I’ve been reading through some of my older ideas on inquiry. Early in my teaching, I took inquiry as a kind of golden ideal — it was hard, but it was the best way to teach math, and I needed to work my way in that direction. I didn’t have a very specific idea of what I meant, but it more or less meant kids figuring out mathematical concepts themselves.
I’ve come to dislike the term inquiry. There’s plenty of good in what is often termed inquiry, but my dogmatic approach led me to some ineffective teaching strategies. Most of all, inquiry is a pretty divisive term, and I think a specific conception of exactly what it is that helps students learn and why is worth figuring out. Here are the issues with an inquiry approach that I’ve had in my teaching.
Inquiry Assumes Kids Remember What They Figure Out
It might be true that figuring something out for yourself leads to better retention, but that’s not the whole puzzle. If I want a student to be able to apply a mathematical idea six months after he or she leaves my class, there are a ton of variables in play. They need to have applied the idea in a number of different ways, spaced over a period of time, tied to what they already know, and in a way that emphasizes essential elements of the concept. Inquiry is one approach here, and it’s not invalid, but I think my assumption that kids figuring things out for themselves is inherently good led to some questionable decisions. I would hypothesize that learning depends less on how an idea is first introduced, and more on the thinking that students do with that idea down the road.
Are They Really Just Playing “Guess What’s In My Head”?
I understand different mathematical ideas in my own idiosyncratic way. Too often when I went into a lesson wanting students to figure something out, I wasn’t actually asking them to figure something out about the math, I was asking them to figure out how I understood the math. It’s a subtle distinction, but it led to long worksheets where students answered questions they didn’t know the meaning of in an attempt to lead them to a very specific understanding that I had envisioned. Even if I do have some superior conception of how a mathematical idea operates, students aren’t doing the hard intellectual work here. Instead, they’re being conditioned that learning math is about guessing what the teacher wants students to say, and being led by the nose until they get to some magical formula that I could’ve just told them to begin with. Students learn by making their own sense of the math they’re encountering, but a classroom where they’re constantly guessing what’s in the teacher’s brain builds beliefs in students that make this learning impossible. I’ve given classes way too many worksheets where they solve what are (to them) a series of simple but unrelated problems. Then I ask them to make an inference about a new problem. A few students offer incorrect ideas, then a student “gets it”. I repeat that student’s answer, have everyone write it down, and I delude myself into thinking that the entire class just learned something.
There Isn’t Just One Big Idea
When I was focused on teaching so that students figured things out for themselves, there was usually a big, magical “key understanding”. A classroom where students are just trying to guess this mysterious “key understanding” in my head seems likely to be unproductive, but I think the illusion of one big idea is damaging as well. Mathematical ideas are messy, and coming to a deep and flexible understanding of math involves dealing with ambiguities, special cases, and unusual applications. If I have some illusion that, just because students found the rule for the Pythagorean Theorem instead of having it told to them, they will be able to apply it in new ways, I’m taking away opportunities for learning. Students need to know the formula — but they also need to understand non-examples, relate right triangles to the coordinate plane, find right triangles in complicated figures, drop auxiliary lines, recognize patterns in Pythagorean triples, relate the Pythagorean Theorem to acute and obtuse triangles, and much more. There isn’t a finite set of key understandings that will get students there — all of those ideas are interconnected, and build a broadly applicable schema, but only if students are wrestling with these interconnections. Inquiry doesn’t make any of this impossible, but oversimplifying inquiry to students figuring out one new big idea obscures the complexity of the relevant mathematical ideas.
Inquiry definitely does some things right. But I think an oversimplified understanding of inquiry chasing the idea that students should figure everything out for themselves leads to a classroom that is centered on students guessing what the teacher is thinking, at the expense of doing original mathematical thinking of their own.
I don’t mean to attack anyone else’s practice, or declare that inquiry is useless, or take a side in the math wars, just to illuminate the issues I’ve encountered. I’ll put some together some thoughts on the good in inquiry in my next post.