“Last year, we learned to keep-flip-change …”

She sees my face wince, but proceeds anyways.

“The way that goes is you keep the original fraction, change the division to multiplication, and then flip the second fraction.”

“OK, but why does it work?”

Shrug.

“Who else learned it this way?”

About three-fourths of the room raises their hand. (The other fourth still seemed to struggle with dividing fractions. More later.)

“OK, out of those of you who learned it this way, why does this work?”

I get a few “because the teacher said so”‘s, “because it just works,” “because I did well on the test” and just kept shaking my head gently while listening. Sure, the easy thing to do at that moment would have been to punch down (metaphorically, people) and tell them that their last teacher was wrong for unveiling a trick without giving them the magic behind it. The harder conversation was sitting right there, and I had it.

“What does it mean to divide?”

A few students had responses that equated to splitting apart.

I then spent about a week just showing (with) them that division is also a matter of regrouping into even pieces. In the first unit, my students determined that we use common denominators for operations with fractions because it’s fair, a simple but deep observation. With division of fractions, I’m asking students to consider multiple approaches to engaging in division. Eventually, we brought back “keep-flip-change,” but we used the term “reciprocal” instead, and only after I laid the foundation of direct division, common denominators, and modeling. (For more, check this and this.)

But once they learn to divide fractions directly, they can’t say they weren’t exposed to it anymore. They now know of its existence, at least. So do thousands of people across the country now because I told them the same thing on and offline. The knowledge isn’t new. In fact, I learned it at NCTM in 2008 (Salt Lake City). The researcher made a convincing argument that direct division helped students gain a deeper conceptual understanding of division as a whole by focusing on these numbers. It took me a couple of years, but eventually, I became fluent in direct division of fractions. Some of it was me worrying about what to do with the remainders, but a larger part of me was unlearning a topic I learned, used, and taught for more than a decade.

If I don’t have this learner orientation as an adult, how could I expect the students I teach to take a similar disposition?

Meanwhile, for years, people deceived themselves into thinking I taught the humanities, asserting that teaching math and advocating for justice were mutually exclusive. They’d say this isn’t a math blog in the way that my while counterparts’ blogs were. On the occasions where they would delve into the national zeitgeist, people would praise my counterparts for their intellect and worldliness. I couldn’t allow for my platform to continue beating the drum of math as a sterile subject while racism, sexism, militarism, and imperialism left humans around the world dead in, around, and outside of America’s borders. I’d write about math from this standpoint, only for the same people who couldn’t casually name non-white male mathematicians to suddenly find non-white male mathematicians in their history books in an attempt to disprove an argument they won for me. In ways large and small, I kept seeing instances where math teachers gave themselves permission to overintellectualize and quarantine the realm of mathematics from the complicated and messy and oppressive rules America created for its inhabitants.

Worse still, these mathematicians and math educators had some of the vaccines necessary to save the rest of humanity and kept pretending the bubble was only reserved for the already quarantined.

But now that teachers, especially those who teach post-elementary school math, have to acknowledge the complication, it’s even more critical to work through that understanding. No longer can we pretend ignorance of the doors we lock up behind us when we could easily place a wedge for more of us to go through those that come after us. No longer can we hide behind “the way we’ve always done things” to limit our students of color from getting the math they deserve. No longer can we shrink from the role mathematicians of color have played across millennia in the development and advancement of mathematics as a field, down to who deserves to chronicle math learning. No longer can we use our varied platforms as instructional coaches, writers, speakers, and upper-level / honors-courses teachers to soak up burgeoning movements for equity when the teachers who never got those platforms have more expertise than us are standing right there. No longer can we pretend that elitism in math education doesn’t exist and that we shouldn’t try our best to eradicate the straining and tracking of students so everyone can get exposure to this math.

That is to say, our students deserve to learn how to divide directly now that we all know they can get that.

And, on the occasion that a student regresses to “keep-flip-change,” I gently nudge them to consider the other ways we could have discussed this. We as a class would do well to remember the fourth of the students who didn’t respond to my original prompt at all. Remember the fourth.