In a recent post to the math-teach@mathforum.org list-serve, Bill Marsh wrote, in part:

I continue to think that guided discovery can be a powerful teaching tool, of definitions, as well as of proofs, but I should emphasize that in GUIDED discovery there will be a teacher in the room who knows the theorems that are coming and some of the pitfalls on the way to getting them.

If you don't know how to use them, power tools can dangerous. If you don't like them or have to use them, you won't use them very much. Unless you use them a lot, you are unlikely to learn how to use them well. If you don't know how to use them well, you may underestimate what can be done with them.

Suppose students arrive in a seventh grade math class to see 1+2+3 = 6 on the board. After a moment or two, the teacher might mention that numbers like six are called perfect and ask if anyone can say what's going on. I'd expect that pretty quickly someone would suggest adding up all the divisors, which can then be tweaked into a good definition. The class could look for another perfect number less than a hundred. If this were done near Valentine's Day, amicable numbers might be mentioned.

At a higher level, I might, on the first day of a real analysis course I was teaching, say that we were going to be looking at things like the epsilon-delta definitions and proofs they saw briefly in calculus. I might ask them to consider, for a real number a, the sequence the intervals defined by |x-a| < dk="" education="" 01330="" pdf="">http://www2.mat.dtu.dk/education/01330/Chapter1Math3.pdf

I don't claim that guided discovery is the best way to teach. I will claim that it is a good way, and that it is an especially good way in K-12. But only for those who like it enough to be willing to try to do it well.

I think Bill's post points to something far more important than "constructivism" (a topic that generates lots of heat and virtually no light here, in my experience). Who is and who isn't a constructivist, and what comprises applications of constructivist learning theory to mathematics lessons is rather useless to talk about amongst folks who can't even approach the slightest agreement about what "constructivism" actually is or what its theorists have to say that is relevant to mathematics teaching and learning.

What Bill's post touches upon, however, is the issue of pedagogical content knowledge. A teacher who proposes to teach a lesson on perfect numbers and offers the example Bill originally posted may well, as Dave Renfro subsequently pointed out, find herself in the middle of a conversation about triangular numbers (though it's quite possible that no one in the classroom, the teacher included, may know that term). Or perhaps (though less likely) the reverse situation takes place, with a lesson on triangular numbers potentially branching off towards a conversation about perfect numbers. One important consideration is, as always, what do teachers do when the unexpected or unplanned for arises? And clearly, one consideration in that regard is whether the teacher has the requisite content knowledge (awareness of the mathematics being pointed to, the mathematics that leads into what's arising, and some of the mathematics that is pointed to by what's under consideration AND being raised by the "surprise" issue). But more than that, the teacher must be able to make quick but reasonable choices as to how to respond to what arises.

Bill's proposed response seems to me to be one exemplary option. And in order to produce it, a teacher would need to be prepared to deal with the math that comes up, and also have a strong sense of whether it is a better idea to continue with the plan or to go down the new road. That means pedagogical content knowledge and knowing the class and what is most likely to serve them well, collectively and individually. If the original plan is chosen, how is the new idea addressed (never? in the next class? by the teacher raising it? by the teacher asking the class to think about it as a homework assignment? by asking the student who raised it to report on it or lead a class investigation/discussion, etc., as an extra credit assignment? some other option?) If the new opportunity is pursued, what happens to the previous plan?

My view here is, of course, that content knowledge alone and general pedagogical knowledge alone will not suffice. While strength in both these areas is necessary and contributes to making good decisions in this and many other situations in math class, they are not sufficient. There is a third kind of knowledge pointed to in what I raise above that is peculiar to this domain of teaching (and so saying does not preclude the idea that similar but different specific pedagogical content knowledge is needed in other subject areas) without which teachers are less likely to make choices that adequately serve the vast majority of students.

None of the above is intended to imply even remotely that there is a single "correct" decision to be made in the situation Bill describes. I'm perfectly comfortable that he made a good one, but there were other possibilities that, depending upon information that only the teacher has access to at the point the decision is made, might have been as good. There's no way to know with certainty what the results of these other choices would have been, but experienced, reflective teachers who have the requisite knowledge and use it actively are the ones most likely to pick from a "menu" of better and more productive options. Naturally, there are worse, less effective choices, based on previous experience, observation of and/or conversations with colleagues, consultation with the literature, interaction with coaches, master teachers, etc. It is naive to think that without reflective practice and competence in the three domains of teaching knowledge mentioned here that teachers will "naturally" make better choices when these situations arise, based simply on content knowledge alone. It is for this reason, among others, that effective teacher education and professional development is vital if we are to improve mathematics teaching and learning in this country.

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