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

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.

My interaction with the students (non-mathematically inclined ones, in the United States) has lead me to suspect that for some reason they are not taught the following two crucial ideas.

1. Mathematical expressions have meaning.
2. The validity of a rule for manipulating mathematical expressions is determined by what those expressions mean. In particular, the rules themselves are derived from what the expressions mean.

Understanding of those two ideas seems to me is the key difference between the students who "get it" (i.e., the ones that simply do things correctly and their mistakes usually boil down to not noticing something) versus the one who don't "get it" (which do only as well as they can memorize a bunch of boring, arbitrary-seeming rules).

Consequently, I am of the opinion that searching for

a particularly clear and memorable explanation [of when a particular rule is applicable] that will stick with students

is not at all the correct approach to this issue (unfortunately, given the structure and expectations of the educational systems . The sad reality (as I perceive it) is that for the majority of (U.S.) students, mathematics is the art of manipulating weird, meaningless strings of symbols according to equally weird and exception-filled rules that they barely have the mental capacity to remember. In essence, the kind of content that students appear to be taught seems much more appropriate for simple-minded, inhumanly precise computer, than a human being with the capacity to reason.

This is why no matter how much we illustrate and explain the rules to them, they keep misusing them: what they are missing is not explanations or illustrations, but the ability and mental habit of determining on their own whether the mathematics they are doing is correct or not (which is still hard for a computer: computer proof assistants are still in their infancy).

I personally have no idea how such this skill of doing mathematics right can be cultivated without awareness of the two facts above, and I believe that what separates the students who do demonstrate this skill is that they have (at least an implicit) understanding of those two ideas. Furthermore, I do believe that exposing them to, making them think about, and making them use the meanings of the symbols they write, and doing it again, and again, and again, and again, will have a much more significant effect than reminding them of one-off examples and illustrations of why a particular manipulation they did is not allowed. The one-offs they will forget and not be able to reproduce because of their infrequency, but the repeated insistence on using the meaning of the expressions to establish the validity of the manipulations will hopefully make it habitual for them.

In terms of implementing this in practice, I think that college is way too late, and also quite difficult because college math (and STEM) courses tend to be mostly about transmitting massive amounts of boring technical content and technical skills, leaving little to no room for actual ideas or ways of thinking. Nevertheless, I do think it would be an interesting experiment to have students keep something akin to a "vocabulary notebook" where they record the meaning (as opposed to the formal definition) of the various kinds of expressions they run in to. For example, a fraction $\frac ab$ is supposed to mean "a number which when multiplied by $b$ gives $a$"; it is short and illuminating work to figure out from this (using distributivity of multiplication over addition, which we definitely want numbers to satisfy) that $\frac ab+\frac cd=\frac{ad+bc}{bd}$, that there is no number meant by $\frac a0$, and that $\frac00$ can mean any number). This of course, presupposes that somebody takes the time and makes sure that the language in which these meanings are explained is coherent, so it would be a lot of work to design a course around this method.

I did in fact once successfully disabuse a(n Honors Calculus) student of "the Law of Universal Linearity" using these ideas. The particular instance concerned manipulating the Fibonacci sequence, and the student had made the error of writing something like $F_x+F_x=F_{2x}$. What I did is explain the stuff above and had the student apply them by analyzing the meaning of the various expressions he had written down was, and then ask whether that equality was justified based on what he knew the expressions meant. That seemed to make an impression on the student, but I personally believe it was an impression made ten years too late...

(This is a rather "soft" answer!)

I don't think there is a solution to this.

In my experience the problem is that math beginners don't understand / assimilate formal laws: they agree that $(a + b)^2 \neq a^2 + b^2$ (because "$2ab$ is missing") but they have no problem writing $(x + 3)^2 = x^2 + 3^2$ two minutes later.

The only "solution" is to take money from them / hit them every time they use the "law of universal linearity", but it takes years to have any effect (and earns you thousands of dollars)

I had a teacher in college who was very fond of repeating phrases like "The Flarn of the Klarp is the Klarp of the Flarn" and "The Flarn of the Klarp is the Twarble of the Flarn." I believe these are from Lewis Carroll. But the way they were incorporated in lecture was like an call-and-response.

For example, the teacher might rapid-fire questions at the student audience such as "The product of the sum is the sum of the product?" followed by "The derivative of the sum is the sum of the derivative?" followed by "The product of the logs is the log of the products?" Just seeing if students would get into a pattern of saying "yes.. yes.." and then whacking them with something to think about. I can imagine this working with trig functions as well.

This teacher would also routinely use small hand-drawn pictures in place of variables like x or y. For instance, I learned about the Taylor series expansion of "e-to-the-doggie" being the sum of "doggie-to-the-n-over-n-factorial". We similarly talked about moment generating functions as "e-to-the-tree-x" with a little tree drawn where the transform variable (usually t or s) would go, and then the moment-generating function's domain was the "tree domain" since that was the independent variable there.

I know this sounds ridiculous, but boy did it work. After a few weeks of acclimating to the sheer bizarreness of it, it really started to make the concept of variables disappear. Rather than fixating on why particular weird non-number symbols like x were showing up, you had to hold onto your butt because it might be a little tulip or a fire hydrant on the test and you were supposed to solve equations and whatnot. It was like there was no time to be confused about symbols because the sheer whimsical arbitrariness of whatever the symbols might be forced you to understand how to manipulate any symbol, which was the whole point.

This was for a first course in calculus-based probability, and eventually we started talking about things like variance, which then naturally became a discussion about how Var(X) = E[X^2] - E[X]^2 is totally a kind of measurement of "non-commuting-ness" between the squaring operation and the expectation operation. So whereas E[X] is linear, (i.e. the flarn of the klarp is the klarp of the flarn), for variance this is not true unless it's a Dirac variable with no variance. For everything else, one measure of central tendency is to say "the flarn of the klarp minus the klarp of the flarn equals ..." so you know just how far off you are from those operations commuting with each other.

I'm not sure if this would work with classes where aptitudes vary considerable, or where there are time constraints to hit materials in time for a standardized test. And it certainly is weird and requires great confidence on the teacher's part (the teacher who taught this to me was a Vietnam veteran who truly didn't give a damn about what students or administration thought of him... he was a bit like the character Walter Sobchak from The Big Lebowski actually). But it seemed to be extremely effective in my class and was one of the big milestones in my own study of mathematics where I went from merely knowing how to compute things when given problem set-ups to really trying to suss out deeper connections, analogies, patterns, etc.