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.
Scores and Rankings
Allen Cheng’s blog post has a great title: “How to Get a Perfect 1600 SAT Score.” He didn’t call it “How to Improve Your SAT Score” or even “Improve Your Score by 160 Points, Guaranteed” (though that guarantee is later in the article). He promises the ultimate result: a perfect score.
To show us he knows what he’s talking about, Cheng posts a screenshot of his SAT score history from the 2004 SAT (two sections, for a maximum of 1600 points), a 2004 SAT subject test in Physics (out of 800), and the 2014 SAT (three sections, for a maximum of 2400). The score report is a grid of six perfect 800’s.
When you’re giving advice about something that has a score, like a standardized test, it helps to have an eye-popping transcript to show off. It’s the same reason competitive programming experts on Quora say “Red on Codeforces” or “ACM-ICPC medallist” in their credentials.
But my topic is discrete math, not SAT math or SAT scores. So let’s take a look at how we can apply some of the article’s study techniques to other types of math practice.
Study techniques can only get you so far without underlying motivation. For the SAT, the motivation is to get a good score and improve your college acceptance chances. But even that can lead to different levels of motivation. A bright student who scores a 1280 (88th percentile) might decide that score is good enough. It’s well above average, and it’s achievable by many students who just stay awake in their classes, do a bit of homework, and never specifically prepare for the SAT. Seriously applying Cheng’s study techniques requires more than just basic motivation, because they take extensive planning and effort.
Most people learn math as part of a class. That provides some motivation. But every student also brings their own motivation for learning a subject, which can drive more elaborate study techniques. For self-study, finding your own motivation is essential. One reason I study math is to practice thinking in logical and quantitative terms. Discrete math in particular is relevant to the style of thinking required to solve programming problems.
Motivation can get you started and remind you why you’re doing the work, but it’s not enough on its own. To enforce the decisions you make, you need good habits. Some of the habits Cheng writes about include:
Finding a distraction-free environment. Focus is especially difficult to maintain as online and app algorithms try to steal our attention. I like to remind myself of Cal Newport’s rule that Work Accomplished = Time Spent x Intensity. You can save a lot of hours by working more intensely.
Allocating enough time. Cheng suggests setting aside 200 hours for serious SAT preparation. This happens to be the same estimate I use for allocating homework time in a one-quarter technical university class (e.g., programming or math).
Working to a schedule. Rather than relying on motivation to decide when to study, rely on a schedule that you construct at the beginning of your project, when motivation is highest.
The benefit of mistakes
At its core, Cheng’s advice focuses on finding and fixing mistakes. The key to learning math is solving math problems. But it’s not enough to just solve a lot of problems. You can make some progress that way, and it’s a lot better than just reading a textbook, but it doesn’t get to the essence of the learning process. To get the best results from problem solving, you need to see each problem as part of a multi-step process:
- Value quality over quantity: Follow this process for every problem you are even slightly unsure about, even if it means doing fewer problems overall.
- Track your mistakes: If you get a problem wrong, or get it right because of a lucky guess, or take too long to solve it, track it in a study notebook.
- Understand your mistakes: For each tracked problem, figure out what you need to do to avoid making the same mistake in the future. Record what the problem is asking, why you missed it or weren’t confident of the answer, and what you plan to do about it.
- Find the root cause: Use the 5 Whys approach to get to the root cause of the mistake, not just the immediate cause. For example, the first why might point to missing math knowledge, but by the time you get to the fifth why, you may realize that the actual problem is an inefficient study habit or a distracting environment.
- Find patterns: As your study notebook gets larger, look for patterns in your weaknesses. That will allow you to target larger areas rather than individual problems. For example, maybe there’s a math topic that you never quite got, and you need to review it from scratch.
Build your own systems
Just as you can write your own textbook that explains concepts in a way that you understand, you can create your own learning system that works best for you. By experimenting with different learning techniques and analyzing your problem notebook, you can figure out what strategy gets results.
Cheng’s article emphasizes one of the key points of deliberate practice theory: not all practice is equal. If you want to practice something seriously, you need goals, a plan, concentration, feedback on your performance, sufficiently difficult problems, and a professionally designed training plan. Fortunately, math study is so ubiquitous that it’s possible to collect these requirements from online sources. But that doesn’t make it easy.
(Image credit: Karla Mora)