Two different statements come up often in mathematical proofs: “for all” and “there exists.” “For all” makes a statement that is true in every case. For instance, for all numbers divisible by both 2 and 3, they are also divisible by 6. “There exists” makes a statement that is true in at least one case, but need not be true in every case. For instance, there exist whole numbers not divisible by any whole number except for 1 and itself.

I think a similar distinction is useful when talking about teaching. There are some things in the “for all” category — things that should always be true, that I should strive to do every class, every day. Academic safety falls into that category. Every student should feel like they can take risks, share ideas, and be wrong, all with unconditional support. I should strive never to compromise on academic safety. I fall short on this all the time, but I need to set “for all” as my goal and work to help every student feel safe every day.

There are other things that fall in the “there exists” category — things that students should experience but don’t need to happen every day. Learning through discovery falls here for me. I think every student should experience mathematical discovery. And it’s hard to get discovery right, so this can’t be a one-off every few months. At the same time, I don’t believe students need to discover something themselves to understand it, or need to experience mathematical discovery every class.

There are other examples. Practice is a “for all” — I value practice, and I want students to practice every mathematical concept they encounter. Representation is “there exists” — I can’t show every student a mathematician who looks like them every day, but I can strive to share mathematicians who represent each of my students several times over the course of our time together.

Discourse around teaching can get lost when we confuse “for all” with “there exists.” I need to hold myself to a high standard around academic safety, every day and for every student. But it would be easy to get defensive and say, “but this other student feels safe, so x student should feel safe too!” Teachers face a constant onslaught of decisions and information; I have to avoid cherry-picking examples to fit my narrative. And it’s easy to make the opposite mistake, to take something that should exist somewhere and assume it has to exist everywhere. The value I place on discovery doesn’t mean that every lesson has to be a discovery lesson, and doing so risks losing sight of my true goals. One of the biggest challenges of teaching is how many decisions I have to make a day, and how quickly I have to make them. I hope distinction can help me to better live out my values and avoid lazy shortcuts.