Grade 7 students' understandings of division : a classroom case study
Creators: name: Rudge Clouthier, Gillian
Publisher: name: University of British Columbia
Year: 2011

description: 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.