What does it mean to be smart in mathematics?

In the last two posts, I discussed the idea of status. First, I talked about why status matters, then I talked about how teachers can see it in the classroom.

Sometimes, after I have explained how status plays out in the classroom, somebody will push back by saying, “Yeah, but status is going to happen. Some kids are just smarter than others.”

I am not naive: I do not believe that everybody is the same or has the same abilities. I do not even think this would be desirable. However, I do think that too many kids have gifts that are not recognized or valued in school — especially in mathematics class.

Let me elaborate. In schools, the most valued kind of mathematical competence is typically quick and accurate calculation. There is nothing wrong with being a fast and accurate calculator: a facility with numbers and algorithms no doubt reflects important mathematical proclivities. But if our goal is to address status issues and broaden classroom participation in an authentically mathematical way, we need to broaden our notions of what mathematical competence looks like.

Again, my naysayers roll their eyes and groan, assuming that I want to “soften” mathematics or dilute the curriculum. But I claim that broader notions of mathematical competence are actually more authentic to the subject.

Let’s cherry pick some nice examples from the the history of mathematics. We see very quickly that mathematical competencies other than quick and accurate calculation have helped develop the field. For example, Fermat’s Last Theorem was posed as a question that seemed worth entertaining for more than three centuries because of its compelling intuitiveness. When Andrew Wiles’s solution came in the late twentieth century, it rested on the insightful connection he made between two seemingly disparate topics: number theory and elliptical curves. Hyperbolic geometry became a convincing alternative system for representing space because of Poincaré’s ingenious half-plane and disk models, which helped provide a means for constructions and visualizations in this non-Euclidean space. When the controversy over multiple geometries brewed, Klein’s Erlangen program developed an axiomatic system that helped explain the logic and relationships among these seemingly irreconcilable models. In the 1970s, Kenneth Appel and Wolfgang Haken’s proof of the Four Color Theorem was hotly debated because of its innovative use of computers to systematically consider every possible case. When aberrations have come up over the years, such as irrational or imaginary numbers, ingenious mathematicians have extended systems of calculation to encompass them so that they become number systems in their own right.

This glimpse into the history of mathematics shows that multiple competencies propel mathematical discovery:

• posing interesting questions (Fermat);
• making astute connections (Wiles);
• representing ideas clearly (Poincaré);
• developing logical explanations (Klein);
• working systematically (Appel and Haken); and
• extending ideas (irrational/complex number systems).

These are all vital mathematical competencies. Surprisingly, students have few opportunities to recognize these competencies in themselves or their peers while in school. Our system highlights the competence of calculating quickly and accurately, sometimes at the expense of other competencies that require a different pace of problem solving.

Evaluating people on one dimension of mathematical competence will rank students from most to least competent. This rank order usually relates to students’ academic status, and students tend to be aware of it. One way to interrupt status is to recognize multiple mathematical abilities. Instead of a one-dimensional rank order, we create a multidimensional competence space. Although some students may have multiple mathematical strengths, more places in which to get better surely exist. Likewise, a student who ranks low on the hierarchy produced when we focus on quick and accurate calculation may have a real strength at making astute connections, working systematically, or representing ideas clearly. We cannot address status hierarchies without emphasizing multiple mathematical competencies in the classroom.

A multiple-ability classroom represents a dramatic shift in the topography of mathematical ability. Instead of lining students up in a row in order of smartness, a multiple-ability classroom has students standing on different peaks and valleys of a hilly multidimensional terrain. No one student is always clearly above another. This structure may unsettle students who are used to being on top, as well as those whose vantage points and contributions have been presumed less valuable. In other words, challenging the status hierarchy by developing a multiple-ability view can provoke strong emotions from students, positive and negative. Teachers should not be surprised to see this response in their classrooms.