# College Math Teaching

## April 8, 2011

### A possible way to explain the contrapositive

Filed under: class room experiment, logic, mathematical ability, mathematics education, media — collegemathteaching @ 2:16 am

Mathematics Education

This post at Schneier’s security blog is very interesting. The gist of the post is this: do you remember the simple logical rule: “P implies Q” is equivalent to “not Q implies not P”. Example: if you have the statement “green apples are sour” means that if you bite an apple and it isn’t sour, then it can’t be green. In my opinion, there is nothing hard about this. We use this principle all of the time in mathematics! As an example, consider how we prove that there is no largest prime: Suppose that there was a largest prime $q_n$ with the previous (finite) primes indexed. Now form the number $p = q_1q_2q_3....q_n + 1$ Now $p$ cannot be prime because it is bigger than $q_n$ So it is composite and therefore has prime factors. But this is impossible because $q_k$ can never divide $p$ because it divides $p - 1$. QED.

The whole structure of the proof by contradiction is the principle that “p implies q” is equivalent “not q implies not p”. Here the q is “there is no biggest prime” and the “suppose there IS a biggest prime” is the “not q” which ended up implying “not p” where p is the true statement that $p$ and $p -1$ are relatively prime.
No mathematician would have a problem using that bit of logic.

But evidently mathematicians are in the minority.

Consider this experiment:

Consider the Wason selection task. Subjects are presented with four cards next to each other on a table. Each card represents a person, with each side listing some statement about that person. The subject is then given a general rule and asked which cards he would have to turn over to ensure that the four people satisfied that rule. For example, the general rule might be, “If a person travels to Boston, then he or she takes a plane.” The four cards might correspond to travelers and have a destination on one side and a mode of transport on the other. On the side facing the subject, they read: “went to Boston,” “went to New York,” “took a plane,” and “took a car.”

So, which card needs to be turned over? Of course, the card has to be “went to Boston” because there is nothing in the rule about going to New York, there is nothing that says that Boston is the only place you can fly to, and turning over the “car card” might reveal “New York” as a destination. Evidently, this problem is hard for most people.
But here is where this gets interesting: if the exact same logical problem is phrased as a “fairness rule”; say “for you to play in a game, you must attend practice” then the problem because very easy for people to solve! Schneier concludes:

Our brains are specially designed to deal with cheating in social exchanges. The evolutionary psychology explanation is that we evolved brain heuristics for the social problems that our prehistoric ancestors had to deal with. Once humans became good at cheating, they then had to become good at detecting cheating — otherwise, the social group would fall apart.

So, maybe I can use the fact that people seem to understand this rule in this setting when it comes to teaching this point of logic?

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