# Open Collections

## UBC Theses and Dissertations ## UBC Theses and Dissertations

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

#### Abstract

This study explores characteristics of students’ repertoires of representations in two mathematical contexts: whole number multiplication and the comparison of common fractions. A repertoire of representations refers to a set of representations which a student can reconstruct as needed. Of particular interest are (1) how multiplicative relationships among units were represented, and (2) whether continuous measurement was an underlying conceptual framework for their representations. In addition, the characteristics of students’ representations and interpretation of units of linear and area measurement were explored. Data were collected through a series of interviews with Grade 5 and Grade 7 students. Some results of the study were as follows. Each repertoire of representations was exemplified by a dominant form of units, either discrete or contiguous. Within a repertoire, all forms of units were related, first through a common system of measurement (either numerosity or area), and second through their two-dimensional characteristic. In the multiplication context, some repertoires were comprised only of representations with discrete units, but others also included some representations with contiguous units. Students sought characteristics in their representations which reflected those based on continuous measurement, however linear or area measurement was not used as a conceptual framework. Instead, all representations were based on the measurement of numerosity. Also, students exhibited different limits in their representation of multiplicative relationships among units. Some represented no multiplicative 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, but the physical characteristics of the units varied. Some repertoires had only contiguous representations of units, others also included representations with discrete units, and a few did not represent fractional units at all. Students’ representations reflected characteristics of area-based representations, however area measurement was not necessarily a conceptual framework. In addition, students’ beliefs about what constituted units of area measurement were variable. As a result, they either represented no multiplicative relationships among units, or fluctuated between representing two-unit and three-unit relationships. Linear measurement was notably absent as a basis for representations in both mathematical contexts. The one-dimensional characteristic of linear measurement did not fit students’ dominant framework for constructing mathematical representations. With respect to measurement, students represented linear units in terms of discrete points or line segments. Counting points and interpreting the count in terms of the numerosity of line segments was problematic for nearly all students. When partitioning regions into units of area, a few students also equated the number of lines with the number of parts. The direct relationship of action and result in counting discrete objects was generalized without consideration of other geometric characteristics. When comparing quantities having linear or area units, numerical reasoning was not always used. Alternatively, either quantities were transformed to facilitate a direct comparison, or only perceptual judgements were made. No students consistently used numerical reasoning to compare fractional units of area. In the latter situations, the part-whole relationship among units seldom was observed. In general, there was no direct relationship between the forms of representations used by students in the two mathematical contexts and the characteristics of their representations of units of the measurement contexts. The development of repertoires of representations appears to be context specific. The repertoires were strictly limited in terms of the forms of representations of which they were comprised.