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.

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.

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

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).

 

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.

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.

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.

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.

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.

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:

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).

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?

Email is a wonderful thing for people whose role in life is to be on top of things. But not for me; my role is to be on the bottom of things.

From Knuth. Between twitter and email, being part of an online community (or at least the way I’ve been doing it) can end up feeling rather exhausting.

Chris (whose writing I love) has an idea:

I’m about to make a suggestion that’s wrong in specific and important ways, but probably right in general. We need GitHub for math curriculum.

GitHub is a tool for software developers. I could be wrong (I am probably wrong) but it’s not a tool that is primarily used for collaboration in a non-online sense. My impression (I am probably wrong) is that GitHub is essentially a more open distribution channel. You can copy the code and make your own version, which is forever associated with the original. This makes new versions a lot like suggestions, and the original might fold your changes into their versions.

I guess this is a sort of collaboration, but my point is that it’s not “hey Chris we’re writing this bit of code together so let’s go to GitHub and work this out” collaboration. It’s more like Wikipedia, sort of, where anonymous people in the crowd are making their small contributions to the effort.

Chris calls this as collaboration, fine I admit it he’s right:

Most crucially (so I’ll break it out of the bulleted list), collaboration is built into the DNA. I said “you” a lot in that list, but only because English lacks a distinct plural second-person pronoun. “You” could be your algebra team, or your math department, or your district curriculum folks, or…you. The whole point of community curriculum repositories would be for a group of education professionals using their collective expertise to take ownership of curriculum in a sustainable way.

I am totally agnostic about whether this is or isn’t helpful. Actually, I bet this would be very helpful to some people. But something happened when I was reading this, and maybe this sometimes happens to you. My first time through, I was nodding vigorously because I just assumed that Chris was describing the exact thing that I want and need. And because Chris is his own human person he was not. I only realized this later.

So, here is the thing that I want and need. (I think I want it. Be careful what you wish for is the law for new tools.)

I want a very simple web page that quickly captures a plan for a sequence of lessons. This might be a unit, or it just might be 5 lessons or whatever. I want a link that I can just share that would say “Developing Understanding of Fraction Addition” and have quick descriptions/images/links to the four activities I’m planning on using next week.

I want a thing that quickly makes these very simple web pages.

I want this because I don’t know a good way to share my units or unit planning on twitter or my blog. I want an object that I can refer to during discussion and collaborative planning. (The same way collaborative planning sometimes happens under the #CthenC hashtag on twitter. An example.)

And then I want to be able to make my own copies of other people’s units and modify them for my own personal record keeping and eventual sharing.

This is not a technological request that would massively change the world or solve any of the problems that Chris is describing. It’s just a modest need that I regularly feel in my online math teaching life. Maybe it’s insanely tricky to design this thing, maybe everybody else in the world is perfectly happy with Google Docs, or maybe nobody particularly wants to share and talk about other people’s units. I don’t know. But this is something that I think I want.

Gary Davies wrote a piece titled “Why Don’t Teachers Engage With Research?” I read it, and I found it interesting, though I disagreed with much of it.

Joe asked me to say more. Here’s the more, Joe.

There is a lot that Gary says in this piece that I agree with. He notes that research in education is not written by or for teachers. That is true and important. Can you imagine a world where classroom teachers collaborated with researchers on joint publications? Forgetting for a moment everything else that would have to change (including the sort of research being done) that would just be a great thing for the profession(s).

I also agree with Gary’s call for open access to studies. This seems like something that just has to change.

Gary is a physicist who is moving towards the classroom. I think this is fantastic. My understanding is that he’s not in a k-12 classroom yet. Gary’s premise is that his expertise on physics research gives him perspective on educational research. I would suggest, though, that in some ways this might lead him astray in this piece. Gary points out convincingly that research in education does not resemble research in physics. He concludes that this is a problem, that education should be more like physics.

I think there are good reasons for why research in education doesn’t look like research in physics. Consider Gary’s first point:

Go and look at any field that is heavily evidence based and relies on research and you will find something striking: the people who do the research are the people who use the research.

Now, I just don’t think this is true. If it were true, though, I don’t see why we would saddle teachers with the responsibility to perform research. There is a striking difference between teaching and physicisting: teaching requires a great deal of relational expertise. You need to know how to navigate these kids in this school with this material. It’s surprisingly difficult to share this knowledge. In physicisting, the work is already abstract and ready to be shared.

A more apt analogy would be if we imagined that physicists were expected not just to publish research on physics but to publish on the act of having a successful science career. What if we wanted every physicist to be doing excellent science? What are the practices that lead to successful science? What sort of graduate education eventually yields the best scientific work?

These are questions that are more like what we deal with in education. Being good at grappling with these questions is quite different from doing good science in your area of expertise!

As Gary goes on, he doubles down on the idea that research on teaching ought to resemble research on physics.

What’s the average length of a research paper in education? I don’t know the answer but I have read many that are 30 pages or more. I’m not sure I’ve ever seen one with fewer than 5 pages. In physics, we typically publish papers 4-10 pages long.

But why should we expect papers to be short? “A paper is an announcement of a discovery or new piece of knowledge. The goal is to explain the significance of your discovery and back it up with evidence as quickly as possible.”

OK, well the obvious answer would be that in research on education it is significantly more difficult to do these things. Teaching is supremely messy, and it has only been an object of research for about one hundred years. The problems with educational research are numerous. To start, different people have different understandings of the outcomes of education. (That’s true even if you limit yourself to the research that assumes we can measure these outcomes by giving kids a test.)

What counts as evidence in teaching? This is also more complex. Different schools exist. There is a need to carefully explain their assumptions and techniques, and this takes longer to do.

In short, your typical education research paper is more complicated than a …

I hope nobody wants to tell me that a typical education research paper is more complicated or sophisticated than a typical physics paper. So if physicists can do it, why can’t education researchers?

Well ok then.

Education research papers tend to also be very boring and full of jargon.

Says you!

Jargon serves only as a barrier-to-entry for those who are not on the “inside” of your little crowd. Many fields do this on purpose (some parts of philosophy, for example) so that only the indoctrinated read the papers, lest their entire field be found out as a con.

Umm well this is getting pretty conjectural for my tastes. The possibility that it’s just harder to express true things in education vs. physics doesn’t seem to be considered.

I really wonder what Gary would think about psychological research in general. Those studies are likewise long, likewise “jargon”-ized. For good reason. It’s hard to talk about human beings. It takes time and care to say the thing that you want to say. And an important part of the work of psychology and educational research is identifying new concepts (jargon) that might provide explanatory power.

Though I have an easy time agreeing with this:

They tend to be boring partly because of their length and partly because no effort has been made into crafting a succinct, well-written, memorable message for the paper. It’s not uncommon for me to get 10 pages into an education research paper and think: Why have they done this? What’s the point in this paper? What are they trying to tell me?

It seems crazy to me that we’ve created a research system that prioritizes the creation of knowledge over its dissemination. We aren’t trying to figure all these things out to put ’em in a jar!

Much of the most interesting and informative educational research is highly quantitative.

And much of the rest of it is qualitative! You need both. Everyone knows this. Different methods are appropriate for different research needs. Psychologists know this. Education researchers know this. It’s like saying consonants are the most informative letters.

How can we fix it? This is the really depressing part, because I can’t really see how things are going to change.

I like Jack Schneider’s ideas. I like Researcher-Practitioner Partnerships. I hold out some hope that online communities can help researchers and teachers come together.

At the end of his piece, Gary suggests that teachers could have a role as writers who review research. I think this is a great idea! I do think that if we can find ways to create a community of teacher-writers that could do a great deal to help us make intellectual progress. How to do that is its own puzzle, though.

Anyway, I’ve said enough. Gary — are you out there? — I’d love to continue the conversation some time. I’m no expert on anything, but the above is my take.

 

Hopefully by putting “Pershan” in the title I’ve scared off anyone from reading on…

…though what a bizarre impulse it is to want to write something publicly that you don’t want anyone to read. On second thought it’s not that I don’t want people to read this, it’s just that I want people to know that I know that this isn’t important stuff. I want to avoid presumption.

Anyway.

I have a blog problem. My first blog was great but then I decided to end it. There were a few reasons for that. Something about my blogger routine constantly drew my attention to my web stats, to the extent that it got distracting. I also wanted to establish that my blog could end, that a blog didn’t have to be a running platform but could be a self-contained project. I thought that my next blog was going to be nice and tight, focused on a single issue. I just had to find the right issue.

Well, that was this blog. I called it “Problem Problems” — are there any good blog titles? — with the thought that I’d focus on problem solving, especially problem solving strategies. One thing led to another, fine, but I still wrote a lot of stuff here about problem solving.

I’m officially calling an end to that experiment. (In a way, this essay is the culmination of the super-focused-blog thought.) The title of this blog has been recast as “Teaching With Problems — seriously, are there? — to emphasize that the problems are with my teaching as much as they are mathematical. And while I hope to avoid sprawling too much, I do want to give myself permission to write about whatever I feel like writing about.

There might be something more here. We go through periods of sprawl and focus. The last few years have felt like a time of doubling down for me. I’ve thought really hard about feedback and about the role of research in my life. I don’t really know where to go from here, though. So…let’s spread out a bit. Sing when the spirit says sing, stomp when the spirit says stomp, etc. (Parenthood, man.)

There really is no road map for a teacher interested in doing stuff that’s professionally relevant. That’s fine. One thing that the last few years has clarified for me is that, primarily, I see myself as a teacher and a writer. That’s what I enjoy the most (more than presenting or writing curriculum) and I think it’s truest to my online activity. That means that, primarily, I should be teaching and writing. That’s as much as I’ve figured out.