How Different Can The Best Teaching Look?

"Is teaching an art or a science?" 

Well, that's a muddled question. For one, "science" and "art" are terribly abstract terms, and it's not clear what we mean by them. So we ought to stop asking this question. Instead, I'd suggest a better formulation of that question is

"Does the best teaching always look the same?"

You can tell that this is getting closer to what we mean to ask since the answer to this question has real consequences. If all teaching looks the same, then we might be able to (1) discover the recipe and (2) prescribe it or at least attempt to (3) convince people to follow the recipe.

What we're talking about here is convergence. We're wondering whether all good teaching converges, whether it all ends up being congruent.

But this question isn't quite right either, because of course all great teaching won't look the same. It's not going to have the same homework collection routines, and sometimes it's going to be in Japanese and sometimes it's going to be in Spanish. It'll almost never be in English har har har har.

We're not actually wondering whether great teaching always looks the same. What we're really wondering is

"How different can the best teaching look?"

Does great teaching always have the same core? Are kids always learning how to multiply via the Whatever Method? Can you learn the same amount in a classroom regardless of whether homework is assigned? Is note-taking always helpful?

I don't know the answer to this question. You don't either. Nobody knows the answer to this question.

But here are a bunch of arguments in support of the answer, "The best teaching can look very different."

• Technique A works great in my classroom. But kids in someone else's classroom find Technique A infantalizing and they aren't willing to give it a shot. It's an interesting theoretical question to ask, "If those kids were more like Michael's, would Technique A be most effective?" but ultimately a meaningless one. You can't abstract the kids out of teaching.
• To make that last argument a bit more explicit: different kids will end up needing different teaching.
• There are no agreed upon goals for a math class. Some people think that the purpose of math class is to prepare kids for a career that uses math. Some people think that the purpose of class is to help kids get through math class and into college. Others want to prepare kids for a life of asking and answering interesting questions. Before we can converge on a definition of the best teaching, we need to converge on a shared purpose for math class. That aint going to happen, so we can't expect any sort of convergence on what best teaching is.
• Let's talk directly here: Can we expect better results if we give kids lots of chances to problem solve in class, rather than use class for lecture? I would say, maybe. Even probably. But I can imagine a uniquely inspiring lecturer, combined with some very hard-working students who are dutiful on their homework. They get help from parents when they need it. They come in with questions, and the whole thing is a productive endeavor. An extremely productive endeavor.
• "But for most teachers, with most kids, lecture won't work." Well, fine. But that's not quite what we're talking about. Instead, that's a pivot to a different question, "What should we be telling teachers to do?"

Like I said, I don't know the answer to this question. But I suspect that things can look very different and still be operating at peak awesomeness for kids. 

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In case you're curious, here's the bit of writing that sparked this post. It's from an interview with George Saunders.

First, let me say that all of the above is true for me – I have no idea that those ideas are more widely applicable. A writer has to figure out what works for him/her and sometimes that bag of tricks is just that: a small bag, full of specially developed tricks that, even as he/she pronounces them (as one is called upon to do when teaching, in interviews, etc) seem crazy or overspecialized or dictatorial. That said...

You can go read the rest here

The post was also inspired by this twitter conversation, so I know that Ilana Horn thinks that the answer is "not so very different."

Naming The World

The way we talk is deeply connected to how we think.

That's the best way I have to sum up what I've learned about teaching over the past few months. It's a simple statement with deep consequences. 

Consequence #1: Every piece of learning worth remembering needs a name. 

It's not important that the name be particularly descriptive. Names aren't usually descriptive. "Michael" doesn't describe my features in any way, but it is a hell of a lot easier to drop "Michael" in conversation than "that red-haired math teacher from NYC."

I almost always ask my students to come up with the names for these things, because it's fun, and that's how we end up with "The Feces Theorem" or "The Friday the 13th Theorem."

Consequence #2: Thinking is having a way of talking. Giving kids new ways to talk helps give them new ways to think.

By the end of our polygons unit we were able to go far beyond talking about this shape as "looking like a rocket". We were talking about pointiness, and the number of points, and which sort of points counted as sharp points, and we were able to talk about acute angles and obtuse angles and all sorts of other things. (Documented here.)

I put in all this work into helping my Trig kids see that the sin(45) is not 0.5 on the Unit Circle. We investigated the Unit Square, and it's "sine" didn't have this property either. This tends to be tough and mind-bending for my students, but the learning is slippery. You know how I can make it more effective, though? By introducing the language of "linear" and "non-linear", and giving them a way to talk about these functions and their properties. 

Having a concept is deeply connecting to having a way to talk about a thing. Not having a way to talk about a thing makes it harder to have a concept.

Consequence #3: You can't learn math from Khan Academy.

Learning from Khan Academy means learning alone, and that means never developing a language to talk about concepts. Which means that you probably aren't really learning them.

(Or do you believe in the possibility of a private language?)

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Credits: 

Trying to implement Christopher Danielson's Orpda and Hexagons lessons sensitized me to the connection between language and concept development.

Naming Infinity is a really interesting book about intersections between mysticism and math that influenced some of my thinking here. Ditto on my philosophy courses in Philosophy of Language, in particular "Two Dogmas of Empiricism," which is arcane but I take the paper to be an attack on the hard distinction between knowledge and language.

The title of this post has been lifted from Bret Anthony Johnston's book.

Come Speak At The Global Math Department, And This Year Let's Get More Diverse

Have you checked out the Global Math Department? You should.

The Global Math Department is fantastic. Every week (Tuesday) a different math teacher leads a short session about something they're fascinated by. Sometimes it's great math, other times it's a teaching tip, and often it's just a tough question, unanswered and maybe unanswerable but tossed around by a bunch of really sharp and curious teachers.

This year, I'm on the planning staff of the Global Math Department. My job is to recruit speakers for our Tuesday sessions, and this post is a Call For Speakers.

With over a year of conferences under our belt, it's becoming clear that we have a diversity problem, i.e. there's a lack of diversity in our speakers, i.e. the vast, vast majority of our speakers have been white. After reading a great piece about how one conference did a better job recruiting people of color (thanks Max!), we're ready to make a concerted effort toward drawing our session leaders from a wider palette.

So...

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[Spread Widely]

Call For Speakers: Global Math Department

We're looking for teachers, consultants, teacher-leaders, writers, and all sorts of people interested in math education to lead sessions at the Global Math Department. Right now most of 2014 is unbooked, and we're looking to find speakers for the new year.

Our Selection Process: This is the second full year of the Global Math Department. We have had sessions run by seasoned, big-name math education speakers, and we've also had sessions hosted by complete novices who have never ran sessions for teachers before. We don't accept every speaker, but when we don't accept proposals its not because of lack of experience. This is a friendly crowd, and if you're excited by something we'll probably want to offer you a session.

We Care About Diversity: Our process for recruiting speakers in the past mostly involved leaning on our friends or people that we knew from the online blogging world. In doing so, we accidentally ended up with primarily white speakers. This is a sort of self-propagating problem, because of the way our past speakers inevitably signal the sort of future speakers we're interested in.

This is trouble for all sorts of reasons. First, because there are people of color who haven't had the chance to have their voices amplified. Second, because our community itself has suffered from a lack of diversity of views and perspectives. Third, because our community has unintentionally mirrored the existing prejudices of the education establishment. This is at odds with our grass-roots effort to empower and connect teachers who are often silenced or ignored in their schools and departments.

With this new year, we're trying to break the cycle. It won't be easy, but we're going to reach beyond our comfort zone to find great speakers.

Nominations, Please!: Do you want to speak at the Global Math Department? Great! Do you have someone who you'd love to hear speak at the Global Math Department? Double-plus good. Let us know, and we'll do the sweet-talking for you and get your nominee into a speaker slot.

So, please nominate others, especially if they're outside of the math twitter/blogging crowd that we (the organizers) tend to run in.

Contact Us: Get in touch with us on twitter (@globalmathdept or @mpershan), or talk to Michael through email (mathmistakes-at-gmail works). You can also drop a comment to this post, and I'll make sure to follow-up.

Why You Need A (Great) Curriculum, Not Just A Bunch Of Lessons

The above is an exercise from my favorite Geometry textbook.

You know when you're going to need to find the area of a 30 degree sector? In Trigonometry, when quickly knowing how to transform a central angle into a proportion of a circle is crucial for quickly moving between degrees and radians.

That's why you need a good curriculum, to start helping these kids get ready for Trigonometry a solid two years before they've even enrolled in the course. That sort of thoughtfulness is so, so hard to achieve while trying to make sure each individual class session is awesome.

Here's another bit of curricular thoughtfulness that amazed me this week.

Do you see it? (Actually, this picture is pretty small so you might literally not be able to see it.)

Question 1: Make an angle that measures 60 degrees.
Question 2: Explain how you know that this is a 60[degree symbol] angle."

This is the sort of little touch that makes this such a lovely 4th Grade curriculum. It's like, forget the progressive, weirdly controversial activities and lessons. Does your curriculum go the extra mile to make sure that kids can actually read mathematical notation?

I didn't even notice this genius move until a little 4th Grade girl asked me what that symbol meant in class.

If you're making your own curriculum -- using all your own worksheets, your own activities and assignments and everything -- you need to take a close look at some of the best work out there in curricular design. Every time I get cocky, I take a look at CME, CMP or TERC and I realize that they're thinking like ten steps ahead of me.

I'm not saying that a great curriculum's activities and tasks are better than mine. Sometimes they're great for my kids, usually I at least have to turn them inside out before using them. But for previewing, reviewing, and a general sensitivity toward common student pitfalls, I think it's going to be a long time before I can do better than these guys.

The Exploding City of Las Vegas, And An App That Helped

That's some pretty remarkable growth, Las Vegas. I asked you guys what questions you had after seeing this image, and a lot of you wanted to know what the population was now, or what it would be like in the future. Those are some great questions, and of course we could go down that route. There are a lot of great population modeling problems, and this could be a great population modeling problem.

Here's another great question: Does the green splotch grow faster, slower, or the same as the population?

We recorded opinions, arguments and predictions. Then we took rough estimates for the actual area of the green stuff in 1973 and 2000.

The problem with finding area is that Greater Vegas Township isn't shaped like a rectangle or anything else we know how to deal with. So we've got a blob, and we need a technique for finding the area of a blob. We talked about it a bit, and they mentioned that grids would be helpful, so I gave them what I had.

There were two different techniques that I remember seeing for approximating the area. Some students wanted to cut out the green splotches and impose it on some graph paper for a better estimate. (Good job, kids: a more fine-grained grid is going to get you closer and closer to the "true" area.) Others went about counting and chopping up the green parts to fill out the squares. (Maybe there were more techniques, but honestly this lesson was a few weeks ago and I've gone rusty on a few of the details.)

In the end, the kids ended up with their approximations. But we wanted to know how good our approximations were. Enter SketchAndCalc, a really solid app. SketchAndCal lets you import pictures and trace out the perimeter of an area, and it'll spit out the area. (A digital planimeter!)

So, I traced Vegas:

Using the info from the app, the area of Greater Vegas grew a bit slower than population. We had a great conversation about why this might be. Do cities tend to grow denser as they grow outward? Why would that be?

Anyway: this was a ton of fun. It's all posted over here, if you want the files.

Improvement, Like So Many Things, Comes Down To What You Enjoy

For purely selfish reasons, I've spent a lot of time worrying about how teachers get better at teaching. That's lead me to write a series of hystericalish posts over the past couple years.

• In July 2012 I argued that we need to find more "drills for teachers", whatever that means.
• In January 2013 I said that taking on tough challenges was the key to continuous teacher improvement. I made a commitment to finding those tough challenges.
• In May 2013 I reported that I'd picked a bad set of challenges, and that I was ditching them for new ones.
• In August 2013 I made the case for writing your own lessons as a high-impact challenge for teachers.

I have a quick thought to add to this story, and it starts with "I was wrong."

I was wrong about how normal people become great at their craft.

I've got a long-term writing project that I'm working on, and it's absolutely terrifying for me to spend time with it. The chances of this project failing are high, and the product will almost certainly suck. It's absolutely crucial that I finish the project, though, because the only way that I'll ever get good enough to do this well is by practicing.

The only way that I can psychologically wrap my head around working on such a difficult, ill-fated project is by finding joy in the process itself. Any investment I have in the result is debilitating, since the result is almost certainly going to be crap. But if I'm enjoying the process then I've got a decent shot of chugging along.

In my thinking about how teachers become great, I've focused on finding routines that will artificially impose challenges and reflection. That makes some sense, since working through challenges is the way that we human folk get better at anything. 

But drills? Artificially imposed challenges? My imagined model for becoming great was the perfectly composed saint, the priest worshiping at the altar of his own self-control and diligence who forces upon himself the routines and habits that lead to greatness. And certainly such priests exist, toiling away and kept to task by some angel or demon that keeps them focused, maybe it's obsession or ambition or maybe it's just desperation.*

* Sorry for the ridiculous sentences. I've been reading a lot of Cormac McCarthy this weekend. He told Oprah that he prefers "short declarative sentences," so apparently Cormac McCarthy is not exactly the biggest fan of Cormac McCarthy.

But normal people can't spend their lives motivated by obsession or ambition or anything else. Normal people don't do things that they hate. They do things that are fun, or interesting, but mostly fun.

The best long-term strategy I can see for continuously getting better is for the process of improving to be fun. If I want to get better at teaching, it's got to be fun for me to do so, because that's the only way for me to stare down the abyss of my current craptitude and the probability of my own immediate failure. That's going to look different for different people, because we've all got different tastes. I enjoy planning lessons, so I spend a lot of time on that. You like giving feedback, so you spend your time on that and you get great at that. I hate it, so I suck at feedback and am decent at curriculum.

I feels a bit silly.  To have spent a year thinking about how to become a great teacher and landed on "Enjoy the process" seems, like, duh, but there it stands.

"Many people enjoy readings cut-and-pasted from reflective emails."

I'm betting that Danielson is wrong on his cut-and-paste theory, but I figure we'll give him a shot.

Earlier this year, I tried to implement Prof. Danielson's hexagon sequence in my Geometry class, and it basically failed. I tried again in my 4th Grade class this past week, and it's been going really well. Part of that is the age difference, but the other part is what I wrote to the kindly Professor about.

I sent him this email tonight. Enjoy, and keep the fantastic comments coming.*

* Kudos to the crew responsible for a great conversation on the last two posts. I'm looking at you, Denise, Chris Painter, mrdardy, Sue, Gregory, Megan, Justin, Christopher, Mike, EdRealist, David and Teresa. You guys rock.

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Hey Professor, this email is too long, but I wanted to share some exciting hexagon moments with you.

Another fun day of hexagons today. After a slow start last week, things have been clicking with the 4th graders over the past few days. In particular, I was surprised by a bunch of coolness that happened today, and I wanted to share it with Prof.

Teresa called her shape "Squashed" because it looked like a squashed hexagon. Cool. Rather than directly ask students to clarify their thinking, I asked them to make little "What's My Rule?" puzzles and had the class try to discover and express the rule. Someone articulated a version of the "squashed" definition, so I turned a squashed shape on its side (so that it was now very narrow), and asked whether it was squashed.

Unanimously, no.

Today I cut out a bunch of shapes (squares, triangles, various hexagons) and got all of the students sitting in a circle on the floor. The idea was to force them to observe the shapes from different perspectives, and to build the notion that our classifications ought to be invariant of viewing perspective.

I put down the shapes in the middle of the circle, one at a time, and asked them to speak up if we already had a name for this shape, or if it was a new one. I accidentally stumbled onto a really great question, because the hexagons that I presented were all similar, but non-identical to, some of your hexagon pieces.*

* I put those to good use as well. Each student got one hexagon piece handed to them, and that was their shape. I was eager to make this more about individual perspective than anything mediated by the group, so I wanted each kid to own a different hexagon.

This, more than anything else, helped drive us to some clarity in our terminology.

I meant to cut a shape that looked a lot like the three-pronged shape from your set, but I accidentally altered it a bit. When I put it in front of the kids, a lot of them started giving it a new name. I asked whether any of our old names applied to it. Someone said it looked like a rocket. Others said it didn't. Then a bunch of stuff clicked for me.

I realized that Lily, who gave us the "rocket" in the first place, owned the shape. I asked her, as the originator of the term, whether this new guy was also a rocket. She said that it also had three points, and then I realized that this was our data. This was the fact, end of the story. So I turned to the rest of the class and said, "Hey, that's really interesting. We weren't sure whether it was a rocket, but Lily said that it counted because it had three points." Then -- thank god for this -- a kid pointed out that my shape had four points. So then this became our puzzle. "Yeah, you're totally right. This does have another point. So how come it looks like a rocket to Lily?"

The big, big reason why this flopped so hard for me earlier was because I was completely unable to find the constraints on the problems involved in this context. If the kids have complete ownership over all these observations and terminology, where's the reality pushing back on their views? What are the problems to solve?

Over the past two sessions I feel as if I've had a breakthrough. The move is to make the particular visions of individual students into problems that the rest of us have to solve. We become students of our various ways of seeing things. The way our friends see the world is non-negotiable, and it's the constraints that push our language to more precision.

Nobody Appreciates Your Lack of Confidence, Even When They Say They Do

I started at a new place this year, and things were generally going really well in the first few weeks. Except for one class, where there was some rockiness. I was trying some weird stuff, and the kids said that they wanted the normal math class experience.*

* When the kids are saying class isn't going well, they're almost always right, but almost always wrong about the reason why. (Credit to Neil Gaiman, who said something like this about readers.)

Anyway, I've got a lot more colleagues and administrators in this new school, and they wanted to know how my first few weeks of classes were going. I told them the truth. I mentioned how it felt like a lot of these kids felt unhappy, and when I just came out of a bum lesson I told people that it was a bum lesson.

The result: a lot of administrators visited my classroom.

Forgive me, but I'm fairly self-deprecating. Lately I've been thinking too much about how this way of carrying myself has affected my career.

When I see conventional success, I tend to find a certain kind of confidence, what my grandmother or rabbi might call chutzpah. Our writers of note are those with answers, so rarely those with questions. There's no room for doubt in a keynote address. Your most popular posts are those with opinions, clearly stated and forcefully held.*

* (OK, here's a strong opinion: David Foster Wallace is overrated partly because of his almost divine levels of chutzpah. He wrote a book about infinity riddled with mathematical errors, and nobody bothered to send it to a mathematician for an edit?)

I'm trying to carry myself with a bit more confidence. When I write, I try to write with a bit more magnitude and direction. (Like a vector!) When people around school ask me how things are going, I don't stutter as much. Things are going well, thanks for asking, though we're all working hard, amiright?

But I'll never really be able to shake who I am, I don't think. I'm always going to be giving myself and my work a hard time. And I've got a feeling that people like me end up in the second row of success, professionally speaking.

What's wrong with being too easy?

A great math activity can't be too hard. That makes sense to me. If I don't think that I'll be able to do the activity, then what's the point of even trying?

A great math activity shouldn't be too easy. If a problem is too easy it holds no interest to us. This is a claim that's all over the comments of Dan's recent post.

But, why not? I mean, we do easy things all the time. Why would a math problem become less interesting if it's too easy.

"Because people have an intrinsic love of challenges."

Really? Is our perception that something would be challenging (but doable) sufficient to get us interested in an activity? Aren't most of our worksheets doable but challenging? And why would some sort of activities be more interesting than others? Do they appear more challenging? More doable? What's the theory?

Drop a comment, and we'll hash this out.

Blogging > Twitter

This is just to say that I spent November tweeting less and blogging more, and it made being on the internet more productive for me than it had been in a long time.

We read poems, short stories and novels differently, and one of the the many reasons why is because of length. What we expect from a piece of writing depends crucially on how long it is, and for good reason. Writing is hard, and we expect people to write something that's roughly as long as it absolutely needs to be. A novel, presumably, couldn't have been a short story. I'd suggest that much of the power of short poems comes from their brevity. Their length announces a sort of immediacy and clarity that ought to come as revelations. (A haiku isn't the sort of thing that's supposed to need argument or evidence.)

All of this to say is that, much to my past frustration, it's very hard to be subtle on Twitter, because the brevity of any tweet makes anything you say come off as a proclamation. That's good for a gal or guy with a lot of confidence, but I found myself just making people angry on Twitter with (what I thought were) speculative comments. These days I find Twitter most helpful when (a) I want to proclaim! or (b) I have a question.

Some more reasons why I prefer blogging to twitter for working out ideas:

• Tweets disappear, blog posts stick around. 
• Every comment on a post is worth ten replies to a tweet.
• Blogging is consistent with me not constantly being near the internet. Twitter supports some of my worst internet habits.

Of course, Twitter is good for many things, and its brevity is a great deal of what makes it great. And, of course, you should do whatever you want to do. But November has been a very happy month for me.