This study is concerned with Grade 7 students' conceptual and procedural understandings of division. Although division is formally introduced in Grade 3 or 4, late intermediate students frequently have difficulty understanding both the concepts and the procedures associated with division. The classroom case study was chosen as the method of investigation for this study. Because the researcher was also the enrolling teacher of the group of 22 Grade 7 students, the conditions of the study were as similar as is possible to regular classroom instruction. The investigation followed a unit of study in division of whole numbers and decimal fractions from the pretest, through instruction, to the posttest. The researcher elicited students' understandings of division in computational and problem-solving situations in a variety of ways. Students wrote a pencil-and-paper pretest which was designed to reveal understandings. Areas of interest identified by the pretest were then investigated through small group and whole class discussions. Instruction was based on eliciting and confronting students' beliefs regarding division, and on strengthening conceptual understanding of both division and decimal fractions. Students viewed division procedurally, attaching little meaning to the processes associated with the division algorithm. Approximately one fourth of the students were uncertain about the meaning of the two forms of notation, and most read "b ÷ a" as "b goes into a." When asked to use manipulative materials to reflect a division question, some students were unable to do so independently. It was found that students relied heavily on the partitive model of division. Although some students demonstrated an understanding of quotitive division, these students also tended to rely on partition and turned to quotition only when it became apparent that partition was not appropriate. This reliance on partition influenced the students' ability to solve story problems requiring division. Students were able to solve story problems which fit the partitive model: the divisor is a whole number and is less than the whole number dividend. In situations where this was not true, students had difficulty. In these cases, students reversed the terms of the question or chose an operation other than division. These results led to an investigation of students' beliefs about division. The belief that "division always makes smaller" was common. This belief stems from partition with whole numbers where it is true. A related belief held by students is that the divisor must be smaller than the dividend. An exception is the case where this would necessitate a divisor less than one. In this case, students preferred a larger whole number as the divisor. Division by a number less than one was seen as illegitimate. Division involving decimal fractions was generally difficult for students. Weak place value concepts, coupled with a belief that whole numbers and decimal fractions were two separate and unrelated number systems, contributed to difficulty when solving problems. Students had few representations for decimal fractions which compounded their difficulty. The dominance of partition and the tendency to overgeneralize whole number rules appear to be partly responsible for this. When solving problems students showed little evidence of planning or looking back. Generally they found the numbers in the problem and performed the operation that seemed appropriate. Decisions about operations were often driven by the relative size of the numbers in the problem and by the beliefs mentioned earlier. Because they omitted the looking back phase of problem solving, students rarely accounted for remainders and did not recognize when an answer was unreasonable. Implications for instruction resulting from this study centre on the assessment of students' understanding of division. This can be accomplished in the regular classroom setting through pencil-and-paper tests, small group work, whole class discussions, and individual interviews. Beliefs which may interfere with learning must be revealed and confronted. Asking students to defend and justify their thinking is part of this process. Students' reliance on partition and their procedural view of division suggest changes in the way in which division is introduced in the early intermediate years. Delaying the formal introduction of the division algorithm to Grade 5 would allow more students time to develop their conceptual understanding of partition and quotition. Students should focus on estimation and reasonableness of responses. Introduction of division involving decimal fractions, including numbers less than one, could be accomplished by using manipulative materials and calculators. Contexts in which the divisor is greater than the dividend should also be introduced in the early intermediate years. Procedures, when finally introduced, should be linked to the concepts. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

This study is concerned with children's conceptions of division in both computational and problem-solving settings. Division was chosen because it is a mathematical topic with which many children have difficulty. Even though division is often introduced in the primary grades and reviewed every year following, late intermediate students still have difficulty understanding this concept. For this investigation, the researcher chose to use a semi-structured individual interview as a means of collecting data about Grade 7 students' conceptions of division in different contexts. During the interview, each student was asked to describe his or her thinking while working through a series of computations and word problems involving division with whole numbers or with decimal fractions. Both whole numbers and decimal fractions were used in the interview items in order to investigate whether or not students' conceptions of division changed as they worked with one, then the other. Twelve students were chosen for this study. It was found that these seventh graders varied in their demonstrations of different meanings of division. Some students demonstrated only the partitive meaning, some the quotitive, some that division is the inverse of multiplication, while others demonstrated a variety of meanings of division. It was noted that students who had an understanding of both the partitive and quotitive meanings of division were more successful solving the problems presented. This could be due in part to the notion that implicit models of division, such as the partitive model, influence problem-solving behaviour. It was also found that some students hold particular mathematical beliefs about division and about the form of the divisor which influence their problem-solving ability. Often these beliefs or misconceptions are a result of an overgeneralization of whole number rules. A student's choice of operation could be influenced by a number of factors including a student's implicit model of division, a student's mathematical beliefs, and the implied action in a problem. Some students used immature strategies such as verbal cues or "try all the operations and see which produces the most reasonable result" to determine the operation. These strategies indicated a lack of understanding of the meaning of the division operation. In some cases, students were able to reason qualitatively but were unable to relate that reasoning to mathematical symbols. Although they could give an approximate answer, they could not perform any further calculations. Implications for instruction resulting from this study include assessing students' conceptual understanding of the division concept and the algorithm through interviews and group discussions prior to and during instruction. Related to this is the notion of teachers ascertaining if students hold mathematical beliefs or misconceptions which may influence new learning and/or the application of knowledge. Teachers must be aware of students' thinking in order to plan instruction which will place those beliefs in context. Instructional activities should be planned which emphasize understanding of the division concept and of the long division algorithm. There should be a linking of conceptual knowledge and procedural knowledge by pairing activities using concrete materials with symbolic representations. The division concept should be discussed in terms of both whole numbers and decimal fractions. The calculator could be used to explore relationships between the divisor, the quotient, the dividend, and the remainder in these different number systems. / Education, Faculty of / Graduate

Literature emphasizes how important it is that procedural and conceptual knowledge of mathematics should be learned in integration. Yet, generally, the learning and teaching in mathematics classrooms relies heavily on isolated procedures. This study aims to improve teaching and learning of partitive and quotitive division, moving away from isolated procedural knowledge to that of procedures with their underlying concepts through the use of manipulatives, visual representation and questioning. Learning and teaching lessons were designed to teach partitive and quotitive division both procedurally and conceptually. The study explored the roles these manipulatives, visual representations and questioning played toward the conceptual learning of partitive and quotitive division. It was found that manipulatives and iconic visualization enhanced learning, and this could be achieved through scaffolding using a questioning approach. It was concluded that manipulatives and iconic visualization need to be properly planned and used, and integrated with questioning to achieve success in the learning of procedural and conceptual knowledge.

This study was aimed at discovering the differences in how children responded to word problems involving an operation in which they had received formal instruction (subtraction) and word problems involving an operation in which they have not received formal instruction. Nineteen children were individually interviewed and were asked to attempt to solve 6 subtraction and 6 division word problems. Their solution strategies were recorded, and analysed with respect to whether or not they were appropriate, as to whether or not they modeled the structure of the problem, and as to how consistent the strategies were, within problem types. It was found that children tended to model division problems more often than subtraction problems, and also that the same types of errors were made on problems of both operations. It was also found that children were more likely to keep the strategies for the different interpretations separate for the operation in which they had not been instructed (division) than for the operation in which they had been instructed (subtraction). For division problems, the strategies used to solve one type of problem were seldom, if ever used to solve the other type of problem. For subtraction problems, children had more of a tendency to use the strategies for the various interpretations interchangeably. In addition, some differences in the way children deal with problems involving the solution of a basic fact, and those involving the subtraction of 2-digit numbers, were found. The 2-digit open addition problems were solved using modeling strategies about half as often as any other problem type. The same types of errors were made for both the basic fact and the 2-digit problems, but there were more counting errors and more inappropriate strategy errors for the 2-digit problems, and more incorrect operations for the basic fact problems. Finally, some differences were noted in the problem-solving behaviour of children who performed well on the basic fact tests and those who did not. The children in the low group made more counting errors, used more modeling strategies, and used fewer incorrect operations than children in the high group. These implications for instruction were stated: de-emphasize drill of the basic facts in the primary grades, delay the formal instruction of the operations until children have had a lot of exposure to word problem situations involving these concepts, use the problem situations to introduce the operations instead of the other way around, and leave comparison subtraction word problems until after the children are quite familiar with take away and open addition problems. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

Yeung Pui Lam. / Thesis submitted in: December 2003. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 99-102). / Abstracts in English and Chinese. / Table of Content --- p.i / List of Tables --- p.iv / List of Figures --- p.v / Abstract --- p.vi / Chapter Chapter 1 --- INTRODUCTION / Research Background --- p.1 / Purpose and Significance of Study --- p.6 / Organization of the Thesis --- p.8 / Chapter Chapter 2 --- CONCEPTS OF UNIT / Introduction --- p.10 / Concepts of Unit as Key Concepts for Transforming Learning from Whole Numbers to Rational Numbers --- p.11 / Development of Unit Concepts across Additive and Multiplication Conceptual Fields --- p.11 / Six Aspects of Unit Concepts --- p.15 / Formation of Units in Rational Number Situation by Partitioning --- p.19 / Formation of Singleton and Composition Unit --- p.22 / A Flexible Concept of Measurement Unit --- p.23 / Decomposition and Composition of Unit --- p.23 / Reconstructing the Unit Whole --- p.24 / Fraction Equivalence --- p.24 / Chapter Chapter 3 --- SITUATIONS OF MULTIPLICATION AND DIVISION / Introduction --- p.26 / Semantics of Rational Numbers --- p.27 / Part-whole Subconstruct / Part-whole Comparison --- p.27 / Quotient --- p.28 / Operators --- p.31 / Measures --- p.32 / Ratio --- p.32 / Structures of Multiplication and Division --- p.34 / Vergnaud's Interpretation on Multiplicative Structures- --- p.35 / Isomorphism of measures --- p.35 / Product of measures --- p.38 / Multiple Proportion --- p.39 / Greer's Classification on Multiplication/Division Situations --- p.40 / Relationship between Unit Concepts and Rational Number Multiplication and Division --- p.46 / The Role of Unit in the Scope of Rational Numberśؤ --- p.46 / Children's Implicit Model of Fraction Multiplication / Division --- p.47 / The Role of Unit in Multiplication/Division Situation --- p.49 / Research Question --- p.54 / Influence of Concepts of Unit in Students' Solving Rational Number Multiplication and Division --- p.55 / The Development of Unit Concepts in the Course of Learning --- p.57 / Chapter Chapter 4 --- METHODOLOGY / Design --- p.58 / Participants --- p.58 / Instruments --- p.59 / Unit concept test --- p.60 / Fraction multiplication and division test --- p.61 / Procedures --- p.62 / Research Hypothesis --- p.63 / Chapter Chapter 5 --- RESULTS / Dimensions of the Unit Concepts --- p.67 / Relationship between Various Concepts of Unit and Fraction multiplication and division Performance --- p.68 / Relationship between unit concepts and rational number multiplication and division situational problem --- p.69 / Differentiated relationship between unit concepts and situational vs. symbolic rational number multiplication and division --- p.71 / Relationship between unit concepts and types of situational rational number multiplication and division- --- p.73 / Relationship between Grade Level and Performance on Measures of Unit Concepts and Rational Number Multiplication and Division --- p.83 / Chapter Chapter 6 --- DISCUSSION / Summary of Findings --- p.87 / Importance of Concepts of Unit --- p.88 / Dimension of concepts of unit --- p.88 / Relationship between concepts of unit and situational problems --- p.89 / Differentiated impacts of concepts of unit on situational problems and symbolic problems --- p.90 / Differentiated impacts of concepts of unit between multiplication and division problems --- p.91 / Implications of the Findings for Development of Concepts of Unit and Mathematics Learning --- p.92 / Research Implication --- p.92 / Instructional Implication --- p.93 / Limitations in the Study --- p.94 / Directions for Future Research --- p.96 / Conclusion --- p.97 / References --- p.99 / Appendices / Appendix 1: The questionnaire used in present study --- p.104 / Appendix 2: The characteristics of items in Questionnaire I- --- p.115 / Appendix 3: The characteristics of situational problems of fraction multiplication and divisionin questionnaire II --- p.116 / Appendix 4: The CFA model of concepts of unit --- p.117 / Appendix 5: Sample cases for illustrating strategiesin solving situational problems --- p.118

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