Be afraid. Be very afraid. For hiding out in a classroom near you there is a maths zombie! You will be able to identify this supernatural algorithm-cruncher by the fact that she can answer complex mathematics questions yet has no understanding whatsoever of what she is doing.
You might however object that ‘understanding’ is latent and cannot be measured. You may in fact suggest that our best guide to mathematical understanding is the ability to answer maths questions. And our zombie has this ability. So what should we do? How can we know?
Well, it turns out that there is a way to make the distinction. Maths zombies will not be very good at the sorts of tasks beloved of constructivists and progressive educators. You know the sort of thing; explaining a mathematical method in words, solving a mundane problem several different ways, making a poster, composing an interpretive dance.
But wait; you may not yet be convinced. Perhaps you might wonder why these other things display a superior understanding of maths than the ability to do complex maths? Surely, a student could be as easily trained to deliver rote explanations or multiple methods without understanding as he could be drilled in the mindless application of a formula? And can we really conclude that a student who struggles to explain his thinking does not ‘understand’? He might just struggle with communication? What about English language learners?
And hang on a minute; exactly how would a traditional, instructivist teacher demonstrate that her students had understanding using her usual methods? She couldn’t. And yet by simply getting students to do the stuff that a constructivist teacher would prioritise, we will demonstrate that those students do have understanding; at least the ones that participate. Ergo, a priori and without any further need for investigation, constructivism wins. It’s about understanding, dude. Deep.
Hmmm… I suppose this is all predicated on the assumption that zombies exist…