On Being a Friendly Mathematician

By Art Duval, Contributing Editor, University of Texas at El Paso

Almost fifteen years later, Lucy Michal still remembers the exact words Phil Daro told the leaders of the El Paso Collaborative for Academic Excellence as they were preparing to launch the K-16 Mathematics Alignment Initiative, which Lucy would direct: “Find a friendly mathematician.”  The goal was to align mathematics in grades K-16, through regular meetings of a working group of a few dozen local teachers of all grade levels.  Phil had many contacts, including national authorities in K-12 mathematics, but, for a project like this, he stressed the need for local mathematics experts.  A “friendly mathematician” would be respected for mathematics, but would also understand the importance of working with both pre-service and in-service teachers.  I became one of those friendly mathematicians.  What did I do to live up to this billing?

I had only started working with pre-service teachers two years before this, teaching a “math content” course for prospective K-8 teachers.  I was still a little stuck in my academic silo, bracing myself before the first meeting to make sure mathematical content wasn’t going to be given the short shrift in this project by pedagogical concerns.  I was also worried what the teachers would think of me, and if they would dismiss me for not having stood in their shoes in a K-12 classroom.

A funny thing happened, though, putting teachers from kindergarten through college in the same room.  I was used to the “if only” chorus you often hear when you get enough calculus instructors together: “if only the high school teachers taught algebra better…”.  I found out high school teachers have their own “if only” chorus: “if only the middle school teachers taught fractions better,” and so on down the line.  But all those “if only”s went away pretty quickly, because we were each sitting next to living, breathing examples of people doing their best to solve the very problem the “if only”s accuse them of causing.

We also found we had a lot more in common than we thought, even regarding the mathematics itself.  A striking moment early on was when we discussed how algebraic thinking is (ideally) developed through all the grade levels, starting in kindergarten, which I did not fully appreciate until that day’s discussion.  I was sitting next to a kindergarten teacher who described how they teach “clap patterns” (for instance, two claps, then three claps, then two, then three, and so on).  To make a long story short, this idea of repeating patterns eventually is extended to the idea of growing patterns, which is an entry point to linear functions and Algebra I.  The kindergarten teacher was surprised to see that this activity led all the way to algebra, and I was surprised to learn that ideas about algebra can be started this early.  This was where I think I first started to understand the real depth of the mathematics that underlies the K-12 curriculum.

In another early session, Phil Daro came back to talk to the whole working group, and the familiar patterns of “odd + odd = even”, etc., came up, as an example of something or another.  I forget if Phil mentioned something like “mathematicians can extend” this idea, but somehow I ended up showing the group, step by step, how modular arithmetic generalizes this whole set up.  Yes, this is standard stuff for math majors, but the elementary teachers had not majored in math, nor had some of the high school teachers.  Even those teachers who had seen it before may have forgotten it, since they don’t teach modular arithmetic in high school, or more importantly, they may not have necessarily made the connection to what they do teach.  So this is one of the things I did to earn my “friendly” stripes:  Find and share the deeper mathematical structures that extend the topics that show up in the K-12 curriculum.

What else did it take to be a “friendly mathematician”?  I wish I could remember more specific examples like the “odd + odd = even” story, but I mostly only remember general behaviors.  I think there were a few other opportunities to connect K-12 math to deeper topics, but many more smaller instances of why some topic in K-12 is important for some later topic in calculus or differential equations.  I ensured statements the group made were mathematically correct, for instance establishing precise mathematical definitions.  Also, once a teacher asked about a new calculus textbook he liked, but wasn’t certain he should use in his AP calculus class because it might not align with what we were doing at the university; it turned out to be exactly the book we were then using at the university.

Of course, there were also basic social norms. I had to do my share of the writing and other work.  I had to listen to the teachers, not just talk at them, and value their expertise.  Indeed, one of the joys of the experience was how everyone valued the different contributions and experiences others brought to the conversation.  Let me mention here that I was joined by my UTEP colleague Mourat Tchoshanov, a mathematics education professor in the Teacher Education department, and he also provided valuable insights, different from mine.

When I asked Lucy Michal recently (in preparation for this post), she said that Mourat and I treated the work seriously, as if it was worth our time and attention.  Indeed, I quickly saw the value of alignment, and it was clear to me that representing a post-secondary mathematical perspective was an important role I could fill in this project.  What surprised me was how much I learned about the K-12 curriculum, and about vertical and horizontal alignment.  I also learned a lot about mathematics and education from the outside experts who were brought in occasionally.

I have since jumped at every chance to work with K-12 teachers (well, within the confines of my busy schedule).  If you are a mathematician, becoming a friendly mathematician may bring you unexpected profound moments, both socially and mathematically.  More broadly, whatever your situation, if you have the opportunity to collaborate with people who have different mathematical expertise, you too can be a friendly mathematician. How do you get this chance?  I was lucky; they called.  But you don’t have to wait.  Find a local school district or professional development effort in your area, and call them — I have no doubt they would be thrilled to work with a friendly mathematician.