Math 120: Calculus II

Last taught in Fall of 2023.

Context

Math 120 fills many roles at Carleton and, as a result, can feel a bit like an everything-in-the-kitchen-sink sort of class. “Vectors, partial derivatives, and iterated integrals in Calculus II?” I hear you ask. “Where are the sequences and series??” Ah, those are in Calculus III! It’s peculiar, and it’s reflected in the vague learning objectives above (I know, I’ll fix that next time I teach this class).

When I design a course, my approach is to ask: “What story can I tell over this term?” The first time I taught Math 120, I was perplexed by the minimal syllabus and wondered how I could develop a coherent arc, where students are repeatedly exposed to rich core concepts and come to know them in deep ways while building to an exciting conclusion. As usual, I needed to understand some history!

Before I joined the faculty at Carleton, the Calculus sequence (among other things) had undergone a substantial overhaul in response to the needs of the department and College community. Why do students see vector algebra or partial derivatives? So the physicists and chemists have some familiarity before those ideas come up in their major courses! What about Lagrange multipliers? Ah, that’s for the economists! Why this aside on differential equations? Well that’s for everyone, especially the application to population growth. Indeed, much of Math 120 is a sort of primer for things to come, with curriculum shuffled and reprioritized as a result. While this course might suffer from a lack of coherent mathematical through-line, I find this context useful for a sort of meta-arc: Calculus II as a calculus, i.e., a sort of elementary methods course in problem-solving. Developing curriculum at liberal arts institutions is hard!

Now, as is true for many Calculus II courses across the world, Math 120 is often the last math class a person will ever take. In 2023, Dave Kung‘s farewell keynote with Project NExT reframed crises in education arising from ChatGPT and other large language models as an opportunity for reevaluating our priorities. What do we hope students learn in our courses and how can our class structure and assessments move us there? As mathematicians, we desire many things for our students: deep understanding beyond (but informed by) calculation, practice in communicating mathematical ideas, and ethical dimensions of our disciplines. These are especially vital as we face disciplinary reconfigurations via ongoing processes of automation. It’s easy to imagine these things for our majors, but more challenging (and, I would argue, just as important) for our sojourners! When developing my version of Math 120, and in the iterations I’ve run since, these problems were at the forefront of my mind. Hence we have…

Special features

Quizzes

These were loosely inspired by an article of Lew Ludwig. The objective of this medium-stakes group project, of which there are roughly four throughout the term, is to challenge participants with the problem of evaluating how someone understands math. Students develop a quiz on course concepts using graphics designed in Mathematica. The project outline details the number of questions in the quiz, the concept to be assessed and constraints on format for each question, and sample code. The group is expected to turn in a quiz (typed) and a solution manual (likely handwritten), and an explanation for how each question assesses the concept. The results are both peer- and instructor-evaluated (now that I’m more practiced, I find these much easier and faster to grade that other assignments) via a rubric organized on three pillars:

• Is the question well-posed? (e.g., do we have the information we need to solve the question, are any included graphs properly labeled and scaled, is it clear what is being asked, etc.)
• Is the question effectively assessing the indicated concept? (e.g., does solving the question require an unnecessary amount of auxiliary knowledge, is the central concept easily circumvented, is the solution needlessly complex, etc.)
• Are the solution and explanation mathematically sound and effectively justified?

This assignment is fantastic (and challenging to pull off) for many of the reasons we learn so much through teaching. Quizzes are designed to be taken without a calculator, which really forces students to think carefully about how to craft feasible but challenging questions. “Okay, we need to write down a separable differential equation for the quiz-taker to solve. What about something simple, like ? Oh… that’s harder to solve than we thought, and it will be annoying to write the next initial value problem question. Well…” This project requires a good deal of scaffolding and I’ve found it most successful when students begin in class, where they can work together at the boards to blueprint their quiz and think through its core elements and I can circulate as a guide.

While I’ve experienced some initial resistance to the unconventional format, students routinely attest to the effectiveness of this assignment in their course journals; I always get a handful of comments expressing new appreciation for the work we professors do on their behalf. Students work together, they think about what makes problems hard or interesting, and they tinker with Mathematica to visualize oodles of curves, surfaces, and vector fields. Next time I teach Math 120, I will include short surveys with these assignments so that students can indicate how work was shared throughout their group.

Fractals

In my first iteration of Math 120, before developing the quizzes, I included weekly readings and discussions in math history and sociology. This culminated in a final project, on a topic of the student’s choice rooted in these discussions: these ranged from historical developments of a specific concept or practice to ethical questions regarding applications of math or participation in mathematical institutions. One of the objectives of this assignment was for students to reflect on what it takes for math to happen—reports about a specific mathematician were disallowed, so as to discourage thinking of breakthroughs as coming from single individuals acting in a vacuum. The formats varied, from papers to podcasts. Having spoken with many of these students in the years since, these projects were clearly a meaningful opportunity to think about mathematics as a living and human discipline. They were also incredibly difficult to grade and took up way too much of the students’ time just prior to finals! I don’t do that anymore.

From all this, one of the units that clearly resonated with students was our unit on fractals. Now, I am a mathematician because of fractals. Due to awful middle school experiences, I came into high school with a strong distaste for math; it was because of an amazing programming class by an extraordinary teacher that I came to know math (via coding fractal visualizers) as joyful and exciting. Fractals are a source of endless wonder and, as I later learned from Ron Eglash’s book and others, a site of contested knowledge witnessing cultural aspects of mathematical practice¹ and interplays of mathematical institutions with colonialism.

After some experimentation, the fractals unit has been developed more thoroughly and incorporated as a warm-up for Math 120. There are a myriad of technical advantages to this: we can challenge students to think of dimension in new ways, review troublesome concepts like logarithms, and recall calculus as springing from limiting processes². I’m always sure to include the Weierstrass function to drive home how fractals defy the assumptions that we build into calculus, and a brief survey of rich and diverse examples in Africanist material cultures. This has become a hallmark of my course that, especially for those whom Math 120 is their first math class in their first term of college, appears to have a lasting impact.

1. See our article on fractals in Africanist music and dance. ↩︎
2. A thought-provoking exercise is asking students to draw a conceptual map of calculus—the big ideas, how they fit together, and so on. You expect, for example, “Differentiation” and “Integration” to feature prominently, maybe connected by an edge labeled “Fundamental theorem” or something of that flavor. I have found, across many institutions and stages of education, that limits (if they are included at all) appear off to the side as an afterthought, usually half-heartedly connected via L’Hôpital’s rule or maybe indefinite integration. Yet, if you ask an instructor to do the same exercise, limits lie at the very heart of the picture! ↩︎