Mathematicians at Work

mathematician    NYTimes excerpt from The Singular Mind of Terry Tao

What does a mathematician do?

Five years ago, I had no idea. Today, I do, which is ever expanding and refining. Learning about Terry Tao a few years ago set the tone for my understanding. The contrast between what has historically characterized school mathematics and what Tao describes as mathematics could not be starker.

In Tracy Zager’s heartening and enlightening new book Becoming the Math Teacher You Wish You’d Had, she explores this dissonance by comparing the words K-12 math teachers use to describe math and the words mathematicians use to describe it.


Zager, Tracy J. 2017 Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms. Stenhouse Publishers: Portland, ME

Care to guess, which words belong to which group?

Curiously, between the Standards for Mathematical Practice (SMPs), and the Principles to Actions, an unknowing soul might believe that K-12 math education in America looked a lot like the world that mathematicians describe above. But our problem is as old as education reform itself: confusing standards for a vision and a viable strategy.

Bring up the SMPs or PTAs with a fellow teacher and you’re likely to get a blank stare. My colleague Nina Sudnick calls us “the notch generation,” teachers asked to teach in a way they never learned nor have ever seen. Thus the SMPs, foreign to us raised in procedural environments, end up either abused, neglected, or largely forgotten.

Yet just last week I saw our K-12 SMPs brought to life by David Butler in his “biggest maths idea I’ve ever had’ post. David works out of the University of Adelaide, and is a fine contributor to the #mtbos on Twitter.

Considering that Butler holds a PhD in Finite Geometry, was tackling imaginary numbers, and my higher level math knowledge ends around algebra, I didn’t expect to parse much from his post. However, as I dug into David’s work “Where the complex points are” I felt comfortable. His writing was clear. I could follow his logic and wonder with him, “where are the complex points?” David’s “biggest maths idea ever” sent my mind into overdrive thinking about what it means to be a mathematician, the Standards for Mathematical Practice, and young mathematicians at work in a K-12 classroom.

As a brief aside, I wonder, how many of our students have ever had a math idea? What about a BIG math idea? If they have had one, do we have classrooms that honor these ideas where they can be explored, connected, dissected?

My inclination is that lots of younger students have math ideas, but there is no place for them in the classroom. Like a muscle, fail to use it, and it atrophies, leaving our upper grades bereft of students with math ideas.

The SMPs and a Mathematician in Action

Like any good mathematical exploration, David’s work is a testament to SMP #1: make sense of a problem and persevere in solving it. His time frame is not explicit but it’s clear he has revisited this problem and spend considerable time crafting his solution. He’s also not done. This exploration is leading him to future explorations and other connections he wishes to make. I saw this transpire in my 4th grade classroom when a student’s conjecture about multiplication with fractions led to several weeks of investigations that kept unraveling from the students’ work.

While it was immediately clear that the mathematical exploration David was taking us on was beyond my circle of competence, going from the real numbers to something I knew in name only, imaginary numbers, David’s writing was a model for SMP #6: attention to precision.

From the opening paragraph I understood the “what” of his post and the purpose (a purpose?) for imaginary numbers. In my classroom, I valued SMP #6 for what I understood it to be, the ability to clearly communicate so that others could understand you.

openingparagraphThe yin and yang of SMP #2: reason abstractly and quantitatively and SMP #4: model with mathematics was on full display as he thought about the equation x^2 + 1 = 0, its complex solutions x = i and x=-i, and wondered “where are the complex points?”


The next practice that jumped at me was SMP #5: use appropriate tools strategically. This is an area of debate I hear throughout elementary math education in schools, at conferences, and at professional development sessions.

Do you give students tools for the precise job you’re asking them to do? Do you give them a bucket of tools they can draw on as needed? Should they have the freedom to improvise and try tools that may not fit the job?

Let’s look at what David did here. He chose to use two coordinate planes, one real, one imaginary. He chose to shade the imaginary plane pink because he knew he wanted to differentiate it from the ordinary coordinate plane.

To better make sense of the complex points and to better model them he used cellophane and a physical coordinate plane to show the complex points literally attach to a real point.


Why cellophane? Because it has a degree of transparency and in his mathematical creation, David believes the imaginary plane should be transparent so that the real plane is visible through it.

My take on tools:

In the debate about SMP #5, using appropriate tools strategically, what experiences are we engineering for our students throughout their mathematical careers to be able to create something novel. If they are always looking to the teacher for the tool, or expecting the tool they need to be in the bucket on their table, it would seem we as teachers are not doing our diligence.

Some of the novelty I witnessed from young mathematicians in my own classroom was because I didn’t know what would or wouldn’t work. In certain areas I was not afflicted by the curse of knowledge, which led to plenty of times when students had space to choose tools and explore putting them to use in their problem solving pursuits.

Well before David jumped into the understanding he gained, he offered us a caveat, which to this reader, looked like an ode to the first part of SMP#3: construct viable arguments and critique the reasoning of others.


I am not sure what I might critique in this argument, but I did pass it along to a friend of mine who is an engineer. I was happy that I was able to explain David’s idea well enough that my friend seemed able to make sense of it. He said “That makes sense. I figured you would have to go to some sort of third dimension to see imaginary coordinates.” At which, point he asked “You do know what imaginary numbers are, right?” I did not. That’s when I learned that ‘i’ is the square root of -1. Much more to explore.

Finally, as I read David’s conclusion, I was encouraged.

“So why am i so very very excited by this idea? Well, it all has to do with what happens when you figure out the complex points that are part of some familiar graphs. So far, I’ve thought a lot about lines and a bit about parabolas. It will take me a while to write about it all, so I’ll save those topics for the subsequent blog posts.”

This isn’t the contrived problem solving found in a Go Math book. This is the problem solving of a mathematician who enjoys what he does. It’s a mathematician showing that math can and should make sense. It’s that hopeful word cloud in Tracy’s book come to life.

Most importantly, students have the power to do this. I saw it often in my own 4th grade classroom. It can happen in high school, at university, in middle school, and even in kindergarten to which Simon Gregg’s wonderful classroom is a testament.

There is so much good happening out there in education and exponentially more potential. It makes me hopeful.

It seems appropriate to leave as we began, with Terry Tao.

That spring day in his office, reflecting on his career so far, Tao told me that his view of mathematics has utterly changed since childhood. “When I was growing up, I knew I wanted to be a mathematician, but I had no idea what that entailed,” he said in a lilting Australian accent. “I sort of imagined a committee would hand me problems to solve or something.” But it turned out that the work of real mathematicians bears little resemblance to the manipulations and memorization of the math student. Even those who experience great success through their college years may turn out not to have what it takes.

The ancient art of mathematics, Tao has discovered, does not reward speed so much as patience, cunning, and perhaps most surprising of all, the sort of gift for collaboration and improvisation that characterizes the best jazz musicians. Tao now believes that his younger self, the prodigy who wowed the math world, wasn’t truly doing math at all. “Its as if your only experience with music were practicing scales or learning music theory,” he said, looking into light pouring from his window. “I didn’t learn the deeper meaning of the subject until much later.”

NYTimes excerpt from The Singular Mind of Terry Tao

An act of originality

Anton’s creative way to derive multiplication facts

“I have no idea what he’s saying,” I thought to myself as I scanned the faces of the other nineteen 4th graders, hoping one of them had a glimmer of understanding in their eye I could latch on to.

The young mathematician persisted.

“You can subtract to multiply” he said as he continued on about a messy, confusing sounding idea.

“Interesting,” I said, a catch all word I used when I wanted to keep a poker face, which at this moment concealed that I had no sense of how this student was reasoning.

I desperately hoped to do two things. First make sense of what this young mathematician was saying. Second, to orchestrate a productive discussion that would help the entire class learn. But this was early in my first year teaching 4th grade, my pedagogical toolbox was sparse.

The faces, I scanned them again, nothing to grasp at. We were still gelling as a class and the trust wasn’t there to put another student on the spot hoping they might have an idea what he meant.

I didn’t know it at the time but Anton’s thinking was original. While I didn’t know much about teaching I did have Steven Reinhart’s Never Say Anything a Kid Can Say, Van de Walle and co., and Lainie Schuster in my head. At the time, ‘student-centered’ was my mindset with a mantra of ‘first do no harm’, which I hear framed today as foster a productive disposition.

Anton’s voice would be heard.

“Come on up to the chart paper and show us how you’re thinking” I said. Was this the right move? The class had disengaged. Even if I could make sense of Anton’s thinking was it worth it to lose the nineteen other students? Should I have refocused our exploration on more conventional ways of deriving multiplication facts?

This was the fall of 2013, my first year really teaching, and what I didn’t know, in regards to pedagogy, content, and math practices, outweighed what I did. However, I knew exactly what I didn’t want to do: lecture, make kids hate math, destroy their sense of worth in math class, build a house of cards based on procedural proficiency without understanding the concepts—in short, run a traditional classroom that serves no one well. I was following the playbook of Charlie Munger and applying the powerful tool of inversion. I knew how to create an awful math class. Start by not doing those things. Instead play in the wide open world beyond the teaching I had experienced.

Experiment. Reflect. Improve. Repeat.

It takes luck…

At the chart paper Anton began scratching out what would become a wonder of creative thought.


Shortly, another young mathematician was on his feet at the chart paper talking through Anton’s procedure with him. I used what pedagogical know-how I had to get them to repeat, clarify, and converse with the class. The beginnings of a procedure were codified.


A trickle of students had retrieved their math journals. Then others jumped up to get their own. Suddenly, we were hitting on all cylinders as the class was engaged in making sense of and applying Anton’s procedure. The tenor of the room had taken a 180 from earlier.

Before long the class had tested all 9’s, then 8’s, then 10’s using the procedure they codified before we ran out of time. I didn’t know it at the time but thanks to Anton’s creativity, we had stumbled into an early version of conjecture and test—finding the boundaries and conditions for when something is true. Just as important, “The Mathematics Laboratory” was born, a foundation for what would emerge as our classroom culture for the year.


In the moment, I was surprised as I watched this unfold. Reflecting on this lesson over the years, I have become even more impressed. Anton’s work was truly original. I have never seen anything like it. It was beyond our scope to do so at the time but wouldn’t it be wonderful if he goes back to his conjecture some day to try and derive why this procedure works?

Luck played a part in this day’s seminal events. I didn’t ask Anton or any other student to “be creative” or “think innovatively” while deriving facts. However, to use the garden analogy, the soil was fertile. Beginning on the first day of school the soil had been tilled, seeds planted, and watered. The luck was that upon seeing a sprout, even when it was ugly and unexpected, it wasn’t confused for a weed and ripped from the dirt. It was allowed to grow. I’m confident if not this day, a similar catalytic event would have happened another day, given I didn’t mess it up.

I think back on this moment often. It ties back to many ideas, big and small that I think  about—not just in regards to teaching and education but about listening, developing people, and culture. On a micro-level this early success, for the students and for me, laid a positive foundation for our year.

It showed me what *was* possible in seeing/hearing, understanding, and using student thinking to drive learning for an entire classroom. Seeing what was possible gave me a new set of questions to ask and led me to the research and practices that laid the foundation for the exponential leap in pedagogy I would make in my second year. Reflecting on this day at a macro-level causes me to rethink the way education is structured.

I’m curious—what creativity/originality/innovation have you witnessed from young mathematicians and under what circumstances did it happen?