If I ever imagine that I can see the edges of teaching, if I ever tell myself that I’m apprehending all of its angles and dimensions, I just call up my friends Sarah Kingston, Ben Spencer, and Megan Snyder at Beach Elementary School and ask them if they’ll let me learn with some elementary school-aged children, an experience which corrects my vision for months.

Most recently, they let me think about time with some second graders — the youngest kids I’ve ever taught and probably ever met, I can’t be sure — and especially how to tell time on an analog clock.

My goal in these experiences is always to find areas of agreement between the teaching of different age groups and different areas of math. Whether I’m learning about time with second graders or about polynomial operations with high schoolers or about teaching with math teachers, I’m asking myself, what’s going on here that crosses all of those boundaries, not one of which is ever drawn as sharply as I first think.

One way teaching second grade is different from teaching high school.

The odds of me stepping on a child go way up, for one.

For another, these students were inexhaustible. Their default orientation towards me and my ideas was rapt engagement and an earnest, selfless desire to improve my ideas with stories about their friends, their pets, and their families.

My tools for curriculum and instruction were forged by students who communicated to me that “none of this matters” and “I can’t do it even if it did.” Those tools seemed less necessary here. Instead, I needed tools for harnessing their energy and I learned lots of them from my friends at Beach Elementary — popsicle sticks for group formation, procedures for dismissing students gradually instead of simultaneously, silent signals for agreement instead of loud ones, etc.

Even still, with these second graders, I tried to problematize conventions for telling time, just as I would with high school students. I asked students to tell me what bad thing might happen if we didn’t know how to tell time, and they told me about being late, about missing important events, about not knowing when they should fall asleep and accidentally staying awake through the night!

I tried to elicit and build on their early language around time by playing a game of Polygraph: Clocks together. I told them I had picked a secret clock from that array and told them I would answer “yes” or “no” to any question they asked me. Then they played the game with each other on their computers.

One student asked if she could play the game at home, a question which my years of teaching high school students had not prepared me to hear.

One way teaching second grade is the same as teaching high school.

I saw in second grade the students I would eventually teach in high school. Students who were anxious, who shrunk from my questions, either wishing to be invisible or having been invisibilized. Other students stretched their hands up on instinct at the end of every question, having decided already that the world is their friend.

Those students weren’t handed those identities in their ninth grade orientation packets. They and their teachers have been cultivating them for years!

Rochelle Gutierrez calls teachers “identity workers,” a role I understood better after just an hour teaching young students.

All mathematics teachers are identity workers, regardless of whether they consider themselves as such or not. They contribute to the identities students construct as well as constantly reproduce what mathematics is and how people might relate to it (or not).

I have wanted not to be an identity worker, to just be a math worker, because the stakes of identity work are so high. (Far better to step on a child’s foot than to step on their sense of their own value.) We wield that power so poorly, communicating to students with certain identities at astonishingly early ages — especially our students who identify as Latinx, Black, and Indigenous — that we didn’t construct school and math class for their success.

I have wanted to give up that power over student identities and just teach math, but as Gutierrez points out, students are always learning more than math in math class.

My team and I at Desmos are forging new tools for curriculum and instruction and we’re starting to evaluate our work not just by what those tools teach students about mathematics but also by what they teach students about themselves.

It isn’t enough for students to use our tools to discover the value of mathematics. We want them to discover and feel affirmed in their own value, the value of their peers, and the value of their culture.

We’ve enlisted consultants to support us in that work. We’re developing strategic collaborations with groups who are thoughtful about the intersection of race, identity, and mathematics. A subset of the company currently participates in a book club around Zaretta Hammond’s Culturally Responsive Teaching and the Brain.

Before undertaking that work, I’d tell you that my favorite part of teaching Polygraph with second graders is how deftly it reveals the power of mathematical language. Now I’ll tell you my favorite part is how it helps students understand the power of their own language.

“Is your clock a new hour?” a second-grade student asked me about my secret clock and before answering “yes” I made sure the class heard me tell that student that they had created something very special there, a very interesting question using language that was uniquely theirs, that was uniquely valuable.