The Limits of “Just Teaching Math”

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!

16 clock faces

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.

About 
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

18 Comments

  1. Reply

    Nice post, Dan.

    Before Uncle Ben told Peter Parker, the French said something very similar in 1793: “they must contemplate that a great responsibility is inseparable from a great power“

    We, as teachers, need to harness that responsibility rather than refute its existence.

  2. Reply

    Such a resonating post! I’ve been contemplating identities ever since reading Impact of Identity last summer and it simultaneously psyches me up and stresses me out.

    I wish more teachers would look at their practice through this lens. Thanks for getting it out there.

  3. Reply

    Love the post. It helps me to think about growth mindset in light of this identity lens. I have a question – with a class how exactly do you demo it to students as you did (where you chose the secret clock). I’ve done it clumsily with other polygraphs with two computers and 2 fake students but sometimes the large screen ends up as the chooser not the guesser, if that makes sense.

  4. Reply

    Thanks for sharing how you’re progressing through this work. I especially identify with this part “I have wanted not to be an identity worker, to just be a math worker.” I realize that it’s been part of my privilege to even see that as something I could opt out of. In reality, it’s really amazing how just a kind word or recognition can make huge differences in how students develop their own identities, and that’s it’s foolish to not take advantage of them.

  5. Reply

    This is one of my favorite things you’ve written. I so appreciate the humility you bring to your experience of learning and acknowledge how much we all have to learn from even the youngest students, before they become jaded and cynical and while their identities are forming. Indeed elementary teachers hold some of the greatest responsibility here. There is probably a special place in heaven for kinder teachers just for that reason…they tend to “get it” more than most others in my experience. This line especially resonates: “students are always learning more than math in math class.” Many teachers recognize this already, but hopefully more and more will as they study and reflection on the great canon of literature out there around identity and agency in math, and more practice. Hammond is a great place to start.
    I’m curious to know, Dan, how your thinking on this intersects with what you wrote in the foreword for Peg Smith’s book about coming to math education either because of your love of math or your love of kids, but how both are necessary? It seems a deeper evolution of that idea.

  6. Reply

    I’m curious to know, Dan, how your thinking on this intersects with what you wrote in the foreword for Peg Smith’s book about coming to math education either because of your love of math or your love of kids, but how both are necessary? It seems a deeper evolution of that idea.

    Thanks for the feedback, Erin. As you’re suspecting, the Venn diagram representing “motivations for the foreword” and “motivations for this post” is a single overlapping circle basically.

  7. Joanne Ward

    June 7, 2019 - 5:39 pm -
    Reply

    Thank you for sharing the concept of the “identity worker.” I agree. As a high school teacher, I see how the atmosphere of welcoming the struggles in a high school math classroom affects students’ attitude toward learning new things in their college life.

    Now after reading this post, I want to create a safe environment that builds healthy identities in my students through constructive struggling.

  8. Reply

    I’ve been re-engaged by your blog recently as it seems to have taken a turn in the direction of disintegrating entrenched power structures. Maybe it’s always had that edge implicitly but lately it has turned explicit and I appreciate it. Also I think number systems is a topic that could really be harnessed for engaging the marginalized population. I’ve taught Mayan Numerals as a one-off lesson the day before a vacation or after a big test, but there’s a lot more potential out there. The Mayans had a base-20 number system. And the Roman Numeral system was inferior b/c it isn’t logarithmic but somehow we still see them everywhere. Their mathematicians or philosophers would convert Roman Numbers into Babylonian Numbers (sexy-gesimal or base 60) to calculate the distance to sun or the circumference of the earth at the equator, and then convert the answers back into Roman Numerals, because the politicians wouldn’t admit their system was inferior. And you can get into binary and computers and even a dozenal society who advocates for switching to a base-12 system. Which will never happen, but it’s cool to see why they advocate it. It’s a nice convergence of math, language, history and current events. Seems like it could be a deep and rich unit in 5th or 6th grade that can be differentiated to engage certain kids while still challenging others.

  9. Reply

    This has to be my favorite post of yours. As someone who has also spent the majority of my time in front of secondary students, going in and teaching elementary-aged kids has been a breath of fresh perspective. Thank you for what you do to bring light to things that need illuminating.

  10. Reply

    What a beautifully thoughtful piece Dan. It goes right to the heart of what makes teaching complex and powerful. Thank you for sharing.

    • Complex and powerful – that’s it! Teachers are actors in so many cultural scenes. Tremendous power. Work that’ll never be finished, never get boring. Thanks for your feedback, Sara.

  11. Jennifer Lagrange

    June 14, 2019 - 3:30 am -
    Reply

    Hi Dan,
    Thanks to Twitter, I came across your blog post. It reminds me of the excellent book: Choice Words: How Our Language Affects Children’s Learning by Peter H. Johnston (2004). I reread it from time to time to remind me of how my interactions with students have a huge impact on them. Identity is one of the big themes in the book.

  12. Chester Draws

    June 15, 2019 - 4:09 pm -
    Reply

    My goal in these experiences is always to find areas of agreement between the teaching of different age groups and different areas of math.

    Intriguing. My aim is generally the opposite.

    There’s many things about teaching that are the same whether the students are big or small, find maths easy or difficult, what their background is etc. What is very hard to do is take apart the differences — just how much assistance is best, whether to use more carrot or stick, whether to work in groups/pairs/individually etc.

    I do tend to ascribe to the view that we are all much more alike than we are different, so that teaching does have lots of similarities regardless of the class, and that’s how we can manage to teach effectively with the ridiculously rudimentary preparation that teacher training gives. But the difference between good and great is spotting the differences surely?

  13. Rishabh Sharma

    July 14, 2019 - 9:07 am -
    Reply

    Hi Dan,

    You mentioned that a big difference between teaching high school math and elementary school math was that the latter were incredibly enthusiastic. I teach high school math, but the few hours I’ve spent volunteering to teach younger students have been similar. They are inexplicably excitable and engaged in a classroom. I was wondering, however, when and why students tend to lose this energy?

    I remember being in elementary school and seeing amazingly gifted students reading at 12th-grade reading levels struggle to graduate high school when they are older. I think this occurs because, as you said, we do so much more than teach math. Teachers cultivate students’ identities throughout their lives by their actions in a classroom. For example, a teacher may shun a gifted student from speaking in a classroom because she’s not adhering to norms, thereby limiting her potential.

    Furthermore, could you elaborate on how your team at Desmos is teaching students about themselves? This sounds fascinating and I would love to hear more about the specifics.

    Best,

    Rishabh Sharma

    • Furthermore, could you elaborate on how your team at Desmos is teaching students about themselves? This sounds fascinating and I would love to hear more about the specifics.

      Thanks for the question, Rishabh. Disclaimers: (1) I need to write more about this. (2) We are early in our work.

      Most math education technology communicates to students that “smart ideas are had by other people – people who are most often older and whiter than you are – and your job is to re-iterate those smart ideas exactly.”

      Re-iterate their same method. Re-iterate their same words defined in the same way. Etc.

      So we create activities that draw out of students their own interesting early ideas and we offer everyone in the class tools to learn from those ideas. Students can write down paragraph responses, draw sketches, create shapes, not just complete multiple choice items. Teachers can snapshot those responses and communicate to students how interesting they are, without premature judgment of correctness. That’s the direction we’re moving.

  14. Kristen San Filippo

    July 21, 2019 - 3:34 am -
    Reply

    I love this! I promise All oh my students on the first day of school every year that they will leave my classroom with more confidence and understanding about mathematics.

  15. Kris Lindeblad

    July 22, 2019 - 10:22 am -
    Reply

    Your reply makes me excited. I do think that Desmos has the potential to create mathematicians and I am very interested in exploring this intentionally and not just hoping for a serendipitous outcome.

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