The Assertion: There are no important open questions in math pedagogy.
The Challenge: Prove me wrong by citing even one on-going discussion, anywhere.
(1) "Open question" means something more than simply someone not knowing the answer to a question, otherwise every question is open all the time because someone, somewhere may not yet know the answer---as in all the high school and college classrooms teaching their various subjects.
(2) "Important" must mean something other than "idle" or "artificial". That is, anyone can concoct a difficult, even unanswerable, question that no one cares about, but I hope we can all agree that the open question should be one that actually matters to at least a few people. And by "matters to", I mean that at least a few people are actively researching the question or, even if they have given up, they would be keen to learn the answer should someone be able to provide it.
One good example of an open question that is now settled is Fermat's Last Theorem, which occupied mathematicians for centuries. Another, is the Unified Field Theory, in physics, which remains open and interests physicists.
I am prompted to reissue my Challenge by a recent, brief, off-list exchange I had with a notable member of this forum:
MPG writes: >Gee, Haim, why not surprise me and other list readers >by NOT mocking the idea that there are a lot of folks >who know you're wrong about the "no open issues of >mathematics pedagogy" bluff?
No there aren't, and you cannot show otherwise. Otherwise you would have already done so, with bells on.
>Is Keith Devlin a heavy enough player?
Devlin explicitly states in his MAA article that he is making a mathematical point, NOT a pedagogical one.
>I really love listening to two people who spend ZERO >time in K-12 classrooms, as far as anyone can tell, >pontificating to the rest of us about what does and >doesn't, should and shouldn't go on in such places. >Can't speak for everyone on this list, of course, but >I'm in a high school math classroom dealing with >everything from elementary arithmetic to geometry to >trig to precalculus (with occasional forays into the >southern foothills of Mt. Calculus and even some topics >from non-Euclidean geometry, discrete mathematics, and >statistics) 8 hours a day, five days a week, 12 months >a year (year-round program). I spend a great deal of my >spare time in contact with K-12 teachers, mathematics >educators, and even the occasional research >mathematician (the ones who actually know and care >about mathematics teaching).
And yet, you never discuss pedagogy. If you did, it would be an awfully simple thing for you to produce an example and publicly humiliate me. How can you resist the temptation? Unless, of course, no such discussions exist. - -------------
One more point about my Challenge. Richard Strausz has come closest, so far, to claiming the the prize by his question, "how should we teach arithmetic to adults? but I am dissatisfied with this question for two reasons.
First, we have to ask if the adults are, in principle, teachable. I.e., are they mathematically tone deaf? Otherwise we can have a whole class of worthless open questions like, "how should we teach calculus to three year olds."
Second, if the adults are indeed teachable, then the basic problem degenerates into a self-inflicted wound. That is, the only reason we are trying to teach arithmetic to adults is because we failed to teach it to them when they were children. And so, Richard's question devolves back into teaching arithmetic to children.
Strictly speaking, a wound is a wound whether it is self-inflicted or not, and Richard may have a legitimate claim. But if we allow Richard's claim, then we leave ourselves open to the kind of hero mentality in which we work to cure the ills that we, ourselves, inflict.
The two most famous examples of this (pathological) mentality are firefighters and nurses. Once every couple of years, we read about a firefighter who sets life-threatening fires and makes sure he is the first on the scene to rescue the people. And every once in a while we read about a nurse who purposely sickens her patients so that she can then rescue them. In both cases, these people want to be perceived as heroes and if the Fates do not cooperate, they will take matters into their own hands.
Similarly, some districts have rules against teachers taking their own students as private pupils. Otherwise, a teacher could fail to do his job properly in class (while still getting paid), and then get paid again for tutoring the children. In other words, people should not be allowed to profit from the ills they inflict on others, and I am afraid that Richard's claim comes dangerously close to that---not that I blame Richard personally. I blame the public school system that is in collapse.
Haim Unashamedly White and Unapologetically Jewish