I’m still absorbing everything from PCMI, but right now I want to capture some nuggets from our conversation with Dylan Wiliam yesterday that make a great deal of sense to me, but I am not putting into practice as effectively as I could. Also, I love this picture of him:

I want to post it above my desk so he can silently remind me to get a little better every day. Is that weird?

**What matters about feedback is what they do with it**

If we put a grade on an assignment, students tend to look at the grade, and then look at the grade of the student next to them. Ok, sure, there are some students who are able to take feedback in addition to a grade, but it’s mostly strong students, and we are diminishing the value of our feedback if students don’t have a concrete next step to move their learning forward with more mathematical thinking.

**Memory is the residue of thought**

This is a quote from Daniel Willingham, and I think it’s a great perspective to look back on a lesson with. While students were doing math, what were they thinking about? Were they copying a procedure? Were they thinking about structure? Were they connecting similar mathematical ideas? Were they flirting with the kid next to them? If I want students to remember something, they need to think about it. And then, after some time passes, they need to think about it again. And again. And again.

**Differentiate based on outcomes**

I’ve been uncomfortable with differentiation that involves different tasks for different students for awhile, and Dylan phrased it in a really great way that also gave me a concrete method to improve my differentiation. Instead of differentiating based on the task I give students, I differentiate based on the outcome. All students try the same task — and I have extensions ready for students to push their thinking further if they are successful. This sounds so simple, but I often don’t put the effort in to make it happen, and to plan these questions in advance so they are worth students’ time.

**Productive struggle**

Dylan said “anything students struggle and persist on is probably good, anything students struggle and give up on is probably bad”. It’s easy for me to check the “productive struggle” box by just giving kids hard problems and asking them to persevere. But what matters is the outcome — if students struggle and succeed, they get a positive feedback loop. If they struggle and fail, negative feelings about math can be reinforced. It’s important for kids to fail sometimes — but if it’s the same kids failing every time, it’s going to be a problem.

**Dangers of simplicity**

Dylan cited a study about two groups reading the same article. One group got a blurred and smudged copy of the article, and the other got a clear copy. On a test of the material after a week, the group that read the smudged copy retained significantly more. This is called a “desirable difficulty” — when things are hard, we expend more cognitive effort, and we retain more (as long as we persevere). I don’t think this is an argument for giving my students smudged handouts. It is an argument against oversimplification — I’ve often heard “I don’t want to show them that method because they’ll just get confused”. This gets at something I wrote about recently that I called the pyramid of abstraction — that students build abstract ideas from looking at connections between a wide variety of examples, rather than simply jumping from concrete to abstract. It’s easy to shy away from two things there — it’s hard giving students a wide variety of problem types and examples, rather than one “perfect” example (that happens to be the only way I ever ask a question on that topic), as well as taking the time and putting in the effort to help students make connections between ideas. This is the kind of knowledge students will be able to apply in the future, and oversimplifying content will only hold them back. This isn’t an argument against clarity — each task or example should be clear so that students have an opportunity to understand. The hard, smudgy, messy work is making those connections, making sense of big mathematical ideas, and applying them in new ways in the future.

**There are no easy answers**

A final point he made is that there are many questions to which there are not easy answers. Teaching relies on experience, judgment, and continuous learning. He had no problem telling us that he didn’t know the answer to certain questions, or that they depended too largely on the specific students’ situation. I feel like I instinctively look for easy solutions to many of these problems — if I just explain integers this way, students will get it. A certain approach to conics will help students graph perfectly every time. But there are no magic bullets, just a new challenge every day, and always more to learn.

Dylan Wiliam is an incredibly articulate thinker about teaching, and hearing him speak was an absolute treat. His conversation with PCMI wasn’t recorded, but to hear him do his thing there’s a great two-part series on BBC watching him implement many of the practices he talks about in actual classrooms. It’s definitely worth checking out if you have a bit of time. And of course, his book Embedded Formative Assessment is solid gold.