Interpreting and Communicating about Phenomena with Negative Rates of Change

摘要

Much research over the last twenty years has documented the difficulties that students encounter when reasoning about and interpreting rates of change. The complexity of such reasoning has proven difficult for high achieving undergraduate mathematics students and students studying physics. To reason about rates of change, students must be able to simultaneously attend to both the changing values of the outputs of a function and changing values of the inputs to the function. In addition, students must be able to distinguish between the values of the outputs of a function and the values of the function's average rate of change over subintervals of the domain. When reasoning about changing phenomena, students often confuse these two quantities. Furthermore, students have difficulty in distinguishing between the amount of change in a function's output value over a subinterval and the average rate of change of the function over that subinterval. To meaningfully interpret the graph of a function that represents two quantities that co-vary, students need to be able to simultaneously attend to and distinguish among three quantities: the value of the output of a function, the change in the values of the function's output over a subinterval, and the change in values of the input to the function. Reasoning about the latter two quantities is a foundational understanding for average rates of change in pre-calculus and instantaneous rates of change in calculus.