The Wason Selection Task, Finale

My 3-year-old can make free-throws. My proof? When I put her on my shoulders close to the basket, she makes it almost every time.

In a sense, this is what Margolis is saying about the Wason Selection Task and the reduced-array selection task. People aren't bad at modus tollens reasoning, because when we alter the wording in the task and/or reduce the number of possible answers, a "clear majority" of people respond correctly:

The most essential aspect of the puzzle does not depend on a presumption that people have difficulty in consciously grasping the underlying abstract logical relations. As it happens, people do have exactly that difficulty. Few readers aside from those who routinely have occasion for using formal logic (because they teach it, for example) are likely to have "gotten" [it] at first glance, without thinking it over a bit . . . But that is not at all the important point of the Wason material.

Rather the important point is this. If the two pairs of tests each involve questions that are exact logical equivalents, as they do, then why should we find one easy and the other hard?

I'd like to answer Margolis's question with an analogy. Suppose Dick and Jane are traveling together in a foreign country. In that country, Language X is spoken. Language X is Dick's first language, and he has been speaking it all his life. It is not Jane's first language, but she learned to speak and understand it in school very well, and she has been conversing in the language for years with friends that also learned the language.

Scenario 1: Dick and Jane travel to a region of the country where a rather strange dialect of Language X—called Low Language X—is spoken. Even though Dick has never encountered the dialect before, he has only a little trouble understanding it. Jane, on the other hand, has considerable difficulty with it. In fact, she only understands about 10% of what is said.

Scenario 2: Next, Dick and Jane travel to a region of the country where a dialect called High Language X is spoken. Both Dick and Jane can understand everything perfectly. In this region, Jane can understand a "clear majority" of the utterances.

These two scenarios have an underlying "logical equivalence" in that, in both situations, the same language is spoken. Is it reasonable, then, to argue that Jane's hypothetical lack of knowledge, as a non-native speaker, of the "underlying abstract logical relations" of the language is irrelevant in explaining her very poor understanding of Low Language X?

Of course not. But if we map Margolis onto this analogy, the answer would be yes. And he gets to this yes through two awkward assumptions, again mapped onto the analogy: (1) Jane should be able to understand Language X (in all its forms) by virtue of her considerable everyday experience with the language:

[The inability to handle modus tollens] has always warranted more suspicion than it has received, since anyone who listens to their children will hear them quite readily make what are functional equivalents of modus tollens inferences. And not very surprisingly, since the world provides us with endless occasions to make such inferences. (If I picked my keys off the desk, they would now be in my pocket. My keys are not in my pocket. So they are probably on my desk.)

And (2) the two scenarios have the same underlying features, so the different surface features make no difference:

If the two pairs of tests each involve questions that are exact logical equivalents, as they do, then why should we find one easy and the other hard?

Reading these assumptions in reverse and in context makes more sense: There is no "logical" difference between understanding High Language X and Low Language X. Therefore, Jane's everyday experience with High Language X should transfer to Low Language X.

These assumptions obviously ignore a central fact: Since Jane does not have knowledge of the "underlying abstract logical relations" of the language, then the "logical equivalence" between the two scenarios is NOT AN EQUIVALENCE AT ALL, to her.

Or, to put this back into the original context of reasoning ability: Since people do not, in general, have knowledge of formal reasoning rules, the "logical equivalence" between the Wason Selection Task and the reduced-array selection task (or the "drinking" version) is NOT AN EQUIVALENCE AT ALL, to people.

The reason why we fail the WST in large numbers but do well on the RAST is because the two tasks are not the same to us. We don't have deep understanding of the abstract logical relations that can transfer between the two tasks. At the very least, that is a possible explanation. Thus, we can't simply dismiss people's knowledge of and access to formal reasoning ability in comparing the results from the WST and the RAST.

Back to Framing

The same two assumptions I mentioned above influence a lot of thinking in elementary mathematics education: (1) children should be able to understand mathematics (in all its forms) by virtue of their considerable everyday experience with it, and (2) both "everyday" mathematics and formal mathematics have the same underlying features (logical equivalence), so the surface features make no difference. As a result of these assumptions, we feel more and more comfortable discounting the relevance of formal mathematical knowledge, access to that knowledge, and correct application of that knowledge in favor of sanctifying students' gut intuitions and misperceptions.

Keith Devlin, though he is by far much more reasonable in many of his other writings, uses the two assumptions above to make his "birds do math" argument:

When we humans try to emulate the navigational feats of lobsters or migrating birds, we have to resort to mathematics. In human terms, those creatures have built-in mathematical ability: they have brains that have evolved to carry out the trigonometrical calculations necessary to determine north from the position of the sun or to set a course based on a knowledge of where the North Pole lies. They are, in short, natural born mathematicians.

Obviously, though, we are not birds or lobsters. If it is indeed the case that our species does not have much in the way of built-in mathematical ability, this is actually a good thing. It means that we can continually improve our ideas, our understanding of and power over our world and our universe. We can erase and start over. We can imagine ourselves beyond what we are capable of today. What it also means is that we are in charge of giving ourselves this ability. It can not be drawn out of us. It doesn't happen by osmosis.

The same can be said for reasoning ability. That we are not hard-wired to reason in formal, abstract ways is a cause for celebration. It means that we can recognize when it is not appropriate and apply it even when it is counterintuitive to do so. But, again, it also means that we have to teach it to ourselves.

In Conclusion

The conclusions I have been building to in this series on the Wason Selection Task are quite obviously simply extensions of those I presented in my framing argument. I have tried to show here, however, that a subtle and powerful excuse that leads to poor teaching (or no teaching)--that students are inherently mathematical--has a sibling in cognitive science--that people are inherently rational. It is my somewhat educated opinion that removing this excuse would be an important first step in making mathematics fluency for all happen.

In his book, Margolis quotes Nisbett and Ross (1980) in reference to people's general inability to handle formal reasoning: "If we're so dumb, then how did we get to the moon?" I like Tom Hanks' (as Jim Lovell) statement as an appropriate response to this question:

From now on, we live in a world where man has walked on the moon. And it's not a miracle, we just decided to go.

Indeed, if we ever see the day when all of our students are fluent in the language of mathematics, it will not be because of any miracle curriculum or teaching method or law. We will just have decided to make it happen.

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