We’ve got to do this again (#tmcnyc16)

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Last week, we organized a small conference in NYC about teaching math. It felt different to me than any other conference I’ve attended, and I think that I can say a bit about why.

My first conference was NCTM Philly in 2012. This was my second year teaching. I had no idea what do to with myself. I was bopping from session to session, feeling very lost. That was my first time meeting Christopher Danielson, who told me he mostly chose to attend sessions of people he knew and found interesting. (Oh, cool, you can know people in math education.) He dropped me off at Kathleen Cramer’s session and then I snuck into back of Dan Meyer’s talk.

I went back to Philadelphia in 2013 to attend Twitter Math Camp. I remember that I offered sessions for the first time and they sort of sucked, but I mostly remember the people that I met — too many to name. In fact, I’ve embarrassed myself since 2013 for forgetting people that I met in Philadelphia. (I really wish I remembered meeting Lani.) I remember excitement and a sort of exhaustion that comes from making so many connections over so few days.

My last big conference was in Nashville, last fall. It was a wonderful time. I’m in a much better place professionally than in 2012 — I know who I am and what I’m into a lot more. Now these big conferences don’t scare me so much, and I know what to do with myself…or I thought I did, except that I found myself in a corner of a hotel with my laptop, wishing for something else that I couldn’t quite put my finger on.

When I got home I tried to identify my feelings. I ranted in a Google Doc. (Thoughts About the Future of NCTM Conferences). I wrote then: “I want NCTM conferences to be places where long-lasting professional relationships are formed. I do not want it to be a place whose primary purpose is for people go to sessions.”

Now, though, I’m wondering if all I wanted was a smaller conference. The little conference we just organized did a lot of the things that I was hoping to get out of the big ones. I met a lot of people who I didn’t know well, I didn’t find it overwhelming, and I didn’t feel lost. I could tell you what I learned, and who I learned it from. I met new people, and can tell you all about them.

The big conferences are big. And these big conferences are going to be overwhelming for the same reason that they’re great. Mush a ton of people together and you’re going to have a chance to begin a lot of conversations you’re unable to finish, expose yourself to many ideas and (if you’re lucky) draw some connections between all of these interactions.

But big conferences shouldn’t be all that we can offer to teachers. These big events can feel overwhelming, the focus on attending sessions can work against having nice, lengthy conversations, and for all the beauty of these conferences they can sort of feel like a zoo.

(And, in case I haven’t been clear, an incredibly vibrant zoo that I am eager to attend!)

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The mini-TMC in NYC was entirely different. It was comprehensible. Mid-way through the second day, I noticed that I was calm. That’s not a word I usually associate with these conferences. But I was calm. I knew where I was, who was in the room. We had fewer session offerings — perfect, because I didn’t fret about my choices. Plenty of time in a relatively quiet room to catch up with friends. When I wanted to keep talking about an idea, I didn’t have to use some twitter back channel or find them in Goldcourt 307a or somesuch. Nah, because that person was right there in the room with me.

And then there are logistics. It’s hard to get to a big conference. A lot of us were trying to balance the conference with childcare. Some people could only come for a day or two, others had to leave early or come late. A number of us couldn’t make it to any big conferences because of money or family.

The other amazing thing was that the conference was local. There’s something beautiful about going home at the end of the day. There’s also something beautiful about staying up late into the evening talking teaching. Is professional learning manageable as part of our daily routines, or do we need to break ourselves out of routines to learn? Part of the pleasure of a local conference was that it didn’t feel as if the learning was cataclysmic. It was just learning.

All of this brings to mind a nice line from Stephen King about writing and desks. “It starts with this,” he writes.

“It starts with this: put your desk in the corner, and every time you sit down there to write, remind yourself why it isn’t in the middle of the room. Life isn’t a support system for art. It’s the other way around.”

It felt like our smaller conference was a conference in a corner. Those big things get in the way of our lives, and while they are important they feel to me like a necessary exception to the rule that things go badly when professional commitments dominate our lives.

So, we have to do this again. I’d like to make sure that our NYC meeting happens again in 2017. It wasn’t so hard to put this together, but I understand NYC is weirdly dense with educators. I hope others can put together other small, local conferences to help us restore some variety to professional meetings.

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Michael’s Standards for Mathematical Practice

I’m having a really interesting conversation with Anna and Dylan about whether we should be trying to help students understand the Standards for Mathematical Practice. (It’s on twitter here.)

My position is a work in progress, but I’m trying to stake out a position that favors teaching kids about math, but doesn’t seek to connect these to the Standards for Mathematical Practice. I’ve been yapping on about how I think the concepts (not just the language) of these SMPs depend on k-12 math knowledge.

It occurs to me that it would be helpful for me to make a quick list of things that I think are attainable (and important) to teach students about math. I’m making this list so that each of my practices are related to the SMPs, but I want to be clear: this is not a kidified version of the SMPs. I don’t know if that’s possible. Instead, this is a kiddified list of things k-12 kids should know about the nature of mathematical work.

Big Math Idea 1:  People often make important contributions, even when they fail to solve a problem.

Big Math Idea 2: In math, understanding what a problem is asking is often really hard.

Big Math Idea 3: Giving reasons and explanations is an incredibly important part of what mathematicians do.

Big Math Idea 4: Math is used by lots of different people in a lot of different ways to understand the world.

Big Math Idea 5: Despite what people think, math actually involves a lot of messy choices.

Big Math Idea 6: A lot of math involves coming up with a definition or name for something that’s hard to describe.

Big Math Idea 7:  Despite what people think, math actually involves a lot of creativity.

Big Math Idea 8: There’s an important sense in which math is the study of patterns.

Stray Thoughts About a Constructions Activity

The activity: connect each construction summary to a comic strip showing that construction.

CR - Summarizing Angle Constructions (Images)
Construction Summaries
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Construction Comic #1
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Construction Comic #2
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Construction Comic #3

I made this Connecting Representations activity today. Some things I thought about while making it:

  • Some mathematical objects roll out over time. Procedures or algorithms are like this: first this, then this, finally that. Constructions are like this. Proofs are also like this.
  • To represent something that rolls out over time you can’t use a single static image.
  • The most sturdy representation we have for things that roll out over time is language.
  • At an abstract level, there are two ways to help make a complex thing more understandable. One is to break it down into parts, and the other is to compare the whole thing to some other whole thing it resembles. (Are there more?)
  • If you ask students to connect SOMETHING to a subset of that SOMETHING, their attention will likely focused on that subset in the SOMETHING. In other words, this gives students the experience of focusing on a very specific part of that complex SOMETHING.
  • If you want students to compare SOMETHING to another thing it resembles, you need to compare a representation of SOMETHING with a simpler representation of that whole SOMETHING.
  • While I don’t think the above task is amazing or anything, to the extent it succeeds it’s because you’re comparing a whole representation of a construction (in the comic form) to a simpler representation of that comic (the summaries).

 

The Difficulty of Representing Proofs

How do you represent a proof, if you want students to think about the proof? I’ve been writing activities to support proof writing over the past two weeks, and it’s a question that I’ve found difficult to answer. I want to share where I’ve started and where I’ve ended up in my work.

The first proof-related activity I wrote was Overlapping Triangles.

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Overlapping Triangles
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Separated Triangles

Like all the activities I’ve been writing, Overlapping Triangles is a Connecting Representations task. The short version: match each of the overlapping triangles with a pair of separated triangles. Oh, you have a leftover pair of triangles? Draw the diagram with overlapping triangles that it represents.

I still like the idea behind this activity, but it focuses attention on an itsy-bitsy subcomponent of writing proofs. There’s a type of problem that this helps with, but we have to zoom out a bit to see actual reasoning.

Next, I made Sequences of Transformations.

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Another fine activity, but it seems to draw attention towards the results of the sequence of transformations and not towards the logic of developing that sequence. Maybe a sequence of transformations counts as a proof, but this is an activity about connecting a procedure to what results from that procedure. We still aren’t representing the proof or reasoning itself.

Mobiles and Equations manages to avoid representing the reasoning itself in a similar way. It’s about connecting puzzles with equations that represent solutions of those puzzles.

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Once again, I’ve avoided representing thinking by asking students to connect a scenario with its final state. I’m tiptoeing around the difficulty of representing a proof in two meaningfully different ways.

What’s next? Realizing that I was skirting the issue, I decided to include congruence proofs in the task itself. This line of thinking led to Givens and Diagrams, where I ask students to connect (you guessed it) givens and diagrams.

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Another failure to capture proof itself. I kept trying.

Givens and Proofs gets closer, right?

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At least there are proofs being represented here. But I was starting to realize the problem. If you try to connect a representation of a proof with anything that is not a representation of a proof, you end up removing the proof from the task. Despite all the arrows and flowchartiness of Givens and Proofs, it really comes down to figuring out which given connects to the line below it. You could excuse yourself from thinking about the logic of the entire proof entirely and simply focus on what each set of givens entails.

At this point (yesterday) I thought, OK, so what if I just showed a flowchart proof and that same proof represented in some other way. The issue is that it then becomes possible to make the connection using superficial features like “does this proof end with congruent triangles” or “is M given in the midpoint in both this flowchart and this paragraph proof” or whatever.

Then I thought, OK, so what if I split a two-column proof in half? And what if I made it three proofs that all use the same diagram? That led to Connecting Statements with Reasons. 

One thing I realized then was that if you’re representing even half of a whole proof, things get very wordy and overwhelming quite quickly. To avoid that, I would be careful to start by showing students just one of these representations at a time. I’d encourage you to figure out as much as you can about these sets of reasons before attempting to do any other thinking.

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After you’ve studied the reasons fairly thoroughly, you might then study these sequences of statements in a similar way.CR - Connecting Statements and Reasons (Images)

Finally, you might try to connect each sequence of statements to a sequence of reasons.

And, after all this, have we finally gotten to thinking about proofs themselves? I think we have — both the sequence of statements and the sequence of reasons are representations of a line of argument — but the task doesn’t feel great because it’s just so wordy.

At the end of today, I made another activity, this time playing with representing a proof in a schematic summary. Here is Connecting Summaries to Proofs:

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What I like about this last one is that the proof summaries draw attention to the structure of the argument, and the student is asked to chunk the flowcharts into that helpful structure. I think this is the most promising activity I have for actually representing the argument itself in two meaningfully different ways, and it manages to use fewer symbols than the Statements/Reasons attempt.

I have a last day of activity design tomorrow before I have to walk away from this project for a bit. How do you represent a proof so that you draw attention to the proof? The best answer I have so far is to ask students to connect a representation of a proof to a simplified representation of the whole proof. The key challenge is to make sure that this simplification is actually still a proof as opposed to a component of the argument (like givens or a diagram).

Teachers and Activists

Ta-Nehisi Coates identifies as a writer, but not as an activist:

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This seems entirely sensible to me. Being a writer does not make you an activist, even though Coates’ writing is obviously politically relevant.

And (I’ll add) it’s good that to have both writers and activists. They answer to different calls and do different work. I wouldn’t want every writer to be an activist. I wouldn’t want every activist to be a writer.

Should every teacher be an activist? (Jose didn’t ask this question, but he got me thinking.)

We live in a world that finds it useful to use mathematical achievement as a loose guideline for how much money you should make. So, yes, there is something inherently politically relevant about teaching math. If you teach well, you have the chance to slip a person through the social machine.

Kids form their identities in our classes. School is part of the government. Every experience that a child has in school either supports or contradicts the hypothesis that their country has their best interests at heart. So there are political stakes to teaching math.

Does that mean that every teacher should be an activist? (Could be an activist? Is an activist?)

Perhaps this is a matter of semantics. What makes someone an activist? I take it the term refers to those who actively agitate for political outcomes. And perhaps that term can include many different forms of agitation, and maybe that can include the act of teaching math itself. Maybe teaching math in a certain way is a form of activism.

Personally, though, I think this runs the risk of mishandling the political energy of math educators. Teaching well is something teachers are already trying to do. What do we gain by seeing this as political activity?

I would rather see a more limited and ambitious use of the term. To be a teacher and an activist is to be a teacher who organizes or campaigns towards a political goal. And more teachers should be activists: they should form groups that push teacher organizations, districts and schools to adopt better policies.

Activism is important. It takes skill, not everyone can do it, and lots of people should. At least, that’s the activism I’m interested in.

 

What I’m (Probably) Teaching Next Year

I got my (tentative) teaching assignment for next year:

  • 9th Grade Geometry (2 periods)
  • 8th Grade Algebra
  • 3rd Grade Math
  • 4th Grade Math

Which is more-or-less what I taught this year. This past year was a pretty happy year for teaching, and I’m not yet at the part of the summer where I think about what I want to do differently. (In short: I need to step up my 8th Grade game, get more organized about 3rd and 4th.) There’s time for all of this later.

Besides, I start teaching at math camp in a week, and I’m hard at work at my summer project. And — have you heard? — we’ve got a conference we’re planning!

That’s what I’m up to this summer and next year. What’s going on with you?