As a math teacher, I spend a lot of time belaboring a certain distinction. To some students, it’s inscrutable hair-splitting, as subtle as John Adams vs. John Quincy Adams. To others, it’s barely worth saying aloud, as obvious as John Adams vs. Amy Adams.
The distinction is wrong vs. weird.
Mathematical notation is a system of communication. Thus, a mathematical statement can be held to two distinct standards: (1) Content: Is it saying something true? (2) Form: Is it clear, concise, and written in observance of all the familiar conventions?
The same dichotomy holds in any language. There are lot of crisp, idiomatic ways to say something wrong, and there are weird, off-putting ways to say something true.
Students don’t always see math this way. For many of them, rules are rules are rules, no matter whether they concern form or content. I’ve seen teachers and textbooks blur the distinction, too. For them, there are no mathematical misdemeanors, no minor slips of the tongue. All sins are mortal sins. An unrationalized denominator is murder most foul.
To me, that’s silly. Mathematics is both (1) a world of ideas, and (2) a language for describing that world. We must not mistake one for the other.
One key testing ground for these ideas is the concept of “simplification.”
I’ve seen exercises that use the word “simplify” in maddening, capricious ways. After all, simplicity is a matter of taste and judgment. Which is simpler: a(b+c), or ab + ac? Depends who you’re talking to, what you’re talking about, where you want to steer the conversation next.
Simplicity is subjective. It’s tangled up in our mathematical customs and conventions.
Should we then banish the word “simplify”? No, no, no! It’d be like English teachers abandoning the idea of an elegant sentence. 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 sanity and hair.
Something subjective can still be real and important.
For example, why do we reduce fractions? To reveal hidden equalities (how else would you notice that 34/51 is the same as 58/87?). Why write coefficients first, followed by variables in alphabetical order? To speed up operations like addition (just try adding 3abc + cb5d + b2ac + dbc7). Why arrange a polynomial by descending powers? So we can know its degree at a glance.
Even where simplification is not a matter of consensus, it’s worth exploring the different possible forms, and debating the advantages and disadvantages of each, just as we’d try on outfits before making a purchase.
Mathematics, at its best, teaches us to distinguish the necessary from the arbitrary, the content from the form. What better place to begin those lessons with the language in which they are taught? I want my students to learn the conventions, and also to learn that the they are precisely that: conventions.