The Development of Year 3 Students Place-Value Understanding: Representations and Concepts

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The Development of Year 3 Students’

Place-Value Understanding:

Representations and Concepts

Peter Stanley Price

Dip.Teach., B.Ed., M.Ed., A.C.P.

Centre for Mathematics and Science Education School of Mathematics, Science and Technology Education

Faculty of Education

Queensland University of Technology

A Thesis

submitted in partial fulfilment of the requirements for the award of the degree of

Doctor of Philosophy

March, 2001

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Keywords

Place value, base-ten blocks, Year 3, mathematical understanding, place-value software, representations of number, conceptions of number, electronic base-ten blocks, conceptual structures for multidigit numbers, feedback, misconceptions of number, independent-place construct, face-value construct, mathematics teaching with technology, number models, Payne-Rathmell model for teaching number topics.

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Abstract

Understanding base-ten numbers is one of the most important mathematics topics taught in the primary school, and yet also one of the most difficult to teach and to learn. Research shows that many children have inaccurate or faulty number

conceptions, and use rote-learned procedures with little regard for quantities represented by mathematical symbols. Base-ten blocks are widely used to teach place-value concepts, but children often do not perceive the links between numbers, symbols, and models. Software has also been suggested as a means of improving children’s development of these links but there is little research on its efficacy.

Sixteen Queensland Year 3 students worked cooperatively with the researcher for 10 daily sessions, in 4 groups of 4 students of either high or low mathematical achievement level, on tasks introducing the hundreds place. Two groups used physical base-ten blocks and two used place-value software incorporating electronic base-ten blocks. Individual interviews assessed participants’ place-value

understanding before and after teaching sessions. Data sources were videotapes of interviews and teaching sessions, field notes, workbooks, and software audit trails, analysed using a grounded theory method.

There was little difference evident in learning by students using either physical or electronic blocks. Many errors related to the “face-value” construct, counting and handling errors, and a lack of knowledge of base-ten rules were evident. Several students trusted the counting of blocks to reveal number relationships. The study failed to confirm several reported schemes describing children’s conceptual structures for multidigit numbers. Many participants demonstrated a preference for grouping or counting approaches, but not stable mental models characterising their thinking about numbers generally. The

independent-place construct is proposed to explain evidence in both the study and the literature that shows students making single-dimensional associations between a place, a set of number words, and a digit, rather than taking account of groups of 10.

Feedback received in the two conditions differed greatly. Electronic feedback was more positive and accurate than feedback from blocks, and reduced the need for human-based feedback. Primary teachers are urged to monitor students’ use of base- ten blocks closely, and to challenge faulty number conceptions by asking appropriate questions.

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List of Tables

TABLE 2.1. Aspects of Place-value Understanding Described in the Literature... 26

TABLE 2.2. Task Performance Illustrating Limited Conceptions in Place-value Understanding... 31

TABLE 3.1. Phases of the Research Design ... 75

TABLE 3.2. Participant Groups for the Main Study... 76

TABLE 4.1. Transcript Notations ... 98

TABLE 4.2. Summary of Participants’ Numeration Skills Identified in two Interviews ... 102

TABLE 4.3. Summary of Numeration Skills Demonstrated by Each Participant and by Each Group... 103

TABLE 4.4. Summary of Place-value Understanding Criteria Achieved by High- achievement-level and Low-Achievement-Level Participants... 105

TABLE 4.5. Summary of Place-value Understanding Criteria Achieved by Participants in Computer and Blocks Groups ... 106

TABLE 4.6. Use of Grouping Approaches for Selected Interview Questions... 113

TABLE 4.7. Use of Grouping Approaches by Each Group... 113

TABLE 4.8. Use of a Counting Approach for Selected Interview Questions... 121

TABLE 4.9. Use of Counting Approaches by Each Group ... 122

TABLE 4.10. Incidence of Face-value Interpretations for Written Symbols after Selected Interview Questions ... 130

TABLE 4.11. Use of Face-Value Interpretations of Symbols by Each Group ... 131

TABLE 4.12. Incidence of Approaches Adopted for Selected Interview Questions.... ... 133

TABLE 4.13. Response Categories for Interview Digit Correspondence Questions ... ... 142

TABLE 4.14. Summary of Digit Correspondence Response Categories... 143

TABLE 4.15. Participants’ Written Responses to Task 27 (b) ... 171

TABLE 4.16. Incidents of Feedback of Each Source per Group ... 177

TABLE 4.17. Percentage of Feedback Compared With Answer Status ... 180

TABLE 4.18. Quality of Feedback Provided for Correct or Incorrect Answers... 181

TABLE 4.19. Percent of Feedback for Correct Answers from Each Source... 182

TABLE 4.20. Percent of Feedback for Incorrect Answers from Each Source ... 183

TABLE 4.21. Feedback Providing Answers from Each Source for Each Group ... 187

TABLE 5.1. Comparison of Results of Digit Correspondence Tasks Between This Study and Ross (1989)... 209

TABLE H.1. Overview of Teaching Program Tasks ... 284

TABLE L.1. Source of Feedback ... 307

TABLE L.2. Effects of Feedback ... 307

TABLE L.3. Responses to Feedback... 308

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List of Figures

Figure 2.1. The face value of each individual numerical symbol, together with its position relative to the ones place, determines the value it represents.

... 18

Figure 2.2. Relationships inherent in base-ten blocks. ... 41

Figure 2.3. Relationships among numbers, written symbols, and concrete materials... 46

Figure 2.4. Conceptual gap between written symbols and concrete materials. .... 48

Figure 2.5. The use of transitional forms to bridge the gap between written symbols and concrete materials... 49

Figure 3.1. Dimensions of research design. ... 65

Figure 3.2. Original graphic images used on regrouping buttons in software used during pilot study... 72

Figure 3.3. Replacement graphic images used on regrouping buttons in software used during main study... 72

Figure 3.4. Sample Representing numbers task... 80

Figure 3.5. Sample Regrouping task... 81

Figure 3.6. Sample Use of numeral expander task. ... 81

Figure 3.7. Sample Comparison task. ... 81

Figure 3.8. Sample Counting task... 82

Figure 3.9. Sample Addition task, including regrouping. ... 83

Figure 3.10. Diagram showing objects used in interviews for Digit Correspondence Task with misleading perceptual cues. ... 85

Figure 4.1. Interview scores compared to use of grouping approaches... 115

Figure 4.2. Interview scores compared to use of counting approaches. ... 122

Figure 4.3. Interview scores compared to use of face-value interpretations of symbols... 132

Figure 4.4. Proportions of feedback from each source for each group. ... 178

Figure 5.1. Column counters in software representation of 248... 232

Figure A.1. Screen view of on-screen tutorial question with block representations. ... 262

Figure A.2. Partial screen image from Rutgers Math Construction Tools, showing block and symbol representations of a number. ... 263

Figure A.3. Screen view of Blocks Microworld showing block representation of a number, nominating a cube as one. ... 264

Figure A.4. Main screen of Hi-Flyer Maths. ... 266

Figure A.5. “Show as tens” feature activated. ... 268

Figure A.6. Number name window and numeral expander displayed... 269

Figure A.7. A block is “sawn” into 10 pieces... 271

Figure A.8. “Add blocks” requester... 272

Figure L.1. Data entry screen for feedback database. ... 306

Supplementary Material

Hi-Flyer Maths Installation Files [CD-ROM] ... 385

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Statement of Original Authorship

The work contained in this thesis has not been previously submitted for a degree or diploma at any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made.

Signed: ________________________________

Date: ________________________________

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Acknowledgments

The completion of a thesis is a drawn-out, sometimes painful task that cannot be done without much assistance, both professional and personal, from many others.

I gratefully acknowledge my indebtedness to the following people for their support over the past six years:

To my principal supervisor, Professor Lyn English, I offer my heartfelt appreciation for her patience, wisdom and unfailing support since I started this journey. Your example to me, Lyn, as an academic and colleague has always been of the highest standard, and I greatly appreciate your patience in leading me to the completion of the thesis. Thank you for believing in me and for giving me the space to finish.

To my associate supervisor, Dr Bill Atweh, I thank you also for your patience, support, and wisdom. Your ability to see past the data to what they reveal has been invaluable in helping me frame the last few chapters and in structuring what was quite a mess and turn it into a coherent account.

To my dear wife and partner, Trish, I can only say that a lesser person would have given up long ago. I deeply appreciate your love and support over what has ended up as a longer time than we could have imagined when I started. This has truly been a partnership, in which you have sacrificed your desires and your time to give me space to study, since 1993. Thank you from the bottom of my heart.

To my lovely, wonderful children, Mary, Andrew and Hannah, I express my deep love and devotion. You too have had to give up time with me, and to put up with your Dad’s frequent absences over a substantial part of your lives. I am

immensely proud of each of you, and I look forward to seeing you grow and develop into the adults God intends.

To my parents, Rev and Mrs Stanley and Eva Price, I express my love and heartfelt thanks for everything you put into raising me. Though we are separated by great distance, I am aware of your constant support and prayers that you have provided all my life. Thanks, Dad and Mum.

To my colleagues and friends at Christian Heritage College, I express my heartfelt thanks and love for accepting me and supporting me in this endeavour. In particular, Dr Robert Herschell has been a constant friend, mentor and source of support over many years. Thanks, Rob, for believing in me, for giving me the chance to follow God’s call to teach others.

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To many colleagues, mentors and friends at the School of Mathematics, Science and Technology Education, QUT, thank you. I have had a very rewarding time at QUT over many years, and appreciate your input into my life and career, including the writing of this thesis. In particular, a sincere “thank you” to Professor Tom Cooper, A/Prof Cam McRobbie and Drs Cal Irons, Ian Ginns, Rod Nason and Jackie Stokes for your wise advice and counsel. And to my fellow PhDers over the past several years—Drs Neil Taylor, Carmel Diezmann, Kathy Charles, Mary Hanrahan, David Anderson, Stephen Norton, Anne Williams and Gillian Kidman—

thank you all for your friendship and support.

Finally, but by no means least, I express my love and appreciation to the Lord Jesus Christ, without whom I could do nothing. My abilities and talents are from Him alone; my prayer is that I walk worthy of the calling He has placed on my life, as a faithful witness to His love and power.

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Chapter 1: The Problem

The development of a competent understanding of place-value concepts by primary students is a prerequisite for the learning of much later content of the school mathematics curriculum. Children need to learn from the early primary school grades¹ how numbers are written in the base-ten numeration system, and to construct accurate mental models for numbers, in order to develop a proficiency with

mathematics that will equip them to solve problems in later life. However, several authors have noted that place-value concepts are difficult both for teachers to teach and for students to learn (G. A. Jones & Thornton, 1993a; S. H. Ross, 1990). The study described in this thesis investigated the teaching and learning of place-value concepts using number representations in two formats: conventional base-ten blocks and a computer software application.

1.1 Recommendations for Changes in Mathematics Education

Several documents published over the past 20 years have recommended important changes in the way mathematics is taught in schools. These documents include Mathematics counts (Cockcroft, 1982), Everybody Counts (National Research Council [NRC], 1989), Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989), A National Statement on Mathematics for Australian Schools (Australian Education Council, 1990), and Principles and Standards for School Mathematics (NCTM, 2000). Three prominent topics in these documents are relevant to this study: (a) the development of mathematical understanding, (b) the development of number sense, and (c) the use of technology in mathematics classes.

The first recommendation for mathematics education identified as relevant to this study, that more emphasis be given to students’ development of mathematical

1 N.B. Queensland primary schools include Years 1-7; the term primary as used in this thesis refers to this range of school class levels, which may be considered to be roughly equivalent to primary and elementary schools in the U.S.

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understanding, underlies the advice contained in the policy documents listed in the previous paragraph. The view of the NCTM (2000) is clear: “Learning mathematics with understanding is essential” (p. 20). The documents embody a view of learning as a sense-making activity (Mayer, 1996; McIntosh, Reys & Reys, 1992), in which learners develop their own personal understandings of concepts to which they are exposed. Thus the act of teaching is seen not as transmitting ready-formed

knowledge from teacher to learner, but rather as encouraging the learner to construct concepts so that they make sense to him or her (Cobb, Yackel & Wood, 1992; NRC, 1989). The view of learning as a sense-making activity has special relevance for the teaching of mathematics, because of its focus on abstract entities that need to be conceptualised by each learner (Davis, 1992). If learners do not form appropriate, accurate mental models of numbers, they will be hindered in attempting to solve mathematical problems in meaningful ways. The literature is replete with

observations of students who, though they can do some computation, do so without understanding the meanings behind the symbols and procedures used (e.g., Kamii &

Lewis, 1991).

Meaningful understanding of numbers is linked to the second

recommendation relevant to this study, that the development of number sense be made a priority for mathematics teaching (McIntosh et al., 1992; NCTM, 2000;

Sowder & Schappelle, 1994). Number sense is regarded by many as an important goal of mathematics education, enabling students to answer flexibly non-routine questions that require a mathematical solution. Traditionally, mathematics was taught so that students could answer routine arithmetic questions accurately, for future employment in retail or manufacturing jobs (NRC, 1989). Today there is a greater need for adults who can think mathematically and who can devise methods of solving numerical questions in novel ways (NCTM, 1989).

The third recommendation for change in the way that mathematics is taught is for the use of technological devices—calculators and computers—to be a matter of course at all school grade levels (Australian Education Council, 1990; NCTM, 2000;

NRC, 1989). The question of how computer technology (referred to in this thesis as

“technology”) can best be incorporated in mathematics education is the subject of some debate. Research such as that described here is needed to help answer questions about the effects of technology on students’ learning. In particular, the computational power and the representational capabilities of computers have the potential to assist

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students to develop more meaningful concepts for numbers (Clements & McMillen, 1996; NCTM, 2000; Price, 1996, 1997). This potential needs further investigation.

1.2 The Learning of Place-Value Concepts

The development of understanding of the base-ten numeration system is foundational to all further use of numerical symbols, both in school and outside the classroom. Thus, understanding how children develop place-value concepts, and the difficulties they face in doing so, is of great importance to mathematics educators.

1.2.1 Conceptual Structures and Difficulties With Place-Value Concepts Children’s difficulties in making sense of the meanings represented by multidigit symbols have been reported widely in the literature (e.g., G. A. Jones &

Thornton, 1993a; Resnick, 1983; S. H. Ross, 1990). In particular, several authors reported students having difficulty linking the abstract realm of numbers and their symbolic and physical referents (e.g., Baroody, 1989; Baturo, 1998; Fuson, 1992;

Hart, 1989; Hiebert & Carpenter, 1992). In describing and analysing these

difficulties, several researchers have postulated children’s conceptual structures for numbers (e.g., Fuson, 1990a, 1990b, 1992; Fuson et al., 1997; Resnick, 1983). A number of conceptual structures, and several limited conceptions for numbers, have been reported as being common among primary-age students. Such conceptual structures feature prominently in much writing about children’s learning of place- value concepts, and are considered by many, including this author, to be of critical importance in understanding how children develop place-value concepts.

This thesis includes an analysis of evidence for conceptual structures for multidigit numbers in the present study, and a comparison between that evidence and reported findings of other researchers. Finally there is a discussion of possible links between conceptual structures and participants’ use of two types of representational material: physical and electronic base-ten blocks.

1.2.2 Use of Number Representations

Physical base-ten blocks.

Physical base-ten blocks, generally known in Queensland schools as

multibase arithmetic blocks [MABs], are regarded by many teachers as particularly useful for helping students to build meaningful conceptual structures for multidigit

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numbers (English & Halford, 1995). Developed by Dienes (1960) 40 years ago, they have become the concrete materials of choice for teaching the base-ten numeration system in many countries, including the USA, the UK, and Australia. Physical base- ten blocks can be thought of as physical analogues of numbers, and mirror the internal structures and relative magnitudes represented by the digits that make up a written symbol (English & Halford, 1995). Students must reason analogically to use the blocks effectively; that is, they must map the relations inherent in the blocks onto the relations in the target realm (Gentner, 1983), the domain of numbers. In order for physical base-ten blocks to be effective in representing numbers, it is important that students’ attention be drawn to the analogical relationships that exist between the blocks and the numbers they represent (Fuson, 1992).

Electronic base-ten blocks.

In light of the difficulties students have making links between numbers and their referents, a number of suggestions have been made of teaching methods that may help students to perceive connections among various forms of number

representation. One such suggestion is the use of computer-generated representations for numbers (Clements & McMillen, 1996, Hunting & Lamon, 1995; NCTM, 2000).

Several software programs have been designed to model base-ten blocks electronically on screen (e.g., Champagne & Rogalska-Saz, 1984; Rutgers Math Construction Tools, 1992; P. W. Thompson, 1992). All use the capabilities of the computer to enhance the number representations available to the user beyond those provided by conventional physical blocks. For example, many of these programs include number representations such as written symbols and representations of regrouping actions on blocks, and link these representations tightly together so that a change in one representation is mirrored by an equivalent change in the other

representations (see Appendix A). At the time the study was conducted, apart from Rutgers Math Construction Tools the author only had access to descriptions of these programs, and not to the programs themselves. Furthermore, none of the programs included all the features that were felt to be desirable for teaching place-value concepts; specifically, the author wanted the software to model multidigit numbers with pictures of base-ten blocks on a place-value chart, to model regrouping actions on the blocks, to show various symbolic representations for the numbers represented by the blocks, and to play audio recordings of the number names. Because of the lack

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of these features in available software, the author developed a new software program for teaching place-value concepts, named Hi-Flyer Maths (described in Appendix A;

installation files available on CD-ROM in Error! Reference source not found.).

Central to this study is the effect of base-ten blocks, both physical and electronic, on Year 3 students’ place-value conceptions of multidigit numbers. The Hi-Flyer Maths software was used in the exploratory teaching study to assess these effects.

1.3 The Research Question

Based on the issues outlined in the previous section, the question investigated in this study is

How do base-ten blocks, both physical and electronic, influence Year 3 students’ conceptual structures for multidigit numbers?

Within the context of Year 3 students’ use of physical and electronic base-ten blocks, the following specific issues were of concern:

1. What conceptual structures for multidigit numbers do Year 3 students display in response to place-value questions after instruction with base- ten blocks?

2. What misconceptions, errors, or limited conceptions of numbers do Year 3 students display in response to place-value questions after instruction with base-ten blocks?

3. Which of these conceptual structures for multidigit numbers can be identified as relating to differences in instruction with physical and electronic base-ten blocks?

4. Which of these conceptual structures for multidigit numbers can be identified as relating to differences in students’ achievement in numeration?

1.4 Overview of Research Methodology

The research questions were investigated using qualitative case studies involving Vygotskian teaching experiments and Piagetian clinical interviews (Hunting, 1983; Hunting & Doig, 1992). The study involved 16 Year 3 students selected from a single primary school, half of each gender, and half of either high or low mathematical achievement level (Table 3.2). The students were assigned to 4

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groups of 4 students, each group comprising 2 boys and 2 girls, all of the same achievement level. One high-achievement-level and one low-achievement-level group were assigned to use physical base-ten blocks, and the other 2 groups used computer software (electronic base-ten blocks). The groups each took part in 10 teaching sessions, involving up to a total of 45 place-value activities designed to develop two-digit and three-digit place-value concepts. Each student was interviewed individually both before and after the teaching sessions, to assess their place-value understanding.

Each teaching session and interview was videotaped and audiotaped for later transcription and analysis. As well, the researcher took field notes and the students’

workbooks were collected. The raw data from the teaching sessions and interviews were transcribed for coding, principally using the grounded theory method described by Strauss and Corbin (1990). Categories for participants’ responses emerged from the data as they were analysed. These categories were compared with a framework of conceptual structures identified in the literature.

1.5 Significance of the study

There have been a number of suggestions for teaching strategies to help students develop good place-value understanding, including the use of some means of “bridging the gap” between numbers and physical number representations (Hart, 1989). One suggestion for bridging this gap is to use computer-generated

representations of numbers (e.g., Clements & McMillen, 1996; P. W. Thompson, 1992). However, there are few reports of in-depth investigation of the use of such software, or of research-informed guidelines for future software development. In particular, there is no evidence of analysis of children’s conceptual structures for multidigit numbers as they use electronic base-ten blocks to learn place-value concepts. Considering both the recommendations to use suitable place-value

software and the money invested in its development and purchase, there is a pressing need for such research.

This study investigates the ideas that students have of numbers, and how those ideas may be affected by the use of either physical or electronic base-ten blocks. The study provides important findings in this field with significance for both the teaching of place-value concepts generally, and the design and use of place-value software.

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1.6 Outline of the Thesis

The thesis has 6 chapters. The current chapter provides an overview of the study. Chapter 2 is a review of literature relevant to the study. Issues addressed are current issues in mathematics education, place-value understanding, cognitive science contributions to understanding of learning of place-value, the use of number representational materials, and the use of computer software for teaching

mathematics. Chapter 3 contains a description of the methodology used in the study, including assumptions and issues underlying the design, a description of the pilot study and the main study, and discussion of validity, reliability, and limitations of the design. Chapter 4 reports results of the study from the teaching sessions and

interviews. Chapter 5 comprises a discussion of the results in the light of other reported research, and includes a description of a previously-unreported category of student response to place-value questions, the independent-place construct. Chapter 6 concludes the thesis with a summary of findings, implications for the teaching of place-value, and suggestions for further research in the area.

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Chapter 2: Review of Literature

2.1 Chapter Overview

This chapter comprises a review of literature relevant to the study, divided into 5 main sections. The broad background to the research questions is related to several current issues in mathematics education. Three issues relevant to this study are (a) the development of mathematical understanding, (b) the development of number sense, and (c) the use of technology in mathematics classes. These three issues are linked in section 2.2 to the teaching of place-value concepts in primary schools. Section 2.3 defines place value and place-value understanding for the purposes of this thesis. This section identifies the skills that children need to develop and introduces the desired mental models of numbers that are an important focus of the study.

The contribution that cognitive science has made to the study of children’s understandings of mathematics, and in particular place value, is summarised in section 2.4. Two areas of cognitive science study in particular are described: mental models and analogical reasoning. First, based on previous research, a framework of four conceptual structures considered necessary for children to learn place-value ideas is proposed, and three common limited conceptions of numbers are listed.

Second, analogical reasoning is an important consideration in the teaching of many mathematical topics, including place-value concepts. Base-ten blocks are analogues of the base-ten numeration system, and mirror the relations among digit places. A focus on understanding of analogical reasoning is therefore important in considering their use as representations of numbers.

Section 2.5 describes the teaching of place-value understanding, including the use of physical models of numbers in teaching place-value concepts. It is shown that there is evidence of a “conceptual gap” in the minds of many children between written symbols and base-ten blocks, which a number of researchers have attempted

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to bridge. One solution introduced in this section is the use of computer-generated manipulatives. Section 2.6 includes a description of capabilities of modern

computers which make them potentially valuable for helping students to make connections within many domains, including mathematics. Specifically, the capability to present different representations of a concept shows promise for representing numbers in several formats, with the aim of helping students to see connections among them.

2.2 Issues in Mathematics Education

Several issues of current concern in mathematics education are particularly relevant to this study. This section describes three of these issues: students’ active involvement in mathematics learning, development of number sense, and the use of technology.

2.2.1 Students’ Active Involvement in Mathematics Learning

The view that students should actively participate in the process of learning mathematics is a comparatively new one. As the NCTM (1998) noted, “the notion of mathematics as something to be deeply understood, so that it can be used effectively, has not always been a valued outcome of school mathematics” (p. 33). A

“traditional” model of mathematics teaching, typical of the first half of the 20th Century, has been widely criticised (NCTM, 1989, 1991; NRC, 1989). This model viewed the teaching-learning process as the transmission of information, and thereby knowledge, from teacher to student. In this model the teacher was perceived to be the source of information, “the sole authority for right answers” (NCTM, 1991, p. 3), and the student was merely a passive recipient of the information. This model owes much to behaviourist views of learning: namely, that “learning is conceived of as a process in which students passively absorb information, storing it in easily

retrievable fragments as a result of repeated practice and reinforcement” (NCTM, 1989, p. 10). In contrast, recent recommendations for mathematics teaching and learning (NCTM, 1989, 1991, 2000; NRC, 1989) portray a very different picture.

First, the learning process is now widely seen as one of individual construction of understanding, in which new experiences are integrated with prior knowledge to form understandings that are meaningful to the student (Simon, 1995). Second, students are seen as “autonomous learners . . . . [who should] take control of their

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learning” (NCTM, 2000, p. 21), to make sense of it for themselves. Third, the teacher’s role is to be a “guide for exploring academic tasks” (Mayer, 1996, p. 152;

see also Sowder, 1994, p. 146), or an “[orchestrator of] classroom discourse in ways that promote the investigation and growth of mathematical ideas” (NCTM, 1991, p. 1).

A critical component of the view of mathematics learning described here is the necessity of students making sense of what they learn (Mayer, 1996). If teachers want their students to develop meaningful understanding of mathematical concepts, then there is a need to consider many aspects of the learning environment that exists in the classroom. One aspect of the learning environment of major relevance to this study is the question of various interactions that take place, described below; this is an important item of interest in the research described in this thesis. As explained by McNeal (1995), “[by] studying classroom interactions, the observer could . . . infer a particular individual’s knowledge . . . from observations of his/her interactions with the objects or with other individuals” (p. 3). The following subsection addresses interactions of three kinds that are of relevance to this study.

Student-teacher interactions.

If the view of learning as a constructive meaning-making activity is accepted, then the interactions between students and teachers are of obvious importance.

“More than any other single factor, teachers influence what mathematics students learn and how well they learn it” (NCTM, 1998, p. 30). Part of a constructivist model of learning is a view that students construct mathematical knowledge as a product of

“interaction in social contexts” (Putnam, Lampert & Peterson, 1990, p. 134). As Cobb and Yackel (1996) stated, “we consider students’ mathematical activity to be social through and through because it does not develop apart from their participation in communities of practice” (p. 180). This idea of a community of practice is implicit in much recent writing about teaching mathematics (NCTM, 1991, 2000; NRC, 1989), and learning in general (Brown, Collins, & Duguid, 1989; Harley, 1993). The NCTM’s (1991) recommendations for what a teacher can do to encourage the

development of a community of practice included: helping students to work together, to rely more on themselves, to reason mathematically, to solve problems, and to connect mathematics and its applications. More recently, the NCTM (2000) stated the view that “teachers’ actions are what encourage students to think, question, solve

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problems, and discuss their ideas, strategies, and solutions. The teacher is responsible for creating an intellectual environment where serious mathematical thinking is the norm” (p. 18). Similar advice was given by the NRC (1989), who described teachers’

actions as denoting supportive interaction with students: “encourage,” “help students verbalise,” “build confidence” (pp. 81-82). Clearly, these authors believed that interactions between a teacher and students are an important aspect of teaching and learning mathematics.

Student-student interactions.

The second type of interaction, between student and student, is closely linked to the first and has also received attention in the mathematics education literature.

One aspect of learning theories that typifies the differences between constructivism and transmission models of learning is the focus on the interactions occurring among students. As mentioned earlier, under the transmission model the student was

expected to receive knowledge passively without questioning it; modern learning theories assert that discussion and debate among students is an essential part of the learning process. Various benefits have been claimed for students learning in a social community, whether in pairs, a small group, or a whole class (Akpinar & Hartley, 1996; Brown et al., 1989; Fox, 1988). These benefits include opportunity for collective problem solving, development of skills of collaboration, development of flexible thinking, and exposure of misconceptions.

Student-materials interactions.

The third type of interaction of interest here is interaction between a student and learning materials, such as blocks or a computer. Interaction with materials is linked with the two previous types of interaction, as the materials are an integral part of the learning environment, and there is assumed to be an “interplay between students’ cognitive activity and physical and social situations” (Nitko & Lane, 1990, p. 5). The connection between learning and the learning environment was mentioned by Kozma (1991), who stated that the learning process involves “extracting

information from the environment and integrating it with information already stored in memory” (pp. 179-180). Writing specifically about computer learning

environments, Kozma stated that the learning process was “sensitive to characteristics of the external environment, such as the availability of specific information at a given moment” (p. 180).

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The idea of situated cognition (Brown et al., 1989) addresses the question of learning and its relation to the learning environment. Brown et al.’s idea, that a learning environment constrains the learning activity of the students in that environment, is important to this research. A central assumption of the situated cognition view is that students “reason with what a situation affords them” (Winn, 1993, p. 16). In other words, the particular capabilities, or affordances (Salomon, 1998), provided by the materials available to a student can have an important influence on the student’s learning. This view is supported by Kozma’s (1994b) statement that

knowledge and learning are neither solely a property of the individual or of the environment. Rather, they are the reciprocal interaction between the learner’s cognitive resources and aspects of the external environment . . . and this interaction is strongly influenced by the extent to which internal and external resources fit together. (p. 8)

The research described here involves the investigation of children’s learning when using one of two types of materials; the author assumed prior to the study that the materials’ different special characteristics would have different effects on the students’ learning.

2.2.2 Number Sense

The idea of number sense is related closely to development of mathematical understanding, as it “typifies the theme of learning mathematics as a sense-making activity” (McIntosh et al., 1992, p. 3). Number sense refers to a student’s familiarity with numbers, and the ability to use numbers in sensible ways to answer

mathematical questions. It lacks a precise definition, but has been likened to “road sense” (familiarity with a particular geographical area; Trafton, 1992) or

“friendliness with numbers” (Howden, 1989). The NCTM (1989) listed five understandings demonstrated by students with good number sense. They “(1) have well-understood number meanings, (2) have developed multiple relationships among numbers, (3) recognize the relative magnitudes of numbers, (4) know the relative effect of operating on numbers, and (5) develop referents for measures of common objects and situations in their environments” (p. 38).

The need for students to possess number sense is extensively argued in the literature (Australian Education Council, 1990; K. Jones, Kershaw, & Sparrow, 1994; McIntosh et al., 1992; NCTM, 1989, 2000; Sowder, 1988, 1992; Sowder &

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Schappelle, 1994). Good number sense is important for making sense of mathematical questions and for working out sensible answers. Perhaps the

characteristic that most easily sums up good number sense is flexibility, in finding solutions to mathematical problems and in being able to see connections among numbers in different ways. Trafton (1992) described a person with number sense as having “a well-integrated mental map of a portion of the world of numbers and operations and [being] able to move flexibly and intuitively throughout the territory”

(p. 79). A similar idea was proposed by Greeno (1991), who likened knowing and learning “as an activity in an environment” (p. 175). Greeno connected number sense with situated cognition, stating his view that “knowing the domain [e.g., the

mathematical domain] is knowing your way around in the environment and knowing how to use its resources” (p. 175). In this view, number sense relates closely to the ability to use available resources to make sense of the domain. Though Greeno was writing specifically of mental resources, this idea is assumed here to apply also to physical resources, as described in an earlier paragraph. In other words, a student with good number sense could be seen as having not only a good idea of the cognitive domain, but also of the physical environment, including how to use available materials to answer mathematical questions.

For teachers to help students to develop good number sense involves helping the students to develop a range of prerequisite understandings of numbers and operations. This is borne out by McIntosh et al.’s (1992) description of number sense:

Number sense refers to a person’s general understanding of number and operations along with the ability and inclination to use this understanding in flexible ways to make mathematical judgements and to develop useful strategies for handling numbers and operations. It reflects an inclination and an ability to use numbers and quantitative methods as a means of communicating, processing and interpreting information. (p. 3)

Prerequisite mathematical skills and understanding needed for good number sense include proficiency with written algorithms, mental computation skills, problem-solving ability, and place-value understanding (see NCTM, 2000, p. 32).

Teaching techniques suggested for developing number sense include the use of calculators to investigate number magnitudes (Bobis, 1991; Schielack, 1991), emphasising and encouraging sense-making (Sowder & Schappelle, 1994), and the use of estimation activities (K. Jones et al., 1994; Lobato, 1993; Sowder, 1988,

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1992). The particular focus of this research, the development of place-value

understanding, is directly relevant to the development of number sense. As explained by the NCTM (2000),

understanding number and operations, developing number sense, and gaining fluency in arithmetic computation form the core of mathematics education for the elementary grades. As they progress from prekindergarten through grade 12, students should attain a rich understanding of numbers—what they are; how they are represented with objects, numerals, or on number lines; how they are related to one another; how numbers are embedded in systems that have structures and properties;

and how to use numbers and operations to solve problems. (p. 32)

2.2.3 Use of Technological Devices

The third issue of particular relevance to this study is the use of technology in mathematics teaching. There is common agreement that technological advances in the general society outside schools lead to the need for different objectives in mathematics education (NRC, 1989). These objectives will be seen in (a) the use of different means of doing mathematics, and (b) having different emphases in the curriculum.

Much has been written about the need to bring the procedures used in school mathematics into line with those expected of workers in the 21st Century. It has been pointed out (NCTM, 1989) that in the industrial age the goal of public schools was to educate future shop assistants and factory workers, and so schools taught their students so-called “shop keeper arithmetic” (Cruikshank & Sheffield, 1992; NCTM, 1989; NRC, 1989). There is no longer the same need for adults to be highly

proficient in written computation procedures; in its place is a need for workers and citizens who possess a broader range of mathematical “concepts and procedures they must master if they are to be self-fulfilled, productive citizens in the next century”

(NCTM, 1989, p. 3). These skills include developing methods to solve a variety of problems; working cooperatively in teams; and reading, interpreting, and critically evaluating quantitative data. The development of these skills is linked to the issues discussed in the previous two sections—meaningful understanding of mathematics and number sense—as well as the use of technological devices in mathematics teaching.

One feature of the mathematics used by adults in homes and workplaces is the use of calculators and computers to assist with a range of mathematical tasks

(Sparrow, Kershaw, & Jones, 1994). These include computation, storage of data, and

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presentation of results of mathematical processes such as in spreadsheets and graphs.

It is assumed by writers of mathematics education policy documents that students in schools will similarly have access to a range of technological devices to assist them in learning mathematics (Australian Education Council, 1990; Australian Association of Mathematics Teachers [AAMT], 1996; Cockcroft, 1982; NCTM, 2000; NRC, 1989). As the AAMT (1996) put it, “mathematics education must reflect the influence of technology upon both mathematics and society” (p. 2). The NCTM (1998) similarly recommended that schools improve their level of use of technology also to match what happens in schools with what employers and others expect of workers in the workforce:

Today’s jobs demand the use of mathematically driven technological tools. If schools do not have a level of technology equivalent to the level found outside schools, and if they do not prepare students appropriately with it, then they are placing their students at a serious disadvantage. (p. 43)

Merely increasing the amount of technology available in classrooms is not sufficient to bring about the desired improvements in mathematics education,

however, and technological devices should not merely be added to existing programs of instruction. Changes are also required in the methods of mathematics instruction, so that appropriate tasks are set to answer with technological devices. As the NCTM (1989) pointed out, “access to [calculator and computer] technology is no guarantee that any student will become mathematically literate. Calculators and computers for users of mathematics . . . are tools that simplify, but do not accomplish, the work at hand” (p. 8). There is a need for detailed knowledge of the effects that calculators and computers have on students using them, especially as they represent considerable investments of finance and time by education departments and teachers. This topic is reviewed further in section 2.6.

Electronic technology has the potential for several important effects on mathematics curriculum. Technology makes mathematical skills such as written computation easier, but it also hides processes that students in earlier times had to consider, such as regrouping required for operations. Technology can also make available to students mathematics learning experiences that previously were not possible. For example, calculators can allow a student to investigate operations on large numbers that would be too time-consuming with other mechanical computation procedures. Computers similarly provide students with considerable computing power, which can be used to represent mathematical and other domains with which

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students can “interact” in ways not possible with any other technology. This study investigates one such interactive learning environment in which a computer is used to represent the domain of numbers for the purpose of developing students’

understanding of place-value concepts.

2.3 Place-value Understanding

One area of the mathematics curriculum where the issues described in the previous section warrant attention is in place value. The teaching of place-value concepts is foundational for understanding of the base-ten numeration system, and is thus central to the primary mathematics curriculum. As the NCTM (2000) stated,

“foundational ideas like place value . . . should have a prominent place in the mathematics curriculum because they enable students to understand other

mathematical ideas and connect ideas across different areas of mathematics” (p. 15).

Issues such as those described in the previous section all have a potential impact on the teaching of place value. It is this author’s view that recommendations for students’ active involvement in learning mathematics and the development of number sense have direct relevance for how place value is taught. Advances in technology have a more indirect influence, through the capabilities they offer in the area of models of numbers for place-value teaching (section 2.6).

The importance of place value in the primary mathematics curriculum and the difficulty teachers experience in teaching place-value concepts to their students are well documented (G. A. Jones & Thornton, 1993a; S. H. Ross, 1990). As Resnick (1983) stated,

the initial introduction of the decimal system and the positional notation system based on it is, by common agreement of educators, the most difficult and important instructional task in mathematics in the early school years. (p. 126)

Teachers have difficulty teaching place-value concepts, and their students have difficulty learning them (S. H. Ross, 1990). One source of difficulty for teachers is that, as Skemp (1982) pointed out, mathematics’ “[conceptual structures]

are purely mental objects: invisible, inaudible, and not easily accessible even to their possessor” (p. 281). Thus, teachers are limited in what they can know of what their students are thinking with regard to mathematical entities. Research investigating place-value understanding must address students’ conceptual structures for numbers and how they are developed. This topic is dealt with in more detail in section 2.4.2.

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2.3.1 Place Value

Place value refers to the feature of the base-ten system of numeration (sometimes called the “Hindu-Arabic” system) in which each digit in a number represents a precise amount, dependent on both the face value of the digit and its position (Baturo, 1998; Miura & Okamoto, 1989). This contrasts with other numeration systems that do not exhibit place-value, such as that of the ancient Egyptians, who wrote a different symbol for each power of 10 (Irons & Burnett, 1994). In the base-ten numeration system the value represented by each digit is equal to the product of the face value of the digit and the value assigned to the digit’s position, relative to the rightmost whole number digit, or ones place (Fuson, 1990a;

Figure 2.1). Thus, though pairs of numbers such as “25” and “52” look very similar (and may be confused by young children), the position of each digit determines its value, giving a unique value to each different written symbol. This idea is at once both simple and very powerful, and can be extended an indefinite number of places to the left (for whole numbers) or the right (for decimal fractions).

Figure 2.1. The face value of each individual numerical symbol, together with its position relative to the ones place, determines the value it represents.

The base-ten numeration system is principally a system of written symbols by which users record physical quantities and use them in calculations. Its importance was emphasised by the NCTM (1998), who commented that “mathematical symbolism and representation is one of the most significant achievements of humankind” (p. 94). Associated with the written symbols are number words that are alternative representations of numerical quantities. Whereas the written symbols

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follow an entirely consistent mathematical system in which a quantity of ten of each place equals one of the place immediately to the left, number words include

inconsistencies relating to the history of the language used. In English, there are inconsistent words for multiples of 10, and for numbers from 11 to 19. Once a child reaches the study of three-digit and four-digit numbers number naming is much more consistent, but until a child reaches that stage the learning of numbers and their names is very difficult. Thus, young children have a very challenging task of developing understanding of a system that in its earliest, numerically simplest, examples contains numerous inconsistencies.

The base-ten numeration system has been named as an “unnamed-value positional value system of written marks” (Fuson, 1990a, p. 343) and a “regular relative positional system” (Fuson, 1992, p. 136). In these two phrases Fuson has captured three essential features of the base-ten numeration system: (a) Place names and values are implicit in written numeric symbols; (b) the system is completely consistent across all symbol positions; and (c) value is assigned according to each digit’s position, relative to the point of reference, the ones place. The structure and rules by which the conventional base-ten numeration system operates are not evident from merely observing the written symbols, even if the meaning of each individual symbol (“1,” “2,” etc.) is known (Fuson, 1992, p. 138). However, to those who are familiar with the scheme’s conventions, each numerical symbol uniquely represents a number. Apart from minor variations of the symbol used for the decimal point (usually “.” or “,”), and leading or trailing zeros, each rational number is represented by a unique symbol, and each sequence of digits stands for a unique number.

In summary, children need to understand the features of the base-ten numeration system (Baturo, 1998; English & Halford, 1995; Fuson, 1990a, 1992;

Hiebert, 1988; Miura & Okamoto, 1989; Sowder & Schappelle, 1994). A number of key features of the base-ten numeration system, in common with all place-value numeration systems, are expressed in the following statements:

1. A discrete set of individual number symbols (base 10: 0 to 9), and a decimal point marker, used in combination can uniquely represent any rational number quantity.

2. Each place represents a power of the base, derived from its position relative to the rightmost whole-number place.

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3. The system is completely consistent across all places, in that the value of each place is equal to the base number times the value of the adjacent place on the right, and the value of the adjacent place on the left divided by the base number.

4. The value represented by a digit is the product of the face value of the digit and the value associated with the place of the digit.

5. The system allows operations on numbers to be represented

symbolically, and these operations work consistently on numbers in all places.

From the previous discussion, definitions can be given for place value and for place-value understanding. This study is concerned with understanding of the base- ten numeration system only; hereafter “place value” will be used to refer only to place value in the base-ten numeration system. For the purposes of this thesis, place value is based on the description given by Miura and Okamoto (1989, p. 109), and defined thus:

Place value is the property of the base-ten numeration system, by which the numerical value represented by each digit of a written multidigit symbol is equal to the product of the digit’s face value and the power of 10 associated with the digit’s position in the numeral.

2.3.2 Place-value Understanding

For the purposes of this thesis, place-value understanding is described in terms of both actions and conceptual structures. This approach is supported by Sfard (1991), who described historical progress made in mathematical understanding as having to do with both “operational” (actions) and “structural” (objects) conceptions, dynamically linked together as professional mathematicians have struggled to

advance knowledge in the field. Sfard stated that “the ability of seeing a function or a number both as a process and as an object is indispensible for a deep understanding of mathematics” (p. 5). Though school students are not involved in the same level of mathematical thinking as professional mathematicians, nevertheless students need to develop both structural and operational conceptions for each new mathematical concept. These two types of conception of numbers link students’ internal conceptual structures of numbers and external physical models of numbers. In other words, students are assumed to possess internal representational structures for numbers and

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processes that are influenced as the students access and manipulate available external representations of numbers (Hiebert & Carpenter, 1992; Putnam et al., 1990).

Internal and external representations and manipulations are closely linked, and in this thesis they are considered together to be involved in students’ place-value

understanding.

The definition of place-value understanding used here takes into account advice given by several authors, relating to both actions and conceptual structures.

This includes statements that children need to “construct number representations that reflect the Base 10 numeration system” (Miura & Okamoto, 1989, p. 109),

“coordinate and synthesize a variety of subordinate knowledge about our culture’s notational system for numbers” (S. H. Ross, 1989, p. 47), “develop flexibility in representing and understanding multidigit numbers” (G. A. Jones, Thornton, & Putt, 1994, p. 122) and “[develop understanding of] the interpretation of numbers as compositions of other numbers” (Resnick, 1983, p. 126).

For this thesis place-value understanding is defined thus:

A student possessing place-value understanding is able to use the place- value features of the base-ten numeration system to form accurate, flexible conceptual structures for quantities represented by written numerical symbols. The student is able to manipulate numerical quantities in meaningful ways to answer mathematical questions.

A student’s place-value understanding must be assessed at a deep level, by probing the student’s conceptual structures for numbers (Skemp, 1982). Research in this area generally uses observation of participants’ behaviour to posit “various cognitive structures and processes believed to produce the behavior” (Putnam et al., 1990, p. 65). The following section describes research into children’s cognition, including the investigation of children’s conceptual structures for numbers.

2.4 The Contribution of Cognitive Science to Mathematics Education

As shown in the previous section, place-value understanding is an internal phenomenon that a teacher or researcher cannot access directly. Research on conceptual structures and analogical reasoning has particular applicability for the study of mathematical understanding, in two respects. Findings about mental

structures and processes can explain how abstract number concepts are represented in

The Development of Year 3 Students Place-Value Understanding: Representations and Concepts