Back in the day, textbooks and classes were the way to learn math. Today, we have abundant online options. I wrote earlier this year about the benefits of practicing on Khan Academy, even if you’re not in its target audience. A similar online offering is Brilliant, which like Khan Academy has online math problems, but which uses a different philosophy of learning.
Khan Academy
The Khan Academy system mirrors the traditional American primary and secondary school approach to math instruction: teach students a process for each problem type, then have them practice using that process to solve problems. But Khan Academy offered two innovations: rather than a live instructor, give students recorded videos they can watch at home as many times as they need to; and rather than a fixed problem set that’s the same for every student, keep giving students problems of the same type until they demonstrate mastery of a concept.
There is much to be said for the traditional approach as interpreted by Khan Academy. The best way to learn math is by practicing it. To practice it, you need problems and solutions. As much as you might think you understand properties of exponents (for example), you don’t know for sure how well you know them until you have solved a lot of problems. Problems solidify the concepts in your mind and verify that you understand the main ideas and edge cases.
But despite the benefits of this approach, it has one major drawback: by focusing on learning a process and matching it with a problem type, students risk learning math as just a collection of procedures. Besides being inaccurate, this is problematic for a couple of reasons: It’s difficult to remember procedures, and procedures probably won’t help when you’re trying to solve more conceptual problems, like proofs. Brilliant tries to address this using a different approach to problems.
Brilliant
To explain math concepts, Brilliant relies on text and images rather than videos. But unlike the text and images found in textbooks, the ones on Brilliant don’t have large sections of prose and math interspersed with examples. Instead, they have a small section of text and images, followed by a short multiple-choice problem, followed by a more detailed explanation once the students submits an answer.
Since my topic this year is discrete math, let’s consider Brilliant’s number theory course.
Brilliant divides its content into top-level topic areas (Math; Science; Computer Science). Each topic area contains courses (e.g., Number Theory; Logic). Each course contains chapters (e.g., Factorization; GCD and LCM; Modular Arithmetic). And each chapter contains quizzes, made up of a set of questions on a related theme. Even if you don’t have a Brilliant subscription — unlike Khan Academy, Brilliant doesn’t offer all its content for free — you can try out the introductory chapter for each course.
The introductory chapter for the Number Theory course has three quizzes. The third one, called Rainbow Cycles, is a very brief introduction to modular arithmetic.
Section 4.1 of Rosen’s discrete math textbook introduces modular arithmetic using a traditional approach: precisely define integer division, prove some theorems using the definitions, use those concepts to define congruence modulo $m$, prove more theorems, and finally define addition and multiplication modulo $m$.
It’s not coincidental that most math instructors use this pattern, since it mirrors how math research works: pick a set of axioms and see where they lead, proving theorems along the way to verify your conclusions. By following along with this process in a textbook, students learn the mathematical reasoning behind every fact, not just facts and procedures.
But despite the formal rigor of the traditional approach, students may come out of it lacking one thing: intuition for why a concept is true. A textbook author can provide that intuition, or students can invent their own, but it’s not built in to the definition-lemma-theorem approach.
Now back to Brilliant’s rainbow cycles. There’s nothing in the problem statements about div or mod or congruence. Instead, every exercise in the quiz uses a grid of integers where each grid cell is colored based on its value modulo $m$ (for a particular $m$). The goal is to start with an intuitive idea of modular arithmetic, rather than a formal idea. The explanations for each exercise get into more mathematical detail, so that’s where intuition encounters a more traditional approach to teaching the concepts. But you can always go back to the intuitive explanation if something is hard to understand.
Brilliant, Khan Academy, and textbooks
For learning math, Brilliant, Khan Academy, and traditional math textbooks have some features in common, but none of the three are perfect, so there’s value in using all of them. The two online platforms have the benefits that come with the online format: gamification, continual updates, automatic grading, and visual progress tracking. Brilliant’s approach to instruction starts with an intuitive approach, which not all textbooks do. However, each Brilliant exercise is hand-crafted and unique. This works for communicating the intuitive idea and practicing it a few times. But a handful of exercises aren’t enough to solidify a concept. Khan Academy’s advantages over both Brilliant and textbooks is that it offers “unlimited” exercises. So you can keep practicing until you completely understand a concept.
One unique Brilliant feature is daily challenge problems, a supplement to its chapters and quizzes. These can be useful for practicing problem-solving skills, but because they cover random topics, they aren’t as relevant to the goal of learning a particular topic from beginning to end.
The main advantage of textbooks is depth. If you pick a good textbook and read it correctly, you can extract knowledge from every sentence. If you can do this for a whole book, or even just a chapter, you can learn a topic in more depth than you can with Brilliant or Khan Academy. For learning proofs, there’s no other option. I haven’t yet seen a good online proof-oriented learning platform.
So the best approach is to take advantage of the strengths of each source: Brilliant for intuition, Khan Academy for more traditional explanations and problem drills, and textbooks for learning from the ground up.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.