As I’ve been preparing to give a talk at NCTM in October, I’ve been thinking about my much-beloved growth rate activity and why I like it so much. Thinking about this helped me name three enduring understandings for derivatives: I want students to
- Understand the difference between change and rate (or ratio) of change.
- Understand slope (or rate of change) as quantity which is dynamic, and is therefore worth viewing as a function in its own right.
- Understand the distinction between slope over an interval and slope at a single point.
Understandings 1 and 2 are, perhaps, the most important to the widest range of students: these are concepts that everyone should understand in order to be an intelligent consumer of quantitative information. And yet understanding 3, which is of interest to a relatively select audience, is what often gets the most emphasis in calculus classes. Moreover, I believe that students have a hard time engaging with number 3 if they don’t first understand numbers 1 and 2. So, I start my class by teaching understandings 1 and 2, then get to understanding 3 later in the year (See here for more thoughts on delaying limits).
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