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An exploratory study of students’ representations of units and unit relationships in four mathematical… Cannon, Pamela Lynne 1992

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AN EXPLORATORY STUDY OF STUDENTS REPRESENTATIONS OFUNITS AND UNIT RELATIONSHIPS INFOUR MATHEMATICAL CONTEXTSbyPAMELA LYNNE CANNONB.A., University of London, 1970M.A., University of London, 1973A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF EDUCATIONinTHE FACULTY OF GRADUATE STUDIES(Department of Mathematics and Science Education)We accept this dissertation as conformingo the required standard.THE UNIVERSITY OF BRITISH COLUMBIADecember 1991Pamela Lynne CannonSignature(s) removed to protect privacyIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)________________Department of &kwd tThe University of British ColumbiaVancouver, CanadaDate 2E-cj /c1/DE-6 (2188)Signature(s) removed to protect privacy11ABSTRACTThis study explores characteristics of students’ repertoires of representations in twomathematical contexts: whole number multiplication and the comparison of common fractions. Arepertoire of representations refers to a set of representations which a student can reconstruct asneeded. Of particular interest are (1) how multiplicative relationships among units were represented,and (2) whether continuous measurement was an underlying conceptual framework for theirrepresentations. In addition, the characteristics of students’ representations and interpretation of unitsof linear and area measurement were explored. Data were collected through a series of interviews withGrade 5 and Grade 7 students.Some results of the study were as follows. Each repertoire of representations was exemplifiedby a dominant form of units, either discrete or contiguous. Within a repertoire, all forms of units wererelated, first through a common system of measurement (either numerosity or area), and second throughtheir two-dimensional characteristic.In the multiplication context, some repertoires were comprised only of representations withdiscrete units, but others also included some representations with contiguous units. Students soughtcharacteristics in their representations which reflected those based on continuous measurement,however linear or area measurement was not used as a conceptual framework. Instead, allrepresentations were based on the measurement of numerosity. Also, students exhibited differentlimits in their representation of multiplicative relationships among units. Some represented nomultiplicative relationships, but most represented at least a multiplicative relationship between two units.Relationships among three units were seldom constructed and difficult to achieve.Common fraction repertoires were based on the measurement of either numerosity or area, butthe physical characteristics of the units varied. Some repertoires had only contiguous representations ofunits, others also included representations with discrete units, and a few did not represent fractionalunits at all. Students’ representations reflected characteristics of area-based representations, however111area measurement was not necessarily a conceptual framework. In addition, students’ beliefs about whatconstituted units of area measurement were variable. As a result, they either represented nomultiplicative relationships among units, or fluctuated between representing two-unit and three-unitrelationships.Linear measurement was notably absent as a basis for representations in both mathematicalcontexts. The one-dimensional characteristic of linear measurement did not fit students’ dominantframework for constructing mathematical representations.With respect to measurement, students represented linear units in terms of discrete points orline segments. Counting points and interpreting the count in terms of the numerosity of line segmentswas problematic for nearly all students. When parLitioning regions into units of area, a few students alsoequated the number of lines with the number of parts. The direct relationship of action and result incounting discrete objects was generalized without consideration of other geometric characteristics.When comparing quantities having linear or area uns, numerical reasoning was not alwaysused. Alternatively, either quantities were transformed to facilitate a direct comparison, or onlyperceptual judgements were made. No students consistently used numerical reasoning to comparefractional units of area. In the latter situations, the part-whole relationship among units seldom wasobserved.In general, there was no direct relationship between the forms of representations used bystudents in the two mathematical contexts and the characteristics of their representations of units of themeasurement contexts. The development of repertoires of representations appears to be contextspecific. The reperLoires were strictly limited in terms of the forms of representations of which they werecomprised.ivACKNOWLEDGEM ENTSI would like to express my deepest appreciation to my family, particularly my parents May andHarry Cannon. Through my late fathers life-long dedication to education I developed the interest inchildren’s learning which brought me to pursue this thesis. As well, without my mother’s unwaveringbelief in my ability to overcome obstacles encountered during the process, this thesis may never havebeen completed.In addition, I would like to express a special thank-you to all my friends who gave meimmeasurable moral and intellectual support. A particular appreciation goes to Edward and BrigittaO’Regan who spent long days and nights discussing the thesis and cheering me along. As well, thecomradery of Zahra Gooya and Darlene Perner contributed considerably to my ability to extend the extraefforts needed for this project. And last but not least, a very special thank-you goes to Dr. Gail Spitler forher long standing friendship and timely advice.I also would like to thank all the teachers and students without whose cooperation andparticipation this study would not have happened. And finally, I thank all of the members of mysupervisory committee, Dr. David Robitaille, Dr Douglas Owens, Dr. Gaalen Erickson, and Dr. PatriciaArlin for all of the support they gave me over the years.VTABLE OF CONTENTSPAGEABSTRACT iiACKNOWLEDGEMENTS ivLIST OF TABLES xiiiLIST OF FIGURES xvCHAPTER 1 OVERVIEWOFTHE STUDY 1Research Questions 4Significance of the Study 4Definition of Terms 8Overview of the Plan of the Study 11Limitations of the Study 11Justification of the Study 1 2Organization of the Dissertation 1 3CHAPTER 2 REVIEW OF RELATED LITERATURE 1 4Repertoires of Representations 1 4The Nature of Mathematical Representations 20Measurement as Content of Forms of Mathematical Representations 23Multiplicative Relationships Between Units 25Conceptions Of Units of Length and Area 28Linear Measurement 28Area Measurement 30Comparing areas of regions 30Partitioning regions 3 1viPAGEStudents’ Representations of Whole Number Multiplication and Common Fractions 33Student-Generated Representations 33Students’ Interpretations of Instructional Representations 35CHAPTER 3 PLAN AND IMPLEMENTATION OF THE STUDY 39Subjects in the Study 40Plan of the Study 41Overview 41Repertories of Representations 42Conceptions of Linear and Area Measurement 44Data Collection 44Generative Interviews 44Generative Interview Tasks 46Procedures for Selecting Tasks 47Materials Used in the Cued Generative Interview 48Interpretive Interviews 48Interpretive Interview Tasks 49Interpretive Interview Procedures 49Measurement Interviews and Test 52Data Analysis 59CHAPTER 4 STUDENTS’ REPRESENTATIONS OF UNITS IN MULTIPLICATIVERELATIONSHIPS 60Relationship of research questions to interview data 60Plan of chapter 6 1Analytical Categories to Classify Representations 6 1viiPAGEAnalytical Categories of Forms of Representations 63Discrete Units 64Discrete units with sets as spatial frameworks: linear, rectangular, or other 64Discrete units with lines or regions as spatial frameworks 65Contiguous Units 67Units are Undefined 69Analytical Categories of Functions of Representations 71Mono-relational Representations of Units 7 1Bi-relational Representations of Units 72Tn-relational Representations of Units 75Students’ Repertoires of Representations: Whole Number Multiplication 76Characteristics of Primary Repertoires of Representations: Whole NumberMultiplication Context 78Characteristics of the Forms of Representations in Primary Repertoires 80Characteristics of discrete forms of representations 80Characteristics of contiguous forms of representations 82Common spatial characteristics of discrete and contiguous forms ofrepresentations 83Functions of Representations in Primary Repertoires 84Summary of Characteristics of Primary Repertoires of Representationsof Whole Number Multiplication 85Characteristics of General Repertoires: Representations of Whole NumberMultiplication in Other Material Settings 86Representations Constructed in the Cued Generative Interview Compared to theUncued Generative Interview 87Influence of material setting on forms of representations generated 87Relationships between discrete and contiguous representations 89Extent of repertoires: attributes of area or length 9 1viiiPAGEFunction of representations in both generative interviews 92Summary of characteristics of repertoires generated in both generativeinterviews 94Comparison of Repertoires Reflected in the Generative Interviews WithResponses in the Interpretive Interview Settings 95Forms of representations in the interpretive interview setting 96Functions of representations in all interview settings 9 8Characteristics of the form of representations and students’ attention toattributes of length and area measurement 99Summary of the Characteristics of Students’ Repertoires of Representations ofWhole Number Multiplication 102Students’ Repertoires of Representations: Common Fractions and Comparisons ofCommon Fractions 103Characteristics of Primary Repertoires of Representations: Common FractionsContext 105Characteristics of the Forms of Representations in Primary Repertoires 107Characteristics of forms of representations based on regions 108Characteristics of discrete forms of representations 115Functions of Representations of Common Fractions in Primary Repertoires 116Summary of Characteristics of Primary Repertoires of Representations ofCommon Fractions 118Characteristics of General Repertoires: Representations of Common Fractions inOther Material Settings 11 9Representations Constructed in the Cued Generative Interview Compared tothe Uncued Generative Interview 120Influence of material setting on forms of representations generated 122Extent of repertoires 124Function of representations in both generative interviews 124Comparison of Repertoires Reflected in the Generative Interviews WithResponses in the Interpretive Interview Settings 126Forms of representations in the linear interpretive interview setting 127ixPAGEForms of representations in the area interpretive interview setting 130Attributes of Area Measurement in General Repertoires of Representations ofCommon Fractions 133Interpretations of common fractions and characteristics of the forms ofrepresentations in general repertoires 134Equality of contiguous units (whole and fractional units) 137Functions of representations in students’ general repertoires ofrepresentations of common fractions 139Summary of the Characteristics of Students’ Repertoires of Representations ofCommon Fractions and Comparison of Common Fractions 143Students’ Representations of Unit Relationships in the Whole Number Multiplicationand Common Fraction Contexts: Common Patterns and Related Themes 147Repertoires of Representations 147The Discrete-Continuous Dichotomy 149Multiple and Nested Relationships of Units 152CHAPTER 5 REPRESENTATIONS AND INTERPRETATIONS OF UNITS OFLINEAR AND AREA MEASUREMENT 156Students’ Representations and Interpretations of Units of Linear Measurement 157Analytical Categories to Classify Responses to Linear Measurement Task 159Students’ Representations of Units of Length 161Variations in Students’ Representations of Units of Length 163Point/Line Segment Conflict: Differences Among Tasks and Solution Strategies 164Reasoning with Units and Number with the Irregular Path Tasks 167Numerical reasoning strategies 169Transformational reasoning strategies 170Summary of the Nature of Students’ Representations and Interpretations ofUnitsof Length 170Students’ Representations and Interpretations of Units of Area Measure 171xPAGEAnalytical Categories to Classify Responses to Area Measurement Task 174Equal Part of a Region: the Partitioning Tasks 177Differences in Success with Different Regions 178Strategies Associated with Partitioning Problems 178Configurations applied inappropriately to a region 180Unsuccessful partitions with appropriate configurations 182Comparing Parts of Regions: the Cake Tasks 184Comparing Non-congruent Fractional Units 186Variations in Reasoning Strategies 188Numerical reasoning strategies 188Perceptual reasoning strategies 189Partitioning Regions and Comparing of Parts of a Region 190The Comparison of Irregular Regions: the Tile Tasks 191Students’ Reasoning with Linear and Area Measurement: an Overview 193Students’ Beliefs about the Representation and Interpretation ofLinear and Area Units 197CHAPTER 6 CONCLUSIONS AND DISCUSSION 199Conclusions 1 99What are the Characteristics of Students’ Repertoires of Representations ofWhole Number Multiplication? 199Forms of Representations of Whole Number Multiplication inStudents’ Repertoires 200Functions of Representations of Whole Number Multiplication 201What are the Characteristics of Students’ Repertoires of Representations ofCommon Fractions? 202Forms of Representations of Common Fractions in Students’ Repertoires 202Area measurement as a basis for representing common fractions 203xiPAGEFunctions of Representations 204What are the Characteristics of Students’ Representations and Interpretationsof Units of Length? 205Representations of Units of Length 205Comparing Lengths and Reasoning with Units of Length 206What are the Characteristics of Students’ Representations and Interpretations ofUnits of Area? 206Representations of Units of Area by Partitioning Regions 207Comparisons of Area: the Interpretations of and Reasoning with Units 208Discussion and Implications for Instruction 209Repertoires of Representations 209Repertoire of Representations as a Construct 210Structure of Repertoires of Representations 210Forms of Representations 211Content of the Form 211Implications for Instruction 213Measurement of Numerosity or Area as Content of the Form 214Implications for Instruction 217Dominance of Forms of Representations in Repertoire 217Implications for Instruction 219Differentiation of Forms of Representations of Units 220Forms of representations in the arithmetic contexts comparedto interpretation of units in the continuous measurement contexts 220Linear measurement as content of the form: a special case. 222Comparisons of representations of units in both repertoires ofrepresentations 225Implications for Research 226xiiPAGERepertoires of Representations as a Construct in the Mathematics LearningProcess 227The Conceptualization of Units of Continuous Quantities 229Integrating Measurement of Continuous Quantities as Content of Formsof Representations 230Concluding Comments 231BIBLIOGRAPHY 232APPENDIX A: Sample Interview Protocols 240APPENDIX B: Measurement Concept Test 316xliiLIST OF TABLESPAGE3.01 Characteristics of the Interviews Designed to Investigate Students’Repertoires of Representations 424.01 Forms and Functions of Representations of Multiplication: Uncued GenerativeInterview 794.02 Extent to Which Spatial Organizations were used to Construct Representationsof Multiplication with Sets (Uncued Generative Interview) 824.03 Form and Function of Representations of Multiplication Tasks During the CuedGenerative Interview with a Summary of the Primary Repertoires 884.04 Extent to which Spatial Organizations were used to Construct Representationsof Multiplication with Sets (Cued Generative Interview) 894.05 Forms and Functions of Representations of Multiplication in All InterviewSettings - 974.06 Extent of Contiguity of Units and Form of Contiguous Units: Multiplication 1004.07 Forms and Functions of Representations of Common Fractions: UncuedGenerative Interview 1064.08 Extent to which Different Regions were used to Construct Representations ofCommon Fractions (Uncued Generative Interview) 1084.09 Form and Function of Representations of Common Fractions: Cued GenerativeInterview with a Summary of Primary Repertoires 1214.10 Extent to which Different Regions were used to Construct Representations ofCommon Fractions (Cued Generative Interview) 1234.11 Form and Function of Representations of Common Fractions: Cued Generative,Linear Interpretive, and Area Interpretive Interviews with a Summary ofInterpretations Over All Interviews and a Summary of Primary Repertoires 1284.12 Criteria Used to Evaluate Beginning Diagrams and the Form and Function ofRepresentations (Area Interpretive Interview) with a Summary ofInterpretations and Repertoires over all Interviews 1324.13 Contiguity and Equality of Units: Representations of Comparison of CommonFraction Tasks in All Interviews 1354.14 General Patterns in the Form of Representations in Students’ GeneralRepertoire in the Multiplication and Common Fractions Contexts. 148xivPAGE4.14 General Patterns in Students’ Representations of Unit Relationshipsin the Multiplication and Common Fractions Contexts. 1545.01 Students’ Representations of Unit of Length and Relationships Between Units 1625.02 Students’ Reasoning Strategies with the Irregular Paths Tasks, and aSummary of their Representations of Units of Length 1685.03 Frequency of the Different Approaches to Numerical Reasoning with LineSegments for Each Irregular Path Task 1705.04 Frequency of Students’ Constructions of Six Equal Parts ofEach Region in the Test and Interview 1785.05 Consistency of Students’ Partitioning of Each Region and a Breakdown of theResults of the Initial and Final Partition of Each Region 1795.06 Frequency of Students’ use of Configurations to Partition Each Region:Initial Partitions and Final a Partitions 1 815.07 Summary of the Reasoning Strategies Used to Solve the Cake Tasks, and theConsistency with which Students Partitioned Regions into Six Equal Parts 1855.08 Students’ Interview Responses and Test Scores for the Tile Tasks 1925.09 Summary of Student Responses to All Linear and Area Measurement Tasks 195xvLIST OF FIGURESPAGE2.01 Counting sets of dots beneath the arrow on the number line 363.01 Beginning diagrams used in the linear interpretive interviews 503.02 Beginning diagrams used in the area interpretive interviews 5 13.03 Linear measurement tasks with explicit reference to units and number 543.04 Linear measurement tasks without explicit references to units and number 553.05 Partitioning tasks: divide each figure into 6 equal parts 563.06 Cake tasks: compare the sizes of the shaded pieces of cake 573.07 Tile tasks: compare the amount of space in each playroom 584.01 Categories for classifying the forms of representations. 634.02 Categories for classifying the function of representations 7 14.03 Examples of standard configurations used to partition regions for commonfraction representations 1094.04 Examples of representations with regions as they relate to interpretations 1104.05 Representations of common fractions with alternative conceptions of equalparts: Edwin (Grade 5) 1144.06 Discrete and contiguous representations of three fourths: Edwin, (Grade 5) 1154.07 Discrete representations of comparisons of common fractions 1164.08 Examples of different representations of three-fourths using a commonframework 1244.09 Examples of differences in mono- and bi-relational representations constructedin each generative interview 1254.10 Examples of students’ transformations of lines into regions 1294.11 Deriving units of length analogously from sectors of a circle (Pete, Grade 5) 1294.12 Representation of muftiplicative algorithm for generating equivalent fractions(Coran, Grade 7) 141xviPAGE5.01 Linear measurement tasks with explicit reference to units and number 1585.02 Linear measurement tasks without explicit references to units and number 1595.03 Analytical categories used to classify student responses to the linearmeasurement tasks 1605.04 Students’ use of discrete points as units to partitioning a line into five units,then drawing a line of three units 1645.05 Students’ use of discrete points as units with the ruler task 1655.06 Students’ use of line segments as units with the ruler task 1655.07 Partitioning tasks: divide each figure into 6 equal parts 1735.08 Cake tasks: compare the sizes of the shaded pieces of cake 1735M9 Tile tasks: compare the amount of space in each playroom 1745.10 Analytical categories used to classify the students’ responses to the areameasurement tasks 1755.11 Comparison of shaded fractional units: Pairs of Cake Tasks 1865.12 Examples of students’ transformations of triangles into quadrilateralsand rectangles into rectangles 1 875.13 Comparisons of fractional units: Cake Task 4 1881CHAPTER 1OVERVIEW OF THE STUDYThere is an extensive history of mathematics educators advocating the use of physicalrepresentations as a means of facilitating children’s learning of mathematics. McLellan and Dewey(1895), Montessori (1912/1964), and Brownell (1928) considered experiences with manipulative andpictorial representations of numerical relationships to be central to meaningful learning of mathematicalrelationships at the elementary level. More recently, Piaget’s formulation of concrete operational andformal operational stages of cognitive development, Bruner’s (1960,1968) formulation of enactive,iconic, and symbolic representational stages, and Dienes’ (1963, 1964) advocacy of the use of multipleembodiments based on perceptual variability and mathematical variability principles have lent support topreviously held beliefs about the importance of children’s experiences with physical representations ofmathematical relationships in their learning process (Post, 1980). In British Columbia during the 1970’selementary teachers and other mathematics educators generally concurred with this view (Province ofBritish Columbia, Ministry of Education, 1978; Robitaille & Sherrill, 1977). Most recently, the importanceplaced on representations in the learning of mathematics has not diminished. In the MathematicsCurriculum Guide 1-8 (Province of British Columbia, 1987) it is stated that, in the early years “all[students] require extensive experiences in concrete manipulations in order to form sound, transferablemathematical concepts,” and in later years “the use of models is beneficial when introducing a topic” (p.2). For the later elementary grades, the Curriculum and Evaluation Standards for School Mathematics(National Council of Teachers of Mathematics, 1989), emphasized the importance of the use ofmathematical representations by students, stating that:the study of mathematics should include opportunities to communicate so that studentscan model situations using oral, written, concrete, pictorial, graphical, and algebraicmethods. (p.78) [Emphasis added.]Despite general beliefs about the importance of representations in mathematics learning,concerns remain about the efficacy of different representations to exemplify mathematical ideas, and2about how students might make sense of the mathematics being represented (e.g., Dufour-Janvier,Bednarz, & Belanger, 1987; Fischbein, 1977; Hart, 1987; Hunting, 1984a, Hunting, 1984b; McLelIan &Dewey, 1895; Payne, 1975, 1984). In this regard, Fischbein (1977) argues that many representationssuch as graphs or Venn diagrams are themselves representations of conceptual structures which aredefined independently of the mathematics they represent. He further contends that an incompleteunderstanding of the conceptual structure and “language” of the representation will result inmisunderstandings of the mathematics being represented. As well, Dufour-Janvier, Bednarz, andBelanger (1987) argue that some representations in elementary instruction might be “inaccessible” tostudents if they are too far removed from the child’s “internal representations” of the situation or therepresentations envisaged by the child of the problem situation” (p. 118). They contend that,the use of representations that are just as abstract for the child as that which is studiedbrings the child to manipulate rules and symbols that are meaningless to her. (p. 118)In the elementary school curriculum, representations of mathematical relationships commonlyare based on conceptual structures which are “defined independently of the mathematics theyrepresent.” For instance, relationships between factors and product might be represented as the sumof a number of equal groups of discrete objects, as the length of a regular number of jumps of linear unitsalong the number line, or as the number of units within a rectangle when units are defined by thepartitioning of each dimension by the factors. These three forms of representation can be differentiatedby the systems of measurement which underlie them. The first representation is based on measures ofthe numerosity. The other two are based on measures of length and area respectively. In the firstrepresentation units are discrete, and in the other two the units are contiguous. As a consequence,they differ in terms of the physical properties which define units, the limits on the spatial configurations ofthe units, and the procedures governing the representation of unit relationships.The primary motivation for using different forms of representations in elementary instrUction is tofacilitate the students’ learning of the mathematics being represented. The students’ learning about aconceptual structure underlying a particular form of representation is, at best, secondary, and may neverbe addressed explicitly by the teacher. Yet, as Fischbein (1977) suggests, the ways in which students3make sense of the form of the representation might influence how students construe the mathematicsrepresented, as well as how the students use such representations to solve mathematical problems.How students might make sense of different types of mathematical representations, the forms ofwhich are assumed by others to be based on different systems of measure, may depend on a number offactors. These may include: (1) the associations students make between forms of representation ofmathematical relationships and measurement systems, (2) the nature of students’ conceptions of unitsand unit relationships in the measurement system used by students as a framework through which tomake sense of the units and unit relationships in the mathematical representation, and (3) thecomplexity of the unit relationships represented. It would not be sufficient for students, given arepresentation based on a particular measure system, simply to conceive of the units of this measuresystem independently of their use in the representation. They would also need to conceive of therepresentation as being based on that system of measure. Thus, students’ conceptions regarding whatconstitutes a representation of different mathematical ideas, what the salient features of a mathematicalrepresentation are, and what attributes of a representation are mathematically significant, would becritical to the ways in which students interpret and construct mathematical representations.There are two purposes to this study, given that students’ interpretations and use of differentforms of representations might depend on: (1) their prior knowledge of the conceptual structureunderlying a form of representation, (2) their previous experiences with mathematical representations,and (3) the complexity of the unit relationships represented or to be represented. One purpose is tocharacterize the repertoires of representations that students use to explain relationships among units. Arepertoire of representations refers to a set of representations which a student can reconstruct asneeded. Do students have different forms of representations in their repertoires to explain mathematicalrelationships? Are there forms of representations commonly used in instruction which are not includedin students’ repertoires of representations? How do students represent multiplicative relationshipsbetween or among different units? The second purpose of this study was to characterize students’representations of units of length and area measurement as well as multiplicative relationships betweendifferent units of length or different units of area.4As Resnick (1983) states, “Learners try to link new information to what they already know inorder to interpret the new material in terms of established schemata.... [They] constructunderstanding,... look for meaning and will try to find regularity and order in the events around them.”The events may include experiencing a variety of different forms of mathematical representations.Students, in making sense of these experiences, may interpret new forms of mathematicalrepresentations in terms of their prior knowledge about mathematical representations, and suchinterpretations may not conform to those intended by mathematics educators. Students’ conceptions ofrepresentations and the representational process, as characterized by their repertoires ofrepresentations, would have implications for how they might construe mathematical ideas embodied indifferent forms of representations.Research QuestionsThe foflowing research questions were used to guide the investigation:1. What are the characteristics of students’ repertoires of representations of whole numbermultiplication?2. What are the characteristics of students’ repertoires of representations of common fractions?3. What are the characteristics of students’ representations and interpretations of units of lengthmeasurement?4. What are the characteristics of students’ representations and interpretations of units of areameasurement?Significance of the StudyWhole number multiplication and common fraction concepts and relationships were selected asthe mathematical contexts within which to investigate the characteristics of students’ repertoires ofrepresentations for a number of reasons: (1) because of their importance as components in theintermediate curriculum (Grades 4 to 7); (2) because they require a different way of thinking about unitsand representing relationships between units, compared to students’ earlier, extensive experience with5representations of additive relationships; and (3) because of the increasing significance of continuousmeasurement as a basis for representing these mathematical concepts and operations.In the primary grades additive relationships of units constitute a major part of the curriculum.Comparing the numerosity of collections of objects, or adding and subtracting with whole numbersinvolve additive relationships of quantities measured with only one unit. However, in the intermediategrades, the curriculum shifts to an emphasis on multiplicative relationships (Hiebert & Behr, 1988;Vergnaud, 1983a). Both whole number multiplication and common fractions involve multiplicativerelationships between different units. Two or more units are related proportionally or through a simpleratio. It is in this sense that Vergnaud (1 983a) considers both multiplication and common fractions to beconnected within a conceptual field of multiplicative structures. Instruction in whole numbermultiplication, which precedes formal instruction in common fraction concepts and relationships, is oneof the first formal experiences children have of multiplicative relationships between different units.Common fraction concepts, relationships, and operations are another major part of the curriculum at theintermediate level.In both of these mathematical contexts, representations based on discrete sets, onmeasurement of length, and on measurement of area are used in instruction. At least some of theseforms of representations would have been used previously to explain simpler, additive relationships.However, with additive relationships, representations embodying number concepts and operationsinvolve units which are all of the same order of magnitude. Representations used to embody additiverelationships in the primary grades are extended to explain multiplicative relationships between differentunits, where one unit is an aggregate of another or, in other words, one unit measures a differentamount than the other.In addition to the increasing complexity of unit relationships inherent in the shift from additiverelationships to multiplicative relationships, there is the increasing importance of instructionalrepresentations which rely on properties of continuous measurement for embodying these multiplicativerelationships. As Osborne (1975) stated:6measurement is ubiquitous, pervading all we do in mathematics and science. Withinmathematics this is reflected by the fact that measure concepts are used so frequentlyas the intuitive base for instruction for nonmeasure concepts and skills But thepurpose of using these measure based embodiments is not to teach about measure;rather the instruction is directed towards nonmeasure teaching. (p. 37)A critical assumption behind the use of representations based on continuous measurement is thatstudents’ conceptions of linear or area measurement are sufficient for these representations to providean intuitive understanding of the mathematics being represented. In questioning this assumption,Osborne (1975) argues for the need to focus on the “nature of the measure base possessed by thestudents” (p. 39) as this might relate directly to their interpretations and use of such representations.Studies of students’ conceptions of length and area measurement suggest significant variabilityin the nature of students’ understanding of these concepts in the intermediate grades (Bailey, 1974;Beilen & Franklin, 1962; Beilen, 1964; Hirstein, 1974; Hirstein, Lamb and Osborne 1975; Wagman,1975). The collective evidence suggests that, for some students, limitations in understanding of linearor area measurement concepts persist throughout the elementary grades.The nature of students’ linear and area measurement concepts may play a role in the extent towhich students use properties of linear and area measurement to represent mathematical ideas. On theother hand, regardless of their conceptions of linear or area measurement, students simply may notconsider properties of linear or area measurement as a framework for thinking about and usingrepresentations of numerical relationships. Their previous experiences with discrete representations inwhich measurement of numerosity played a singular role in defining the form of representations maypersist as their framework for thinking about all representations. However, with the introduction anddevelopment of rational number concepts and relations, the dichotomy between representations basedon the measurement of numerosity and representations based on the measurement of continuousquantities is highly significant to the learning process. Representations of fractions based oncontinuous measurement would no longer convey the intended meaning of fractions it interpreted asdiscrete representations. At this stage, students who think about all representations as only collectionsof units without special space-filling characteristics would construct different meanings for common7fractions than students who incorporate properties of linear or area measurement within their conceptionof the form of the representations.Until recently, little research had been conducted which directly investigated students’knowledge and use of different forms of representations. Many studies were conducted which comparethe effectiveness of representations within different instructional treatments (Fennema, 1972; Suydam& Higgins, 1977; SowelI, 1985); but none was designed directly to examine the students’ knowledge ofdifferent forms of representations. A few studies have reported on students’ interpretation or use ofspecific forms of representations in the common fractions context (Behr, Lesh, Post, & Silver, 1983;Bright, Behr, Post, & Wachsmuth, 1988; Hart, 1987; Hasemann, 1981; Novillis Larson, 1980, 1987;Payne, 1975), but these investigations did not explore students’ personal repertoires of representation.Instead, they focused on the difficulties students have with interpreting forms of representationsdetermined by others in a test or interview situation.- That different forms of representations might not be equafly understood by students has beenrecognized. The number line has been identified as one form of representation which is particularlyproblematic for students to interpret and use in a wide variety of mathematical contexts, including thecommon fractions context. However, representations whose form is based on the area of regions or thenumerosity of sets also may present interpretive problems to students, particularly when multiple unitsare embodied within the representation. (Behr et al., 1983; Bright et al., 1988; Dufour-Janvier et al.,1987; Hiebert and Tonneson 1978; Hunting 1 984a; Novillis Larson, 1980, 1987; Payne, 1975; Post etal., 1985; Sowder, 1976: Vergnaud, 1983b)As a result of observing the difficulties and misinterpretations associated with children’s use ofinstructionally-imposed representations, Dufour-Janvier et al. (1987) assert that an instructionalrepresentation which is too distant from a child’s internal representations renders the instructionalrepresentation inaccessible to the child, and “brings the child to manipulate rules and symbols that aremeaningless to him.” They propose that “there is a need to examine how children use objects, how theyact, and the representations that they construct.” (pp. 118-119) The exploration into the characteristics8of students’ repertoires of representations contributes to an illumination of the issues related tomathematical representations.A premise which guided the formulation of this study was that students construct their ownconceptions about representations, and that the nature of their representations is not well understoodby researchers and teachers. The intention therefore was to identify characteristics of students’repertoires of representations which they use to explain whole number multiplication and commonfraction relationships and thereby contribute to an understanding of students’ conceptions ofrepresentations and the representational process. With the introduction of common and decimalfraction concepts and operations as major components of the curriculum during the intermediate grades,students’ interpretations of representations based on continuous measurement become critical to thelearning process. Students’ representations of units of linear and area measurement and relationshipsbetween units were also investigated because of this increasing importance of continuousmeasurement as a basis for mathematical representations. - -Definition of TermsBeginning Diagram refers to an imposed framework, such as a partitioned line or region, within which thestudent constructs a final diagrammatic representation of mathematical relationships. (see Figures3.01 and 3.02)Dominant Form of Representation refers to the form of representation generated most frequently by astudent.Conception refers to the internal representations of a mathematical relationship.Extensiveness of a Repertoire of Representations refers to the number of different forms ofrepresentations included in a student’s repertoire or representations. The extensiveness wasdefined positively in terms of the forms of representations generated by a student, and negativelyin terms of the forms of representations overtly rejected by a student.Intermediate Level refers to students in Grades 4 to 7.Interview. Generative Students were required to generate as many different representations as theycould for any particular interview task.Interview. Generative. Cued A collection of materials was available from which students independentlyselected materials with which to construct representations of the mathematical tasks. It wasanticipated that the presence of the materials might cue some students to construct differentforms of representations from those constructed in the uncued generative interview.9Interview. Generative. Uncued Students had no materials to suggest different types of representationswhich could be used to explain the mathematical tasks. They were presented with a mathematicaltask and asked to explain the meaning of the task through the drawing of diagrams or roughpictures. The questioning was designed to elicit as many different kinds of diagrams as thestudents could think of to explain the same task. All of the representations generated in this wayrepresent a student’s primary repertoire of representations of whole number multiplication.Interview. Interpretive Students were required to interpret and evaluate a series of specific beginningdiagrams as to their appropriateness as a framework for constructing a representation to explainthe mathematical ideas. Students also were asked to construct representations of themathematical task, using those beginning diagrams which they deemed to be appropriate.Interview. Interpretive. Area An interview designed to investigate the extent to which students attend toproperties of area measurement and other geometric characteristics as critical features in theirrepresentations. To this end, geometric and measurement properties associated with area-basedrepresentations were identified and systematically distorted within a series of beginning diagrams.Interview. Interpretive. Linear An interview designed to investigate the extent to which students attendto properties of linear measurement and other geometric characteristics as critical features in theirrepresentations. To this end, geometric and measurement properties associated with linear-based representations were identified and systematically distorted within a series of beginningdiagrams.Mathematical Context refers to the general mathematical domains of interview tasks. The fourmathematical contexts were whole number multiplication, common fractions, linear measurement,and area measurement.Repertoire is defined in the Shorter Oxford English dictionary on historical principles as, “A stock ofdramatic or musical pieces which a company or player is accustomed or prepared to perform; one’sstock of parts, tunes, songs, etc.” The term repertoire underscores the notion of a set of activeproductions, that which you may re-create and perform, not that which you simply may recognizein the performance of others. Applying the term in this mathematical context, repertoire has thesense of a stock of representations which a person is accustomed to reconstructing or preparedto reconstruct. Repertoires can have set-subset relationships. A person’s repertoire ofrepresentations of common fractions is a sub-set of a person’s repertoire of mathematicalrepresentations.Repertoire. General. .A general repertoire of representations refers to all forms of representation that aperson evokes with or without external cues and prompting. The general repertoire is defined inboth positive and negative terms, that is in terms of those forms of representations a personevokes as well as in terms of those forms of representations that a person rejects as a means ofexplaining the mathematics.Repertoire. Primary A primary repertoire of representations of multiplication or common fractions refersto the variety of forms of mathematical representations that a person evokes spontaneously, in theabsence of external cues or suggestions.Representation When used without modifiers the term refers to what others have called “externalrepresentations, or embodiments” such as manipulative models and diagrams used to embodymathematical ideas.(see for example, Janvier, 1987) Written symbols also are considered to beexternal representations of mathematical ideas, but in this study they would be referred to as“symbolic representations,” not “representations” alone. Internal reDresentations” are referred toas conceptions.10Representation. Form of refers to the physical characteristics of a representation which distinguish itfrom other representations of the same mathematical ideas. Forms of representations aredifferentiated by the spatial organization and geometric characteristics of the units.Representation. Function of refers primarily to students’ interpretations of relationships betweendifferent units as expressed through their representations: that is, the extent to which arepresentation served to express relationships between different units.Setting. Interview refers to material and procedural characteristics of an interview. It does not refer to themathematical content of the interviews. The mathematical content was relatively consistent acrossall interviews, but the setting of the interviews differed.Setting. Material refers to the objects, materials or diagrams available to the student with which toconstruct mathematical representations. This changed from one interview to another.Spatial Framework By spatial framework of a representation is meant the general structure within whichthe units are represented, whether geometric regions, lines, or sets. Within a spatial framework,units may be represented in different forms. For example, a line may be used as spatialframeworks to represent discrete or contiguous units, or may be used to represent quantitieswhich were not defined explicitly with units. Units within the spatial framework of sets arenecessarily discrete.Unit. Aggregate refers to a unit which is also a collection of smaller units. For example, in the place valuecontext 1 ten would be an aggregate of 10 ones. This term was adopted in this study from- Gal’perin and Georgiev(1960/69): Aggregate unit issynonymous to derived unitas used byMcLellan and Dewey (1895), and composite unit as used by Steffe and von Glasersfeld (1983).Unit. Contiguous refers to a unit which is touching or adjoining another unit.Unit. Discrete refers to a unit which is spatially separate from other units.Units. Bi-relational Whole number multiplication as well as rational number concepts minimally involve arelationship between two different units. These units are related one to the other by a simpleratio; one unit is either an aggregate or a part of the other. Such a relationship between two unitshas been termed bi-relational.Units. Mono-relational A representation of unit relationships in which all units are treated as equivalent.Additive situations, such as comparing, adding, or subtracting whole numbers involve measuresof only one unit.Units. Tn-relational A representation of units relationships among three different units. In such arepresentation the three units would be “double-nested.” That is, of the form A groups of Bgroups of C. In the multiplication context it would occur as the representation of the product ofthree factors. In the common fraction context it would occur as the representation of two fractionalunits nested within a whole unit, such as the representation of thirds and ninths within a singleregion.11Overview of the Plan of the StudyTo identify the characteristics of repertoires of representations which Grade 5 and Grade 7students use to explain whole number multiplication and common fraction concepts and relations,clinical interview methods were employed. Two pilot studies were conducted to develop and refine theinterview tasks and procedures.Six Grade 5 students and nine Grade 7 students from two schools were selected by theirclassroom teachers for the main study. The students within each grade were considered by theirteachers to represent a range of school mathematics achievement.Four interviews were designed to explore the characteristics of students’ repertoires ofrepresentations. The interviews differed in their material setting. In the first interview the students hadonly paper and pencil available with which to construct their representations. In the second interview avariety of materials was available. In the third interview the students were presented with beginningdiagrams related to properties of length measurement. In the fourth interview the students werepresented with beginning diagrams related to properties of area measurement. In each of theseinterviews, students produced representations to explain some numerical mathematical tasks. It wasassumed that by changing the material setting within which the students were asked to constructrepresentations of mathematical tasks, forms of representations which students might not use in onesetting might be used by them in another.Data to explore the nature of students’ representations of units of linear and area measurementwere derived primarily from Iwo interviews: one for linear measurement and the other for areameasurement. One of these two measurement interviews directly followed the third or fourth interviewson representations.Limitations of the StudyAs an exploratory study, 15 subjects from Grade 5 and Grade 7 were selected to represent arange in achievement and a range in formal instructional experience in order to characterize variations in12students’ knowledge and use of mathematical representations of units and multiple unit relationships. Itwas not the purpose of this study to determine the extent to which such conceptions of mathematicalrepresentations are held by students in general. Rather, the study was designed to characterizepossible conceptions of mathematical representations held by students in the intermediate grades.Thus, subjects were not selected as a representative sample of a population.The characterization of students’ conceptions of representations of units and unit relationshipsof measurement was derived from a categorization of students’ responses to a limited number of tasks.The study investigated characteristics related only to students’ translations of symbolic representationsof whole number multiplication or common fractions into manipulative or diagrammatic representations.Additional characteristics of students’ representations may have been derived if the tasks had involvedreal world problems rather than only symbolic expressions. The results therefore are limited to acharacterization of these students’ responses within the constraints of the tasks used and are notconsidered to be exhaustive.Justification of the StudyTeachers use a variety of representations during mathematics instruction which are assumed toillustrate and clarify abstract mathematical relationships for students. During instruction it is assumed thatthe interpretations imposed on the representations by teachers are accessible to the students. That is,it is assumed that teachers and students share a common knowledge about the nature of the units andunit relationships in the representations such that the meaning of the mathematics being represented isinterpreted appropriately by the students. If students hold conceptions of units which differ from thoseassumed by the teacher, the students’ interpretation of the mathematics being represented may differfrom the meanings assumed to be communicated by the teacher. This study characterizes students’representations of units and unit relationships in four mathematical contexts, each of which is animportant component in the elementary mathematics curriculum. The nature of students’representations of units is likely to play a significant role in the ways in which they interpretrepresentations commonly used in instruction. This study raises questions about the ways in which13different representations, which are assumed to communicate ideas about mathematical relations, mightbe interpreted by students.These questions are particularly critical at the intermediate grade level. When the number ofrelationships between units contained within representations increases, and when properties of linear orarea measurement are used more extensively as the framework for constructing representations, thestructure of instructional representations becomes more complex. These sources of complexity maydecrease the accessibility of instructional representations for the students.Organization of the DissertationThere are five remaining chapters to this dissertation. Literature related to the study is reviewedin Chapter 2. The plan and implementation of the study is presented in Chapter 3. The results anddiscussion are reported in two chapters. Chapter 4 explores the the characteristics of students’representations of who[e number multiplication and common fractions. Students’ representations oflinear and area measurement are explored in Chapter 5. The conclusions and implications for futureresearch and instruction are presented in Chapter 6.14CHAPTER 2REVIEW OF RELATED LITERATUREThe unifying focus of this study is the investigation of students’ representations of units andunit relationships in a number of different mathematical contexts: whole number multiplication, commonfractions concepts and comparisons, length measurement, and area measurement. As little researchhas been directed toward the issue of students’ personal representations of mathematical relationships,the review of the related literature is divided into four main themes of discussion, each of which raisedissues which contributed to the formulation of the study. Some of these themes draw upon theoreticaldiscourses in the literature, while others draw upon some empirical studies. The four themes are:1. The construct of repertoires of representations.2 The nature of mathematica[ representations. -3. The role of measurement as a framework for representing mathematical relationships. Includedin this theme are two issues:(1) representations of multiplicative relationships between units, and(2) students’ conceptions of units of continuous measurement.4. Students’ representations of whole number multiplication and common fractions. Included inthis theme are two issues:(1) The nature of student-generated representations.(2) Students’ interpretations of instructional representations.Repertoires of RepresentationsAlthough little was said a decade ago about students’ representations of mathematics, muchattention was paid to the need to use manipulative and diagrammatic representations during instruction.Core features of elementary instruction were that teachers should use “a wealth of manipulativeexperiences through which concepts and relations are understood at an intuitive level,” and that15“mathematics as a discipline, as a formal structure, must be built upon a sound foundation of concreteexperiences” (Province of British Columbia, 1978, P. 1). The National Council of Teachers ofMathematics (1989) stated as part of their “Standard 2: Mathematics as Communication” that:In grades 5 to 8, the study of mathematics should include opportunities to communicateso that students can model situations using oral, written, concrete, pictorial, graphicaland algebraic methods. (p.78)Where, in 1978, the British Columbia Ministry of Education spoke of manipulative and concreteexperiences as simply a means to attain an objective, the objective being the formal study ofmathematics, in 1989 the N.C.T.M. speaks of students’ use of concrete, pictorial or graphicalrepresentations of mathematics as objectives in themselves. The N.C.T.M.’s statement about theimportance of these forms of communication closely resembles the model of “representation systems”proposed by Lesh (1979). Lesh argued that students should be able to translate mathematical ideasbetween and within all modes of representations, namely real world situations, oral and writtenlanguage, manipulative-concrete, pictorial or graphical, and algebraic or symbolic methods. Theassumption behind this model is that students should be able to both interpret and constructmathematical representations in and between these modes of representation. The representationalmodes are to be considered both expressive and receptive (Clements & Lean, 1988).In this study the focus is on the nature of students’ expressions of mathematical ideas as theyare represented with concrete materials, pictures or diagrams. As was explained in Chapter 1, the termrepresentations is used in this report to mean concrete or diagrammatic expressions unless otherwisequalified. The term “repertoire of representations” was defined by the investigator of this study to referto a set of representations which a student could construct as a way to explain a mathematical concept,relationship or operation.Proponents of instruction of mathematical concepts or relationships with multiple forms ofrepresentations, such as Bruner (1968) and Dienes (1967), were some of the sources which influencedthe thinking about students’ representations and questions about possible repertoires ofrepresentations in this study. Bruner (1968) proposes that “by giving the child multiple embodiments ofthe same general idea expressed in a common notation we lead him to “empty” the concept of specific16sensory properties until he is able to grasp its abstract properties” (p. 65). The “Perceptual VariabilityPrinciple” elaborated by Dienes (1967) expressed the same belief in the role of multiple embodimentsin the learning of abstract mathematical concepts. Dienes argued that “the same conceptual structureshould be presented in the form of as many perceptual equivalents as possible” in order “to allow asmuch scope as possible for individual variations in concept-formation, as well as to induce children togather the mathematical essence of an abstraction.” (p. 32). The goal motivating these proposals forinstruction with multiple forms of representations was to create an environment which would facilitatechildren’s abstract mathematical thinking. However, both Bruner and Dienes observed otherphenomena in relation to children’s thinking with and about different forms of representations.Bruner (1968) and Bruner and Kenney (1965) report that, when children explored mathematicalrelationships with multiple forms of representations, they were observed to have constructed “a store ofconcrete images that served to exemplify the abstractions.” They reached the tentative conclusion that,“it is probably necessary for a child learning mathematics not only to have as firm a sense of theabstraction underlying what he was working on but, also, a good stock of visual images for embodyingthem. For without the latter, it is difficult to track correspondences and to check what one is doingsymbolically.” (Bruner, 1968, pp. 65-66; Bruner & Kenney, 1965, pp. 56-57) The “stock of visualimages” is analogous to a “repertoire of representations” as used in this study.Bruner and Keriney (1965) also reported that children were observed to have “equated’concrete features of one form [of representation] with concrete features of another” as a means ofmaking sense of new forms of representations (p. 57). This would suggest that analogies betweenfamiliar and new forms of representations may play an important role in the way in which children interpretdifferent forms or representations, regardless of critical differences in the physical attributes of eachform of representation. Furthermore, Dienes (1964) suggested that, through such analogous thinkingabout different forms of representations, students might come to conceive of one situation as aprototype of all others in their previous experience; that is, that one representation might come to serveas a “set of symbols” for their repertoire of representations (p. 142).17Dienes’ association of students’ multiple representations with a prototype or representativeexemplar is similar to Vinner’s and Hershkowitz’ construct of a concept image (Vinner & Hershkowitz1980; Hershkowitz & Vinner 1984; Hershkowitz, 1987). The construct of a concept image was derivedand elaborated through a number of studies which explored students’ and teachers’ construction oridentification and interpretation of representations of geometric concepts.Vinner and Hershkowitz argue that a concept image develops from a collection of realdiagrammatic or manipulative experiences. A student’s interpretation of the significance to be attachedto features common to these collective experiences serves to construct a generalized concept image.In other words, a process is involved whereby salient features among a variety of representations aredefined, attended to, and used by a student to construct general concept images. Furthermore, theysuggest that a student’s perception of which features are salient to the task at hand may includenon-critical features or exclude critical features, and thereby result in a concept image which is distortedor incomplete compared to the intended objectives of instruction. For example, for many students, theconcept image of obtuse angles included only angles with a horizontal ray. The horizontal characteristicof the ray, though a non-critical feature of obtuse angles, was interpreted as a critical attribute of thesestudents’ conceptions of obtuse angles. Even when a formal definition of a concept such as thealtitude of a triangle was provided, both elementary teachers and students were found to respond onthe basis of a concept image rather than on the basis of the defined attributes of the concept. Thepredominant concept image of altitudes of triangles was limited to images in which the altitude fell withinthe interior of a triangle. A concept image appeared to be unrelated or indirectly related to a formaldefinition of the attributes of a concept. (Vinner & Hershkowitz, 1980; Hershkowitz & Vinner, 1984)Hershkowitz (1987) related concept images to that of Rosch’s (1977; 1978) formulation ofprototypes of categories. Hershkowitz stated,Every concept has a set of critical attributes and a set of examples. In the set of conceptexamples there are the “super” examples: - the prototypes; that is the popularexamples. In other words, all the concept-examples are mathematically equal, becausethey conform to the concept definition and contain all its critical attributes, but they aredifferent one from the other psychologically. (p. 240)18In these terms, the set of examples were those defined from a mathematical point of view. From thestudents’ point of view, often only a subset of the examples was associated with a concept. This subsetwas considered to contain the “super” examples or the prototypes of the concept.A distinction needs to be made between people interpreting someone else’s representationand people constructing their own representation. Tasks in which a person need only recognize oridentify representations are less complex than those in which a person must reconstructrepresentations (Clements, & Del Campo, 1987; Sinclair, 1971 b). Vinner and Hershkowitz used bothforms of data to support their construct of concept images. However, they relied more heavily upon datawhich derived from students’ and teachers’ interpretations and judgements of the correctness of aseries of predetermined representation rather than data derived from the students’ and teachers’ ownrepresentations of a concept. There was some evidence of concurrence between the image mostcommonly associated with a concept in the interpretive and constructive condition. The distinctionbetween interpretation and construction of representations is significant if one is concerned about arepertoire of representations. Physical images which people may recognize as a representation of amathematical idea still may not be evoked by them independently. What a teacher’s or student’srepertoire of physical images would be was not investigated.Vinner and Hershkowitz focused primarily but not solely upon the interpretation ofrepresentations rather than their construction. However, a concept image is defined as “all mentalpictures and associated properties and processes of a concept built up by a person over the yearsthrough experiences of all kinds” (Tall & Vinner, 1981). As such a concept image can seen to includethe notion of a repertoire of representations of some mathematical idea. A repertoire, rather than asingle representation, is assumed in this study to illuminate more completely a student’s conception ofthe mathematical concept under consideration.There are a variety of other terms used to denote constructs similar to that of a concept image,as well as variations in the meanings associated with these terms. Janvier (1987) speaks ofschematizations within a representation. Schematizations are the variety of ways in which amathematical relationship may be expressed externally, and “representation” is used as synonymous19with the term “conception” in this study. Included within the schematizations would be a repertoire ofphysical representations evoked to explain a mathematical relationship. Dufour-Janvier, Bednarz andBelanger (1987) make a distinction between external representations and internal representations inthe context of discussing manipulative and diagrammatic instructional representations. TheirexternaL1in al dichotomy would parallel the use of representations and conceptions in this study.Further, Lesh, Post, & Behr (1987) use a model of modes of representations (written representations,symbolic representations, representation through spoken language, static pictorial representations,and manipulative representations). They define conceptual understanding in terms of translationswithin and between these modes. In such a case, “representation” is used to refer to all expressions ofa person’s conceptions. Despite apparent differences in these uses of terms, there are a number ofassumptions shared by these constructs. These include a constructivist view of cognitive functioning, adichotomy between internal and external representational forms, and variabilities within internal andexternal representations. None would assume that there would be a single representation which mightcapture all facets of a person’s conception of a mathematical idea or which would serve as a conceptdefinition. In all of these cases, repertoires of manipulative and diagrammatic representations, exploredin this study, would fit as a sub-structure within the proposed models.From the observations of Dienes, and Bruner and Kenney, there are a number of elementswhich might characterize students’ repertoires of representations. First, a repertoire might consist of avariety of “concrete images” of a particular mathematical relationship which students have constructedthrough their experiences with different forms of representations. Second, the variety of “concreteimages” might be equated by the students in terms of their analogous features. Third, there might be aparticular form of representation which could be considered to act as a prototype of their repertoire ofrepresentations. Such prototypes might be derived by students analogously equating commonfeatures of the representations in their repertoire. The prototypes would then serve as a generalizedform for representing the mathematical relationship.Where Bruner, Kenney and Dienes were concerned with the learning of mathematical conceptsor relationships which were not intrinsically geometric but for which geometric representations might be20constructed, Vinner and Hershkowitz were concerned with geometric concepts which were directlyrepresentable as objects or diagrams. The differences in their formulation of “concrete images” asopposed to “concept images” appear to be more a question of levels of abstraction rather than aquestion of competing premises about representations.The Nature of Mathematical RepresentationsPost (1980) defines manipulative materials and diagrams as partial isomorphisms or isomorphicstructures which represent the more abstract mathematical notions children are to learn. He states that,.if a parallel structure that was more accessible and perhaps manipulable could beidentified having the same properties as the set of whole numbers, then it would bepossible to operate within this accessible (and isomorphic) structure and subsequentlymake conclusions about the more abstract system of number. (p. 113)The assumption is that the intervention of physical representations in the learning situation provides aconcrete environment within which children may think and learn about mathematical relationships. Theother assumption is that such concrete representations are accessible to children.On the other hand, Post (1980) also acknowledges that these physical interpretations orembodiments of abstract mathematical notions are also “artificially constructed systems” (p. 113). Theartificial construction of physical representations as models or embodiments of mathematicalrelationships implies that they are themselves abstractions relative to real world situations. They aremeant to “simplify and generalize” mathematical relationships interpreted from real world situations(Lesh, 1979). In other words, they are “generally a simplified version of the original, which permits aneasier and more complete control of a set of variables” which may serve to facilitate the interpretation ofcertain given facts (Fischbein, 1977, p. 155). Physical representations could be considered one levelof abstraction from which more abstract representations of concepts and operations are derived bychildren. As Von Glasersfeld (1983) stated,Concepts and operations involved in mathematics are not merely abstractions, but mostof them are the product of several levels of abstraction. (p. 64)How accessible different physical representations are to children is a question considered bynumerous authors. (E.g., Dufour-Janvier, Bednarz, & Belanger, 1987; Ernest, 1985; Fischbein, 1977;21Bell & Janvier, 1981; Sowder, 1976; Vergnaud, 1983b, 1984) Underlying this question is a recognitionthat some forms of physical representations are more abstract than others. Furthermore, there is arecognition that different types of physical representations are based on different assumptions, rules,and conventions regarding what and how attributes of the representations embody and conveymathematical meanings. There is also a recognition that children must understand the assumptions,rules, and conventions associated with a particular form of representation in order to interpret or use thephysical representation appropriately.As an example, Fischbein (1977) states that visual images used to represent mathematicalrelationships, such as Venn diagrams, tree diagrams, and graphs, are not primitive images which mapdirectly to real world situations. Instead, he terms them “conceptualized images, controlled by abstractmeaning” (p. 154). For example, the image of a curve on a graph is at least a third-order abstraction. Itsproduction involves translations from the physical situation of the function to pairs of numbers, to theplotting of points, to the drawing of the curve. Fischbe[n argues that “the child has to learn the languageof the image, not relating it directly to the physical process, but indirectly to the conceptual language ofcoordinates. A short circuit between the graph and the original, physical phenomenon, will result inmisunderstandings” (p. 154).Doufour-Janvier et al. (1987) presented a similar argument about using an abacus to representthe place value system. An abacus does not directly represent the image of the successive groupingsin the place value system. Attributes such as colour and position are used to distinguish the fact thatone disc represents “one” and another disc represents “ten.” The basis for the grouping rule is notapparent in the representation, and unless its meaning is understood the operations illustrated on theabacus would not be understood in terms of the place value system. The child would need to interpretthe form of the representation in terms of the meaning of its structure and language in order toapprehend the place value relationships and operations represented. In conclusion, Dufour-Janvier etal. (1987) stated,The use of representations that are just as abstract for the child as that which is studiedbrings the child to manipulate rules and symbols which are meaningless to him... .Thechild is forced to “learn” the representation that is submitted to him: the rules of usage,22the conventions, the symbols, and the language linked to the representation... .Theuse of such nonaccessible representations encourages a play on symbols, puts theemphasis on the syntactical manipulation of symbols without reference to the meaning.(pp. 118-119)A physical representation, based on a framework such as a number line, an a x b rectangularmodel, an abacus, or base ten blocks, can be thought of as a duality comprised of (1) therepresentation’s form with associated attributes, structure and language, and (2) the representation’sfunction as a potential representation of diverse mathematical relationships. Similar dualities have beenproposed in the analysis of symbolic representations of mathematics (flyers & Erlwanger, 1984; Hieberl,1984; Resnick & Omanson, 1987; Skemp, 1982). Byers and Erlwanger (1984), analyzing mathematicsin terms of its form and content, state that “the content of mathematics consists of ideas embodied in itsmethods and results; mathematical form includes symbolic notation and chains of logical arguments(Byers & Herscovics, 1977)” (p. 260).Byers and Erlwanger (1984) acknowledge a further, subtle distinction in the analysis ofmathematics in terms of form and content. Using place value notation as an example, they argue that,on the surface, this notation would be considered to belong to the form of mathematics in contrast tooperations which would be considered to be the content. Yet when learning two-digit addition, placevalue notation becomes the content to be learned. They conclude that mathematical forms havecontent of their own. The understanding of the content of mathematical form is important to be able touse the mathematical forms appropriately to represent other mathematical content. In other words,there is a duality within a duality.Byers and Erlwanger analyzed symbolic notation in terms of form and content, but theirarguments parallel that of Fischbein’s (1977) analysis of the nature of physical representations. Physicalrepresentations also have their own content independent of the mathematics which is represented.This content needs to be understood by children in order for the mathematical relationships embodiedin the representations to be accessible. In this context, accessibility refers to children being able tointerpret the meaning of attributes of the form of the representation in order to interpret the mathematicsrepresented. Accessibility also refers to children being able to use these physical representations toembody their own mathematical ideas. The duality of physical representations has been used as a major23framework for considering the characteristics of students’ representations in this study. This duality isexpressed in terms of the form of a representation and the function of a representation as anembodiment of a mathematical relationship. However, that the form of a representation has contentindependent of the mathematics represented is also a significant analytical element within theframework of the current study.Measurement as Content ofForms of Mathematical RepresentationsThe concept of number would not even exist if man had not met problems ofmeasurement;. ..Numerical situations usually deal with quantities that are not purenumbers but magnitudes of various kinds... .Numbers are undoubtedly central inmathematics but it is impossible to understand what difficulties children meet withnumbers if you do not look at them as magnitudes of different sorts, transformations, orrelationships.(Vergnaud, 1980, pp. 264, 265)Implied in Vergnaud’s assertion is a broader meaning of measurement than that which isnormally associated with measurement in elementary mathematics teaching. It implies not onlymeasured magnitudes of continuous quantities, but also measures of discrete quantities. That thecommon distinction between counting objects and measuring continuous quantities is a falsedichotomy has been argued by others such as Blakers (1967) and McLeIlan and Dewey (1895). Blakersrefers to “the measurement of numerosity,” stating that,the ideas involved in the empirical measurement of numerosity ... are so simple, and thestructure of the most appropriate value set (the set of positive integers, whoseprehistorical development was undoubtedly due to the empirical properties of themeasurement operation known as ‘counting’) is so closely related to the attribute thatwe wish to measure, that numerosity measurement is ignored in many treatments of thegeneral subject of measurement. (pp. 77-78).In a similar vein, McLellan and Dewey (1895) asserted thatall counting is measuring, and all measuring is counting.. ..When we count up thenumber of particular books in a library, we measure the library... .The only way tomeasure weight is by counting so many units of density; distance by counting so manyparticular units of length;... (p. 48)It has long been recognized that “the character of a child’s first encounters with numerousnessis in the sense of measure applied to discrete sets of objects” (Osborne, 1975, p. 38). The use of24continuous measure systems as bases for representations of mathematical relationships and operationsis qualitatively different from children’s earlier experience with discrete measures because of theadditional attributes involved in defining the properties of the units. For example, the space-fillingcharacteristic of units of area, the definition of units of length as congruent line segments, as well as thecontiguity of units of length and area are particular attributes which differ from discrete units (Osborne,1975). Nonetheless, measurement of discrete objects, length, and area are used as a basis forrepresenting numerical operations and relationships in elementary mathematics curricula withoutnecessarily attending to the qualitative differences between properties of the units and the attributesbeing measured.The lack of explicit attention to the qualitative differences between the attributes and propertiesof the units in different forms of representations is to be found in elementary school textbooks. InEicholz, O’Daffer, & Fleenor, (1974), both the number line and discrete sets are used to represent theoperation of multiplication. The distributive property of multiplication is introduced with representationsbased on the area of rectangles, then represented with discrete rectangular arrays at the Grade 3 level.Yet, each of these forms of representations is associated with a common, abstract language, “X of Y,”without referring to other attributes of the units. Similarly, at the Grade 4 level, common fractionconcepts and relations are introduced primarily with representations based on area measurement, andsecondarily with discrete sets. Yet, both forms of representations are associated with the samelanguage, “X out of Y parts.” Later, representations of common fractions on the number line are usedextensively. Notions of distance on the number line is associated cursorily with the ruler and runningraces, but in the exercises only the language of points is used: explicit reference to distances from theorigin, or units of length do not occur. In the previous examples, the assumption appears to be that thestudent will discriminate intuitively between the properties of units in different forms of representations.As Osborne stated:It is assumed that measure is intuitive and sufficiently possessed and understood bythe learner to serve as an intuitive representation for explaining numerical operations.This assumption should be questioned. (Osborne, 1975, p. 42)25There are several issues, related to this study, which arise from the fact that both discrete andcontinuous measurement are used as a basis for representations of whole number multiplication andcommon fractions. One issue is how students conceive of one unit to be a multiple or a part of anotherunit regardless of the basis of a representation. A second issue is whether students have an intuitiveunderstanding of the measurement of length or area with which they might interpret and constructcontinuous measurement representations appropriately. The literature related to each of these issueswill be discussed in turn.Multiplicative Relationships Between Units.As early as 1895, McLellan and Dewey argued that number represents a ratio or measure of aunit in relationship to a quantity or “unity.” Furthermore, the unit should be conceived of also as aquantity measurable by smaller units.More than half of the difficulty of the teacher in teaàhing, and the learner in learning, isdue to the misconception of what the unit really is. It is not a single unmeasured object;it is not even a single defined or measured thing; it is any measuring part by which aquantity is numerically defined;.... (p. 158)As necessary to the growth of the true conception of unit as a measuring part,the idea of the unit as a unity of measured parts must be clearly brought out. The givenquantity is measured by a certain unit; this unit itself is a quantity, and so is made up ofmeasuring parts. This idea must be used from the beginning; it is absolutely essentialto the clear idea of the unit, and of number as measurement of quantity. (p.160)They expressed the view that, by limiting representations of numbers to ones in which oneobject within a set always represented the unit, the opportunity for children to construct acomprehensive understanding of the meaning of units, relationships between units, and subsequentlythe meaning of number is artificially restricted.Count by ones, but not necessarily by single things; in fact, to avoid the fixed unit error,do not begin with counting single things. (McLellan & Dewey, 1895, p. 160)From an experience of variable representations of units used to express measures of quantitiesnumerically, McLellan and Dewey argued that children would conceive of numerical concepts in bothadditive and multiplicative terms. Derived units were made up of a number of smaller units, called primary. These units were to to be central to children’s earliest, formal experiences of number concepts26and relationships. From these experiences of the meaning of number, the operation of multiplication,therefore, was represented as “the complete expression of any measured quantity [through] (1) thederived unit of measure, (2) the number of such units, and (3) the number of primary units in the derivedunit of measure” (p. 112). Also, from these initial experiences of units and number, the “measuremeaning” of common fractions would follow as an expression of a relationship between two variable,designated units of measure. It should be noted that McLellan and Dewey’s formulation of commonfractions as measures is more general than that defined by Kieren (1975). Kieren refers to one of sevenpossible interpretations of rational numbers as “measures or points on a number line” (p. 103),McLellan and Dewey refer to any measures of discrete and continuous quantities.Even Thorndike (1924) gave explicit support to McLellan and Dewey. He argued that “desirablebonds” would be neglected if the numbers were notconnected ... each with the appropriate amount of some continuous quantity like lengthor volume or weight, as well as with the appropriate sized collection of apples, counters,blocks and the like.... Otherwise the pupil is likely to limit the meaning of, say, four tofour sensibly discrete things and to have difficulty with multiplication and division. (p. 75)During the 1950’s McLelJan and Dewey’s (1895) ideas on the role of units and unit relationships on thechild’s development of number concepts and relationships was a significant influence on the work ofGal’perin., & Georgiev (1969) in the Soviet Union. In that study, McLellan and Dewey’s analyses areseen to be directly relevant to students’ representations of relationships among units.In the educational setting of this study, whole number concepts and relationships, as well asaddition and subtraction, are introduced primarily, if not solely, through the counting of individual units.Experience with units as aggregates arises for students, at the primary level, in the context of teachingplace value notation, and the operations of whole number multiplication and division, and at theintermediate level, with common fraction concepts and relations. (Province of British Columbia, 1987)That students at the intermediate grade level have difficulty interpreting units as aggregates inthe common fractions context is attested to by several reports in the literature. Behr, Lesh, Post andSilver (1983) report results of a study in which Grade 4 students, after normal classroom instruction,were tested on the representation of common fractions with regions, number lines, or discrete sets.27Students had more difficulty completing the task when the number of units presented was a multiple ofthe denominator, than when the number of units presented was equal to the denominator, or when thepresentation of the units was incomplete. For example, given three rectangles, (a) one with four equalparts marked, (b) one with one part equal to a fourth marked, and (c) one with eight equal parts markedand asked to represent 3/4, there was an order of increasing difficulty from a to c. The idea of a unit,which results in a measure of four, being an aggregate of two smaller units, was considerably moredifficult for students to conceive.Behr et al. (1983), reporting on responses to similar tasks in clinical interview settings duringtheir teaching experiment, found similar difficulties in the ability of students to interpret representationsin terms of aggregate units. For example, some students were unable to consider a fourth of a circlepartitioned into three as a representation of one-fourth even though they were able to explain that anequivalent unpartitioned sector in the same circle was one-fourth.Bright, Behr, Post and Wachsmuth (1988), reporting on a part of the same teaching eerirnentproject directed towards the identification and ordering of common fractions on the number line withstudents in Grades 4 and 5, concluded that “when a representation is given in unreduced form,students have difficulty choosing the correct reduced symbolic fraction” (p. 227). Similar results werereported for Grade 7 students by Novillis-Larson (1980). Comparing subtest scores of different types ofnumber line tasks, Novillis-Larson found that unreduced representations were more difficult to interpretthan reduced representations.Collectively, these studies suggest that some intermediate grade students may conceive ofunits generally as only single objects or parts; that is, students may not conceive of units as aggregatesin other numerical contexts besides the common fraction context. No studies were located whichexplored students’ representations of units in the whole number multiplication context. The questionremains whether some students construct a “fixed unit error,” as McLellan and Dewey (1895) termed it,based on their extensive experience of number represented with units as only single objects or parts.28Concertions of Units of Length and Area.Carpenter (1 975a) in his review of research on measurement, observed that most studies hadfocused on the development of the pre-measurement concepts of conservation and transitivity, andthat fewer studies had dealt with measuring (p. 47). Furthermore, many of the studies whichinvestigated children’s conceptions of measurement and units of measure, or the acquisition ofmeasurement concepts, have been conducted at the primary grade level (Bailey, 1974; Beilen &Franklin, 1962; Carpenter, 1 975b; Carpenter & Lewis, 1976; Daehler, 1972; Heraud, 1987; Hiebert,1981; Sinclair, 1971a). Few studies have been identified, at the intermediate grade level, whichinvestigated students’ conceptions of units of measurement and their thinking about the measurementprocess.Beilen and Franklin (1962), conducted a Piagetian training study involving the acquisition oflogical operations in area and length measurement, with students from Grades 1 and 3. On the basis ofthe full results of the study, they concluded that length and area measurement are achieved insuccessive order and that, “the constituent operations of measurement (transitivity, subdivision,change in position etc.) are applied more easily first to a single dimension, then to two dimensions, andthen to three” (p. 617). At the Grade 3 level, 27 percent of the students were at a concrete operationallevel for area measurement, and 82 percent were at a concrete operational level for linear measurement.These findings would suggest that at the intermediate level, students would be more likely to have aconception of linear measurement that would be consistent with properties of that measurement systemthan would be the case with area measurement.Linear Measurement.Beilen and Franklin’s (1962) results suggest it is likely that students in the intermediate gradeswould conceive of linear measurement in terms of unit iteration. On the other hand, the linearmeasurement tasks in their study required students to iterate a single unit to compare quantities oflength. They did not require students to use compensatory reasoning about the inverse relationship29between the size of the unit and the value of a measure of a quantity of length, or identify and interpretunits on a line. In contrast, there are two studies, Bailey (1974) and Hirstein, Lamb, and Osborne(1978), which indirectly suggest that students at the intermediate grade level may not consistentlyconceive of units as congruent line segments in a length measurement context, or may not conceive ofthe logical, inverse relationship between the size of the unit and the value of a measure of a quantity oflength.Bailey (1974) conducted a study in which children from Grades 1 to 3 were required to comparepolygonal paths in which the size and number of line segments were varied. Four tests were individuallyadministered to students judged to be in the top two thirds of Grades 1, 2, and 3. Test 1 involved twosymmetric paths with an equal number of segments but unequal segment length; Test 2 involved twopaths with equal segment lengths but unequal numbers of segments; Test 3 involved two paths inwhich one path had longer but fewer segments than the other, and Test 4 involved two paths withsegments of equal number and length. None of the students was able to come to a logical conclusion inTest 3. With respect to the other tests, Bailey reports that only three of the 90 students, all of themGrade 3 students, indicated an ability to use the dimensions of length of units and number of iterationssimultaneously to establish the length relationship between two polygonal paths. The other childrencentred either on the number of segments, or the length of the segments to explain their comparisons.He concludes that “perhaps children older than nine years will experience the same problem” (p. 524).Hirstein, Lamb and Osborne (1978) conducted a study in which 106 children from Grades 3 to 6were interviewed on a variety of area-measurement tasks designed to investigate how childrenincorporate number into their area measurement judgements. In one set of tasks, the rectangles weremarked along the dimensions to indicate a grid. In this context, they observed that students frequentlycounted points rather than line segments along the dimensions. Although they did not report whetherthis behaviour occurred more frequently with younger children, or was observed with equal frequencythroughout the grades, they concluded that these children had no sense of a linear unit.These studies suggest that some students at the intermediate grade level are likely to haverestricted conceptions of units of length, and have difficulty logically coordinating the inverse30relationship between size and number of units. These factors would influence the way in whichstudents interpret and use mathematical representations based on length measurement.Area Measurement.There are two aspects of area measurement and students’ conceptions of units of area whichrelate to students’ interpretations and use of representations based on area. The first aspect is howstudents interpret and use units to compare the area of region, and the second is how studentsconstruct equal units within a region. The literature related to these two issues will be discussedseparately.Comparing Areas of Regions. Beilen (1964) conducted a study with students fromKindergarten to Grade 4, in which he evaluated their ability to make equality and inequality judgementsof areas inscribed with square units. The equality tasks involved the comparison of a 2 X 2 or 3 X 3square with an irregular polygon of the same area. To resolve the task, the student would have to eithercompare the number of units in each region or imagine the transformation of one region into another.Some students who failed to determine the equalities established that the numbers of “boxes” wereequal but still judged the space to be unequal because a piece was “sticking out.” Only 50 percent ofthe Grade 4 students determined the equality of the regions with four square units, and only 38 percentdid so with the regions with nine square units.Wagman (1975) found, with a unit-area task in which the measure of a quantity was determinedby tiling different units, that 42 percent of 11-year olds (Grade 5) were able to understand consistentlythe notion of the unit, take the differences in the size of the units into account, and predict the measurewith a unit given its relationships to another larger or smaller unit. The percentage of younger studentswho were able to reason consistently about the unit relationships decreased with age.In Wagman’s study the two units were a square and a triangle which was half the size of thesquare. Once the area measure was determined by the square, it was possible for children to reasonthat two triangles were equal to a square, and therefore the measure would be double with the triangularunits. In contrast to this task, Hirstein, in an unpublished pilot study in 1974 (Steffe & Hirstein, 1976)31used a task in which two units, which were equal in area but not congruent, were tiled within congruentrectangles, that is, six of each unit tiled a rectangle. This task would require the student to either reasonfrom the premise that an equal number of measures of the same quantity implies equal units of area,regardless of incongruency, or use imagined transformations to compare the two units. With this task nostudents below the age of 10 were able to determine that the non-congruent units were equal in area.Bergeron and Herscovics (1987) interviewed students from Grades 3 to 6 with a task similar tothe one used by Hirstein (Steffe & Hirstein, 1976). The students were asked to compare the area ofone fourth of two congruent squares when the parts in one square were not congruent to the parts inthe other. Even after students had established that one part in each square was one fourth of thesquare, some students in all grades (7 out of 13 in Grade 6) later used visual transformations of one unitinto another to justify their judgements that the parts in both squares were equal in area. The numericalreasoning used to name the fraction of each square was not related to reasoning about the comparisonof the area of the non-congruent parts. - -Hirstein, Lamb and Osborne (1978) report a number of alternative strategies which studentsfrom Grades 3 to 6 used to compare quantities of area. They interviewed 106 students to study howchildren incorporate number into their comparative judgements of areas of regions. They observed thatcentering on only one dimension was a strategy used by some students at all grade levels. Others usedprimitive compensation methods, reasoning that one region is longer but the other region is wider.Partitioning and re-combining was a third alternative strategy used which did not involve units.These studies all indicate that in the intermediate grades, children’s notions about units of area,and the relationship between the number of units, the size of the units, and the measure of a quantitymight vary greatly. That students incorporate number into their reasoning about comparisons of area,particularly when units are not congruent, cannot be assumed.Partitioning Regions. Investigations into the development of children’s conceptions of equalsubdivisions of areas have revealed a complex process in which differences in the number beingconsidered, and differences in the shape of the regions being partitioned, affect the success of youngchildren’s partitions. Rectangles were found to be easier to subdivide than squares and both were32easier to subdivide than circles. Subdivisions of powers of two (e.g 2, 4, 8...) are generally easier toconstruct than subdivisions of odd numbers. (Hiebert & Tonnessen, 1978; Piaget, Inhelder &Szeminska, 1948/1964; Pothier & Sawada, 1983)Piaget, Inhelder and Szeminska (1948/1964) found that subdivisions of six were constructedsuccessfully by children at a much later age than subdivisions of three and subdivisions of powers oftwo, a result which they found surprising (p. 326). This delay may be explained by the findings ofPothier and Sawada (1983). They found that “number-theoretic” concepts such as even and odd arecritical factors in children’s differentiation of algorithms for partitioning with different values. Oddnumbers eventually are recognized as “hard,” become disassociated from the algorithm of successivehalving, and children anticipate the need to use different partitioning strategies with these odd values.On the other hand, the even-odd differentiation is insufficient to accommodate the subdivision of anarea into an even number of parts which is not a power of two. For example, although six is an evennumber, one of its factors is odd; successive haMng therefore will not result in six equal parts. Not onlymust children eventually differentiate between even and odd number, but also they must differentiatebetween even numbers that are powers of two and those that are not. Pothier and Sawadahypothesized that at a later level children, to become efficient partitioners, would have to construct amultiplicative algorithm for partitioning with factors. This algorithm would allow for the efficientpartitioning of areas by all composite numbers which are not powers of two. Whether students at theintermediate grade level use number-theoretic concepts as a basis for partitioning, along with otherpartitioning strategies, is unclear.A distinction must be made between knowing or not knowing techniques for achieving anynumber of equal partitions and believing that parts are equal when they are not (Streefland, 1978).Students may believe that the parts are equal in area when they are not, and by extension believe thatthat a technique which results necessarily in unequal parts is adequate. On the other hand, they may beaware of the inadequacy of their technique and the resulting inequality of the area of the parts, but maybe unable to devise an alternative technique to achieve their goal. Hence, there are two factorsinvolved in successfully partitioning regions into parts of equal area. The first is the understanding of33what constitutes parts of equal area in different types of regions. The second is the understanding oftechniques for constructing any number of parts of equal area in different types of regions. Both involveconceptions of geometric characteristics of different regions. But “number-theoretic” concepts arerelated primarily to the development of partitioning techniques. Students’ conceptions of whatconstitutes parts of equal area may not be reflected fully in the partitioning techniques available to them.Students’ Representations of Whole Number Multiplicationand Common FractionsThere are few studies which have explored, either directly or indirectly, the forms ofrepresentations students generate independently to illustrate or explain symbolic mathematicalexpressions. The studies that have been located deal with students’ representations of commonfraction concepts or operations, not with whole number multiplication. More common are studies in thecommon fractions context which report ways in which students interpret and use different forms ofrepresentations when the representation used was determined by someone else in a test or interviewsituation.Student-Generated Representations.Previous studies in which students were asked to represent common fraction concepts oroperations suggest that parts of regions, particularly parts of circles, is a pervasive image. Peck andJenoks (1981) report the results of interviews with 20 Grade 6 students in which the students wereasked to explain their ideas about common fraction concepts, relations, and operations with the use ofdiagrams. They also report general observations based on a large number of interviews of students inGrades 6, 7 and 9. The forms of representations constructed by students were invariably based onparts of a region, and the “large majority of students used pies (circles) as their only model for fractions”(p. 347). Clements and Lean (1988) report that, of the 59 students from Grades 4, 5 and 6 interviewedabout the meaning of unit fractions, all constructed representations based on regions and all but oneused either circles or squares. These results are similar to those found by Silver (1983) when34interviewing community college students. Fifteen of the 20 college students first represented acommon fraction with circles, and 10 of these students could not extend their thinking to other images.In addition, Silver found that the more restricted the student’s representations of common fractionswere, the less able the student was to comprehend numerical relationships between and operationswith common fractions.The students in Peck and Jencks’ study based common fractions representations on parts ofregions, but over half did not interpret their representation in terms of measures of area. For thesestudents, the number of parts determined by the denominator was the criterion used for comparison. Inthese cases, Peck and Jencks observed that many students had consistent but incorrect strategies forpartitioning the circle which resulted in unequal parts. They also did not believe that the parts ought tobe equal. A lack of adequate partitioning techniques alone did not account for the form of theirrepresentations. Clay and KoIb (1983) reported that some Grade 4 students, even when theyconstructed representations of common fractions with equal parts of a region, did not believe in theneed for equal parts. One student, when asked about why he drew equal parts after he agreed thatunequal parts were adequate in a representation, replied, ‘that’s the way my teacher does it, ... becauseit is easier or it looks better” (p. 245).In the context of comparing common fractions, Peck and Jencks (1981) report that, consideringthe interviews across all grades, fewer than 10 percent were able to provide an appropriate rationalebased upon a diagram. Among the 20 Grade 6 students interviewed, 13 were unable to do so. Somestudents had constructed representations of common fractions with equal parts, but when using suchrepresentations to compare common fractions only the number of “pieces left over” were compared.In these studies, two characteristics of the forms of representations students constructed arenotable. The most obvious characteristic is the exclusive use of regions, particularly circles by studentsfrom the intermediate grades up to college level. However, only in Silver’s study were students askedfor alternative forms of representations. It therefore is uncertain in the other cases whether parts ofregions is their only or simply their most common form of representation. The other notablecharacteristic is that measures of numerosity rather than measures of area appear to provide the basis for35many of these students’ representations, despite the fact that some students constructed equal partsof regions to explain the meaning of a common fraction. This would suggest that the “equal parts”requirement in representations with regions may be conceived by students to be unconnected to aconception of units as measures of area.Recognition memory and memory which requires the evocation of situations alreadyencountered but not present are distinguishable in terms of their cognitive complexity (Sinclair, 1971 b).The limitations on the forms of representations of common fractions generated by the students in theprevious studies do not imply that other forms of representation would not be recognized by thestudents as constituting physical embodiments of fractions. However, alternative forms ofrepresentations of common fractions may not be part of students’ repertoires of evokablerepresentations.Students’ Interpretations of Instructional Representations.At the elementary and lower secondary levels, the number line is a form of representation whichstudents have difficulty interpreting and using appropriately in a wide range of mathematical contextsfrom whole number addition to operations with integers (Behr, et al., 1983; Bright et al., 1988;Dufour-Janvier et aL, 1987; Ernest, 1985; Hart, 1981; Novillis-Larson, 1980, 1987; Payne, 1975, 1984;Sowder, 1976: Vergnaud, 1 983b). Nonetheless, it is a common form of representation for theinstruction of whole number multiplication and common fraction concepts and relations.Dufour-Janvier et al. (1987) describe some students’ naive conceptions of the number line. Inone case some primary students thought of the dots on the line to be stepping stones which arecounted in a manner similar to the counting of discrete objects. In another case, students interpretedcurved arrows marking the jumps of units along a number line in an addition question as boundariesmarking off sets of dots (see Figure 2.01). The dots which were counted for the addition question werethose which were beneath the curved arrow. The beginning point and end point of the arrows were notcounted in the representation of the addends.3601234567 8 9 10 112+3=Figure 2.01 Counting sets of dots beneath the arrow on a number line.Similar misinterpretations are reported by Sowder (1976). Whether such interpretations ofrepresentations of whole number operations on a number line persist with students in the intermediategrades is not known.In the common fraction context, number line representations have been reported to be moredifficult than representations based on the area of regions. Payne (1975), reporting on a series ofteaching studies conducted at the University of Michigan, summarized the findings that students fromGrades 1 to 5 had difficulty with number lines. Behr, et al. (1983) report that two to three times as manystudents made errors when representing proper fractions on a number line than with sets, regardless ofthe number of consistent or inconsistent cues embodied on the line or in the sets. Discrete sets wereonly as difficult as the number line when students were required to represent improper fractions.Novillis-Larson (1987) also found, from results of a test administered to Grade 5 and 6 students on theirability to associate proper fractions, improper fractions, and mixed numerals with representations basedon the number line, ruler, and regions, that the mean score for number line items was much lower thanfor regions or rulers. In general, these results suggest that the number line is the most difficult form ofrepresentation for students to interpret and use in a common fraction context.Representations of common fractions based on discrete sets have also been identified as moredifficult for students to interpret and use than area-based representations (Behr, et al., 1983). Similarly,Payne (1975), reported that, when teaching initial fraction concepts and multiplication of fractions, theuse of instructional representations based on sets resulted in lower achievement than the use ofrepresentations based on the area of regions. Partitioning a discrete set is easier than partitioningcontinuous quantities because it can be done through a partitive division process without having to plan37and anticipate the full partitioning process (Hiebert & Tonnessen, 1978). But Post, Wachsmuth, Lesh,& Behr (1985) argue that one distinction between the construction of area based and discreterepresentations of common fraction relationships is that the region predetermines the whole unit,whereas with discrete sets no whole unit is predetermined. To construct representations with discretesets to compare common fractions with different denominators, knowledge about commondenominators has to be used in the planning process. Hart (1987), observing 10- to 11-year olds’failures to derive unknown equivalent fractions with sets, concluded that the task seems “impossible,since one does not know how many bricks to select for the demonstration” (p. 7). Hart concluded,“Although we may use materials and diagrams to make the rule seem reasonable, if we expect childrento reconstruct these concrete models we must probably explicitly teach them a method for doing so” (p.7).Rules and conventions related to the structure and use of different forms of representationsdiffer, and appear to influence the facility with which students are able to use them to interpret andrepresent common fraction relations. Bright et al. (1988), in their analysis of the differences betweenrepresentations of common fractions on the number line and those based on sets or regions, illustratesome of the ways that different physical representations each have unique norms and conventions.First, the measurement constructs of each form of representation differ: length, numerosity and area.Second, the spatial characteristics of units differ in terms of their visual discreteness: with discrete setsall units are separate, with regions units representing the whole generally are separate, with numberlines none of the units are separate. With each form of representation different physical attributes aresignificant in representing the meaning of common fraction relationships.The nature of students’ interpretation and use of physical representations is recognized as acurrent problem in mathematics education. Even the “simple” matching of a pictorial representation witha symbolic expression has been found to be not a “simple” process. Among fraction and ratio taskswhich required students to match one mode of representation to another, it was found that matchingpictures to symbols was the most difficult task (Lesh, Landau, & Hamilton, 1983; Lesh, Behr & Post,1987). Lesh, Post, and Behr (1987), conclude that:38Not only do most fourth through eighth grade students have seriously deficientunderstandings in the context of “word problems” and “pencil and papercomputations,” many have equally deficient understanding about the models andlanguage(s) needed to represent (describe and illustrate) and manipulate these ideas.Furthermore, we have found that these “translation (dis)abilities” are significant factorsinfluencing both mathematical learning and problem solving performance.... (p. 36)This conclusion and the research cited previously suggest that the nature of students’interpretations of instructional representations cannot be taken for granted. Furthermore, there isreason to believe that as students attempt to construe meaning from instructional representations, theirown conceptions as to the significance of attributes in the representation are important to their thinkingprocess. In this regard, Dufour-Janvier et.al (1987) have asserted that:The imposition of representations which are too distant from the child results in the childreacting negatively or causes him difficulties. If one wants to use an externalrepresentation in teaching, he needs to take into consideration that it should be asclose as possible to the children’s internal representations. (p. 117-118)One premise of this assertion appears to be that teachers need to know about the ways in whichstudents might represent mathematical ideas in order to make instructional decisions about how torepresent mathematics for the students.A premise of this study is that the distance between children’s personal, intuitiverepresentations of mathematical relationships and forms of physical representations used in instructionhas implications for the learning process. It is widely held that students are not passive receptois of“knowledge” but are actively involved in their own construction of knowledge based upon thereconstructed meanings they draw from their present and past experiences (Sinclair, 1987; Vergnaud,1983b; Von Glasersteld, 1983). This study was designed to explore the nature of students’representations of units and unit relationships in the context of whole number multiplication andcommon fractions, as well as length and area measurement, in order to contribute further tounderstanding of the ways in which students might represent their mathematical ideas.39CHAPTER 3PLAN AND IMPLEMENTATION OF THE STUDYThe study was designed to explore the characteristics of students’ representations ofmultiplicative relationships of units, and the extent to which properties of length or area measurementwere used as a framework for representations within their repertoires. The representations constructedby students were translations of numerical expressions of whole number multiplication and commonfraction concepts and relations. The students constructed these representations during a series of fourclinical interviews.The interview tasks included whole number multiplication and common fractions concepts andcomparisons. Because both are multiplicative in structure (Vergnaud, 1 983a), they were selected as thedomain of study in order to explore the students’ representations of relationships between differentunits. They were also selected because it was likely that students would have encounteredmeasurement-based representations in one or both of these mathematical contexts, and becausemeasurement-based representations play a critical role in the instruction of the part-whole meaning ofcommon fractions.Students’ representations of units of measurement were also investigated in order to interpretthe extent to which students incorporated properties of linear or area measurement in light of theirconceptions of units of measurement. Their representations of units of measurement were investigatedthrough their responses to a series of measurement tasks in a written test and two interviews.The plan of the study, along with a description of the interview tasks and procedures designedto investigate students’ representations of whole number multiplication, common fractions, and units oflength and area are described in this chapter. The analytical categories used to characterize students’responses during the interviews are defined in Chapters 4 and 5, immediately preceding thepresentation and discussion of the results of the analyses of the data.40Two pilot studies were conducted before the plan of the study was finalized. These weredesigned to evaluate or redefine elements of the plan such as the appropriateness of selecting subjectsof particular grade and achievement levels, the interview tasks and procedures, and the means forinvestigating students’ representations of units of measurement. The plan of the study described in thebalance of this chapter was derived from evaluations of the pilot studies.Subjects in the StudyStudents were selected to represent a range of achievement in order to maximize the likelihoodof variations in their responses to the interview tasks. Teacher assessments were used to selectstudents to represent a range of achievement. Criteria used by each teacher varied, but this was notconsidered to be a limitation of the study since questions about relationships between levels ofachievement and students’ conceptions of representations were not addressed. A range ofachievement was sought only to increase the likelihood that students would generate a range ofresponses to the interview tasks.Grades 5 and 7 were selected to represent a variation in students’ formal mathematicalexperiences. According to the prescribed mathematics curriculum for all school districts in the Provinceof British Columbia, students in Grade 5 should have received the first systematic unit of instruction onthe comparison and equivalence of common fractions in Grade 4. For the Grade 7 students, systematicinstruction should have occurred with common factions during each of the previous three years.The prescribed curriculum was followed in both schools that participated in the pilot studies,which were in two different school districts. The students who participated in the final study were fromschools in a third school district. Teachers of the Grade 5 students in the final study had decidedinformally to delay instruction on common fractions. This was not known prior to the commencement ofthe study. The assumptions about students’ previous formal instruction with common fraction conceptsand relations at different grade levels, therefore, did not apply to the subjects in the study. The studentsin Grade 5 had received no systematic unit of instruction on common fraction concepts or the41comparison of common fractions. The students in Grade 7 had received at least one systematic unit ofinstruction in Grade 6.Fifteen students from two schools participated in the study: six in Grade 5 and nine in Grade 7.These students represented a range of achievement in each grade. The original plan to have an equalnumber of students per grade was not met because of technical failures with the video equipment.Plan of the StudyThe plan of the study is presented in four parts. The first part is an overview of the study. This isfollowed by more detailed descriptions of (1) the generative interviews, (2) the interpretive interviews,and (3) the measurement interviews and measurement concepts test.OverviewIn this section only the general characteristics of the different interviews and a general outline ofthe plan of the study are described. A more detailed description of interview procedures and tasksfollows in subsequent sections of the chapter.There were two areas of investigation in this study. The nature of students’ repertoires ofrepresentations of multiplicative relationships was the primary area of investigation. The secondary areaof investigation was the nature of students’ conceptions of units of linear and area measurement. Therepertoires of students’ representations were investigated through a series of four different interviews,the characteristics of which are summarized in Table 3.01. Students’ conceptions of units of linear andarea measurement were investigated primarily through two interviews. One dealt with linearmeasurement tasks, and the other dealt with area measurement tasks. A measurement concepts testalso was administered to all students prior to the interviews. Hence, there were six different interviewsplanned for the study. These interviews are summarized in this section in two parts. The first summarydeals with the four interviews designed to investigate students’ repertoires of representations in thewhole number multiplication and common fraction contexts. The second summary deals with the twointerviews designed to investigate students’ conceptions of linear and area units.42Repertoires of RepresentationsThere were four facets of students’ repertoires of representations which four different interviewswere designed to investigate (see Table 3.01). The first facet was the nature of the spontaneousrepertoires students generated, that is the representations that students are likely to generate whennothing in the material setting cues them to consider a particular form of representation. The secondTable 3.01Characteristics of the Interviews Designed to Investigate Students’ Repertoires of RepresentationsGenerative Interviews Interpretive InterviewsCharacteristics Uncued Cued Linear AreaOrder 1st 2nd 3rd 4thMaterial Paper & Variety of Distorted DistortedSetting pencil materials linear-based area-baseddiagrams diagramsStudent Generate Generate Evaluate EvaluateAction variety of variety of diagrams & diagrams &represent- represent- represent representations for ations for tasks taskseach task each taskFacets Spontaneous, Extension of Attention to Attention toExplored primary repertoires properties of properties ofrepertoires length as area ascritical criticalfeatures? features?facet was the nature of the repertoires of representations students generated when the material settingprovided cues which might lead them to consider the construction of representations which they did notgenerate spontaneously. The third and fourth facets were the nature of students’ attention to propertiesof linear and area measurement as critical features of representations.43In all four interviews students constructed representations of whole number multiplication andcommon fraction concepts and relations. But the two generative interviews are distinguished from thetwo interpretive interviews by particular differences in the demands placed on the students. The first twointerviews were called generative interviews because the students were required primarily to generate asmany different representations as they could for any particular task. The last two interviews were calledinterpretive interviews because students were required to interpret and evaluate a series of specificbeginning diagrams (see Figures 3.01 and 3.02 later in this chapter) as to their appropriateness as aframework for constructing a representation to explain the mathematical ideas. The students imposedtheir own interpretation on these beginning diagrams and identified which physical features were criticalto their evaluation.The exhaustive generation of different representations required in the generative interviewswas not a part of the procedures for the interpretive interviews. The interpretive interviews weredesigned to investigate students’ attention to properties of linear or area measurement. As such,representations were constructed by students as part of their explanations of why and how particularbeginning diagrams were or were not appropriate for explaining the mathematics. On the other hand,the direction imposed by the diagrams in the interpretive interviews was not a part of the generativeinterview procedures. The only direction involved in the generative interviews was that studentsconstruct representations, the nature of which was left up to them.The order of the four interviews for investigating students’ repertoires of representations wasestablished to minimize the possibility that responses in some interviews would be confounded bystudents’ experiences with materials in other interviews. Hence, the generative interviews, which werenon-directive or less directive with regard to students’ consideration of length or area-basedrepresentations, preceded the interpretive interviews. The uncued generative interview preceded thecued generative interview in order to elicit forms of representations students would be more likely toconsider spontaneously without the benefit of cues.44Conceptions of Linear and Area Measurement.There were two separate interviews: one on the measurement of length and another on themeasurement of area. The length and area measurement interviews were conducted directly after eachof the linear and area interpretive interviews, respectively. The measurement interviews were placedafter rather than before each interpretive interview so that the tasks of the measurement interviews didnot sensitize the students to consider measurement attributes during the interpretive interviews.Data CollectionThe students were interviewed individually in a private room within their school. Each interviewwas recorded on videotape. The order of the data collection was as follows:1 2. 3. 4. 5.Measurement Uncued Cued a Linear a AreaConcepts Generative Generative Interpretive InterpretiveTest - Interview Interview Interview Interviewb Linear b AreaMeasurement MeasurementInterview InterviewEach of the four interview periods took about two weeks to complete with all students. A breakof approximately five weeks occurred in the schedule during October-November. This break occurreddirectly after the cued generative interviews. For each student, at least two weeks elapsed betweeninterview periods.Generative InterviewsIn the uncued generative interview the students had no materials to suggest different types ofrepresentations which could be used to explain the mathematical tasks. They were presented with amathematical task and asked to explain its meaning through the use of diagrams or rough pictures.During the cued generative interview, a collection of materials was available from which studentsindependently selected materials with which to construct representations of the mathematical tasks. It45was anticipated that the presence of the materials might cue some students to construct different formsof representations from those constructed in the uncued generative interview. During both interviews,the student was asked to produce a “completely different” representation upon completion of a previousrepresentation. This request deliberately left undefined the meaning of “completely different” in orderto allow the students to determine their own meaning of differences between representations.Students were asked to adopt the role of teacher because it made more explicit the request forthem to explain the meaning of the task and not simply to derive an answer. The teacher’s role alsoprovided a rationale for the repeated requests for completely different diagrams or for completelydifferent ways of using the materials. It was explained that teachers often use a variety of diagrams orrough pictures to help to make the mathematical ideas clear to the students and that they, therefore,would draw diagrams or rough pictures to explain the mathematics clearly. At times, the interviewerplayed the role of a confused student to encourage the student to clarify his or her meaning or toconstruct different explanatory diagrams The teaching role has been used to motivate students toexplain their mathematical meanings by other investigators such as Vergnaud (1 983a), and Hiebert andWearne (1986).The request for different diagrams to explain a task was repeated until one of two conditions wasobtained: (a) the student declared that he or she could not think of any more “completely different”diagrams, or (b) the student generated a series of representations which were structurally the same butperceptually different. An example of the latter case would be a situation in which a student generated aseries of discrete representations with the differences between representations being only changes inthe objects within the sets, for example only changing from triangles to stars.In addition, in the cued generative interview the student was asked to indicate which of theunused materials could not be used to explain the task and why. The student also was asked to explainhow remaining materials not classified by him or her as inappropriate would be used. In the latter case thestudent was asked whether the materials would be used to construct representations which weredifferent or similar to those already generated. The purpose of these questions was to explore the limitsof the forms of representations within a students’ repertoire of representations.46Generative Interview TasksThree sets of mathematical tasks were devised for the generative interviews: a set of wholenumber multiplication tasks, a set of common fraction concept tasks, and a set of comparison of commonfraction tasks. The results of the second pilot study indicated that a student had to be sufficiently familiarwith the mathematics for any response to be forthcoming. In order to anticipate differences in students’mathematical experience, the tasks within each set varied in terms of the complexity of unit relationships.It was assumed that, by Grade 7, students would be sufficiently familiar with comparisons of commonfractions to offer an explanation, but that not all Grade 5 students might be as familiar with these tasks.Hence, the set of common fraction concept tasks was designed for the Grade 5 students as a lesscomplex set of tasks.Tasks from these sets were presented individually to a student, and the student was asked toexplain the meaning of an individual task by constructing representations with diagrams or concretematerials. Procedures for selecting tasks from within a set are described in the next section.Muftiplication Tasks (Grade 5 and Grade 7)1. 4X5=_2. 18X3=_3. 2X3X4=_4. 3X(4+5)=(3X4)+(3X5)Concept of common fractions (Grade S only)The planned set of fraction concept tasks were:1. 3/42. 4/53. 11/64. 7/65. 17/547Unit fractions were not planned as part of this set but were used with some students because of theirinability to respond to any of the tasks above.Comparison of common fractions (Grade 5 and Grade 7)1. 113vs2162. 2/4vs4/83. 2/3vs3/44. 5/9vs2135. 5/8vs7/12Procedures for Selecting Tasks.It was not intended that all tasks would be used with each student in both generative interviews.A selection procedure was devised to determine which tasks within a set would be presented to astudent. The objectives of the procedure were twofold:1. to select tasks which were the most complex within a set to which a student was able to respond,2. to limit the number of tasks from each set presented to a student in order to keep the length ofthe interview within reasonable bounds.It was not possible to anticipate the ability of a student to respond to more or less complex tasks.For example, in the second pilot study one “low achieving” student was able to respond more fully to thetasks than some “high achieving” students. Therefore the interviewer made decisions regarding whichtasks to present to a student on the basis of the extent to which the student was able to respond toprevious tasks.The only time that this flexible selection procedure was not followed was when the first set oftasks, whole number multiplication, were used in the uncued generative interview. Since this was thefirst interaction with a student, the student’s reaction to the interview situation or individual interviewtasks could not be anticipated. A simple to complex order was used to provide general information aboutstudent’s ability to respond to more or less complex tasks. Thereafter, the tasks for each student wereselected flexibly to maximize the chance of presenting the most complex task to which a student could48respond and minimize the number of tasks required to determine this threshold. In the cued generativeinterviews the first task of each set was the most complex task within that set to which the student wasable to respond in the uncued generative interview. In both generative interviews, when the flexibleselection procedure was used, second tasks may have been more complex or less complex, dependingon the responses of the student to the first task. As a result, the interview tasks and their order ofpresentation varied among students because of the range of achievement represented by the studentsin the study.Materials Used in the Cued Generative InterviewThe materials available to the students during the cued generative interview were as follows:Multilink blocks Coloured DisksYellow wooden hexagons Base ten blocksCentimetre dotted paper Centimetre squared paperCircular filter paper Inch squared paperColoured string loops Coloured felt pensRuler with solid coloured Blank line on a sheet of paperunit markingsThese materials were either common in elementary classrooms or were used as illustrations in aprescribed textbook. Some also had a potential for multiple use. For example, the multilink blocks couldbe used to construct discrete representations in random, linear or rectangular configurations, orcontiguous representations in linear, rectangular or cubic configurations. Particular care was taken toselect a variety of materials which might cue students to construct measurement-based representations.Interpretive InterviewsThe linear interpretive interview was designed to investigate the extent to which students attendto properties of linear measurement, and the area interpretive interview was designed to investigate theextent to which students attend to properties of area measurement as critical features in theirrepresentations. In both interviews students’ attention to measurement properties was investigated by49exploring students’ tolerance for features which did not conform to properties of linear or areameasurement, for example, their tolerance for unequal line segments marked as units along a line. Aswell, the interviews explored other geometric characteristics of linear-based and area-basedrepresentations that may or may not have been considered as critical features by the students. To thisend, geometric and measurement properties associated with either linear or area-based representationswere identified and systematically distorted within a series of diagrams (see Figures 3.01 and 3.02). Forexample, with the number line, properties such as equality of units, horizontality, straightness, orcontinuity were subject to distortion. With the rectangular representations, properties such as equality ofunits and squareness of units were subject to distortion.The Interpretive Interview TasksAll of the mathematical tasks in the interpretive interviews were parallel or identical to tasks usedin the generative interviews. In order to accommodate the different achievement levels of the students,each mathematical context contained two tasks which differed in complexity. For example, multiplicationof whole numbers was represented by the following tasks:a) 2X3X4=_b) 7 X 3In general only one of a pair of mathematical tasks was used with a student, and the selection of a task fora particular student depended on their responses in previous interviews.As can be seen in Figures 3.01 and 3.02, five diagrams were designed for each of the linear andarea interpretive interview. Each of these diagrams was associated with each mathematical task. Forexample, when 2 X 3 X 4 = was a task used with a student, then the student was asked to evaluateeach of the linear or area diagrams with respect to its appropriateness for constructing a representation toexplain 2X3X4=_.Interpretive Interview ProceduresThe introductory phase of these interviews was the same as that of the generative interviewsexcept that students were told that they would be shown some different diagrams and would be asked to50i.0I i I I I I I I I I I03.4.6.Figure 3.01 Beginning diagrams used in the linear interpretive interviews (Reduced 1048 percent).2a)L()0C.)U)ccI2-a)CUCUci)--U)CU0ECUU)toc’jI0II-I.-——“5t152decide if each one would be a “good beginning diagram” to use for explaining the mathematics. Anemphasis was placed upon the notion of a “beginning” diagram so that it is clear that they were free toadd to or alter the diagram as they saw fit.When a student responded positively to the question as to whether the diagram was a “goodbeginning diagram”, he or she was requested to show how he or she would use the diagram to explainthe mathematics. When a negative response occurred the student was asked for the reasons why thebeginning diagram would not be good for constructing a representation of the mathematics. Thestudent also was asked to explain what features should be changed in order for the beginning diagram tobe used. Furthermore, because the reasons for rejecting a diagram could be based upon eithermathematical or aesthetic criteria, the students were asked directly whether such changes weremathematically necessary or only preferred for other reasons.Measurement Interviews and TestThe Measurement Concepts Test was designed before the second pilot study. Test items fromBabcock’s (1978) “Test of Basal Measurement Concepts” were used as the initial framework for theconstruction of the test. Some test items, such as questions related to the use of the ruler, were added,while others were modified. (See Appendix B.) From the results of the second pilot study, it was foundthat students’ reasoning about the nature of and relationships between units of measurement could notbe inferred from the results of a paper and pencil test. It was decided that students’ representations ofunits of linear and area measurement would be investigated primarily through two individualizedinterviews: one for linear measurement and the other for area measurement.Items from the Measurement Concepts Test were selected as tasks for the linear and areameasurement interviews. Students’ responses to these interview tasks constituted the primary sourceof data regarding students’ representations of units of length and area measurement. However, thestudents’ test responses to a few of the tasks which were amenable to analysis with regard to inferredstrategies also were used as a secondary source of data. The test and interview items used toinvestigate students’ representations of units of linear measurement are presented in Figures 3.03 and533.04. The test and interview items used to investigate students’ representations of units of areameasurement are presented in Figures 3.05 to 3.07.The linear measurement tasks were designed to investigate students’ representations and useof units from a number of points of view. All of the tasks required students to construct or identify units oflength to resolve the tasks, but the problem contexts differed. With the tasks in Figure 3.03, directreference was made to the use of units and number, but relationships between units differed. Thesetasks were designed to investigate two different aspect of students’ thinking about units of length: (1)the consistency with which they would identify or construct units of length as line segment, and (2) theextent to which they would reason appropriately about relationships between different sized units. Withthe tasks in Figure 3.04, no direct reference was made to the use of units or number. These tasks weredesigned to investigate whether students would identify or construct units of length and use appropriatenumerical reasoning to resolve the comparisons.The irregular path tasks are modifications of items in Babcocks (1978) test. Babcocks test itemswere derived from interview tasks used by Bailey (1974) to investigate the extent to which primary schoolchildren coordinated length and numerical cues, and used compensatory reasoning about differences inthe number and sizes of units when comparing lengths.In Bailey’s study the differences in the lengths of line segments in the paths were peiveptuallyless obvious than in the tasks in the current study. Also, in Bailey’s study the children had movable unitswhich they could use to evaluate the equality of the line segments in the paths. In this study nomeasuring device, apart from fingers and pencils, was available to students. During the interviews in thesecond pilot study students were able to perceive the differences in lengths of line segments and usecompensatory reasoning about differences in the number and sizes of units when comparing thelengths of the paths. It therefore was expected that students in the final study would be able to perceivethe differences in the lengths of line segments within the paths, and make comparative judgementsabout the overall length of the paths without the assistance of measuring devices.54A. Ruler task(Interview & Test Task)This ancient ruler ‘.asures length is “TLUGS. On. “TLUG is he sane astwo centixacres. Draw a in. ahoy. the ruier chat is 6 cencinecr.s long.3 4LFuB. Aggregate unit task(Interview & Test Task)The lix. b.ia, is ‘ onits long. Draw a line that is L2 mtcs long.C. Partitioning tasks1. (Test Task)Thi path is six units long, a) .rk the six units on the path.b) Drv anor..r path 5 units long.2. (Interview & Test Task)This path is 5 unitS lone. a) iark the 5 its on the path.b) Draw another path 3 units long.Figure 3.03 Linear measurement tasks with explicit reference to units and number. (Reduced by 35percent)55D. Irregular path task (Interview only)V ‘J V AHT ‘A’ path ii 1øn.r —HT ‘3’ path is 1cn.r —Th.y sri the sea. laugthE. Irregular path task (Interview only)At ‘A’ path is 1.oL*.r —IT‘‘path La Lanssr —They sri the Isaith____IFigure 3.04 Linear measurement tasks without explicit references to units and number. (Reduced by 35percent)The area measurement tasks in Figures 3.05 to 3.07 were designed to investigate severalaspects of students’ representations and use of units of area. With the partitioning tasks in Figure 3.05,the students were asked to construct six equal parts in order to investigate the extent to which studentsrelied on a successive halving algorithm, an algorithm which would not result in 6 equal parts. Differentgeometric regions were included in order to investigate how the geometry of the regions might influencethe way in which students approached the partitioning problem.Both the cake and tile tasks in Figures 3.06 and 3.07 were designed to investigate the extent towhich students identified appropriate units and used numerical reasoning with different units to resolvethe comparisons. In particular, the cake tasks in Figure 3.06 were designed to investigate the extent to56which students considered parts of each region to be fractional units of area and used part-wholerelationships to evaluate the comparison of the fractional units. The tile tasks in Figure 3.07 weredesigned to investigate the extent to which students attended to differences in the space-filling qualityof units or simply counted units unequal in area as equivalent.(Interview & Test Tasks)Figure 3.05 Partitioning tasks: divide each figure into 6 equal parts. (Reduced by 35 percent)571. 4.A ‘I 1. A D41 ‘12. (Interview Task) 5. (Interview Task)A__A_ I_3. (Interview Task) 6. (Interview Task)A_ __Ar_____Figure 3.06 Cake tasks: compare the sizes of each shaded piece of cake. (Reduced by 35 percent)1. (Test task) 2. (Test task)58I I i1LII3. (Test Task)AHH4. (Test Task)ILHHri a I F II I5. (Interview & Test Task) 6. (Interview & Test Task)Aa-rt I I I II I I I iIAIF IEL[LF I IaAl Ia——I I I—— I •••r••••••i iI II I I I III I I II II I IIIFigure 3.07 Tile tasks: compare the amount of space in each playroom. (Reduced by 40 percent)59Data AnalysisThe principal sources of data for this study were transcripts of the six different interviews: theuncued generative interview, the cued generative interview, the linear interpretive interview, the areainterpretive interview, the linear measurement interview and the area measurement interview. Theseinterviews were designed to explore students’ representations of units and unit relationships in fourmathematical contexts: whole number multiplication, common fractions, linear measurement, and areameasurement at the Grade 5 and Grade 7 levels.Categories used to analyse the form and function of students’ representations were not alldetermined before the implementation of the study. Some categories were suggested from theliterature or became apparent during the pilot studies, but others were identified from the data of the finalstudy. The identification and definition of categories with regard to students’ conceptions of linear andarea measurement followed a similar progression over time.The final sets of categories are defined and illustrated at the beginning of each results section.Categories used to characterize students’ representations of units in the whole number multiplicationand common fractions context are defined and illustrated at the beginning of Chapter 4. In the case oflinear and area measurement, the categories are defined and illustrated at the beginning of theirrespective sections of Chapter 5.60CHAPTER 4STUDENTS’ REPRESENTATIONS OF UNITS INMULTIPLICATIVE RELATIONSHIPSThe characteristics of the repertoires of representations students constructed to explain wholenumber multiplication, common fractions, and comparisons of common fractions are explored in thischapter. The analysis of the characteristics of students’ repertoires of representations in each of thesemathematical contexts (whole number multiplication and common fractions) was guided by the followingsubset of questions:1. What are the characteristics of students’ primaly repertoires of representations?2. To what extent are the characteristics of representations in students’ repertoires influenced bythe material settings within which representations are constructed?3. To what extent do students’ repertoires of representations include forms of representationsbased on attributes of length or area?4. To what extent do students’ representations have the function of representing relationshipsbetween different units?Relationship of research questions to interview data. The characterizations of students’representations of whole number multiplication and common fractions were based on an analysis of therepresentations they constructed during four different interviews. These were the uncued generative,cued generative, linear interpretive, and area interpretive interviews described in Chapter 3. The fourinterviews focused, in different ways, on particular questions related to the characteristics of students’repertoires of representations of whole number multiplication and common fractions.1. The uncued generative interview was designed to elicit the representations which the studentswere most likely to construct spontaneously (see Question 1). A primary repertoire of representations ofmultiplication or common fractions refers to the variety of forms of representations that a person evokesspontaneously, in the absence of external cues or prompts.612. The cued generative and both interpretive interviews, in which materials or different beginningdiagrams were presented to the students, were designed to explore the influence of changes in thematerial setting on the characteristics of students’ representations (see Question 2).3. The linear and area interpretive interviews were designed to focus directly on the question ofwhether attributes of length or area are used by students as a basis for their construction ofrepresentations (see Question 3). In addition, characteristics of students’ representations constructedin the generative interviews relate to Question 3.4. Question 4, which refers to students’ representation of relationships between different units, isaddressed through the analysis of all interview data.Collectively, the different forms of representations constructed by students during these interviews toexplain whole number multiplication, or common fraction concepts and relationships are considered torepresent their general repertoires of representations of these multiplicative relationships.Plan of the chapter. This chapter is comprised of four major sections. First, the analyticalcategories used to classify students’ representations are defined and illustrated. Second, the analysesof the characteristics of students’ repertoires of representations of whole number multiplication arepresented and discussed. Third, the analyses of the characteristics of students’ repertoires ofrepresentations of common fraction relationships are presented and discussed. The last sectionconsists of a discussion of common patterns and related themes in students’ representations of units inboth mathematical contexts.Analytical Categories to Classify RepresentationsTwo general elements are considered to constitute a representation: one is its form, and theother is its function. These elements, the form and the function of the representations, are used asmajor constructs to characterize students’ representations.The form of a mathematical representation refers to general physical characteristics whichdistinguish it from other representations of the same idea, just as a comic strip and an animated cartoon,which have different characteristics of form, may be used to tell the same story. For example, a62representation of multiplication constructed with collections of objects differs in its physicalcharacteristics from one constructed on the number line. The spatial relationships of the units withinthese two forms of representations and the formalities governing their construction are based upondifferent conceptual structures: measurement of numerosity in the first instance, and measurement oflength in the second. However, both forms of representation may be used to communicate the samemathematical idea.The function of a representation refers to the mathematical ideas communicated through therepresentation. Just as two comic strips, which share characteristics in their form, may have the functionof communicating different meanings of an event, so also two representations which sharecharacteristics in their form may have the function of communicating different mathematical meanings interms of relationships among units. For example, Diagram A may have the function of representing 2groups of 3 units, 3 groups of 2 units or simply 6 units. Each may represent an interpretation of 2 X 3.Diagram B is also similar in form to Diagram A, but may have the function of representing an operation ona number of single units such as 6- 2, or a relationship between two different units such as 2/6 of therectangle. Just as the function of the comic strip reflects the artist’s interpretation of essentialrelationships in the event, the function of the representation reflects the student’s interpretation ofessential relationships in a mathematical situation.Diagram A Diagram BAnalytical categories used to classify students’ representations are discussed in two parts: (1)categories used to classify the form of students’ representations, and (2) categories used to classify thefunction of students’ representations. The distinction between the form and function ofrepresentations has not altered since the inception of the study, but the development of categories bywhich to analyze the interview data evolved over time. Some categories were identified during the63planning and piloting stage and refined further during the transcription of the interview protocols, whileothers were defined during the initial analysis of the interview data. These analytical categories aredefined in the next two sections. They are illustrated with selections from students’ interview protocols.Analytical Categories of Forms of RepresentationsThere are two levels of categories related to the form of a representation. The first level is thespatial framework of the representation, and the second level is the within the spatial framework(see Figure 4.01). By spatial framework of a representation is meant the general structure within whichthe units are represented, whether geometric regions, lines, or sets. Within a spatialSpatialFrameworkUnitsLinesEqual UnequalIPartial TotalUndefinedFigure 4.01 Categories for classifying the forms of representations.framework, units may be represented in different forms. For example, a line may be used as spatialframeworks to represent discrete or contiguous units, or may be used to represent quantities whichwere not defined explicitly with units. Units within the spatial framework of sets are necessarily discrete.Representations ofMultiplicative RelationsSetsILinear Rectangular IrregularRegionsDiscrete ContiguousI 164The explanations of the categories which follow are organized in three sections by types ofunits: discrete, contiguous, and undefined. Within each section, examples of the variations in spatialframeworks are presented. Also, in the section on contiguous units, the other characteristicsassociated with this form of units are explained (i.e., equal, unequal, partial, and total).Discrete UnitsThe over-riding characteristic of representations with discrete units is the complete separationof all units within the representation. This form of units was organized in every spatial frameworkobserved in this study. These different spatial frameworks are explained and illustrated as theypertained to discrete units.Discrete units with sets as spatial frameworks: linear, rectangular. or irregular. In a linearconfiguration, the discrete units were in an ordered sequence as shown in Examples 1 and 2. In arectangular configuration, the discrete, primary units or aggregate units were arranged to create orapproximate a rectangular array as shown in Example 3. “Irregular” applied to configurations of the unitsin sets which did not follow an obvious pattern.Example 1: Discrete units in sets linear configuration.Task: Compare 1/3 and 1/2“They’d be the same.” (i.e., 1/3 = 1/2)“Well we take 4 for this one [drew 4 for 1/3] and 3 for this one [drew 3 for 1/2]”[1/2] [1/3]“and I take 2 of these [from 1/2] and 3 of these [from 1/3].”[The value of each set is derived by adding numerator and denominator, hence 1/2 = set of 3objects. The equality of 1/2 and 1/3 is derived by comparing what is left in each set when thevalues of the denominators are subtracted from the set.]James (Giade 5)65Example 2: Discrete units in sets with a linear configuration.Task: 4X5=“Well, maybe you’d go like this. Maybe going this way.”Edwin (Grade 5)Example 3: Discrete units in sets with a rectangular configuration.The student in this example organized the four sets into a 2 X 2 array, and approximated arectangular form to represent five in each set.Task: 4X5=x x x xx xx x x xx x x xx xx x x x“Okay, take 4 groups with 5 in it, and then you would count all of them, and when you’vefinished counting, you’d see what’s the number, and then it should be the answer.”Dalia (Grade 7)Discrete units with a line or regions as spatial frameworks. When a line was a spatial framework,the discrete units invariably were marks along the line as shown in Example 4. The discrete units inregions were lines which generally gave the appearance of a partitioned region. However, the linesrather than the spaces were counted by the student as units as shown in Example 5.Example 4: Discrete units on a line.Task: 3X(4+5)=‘Four lines, then a little ways apart 5 lines, then make two more groups the same.”66IIIj 11111 liii 11111 liii 1—Ilil >Pete (Grade 7)Example 5: Discrete units in regions.Task: Compare 1/3 and 2/6“There is the two triangles right there .... They are the same, and you draw a line between bothof them.”“One, ... one, two, three and one,” [drew 3 horizontal lines, then 1 horizontal line in the firsttriangle to represent 113]; “two ... one, two, three, four, five, six” [drew 2 horizontal lines, then 6horizontal lines in the second triangle to represent 2/6]14\“Now, this one is just the same as this one here [i.e the triangles are the same] but this has oneinstead of two [comparing the number of horizontal lines in the right side of each triangle] andthis has three instead of six [comparing the number of horizontal lines in the left side of eachtriangle]. That [i.e., 1/3] equals 4 halves and that [i.e., 2/6] equals 8 halves.”Marlene (Grade 5)67Contiguous UnitsThe over-riding characteristic of representations with contiguous units is that some. it not all.units within the representation “touch” each other or share a common boundary. These units wereorganized wiLh a line as a spatial framework (see Example 1) or with a region as a spatial framework (seeExample 2). Since not all units in a representation were necessarily contiguous, the extent of contiguitywas classified as partial (see Example 2) or total (see Examples 3 and 4). In a partially contiguousrepresentation aggregate units are discrete.Representations of contiguous units were also classified with regard to the equality of the units.The units in the representations were classified as equal in length or area either when units appeared tobe approximately equal, or when a student had difficulty constructing equal units but stated that theunits should be equal.ExamDle 1: Contiguous units along a line.Task: 2X3X4 =[reduced photocopy of her number line]“To have one group of six you have to go up like that.” [drew a jump from zero to the end of thesixth space] That would be one group of six, another group, another group, and anothergroup - and then you leave it and when you are all finished there are 24 spaces.”Tammy (Grade?)68Example 2: Partially contiguous units in regions.Task: 4X5=F? I H[liii[liiiLilt]F! [I]“There are five groups of one, two, three, four blocks.”Coran (Grade 7)Example 3: Totally contiguous units in a region.Task: 2X3X4=H H“Here, these things are all six and 1, 2, 3, 4.” [counting the rows] “There are 4 here and fourtimes six ... twenty-four.”“Just using squares and all that- well it’s got sets right up to here and then it seems like adding.”[indicating the rows in his representation]“Or you can do a different way of adding. Like 2 times 4, and then add them together makes Btimes 3 is 16, 24.” [indicating columns and pairs of columns in his representation]Edwin (Grade 5)69Example 4: Totally contiguous units in a region.Task: Compare 2/4 & 4/8Step 1.“This is a whole and there’s 1, 2, 3, 4 in that...”“and that says 8” [i.e., 418] “well, that one says 4, so you need four-TMStep 2.“and that one says 8, so you make like that and that.” [partitions the fourths with two diagonalsto make eighths] “and if it says 4, well 2, well you’d have - well say these lines aren’t there - 2,you’ve got these two big pieces” [imagine removal of the additional diagonals think of 2/4] -“and 4, you’ve got 1, 2, 3, 4.” [indicates four of the eighths within the 214] “It’s just the same.”Tamrrry (Grade 7)Units are UndefinedUnits were classified as undefined when a student represented a value as a global quantity.The student did not construct units to represent a relationship between the numbers in the task. In thefollowing examples, the student’s general notion about the relative size of common fractions wasrepresented as a global quantity of a line (see Example 1) or within a region (see Example 2). Neitherthe values of numerators or denominators, nor the relationship between numerators and denominatorsare represented with units.70Example 1: Units are undefined in a line.Task: Explain meaning of 3/4Step 1.halfStep 2.Ihalf“That’s three quarters right there.” [between right-hand end and the first mark to the left] “andthat’s a half.” [between the first mark to the left and the middle mark]“Oh! That’s wrong!”13“This has got to be on this side ....“ [Points to a position closer to but still to the right of the halfmark] “and that would be one-quarter.” [Indicates the space previous called three-quarters.Wrote 1/3.]Example 2: Units are undefined in a region.Task: Compare 1/3 and 2/6This would be smaller (2/6) sO ... one third and two sixths.Step 3.halfBiock (Grade 5)<— 1/3 —><-2/6->Brock (Grade 5)71Analytical Categories of Functions of RepresentationsThe categories related to the function of students’ representations classify the extent to whichrelationships between different units were represented (see Figure 4.02).Representations ofMultiplicative RelationshipsUnit IRelations Mono- Bi- TnRelationalFigure 4.02 Categories for classifying the function of representationsMono-relational Representation of Units“Mono-relational” is used in this study to describe situations which involve quantities measuredwith only one unit. For example, when comparing, adding, or subtracting whole numbers all quantitiesare represented by only one unit. Within any particular case of these operations, all of the numeralsrepresent a measure of the same unit. Hence, for 4 + 5 = 9 to be true, the addends and the sum aremeasures of the same unit.By definition, units in a multiplicative relationship are not all equivalent. However, somestudents constructed mono-relational representations of the units to explain the tasks. All units wererepresented equivalently. Such representations were classified as mono-relational. In the multiplicationcontext, mono-relational representations generally were of the form presented in Example 1. All factorswere represented by sets of equivalent units. In Example 2, the final comparison of the commonfractions was based on measures of the numerosity of the sets. Each object in the sets was treated asequivalent units.72Example 1: Mono-relational representation of units.Task: 4X5=“I already know what to do. You draw .... Hmm. And you count them out and it equals that.”0 0 0 0 x 0 0 0 0 o = 20“Twenty.”“Well you go four times five and it equals twenty.” [Points to the set of 4 circles and then the setof 5 circles]Connie (Gide 7)Example 2: Mono-relational representation of units.Task: Compare 2/3 and 5/9“Well, I’d draw holes.... holes in cheese and mouses live, and mouses live inside ... only fourmouses came and there are five cheese left.”0“Two thirds. Well, these are all donuts ... these are the two thirds that have holes and this onedoesn’t have a hole. See there’s 9 cheeses and here’s the 3 donuts and these two are theones that have holes [2 donuts with holes], and these ones the five things that are left alone[cheeses without mice] .... that would be 5 and 2; 9 and 3.” [comparing numerators and thendenominators]Edwin (Giade 5)Bi-relational Representations of UnitsWhole number multiplication as well as rational number concepts minimally involve a relationshipbetween two different units. These units are related by a simple ratio; one unit is either an aggregate ora part of the other. Such a relationship between two units has been termed bi-relational.Although comparisons of common fractions with unlike denominators are minimally tn-relational(whole unit, 1St fractional unit, and 2nd fractional unit) bi-relational representations which omitted the73relationship of one of the units were constructed as shown in Example 1. In this example the fractionalunits of ninths and thirds are inter-related, but they are not related to a consistent representation of awhole unit.Other students constructed representations which appeared to account for relationshipsbetween three rather than two units, but interpreted their representation only in terms of two units. InExample 2, the student represented the part-whole relationship of eighths and twelfths but did notattend to the difference between eighths and twelfths. In Example 3, the student constructed arepresentation in which 2 groups of 3 groups of 4 are discernible, but to the student it only represented6 groups of 4. These representations were classified as bi-relational.Example 1: Bi-relational representation of units.Task: Compare 2/3 and 5/9“I could use a pie like this. Then i’d have nine pies and whoever was eating the five pies, fivepieces of pies out of the nine pieces of pie, would have eaten this much of the pie” [shadedpieces below] “. ..except for the person who has to eat four pieces of pie, okay, the leftovers.”“And then that [2/3] would be something like that” [see diagram below]. “This is a jumbo pieand the person out of two-thirds pie would only eat two out of three pies. Those two” [shadedpieces below] -.and then he’d take these and he’d ... those [thirds] are bigger than these [ninths]. So I’d askhim to half the pie and he’d go up . ..“ [partitioning of thirds into two] “.. .and those would beequal to two of the smaller pies, and he’d do the same for this one, so he’s eaten four ...“ [i.e.,two thirds means four-ninths are eaten] “. ..and he’d only eaten half the pie over here ...“ [4ninths is half of the ninths pie] “but the person who ate five ninths of the pie had one moreslice.”Pete (Grade 7)74Example 2: Bi-relational representation of units.Task: Compare 5/8 and 7/12“and then you can see, like half of this circle would be four, 1, 2, 3, 4; so there would be half anda little bit - and then half of this one would be six, 1, 2, 3, 4, 5, 6, and then you shade in this one -so they would be equal. There is a little bit of each here.”Lara (Grade?)Example 3: Bi-relational representation of units.Task 2X3X4=“You could put boxes, four in each ... four six times ... and count up all the boxes.”UI Hi [‘[1[ii HI HIInterviewer: Where in the picture is two times three?[Fanya was unable to indicate the units associated with these factors. Instead she drew aseparate representation for this part of the numerical expression.]“Well two sections in three so that equals up to six.”RRHFanya (Grade 7)75Tn-relational Representation of UnitsRelationships between three different units underlies some complex whole numbermultiplication and the comparison of common fractions. In such a representation in the whole numbermultiplication context, the three units would be “double-nested.” That is, of the form such as A groupsof B groups of C. In the multiplication context it would occur as the representation of the product ofthree factors. In Example 1, a single representation embodies the relationships between all unitsrepresenting the factors in 2 X 3 X 4 = —. In the common fraction context double-nested unitswould occur as the representation of two fractional units nested within a whole unit, such as therepresentation of thirds and ninths within a single region. However, tn-relational representations in thewhole number context did not necessarily need double-nest units. In example 2, the studentconstructed a tn-relational representation of units without having the units double-nested.Example 1: Tn-relational representation of units.Task 2X3X4=010 0010 0I + 12010 0010 0+010 0010 0I + 12010 0010 0You could say there is 4 two times threes, and two times three is six, so four of those. You couldsay two times twelve, ... four times three is twelve there and four times three is twelve there.Kit (Grade 5)76Example 2: Tn-relational representation of units.Task: Compare 2/3 and 5/9[Drew two squares, one above the other. Partitioned one into thirds and the other into ninths.Shaded 2/3 and 5/9, then compared the shaded amounts in both squares.]J/yJ;I\\ ‘s’%“That is cut into thirds, and that is cut into ninths. So you have all of that shaded in - two-thirdsand five ninths.”Laia (Giade 7)In summary, the major elements or constructs used to analyze the characteristics of students’representations are their form and their function. Categories related to the form of representations aredefined according to (1) the spatial framework of the representations, and (2) the physical characteristicsof the units constructed within the spatial framework. Categories related to the function ofrepresentations are defined according to the number of different units explicitly inter-related bystudents in their representations of the mathematical tasks.Students’ Repentoires of Representations:Whole Number MultiplicationThe four interviews (uncued generative, cued generative, linear interpretive, and areainterpretive) focused, in different ways, on questions related to the characteristics of students’repertoires of representations of whole number multiplication. The uncued generative interview wasdesigned to explore the characteristics of students’ primary repertoire of representations. The first part77of this section is devoted to an analysis of the characteristics of students’ primary repertoires ofrepresentations.In the second part, students’ primary repertoires are compared and contrasted with theirrepresentations constructed in the subsequent interviews. Of particular interest was whether, bysuccessively changing the interview setting, alternative forms of representation might be used thatwere not generated in the uncued interview, or whether their primary repertoire characterizes theirgeneral repertoire of representations.There are two general characteristics of the forms of representations in students’ repertoireswhich are explored through the sequential analyses of the interview data.1. Whether there is a dominant form of representations in their repertoires.2. The extensiveness of their repertoires. Extensiveness is considered in two ways: (1) in termsthe variety of forms of representations in students’ repertoires, and (2) in terms of the forms ofrepresentations rejected by students as a means for representing whole number multiplication.The general organization of the analysis is as follows. Two general characteristics of repertoires,the form of representations as well as the function of the representations, provide the generalframework within which students’ repertoires of representations of whole number multiplication arecharacterized. Within this framework, the results of the analysis of the data from the uncued intervieware considered to represent the characteristics of students’ primary repertoires of representations. Theprimary repertoires are then compared and contrasted with students’ responses in the other interviewsettings with regard to the forms and function of students’ representations. Through the successivecomparisons of students’ primary repertoires with their responses in the other interview setting, theissue of the extent of their general repertoires will be addressed. Finally, the over-all analysis will bedrawn upon to address the general research question, “What are the characteristics of students’repertoires of representations of whole number multiplication?”78Characteristics of Primary Repertoires of Representations:Whole Number Multiplication ContextThe questions addressed in this section are:1. What are the characteristics of students’ primary repertoires of representations used to explainthe meaning of whole number multiplication? In particular:a. Is there a form of representation which dominates within their primary repertoires?b. How extensive are their primary repertoires with regard to the variety of forms ofrepresentations generated to explain whole number multiplication tasks.2. To what extent do the representations of whole number multiplication in students’ primaryrepertoires have the function of representing relationships between different units?Table 4.01 was designed to illustrate the nature of students’ primary repertoires ofrepresentations of whole number multiplication. The major categories for classifying the form of theirrepresentations are represented in the columns of this table. These categories include the spatialframework of the representations, (sets, lines and regions), and the form of units within the spatialframeworks, (discrete, contiguous, and undefined). For example the first column represents discreteunits (D) within a spatial framework of sets (S). For each different form of representation constructed bya student, the function of their representations with regard to unit relations (M, B, or T) is coded withinthe table. In addition, a dot in the body of the table indicates that the form of representation was notgenerated by the student. Blanks in the body of the table indicate that the task was not presented tothe student.Most students constructed more than one representation to explain one or more tasks.However, when the form and function of their multiple representations were the same, only the generalpattern of their responses is codified. For example, Connie generated two representations for eachtask, representations which she considered to be completely different. All of her representationsconsisted of discrete mono-relational units in sets. Hence, there is one entry per task for Connie (M incolumn DS) which codes the form and function of all of her representations per task.Table 4.01Forms and Functions of Representations of Multiplication: Uncued Generative Interview.Interview tasks by form of representationAXB= AXBXC= AX (B÷C) =Dominanceof Forms DS DL DR CL CR DL DR DS DL DR CL CR DL UR D$ DL DR CL CR UL URExclusivelyDiscrete1 Connie M MMarlene B M2 Brock B MDerek B BMEdwin B M BJames B MT3 Dahlia B B BKasey B BPete B B BTammy B B BLara B B BDominant lvDiscreteLolande B . . . B . . B BCoran B . . . B . . BT BB . . . B . . T MTDominant lvCont iouousFanya B . . . B . . B . . . BNames of grade 5 students are underlined.Dots within the table mean that the form was not used.Blanks within the table mean that the task was not given.Spatial Framework flhIs Unit RelationsS = sets D = discrete M = mono-relationalL = lines C = contiguous B = bi-relationalR = regions U = undefined T = tn-relational7980There were two criteria used to position the students within the table. First they were groupedaccording to the dominance of a form of representation within their repertoire. A form ofrepresentations was considered to be dominant when a majority of the representations generated by astudent were of that form. Second, they are ordered within the group according to the extent to whichthey represented relations between different units.Characteristics of the Forms of Representations in Primary Repertoires.As can be seen in Table 4.01, in all but one case (Fanya), students’ primary repertoires weredominated by representations based on a spatial framework of sets in which all units were discrete.1Despite the exclusive use of discrete forms of representations by two-thirds of the students, all but onestudent (Brock) constructed at least two representations for a task which they considered to be“completely different.” Students’ criteria for defining “completely different” were not related necessarilyto differences between the proximity of contiguous and discrete units.Characteristics of discrete forms of representations. There were three ways in which studentsconstructed what they considered to be “completely different rough pictures or diagrams” to explain amultiplication task, while still using only sets as the spatial framework.1. Change the type of objects within the sets. For example, Marlene represented 4 X 5 = firstasL\&L\ X 0 0 0 0 0 = 20then as1. Descriptions of a student’s responses serve only to illustrate patterns in responses. They are notintended to be read as part of indMdual case studies. Students’ names are included only to assist thereader to locate data in the tables. Appendix A contains sample transcripts of each type of interviewused in this study. Included in the appendix are selections from 10 of the 15 students’ transcripts.812. Change the spatial organization of the sets. For example, James’ two representations of4 X 5 = were “four groups,” then “four rows with 5 in each.”A. I I I I B. XXXXXI I II I thenas XXXXXI I I I xxxxxI I I I II xxxxx3. Change the function of the representation by using the commutative or associative principles.For example, Lara represented 6 X 4 = first asA. A.AA B.thenasThese three variations in changes to discrete representations with sets, that is, (1) changes intype of objects, (2) changes in spatial organization of units, and (3) changes in the function of therepresentation, were used in different combinations by students to construct “completely different”representations of a task. Most students changed at least two of these three features to construct a“completely different” representation.A rectangular rather than a linear or irregular spatial configuration was used exclusively ordominantly by 11 of the 15 students when they constructed representations based on sets (see Table4.02). The students used factors in an A X B array to organize primary and aggregate units, orapproximated a rectangular organization of units when the number of units was prime or odd. A linearconfiguration was used only with mono-relational representations, not to represent a multiplicativerelationship between two or more different units. Hence, when students were representing unit82relationships as bi-relational or tn-relational, they tended to represent the product as a rectangularconfiguration with factors as the dimensions, or to represent the cardinal values of the factors in arectangular configuration. In the latter case, 4 groups of 5 would be represented with the 5 primary unitsas a (2 X 2 array) +1, and the 4 aggregate uniLs as a 2 X 2 array.Table 4.02Extent to which Spatial Organizations were used to construct Rerresentations of Multiplication withSets (Uncued Generative Interview).Spatial OrganizationsExtent1 Linear Rectangular IrregularExclusive 9 2 7Dominant 4 4Secondary 6 4 2 2Note. 1. Exclusive means all, dominant means more than half,secondary means half or less of a student’s discreterepresentations used the configuration.2. Total n = 15 Students are counted more than once in the“dominant” and “secondary” categories.Characteristics of contiguous forms of representations. Four students generated primaryreperloires which included both discrete and contiguous representations. There were two commoncharacteristics of their contiguous representations.1. Only regions were used as the spatial framework, not lines (see CL & CR in Table 4.01).2. The aggregate units were only partially contiguous. Each aggregate unit was represented as aseparate partitioned region, not as a contiguous unit within a partitioned region. For example,4 X 5 = was represented as:83or__notThe spatial separation of aggregate units meant that the features of the representation which embodiedeach of the two factors were visually distinct. To interpret the final representation in terms of bothfactors, less analysis would be required than if the units were fully contiguous.The discrete and contiguous forms of representations in their primary repertoires, which theyconsidered to be “completely different,” were distinguished by the fact that the units within anaggregate unit were either discrete of contiguous. Aggregate units were invariably discrete.Common spatial characteristics of discrete and contiguous forms of representations. Therewere two spatial characteristics common to both discrete and contiguous representations whichappeared to serve to reduce the perceptual complexity of the representations.1. A tendency for most students to organize their discrete and contiguous representationssystematically within two dimensions rather than one dimension.2. The consistent spatial separation of the aggregate units.Characteristics of contiguous and discrete representations were analogous in an number ofways. Aggregate units were constructed as separate, partitioned regions or separate sets. The visualdistinctiveness of each contiguous or discrete aggregate unit sometimes was emphasized by theinsertion of operation signs between aggregate units. It therefore was unnecessary to reconstruct orimagine the aggregation of units from a collection of primary units. The multiplicative relationshipsamong the units in the representation were visually explicit. Within a general, two-dimensional spatialorganization, primary units within an aggregate unit were either contiguous or discrete.84Functions of Representations in Primary Repertoires.As can be seen in Table 4.01, mono-relational representations of A X B tasks wereconstructed by only two students (Connie & Marlene), but the incidence of mono-relationalrepresentations increased with the more complex tasks. A majority of the students constructed a seriesof bi-relational representations for some or all of the more complex tasks. The bi-relationalrepresentations of these more complex tasks illustrated separate steps in a calculation procedure, notthe inter-relationship of all units involved in the task. Only three students managed to construct any tnrelational representation.Students who constructed only mono-relational representations were able either to determinesome products with a skip counting or counting on procedures. They showed no indication that theseprocedures were associated with a representation of consecutive groups of units. For example, whenMarlene “knew” a skip counting sequence, a representation such as liii X 11111 = 20 provided a way tokeep track of count. When counting by five’s, she marked each item in the set of four with each count.When counting sequences were not known she did not use sets in her representation to keep track ofher counting-on sequence. Instead she laboriously and often unsuccessfully used a verbalcounting-on procedure while keeping track of the count on her fingers. The procedures (skip countingor repeated counting on) were not associated with a concrete representation of consecutive aggregateunits. Instead, these mono-relational representations were, at most, procedural tools. There was noexplicit representation of their skip counting or repeated addition procedures in which the product was asynthesis of the procedures.There were two circumstances under which other students constructed mono-relationalrepresentations with only the more complex tasks. In the first circumstance, some students could notinterpret the symbolic notation and determine the final product numerically or with mental arithmetic.They constructed mono-relational representations as a minimal response. In the second circumstance,some students determined the product, and appeared to establish the representation of relationshipsamong all factors as a final goal. However, they had difficulty translating their thinking processes into a85tn-relational representation. Their tn-relational representations were preceded by either mono-relationalor bi-relational representations of the task. The preceding mono- or bi- relational representationassisted the student in thinking through the problem of representing units relationships among allfactors when the product was already known. Reflection on their earlier mono- or bi-relationalrepresentations eventually resulted in the construction of a tn-relational representation.In summary, there were different limits in the complexity of unit relations which students couldrepresent. Two students were limited to mono-relational representations. Most other students werelimited to bi-relational representations of units. Nearly all tn-relational representations were preceded byone or more trials.Summary of Characteristics of Primary Repertoires of Representations of Whole Number Multiplication.Considering the representations constructed by the students during the uncued generativeinterview, a number of features characterized their primary reperloires of representations ofmultiplication. With regard to the form of representations, characteristics were:1. The general dominance of discrete units in sets, and the more limited use of partitioned regionsas spatial frameworks.2. The general use of a 2-dimensional spatial organizations of units.3. The spatial separation of aggregate units.With regard to the functions of the representations, characteristics were:1. Limits in the complexity of the unit relations fluently represented by students. With fewexceptions this was either mono- or bi-relational representations of the units.2. Tn-relational representations were not constructed directly. Students seemed to derivestrategies for constructing such representations from a reflection on preliminary mono-relational orbi-relational representations.In general, the spatial frameworks and organization of the form of the representation appear tofunction to reduce the perceptual complexity of the representations. In this regard, the spatial86organization of contiguous representations was analogous to that of discrete representations in thataggregate units were spatially separated, and organized within a two-dimensional configuration.Characteristics of General Repertoires:Representations of Whole Number Multiplicationin Other Material SettingsThe questions addressed in this section are:1. To what extent are the characteristics of representations in students’ repertoires influenced bythe material setting within which representations are constructed?2. How extensive are their general repertoires with regard to the variety of forms ofrepresentations used to explain whole number multiplication tasks.a. To what extent are particular forms of representations excluded from students’ generalrepertoires?b. To what extent do students’ general repertoires of representations include forms ofrepresentations based on attributes of length or area?In this section, the students’ responses to the multiplication tasks during the other threeinterviews (cued generative, linear interpretive, and area interpretive) are compared and contrasted tothe characteristics of the students’ primary repertoires of representations generated during the uncuedgenerative interview. The forms of representations of multiplication tasks constructed during the cuedgenerative interview which differed from those spontaneously generated during the uncued interviewwere considered to indicate extensions within students’ repertoires. As well, materials which studentsrejected during the cued generative interview were considered to be one indication of limits in theirrepertoires. The students’ acceptance or rejection of beginning diagrams during the linear and areainterpretive interviews was considered to be another indication of limits in their repertoires.This section is organized in two parts. In the first part, the primary repertoires generated duringthe uncued generative interview are compared to the forms and functions of the representationsconstructed during the cued generative interview. In the second part, the responses in all of theinterview settings are compared and contrasted.87Representations Constructed in the Cued Generative Interview Compared to the Uncued GenerativeInterview.Table 4.03 was designed to illustrate students’ representations constructed during the cuedgenerative interview. Also included is a summary of their responses during the uncued generativeinterview which represent their primary repertoires. The format of the table is similar to that of Table4.01. The major categories for classifying the form of their representations are represented in thecolumns of this table. These categories include the spatial framework of the representations, (sets,lines and regions), and the form of units within the spatial frameworks, (discrete, contiguous, andundefined). For each form of representation constructed by a student during the cued generativeinterview, the function of the representation with regard to unit relations is coded within the table.As with the previous uncued generative interview, most students constructed more than onerepresentation for one or more tasks. When the form and function of these multiple representationswere the same, only the general pattern of the responses is indicated. In addition, a dot in the body ofthe table indicates that the form of representation was not generated by the student, and blanksindicate that the task was not presented to the student.The summary of the characteristics of students’ primary repertoires (as reflected in the uncuedinterview) is recorded to the far left with the students’ names. The functions of their representationsfrom the uncued generative interview precede each student’s name. The students are groupedaccording to the dominance of a form of representation within their primary repertoire as in Table 4.01.Within these groups they are ordered according to the extent to which they represented relationsbetween different units during the cued generative interview.Influence of material setting on forms of representations generated. For most students, thepatterns of response during the cued generative interview mirrored those of the uncued generativeinterview. Despite the presence of materials that might have cued students to construct contiguousforms of representations, only four students (Dahlia, Pete, Lara, and Fanya) constructed contiguousrepresentations as their most frequent form of representation during the cued generative interview.Discrete units were still the dominant or only form of representation for 11 of the 15 students. However,88Table 4.03Form and Function of Representations of Multiplication Tasks During the Cued Generative Interviewwith a Summary of the Primary ReDertoires.Cued interview tasks by form of representationSummaryof A X B = A X B X C = A X (B + C) =PrimaryRepe r to ire 5(Uncued) DS DL DR CL CR UL UR DS DL DR CL CR DL UR DS DL DR CL CR DL DRExclusivelyDiscrete1M Connie N MM.....M Marlene M MBM Brock B MBM Derek MB MBMT James B NRB Tammy B . . . BB Dahlia . . . . B B1B Pete . B . . B1 1BM Edwin T B . . BT . BB Lara . . . . BT BB Kasey T . . . BDominantlyDiscreteB Lolande BBT Coran BT1BTMJj. T B . . TDominant lvCont iciuousB Fanya . . . . BNote. Names of Grade 5 students are underlined.Dots mean the form was not used; blanks mean the task was not given.1. Representation constructed only when questioned directly about materials.Spatial framework Units Unit RelationsS = sets D = discrete M = mono-relationalL = lines C = contiguous B = bi-relationalR = regions U = undefined T = tn-relationalNR = no response89six students (Tammy to Kasey in Table 4.03), whose primary repertoires included only representationsconstructed with sets of discrete units (DS), constructed at least some representations with contiguousunits in regions (CR). Hence, there was an increase in the incidence of contiguous representationsduring the cued generative interview, but most students continued to favour discrete representationsbased on sets.As was the case with the responses in the uncued generative interview, most students used arectangular configuration for the organization of sets and units within sets during the cued generativeinterview (see Table 4.04). Similarly, the bi-relatiorial and tn-relational representations were invariably ina rectangular configuration, whereas linear or irregular configurations were used only for mono-relationalrepresentations.Table 4.04Extent to which Spatial Organizations were used to Construct Representations of Multiplication withSets (Cued Generative Interview).Spatial OrganizationExtent1 Linear Rectangular IrregularExclusive 10 8 2Dominant 2 2Secondary 2 2Note. 1. Exclusive means all, dominant means more than half,secondary means half or less of a student’s discreterepresentations used the configuration.2. Total n = 12 Students are counted more than once in the“dominant” and “secondary” categories.Relationship between discrete and contiguous representations. As was the case withcontiguous representations generated during the uncued generative interview, the contiguous90representations constructed with materials were limited in a number of ways. All contiguousrepresentations were based on regions. Generally aggregate units were separated, and the spatialorganization of the aggregate units was rectangular.Students constructed contiguous aggregate units with the materials by putting a set ofseparate units together rather than by partitioning regions. Aggregate units which the students hadrepresented previously as a discrete set now were pushed together and represented contiguously.This suggested a close relationship between students’ discrete and contiguous representations. Thecontiguous representations appeared to be derived through simple transformations in the proximity ofprimary units within each aggregate unit. For example:Task:3X(4+5’)=(3X4)+(3X5)Uncued Interview3X(4+5) (3X4)+(3X5)o o 0 0 0 0 0 0 0 LILILILI LIL\LIIIL\o o 0 0 0 0 0 0 0 LILILXLI LILILIL\LXo o 0 0 0 0 0 0 0 LiLXLXi âL1L1LL\27 27Lara (Grade 7)Cued Interview: Partially Contiguous3X(4-i-5) (3X4)+(3X5)_________!TI!1______________—!I I I I I I I IIF II Full [1111.1LI HLara (Grade 7)91Extent of repertoires: attributes of area or length. Two-thirds of the students constructeddiscrete and contiguous representations during the two generative interviews. All contiguousrepresentations were based on parts of regions, not line segments. Linear measurement was not aframework used for the construction of any representations. There are four factors which suggest thatattributes of area measurement also were not considered by students as a basis for their contiguousrepresentations.1. As described above, students appeared simply to have altered the proximity of discrete units sothat some units were contiguous.2. The representations were partially contiguous. In all but one instance, A X B was notrepresented as a multiplicative relationship between units within a single region.3. The units in contiguous representations were described by students as being analogous tounits with discrete materials rather than analogous to units constructed by partitioning regions.4. None indicated that parts within a region should be equal or adjusted their representations in amanner which would suggest such a concern during the uncued generative interview. During the cuedgenerative interview, none of the students partitioned regions into units.The line was not used independently as a spatial framework with which to construct arepresentation of a multiplication task. Four students (Connie, Edwin, Pete, Kit), when later asked if theline could be used to explain a task, agreed that they could do so. However, units of length were notused by these students when they represented multiplication on the line. Instead, they representedunits as discrete marks along the line. For example:Task: 3X(4i-5)=(3X4)+(3X5)“Four lines, then a little ways apart 5 lines, then make two more groups the same.”.c:I—III Ill—Il liii 11111 liii 11111Pete (Grade 7)92Other students either rejected the blank line altogether, or defined the whole line or the whole sheet onwhich the line was drawn as one among many discrete units.The absence of representations based upon linear measurement within students’ repertoireswas further emphasized by students’ responses to questions about the usefulness of a centimetreruler. Despite the fact that students and interviewer all used the term “ruler”, the three students(Tammy, Dahlia, & Edwin) who attempted to use the ruler perceived it to be a set of discrete green units,ignoring the unpainted spaces (see diagram). The ruler was then rejected because there were aninsufficient number of units. No explicit association of the ruler with units of length was made by anystudents in this mathematical context.Ruler used during Cued Generative Interview (Reduced by 50 Percent).In summary, measurement of numerosity appeared provide the sole basis for defining unitsregardless of whether the units were contiguous or discrete. Alternative materials which could haveprovided a framework for representations based on linear or area measurement were evaluated in termsof units of numerosity alone.Function of representations in both generative interviews. As can be seen in Table 4.03, moststudents constructed representations during the cued generative interview which had the samefunction as those they constructed during the uncued generative interview. However, three students(Edwin, Lara & Kasey), who previously had constructed either mono-relational or bi-relationalrepresentations, constructed tn-relational representations of A X B X C =.... during the cuedgenerative.The process of constructing a tn-relational representation was a complex problem for thosestudents who were successful. The most common interpretation of a numerical expression ofmultiplication was that A X B = was read from left to right as “A groups of B.” With this93interpretation, the task 2 X 3 X 4 was read as [(2 X 3) X 4] ‘2 groups of 3”, then “6 groups of4.” This interpretation resulted in two bi-relational representations: one that represented 2 groups of 3,and one that represented 6 groups of 4. This interpretation follows normal procedures for thecalculation of the product.To construct a tn-relational representation of 2 X 3 X 4 the associative or commutativeproperties have to be applied in ways which conflict with the common interpretation described above. Inone thinking strategy the commutative principle was applied to interpret the expression as(a) (2 X 3) X 4 then (b) 4 X (2 X 3), or “4 groups of 2 groups of 3” or “2 groups of 3, repeated 4times.”I I I I I I I I I II I I I I I I IIn another thinking strategy the associative principle was applied to interpret 2 X 3 X 4 as2 X (3 X 4) or “2 groups of 3 groups of 4” constructed as 3 groups of 4, repeated twice.I I I I I I I II I I I I II I I I I I IWith the use of the associative and commutative principle these cases are generalizable to anycombination of factors in the form X groups of Y repeated Z times.Difficulties encountered by students who tried to represent the relationships among threeunits, are exemplified by Lara’s responses during the cued generative interview. She began by tryingto construct a representation of 4 X (2 X 3) to explain 2 X 3 X 4. She finished by constructing arepresentation of 2 X (3 X 4).94“You would have 2 times 3 and then that is a group of 6” (made 2 groups of 3). “And then youmultiply that 2 times 3 by 4 more groups” (added 4 more groups of 3)ii 11 LiiiLi Li H iiLIHUJT“I blew it! That will be two times three. It will equal 6 and then it is multiplied by 4. One has toequal that first to the 6 and then you have 4 more groups of that Okay, I’m going to multiplythat first” (i.e 3 X 4) (made 3 groups of 4) “and that equals 12. and you have to ... have toOkay, the 2 groups of 3 times 4. You have 2 groups, so there is that one and that one and youadd them together and you get 24.”Lt! ii 1 I I IiLJHI IIJLIt cit tiLara’s purpose from the outset was to construct one representation which would function to explain therelationship between three factors, having already determined that the product was 24. Herinterpretation of “X 4” as “four more”, and her association of this action with the group of 3 rather thanthe group of 6 led to a representation of (2 X 3) + (4 X 3).Lara had to change her thinking strategies in order finally to construct a representation of thismultiplicative process. Other students followed an even more protracted series of trials beforeachieving their goal of a tn-relational representation. All but one student who constructed a tn-relationalrepresentation had occasion to construct a series of bi-relational representations as steps towardsrepresenting the product. They then either reinterpreted their representation of 6 groups of 4 in termsof the three factors, or changed their thinking strategy to construct another representation in which theunits were tn-relational.Summary of characteristics of repertoires from both generative interviews. Some differenceswere observed with respect to the forms or functions of representations constructed in the uncued andcued generative interviews. Some of the students whose primary repertoires were exclusively discretegenerated contiguous representations during the cued generative interview. In addition, some95students constructed tn-relational representations whose representations previously had been onlybi-relational.Besides the differences between interviews in the form or function of some individual students’representations mentioned above, the general characteristics of the variety of representationsgenerated remained the same. The students’ bi-relational or tn-relational representations of wholenumber multiplication still were organized around rectangular spatial configurations within which unitswere more or less discrete. Discrete representations continued to be dominant for a majority of thestudents and a third of the students persisted in constructing only discrete representations with sets.Furthermore, the complexity with which different students were able to explain the multiplication tasksfluently continued to be limited to either mono-relational or bi-relational representations of units.The extent of repertoires of representations appear to be limited as follows.1. Representations based on regions were only partially contiguous with the aggregate unitsrepresented as separate regions.2. Attributes of area measurement did not appear to be associated with contiguousrepresentations. Instead, students described contiguous representations as being analogous torepresentations with discrete units in sets.3. Attributes of length were not used as a basis for representing units and none of the studentsindependently generated representations with a line as the spatial framework.4. Measures of numerosity appeared to provide the quantitative framework for all forms ofrepresentations.Comparison of Repertoires Reflected in the Generative Interviews With Responses in the InterpretiveInterview SettingsThe interpretive interviews were designed to explore more directly whether attributes of lengthor area measurement might be considered by students as salient features in their representations ofwhole number multiplication. To this end, attributes associated with representations based on linear orarea measurement were distorted in beginning diagrams (see Figures 3.01 & 3.02). These distortionswere designed to identify attributes students might consider to be critical for constructing a96representation of whole number multiplication. Students constructed representations with thebeginning diagrams they judged to be appropriate, and justified their rejection of other beginningdiagrams.Table 4.05 was designed to illustrate the general forms and function of the students’representations constructed in response to the interpretive interview tasks and allows for a comparisonof these responses with students’ responses during the generative interviews. The format of the tableis similar to that used for Tables 4.01 and 4.03. Students constructed more than one representationduring the linear or area interpretive interviews. When the form and function of these multiplerepresentations were the same, only the general pattern of the responses is indicated.The summary of the characteristics of students’ primary repertoires is recorded to the far leftalong with the students’ names. The functions of their representations from the uncued generativeinterview precede each student’s name. The responses indicated in the table for the cued generativeinterview are a summary of the responses in Table 4.03. The responses indicated for the interpretiveinterviews are a summary of students’ responses during those interviews. In the interpretive interviewsonly one of two forms of multiplication tasks were used: either A X B = — or A X B X C = —.The students are grouped according to the dominance of a form of representation within theirprimary repertoire as in Table 4.01. Within these groups they are ordered according to the extent towhich they represented relations between different units during all interviews. Numbered groupsindicate students with common characteristics in the function of their representations, regardless oftheir forms of representations. For example, students in Group 3 (Dahlia and Fanya) constructed onlybi-relational representations throughout the four interviews. Students in Groups 4a to 4c constructedboth bi- and tn-relational representations, but these groups differ in the frequency with whichtn-relational representations were constructed.Forms of representations in the interpretive interview setting. During the linear interpretiveinterview only one student (Tammy) represented the multiplication task with line segments as units (CL).All of the other students circled or marked off sets of points to represent units. The most salient featureof these linear beginning diagrams was the marks along the line and not the spaces between the marks.Table 4.05Forms and Functions of Representations of Multiplication in All Interview Settings.Interview settings by forms of representationSummaryof Cued Generative Linear Interpretive Area InterpretivePrimaryRepertoire(Uncued) DS DL DR CL CR DL UR DS DL DR CL CR DL DR DS DL DR CL CR DL DRDiscrete1.mM Connie mM M . . m m m . mmM Marlene mM . m m2.bM Brock bM . b . bbM Derek mbM b mb . b3.bB Dahlia . BB B B B4a.bB Lara .BTB - . B BbMB Edwin TB B . BT . B B4b.bB Kasey T . B B B TbB Pete B . B . T . TbB Tammy B . B . - T TB4c.bMTJames b . .BT BTDiscreteDominant4a.bB Lolande B . B T4c.bBTCoran BT . B TbMTj. T B . . T . T B BContinuousDominant3.bB Fanya . . B . . . B . . . . - BNote. Names of Grade 5 students are underlinedDots within the table means that the form was not usedUnit relations: lower-case indicates task A X B, upper-case indicates a complex task.Framework of units Units Unit RelationsS = sets D = discrete M = mono-relationalL = lines C = contiguous B = bi-relationalR = regions U = undefined T = tn-relational9798Four students, (Derek, Dahlia, Kasey, & Kit) initially rejected the segmented line as a goodbeginning diagram in general. They argued for representations based on discrete sets in a rectangularconfiguration (DS), even though they eventually represented units as points on a line (DL). Dahlia, themost adamant of these students, stated unambiguously, “I don’t like lines,” and rejected all subsequentdiagrams after her first representation using points on the line as units. The dominance of two-dimensional configurations in their repertoires conflicted with the task of constructing a one-dimensional representation, and made the use of these linear beginning diagrams strongly improbableto the students.During the area interpretive interview, the dominance of contiguous representations contrastsmarkedly with the dominance of discrete representations during all other interviews. Only threestudents (Connie, James & Dahlia) constructed representations with discrete units. All others usedpartially or totally contiguous units in their representations. However the use of contiguous units did notimply necessarily that the students were basing their representations on notions of area measurement.This issue is addressed in a later section.Functions of representations in all interview settings. Considering the function of therepresentations constructed by students in all interviews, some patterns are discernible which suggestdifferent limits in the extent to which students represent relationships between different units. As canbe seen in Table 4.05, some students were relatively stable in the manner in which unit relationshipswere represented. Some constructed only mono-relational representations (see Group 1) regardless ofthe tasks, or only mono-relational representations of the more complex tasks (Group 2) Othersconstructed only bi-relational representations (Group 3). However, the balance of the students wereunstable in their representations of unit relationships when explaining the more complex tasks,fluctuating between bi-relational and tn-relational representations (Groups 4a, 4b, & 4c).Students’ fluctuations between bi- and tn-relational representations reflect two factorsassociated with the representational process. The first factor was that both tn-relational and bi-relationalrepresentations of units were valid in terms of different objectives. One objective was to explain aprocedure for calculating A X B C. This would result in bi-relational representations of A X B = D99and D X C = E. Another objective was to explain the relationships among all factors and the product.This would result in a tn-relational representation. In some cases, fluctuations were associated with astudent’s change in objective.The second factor was that, when students set the construction a representation of therelationships among all factors and product as their objective, tn-relational representations were difficultfor most students to achieve. To plan and construct a tn-relational representation requires an analysis ofthe relationships among the factors which does not follow that of normal calculating procedures.Instead of the calculating steps (A X B) X C, either A X (B X C) or C X (A X B) has to beconstructed. As a result, bi-relational representations at times preceded or substituted for attempts toconstruct tn-relational representations. Only three to five students constructed tn-relationalrepresentations in any one interview and no student was consistent in the construction of tn-relationalrepresentations across all interviews.Characteristics of the form of representations and students’ attention to attributes of length andarea measurement. The extent to which units were represented as contiguous and the extent to whichstudents attended to attributes of length or area measurement are illustrated in Table 4.06. Thecolumns code three degrees of contiguity in the students’ representations: none, partial and totalcontiguity. In the body of the table students’ attention to the equality or inequality of the area ofcontiguous regions in students’ representations in the generative interviews are coded (E, U, #). Thelatter code (#) indicates that the attributes of equality or inequality were not determinable. The studentsare grouped within the table first according to their tendency to require that beginning diagrams haveline segments or regions which were equal in length or area, and second, according to their use ofequal contiguous units during the generative interviews.All students rejected some beginning diagrams and gave aesthetic rather than quantitativereasons for doing so. However, students expressed different tolerances for attributes in the beginningdiagrams which are not associated with either a number line or a rectangular grid of square units. Thebeginning diagrams in which attributes were most distorted were rejected by more than two-thirds of thestudents (see Figure 3.01, #4, and Figure 3.02, #3). They were “messy”, “too hard to see”, “too100Table 4.06Extent of Contiguity of Units and Form of Contiguous Units: MultiplicationInterview contexts by extent of contiguity of unitsUncued Cued Linear AreaGroups N PT N PT N PT N PT1.Connie . . # . . U . . # UMarlene # . . # . . U . . . UBrock # . . . . U . . . . UDerek # . . # . . U . . . . U2.Kasey . . # E E # . . UTammy . . # E . . . U . . U3.Dahlia . . . U . E . . UFanya # # . . U . U . . . . ELolande # # . - . U . . . . E4.Coran # E . # . . U . . . . EPete # . . # E . U . . . . EEdwin # . . . E U . . . . E5.James # . . . . E . . ELara . . . E - E - . . E# E . # E . E . . . . ENames of grade 5 students are underlined.Extent of ContiguityN = none contiguous E = equal spaceP = partially contiguous U = unequal spaceT totally contiguous # = equality indeterminablebumpy,” “too squiggly,” or “confusing.” Whereas, fewer students questioned the appropriateness ofother beginning diagrams which had less distortion. In addition, regularity in verticaVhorizontal101orientation of units was sought by some students. There is less distortion between squares and rhombithan between squares and irregular polygons (see Figure 3.02, #2 & #4). Yet, the sense of imbalanceof the rhombi off the vertical was tolerated less than the irregularity of polygons (Figures 3.02, #1 & #4).For some, the congruency of units was necessary but not sufficient; for others horizontal/verticalorientation took precedence over congruency. In general, perceptual orderliness appeared to be acritical characteristic sought in different ways by students.The use of contiguous units did not imply that continuous measurement was used as aframework for the representations. First, students constructed partially or totally contiguousrepresentations without the congruency of the units being of importance to them (see Groups 1 & 2).The only student to use contiguous units during the linear interpretive interview (Tammy) did so withunequal line segments. Congruency was not a basis on which these students accepted or rejectedbeginning diagrams. Second, students in Groups 3, 4 & 5 generally were more consistent in their needfor equal parts when using contiguous units, but the congruency was not related to ideas about equalmeasures of space. Instead, the need for perceptual orderliness was used to justify their rejection ofbeginning diagrams in which parts were not congruent.The need for perceptual orderliness did not imply that the students’ were making judgementson the basis of properties of length or area measurement. Even students who insisted that parts beequal in both interviews used only discrete units during the linear interpretive interview (see Group 5).In the most extreme case, James constructed only discrete representations during all interviews, yetaccepted only beginning diagrams with equal parts during both interpretive interviews. They were notassociated with representations based on the comparisons of measures of one or two-dimensionalspace. Perceptual orderliness appeared to be a characteristics of importance to students inrepresentations in general. However, the importance placed on perceptual orderliness at timesresulted in beginning diagrams with congruent parts being selected in preference to ones withoutcongruent parts. Applying the most restricted criteria for perceptual orderliness, students couldconstruct unwittingly representations of whole number multiplication which would be identical toinstructional representations based on area measurement.102In summary, students had clearly defined criteria regarding appropriate characteristics ofrepresentations of whole number multiplication. Their criteria often resulted in a replicationcharacteristics of units in instructional representations based on length or area measurement. However,the correspondence of students’ judgements with attributes of units of length or area measurement didnot imply that students used length or area measurement as a framework for representing wholenumber multiplication. Students evaluated beginning diagrams in terms of aesthetic criteria alone.Some students’ may have developed ideas about appropriate physical characteristics of arepresentation from experiences with instructional representations based on linear and areameasurement. However, in the context of whole number multiplication, these physical characteristicsare not associated by them with linear or area measurement.Summary of the Characteristics of Students’Repertoires of Representationsof Whole Number Multiplication.The students’ primary repertoires, with few exceptions, also characterized the representationswhich students constructed when materials were present. Students generally used or evaluatedmaterials in terms of their dominant form of representation in their primary repertoire. In all but one case,the dominant form of representation was discrete units within sets.Contiguous representations included in students’ repertoires were based on regions. For themost part, the extent to which the units within a representation were contiguous was limited. Theserepresentations were more closely associated with characteristics of discrete representations thancharacteristics of representations based explicitly on area measurement. Through a simpletransformation of the proximity of units, one form of representations could be derived from the other.There was no direct evidence that area measurement was thought to provide a framework for definingunits in contiguous representations. Instead, attributes of units of area measurement such ascongruency of units were associated by students with perceptual orderliness rather than associatedwith measures of two-dimensional space. Regardless of the regularity of units in a representation,103contiguity seems to be an incidental characteristic of discontinuous units which measure onlynumerosity.Students’ repertoires were limited in at least one of three ways.1. Mono-relational or bi-relational representations were limits at which students could fluentlyconstruct representations of unit relationships. No student consistently constructed tn-relationalrepresentations, and only one student was fluent in her construction of such representations2. The extent to which units were contiguous was limited in some students’ repertoires. In onecase units were always discrete, in five cases the units were only discrete or partially contiguous.3. Forms of representations were excluded from some or all students’ repertoires.Representations based on linear measurement were excluded from all repertoires, and contiguousrepresentations were excluded from individual repentoires.Students’ Repertoires of Representations:Common Fractions and Comparisonsof Common FractionsThe same general research questions regarding the characteristics of students’ repertoires ofrepresentation of whole number multiplication guided the analysis of the characteristics of repertoires ofrepresentations of common fractions and their comparisons. These are:1. What are the characteristics of students’ primary repertoires of representations?2. To what extent are the characteristics of representations in students’ repertoires influenced bythe material setting within which representations are constructed?3. To what extent do students’ repentoires of representations include forms of representationsbased on attributes of length or area?4. To what extent do students’ representations have the function of representing relationshipsbetween different units?The general organization of the analysis of students’ repertoires of representations in thecommon fraction context is the same as that which was followed in the whole number multiplicationcontext. The two general characteristics of repertoires, the fQni of the representations, and the104function of the representations within repertoires, provide the general analytical framework. Thisframework guided the analysis of the characteristics of students’ repertoires of representations ofcommon fractions and the comparisons of common fractions. Within this framework, the results of theanalysis of the data from the uncued interview are considered to represent the characteristics ofstudents’ primary repertoires of representations. The primary repertoires are compared and contrastedwith students’ responses in the other interview settings with regard to the forms and function ofstudents’ representations. Through the successive comparisons of students’ primary repertoires withtheir responses in the other interview setting, the issue of the extent of their general repertoires will beaddressed. Finally, the over all analysis will be drawn upon to address the question of the characteristicsof students’ repertoires of representations of common fractions and the comparison of commonfractions.Students in both grades were presented with tasks which required them to compare twocommon fractions with unlike denominators. Because Grade 5 students would have had less formalexperience with relationships between common fractions than Grade 7 students, they were given tasksto explain the meaning of a common fraction before being presented with comparative common fractiontasks. The selection of tasks used in these interviews are in Chapter 3, p. 46-47.A variety of interpretations of common fractions were used by students in their explanations oftheir representations. These interpretations are used along with the categories related to the form andfunction of representations to characterize students’ representations of common fraction concepts andcomparisons. The interpretations are:1. Cardinal values interpretation (Cv): Students represented only the cardinal values ofnumerators or denominators. There were two ways in which students expressed thisinterpretation. One way was that “a/b “was represented as the value of “a” as well as the valueof “b.” The other way this interpretation was expressed was that “a/b “was either representedas the value of “a” represented as the value of “b”. Comparisons were based on the directcomparisons of cardinal values, or on comparison of the sums of the values of numerators anddenominators. In the latter case, a/b vs c’d would translate into (a+b) vs (c+d).2. Half of a number interpretation (Hn): The student expressed 1/b as half of any number. Forexample 1/2 might mean 3 out of 6, and 1/4 might mean 2 out of 4, or half of 4. Regardless ofthe denominator, the reference was to represent a half.1053. Relative size interpretation (Rs): The student compared common fractions with notions aboutrelative sizes of common fractions. There was no explicit reference to any relationship involvingnumerators or denominators.4. Inverse size of the denominators interpretation (Id): The student made comparisons only byreasoning about the denominators. The premise of the representations was that the larger thevalue of the denominator, the smaller the size of the part. Hence, the common fraction with thelarger denominator was the smaller fraction regardless of the values of the numerator. Forexample, 3/4 was less than 2/3 because 1 part would be a smaller piece than the other.5. Multiplication interpretation (Mu): The student expressed “a/b” as “a x b” and compared thevalues of the products. For example, 2/6 was 2 groups of 6, or 2/3 was 3 groups of 2.Therefore 2/6 was greater than 2/3 because 12 was greater than 6.6. Take away interpretation (Ta): The student expressed “a/b” as “b-a” or as “b-c=a” and relatedthe meaning to a removal action. For example, 3/4 might be explained by “4 pieces of pie and 3were eaten” or “4 chairs, one was broken, and 3 good chairs were left.” The use of thisexplanation did not preclude the representation of part-whole relationships of bi- or tn-relationalunits.Characteristics of Primary Repertoires of Representations:Common Fraction ContextThe questions addressed in this section are:1. What are the characteristics of students’ primary repertoires of representations used to explainthe meaning and comparisons of common fractions? In particular:a. Is there a form of representation which dominates within their primary repertoires?b. How extensive are their primary repertoires with regard to the variety of forms ofrepresentations generated to explain common fraction tasks and comparison ofcommon fractions tasks.2. To what extent do the representations of common fractions and the comparison of commonfractions in students’ primary repertoires have the function of representing relationshipsbetween different units?Table 4.07 was designed to illustrate the nature of students’ primary repertoires ofrepresentations of common fractions and comparisons of common fractions. The general format of thistable is the same as that used for Tables 4.01, 4.03 and 4.05. The major categories for classifying theform of their representations are represented in the columns of this table. These categories include theTable 4.07Forms and Functions of Representations of Common Fractions: Uncued Generative InterviewInterview tasks by form of representationA/B A/B vs C/DDominance ofForms &Interpretations DS DL DR CL CR UL UR DS DL DR CL CR UL URUndefined UnitsRs Brock N NId Lolande NDiscrete &ContiguousCv Marlene . . . . M . . . . M .Mn James M . . . M . . NRTaCv Edwin M . . . M . . M . . . MTa Pete B . . . BTaMu Dahlia M . . . M . . B . . . BT . *ExclusivelyContiguousCv Derek . . * . M * * NRCvRs Connie . . . . MMu Fanya . . * * BTa Kit . . . . B * * NRTa Kasey . . * * B *Ta Lara . . . . TBTa Coran . . . . TTa Tammy . . * . TNames of grade 5 students are underlinedDots mean the form was not used; blanks mean the task was not given.Units Spatial Framework Unit RelationsD = discrete S = sets M = mono-relationalC = contiguous L = lines B = bi-relationalU = undefined R = regions T = tn-relationalN = no unitsNR = no responseInterpretationsCv = cardinal values (numerator Id = inverse denominatoror denominator) Mu = multiplication, a/b = ax bHn = 1/2 of a number Ta = take-away, a/b = b - aRs = relative sizes106107spatial framework of the representations, (sets, lines and regions), and the form of units within thespatial frameworks, (discrete, contiguous, and undefined). For each form of representation constructedby a student, the function of the representation with regard to unit relations (M, B, T, N) is coded withinthe table. The latter symbol (N) indicates representations in which units are undefined. Theinterpretations used to explain the meaning of common fractions during the uncued generativeinterview are coded in the first column of the table to the left of the students’ names.Students constructed more than one representation for one or more tasks. When the form andfunction of these multiple representations were the same, only the general pattern of the responses isindicated. In addition, a dot in the body of the table indicates that the form of representation was notgenerated by the student. Blanks indicate that the task was not presented to the student.There are two criteria used to position the students within the table. First, they are groupedaccording to the dominance of a form of representation within their repertoire. Second, they areordered within the group according to the extent to which they represented relations between differentunits.Characteristics of the Forms of Representations in Primary Repertoires.As can be seen in Table 4.07, more than half of the students generated repertoires ofrepresentations in which units were represented only as contiguous parts. Two students generatedrepertoires of representations in which no units were defined. The balance of the students generatedrepertoires in which no single form of units dominated. Instead they generated discrete and contiguousrepresentations of units with equal frequency.Regions were the most common spatial framework for representations of common fractions(DR, CR, & UR), regardless of the form of the units represented (whether discrete or contiguous). Theywere used by all students in at least one instance, and exclusively by two-thirds of the students. Evenwith students who generated a representation based on sets, their first representation of commonfraction tasks was generally with regions. No students used a line as a spatial framework for their108representations. Hence, for nearly all students, regions played a prominent role as a primary spatialframework for constructing representations of common fractions or comparisons of common fractions.Table 4.08Extent to which Different Regions were used to Construct Representations of Common Fractions(Uncued Generative Interview).Types of RegionsExtent1 n2 Circle Rectangle IrregularExclusive 7 6 1Dominant 6 4 1 1Secondary 8 4 6 21. Exclusive means all, dominant means more than half,secondary means half or less of a student’s discreterepresentations used the configuration.2. Total n = 15 Students are counted more than once in the“dominant” and “secondary” categories.The types of regions used as a spatial framework for representations were limited. As can beseen in Table 4.08, circles were used most frequently when representations were based on a spatialframework of regions. Furthermore, of the 14 students who used circles, thirteen used one as thespatial framework for their first representation of common fractions. For most students, the responsepattern during the interview was: (1) represent a task with circles and (2) in some cases, constructrepresentations with other regions (generally a rectangle) or with sets.Characteristics of forms of representations based on regions. Students encounteredconsiderable difficulty in partitioning regions into equal parts. Efficient partitioning procedures requiretwo forms of preliminary analysis. First is an analysis of the geometric characteristics of the region whichdetermines what kinds of configuration will partition a region into equal parts (see Figure 4.03). Secondis an analysis of number-theoretic characteristics of the task. No students consistently evidenced109preliminary analyses of all relevant characteristics, particularly when partitioning circles. The successivehalving algorithm was used with the radial configuration to partition circles, regardless of the value of thedenominator (see Lara, Pete, and Kasey in Figure 4.04). Others displayed difficulties in anticipating theoutcome with the hatched configuration when applied to rectangles (see Dahlia and Pete, Figure 4.04).These configurations required students to use factors when denominators were neither prime norpowers of two. However, some students had a variety of partitioning techniques including multiplicativepartitioning with factors, but these techniques were not well defined in terms of the conditions in whichthey could be applied most effectively.Radial Hatched VerticalFigure 4.03 Examples of standard configurations used to partition regions forcommon fraction representations.Although students’ partitioning techniques might have been an important factor in the extent towhich students constructed equal parts within a region, this was generally not the case. In the followingthree cases, the students encountered similar partitioning difficulties, yet “resolved” the problemdifferently.Case 1. Number of parts more critical than equal parts: first partitioned the circle with thesuccessive halving algorithm into 8 instead of 9 equal parts, then one of the eighths was halved to make9 parts altogether (See Kasey, Figure 4.04). No attempts made to adjust the partitions. Thecomparison of thirds and ninths was confidently based on the actual space shaded in each circle,including the 2/l6ths as 2/9ths.110Relative sizeEHBrock 3/4Cardinal values of numerator and/or denominator‘SMarlene1/2 > 3/4 > 2/3A1/3 <2/6ConnieDerek3/4 > 2/3CD4/4>RelativeSize3/42/6> 1/3Inverse size of denominatorLolandeFigure 4.042/3> 3/4[JEIP1/3 3/4 4/5Examples of common fraction representations with regions as they relate tointerpretations. (Continued on next page)111MultiplicationFanya 2/3 <3/4 3/4 = 2/6_ ___k”‘-izi—> W1fr__ __Takeaway 1”1’I ‘1 hI 1-1 - L&Ei I I____ _ _ ____ __________114’J”F’ I IDahlia 2/3 > 3/4 5/8> 7/12Take awayX1\ cr1 i4IjjD2J P// V?Z,Kasey 2/3 > 5/9i1 lIPeteI :‘(1I5/9 > 2/3I “Flu’ITQI,_ _ __ _ _?f> I,—I,,Lara1//iT I[fflIUi2/3 > 5/9TammyciII314? 2/3trackEl I (lost/,, /, of 2/3)2/3 = 5/91/3 = 2/65/8 = 7/122/4 = 4/8Figure 4.04 (Continued) Examples of common fraction representations with regions as they relate tointerpretations.112Case 2. Equal parts are critical, but equal regions are not: partitioned a rectangle into 8instead of 9 parts, so an additional but separate ninth piece was drawn the same size as those within therectangle. The parts were then compared proportionally, but not the regions (See Pete, Figure 4.04).Case 3. Equal parts and equal regions are critical: partitioned a circle with the successivehalving algorithm into 8 instead of 9 parts, or a square with the hatched configuration into 9 instead of 8parts (See Lara & Dahlia Figure 4.04). Either the type of region or the partitioning strategy was thenaltered in order to achieve the desired number of equal parts.The students in the first two examples did not attempt to alter the partitions further. Nor did theycomment on the inequality of the parts or the regions. They simply did not appear to consider theirsolution to be problematic. Differences in students’ solutions suggested differences in their beliefsabout which characteristics of units were critical in a representation of common fraction comparisons,rather than differences in technical abilities to partition the regions.The ways in which regions were used to represent common fractions varied widely (see Figure4.04). However, individual students appeared to hold relatively consistent beliefs about criticalattributes of representations based on regions. These beliefs seemed to result in representational“procedures” or algorithms. For example:1. Students had consistent procedures regarding how regions were used as a spatial frameworkfor representing units (E.g., Marlene, Connie, Derek & Fanya, Figure 4.04).2. Students expressed a belief that equal regions were important when comparing commonfractions. However, units within the regions were interpreted as measures of numerositv (E.g., Marlene& Fanya).3. Students’ expressed beliefs that equal parts were important when representations wereinterpreted as measures of numerosity (see Connie & Derek). As well, equal parts were compared interms of quantities of area without including comparisons of equal regions (see Pete).None of the students in the previous examples constructed a standard interpretation ofcommon fractions based on measurement of area. Yet each student imposed some procedures on theform of their representations that harkened to attributes emphasized in area-based instructional113representations. Beliefs about the form of representations were disassociated from the measurementframework on which instructional representations were based, yet their beliefs mirrored selectivecharacteristics of such instructional representations.Whether students thought in terms of measures of numerosity or measures of area was notalways explicit, particularly when individual common fraction was represented with the take-awayinterpretation of a common fraction. The language of the take-away interpretation is sufficientlyambiguous that the verbal explanation of the representation could imply measures of numerosity orarea. Likewise, a representation might contain all physical characteristics consistent with a standardarea-based representation, but be thought of by the student in terms of units of numerosity alone.Edwin’s contiguous representations of individual common fractions and comparisons ofcommon fractions provide an example of the ambiguities associated with representations based onregions and the take-away interpretation. He consistently expressed a take-away interpretation in acontext of sharing pie (see Figure 4.05). Food was shared equally, and pieces were eaten and left over.However, his apparent belief in the need for all parts to be equal when representing a single commonfraction (4/5) was not reflected in the comparison task. Neither the size of the left-overs nor the equalityof the original pies was a concern. However, the notion that people in a group receive fair shares wasconsistent. The fair shares of pies were not related to his final comparison of 3/4 and 2/3. Instead, 3/4was larger simply because of its greater numerosity. His social meaning of common fractions wasdisassociated from his mathematical meaning of common fractions. This could not have beenanticipated from his take-away explanation and contiguous representation of four fifths. Hisrepresentation of 4/5 and the language he used to explain the representation had the appearance of astandard representation and interpretation of a common fraction.114Task: Explain 415you have five pieces of a pie and you take four out.But one piece is like big (i.e., he drew it too big), but it is just the same size. You have five pieces hereand one guy eats this piece, so this one is gone (shades pieces as he talks). Another guy eats thispiece so this one goes. Another one eats this so this one goes. Another one eats this one andanother one eats this one. All the people have aten (sic) it so they give it to the cat (i.e., the left-overpiece).”Task: Compare 2/3 and 3/4“...they only got a half a piece of pie, like the man made a mistake at the wrong house, so they gavehim only a half a pie There was only four people in the family so they had to make it like this. So theygo and eat three and a guy wasn’t hungry so he didn’t eat his piece. That would be three fourths.Three fourths Two thirdsHere is the two thirds. They delivered a whole pie to the other house. So they just split it in half andthen two thirds. So they ate this piece (amount shaded).”Figure 4.05 Representations of common fractions with alternative beliefs of equal parts:Edwin (Grade 5)In summary, it could not be assumed that area measurement played a role in representingquantitative relationships between units even when a student’s contiguous representation of acommon fraction conformed to standard area-based representations. Students expressed beliefs thatthe parts, the regions, or both ought to be equal in size even when measurement of area played noquantifying role. These ideas about the form of the representation mirrored characteristics ofcontiguous representations probably stressed during instruction. Only when comparisons of commonfractions were represented was the quantitative role of measurement of numerosity or area explicit. Inthe comparative context, some students integrated characteristics of units of area measurement as115attributes critical to their representations of common fractions, but the consistency with which theyattended to these attributes varied from task to task.Characteristics of discrete forms of representations. Discrete representations of a commonfraction generally mirrored characteristics of the student’s contiguous representations (see Figure4.06). When representing a common fraction with a take-away interpretation, the language, the actions,and the results of discrete and contiguous representations are the same. For example, in Edwin’srepresentations of 3/4 in Figure 4.06, the action associated with the meaning of the units was the same:4 pieces of pie or 4 chairs, 3 were eaten or used, and one remained. These actions were expressed bythe same phrases: “3 of the 4 — were .“ Whether a student considered measures of numerosityand measures of area was of no immediate consequence when explaining the meaning of an individualcommon fraction with a take-away interpretation. The same could be said of these forms ofrepresentations applied to comparisons of common fractions with like denominators. Changes in theproximity of the units still would not alter the result of the comparison.*Figure 4.06 Discrete and contiguous representations of three fourths: (Edwin, Grade 5)Students’ discrete and contiguous representations of comparisons of common fractions withunlike denominators also differed only in the proximity of the units. No students representedcomparisons with discrete sets by establishing the whole unit as a denominator common to bothfractions (see Figure 4.07). Considering those who used a take-away interpretation, each student wasable to rationalize their comparisons with discrete representations without facing any conceptualconflict. Edwin simply compared the numerosity of units, regardless of the form of the representation.Pete incorporated attributes of area in the discrete units to maintain a 2:1 part-part comparison in his116representations. Dahlia interpreted both representations as equivalent to a half and thereby ignoredquantitative disparities in her representation.zir:jØ HHHVCDcc H H V WDDOD2/4 = 4/8 3/4 > 2/3 2/3 < 5/9Dahlia Edwin PeteFigure 4.07 Discrete representations of comparisons of common fractions.Functions of Representations of Common Fractions in Primary Repertoires.The function of representations in students’ primary repertoires was categorized in Table 4.07by (a) the extent to which a representation expressed relationships between different units (M, B, T orN), and by (b) the interpretation students gave to their representations. There was no singularrelationship between some interpretations students used and the extent to which studentsrepresented relationships between multiple units. Instead, a one-to-many relationship existed betweensome interpretations and representations of unit relationships. For example:1. Students’ cardinal values and multiplication interpretations were explained necessarily withmono-relational and bi-relational representations respectively. In contrast, comparisons with a take-awayinterpretation were explained with either mono-, bi-, or tri-relational representations, as in the followingexamples:hhhhhhhMono-relational Bi-relational Tn- relational3/4>2/3 5/8=7/12 2/4=4/8(Edwin) (Lar) (Tammy)1172. Similarly, tn-relational representations necessarily expressed students’ comparisons with a take-away interpretation. In contrast, mono-relational representations expressed cardinal values, ortake-away interpretations. Bi-relational representations expressed either a multiplication or take-awayinterpretation as in the following examples.“4/4 equals 1 whole”Bi-relational Bi-relationalMultiplication Take-away3/4 (Fanya) 3/4 (Kit)Six students constructed mono-relational representations of units, five of whom did soregardless of the task. For most of these students their mono-relational representations wereassociated with a cardinal values interpretation of common fractions. However, two students (Edwin andDahlia, Table 4.07) also constructed mono-relational representations with the take-away interpretationby comparing the number of pieces removed or left over.Bi-relational representations of the comparison of common fractions with the take-awayinterpretation took on one of two forms: (a) the student did not attend to the relationship betweendifferent sizes of the unit fractions, or (b) the student did not attend to the relationship between the twowholes, as in the following examples.“Equal, both 1/2 plusabit mor& 1/3=2/9, 5/9 >2/3a. (Lara) b. (Pete)For two students, the bi-relational representations were associated with a multiplicationinterpretation. The explanation from these students was insufficient to determine a genesis for thisinterpretation. Possibilities are that it might derive from their experience with (1) the multiplicativec)118algorithm for generating equivalent fraction or (2) the cross-multiplication algorithm for comparingcommon fractions.2When students constructed tn-relational representations for the comparison of fractions,generally each of the common fractions was represented with a separate region. The approximateequality of two regions and units within each region established a basis for judging the comparativedifference in the area representing each common fraction. This method limited the need to representunits nested in units, and the need to relate fractional units to a common denominator. Only onestudent (Tammy) represented fractional units nested in fractional units within a single region.Not all students represented common fractions with units. Instead, they (Brock, & Lolande)used gross quantities within a region to express ideas about the relative sizes of common fractions. Ineach case there was a rudimentary notion about common fractions being part of something. However,the representation of the whole unit was not a critical feature for expressing the differences in the sizesof common fractions. Instead separate regions of different sizes were sufficient to represent ideasabout the comparisons (see Brock, Figure 4.04).Summary of Characteristics of Primary Repertoires of Representations of Common Fractions.Considering the forms and the functions of the representations constructed by students duringthe uncued generative interview, a number of features characterized students’ primary repertoires ofrepresentations of common fractions.With regard to the form of representations, characteristics were:1. The exclusive or prominent use of regions, in particular circles as a spatial framework.2. Wide variations in the ways in which regions were used as a framework to represent comparativequantities of common fractions, including the representation of discrete or contiguous units as well asrepresentations without units.2. A similar procedures was used by a student in Hunting (1983, pp. 193-194). However, Huntingdoes not identify the procedures as an alternative interpretation of common fractions derived from someprevious experience. Instead he explains the behaviour as a “freely invented algorithm.”1193. Physical attributes such as equal regions or parts were believed to be important regardless ofthe interpretations of common fractions and the measurement system referenced in the representation.4. Selective and variable attention to attributes of units area measurement when constructing andinterpreting representations of common fractions.5. Similarities between a student’s discrete and contiguous representations such that the onlysubstantial difference was in the proximity of the units.With regard to the functions of the representations constructed:1. Interpretations of common fractions influenced but did not always determine the extent towhich students represented multiple relationships between units.2. The implicit relationship of fractional units to the whole unit was not made explicit in theirexplanations when using a take-away interpretation. Explanations of the meaning of single commonfractions were given generally with mono-relational representations.3. Comparisons of common fractions were explained with take-away interpretation by just over halfof the students. However, mono-. bi- and tn-relational representations were constructed to explaincomparisons with this interpretation.4. Explanations with other interpretations necessarily involved mono-relational or bi-relationalrepresentations, or representations without explicit units.Characteristics of General Repertoires:Representations of Common Fractionsin Other Material SettingsThe questions addressed in this section are:1. To what extent are the characteristics of representations of common fractions in students’repentoires influenced by the material setting within which representations are constructed?2. How extensive are their general repertoires with regard to the variety of forms ofrepresentations used to explain common fraction tasks.a. To what extent are particular forms of representations excluded from their repertoires?120b. To what extent do students’ general repertoires of representations include forms ofrepresentations based on attributes of length or area?The other three interview settings (cued generative, linear interpretive, and area interpretiveinterviews) were designed to explore whether or not the primary repertoires of representationsgenerated in the uncued interview represented the extent of students’ repertoires. In this section, thestudents’ responses to the common fraction tasks during the other three interviews are compared andcontrasted to the characteristics of the students’ primary repertoires of representations generatedduring the uncued generative interview.The representations of common fraction tasks constructed during the cued generativeinterview which differed from those generated during the uncued interview were considered to indicateextensions of students’ repertoires. As well, materials which students rejected during the cuedgenerative interview were considered to be one indication of limits of repertoires. Students’acceptance or rejection of beginning diagrams (see Figures 3.01 and 3.02) during the linear and areainterpretive interviews was considered to be another indication of limits of repertoires.This section is organized in two parts. In the first part, the primary repertoires are compared tothe forms and functions of the representations constructed during the cued generative interview. Inthe second part, the responses in all of the interview settings are compared and contrasted.Representations Constructed in the Cued Generative Interview Compared to the Uncued GenerativeInterview.Table 4.09 was designed to illustrate students’ representations constructed during the cuedgenerative interview. Also included is a summary of their responses during the uncued generativeinterview which represent their primary reperloires. The format of the table is similar to that of Table4.07. The major categories for classifying the form of their representations are represented in thecolumns of this table. These categories include the spatial framework of the representations, (sets,lines and regions), and the form of units within the spatial frameworks, (discrete, contiguous, andundefined). For each form of representation constructed by a student during the cued generativeinterview, the function of the representation with regard to unit relations is coded within the table.121Table 4.09Form. Function and Interpretation of Representations of Common Fractions: Cued Generative Interviewwith a Summary of Primary RepertoiresPrimary Cued Interview tasks by form of representationRepertoire s(Uncued)A/B A/B vs C/BForms,Functions & Cued InterviewInterpretations Interpretations DU DL DR CL CR UL DR DS DL DR CL CR DL DRUndefined UnitsRs Brock Rs N N NId Lolande Id NDiscrete &ContiguousB Rn James HnTaCv B . . . M . . MB Cv Marlene Cv M . . . M . . M . . . BB TaCv Edwin TaCv B . . . M . . MB Ta Pete Ta B . . . BBT TaMu Dahlia TaMu B . . . TBExclusivelyContiguousM CvRs Connie Cv MM Cv Derek Cv M . . . M . . B . . . MB Mu Fanya Mu? . . . . MB Ta £j.. Ta . . . . BB Ta Kasey Ta . . . . BTBTa Lara Ta . . . .BTT Ta Coran Ta . . . . TBT Ta Tammy Ta . . . . TNames of grade 5 students are underlined.Dots mean that the form was not constructed; blanks that the task was not given.Units Spatial Framework Unit RelationsD = discrete S = sets M = mono-relationalC = contiguous L = lines B = bi-relationalU = undefined R = regions N = no unit relationshipInterpretationsCv = cardinal values (numerator Id = inverse denominatoror denominator) Mu = multiplication, a/b = a x bHn = 1/2 of a number Ta = take-away, a/b = b - aRs = relative sizes122The summary of the characteristics of students’ primary rerertoires is recorded to the far left ofthe students’ names. The function and interpretations of their representations from the uncuedgenerative interview precede each student’s name.The students are grouped according to the dominance of a form of representation in theirprimary repertoire as in Table 4.07. Within these groups they are ordered according to the extent towhich they represented relations between different units during the cued generative interview.Influence of material setting on forms of representations generated. As can be seen in Table4.09, most students used forms of representations during the cued generative interview which wereconsistent with the forms of representations in their primary repertoires. Only 3 students (Brock,Connie, & Derek) generated a form of representations which was not used in their primary repertoire.With few exceptions, the importance of regions as a spatial framework continued despite thepresence of a variety of materials which might have suggested alternative forms of representations.Furthermore, of the 14 students who constructed representations with regions as the spatialframework, 12 students used circles at least once and 7 students used circles exclusively ordominantly (see Table 4.10). These response patterns are similar to those observed during the uncuedgenerative interview.123Table 4.10Extent to which Different Regions were used to Construct Representations of Common Fractions(Cued Generative lnterview.Types of RegionsExtent n2 Circle Rectangle IrregularExclusive 7 5 2Dominant 2 2Secondary 7 5 7Note. 1. Exclusive means all, dominant means more than half,secondary means half or less of a student’s discreterepresentations used the configuration.2. Total n = 15 Students are counted more than once in the“dominant” and “secondary” categories.When different forms of representations were generated by a student to explain a task, therepresentations generally shared a common measurement framework within which some spatialfeatures changed, but the essential attributes attended to by the students did not change. Studentswho attended only to number and not to sizes of parts in contiguous representations, defined discretesets only numerically. For example, Marlene independently generated the first two representations inFigure 4.08. When asked later if she could use the line, Marlene proceeded to construct the thirdrepresentation. These representations are built around a common framework, and are distinguishedonly by changes in the spatial arrangement of units or the objects used as units, not by differences inattributes of measurement. Conversely, those who attended to differences in the size of the parts withcontiguous representations, include this property in their discrete representations. Larger objects wereused in the set which represented the fraction with the smaller denominator. There was no measuredrelationship between the sizes of objects which would represent a ratio between the two fractionalunits, but the attribute of differences in sizes of fractional units still was critical to the discrete1 2 4r e p r e s e n t a t i o n s . D i s c r e t er e p r e s e n t a t i o n s c o u l d b ed e r i v e df r o m t h e s e p a r a t i o n o fu n i t sw h i c h w e r ep r e v i o u s l yc o n t i g u o u s .I n t h es i m i l a r v e i n , B r o c k ’ s i n c l u s i o no f r e p r e s e n t a t i o n s b a s e do n a l i n e r e l a t e da l s o t o af r a m e w o r kc o m m o n t o a l l o fh i s o t h e r r e p r e s e n t a t i o n sb a s e d o nr e g i o n s . Q u a n t i t i e sw i t h i nr e g i o n si n v a r i a b l yw e r e c o m p a r e db y j u d g i n g t h ed i s t a n c e a l o n gt h eh o r i z o n t a l d i m e n s i o n .Q u a n t i t i e so fh o r i z o n t a ll e n g t h p r o v i d e da c o m m o nf r a m e w o r k f o r h i sr e p r e s e n t a t i o n sw h e t h e r b a s e d o n r e g i o n so rl i n e s .0o or m .-ggF i r s tS e c o n dT h i r dF i g u r e 4 . 0 8E x a m p l e so f d i f f e r e n tr e p r e s e n t a t i o n so ft h r e e - f o u r t h su s i n gac o m m o nf r a m e w o r k :( M a r l e n e , G r a d e 5 )E x t e n to fr e p e r t o i r e s .T h e r ew e r e s e v e r a lw a y s i nw h i c hs t u d e n t s ’ r e p e r t o i r e sw e r el i m i t e di nt h e s e n s et h a t p a r t i c u l a rs p a t i a lf r a m e w o r k sa n d f o r m s o fr e p r e s e n t a t i o n sw e r ee x c l u d e d b ys t u d e n t s .M o s ts t u d e n t sp o s i t i v e l yr e j e c t e dt h el i n e a s as p a t i a lf r a m e w o r kw h e nd i r e c t l yq u e s t i o n e d .T h o s ew h os a i d t h e yc o u l du s et h e l i n ec o n s t r u c t e da r e p r e s e n t a t i o ne i t h e r w i t hd i s c r e t eu n i t so n t h e l i n e ,o r w i t ht h el i n e a s au n i t i na s e t o fl i n e s ,o r w i t h t h el i n e a s ab o u n d a r yo fa r e g i o n . N os t u d e n t su s e dl i n es e g m e n t sa s u n i t s .M a t e r i a l sw e r er e j e c t e do r a c c e p t e di nt e r m so ft h ee x t e n tt o w h i c ha t t r i b u t e s o ft h em a t e r i a l sc o u l d b eu s e d t of i t t h e f o r m so f r e p r e s e n t a t i o n ss t u d e n t sa l r e a d yh a d g e n e r a t e di n d e p e n d e n t l y .I fp r e v i o u sr e p r e s e n t a t i o n sw e r e d i s c r e t et h e n m o s tm a t e r i a l sw e r ev i e w e da s p o t e n t i a ld i s c r e t e u n i t s .H o w e v e r ,i fp r e v i o u sr e p r e s e n t a t i o n sw e r eo n l yc o n t i g u o u s ,t h e nl i t t l e o f t h em a t e r i a l sw e r e“ s u i t a b l e ”e x c e p tt h ec i r c u l a rf i l t e r p a p e r s .I n t h e l a t t e rc a s e b o t hd i s c r e t er e p r e s e n t a t i o n sa n dr e p r e s e n t a t i o n sb a s e do n o na l i n e a rf r a m e w o r kw e r e e x c l u d e db yt h es t u d e n t s .F u n c t i o no fr e p r e s e n t a t i o n si nb o t hg e n e r a t i v ei n t e r v i e w s .M o s t s t u d e n t sw e r ec o n s i s t e n ti nt h e i ri n t e r p r e t a t i o n so f c o m m o nf r a c t i o n sd u r i n g b o t hi n t e r v i e w s( s e e T a b l e4 . 0 9 ) . T h e ya l s o125constructed representations which expressed relationships between different units to the same extentduring both interviews. The extent to which students’ constructed either mono- or bi-relationalrepresentations were relatively stable. Only the incidence of tn-relational representations variedsomewhat from one interview to the other.Take-away/Cardinal ValuesNumerator as SubsetsUncuedCardinal ValuesNumerator as Separate SetsCuedBi-relationalLara(7/12 vs 5/8)Partitioned CirclesUncuedf (ii //1 ‘iI”J (ii__v/Inch Squared PaperCuedFigure 4.09 Examples of differences in mono- and bi-relational representations constructedin each generative interviewSome characteristics of students’ representations of unit relationships differed from oneinterview to the next even when the general unit relationships did not differ. Differences related eitherto the instability of a student’s interpretations of common fractions (see Edwin, Figure 4.09), or tocharacteristics of materials which placed constraints on the results of the representational process (seeLara, Figure 4.09).Mono-relationalEdwin(5/9 vs 2/3) @© ,coco cc —0 Cco1261. Instability of a student’s interpretations. In the uncued generative interview Edwin used thetake-away interpretation to represent each common fraction then based his comparisons on a cardinalvalues interpretation (See Figure 4.09). However, the take-away interpretation was not used during thecued generative interview. Instead, the numerators and denominators of each fraction to be comparedwere represented as separate sets. These shifts in interpretations, reflected in the structures of therepresentations, did not mean that the function of the representations differed. In both cases, the unitswere represented mono-relationally and final comparisons were based on the same rationale.2. Characteristics of materials. The processes of partitioning regions into units and combiningdiscrete units into regions were assumed by students to be equivalent and interchangeable whenrepresenting comparisons of common fractions with unlike denominators. Other experience ofrepresenting whole number relationships, and of representing individual common fractions wouldsupport the student’s suppositions about the equivalence of the two processes. However, thesematerials were governed by properties of discrete representations in common fraction contexts. Thesize of each part is pre-determined, not the size of the regions and therefore a cube or square could notrepresent two different unit fractions. Unless a common denominator was represented, all suchrepresentations would be bi-relational (see Lara’s cued generative representation in Figure 4.09).Comparison of Repertoires Reflected in the Generative Interviews With Responses in the InterpretiveInterview SettingsOf particular interest in this section is the ways attributes of length or area measurement mightbe considered by students to be critical features in representations of common fractions. During thegenerative interviews representations based on length measurement were notable by their absencefrom students’ repertoires. Furthermore, attributes of area-based representations were referenced bymost students but were attended to inconsistently. The question remains as to how students wouldinterpret and use beginning diagrams (see Figures 3.01 and 3.02) which were related to length or area.What would be the critical features of the beginning diagrams to which students would attend whenevaluating their usefulness for representing common fractions and their comparisons, and would thestudents interpret and use the beginning diagrams with reference to attributes of length and area?127Table 4.11 was designed to illustrate the forms and functions of the students’ responses to theinterpretive interview tasks. It also allows for a comparison of these responses with summaries ofstudents’ responses during the generative interviews. The format of the table is similar to that used inTable 4.09. The summary of characteristics of students’ primary repertoires, including the dominance offorms of representations as well as the functions of their representations, is recorded to the far left ofthe students’ names. The interpretations used in all interviews are summarized after each student’sname.Responses to the common fractions tasks from the cued generative and the two interpretiveinterviews are presented in the body of the table. Responses to the single common fraction tasks areindicated by lower case letters, and responses to the comparison of common fraction tasks are indicatedby upper case Petters.The students are grouped according to the dominance of a form of representation in theirprimary repertoire as in Table 4.07. Within these groups they are ordered according to the extent towhich they represented relations between different units during the four interviews.Forms of representations in the linear interpretive interview setting. As can be seen in Table4.11, 8 of the 15 students accepted linear beginning diagrams as a possible spatial framework forconstructing a representation of the common fraction or comparison of common fraction tasks (DL, CL &UL). Most students who used linear beginning diagrams interpreted units on the line to be discretepoints (DL). In addition, one student (Pete) shifted between defining units as points and line segments(CL).The presence of the line itself seemed to conflict with some students’ beliefs about commonfraction representations. Regardless of how students used the beginning diagrams, only two students(James & Fanya) did so without question. Some used the line to construct a discrete representationwhile expressing a preference for regions as a spatial framework; others constructed alternativerepresentations without a line at some point during the interview. In addition, seven students eithertransformed the lines into regions (see Figure 4.10), or drew all representations on other parts of the128Table 4.11Form and Function of Representations of Common Fractions: Cued Generative. Linear Interpretive.and Area Interpretive Interviews with a Summary of Interpretations Over All Interviews and a Summary ofPrimary ReDertoires.Interview setting by forms of representationsSummaryof Summary Cued Generative Linear Interpretive Area InterpretivePrimary ofRepertoire Interpre(Uncued) tations DS DL DR CL CR UL DR DS DL DR CL CR DL UR DS DL DR CL CR DL DRUndefinedUniteBrock Rs nnN n nLolande Id N N N NDiscrete &Contiuousm James HnCvTa mM . . . m . . . m m . . . mMmM Marlene Cv mM . . . mM . . mM . . . m . . . . m . NM Edwin TaCv mM . . . m . . . mbM . . b . . . . . . mMTmBT Dahlia TaMuId B . . . TB . . . . . . B . . . . . . BB Pete Ta B. . . B. . . B. T T . . . . . .BTContiuousM Connie Cv m N M . . M .mM Derek Cv mM . mM . . . . . M . . . . . mMB Fanya MuTa . . . . M . . . mM . . . . BB Kasey Ta . . . . B . . . B . B . . . . TT Coran Ta . . . . TB . . . B . . . . BTB Lara Ta . . . . BT . . . . T . . . . Tb Jj Ta . . . * b . . . .bT . . . . TT Tammy Ta . . . . T . . . . T . . . . TNote. Names of Grade 5 students are underlined.Unit Relations: upper case indicates comparison tasks, lower case indicates single fraction tasks.Units Spatial Framework Unit RelationsD = discrete S = sets M(m) = mono-relationalC = contiguous L = lines B(b) = bi-relationalU = undefined R = regions T(t) = tn-relationalN(n) = no unitsInterpretationsCv = cardinal values (numerator Id = inverse denominatoror denominator) Mu = multiplication, a/b = ax bHn= 1/2 a number Ta=take-away,aib=b-aRs = relative sizespaper. These responses confirmed a general absence of length as a spatial framework forrepresentations of common fraction in some students’ repertoires.129()t_1 I I I3 2/3 (Kit)LrL’’’EZEI1/3 compared to 1/4 (Derek)kV/M/,V//(1 z 3L_t• ‘ ) I I J •‘Figure 4.10 Examples of student’s transformations of lines into regions.Figure 4.11 Deriving units of length analogously from sectors of a circle (Pete, Grade 7)The general absence of length measurement as a form of representation in students’repertoires is underscored by some students’ responses to linear beginning diagrams compared to theforms of representations they previously generated. First, the students who constructed onlycontiguous representations based on regions during the generative interviews responded to the linearbeginning diagrams in one of three ways. They either (1) used the marks along the line as discreteunits, (2) transformed the line into a region by drawing boxes around it (Figure 4.10), or (3) drewpartitioned regions elsewhere on the paper. They did not associate a contiguous unit oftwo-dimensional space analogously to a contiguous unit of one-dimensional space. Second, the only130student (Brock) to use a direct comparison of length as the critical feature in his representations ofcomparisons of common fractions during the generative interviews rejected the linear beginningdiagrams altogether. His representations of common fractions did not include a notion of units, hence,length measurement as opposed to direct comparisons of length did not fit his repertoire ofrepresentations. Third, the one student (Pete) who eventually identified units as line segments usedan indirect argument to do so. The line segments described as analogous to sectors of a circle ratherthan described solely on the basis of units of linear measurement (see Figure 4.11). Pete also relateddiscrete units analogously to parts of regions, incorporating attributes of area in his discrete units.Hence, no students directly considered units of length to be a form of representation for commonfractions. Their responses to the linear beginning diagrams were grounded on the characteristics oftheir repertoires of representations generated in the previous interviews.The patterns of responses during the linear interpretive interview, collectively, confirmed theevidence from the generative interviews that suggested that forms of representations based on lengthmeasurement were not part of students’ repertoires, and that lines as a spatial framework similarly wereexcluded from the repertoires of nearly half of the students.Forms of representations in the area interpretive interview setting. Turning now to thestudents’ responses during the area interpretive interview, all students accepted some beginningdiagrams and constructed representations with regions as a spatial framework (Table 4.11, DR, CR, &UR). However, the use of regions as a spatial framework for common fraction representations did notimply necessarily that the representations were based on area measurement. Measures of numerosityformed the basis for mono-relational representations of the cardinal values and take-awayinterpretations of common fractions, and featured in bi-relational representations of comparisons ofcommon fractions. In most instances these representations were constructed with regions as thespatial framework.There were four criteria used by students to support their judgements of the appropriateness ofthe beginning diagrams for constructing representations of common fractions. The first two criteriarelated to limits students imposed specifically on characteristics of the whole unit. The second two131criteria related to limits students imposed on the characteristics of units in general, whether used aswhole units or parts of whole units. Patterns of responses related to these criteria as well as the formand function of representations constructed during the area interpretive interview are illustrated in Table4.12.1. Whole units must be separate. not be regions nested in regions (S in Table 4.12). Thebeginning diagrams were judged to be inadequate for the comparison of common fraction tasksbecause there were not two separate regions, one for each common fraction. There were two ways inwhich students circumvented this problem. Some students partitioned a whole diagram to represent acommon fraction, ignoring the potential units already in the diagrams then drew a second region torepresent the other fraction in the comparison task. Others removed parts of the beginning diagramwhich were extraneous to the representation.2. Whole units must initially be empty regions (Em in Table 4.12). Representations of commonfractions could only be constructed by partitioning an empty region into fractional units not bycombining fractional units to make a whole unit. Students argued that the lines in the diagram should beremoved to create an empty region, or they partitioned a small region within the beginning diagram torepresent a common fraction.3. Units should be orderly for aesthetic reasons (A in Table 4.12). Students rejected beginningdiagrams with reasons such as “it was too wiggly,” or “it needs to be squared up,” or “it could be usedbut it’s a really bad idea.” However, in these cases, no direct reference was made to problems ofrepresenting quantities with unequal units. Perceptual orderliness was being sought.4. Units should be equal for quantitative reasons (Qu in Table 4.12). Students rejected beginningdiagrams with unequal parts with statements which directly referenced problems in comparing quantitiessuch as, “Should be even in order to compare,” or “You couldn’t really compare them because .. .theywould be all different sizes.”As can be seen in Table 4.12, some students used none of these criteria. They simplyaccepted all beginning diagrams as a reasonable framework for constructing their representation ofcommon fractions (N). Others used aesthetic criterion intermittently for evaluating general unitsCriteria: Whole UnitsS = separate regionsEm = empty regionsSpatial FrameworkCriteria: General UnitsN = not reject unequal unitsA = aesthetic criteriaQu = quantitative criteriaUnit RelationsM(m) = mono-relationalB(b) = bi-relationalT(t) = tn-relationalN(n) = no unitsUnitsD = discreteC = contiguousU = undefinedInterpretationsCv = cardinal values (numeratoror denominator)Hn = 1/2 a numberRs = relative sizesId = inverse denominatorMu = multiplication, a/b = a x bTa = take-away, a/b = b - a132Table 4.12Criteria Used to Evaluate Beginning Diagrams and the Form and Function of Representations (AreaInterDretjve Interview) with a Summary of Interpretations and Reøertoires over all Interviews.Surnniary Evaluative Criteria RepresentationsofGeneral InterpreRepertoire tations S Em N A Qu DS DL DR CL CR UL URUndefinedUnitsBrock Rs . . nLolande Id . . # # NDiscrete &ContiguousConnie Cv . . . M . .Marlene Cv . . . m . MJames HnTaCv . . # # m . mMEdwin TaCv . #1 # # . . . . . rnMTDerek Cv . . # . . . . .rnMFanya MuTa . #1 . # . . BDahlia TaMuId . # . BCoran Ta #1 . . . . . . . BPete Ta . #1 . . . . . .BTKasey Ta . . . . # . . . . TContiguousTa . . . . . . . TLara Ta . . . . # . . . . TTaimriy Ta # . . # . . . . TNote. Names of Grade 5 students are underlined.1. Criteria was not applied throughout interview.Lower case indicates single fraction tasks, Upper case indicates comparison tasks.# indicates the student used the evaluative criteria.S = setsL = linesR = regions133(e.g., James). As well, the criterion which limited the whole unit to an empty region was imposed bysome students only during part, not all of the interview.The evaluative criteria applied to general units (N, A & Qu) seemed to relate to students’interpretations of common fractions and the functions of their representations. Those who did not usea quantitative criterion for evaluating the beginning diagrams constructed either mono-relationalrepresentations or representations which did not incorporate units. They interpreted comparisons ofcommon fractions in terms of the cardinal values of numerators or denominators, or in terms of theirrelative size. The equality of regions or parts of regions were not critical to their interpretations. Hence,their evaluations of beginning diagrams were consistent with their representations and interpretationsof common fractions. They either did not discriminate between physical characteristics, or discriminatedonly from an aesthetic point of view.Students who used a quantitative criterion to support their selection and use of beginningdiagrams constructed bi- or tn-relational representations. These students explained and representedthe comparison tasks generally as a comparison of quantities of area. All but one student (Fanya)represented a take-away interpretation of common fractions during this interview. However, as thebi-relational representations indicate, not all of these students were consistent in their attention toattributes of area measurement. Some students attended to equal regions but not equal parts, whileothers attended to equal parts but not equal regions in their representations or interpretations of therelationships between the different units.Attributes of Area Measurement in General Repertoires of Representations of Common Fractions.Table 4.13 was designed to examine the extent and consistency with which attributes of areameasurement were critical characteristics of representations in students’ general repentoires ofrepresentations of comparative common fraction tasks. For this purpose, students’ responses to thecomparative tasks in each of the four interviews were coded in terms of the degree to which all units intheir representations were contiguous (Columns N, P, T). The extent to which contiguous units wereapproximately equal in area is coded in the body of the table. The degree to which students maintained134the equality of whole units is indicated by upper case letters, and the degree to which studentsmaintained the equality of fractional units is indicated by lower case letters. In addition, the criteriastudents used to evaluate the characteristics of beginning diagrams with regard to the equality of unitsduring the area interpretive interview are summarized in the right hand column. Summaries of students’interpretations of common fractions and their general repertoires of representations also are included inthe table.Students are grouped in the table on the basis of two criteria: first, according to the extent towhich contiguous representations characterize their repertoire of representations of comparative tasksevidenced during the generative interviews; and second, according to the extent to which studentspersisted in constructing contiguous representations during the two interpretive interviews. Forexample, during the generative interviews students in Group 3 independently constructed bothdiscrete and contiguous representations, whereas those in Group 4 constructed only contiguousrepresentations. On the other hand, students in Group 4 constructed discrete representations in oneof the interpretive interviews whereas those in Group 5 persisted in constructing contiguousrepresentations in all interview settings.Table 4.13 serves to illustrate two general patterns related to the the extent and consistencywith which attributes of area measurement were critical characteristics of representations of comparativetasks in students’ repertoires. The first pattern concerns the association of interpretations of commonfractions and characteristics of the forms of representations in students’ general repertoires. Thesecond concerns the consistency with which students constructed equal contiguous units to representboth whole and fractional units.Interpretations of common fractions and characteristics of the forms of representations ingeneral repertoires. All students included representations based on regions while some also used setsduring the uncued generative interview. However, considering general repertoires, there werestudents whose representations tended to be based on discrete units in sets rather than contiguousunits in regions (see Group 2, Table 4.13). When these students did construct representations withcontiguous units, attributes of area measurement were generally ignored or attended to inconsistentlyTable 4.13135Contiguity and Equality of Units: Representations of Comparison of Common Fractions Tasks in AllInterviewsInterview settings by extent of contiguity of unitsUncued Cued Linear AreaSummary On Line Off Lineof CriteriaGeneral Interpre- forRepertoire tations N P T N P T N P T N P T N P T EqualityUndefinedUnits1. Brock Rs N . . N . . . . . N . . N . . NLolande Id N. . N. . N. . N. . N. . NADiscrete &Cont iouous2.Connie Cv .eE . ft . . ft . . ft . . ft . . NMarlene Cv E . . ft uU . . . . ft . . . uU . N11 1James HnTaCv ft u . ft . . ft . . . . . . 1U . NAEdwin TaCv ft iU . ft . . ft . . . . . . ii . NA13. Derek Cv . eE . ft ii . . . . . e . . iE . ADahlia TaMuId . IE . eU eE . . . . . eT . . uE . QuPete Ta eU eU . eU eE . eU eE . . . . . el . Qu4. Fanya MuTa . eU . . eU . ft . . . . . . eU . QuKasey Ta . ii . . IE . iE . . . jU . . eE . QuCoran Ta . eE . . eE . el . . . . . . el . QuContiguous1 15.Jj Ta . e . . a . . . . .iE . .eE . QuLara Ta . eE . . el . . . . . eE . . eE . QuTammy Ta . . eE . . iE . . . . eE . . . eE QuNQI. Names of Grade 5 students are underlined.1. Common fraction task only, no response to comparative tasks.Contiguity of UnitsN = none contiguousP = partially contiguousT = totally contiguousInterpretationsCv = cardinal valuesId = inverse denominatorHn = 1/2 a numberRs = relative sizesCriteria for equal units(Area Interpretive)N = no attentionA = aestheticQu = quantitativeMu = multiplicationTa = take-awayCharacteristics of UnitsE = equal whole unitU = unequal whole unitI = inconsistente,u,i = fractional units# = no area referenceN = no units136(see codes i, u, I, U). In addition, when equal units were sought by these students, aesthetic rather thanquantitative criteria were used to support their judgements (see code NA). In nearly all instances, thesestudents interpreted the comparison of common fractions as comparisons of the numerosity of units inthe representations of numerators, denominators, or differences between numerators anddenominators. Units of area played no quantitative role in their representations. When contiguousunits were represented they were, in essence, discrete units which were incidentally contiguous.Hence, characteristics of the forms of representations in their general repertoires were consistent withtheir interpretations of common fractions.In contrast, students who consistently used a take-away or multiplication interpretation ofcommon fractions (see Groups 3, 4, & 5) more consistently constructed representations based oncontiguous units in regions. As well, they used quantitative rather than aesthetic criteria to justify theirrejection of diagrams with unequal parts (see code Qu), and were more consistent in their attention toattributes of area measurement. Among these groups (3, 4, & 5), however, the extent to whichrepresentations with contiguous units dominated in their general repertoires varied. Only the studentsin Group 5 excluded discrete representations from their general repertoires. The others did includesome discrete representations in their general repertoires.Contrasts between the general repertoires of students in Group 2, those in Groups 3 and 4,and those in Group 5, suggests a differentiation in students’ beliefs about forms of representations forcomparing of common fractions. The students in Group 2 essentially represented common fractionswith discrete units since contiguous representations were interpreted through attributes of discretemeasures. Measures of numerosity underlay all their forms of representations of units even thoughsome of these students explicitly stated that units in contiguous representations should be equal.There was generally no differentiation between systems of measurement regardless of thediscreteness or contiguity of units.In direct contrast, students in Group 5 used only regions as a spatial framework for area-basedrepresentations of comparisons of common fractions. They clearly differentiated their representations137from other possible forms of representations. Discrete as well as linear representations were explicitlyexcluded from their general repertoires.With the students who primarily used a take-away interpretation in Groups 3 and 4, their majorform of representation was one based on comparisons of area in partitioned regions. As well, images ofdifferences in sizes of contiguous units were used in discrete representations of comparisons torationalize unit relationships (E.g. Pete, Table 4.12). These students interpreted representations,regardless of their form, through some notions about measures of area. For them, units wereessentially represented as quantities of area which at times were incidentally discrete. This pattern ofresponses parallels and contrasts with repertoires of students in Group 2. In both cases, theirrepertoires contained different forms of representations most of which were interpreted through asingle system of measurement, numerosity in one case and area in the other.In summary, students’ repertoires included different forms of representations but these werenot differentiated clearly in terms of units and unit relations appropriate to each system of measurement.Instead, students essentially represented and interpreted comparisons of common fractions through asingle measurement system, either numerosity or area. Even though most general repertoires includedboth discrete and contiguous representations of comparisons of common fractions, both forms ofrepresentations generally were quantified in terms of the dominant system of measurement in astudent’s repertoire.Equality of contiguous units (whole and fractional units). The second patterns concerns theconsistency with which students constructed equal contiguous units to represent both whole andfractional units. As can be seen in Table 4.13, none of the students were totally consistent inconstructing contiguous representations in which all units conformed to properties of areameasurement. Nonetheless, there were differences among the students in the extent to which theyattended to ideas about equal measures. There are two different cases to be considered in this regard:(1) students whose representations were based essentially on measures of numerosity, and (2)students whose representations were based essentially on notions about measures of area.138First, considering the students whose representations were based on measures of numerosity,all of these students stressed that regions, parts of regions, or both should be equal, and includedthese characteristics in their representations at some instance during the four interviews. However,none of these students justified the need for equal parts quantitatively, and all but Derek attendedseldom or very inconsistently to these characteristics in their contiguous representations ofcomparisons. Nonetheless, they all had some notions about characteristics of contiguousrepresentations which reflected attributes stressed in instructional representations based on areameasurement. Students adopted and attended to characteristics of this form of representation withoutassociating them with the quantitative meaning assumed to be represented by educators.Considering the case of students whose representations were based on some notions of areameasurement, there were three ways in which students differed in their attention to the equality ofunits. First there were differences in the general consistency with which students attended to theequality of whole and fractional units. The students in Group 5 were most consistent. Each constructedunequal units in only one instance. Nearly all of the other students were inconsistent in at least threeinterview settings.Second, there were different patterns in the manner in which students were inconsistent. Forexample, Coran’s inconsistency in equating whole units was associated only with his representations ofthe algorithm for generating equivalent common fractions, Dahlia was inconsistent about the equality offractional units with the take-away representations and about the equality of whole units with themultiplication interpretation, and Kasey was more generally inconsistent with regard to whole andfractional units. These students appeared to have integrated elements of area measurement into theirforms of representations based on regions, but had not fully coordinated the relationships betweenmeasures of the whole units and measures of the fractional units when comparing common fractions.Third, some students did not construct parts of equal area but still attended to equal measures.When partitioning circles with a vertical configuration, students in Group 5 focused on equal distancesbetween the vertical lines as the representation of equal measures. With students in the other groups,139their inconsistent attention to equal whole or fractional units reflected different beliefs about thecharacteristics of representations necessary for comparisons of common fractions.Functions of Representations in Students’ General Repertoires.Turning again to Table 4.11, there are patterns in the functions of students’ representationsacross all interviews. These suggest limits in the complexity of unit relations which different studentsmight consider and represent for comparisons of common fractions.3 There were eight studentswhose representations fell invariably into one of three categories: no units, mono-relational units, &tn-relational units. The other seven students varied in the extent to which they represented unitrelationships.The six students whose repertoires were limited to mono-relational representations orrepresentations without units, represented units in a manner consistent with their interpretations ofcommon fraction. Their interpretations were either relative values, half a number, inverse denominator,or cardinal values interpretation. The two students who constructed only tn-relational representations toexplain the comparison tasks did so within a restricted repertoire of representations based only onpartitioned regions. Neither discrete nor linear units were accepted by them. This meant that they didnot face the problem of rationalizing representations of comparisons of common fractions with discreteunits with representations based on measures of area. Whether these students would haveconstructed tn-relational representations with discrete units is not known.The other seven students were unstable in their representations of unit relationships. Oneshifted between mono- and bi-relational representations as she shifted between a take-away and amultiplication interpretation. Another shifted from mono-relational representations to a tn-relationalrepresentation. The remaining five students shifted between constructing bi- and tn-relationalrepresentations of the comparative tasks.3. Only the comparison of common fraction tasks are discussed in this section because they are lessambiguous as a bench-mark of students’ representations and interpretations of unit relations.140There were four general features in students’ approaches to representations of comparisons ofcommon fractions associated with the instability of students’ representations of units relations. Thesewere (1) changes in interpretations of common fractions, (2) notions about the algorithm forgenerating equivalent common fractions, (3) different complexities in the denominators of comparisontasks, as well as (4) shifts in between measures of area and measures of numerosity. Each of these willbe discussed in turn.1. Shifts in interpretations of common fractions. The multiplication interpretation invariably implieda bi-relational representation, whereas the take-away interpretation was associated with all levels of unitrelationships. All of Fanya’s and some of Dahlia’s bi-relational representations were associated with themultiplication interpretation. Their mono- or tn-relational representations expressed a take-awayinterpretation of common fractions.2. Notions about the multiplicative algorithm for generating equivalent fractions. There were twoways in which multiplication was incorporated in representations of comparisons of common fractions.One was with the multiplication interpretation, and the other was with the representation of themultiplicative algorithm for generating equivalent fraction. Attempts to represent the equivalent fractionalgorithm resulted in a distortion in representation of relationships between fractional units. Themultiplication of numerators and denominators was associated with the repeated addition of units ratherthan the repeated partitioning of units. As a result, the second equivalent common fractions comparedto the first was larger in terms of both the numerousness of its fractional units arid the actual size of itswhole unit. For example, in Coran’s comparison of 5/9 and 2/3 during the area interpretiveinterview, he represented 5/9 and 2/3 followed by a representation of 6/9 as equivalent to 2/3 (seeFigure 4.12). The 2/3 was transformed into 6/9 by multiplying each numeral by three. He did notperceive as a conflict the fact that 2/3 and 6/9 were represented as different amounts of area while hestated that the fractions were the same. In contrast, when not considering the algorithm, he generallyconstructed tri-relational representations based on measures of area.141“ ‘.‘ /4’ / / If1 ‘Ii f , (1 ,‘6/92/3Figure 4.12 Representation of multiplicative algorithm for generating equivalent fractions(Coran, Grade 7)3. Different complexities in the denominators of comparison tasks. Some representationstrategies applied to tasks in which one denominator was a multiple of the other (e.g. 2/3 vs 5/9) or tasksin which denominators were not so related (e.g. 7/12 vs 5/8) were associated with some students’ shiftsbetween bi- and tn-relational representations. When students explicitly constructed representationswith common denominators, they did so only with tasks in which one denominator was a multiple of theother. There were two procedures used to explain the part-part relationship (3:1 for ninths and thirds,2:1 for eighths and fourths). First, there was the procedure through which the part-part relationship wasexplained indirectly with a numerical algorithm which resulted in a bi-relational representation. Second,there was the procedure through which the part-part relationship was explained directly with thepartitioning of one or two equal regions which resulted in a tn-relational representation. Withcomparisons of the form a/b vs c/d, students directly compared representations of each commonfraction. When differences in sizes of fractional units were relatively small, reliance on only perceptualevaluations of the representations led to bi-relational interpretations of units. With other tasks,perceptual evaluations led more easily to tn-relational interpretations of units. As well, when directperceptual judgements were accompanied by reasoning about relative sizes of denominators andrelative sizes of parts, students’ comparisons were more reliable.1424. Shifts between measures of numerosity and measures of area. The most stark example of theassociation of shifts between systems of measurement and shifts in the representation of unitrelationships is provided by Edwin. After constructing a mono-relational representation whichsupported his judgement that 4/8 was “much bigger” than 2/4 because the units are more numerous,he then determined that they were the “same amount of pie”. When questioned about the twoconclusions the following dialogue ensued:I: But you just told me that 4 eighths is bigger than 2 fourths?It is, but if the pie is the same size it is a quantity - well you have more of the quantity of 8 - well,you have 8 pieces but if the pie is the same size and 8 pieces it really doesn’t matter.I: It doesn’t matter?E: Yah. It could be like 4 in a pie or 8 in a pie. It really doesn’t matter if the pies are the same -that’sall.But could you still say that 4 eighths is bigger than 2 fourths?E: Yah.I: So it just depends, eh?Yah, if it’s a pie - but if it’s different it would be bigger.I: It would be bigger?Yah, if its not food.I: So what kinds of things would you use to show when its bigger?E: Okay, you could use lines (drew 8 lines) and take away 4 of them to make something - so youhave 4 here. And 1, 2, 3, 4, (drew 4 lines) and then you would take away two because youmade something.The mono-relational representation comparing measures of numerosily was as intelligible to Edwin asthe tn-relational representation comparing measures of area. In each case the take-away interpretationwas applied, but differences in the situations were sufficient to justify the different results. The meaningrested for Edwin in characteristics unique to each situation, not in general characteristics of commonfractions’ multiple unit relationships which apply regardless of the system of measure.Changes in interpretations of common fractions, notions about the algorithm for generatingequivalent common fractions, different complexities in the denominators of comparison tasks, as well as143shifts between measures of numerosity and area were inter-independent features of students’approaches to comparisons of common fractions. For example, shifts in attention between measures ofarea and numerosity occurred when multiplicative notions of the equivalent fractions algorithm wereincluded in representations. As well, such shifts occurred directly when students simply changed theform of their representations from contiguous to discrete units, or changed the materials used toconstruct contiguous representations from empty regions to squared paper or multilink blocks. Indeed,shifting attention between measures of area and numerosity often was a consequence of the otherthree features, and underlay much of the instability in the representation of unit relations evidenced bythese students. Students’ beliefs about the characteristics of units of area as a basis for representingthe comparison tasks alone are insufficient to explain the instability in the representation of unitsrelationships. The other features in students’ approaches to the representational tasks also contributedto the instability.In summary, more than half of the students were stable in the extent to which they representedunits relationship over all interviews. However, for those who were unstable in their representation ofunit relationships, the instability was generally associated with shifts in attention between measures ofarea and numerosity, as well as with other beliefs about interpretations and representations ofcomparisons of common fractions.Summary of the Characteristics of Students’Repertoires of Representations of Common Fractionsand Comparisons of Common FractionsA number of general patterns were evident in students’ repertoires of representations. Theseare presented in three parts. The first concerns the extent of students repertoires of representations ingeneral. The second concerns the inclusion of representations based on length or area measurement.The third concerns the extent to which representations functioned to express multiple relationshipsamong units.144First, regarding the extent of their repertoires, the following patterns were evident:1. Prominence of regions in primary and general repertoires. All students’ primary repertoiresincluded representations based on regions, regardless of whether units were discrete, contiguous orundefined. More particularly, circles were generally the region of choice. In addition, representationsbased on sets were included by some students as an alternative to their first representations withregions. During the other interviews, discrete representations were included by students whoseprimary repertoires had been based exclusively on regions. However, there were still three studentswhose general repertoires were based completely on regions partitioned into contiguous units.2. Explicit limits were imposed on the variety of forms of representations in students’ repertoires.All students imposed some limits on which forms of representations they deemed acceptable as ameans to explain the common fractions. Most commonly, students rejected the line or transformed itinto regions. Less commonly, students rejected discrete materials, required empty regions in which toconstruct units, or needed separate regions to represent each common fraction. Students reacted toalternative materials in a manner consistent with the forms of representations generated independently.The forms of representations finally included in their general repertoires represented a limit in the typesof situations which students associated with common fraction representations.3. A student’s discrete and contiguous representations were related one to the other in terms of acommon quantitative framework. Students who primarily focused on measures of numerosityconstructed representations of comparisons with contiguous units. However, the contiguous unitswere treated simply as units of numerosity which were incidentally contiguous. Similarly, students whoprimarily focused on measures of area constructed representations with discrete units. However, thediscrete units were treated simply as units of area which were incidentally discrete. In general,differences between discrete and contiguous representations lay only in the spatial features of theunits. Students tended to interpret representations of units through a single measurement system,either numerosity or area.Second, regarding the extent to which attributes of linear or area measurement were integratedin students’ contiguous representations, the following patterns were evident:1451. Representations were not one-dimensional. Units of length were not used directly to representcommon fractions. However, they were used indirectly to equate fractional units along a dimension intwo-dimensional representations. Similarly, direct comparisons of the lengths of a dimension in regionswere also the basis on which common fractions were compared. However, the spatial framework of therepresentations was two-dimensional.2. Beliefs about the importance of units of equal area were not necessarily associated withquantitative characteristics. All students who constructed representations based on contiguous units attimes expressed beliefs about and attended to the need for units of equal area. However, this did notmean that measures of area necessarily played a quantitative role in their representations ofcomparisons of common fractions. For some students, quantitative comparisons in their contiguousrepresentations were based on measures of numerosity. Their need for equal regions or parts ofregions was justified by aesthetic criteria alone. These students imposed procedures on the form oftheir representations which conformed to some characteristics of area-based instructionalrepresentations, but which were disassociated from an area measurement framework.3. No student consistently attended to the equality of the area of fractional and whole units. Theextent to which students were consistent appeared to be influenced by one or more of the following:a. The measurement framework of interpretations of common fractions Some students primarilyinterpreted comparisons of common fractions in terms of the numerosity of the parts. Becausemeasurement of numerosity was the framework for their representation, their comparisons wereinternally consistent regardless of the equality of the area of regions or parts of regions. Otherswho interpreted these comparisons in terms of quantities of area were more likely to attend tothe equality of the area of regions or parts of regions.b. Beliefs about critical physical characteristics of contiguous representations. There was a varietyof reasons associated with students’ beliefs that units should be equal in area. Even amongstudents who used quantitative criteria to justify the need for these characteristics, someemphasized equal fractional units but not regions and vice versa. Students had different levelsof tolerance with respect to inequalities of units in contiguous representations.146c. Interaction between partitioning problems and strengths of beliefs about the imoortance ofequal units of area. Student& algorithmic approaches to the partitioning of regions (particularlycircles) inhibited their ability to achieve equal parts with particular denominators. This, combinedwith variable beliefs in the importance of equal measures, led some students to violateproperties of area measurement in some but not all instances.Third, regarding the function of students’ representations of the comparisons of commonfractions, the following patterns were evident:1. Students for whom the function of their representations was consistent did not construct birelational representations of units. Their representations were either all with mono-relational units, allwith tn-relational units, or all without defined units. In each case, the function of a student’srepresentations were associated consistently with their interpretation of common fractions and themeasurement system on which the representations were based.2. Students who represented unit relations inconsistently generally shifted between bi-relationaland tn-relational representations. Regardless of how students shifted in the representations of unitrelationships, one or more of the following features were associated with these shifts:a. Shifts in the interpretations of the comparisons.b. Shifts from direct to indirect comparison of common fractions through a representation of themultiplicative algorithm for generating equivalent fractions.c. Shifts in representational strategies associated with different pairs of denominators whichpresented different complexities in the tasks.d. Shifts between a focus on equal parts to a focus on the number of parts.147Students’ Representation of Unit Relationships in theWhole Number Multiplication and Common Fraction Contexts:Common Patterns and Related Themes.In this section general patterns and themes are discussed concerning students’ repertoires ofrepresentations of units and unit relationships in the contexts of whole number multiplication andcommon fractions.Repertoires of RepresentationsIn both mathematical contexts, students demonstrated a variety of ways in which they couldrepresent the mathematical tasks. They also clearly indicated limits in the forms of representationswhich they considered could be used to represent the tasks. It was in the sense of the forms ofrepresentations a student generated as well as the forms of representations a student rejected that theconstruct of a repertoire of representations was defined. Students’ repertoires of representations inboth mathematical contexts can be characterized by the six groups presented in Table 4.14.Central to a student’s repertoires was a dominant form of representation. This was the formwhich the student used most frequently, and the form through which the student often interpretedother materials and beginning diagrams. In the case of whole number multiplication the dominant formofrepresentations was most commonly that of discrete sets. In the case of common fractions thedominant form was most commonly that of regions with contiguous or no defined units.In addition to the dominant form in a student’s repertoire, most often there was a secondaryform of representation. The most common secondary form in repertoires of multiplicationrepresentations was that of contiguous units in regions. For repertoires of common fractionrepresentations it was discrete units in sets. There was commonly an asymmetrical relationshipbetween the dominant and secondary forms of representations of each mathematical context. The4. In order to focus on major patterns of response, the incidental references to line segments as unitshave not been presented graphically. Only 2 students referred to line segments and then not morethan more than once each.Table 4.14General Patterns in the Form of Representations in Students’ GeneralReportoires in the MultiDlication and Common Fractions Contexts.148Multiplication Common Fractions(Complex Tasks) (Comparison Tasks)Inter-Part All Part All No pretGroups Disc Contig Contig Disc Contig Contig Units ationsI I IJames *&**i1 I I I HnCvTa2I I I I I2 I I I I IConnie **&&I* I *&*I* I I CvMarlene &I& I ***I** I I CvEdwin ***l I I TaCvI I I I I3 I I I I IBrock **&I& I I I**** RsLolande ***I 1* I I iaI I I I4. I I I IDerek I I I CvDahlia ***I* *I**** I I TaMuIdKasey ***I** 1* *I**** I I TaPete ***I* 1* ***I**** I I TaCoran ***I 1* **** I I TaI I I I5 I I II ***I* 1* I&& I I TaLara I I I TaTammy **I* I *I*** I TaI I I I I6 I I I I IFanya ** ** * * *** I MuTaI I I IThe number of symbols (*/&) indicates the number of interviews in which the form wasused by the student.Response only to the less complex task is indicated with an ampersand.The order of the common fraction interpretations reflects the frequency of use.Form of Units Common Fraction InterpretationsDisc = discrete units in sets HnContig = contiguous units in regions CvNo units = regions without units Id= half of a number Rs = relative size= cardinal value of numerals Ta = take-away= inverse denominatorMu = multiplication149most common secondary form in one mathematical context was the most common dominant form in theother, and vice versa (see Groups 3 to 5 in Table 4.14).There were some students whose repertoires in both mathematical contexts were baseddominantly on discrete units (see Groups 1 & 2). The number of units, not the area of units was thecritical attribute to which they attended in their representations. A child’s dominant informal experiencewould be with discrete units in sets, whether a single or aggregate unit, with the fingers probably beingthe most common representations of units. The preponderance of discrete representations of unitswith a whole number operation fits the more general experiences of children’s representations of wholenumbers from early childhood to the intermediate grades. For these students it these discretenumerical experiences appear still to provide the basic framework for representing about numericalrelationships in general. There was little or no differentiation between their two repertoires in terms ofthe dominant form of the representations, or the measurement framework through which the units weredefined.The contiguous-discrete dichotomy of dominant forms of representations in common fractionrepertoires appears to distinguish the more proficient from the less proficient students in representingcomparisons of common fractions. However, as the cases of Derek and Fanya indicate, the dominanceof contiguous forms of representations does not necessarily mean that the students considered partsof a region to represent units of area. The relationship between forms of units and measurementsystems the students referenced was equivocal.The Discrete-Continuous Dichotomy.Conventionally, whole number multiplication and common fractions are seldom representedexplicitly in a continuous form even though units may be based on measures of continuous quantities.As Ohlsson (1988) stated,Partitioning replaces a [continuous] quantity with a set, namely the set of its parts. Byreducing the continuous to the discrete case, partitioning enables us to assignnumerical values to continuous quantities through counting. (P.74)150Once regions are partitioned, the continuity of the quantity is not the most salient characteristics of therepresentation. Instead, the contiguity of the units and their potential separateness are the morepronounced characteristics.5However, when instructional representations are based on partitionedregions, it is assumed that contiguous units should be interpreted to mean measures of the continuousquantities.The physical characteristics of units, whether discrete or contiguous, do not automatically meanthat units must be interpreted as measures of numerosity or measures of area in order for the unitrelationships embodied in the representation to be mathematically valid. This is universally the case inthe whole number multiplication context. Contiguous representations need not have units of equalarea to embody the unit relations between factors and product. It would only be when othercharacteristics of the problem situation impose a continuous measurement context that units in arepresentation need conform to particular attributes of a measurement system.Likewise, when representing a common fraction with contiguous units, the unit relationshipsembodied in the partitioned region may be based either on numerosity or on area. “A out of B pieces ofpie” may mean A/Bths of the pieces (of pie) or may mean NBths of the pie. The contiguous units maybe measuring discrete attributes or continuous attributes. Both interpretations of the representationare valid mathematically. The contiguity of the units does not, in itself, define the mathematical situationrepresented.Which measurement systems are associated with discrete or contiguous units isinconsequential when representing whole number multiplication, single common fractions, andcomparisons of common fractions with like-denominators. For example, the comparison of 5/9ths and6/9ths could as easily be a difference of 1/9th of the pieces as 1/9th of the pie. It is only because therepresentation of comparison of common fractions with unlike denominators cannot be achieved in the5. Ohlsson (1988) associates the term discrete with all units regardless of their form. In this report adistinction is made between contiguous, referring to units which share a boundary, and discrete,referring to units which are spatially separate. These differences in definitions probably reflect the factthat Ohlsson was writing from a theoretical point of view rather than from the view of studentsmathematical thinking.151discrete case that units of equal area become a necessary characteristic. Except when commonfractions with unlike denominators are compared, there would be no conceptual conflict regardless ofwhether students represented or interpreted the units as measures of continuous quantities ormeasures of numerosity. As such, to assume that a student who attends to the equality of units is alsoconsidering area as the quantitative attribute may be invalid in these mathematical situations.The ambiguity concerning which system of measurement defined different forms of units wasreflected in the manner in which students considered discrete and contiguous representations to beanalogous. Considering either mathematical context, a student’s discrete and contiguousrepresentations essentially differed only in the proximity of their units. The student defined criticalattributes of the units and interpreted the unit relationships through the same measurement framework.Even when the problem of representing a comparison of common fractions with unlike denominatorswas encountered, students who based their contiguous representations on units of area representeddenominators with different sizes of discrete units. Their “discrete” representations were simplyrepresentations based on an approximation of quantities of area in which the units were no longercontiguous. In this way, there was a sense in which the contiguity and discreteness of units wereconsidered to be reversible without altering the mathematical relationships being represented.The ambiguity regarding which system of measurement is represented by contiguous units alsowas reflected in the students’ use of aesthetic and quantitative criteria for evaluating contiguousbeginning diagrams. In the multiplication context, that units be of equal area was incidental to themeaning of the mathematical relationships, even though some students imposed this characteristic ontheir representations. In that case, students used aesthetic criteria in the multiplication context to justifytheir rejection of diagrams with unequal units. In contrast, when presented with the comparison ofcommon fractions, some used quantitative criteria to justify their rejection of diagrams with unequalunits. These students were reflecting a subtle distinction between normal, preferred characteristics ofunits in one mathematical context and necessary characteristics of units in another. As would beexpected, students who continued to base their comparisons of common fraction on measures ofnumerosity sought equal sized units for aesthetic reasons alone.152Students’ repertoires of representations in either mathematical context generally werecomposed of discrete and contiguous representations with one or other form dominant. These formsof representations could be considered to be simply variations of representations of units within afundamentally two-dimensional world. From the earliest experiences of counting discrete object topushing discrete units together or fracturing a quantity into parts, the informal and formal experiencesare essentially with two-dimensional or three-dimensional materials.The general absence of linear measurement as a framework for representations should beconsidered in the context of the associations between discrete sets and contiguous regions inrepertoires of representations in both mathematical contexts. The one-dimensional framework of theline stands in direct contrast to the two-dimensional characteristics discrete sets or contiguous regions.Thus, students transformed the line into two-dimensional representations, identified and used the dotson the line as analogous to discrete objects in sets, or rejected the line altogether in favour of sets orregions. In all cases, their avoidance or rejection of a line as a spatial framework and line segments asunits could be related to a general two-dimensional characteristic of their repertoires of representations,regardless of the mathematical context.The forms of representations in students repertoires were circumscribed in all cases. Studentsdid not consider units in the general case in which different systems of measurement were potentialbases for representations. They did not use a multiplicity of measurement frameworks between whichthe physical attributes of the units changed. Instead, their representations were formulated around acore image of units which were incidentally contiguous or discrete, but which primarily referenced asingle system of measurement, either that of numerosity or area.Multiple and Nested Relationships of UnitsThe representation of double-nested units was clearly a difficult task. Double-nested unitswere of the form A groups of B groups of C. In the multiplication context it would occur as therepresentation of the product of three factors. In the common fraction context it would occur as therepresentation of two fractional units nested within a whole unit, such as the representation of thirds153and ninths within a single region. In each case the representations would be categorized astn-relational. In the multiplication context, most of such representations were achieved only after somepreliminary trials. In the common fraction context, the comparative relationships among the whole unitand two fractional units could be achieved without recourse to double nesting. Students generally didnot encounter the double-nesting problem because they directly compared separate representationsfor each common fraction. There was only one instance when a student compared two commonfractions with a tn-relational representations in which both fractional units were nested within a singleregion.There were situations in both contexts in which students constructed representations whichwere potentially double-nested, but they did not “see” the units nested in units. In the occasionalinstances when thirds were transformed into ninths, few students saw aggregates of three-ninths alsoas thirds. Similarly, with multiplication, some students did not perceive two three’s in representationsof 2 X 3 X 4 as four sixes. In the most extreme example, a student placed two discrete marks ineach of six circles to represent 2 X 2 X 3, then interpreted the representation as a product of 18.The meaning of his own nesting procedure was lost in his interpretation that all marks and circlesrepresented equivalent units.Table 4.15 presents a graphic summary of the unit relationships the students represented inboth mathematical contexts. As can be seen in this table, in the multiplication context students weregenerally as successful or more successful in representing multiple unit relationships than in thecommon fractions context. There were students whose representations of unit relationships wererelatively consistent in both mathematical contexts. For some, units were consistently mono-relational.It would appear that these students do not have a general representation of units as aggregates. Forothers, unit relationships fluctuated between being bi- and tn-relational in both mathematical contexts.As well, there were students for whom unit relationships were mono-relational in the common fractionscontext but whose representations were bi- or tn-relational in the multiplication context.Table 4.15154General Patterns in Students Representations of Unit Relationships in the Multiplication and CommonFraction Contexts.Multiplication Common Fractions(Complex Tasks) (Comparison Tasks)_________________________________InterI I I I pretaMono Bi I Tn None I Mono I Bi I Tn tionsI I II I I I IConnie **&& I I I I I CvMarlene **&& I I I I CvDerek I **** I CvBrook I I I I RsI I I I I2 I I I 1 IFanya I ** ** MuTaDahlia I **** I TaMuIdEdwin *1 ****I* I 1* TaCvLara I ****I* I I **l**** TaLolande I ***I* **** IdKasey I ***I** I TaPete I **I** I I ****I** TaTarnmy ***I** I I TaJames *1 **l*** **&* I I HnCvTaCoran I ***I*** I I **I** TaKLt *1 **I*** I I &&l TaI I I INote: The number of symbols (*,&) indicates the number of interviews in which the unitrelationship was represented at least once by the student.Response only to the less complex task is indicated with an ampersand.1. The bi-relational representations occurred only with the multiplication interpretation.Unit RelationsNone = units were not definedMono = mono-relational unitsBi = bi-relational unitsTn = tn-relational unitsCommon Fraction InterpretationsHn = half of a numberCv = cardinal value of numeralsId = inverse denominatorRs = relative sizeMu = multiplicationTa = take-away155In summary, students’ repertoires of representations in both mathematical contexts werecharacterized in terms of the forms of representations which were included and excluded. As well,within a student’s repertoire of representations the discrete and contiguous representations wereanalogously related one to the other through a single measurement framework. Which characteristicsare quantitatively necessary to represent unit relationships is ambiguous in many mathematicalsituations including whole number multiplication and common fractions. As a result, students whointerpret contiguous representations in terms of numerosity are not challenged until faced with theproblem of representing comparisons of common fractions with unlike denominators. In this specificmathematical context, a student’s beliefs about units of area measurement become critical to theirrepresentations with contiguous units. It is no longer adequate for a student to consider unitrelationships with common fractions as mono-relational.156CHAPTER 5STUDENTs’ REPRESENTATIONS AND INTERPRETATIONSOF UNITS OF LENGTH AND AREAThe general questions which guided the analysis in this chapter were:1. What are the characteristics of students’ representations and interpretations of units of length?2. What are the characteristics of students’ representations and interpretations of units of area?It was expected that students at Grades 5 and 7 would be more likely to operate appropriatelywith units of linear measurement than with units of area measurement given that “the order ofachievement is a function of added dimensions” (Beilen & Franklin, 1962, p.617). On the other hand, itwas not assumed that all students would operate appropriately with units of linear measurement. Noclinical studies have been located that directly investigated children’s conceptions of linearmeasurement beyond the age of 9 years.Compared with linear measurement tasks, area measurement tasks present greater perceptualand conceptual complexities to the student. These complexities not only derive from the need tocoordinate relationships between two dimensions, but also derive from the variations in geometricproperties of different regions which must be attended to when measuring areas. There is evidencethat students in Grades 5 and 7, when presented with area measurement tasks, exhibit a wide range ofbehaviours suggesting a variety of conceptions about area measurement (Hirstein, 1974 unpublishedpilot study cited in Steffe & Hirstein, 1976; Hirstein et al., 1978; Wagman, 1975).The chapter is divided into three main parts. In the first part, the analysis of the students’responses to the linear measurement tasks is presented. In the second part, the analysis of thestudents’ responses to the area measurement tasks is presented. In the third part, an overview of allmeasurement tasks is presented.157Students’ Representations andInterpretations of Units of LengthThe research question concerning the characteristics of students’ representations andinterpretations of units of length was particularized by the following questions:A. To what extent do students interpret or construct units of length in a manner consistent withproperties of linear measurement?B. What are the characteristics of the reasoning strategies students use to compare quantities oflength?Each of the questions is addressed through the analysis of different subsets of the tasks on thetest which were used during the interview. The ruler, aggregate unit and partitioning tasks (see Figure5.01) were designed primarily to address Question A. All of these tasks directed students to constructand use units of linear measurement. The ways in which students did so provided evidence from whichtheir representations of units were characterized. Question B was explored through students’responses to the irregular path tasks (see Figure 5.02). The irregular path tasks did not contain explicitreference to units or number.The balance of this section on linear measurement is organized in the following manner. First,the analytical categories used to classify students’ responses to the linear measurement tasks arepresented and defined. Second, the analysis of the responses to the ruler, partitioning and aggregateunit tasks is presented. The responses to these tasks are directly relevant to Question A. Third, theanalysis of the responses to the irregular path tasks is presented. The responses to these tasks aredirectly relevant to Question B.A. Ruler taskrbia auci.t IJ.I ur* Lah ia ‘TLUG. Qua asvo sv a LLu aboa. c. rui.r a: 1., 6 cIacc.s u;.I 4 1I_l.I I. iB. Aggregate unit taskLa 4 iss 3.aui. DTw a LLua ac La L2 iu Lou;.C. Partitioning taskThis pa La 5 i:s Loul. a) lark ha 5 ies h. pau’.b) Dx acar ;ac 3 •mis Loug.Figure 5.01 Linear measurement tasks with explicit reference to units and number.158D. Irregular path taskA1/fyE. Irreciular Dath task$Alit ‘A’ pach Li L05ç.rAlr ‘V path Li______Th.y sri the iei —A ‘A’ path Li_—Alit ‘j’ path Li__—They are ha 1.ea;th —159Analytical Categories to Classify Responsesto Unear Measurement TasksThe categories described in this section and outlined schematically in Figure 5.03 were derivedfrom an integrated analysis of the tasks and the students’ responses to the tasks during the interviews inthe second pilot study and final study.AFigure 5.02 Linearmeasurement tasks without explicit references to units and number.Definition of UnitsA. Undefined Students directly compared the total lengths of two lines and did not define a unit.B. Discrete points Students drew or counted points as the units along a line; or countedbeginning, end-points and junctions between line segments along a path of line segments.C. Line segment Students counted line segments as the unit.160Task ResnonseUnits Line Discrete UndefinedSegments PointsI IReasoning Numerical Transformational PerceptualStrategy IActual ImaginedUnit Bi- Mono—Relations Relational ThinkingFigure 5.03 Analytical categories used to classify student responses to the linearmeasurement tasks.Reasoning StrategiesA. Perceptual strategies Students did not describe imagined actions or use external actions tojustify their judgements. Their judgements were based on the appearance of the paths andwere derived by:1. comparing the relative positions of the ends of the paths, or2. evaluating the shapes of the paths with regard to the degree to which each path “goesup and down” to estimate the comparison of the total lengthsB. Transformational strategies Students transformed the paths in the following ways:1. Imagined The students imagined actions to:a. stretch each path straight “in their mind” and compare the results, orb. “shrink” one path “in their mind” and compare the shapes of the paths, that isimagine one path squeezed up thereby reducing the angles at the junctions ofthe line segments.2. Actual The student transformed the configuration of the paths by:a. redrawing one path into the shape of the other, orb. redrawing each path into a single line segment and comparing the total lengths.161C. Numerical strategies Students identified or constructed units. Mono-relational and bi-relationalreasoning strategies were used to interpret the units. The relationships between units wereexpressed within their reasoning strategy in the following ways.1. Mono-relational thinkinga. Students defined units to be equivalent regardless of differences in theirlength and counted them accordingly.b Students used only one of the units to measure off the length of both pathsand compared the counts of the measures.2. Bi-relational thinkinga. Students matched units in one path in one-to-one correspondence with unitsin the other, then reasoned about the comparison in terms of the numerosityand size of units.b. Students counted the number of short and long units and used compensatoryreasoning with regards to the differences in the sizes and counts of the units.Not all categories were applicable to all tasks. Distinctions between the students’ use ofreasoning strategies (numerical, transformational and perceptual) only applied to the irregular path tasksbecause number was an explicit part of the other tasks. Also, distinctions in students’ reasoning withmultiple units (mono-relational versus bi-relational thinking) did not apply to the partitioning task. Onlyone unit was used in this task; all of the responses were necessarily mono-relational.Students’ Representations of Units of Length.This section addresses the question of the extent to which the students’ representations ofunits were consistent with properties of linear measurement. For this purpose student responses (inboth the interview and linear measurement test) to the following tasks were analyzed: the ruler tasks, theaggregate unit task and the partitioning task. These data are presented in Table 5.01.Table 5.01 is designed to explore students’ representations of units in several ways. First itpermits us to determine the general extent to which students constructed units which were eitherdiscrete points or line segments. Second, it allows us to compare each of the tasks with regard to theextent to which discrete points or line segments were constructed by students. And finally it reveals theextent to which their reasoning about the relationship between different units was mono or bi-relational.Table 5.01Students’ Representations of Units of Length and Relationships Between Units.Ruler Task Aggregate Unit Task Partitioning TaskTest Interview Test Interview Test InterviewResponseGroups D Mi L D Mi L D Mi L U Mi L D Mi L U Mi LDominant lvDiscreteJames M .. B . . B . . . . B M . . . MMarlene . . . . . - . . M M - . * MLolande M . . . . B B . - B . . . - ? MConnie NR . . M —-> B . . ? B . * M . . . MFanya B . . . . B . . B . B * . . . MDerek . . . . B B . . . . MB M . . M -Dominant lvLine Seciments . -Edwin B. . B. . . . B . . B . . M . . MDahlia B. . . . B . . B . . MB . . M . . MTammy B . . . . B . . B . . B . . M . . MLara B . . . . B . . B . . B . . M . . MKasey B. . . . B . . B . . B . . M . . MLine seomentBrock * . B . . B . . B . . B . . . . MKit * . B . . B . . B . - B . . M . . MCoran . .. B . . B . . B . . B . . M . . MPete . . B . . B . . B . . B . . M . . MNote. The names of the students in Grade 5 are underlined.Units Unit relationsD = discrete points M = mono-relationalMi = mixed, points & line segments MB = mono- then bi-relationalL = line segments B = bi-relational? = no defined unitsNR = no response162163The students have been grouped in Table 5.01 according to the extent to which theyrepresented units as discrete points or line segments: (a) dominantly discrete points, (b) dominantly linesegments, and (c) consistently line segments.1 All students who represented units dominantly asdiscrete points, did so in response to the partitioning task. With the aggregate unit and ruler tasks, theform of units was more variable. In contrast, other students used discrete points as units only with thewier task. All of the students who dominantly or consistently used line segments as units used birelational reasoning for their final responses to the ruler and aggregate unit task. They accounted forthe differences in size of units within their solutions. Of the students whose units were dominantlydiscrete points, four constructed a mono-relational representation with one or other of the tasks, butonly one student (Marlene) did so in all instances. Marlene (Grade 5) was exceptional in that shegenerally did not reference units at all in the tasks which required the inter-relationship of two differentunits.In summary, most students represented some units as discrete points. However, none did soexclusively, and a few students never did so. Instead, they consistently represented units as a linesegment. The ruler task was the context in which the greatest number of students were inconsistent intheir representation of units. But for those students who dominantly defined units as discrete points,the partitioning tasks was the context in which they did so most consistently. Nearly all studentsdemonstrated an ability to reason appropriately about the relationship between two different linear units,although some students were inconsistent. In the balance of this section the manner andcircumstances in which units were represented as discrete points are explored in more detail.Variations in Students’ Representations of Units of LengthFigure 5.04 illustrates ways in which discrete units were constructed for the partitioning task.The first two students marked five points and three points along each line. However, James gave amixed response. He partitioned the line into five line segments, then counted three points to1. Students in the “predominantly discrete” group used discrete points as a unit or did not define unitsin more than half of the tasks in the test and interview setting.164determine the length of the second line. Similar variations occurred with the aggregate unit task whenstudents used discrete points as the unit.1. (Lolande, Grade 7)2. (Derek, Grade 5)3. (James, Grade5)I I II II II I II I I II I IFigure 5.04 Examples of students’ use of discrete points as units to partitiona line into five units then drawing a line of three units.Figure 5.05 illustrates ways in which discrete units were used with the ruler task. The reasoningbehind these responses differed. In the first example, the relationship between the size of thecentimetre and flug was ignored. The points with each numeral determined the length of the line drawn.In the second example, the student attended to the 2:1 relationship between centimetres and flugs butcounted the beginning and end points of the line segments as the units, beginning with the pointassociated with the 1 on the ruler. In the third example, the student converted centimetres to flugsusing mental arithmetic and then represented the 3 flugs to correspond with the numerals on the rulerand not the length of units.The ways in which students used line segments to represent the units with the ruler task areillustrated in Figure 5.06. The only difference in the two cases was whether or not the line wasextended from the beginning of the ruler or from the point associated with the numeral “1Point/Line Segment Conflict: Differences Among Tasks and Solution StrategiesIt was observed earlier that there were differences among the tasks with regard to the extent towhich students represented units as contiguous line segments or discrete points. The ruler task was165the one with which the largest number of students were inconsistent in their representation of units. Inthe partitioning task, discrete units were used most extensively by the students in the1. (James, test)I I I I I1 2 3 4 5 6flugs2. (Kasey, test) I I II I I I I1 2 3 4 5 6flugs3. (Edwin, interview)I I I I1 2 3 4 5 6flugsFigure 5.05 Students’ use of discrete points as units with the ruler task.1. (Derek, interview)I I I I I1 2 3 4 5 6flugs2. (Fanya, interview)I I I I I I1 2 3 4 5 6flugsFigure 5.06 Students’ use of line segments as units with the ruler task.166“dominantly discrete” group. There were a number of characteristics in students’ responses whichsuggest that counting actions associated with different procedures for drawing a unit might haveinfluenced the manner in which some students attended to line segments or discrete points as units.When students who represented units dominantly as discrete points used a partitioningprocess to resolve the aggregate units task, they defined units as points. However, those who resolvedthe aggregate unit task by iterating line segments rather than partitioning a line faced no ambiguity aboutwhether to determine the measure by the count of the points or the line segments. The contrast in thetwo approaches to the aggregate unit task suggests that the different counting actions influencedthese students’ attention on line segments or discrete points as units. It would appear that the processof partitioning a line led these students to attend more to the points rather than the line segments.With the ruler task there was the additional feature that points were juxtaposed with numerals.This juxtaposition further emphasized a counting relationship between points and numerals. In thisregard, students interpreted the “1” as the beginning marker of their representations of 6 centimetres,not as the end marker of the first “flug” unit (see Figures 5.05 & 5.06, 2) In all instances when studentsused discrete points as units and some when students used line segments as units, theirrepresentation of 6 centimetres began from the “1 .“ The general structure of the ruler and the meaningof the numerals implied by that structure did not guide the students’ representation of 6 centimetres.The point/line segment conflict, which nearly all students demonstrated to some extent,appears to be influenced by perceptual and conceptual factors. The points are perceptually salient tothe ruler and partitioning tasks, and attention is centred on them during their resolution. They are thecomponent of the representation acted on synchronously with the verbal count, exactly the sameactions as counting discrete units. However, it is indirectly through the points that line segments aredefined as linear units. One has to attend to the points, think about line segments, and keep track ofthe relationship between the count of points and the number of line segments.There is not a single, direct relationship between the count of points and the number of linesegments. Depending on whether one counts all beginning and end-points, only end-points, or onlyinternal points between line segments, X line segments would be represented by a count of X + 1,167X, or X - 1 points. However, earlier experiences of counting discrete units establishes that thevalue of the count always equals the number of units. In addition, the common use of the rulerreinforces the notion that there is a direct relationship between the count of points and the number ofunits. Having placed a ruler correctly, only the points and numerals have to be attended to to “read” thelength. As a result, the need to attend to other factors besides the count of the points whenrepresenting or interpreting units in some linear measurement situations does not appear to berecognized universally by the students.In summary, only four students consistently represented units as line segments in the linearmeasurement context. Among the rest of the students, the extent to which they considered the unitsto be line segments varied with each task For some, discrete units were used only with the ruler task,but for others inconsistencies in what constituted the unit occurred with all of the tasks. Some studentsdo not appear to have a clear sense of the significance of the invariance of the attribute of length beingconsidered. For these students their earlier experiences with number and counting as a measure ofdiscrete units still appears to influence their representation and interpretation of units in a linearmeasurement context. From an instructional point of view, it cannot be assumed that students, even atthe upper intermediate grades, will represent or interpret units in a linear measurement contextinvariably as line segments.Reasoning with Units and Number withthe Irregular Path TasksThis section addresses the question of the characteristics of the reasoning strategies studentsuse to compare quantities of length. For this purpose student responses to the irregular path tasks (seeFigure 5.02) were analyzed. A secondary question is whether there appears to be any consistencybetween students’ representations of units (dominantly discrete, dominantly line segments, orconsistently line segments) and the types of reasoning strategies they used to compare the lengths ofthe paths. These data are presented in Table 5.02.Table 5.02 presents the responses of students to each irregular paths task during the interview.The characterization of each student’s representations of units as derived in the analysis of the previoussection are also included: dominantly discrete points, dominantly line segments, and solely line168segments. The intent of the table is threefold. First, it illustrates the extent to which different reasoningstrategies were used by the students for each task. Second, it permits a comparison of theTable 5.02Students’ Reasoning Strategies With the Irregular Path Tasks and a Summary of Their Representationsof Units of LengthTasks by reasoning strategiesRepresentat ionsof units Task D Task EStudents D PL L P TI TA ND Ni N2 P TI TA ND Ni N2Connie # . . # . . . . . . . .Fanya . . # . . . . . .Marlene # . . . . . . . . # . .James # . . . . . . . . . .Lolande # . . . . . . . . . .Derek . . . . . . . . # . .Dahlia . . . . . . . . . .Brock . . # . . . . . . # . .Tamrny . . . # . . . . . . . . .Kasey . # . . . . . # . . . . . . #Lara . # . . . . . . # . . . .Edwin . # . . . . . . # . . . . .Ki.t . . # . . . . . # . . . . .Coran . . . . . . . . . . . . #Pete . . # . . . . . # . . . . . #Note The names of students in Grade 5 are underlinedAsterisk indicates the category of a student’s responsesRepresentations of unitsD = dominantly discrete points.PL = dominantly line segments.L = line segments only.Reasoning strategiesP = perceptual.TI = transformational (imagined)TA = transformational (actual).ND = numerical (discrete units)Ni = numerical (with one linear unit)N2 = numerical (with both linear units)169consistency in the reasoning strategies used by a student to resolve both comparisons. Finally, it allowsfor a comparison between the students’ general representations of units and the reasoning strategiesthey used with these comparative tasks.As can be seen in Table 5O2, the same number of students (eight) used a numerical reasoningstrategy as used a transformational reasoning strategy to compare the lengths of the paths. Whetherstudents used numerical or transformational reasoning strategies, nearly all were consistent in thestrategies they used for both comparative tasks. In addition, there were two general response patterns:(a) students who compared the lengths of these irregular paths with appropriate numerical reasoningstrategies also represented units dominantly as line segments during the first set of tasks, and (b) mostof the students who used transformational strategies had represented units dominantly as discretepoints with the first tasks.. Students in the latter case did not count discrete points as a final strategy,even though this strategy would have been consistent with their representations of units in other tasks.It is possible that these students did not consider numerical strategies to be reliable and used thestrategy on which they attached the greatest confidence, namely a direct comparison of lengths withoutrecourse to units.Numerical reasoning strategies. Generally, when numerical reasoning strategies were used, thestudents identified and used units appropriately. There were three ways in which the studentsapproached the comparative reasoning with units as line segments: (a) they measured both paths withonly one unit and compared the measures, (b) they counted the long and short units separately andreasoned about the inverse relationship between the numerosity and sizes of the units, and (c) theyevaluated the relative numerosity of the different units through one-to-one correspondence beforereasoning about the inverse relationship between the numerosity and sizes of the units. The frequencywith which each of these approaches was used for the different tasks is presented in Table 5.03. Theapproaches used most frequently to resolve Task D and Task E were the most efficient ones, given thecomplexities of the tasks.170Table 5.03Frequency of the Different Approaches to Numerical Reasoning with Line Segments for Each IrregularPath TaskCount Measure Count units One-to-onediscrete with separately correspondenceunits one unit and compare and compareTaskD 6 1 5TaskE 8 1 1 1 5Transformational reasoning strategies. Students imagined (and a few drew) the effect ofchanging the configuration of the paths. In so doing they sought to compare the lengths of the pathsdirectly. In some cases, the differences in the heights and drops of paths were mentioned, or theconfigurations of the paths compared, while in others the “stretching” action was the only verbalrationale. They believed that a path would be longer or shorter if each were straightened, or that suchwas the case as they straightened each in their imagination. They appeared to be reasoning from thepremise that the transformed length would be equivalent to the length of the original path. The mannerin which students used imagined transformational strategies implied a belief in their own ability to makecomparative judgements on the basis of very rudimentary approximations of each path’s length.Summary of the Characteristics of Students’ Representationsand Interpretations of Units of LengthThere was little difficulty encountered when tasks were solved by iterating a line segment as aunit to define a length. However, tasks were less easily solved when they involved representing orinterpreting units within a pre-defined length. The relationship between action, language, and numberis more complex when representing or interpreting units within a pre-defined length. When interpretingthe relationship between the count of points and the number of line segments, the orientation of pointsto line segments also must be attended to. There were students for whom the number of points on a171line was often associated as the number of units determining the measure of the line, regardless of theorientation of points to line segments. Their assumption of a one-to-one relationship between countingaction and number probably derives from the extensive experience of counting in a discrete context inwhich action, language and number are universally synchronous.2 A belief that the number of pointsuniversally equals the number of line segments matches the numerous situations in which this conditionis true in a linear measurement context.When not directed to consider units as a means of comparing lengths, over half of the studentscompared lengths directly through perceptual or transformational reasoning strategies rather thanindirectly through numerical reasoning strategies. These students either (1) were more confident withdecisions based on transformational rather than numerical strategies, or (2) did not interpret the problemsituation as one involving units of measure. Most students who used direct comparison strategies werealso less consistent in representing units as line segments in other linear measurement contexts. Formost of these students, measurement with units of length was not associated generally with theenumeration of congruent line segments. In contrast, most students who compared lengths indirectlythrough numerical reasoning strategies were more consistent in their representation of units of lengthas line segments and associated measurement of length with the enumeration of congruent linesegments.Students’ Representations andInterpretations of Units of AreaThe research question concerning the characteristics of students’ representations andinterpretations of units of area was particularized by the following questions:A. To what extent do students interpret or construct units of area in a manner consistent withproperties of area?2. The importance of the synchronous relationship of action, language, and number in the counting ofdiscrete units, and the difficulties younger children encounter in this regard is well documented (e.g.,Gelman & Gallistel, 1978). In linear and area measurement contexts this synchronous pattern can nolonger be assumed. Hence, different relationships between partitioning actions, counting actions andthe units have to be accommodated in the representation of linear and area units.172B. What are the characteristics of the reasoning strategies students use to compare quantities ofarea?These questions led to the identification of a number of sub-questions related to particularaspects of students’ responses to the area measurement tasks. Some of these sub-questions areoutlined in this introduction, while others are referenced within the sections specific to different tasks.Area measurement is geometrically more complex than linear measurement. The additionaldimension introduces perceptual and conceptual variations into the characteristics of the unit. Regionsor part of regions of equal area are not necessarily congruent. As a consequence, direct comparisonsof areas are difficult if not impossible to achieve effectively. In addition, units themselves are subject tothe many geometrical variations encountered in the regions to be compared. While only squares areused in the formal measurement of area, a more general definition of units is any polygon that cantessellate a region. This latter definition applies more generally to area-based representations ofcommon fractions. Regions are partitioned into triangles, squares, rectangles or sectors of a circle torepresent the same fractional relationships. Students are expected to comprehend the general casethat when a region is partitioned into n parts of equal measure, the measure of each part is 1/n,regardless of the shape of the part (i.e. unit fraction piece). Whether students’ conceptions of units ofarea might include such a generalization is a question explored in this section.The area measurement tasks were designed to explore a number of aspects of students’representations of area units. The first aspect was how students partition different regions when askedto construct equal parts (see Figure 5.07). Students’ representations of equal parts have directimplications on the ways in which they might represent and interpret area-based representations ofcommon fractions. It also has implications for their general conception of units of area. The secondaspect was how students would compare fractional units of regions (see Figure 5.08). Of particularinterest was whether students would perceive a part-whole relationship between units as a basis forreasoning about these comparisons. While the third aspect was whether students would compareirregular regions by using numerical reasoning strategies with units which were consistent withproperties of area measurement (see Figure 5.09).173IFigure 5.07 Partitioning tasks: divide each figure into 6 equal parts.1. 4.A[ 1I 1 I2. (Interview Task) 5. (Interview Task)A k13i 1 A‘ HH3. (Interview Task) 6. (Interview Task)A[j 8Li A1_HVFigure 5.08 Cake tasks: compare the sizes of the shaded pieces of cake.I II I I5. (Interview Task)ILiliwI ———I I I I I EA::c:IE:::::iI hiI I lit I I‘r i iiiiiFigure 5.09 Tile tasks: compare the amount of space in each playroom.Analytical Categories to Classify Responsesto Area Measurement TasksThe analytical categories described in this section and schematically represented in Figure 5.101742.__-I-II —1.AbHHIA4.:::: Hill$[III II I I I IA I I I ‘ F I ‘ I F F II I I I I I6. (Interview Task)were derived from an analysis of the tasks and an analysis of the students’ responses to the tasks during175the second pilot study and the final study. The categories are analogous to those used to classify theresponses to the linear measurement tasks. As was the case with the linear measurement categories,not all categories apply to all tasks because of differences in the demands of the tasks.Task ResponseI IUnits Contiguous Line UndefinedRegions SegmentsI 1Reasoning Numerical Transformational PerceptualStrategies I•1Actual Imagined1Unit Tn- Bi— Mono-Relations_______IIPartitioning Hatched Vertical Radial SlantedConfigurationsFigure 5.10 Analytical categories used to classify the students’ responses to the area measurementtasksI. Definition of UnitsA. Undefined Students did not construct or use units to compare areas.B. Line segments Students used discrete line segments as units when they partitioned a region.C. Regions Students constructed or used regions as units.176Reasoning StrategiesA. Perceptual strategies Students’ judgements were based on either1. evaluations of only one of the dimensions in each region, such as stating that oneregion is taller therefore it is bigger than the other, or2. global evaluations of the size of the regions that were not substantiated further, such asstating that one region “just looks bigger.”B. Transformational strategies The students transformed one or more regions in the followingways:1. Imagined transformations Students imagined primitive compensations of the twodimensions such as describing one dimension stretching as the other shrinks or byimplying compensations by stating that “one shape is wider but the other is longer.”2. Actual transformations Students performed actions which transformed regions by:a. overlaying (drawing) one region on the other,b. cutting and moving parts of one region in order to approximate the shape of theother region or to equalize both regions along one dimension, orc. partitioning each region into different sized subregions and comparing eachsubregion, one to one.C. Numerical strategies Students identified or constructed units. The relationships betweenunits were expressed within their reasoning strategy in the following ways.1. Mono-relational thinking Students defined the units as equivalent regardless of theirsize and counted them accordingly.2. Bi-relational thinking Students identified two different units and included therelationship between the two units within their reasoning strategy by:a. translating one unit into the other with reference to the ratio between the unitsand counting the equivalent units, orb. counting the number of small and large units and using compensatoryreasoning with regards to the differences in the size of the units and thecounts.3. Tn-relational thinking Students identified three different units and included therelationship between the three units within their reasoning strategy.177III. Partitioning configurationsA. Hatched Configuration: involved the crossing of vertical and horizontal lines to produce arectangular grid of subregions.H 1B. Radial configurations involved the crossing of diagonals at a midpoint or the extension of radii toor from a midpoint, or the crossing of diagonals and bisectors of sides of the square at amidpoint, such that all lines radiate from the midpoint of the region.C. Vertical configurations involved the drawing of vertical parallel lines to produce six consecutiveparts along one dimension of the region.___rn iiiD. Slanted configuration involved the drawing of slanted, parallel lines to produce 6 consecutiveparts.Both the test and interview responses were analyzed with regard to the types of strategies usedby the student to solve the problems and the consistency of a student’s responses to similar tasks. Inmost cases strategies could not be inferred from the test papers; therefore, except where indicated, theclassification of a student’s strategies refers only to those used during the interview.Equal Parts of a Region: The Partitioning TasksThis section addresses the question of the extent to which students represent units of area in amanner consistent with properties of area. For this purpose, students’ responses to the partitioningtasks were analyzed. Two secondary questions are addressed in this analysis.1. Are there differences in students’ success at partitioning each of the three regions?1782. If there are differences in the success with which students partition different regions into sixequal parts, what might account for the differences?Differences in Success with Different RegionsTable 5.04 illustrates the frequency with which students were successful at partitioning each ofthe regions into 6 approximately equal parts in both the test and interview setting. As can be seen inthis table, the rectangle was consistently partitioned into 6 equal parts by the largest number of studentsand the circle by the least number of students. There was little difference in the number of studentswho were successful at partitioning the square and the rectangle at least once, but fewer students wereconsistent in their partitioning of the square.Table 5.04Freauencv of Students’ Constructions of Six Eciual Parts of Each Region in the Test and Interview.Number of settingsRegions Both One NeitherRectangle 10 5 0Square 6 8 1Circle 5 6 4Strategies Associated with Partitioning ProblemsTable 5.05 was designed to explore patterns in students’ partitioning strategies which mightinfluence the partitioning results. The results of student’s partitioning strategies with each region in thetest and interview setting are presented in this table. In the body of the table, first and final trials havebeen coded, indicating the type of configuration used in each instance: hatched, vertical, radial orslanted. The students are grouped within the table according to the number of different regions whichTable 5.05179Consistency of Students’ Partitioning of Each Region and a Breakdown of the Results of the Initial andFinal Partitions of Each Region.3 RegionsNote. The names of the students in Grade 5 are underlined.Result of PartitionN = number not equal to six.U =6 unequal partsE =6 equal partsConfigurationsR = radial configurationH = hatched configurationV = vertical configurationS = slanted configuration*= first partitionCircle Square RectangleTest Interview Test Interview Test InterviewN U E N U E N U E N U E N U E N U E0 RegionsMarlene H . . . H . H . . . . H H . . . VDahlia *R H . . . R *HH . . . . H H . . HDerek R *R . R R . . . . H V. . *V . VJames . . R *RH . H . . *H . H V . . *\7 . H1 RegionEdwin R . . . R . R . . . R . . . H . . HConnie . H. . R . . . H . . V . . V . SLolande . H . . *R R . R . . *R H . *V H . . VTammy . . R . V. H. . . . V . . V .*V V2 RegionsBrock . H . . H. . . H . . H . . H . . HKit .*R R . V. . . V . . V . . V .*V VKasey . *R R . *R R H . . . . H . *H H . . HFanya . . R . . R . *R V . R . . . V . *V VCoran . . R . . R . . V . *R V . *V H . . VLara . . R .*R R . . H . . H .*V V . . VPete .*R R .*R R . . H . . H . . H . . VTotalsInitial 465 1113 726 1 410 357 447Final 348 078 717 0 213 4 011 0 114180they successfully partitioned both on the test and in the interview. These groups are labelled “0regions”, “1 region”, “2 regions”, and “3 regions” and indicate the number of regions with which thestudents were consistently successful.A number of patterns are discernible in this table. First, some problems of planning andexecuting partitions with different regions were encountered by students in all groups (see asteriskedentries in table). However, production problems were not associated simply with regions with whichstudents were less successful. Half of the students who successfully partitioned the rectangle duringthe interview required more than one trial to do so. The inconsistency with which some studentspartitioned regions arose not only because of technical production problems, but also because ofcharacteristics in their solution strategies.Students who failed to achieve six approximately equal parts for a region (see columns N & U)proceeded to partition the region in one of two ways.1. Students used a configuration with which they could not partition the region into six equal parts(e.g., the radial configuration applied to the square).2. Students used an appropriate configuration in ways which did not result in 6 equal parts (e.g.,successive halving with the radial configuration applied to the circle).Both patterns occurred with circles and squares, but only the second generally applied to rectangles.Each of these solution patterns will be discussed in turn.Configurations applied inappropriately to a region. As can be seen in Table 5.05, nearly allstudents used standard configurations which had the potential to result in equal parts if appliedappropriately. However, some students appeared to assume that a standard configuration wouldproduce equal parts regardless of the geometric properties of a region to which it was applied. Twothirds of the students applied the radial configuration to the square, or the vertical or hatchedconfiguration to the circle (e.g. Lolande in Table 5.05). Five of these students applied oneconfiguration to all three regions (e.g. Tammy & Brock). In these cases, notions of what constitutesequal parts for different regions appear not to be founded on measures of area in the usual sense.181Instead, the configurations appeared to function as over-generalization algorithms for partitioningregions.Students applied configurations to different regions in a relatively systematic pattern. In Table5.06, it can be seen that the initial configuration used most frequently by students differed with eachregion: radial with the circle, hatched with the square, and vertical with the rectangle. This suggests thatvariations in geometric properties of regions might influence students’ partitioning procedures. Inparticular, the rotational symmetry of each region, and the dominance or lack of dominance of adimension in the quadrilaterals might have had a bearing on students’ solution strategies.Table 5.06Frequency of Students’ use of Configurations to Partition Each Region: Initial and FinalPartitions.Circle Square RectangleTest Interview Test Interview Test InterviewInitialRadial 11 11 4 4Hatched 4 2 9 8 5 3Vertical 2 2 3 10 11Slanted 1FinalRadial 10 10 3 2Hatched 5 3 9 9 8 5Vertical 2 3 4 7 9Slanted 1The hatched and radial configurations, which themselves have rotational symmetry, were thedominant configurations used with the square and circle in turn. Furthermore, the hatchedconfiguration was also the second most frequent configuration used with the circle and the radialconfiguration was the second most frequent configuration used with the square. The more extensive182rotational symmetries of the square and the circle may account for students1over-generalization of theradial configuration to the square and the hatched configuration to the circle. The limited rotationalsymmetry of the rectangle might account for the absence of the radial configuration, while thedominance of the horizontal dimension of the rectangle might account for the prominence of the verticalconfiguration for initial partitions of the rectangle. It would appear that, as a result of the differences inthe salience of such geometric characteristics, some students treated the rectangle and the square asseparate partitioning problems.In summary, standard configurations appear to have functioned as algorithms which somestudents over-generalized. Furthermore, geometric properties of regions which are not necessarilyrelated to the partitioning problem might have influenced the types of configurations students use withdifferent regions, and the ways in which configurations are over-generalized.Unsuccessful partitions with approøriate configurations. There were different problemsassociated with the ways in which students applied each of the standard configurations. First, thesuccessive halving algorithm was used frequently as the means of applying the radial configuration. Itaccounted for all unsuccessful cases (see Table 5.05). There were few instances where the applicationof the successive halving algorithm did not precede a successful application of this configuration. Eventhree of the four students, who used a multiplicative partitioning procedure with the factors of 6, initiallyused the successive halving algorithm. At the outset, students generally did not differentiate betweenpartitioning strategies for powers of two and other even numbers.The hatched configuration, requires a multiplicative procedure to partition a region into six parts,but this multiplicative procedure was not invariant for some students. Instead, they used an additiverelationship of 3 +3 =6 to plan the partitions. This was so in most cases when the square waspartitioned with the hatched configuration into more than six parts. The result was either 9 or 16 parts,depending on whether the students counted lines or spaces.With the vertical configuration major difficulties occurred only when students counted 6 linesrather than 6 spaces, resulting in 7 parts. This problem of count lines rather than spaces was mostcommon among the students who did not partition any of the regions consistently. These students183appeared to assume a one-to-one relationship between the count of the lines and the number ofspaces in a manner analogous to the counting of discrete points with the linear measurement tasks.Multiplicative procedures based on factors were used in few instances except with the hatchedconfiguration. Three students also used it with the radial configuration, and one with both the radial andvertical configurations. Even so, three of these students applied this procedure with the radialconfiguration only after the failure of the successive halving algorithm. There was little evidence that thismultiplicative procedure was treated by students as a general algorithm for partitioning regions.In summary, the inconsistencies in the success with which the different students partitioned theregions into six equal parts appeared to be related to one of several beliefs about procedures fordetermining equal parts, all of which could be considered to be algorithmic in nature. These commonlywere: (a) the count of the lines is equivalent to the number of regions, (b) successive halving with theradial configuration results in equal parts, and (c) a configuration which will partition one region into equalparts will do so with others regardless of their different geometric characteristics. For final partitions, themost common source of error amongst these was the latter, the over-generalization of a configuration.In this regard, the square appears to be more difficult to partition successfully than the rectanglebecause the two quadrilaterals were treated as different partitioning problems. Instead, the salience ofgeometric properties shared by the square and the circle probably influenced students’ to differentiatebetween the two quadrilaterals.Algorithmic configurations provide a means to circumvent the global problem of directlyestimating and comparing the areas of the parts. Given a belief that the configuration will result in partsof equal area, judgements could be based on spatial relationships of the lines alone. For example,James used the hatched configuration to partition the circle. He adjusted the position of the two parallellines so that the three segments of the intersecting line were equal, with the belief that this wouldestablish the equality of the parts. In this sense, linear measures within a configuration indirectlydetermine the relative areas of parts. However, until a student relates the geometric properties of theregions to the choice of configuration, and relates the properties of the number to be partitioned to the184procedure for partitioning with a configuration, inconsistencies in the student’s partitioning of regionswill occur.Comparing Parts of Regions: The Cake TasksThis section addresses the question of the characteristics of the reasoning strategies studentsuse to compare the area, in particular the area of fractional units. Of particular interest was whetherstudents would consider a part-whole relationship between units as a basis for reasoning about thesecomparisons. For this purpose student responses to the cake tasks were analyzed. The students’responses to the four cake tasks presented during the interview were clustered because of differencesin the complexities of the tasks (see Figure 5.11). The data for each pair of tasks are presented in Table5.07.Table 5.07 was designed to reveal a number of response patterns. First, it allows the extent towhich students used different reasoning strategies to be examined with reference to each pair of tasksand all tasks in general. Students with similarities in their responses to the Cake Tasks have beengrouped within the table in order to assist the discussion of patterns in students’ responses to bothpairs of tasks. And finally, it is designed to explore the relationship between the consistency with whichstudents partitioned regions into equal parts and the extent to which they used reasoning strategieswhich referenced attributes of area to compare parts of regions. For this purpose the consistency withwhich each student partitioned the three regions into 6 equal parts, that is whether the studentpartitioned 0, 1, 2, or 3 regions consistently is also entered in the table.185Table 5.07Summary of the Reasoning Strategies used to Solve the Cake Tasks and the Consistency withWhich Students Partitioned Regions into Six Equal Parts.Cake tasks by reasoning strategiesNo.Consistent correctpartitions tasks Tasks 2 & 6 Tasks 3 & 5No.Groups regions 2&6 3&5 P P1 Ti Ti T2 N P Pi Ti Ti T2 N1.Derek 0 0 1 # . . . . . . # .Connie 1 0 2 # # . . . . . .Marlene 0 0 1 # . . . . . # # .2.Lolande 1 0 0 # # . . . . . . #Brock 2 0 1 # . . . . . . . . . #3.Dahlia 0 1 0 . . . . . .Tanimy 1 2 0 . . . . . # . . #4.Pete 3 2 0 . . . . . . .Coran 3 2 0 . . . . # . . . .James 0 1 1 . . . . . . . . # #5.Kasey 2 2 2 . . # . # . . . . . #Fanya 2 1 2 . . . . . . . . . #Edwin 1 2 2 . . . . . . . . . #6.Lara 3 2 2 . . . . . . . . .2 2 1 . . . . # . . . . # #NQI. The names of the students in Grade 5 are underlined.Reasoning strategiesP = perceptual strategy with a global judgement.P1 = perceptual strategy comparing one dimension.Ti = transformational strategy (imagined) compensating on two dimensions.Ti = transformational strategy (actual) comparing only one dimension.T2 = transformational strategy (actual) comparing two dimensions.N = numerical strategy.186AjH Ix AJ1 a[J!I 1 IAj ) ‘V1’liTasks 2 & 6 Tasks 3 & 5Figure 5.11 Comparisons of shaded fractional units: Pairs of cake tasks.Comparing Non-congruent Fractional Units.Part-whole relationships among units of area were seldom used as a basis for comparing theareas of the “pieces of cake.” More than two-thirds of the students did not interpret any of the tasks tobe problems of part-whole relationships of units. This was the case even with the commonconfigurations of fourths in Tasks 2 and 6. Instead, the problems generally were interpreted as ones inwhich pieces of cake were compared directly without regard to the regions of which the pieces were apart.Only a third of the students were relatively successful at comparing fractional parts withreasoning strategies which effectively addressed the two dimensional character of the task (see Groups5 and 6). The other students’ comparative judgements were subject to varying degrees of perceptualbiases unrelated to the space-filling quality of the parts. The most extreme case of such biases occurredwith the students in Group 1 who used only perceptual strategies.Some students responded in a relatively stable manner to all four tasks. They used eitherperceptual reasoning strategies alone, or only transformational strategies based on two-dimensionalreasoning(see Groups 1 and 5). However, for a majority of the students, the reasoning strategiesgenerally differed between the pairs of tasks. This suggested that changes in strategies might berelated to differences in the characteristics of the pairs of tasks. There appears to have been aninter-play between geometric properties of the tasks and ways in which students’ reasoned about area.187The comparisons of rectangles (Tasks 3 & 5) were resolved by most students through some form of atransformational reasoning strategy (see Figure 5.12). In contrast, the comparisons of rectangles totriangles were resolved more frequently through perceptual or numerical reasoning strategies. Theconfigurations in these latter tasks were less complex, which accounts for the larger number of studentsusing numerical strategies. However, the transformation of triangles into rectangles is geometricallymore complex. This may account for the smaller number of students using transformational strategieswith these tasks.Students in Group 2, 3, 4 and 6 changed their reasoning strategies in a manner consistent withthe complexity of the tasks. Tasks 3 and 5 were more complex for students who used numericalreasoning strategies, hence the shift generally to transformational strategies with these task. Tasks 2and 6 were more complex for students who did not use any numerical reasoning strategies, hence theirshift to perceptual reasoning strategies with these tasks.1.A_____2._________B_________A____BAfl]Figure 5.12 Examples of students’ transformations of triangles into rectangles andrectangles into rectangles.An exception to these general patterns was presented by the students in Group 4. Thesestudents successfully compared triangles and rectangles in Tasks 2 and 4, yet failed to compare the5.A B188rectangles in Tasks 3 and 5 correctly. That transforming rectangles was easier than transforming atriangle into a rectangle did not seem, on the surface, to apply to these students. However, the actualtransformations still were easier with the rectangles, but the act of transforming the rectangle appearedto fix their attention on the dimension on which they were equalizing the rectangles (see Figure 5.12).Their final judgements therefore were based on comparisons of only one dimension.In summary, there were different limitations in the types of reasoning strategies used bystudents, and the problem contexts in which different reasoning strategies were applied. Geometriccharacteristics of the parts to be compared, as well as characteristics of the partitioning configurationsappeared to affect the complexity of the tasks, and influence the types of reasoning strategies to whichstudents resorted.Variations in Reasoning Strategies.There were two characteristics of some students’ comparative strategies which might occurmore generally than the interview data suggest. The first involved differences in the unit relationshipsconsidered in students’ numerical reasoning strategies, and the second involved the application of aperceptual reasoning strategy to compare a triangle and rectangle.Figure 5.13 Comparisons of fractional units: Cake Task 4.Numerical reasoning strategies. There were two numerical reasoning strategies used bystudents. One involved tn-relational thinking about units, and the other involved bi-relational thinkingabout units. In the tn-relational case, the student reasoned that if two regions were equal in area, andboth regions were partitioned into 4 equal parts, then the shaded parts within each region would beequal regardless of their shape. They also recognized conditions when the fractional units were notequivalent, even though they were all fourths of a region. In the bi-relational case, the student189reasoned that each subregion was equal because they both represented 1/4 of a region, regardless ofthe relative sizes of the regions. This restrictive bi-relational reasoning about units might have beenused by other students to compare the subregions of Task 4 on the test (see Figure 5.14). Four of the5 students who incorrectly compared these areas, judged the parts to be equal despite the fact that thepartitioned regions were different sizes. Equality in such cases might have referred only to an abstractrelationship between fractional numbers, not to measures of the space within each part.PerceDtual reasoning strategies. Perceptual strategies were used most often to compare atriangle with a rectangle, not to compare two rectangles. In such cases, students invariably concludedthat the triangle was larger. A number of other students who eventually resolved the comparisons with atransformational strategy, also made this perceptual judgement initially, but on reflection compared theareas more systematically. In addition, on the test 9 students judged the triangle to be larger in Task 2,and 12 students did so with Task 6. This suggests that most of these students might consider thisperceptual judgement to be a reliable comparison of areas of the regions unless prompted to reflectfurther on their initial judgement.There are two possibilities which might account for the rather general view that a triangle, equalin area to a rectangle is judged to be larger. First, there is the sense that a triangle appears to be moreextensive than a rectangle. There is a greater distance between the geometric centre and the cornersof the triangle than of the rectangle. Without attention to other geometric characteristics of the tworegions, this “appearance” would led to the conclusion that the triangle is larger.A second explanation might rest in procedures followed to compare regions with bothperceptual (P1) and transformational strategies. In both cases, final judgements of the comparisonsgenerally were based on the comparison of the horizontal or vertical dimensions of the fractional parts.What differentiated strategies which were successful and unsuccessful in addressing the twodimensional character of the tasks was the actions or reasoning which preceded this final comparison.Students either (1) ensured that the regions were equal on the other dimension, (2) assumed thatregions were equal on the other dimension, or (3) did not attend to the question of the equality of theother dimension before they made the final comparative judgement. With the triangle-rectangle190comparisons, students might have (1) attended only to the comparison of the differences in onehorizontal or vertical “dimension,” or (2) assumed that a horizontal or vertical “dimension” of both regionswas already “equal” and concluded that the triangle was larger because it was longer on the “otherdimension.”. Procedures of equalizing and comparing horizontal or vertical dimensions, which appliesto comparisons of rectangles, might be over-generalized without regard to other spatial characteristics ofthe regions.Partitioning Regions and Comparing Parts of a RegionA feature common to most students’ reasoning about the comparative problems was that theydid not consider the relationship of parts to the region as a whole even when each region waspartitioned into congruent parts. Regions simply were not considered to be a unit that had beenpartitioned into fractional units of equal area. This was the case regardless of the extent to which thesestudents consistently partitioned regions into equal parts. For example, 2 of the 3 students (Coran &Pete) who consistently partitioned the three regions into 6 equal parts used transformational strategiesfor all of the cake tasks. The other student (Lara) was the one student who, when using a numericalreasoning strategy, used bi-relational reasoning without full regard to the relative size of regions.One critical difference between these two types of tasks is the congruent/non-congruentcharacteristics of the parts being considered. Partitioning a region into X equal, congruent parts doesnot require a student to consider the more general case, that is, that parts associated with allconfigurations which partition that region into X equal parts would be equal even when non-congruent.Students who consistently partitioned 2 or more regions into congruent parts used reasoning strategieswith the cake tasks which expressed one or more of the following beliefs:1. Belief in the reliability of perceptual comparisons of non-congruent regions based oncharacteristics unrelated or only partially related to comparison of areas.2. Belief in a compensatory relationship between dimensions and areas of rectangular regionswithout further means to act on their beliefs.1913. Belief that areas of regions are constant under transformations. Comparisons of non-congruentregions involved trials designed to reduce or eliminate the non-congruent characteristics of thefractional units by transforming one into a region more congruent with the second.4. Belief that non-congruent regions are equal in area under specific conditions of unitrelationships. The extent to which this belief was applied depended on the geometric and arithmeticcomplexity of the unit relations.Except in the last instance, the two types of tasks, partitioning a region into congruent fractional unitsand comparing non-congruent fractional units, appear to have been treated as independent problems.The Comparison of Irregular Regions: Tile TasksThe third point of view from which students representations of units of area was explored, waswhether students would compare irregular regions by reasoning with units of area measurement. Forthis purpose, student responses to the tile tasks were analyzed. These data are presented in Table5.08.Table 5.08 allows for a comparison of the reasoning strategies students used to compare areasin both interview tasks along with the students’ success at comparing the areas in these tasks on thetest. The reasoning strategy used by a student for each task is indicated by a hatch mark except in thecase where a student used a numerical reasoning strategy to resolve a task. These numerical reasoningstrategies are coded in the body of the table as mono or bi-relational. Also included in the left handcolumn are the test scores for all tile tasks. The students have been placed into three groups. The firstgroup contains students who used reasoning strategies during the interview which did not focusexplicitly on two dimensions, or whose test results strongly suggested that they treated units asequivalent regardless of their size. The second group contains students who used transformationalreasoning strategies which focused on both dimensions of the regions. The third group containsstudents who used a numerical reasoning strategy.In general, there is a consistency between the test results and the students’ responses duringthe interview. Students who were more successful on the test also reasoned numerically withTable 5.08Students’ Interview Responses and Test Scores for the Tile Tasks.Interview tasks byreasoning strategiesTile Task 5 Tile Task 6TestscoreGroups Max=6 P P1 Ti T2 N P P1 Ti T2 N1.Connie 3 # . . . . # .Lolande *2 . # . . . . #Brock 2 . . # . . . . #Marlene *2 . . . . MB . . . . BJames *2 . . . . B . . . . B2.Edwin 4 . . . # . . . #Derek 5 . . . # . . . . #3.Fanya 4 . . . . B . . . . BDahlia 5 . . . . B . . . . BKasey 5 . . . . B . . . . BPete 5 . . . . B . . . . BTammy 6 . . . . B . . . . BCoran 6 . . . B . . . . BLaia 6 . . . . B . . . . B11I 6 . . . B . . . . BNote. The names of the students in Grade 5 are underlined.* indicates that the error pattern on test suggests unitswere treated as equivalent regardless of their size.# indicates the strategy was used by the student.Reasoning strategiesP = perceptual strategy with a global judgement.P1 = perceptual strategy comparing one dimension.Ti = transformational strategy (actual) comparing one dimensionT2 = transformational strategy (actual) comparing two dimensions.N = numerical strategy.Unit RelationshipsMB = mono-relational then bi-relational; counted units asequivalent then used 4:1 ratio.B = bi-relational; related the units in a 4:1 ratio.192193appropriate units of area, or who used transformational reasoning strategies which referenced twodimensions during the interview. Exceptions to this pattern occurred with students who, on the test,treated units as equivalent regardless of their size, yet during the interview reasoned with bi-relationalunits. On the test these students gave a correct response only to tasks in which the units wereequivalent, and judged the region with the smaller units to be greater in all other tasks (see Figure 5.09).It was inferred that they were comparing regions on the basis of measures of numerosity. Theirinterpretations of units in an area measurement context were relatively unstable.The students who reasoned with units aggregated the smaller units into a larger unit, thenfollowed one of two procedures: (1) directly counted this larger unit within each region, or (2) matchedlarger units within each region in a one-to-one correspondence to compare the numerosity of the unitsin each case. When difficulties were encountered with these procedures, they invariably occurred withTask 6 which was perceptually more complex.In general, more students were successful with these tasks than with the partitioning or the caketasks. Two-thirds of the students interpreted the tile tasks as problems involving the comparison ofmeasures of congruent units of area. The facility with which most students solved these tasks wasprobably due to the