terminology is important too
It's been a busy day, spent revamping a presentation on matrix algebra for Linear Algebra and planning out a Minitab tutorial for GE 103, both for tomorrow — plus dealing with a committee-related crisis and capped off by, of course, grading. It didn’t look like much grading at first — just a handful of Linear Algebra homework assignments — but it turned into a headache-producing session of trying to untangle some incredibly tangled thinking. And it all had to do with terminology – and the misuse thereof.
Before I say anything else, I will reiterate that I enjoy my Linear Algebra class. It may seem unfair that I have picked on them before and I am going to pick on them again now. But what I am about to mention is pandemic to all math classes at all levels. It’s not just them, and it’s not all of them either. But I have to have an example to explain the general message.
There are two big problems when misusing terminology in mathematics or any technical subject:
(1) Using terms interchangeably when they are related but not synonyms. For example, the words car and road are related (they refer to similar ideas) but not synonymous. The sentence “I got into my road today and drove down the car to the store” doesn’t make any sense. Being related isn’t good enough to admit swapping. Likewise, the meaning of an explanation falls apart when we use the following words interchangeably: line, vector, matrix, equation, system. In Linear Algebra all these things are related. BUT THEY REFER TO DIFFERENT THINGS. It makes it extremely hard to discern whether a student has the right idea about a proof or explanation when they say things like “The matrix has infinitely many solutions” (Did they mean to say “system”?) or “These equations all lie on the same line” (WTF does that mean at all?). Look: The goal here is to make the reader of an explanation DO NO WORK WHATSOEVER. When the reader has to imagine all the things that you might have meant when you wrote your explanation, it’s trouble. Especially if you’re an education major and explaining things clearly is what they will pay you hundreds of thousands of dollars a year to do.
Anybody who has taught calculus knows that this happens all the time there too. How many times have we had students equating the terms function, equation, line, and graph?
(2) Modifying a noun with an adjective that doesn’t apply to the noun. For example, the statement “My drive in to work today was purple” doesn’t make sense because a commute cannot be “purple” in any reasonable sense. Names of colors do not apply to descriptions of commutes. Neither does saying “This matrix is linearly independent“, because the term “linearly indepnendent” does not apply to matrices. (It can apply to the columns of a matrix, but it can also apply to the rows as well, so what’s the real thing you meant to say?) This may just be another way of interchanging two nouns that are related but not equal, c.f. “These equations lie on the same line.” What? Equations don’t lie on lines; points do, and vectors do, but equations?
This is about as far from nitpicking as you can get. Mathematics is not really a language, but it is like one in the sense that it uses specialized words that convey highly specific bits of information, and in a way the whole point of mathematics is to convey that information as clearly as possible. Screwing up terminology is more than just a vocabulary error, it’s a flaw at the heart of understanding the subject and needs to be worked on.
Next time I may rant about the use of the pronoun “it” without any kind of antecedent. As in: “I took it and plugged it into it, and its graph looked just like it was supposed to except it didn’t look like it’s graph.” (Apostrophe added)