Constructivism

I just finished reading Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. Dr Paul Lockhart is a real bomb-thrower; his aim is clearly to provoke, not engage in a balanced or nuanced take on the topic. His point is, Throw out the existing math curriculum, especially the traditional approach that deadens any sense of inquiry or exploration. Dr Lockhart argues that "here's the formula", divorced from any context, baffles students and mystifies the whole nature of mathematical thought. While not realizing it, he makes Dewey's point about psychological order being more important than logical order when he states that a genuine problem context should come before defining terms. (The opposite, logical order where definitions are first is how proofs are written -- and how traditional instruction is ordered.) Dr Lockhart continues this theme of emphasizing curiosity and psychological motivation over logic by bringing in Geometry for special criticism. Part of his disgust is that Geometry proofs are ugly, tedious, and do not look like real mathematical proofs anyway. So part of his problem is that Geometry proofs have corrupted the good of real proofs. However, his real objection is that these ersatz proofs hide the beautiful, the simple, and the intuitive. Dr Lockhart gives an absolutely lovely example of a student proof, which conveyed the idea clearly but without rigid formalism. His preference for the psychological -- what motivates, what communicates -- over the strictly logical is clear.

Dr Lockhart proposes something close to anarchy, where students simply investigate interesting problems, under the watchful eye of a teacher who is skilled enough to critique their work and to ask them to refine key steps in their work. Sounds a lot like grad school! While there is certainly dreck in the curriculum, I think reform is better than his revolution. If you eliminate inessential topics and repetition, then there's plenty of time for exploring topics in a more genuine way. For example, what is essential in Geometry? Distance, slope, area, angles, triangle properties, circle properties, polygon properties -- but really the first four. There is plenty of time to have students generate a rich, deep understanding, without abandoning the whole concept of any set curriculum for the course. I would be curious to know what he would think of the Geometry book I wrote: nothing but problems, with almost no proofs; plenty of interesting properties to explore, with a set of problem leading students through the idea-formation, generalization, limitation of the generalization (by counter-examples), and explanation (but not proof) stages. I tried to eliminate everything I believe my students could not get a really good intuition for -- and anything that was not interesting. But it is still a curriculum; at the end of the year, the students were supposed to know certain things. Not anarchy. Just slower, more focus on learning for themselves, and much less handed down from on high.

I'm reminded of a book on constructivist teaching over a decade ago. I agree with the general principle that students have to construct the knowledge on their own; you cannot simply "beam" ideas into their heads, no matter how effective your explanations are. This is especially true in math. I can tell you how a geometry theorem applies, but until you play around with it on your own, getting the sense of how it works and when it doesn't, we can't really have a meaningful discussion about it. I'll just be talking over your head -- which is what happens in most math classrooms. My big complaint about constructivism, as a teaching philosophy, is the focus on presentation: Don't present. The "let students explore" credo is dogmatic. But sometimes a little introduction is a great way to set the students up to DO exploration. And besides, the real question in designing a curriculum is, "How do we set it up to guide students through, to get them to intuit the big ideas, to make the key connections, and to continue to think about the ideas conceptually?" The continuation is the big goal. The brain looks for shortcuts; once students have an idea, they will start using that idea unthinkingly. It doesn't matter whether the idea was one they acquired from a teacher's explanation or from exploring the subject themselves. The brain looks for shortcuts, so a good curriculum needs to constantly unsettle them. Think you understand right triangles? Then BAM, here's a question that unsettles your notions and requires you to go back to basics. That's how you build and retain really solid understandings.

The worst example in this constructivist book (not Mathematician's Lament) was about a teacher trying to figure out how to frame a lesson for her students -- before realizing that the "correct" answer was that they should figure out how to frame it themselves! That's how to be a real constructivist teacher. To me, this is a silly way to evaluate the strength or weakness of a lesson. The real questions are: (1) What is the intellectual work the students are just capable of, and (2) Is the lesson set up so they can effectively do them, or is the teacher taking too much of the students' work for him- or herself? Maybe how to frame the lesson simply gets them set up and rolling along and is not their key intellectual work. Then the teacher should frame it and get on with it. Maybe framing the lesson IS the work the students should be doing, but this seems like a rather odd occurrence.