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

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1 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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3 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. i

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5 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 baseten blocks closely, and to challenge faulty number conceptions by asking appropriate questions. iii

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7 Table of Contents Keywords...i Abstract... iii Table of Contents...v List of Tables...ix List of Figures...x Supplementary Material...x Statement of Original Authorship...xi Acknowledgments... xiii Chapter 1: The Problem Recommendations for Changes in Mathematics Education The Learning of Place-Value Concepts Conceptual Structures and Difficulties With Place-Value Concepts Use of Number Representations The Research Question Overview of Research Methodology Significance of the study Outline of the Thesis...7 Chapter 2: Review of Literature Chapter Overview Issues in Mathematics Education Students Active Involvement in Mathematics Learning Number Sense Use of Technological Devices Place-value Understanding Place Value Place-value Understanding The Contribution of Cognitive Science to Mathematics Education Understanding Mathematics Mental Models Analogical Reasoning Teaching Place-value Understanding Teaching Approaches Building Place-Value Connections Use of Concrete Materials Computers and Mathematics Education Claimed Benefits of Computers Cognitive Aspects of Computer Use Chapter Summary; Statement of the Problem...59 Chapter 3: Methods Chapter Overview Aims of the Study Variables...62 v

8 3.3.1 Mathematical Achievement Level Number Representation Format Data collection and analysis Design Issues Assumptions Theoretical and Methodological Stance Pilot Study Purposes of the Pilot Study Selection of Pilot Study Participants Pilot Study Procedures Pilot Study Data Collection and Analysis Changes Made to Study Design After Pilot Study Main Study Selection of Participants Teaching Program Instruments - First and Second Interviews Administration Procedures Data Collection and Analysis Validity and Reliability Limitations Chapter Summary...94 Chapter 4: Results Chapter Overview Restatement of the Research Question Transcript Conventions Used in this Thesis Place-Value Task Performance Revealed in Interview Results Methods used to Analyse Interview Data Overview of Interview Results Students Conceptions of Numbers Grouping Approaches Counting Approaches Face-Value Interpretation of Symbols Summary of Approaches to Interview Questions Changeability of Participants Number Conceptions Digit Correspondence Tasks: Four Categories of Response Category I: Face-Value Interpretation of Digits Category II: No Referents For Individual Digits Category III: Correct Total Represented by Each Digit, but Tens not Explained Category IV: Correct Number of Referents, Tens Place Mentioned Summary of Responses to Digit Correspondence Tasks Errors, Misconceptions, and Limited Conceptions Counting Errors Blocks Handling Errors Errors in Naming and Writing Symbols for Numbers Errors in Applying Values to Blocks Use of Materials to Represent Numbers vi

9 4.7.1 Counting of Representational Materials Use of Trial-and-Error Methods Handling Larger Numbers Interpreting Non-Canonical Block Arrangements Face-value Interpretations of Symbols Predictions About Trading Feedback Using Blocks To Discover Number Relationships Chapter Summary Chapter 5: Discussion Chapter Overview Participants Ideas About Multidigit Numbers Participants Preferences for Grouping or Counting Approaches Comparison of Grouping and Counting Approaches Difficulties With Existing Conceptual Structure Schemes Face-value Interpretations of Symbols Independent-Place Construct Description & Definition of the Independent-Place Construct Comparison of the Independent-Place Construct and the Face-Value Construct Evidence for the Independent-Place Construct in This Study Evidence of the Independent-Place Construct in the Literature Written Computation and the Independent-Place Construct Place-Value Tasks and the Independent-Place Construct Participants Construction of Meaning Organic Understanding Participants Invented Answers Teaching, Learning, and Constructing Meaning Effects of Physical or Electronic Base-Ten Blocks on Place-Value Understanding Differences in Learning of Participants Using Physical or Electronic Blocks Sensory Impact of Physical or Electronic Blocks How Numbers Are Represented by Physical or Electronic Blocks Development of Links Among Blocks, Symbols, and Numbers Support for the Development of Number Concepts Place-Value Understanding Demonstrated by High- and Low-Achievement- Level Participants Similarities in Place-Value Understanding of High- and Low-Achievement- Level Participants Differences in Place-Value Understanding of High- and Low-Achievement- Level Participants Chapter 6: Conclusions Chapter Overview Conclusions About Answers to Research Questions Conceptual Structures for Multidigit Numbers Evident in Participants Responses Misconceptions, Errors, or Limited Conceptions Evident In Participants Responses vii

10 6.2.3 Effects of the Two Materials on Students Learning of Place-Value Concepts Differences Between Place-Value Understanding of High- and Low- Achievement-Level Participants Implications for Teaching Implications of Using Physical Base-Ten Blocks to Teach Place-Value Concepts Implications of Using Electronic Base-Ten Blocks to Teach Place-Value Concepts Implications of the Independent-Place Construct for Teaching Mathematics Implications of Construction of Meaning for Teaching Mathematics Recommendations for Further Research Appendix A Design of Software used in the Study Appendix B - Overview of Teaching Session Content for Interviews and Teaching Phase of Pilot Study Appendix C Summary of Pilot Study Teaching Program Appendix D - Excerpt of Teaching Script of Pilot Study: Session Appendix E Audit Trail Example Appendix F Results of The Year Two Diagnostic Net, Used to Select Participants for the Main Study Appendix G List of Participants Appendix H - Main Study Teaching Program Appendix I - Main Study Interview 1 Instrument Appendix J - Main Study Interview 2 Instrument Appendix K Letter Requesting Consent by Parents or Guardians of Prospective Participants Appendix L Coding Teaching Session Transcripts for Feedback Appendix M Descriptions of Numeration Skills Targeted by Interview Questions and Criteria for Their Assessment Appendix N Transcript of Interview 1 Question 6 (a) with Terry Appendix O Transcript of Interview 2 Question 6 (a) with Hayden Appendix P Transcript of Low/Blocks Group Answering Task 28 (a) Appendix Q Transcripts of Task 4 (a) from 4 groups Appendix R Transcript Excerpts Showing Participants Predicting Equivalence of Traded Blocks Appendix S Transcript of Task 4 (d) from Low/Blocks Group Appendix T Comparison Between Ross s (1989) Model and a Proposed Model for Categories of Responses to Digit Correspondence Tasks Appendix U Sample Coding of Transcript for Feedback References Supplementary Material Hi-Flyer Maths Installation Files [CD-ROM] viii

11 List of Tables TABLE 4.5. TABLE 2.1. Aspects of Place-value Understanding Described in the Literature TABLE 2.2. Task Performance Illustrating Limited Conceptions in Place-value Understanding TABLE 3.1. Phases of the Research Design TABLE 3.2. Participant Groups for the Main Study TABLE 4.1. Transcript Notations TABLE 4.2. Summary of Participants Numeration Skills Identified in two Interviews TABLE 4.3. Summary of Numeration Skills Demonstrated by Each Participant and by Each Group TABLE 4.4. Summary of Place-value Understanding Criteria Achieved by Highachievement-level and Low-Achievement-Level Participants Summary of Place-value Understanding Criteria Achieved by Participants in Computer and Blocks Groups TABLE 4.6. Use of Grouping Approaches for Selected Interview Questions TABLE 4.7. Use of Grouping Approaches by Each Group TABLE 4.8. Use of a Counting Approach for Selected Interview Questions TABLE 4.9. Use of Counting Approaches by Each Group TABLE Incidence of Face-value Interpretations for Written Symbols after Selected Interview Questions TABLE Use of Face-Value Interpretations of Symbols by Each Group TABLE Incidence of Approaches Adopted for Selected Interview Questions TABLE Response Categories for Interview Digit Correspondence Questions TABLE Summary of Digit Correspondence Response Categories TABLE Participants Written Responses to Task 27 (b) TABLE Incidents of Feedback of Each Source per Group TABLE Percentage of Feedback Compared With Answer Status TABLE Quality of Feedback Provided for Correct or Incorrect Answers TABLE Percent of Feedback for Correct Answers from Each Source TABLE Percent of Feedback for Incorrect Answers from Each Source TABLE Feedback Providing Answers from Each Source for Each Group TABLE 5.1. Comparison of Results of Digit Correspondence Tasks Between This Study and Ross (1989) TABLE H.1. Overview of Teaching Program Tasks TABLE L.1. Source of Feedback TABLE L.2. Effects of Feedback TABLE L.3. Responses to Feedback ix

12 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 Figure 2.2. Relationships inherent in base-ten blocks Figure 2.3. Relationships among numbers, written symbols, and concrete materials Figure 2.4. Conceptual gap between written symbols and concrete materials Figure 2.5. The use of transitional forms to bridge the gap between written symbols and concrete materials Figure 3.1. Dimensions of research design Figure 3.2. Original graphic images used on regrouping buttons in software used Figure 3.3. during pilot study Replacement graphic images used on regrouping buttons in software used during main study Figure 3.4. Sample Representing numbers task Figure 3.5. Sample Regrouping task Figure 3.6. Sample Use of numeral expander task Figure 3.7. Sample Comparison task Figure 3.8. Sample Counting task Figure 3.9. Sample Addition task, including regrouping Figure Diagram showing objects used in interviews for Digit Correspondence Task with misleading perceptual cues Figure 4.1. Interview scores compared to use of grouping approaches Figure 4.2. Interview scores compared to use of counting approaches Figure 4.3. Interview scores compared to use of face-value interpretations of symbols Figure 4.4. Proportions of feedback from each source for each group Figure 5.1. Column counters in software representation of Figure A.1. Screen view of on-screen tutorial question with block representations Figure A.2. Partial screen image from Rutgers Math Construction Tools, showing block and symbol representations of a number Figure A.3. Screen view of Blocks Microworld showing block representation of a number, nominating a cube as one Figure A.4. Main screen of Hi-Flyer Maths Figure A.5. Show as tens feature activated Figure A.6. Number name window and numeral expander displayed Figure A.7. A block is sawn into 10 pieces Figure A.8. Add blocks requester Figure L.1. Data entry screen for feedback database Supplementary Material Hi-Flyer Maths Installation Files [CD-ROM] x

13 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: xi

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15 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 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. xiii

16 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. xiv

17 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 grades 1 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. 1

18 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 2

19 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 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 placevalue 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 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 3

20 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 baseten 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 4

21 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 baseten 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 5

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