As a math teacher, I spend a lot of time working on a particular distinction. For some students it is an unfathomable hair-splitting act, as subtle as John Adams vs. John Quincy Adams. For others, it’s barely worth it out loud, as obviously as John Adams vs. Amy Adams.
The difference is not correct vs. strange.
Mathematical notation is a communication system. Thus, a mathematical statement can be made according to two different standards: (1) Content: Does it say something true? (2) Form: Is it written clearly, concisely and in compliance with all known conventions?
The same dichotomy applies in every language. There are many clear, idiomatic ways of saying something wrong and there are strange, disgusting ways of saying something true.
Students don’t always see math that way. For many of them, rules are rules, regardless of form or content. I’ve also seen teachers and textbooks blur the distinction. For them there are no mathematical mistakes, no small slip of the tongue. All sins are deadly sins. An unrated denominator is the worst murder.
To me this is silly. Mathematics is both (1) a world of ideas and (2) a language used to describe that world. We must not confuse one with the other.
An important test field for these ideas is the concept of “simplification”.
I’ve seen exercises where the word “simplify” is used in insane, capricious ways. Because simplicity is a matter of taste and judgment. Which is easier: a (b + c) or ab + ac? Depends on who you’re talking to, what you’re talking about, where you want to take the conversation next.
Simplicity is subjective. It’s tangled up in our mathematical habits and conventions.
Should we then ban the word “simplify”? No no no! It would be like giving up the idea of an elegant sentence as an English teacher. Just as a sentence can be true and also hideous, a mathematical statement can be correct and also a hard-to-interpret mess that costs me my mind and my hair.
Something subjective can still be real and important.
For example, why do we reduce fractions? To expose hidden parallels (how else would you know 34/51 equals 58/87?). Why write coefficients first, followed by variables in alphabetical order? To speed up operations like adding (just try adding 3abc + cb5d + b2ac + dbc7). Why order a polynomial in descending powers? So we can see its degree at a glance.
Even if simplification is not a matter of consensus, it is worth exploring the various possible shapes and discussing their advantages and disadvantages, just as we would try on outfits before buying.
At best, mathematics teaches us to distinguish the necessary from the arbitrary, the content from the form. What better place to start these lessons with the language in which they are being taught? I want my students to learn the conventions and also learn that they are just that: conventions.