The process of learning math involves mastering thousands of small skills. Khan Academy has exercises that help you practice the first 1500 or so of these skills. But as I discussed last week, the Khan Academy mastery system only gets you to an initial level of mastery. The topic for this week: how to continue using Khan Academy to increase your skill mastery after you have officially “mastered” the skill.

## Unconscious Competence

Khan Academy provides a way to prepare for studying a specialized math topic like discrete math by first mastering more fundamental math skills. Knowing the fundamentals frees up your mind when you’re learning advanced skills. For example, some discrete math topics use properties of exponents. It’s easier to focus on learning those topics if you know exponent properties as completely as you know your multiplication tables. The goal is unconscious competence: you want to be good enough at a skill that you can use it automatically without thinking about it.

A potential trap with practicing fundamental problems is trying to learn each skill equally well. Although it’s good to work on the basics, it’s not efficient to practice skills you’re already sufficiently competent at. Your target competence level will differ for each skill based on your goals. For discrete math, it’s valuable to be intimately familiar with properties of exponents. But you probably don’t need to be as good at solving integrals using trig substitution.

Even for skills where you want high competence, you must be selective about which problems to use for practice. You can’t keep getting better forever by solving just one problem type. Although Khan Academy can generate unlimited problems for each skill, the problems follow a few predictable pattern. That means they eventually become too familiar, despite the variations that the problem engine generates. Once you reach that point, it’s time to find another source of practice problems. One option is to look for a more advanced skill that implicitly uses the skill you’re trying to get better at. Or you could find a source that approaches the skill differently than Khan Academy does, perhaps by using an abstract proof-based approach.

One other way to combat excessive familiarity with a problem type is just to wait awhile. After a few weeks or months, most problem types will be more difficult than they were when you were practicing them regularly. The Khan Academy mastery system knows about this technique, but once you achieve 100% mastery it’s best to keep your own records of target skills, since the built-in system doesn’t work as well at that point.

## Mastery Level Examples

Last week, I proposed a list of four mastery levels to track your competence at skills that Khan Academy says you have mastered. Here are some examples of skills that might appear at each level.

**Trivial**

A trivial problem requires no effort in any problem-solving step: reading and understanding the problem statement, solving the problem, and checking the answer. You can do it all in your head or you can write the answer as you’re solving the problem.

Early math problems, like basic arithmetic and geometry, are trivial for most adults. But for someone with an aptitude for or interest in math, many other skills can also be trivial. For example, the negative exponents skill requires you to know that $a^{-n} = 1 / {a^n}$. Once you know and remember that identity, any problem in that Khan Academy skill will meets the criteria for a trivial rating.

Even calculus problems can be trivial. For someone who has never studied calculus, the problem “Given $f(x) = 5x^2+3x+2$, find $f'(x)$ ” might be incomprehensible. But even an average calculus student shouldn’t need more than one line on a paper to solve it.

If you truly find a skill to be trivial when you come back to it after a few weeks or months, there’s no need to continue practicing it. There’s no room for improvement, so there’s no point in spending time on it. Find another skill that needs more work.

**Easy**

If you rate a problem *easy*, you must know how to solve it without notes or references, and you must be able to perform each step of the solution promptly, without pausing to think about how the solution process works or derive a formula from first principles.

In theory, you could practice every Khan Academy World of Math skill until you could rate every problem as *easy*. Khan Academy’s problem writers design problems so you can solve them with well-defined steps. The problems require little creativity once you learn the process. But it may not be worth your time to do this for every skill.

An example of an *easy* skill might be interpret change in exponential models: changing units. Problem in that skill start with a short word problem like:

A virus is infecting computers on a network. The relationship between the elapsed time, $t$, in days, since the virus was released, and the total number of computers infected, $C(t)$, is modeled by $C(t) = 310 \cdot (1.31)^t$. Every week, the number of infected computers grows by a factor of ____ (round to two decimal places).

The calculation part of this problem only takes one step. But before you can do that calculation, you must read the problem carefully to be sure what units it’s asking about. And the numbers are usually chosen so you need a calculator to get the answer. So although there’s not much writing involved in this problem, you do have to concentrate to get the answer. Those characteristics move it out of the *trivial* category and into *easy* territory.

**Moderate**

A *moderate* problem is one that is familiar to you, but which you can’t solve from memory. If you had to solve a moderate problem on an exam, you wouldn’t get full credit unless you made a lucky guess. In the Khan Academy environment, you would need to check your notes or consult a reference to solve a moderate problem.

The equation of a hyperbola from features skill is a typical source of moderate problems. Although the process to solve these problems isn’t complicated, it requires knowing a few formulas, or deriving them. And unlike properties of exponents, these formulas (which relate the vertices and foci of a hyperbola) aren’t ones you would regularly practice in other contexts.

**Hard**

*Hard* problems are problems you haven’t practiced enough yet. Even for hard problems, my assumption is that you have solved at least a few similar problems, since in this article I’m only considering skills where you’re rated as *mastered* on Khan Academy. But some problems types are complicated enough that they require substantial practice spread out over weeks or months to get to the point where you can solve them from memory.

When I was going through the World of Math, the hardest problem type I found was volumes with cross sections: triangles and semicircles. It requires visualizing a moving graph and evaluating integrals to get the volume of the resulting three-dimensional shape.

For World of Math skills, it’s usually worth getting past *hard* and achieving *moderate* or *easy* level. These skills provide an overview of math fundamentals and a good background for future studies.

*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.*