The Form of the Problem


What’s the best way to teach math?

It’s a big question. Parents, teachers, scientists, politicians, and tech entrepreneurs all seem to have their own ideas. And, since it’s a matter of intense national interest, there’s a legion of government initiatives dedicated to figuring this out.

One obvious component of math education is the curricula. A lot of research has been done in this area: the ideal problems to use when first introducing long division or fraction multiplication, for example, or the order in which new math concepts are introduced. In a sequence of exercises, should difficulty ramp up fast or should it ramp up gently? Some even advocate for a difficulty spike early on that lets up after a while. There are many questions like this.

You can’t stop there, though. As many parents and teachers know, there are a thousand other factors that affect student performance in math. One of those is attitude: when a teacher clearly dislikes math themselves, or gives math problems as punishment, that sends a message to the students about the value of math. Another is the physical classroom: is it bright and airy? Dank and crowded? Or what about cultural factors– do the children live in a culture that celebrates achievement in math, or a culture that denigrates those good at math as uncool nerds? Singapore achieved great results with a homegrown curriculum, but if the Singapore math approach is subsequently introduced to Seattle (as is actually happening now), will it have the same wondrous effect? It’s difficult to do real controlled studies on this kind of thing: no two classrooms are alike, even in the same school, let alone when you start varying things across languages and cultures. 

Then there are the more existential questions: does the ability to pass a math test really mean the student truly understands math? Why are we so focused on test scores, anyway? 

To bring it back to the classroom: teachers struggle every day to try to give their students a basic facility with math. They are the footsoldiers on the front line of a neverending battle to teach students more, better, and faster than ever before. And if you watch them you’ll see they use a wide variety of techniques. There’s motivating students with gold stars or other rewards. There’s doing exercises with real money or other objects like candy in order to connect the abstract numbers to the real world. Some teachers write their own curricula and incorporate details they know their students will like. There are a lot of ways to approach it and at the moment nobody is really able to definitively claim one specific way is objectively better than another.

* * *

So consider this:

5 × 30 - 17 = 

As opposed to this:

Creeping unnoticed through the evil wizard’s dungeon, the wily thief Esmerelda found five treasure chests, each with thirty gold coins inside. Suddenly the wizard’s pet dragon spotted her and roared a terrible roar! Esmerelda ran for her life and dropped seventeen coins as she escaped. How many coins did Esmerelda get from her dungeon adventure?

These two problems are fundamentally “the same,” yet they are not the same at all. Each has its own effect. Students who are trying to solve as many problems as possible in a limited amount of time may not welcome the inclusion of the useless and clearly contrived story elements, which only serve to slow them down, but others may find that it’s only the excitement of Esmerelda and her adventures that interests them enough to put up with the math at all.

The examples are also rather different from a pedagogical standpoint. Only the latter tests reading comprehension– the ability to understand how the words of the story form its meaning– and modelling– the ability to arrange the important terms from the story (five chests of thirty, seventeen dropped coins) into the correct mathematical relationship.

In other words, the problem has changed in form, even though all I did was wrap some content around it.

* * *

If you want to investigate the open question of math education seriously, you need to acknowledge the significant effects that factors like this have on learning outcomes. Which is not to say that research on the individual components of this ecosystem has not been invaluable in building an understanding of what elements lead to successful learning. Obviously, when we play with the individual parts of the whole, we start to learn how everything works together. But the danger lies in believing that the answer to a small question is also the answer to the big question.

If we wish to keep that big question in our sights, we’re doing ourselves a disservice if we don’t consider all of its pieces with comparable weight– all the way from the basic forms of the math problems themselves, to how those problems are framed, and on up to the baggage that we, as a culture, carry around math. We should concern ourselves with both what we would call “the math” and how we present the math. It’s only then that the full picture can emerge, and only then that a relevant and useful generalized theory of math education will be possible.

3 Comments