# Open Collections

## UBC Theses and Dissertations

### Grade 7 students' conceptions of division Shandola, Darlene

#### Abstract

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.