Cakes and bicycles

Why teaching mathematics is not like teaching anything else

How would you teach a child to bake a cake? To ride a bicycle? For the cake I’d probably start by showing how to measure out the ingredients, and then invite the child to try it. Maybe let the child do some mixing. In other words, explicit instruction: start with modeling, then gradual release of responsibility. For the bicycle I’d start by getting the child to sit on the bicycle, get a feel for the pedals and balance, and try a short run while I ran alongside. In other words, guided discovery: start with a bit of productive struggle, then provide strategic support along the way. These examples are why I find the debate about these topics unhelpful, at least for mathematics, as I explained in my post about choosing the right tool for the job.

You could also try teaching about cakes and bicycles the other way—discovery learning for the cake, explicit instruction for the bicycle—and you’d probably be fine. You’d get a messy kitchen and a slightly bewildered child, but it would all work out in the end because the child already knows what a cake is and what a bicycle is. The object of the lesson is already a real thing for them. Not to mention that they probably want to make or ride one, but motivation per se is not my point here. Rather, my point is that the reality of the object you are teaching them about is a huge advantage.

And mathematics doesn’t have that advantage. Other disciplines are about things that students already see as real. Stories are real, as are the books that contain them; birds and falling stones are real; even historical events can be made real. But in mathematics, as the abstraction grows, objects become less tangible. Are quadratic equations real? Only if the lesson makes them so—they don’t arrive that way. Are ratios real? You can make them real with carefully chosen examples. There is something constant about what happens when you mix two parts red paint to one part blue, in whatever quantities. It’s called magenta. That’s the reality of the ratio.

The question is not how I would teach completing the square (probably by explicit instruction) or the concept of a ratio (probably by guided discovery), but how I get the student to see a quadratic expression or a ratio as a real thing in the first place.

What are your examples? What are mathematical objects you’ve struggled to see as real, or to get your students to see as real?

On Friday I want to dig into this issue. I want to think about how we get a student to pay attention to something they can’t yet see.