Two thoughts on competence, reflecting on time at PCMI. First:
We’ve been thinking about Lani Horn’s book Motivated, and talking about the idea of competence. I had been thinking of competence as a student’s need to feel successful in doing mathematics. If students don’t feel like they are successful in doing math they are unlikely to engage in unfamiliar or challenging problems, take risks by sharing ideas, or persevere when learning feels hard.
We spent some time with Peg Cagle, and Peg offered a different take. She described competence as a student’s need to recognize their success in doing mathematics. I see this as an important shift. While it’s just language, the word recognize makes an assumption — that all students bring meaningful mathematical ideas and mathematical thinking skills to class. My job as an educator is to create structures and space to help students recognize those competencies. Horn offers a partial list of mathematical competencies that teachers can value beyond what is traditionally valued in math class — fast and accurate computation. Those broader competencies are:
- making astute connections
- seeing and describing patterns
- developing clear representations
- being systematic
- extending ideas
Peg shared this image from her classroom as well:
I’m sure other educators could expand on these. The point is that there are lots of ways for students to be mathematically competent. If my goal is for them to feel competent, I might work to prop up their confidence or self-esteem without changing any structures in my classroom. If, instead, my goal is to help students recognize the ways they are mathematically competent, I am obligated to find ways to surface and highlight broader competencies that create avenues for every student to recognize their successes. Those are two very different classrooms; the second is much more responsive to the needs of particular students, and develops a much richer idea of what it means to do mathematics.
Second idea. In talking about competence yesterday, we also talked about the challenges of grades. What does it mean to value broader competencies like extending ideas when we are obligated to put a letter grade on a transcript at the end of the term? Those grades seem to be all many students care about. Do grades erase any work we do to assign competence in broader ways? I can’t realistically improve at more than one or two areas of my practice at a time. In the context of schools unwilling to change grading policies, would I be smart to put my effort somewhere else?