[This is my contribution to The Virtual Conference of Mathematical Flavors. To see what that is all about, and read other great blogposts that take a zoomed out view of their classrooms, click on that link! The fundamental question that the conference has people musing on is: How does your class move the needle on what your kids think about the doing of math, or what counts as math, or what math feels like, or who can do math?]
When introducing this virtual conference of mathematical flavors at TMC18, I went through a little rant about how much was expected of us, and how much we expect of ourselves. I went on a mini-rant about this:
Here’s the thing. We all are told to do a thousand things in our classrooms.
Build risk-takers. Have kids work collaboratively. Provide rich tasks. Utilize Vertical Non-Permanent Surfaces. High five kids each day. Do random grouping. Call on students with the Popsicle Sticks of Destiny. Instill grit in our kids. Develop growth mindsets. Provide formative feedback and alter our lessons based on it. Do Standards Based Grading. Engage kids with project-based learning. Obviously we have to do lagging homework… or… is it no homework? Be sure to interleave concepts so you can make it stick, amiright? And make sure kids have facility with technology.
But the thing is: we don’t all do all of that. It would be CRAZY to do that.
We pick and choose what makes sense to us based on where we are, who our kids are, what we decide our goals are for our kids.
And by doing that, we are actually shaping what kids think mathematics is all about.
I came up with this conference because I was genuinely curious. Every day I see all the little pieces, the beautiful flotsam and jetsam that people post on their blogs and on twitter about things they do in their classrooms. But each person has a different flavor, and each person’s classroom has some sort of impact on their kids and how they think about mathematics. I wanted to see how people thought about the work they were doing on a broader scale.
And I was relieved, because I have no idea how I’d answer that question myself. And if I were hosting the conference, it would be gauche for me to also contribute to it, right? I’m the organizer! But then stupidhead Rebecka Peterson — who clearly forgot that I asked her to keynote and now she was being super mean — tweeted:
Followed quickly by another keynote speaker and consummate jerk:
And so, bullied, I felt compelled to write a contribution to the virtual conference. But the hard thing for me is that I don’t want to profess my classroom moves the needle on how kids think about mathematics when I’m not sure it does. I know what I intend to do, but with so many intangibles, how do you know if it actually worked? So I started thinking if I had any evidence that could help me figure this out and I realized I have the perfect thing. It’s totally biased, but for some subset of kids, it can help me answer how I move the needle. You see, I historically have taught a lot of juniors, so I’ve written a lot of college recommendations. And when I agree to write one, kids have to fill out a questionnaire. Those reflections often tell me how our class has had an impact on the kid. Now granted, these are reflections from kids who asked me to write a college recommendation — so a total skewed sample. But I’ll take it. Because even though I don’t know how our class affected a lot of kids, I can say with some confidence I know how it affected some kids because they chose what to write about. I’m going to find some old reflections and see if I can’t maybe come up with a thesis after seeing them. [In case the gallery isn’t showing the student quotations, I’ve included them at the very bottom of this post, so scroll down!]
I’m so glad I went through those (and others). It helped me see there are three main things that I think our time together has an impact on. Again, not all kids… I know I’m not one of those life-changing teachers that entire generations of kids will remember… but for some kids, our time together has changed them in some way.
- Our class helps kids recognize the importance of sense-making and drawing connections as the heart of what mathematics is all about. And in that, there is excitement and beauty.
- Our class often provides a shock for kids, where they are initially unsuccessful and have to undergo some sort of transformation that pushes them, that I help them along, so they can see they are more capable than they thought. (In general, I try to push kids to do something juuuust a tiny bit past what they think they’re able to do.) Often times, this aspect of the class leads to kids telling me how they learned to believe in themselves and have confidence in their own abilities in a way they hadn’t before.
- Our class helps kids recognize the value that others students have in their own learning process, and that working collaboratively can be much so much more fun and much more generative than working independently. With the combination of all our knowledge, we can do so much more than we could do alone.
So here’s the tough part of the question — but also the whole point of the conference. How is my class designed so it can move the needle in these ways? What do I do to make this a reality?
But I guess if someone was going to get close to the answer, I’d probably have a better shot at it than you, since you know, it’s my classroom and all.
- I write my own curriculum. I do this because I want my kids to be mathematicians (as much as I can do in a school setting with a list of topics I need to cover). I don’t know how to do this with a textbook. I write curriculum because I don’t know how else to get across the insane interconnectedness of everything that mathematics is, that I want them to slowly build throughout the year a large woven superstructure and at the end of the year recognize “Whoa, yeah, we did that. My teacher might have been a guide, but we did all the heavy lifting.”
In general, I start with them playing around with a problem or an idea, and then they start to codify and see a structure, then they try to articulate in words (and diagrams! and with examples!) why their approach/algorithm/solution works, and then I tend to ask questions that extend their understanding or force them to relate what they’ve done to something they’ve done before.
And central to this superstructure is one question: why. For many of them, our course is the first time they have to do that regularly and deeply. (I’ve come to realize how incredibly challenging and anxiety-producing this can be for kids who have never done this before.) And for some of them, this real understanding opens up for them a sense of beauty and interconnectedness that they didn’t quite see before.
- I am a “warm demander.” I only heard this term a year or two ago, I think from Sara Van Der Werf, but I think it suits me better than any other lingo I’ve been exposed to.
One quotation: Warm demanders “expect a great deal of their students, convince them of their own brilliance, and help them to reach their potential in a disciplined and structured environment.”
A second quotation: “Warm demanders approach their students with unconditional positive regard, knowing students and their cultures well, and insisting that students perform to a high standard. Students have told researchers that they want teachers who communicate that they are ‘important enough to be pushed, disciplined, taught, and respected.'”
- I have kids work in groups almost every single day. I try to build a classroom environment which is both relaxed and focused. I let things just unfold and don’t try to put time pressure on kids — each class is different and will get through things at different paces. What this looks like, if you walked in, would be groups of 3 (and maybe one of 4) sitting at tables working on a packet. When the packet is pushing them, kids will be talking a lot and there might be some frustration that comes before excitement when someone has a breakthrough. Sometimes when they are working on more formulaic practice (seeing if they can apply what they just learned to more problems, which may change slightly in difficulty but not much), there tends to be less noise and chaos. Music often will be streaming softly from my computer (from a playlist kids have created). I will be walking around checking in on groups, asking some probing questions for groups who seem to know what they’re doing, giving hints to groups which are genuinely stuck. And occasionally, I’ll bring us all together to talk as a whole class about places people were getting stuck and to help tie some things together.
To be successful, with a few exceptions, kids need each other in my class. And I try to make clear that most classtime will be groups working towards collective understanding. That means you’re thinking not only of yourself but your group members. What are their needs? And they’re thinking of you, and what you might need. When kids are at home, they can put their focus on their individual understanding.
But I don’t just expect this to happen. I have kids talk about what they need in a group, and what their strengths/weaknesses in groupwork are. They often make a list of group norms. Sometimes we do a bonding activity. I have a way for groups to check-in with each other and have honest conversations about how things are going. At the beginning of the year, I try to give explicit feedback to positive things I see people doing in groups (“I really love how you asked X if he felt like she understood…” or “I love how you all are leaning in and listening to each other” or “I totally see that you’re getting frustrated about not quite getting it, but you haven’t decided to ask for help yet and you’ve kept on trying different things… that’s awesome.”) So kids can really learn how to work together, I keep groups together for four to six weeks. At least for me, I’ve found this gives my kids enough time to develop real relationships and get into a rhythm.
Most importantly, though, working in groups (especially ones that feel safe) lowers the cost of entry into mathematics. Kids aren’t talking aloud to the whole class and feeling totally self-conscious. They are working with two or three other kids so it’s less scary to say “I’m confused.” And the flip side is true. When working and there’s a moment of celebration and kids might think (gasp!) math is cool, it’s more okay to share that excitement with those who have been on the journey with them. It also is just more fun when math can be a social activity, where emotions can be shared. Frustrations, elations, and everything in between. And since the bar is always set a bit higher than they think, these emotions are important.
Now it doesn’t always work. My classroom doesn’t always work. This isn’t a recipe that I can follow mindlessly and transform what every kid think about math. It’s just the best I have right now. I remember some years ago I was teaching two sections of the same class where I was using the same curriculum, I was being a warm demander, and I had kids work in groups. One class was filled with joyful mathematical noise and the other was a slight and occasional murmur. Night and day, though everything else was the same except the kids. I truly struggled that year. And that’s the thing. My mathematical flavor — the thing I do to move the needle and change students’ perceptions about math and who could do it — was delicious for one class but just meh for another. Which is why I keep on iterating and changing and trying new things. Because no two classes will be the same.
In case the little gallery isn’t working above, here are the student quotations: