Prepared to Do Something About It

The seed of social reconstructionism in education that Counts had planted in the mid- and late 1920s flowered. By the early 1930s, his sentiments were being echoed by educational leaders throughout the country. Even some of those who had championed the child-centered movement, like Kilpatrick, were drawn wholeheartedly into the new orbit. Somehow the long unemployment lines and the soup kitchens dampened the spirit of optimism that had earlier prevailed, not only about the future of capitalism, but with respect to romantic ideas about the natural development of the child in the school setting. Both social efficiency–fitting the individual into the right niche in the existing social order–and developmentalism, with its emphasis on freedom and individuality for children and adolescents, gave ground to the feeling that the schools had to address ongoing social and economic problems by raising up a new generation critically attuned to the defects of the social system and prepared to do something about it.

Kliebard, The Struggle for the American Curriculum p. 157

Open Curricular Frameworks

There is a lot of talk about open curricular materials, but I hear significantly less talk about open curricular frameworks.

Consider this free-but-sadly-not-open 7th Grade curriculum from the charter network Match. I am sure that it’s going to be helpful for some people, and I expect that it will be helpful for me in my 8th Grade class at certain points of the year. But what would be REALLY helpful is if I could just delete all the content and links from that curriculum and use it to share curricular resources with the other high school geometry teachers at my school.

Or consider Julie Wright’s Math and Social Justice site. It’s great, but there are limits to the way you can organize curricular materials on Google Sites — links pile up, it’s hard to tag things by grade, and there’s the danger that with further accumulation of resources things get harder and harder to find.

Imagine if there was something like the Mathalicious framework that Julie could use to organize all those resources. That would make it much easier to collectively develop open curricular resources online, even without any of the baked-in collaborative technology that others have imagined.

A dream I have: that just as there is a team of organizers for the Global Math Department and Twitter Math Camp there could be a team of benevolent web programmers who work on projects of communal importance. They would work on developing the web with the needs of math (fine and other) teachers in mind.

I lack any of the programming capacity to manage such a group, but I did just guzzle a cup of coffee and in my caffeine high let me say that if you’re interested in being part of such a team drop a comment or let me know in some other way.

On Amateurism

A writer is, mostly, a professional amateur. – Ta-Nehisi Coates

Public amateurs can have exceptional social value, not least because they dare to question experts who want to remain unquestioned simply by virtue of accredited expertise; public amateurs don’t take “Trust me, I know what I’m doing” as an adequate self-justification. But perhaps the greatest contribution public amateurs make to society arises from their insistence — it’s a kind of compulsion for them — on putting together ideas and experiences that the atomizing, specializing forces of our culture try to keep in neatly demarcated compartments. – Alan Jacobs

The point is not to replace specialists, but to open the hermetic quarters of specialized knowledge to public forms of interrogation. So it is almost an anarchist position: people should be entitled to learn what they need to learn and to contribute to the decisions that affect them. It’s a question of cognitive sovereignty. These positions and methods: amateur research, self experimentation, collective experimentation, unregulated discourse, exposure of interest and transparency, collaboration with non-conforming scientists or experts, are meant to invade specialties with questions of value, questions that most specialization is designed to eliminate. Since art is a realm where values are debated, one of the points of the artist as public amateur is to perform learning publicly to bring the question of value to the production of knowledge. – Claire Pentecost

It doesn’t take long for a conversation between teachers to include something sarcastic about the fad du jour. By being sarcastic, we put up an umbrella to try protect our sanity from the ideas raining on us from administrators, academics, and yes, even colleagues. I will go further, and boldly say to the proponents of the current pedagogical panacea: I’m sorry, but whatever “evidence-based” product you’re selling today, I’m not buying. The research it is based on is flawed. The anecdotes that support it only apply to specific circumstances which are not easy to replicate. In short, as I have written before:nothing works. – Henri Picciotto

Looking for ways to expand your career while staying in the classroom is a real trick. What I’m wondering is if part of the mismatch has to do with a tension between amateurism and professionalism. A teacher is a professional amateur and every other role in education calls for expertise. Teacher-researcher, teacher-speaker, teacher-leader, all of these are amateur-expert pairings, and maybe for that reason they make a mismatch.

But if a writer is a professional amateur, and if that’s what teachers are too, well?

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


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 attended 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 the 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. (Apparently I had a conversation with Lani?) I remember excitement and a sort of exhaustion that comes from making so many connections over so few days.

My most recent 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!)


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 ask someone a question, I didn’t have to use some twitter backchannel 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. 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 in 2017. I understand NYC is weirdly dense with educators, but I hope others can also put together other small, local conferences to help us restore some variety to professional meetings.


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

ConRep - Overlapping Triangles
Overlapping Triangles
ConRep - Overlapping Triangles (1)
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.

Cpaq9hZXEAAPFCECpaq9haWcAAuB6a (1)

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.

CR - Systems of Mobiles and Equations (Images)


CR - Systems of Mobiles and Equations (Images) (1)

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.

Screenshot 2016-08-11 at 8.28.35 PMCpgttviWYAA_Mk0Screenshot 2016-08-11 at 8.28.57 PM

Another failure to capture proof itself. I kept trying.

Givens and Proofs gets closer, right?


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.


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:

CR - Connecting Flowcharts with ExplanationsCpmlx4OWgAEoknxCpmlzGkWgAANakC

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).