Just simply an obvious, elementary, clear, basic way to talk math. Any questions?

Luke Walsh describes catching himself saying “These are basic properties of triangles” where he was meaning basic as ‘fundamental‘. But basic also can mean simple. Novices and experts may disagree a lot on simplicity!

Simplicity is relative. To the great majority of mankind it is a simple fact, for instance, that 17 x 17 = 289, and a complicated one that in a principal ideal ring a finite subset of a set E suffices to generate the ideal generated by E. For others among a select few, the reverse is the case. (Mathematics Made Difficult, Linderholm)

It reminded me of one of my college math professors who (apparently against the grain, given my other classes) wanted to remove the words “clearly” and “obviously” from all math proofs.

The statement is either obvious or not. If it is obvious, why write it at all? If it is not obvious, why describe it as such?

We, as teachers and experts, should not attempt to describe ideas, lessons, concepts as simple, obvious, or clear. Nor should we describe things as tricky, complicated, or difficult. Those should be the interpretations of the learner, and they will vary depending on the learner’s prior knowledge and experiences. If we label something as simple and the student does not understand it, how does the student feel? “I don’t even understand the simple stuff!”

I get the intent. We feel some need to convey information about how this idea fits within the larger structure. Or we want to set some expectation for the learner so that they do not fret about difficulty. Perhaps we are cajoling the student with the juxtaposition: “you don’t understand it, but it is simple… so push a little bit and you can get it soon.” However, these pieces of information are a non mathematical crutch– they are a transmission of an experts interpretation, in place of letting the learner put in the work to make their own interpretation. It doesn’t let the learner have agency over the conceptual development, precisely because the intent is to speed things along through the expert’s development of the concept.

Clearly and obviously serve a similar goal: moving people through an idea. “Clearly, 60 has more divisors than any smaller natural number” The sample statement about divisors may have intended, “this is easy to check, but just trust me, it will be faster.” Perhaps it is trying to guide the reader towards meatier contributions of the author later on. But in the process, it may alienate readers who don’t see it immediately. I believe other language or layouts can serve to structure an argument without needing to proscribe a difficulty or complexity interpretation. “To introduce my argument, consider that 60 has more divisors than any smaller natural number.”

What can we replace these problematic terms with? What can we say instead?

Just / Simply / Obvious / Clear
“You just set the equations equal to each other”, “The distance formula is just the Pythagorean theorem”, “The solution becomes obvious”
Instead: Try just (ha ha) removing the word from the sentence, or removing the sentence if appropriate. If it sounds too declarative afterwards, maybe that’s a clue on when such a declaration should be made!

Simple / Basic / Elementary
“This is a simple problem”, “These are the basic properties of triangles”, “Elementary number theory tells us…”
Instead: Descriptions of simplicity can probably just be removed. For basic, I like Luke’s substitution of fundamental. Elementary is sometimes used to try to be more specific about the concept being referred to, but it sounds condescending and is also not necessary. Either remove it or replace with something like fundamental if that is what is meant.

Finally, the phrase “any questions?” The discriminating factor to this phrase is the preparation for it. Are you, as a teacher, expecting the students to have questions? What might they ask? How might you answer? Have you left time in the class period to answer questions that do arise? If you don’t have those things, the phrase “any questions?” actually means “I’m done. You should know it now. We can all move on if nobody speaks up.” No matter how else you might try to set expectations about question-asking and inquiry, actions will speak louder than words. Only some students– those already near the same page as the teacher– have the social capital in class to ask a question after the teacher’s “I’m done” signal. Anyone who is confused faces a choice: publicly reveal the confusion at a time when the teacher is ready to move on, or passively wait and let everyone move on. As teachers, we should ask for and encourage questions not only with our words, but with our preparation and planning.

These words and phrases are habitual. It takes careful attention to your speech patterns to reduce how often you say them. But the less we speed students artificially through expert interpretations and evaluations the more the students get practice making their own interpretations and evaluations.

What other phrases do you hear like the ones from this post? What have you used to substitute for them?