The ideas of measurement and measurement comparisons (e.g., fractions, ratios, quotients) are introduced to students in elementary school. However, studies report that students of all ages have difficulty comparing two quantities in terms of their relative size. Students often understand fractions such as 3/7 as part-whole relationships or “three out of seven.” These limited conceptions have been documented to have implications for understanding the quotient as a measure of relative size and when learning other foundational ideas in mathematics (e.g., rate of change). Many scholars have identified students’ ability to conceptualize the relative size of two quantities values as important for learning specific ideas such as constant rate of change, exponential growth, and derivative. However, few researchers have focused on students’ ways of thinking about multiplicatively comparing two quantities’ values as they vary together across select topics in precalculus. Relative size reasoning is a way of thinking one has developed when conceptualizing the comparison of two quantities’ values multiplicatively, as their values vary in tandem. This document reviews literature related to relative size reasoning and presents a conceptual analysis that leverages this research in describing what I mean by a relative size comparison and what it means to engage in relative size reasoning. I further illustrate the role of relative size reasoning in understanding rate of change, multiplicative growth, rational functions, and what a graph’s concavity conveys about how two quantities’ values vary together. This study reports on three beginning calculus students’ ways of thinking as they completed tasks designed to elicit students’ relative size reasoning. The data revealed 4 ways of conceptualizing the idea of quotient and highlights the affordances of conceptualizing a quotient as a measure of the relative size of two quantities’ values. The study also reports data from investigating the validity of a collection of multiple-choice items designed to assess students’ relative size reasoning (RSR) abilities. Analysis of this data provided insights for refining the questions and answer choices for these assessment items.

This dissertation reports three studies of students’ and teachers’ meanings for quotient, fraction, measure, rate, and rate of change functions. Each study investigated individual’s schemes (or meanings) for foundational mathematical ideas. Conceptual analysis of what constitutes strong meanings for fraction, measure, and rate of change is critical for each study. In particular, each study distinguishes additive and multiplicative meanings for fraction and rate of change.

The first paper reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. Most teachers conveyed primarily additive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topics of targeted professional development.

The second paper reports the quantitative part of a mixed method study of 153 calculus students at a large public university. The majority of calculus students not only have weak meanings for fraction, measure, and constant rates but that having weak meanings is predictive of lower scores on a test about rate of change functions. Regression is used to determine the variation in student success on questions about rate of change functions (derivatives) associated with variation in success on fraction, measure, rate, and covariation items.

The third paper investigates the implications of two students’ fraction schemes for their understanding of rate of change functions. Students’ weak measurement schemes obstructed their ability to construct a rate of change function given the graph of an original function. The two students did not coordinate three levels of units, and struggled to relate partitioning and iterating in a way that would help them reason about fractions, rate of change, and rate of change functions.

Taken as a whole the studies show that the majority of secondary teachers and calculus students studied have weak meanings for foundational ideas and that these weaknesses cause them problems in making sense of more applications of rate of change.