes requires consideration of the kind of information being processed, the strategies the learner employs to manipulate the information, and the degree to which information can be processed given limits imposed by the capacity of memory.

Information processing models are used to demonstrate how information is incorporated into the learner's knowledge base. Such an occurrence is a fluid operation that occurs in a systematic manner. Many models have been proposed that illustrate how learners construct knowledge (Mercer, 1992).

This study was based on two such models. The first provides an overview of information processing theory in general. The second provides detail to describe the learning process within one element of the general model. Both models are discussed in the sections that follow.









Model Providing an Overview of Information Processing

The model shown in Figure 1 was based on the work of Keith Lentz and presented by Mercer (1992). It is illustrative of the manner in which information moves through the processing system. The process begins with a sensory stimulus, continues by moving through any number of possible manipulations at various levels of understanding, and leads to a response by the learner. Arrows are used to illustrate that the learning sequence is not entirely linear. Information can move back and forth through integrative and executive processes and longterm memory to undergo a variety of manipulations. In addition, the learner must function within a specific context. Factors including the type of stimulus, the teacher, materials used, and the response type alter what is required of the learner. Such conditions are represented in Figure 1 as contextual demands.

The types of processing included in the ovals at the top of the figure and in the integrative processing section represent the general procedures for assimilating information. In pre-perceptual processing, the learner detects visual and auditory features of a stimulus but does not attach meaning. At the perceptual processing level, symbols (e.g., phonemes, numerals) are perceived as meaningful. Perceptions at this level are automatized as skill levels increase. In recognition processing, perceptual units are combined to create meaning in the form of concepts.

Integrative processing is the term used to describe the learner's

comprehensive approach to a task. An individual's prior knowledge and choice of strategies are important elements of the integrative processing component of the model.

























Executive Processing


Contextual Demands
Stimulus Factors I Teachers, Materials, Settings I Response Factors
Figure 1. Information Processing Model


Declratie CoditinalProcedural KnowedgeKnowedgeKnowledge

Long-term Memory






16

Short-term memory is also represented by an oval at the top of the figure. However, it is not connected directly to the ovals representing the general processes. The short-term memory area serves as a place where information is held, manipulated, or synthesized in order to construct new information. To get to short-term memory, information progresses through a series of procedures beginning with preperception. From there, information may move between the general procedures or go directly to the integrative processes. The unit of information may move among the general procedures, the integrative processes, and short-term memory as assimilation occurs. Knowledge from long-term memory can be retrieved and held in short-term memory to be incorporated into integrated units to develop new meaning.

The executive processes and long-term memory form the foundation of

the model and support the active stages of processing. Executive processes are metacognitive in nature and include such skills as maintaining attention and providing internal feedback. Long-term memory is foundational in the model because learning is based on prior knowledge and beliefs. Three categories of knowledge are contained in long-term memory. Declarative knowledge covers facts and concepts. Procedural knowledge includes steps for performing tasks. Knowing when and how to integrate declarative and procedural knowledge is the function of conditional knowledge.

This model is appropriate for the proposed research in that knowledge present in long-term memory constitutes the basis for developing multiplication facts to the declarative level. Knowledge from all three categories of long-term memory will be retrieved to be held and used in short-term memory. For example, declarative knowledge will be used in the form of numbers and groups.








Procedural knowledge of the steps for solving multiplication problems and conditional knowledge of how to use the declarative and procedural elements to complete the multiplication task is needed also. The manipulation that occurs in short-term memory is illustrated in the model by Baumeister and Kellas (as cited in Mercer & Snell, 1977).


Acquisition Strategies Model

The model of acquisition strategies was adapted from a model proposed by Baumeister and Kellas (in Mercer & Snell, 1977) to compare strategies used by students with and without mental retardation to learn paired associates. According to this model, the student receives a stimulus then selects a strategy for learning the pair. A variety of coding rehearsal strategies for learning the pairs is available. Coding strategies are followed by rote repetition, decoding, and finally output.

Figure 2, the acquisition strategies model, is the basis for the proposed

research. Adaptations are made to the coding strategies portion for the purpose of including procedures relevant to the concrete-to-representational-to-abstract instructional sequence. The abstract portion of the sequence is presented in rote repetition. Additionally, the decoding element is renamed recoding to more accurately describe synthesis as included in current literature.

Movement from the strategy selector includes choices from a variety of

processing sequences. The concrete and representational coding strategies can be used within the same sequence or independently of each other. After the selected coding strategies are implemented, rote repetition at the abstract level takes place.

















Figure 2. Acquisition Strategies Model


C oding Strategies Repetition) [Representational


E i Itk ' Strategy






19

During the recoding stage, the learner uses the information and strategies to solve a math problem. Finally, the solution is made known by means of a motor output (e.g., speaking, writing, pointing).

The full concrete-to-representational-to-abstract sequence begins with the concrete coding strategy then continues through the coding strategies to the representational strategy. The remainder of the sequence continues through the abstract level or rote repetition, recoding, and output. When either the concrete or the representational strategy is omitted, the concrete-to-abstract or the representational-to-abstract sequence results.


Theoretical Basis for Concrete, Representational, and Abstract Instruction

Piaget (1960) characterized children's intellectual development as having four levels. The first stage described is the sensorimotor stage, which begins at birth and lasts until about age two. During the sensorimotor stage, children learn to interact with and respond to their environment. The preoperational stage is second and lasts until about age seven. It is during this stage that children begin to use representational thought. Their thoughts, however, are based on their perceptions of concrete experiences. The concrete operational stage is next and lasts until about age 11. During this stage, children develop the ability to focus on more than one attribute of a situation at a time. It is also during this period that logical thought begins, though primarily at the concrete level. Finally, children in the formal operational stage "are able to think abstractly without reference to actual objects or actions in the real world" (Fuys & Tischler, 1979, p. 34).








Bruner (1966) described similar stages of understanding. His enactive, iconic, and symbolic levels parallel Piaget's levels of intellectual development. Sovchik (1989) described activities that would be characteristic of children in each of Bruner's stages. Children in the enactive stage might be able to perform a motor task, but not be able to describe how it is done. At the iconic level, mental images of concrete objects are developed. When employing symbolic knowledge, children are able to use language and symbols. For example, generating a number sentence to go with a word problem in math requires symbolic knowledge.

In 1977 Underhill described three developmental levels of learning

experiences related to mathematics. The goal of the first two levels, concrete and semiconcrete, is to provide students with opportunities to practice math skills. At the concrete level, students learn through the use of visual, kinesthetic, and tactile practice. Semiconcrete experiences are only visual. Abstract experiences are not manipulative or visual. Instead, "it is assumed that earlier concrete experiences have enabled the learner to think iconically when using the abstract symbols" (Underhill, Uprichard, & Heddens, 1980, p. 30).


Concrete Level Instruction

Students functioning at Bruner's (1966) concrete level need to be taught using examples and methodology at the concrete level. Such instruction necessitates that the learner engage in problem-solving using three-dimensional objects. The learner attends to the objects being manipulated as well as the mathematical procedures they represent (Mercer, 1997). In concrete level








instruction, a student is given a problem to solve (e.g., 2 x 4). In solving this problem, the student uses two cups to represent groups and places four objects into each cup. After counting the number of objects in the cups, the student is able to state that two groups of four objects equals eight objects. Representational Level Instruction

Instruction at the representational level involves using pictures, tallies, or other items that stand for concrete objects in order to solve math problems (Mercer, 1997). Such instruction coordinates with the semiconcrete level of understanding (Bruner, 1966). A representational lesson would entail the student making a drawing to solve a math problem. For example, when presented with the problem 2 x 4, the student draws two circles to represent two groups, then draws four tallies in each circle. After counting all the tallies, the student will be able to state that two groups of four tallies equals eight tallies.


Abstract Level Instruction

Lessons taught in accordance with Bruner's (1966) abstract level are characterized by the absence of manipulatives or graphic representations. Instruction at the abstract level requires the use of numerals (Mercer, 1997). Mercer points out that students who have difficulty with math need concrete and representational instruction before they work math problems abstractly.


Math Achievement for Students with Learning Problems

Students with learning problems achieve less in math than their typically achieving grade-level peers. Deficits in achievement among students with math






22

disabilities have been demonstrated in studies over the last 30 years. Studies of overall math achievement show students within this population make less than the expected amount of growth in math skills. Poor performance on measures of minimum skills is common among students with learning problems (Algozzine, O'Shea Crews, & Stoddard, 1987).

The effects of math deficits are cumulative. As Cawley, Parmar, Yan, & Miller (1996) found, discrepancies in math performance of students with learning problems increase as students get older. Students with mild learning difficulties were found to make half the growth in math skills as other students for the same amount of instruction (Cawley et al., 1996; Cawley & Miller, 1989). Two years of instruction resulted in only one year of growth. This level of achievement is consistent with earlier findings in which students with learning problems exiting high school were determined to be functioning at fifth to sixth grade levels in math (Cawley, Kahn, & Tedesco, 1989). Similarly, McLeod and Armstrong (1982) found that intermediate and secondary school students with learning problems performed at third and fourth grade levels in math.

Reports of data from numerous studies indicate that students with

learning problems have deficits in skills with basic math facts. For example, Goldman, Pellegrino, and Mertz (1988) found students with learning problems responded more slowly then nondisabled students when computing basic addition facts. Studies examining computation skills found that nondisabled students attempted more problems and answered more problems correctly than students with learning problems when computing basic math facts (Garnett & Fleischner, 1983; Fleischner, Garnett, & Shepherd, 1982).








Causes of Mathematics Disabilities

Causes of problems in mathematics for students with learning problems can be classified as student factors or instructional factors. Student factors include identifiable disabilities such as visual-spatial disorders and slow processing speed (Garnett, 1992). Poor verbal and language skills (Miller & Mercer, 1997) also contribute to math disabilities. Other student factors are related to differences in the way students process information. Students with learning difficulties approach math problems in qualitatively different ways than typically achieving students (Montague & Applegate, 1993; Kulak, 1993). Such differences suggest that students with learning problems have different learning styles (Deshler, Schumaker, Alley, Warner, & Clark, 1982), which might prevent them from processing information at the same rate as normally achieving students. Other student factors that might result in math disabilities are related to students' inability to perform basic operations and to plan strategies for completing problems. Students who cannot execute basic operations (Kirby & Becker, 1988) or who have not been able to automatize skills (Garnett, 1992) have difficulty learning math skills. Miller and Mercer (1997) found that students with learning problems often lack the skills needed to choose appropriate strategies to complete problems.

Instructional factors related to math deficits for students with learning

problems are associated with instruction and content presentation. For example, too little time spent on arithmetic instruction and not making connections between math concepts and language, written symbols, and practical applications are found to be part of the problem of poor student performance in






24

math (Garnett, 1992). Fixed and spiral mathematics curricula also make learning more difficult for many students (Jones, 1982; Englemann, Carnine, & Steely, 1991; Miller & Mercer, 1997). Fixed curriculum refers to a predetermined amount of content that must be covered within a school year. Teachers often continue through the content at a pace that does not allow low achieving students to grasp concepts fully before a new one is introduced. Similarly, teachers often present lessons by simply writing on the board and talking about concepts so that as much material can be covered in a class period as possible (Pieper & Deshler, 1985). Such presentation can result in concepts remaining unconnected to student experiences. Math instruction using a basal series is designed in a spiraling format. Spiral curriculum provides for cursory coverage of many concepts from year to year without allowing sufficient practice for struggling students to master skills.

Instructional factors contributing to math skill deficits for problems are

further exacerbated by the fact that strategies for teaching math skills to students with learning problems have received little attention from researchers (Fleischner & Manheimer, 1997). Without research-based data, instructional strategies cannot be identified as effective or ineffective. Additionally, relying on methods described in math texts and other commercially obtained materials does not ensure the use of research-based best practices in math instruction. Finding appropriate materials for teaching math is complicated (Miller & Mercer, 1997) by the fact that only 3% of materials are actually tested with students before being released in texts (Sprick, 1987).








Results of Math Deficits

Students who experience math problems are at risk for developing more serious learning problems as they progress through school. Students who are unable to acquire and maintain basic math facts have little foundation upon which to build other skills (Fleischner, Garnett, & Shepherd, 1982). As a result, their "difficulties are manifested in the inability to acquire and apply mathematical skills and concepts, to reason, and to solve mathematical problems" (Rivera, 1997, p. 20).


Criteria for Selecting Concrete-to-Representational-to-Abstract Studies for Review


Studies reviewed in the following section were selected according to the following criteria:

1. The study was conducted to examine the effects of concrete,

representational, and/or abstract instruction on acquiring basic math facts.

2. Performance on math skills was the dependent variable.

3. Subjects in the studies were students in elementary or middle schools.

4. Studies provided quantitative data and included descriptions of subjects, procedures, and results.

5. Studies were conducted between 1970 and 2000.

Literature searches were conducted using Education Resources

Information Center (ERIC), Education Abstracts, and Library User Information Service (LUIS). In addition, ancestral searches of some reference lists were conducted.








Review of Relevant Concrete-to-Representational-to-Abstract Literature

A total of 15 studies meeting the selection criteria were identified. Five of the studies were conducted using all three elements of the concrete-torepresentational-to-abstract (CRA) instructional sequence. One study was designed to compare the effectiveness of CRA instruction to instruction using only the abstract portion of the sequence. In six investigations, researchers looked at differences in results of concrete only, representational only, and abstract only instruction. In one study the order of concrete and abstract of instruction considered. Finally, the effectiveness of using only parts of the CRA sequence (i.e., CA and RA sequences) to teach math skills was the focus of two studies.

Researchers use a variety of terms synonymously with the terms

concrete, representational, and abstract as defined in Chapter 1. "Manipulative" is the term used in place of concrete by Armstrong (1972), Evans and Carnine (1990), Marsh and Cooke (1996), Prigge (1978), and Scott and Nuefeld (1976). The representational level is referred to as "semiconcrete" (Hudson, Peterson, Mercer, & McLeod, 1988; Miller & Mercer, 1993; Peterson, Mercer, & O'Shea, 1988; St. Martin, 1975), "symbolic" (Fennema, 1972), "pictorial" (Scott & Neufeld, 1976), and "graphic" (Smith, Szabo, & Trueblood, 1980). Abstract level instruction was referred to as "nonmanipulative" by Prigge (1978).

The sections that follow contain analyses of each of the studies with

consideration for skills being taught, characteristics of subjects, research design and measurement procedures, and results. The studies are presented according to the topics of the investigations as described above.








Studies of the CRA Sequence

The five studies in this category were designed so that the entire CRA

instructional sequence was used. In three of the studies, the effectiveness of the CRA sequence to teach specific math skills was investigated. Hudson, Peterson, Mercer, and McLeod (1988) focused their study on place value while Harris, Miller, and Mercer (1995) and Miller, Harris, Strawser, Jones, and Mercer (1998) were interested in the viability of the CRA sequence to teach multiplication. Sealander (1991) and Miller and Mercer (1993) concentrated their efforts on a phenomenon they call crossover effect. Crossover is the point at which students receiving CRA instruction answered more problems correctly than incorrectly. Subjects in both studies were taught basic addition, subtraction, or division facts or coin sums.

Characteristics of subjects. Two of the five studies (Harris et al., 1995; Miller et al., 1998) involved second grade students identified as LD, emotionally handicapped (EH) or low achievers. Hudson et al. (1988) described their subjects as having LD and being ages eight and eleven years. Miller and Mercer (1993) also studied students with LD as well as students at risk for LD, and students identified as educable mentally handicapped (EMH). Sealander's (1991) subjects were in first or second grades and were not described as having disabilities.

Research design and measurement procedures. Miller et al. (1998) used a group design to compare performances of 123 subjects including students with disabilities, low achievers, and normally achieving students. Data were analyzed using a repeated measures multivariate analysis of variance. The remaining four








studies (Harris et al., 1995; Hudson et al., 1988; Miller & Mercer, 1993; Sealander, 1991) are single subject investigations using multiple baseline designs. Measurements of the dependent variables include counting the number of correct and incorrect responses (Harris et al., 1995; Hudson et al, 1988; Miller & Mercer, 1993), pre and/or posttest scores (Harris et al, 1995; Hudson et al., 1988; Sealander, 1991), and visual analysis and celeration slopes (Sealander, 1991).

Implementation. Teachers participating in the Miller et al. (1998) and Harris et al. (1995) studies were trained to use Multiplication Facts 0 to 81 (Mercer & Miller, 1992a) before beginning instruction. Miller et al. reported that teachers implemented 21 lessons in regular education classrooms. Harris et al. used the first 10 of the 21 lessons (three concrete, three representational, one mnemonic device, and three abstract) in three classrooms and the last 11 lessons (abstract lessons designed to teach solving word problems and to increase rate of computation) in the remaining two classrooms. Sealander (1991) used individually implemented lessons composed of an advanced organizer, demonstration and modeling of the skill, guided practice, and independent practice. Similarly, Hudson et al. (1988) used individually implemented lessons that included modeling, guided practice independent practice, and demonstration of mastery. Miller and Mercer (1993) used scripted lessons at the concrete, representational, and abstract levels and one-minute probes for pre and posttests.

Results. Harris et al. (1995) found that students with disabilities made similar gains as nondisabled students in all skills covered in the study except






29

word problems. In the study by Miller et al. (1998), normally achieving students outperformed students with LD and low achievers while no differences were found between the latter groups. All three of Hudson's et al. (1988) subjects made significant gains in place value skills. In the studies of crossover effects, positive results were reported. Sealander (1991) found that (a) crossover occurred in the concrete and representational phases, (b) skills continued to improve after crossover as determined by the number of problems completed correctly in one minute, and (c) students acquired, maintained, and generalized subtraction skills when CRA instruction was discontinued at crossover. Comparison of CRA and Abstract Only Instruction

Like the studies described above, Peterson, Mercer, and O'Shea (1988) also examined the results of instruction using the whole CRA sequence. Their investigation, however, compared performance of students receiving three levels of instruction (i.e., concrete, representational, and abstract) in place value to that of students receiving only abstract instruction in the same skills.

Characteristics of subiects. Twenty males and four females were included in this study. Subjects' ages ranged from 8 to 13 years. All subjects received math instruction in special education classes.

Research design and measurement procedures. A group design was

used to compare the effects of CRA and abstract only instruction on acquisition and generalization of place value skills. The experimental group received three lessons-each at the concrete, representational, and abstract levels. The control group received nine lessons at the abstract level only. All lessons were






30

structured to include an advance organizer, demonstration and modeling of the skill, guided practice, and independent practice. Teachers were trained to conduct the lessons. Multivariate analysis of variance was carried to determine the effects of the different types of instruction on student acquisition and generalization of the target skill.

Results. Students receiving CRA instruction performed significantly higher on acquisition than students receiving abstract only instruction. No differences between the groups were present on measures of generalization. Comparisons of CRA to RA Instructional Sequences

St. Martin (1975) and Sigda (1983) compared the effects of CRA and RA sequences. St. Martin tested the effectiveness of the same sequences for teaching multiplication and division of fractions. Students in Sigda's study were learning multiplication skills.

Characteristics of subjects. The subjects in these two studies were all

general education students. The 99 students in St. Martin's (1975) investigation were fifth graders. Sigda (1983) examined 52 third graders.

Research design and measurement procedures. St. Martin (1975) set up a 2-by-2 factorial design to compare the results of CRA and RA sequences, effects of Piagetian levels, and interactions of the instructional and developmental levels. Sigda (1983) used a posttest only control group design and analyzed data using t-tests. In addition, analysis of variance was used to test for interactions between treatments, gender, and IQ levels.






31

Implementation. Subjects in both studies received group instruction lead by the researcher (St. Martin, 1975) or by the teachers of the classes in which the research subjects were enrolled (Sigda, 1983). Teachers delivering instruction received training before the study began and participated in weekly meetings while the treatment was ongoing. The treatments were implemented for periods of 27 days (St. Martin, 1975) to 38 days (Sigda, 1983).

Results. Findings of these studies are mixed. St. Martin (1975) found the CRA and RA sequences equally effective for acquisition and retention of multiplication and division of fractions. In Sigda's (1983) study, students who received concrete level instruction did better than those receiving only representational or abstract instruction. Comparisons of Concrete Only, Representational Only, and Abstract Only

Instruction

Smith, Szabo, and Trueblood (1980) made comparisons of each of the levels of instruction when used to teach linear measurement skills. Fennema (1972) and Marsh and Cooke (1996) compared concrete only to abstract only instruction for teaching multiplication as the union of disjoint sets and identification of appropriate operations for solving word problems respectively. Additional studies in which concrete only and representational only instruction were compared are also included in this category. Armstrong (1972) conducted two studies of concrete only and representational only instruction that targeted the following skills: (a) numeral-quantity association, (b) conversion of quantities,

(c) numeral identification, and (d) counting. Prigge (1978) taught geometric









concepts and Scott and Neufeld (1976) worked on multiplication skills using concrete only and representational only instruction.

Characteristics of subiects. The 95 subjects in Fennema's (1972) study and the 145 subjects Scott and Neufeld (1976) studied were all second grade students in general education classrooms. Marsh and Cooke (1996) worked with third grade boys with LD. The third grade students in Prigge's (1978) study did not have disabilities. Smith et al. (1980) studied the learning of first and second grade students. Armstrong's (1972) studies included 87 students with mental retardation.

Research design and measurement procedures. Marsh and Cooke

(1996) used a single subject design with multiple baseline across subjects. Data were analyzed according to the number of correct responses. The remainder of the studies in this group followed group designs to compare results according to the type of instruction (i.e., concrete or abstract) students received. Data were analyzed using analysis of variance (Fennema, 1972; Prigge, 1978; Smith, Szabo & Trueblood, 1980), multivariate analysis of covariance (Armstrong, 1972), and analysis of covariance (Scott & Neufeld, 1976).

Implementation. During the baseline phase of the study by Marsh and

Cooke (1996), only abstract instruction in the form of a verbal strategy was used. Concrete instruction was implemented during the treatment phase. In the studies by Armstrong (1972), Fennema (1972), and Scott and Neufeld (1976), each group received only one level of instruction. In other studies (Prigge, 1978; Smith et al., 1980) groups received different types of instruction.








Fourteen lessons were implemented with each of the groups in

Fennema's (1972) study. Prigge's (1978) study included 10 lessons, and Scott and Neufeld's (1976) had 20 lessons. The subjects of Smith et al. (1980) received 4, 8, or 12 instructional sessions depending upon the treatment group they were in. The first of Armstrong's (1972) studies included 20 lessons delivered using slides and tapes. Teachers who were rotated among the students to control for teacher effects implemented lessons in the second study. The number of lesson in this study was not provided.

Results. The results of the studies in this group are mixed. Fennema (1972) found concrete and abstract instruction were effective for recall, but transfer of the skills was higher in students taught using abstract instruction. Marsh and Cooke (1996) found that students' scores increased by 58% to 77% from baseline during the concrete phase of the study. Smith et al. (1980) also found concrete instruction more effective than representational instruction for acquisition of skills, but no differences were found between concrete and abstract strategies. In Armstrong's (1972) study, students receiving concrete instruction did better when the targeted skill required representational thought. Concrete and representational strategies were equally advantageous in Scott and Neufeld's (1976) work. Low achievers in Prigge's (1978) research, students made more progress with concrete than with representational instruction. Study of the Order of Concrete and Abstract Instruction

One study was conducted to examine the effects of changing the order of concrete and abstract instruction. Evans and Carnine (1990) compared the






34

effects of a concrete-to-abstract sequence and an abstract-to-concrete sequence to teach subtraction skills.

Characteristics of subjects. Evans and Carnine (1990) carried out their investigation using 26 second and third grade students. All of the students were served in special education classes or received Chapter One services for at-risk students.

Research design and measurement procedures. A group design was

employed to compare performances of students receiving different instructional sequences. Data were analyzed using a two-way repeated measures analysis of variance.

Implementation. Scripted lessons were used to deliver instruction. Three graduate students served as instructors and rotated among the groups. During concrete instruction, students learned a manipulative strategy for solving subtraction problems. Abstract instruction consisted of an algorithm for solving the same type of problems.

Results. No significant differences were found between groups.

However, the researchers concluded that abstract-first instruction is more efficient than beginning with concrete lessons. Conversely, the concrete-toabstract sequence was found to be better for transfer of skills.


Summary

The fact that students with learning problems have demonstrated deficits in math skills has been present in the literature for 30 years. It is important for educators to identify effective and efficient teaching strategies to help students






35

with learning problems increase math skills and achieve at levels commensurate with their age, ability, and years of instruction.

Literature reviewed in this chapter provides insight into the efficacy of the acquisition strategies model illustrated in Figure 2 as part of the information processing procedure. All the research reviewed in this chapter tested the acquisition strategies model to determine if the selected coding strategies (i.e., concrete and representational instruction) and rote repetition at the abstract level lead to output of multiplication facts for students with learning problems.

The strategies tested in this investigation were examined in the studies reviewed. The effectiveness of the strategies for students with and without learning problems was studied. Subjects in nine of the 15 studies reviewed included students with disabilities. All the studies were carried out with elementary school students. One study also included middle school students. Concrete, representational, and abstract strategies used in isolation were the subject of six studies in this review. Research results indicate that concrete instruction was more effective than abstract instruction for teaching multiplication skills (Fennema, 1972) and word problems (Marsh & Cooke, 1996). Similarly, concrete instruction resulted in greater gains on skills including counting and numeral identification (Armstrong, 1972), geometric concepts (Prigge, 1978), and linear measurement (Smith, Szabo, & Trueblood, 1980) than did representational instruction. Concrete and representational strategies were equally effective for teaching multiplication skills (Scott & Neufeld, 1976).

Several studies examined the effectiveness of using the CRA sequence to teach a variety of math skills. The CRA sequence was found to be effective for








teaching place value (Peterson, 1988; Hudson, Peterson, Mercer, & Miller, 1988), multiplication (Harris, Miller, & Mercer, 1995; Miller, Harris, Strawser, Jones, & Mercer, 1998; Sidga, 1983), subtraction (Sealander, 1991), and basic facts and coin sums (Miller & Mercer, 1993).

The effects of partial sequences cannot be defined based on the current literature. St. Martin (1975) found RA instruction as effective as CRA instruction for teaching computation skills with fractions. The results of Sigda's (1983) study, however, indicate that the concrete element makes instruction more efficacious. More research is needed to determine the viability of CA and RA sequences as acquisition strategies appropriate for processing information leading to increased math skills for students with disabilities. Currently, the literature contains little data about the effectiveness of concrete, representational, and abstract strategies for teaching math skills to middle school students.

This study was designed to test the strategies acquisition model. Concrete and representational coding strategies and rote repetition were employed to process multiplication facts.













CHAPTER 3
RESEARCH METHODOLOGY

Students with disabilities make about half the progress in math per year as do their nondisabled peers (Cawley, Parmar, Yan, & Miller, 1996). Therefore, it is important that math instruction for this population be effective and efficient and based on established theories of information processing. The investigation was conducted to determine the effectiveness of various combinations of concrete, representational, and abstract level strategies to facilitate acquisition, retention, and retrieval of multiplication facts in middle school age students with learning problems. Specifically, within group differences were examined to determine the effectiveness of each of the instructional sequences (i.e., concrete-to-representational-to-abstract (CRA), concrete-to-abstract (CA), representational-to-abstract (RA), and abstract only (A)). Efficiency of instruction was demonstrated by looking at between group differences.

The methodology of the proposed study is described in the following

sections. The instructional procedures are outlined. Descriptions of the settings, materials used, and subjects are included. Finally, methods of statistical analysis are discussed.


Instructional Procedures

Concrete and representational level lessons were conducted by the

researcher. The researcher and participating teachers implemented abstract level lessons to all groups. Four groups were included in the study. Each group

37








received only CRA, CA, RA, or A instruction. Concrete, representational, and abstract level lessons are described below. Concrete Level Lessons

Concrete instruction required the manipulation of three-dimensional

objects to solve math problems. Students receiving concrete instruction were presented with multiplication facts and were asked to find the solution to the fact using objects to illustrate the problems.

Concrete level lessons followed these steps. The first session began with the researcher providing a rationale for the lesson. To give students a rationale for the lessons, the researcher stated that it is necessary to understand multiplication and using objects helps to build understanding. The rationale was established in the first session. Subsequent lessons did not include this step.

Next, the researcher explained the factors and signs in a multiplication fact (e.g., 3 x 2). The first number (3) represents the number of groups. The multiplication sign (x) means "of." The second number (2) tells how many objects are in each group. Therefore, the multiplication fact 3 x 2 means "three groups of two objects." The researcher then told the subjects to count the number of objects in all the groups to solve the problem (three groups of two objects equals six objects).

After explaining each of the parts of a multiplication fact, the researcher demonstrated how to use objects to solve a problem (e.g., 3 x 2). During this portion of the session, the researcher began by pointing to the first number (3) and placing three paper plates on the desk to represent three groups. Next, the






39

researcher read the multiplication sign (of). Then the researcher pointed to the second number (2) and placed two objects into each group. Next, the researcher counted all the objects in the groups and said the answer. Finally, the researcher read the multiplication fact with the answer (three groups of two objects equals six objects).

The researcher conducted four concrete level sessions with each subject assigned to groups receiving concrete instruction. The fifth concrete session served as an evaluation of participants' levels of mastery of multiplication facts at the concrete level. The same procedures described above were used. Subjects reached mastery when they correctly solved five of five multiplication facts at the concrete level without pausing. After achieving mastery, subjects moved to either representational or abstract instruction depending upon the group to which they were assigned. Subjects who did not achieve mastery continued concrete instruction with evaluations after every session.


Representational Level Lessons

Instruction at the representational level followed the same format as the concrete lessons described above. However, circles were drawn to represent groups and three-dimensional objects were replaced with tallies.

The first representational level lesson began with the researcher providing a rationale for the lesson. To give participants a rationale for the lessons, the researcher stated that it is necessary to understand multiplication and using drawings helps build understanding. The rationale was established in the first session. Subsequent lessons did not include this step.








After the rationale was established, the researcher explained the factors and signs in the multiplication fact (e.g., 3 x 2). The first number (3) represents the number of groups. The multiplication sign means "of." The second factor (2) tells how many tallies are in each group. Therefore, the multiplication fact 3 x 2 means "three groups of two tallies." The researcher then told the subject to count the number of tallies to solve the problem (three groups of two tallies equals six tallies).

After explaining each part of the multiplication fact, the researcher

demonstrated how to use tallies to solve a problem (e.g., 3 x 2). During this portion of the session, the researcher began by pointing to the first number (3) and drawing three circles to represent three groups. Next, the researcher read the multiplication sign (of). Then the research pointed to the second number (2) and drew two tallies in each circle. Next, the researcher counted all the tallies in the circles and said the answer. Finally, the researcher read the multiplication fact with the answer (three groups of two tallies equals six tallies).

The researcher conducted four representational lessons with each subject assigned to groups receiving representational instruction. The fifth representational lesson served as an evaluation of the subjects' mastery of multiplication facts at the representational level. The same procedure described above was used. Subjects achieved mastery when they correctly solved five of five multiplication facts at the representational level. Subjects who did not achieve mastery continued representational instruction with evaluations after each session. Subjects who correctly solved five of five multiplication facts at the representational level began abstract instruction.








Abstract Level Lessons

At the abstract level, participants were asked to solve multiplication facts without the aide of manipulatives or tallies. During abstract instruction, students were presented with multiplication facts and were asked to solve them using only the numerals on the probe sheet. Subjects participated in eighteen to 22 abstract sessions.

Abstract lessons were conducted by the researcher and by participating teachers and included the following steps. The researcher or teacher began by providing a rationale for the lessons. To give students a rationale for the lessons, the researcher or teacher stated that being able to solve multiplication facts quickly makes learning other math skills easier. Practicing multiplication facts regularly increases the speed at which the facts can be solved. The rationale was established in the first session. Subsequent sessions did not include this step.

The next step of abstract level lessons was for the researcher to remind students of rules of multiplication. For example, any number times zero is zero. Students were then told they would practice multiplication facts and were given a page of 60 facts. The 60 facts were selected to be representative of all the multiplication facts through 9 x 9. Students were instructed to write the answers to as many facts as they could in one minute. After one minute, the researcher or teacher said "stop." The researcher recorded the total number of digits each participant answered correctly and incorrectly for each probe sheet.








Setting

The study was conducted in four local middle schools. Math classes in which only special education students or students at risk for dropping out of school are served were studied exclusively. Concrete and representational level lessons were implemented in small groups outside the classroom setting. Abstract level sessions were conducted in the classroom as large group activities.


Subjects

Study participants were selected according to the following criteria.

Subjects had to attend middle school in a local school district and receive math instruction in special education classes or in classes serving low achieving students at risk for dropping out of school. Also, participants were required to have parental consent to be included in the study. Assent from the participants themselves was also required.

The final selection criterion was related to students' proficiency with multiplication facts. No clearly defined fluency rate is established in the literature. Estimated rates are as low as 40 digits per minute (Starlin & Starlin, 1973) to as high as 100 digits per minute (Johnson & Layng, 1992). The higher estimate was chose as the selection criterion for this study. Students who answered fewer than 100 digits correctly in a minute were included in the sample.









One hundred eleven students met the selection criteria. One hundred

students participated in the study through the posttest stage. Of that number, 96 completed the follow-up measure.

Each of the subjects was randomly assigned to one of four treatment groups within the school. Participants' names were listed then numbered one through four. Each number was assigned to one group for a specific type of instruction. See Table 1 for demographic information on each of the treatment groups.


Table 1.


Descriptive Information for Groups


CRA CA RA A Total
Gender Male 12 10 9 14 45 Female 11 16 14 10 51
Black 13 26 14 16 59 Race White 12 9 6 8 35 Hispanic 0 1 1 0 2
5 0 0 1 0 1 6 11 16 14 10 51 7 9 4 5 9 26 8 6 6 1 5 18

11 3 4 6 3 16 12 7 11 10 9 37 Age 13 9 6 3 6 24 14 5 4 2 4 15 15 1 1 0 2 4
Drop-Out-Prevention 8 8 6 8 32 Classification Learning Disabled 14 15 15 16 60
Emotionally Handicapped 3 2 0 0 5
Severely Emotionally Disturbed 0 1 0 0 1

1 5 5 5 2 17 School 2 10 12 7 13 43 3 3 4 4 3 14 4 6 5 5 6 22








Materials

Lesson materials were provided by the researcher. Probe sheets used for pre and posttesting and for abstract level lessons were provided. Each probe consisted of one page with 60 multiplication facts that were selected to be representative of multiplication facts to 81. The facts were randomly ordered on each page. The probe sheets were arranged with alternating lines presenting facts vertically and horizontally. An example of the probe sheets is provided in Appendix A. All pretests, posttests, and abstract level probes were scored by the researcher.


Statistical Analysis

The instructional sequence was the independent variable for the study with the CRA, CA, RA, and A only instructional sequences representing four levels of the variable. Acquisition and retention of multiplication facts were the dependent variables measured. Subjects were screened to ensure that the sample did not include participants who had already mastered the target skill as defined by the selection criteria. Pretests were administered to determine subjects' levels of mastery of multiplication facts. Each group's progress was measured by comparing their posttest and retention scores to their pretest performances. Acquisition of multiplication facts was the dependent variable and was measured by a posttest administered at the end of the treatment period. The dependent variable was measured a second time with a posttest administered one to two weeks after the treatment period ended.






45

Data were analyzed using a split plot design. Within subject differences demonstrated the effectiveness of each instructional sequence. Between subject comparisons showed whether or not a particular type of instruction was superior for facilitating growth in multiplication facts or if students made similar progress with different types of instruction. Results are presented in Chapter 4.













CHAPTER 4
RESULTS

This investigation was conducted to examine the impact on acquisition,

retention, and retrieval of four instructional sequences for teaching multiplication facts to middle school students with learning problems. The concrete-torepresentational-to-abstract (CRA) sequence was one of the methods used. Manipulations of the CRA sequence (i.e., concrete-to-abstract (CA), representational-to-abstract (RA), and abstract only (A)) comprised the remaining instructional strategies. Five research questions were asked. The first four questions addressed the effectiveness of the above mentioned instructional sequences individually. Is each sequence effective in facilitating acquisition and retention of multiplication facts for middle school students with learning problems? The fifth question is the basis of a comparison of the effectiveness of the four sequences. Are CRA, CA, RA, and A instruction equally effective in facilitating acquisition and retention of multiplication facts for middle school students with learning problems?

Four treatment groups were used to answer the research questions.

Participants in each group received multiplication facts instruction in which one of the specified sequences was used. Each groups' performance was measured with a pretest, a posttest, and a follow-up test. Within group differences were examined to determine the effectiveness of the individual instructional






47

sequences. Between group differences were analyzed to determine the relative effectiveness of each sequence. This chapter provides analyses of these data.


Analyses of Data

Means for pretest, posttest, and follow-up test scores for all groups were calculated (Table 2). Adjusted means for posttest and follow-up measures were also determined (Table 3). Repeated measures analyses of variance (ANOVA) were used to determine whether or not significant differences exist between test scores for each treatment group (Table 4). Family-wise comparisons were made to determine what differences were significant with each test group being treated as a family. All family-wise comparison tests were conducted using Bonferoni adjusted levels of significance when the ANOVA tests indicated differences exist between test scores. Results of family-wise comparisons are reported in Table

5. Finally, data were analyzed using a repeated measures analysis of covariance (ANCOVA) design (Tables 6 and 7) to determine if there were relative differences between the effectiveness of the sequences. Results for each group and for the full sample are presented in the following sections.


Concrete-to-Representationa I-to-Abstract Group Results

The first research question addressed the effectiveness of CRA

instruction. Is the CRA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Results of the repeated measures ANOVA revealed a significant difference among test scores (F = 12.15; p = .0002). All possible comparisons of the pretest, posttest, and follow-up scores were made to determine where









differences exist. Posttest and follow-up scores were significantly higher than the pretest scores (F = 13.34; p = .0012). Bonferoni family wise comparisons resulted in an adjusted p value of .017. Posttest and follow-up scores were still significantly different from the pretest at this level. No difference was found between the posttest and follow-up scores.


Table 2. Test Score Means and Standard Deviations by Group

Group Pretest Posttest Follow-up Mean SD Mean SD Mean SD
CRA 30.48 19.32 41.96 26.00 42.2 22.15 CA 34.00 18.18 47.00 24.46 46.54 24.17 RA 28.57 14.03 42.10 18.73 42.10 18.73 A 32.46 15.28 43.08 23.88 44.88 21.37


Table 3. Test Score Adjusted Means by Group Group Posttest Follow-up CRA 41.56 42.03 CA 43.06 43.17 RA 44.56 40.70 A 40.24 42.49


Table 4. Repeated Measures ANOVA of Test Effect by Group

Group N df F p CRA 25 2,48 12.15 .0002* CA 26 2,50 17.75 .0001* RA 21 2,40 16.34 .0001* A 24 2, 46 9.02 .0005*
*Significant at the p < .05 level.









Table 5. All Possible Family-wise Comparisons of Tests by Group


Pre/Post Pre/Follow Post/Follow Group N df F p N df F p N df F p CRA 26 1,25 13.34 .0012* 25 1,25 25.04 .0001* 25 1,24 0.02 0.8923 CA 26 1,25 28.52 .0001* 26 1,25 18.83 .0002* 26 1,25 0.05 0.8219 RA 24 1,23 30.94 .0001* 21 1,20 17.59 .0004* 21 1,20 2.29 0.1457 A 24 1,23 9.22 .0059* 24 1,23 20.10 .0002* 24 1,23 0.32 0.5777
*Significant at the p < .017 level.








Table 6. Repeated Measures ANCOVA of Full Model


Source Type III Mean Sums of Squares df Square F p Instructional
Strategy 10.263 3 3.421 .01 .998 Pretest 40211.275 1 40211.275 131.50 .0001* Pretest*
Instructional
Strategy 195.491 3 65.164 .21 .89 School 1359.603 3 453.201 1.48 .226 School*
Instructional
Strategy 3166.238 9 351.804 1.15 .339
*Significant at the p < .05 level.


Table 7. Repeated Measures ANCOVA of Instructional Strategy, Pretest, and
School

Type III Mean
Source Sums of Squares df Square F p Instructional
Strategy 91.137 3 30.380 .10 .960 School 1385.962 3 461.987 1.53 .214 Pretest 48304.131 1 48304.131 159.50 .0001*
*Significant at the p < .05 level.


Concrete-to-Abstract Group Results

The second research question addressed the effectiveness of CA

instruction. Is the CA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Repeated measures ANOVA indicated differences in test scores were significant (F = 17.75; p = .0001). Scores on the posttest and follow-up measures were








found to be significantly higher than pretest scores at .0001 and .0002 levels respectively when all possible comparisons were made. Significance was determined with the Bonferoni adjusted p value of .017. Posttest and follow-up scores were not significantly different. Representational-to-Abstract Group Results

The third research question addressed the effectiveness of RA instruction. Is the RA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Significant test differences were indicated by repeated measures ANOVA (F = 16.34; p = .0001). Posttest (F = 30.94; p = .0001) and follow-up measures (F = 17.59; p = .0004) were significantly higher than pretest scores for the RA group. Posttest and follow-up scores were also significantly higher than the scores on the pretest when the Bonferoni family wise comparison was used to adjust the level of significance (p = .017). No difference was found between the posttest and follow-up scores.


Abstract-Only Group Results

The fourth research question addressed the effectiveness of A-only

instruction. Is A-only instruction effective in facilitating acquisition and retention of multiplication facts for middle school students with learning problems? Results of the repeated measures ANOVA revealed significant test differences (F = 9.02; p = .0005). Comparisons of test scores indicate that posttest (F =

9.22; p = .0059) and follow-up scores (F = 20.10; p = .0002) increased






52

significantly over pretest scores. Significance was still present after adjusting the p value (.017) with Bonferoni family wise comparisons. Posttest and follow-up scores were not significantly different. Relative Effectiveness of Instructional Sequences

A repeated measures ANCOVA was run including all scores across treatment groups to determine if instruction, pretest scores, or schools participants attended were predictors of posttest or follow-up results after controlling for pretest scores and the school participants attended. Between subjects interactions were included in the model as well. Table 7 contains the findings from the ANCOVA. Results indicated that no interactions were present and that neither school nor instructional sequence predicted posttest or follow-up scores. Only the pretest was found to be significant as a predictor of participants' future performances (F = 131.50; p = .0001).

Since no interaction effects were present, they were eliminated from consideration and a second ANCOVA was run with instruction, school, and pretest scores as the between subjects factors. Instruction (F = .10; p= .9695) and school (F = 1.53; p = .2135) were not found to be significant. Results revealed pretest scores to be the only significant indicator of participants' performances on posttest and follow-up measures (F = 159.50; p = .0001).

Within subjects factor differences were also examined. Posttest and

follow-up scores were not found to interact with instructional sequence (F = .91; p = .4417), school (F = .26; p = .85), or pretest scores (F = 1.17; p = .2819). No







53

test effect was found (F = .80; p = .3726) indicating there were no differences in the posttest and follow-up test scores. See Table 8 for within subjects data.


Table 8. Repeated Measures ANCOVA of Tests and Test Interactions Type III Mean
Source Sums of Squares df Square F p Test 62.704 1 62.704 .80 .373 Test*
Instructional
Strategy 212.135 3 70.712 .91 .442 Test
*School
60.888 3 20.296 .26 .854 Test
*Pretest 91.508 1 91.508 1.17 .282
*Significant at the p < .05 level.


Pretest scores ranged from 3 to 89, creating a lot of variability in the data. To determine if results would differ without such a large range of pretest scores, the repeated measures ANCOVA was run without data on participants whose pretest scores were greater than 40. Results were consistent with the original test including all participants.


Summary

This study was conducted to determine the effects of CRA, CA, RA, and A instructional sequences for facilitating acquisition and retention of multiplication facts for middle school students with learning problems. Additionally, the relative effectiveness of the sequences was considered.









Results revealed the presence of within group differences for all groups. Posttest and follow-up test scores were significantly higher than pretest scores regardless of the treatment. Such differences indicate that all four treatments were effective for acquisition and retention of multiplication facts for students in this study.

No instructional strategies differences were found. All groups performed equally well on posttest and follow-up measures after accounting for initial differences on pretest scores and school. The absence of an instructional strategy effect indicates that each of the strategies were equally effective for the subjects being studied.

There were no differences between posttest and follow-up measures after accounting for initial differences in pretest scores. Follow-up scores remained at the same level as posttest measures. Such data reveal that students retained the skills they developed.













CHAPTER 5
DISCUSSION

Academic demands placed on all students are increasing as standards for student performance become more rigorous. Students with learning problems are expected to meet the same levels of performance as their normally achieving peers. Unfortunately, students with learning problems often fall short of meeting such expectations. Students with learning problems perform poorly on measures of minimum skills in math (Algozzine, O'Shea, Crews, & Stoddard, 1987) and achieve only half of the expected growth for each year they are in school (Cawley, Parmar, Yan, & Miller, 1996).

Efforts to alleviate math deficits in students with learning problems are essential to their future academic success. Teachers need to know and use effective and efficient strategies to help students learn more and do so as quickly as possible. However, research regarding effective math instruction has not received a great deal of attention.

Information processing theories explain how learners incorporate information into their knowledge base. Instructional strategies need to be designed based on such models. The acquisition strategies model proposed by Baumeister and Kellas (as cited in Mercer & Snell, 1977) is such a model and provided the basis for this study. According to this model, learners select strategies to code information, then practice and recode the information into a








form for output. In this investigation, participants coded information with concrete and representation level strategies then participated in abstract level practice which culminated in written output.

The findings and implications of the effectiveness of concrete-torepresentational-to-abstract (CRA), concrete-to-abstract (CA), representationalto-abstract (RA), and abstract only (A) strategies for teaching multiplication facts to middle school students with learning problems are discussed in this chapter. The research questions and results are reviewed. Next, the relevance of the findings with respect to the acquisition strategies model is addressed. Limitations of the study are considered. Finally, implications for further research are discussed.


Summary of Research Questions and Findings

This study was conducted to answer questions regarding four instructional sequences comprised of concrete, representational, and abstract strategies. Specifically, are CRA, CA, RA, and A instruction each effective methods for facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? The relative effectiveness of the sequences was also examined to answer the final research question. Are CRA, CA, RA, and A instruction equally effective strategies for facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Each of the treatment groups performed significantly higher on posttest and follow-up measures than on pretests. Such data indicate that CRA, CA, RA, and A instruction were all effective for facilitating acquisition,









retention, and retrieval of multiplication facts. Furthermore, each of the treatment groups demonstrated very similar growth on posttest and follow-up measures. The parallel nature of the results demonstrates that the treatments were equally effective. No treatment effects were found.


Relevance of Findings to the Acquisition Strategies Model

Previous studies support the efficacy of the acquisition strategies model for teaching multiplication skills to elementary school students. Results of studies by Harris et al. (1995) and Miller et al. (1998) demonstrated the effectiveness of the full sequence of concrete and representational coding strategies and abstract rote repetition for developing multiplication skills with students with learning problems. Sigda (1983) compared the effectiveness of the full CRA sequence with a sequence in which only the representational coding strategy was implemented prior to rote repetition. Results of this study indicated that students whose instruction included both concrete and representational coding strategies achieved more. Fennema (1972) and Scott and Neufeld (1976) compared groups who were instructed with only one strategy. Results of these studies indicated that concrete instruction is more advantageous than abstract-only and equally effective as representational-only instruction.

The current study also supported the efficacy of the acquisition strategies model. This research, however, studied students in middle school rather than elementary grades. The results of this study differ from previous studies in that all sequences tested were equally effective for teaching multiplication facts to students with learning problems. Use of concrete, representational, and abstract








strategies lead to increased skills in multiplication. Coding strategies at the concrete and representational levels in combination with rote repetition at the abstract level comprised effective instructional strategies (e.g., CRA, RA, CA). The model also demonstrates that learners can bypass coding strategies and use only rote repetition to acquire knowledge. The performance of the group receiving A-only instruction supported this portion of the model. Proceeding directly to rote repetition proved as effective as teaching one or both of the initial selected coding strategies prior to practice at the abstract level. CRA, CA, and RA sequences were more time and labor intensive for the teacher and resulted in no enhanced learning for these students.


Limitations

The study has the following limitations. Because only one Florida school district was represented in the sample, the results should not be generalized to other geographic areas. Further, since acquisition and retention of multiplication facts were the targeted skills, the results of this investigation should not be generalized to other math skills. Also, subjects were all middle school students served in special education or drop out prevention programs in public schools. Therefore, generalization to older or younger students or to students served in other educational programs should not be made. It is reasonable to assume that participants had previously received instruction at the concrete and representational levels because such strategies are often used with elementary school students. Therefore, participants' past instruction might limit the study. Finally, criteria for selecting participants (i.e., fewer than 100 digits correct on the







59

screening) is less stringent than would be accepted by some and might include subjects who did not need instruction.


Implications

Given the deficits in math skills experienced by students with learning problems, it is essential for math instruction to be effective and efficient to optimize learning. The results of this study demonstrate that the acquisition strategies model is a viable design upon which to base instruction. Concrete, representational, and abstract strategies are effective elements of instructional sequences which resulted in increased multiplication skills.

The most notable result of this study is that no treatment effect was demonstrated. All four of the instructional sequences were equally effective. Since the A-only group achieved as much as the other three treatment groups, it can be concluded that concrete and representational strategies are not necessary for middle school students with learning problems to make progress in learning multiplication facts. It is possible that this is due to previous concrete and representational level instruction. Such knowledge is important to ensure the most efficient use of instructional time. Class time does not need to be spent on concrete or representational strategies to help students learn multiplication facts even if students have learning problems in the area of math.

Future research should be conducted to determine the effectiveness and efficiency of CRA, CA, RA, and A instruction on other math skills. Investigations into the effects of the four instructional sequences on skills with students of other ages and grade levels who have not had CRA instruction would also be






60

warranted since all the participants in this study had approximately four years of concrete, representational, and abstract instruction in multiplication facts. Also, given the ease with which abstract instruction can be implemented, it would also be valuable to test the results of such instruction when implemented by paraprofessionals, parents, and school volunteers. If noninstructional personnel can successfully administer abstract level sessions, opportunities for working with students in various settings (e.g., after-school programs; at home) are opened.

Future research should also address the issue of expanding the

acquisition strategies model to present a more detailed illustration of how information is processed. For example, additional boxes representing retention and retrieval of data could be added to the model between recoding and output. Subsequent investigations could then be designed to determine under what conditions students best retrieve data.


Summary

Students with learning problems often make poor progress in math. Efforts to increase their skills need to include the use of research proven methods. This study was conducted to examine the effectiveness and efficiency of CRA, CA, RA, and A instruction for acquisition and retention of multiplication facts with students with learning problems. The methods had been reported in the literature as necessary, but no data were available as to what strategies are most viable. Results of the study indicate that all of the instructional sequences are equally effective. Therefore, it can be concluded that abstract strategies in






61

the form of one minute probes as implemented in this study are the most efficient of the strategies tested in teaching multiplication facts to middle school students with learning problems.














APPENDIX A
SAMPLE PROBE SHEET






Name


8x5=


2x8=



5
x5


5x3=


8x4=


7x3=


1 x4=


6x6=


3x4=


8x8=


0x8=


4x6=


5x4=


4x3 =_


9x5=


8x7=


7x5=


7x8=


6x7=


3x7=


5x9=


2x3=


3x9=


9x8=


2x6=


7x9= 6


2 x5 =


5 x7 =


6 x 3 =


6 x 8 =


9 x4 =













APPENDIX B
INFORMED CONSENT LETTER

Dear Parent/Guardian:

During the next two months, some graduate students from the Department of Special Education at the University of Florida will be visiting your child's school to work with students who receive math instruction in special education classes. The graduate students will be using three different types of instruction to help students learn multiplication facts. The first type of instruction is concrete. During concrete lessons students will use objects to solve multiplication facts. The second type of instruction is called representational. During this type of instruction, students will draw tallies (or make marks on paper to represent objects) to show multiplication facts. Finally, abstract lessons will be used. At this point, students will use only a page of multiplication facts to solve problems.

Some students will do all three types of lessons. Some students will do concrete and abstract lessons. Other students will do only representational and abstract lessons. Other students will do only abstract lessons. After all the lessons have been completed, students will be tested to see if their multiplication skills have increased. They will be tested again two weeks later to see if they have retained the skills they learned during the lessons.

Results for students receiving each type of instruction will be compared. We hope to get information that will help determine what kind of instruction is best for middle school students who have difficulty with math.

Results of the project will be shared with professionals within the field of education who are interested in improving instruction for students with difficulty in math However, your name and your child's name will be kept confidential to the extent provided by law.

Participation in this project is voluntary. Whether or not your child participates in this project will not effect his/her placement in any programs and will not effect his/her grades. If you choose to allow your child to participate, you and your child have the right to withdraw consent for participation at any time without consequence. A possible benefit to you child for participating in the project is increased skills in multiplication. There are no know risks and no compensation for taking part in this project.








The project wil begin in February 2001 and will continue through March, 2001. Results of the project will be available after the project is completed.

If you have any questions about this project, please contact me at (352) 3920701 ext. 261. Questions of concerns about the rights of students participating in this project may be directed to the UF Institutional Review Board office, University of Florida, P. 0. Box 112250, Gainesville, FL 32611 (352-392-0433).

Sincerely,

Penny R. Cox, M.Ed.



I have read the procedures described above. I voluntarily give consent for my child, , to participate in this project.


Parent/Guardian Signature


Date













APPENDIX C
ASSENT SCRIPT READ TO PARTICIPANTS


I am going to be working with students during their math class to help them learn multiplication skills, We'll do some different types of lessons that will help you remember the multiplication facts. You do not have to participate unless you want to. If you do participate, you may quit at any time.

Do you want to participate in these multiplication lessons?















APPENDIX D
PRETEST, POSTTEST, AND FOLLOW-UP SCORES FOR PARTICIPANTS Participant Pretest Posttest Follow-up Number School Age Grade Classification Group Score Score Score
1 2 11 6 DOP RA 21 38 34 2 2 12 6 DOP CA 83 105 114 3 2 12 6 DOP CRA 25 44 42 4 2 11 6 DOP CRA 47 73 65 5 2 12 6 DOP RA 40 62 43 6 2 12 6 DOP CA 37 35 39 7 2 11 6 DOP RA 23 34 45 8 2 13 6 DOP CRA 23 36 40 9 2 11 6 DOP CRA 27 33 35 10 2 11 6 DOP CA 41 46 45 11 2 11 6 DOP RA 28 26 27
12 2 12 6 DOP CRA 33 36 32 13 2 12 6 DOP RA 33 47 34 14 2 12 6 DOP CA 31 34 48 15 2 12 6 DOP CA 11 24 36 16 2 12 6 DOP A 41 51 48

17 2 12 6 DOP A 37 2 29 18 2 13 7 DOP CA 33 25 25 19 2 13 7 DOP RA 46 72 50 20 2 13 7 DOP A 20 30 32 21 2 13 7 DOP CRA 50 71 61 22 2 12 7 DOP A 41 39 47
23 2 12 7 DOP A 65 104 86









Participant Pretest Posttest Follow-up Number School Age Grade Classification Group Score Score Score

24 2 14 7 DOP A 29 44 34 25 2 12 7 DOP RA 16 33 N/A 26 2 13 7 DOP A 32 51 21 27 2 13 7 DOP CA 61 82 52 28 2 13 7 DOP CRA 24 44 N/A 29 2 15 8 DOP A 33 55 61 30 2 14 8 DOP CRA 59 100 75 31 2 13 8 DOP RA 34 60 N/A 32 2 15 8 DOP CRA 51 53 61 33 2 13 8 DOP CA 53 84 104 34 2 14 7 LD CA 23 55 55 35 2 13 7 LD CRA 5 18 20 36 2 14 7 LD A 6 2 22 37 2 14 8 LD CRA 10 16 18 38 2 13 7 LD CRA 28 92 67 39 2 14 8 LD A 33 60 56

40 2 13 8 LD CA 8 7 22 41 2 15 8 LD CA 17 42 42 42 2 14 8 LD A 43 73 56
43 2 13 7 LD CA 31 66 65 44 2 13 7 LD RA 23 54 70
45 2 15 8 LD A 39 43 50 46 2 13 7 LD A 15 13 23 47 4 12 6 LD RA 23 35 44 48 4 11 6 EH CA 29 41 56 49 4 12 6 LD CRA 23 35 44 50 4 11 6 EH CRA 29 41 56 51 4 12 6 LD A 24 58 49








Participant Pretest Posttest Follow-up Number School Age Grade Classification Group Score Score Score
52 4 12 6 LD CA 54 86 67 53 4 11 6 LD RA 18 29 21 54 4 11 6 LD A 56 36 80 55 4 13 6 LD CA 22 31 27 56 4 12 6 LD RA 66 84 76 57 4 13 7 LD A 8 29 27 58 4 13 7 LD CRA 12 37 11 59 4 13 7 LD CRA 28 29 29 60 4 12 6 LD RA 41 66 52 61 4 12 6 LD A 45 41 33 62 4 14 8 SED CA 55 72 67 63 4 12 6 LD CRA 89 84 84 64 4 12 6 LD A 46 58 63 65 4 12 6 LD CA 26 50 56 66 4 14 8 LD CRA 25 32 41 67 4 12 6 LD RA 27 14 32 68 4 13 8 LD A 43 73 84 69 3 12 6 LD A 19 32 24 70 3 12 6 LD RA 21 34 34 71 3 12 6 LD CA 33 43 40 72 3 12 6 LD CRA 17 17 15 73 3 11 6 LD CA 25 35 30 74 3 11 6 LD RA 17 36 27 75 3 11 6 LD CA 7 7 8 76 3 12 6 LD RA 15 22 23 77 3 12 6 LD CRA 27 10 20 78 3 11 6 LD A 16 19 23 79 3 12 6 LD CA 36 49 53








Participant Pretest Posttest Follow-up Number School Age Grade Classification Group Score Score Score

80 3 12 6 LD CRA 20 10 20 81 3 12 6 LD RA 15 28 18 82 3 12 6 LD A 10 20 30 83 1 13 8 LD CRA 14 40 44 84 1 14 8 EH CA 52 47 57 85 1 11 5 LD RA 38 52 55 86 1 11 6 LD CA 12 14 13 87 1 14 8 EH CRA 57 67 80 88 1 13 7 LD A 46 67 76 89 1 12 6 LD CA 23 46 36 90 1 12 7 LD RA 34 56 52 91 1 13 7 EH CRA 30 35 36 92 1 14 8 LD RA 12 40 20 93 1 14 7 LD CRA 3 3 7 94 1 13 8 LD CA 35 37 26 95 1 11 6 LD A 32 34 23 96 1 12 6 LD CA 49 61 52 97 1 12 7 LD RA 24 40 31 98 1 13 7 LD CRA 30 35 52 99 1 14 7 LD RA 10 12 19 100 1 11 5 EH RA 25 20 N/A













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BIOGRAPHICAL SKETCH

Penny R. Cox was born in Jacksonville, Florida, on March 16, 1961. She graduated from Englewood High School in 1979. She attended the University of North Florida where she received her B.A.E. degree in December, 1981, and M.Ed. degree in December, 1995.

Penny taught elementary and special education classes in Duval and Volusia counties for a total of 17 years. During that time, she also fulfilled a variety of additional responsibilities. She was a member of her school's school improvement plan writing team, assisted beginning teachers in completing professional orientation programs, supervised numerous practicum students and interns, and served as the professional development facilitator for her school. While completing her doctoral studies, Penny has taught courses, supervised graduate and undergraduate students, and has served as the assistant to the graduate coordinator for the Department of Special Education. She is currently coordinating the Comprehensive System of Personnel Development grants for the Department of Special Education in partnership with surrounding school districts.








I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



Mary KO Dykes, hair ,
Professor of Special Education



I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



CecilD. D ercer
Distinguished Professor of Special Education



I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



Maureen Conroy
Assistant Professor of Special
Education



I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



M. David Miller
Professor of Educational Psychology








This dissertation was submitted to the Graduate Faculty of the College of Education and to the Graduate School and was accepted as partial fulfillment of the requirements of the degree of Doctor of Philosophy.


August 2001


Dean, College of EducatMT


Dean, Graduate School




Full Text

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THE EFFICACY OF AN ACQUISITION STRATEGIES MODEL FOR MIDDLE SCHOOL STUDENTS WITH LEARNING PROBLEMS By PENNY R. COX 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 2001

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Copyright 2001 by Penny R. Cox

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ACKNOWLEDGMENTS The efforts of many people have contributed to the success of this research. I would like to express my appreciation to all who had a part in making this project possible. I want to thank the students who participated in this study and to thank their parents for permission to work with their children. Without them, none of this could have happened. I extend special thanks go to Lynn Jamison, Tarcha Rentz, Sari Heipp, Christina Zamora, Jody Joseph, and Darby Desmond, the teachers who so graciously allowed me to come into their classrooms and who offered their assistance and support in collecting data. I thank them for their interest in this project and their help in making it happen. I greatly appreciate their kindness in putting up with my interruptions and numerous requests for "just one more" probe sheet or piece of information. I would also like to thank the members of my committee for their support in the preparation, planning, and implementation of this study. Dr. Mary Kay Dykes, Dr. Cecil Mercer, Dr. Maureen Conroy, and Dr. David Miller have provided valuable assistance and advice. Dr. Mary Kay Dykes has proved to be a source of great encouragement throughout my doctoral program. She has been my committee chair, mentor, advisor, and friend. I thank her for believing in me. iii

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I give special tlianl
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' < • : TABLE OF CONTENTS page ACKNOWLEDGMENTS ABSTRACT vii CHAPTERS 1 INTRODUCTION 1 Standards for Mathematics Learning 3 Statement of the Problem 6 Definitions ^ Delimitations ''O Limitations ''O Summary ''0 2 REVIEW OF RELEVANT LITERATURE 12 Information Processing Theories 13 Theoretical Basis for Concrete, Representational, and Abstract Instruction 19 Math Achievement for Students with Learning Problems 21 Criteria for Selecting Concrete-to-Representational-to-Abstract Studies for Review 25 Review of Relevant Concrete-to-Representational-to-Abstract Literature 26 Summary 35 3 RESEARCH METHODOLOGY 37 Instructional Procedures 37 Setting 42 Subjects 42 Materials 44 Statistical Analysis 44 4 RESULTS 46 Analyses of Data 47 Summary 53 V

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5 DISCUSSION 55 Summary of Research Questions and Findings 56 Relevance of Findings to the Acquisition Strategies Model 57 Limitations Implications ^ Summary APPENDICES A SAMPLE PROBE SHEET 63 B INFORMED CONSENT LETTER 64 C ASSENT SCRIPT READ TO PARTICIPANTS 66 D PRETEST, POSTTEST, AND FOLLOW-UP SCORES FOR PARTICIPANTS 67 REFERENCES 71 BIOGRAPHICAL SKETCH 76 vi

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Abstract of Dissertation Presented to the Graduate Sciiool of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE EFFICACY OF AN ACQUISITION STRATEGIES MODEL FOR MIDDLE SCHOOL STUDENTS WITH LEARNING PROBLEMS By Penny R. Cox August 2001 Chair: Mary Kay Dykes Major Department: Special Education This study tested the efficacy of an acquisition strategies model for facilitating the acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems. The effectiveness and efficiency of concrete-to-representational-to-abstract, concrete-to-abstract, representational-to-abstract, and abstract-only instructional sequences were compared. Ninety-six participants were randomly assigned to four treatment groups within their schools. Each group received instruction in the form of one of the sequences being examined. Concrete and representation-level instruction was implemented in small groups. Abstract level lessons were administered as whole-class activities. Participants were enrolled in middle schools in one north central Florida school district. Instruction took place within those schools and was implemented vii

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by special education teachers. Dependent variables were posttest and follow-up test scores. Statistical analyses of the data revealed that posttest and follow-up test scores were significantly higher than pretest scores. No differences were found between posttest and follow-up measures. There were no differences between groups when pretest differences were controlled, indicating that all treatments were equally effective. The various acquisition strategies of the model are supported by the results of this investigation as effective and efficient instructional interventions. More involved and longer teaching interventions based on the model produced basically the same results but were time and personnel intensive. Abstract-only instruction produced results similar to those of the longer CRA, CA, and RA strategies but required less class time and minimal effort on the part of the teacher. Therefore, daily abstract-only level practice of multiplication facts was effective in improving the retention and retrieval of multiplication facts for this population. viii

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CHAPTER 1 INTRODUCTION Demands for more stringent academic standards for students are increasing. More states are requiring students to demonstrate grade level performance before being promoted to the next grade. The public is also more concerned about how American students compare to their peers in the world community. Students with learning problems are expected to meet the same academic requirements as their normally achieving peers. Such demands can be especially arduous for this group since it is well documented that students with learning problems experience academic difficulties. The difficulties they encounter result in poor reading, writing, math, and social/emotional skills. Such difficulties persist into adulthood placing students with learning problems at risk for further problems on their jobs, in their communities, and in their homes and relationships (Cronin & Patton, 1993). Efforts to alleviate such problems for students with learning problems must begin as early as possible in the school experience. Teachers need to know and use effective and efficient teaching methods to help students with learning problems to not only learn more but also to acquire skills after as little instructional time as possible. Such instructional demands make it especially important that the nature of effective instruction be understood. Knowing how to 1

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2 teach students to maximize acquisition and retention of content is necessary for students to achieve academic success. Instruction should be based on information processing models that explain how individuals incorporate information into their knowledge base. Generally, information that is received undergoes various types of processing at different levels of understanding. The use of effective processes transforms the information to create new understanding. Individuals then store the new knowledge in memory to be used during processing of more information in the future. Existing understanding combines with new information and processing strategies to build the individual's knowledge base. The acquisition strategies model proposed by Baumeister and Kellas (as cited in Mercer & Snell, 1977) provides a format for instruction by which occurrence of memory and retrieval of data are facilitated by using strategies to enhance memory, retention and retrieval. According to this model, learners select strategies to code information that is received. After coding the information, it is practiced then receded into a form for output. For the purpose of this research, concrete and representational coding strategies were used. Concrete strategies require subjects to manipulate three-dimensional objects to solve math problems. Representational strategies use drawings in place of objects. Concrete and representational strategies can be used within the same instructional sequence or independently of each other. After concrete and/or representational coding strategies are implemented, practice takes place. In this investigation, practice was done at the abstract level. Abstract strategies involve solving problems without the aid of manipulatives of drawings. During the

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3 receding phase, the learner uses the infornnation and strategies to solve a math problem. Finally, the solution is made known by means of a motor output (e.g., speaking, writing). It is important to identify coding strategies that are effective for helping students with learning problems increase the rate at which they acquire, retain, and retrieve data. Research to validate the effectiveness of concrete and representational coding strategies and abstract level practice is needed to help determine whether or not the use of such strategies constitutes effective, efficient instruction for students with learning problems. Standards for Mathematics Learning The National Council of Teachers of Mathematics (NCTM), the professional organization of math teachers, addressed the issues of determining math curriculum and identifying effective teaching strategies for math skills. The NCTM, in collaboration with professionals in areas requiring math skill (e.g., engineers, information science specialists, scientists) developed and published a set of standards outlining math skills that students need at each level of instruction. The standards include descriptions of how the skills should be taught. The NCTM recommends that all students be guided to construct mathematical concepts through activities designed to facilitate discovery through problem solving. The NCTM standards are supported by professionals within mathematics. The fact that the standards were developed by professional groups with expertise in mathematics and related areas is viewed as a strength (Giordano,

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1993). The instructional methods recommended in the standards are supported as best practices because they are in the math education literature. However, such practices have not been validated for use with students with learning problems. Support for NCTM standards outside the field of mathematics, specifically within the field of special education, has not come so readily. Objections to the NCTM standards are many. First, professional collaboration is not a sufficient basis for setting national standards (Rivera, 1993). Instead, research-based data are necessary to determine what skills and instructional practices should be recommended not just inclusion of ideas as best practice. Second, the absence of references to students with disabilities and other diverse backgrounds is a stated concern of special educators and researchers (Hofmeister, 1993; Hutchinson, 1993; Mercer, Harris, & Miller, 1993; Rivera, 1993). A related objection was expressed by Hutchinson (1993) who stated that it is more important to focus on the quality of teaching and learning for students with disabilities rather than on access to the same curriculum as their nondisabled peers. Hutchinson stated reservations about the NCTM's recommended instructional methods since there are no data to show that students develop concepts and acquire skills because they are exposed to mathematical experiences. Special educators and researchers suggest that different teaching approaches (especially for lower level skills) need to be used with students with learning problems. Research needs to be conducted to identify and validate models of curriculum and instructional practices that give rise to effective

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5 instructional strategies for students with learning problems. Presently, no such model has been validated. Mathematics Deficits in Students with Learning problems Students with learning problems achieve below the level of their typically achieving peers in math. In studies conducted during the 1980s and 1990s, students with learning problems performed poorly on measures of minimum skills (Algozzine, O'Shea, Crews, & Stoddard, 1987 and achieved only half the expected growth for each year of school (Cawley, Parmar, Yan, & Miller, 1996). Causes of Mathematics Disabilities Mathematics deficits in students with learning problems have been reported to be due to factors that can be grouped into two categories. One category is student factors. These factors are directly related to how students go about working math problems and how well they use the strategies and skills they have learned. For example, students with learning problems often choose incorrect or inefficient strategies to solve problems. In addition, students with learning problems work slowly and inaccurately. The second category of factors responsible for deficits in math performance is related to instruction. The first instructional factor contributing to poor math achievement concerns the lack of class time spent teaching students math concepts. Second, teachers often move from one concept to another before students have achieved mastery. These problems are exacerbated by the fact that no model has been determined as the basis for effective instructional practices.

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6 Statement of the Problem The effects of math deficits are cumulative. Students who have difficulty learning math facts are at risk for developing more and greater learning problems as they progress through school. Students who cannot develop strong basic skills have no foundation upon which to build more complex skills (Fleischner, Garnett, & Shepherd, 1982). It is essential that instruction be effective in order to maximize student learning. The development of a validated model of instruction which has demonstrated a positive, effective, and predictable impact on learning of math facts is a priority in order to provide efficient teaching practices. No such model is currently found in the literature. Given the low level of math performance of students with learning problems, it is important to identify instructional strategies that result in efficient and predictable learning. The acquisition strategies model proposed by Baumeister and Kellas (as cited in Mercer & Snell, 1977) provides a framework for instruction and test-specific coding strategies. The model provides flexibility in testing strategies in isolation or in combined with other strategies. The effectiveness of each coding strategy or combination of strategies can be determined by comparing the output of learners using various strategies. Concrete and representational coding strategies and rote repetition in the form of abstract level practice are strategies that have been used in math instruction with students with learning problems. The concrete-torepresentational-to-abstract (CRA) instructional sequence has been effective for teaching math skills to students with learning problems (Mercer & Miller, 1992b?). However, research is needed to pinpoint more effective and more

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7 efficient ways to teach students with learning problems. Toward that end, this investigation examined manipulations of the CRA instructional sequence to determine if both coding strategies and the abstract level practice are all necessary for efficient student learning. The experimental questions were as follows: 1. Is the CRA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? 2. Is the CA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? 3. Is the RA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? 4. Is an A-only model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? 5. Are CRA, CA, RA, and A models equally effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Comparisons of individual and group performance levels will help determine if middle school students with learning problems acquire, retain, and retrieve math facts efficiently when their instruction does not include all three elements of the CRA sequence. Definitions Definitions for terms used in this research are provided below. Abstract level of instruction "involves the use of numerals" (Mercer, 1997, p. 581). During abstract instruction, mathematical problems are solved without

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8 the aid of objects or drawings. While this is the most time-efficient strategy, the effectiveness of isolated abstract level practice is not known. In this research, accuracv refers to the number of correct responses on untimed practice activities at the concrete level and on timed probe sheets used during abstract instruction. Arithmetic involves (a) real numbers in terms of their characteristics and relationships and (b) computations (primarily addition, subtraction, multiplication, and division) using real numbers (Gove, 1986). Basic facts are addition facts with single digit addends, subtraction facts with minuends to 18 and subtrahends to nine, multiplication facts with factors to nine, and division facts representing inverse operations for basic multiplication facts. The concrete level of instruction "involves the manipulation of objects (Mercer, 1997, p. 581). Students receiving concrete instruction solve mathematical problems using three-dimensional objects. Concrete coding strategies are basic processes for all initial tasks and instruction in math in the acquisition-strategies model proposed by Baumeister and Kellas (in Mercer & Snell, 1977). A criterion is a predetermined level of performance for a task. Students with low academic achievement who are not served in special education programs are at risk for dropping out of schools. The academic focus of drop out prevention programs is remediation of deficits in basic skills.

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9 Students with learning problems are those students served in special education and drop out prevention programs whose academic achievement is below expected levels. A probe samples a target behavior. In this investigation, probes of basic math facts consist of a single page of 60 multiplication facts. Students had one minute to complete as many multiplication facts as possible. The number of times a behavior occurs during a specified period is rate. For this research, rate refers to the number of math facts correctly completed in one minute. The phrase representational level of instruction is used synonymously with the terms semiconcrete and semiabstract. This level of instruction "involves working with illustrations of items in performing math tasks (Mercer, 1997, p. 581). Such illustrations can include pictures, dots, or tallies. Representational activities typically follow concrete strategies in math instruction. They are included in the acquisition strategies model by Baumeister and Kellas (as cited in Mercer & Snell, 1977) as coding strategies. Special education consists of services for students with disabilities if such disabilities result in a need for specially designed instruction or related services (Heward, 2000). Specific learning disabilities disorders "in one or more the basic psychological processes involved in understanding or in using language, spoken or written, which may manifest itself in an imperfect ability to listen, think, speak, read, write, spell, or to do mathematical calculations" (U.S. Office of Education, 1977, p. 65083).

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10 Delimitations This investigation has the following delimitations. First the study was limited to students served in one north central Florida school district. Second, it includes only middle school students in public schools. Third, only students receiving math instruction in special education and dropout-prevention classes were included. Fourth, multiplication facts were the only operation taught. Limitations Caution should be exercised in generalizing results of this research. Since only students in some middle schools in one Florida school district were represented, results should not be extended to students in other areas. Further, generalizing findings to students in other types of educational settings or with other types of learning problems should be done carefully. Finally, multiplication facts were the focus of this investigation, so results should not be generalized to other skills. Summary Deficits in math performance for students with learning problems are well documented. Efforts to increase math skills of students with learning problems should include identification of effective instructional strategies. Additionally, examination of such strategies is needed to determine their most efficient uses. This research examined the effectiveness of acquisition-strategies model proposed by Baumeister and Kellas (in Mercer & Snell, 1977). Manipulations of the CRA sequence were tested. Specifically, CA, RA, and A sequences were examined and compared to each other and to the CRA sequence to determine

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11 their potential as effective and efficient interventions for increasing retention and retrieval of multiplication facts. Comparisons of student achievement determined how well students with learning problems acquired math facts with the shortened instructional sequences. Such comparisons were useful in deciding if the CA, RA, and/or A strategies were more efficient ways of helping students with learning problems acquire math facts. A review of literature relevant to this research is presented in Chapter 2. The methodology used for the investigation is described in Chapter 3. Results of the study are reported in Chapter 4 and discussed in Chapter 5.

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CHAPTER 2 REVIEW OF RELEVANT LITERATURE As discussed In the previous chapter, students with learning problems do not achieve at the same levels in math as nondisabled students (Algozzine, O'Shea, Crews, & Stoddard, 1987). To alleviate such problems for this population, it is necessary to identify instructional strategies that are effective and efficient for enhancing retention and retneval of information for students with learning problems. This study was designed to identify such strategies. The purpose of this chapter is to discuss the theoretical basis for the investigation and to present a review of related professional literature. Specifically, literature regarding use of the concrete-to-representational-toabstract (CRA) instructional sequence to enhance retention and retrieval of information with students with learning problems will be reviewed. This chapter includes five sections. First, the information processing theories providing the foundation for this research are discussed. Second, the theoretical basis of concrete, representational, and abstract retention and retrieval strategies are discussed. Third, math achievement for students with learning problems is considered. Next, the criteria for selecting studies to be reviewed are listed. Finally, studies in which math instruction included elements of the concrete-to-representational-to-abstract instructional sequence are reviewed. 12

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13 Information Processing Theories The proposed research was designed to investigate the effects of the concrete-to-representational-to-abstract instructional sequence on the acquisition, retention, and retrieval of multiplication facts by students with learning problems. The study was based on information processing models that propose explanations for how information is assimilated by learners. Such models have been produced in attempts to understand what information individuals learn and how they acquire the information (Mercer, 1992). The implications for the use of such models in the field of education are obvious. A better and more complete understanding of how students learn is required to design effective and efficient instruction that capitalizes on the natural processes of assimilation of new knowledge. According to Siegler (1988) understanding such processes requires consideration of the kind of information being processed, the strategies the learner employs to manipulate the information, and the degree to which information can be processed given limits imposed by the capacity of memory. Information processing models are used to demonstrate how information is incorporated into the learner's knowledge base. Such an occurrence is a fluid operation that occurs in a systematic manner. Many models have been proposed that illustrate how learners construct knowledge (Mercer, 1992). This study was based on two such models. The first provides an overview of information processing theory in general. The second provides detail to describe the learning process within one element of the general model. Both models are discussed in the sections that follow.

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14 Model Providing an Overview of Information Processing The model shown in Figure 1 was based on the work of Keith Lentz and presented by Mercer (1992). It is illustrative of the manner in which information moves through the processing system. The process begins with a sensory stimulus, continues by moving through any number of possible manipulations at various levels of understanding, and leads to a response by the learner. Arrows are used to illustrate that the learning sequence is not entirely linear. Information can move back and forth through integrative and executive processes and longterm memory to undergo a variety of manipulations. In addition, the learner must function within a specific context. Factors including the type of stimulus, the teacher, materials used, and the response type alter what is required of the learner. Such conditions are represented in Figure 1 as contextual demands. The types of processing included in the ovals at the top of the figure and in the integrative processing section represent the general procedures for assimilating information. In pre-perceptual processing, the learner detects visual and auditory features of a stimulus but does not attach meaning. At the perceptual processing level, symbols (e.g., phonemes, numerals) are perceived as meaningful. Perceptions at this level are automatized as skill levels increase. In recognition processing, perceptual units are combined to create meaning in the form of concepts. Integrative processing is the term used to describe the learner's comprehensive approach to a task. An individual's prior knowledge and choice of strategies are important elements of the integrative processing component of the model.

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15

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16 Short-term memory is also represented by an oval at the top of the figure. However, it is not connected directly to the ovals representing the general processes. The short-term memory area serves as a place where information is held, manipulated, or synthesized in order to construct new information. To get to short-term memory, information progresses through a series of procedures beginning with preperception. From there, information may move between the general procedures or go directly to the integrative processes. The unit of information may move among the general procedures, the integrative processes, and short-term memory as assimilation occurs. Knowledge from long-term memory can be retrieved and held in short-term memory to be incorporated into integrated units to develop new meaning. The executive processes and long-term memory form the foundation of the model and support the active stages of processing. Executive processes are metacognitive in nature and include such skills as maintaining attention and providing internal feedback. Long-term memory is foundational in the model because learning is based on prior knowledge and beliefs. Three categories of knowledge are contained in long-term memory. Declarative knowledge covers facts and concepts. Procedural knowledge includes steps for performing tasks. Knowing when and how to integrate declarative and procedural knowledge is the function of conditional knowledge. This model is appropriate for the proposed research in that knowledge present in long-term memory constitutes the basis for developing multiplication facts to the declarative level. Knowledge from all three categories of long-term memory will be retrieved to be held and used in short-term memory. For example, declarative knowledge will be used in the form of numbers and groups.

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17 Procedural knowledge of the steps for solving multiplication problems and conditional knowledge of how to use the declarative and procedural elements to complete the multiplication task is needed also. The manipulation that occurs in short-term memory is illustrated in the model by Baumeister and Kellas (as cited in Mercer & Snell, 1977). Acquisition Strategies Model ^ ' ' The model of acquisition strategies was adapted from a model proposed by Baumeister and Kellas (in Mercer & Snell, 1977) to compare strategies used by students with and without mental retardation to learn paired associates. According to this model, the student receives a stimulus then selects a strategy for learning the pair. A variety of coding rehearsal strategies for learning the pairs is available. Coding strategies are followed by rote repetition, decoding, and finally output. Figure 2, the acquisition strategies model, is the basis for the proposed research. Adaptations are made to the coding strategies portion for the purpose of including procedures relevant to the concrete-to-representational-to-abstract instructional sequence. The abstract portion of the sequence is presented in rote repetition. Additionally, the decoding element is renamed recoding to more accurately describe synthesis as included in current literature. Movement from the strategy selector includes choices from a variety of processing sequences. The concrete and representational coding strategies can be used within the same sequence or independently of each other. After the selected coding strategies are implemented, rote repetition at the abstract level takes place.

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19 During the recoding stage, the learner uses the information and strategies to solve a math problem. Finally, the solution is made known by means of a motor output (e.g., speaking, writing, pointing). The full concrete-to-representational-to-abstract sequence begins with the concrete coding strategy then continues through the coding strategies to the representational strategy. The remainder of the sequence continues through the abstract level or rote repetition, recoding, and output. When either the concrete or the representational strategy is omitted, the concrete-to-abstract or the representational-to-abstract sequence results. Theoretical Basis for Concrete. Representational, and Abstract Instruction Piaget (1960) characterized children's intellectual development as having four levels. The first stage described is the sensorimotor stage, which begins at birth and lasts until about age two. During the sensorimotor stage, children learn to interact with and respond to their environment. The preoperational stage is second and lasts until about age seven. It is during this stage that children begin to use representational thought. Their thoughts, however, are based on their perceptions of concrete experiences. The concrete operational stage is next and lasts until about age 1 1 . During this stage, children develop the ability to focus on more than one attribute of a situation at a time. It is also during this period that logical thought begins, though primarily at the concrete level. Finally, children in the formal operational stage "are able to think abstractly without reference to actual objects or actions in the real world" (Fuys & Tischler, 1979, p. 34).

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20 Bruner (1966) described similar stages of understanding. His enactive, iconic, and symbolic levels parallel Piaget's levels of intellectual development. Sovchik (1989) described activities that would be characteristic of children in each of Bruner's stages. Children in the enactive stage might be able to perform a motor task, but not be able to describe how it is done. At the iconic level, mental images of concrete objects are developed. When employing symbolic knowledge, children are able to use language and symbols. For example, generating a number sentence to go with a word problem in math requires symbolic knowledge. In 1977 Underbill described three developmental levels of learning experiences related to mathematics. The goal of the first two levels, concrete and semiconcrete, is to provide students with opportunities to practice math skills. At the concrete level, students learn through the use of visual, kinesthetic, and tactile practice. Semiconcrete experiences are only visual. Abstract experiences are not manipulative or visual. Instead, "it is assumed that earlier concrete experiences have enabled the learner to think iconically when using the abstract symbols" (Underbill, Uprichard, & Heddens, 1980, p. 30). Concrete Level Instruction Students functioning at Bruner's (1966) concrete level need to be taught using examples and methodology at the concrete level. Such instruction necessitates that the learner engage in problem-solving using three-dimensional objects. The learner attends to the objects being manipulated as well as the mathematical procedures they represent (Mercer, 1997). In concrete level

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instruction, a student is given a problem to solve (e.g., 2 x 4). In solving this problem, the student uses two cups to represent groups and places four objects into each cup. After counting the number of objects in the cups, the student is able to state that two groups of four objects equals eight objects. Representational Level Instruction Instruction at the representational level involves using pictures, tallies, or other items that stand for concrete objects in order to solve math problems (Mercer, 1997). Such instruction coordinates with the semiconcrete level of understanding (Bruner, 1966). A representational lesson would entail the student making a drawing to solve a math problem. For example, when presented with the problem 2x4, the student draws two circles to represent two groups, then draws four tallies in each circle. After counting all the tallies, the student will be able to state that two groups of four tallies equals eight tallies. Abstract Level Instruction Lessons taught in accordance with Bruner's (1966) abstract level are characterized by the absence of manipulatives or graphic representations. Instruction at the abstract level requires the use of numerals (Mercer, 1997). Mercer points out that students who have difficulty with math need concrete and representational instruction before they work math problems abstractly. Math Achievement for Students with Learning Problems Students with learning problems achieve less in math than their typically achieving grade-level peers. Deficits in achievement among students with math

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22 disabilities have been demonstrated in studies over the last 30 years. Studies of overall math achievement show students within this population make less than the expected amount of growth in math skills. Poor performance on measures of minimum skills is common among students with learning problems (Algozzine, O'Shea Crews, & Stoddard, 1987). The effects of math deficits are cumulative. As Cawley, Parmar, Yan, & Miller (1996) found, discrepancies in math performance of students with learning problems increase as students get older. Students with mild learning difficulties were found to make half the growth in math skills as other students for the same amount of instruction (Cawley et al., 1996; Cawley & Miller, 1989). Two years of instruction resulted in only one year of growth. This level of achievement is consistent with earlier findings in which students with learning problems exiting high school were determined to be functioning at fifth to sixth grade levels in math (Cawley, Kahn, & Tedesco, 1989). Similarly, McLeod and Armstrong (1982) found that intermediate and secondary school students with learning problems performed at third and fourth grade levels in math. Reports of data from numerous studies indicate that students with learning problems have deficits in skills with basic math facts. For example, Goldman, Pellegrino, and Mertz (1988) found students with learning problems responded more slowly then nondisabled students when computing basic addition facts. Studies examining computation skills found that nondisabled students attempted more problems and answered more problems correctly than students with learning problems when computing basic math facts (Garnett & Fleischner, 1983; Fleischner, Garnett, & Shepherd, 1982).

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23 Causes of Mathematics Disabilities Causes of problems in mathematics for students with learning problems can be classified as student factors or instructional factors. Student factors include identifiable disabilities such as visual-spatial disorders and slow processing speed (Garnett, 1992). Poor verbal and language skills (Miller & Mercer, 1997) also contribute to math disabilities. Other student factors are related to differences in the way students process information. Students with learning difficulties approach math problems in qualitatively different ways than typically achieving students (Montague & Applegate, 1993; Kulak, 1993). Such differences suggest that students with learning problems have different learning styles (Deshler, Schumaker, Alley, Warner, & Clark, 1982), which might prevent them from processing information at the same rate as normally achieving students. Other student factors that might result in math disabilities are related to students' inability to perform basic operations and to plan strategies for completing problems. Students who cannot execute basic operations (Kirby & Becker, 1988) or who have not been able to automatize skills (Garnett, 1992) have difficulty learning math skills. Miller and Mercer (1997) found that students with learning problems often lack the skills needed to choose appropriate strategies to complete problems. Instructional factors related to math deficits for students with learning problems are associated with instruction and content presentation. For example, too little time spent on arithmetic instruction and not making connections between math concepts and language, written symbols, and practical applications are found to be part of the problem of poor student performance In

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24 math (Garnett, 1992). Fixed and spiral mathematics curricula also make learning more difficult for many students (Jones, 1982; Englemann, Carnine, & Steely, 1991; Miller & Mercer, 1997). Fixed curriculum refers to a predetermined amount of content that must be covered within a school year. Teachers often continue through the content at a pace that does not allow low achieving students to grasp concepts fully before a new one is introduced. Similarly, teachers often present lessons by simply writing on the board and talking about concepts so that as much material can be covered in a class period as possible (Pieper & Deshler, 1985). Such presentation can result in concepts remaining unconnected to student experiences. Math instruction using a basal series is designed in a spiraling format. Spiral curriculum provides for cursory coverage of many concepts from year to year without allowing sufficient practice for struggling students to master skills. Instructional factors contributing to math skill deficits for problems are further exacerbated by the fact that strategies for teaching math skills to students with learning problems have received little attention from researchers (Fleischner & Manheimer, 1997). Without research-based data, instructional strategies cannot be identified as effective or ineffective. Additionally, relying on methods described in math texts and other commercially obtained materials does not ensure the use of research-based best practices in math instruction. Finding appropriate materials for teaching math is complicated (Miller & Mercer, 1997) by the fact that only 3% of materials are actually tested with students before being released in texts (Sprick, 1 987).

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Results of Math Deficits Students who experience math problems are at risk for developing more serious learning problems as they progress through school. Students who are unable to acquire and maintain basic math facts have little foundation upon which to build other skills (Fleischner, Garnett, & Shepherd, 1982). As a result, their "difficulties are manifested in the inability to acquire and apply mathematical skills and concepts, to reason, and to solve mathematical problems" (Rivera, 1997, p. 20). Criteria for Selecting Concrete-to-Reoresentational-t o-Abstract Studies for Review Studies reviewed in the following section were selected according to the following criteria: 1 . The study was conducted to examine the effects of concrete, representational, and/or abstract instruction on acquiring basic math facts. 2. Performance on math skills was the dependent variable. 3. Subjects in the studies were students in elementary or middle schools. 4. Studies provided quantitative data and included descriptions of subjects, procedures, and results. 5. Studies were conducted between 1970 and 2000. Literature searches were conducted using Education Resources Information Center (ERIC), Education Abstracts, and Library User Infonnation Sen/ice (LUIS). In addition, ancestral searches of some reference lists were conducted.

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26 Review of Relevant Concrete-to-Representational-to-Abstract Literature A total of 15 studies meeting the selection criteria were identified. Five of the studies were conducted using all three elements of the concrete-torepresentational-to-abstract (CRA) instructional sequence. One study was designed to compare the effectiveness of CRA instruction to instruction using only the abstract portion of the sequence. In six investigations, researchers looked at differences in results of concrete only, representational only, and abstract only instruction. In one study the order of concrete and abstract of instruction considered. Finally, the effectiveness of using only parts of the CRA sequence (i.e., CA and RA sequences) to teach math skills was the focus of two studies. Researchers use a variety of terms synonymously with the terms concrete, representational, and abstract as defined in Chapter 1. "Manipulative" is the term used in place of concrete by Armstrong (1972), Evans and Carnine (1990), Marsh and Cooke (1996), Prigge (1978), and Scott and Nuefeld (1976). The representational level is referred to as "semiconcrete" (Hudson, Peterson, Mercer, & McLeod, 1988; Miller & Mercer, 1993; Peterson, Mercer, & O'Shea, 1988; St. Martin, 1975), "symbolic" (Fennema, 1972), "pictorial" (Scott & Neufeld, 1976), and "graphic" (Smith, Szabo, & Trueblood, 1980). Abstract level instruction was referred to as "nonmanipulative" by Prigge (1978). The sections that follow contain analyses of each of the studies with consideration for skills being taught, characteristics of subjects, research design and measurement procedures, and results. The studies are presented according to the topics of the investigations as described above.

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27 Studies of the CRA Sequence The five studies in this category were designed so that the entire CRA instructional sequence was used. In three of the studies, the effectiveness of the CRA sequence to teach specific math skills was investigated. Hudson, Peterson, Mercer, and McLeod (1988) focused their study on place value while Harris, Miller, and Mercer (1995) and Miller, Harris, Strawser, Jones, and Mercer (1998) were interested in the viability of the CRA sequence to teach multiplication. Sealander (1991) and Miller and Mercer (1993) concentrated their efforts on a phenomenon they call crossover effect. Crossover is the point at which students receiving CRA instruction answered more problems correctly than incorrectly. Subjects in both studies were taught basic addition, subtraction, or division facts or coin sums. Characteristics of subiects. Two of the five studies (Harris et al., 1995; Miller et al., 1998) involved second grade students identified as LD, emotionally handicapped (EH) or low achievers. Hudson et al. (1988) described their subjects as having LD and being ages eight and eleven years. Miller and Mercer (1993) also studied students with LD as well as students at risk for LD, and students identified as educable mentally handicapped (EMH). Sealander's (1991) subjects were in first or second grades and were not described as having disabilities. Research design and measurement procedures. Miller et al. (1998) used a group design to compare performances of 123 subjects including students with disabilities, low achievers, and normally achieving students. Data were analyzed using a repeated measures multivariate analysis of variance. The remaining four

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28 studies (Harris et al., 1995; Hudson et a!., 1988; Miller & Mercer, 1993; Sealander, 1991) are single subject investigations using multiple baseline designs. Measurements of the dependent variables include counting the number of correct and incorrect responses (Harris et a!., 1995; Hudson et al, 1988; Miller & Mercer, 1993), pre and/or posttest scores (Harris et al, 1995; Hudson et al., 1988; Sealander, 1991), and visual analysis and celeration slopes (Sealander, 1991). Implementation. Teachers participating in the Miller et al. (1998) and Harris et al. (1995) studies were trained to use Multiplication Facts 0 to 81 (Mercer & Miller, 1992a) before beginning instruction. Miller et al. reported that teachers implemented 21 lessons in regular education classrooms. Harris et al. used the first 10 of the 21 lessons (three concrete, three representational, one mnemonic device, and three abstract) in three classrooms and the last 1 1 lessons (abstract lessons designed to teach solving word problems and to increase rate of computation) in the remaining two classrooms. Sealander (1991) used individually implemented lessons composed of an advanced organizer, demonstration and modeling of the skill, guided practice, and independent practice. Similarly, Hudson et al. (1988) used individually implemented lessons that included modeling, guided practice independent practice, and demonstration of mastery. Miller and Mercer (1993) used scripted lessons at the concrete, representational, and abstract levels and one-minute probes for pre and posttests. Results. Harris et al. (1995) found that students with disabilities made similar gains as nondisabled students in all skills covered in the study except

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word problems. In the study by Miller et al. (1998), normally achieving students outperformed students with LD and low achievers while no differences were found between the latter groups. All three of Hudson's et al. (1988) subjects made significant gains in place value skills. In the studies of crossover effects, positive results were reported. Sealander (1991) found that (a) crossover occurred in the concrete and representational phases, (b) skills continued to improve after crossover as determined by the number of problems completed correctly in one minute, and (c) students acquired, maintained, and generalized subtraction skills when CRA instruction was discontinued at crossover. Comparison of CRA and Abstract Onlv Instruction Like the studies described above, Peterson, Mercer, and O'Shea (1988) also examined the results of instruction using the whole CRA sequence. Their investigation, however, compared performance of students receiving three levels of instruction (i.e., concrete, representational, and abstract) in place value to that of students receiving only abstract instruction in the same skills. Characteristics of subjects. Twenty males and four females were included in this study. Subjects' ages ranged from 8 to 13 years. All subjects received math instruction in special education classes. Research design and measurement procedures. A group design was used to compare the effects of CRA and abstract only instruction on acquisition and generalization of place value skills. The experimental group received three lessons.each at the concrete, representational, and abstract levels. The control group received nine lessons at the abstract level only. All lessons were

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30 structured to include an advance organizer, demonstration and modeling of the skill, guided practice, and independent practice. Teachers were trained to conduct the lessons. Multivariate analysis of variance was carried to determine the effects of the different types of instruction on student acquisition and generalization of the target skill. Results. Students receiving CRA instruction performed significantly higher on acquisition than students receiving abstract only instruction. No differences between the groups were present on measures of generalization. Comparisons of CRA to RA Instructional Sequences St. Martin (1975) and Sigda (1983) compared the effects of CRA and RA sequences. St. Martin tested the effectiveness of the same sequences for teaching multiplication and division of fractions. Students in Sigda's study were learning multiplication skills. Characteristics of subiects. The subjects in these two studies were all general education students. The 99 students in St. Martin's (1975) investigation were fifth graders. Sigda (1983) examined 52 third graders. Research design and measurement orocedures. St. Martin (1975) set up a 2-by-2 factorial design to compare the results of CRA and RA sequences, effects of Piagetian levels, and interactions of the instructional and developmental levels. Sigda (1983) used a posttest only control group design and analyzed data using t-tests. In addition, analysis of variance was used to test for interactions between treatments, gender, and IQ levels.

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31 Implementation. Subjects in both studies received group instruction lead by the researcher (St. Martin, 1975) or by the teachers of the classes in which the research subjects were enrolled (Sigda, 1983). Teachers delivering instruction received training before the study began and participated in weekly meetings while the treatment was ongoing. The treatments were implemented for periods of 27 days (St. Martin, 1975) to 38 days (Sigda, 1983). Results. Findings of these studies are mixed. St. Martin (1975) found the CRA and RA sequences equally effective for acquisition and retention of multiplication and division effractions. In Sigda's (1983) study, students who received concrete level instruction did better than those receiving only representational or abstract instruction. Comparisons of Concrete Only. Representational Only, and Abstract Onlv Instruction Smith, Szabo, and Trueblood (1980) made comparisons of each of the levels of instruction when used to teach linear measurement skills. Fennema (1972) and Marsh and Cooke (1996) compared concrete only to abstract only instruction for teaching multiplication as the union of disjoint sets and identification of appropriate operations for solving word problems respectively. Additional studies in which concrete only and representational only instruction were compared are also included in this category. Armstrong (1972) conducted two studies of concrete only and representational only instruction that targeted the following skills: (a) numeral-quantity association, (b) conversion of quantities, (c) numeral identification, and (d) counting. Prigge (1978) taught geometric

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32 concepts and Scott and Neufeld (1976) worked on multiplication skills using concrete only and representational only instruction. Characteristics of subjects. The 95 subjects in Fennema's (1972) study and the 145 subjects Scott and Neufeld (1976) studied were all second grade students in general education classrooms. Marsh and Cooke (1996) worked with third grade boys with LD. The third grade students in Prigge's (1978) study did not have disabilities. Smith et al. (1980) studied the learning of first and second grade students. Armstrong's (1972) studies included 87 students with mental retardation. Research design and measurement procedures. Marsh and Cooke (1996) used a single subject design with multiple baseline across subjects. Data were analyzed according to the number of correct responses. The remainder of the studies in this group followed group designs to compare results according to the type of instruction (i.e., concrete or abstract) students received. Data were analyzed using analysis of variance (Fennema, 1972; Prigge, 1978; Smith, Szabo & Trueblood, 1980), multivariate analysis of covariance (Armstrong, 1972), and analysis of covariance (Scott & Neufeld, 1976). Implementation. During the baseline phase of the study by Marsh and Cooke (1996), only abstract instruction in the form of a verbal strategy was used. Concrete instruction was implemented during the treatment phase. In the studies by Armstrong (1972), Fennema (1972), and Scott and Neufeld (1976), each group received only one level of instruction. In other studies (Prigge, 1978; Smith et al., 1980) groups received different types of instruction.

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33 Fourteen lessons were implemented with each of the groups in Fennema's (1972) study. Prigge's (1978) study included 10 lessons, and Scott and Neufeld's (1976) had 20 lessons. The subjects of Smith et al. (1980) received 4, 8, or 12 instructional sessions depending upon the treatment group they were in. The first of Armstrong's (1972) studies included 20 lessons delivered using slides and tapes. Teachers who were rotated among the students to control for teacher effects implemented lessons in the second study. The number of lesson in this study was not provided. Results. The results of the studies in this group are mixed. Fennema (1972) found concrete and abstract instruction were effective for recall, but • transfer of the skills was higher in students taught using abstract instruction. Marsh and Cooke (1996) found that students' scores increased by 58% to 77% from baseline during the concrete phase of the study. Smith et al. (1980) also found concrete instruction more effective than representational instruction for acquisition of skills, but no differences were found between concrete and abstract strategies. In Armstrong's (1972) study, students receiving concrete instruction did better when the targeted skill required representational thought. Concrete and representational strategies were equally advantageous in Scott and Neufeld's (1976) work. Low achievers in Prigge's (1978) research, students made more progress with concrete than with representational instruction. Study of the Order of Concrete and Abstract Instruction One study was conducted to examine the effects of changing the order of concrete and abstract instruction. Evans and Carnine (1990) compared the

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34 effects of a concrete-to-abstract sequence and an abstract-to-concrete sequence to teach subtraction skills. Characteristics of subiects. Evans and Gamine (1990) carried out their investigation using 26 second and third grade students. All of the students were served in special education classes or received Chapter One services for at-risk students. Research design and measurement procedures. A group design was employed to compare performances of students receiving different instructional sequences. Data were analyzed using a two-way repeated measures analysis of variance. Implementation. Scripted lessons were used to deliver instruction. Three graduate students served as instructors and rotated among the groups. During concrete instruction, students learned a manipulative strategy for solving subtraction problems. Abstract instruction consisted of an algorithm for solving the same type of problems. Results. No significant differences were found between groups. However, the researchers concluded that abstract-first instruction is more efficient than beginning with concrete lessons. Conversely, the concrete-toabstract sequence was found to be better for transfer of skills. Summary The fact that students with learning problems have demonstrated deficits in math skills has been present in the literature for 30 years. It is important for educators to identify effective and efficient teaching strategies to help students

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35 with learning problems increase math skills and achieve at levels commensurate with their age, ability, and years of instruction. Literature reviewed in this chapter provides insight into the efficacy of the acquisition strategies model illustrated in Figure 2 as part of the information processing procedure. All the research reviewed in this chapter tested the acquisition strategies model to determine if the selected coding strategies (i.e., concrete and representational instruction) and rote repetition at the abstract level lead to output of multiplication facts for students with learning problems. The strategies tested in this investigation were examined in the studies reviewed. The effectiveness of the strategies for students with and without learning problems was studied. Subjects in nine of the 15 studies reviewed included students with disabilities. All the studies were carried out with elementary school students. One study also included middle school students. Concrete, representational, and abstract strategies used in isolation were the subject of six studies in this review. Research results indicate that concrete instruction was more effective than abstract instruction for teaching multiplication skills (Fennema, 1972) and word problems (Marsh & Cooke, 1996). Similarly, concrete instruction resulted in greater gains on skills including counting and numeral identification (Armstrong, 1972), geometric concepts (Prigge, 1978), and linear measurement (Smith, Szabo, & Trueblood, 1980) than did representational instruction. Concrete and representational strategies were equally effective for teaching multiplication skills (Scott & Neufeld, 1976). Several studies examined the effectiveness of using the CRA sequence to teach a variety of math skills. The CRA sequence was found to be effective for

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36 teaching place value (Peterson, 1988; Hudson, Peterson, Mercer, & Miller, 1988), multiplication (Harris, Miller, & Mercer, 1995; Miller, Harris, Strawser, Jones, & Mercer, 1998; Sidga, 1983), subtraction (Sealander, 1991), and basic facts and coin sums (Miller & Mercer, 1993). The effects of partial sequences cannot be defined based on the current literature. St. Martin (1975) found RA instruction as effective as CRA instruction for teaching computation skills with fractions. The results of Sigda's (1983) study, however, indicate that the concrete element makes instruction more efficacious. More research is needed to determine the viability of CA and RA sequences as acquisition strategies appropriate for processing information leading to increased math skills for students with disabilities. Currently, the literature contains little data about the effectiveness of concrete, • ' representational, and abstract strategies for teaching math skills to middle school students. This study was designed to test the strategies acquisition model. Concrete and representational coding strategies and rote repetition were employed to process multiplication facts.

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CHAPTER 3 RESEARCH METHODOLOGY Students with disabilities make about half the progress in math per year as do their nondisabled peers (Cawley, Parmar, Yan, & Miller, 1996). Therefore, it is important that math instruction for this population be effective and efficient and based on established theories of information processing. The investigation was conducted to determine the effectiveness of various combinations of concrete, representational, and abstract level strategies to facilitate acquisition, retention, and retrieval of multiplication facts in middle school age students with learning problems. Specifically, within group differences were examined to determine the effectiveness of each of the instructional sequences (i.e., concrete-to-representational-to-abstract (CRA), concrete-to-abstract (CA), representational-to-abstract (RA), and abstract only (A)). Efficiency of instruction was demonstrated by looking at between group differences. The methodology of the proposed study is described in the following sections. The instructional procedures are outlined. Descriptions of the settings, materials used, and subjects are included. Finally, methods of statistical analysis are discussed. Instructional Procedures Concrete and representational level lessons were conducted by the researcher. The researcher and participating teachers implemented abstract level lessons to all groups. Four groups were included in the study. Each group 37 . i

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38 received only CRA, CA, RA, or A instruction. Concrete, representational, and abstract level lessons are described below. Concrete Level Lessons Concrete instruction required the manipulation of three-dimensional objects to solve math problems. Students receiving concrete instruction \Nere presented v\/ith multiplication facts and were asked to find the solution to the fact using objects to illustrate the problems. Concrete level lessons followed these steps. The first session began with the researcher providing a rationale for the lesson. To give students a rationale for the lessons, the researcher stated that it is necessary to understand multiplication and using objects helps to build understanding. The rationale was established in the first session. Subsequent lessons did not include this step. Next, the researcher explained the factors and signs in a multiplication fact (e.g., 3 x 2). The first number (3) represents the number of groups. The multiplication sign (x) means "of." The second number (2) tells how many objects are in each group. Therefore, the multiplication fact 3x2 means "three groups of two objects." The researcher then told the subjects to count the number of objects in all the groups to solve the problem (three groups of two objects equals six objects). After explaining each of the parts of a multiplication fact, the researcher demonstrated how to use objects to solve a problem (e.g., 3 x 2). During this portion of the session, the researcher began by pointing to the first number (3) and placing three paper plates on the desk to represent three groups. Next, the

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39 researcher read the multiplication sign (of). Then the researcher pointed to the second number (2) and placed two objects into each group. Next, the researcher counted all the objects in the groups and said the answer. Finally, the researcher read the multiplication fact with the answer (three groups of two objects equals six objects). The researcher conducted four concrete level sessions with each subject assigned to groups receiving concrete instruction. The fifth concrete session served as an evaluation of participants' levels of mastery of multiplication facts at the concrete level. The same procedures described above were used. Subjects reached mastery when they correctly solved five of five multiplication facts at the concrete level without pausing. After achieving mastery, subjects moved to either representational or abstract instruction depending upon the group to which they were assigned. Subjects who did not achieve mastery continued concrete instruction with evaluations after every session. Representational Level Lessons Instruction at the representational level followed the same format as the concrete lessons described above. However, circles were drawn to represent groups and three-dimensional objects were replaced with tallies. The first representational level lesson began with the researcher providing a rationale for the lesson. To give participants a rationale for the lessons, the researcher stated that it is necessary to understand multiplication and using drawings helps build understanding. The rationale was established in the first session. Subsequent lessons did not include this step.

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40 After the rationale was established , the researcher explained the factors and signs in the multiplication fact (e.g., 3 x 2). The first number (3) represents the number of groups. The multiplication sign means "of." The second factor (2) tells how many tallies are in each group. Therefore, the multiplication fact 3x2 means "three groups of two tallies." The researcher then told the subject to count the number of tallies to solve the problem (three groups of two tallies equals six tallies). After explaining each part of the multiplication fact, the researcher demonstrated how to use tallies to solve a problem (e.g., 3 x 2). During this portion of the session, the researcher began by pointing to the first number (3) and drawing three circles to represent three groups. Next, the researcher read the multiplication sign (of). Then the research pointed to the second number (2) and drew two tallies in each circle. Next, the researcher counted all the tallies in the circles and said the answer. Finally, the researcher read the multiplication fact with the answer (three groups of two tallies equals six tallies). The researcher conducted four representational lessons with each subject assigned to groups receiving representational instruction. The fifth representational lesson served as an evaluation of the subjects' mastery of multiplication facts at the representational level. The same procedure described above was used. Subjects achieved mastery when they correctly solved five of five multiplication facts at the representational level. Subjects who did not achieve mastery continued representational instruction with evaluations after each session. Subjects who correctly solved five of five multiplication facts at the . 1 representational level began abstract instruction. j i I I

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41 Abstract Level Lessons At the abstract level, participants were asked to solve multiplication facts without the aide of manipulatives or tallies. During abstract instruction, students were presented with multiplication facts and were asked to solve them using only the numerals on the probe sheet. Subjects participated in eighteen to 22 abstract sessions. Abstract lessons were conducted by the researcher and by participating teachers and included the following steps. The researcher or teacher began by providing a rationale for the lessons. To give students a rationale for the lessons, the researcher or teacher stated that being able to solve multiplication facts quickly makes learning other math skills easier. Practicing multiplication facts regularly increases the speed at which the facts can be solved. The rationale was established in the first session. Subsequent sessions did not include this step. The next step of abstract level lessons was for the researcher to remind students of rules of multiplication. For example, any number times zero is zero. Students were then told they would practice multiplication facts and were given a page of 60 facts. The 60 facts were selected to be representative of all the multiplication facts through 9x9. Students were instructed to write the answers to as many facts as they could in one minute. After one minute, the researcher or teacher said "stop." The researcher recorded the total number of digits each participant answered correctly and incorrectly for each probe sheet.

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Setting The study was conducted in four local middle schools. Math classes in which only special education students or students at risk for dropping out of school are served were studied exclusively. Concrete and representational level lessons were implemented in small groups outside the classroom setting. Abstract level sessions were conducted in the classroom as large group activities. Subiects Study participants were selected according to the following criteria. Subjects had to attend middle school in a local school district and receive math instruction in special education classes or in classes serving low achieving students at risk for dropping out of school. Also, participants were required to have parental consent to be included in the study. Assent from the participants themselves was also required. The final selection criterion was related to students' proficiency with multiplication facts. No clearly defined fluency rate is established in the literature. Estimated rates are as low as 40 digits per minute (Starlin & Starlin, 1973) to as high as 100 digits per minute (Johnson & Layng, 1992). The higher estimate was chose as the selection criterion for this study. Students who answered fewer than 100 digits correctly in a minute were included in the sample.

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One hundred eleven students met the selection criteria. One hundred students participated In the study through the posttest stage. Of that number, 96 completed the follow-up measure. Each of the subjects was randomly assigned to one of four treatment groups within the school. Participants' names were listed then numbered one through four. Each number was assigned to one group for a specific type of instruction. See Table 1 for demographic information on each of the treatment groups. Table 1 . Descriptive Information for Groups CRA CA RA A Total Gender Male Female 12 11 10 16 9 14 14 10 45 51 Race Black White Hispanic 13 12 0 26 9 1 14 6 1 16 8 0 59 35 2 Grade 5 6 7 8 0 11 9 6 0 16 4 6 1 14 5 1 0 10 9 5 1 51 26 18 Age 11 12 13 14 15 3 7 9 5 1 4 11 6 4 1 6 10 3 2 0 3 9 6 4 2 16 37 24 15 4 Classification Drop-Out-Prevention Learning Disabled Emotionally Handicapped Severely Emotionally Disturbed 8 14 3 0 8 15 2 1 6 15 0 0 8 16 0 0 32 60 5 1 School 1 2 3 4 5 10 3 6 5 12 4 5 5 7 4 5 2 13 3 6 17 43 14 22

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44 Materials Lesson materials were provided by the researcher. Probe sheets used for pre and posttesting and for abstract level lessons were provided. Each probe consisted of one page with 60 multiplication facts that were selected to be representative of multiplication facts to 81 . The facts were randomly ordered on each page. The probe sheets were arranged with alternating lines presenting facts vertically and horizontally. An example of the probe sheets is provided in Appendix A. All pretests, posttests, and abstract level probes were scored by the researcher. Statistical Analvsis The instructional sequence was the independent variable for the study with the CRA, CA, RA, and A only instructional sequences representing four levels of the variable. Acquisition and retention of multiplication facts were the dependent variables measured. Subjects were screened to ensure that the sample did not include participants who had already mastered the target skill as defined by the selection criteria. Pretests were administered to determine subjects' levels of mastery of multiplication facts. Each group's progress was measured by comparing their posttest and retention scores to their pretest performances. Acquisition of multiplication facts was the dependent variable and was measured by a posttest administered at the end of the treatment period. The dependent variable was measured a second time with a posttest administered one to two weeks after the treatment period ended.

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Data were analyzed using a split plot design. Within subject differences demonstrated the effectiveness of each instructional sequence. Between subject comparisons showed whether or not a particular type of instruction was superior for facilitating growth in multiplication facts or if students made similar progress with different types of instruction. Results are presented in Chapter 4.

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CHAPTER 4 RESULTS This investigation was conducted to examine the impact on acquisition, retention, and retrieval of four instructional sequences for teaching multiplication facts to middle school students with learning problems. The concrete-torepresentational-to-abstract (CRA) sequence was one of the methods used. Manipulations of the CRA sequence (i.e., concrete-to-abstract (CA), representational-to-abstract (RA), and abstract only (A)) comprised the remaining instructional strategies. Five research questions were asked. The first four questions addressed the effectiveness of the above mentioned instructional sequences individually. Is each sequence effective in facilitating acquisition and retention of multiplication facts for middle school students with learning problems? The fifth question is the basis of a comparison of the effectiveness of the four sequences. Are CRA, CA, RA, and A instruction equally effective in facilitating acquisition and retention of multiplication facts for middle school students with learning problems? Four treatment groups were used to answer the research questions. Participants in each group received multiplication facts instruction in which one of the specified sequences was used. Each groups' performance was measured with a pretest, a posttest, and a follow-up test. Within group differences were examined to determine the effectiveness of the individual instructional 46

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47 sequences. Between group differences were analyzed to determine the relative effectiveness of each sequence. This chapter provides analyses of these data. Analyses of Data . Means for pretest, posttest, and follow-up test scores for all groups were calculated (Table 2). Adjusted means for posttest and follow-up measures were also determined (Table 3). Repeated measures analyses of variance (ANOVA) were used to determine whether or not significant differences exist between test scores for each treatment group (Table 4). Family-wise comparisons were made to determine what differences were significant with each test group being treated as a family. All family-wise comparison tests were conducted using Bonferoni adjusted levels of significance when the ANOVA tests indicated differences exist between test scores. Results of family-wise comparisons are reported in Table 5. Finally, data were analyzed using a repeated measures analysis of covariance (ANCOVA) design (Tables 6 and 7) to determine if there were relative differences between the effectiveness of the sequences. Results for each group and for the full sample are presented in the following sections. Concrete-to-Representational-to-Abstract Group Results The first research question addressed the effectiveness of CRA instruction. Is the CRA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Results of the repeated measures ANOVA revealed a significant difference among test scores (F = 12.15; p = .0002). All possible comparisons of the pretest, posttest, and follow-up scores were made to determine where

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differences exist. Posttest and follow-up scores were significantly higher than the pretest scores (F = 13.34; p = .0012). Bonferoni family wise comparisons resulted in an adjusted p value of .017. Posttest and follow-up scores were still significantly different from the pretest at this level. No difference was found between the posttest and follow-up scores. Table 2. Test Score Means and Standard Deviations by Group Group Pretest Posttest Follow-up Mean SD Mean SD Mean SD CRA 30.48 19.32 41.96 26.00 42.2 22.15 CA 34.00 18.18 47.00 24.46 46.54 24.17 RA 28.57 14.03 42.10 18.73 42.10 18.73 A 32.46 15.28 43.08 23.88 44.88 21.37 Table 3. Test Score Adjusted Means by Group Group Posttest Follow-up CRA 41.56 42.03 CA 43.06 43.17 RA 44.56 40.70 A 40.24 42.49 Table 4. Repeated Measures ANOVA of Test Effect by Group Group N df F P CRA 25 2,48 12.15 .0002* CA 26 2, 50 17.75 .0001* RA 21 2, 40 16.34 .0001* A 24 2, 46 9.02 .0005* *Significant at the p < .05 level.

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o 5s o Q. O 0) to o 0. 0) Q. O O n (J) IN 1 r\ O) CM 00 CO in o o O o in CD CM O q CO o CD CM O in O 00 CM CM CM t — in CO CN CM CM CM * * * * CM CM o O O O o o o o 1 — > — ' { — 1 t — \ V 1 CO o o 00 in IT) 00 CD CM CM U5 in o CO CM CM CM CM in CO CN CM CM CM * * « « CM o O in Q Q q q q q CM CM CO in q CM CO 00 CD oi CO in in CO CO CM CM CM CM T — CD (D CM CM CM CM < < < o o < > q V Q. 0) -«— ' CD c 8 c D) C/5

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50 Table 6. Repeated Measures ANCOVA of Full Model Source Type III Sums of Squares df Mean Square F P Instructional Strategy 10.263 3 3.421 .01 .998 Pretest 40211.275 1 40211.275 131.50 .0001* Pretest* Instructional 111 \^ \ 1 \^ \^ K 1 \^ 1 1 \^ I Strategy 195.491 3 65.164 .21 .89 School 1359.603 3 453.201 1.48 .226 School* Instructional Strategy 3166.238 9 351.804 1.15 .339 *Signjficant at the p < .05 level. Table 7. Repeated Measures ANCOVA of Instructional Strategy, Pretest, and School Source Type III Sums of Squares df Mean Square F P Instructional Strategy 91.137 3 30.380 .10 .960 School 1385.962 3 461.987 1.53 .214 Pretest 48304.131 1 48304.131 159.50 .0001* *Significant at the p < .05 level. Concrete-to-Abstract Group Results The second research question addressed the effectiveness of CA instruction. Is the CA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Repeated measures ANOVA indicated differences in test scores were significant (F = 17.75; p = .0001). Scores on the posttest and follow-up measures were

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found to be significantly higher than pretest scores at .0001 and .0002 levels respectively when all possible comparisons were made. Significance was determined with the Bonferoni adjusted p value of .017. Posttest and follow-up scores were not significantly different. Representational-to-Abstract Group Results The third research question addressed the effectiveness of RA instruction. Is the RA model effective in facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Significant test differences were indicated by repeated measures ANOVA (F = 16.34; p = .0001). Posttest (F = 30.94; p = .0001) and follow-up measures (F = 17.59; p = .0004) were significantly higher than pretest scores for the RA group. Posttest and follow-up scores were also significantly higher than the scores on the pretest when the Bonferoni family wise comparison was used to adjust the level of significance (p = .017). No difference was found between the posttest and follow-up scores. Abstract-Only Group Results The fourth research question addressed the effectiveness of A-only instruction. Is A-only instruction effective in facilitating acquisition and retention of multiplication facts for middle school students with learning problems? Results of the repeated measures ANOVA revealed significant test differences (F = 9.02; p = .0005). Comparisons of test scores indicate that posttest (F = 9.22; p = .0059) and follow-up scores (F = 20.10; p = .0002) increased

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52 significantly over pretest scores. Significance was still present after adjusting the p value (.017) v\/ith Bonferoni family wise comparisons. Posttest and follow-up scores were not significantly different. Relative Effectiveness of Instructional Sequences A repeated measures ANCOVA was run including all scores across treatment groups to determine if instruction, pretest scores, or schools participants attended were predictors of posttest or follow-up results after controlling for pretest scores and the school participants attended. Between subjects interactions were included in the model as well. Table 7 contains the findings from the ANCOVA. Results indicated that no interactions were present and that neither school nor instructional sequence predicted posttest or follow-up scores. Only the pretest was found to be significant as a predictor of participants' future performances (F = 131.50; p = .0001). Since no interaction effects were present, they were eliminated from consideration and a second ANCOVA was run with instruction, school, and pretest scores as the between subjects factors. Instruction (F = .10; p= .9695) and school (F = 1 .53; p = .2135) were not found to be significant. Results revealed pretest scores to be the only significant indicator of participants' performances on posttest and follow-up measures (F = 159.50; p = .0001). Within subjects factor differences were also examined. Posttest and follow-up scores were not found to interact with instructional sequence (F = .91; p= .4417), school (F = .26; p = .85), or pretest scores (F = 1.17; p = .2819). No

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53 test effect was found (F = .80; p = .3726) indicating there were no differences in the posttest and follow-up test scores. See Table 8 for within subjects data. Table 8. Repeated Measures ANCOVA of Tests and Test Interactions Source Type III Sums of Squares df Mean Square F P Test 62.704 1 62.704 .80 .373 Test* Instructional Strategy 212.135 3 70.712 .91 .442 Test *School 60.888 3 20.296 .26 .854 Test *Pretest 91.508 1 91.508 1.17 .282 •Significant at the p < .05 level. Pretest scores ranged from 3 to 89, creating a lot of variability in the data. To determine if results would differ without such a large range of pretest scores, the repeated measures ANCOVA was run without data on participants whose pretest scores were greater than 40. Results were consistent with the original test including all participants. Summarv This study was conducted to determine the effects of CRA, CA, RA, and A instructional sequences for facilitating acquisition and retention of multiplication facts for middle school students with learning problems. Additionally, the relative effectiveness of the sequences was considered.

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54 Results revealed the presence of within group differences for all groups. Posttest and follow-up test scores were significantly higher than pretest scores regardless of the treatment. Such differences indicate that all four treatments were effective for acquisition and retention of multiplication facts for students in this study. No instructional strategies differences were found. All groups performed equally well on posttest and follow-up measures after accounting for initial differences on pretest scores and school. The absence of an instructional strategy effect indicates that each of the strategies were equally effective for the subjects being studied. There were no differences between posttest and follow-up measures after accounting for initial differences in pretest scores. Follow-up scores remained at the same level as posttest measures. Such data reveal that students retained the skills they developed.

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CHAPTER 5 DISCUSSION Academic demands placed on all students are increasing as standards for student performance become more rigorous. Students with learning problems are expected to meet the same levels of performance as their normally achieving peers. Unfortunately, students with learning problems often fall short of meeting such expectations. Students with learning problems perform poorly on measures of minimum skills in math (Algozzine, O'Shea, Crews, & Stoddard, 1987) and achieve only half of the expected grov\/th for each year they are in school (Cawley, Parmar, Yan, & Miller, 1996). Efforts to alleviate math deficits in students with learning problems are essential to their future academic success. Teachers need to know and use effective and efficient strategies to help students learn more and do so as quickly as possible. However, research regarding effective math instruction has not received a great deal of attention. Information processing theories explain how learners incorporate information into their knowledge base. Instructional strategies need to be designed based on such models. The acquisition strategies model proposed by Baumeister and Kellas (as cited in Mercer & Snell, 1977) is such a model and provided the basis for this study. According to this model, learners select strategies to code information, then practice and recode the information into a 55

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56 form for output. In this investigation, participants coded information with concrete and representation level strategies then participated in abstract level practice which culminated in written output. The findings and implications of the effectiveness of concrete-torepresentational-to-abstract (CRA), concrete-to-abstract (CA), representationalto-abstract (RA), and abstract only (A) strategies for teaching multiplication facts to middle school students with learning problems are discussed in this chapter. The research questions and results are reviewed. Next, the relevance of the findings with respect to the acquisition strategies model is addressed. Limitations of the study are considered. Finally, implications for further research are discussed. Summary of Research Questions and Findings This study was conducted to answer questions regarding four instructional sequences comprised of concrete, representational, and abstract strategies. Specifically, are CRA, CA, RA, and A instruction each effective methods for " facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? The relative effectiveness of the sequences was also examined to answer the final research question. Are CRA, CA, RA, and A instruction equally effective strategies for facilitating acquisition, retention, and retrieval of multiplication facts for middle school students with learning problems? Each of the treatment groups performed significantly higher on posttest and follow-up measures than on pretests. Such data indicate that CRA, CA, RA, and A instruction were all effective for facilitating acquisition,

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57 retention, and retrieval of multiplication facts. Furthermore, each of the treatment groups demonstrated very similar growth on posttest and follow-up measures. The parallel nature of the results demonstrates that the treatments were equally effective. No treatment effects were found. Relevance of Findings to the Acquisition Strategies Model Previous studies support the efficacy of the acquisition strategies model for teaching multiplication skills to elementary school students. Results of studies by Harris et al. (1995) and Miller et al. (1998) demonstrated the effectiveness of the full sequence of concrete and representational coding strategies and abstract rote repetition for developing multiplication skills with students with learning problems. Sigda (1983) compared the effectiveness of the full CRA sequence with a sequence in which only the representational coding strategy was implemented prior to rote repetition. Results of this study indicated that students whose instruction included both concrete and representational coding strategies achieved more. Fennema (1972) and Scott and Neufeld (1976) compared groups who were instructed with only one strategy. Results of these studies indicated that concrete instruction is more advantageous than abstract-only and equally effective as representational-only instruction. The current study also supported the efficacy of the acquisition strategies model. This research, however, studied students in middle school rather than elementary grades. The results of this study differ from previous studies in that all sequences tested were equally effective for teaching multiplication facts to students with learning problems. Use of concrete, representational, and abstract

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58 strategies lead to increased skills in multiplication. Coding strategies at the concrete and representational levels in combination with rote repetition at the abstract level comprised effective instructional strategies (e.g., CRA, RA, CA). The model also demonstrates that learners can bypass coding strategies and use only rote repetition to acquire knowledge. The performance of the group receiving A-only instruction supported this portion of the model. Proceeding directly to rote repetition proved as effective as teaching one or both of the Initial selected coding strategies prior to practice at the abstract level. CRA, CA, and RA sequences were more time and labor intensive for the teacher and resulted in no enhanced learning for these students. Limitations The study has the following limitations. Because only one Florida school district was represented in the sample, the results should not be generalized to other geographic areas. Further, since acquisition and retention of multiplication facts were the targeted skills, the results of this investigation should not be generalized to other math skills. Also, subjects were all middle school students served in special education or drop out prevention programs in public schools. Therefore, generalization to older or younger students or to students served in other educational programs should not be made. It is reasonable to assume that participants had previously received instruction at the concrete and representational levels because such strategies are often used with elementary school students. Therefore, participants' past instruction might limit the study. Finally, criteria for selecting participants (i.e., fewer than 100 digits correct on the

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59 screening) is less stringent than would be accepted by some and might include subjects who did not need instruction. Implications Given the deficits in math skills experienced by students with learning problems, it is essential for math instruction to be effective and efficient to optimize learning. The results of this study demonstrate that the acquisition strategies model is a viable design upon which to base instruction. Concrete, representational, and abstract strategies are effective elements of instructional sequences which resulted in increased multiplication skills. The most notable result of this study is that no treatment effect was demonstrated. All four of the instructional sequences were equally effective. Since the A-only group achieved as much as the other three treatment groups, it can be concluded that concrete and representational strategies are not necessary for middle school students with learning problems to make progress in learning multiplication facts. It is possible that this is due to previous concrete and representational level instruction. Such knowledge is important to ensure the most efficient use of instructional time. Class time does not need to be spent on concrete or representational strategies to help students learn multiplication facts even if students have learning problems in the area of math. Future research should be conducted to determine the effectiveness and efficiency of CRA, CA, RA, and A instruction on other math skills. Investigations into the effects of the four instructional sequences on skills with students of other ages and grade levels who have not had CRA instruction would also be

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60 warranted since all the participants in this study had approximately four years of concrete, representational, and abstract instruction in multiplication facts. Also, given the ease with which abstract instruction can be implemented, it would also be valuable to test the results of such instruction when implemented by paraprofessionals, parents, and school volunteers. If noninstructional personnel can successfully administer abstract level sessions, opportunities for working with students in various settings (e.g., after-school programs; at home) are opened. Future research should also address the issue of expanding the acquisition strategies model to present a more detailed illustration of how information is processed. For example, additional boxes representing retention and retrieval of data could be added to the model between receding and output. Subsequent investigations could then be designed to determine under what conditions students best retrieve data. Summary Students with learning problems often make poor progress in math. Efforts to increase their skills need to include the use of research proven methods. This study was conducted to examine the effectiveness and efficiency of CRA, CA, RA, and A instruction for acquisition and retention of multiplication facts with students with learning problems. The methods had been reported in the literature as necessary, but no data were available as to what strategies are most viable. Results of the study indicate that all of the instructional sequences are equally effective. Therefore, it can be concluded that abstract strategies in

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61 the form of one minute probes as implemented in this study are the most efficient of the strategies tested in teaching multiplication facts to middle school students with learning problems.

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APPENDIX A SAMPLE PROBE SHEET

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Name 7 0 1 9 5 6 )M )l5 x9 x3 )L8 x9 8x5 = _ 2x8 = _ 8x8 = _ 4x3 = _ 7x8 = _ 2x3 = _ 9 5 3 6 4 4 x7 x_5 x5 )L4 )L4 x9 5x3 = _ 1x4 =_ 0x8 = _ 9x5 = _ 6x7 = _ 3x9 = _ 9 4 5 8 4 2 x6 )l8 x6 x_5 x6 8x4 = _ 6x6 = _ 4x6 = _ 8x7 = _ 3x7 = _ 9x8 = _ 7 2 3 8 '1 9 x6 )l9 JL§ -'2L2a, ; • x7 )l9 7x3 = _ 3x4 = _ 5x4 = _ 7x5 = _ 5x9 = _ 2x6 = _ 6 4 3 8 7 8 >l5 x_7 x6 )l2 2
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APPENDIX B INFORMED CONSENT LETTER Dear Parent/Guardian; During the next two months, some graduate students from the Department of Special Education at the University of Florida will be visiting your child's school to work with students who receive math instruction in special education classes. The graduate students will be using three different types of instruction to help students learn multiplication facts. The first type of instruction is concrete. During concrete lessons students will use objects to solve multiplication facts. The second type of instruction is called representational. During this type of instruction, students will draw tallies (or make marks on paper to represent objects) to show multiplication facts. Finally, abstract lessons will be used. At this point, students will use only a page of multiplication facts to solve problems. Some students will do all three types of lessons. Some students will do concrete and abstract lessons. Other students will do only representational and abstract lessons. Other students will do only abstract lessons. After all the lessons have been completed, students will be tested to see if their multiplication skills have increased. They will be tested again two weeks later to see if they have retained the skills they learned during the lessons. Results for students receiving each type of instruction will be compared. We hope to get information that will help determine what kind of instruction is best for middle school students who have difficulty with math. Results of the project will be shared with professionals within the field of education who are interested in improving instruction for students with difficulty in math However, your name and your child's name will be kept confidential to the extent provided by law. Participation in this project is voluntary. Whether or not your child participates in this project will not effect his/her placement in any programs and will not effect his/her grades. If you choose to allow your child to participate, you and your child have the right to withdraw consent for participation at any time without consequence. A possible benefit to you child for participating in the project is increased skills in multiplication. There are no know risks and no compensation for taking part in this project. 64

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t 65 The project wil begin in February 2001 and will continue through March, 2001 . Results of the project will be available after the project is completed. If you have any questions about this project, please contact me at (352) 3920701 ext. 261 . Questions of concerns about the rights of students participating in this project may be directed to the UF Institutional Review Board office, University of Florida, P. 0. Box 112250, Gainesville, FL 32611 (352-392-0433). Sincerely, Penny R. Cox, M.Ed. I have read the procedures described above. I voluntarily give consent for my child, , to participate in this project. Parent/Guardian Signature Date

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APPENDIX C ASSENT SCRIPT READ TO PARTICIPANTS I am going to be working with students during their math class to help them learn multiplication skills, We'll do some different types of lessons that will help you remember the multiplication facts. You do not have to participate unless you want to. If you do participate, you may quit at any time. Do you want to participate in these multiplication lessons? 66

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APPENDIX D PRETEST, POSTTEST, AND FOLLOW-UP SCORES FOR PARTICIPANTS Participant Number School Aqe Grade Classification Group Pretest Score Posttest Score Follow-up Score 1 2 11 6 DOP RA 21 38 34 2 2 12 6 DOR CA 83 105 114 3 2 12 6 DOP CRA 25 44 42 4 2 11 6 DOP CRA 47 73 65 5 2 12 6 DOP RA 40 62 43 6 2 12 6 DOP CA 37 35 39 7 2 11 6 DOP RA 23 34 45 8 2 13 6 DOP CRA 23 36 , 40 9 2 11 6 DOP CRA 27 33 35 10 2 11 6 DOP CA 41 46 45 11 2 11 6 DOP RA 28 26 27 12 2 12 6 DOP CRA 33 36 32 13 2 12 6 DOP RA 33 47 34 14 2 12 6 DOP CA 31 34 48 15 2 12 6 DOP CA 11 24 36 16 2 12 6 DOP A 41 51 48 17 2 12 6 DOP A 37 2 29 18 2 13 7 DOP CA 33 25 25 19 2 13 7 DOP RA 46 72 50 20 2 13 7 DOP A 20 30 32 21 2 13 7 DOP CRA 50 71 61 22 2 12 7 DOP A 41 39 47 23 2 12 7 DOP A 65 104 86 67

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68 Participant Number School Age Grade Classification Group Pretest Score Posttest Score Follow-up Score 24 2 14 7 DOP A 29 44 34 25 2 12 7 DOR RA 16 33 N/A 26 2 13 7 DOP A 32 51 21 27 2 13 7 DOP OA 61 82 52 28 2 13 7 DOP CRA 24 44 N/A 29 2 15 8 DOP A 33 55 61 30 2 14 8 DOP CRA 59 100 75 31 2 13 8 DOP RA 34 60 N/A 32 2 15 8 DOP CRA 51 53 61 33 2 13 8 DOP CA 53 84 104 34 2 14 7 LD CA 23 55 55 35 2 13 7 LD CRA 5 18 20 36 2 14 7 LD A 6 2 22 37 2 14 8 LD CRA 10 16 18 38 2 13 7 LD CRA 28 92 67 39 2 14 8 LD A 33 60 56 40 2 13 8 LD CA 8 7 22 41 2 15 8 LD CA 17 42 42 42 2 14 8 LD A 43 73 56 43 2 13 7 LD CA 31 66 65 44 2 13 7 LD RA 23 54 70 45 2 15 8 LD A 39 43 50 46 2 13 7 LD A 15 13 23 47 4 12 6 LD RA 23 35 44 48 4 11 6 EH CA 29 41 56 49 4 12 6 LD CRA 23 35 44 50 4 11 6 EH CRA 29 41 56 51 4 12 6 LD A 24 58 49

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69 Participant Number School Age Grade Classification Group Pretest Score Posttest Score Follow-up Score 52 4 12 6 LD CA 54 86 67 53 4 11 6 LD RA 18 29 21 54 4 11 6 LD A 56 36 80 55 4 13 6 LD CA 22 31 27 56 4 12 6 LD RA 66 84 76 57 4 13 7 LD A 8 29 27 58 4 13 7 LD CRA 12 37 11 59 4 13 7 LD CRA 28 29 29 60 4 12 6 LD RA 41 66 52 61 4 12 6 LD A 45 41 33 62 4 14 8 SED CA 55 72 67 63 4 12 6 LD CRA 89 84 84 64 4 12 6 LD A 46 58 63 65 4 12 6 LD CA 26 50 56 66 4 14 8 LD CRA 25 32 41 67 4 12 6 LD RA 27 14 32 68 4 13 8 LD A 43 73 84 69 3 12 6 LD A 19 32 24 70 3 12 6 LD RA 21 34 34 71 3 12 6 LD CA 33 43 40 72 3 12 6 LD CRA 17 17 15 73 3 11 6 LD CA 25 35 30 74 3 11 6 LD RA 17 36 27 75 3 11 6 LD CA 7 7 8 76 3 12 6 LD RA 15 22 23 77 3 12 6 LD CRA 27 10 20 78 3 11 6 LD A 16 19 23 79 3 12 6 LD CA 36 49 53

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70 Participant Number School Age Grade Classification Group rreiesi Score r USUcol Score 1 UHUW~ufJ Score 80 3 12 6 LD CRA 20 10 20 81 3 12 6 LD RA 15 28 18 82 3 12 6 LD A 10 20 30 83 1 13 8 LD CRA 14 40 44 84 1 14 8 EH CA 52 47 57 85 1 11 5 LD RA 38 52 55 86 1 11 6 LD CA 12 14 13 87 1 14 8 EH CRA 57 67 80 88 1 13 7 LD A 46 67 76 89 1 12 6 LD CA 23 46 36 90 1 12 7 LD RA 34 56 52 91 1 13 7 EH CRA 30 35 36 92 1 14 8 LD RA 12 40 20 93 1 14 7 LD CRA 3 3 7 94 1 13 8 LD CA 35 37 26 95 1 11 6 LD A 32 34 23 96 1 12 6 LD CA 49 61 52 97 1 12 7 LD RA 24 40 31 98 13 7 LD CRA 30 35 52 99 14 7 LD RA 10 12 19 100 11 5 EH RA 25 20 N/A

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REFERENCES Algozzine, B., O'Shea, D. J., Crews, W. B., & Stoddard, K. (1987). Analysis of mathematics competence of learning disabled adolescents. Ihe Journal of Special Education. 21, 97-108. Armstrong, J. R. (1972). Representational modes as they interact with cognitive development and mathematical concept acquisition of the retarded to promote new mathematical learning. Journal for Research in Mathematics Education. 3 (1), 43-50. Bruner, J. S. (1966). Toward a theory of instruction. Cambridge: The Belkap Press Of Harvard University Press. Cawley, J. S., & Miller J. H. (1989). Cross-sectional comparisons of mathematical performance of children with learning disabilities: Are we on the right track toward comprehensive programming? Journal of Learning Disabilities. 22, 250-254. Cawley, J. F., Parmar, R. S., Yan, W. F., & Miller, J. H. (1996). Arithmetic computation abilities of students with learning disabilities: Implications for instruction. Learning Disabilities Research & Practice. 11. 230-237. Cawley, J. S., Kahn, H. & Tedesco, A. (1989). Vocational education and students with learning disabilities. Journal of Learning Disabilities. 22. 630-634. Cronin, M. E., & Patton, J. R. (1993). Life skills for students with special needs: A practical guide for developing real-file programs. Austin. TX: PRO-ED. Deshler, D. D., Schumaker, J. B., Alley, G. R., Warner, M. M., & Clark, F. L. (1982). Learning disabilities in adolescent and young adult populations: Research implications. Focus on Exceptional Children. 15 (1). 1-12. Engelmann, S., Carnine, D., & Steely, D. G. (1991). Making connections in mathematics. Journal of Learning Disabilities. 24. 292-303. Evans, D., & Carnine, D. (1990). Manipulatives~The effective way. AD! News. 10 (1). 48-55. 71

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72 Fennema, E. H. (1972). The relative effectiveness of a symbolic and a concrete model in learning a selected mathematical principle. Journal for Research in Mathematics Education. 3. 233-238. Fleischner, J., & Manheimer, M. A. (1997). Math interventions for students with learning disabilities: Myths and realities. The School P sycholoav Review. 26(3), 397-413. Fleischner, J. E., Garnett, K., & Shepherd, M. J. (1982). Proficiency in arithmetic basic fact computation of learning disabled and nondisabled children. Focus on Learning Problems in Mathematics. 4 (2). 47-56. Fuys, D. J., & Tischler, R. W. (1979). Teaching mathematics in the elementary school. Boston: Little, Brown and Company. Garnett, K. (1992). Developing fluency with basic number facts: Intervention for students with learning disabilities. Learning Di sabilities Research and Practice. 7. 210-216. Garnett, K., & Fleischner, J. E. (1983). Automatization and basic fact performance of normal and learning disabled children. Learning Disabilitv Quarterlv. 6. 223-230. Giordano, G. (1993). The NCTM standards: A consideration of the benefits. Remedial and Special Education. 14 (6). 28-32. Goldman, S. R., Pellegrino, J. W., & Mertz, D. L. (1988). Extended practice of basic addition facts: Strategy changes in learning-disabled students. Cognition and Instruction. 5. 223-265. Gove, P. B. (Ed.). (1986). Webster's third new international dictionary of the English language unabridged. Springfield, MA: Merriam-Webster. Harris, C. A., Miller, S. P., & Mercer, C. D. (1995). Teaching initial multiplication skills to students with disabilities in general education classrooms. Learning Disabilities Research & Practice. 10 (3). 180-195. Heward, W. L. (2000). Exceptional children: An introduction to soecial education (6th ed.). Upper Saddle River, NJ: Prentice Hall. Hofmeister, A. M. (1993). Elitism and reform in school mathematics. Remedial and Special Education. 14 (6). 8-13. Hudson, P. J., Peterson, S. K., Mercer, C. D., & McLeod, P. (1988). It worked in my classroom: Place value instruction. Teaching Exceptional Children. 20(3), 72-74. i 1

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73 Hutchinson, N. L. (1993). Students with disabilities and mathematics education reform-Let the dialogue begin. Remedial and Special Education. 14(6), 20-23. Johnson, K. R., & Layng, T. V. J. (1994). The morningside model of generative instruction. In R. Gardner, D. Sainato, J. Cooper, T. Heron, W. Heward, J. Eshleman, & T. Grossi (Eds.), Behavioral analysis in education: Focus on measurably superior instruction (pp. 173-197). Belmont, CA: BrooksCole. Jones, S. M. (1982). Don't forget math for special students: Activities to identify and use modality strengths of learning disabled children. School Science and Mathematics. 82 (2). 118-127. Kirby, J. R., & Becker, L. D. (1988). Cognitive components of learning problems in arithmetic. Remedial and Special Education. 14 (6), 7-16. Kulak, A. G. (1993). Parallels between math and reading disability: Common issues and approaches. Journal of Learning Disabilities, 26. 666-673. Marsh, L. G., & Cooke, N. L. (1996). The effects of using manipulatives in teaching math problem solving to students with learning disabilities. Learning Disabilities Research & Practice, 11 (1). 58-65. McLeod, T. M., & Armstrong, S. W. (1982). Learning disabilities in mathematics: Skill deficits and remedial approaches at the intermediate and secondary level. Learning Disability Quarterly, 5, 305-31 1 . Mercer, C. D. (1992). Students with learning disabilities (4'^ ed.). New York: McMillan. Mercer, C. D. (1997). Students with learning disabilities (5*^ ed.). Upper Saddle River, NJ: Merrill. . • Mercer, C. D., Harris, C. A., & Miller, S. P. (1993). Reforming reforms in mathematics. Remedial and Special Education, 14 (6). 14-19. Mercer, C. D., & Miller, S. P. (1992a). Multiplication facts 0 to 81. Lawrence, KS: Edge Enterprises. Mercer, C. D., & Miller, S. P. (1992b). Teaching students with learning problems in math to acquire, understand, and apply basic math facts. Remedial and Special Education, 13 (3), 19-35, 61. Mercer, C. D., & Snell, M. E. (1977). Learning theory research in mental retardation: Implications for teaching. Columbus: Charles E. Merrill.

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74 Miller, S. P., Harris, C. A., Strawser, S., Jones, W. P., & Mercer, C. D. (1998). Teaching multiplication to second graders in inclusive settings. Focus on Learning Problems in Mathematics. 20 (4), 50-70. Miller, S. P., & Mercer, C. D. (1993). Using data to learn about concretesemiconcrete-abstract instruction for students with math disabilities. Learning Disabilities Research & Practice, 8 (2), 89-96. Miller, S. P., & Mercer, C. D. (1997). Educational aspects of mathematics disabilities. Journal of Learning Disabilities. 30. 47-56. Montague, M., & Applegate, B. (1993). Middle school students' mathematical problem solving: An analysis of think-aloud protocols. Learning Disabilitv Quarterly. 16, 19-30. Peterson, S. K., Mercer, C. D., & O'Shea, L. (1988). Teaching learning disabled students place value using the concrete to abstract sequence. Learning Disabilities Research. 4 (1), 52.56. Piaget, J. (1960). The psvcholoov of intelligence. Patterson, NJ: Littlefield, Adams. Pieper, E., & Deshler, D. D. (1985). Intervention consideration in mathematics for the LD adolescent. Focus on Learning Problems in Mathematics, 7 (1), 35-47. Prigge, G. R. (1978). The differential effects of the use of manipulative aids on the learning of geometric concepts by elementary school children. Journal for Research in Mathematics Education, 9. 361-367. Rivera, D. M. (1993). Examining mathematics reform and the implications for students with mathematics disabilities. Remedial and Special Education, 14(6), 24-27. Rivera, D. P. (1997). Mathematics education and students with learning disabilities: Introduction to the special series. Journal of Learning Disabilities, 30, 2-19. St. Martin, A. H. (1975). An analysis of the relationship between two alternate procedures for the utilization of teaching aids in Piaget's developmental theory during the initial introduction of selected fifth grade mathematical topics. Dissertation Abstracts International. 35 . 7037A-7038A. (University Microfilms No. 75-10, 740) Scott, L. F., & Neufeld, H. (1976). Concrete instruction in elementary school mathematics: Pictorial vs. manipulative. School Science and Mathematics. 76 (1). 68-72.

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75 Sealander, K. A. (1991). Discontinuance of the concrete to abstract mathematical instructional sequence minuends 0-9 with mildly handicapped learners (Doctoral dissertation, University of Florida, 1990). Dissertation Abstracts International. 52 . 132A-133A. Siegler, R. S. (1988). Individual differences in strategy choices: Good students, not-so-good students, and perfectionists. Child Development. 59 . 833851. Sigda, E. J. (1983). The development and evaluation of a method for teaching basic multiplication combinations, array translation, and operation identification with their grade students. Dissertation Abstracts International. 44 . 1717A. (University Microfilms No. DA 8322597). Smith, S. R., Szabo, M., & Trueblood, C. R. (1980). Modes of instruction for teaching linear measurement skills. Journal of Educational Research, 73. 151-154. Sovchick, R.J. (1989). Teaching mathematics to children. New York: Harper & Row. Sprick, R. S. (1987). Solutions to elementarv discipline problems [Audiotapes]. Eugene, OR: Teaching Strategies. Starlin, C. M., & Starlin, A. (1973). Guides to decision making in computational math. Bermidji, MN: Unique Curriculums Unlimited. Underhill, R. G., Uprichard, A. E., & Heddens, J. W. (1980). Diagnosing mathematical difficulties. Columbus: Charles E. Merril. United States Office of Education. (1977). Procedures for evaluating specific learning disabilities. Federal Register. 42. 65082-65085.

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BIOGRAPHICAL SKETCH Penny R. Cox was born in Jacksonville, Florida, on March 16, 1961. She graduated from Englewood High School in 1979. She attended the University of North Florida where she received her B.A.E. degree in December, 1981, and M.Ed, degree in December, 1995. Penny taught elementary and special education classes in Duval and Volusia counties for a total of 17 years. During that time, she also fulfilled a variety of additional responsibilities. She was a member of her school's school improvement plan writing team, assisted beginning teachers in completing professional orientation programs, supervised numerous practicum students and interns, and served as the professional development facilitator for her school. While completing her doctoral studies, Penny has taught courses, supervised graduate and undergraduate students, and has served as the assistant to the graduate coordinator for the Department of Special Education. She is currently coordinating the Comprehensive System of Personnel Development grants for the Department of Special Education in partnership with surrounding school .... .1 districts. 76

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Mary Dykes,<^hair Professor of Special Education I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Cecil D. IVfercer Distinguished Professor of Special Education I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. \/laureen Conroy j Maureen Conroy Assistant Professor of Special Education I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. M. David Miller Professor of Educational Psychology

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This dissertation was submitted to the Graduate Faculty of the College of Education and to the Graduate School and was accepted as partial fulfillment of the requirements of the degree of Doctor of Philosophy. August 2001 Dean, College of Educati Dean, Graduate School