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Critical Thinking mini-lesson 4

The Wason Card Problem Revisited

I received several responses to my analysis of the Wason problem. Mathematician and author Jan Willem Nienhuys wrote from the Netherlands:

I don't think that the card problem as presented is compatible with the beer over 21 problem. What would happen if you said "vowels and odds are forbidden to go together on one card" and ask someone to check whether there are cards that are forbidden. That's the beer over 21 problem. Another problem with the example is that the beer problem has a known social setting. If you made some kind of funny restriction, like 'over 22 must drink coke', it's much harder, or you can make a restaurant setting, with a completely strange restriction like 'girls (or people with a polysyllabic name) must order broccoli', then it's much more difficult, for the problem solvers must then keep an odd fact in mind while analyzing several cases. The less unfamiliar facts one has to keep at same time ready in the mind, the easier it is. (And it is quite possible that not everybody knows what's an even number or what's a vowel, or that people with slightly deficient knowledge know at most one of these concepts; you'd be surprised how deficient people's knowledge is).
 

I replied to Jan that, unless I'm mistaken, both problems imply that two cards are forbidden together (vowel and odd number; beer and 19-years or under). I think I will try the problem on my classes with Jan's suggested instruction and see if the results vary significantly. (I'll send him the results and he, the mathematician, can tell me whether the difference, if any, is significant!) The social setting would be part of what I'm calling the context that might be why the beer problem is easier to solve for most people. It had not occurred to me that part of the problem might be in understanding the meaning of words like "vowel" and "even," but that is a consideration that should not be taken lightly (unfortunately) and maybe I should try the test with some set-up questions to make sure those taking it understand such terms.

Jan replied:

I will be very interested in what you find. You might try variations like: if there are two primes on one side, the other side must show their product. This means that if a card shows a single number that is the product of two primes, you don't have to turn it around. If it shows two numbers that aren't primes, you also don't have to turn it around. Obviously the difficulty is that lots of people don't know what are primes, and even if they do so theoretically, some know their tables of multiplication so poorly, that they are at loss what to do when the card shows 42 or 49 or 87 or 36 or 39. Or 10.

Yikes! Jan, I teach a general course in logic and critical thinking, not math! My students would lynch me if I posed such a problem to them.

I do think that one of the problems with solving this problem (and many others!) has to do with how one reads or misreads the instructions. (For those who don't recall the exact instructions, here they are again: Four cards are presented: A, D, 4, and 7. There is a letter on one side of each card and a number on the other side. Which card(s) must you turn over to determine whether the following statement is false? "If a card has a vowel on one side, then it has an even number on the other side."

One reader wrote:

My solution to the problem is to check all cards (or a random sample if there are a large number of them) - Sometimes it's best to see what rules apply. (Sometimes "if" means if and only if...)

This approach represents a common mistake in problem-solving: self-imposed rules. The instructions do not imply that there are more than four cards, nor does "if" mean "if and only if." (See James Adams' Conceptual Blockbusting for a good discussion on common hindrances to problem-solving.)

The reader continues:

A simpler explanation for people choosing A and 4: Given that people tend to satifice, it makes sense that many will just check the cards where they see a vowel or an even number. It's a quick solution made with the immediate data on hand, requiring no additional thought (about the implications of the statement or anything else). Classic satisficing behavior.

Whether this solution is satisficing or satificing, it's wrong.

Another reader, Jack Philley, wrote:

Thanks for a great newsletter. I am a safety engineer and incident investigator. I also teach a segment on critical thinking in my incident investigation course, and I have been using the Wason card challenge. I picked it up from Tom Gilovich's book How We Know What Isn't So. About 80 % of my students get it wrong and some of them become very angry and embarrassed and defend their logic to an unreasonable degree. I use it to illustrate our natural talent to try to prove a hypothesis and our weakness in thinking about how to disprove a suspected hypothesis. This comes in handy when trying to identify the actual accident scenario from a set of speculated possible cause scenarios.

For those who haven't read Gilovich (or have but don't remember what he said about the Wason problem), he thinks that people turn over card "2" even though it is uninformative and can only confirm the hypothesis because they are looking for evidence that would be consistent with the hypothesis rather than evidence which would be inconsistent with the hypothesis. He also finds this behavior most informative because it "makes it abundantly clear that the tendency to seek out information consistent with a hypothesis need not stem from any desire for the hypothesis to be true (33)." Who really cares what is true regarding vowels and numbers? Thus, the notion that we seek confirmatory evidence because we are trying to find support for things we want to be true is not supported by the typical results of the Wason test. People seek confirmatory evidence, according to Gilovich, because they think it is relevant.

As to the notion I put forth that it is because of the context that people do better when the problem is in terms of drinking beer or soda and age, Gilovich notes that only in contexts that invoke the notion of permission do we find improved performance (p. 34 note). This just shows, he thinks, that there are some situations where "people are not preoccupied with confirmations."

lesson 5: logical fallacies