And what it means for teaching every student.
A few weeks ago, Atlanta math coach Lara Metcalf invited me to guest teach a lesson on prisms in a class of deaf geometry students, an experience that helped me understand better than any previous teaching experience the truth that:
What students know.
What students can communicate about what they know.
What I understand about what they're communicating about what they know.
… are very different sets that sometimes only barely overlap:
I had assumed the American Sign Language vocabulary was a subset of spoken English vocabulary. For example, I can speak the word cat or use my hands to sign the word cat—a direct translation. But ASL is, in fact, an entirely separate language from English and, as with other languages, some spoken English words don’t have a direct translation in ASL and vice versa. Deaf people might then combine signs for other known words, use grammatical structures that don’t exist in English to communicate a word’s meaning, or spell the word out letter by letter.
I had also assumed that ASL was communicated through hands alone—that the static pictures of signs you may have seen are the entire meaning of the word one communicates in ASL. Lara clarified, however, that ASL is communicated through hands moving multidimensionally through space, at different speeds, attached to a person with a face making expressions, the sum of which can endow communication in ASL more nuance and meaning than the same words in spoken English. When someone signs the word cat, they can communicate more than just the idea of a cat.
“I find English explanations of math concepts can actually be really clunky compared to ASL explanations,” Lara said. “I might argue that it’s a superior language for teaching math because of its visual nature.”
Mathematics is a discipline that delights in its nuances and distinctions. (It isn’t just a triangle, it’s an obtuse triangle. It isn’t just a number, it’s a rational number.) And my students and I were going to have very different languages for communicating nuance and distinction.
So in the days preceding my lesson on prisms and their volume, I sent my instructional materials to an interpreter Lara recruited so she’d have some familiarity with the nuances I might try to communicate.
I wanted to know the kinds of distinctions my students would make, so I grabbed a bunch of 3-D shapes and asked them to contrast the shapes as "Prism" or "Not Prism" and write about how they were separating the two. This didn't get me as close to understanding their thinking as I had hoped.
Lara told me that deaf kids often suffer from language deprivation since many deaf children are not exposed to an accessible language early in life. While hearing babies are constantly absorbing and learning language just by existing in a world full of spoken words, deaf babies and children often don’t receive the same resources, informally or even formally. With rare exceptions, hearing students receive 12 years of formal language education in the United States that build on the informal language education they received just from living and hearing in the world. Deaf students do not often receive 12 years of ASL language education because they often don’t get the opportunity to learn ASL until later in life.
One consequence of that deprivation is that my deaf students could more accurately express their thinking about prisms in ASL, a language I don’t understand, than they could in written English, a language I do understand. So the students would express their ideas in ASL to their interpreter who would express them verbally to me—a game of telephone with everyone using different languages and having different kinds of knowledge about the subject they're trying to discuss.
I tried to communicate several ideas to the students through that noise:
I loved their thoughts about prisms. Many of the ways they thought about prisms had never occurred to me.
A group of mathematicians have defined prisms in a particular way.
I still loved their thoughts about prisms.
An advantage of having a common definition of prisms is that prisms have particular properties we can use to get stuff done, like these goofballs who constructed the world's largest coffee cup (2,010 gallons, a record they set in 2010). That feat was only possible because the dimensions of prisms relate to their volume in a particular way.
I said throughout the lesson, as much to remind myself as to communicate it to students, "Other mathematicians have defined prisms differently from you folks, but your mind and your eyes are working together to produce some really interesting thinking."
Maybe it's easy to read this and think, "It must be hard to teach students who use a different language from their teacher." But what I'm suggesting is that those are the conditions under which everyone teaches.
We're always teaching students whose ideas about mathematics—the patterns they're noticing, their estimations about quantities and shapes, etc.—outstrip their ability to express them. It's perhaps more obvious when students haven’t received adequate instruction in written English or when their primary language is different from ours. But I'm grateful for the reminder that there is always more going on in a student's mind than meets our eyes or ears.
Much gratitude to Lara Metcalf for her feedback before and after my substitute teaching experience, and for her feedback on drafts of this post. All errors are my own.
Desmos released a free activity based on the classic Bridges of Konigsberg problem. The best part IMO is it has a Challenge Creator so students can create their own problems themselves. Feel free to use it as the year winds down or tuck it away for a warm-up when the new year begins.
Robert Banks IV continues to create some of the most fantastic and instructionally useful math visuals in the game. Head to this link and change the f(x) expression and watch the result. [via the Desmos Educators Facebook group]
Given the general horrific nature of this last school year, I appreciated the final exam Drew Lewis offered his students.
We saw huge growth in the use of our free activity creation tool this last year and people frequently asked us for resources for advanced usage. Better late than never: we’re now developing a multi-week, cohort-based online course called Activity Builder Academy. We’ll kick it off late summer / early fall. Sign up for more information.
Ben Orlin wrote a very good guide for math teachers joining Twitter.
Idil Abdulkadir in the comments of my last email:
If pandemic schooling has taught us anything it's that schools + teachers are tasked with all manner of things that the state could do, should do, but actively decides not to do. I've been having this conversation about letting things fail & not saving broken systems on and off all year - and it's a hard sell for many teachers (including me, until recently). Teacher culture is making impossible, unreasonable things work and being called a hero if you're lucky.