G was a sixth grader in my first full-time student-teaching stint in summer school. She didn’t conform well to typical classroom norms. She struggled to sit still for extended periods of time or raise her hand when she had something to say. And G had never felt successful in math class. I remember her saying, “I can do addition, but I *haaate* subtraction.” She would quickly become frustrated with any math that wasn’t easy for her, or just scribble down the first thing that came to mind. But she was good at addition; she could add multi-digit numbers incredibly quickly and accurately, and she loved doing so. She would have been happy to hang out and solve addition problems all day if I left her to it. Unfortunately, it was my job to teach her subtraction.

I thought of G reading Michael Pershan’s recent post on ancient Greek mathematics. He explores the two ways of practicing mathematics in ancient Greece: the theoretical, abstract, and logic-oriented branch, and the practical, social, procedural branch. We often associate Greece with the culture of theoretical mathematics, yet there was a large and thriving group largely interested in solving practical problems and finding new ways to calculate, measure, and count. Michael writes:

Looking at the ancient Greek example makes me think that we really

oughtto respect practical mathematics — which by definition is mathematics that is not concerned with the “why.”And yet there is so often disdain among some teachers for “mindless” calculation or “thoughtless” problem-solving. That seems unfair to me.

Here’s a quote from a recent piece of mine that seems to argue the opposite perspective:

Lani Horn writes that “Schooling favors one type of mathematical competence: quick and accurate calculation” (Motivated, p. 61). Horn argues that we can value broader mathematical competencies — making astute connections, seeing and describing patterns, developing clear representations, being systematic, extending ideas, and more.

I’ve been interested in finding ways to broaden what we think of as school mathematics, finding new ways for students to recognize their mathematical competence. But the key word there is broaden: it seems silly to find new ways of being mathematically smart if at the same time I devalue others.

I was a pretty inept teacher when I met G, and I missed opportunities to build from her competencies toward new ways of mathematical thinking. But if I only saw mathematical competence as making connections, seeing and describing patterns, and the rest of Lani’s list, I would have missed the most important asset G brought to class, and the one she identified as her greatest strength: quick and accurate calculation. I don’t see any reason to diminish G’s self-efficacy by telling her that math is really about understanding why or explaining ideas or making connections, and not calculating things. She didn’t see those other competencies as her strengths. And students in math class desperately want to feel smart and successful in their learning; G needed to find ways to build from the skills she had toward those I wanted her to learn.

There is a tricky balance to walk here. Quick and accurate calculation is the dominant competence that is valued, implicitly and explicitly, in many classrooms. Given its prevalence, I see value in putting extra effort toward seeking out different ways of being mathematically smart that broaden students’ conceptions of what it means to practice mathematics and helps more students feel valued in math classrooms. But its prevalence also means that there are lots of students, like G, who come to class with strong skills in computation. And if I choose to tell them that those skills aren’t “real math” and are less important than other skills I prefer, I am reframing those strengths as deficits exactly as I am trying to find broader ways to see students’ strengths in math class.

Michael PershanAwesome reflection, all truths.

Quick note. You write that there was a group interested in “solving practical problems and finding new ways to calculate, measure, and count.”

On my read, that’s not quite what was going on. They weren’t interested in INNOVATING and finding new ways to do all these things. They had a recipebook (quite literally, a book of recipes) for solving a variety of problems. That repertoire was handed down to apprentices and was then applied to a variety of situations. I don’t think they were getting together, trying to figure out more efficient ways to measure volume. No, they were professional computers, much as many “hidden figures” of astronomy were computers.

Michael Pershandarn it, forgot to subscribe to new comments

dkane47Post authorGot it, makes sense. And obviously there’s a lot of middle ground between those two (like you say in your piece, the distinctions don’t map perfectly onto contemporary mathematics). That seems like an interesting area to explore. For instance, the math of gerrymandering has been a hot topic recently. It’s very practically oriented, but also values understanding exactly what different measures of gerrymandering are, well, measuring. And there are different subareas interested in different pieces of that. Yet the courts have at times basically said “we don’t understand this, we can’t use these numbers if we don’t understand where they’re coming from.

Just some musings. Thanks for writing your piece!

howardat58Have a look at the girl and addition, and then check my latest post:

https://howardat58.wordpress.com/