Pedagogy: How to cure students of the “law of universal linearity”?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”:

1a+b“=”1a+1b

23“=”23

sin(5x+3y)“=”sin5x+sin3y

and so on. Slightly more precisely, I’d call it the tendency to commute or distribute operations through each other. They don’t notice that they’re doing anything, except for operations where they’ve specifically learned not to do so.

Does anyone have a good cure for this — a particularly clear and memorable explanation that will stick with students?

I’ve tried explaining it several ways, but never found an approach that I was really happy with, from a pedagogical point of view.

Answer

I think this is a symptom of how students are taught basic algebra. Rather than being told explicit axioms like a(x+y)= ax+ay and theorems like (x+y)/a = x/a+y/a, students are bombarded with examples of how these axioms/theorems are used, without ever being explicitly told: hey, here’s a new rule you’re allowed to use from now on. So they just kind of wing it. They learn to guess.

So the solution, really, is to teach the material properly. Make it clear that a(x+y)=ax+ay is a truth (perhaps derive it from a geometric argument). Then make it clear how to use such truths: for example, we can deduce that 3 \times (5+1) = (3 \times 5) + (3 \times 1). We can also deduce that x(x^2+1) = xx^2 + x 1. Then make it clear how to use those truths. For example, if we have an expression possessing x(x^2+1) as a subexpression, we’re allowed to replace this subexpression by x x^2 + x 1. The new expression obtained in this way is guaranteed to equal the original, because we replaced a subexpression with an equal subexpression.

Perhaps have a cheat-sheet online, of all the truths students are allowed to use so far, which is updated with more truths as the class progresses.

I think that, if you teach in this way, students will learn to trust that if a rule (truth, whatever) hasn’t been explicitly written down, then its either false, or at the very least, not strictly necessary to solve the problems at hand. This should cure most instances of universal linearity.

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