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?

beckyWhat would happen if you started giving grades at PCMI?

dkane47Post authorI would guess that, as a start, it would increase competition, decrease collaboration, and decrease the joy that comes out of the math because the grading sends a value that individual success is more important than collective success, which I think is an important part of what makes morning math work. I don’t even want to go in on RoP and working groups.

I also think it would distract a ton of energy to conversations about “this isn’t fair” or “I think it should be this way” rather than focusing on learning.

Pete HorschCheers to not giving grades at PCMI. You’d also probably decrease attendance. What if you had to (were able to) give PCMI participants feedback on their growth, engagement, and performance during morning math? What if any grade a student earned truly reflected these three things?

dkane47Post authorI wonder if trying to measure (or even give feedback on) things as subjective and internal as growth and engagement breaks trust between teacher and student or creates incentives to game the system. I really worry about being perceived as judging things happening in someone else’s head.

annablinsteinGreat post, Dylan. I have the exact same question regarding the effect of grading on trying to broaden our definition of mathematical competencies. I also wonder if it’s better to include these broader competencies as part of the overall grade (grade what we value) or if that would add just another layer of extrinsic motivation to something we want to intrinsically motivate. I lean towards the former, but wonder what others think.

dkane47Post authorYea. I’ve struggled with trying to grade broader competencies though because any time I try to put a rubric to them or something along those lines I feel like I’m cheapening and shortcutting the most important pieces. I also think grading is a complex system and there are lots of moving parts, by which I mean grading or not grading is less important to me than how do you do whatever you do in a way that authentically values broad mathematical competencies as much as possible.

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