I’m introducing integration to my calc students right now. It’s my sixth year teaching calculus, and I’m convinced that the notation we use for integration is a terrible for beginners. I’ve only convinced myself I understand it because I have spent so much time using it. I’ve forgotten what it’s like to be new to integration, and I ignore all the confusing bits.
Seriously. What does dx mean? Really, what is it? A little bit of x? How much? How can I multiply a function by dx? Where does the dx go when I integrate? How can plugging in two numbers account for all the area between them? Wait, what is negative area? But negative area becomes positive when those numbers are in the other order? How is this the same as a Riemann sum?
I understand that calculus teachers reading this post have some way of explaining what integration is and why it makes sense. I’m not looking for better explanations. I’m trying to find better ways to validate my students when they ask these questions, and when they feel dissatisfied with my answers.
Here’s the thing. The most successful students in this part of a calculus course are often the students who can avoid asking what all that notation means and just plug through the algebra. And I think that sucks. I want students to ask questions and to be curious. I want to validate that curiosity. I have a hard time validating curiosity about integration.
And all this trouble reflects the history of calculus. Ben Blum-Smith’s brief but hilarious play “Honor Your Dissatisfaction” gets at many of these themes. For centuries — yes, literally centuries — mathematicians used the machinery of calculus without really understanding why it worked. It just worked, and that was good enough.
I see this same phenomenon, though to a lesser degree, elsewhere in the math curriculum. We like to pretend sometimes that math is purely logical and is some bastion of reason and rationality. But the reality is that there are many more cracks in that artifice than we like to talk about. Notation is ambiguous. Convention is arbitrary. I paper over the inconsistencies so students can compute correct answers, and I worry that I squash curiosity for expediency.
I want to find a balance here. Math is ambiguous and arbitrary in lots of ways. Spending too much time emphasizing those ambiguities and arbitrarities risks sending a message to students that math is worthless and designed to confuse teenagers. But engaging honestly with these tensions validates the student experience. I know I can do more to help students see their struggles as a legitimate part of the learning process, rather than a reflection of their own shortcomings.
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