The questions that guide my writing and teaching are: what content and habits do I want students to learn, and how can I keep them engaged as I help them get there? I want to center students and their needs rather than teachers and their expectations. Here are a few examples:
- I use the phase “area between the curve and the x-axis” rather than “area under the curve” because it is simply more accurate. As calculus teachers, we all know what we mean by “under the curve.” However, the fact is that integrating a negative function finds an area above the curve. In addition, calling it “area between the curve and the x-axis” makes it easy to switch to “area between the curve and the y-axis” without having to talk about vertical areas as opposed to horizontal ones. Our students benefit when we switch to more accurate terminology rather than jargon.1
- I start out by referring to “rate of change over an interval” rather than “average rate of change.” I want students to be focused on the essential distinction between slope at a point and slope over an interval, rather than being distracted by what the word “average” means in this context. I start using the term “average rate of change” after defining what “average” means in the context of calculus. And then the terminology makes total sense: the average rate of change of a function is the average value of its derivative2.
- I try to sequence content in such a way that students are always building on what they already know. For example, I introduce integration using accumulation problems, before moving on to area. This helps students develop intuition about integration, by drawing on common understanding about accumulation in daily life. It also allows for a deeper and more intuitive understanding of the fundamental theorem.
- I delay the introduction of limits, beginning instead with numerical calculus. When we start with limits, students don’t really have anything to build on (other than, perhaps, a brief exposure in precalculus). In contrast, starting with numerical derivatives and integrals, one can draw on students’ intuition regarding accumulation and rate (rainfall, speed, population growth, etc) and tie these ideas to prior knowledge regarding slope and area. Having done this, limits can be motivated by the desire to make numerical derivatives and integrals more accurate.
- I spend serious time making sure that students understand the distinction between “change” and “rate of change,” with or without a limit involved. I used to have students say things like “speed and displacement are the same, because they are both change in position.” This struck me as a fundamental misconception about the nature of slope and I gradually came to realize that it was worth spending some time addressing this issue beginning on day 1
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- Yes, students need to know the phrase “area beneath the curve” because everyone uses it, but I just tell them that it is a common shorthand for the longer but more accurate phrase “area between the curve and the x-axis.” ↩︎
- This follows immediately from the Fundamental Theorem and the definition of average value. ↩︎