TITLE: Pedagogy: How to cure students of the "law of universal linearity"?
QUESTION [388 upvotes]: 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”:
$$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$
$$ 2^{-3} \mathrel{\text{“=”}} -2^3 $$
$$ \sin (5x + 3y) \mathrel{\text{“=”}} \sin 5x + \sin 3y$$
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.

REPLY [3 votes]: Much has been said about inappropriate pattern matching, and I agree with much of that opinion. But I really think that what is going on with many of these students is that instead of seeing mathematical patterns (albeit incorrectly), they are actually seeing  equations and expressions as lexical patterns. $a(x+y)=ax+ay$ , $\frac{a}{x+y}=\frac{a}{x}+\frac{a}{y}$, $2(-(x+y))=-2(x+y)$, and $2^{-(x+y)}=-2^{x+y}$ become the basic designs that a student can tessellate his or her test paper with. One must certainly admit that $\frac{a}{x}+\frac{a}{y}$ has a much more aesthetic quality than $\displaystyle\frac{1}{\frac{x}{a}+\frac{y}{a}}$. I asked one student why, in the equation $y=a(x-c)^2$, does a change in $a$ stretch every point except the vertex. He replied "because the $a$ is closer to the $x$ than the $c$ is".
I'm not sure what can be done about the lexical problem, except to nullify the tendency by teaching why the rules work the way they do. For example, how many students in high school can show why $\frac{a}{c}\cdot\frac{b}{d}=\frac{a\cdot b}{c\cdot d}$? They should be able to, certainly.
One thing that needs to improve in our schools is following through on the "making connections" blurb that we find in every curriculum document across Canada and the United States. For any arbitrary concept we need to teach that there are multiple interpretations of that concept and that students should learn flip back and forth between those multiple interpretations depending on the situation. For example, some possible interpretations of the fraction $p/q$ may be

$p/q$ is the solution to this equation $qx=p$
if we partition something of size $p$ into $q$ partitions, $p/q$ is the magnitude of one partition
it is $p$ multipied by the magnitude of one of the partitions of $1$ partitioned into $q$ equal pieces
if $p$ is partitioned, and $q$ is the magnitude of each partition, then $p/q$ is the number of partitions

When trying to evaluate $\frac{x+y}{a}$, a student familiar with the multiple interpretations of the fraction concept would find the 3rd interpretation useful
$$\frac{x+y}{a}=(x+y)\cdot\frac{1}{a}=x\cdot\frac{1}{a}+y\cdot\frac{1}{a}=\frac{x}{a}+\frac{y}{a}$$
Of course, no useful interpretations can be found for something like
$$\frac{a}{x+y}$$
We can only hope that the student doesn't find any interesting tessellation for such a fraction.
As for other Linearities, perhaps the best thing to do is just to teach fundamental (such as the reasons behind the exponent rules), and perhaps to select a smörgåsbord of exercises that don't repeat the same problem ad nauseam.