Deliberate practice techniques for software developers
If you’re studying a high school or undergraduate college math topic and you have a question, the answer is probably somewhere online. Finding it is just a matter of coming up with the right search terms. Or if searching doesn’t work out, you can always ask on Quora. But another way to use Quora is for an overview of what math concepts are available to study.
Math problems on standardized tests have short, simple answers that fit in a multiple-choice format. But college-level math problems require more detailed explanations. In How to Write a Math Solution, Richard Rusczyk and Mathew Crawford of Art of Problem Solving present a detailed checklist for ensuring that your proofs and solutions communicate your thinking as clearly as possible. Here’s some advice from the article that I found especially useful.
Last week, I considered what we can learn from the standardized test approach to studying math. This week, I’m continuing that line of thinking with some advice from another PrepScholar article.
One way to find out how well you know math fundamentals is to take a timed math test. In the U.S., over two million students per year take the SAT (which includes a math portion) as part of the college admissions process. A multitude of online resources, including Khan Academy, are available to help with SAT preparation. One resource that caught my eye is a blog post by Allen Cheng, founder of an SAT prep company called PrepScholar. Although the post targets SAT preparation, I found parts of it useful for my current math project.
When you’re studying a topic, a course acts as a filter for the material: Rather than having to consider every reference on a subject, you get a nicely curated subset that the instructor believes fulfills the goals of the course. Rather than having to read an entire textbook, you get a list of sections, or maybe just notes and lectures. Rather than having to solve every problem in the book, you get a list of suggested problems, or a standalone problem set.
But a textbook without a course is also a filter. Rather than having to read through all the original research papers on a topic, you get an expert’s distillation of the important results, combined with a set of problems to help you learn and understand the topic. Since I’m using the textbook approach rather than the course approach, here are some ideas about how to select problems from a textbook.
When you’re studying math on your own, a question you must continually ask yourself is how much time to spend on the problem you’re working on. That question also comes up when you’re taking a traditional class, though your options are more limited in that case because of the fixed class schedule. I wrote about this topic last year in the context of competitive programming problems, but there are some special considerations for math problems.
A mathematical proof can be the size of a novella. For example, Andrew Wiles’s published his famous proof of Fermat’s Last Theorem in two journal articles covering 129 pages. But proofs in introductory textbooks like Rosen often contain just a few sentences. It might seem obvious that these proofs are short because they’re easy. But that’s not exactly right. Last week, I wrote a moderately long post covering an “easy” proof about how the div operator behaves when the first argument is negative. Why was it so much longer than the version in the solution guide?
Last week, I suggested a process for getting better at writing proofs. To illustrate that process, here’s an example of how to use it to prove a theorem from Rosen.
Proof-writing skills are important for all college-level math. But there’s a special relationship between proofs and discrete math. In the “Goals of a Discrete Mathematics Course” section in the preface to his textbook, Rosen puts Mathematical Reasoning first in the list. He writes:
Students must understand mathematical reasoning in order to read, comprehend, and construct mathematical arguments [proofs]. This text starts with a discussion of mathematical logic, which serves as the foundation for the subsequent discussions of methods of proof. Both the science and the art of constructing proofs are addressed.
Rosen Chapter 1 is “The Foundations: Logic and Proofs,” and that chapter ends with sections on “Introduction to Proofs” and “Proof Methods and Strategy.” A textbook that specifically covers proof techniques, Daniel J. Velleman’s How to Prove It, begins with chapters on these same topics, and includes chapters on logic and on mathematical induction which Rosen also covers. So it’s not surprising that many of the exercises in Rosen ask for proofs. Here is a process I use to get the most out of these exercises.
If you search for advice on how to read a math textbook, you’ll find plenty to choose from. I’ll link to some of my favorites at the end of this article. But first, here’s an overview of my experience reading math books, and what techniques I found to be useful.