Why students come across as not knowing something that we know they’ve been taught.
In a recent interview of Sir Micheal Atiyah (famous mathematian, etc.) he is quoted as saying
When someone tells me a general theorem I say that I want an example that is both simple and significant. It’s very easy to give simple examples that are not very interesting or interesting examples that are very difficult. If there isn’t a simple, interesting case, forget it.
And it reminded me of a conversation that I had a while back with a colleague about whether or not students in precalculus know the commutative, associative and distributive laws. My colleague said that student’s haven’t seen those laws, and so they don’t know them (and they should, and that’s what’s wrong with education, blah, blah, blah). I didn’t have the chance at the time, nor the desire to make a big deal out of it, and so I didn’t get to respond. But one advantage of a blog is that I can always go back and rewrite history, or at least post my response here and will tie it neatly to Atiyah’s quote.
My fundamental belief is that students appear to not know something that they have studied in the past, because either they don’t recognize it in the given context or it was not presented to them in a way that made it significant enough to remember. The example of the first case that I come across most often in my discussions with instructors is in calculus when students have taken the derivative of a cubic polynomial and don’t know how to solve the resulting quadratic equation to find the critical points. In most cases, a quick reminder/review is all that is needed to get the students back on track. In this case, they know how to solve the problem when it is presented in the context of ‘Solving Equations’ but it doesn’t appear the same when it is ‘Find the Critical Points’. The example of the second is the point of the earlier discussion about the commutative, etc. laws. Students can and do use the laws, but without any recognition of what they are doing. In our proofs course where we go through the field axioms carefully, I always have the students expand and denote all the axioms that they use. It usually takes about 20+ steps and we almost always miss at least one axiom. (As a challenge to the reader, how many times, if any, are each of the distributive, associative and commutative laws used in expansion? (Answer below)).
A solution to these issues is connected to Atiyah’s quote. We need to raise the cognitive awareness of the things we teach, and this comes from increasing the variety and significance of contexts in which we see (and don’t see) the results. If we only see the laws in the context of normal arithmetic, where students have been using such things for years, then it shouldn’t surprise us that they don’t remember them. I suspect that most people that studied math, didn’t really get those laws until they were introduced to some examples where they didn’t work (like matrix multiplication). But just giving a new context is not enough. Advanced math students are taught that continuity does not imply differentiability, usually with an example like for pointwise. But for the general case, we often go through all the details of constructing the Weierstrass Function or something similar based on the Cantor Set. This does provide a new context, but it is so complex and apparently isolated, that it doesn’t surprise me to see advanced students still thinking that continuity does imply differentiability. Later when the example(s) have been repeated (and maybe taught) does it finally sink in. In general, one example or counterexample is not enough to seal the deal. And, as a final note, I’d like to adopt the last part of Atiyah’s quote in that if we give simple but interesting examples, i.e. various and significance contexts, then maybe we should just not teach that topic until we can.
Solution: Distributive: 4, Commutative: 1+, Associative: 3+. (The + depends on whether you’ve defined the distributive law as left or right side only, and whether or not I accurately counted all the uses; associative is the hardest because we often don’t track the grouping of 3 or more terms in a sum or product)