Personal Strategies for Math Learning

Michael Hartl • Aug 21, 2023

In Personal Goals for Math Learning, I alluded to strategies and resources for my math learning project. This is the post about strategies; see here for the first post on resources.

It’s worth noting that these are my personal strategies; the same strategies might not work for everyone. But I offer them here in the hope that they might be useful to some. (If you’d like to discuss things further, hit me up on Twitter and let me know what you think.)

1 Flying solo

While I’m certainly open to discussing math with other people, my math learning project is primarily a solitary activity. Several aspects of my planned strategy are affected by this.

Doing a solo project certainly has some disadvantages. For example, collaborating with other learners can be a real help when tackling difficult material. Taking a more structured course can be good for motivation and often gives access to an instructor for Q&A.

On the other hand, working alone has some big advantages as well. I can cover exactly the topics I want to without getting permission or buy-in from other people. I can move at whatever pace I want, spending more time on things that confuse me and taking breaks as necessary (or as life intervenes). And with things like social media and math Q&A sites, I can get many of the benefits of collaborating with others even when working mostly alone. In addition, I always have the option of recruiting fellow learners, or maybe even hiring a tutor, at a later date.

2 Breadth-first

I feel like the mathematics curriculum lacks sufficient breadth pretty much from top to bottom, starting in elementary school and going through graduate school. For example, there’s simply no good reason why topics like number theory or groups need to be deferred until they are covered in depth as part of upper-division university courses. Far better to get a taste of as many subjects as possible and then go into more depth later on.

This suggests a “breadth-first” approach (a reference to breadth-first searches in computer science), which involves starting with a broad base of topics covered in less depth initially, with greater depth planned for later. This is what’s sometimes known in the education biz as the spiral approach; it’s used to a certain extent in the standard math curriculum, but I believe it could be much more broadly applied. Since I’m writing this math curriculum myself, I’m proceeding accordingly.

3 Focus on foundations

One of the TAs1 in my freshman math class2 once told us that her favorite course as an undergraduate was abstract algebra, because “it was the first course that didn’t lie to us.” What she meant was that her other courses had always left some things vague or undefined (such as being unclear on what exactly a “real number” is), whereas abstract algebra proceeded from rigorous axioms and definitions.

Practically speaking, what this means is initially focusing on foundational topics like logic, set theory, number theory, abstract algebra, and real analysis (including properly constructing the various number systems used in mathematics), with a focus on rigorous proofs. It means doing things like proving that \( 0 \cdot x = 0 \) or showing in gory detail that the complex numbers form a field. Having already gotten started a little down this path, I can testify that proving such seemingly elementary results can be very deeply satisfying in a way that’s difficult to explain.

4 As easy as possible (but no easier)

While there is sometimes virtue in choosing an especially challenging resource—a sink-or-swim, “you’re learning more than you realize” approach—I find that I learn best when I err on the side of easier resources that allow me to achieve near-total mastery of their material. Basically, I want to be marginally overqualified for every resource I use, which in my experience is better for motivation and total amount learned and retained (Box 1).3

Note that one benefit of a breadth-first approach is that it makes this strategy easier to arrange. If you’ve already covered a couple of chapters on, e.g., real analysis as part of a first round of learning, a full-length book on the subject is naturally going to be easier.

Box 1. Learning while overqualified

One of my most satisfying learning experiences was taking Applied Mathematics 105b (Ordinary and Partial Differential Equations) my senior spring in college. In nearly all previous math and science courses, the problem sets were challenging enough that virtually everyone I knew (including me) needed to collaborate in order to solve them. There is sometimes virtue in this—you can end up learning a lot from your fellow students, as well as bonding with them in many cases—but the downside is that you can lose self-sufficiency along the way.

What made Applied Math 105b different was that I was borderline overqualified for it—the course was aimed mostly at sophomores and juniors, so as a senior I had an extra year or two of coursework on most of the class. I initially underestimated the implications of this, though, and I was absolutely astonished to discover that I could solve all the problem sets myself, without any collaboration whatsoever. But even though I’d seen a lot of the material before, it was still challenging, and I learned a ton. (It didn’t hurt that the instructor for the course, Howard Stone, was just fantastic.) I ended up solidifying and synthesizing a bunch of material I’d seen only piecemeal in my other courses.

As a result, my positive experience in Applied Math 105b contributed to my belief that “as easy as possible (but no easier)” is often a good fit for my personal learning preferences and goals.

One potentially surprising aspect of this strategy is just how difficult it is to find something that is clearly “too easy”. For example, in Personal Goals for Math Learning I mentioned starting with “high-school level” contest math, but you might be surprised at how hard “high school” math can be. For example, do you know how to write

\begin{equation} \label{eq:aops_cubes} \frac{1}{\sqrt[3]{1} + \sqrt[3]{2} + \sqrt[3]{4}} + \frac{1}{\sqrt[3]{4} + \sqrt[3]{6} + \sqrt[3]{9}} + \frac{1}{\sqrt[3]{9} + \sqrt[3]{12} + \sqrt[3]{16}} \end{equation}

in simplest terms? It’s pretty tricky, and yet it qualifies as “high school” for the purposes of contest math. (If you want to try your hand at solving the problem, be warned that it is quite difficult unless you have a background in contest math; see the footnote for a hint if you get stuck. A full solution appears in Solution to a Contest Math Problem.4)

In addition, it’s so easy to make math harder that it makes sense to err on the side of “easy”. For example, my early work in contest math has included factoring and solving quadratic equations like this one:

\begin{equation} \label{eq:contest_math_quadratic_equation} 49x^2 - 316x + 132 = 0. \end{equation}

I have a lot of experience with quadratic equations, so I was able to solve Eq. (2) fairly easily by factoring it directly:5

\[ (49x - 22)(x - 6) = 0 \Rightarrow \text{$x = \frac{22}{49}$ and $x = 6$.} \]

Realizing that the problem was too easy for me, I made it harder by solving it in Sage as well.6 This involves using the intuitively named functions factor() and solve() (Listing 1), but I didn’t know those offhand, so I definitely learned something. Moreover, since even that step was pretty easy, I gave myself the extra challenge of being able to extract the values themselves, which involves the much less intuitive subs() function (Listing 2).

Listing 1: Factoring and solving a quadratic equation in Sage.
>>> factor(49*x^2 - 316*x + 132)
(49*x - 22)*(x - 6)
>>> solve(49*x^2 - 316*x + 132 == 0, x)
[x == 6, x == (22/49)]
Listing 2: Extracting the values from the result of solve().
>>> solutions = solve(49*x^2 - 316*x + 132 == 0, x)
>>> for solution in solutions:
...    print(x.subs(solution))

5 Parallel or series?

Should I learn several topics in parallel or study one and then another in series? Or maybe it’ll be some mix of the two? These are open questions, and I’m still experimenting to find the strategy that works best for me. I’ll plan to post updates if I have any insights I think are worthy to share.

6 Repetition and reinforcement

Repetition is a severely underrated learning strategy in my view. Sometimes this is as simple as reading a book chapter and then just reading it again. Sometimes it means using multiple books for the same subject. And sometimes it means using different media for reinforcement, such as videos that cover the same basic material as a book. (I have more to say about these possibilities in the posts on resources for math learning; see here for the first one.) It can also mean reviewing your own notes and answers; this is especially easy and fun when you’ve got nice typeset versions to look at. This brings us to…

7 LATEX all the things!

Finally, while I’m using a paper notebook for scratch work, I decided early on to write up nearly all my notes and solutions as properly typeset LATEX documents.7 This adds a significant amount of friction, often greatly expanding the amount of time it takes to write down problem solutions, but there are lots of benefits. Indeed, one surprising and unintended benefit was this very blog: using MathJax and jekyll-latex, I can easily paste things like Eq. 1 and Listing 2 into a post like this one.

Writing up careful notes and solutions is in effect “teaching” someone else (even if only your future self), which is a great way to learn. Additionally, it’s satisfying to have a polished product to show, again even if the only audience is yourself, and makes it much easier and more fun to review what you’ve done. It also serves as a reminder that you’re actually accomplishing something. For example, as I was taking the first tentative steps in this project earlier this year, I felt like I was making hardly any progress, then noticed I’d written over 200 pages of notes and solutions in only a few months. And I’m just getting started!

1. Technically a “TF” since at Hahvahd they have “teaching fellows” instead of mere “teaching assistants”.
2. Mathematics 22, “a course in mathematics for students of physics”, which covered advanced calculus and linear algebra, with a particular emphasis on the calculus of differential forms.
3. One useful heuristic is that if a resource says something like “Regarding prerequisites, we assume familiarity with Subject X, and Subject Y is useful but not required”, I want to know Subject X cold and have a solid grasp of Subject Y.
4. Apply the identity \( (a - b)(a^2 + ab + b^2) = a^3 - b^3 \) and observe that the result is a telescoping sum.
5. Even if I hadn’t been able to factor it by hand, I could have used the quadratic formula if necessary.
6. I mentioned Sage in Personal Goals for Math Learning when discussing computational math.
7. Quick pronunciation note: I prefer to say “lay-tech”, and I’m pleased to observe that macOS agrees.