Math Education and One True Wayism

An open tool box showing some tools
Photo by Tekton on Unsplash

This is a common criticism of Common Core (CCSS): It offers these strange new methods that students must use.

Except… only the first part of that is true. CCSS does offers some new strategies, but it doesn’t say that students have to use them.

This article isn’t a defense of CCSS, by the way. It’s far from perfect; it has plenty of problems. But one of its problems isn’t dogmatic adherence to specific methods.

So why do people think it does that? That’s been simmering in the back of my mind for a while, but it’s finally coming forward: Because teachers, texts, and curricula do.

I’m a big proponent of “whatever works for you, kids.” For instance, when reinforcing rewriting linear binomials to a quadratic trinomial (wow, that’s mathy sounding, isn’t it?), I present it two different ways: the Distributive Property and the Box Method.

For example, write (2x-1)(x+3) as a trinomial. Here I apply the Distributive Property, although I’m giving extra detail that I generally skip:

Multiplying two binomials using the Distributive (FOIL) method.

It’s easy to get lost in this method, easy enough that we have a mnemonic specifically for multiplying two binomials: FOIL (First, Outside, Inside, Last). I’m not a fan of the mnemonic, personally, because it’s limited to only this case, but if it truly helps students remember, okay.

The Box Method (which has other names) involves putting the binomials on the side and top of a grid and multiplying into each cell:

Multiplying two binomials using a table, often called the Box Method.

There are a few advantages to this method. For one thing, we can see immediately if we’ve taken care of all cases: Any empty cells represent something we forgot. It also reinforces the area model explanation of why we have to do all this multiplying in the first place:

A hand-drawn diagram of the area model showing how multiplying two binomials creates four rectangles.

I could even argue that the Area Model, which I show students but don’t explicitly offer as a strategy, represents a third strategy.

At some point in this discussion, in nearly every class, I am asked: “Do I have to use the Box Method? I hate it.”

Answer: No, use what works.

“This FOILing confuses me. Can I use the Box Method?”

Answer: Yes, use what works.

I have gotten questions like these with enough regularity that, coupled with stories my students have told me, I can identify their source.

At some point, probably multiple points in their past, they have had a math teacher that has docked them points, scolded them, or otherwise made them feel bad about not using the teacher’s method.

I have had it argued to me that there are times in mathematics class where we’re teaching a particular procedure, and that we want our students to use that specific procedure to demonstrate mastery of it.

Okay, why?

If the procedure solves problems that can’t be solved any other way, you should be able to construct problems that force students to use it.

And if the procedure is only one possible way to solve a problem, then what’s wrong with students using a different procedure?

Consider this system of linear equations:

4 x minus 3 y = 33 and x plus 4 y = -19

How could we solve this?

We usually teach two strategies: Substitution and elimination. We tell students to use one or the other. We might even have a question on a test, “Solve this system using substitution. Show your work.”

Substitution will work 100% of the time. Elimination will work 100% of the time. Neither of these, alone, is necessarily the best strategy for solving this particular system.

A better strategy for this specific case is a hybrid system that I’ve never personally seen taught. First, add the two lines together:

The lines add to 7 x plus y = 14

Then, solve for y, and use substitution to finish up:

Several lines of math show that x = 3 and y = negative 7

For comparison, here’s how it would be solved with only substitution:

The same system solved using only substitution, with the same result.

and with only elimination:

The same system solved using only elimination, with the same result.

This can also be solved using a matrix, or a graph, or a table of values.

We don’t generally teach a hybrid method because, even if it’s the most efficient in many cases, it’s also more complicated. We like to teach simple algorithms.

But if a student solved that particular system using the hybrid method that I showed, I would be overjoyed. I would not dock them points because they didn’t solve it “my way”.

Here’s an example from Calculus: f(x) = (x² + 3)². Find the first derivative.

We could use the Power Rule:

The derivative is 4 x cubed plus 12 x, according to the Power Rule.

We could use the Product Rule:

The derivative is 4 x cubed plus 12 x, according to the Product Rule.

We could use the Chain Rule:

The derivative is 4 x cubed plus 12 x, according to the Chain Rule.

Each time, we get the same result because there is only one derivative of f(x). So which one should we use?

In this case, it doesn’t really matter. Whichever is simpler. Whichever makes the most sense to the student. Whichever gets the job done.

Are there cases where only the Chain Rule will work? Absolutely. This isn’t one of them. If I want to force my students to use the Chain Rule, I should construct a problem that only really works with the Chain Rule.

The message I give to my students, repeatedly: When I go over a particular strategy, I’m giving you tools for your toolbox. It’s up to you to decide which ones to use.

When we demand that students use the method we’re teaching this week, we’re telling our students to lock up that toolbox and only use the one tool we’re providing. Maybe they see the problem and instantly think of a screwdriver, but sorry, this week, we’re working with the claw hammer.

I get the impulse to think that way, as a teacher: I want my students to show that they know how to use a particular tool. But, again, if I haven’t constructed a problem so that that one tool is the most obvious choice, that’s on me, not the student.

So, please, if you’ve been giving students the message that they have to do it your way, and especially if you’re docking students points for using an effective method different than what you want: Stop. Now. Immediately.

You’re contributing to math anxiety, you’re communicating that mathematics is a set of rigid rules instead of a playground, and you’re making my job that much harder.

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