Running Head: Images of Rate

have presented details of a theory that addresses the reasoning that might underlie a person’s competent, flexible use of visible algebra. We have called this theory a theory of quantitative reasoning. We should note something about the word quantity. In mathematics, it is common for “quantity ” to be used loosely, almost synonymously with “number. ” We use the word quantity here more in the physical sciences tradition, as something to be measured. Quantitative reasoning, then, is reasoning about things and their attributes. This is not a precise representation of the theory, but it is sufficient for now. In this paper we will focus on the concepts of ratio and rate, approaching them from the point of view that they are constituted by mental operations grounded in images of multiplicative comparisons and dynamic change. We will then explore the idea that there is a developmental, reflexively constitutive relationship between students ’ images of (what we see as) rate-like phenomena and their mental operations entailed in proportional reasoning. We will also explore potential relationships between students ’ understandings of rate and their understandings of function and change in calculus. Ratio and Rat