One of my springtime habits this year has been to stop by Robert Talbert‘s office at 3:35 pm as I make my way back from teaching a linear algebra course (that runs in 6 weeks’ time, incidentally!). He’s been teaching calculus I over the same period, using Active Calculus, and we’ve had frequent conversations about the teaching of calculus and linear algebra. Our chat yesterday inspires today’s post.

In talking about the goals and purpose of calculus, we both expressed a desire to introduce students to mathematics different from calculus that is more modern and, in many ways, more important. We were in agreement: if it was curricularly possible, we’d both make radical changes to the three-semester calculus sequence. Instead, we’d follow the West Point curriculum for the first two years; or teach linear algebra, discrete math, and statistics; or … something. It reminded me that I’ve always liked what Gil Strang says: “Too much calculus!“

But, with so many client disciplines relying on calculus in their programs, it’s probably not curricularly possible to make a radical departure from calculus. (Whew, says the textbook author in me …) So, what should be the goals and purpose of calculus? Robert had a great phrase: “*calculus should be an introduction to doing mathematics in the style of a professional*.”

Now, as Robert points out, that begs a question: “what does it mean to do mathematics in the style of a professional?” That question merits further thought, thought that I hope to flesh out through additional posts and the comments. But for now, here are some basic thoughts we have on “doing mathematics like a professional.” For instance, students should have to think critically about hard problems, to write well and technically, and to use technology in a meaningful and responsible way. In thinking about further goals or purposes for calculus, I would add this: knowing calculus is part of being liberally educated in mathematics.

To expand a bit:

- Calculus students should have to think critically about hard problems.With the computing technology of today, the main point of calculus cannot be solving small, isolated, algorithmic problems. WolframAlpha (The Wolf!) exists. Deal with it. Students need to develop the skills to handle problems a computer cannot: ones with ambiguity, with multiple questions to approach, and that require critical thinking. If all we are expecting of students is that they can replicate tasks that WolframAlpha can do in milliseconds, we are failing them.
- Calculus students should write well, and technically. In addition to being good problem-solvers, students should be strong communicators. Here, too, calculus offers a great opportunity: to improve their written prose, to write technically about difficult content matter, and to learn to use notation, syntax, and mathematical language in ways that explain deep ideas in their own words. In a world of fast-changing and complicated ideas, clear written (and oral) communication skills are incredibly valuable.
- Calculus students should use technology in meaningful ways. One of the beautiful and powerful things about mathematics is the fact that viewing the same concept from multiple perspectives often illuminates the idea. Using technology in a way that generates additional perspective and insight is another valuable skill that students should acquire. In addition, any student who needs to actually do mathematics as part of her professional life will need to understand how to apply and make sense of results that come from using computational technology: spreadsheets,
*Geogebra*, the Wolf,*Maple*,*Mathematica*, and more. - Students who are liberally educated in mathematics should know calculus. Calculus is one of the great intellectual achievements of humankind, and it explains the behavior of moving and changing quantities in our physical world. It’s worth the time to learn, even if sometimes it seems like we spend too much time on calculus and its ilk.

As I reflect on some of the struggles I see students encounter when they transition from the more algorithm-based subject of calculus to more abstract and advanced mathematics, I think that calculus should also be an opportunity for students to see more of what mathematics is really like: beautiful, challenging, open-ended, interesting, technical, and more. We can use that perspective to make calculus a course where students can really strive to develop some of the most valuable skills of a liberal education, including being a creative and independent problem solver, being a strong communicator, and absolutely knowing how to work hard.

What am I missing? What do you think is the purpose of calculus?

PS: if you aren’t reading Robert’s blog over on the Chronicle’s site, you should.

I guess I don’t see it as having one overriding purpose. Reading this made me think of one more way to assess my students’ understanding of derivative. I’d like to ask them to explain it to a younger sibling or cousin, and write up what happened.