Some people have a knack for quickly learning and mastering new concepts or skills. As one of these fortunate individuals, and also as someone who has worked with dozens of students in an attempt to help them master new concepts, I possess a body of experience that leads me to conclude the following: effective learning requires the process of **self-verification**, or, the process of answering the question “how do you KNOW that you know?”

## How Do You KNOW That You Know?

Let’s begin with a practical example to illustrate 3 powerful methods for self-verification. This example happens to be a math problem, but the methods apply equally well to problems in any subject, and even to real-world problems.

**Example Problem:**

A friendly company distributes its profits evenly among all its employees. Two thirds of the employees spend a portion of their bonus. Their total expenditures add up to two fifths of the company’s profits. For the employees that spend a portion of their bonus, what fraction of it do they spend on average?

### Method #1: Multiple Paths

This problem, like most problems, can be solved in many different ways. You could set up a system of equations, use logic and your intuition about the context, pick numerical values for various quantities, or use any number of other strategies.

**The most straightforward method for self-verification is to solve a problem in at least two different ways.** If you get the same answer both times, you can be reasonably confident you have the correct answer. If you’ve made any simple mistakes along the way, this method will almost always reveal their existence.

### Method #2: Sanity Check

Always check that your answer *makes sense*. In our example, the question asks for the fraction of the bonus that each employee spends, on average. So any answer that is negative or greater than 1 would not make sense in this context.

It often helps to put your answer back into the original wording of the problem to see if it makes sense. The answer to our example is 3/5, so doing so would mean saying something like the following to ourselves. “Two-thirds of the employees each spend three-fifths of their bonuses, on average. It makes sense that two-fifths of the total value of the bonuses gets spent, because two-thirds times three-fifths is two-fifths.”

### Method #3: View From Different Angles

Don’t get attached to viewing a problem in the way you first “read” it. Consider this optical illusion:

Sometimes a problem or situation has multiple, equally valid interpretations. How do you apply this to an example like the problem we’ve been working with? Well, the problem says that the employees’ “total expenditures add up to two fifths of the company’s profits,” but since it also says that the company “distributes its profits evenly among all its employees,” we can use “profits” and “bonuses” interchangeably. This allowed us to reword as “two-fifths of the total value of the bonuses gets spent.”

## Self-Verification In Action

When in doubt about whether you really know what you think you know, keep these three strategies in mind.

- Use an alternate path to solve a problem; check that you come to the same result.
- Perform a sanity check; check that everything makes sense.
- Interpret things from another perspective; check that you come to the same conclusion.

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