I have had an absolutely incredible time at the Park City Mathematics Institute the last three weeks. I will hopefully continue reflecting and have a chance to write more in the future. Today, on the last day of our session “Reflecting on Practice”, on formative assessment, we videochatted with Dylan Wiliam.
He is incredibly insightful about just about everything in education, and we toured through distributed practice, memory, learning styles, rich tasks, rubrics, criteria for success, self assessment, and more. While watching him speak and reflecting on my experience the last few weeks really brought together some of my thinking about my practice, and in particular about my strengths and weaknesses as a teacher.
One of the things I’ve been thinking a great deal about is my experience giving rich formative assessment tasks in my class. I haven’t done it enough, but even so, it has been really difficult for me as a teacher. Almost uniformly, I learn that students who I thought understood very well actually knew far less than I thought, and far more students were completely floundering than I had suspected.
PCMI has felt really similar for me in a lot of ways — it has been endlessly humbling as I have been surrounded by great teachers and great ideas that I have so much to learn from, and constantly exposing myself to new ideas and new perspectives on what great teaching looks like.
For me, one of the fundamental benefits of formative assessment is finding out what students know, really honestly, without skewing the data because I just taught a procedure for that problem, getting deep into understanding, and with as little confirmation bias as possible. This reminder — that as a human being who wants to be a good teacher, I will always be biased around what my students are actually learning, and I need to work constantly to see, from a new perspective, what is happening in my students brains, and what math they can and cannot do. Dylan Wiliam was really articulate about this — and about some of the illusions that fool us into thinking students know when they actually don’t.
I feel like this constantly humbling perspective has been a strength of mine while here at PCMI, really just out of necessity. I have learned so much about mathematics I thought I already understood from Darryl and Bowen’s morning sessions. I have gotten quieter and quieter in Reflecting on Practice as the course moved on, trying to listen more and learn more from those around me, soaking up new ideas and perspectives. In my afternoon working group, I’ve definitely frustrated my partner at times by constantly saying “I put this together, but I don’t like it very much, please change as much as you want” — not that that’s bad, but I think constantly being humbled, and working with an incredibly competent partner, made me think my ideas were less valuable than they were.
Big Takeaways
Thinking about my takeaways from Reflecting on Practice, it really comes down to one big idea. I want to ask myself, as often as possible and as honestly as possible, what my students actually know. Having the sand to follow through with this — whether I hear what I want to hear from my students or not — is what I need to take with me from PCMI. Everything else is a way to do this better — a perspective on what I can improve about the way I am listening to student ideas, or a tool to get this information quickly and efficiently, or a technique for further probing this student thinking. I am hoping to write more in the coming weeks about some of these tools and perspectives that I hope to come back to over the course of the year, but this, for me, is a concrete goal that I can return to over the course of the year, and make my teaching a little bit better.
This seems simple, but I ignore this information in small ways every day. I don’t look at student work systematically and naturally get a biased view. I don’t give enough tasks that truly elicit this thinking from students in new ways. I give tasks and don’t take the time to look at student work. I call on kids with their hands raised who are more likely to know the answer. I take what a student says and “round up” — deluding myself that they understand when they actually don’t. And I forget that students forget — if they can do it today, that does not mean they will be able to do it tomorrow, or next week, or next month.
Next Steps
To hold myself accountable, I’m setting a reminder for September to give my students something rich and worthwhile that will elicit their thinking, and reflect and write about how it went and what I learned. I hope to make this a habit, but I have so many ideas this summer that I need to focus my attention on some specific, concrete actions I can take that will start to build these skills.
Five Twelve Thirteen 
To find out if they know what they are doing, or have just done, you have to ask questions. The common student approach tp solving problems is to look at the problem description, guess which of the most recently “learned” techniques is appropriate, apply it, and get an answer, then wait to be told whether it is correct or not. This does not require much more than memory (to use the chosen technique correctly) and luck (to have picked the correct strategy).
So “Can you explain how you used the method/strategy you chose?” is not going to reveal ANYTHING about understanding.
Questions need to be such as
“How did you choose your method?”
“How much faith do you have in your choice?”
“Is your answer an answer to the original problem?”
“How do you know if it is right?” and “If you don’t know, what good is it?”
“You have written an equation. What does it say?”
There are probably more like this.
Thanks Howard — I think you’re absolutely right about the questions, but they also get a lot more mileage when the task is something that requires high cognitive demand. One of my big goals for this year is to push my students more in that direction, in addition to what you’re talking about.
This was a fantastic post to read, even for someone who didn’t know what PCMI stood for till a few days ago. I think you are very insightful about how brave we have to be to really question for understanding, because it can be so crushing to find out what they DON’T understand.
One of the first things I remember reading through/on the MTBoS was this: http://robertkaplinsky.com/why-does-depth-of-knowledge-matter/ . It made me realize that reaching the point where students understand math deeply is hard (what Barbie SHOULD have said?), and I should quit beating myself up when they don’t and instead look for support and ideas to solve the problem instead. Sort of like how I hope my students will learn to deal with difficulties themselves, in fact… The MTBoS is a supportive and nonjudgmental space to analyze the problems and figure out what to do next.
I really like your point here that the reason students don’t understand when we think they do is less due to shortcomings of teachers, and more due to the difficulty of building real, broadly applicable understanding. I’d love to dig more into that.
Hi Dylan, You may not always know who has been reading your posts. But I just wanted you to know that I’ve read all of your posts from PCMI and I have found each one really thoughful. We are as teachers perpetual learners. It’s a good thing when we learn things that surprise us, that make us stop in our tracks, and that inspire us to try new things. Thanks for sharing. Elham
Thanks Elham! Glad you enjoyed it.
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This post was a good reminder. Your posts about reflection on practice and formative assessment totally hit the weaknesses I saw in my first year of teaching. You said, “as a human being who wants to be a good teacher, I will always be biased around what my students are actually learning, and I need to work constantly to see, from a new perspective, what is happening in my students brains, and what math they can and cannot do.” This quote illustrates the kind of thinking we all need to strive for and I love the confession of personal bias that so many teachers overlook. I’m hoping to rethink and rework my approaches to formative assessment this year and I hope to read more of your ideas. P.S. I wish I was at TMC.