I got to make my first trip to Asilomar this weekend. The first of many, I hope. I found CMC-N to be just as warm and retreat-like as people have described. It’s a special conference in a beautiful place with great people. What more can you ask for?
I launched a new talk yesterday, and I get a kick of seeing which ideas or images or phrases resonate or cause reactions, and which fall flat. There are always a few surprises, which is part of the fun of this work.
The talk was about mathematical intuition, and it’s a big sprawly topic I love a whole lot. I wrote a long chapter about it and excerpted a few sections of it for this talk. Here are the slides and handouts part 1 part 2 if you’re interested in the larger arguments and context for this post. I’m going to focus in on one aspect here.
In a little section of the talk, I gave what I hoped would be a gentle but provocative nudge to my colleagues. Based on the response I heard in person yesterday and online, I think I succeeded somewhat in that effort, but I think I need to add a little clarity. This morning, at 3:30AM, as I lay in my hotel room wide awake and jet-lagged, I found myself unable to remember which of the many possible ways I’d planned to say things I’d actually said. (Does that happen to other people besides me? Practice and reality blur together for me in the aftermath, especially the first time I give a talk.) So I thought I’d take to the blog here and lay down what I meant to say in pixels. While I’m at it I’ll take a deeper dive because, well, I’m spending the day stuck on planes on snowy tarmacs and in O’Hare and it’s a good time to make sense through writing. Get comfy.
Over the past several years, I have watched and listened and learned a ton from my colleagues who have been thinking hard about engagement and sense-making and problem-solving. I’m thinking about Phil Daro teaching us all to delay answer-getting. I’m thinking of my friends Max and Annie at the Math Forum, who have taught me so much about slowing down in order to emphasize sense-making when we launch problems. I’m thinking about the idea of low-floor tasks, of removing barriers to access, of the brilliant desmos articulation of making problems easy to start but difficult to finish.
And I’m thinking mostly about Dan Meyer‘s years of work on co-developing the question with students. Nobody has pushed my thinking about this topic more. If you haven’t dug into his work yet, start with these essentials:
Tangent. Bear with me. Last week, I was thinking about Dan’s influence on me and the teachers I support. I met with the 4th/5th grade teachers who had developed a lovely lesson around the open, clear-yet-ambiguous, informal question, “How big is New Hampshire?”
They started with an image like this.
The kids had a great conversation about how big New Hampshire is, and how they might measure it. They dug into the difference between the perimeter and area. They had suggestions for ways to measure the size of New Hampshire using a map that involved string or bendy rulers or graph paper or cubes. Their conversation was gathering steam when one student said, “We could take a string and measure all the way around New Hampshire by putting the string on the line. Then we could take the string off New Hampshire and measure the string with a ruler, and that’s how big New Hampshire’s perimeter is.”
Awesome.
Becky Wright was leading this lesson, and she did a masterful job questioning this student until the student realized that, while she had dome some great thinking, something about her plan didn’t make sense. The length of that string wasn’t big enough to be the length of New Hampshire’s perimeter. I could see light bulbs going off over kids’ heads, one by one, as they realized they needed to know something they don’t yet know about the relationship between the length of the lines on the paper and the distances on earth. A student raised his hand and said something like, “We need one of those maps that has a line that tells you how long the lines really are. So, like, if an inch on the map is 20 miles or something.” Becky revealed the google map she’d queued up for them, and they said, “Yeah! That’s what we need!”
It was a lovely, student-generated lead-in to the huge concepts of scale and proportional reasoning. I stood in Becky’s room, leaning against the wall, thinking about the presentation I saw Dan give in Boston in 2015 in which he talked about the ways technology makes what Becky did possible. Check out this visual that Dan was kind enough to send over. It’s stayed with me.
Because Becky’s maps weren’t constrained by the cost and weight and permanence of textbooks, she could space them out and control when they appeared. Rather than present the entire problem typed up and printed in one big jumble of overstimulating text and images and given information, she could start with the cleanest image and build the question in layers with her kids. She could hold back the map with a scale until the kids had time to figure out what information mattered, and then ask her for it.
Aside from the great mathematical modeling work that was happening here, I was busy noticing how the kids were working. As Dan has argued, starting with informality before formality, sketches before measurements, and so on, has a tangible impact on the way the class participates. The kids started out sleepy, but as the discussion progressed, more and more students joined in, rather than the more typical pattern of losing kids the longer the discussion goes. Engagement went up. Access was great. Students felt ownership. Everyone could contribute. Nobody was boxed out or shut down or denied meaningful mathematics because of an excessively formal problem. All the kids in the class understood the question and had started making sense.
I smiled, ducked out, and walked down the hall to Kitri‘s classroom, where kids were a day or so ahead. They were deep in the work of calculating the area of New Hampshire. A few examples:
So fascinating. Kitri had given students no constraints on which tools they could use. She was looking forward to a discussion about which of these tools made life easier by yielding a decent estimate of the area of New Hampshire without a ton of agonizing calculations. What would the students who had chosen centimeter cubes think when they tried to work with the scale based on inches?
This piece of work stood out to me most of all, though.
There’s so much good thinking here, right? The decomposition, the space, the thinking about arrays, the care not to double-count. There’s tons to build on. But what is happening with those numbers? If we put aside the assumption this student made that the graph paper dimensions matched the scale of the map (they didn’t) and focus on what he did here, what’s going on?
I asked him to tell me about his work. He showed me how he counted up the arrays, and how he was now working on putting together all the fractional bits. I asked him, “What does that 20 inside the rectangle mean?”
He said, “There are 20 squares there because it’s 4 x 5.”
I asked, “But what does that 20 mean in New Hampshire?”
He was stumped. “Twenty inches? Twenty miles? No, twenty meters. I think?”
This is the kind of moment that gives me pause. That led me to make the picture I shared at CMC.
Kitri and Becky both started the kids out in this informal, intuitive, sense-making way. In Kitri’s class, before they got the cubes out, they ballparked the numbers some, so they’d dabbled in magnitude. But somehow, once the cubes and graph paper and numbers came out and this student was in the midst of the problem, he got calculating. He got doing. He got executing. And he stopped asking himself, “Does this make sense?”
What I attempted to argue at CMC is that this powerful work to engage students through intuition at the beginning of a problem is absolutely necessary, but not sufficient. We need to keep going with it, to build off it, and to develop our strategies for teaching students how to stay in touch with their intuitions throughout the problem, even in the midst of calculations. I want students to work a little, then check back in with their guts. To think, “What’s going on here? Does this make sense? How does this jive with where I thought this was going?” I want them to lift themselves out of the details periodically and think, “Where am I?” To jump from street view to birds-eye view and get re-oriented. If they’re where they thought they should be, they can get back to work. If they realize they’re lost, they need to figure out what’s going on, readjust, and recalibrate before diving down into the details again. They need to take time to refine their techniques based on their experiences. These little cycles of do a little work, gut check, reflect, do a little work, gut check, reflect, are what mathematicians do, and what I was representing with the blue dream line.
How to teach students to do this is the kind of question that jazzes me up. I offered a few practical strategies yesterday, and there are more in my book. I am definitely about the how, not just theory, and I hope the classroom teachers and coaches in the crowd found useful ideas yesterday. They’re my first priority, always.
My subterranean agenda yesterday, though, was to suggest to my colleagues who think about and work on these ideas at scale that they increase their focus on supporting intuitive thinking during the problem. I wanted to nudge it toward the fronts of their minds when they design tasks and think about pedagogy and teach pre-service teachers and run professional development, so we can build more ideas and routines and instructional techniques that encourage the synergistic relationship between intuition and logic. For example:
- In a 3-Act task (a brilliant structure from Dan), how do we build on the beautiful, intuitive thinking in Act 1 when we move into the doing of Act 2? What can the teacher do? What’s part of the task design? What’s the pedagogy here?
- When we activate sense-making and delay answer-getting by withholding some information (the question, the numbers, the given information, the details), how do we keep the students making sense even after they have that information?
- If we “dial down the math” at the beginning by spending time in the informal world of sketches and estimates and descriptions, how do we maintain and build on that intuitive thinking even as we dial up the formal and abstract and symbolic?
- If we’re working with beautiful technology that makes it easy to offer iterative feedback to kids, (cough, desmos, cough), how do we use that technology to its fullest potential here? I want my des-friends pondering how they can flex their tech muscles to promote intuition all the way through problems, even after the variables and gridlines and labels come out. Iterative feedback is essential to developing intuition, so they’ve got a big advantage right from the start. Let’s make it count.
I think these questions are worth thinking about. It seems a shame to do all that work to get kids engaged and invested and making sense in act 1 of a problem, but let that intuitive thinking fall off when we start figuring in act 2. I’m in no way criticizing the intuitive work at the beginning of the problem. I’m saying “Yes to all of that, and…”
Dan teased me relentlessly for most of yesterday afternoon and evening, saying my little nudge to him (and others!) was a knife to the ribs. But I also know that he and I agree that this exchange of ideas–this learning from and reacting to and building on each other’s work–is why we have professional guilds. It’s why I crossed the country all day Friday and came back again all day Sunday. It’s why math teachers read and listen and watch and talk and go to sessions. It’s why we travel to conferences and present and blog and tweet and write books and publish articles and teach courses and do podcasts. We listen to other people’s ideas, and we put our own ideas out into the world. We take serious ideas seriously and think about them and react to them. We tell each other when we’re off-base, redirect each other when we’re close, and encourage each other to keep going when we’re on to something good. All this is to say, if you’re not putting your ideas out there yet, you’re missing out on some pretty great feedback.
In our post-session conversation yesterday, Dan asked questions about my talk that I’ve never thought of. His perspective as a designer of tasks and problems and routines is different than my perspective as a coach and author. He pressed and made me think and I appreciated it.
Dan’s work on co-developing the question with students is some of the most provocative and best work I’ve seen. It has led to long and deep thinking in me, and substantial changes in my practice and the practice of my collaborating teachers. That’s the second-highest compliment I can give him.
What’s the highest, you ask?
A nudge to keep going.