# Abstraction is Hard

A few weeks ago I read Malcolm Gladwell’s book The Tipping Point. The book doesn’t really deal with mathematics, but it does contain one math-related anecdote. This anecdote demonstrates an interesting principle of learning mathematics, so I wanted to share it.

Problem 1. We have a deck of cards, such that each card contains a digit on one side and an English letter on the other. We are dealing four cards, with the following result:

We are told that behind every vowel there is an even digit. Which of the four cards should we flip to check whether this claim is true?

I suspect that most mathematicians would only need a few seconds to answer this. On the other hand, people who are not used to mathematical logic would probably need a minute to get to the correct answer.

Problem 2. A cop walks into a bar to check for underage drinking and sees four people drinking. From a first glance, the cop notices that the first is a teenager, the second drinks water, the third is middle aged, and the fourth is drinking beer. Which of the four people should the cop check more carefully?

Most people would probably answer this problem immediately. This is interesting, since the two problems are completely identical! The only difference is that the second problem has a social context while the first is abstract. Even experienced mathematicians probably need to think for a couple of seconds to solve the first problem, but not the second.

This is a nice demonstration of why it is difficult to start studying advanced math. I will definitely use it when teaching proofs classes.

Here is another example of abstract things being hard to get:

## 3 thoughts on “Abstraction is Hard”

1. I didn’t get problem 2 at all. What is the cop supposed to check? First I thought that there should be some crime the cop is after and kept on reading through the description again and again… Now I’ve realized that in american culture the cop is obviously after underage drinkers, right?

• Hi Dömötör,

Yes. That’s an excellent point. I’ve been in the US for over 5 years and I guess I’m getting used to the culture! I wonder if Malcolm Gladwell also got complaints about this.

I revised the problem. Hopefully now it’s clearer.

2. abstract things being hard to get: I don’t get it! Guess that was the point.