At PCMI we’ve been talking a lot in our “reflection on practice” sessions about *assessments*. Starting broadly, we wandered in the realm of epistemology and started talking about what we mean by knowledge, and how we know if we know something. Which then led to conversations about how we have *evidence* of knowing (and why it might matter that we teachers think about this and reach a consensus on this).

From this more abstract beginning (where we did try to bring it down to the classroom level), we then started to get more concrete. A very generative question was “what does it mean to be mathematically competent/proficient?” Everyone interpreted the question in a different way, and initially I was stuck – do we mean in a particular skill? or overall, when can we call a person’s body of work “mathematically competent/proficient”? procedurally or conceptually? isn’t competent < proficient? So many layers to unpack in the question itself. However I still started brainstorming and the very first thing I came up with was “can see the idea in a larger web of ideas/connections.” I often think that’s a hallmark of a strong mathematical thinker – where things aren’t this hodgepodge of ideas and procedures but they are tied together in a larger web. And what’s lovely is that one of the most powerful parts for me of Francis Su’s *Mathematics for Human Flourishing* book so far was in the Meaning chapter:

mathematical ideas, too, are metaphors. Think about the number 7. To say anything interesting about 7, you have to place it in conversation with other things. To say that 7 is a prime is to talk about its relationship with its factors: those numbers that divide evenly into 7. To say that 7 is 111 in binary notation is to have it dialogue with the number 2. To say that 7 is the number of days in a week is to make it converse with the calendar. Thus, the number 7 is both an abstract idea and several concrete metaphors: a prime, a binary number, and days in a week. Similarly the Pythagorean theorem is a statement relating the three sides of a right triangle but it is also, metaphorically, every proof you learn that illuminates why it is true and every application you see that shows you why it is useful. So the theorem grows in meaning for you each time you see a new proof or see it used in a new way.

Every mathematical idea carries with it metaphors that shape its meaning. No idea can survive in isolation — it will die.(37) [italics mine]

In addition to this being part of being mathematically competent (which, overall, I think I ascribed to a body of work that a student did instead of with a single skill), I also immediately thought of various mathematical habits of mind.

Something that struck me, after my brainstorming, was that much of what I wrote was about a world of ideas, but it was disembodied from the physical world… and so I ended my brainstorming: “One thing I’m thinking about when I think about “mathematical competency/proficiency” is that those terms seem to be pretty clinical… I’m wondering if joy and appreciation and an emotional connection to math would exist in this or not.” Looking back at this reflection of mine, I think of Rochelle Gutierrez’s eight dimensions of rehumanizing math, and my wondering seems to fit squarely in the “Body/Emotions” dimension.

As I’m writing this blog post, I’m enjoying seeing how lots of different ideas in this conference seem to all be strung together loosely and I’m only now seeing them braid together.

One great exercise we had was just writing down all the different ways we collect evidence of student understanding. All the ideas we came up as a group are here, and a word cloud of what we came up with is:

We talked about bias in various ways we collect and interpret evidence, the ease/difficulty of various ways we collect evidence, and how confident we were in our interpretation. What I liked about this is that we went beyond “paper tests” and “exit slips” and allowed the true range of things that we as real teachers do to get a sense of what a student or class knows… I mean, I do actually use facial expressions to help me get a sense of student understanding, but I also recognize it’s not always the best indicator. So this broad list we generated didn’t seem like a trite exercise because it valued all the ways we as teachers do truly get a sense of things on a day-to-day basis. In other words, it broadened our sense of “assessment” to go beyond “the things that we grade, that often are done on paper.” Assessing is just getting a sense of what kids know – whether formative, summative, or something else (e.g. self-assessing their own confidence on something).

We read a couple articles on assessments as well as having kids self-assess or peer-assess. I liked NCTM’s section on assessments from their *Principles to Action* — reframing assessments to be something that provides feedback to students but also informs instruction moving forward.

Shifting the primary focus and function of assessment from accountability to effective instructional practice is an essential component of ensuring mathematical success for all students (p. 98)

For me, the big reminder was that we teachers traditionally think about assessment as a noun. That paper thing we give. A static snapshot. But we should think about it as a verb, assessing, and that is part of a learning cycle, a journey — both for student and teacher. However, of course, there’s the ideal and there’s the reality.

There are ways I’ve thought more as a verb, like when I’m did standard-based grading when I taught calculus. But usually, when I give summative tests (which is what I normally do, and I don’t really do projects), I don’t think of them as a part of a cycle or journey. They are the static snapshot.

But I also know in reality, I do lots of assessing in small ungraded ways (self-assessing, start of class problems and walk around, listen intently to students talking, looking at nightly work, etc.), and use that to inform my next teaching move or plan my next class. For one example, I have a general idea of what the nightly work will be each day, but based on what I see in class, I often will alter it based on what makes sense… or generate some new problems to address a misconception or gap I’m seeing… or to have students think about a particular insight that came up in class.

I’m now getting tired, so I want to end with three things before I lose steam.

First, we watched a video of Max Ray-Riek on why 2 > 4. It is an ignite talk, and I’d seen it before, and I realized after watching it that during then pandemic, I started *listening for* instead of *listening to *because I felt so stressed for time. I think this is a video every math teacher should watch!

Second, we brainstormed ways that we get real-time evidence of student understanding in our normal everyday classes. And then we each chose a few to share out. I like this document of collective knowledge that we generated, and I want to come back and read through it again to be inspired.

Third, we talked about ways we had students *self-assess* themselves and their own understanding as a way to become more independent learners, and think metacognatively. I really enjoyed brainstorming this individually and listening to everyone else’s ideas. I wish we had more time for this activity — because I love talking about concrete things we do in the classroom so I can get more ideas and rethink things I already do!

And with that, I’m done! Tomorrow begins our third and final week at PCMI!