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.

## Aiming for Perfection

Practice doesn’t always make perfect, because for many things in life, it’s not possible to attain perfection. But for standardized tests, perfection is an option. Allen Cheng of PrepScholar wrote an article with advice for achieving perfection on the SAT math test.

Even when perfection is possible, not everything is worth studying to that level. “Good enough” is good enough for many topics, because you don’t have unlimited time. But for topics that are important to your overall goals, or for high-stakes tests, it’s worth putting in the time and using strategies that get you beyond “good enough.”

## Strategies

**Speed**

Cheng’s *Math 800* article includes a process for calculating how much of your distance from a perfect 800 score is due to lack of knowledge, and how much is because you’re working too slowly. Here’s the process: take a test with a timer running, but don’t stop when you get to the time limit. Instead, work as long as necessary to finish all the questions, and mark the ones that you finished after the time limit. Then calculate your “under time” score and your “over time” score. Based on the two absolute scores and the difference between them, you can decide where to apply your practice time: on learning new skills, or on getting faster at applying existing skills. For example, if your “over time” score is 750 and your “under time” score is 650, then you have an efficiency problem. But if both scores are under 700, you have content gaps to fill.

Independent of test-taking skills, there’s a relationship between how well you know a topic and how fast you can solve related problems. You can get faster at doing problems just by practicing more problems of the right type, or by studying related topics. But just as analyzing the problems you got wrong is a more efficient path to improvement than just doing more problems of a similar type, the best way to get faster at solving problems is to think specifically about efficiency. Two examples from the article: 1) Factoring an equation to find a solution can be faster (and more accurate) than plugging in values, even on a multiple-choice test; 2) Solving problems with pencil and paper can be faster (and more accurate) than using a calculator, even when the test allows one. So it’s worth thinking about whether you’re using the most efficient problem-solving techniques.

You might think solving problems faster leads to more careless errors, but on a timed test the opposite can be true. Cheng says when he was preparing for SAT math, he practiced until he could finish each section with 40% of the allowed time remaining. This gave him time to solve most problems twice, using a different technique the second time and comparing his two answers. The same idea applies to other kinds of work. If you have the skills to finish your work quickly, you’ll have time to double-check the details and make sure the quality meets your standards. So it’s worth looking for speed improvements even if you don’t have a test to pass.

**Content gaps**

To get faster at a skill, you first need to understand the fundamentals. This gives you the background knowledge to solve practice problems and search for faster techniques. Checking your answers on a test tells you where your content gaps are, so you can work on filling them. In the article, Cheng uses the analogy of a cavity in a tooth, comparing it to a content gap that you need to fill. To fix a cavity, a dentist doesn’t just patch it. They drill it out, sterilize it, and then add the filling. Similarly, a content gap is probably not isolated to the specific problem you missed on a practice test. Unless it was a careless error, a missed question is probably a symptom of underlying weakness in a content area. To fix it, don’t just study the solution to one problem. Find other related problems to make sure you understand everything around the content gap. Build a strong network of related concepts.

**Trying again**

Learning math is a continuous cycle through these steps:

- Study a concept
- Test yourself on that concept by solving problems
- Uncover your weaknesses by checking your answers

Once you know which skills need work, you can go back and study again, then test yourself again, then see which weaknesses remain, then study again, and so on.

In Step 3, you have a choice to make about checking your answer. In the ideal case, you get someone else to check your answer and tell you only whether it’s right or wrong. A computer-based system like Khan Academy provides this capability. Since the system doesn’t reveal the answer, you can continue to work on the problem to see if you can figure out where you went wrong, without the extra hint of knowing the correct answer.

If you’re using a paper practice test and you don’t have a study partner, you can’t do precisely the same thing, but you can approximate it. The *Math 800* article suggests: “if you miss a question, try it again before reading the explanation.” If you use this approach, you know the answer but you don’t know the steps to arrive at it. By discovering the process for yourself rather than just reading the steps, you’re likely to remember it better.

Another PrepScholar article, on the best way to review your mistakes, has the opposite advice: “Don’t immediately go back and try to re-do missed questions (or if you do, don’t let that be the only time you re-do them).” I think solving missed problems immediately is an efficient strategy because your brain is already loaded with the problem’s context. But there’s also a spaced repetition benefit to coming back to a problem after some time. So it’s good to do both. An even better option is to come back to a different problem that tests the same concept, since it eliminates the drawback of answering a previously solved question from memory rather than solving it from scratch.

## Standardized Tests

Like programming contests, standardized math tests take a complex subject and reduce it to a simple evaluation for which you can be assigned a score. The SAT isn’t the last word on your math skills, and a programming contest isn’t the only way to evaluate programming skills. But the advantage of these constrained scenarios is that they give you clear goals and a well-defined curriculum to practice. The skills learned in practicing for artificial targets can then form the basis for learning more advanced skills.

(Image credit: Fabian Pittroff)

*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.*