Teach Wason’s Test

Wason’s Test is a good way to teach the contrapositive and to demonstrate that it is not obvious. The test is essentially a test of logic. There are 4 cards, each has a number on one side and a colour on the other. The picture below shows this set up with the playing cards I use in my lectures. (Playing cards are good as you don’t need to prepare them. These days I use a visualiser to display them but in the past I used jumbo sized playing cards.

Cards for Wason's Test

Cards for Wason’s Test

Now consider the statement “If the card is even, then the other side is red.” Which cards do you need to turn over to determine whether this is true or false?

If you have never seen this before, then take a moment to find the solution.

For many years I have set this as a multiple choice exercise during a lecture (the students vote using Socrative through their smartphones, tablets, etc.). The class response is typically the following:

Class results for Wason Test

Class results for Wason Test

As you can see most get the correct answer (in green) but a large proportion (about quarter) make the common mistake of saying red and 8 instead of blue and 8. The blue coloured card is the tricky part. The 8 card is straightforward and to complete the test we need to realise that turning over the 3 and the red are not necessary. These non-blue cards are easily dealt with and so I don’t spend much time on them in the lecture.

Dealing with the blue card is easier if we consider the contrapositive. The contrapositive statement is “If the card is not red, then the back is not even”. This, of course, is the same as “If the card is blue, then the number is odd.” Hence, we need to turn over the blue card and verify that the other side is odd. If it it isn’t, then the original statement is not true.

If students have accepted the contrapositive, then (hopefully) that reasoning is clear to them. However, accepting the contrapositive is difficult — my belief is that it is counterintuitive. The results of the Wason Test — which I do after a few weeks of logic and teaching about the contrapositive — demonstrate this.

One important aspect of the test is that it is very dependent on context. A follow up experiment by Griggs and Cox, The elusive thematic-materials effect in Wason’s selection task (1982), tried a less abstract form of the test. This time the four cards are about ages and drinks. The faces of the cards say Drinking a beer, Drinking a coke, 16 years of age, and 22 years of age. This time the statement to be checked is “If a person is drinking beer, then the person must be over 19 years of age”. Logically speaking, this is the same test as above but this time many more people give the right answer. The familiar context appears to help.

In replications of the forms of the test it is abstract situations people have trouble (even mathematicians) but in more familiar situations, they have less difficulty. The conclusion for teaching is that students are likely to have more trouble with contrapositive statements in unfamiliar situations and likely to have less trouble with contrapositive statement in familiar situations.

A similar problem occurs when teaching propositional calculus. Teach this — it is claimed — and students learn how to think logically. However, there is no proof that it does. Once students have learned the contrapositive in propositional calculus they can happily understand and use it in propositional calculus proofs. However, outside this context they find it hard to understand and apply the contrapositive method.

An important part of teaching the contrapositive is to see it used in real proofs. First by explicitly pointing out that the contrapositive has been used and then by setting exercises where students have to find the proof methods used. This will help see it used in context and that should lead students to absorb its importance and even begin to use it themselves.

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