# Big M, little m

Last night, during my first class as a PhD student, the instructor* of the course — a seminar giving  a broad overview of mathematics education research — posed the following question:

Imagine a continuum from big-M Mathematics to small-m mathematics, where the big M is the mathematics considered powerful enough to be taught in schools, and the little m is more like grocery store math, or mathematics that you or others do in everyday life, even if it isn’t always acknowledged as math. Where along the continuum would you place your interests, and why?

She asked us to get up and move to an area of the room representing our position. When it all shook out, there were 11 people all toward the big-M side of the scale. Then there was me, all by myself near the little m.

This surprised me for two reasons. First, there was the fact that I was the only one on the little-m side. This ran contrary to my expectations. I thought that in a research setting, there would be more variation, and maybe even a trend toward the little m. As I saw it, the big-M side of the scale is reasonably straightforward, but the little-m side was ripe for investigation.

Then there was the fact that I, of all people, was the one on the little-m side. I have spent my whole career developing school curriculum. I work for the University of Chicago School Mathematics Project. Although I don’t know where the next four years will take me, I think curriculum development is work I’ll return to one day. All of this speaks to big M. Why was I hanging out on the little-m side?

As an icebreaker, each of us had to explain why we chose the position we did. I had to ponder this a bit, and struggled to verbalize my thoughts, but eventually said something like this:

Although I have worked for a long time developing materials to guide mathematics teaching in schools, I have become less interested over time in the specifics, and more interested in helping children to become flexible mathematical thinkers. I don’t really care what mathematics they are doing, so long as they learn to approach problems creatively. I don’t want to develop more effective ways teach kids fractions or probability or decimal division — I want to develop better ways to teach them to think. Little-m mathematics seems to me to be about thinking.

It was a bit of a strange experience to hear myself say this, but it rang true. I’m tired of big M and the associated arguments about what deserves to be under the capital M. Too much focus on the big M, in my view, leads to loss of sight of the bigger picture.

I’ve been thinking since last night about what that bigger picture really is. It’s easy to say that I want to figure out how to best develop mathematical thinking in kids, but it’s a vision that lacks traction. I have been in this line of work long enough to know that good instruction needs better framing than that. The problem is, if the frame isn’t some set of mathematical objectives — add these fractions, identify these shapes, and so on — then what is it?

This afternoon I read a blog post by Dr. Mark Guzdial, a computer science education researcher at Georgia Tech. He mentioned a paper I recently presented at the International Computing Education Research (ICER) conference that focuses on learning trajectories I developed (with my colleagues at UChicago STEM Education) for computer science in elementary school. Dr. Guzdial had the following (very flattering) comments about the work:

“Rich et al.’s paper is particularly significant to me because it has me re-thinking my beliefs about elementary school computer science. I have expressed significant doubt about teaching computer science in early primary grades … . If a third grader learns something about Scratch, will they have learned something that they can use later in high school? Katie Rich presented not just trajectories but Big Ideas. Like Big Ideas for sequential programming include precision and ordering. It’s certainly plausible that a third grader who learns that precision and ordering in programs matters, might still remember that years later.” (2nd. para.)

I was struck by this comment, because it seemed parallel to the issue I was struggling with when thinking about mathematics in elementary school. The learning trajectories in the paper are full of small bits of computer science learning, akin to to the lists of mathematics topics captured in mathematics standards documents. It seems Dr. Guzdial isn’t convinced of the value of any particular computer science topic being taught in elementary school, the same way I’m not so enthusiastic these days about particular mathematics topics. But Dr. Guzdial does see the value in the big ideas. So maybe I could benefit from a higher level of zoom when it comes to thinking about elementary school mathematics.

Organizing an elementary mathematics curriculum around big mathematical ideas isn’t a new idea, of course. I know quite well that Everyday Mathematics, for example, has lessons and games focused on the big idea that numbers have many names. But still, in light of the big-M – little-m contrast, that big idea feels unsatisfying to me. We’re not quite to the thinking.

Although my colleagues and I framed the big ideas in our Sequence learning trajectory simply as precision and order, I realize now that Dr. Guzdial expressed them better than we did. The big ideas aren’t precision and order. They are: “precision and ordering in programs matters” (2nd para.). Matters is the key word.

Perhaps the big mathematical idea to be emphasized isn’t just numbers have many names, then. Perhaps it’s better stated as It matters that numbers have many names.

Or perhaps it’d be even more powerful for the big idea to express why it matters: I can give new, equivalent names to numbers to make problems easier to solve.

This Big Idea could be traced through a lot of mathematics:

“Borrowing” in base-10 subtraction is an act of renaming the minuend to make subtraction easier.

Finding like denominators is an act of renaming fractions to make addition and subtraction easier.

Factoring polynomials is an act of renaming them to make them easier to solve.

And so on.

Maybe I could get on the side of the big M if it stood not just for Mathematics, but rather for why mathematics Matters.

What do you think? Could reframing mathematics instruction around Big ideas and why they matter improve kids’ experiences with school math? Do you know of projects that do this well?

*Credit to Dr. Beth Herbel-Eisenmann for posing this very interesting icebreaker question during our first class. I should note that we also had to place ourselves along an orthogonal continuum from big-E education to little-e education — but more details on that are better suited to another post.