Enduring Understandings For Derivatives

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

1. Understand the difference between change and rate (or ratio) of change.
2. Understand slope (or rate of change) as quantity which is dynamic, and is therefore worth viewing as a function in its own right.
3. 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).

Understanding number 1 is something I would have hoped that most of my students learned in 6th or 7th grade. But experience has taught me otherwise: in my first years teaching calculus I often got asked “what is the difference between displacement and speed? Aren’t they both just change in position?” It occurred to me at some point that a student asking this question lacks essential background information to engage with calculus. If a student thinks that “change” and “rate of change” are the same thing, how can they possibly understand the idea of finding a change in some quantity by integrating a rate of change in that quantity?

I think many students have a vague understanding of how velocity differs from displacement, but calculus really forces the issue in a way that nothing before it does. Spending a small amount of time addressing this distinction at the start of my class has been one of the lowest-effort, highest-yield changes I’ve made in teaching calculus. And the growth rate activity really gets students to think about this: there are parts of that activity that are about “how much someone grew” and parts that are about “how fast they grew.” Anyone who doesn’t get that distinction will immediately identify themselves so that I can have a conversation with them about this concept.

Understanding Number 2 is often glossed over in calculus classes; an understanding of this concept is implicit in the definition of a derivative and the concept of finding slope at a point. Yet many students come into my class thinking of slope as a single number. What is slope? It’s m, the slope of the line. It takes some thinking to grasp that a function (if it’s not a line) has different slopes at different points. And even more thinking to see that this means that we can think of slope as a function in its own right, and that this is actually a valuable thing to do. Again, I like the growth rate example because it highlights the value of making an entirely new graph showing variations in the slope of some other graph. And it gives students the chance to engage with the concept of slope as a function before they engage with limits.

I would argue that a student can’t possibly engage with understanding number 3 unless they have first grasped understanding number 2: if a student doesn’t think of slope as a dynamic quantity (one which is different at different points), how can they possibly comprehend the idea that slope on an interval is different than slope at a point?

Understanding 3 is where limits come in, and this is where most calculus classes start. But, important as this idea is, it really only makes sense to discuss it after students have had some explicit exposure to understanding 2. And to understand why we might care about slope at a point in applications, understanding 1 is also essential background. Which is why I put it third in the list: If we are making a real effort to meet students where they are, then this concept must be taught after the other two.

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