The Wason Selection Task, Part IV

Yes, my friends, it's sad but true. I must here begin to wrap up my Wason Selection Task (WST) series. Before I start, however, let's recap.

In the first post in the series, I introduced the formal version of the WST:

Each card has a letter on one side and a number on the other, but you can see only one of these for each card. Here is a rule: "every card that has a D on one side has a 3 on the other." Your task is to select all those cards, but only those cards, which you would have to turn over in order to discover whether or not the rule has been violated.

Research has consistently found that only around 10% of the general population finds the correct answer to this task (D and 7). And Inglis and Simpson (2004) (referenced in Part I) found that less than half of the mathematics academic staff in their study came up with the correct answer. Most people either choose the D card alone or the D and 3 cards.

In the second part of the series, I mentioned that the WST is considered to be a conditional reasoning task, and I analyzed it according to the structure of conditional reasoning arguments. In short, a basic conditional reasoning argument starts with an "if P, then Q" statement, which is followed by a statement about the truth or falsity of P or Q. The final statement is the conclusion. In the WST, the rule given is considered to be the "if P, then Q" statement, and each of the four cards is one of the four possible second statements (P is true, P is not true, Q is true, Q is not true). However, as I noted briefly in my second post, the WST is not as straightforward as a typical conditional reasoning task—a fact which, while likely worthy of analysis, is mostly irrelevant to and beyond the scope of this series.

Finally, in Part III, I briefly described some explanations that have been given for why subjects fail the WST in large numbers. These include the matching bias and the confirmation bias.

In this post, I'd like to do one thing: review Margolis's explanation for subjects' failures to find the correct answer to the WST. In the following post, I will (1) finish up Margolis and take a look at Myrtle's "birds do math" idea, and (2) connect some conclusions regarding the WST with some thoughts about math education.

Margolis's "Categories" Explanation

The cards most often chosen by subjects in the WST are the D card alone (P) or the D and 3 cards (P and Q), whereas the correct answer is to choose the D and 7 cards (P and not Q). In the language of conditional reasoning, a majority of subjects correctly identify modus ponens (D) as valid but incorrectly identify affirming the consequent (3) as valid. And while they hardly ever incorrectly identify denying the antecedent (K) as valid, they almost always never correctly identify modus tollens (7) as valid.

Margolis uses this fact to begin his explanation and to introduce what is known as the reduced-array selection task (RAST):

So there seem to be two easy cards: [D], which is rarely missed, and [K], which is rarely chosen; and two hard cards: "3" and "7", which supply nearly all the errors. Overall, about 90% of subjects in fact do make errors. So what will happen if subjects are shown only what Wason called a "reduced array". Delete the two easy cards, and have subjects judge only the two hard cards. One might suppose, since essentially all errors are caused in relation to the hard cards, that subjects will continue to do badly.

But they don't! If this test is run on a group of reasonable size (say a class), those asked to respond to the 4-card version will typically return the usual 10% correct responses. But those given the reduced array will return a clear majority of correct responses! What can possibly account for this large improvement, related to merely removing the two cards that are ordinarily judged correctly anyway?

Margolis reasons that if subjects have difficulty with affirming the consequent (3) and modus tollens (7) in the 4-card version, then they should continue to have problems in the reduced-array selection task, or RAST. But since they don't, subjects' difficulty with these forms of arguments cannot explain their improvement in the RAST. Something else must:

This odd, even bizarre, improvement can be explained if subjects are seeing the cards not as particular cards but as indicating categories of cards. If explicitly asked, subjects understand the intended meaning of the question. But their responses make logical sense only with respect to a drastic misreading of the question. The question is misread as being about which categories of cards should be examined; for example, any cards with a "D" on either side; rather than about the particular card shown with a "D" on its upside.

The category idea is difficult to grasp, so let me see if I can make it clearer, using a very strange analogy:

Imagine you are in a room with a number of boxes. You are told that in each box is a dog or some other animal, and each box has a 3 or some other number written on the back of it. Here is a rule: If it is a dog, then its box reads "3." How would you proceed to identify if the rule has been violated?

In this situation (given the more common "if, then" reading rather than the "if and only if" reading), you can simply check all the dogs' boxes (D) for 3's. There is no need to check all the boxes with numbers other than 3 written on them (7), because, having already checked the category of dogs—i.e., all the dogs—you can tell whether or not the rule has been violated. According to Margolis, in the WST, this kind of thinking leads subjects to choose the D alone, and the "if and only if" reading of the rule, along with the matching bias, leads them to choose D and 3.

However, in our scenario, if you were able to check only the numbers on the boxes, then the only correct response—even when reasoning with categories (and, again, given the more common "if, then" reading)—would be to check all the boxes with numbers other than 3 written on them (7). This, says Margolis, explains why subjects make the correct response in the RAST—given the two "hard" cards, 3 and 7:

Subjects still seem to be misinterpreting the cards as categories. But with the reduced array, the only available correct response for the "category" reading is also the correct response for the intended reading! [D] is still salient in the question, but since it is no longer available, subjects must pick the "7".

Wason Task: Part I | Part II | Part III | Part IV | Part V | Part VI