Senior Middle School Students' Understanding About The Rate Of Change

This paper is a report of an empirical study of middle school students' understanding about the rate of change of a function. In 2005, a test on the rate of change was conducted to over 300 students of grades 1, 2 and 3 in a senior middle school where the author teaches, and 14 students were interviewed whose answers to the test were typical.From findings of the test and the interview, it is revealed that students' understanding about the change of rate are greatly influenced by the contexts of the problems and that their strategies used to understand and identify the rate of change are different with the contexts and are instable, resulting from the difference and incompleteness of development of the covariational reasoning abilities. The main results are as follows.(1) The students' understanding about the rate of change are greatly influenced by the contexts. Different terms such as the average rate of change, instantaneous rate of change, average velocity, instantaneous velocity, the slopes of a secant line and a tangent are usually given different mathematical meanings.(2) The students' understanding about the rate of change are influenced by the behaviors of a function. While identifying the changing patterns of a continuous function and a discrete exponential function, the students usually use different strategies. With the age of the students, the level of strategies goes up, but the notion of difference plays a dominant role in all three grades.(3) Generally speaking, the level of students' understanding about the rate of change is low, which is embodied in students' covariational reasoning ability. The difficulties in understanding the instantaneous rate of change lie in the explanation of the meanings of the point of inflection and the reason why a curve is smooth. From the view of the cognition psychology, the difficulties lie in construction of the image of continuous rate of change.At the end of the paper, some suggestions are given on the teaching of the rate of change.