# Prospective teachers' development of whole number concepts and operations during a classroom teaching experiment

T ABLE OF CONTENTS

LIST OF FIGURES

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xi

LIST OF TABLES

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xiv

LIST OF ACRONYMS/ABBREVIATIONS

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xv

CHAPTER ONE: INTRODUCTION

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1

Statement of Problem

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4

Significance of Study

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5

Conclusion

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CHAPTER TWO: LITERATURE REVIEW

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Children’s Development of Whole Number Concepts

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10

C hildren’s Development of Whole Number Operations

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Prospective Teachers’ Development of Whole Number Concepts and Operations

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Hypothetical Learning Trajectory

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33

Conclusion

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CHAPTER THREE: METHODOLOGY

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Design Research

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Participants and Setting

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Data Collection

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Instructional Sequence

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Instructional Tasks

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Interpretive Framework

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Items from the CKT - M Measures Database

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Data Analysis

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Documenting Social and Sociomathematica l Norms

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Documenting Classroom Mathematical Practices

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Analyzing Items from the CKT - M Measures Database

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Trustworthiness

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Limitations

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Conclusion

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CHAPTER FOUR: FINDINGS

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Social Norms

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Students Explaining and Justifying Solutions

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Making Sense of an Explanation Given by Another Student

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Sociomathematical Norms

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Acceptable Solution

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Different Solution

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Efficient Solution

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Classroom Mathematical Practices

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Phase One of the Instructional Sequence

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Developing Small Number Relatio nships Using Double 10 - Frames

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Developing Two - Digit Thinking Strategies Using the Open Number line

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103

Phase Two of the Instru ctional Sequence

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Flexibly Representing Equivalent Quantities Using Pictures or Inventory Forms

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Phase Three of the Instru ctional Sequence

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Developing Addition and Subtraction Strategies Using Pictures or Inventory Forms ................................ ................................ ................................ ................................ ....

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Conclusion

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Results from Content Knowledge for Teaching Mathematics (CKT - M) Measures

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CHAPTER FIVE: CONCLUSION ................................ ................................ .................

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Implications

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Conclusion

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APPENDIX A: TASKS FORM THE INSTRUCTIONAL SEQUENCE

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Base - 8 100’s Chart

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Counting Problem Set #1

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Counting Problem Set

#2

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Candy Shop 1

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Torn Forms

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Candy Shop Inventory

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Candy Shop Addition and Subtraction

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Inventory Forms for Addition and Subtraction (In Context)

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Inventory Forms for Addition and Subtraction (Out of Context)

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Broken Machine

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Mult iplication Scenario

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Multiplication Word Problems

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Division Word Problems

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Create Your Own Base - 8 Problems

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APPENDIX B: SAMPLE ARGUMENTATION

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APPENDIX C: INSTITUTIONAL REVIEW BO ARD FORMS

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APPENDIX D: STUDENT INFORMED CONSENT LETTER

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LIST OF REFERENCES

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LIST OF FI GURES

Figure 1: First subtraction procedure reported by Madell (1985).

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Figure 2: Second subtraction procedure reported by Madell (1 985).

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Figure 3: Subtraction procedure reported by Kamii et al. (1993).

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Figure 4: First addition strategy reported by Kamii

et al. (1993)

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Figure 5: Second addition strategy reported by Kamii et al. (1993)

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Figure 6: Third addition strategy report ed by Kamii et al. (1993)

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Figure 7: John’s subtraction strategy reported by Huinker et al. (2003).

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Figure 8: Jamese’s subt raction strategy reported by Huinker et al. (2003).

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Figure 9: Keisha’s subtraction strategy reported by Huinker et al. (2003).

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Figure 10: DeJuan’s subtraction strategy reported by Huinker et al. (2003).

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Figure 11: Robert’s subtraction strategy reported by Huinker et al. (2003).

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Figure 12: Cleo’s doubling strategy reported by Fosnot and Dolk (2001b).

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Figure 13: Double 10 - Frames representing 10.

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Figure 14: Instructor’s documentation of a student’s thinking on an open number line

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Figure 15: Box, roll, and piece

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Figure 16: Inventory Form

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Figure 17: Common knowledge of content item identified by Ball, Hill, and Bass (2005) ................................ ................................ ................................ ................................ ...........

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Figure 18: Specialized knowledge of content item identified by Ball, Hill, and Bass (2005)

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Figure 19: Double 10 - Frames

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Figure 20: Edith’s Solution to 51 -

22.

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Figure 21: Double 10 - Frames representing 10

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Figure 22: Instructor’s d ocumentation of a student’s thinking on an open number line

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Figure 23: Double 10 Frames

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Figure 24: Double - 10 Frames contai ning 11

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Figure 25: Two ways students describe 10 in Double - 10 Frames

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Figure 26: Double 10 - Frame containing 15

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Figure 27: Double 10 - Frame containing 11

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Figure 28: Instructor’s documentation of a student’s thinking on an open number line

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Figure 29: Cordelia’a solution to 12 + 37

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Figure 30: Claire’s method to solve 12 + 37

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Figure 31: Edith’s Solution to 51 -

22.

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Figure 32: Boxes, roll, and pieces

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Figure 33: Inve ntory Form

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Figure 34: 246 candies represented by 2 Boxes, 4 Rolls, and 6 Pieces

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Figure 35: Cordelia’s representation of 246

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Figure 36: Nancy’s representation of candies of 246 candies

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Figure 37: Inventory Form with 1 Box, 3 Rolls, and 4 Pie ces

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Figure 38: Inventory Form with 1 Box and 34 Pieces

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Figure 39: Inventory Form representing 457 candies

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Figure 40: Instructor’s written record of Claire’s solution

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Figure 41: Instructor’s written record of Edith’s solution

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Figure 42: Instructor’s written record of Claire’s solution

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Figure 43: Instructor’s written record of Caroline’s solution

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Figure 44: 417

253 using Inventory Forms

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L IST OF TABLES

Table 1: HLT for Place Value and Operations

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Table 2: Interpretive Framework

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Table 3: Paired Sample Statistics

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Table 4: Paired Samples Test

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Table 5: Actua lized Hypothetical Learning Trajectory

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L IST OF ACRONYMS/ABBREVIATIONS

CKT - M

Content Knowledge for Teaching -

Mathematics

CTE

Classroom Teaching Experiment

HLT

Hypothetical Learning Trajectory

MKT

Mathematical Knowledge for Teaching

N CTM

National Council of Teachers of Mathematics

PUFM

Profound Understanding of Fundamental Mathematics

TDE

Teacher Development Experiment

1

CHAPTER ONE: I NTRODUCTION

In its 2000 publication Principles and Standards for School Mathematics , the Nat ional Council of Teachers of Mathematics (NCTM) identified gaining understanding of numbers and operations, developing number sense, and attaining fluency in arithmetic operations as the core components of mathematics education in elementary school. Emphas izing the long - lasting importance of studying number concepts and operations, Kilpatrick, Swafford, and Findell (2001) acknowledged,

Number is a rich, many - sided domain whose simplest forms are comprehended by very young children and whose far reaches are still being explored by mathematicians. Proficiency with numbers and numerical operations is an important foundation for further education in mathematics and in fields that use mathematics. (p. 2)

Despite the importance of numerical concepts and operatio ns, American teachers lack a deep understanding of elementary number concepts and operations, and, furthermore, this deficit is often conveyed to their students (Ma, 1999). Equally concerning is that prospective teachers are entering the profession without

a firm understanding of mathematics, leaving them ill prepared to depart from the dull, rule - based mathematics instruction promoted during their own schooling (Ball, 1990). These factors lead to a workforce of teachers not adequately prepared, having at b est a marginally sufficient understanding of mathematics (Kilpatrick et al., 2001). As a consequence, some teachers only instruct students how

to perform arithmetic computations in the early elementary grades. However, only emphasizing students’ rote, proc edural knowledge of these operations neglects a deeper conceptual understanding of

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why

the operation is mathematically valid

(Ma, 1999). Neglecting to stress the need for a deeper conceptual understanding of mathematics leads to the perception that element ary mathematics requires little more than the teaching of the utility of numerical operations (Cobb, 1991). However, research has documented that there is more to teaching elementary mathematics than just straightforward procedural computations (Ball 1990;

Ma, 1999).

In his influential research on teacher content knowledge, Shulman (1986) identified subject matter content knowledge and

pedagogical content knowledge as types of understanding essential for teaching. Shulman explained that possessing subject - m atter content knowledge includes an understanding of both the how and why

of a subject whereas possessing pedagogical content knowledge leads to understanding of the knowledge unique to teaching a specific subject. Shulman asserted that in order to render subject matter content understandable to others, a teacher’s pedagogical content knowledge should include useful representations, illustrations, examples, and explanations that help facilitate learning. According to Shulman, a teacher possessing these type s of knowledge can best promote students’ learning.

After Shulman published his research, other researchers began to examine the specific subject - matter content knowledge needed for teaching. Of particular interest to the current study is the subject - matte r content knowledge required to teach elementary mathematics. Ball (1988, 1990) introduced the term substantive knowledge of mathematics

to describe the necessary mathematical knowledge required for the teaching profession. Substantive knowledge of mathema tics includes knowledge of underlying mathematical principles and meanings, attention to correct procedures and concepts, and understanding of the relationships and connections found within mathematics. In

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addition, teachers require a depth of knowledge of

mathematics

that includes understanding what it means to know mathematics and perform mathematical operations, as well as possessing knowledge about mathematics as a field (Ball, 1990).

Ball's (1990) notions of substantive knowledge of mathematics and kn owledge of mathematics became the foundations for what she and her colleagues later described as the mathematical knowledge for teaching

(MKT) (Ball, Sleep, Bass, & Goffney, 2006). Ball et al.’s MKT concept further delineated the subject matter content kno wledge defined by Shulman (1986) into three domains: common content knowledge, specialized content knowledge, and

knowledge at the mathematical horizon . Common content knowledge is the mathematical knowledge most educated adults possess whereas specialized

content knowledge is the mathematics knowledge teachers require to teach mathematics successfully (Hill, Schilling, & Ball, 2004). Knowledge at the mathematical horizon consists of the mathematics content that connects a student’s current experiences to f uture mathematical experiences (Ball, 1993). Together, these three domains emphasize an understanding of mathematics unique to the needs of teachers.

In the process of describing a teacher’s mathematical understanding that is ―deep, broad, and thorough,‖ (Ma 1999, p. 120) Ma coined the concept of profound understanding of fundamental mathematics . Ma asserted that improving mathematics education for all children requires emphasizing future teachers’ profound understanding of fundamental mathematics during t eacher preparation.

By describing the development of prospective teachers’ understanding of whole number concepts and operations, the current study focuses on mathematical understanding as described by Shulman (1987), Ball (1990), and Ma (1999). In this s tudy, the emphasis of whole number tasks during a ten - day instructional unit is what

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Shulman (1987) coined subject matter content knowledge. Research team members selected these tasks in order to deepen the participants’ profound understanding of fundament al mathematics, prior to the participants entering the teaching profession as Ma (1999) dictated. By emphasizing whole number concepts and operations in a meaningful manner, it was imagined that the tasks would cultivate the prospective teachers’ specializ ed content knowledge and would support the knowledge necessary to teach children in the future (Ball, 1990).

Statement of Problem

Although there has been much research into children’s development and understanding of whole number concepts and operations (K amii, Livingston, & Lewis, 1996; Piaget, 1965; Steffe & Cobb, 1988), there has been limited research into prospective teachers’ development and understanding of these concepts. Prior research largely focused on concepts connected to division (Ball, 1990; Graeber, Tirosh, & Glover, 1989; Simon, 1993) although some research does exist related to place value and operations (Andreasen, 2006; McClain, 2003). In an attempt to increase understanding of prospective teachers’ development of whole number concepts an d operations, the following research questions were investigated during a classroom teaching experiment:

1.

What classroom mathematical practices become normative ways of thinking by

prospective elementary school teachers during a whole number concepts and op erations instructional unit?

2.

Is there a statistically significant difference in group mean raw scores between prospective teachers’ pre -

and post -

whole number concepts and operations

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situated in base - 8 instructional unit administration of items selected f rom the Content Knowledge for Teaching Mathematics (CKT - M) Measures database?

Qualitative and quantitative data were collected to document prospective teachers’ mathematical development. The qualitative data included transcripts of videotaped class session s; field notes of research team members; audio - taped conversations of research team meetings; and prospective teachers' work samples, including class work assignments, homework assignments, and tests. The quantitative data included items selected from the Content Knowledge for Teaching Mathematics (CKT - M) Measures database developed by Hill et al. (2004).

Significance of Study

The purpose of this study was to document prospective teachers’ development of whole number concepts and operations. Findings from F oundations for Success: The Final Report of the National Mathematics Advisory Panel

(2008) emphasized the importance of studying prospective teachers’ development of whole numbers.

Since whole number concepts and operations are fundamental topics used by c hildren to understand other mathematical topics, the panel recommended that teacher education programs should stress the importance of whole numbers. In addition, the panel expressed the importance that prospective elementary teachers’ understand the mathe matical content they are responsible for teaching prior to them entering the classroom.

The preparation of teachers is paramount to the success of their students (Ma, 1999). As the panel noted, teachers’ mathematical content knowledge is linked to

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student s’ achievement, and the impact of a series of effective or ineffective teachers dramatically impacts that achievement (National Mathematics Advisory Panel, 2008).

In order to study prospective teachers’ development of whole number concepts and operations a

classroom teaching experiment was conducted in an elementary - education mathematics content course at a major university located in the southeastern United States. Cobb (2000) identified three major reasons to engage in classroom teaching experiments (CTE)

includes the ability to observe: (a) instructional design and planning, (b) ongoing analysis of the classroom, and (c) retrospective analysis of data sources generated during the course of teaching. Classroom teaching experiments involve the creation and development of instructional sequences as well as the investigation of teaching and learning as it naturally occurs in the classroom setting. Researchers then use the documented results to help inform subsequent instructional plans and instructional decisi ons. Using this process, ―theory is seen to emerge from practice and to feed back to guide it‖ (p. 308). A prior whole number instructional sequence described by Andreasen (2006) was influential in developing the tasks used during the current study. Furthe rmore, the items selected from the CKT - M add a quantitative dimension that documents learning by comparing group means raw scores at the beginning and the end of the instructional sequence.

Cobb (2000) summarized the underpinnings of classroom teaching exp eriments by articulating an emergent perspective . This interpretive framework allowed learning to be viewed simultaneously from the psychological perspective of the individual and the social perspective of the class. More specifically, the domains of the s ocial perspective provide evidence of and describe the social dynamic in the classroom (Cobb & Yackel, 1996). The emergent perspective permits researchers to analyze data retrospectively in

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order to identify learning that occurred and describe significant instances during the learning process (Yackel, 2001).

The current study used two methodologies for documenting prospective

teachers’ development of whole number concepts and operations. The constant comparative method

was used to identify and describe the

regularities in classroom activity or social norms

and the mathematically - based classroom activities or sociomathematical norms

that occurred during a 10 - day instructional unit (Glasser & Strauss, 1967). An argumentation analysis

based on Toulmin’s argume ntation model (1969) and a three - phase methodology articulated by Rasmuss en and Stephan (2008 ) was conducted to identify normative classroom mathematical practices that shift in function in an argument or no longer need mathematical justification and as a result have become taken - as - shared . In addition, the prospective teachers were administered items from the CKT - M Measures database both prior to and following the instructional sequence. A dependent t - test was performed to analyze the group means ( ) of th e two instances in time, which in this study were the pre -

and post -

instructional unit administration of items from the CKT - M Measures database (Hair, Anderson, Tatham, & Black, 1998). Performing this type of quantitative analysis allowed to for the resea rch team to use an instrument in base - 10 before and after instruction in base - 8.

Conclusion

Instruction that creates a deep conceptual understanding of whole number concepts and operations is an essential part of the K - 6 curriculum. Indeed, such an underst anding serves as an important foundation for more advanced mathematical concepts (NCTM, 2000). The importance of children’s understanding of whole number

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concepts and operations is well documented in the research literature. Despite the importance of child ren understanding whole number concepts and operations, the understanding of whole number concepts and operations that prospective teachers possess prior to teaching children has not been documented nearly as often. This study attempted to narrow the resea rch gap between the limited research into prospective teachers’ knowledge of whole number concepts and operations and the greater quantity of research into elementary school children’s knowledge of the same concepts.

CHAPTER TWO

discusses the literature a ddressing both children’s and prospective teachers’ development of whole number concepts and operations. Prior research, including results from previous classroom teaching experiments, is discussed as it relates to the current study and to refining, revisi ng, and implementing the instructional sequence. CHAPTER THREE

describes the methodology used in this study, which in addition to quantitative methods included measuring qualitative aspects of a classroom teaching experiment while grounding the findings in

an interpretive framework based upon the emergent perspective (Cobb & Yackel, 1996). CHAPTER FOUR

documents the qualitative and quantitative results of the study and CHAPTER FIVE

discusses the implications of the study.

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C HAPTER TWO: LITERATURE REVIEW

His torically, whole number concepts and operations have been taught for memorization rather than understanding. In his classic article on the meaning of arithmetic, Brownell (1947) argued,

To classify arithmetic as a tool subject, or as a skill subject, or as

a drill subject is to court disaster. Such characterizations virtually set mechanical skills and isolated fact as the major learning outcomes, prescribe drill as the method of teaching, and encourage memorization through repetitive practice as chief or so le learning process. In such programs, arithmetical meanings of the kinds mentioned above have little or no place. Without these meanings to hold skills and ideas together in an intelligible, unified system, pupils in our school for too long a time have ― mastered‖ skills which they do not understand, which they can use only in situations closely paralleling those of learning, and which they soon forget. (p. 11)

Almost 30 years later, Skemp (1976) distinguished between instrumental understanding

and relatio nal

understanding . Skemp emphasized that when individuals possess instrumental understanding, they blindly follow mathematical ―rules without reasons,‖ unlike individuals who possess relational understanding or the ability of knowing ―what to do‖ and ―why it is done‖ mathematically (p. 20).

Despite past attempts to call attention to Brownell and Skemp’s findings, many elementary school teachers continue to lack an understanding of whole number concepts and operations (Ma, 1999). In her influential compariso n of Chinese and American elementary school teachers, Ma reported that teachers with procedural understanding

of mathematical topics teach their students algorithmically, failing to make connections

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among mathematical topics. When these teachers attempt to

make connections, they do so without substantial mathematical arguments. For example, although teachers may know how to subtract or multiply, they do not display mathematical understanding beyond the actions required to perform these operations. Such teac hers directly contrast with those described as possessing conceptual understanding. Teachers possessing conceptual understanding can coordinate mathematical concepts, operations, and relationships to create a deep understanding of a mathematical topic. The se teachers emphasize not only the actions required to perform mathematical operations but also their significance.

Because whole numbers and operations comprise much of elementary school mathematics and the importance of developing this knowledge base (NC TM, 2000), this chapter begins by addressing research into children’s development of whole number concepts and operations. The chapter then continues by addressing prospective teachers’ development of whole number concepts and operations. Finally, a hypot hetical learning trajectory

(HLT) or predicted pathway through which whole number concepts and operations may be acquired is introduced (Simon, 1995).

Children’s Development of Whole Number Concepts

Number development and counting are important mathematic al foundations in children’s understanding of whole number concepts. Early explorations are often classified by intuitive, direct, and concrete experiences that eventually progress to more elaborate and mathematically sophisticated ways of using numbers, t ypically coinciding with symbolic notation manipulation at an abstract level (Kilpatrick et al., 2001).

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Important landmarks of early number development include one - to - one correspondence, hierarchical inclusion, compensating, and part - whole relationships (F osnot & Dolk, 2001). Based on her work with elementary students, Kamii (1985) defined the following important mathematical concepts: (a) cardinality as the concept that a number tells an individual the total amount because it is the last number counted (b)

one - to - one correspondence

as the concept of accounting for each object only once, and (c) hierarchical inclusion

as the concept that numbers increase by exactly 1 each time while nesting within each other by this amount. Fuson, Grandua, and Sugiyama (2001 ) described children’s numerical connections between oral numbers, written numbers, and numerical quantities. These connections are made by strategies including disorganized counting, counting on fingers, recognizing patterns, relating words and numerals, and using manipulatives. When children attempt to understand quantity, finger usage often shows an initial correspondence between objects and quantity. Eventually, children continue to count with one - to - one correspondence and then transition into other cou nting strategies.

In their research, Steffe, Cobb, and von Glasserfeld (1988) indicated children’s early counting progresses through five distinct stages of activity, beginning with the counting of perceptual unit items and ending with the counting of ab stract unit items. This progression starts with the creation of perceptual unit items

as children use their perceptions to count every item as a unit followed by figural unit items

as children begin counting items outside their immediate perceptual range. Third, children create motor unit items

to coordinate their motor acts with either perceptual or figural unit items. Next, children create verbal unit items

to coordinate the production of a unit item and the simultaneous speaking of a number word. When ch ildren are able to consider the

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previously described units as objects that are counted, they are engaged in counting abstract

unit items . These stages lead to the strategies used to perform addition and subtraction.

While working with kindergarten childr en, Baroody (1987) found that the children used both concrete counting strategies

and mental counting strategies . Concrete counting strategies are used when countable objects are counted for each addend before all of the objects are counted for the total. Mental counting strategies are strategies that keep track of how far one should count from the cardinal number of the first addend. These strategies eventually evolve into addition and subtraction strategies, including those of counting backwards, skip cou nting, and taking steps of 10 (Fosnot & Dolk, 2001).

Children’s Development of Whole Number Operations

When faced with a computation, children choose among various methods to solve the problem, including using manipulatives to model the situation, inventi ng written procedures, drawing a picture, performing a mental calculation, choosing a known paper - and - pencil algorithm, or using technology (Carroll & Porter, 1998). Bass (2003) asserted that it is reasonable to use a single, clearly described generic solu tion method to solve

mathematical problems that occur repeatedly. The use of written algorithms,

or ― procedures that can be executed in the same way to solve a variety of problems arising from different situations and involving different numbers ‖ , is neces sary for a myriad of reasons

(Kilpatrick, et al. 2001, p.7) . Because computations become more difficult to perform as numbers increase, it becomes necessary to keep track of the computation. Although calculators can often be used, paper - and - pencil algorith ms are the most