Parlez-vous Algebra?

In which I reveal the point of my previous post

Encountering the reality of algebraic experience¹

Try as I might I could not get ChatGPT to assign the speech balloons correctly, which is kind of relevant to the point of this post

In my last post I told two stories of my encounters with the French language: one about trying to buy car insurance without understanding half of what was said, but managing to pick the right answer often enough to get through; the other about battling teachers at the local French school to get my daughter into the right class, and really needing to understand and be understood.²

Expressions as abstract art

I think that students in algebra are often in the car insurance situation. They find themselves in a linguistic fog. They’re getting by — answering questions, following procedures, passing tests — without understanding much of what’s going on. They might be doing perfectly fine and then, one day, they simplify (x + y)² as x² + y², and the teacher wonders what happened. From the teacher’s point of view they seemed to be doing alright (just as the insurance agent probably conducted the whole transaction in blissful ignorance of my incomprehension). This disconnect is illustrated beautifully in a video on Dan Meyer’s substack, where a teacher is trying to get students to translate “6 less than a number” into an algebraic expression. One student says 6 − n, perhaps just following the word order in the teacher’s prompt, another says n − 6, correctly but a little uncertainly.

There is a crucial difference between French and algebra, however. When I was in that insurance office, I never doubted that the French meant something — I just couldn’t always follow what it was. Many algebra students don’t even have that. When I look at an algebraic expression I see a language object — a phrase or sentence expressing meaning. I think that students often see expressions as something different. Possibly as pictorial objects, like puzzling pieces of abstract art. Or possibly as something that needs to be poked at and prodded until it gets into the shape the teacher wants. But the idea of looking at it for meaning, of asking what operations it is describing, what it is telling you about a sequence of operations with numbers, is not always there.

There’s evidence for this in the language we use. We teach students to “cross-multiply,” “cancel from top and bottom,” “move things across the equals sign” — all spatial metaphors, about position on the page, not about what the symbols mean. And FOIL is the worst offender: First, Outer, Inner, Last is entirely about where things sit, with zero mathematical content. You could explain this to readers without asking them to do any math — it’s about the positions of the symbols, not the algebra itself.

Expressions as meaningful language

There is a great debate in mathematics education which takes many forms: procedure versus understanding, drill versus exploration, direct instruction versus discovery learning. I think these are simultaneously useful distinctions and false dichotomies,³ and I’ll take them up in future posts. Here I want to concentrate on a necessary condition for all approaches. The argument between “drill the rules” and “teach for understanding” is less important than whether students approach algebra as something that has meaning in the first place. Different approaches can work if that belief is there. None work if it isn’t. My car insurance story is an example of rules before understanding; my school story is an example of understanding then figuring out the rules. It worked out fine in both cases because I knew that French was a real language that conveyed real meaning.

If algebra is approached as a subject that makes sense, then students can make progress, even if they don’t quite understand it right now. Nothing will work if instead they see algebraic expressions as essentially meaningless objects. If a variable arrives empty of purpose, not functioning as “a number whose identity I don’t know yet” or “any number I might want to test this with,” then you have already lost the game.

What is to be done?

I have thoughts about this which I might elaborate in a future post, but first I want to know what you think should be done, in the classroom, with real students. What have you tried that works, or doesn’t work?

To make this question crisper, let me give you an example of what I am looking for. Consider solving the equation 3x + 6 = 8:

\(\begin{aligned} 3x+6 &= 8\\ 3x &=2\\ x &= \frac{2}{3}. \end{aligned}\)

A bare equation by itself has no meaning; it has to be clothed in words. The same applies to a bare list of equations. Here is the same solution, clothed:

If x is a number such that 3x + 6 = 8, then 3x = 2, because if two numbers are equal, then the results of subtracting 6 from them are also equal. If 3x = 2, then x = 2/3, because if two numbers are equal, then the results of dividing them both by 3 are also equal.

Notice that every step is justified by a statement about numbers, not about positions on the page. You don’t “move the 6 to the other side” — you subtract 6 from two equal numbers. You don’t “divide across” — you divide two equal numbers by 3. It’s just as easy to say “subtract 6 from both sides” as to say “move the 6 to the other side.” The first is about math, the second is about pictures.

To be clear, I am not suggesting that a teacher or student should say all these words every time they solve an equation. However, I am suggesting that teaching strategies should endeavor to bring about a state of mind in students where there is mathematically meaningful clothing, at least implicitly.⁴ I am reminded of what I said in my post about 8 + 5 = 13:

I am incapable of thinking of that addition fact without something else flashing through my head. It’s a very rapid decomposition of 5 into 2 + 3, plus a recombination of 8 and 2 into 10, plus a final addition of 3 to get 13.

I’d like the same sort of thing to happen when students solve equations. There should be a similar flow of mathematical reasoning, monitoring for the operations being conducted rather than for the symbols being moved.⁵

I’d love to hear your ideas on how we can make algebra more meaningful for students. How do we help them see the clothing of words, not the abstract positioning of symbols? Please let me know in the comments.

1. “I go to encounter for the millionth time the reality of experience and to forge in the smithy of my soul the uncreated conscience of my race.” James Joyce, Portrait of the artist as a young man. I’ve always thought that “forge in the smithy of my soul” was a particularly poetic description of constructivism.

2. I learned later that parents arguing with teachers is very much not done in France. Once I wandered into the school to pick up my daughter and was told I could wait outside the gate with all the other parents. I was like, “Oh, that’s what that crowd of people was.”

3. Direct instruction versus discovery learning is not so much a dichotomy as two ends of a spectrum, with the loudest voices at each end.

4. Jim Madden’s comment on my Wednesday post is relevant here: “a “fundamental principle of all communication [is to] understand that you’re trying to affect another agent.” [My emphasis.] My desire to have students think mathematically rather than pictorially while solving equations has consequences for how I communicate with them.

5. The principle of using mathematical language over position language applies elsewhere. When we tell students to “cancel 6 from top and bottom,” the word “cancel” has some mathematical meaning here — multiplying by 6 cancels multiplying by 1/6 in a meaningful sense. But the words “top” and “bottom” are strictly positional, and don’t make sense when students start seeing fractions written horizontally. The words “numerator” and “denominator” are more future-proof, and will come in handy when students start seeing algebraic expressions with nested numerators and denominators.