This Year, More Problem Posing

A few months ago, I sat down with pad and paper and started looking for some math. It was an open-ended investigation that didn't really start with anything other than the idea that polygons were cool and I knew little about them. 

It was a blast for me. My week was consumed by polygons, and I passed through hexagons to sequences of inscribed polygons, which are actually really fantastic mathematical things. I looked into regular polygons. 

Seriously, it was great fun, and it was great fun because I was investigating, like really investigating, all on my own just mining some rich mathematical vein. It was research. I felt like a mathematician, which I don't always feel like.

One thought that's been with me all summer is, How nice would it be to share this with kids?

Nice, but would they know what to do with a wide-open exploration, if I just handed it to them? Maybe a few, but not most.

Why not? What's missing?

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"If you can't solve a problem, then there is an easier problem you can solve: find it."

That's Polya. 

Here's more from his How To Solve It: 

“If you cannot solve the proposed problem do not let this failure afflict you too much but try to find some consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again...You should now invent a related problem, not merely remember one.”

Polya is pointing to "Solving an easier problem" as a move that's helpful in solving a problem. But that invariably involves inventing an easier problem. But what if you aren't used to posing problems? 

Problem posing seems like the hard part of this problem solving technique.

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And then I was reading Professor Triangleman’s blog, and he said something really smart. To wit,

This coming school year, I will characterize learning—for myself and for students—in the following way: Learning is having new questions to ask. If I have learned something, it is because I can ask questions that I previously could not.

Having questions means that you've learned something. That's great. The kindly professor surely wouldn't say that this is the only way to characterize learning, but it’s a good and important way.

Of course, if you don't know how to ask questions, that you can't really use questions show what you've learned. But having questions is the way that we express understanding so much of the time -- it's a shame for this to be locked away for students in math class.
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I'm going back to investigations. I had a great time researching, doing investigations this summer with polygons.

But, what is investigation really? I'd say: an investigation is a sequence of posed problems, one following another. Investigation just is problem posing and problem solving, coming together in a messy bundle.

In class, I've been spending a lot of time helping my students become better problem solvers. But that's not enough to sustain an investigation. If I want to share investigation with my students, I need to help them become better problem posers.

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One last piece of the puzzle is from Jo Boaler:

Through my work with teachers and students in different schools, I have come to appreciate three acts that are critical to the development of quantitative literacy as well as a higher level of mathematical fluency. These three acts -- questioning, reasoning and representing -- are often thought of as important aids to learning, but they are all important practices to learn in their own right.

I think that she's right that questioning, reasoning and representing are important all on their own to mathematics. 

I want to do more than help my kids master content. I want to share things that I love with them. I also want them to see what's great about math. And there's no principled way to limit math to something like problem solving. Mathematicians regularly pose problems, and many of them are remembered for the questions they asked more than those they solved.

(Many mathematicians are also remembered for their representations too.)

In English class you ought to be able to be good at writing stories, or analyzing poems, or writing essays, or performing or reading. I want there to be more than one way to be good at math in my classroom because that's just the way math is.

(Good god that last one was a dud of a sentence. Get it together, Pershan.)

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Quick, as many spin-offs as possible! One minute! Go!

• What if this went out to 100 circles? What about n circles?
• What if there were triangles instead of squares in this?
• What if there were n-gons instead of squares?
• Wait, would you always be able to get a circle around those polygons?
• What if the polygons are weird shaped?
• What if it's not circles? Can you always embed polygons in polygons?

I cheated more than a bit: this was the list that emerged as I was solving the initial problem, and then solving the first problem that I posed, and then the next question emerged from the that one, and ...

I care about whether kids can do this. 

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The way I see it now, encouraging problem posing hits four of my major needs.

1. It will help my students pursue open-ended investigations, which I love and want to share.
2. It will give me another way to assess when my students have really understood something.
3. I think that my students ought to value mathematical activities besides problem solving.
4. I’m also betting that it’ll help them solve problems, ala Polya.

I want to incorporate problem-posing into every class that I teach. (By the way, that's 4th Grade, Geometry and Trigonometry right now.)

My extension to every task this year is going to be "Look for an interesting spin-off problem."

We'll see how it goes, obviously. Any thoughts on this all would be appreciated, especially if you disagree.

A Thought Experiment

Imagine your Favorite Author. She writes novels about mothers and sons and the way a city can suffocate its locals and also the Iraq War. Critics adore her work. High Schools can't decide whether to assign or ban her book. Commuters read her on their iThings. She's fantastic.

Your Favorite Author wakes up. She fumbles down the steps and lands in front of a cup of coffee that her husband left her. ("Thanks Dave," she thinks.) She takes a long sip and opens up the latest issue of the Journal for Research in Fiction Writing, which is lying on the table. She flips through it.

A title strikes her: "Characters that Stick: An Exploration of Backstory Tragedy in the Likeability of Supporting Characters." Interesting. She's working on a story now, one with a woman who, as a little girl, watched her parents die. (Spontaneous combustion, a metaphor for the ways in which life and time stories the inability to communicate in the digital era and recession.) She turns to the piece.

She reads the abstract:

This study investigated whether a character produced with a tragic background contributed to 236 randomly selected readers' literary pleasure, that is, the effectiveness of a story at inducing a passionate reaction. An exploratory quasi-experimental study was conducted with a pretest-posttest-control-group design. Men were found to derive more literary value out of a character with tragic backstory than women. Implications of these results for novel-writing are discussed.

Fascinating. Her next story would be better. She takes another sip, takes out her reading glasses. She digs in.

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Here's what I'm curious for in the comments:

• The above scenario strikes me as absurd. Do you share this reaction? 
• If you think it's ridiculous to imagine novelists improving their craft in this way, then do you also think it's ridiculous to imagine teachers improving their craft in this way?

Here are some points that I think are worth making:

• Research about how students learn isn't the same thing as research about what makes effective teaching. As an analogy, doctors regularly keep up on research about medications, procedures or treatments. But this is different than research concerning how to make a good diagnosis. It's the difference between research that's relevant to practice, and research on that practice.
• You might be tempted to say that this comes down to whether teaching is an art or a science, and then (in the spirit of even-handedness) you'll want to say that it's a little bit of both. But you'll have to admit that artists have a wide base of knowledge from which they draw on, and that there is widespread (if far from complete) agreement that at least some art is really good, and some art aint so great. So it's not like facts and knowledge and correct judgement are foreign to art.
• Besides, what would it mean to say that teaching is a science? Physics is a science; it's the study of the fundamental properties of the natural world. Biology is a science; it's the study of the living world. If teaching is a science, would that mean that it's the study of the something? But teaching, whatever it is, is just not the study of some subject. Maybe instead of "teaching is a science" we mean to say something closer to "there is objective, testable knowledge about what good teaching looks like"?

Looking forward to a spirited, late-August debate here. See you in the comments.

"Beg, Borrow, and Steal" isn't great advice

Table of Contents

1. What BB&S Stands For
2. How Artists Get Great
3. Imitating Dan
4. Good vs. Great
5. Call For Comments

1. What "Beg, Borrow, and Steal" Stands For

"Beg, borrow, and steal."

This is how you're supposed to get good at teaching. Especially during your first few years.

Implicit in this advice is that the difference between a lousy teacher and a good one is the stuff that she has at her disposal. It's curricular resources that you're supposed to beg, borrow and steal.

How do teachers get better? If you believe in "Beg, borrow, and steal" then your answer is that when a teacher starts out, she has very few good lessons at her disposal. With time, she accumulates good material. Some of it she creates, but much of it she takes from colleagues or books.*

* It's not hard to imagine a day when she finds herself completely satisfied by her collection of curricular resources. Then it's just a matter of plucking lessons out of her binder for the kids.

This belief is widely held.

2. How Artists Get Great

I've been reading "The Great Jazz Pianists," a book of interviews with (you guessed it) great jazz pianists. There's a line that keeps popping up when these musicians talk about how they got good:

"You have to have a model to play from and start by imitating." - John Lewis

"The pianist has to be a careful artist. It's like a painter trying to duplicate the work of an old master. He must get every detail and every fine line. Of course, a true artist wouldn't want to duplicate someone else's work, but he would be capable of it." - Sun Ra 

"In your formative years the influences you're absorbing possess you." - Horace Silver

These musicians recommend learning by imitation. As Sun Ra mentions, this is similar to the process that the old masters used to train new painters. It's also similar to the way that some fiction writers describe their formative years.

George Saunders wrote the book that the New York Times called "the best book that you'll read this year." By his own telling, imitation played a crucial role in his development as a writer:

“If I got tired of [Hemingway], I did a Carver imitation, then a Babel imitation. Sometimes I did Babel, if Babel lived in Texas. Sometimes I did Carver, if Carver had worked (as I had) in the oil fields of Sumatra. Sometimes I did Hemingway, if Hemingway had lived in Syracuse, which, to me, sounded like Carver.”

Across several creative fields, imitation is mentioned as a crucial stage in the development of an expert.

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Imitation is different than appropriating or remixing the material of others. If piano players believed in "Beg, borrow and steal" they'd be taking the piano lines of others and memorizing them, incorporating them wholesale into their solos. But they weren't doing that. Instead, the great jazz pianists intensively studied the works of their elders and attempted to imitate them.* As a result, their styles inevitably resembled their master's, until eventually they found their own voices.

If George Saunders "beg, borrowed, and stole," then he'd be using the plots of the great writers, using their phrases and sentences for his own purposes. He didn't do that. Instead, he used imitation as a way to attempt to create stories that he'd love as much as those he'd read.

People don't get better at creative activities by stealing. They get great through imitation.**

* Well, sort of. To be clear, what they were often doing was attempting to play the original work note-for-note, and then later imitating their style.

** Just a quick, very parenthetical, note: there are two ways to talk about your career. You can talk about "being great" or about "making great things." I prefer the latter, and I think that the former is often unhelpful and sort of creepy. But I fall into that language because it's more concise, and I'm sometimes stuck in the "be great" model. On reflection, though, I prefer "make great" to "be great."

3. Imitating Dan

A few months ago, Andrew Stadel posted a snapshot of an exchange rate board and asked if anyone could design a lesson around it. I decided to give it a go.

I started thinking about what I could do with this picture. Since it was an digital, "real-world" picture, I thought of Dan Meyer, and then my mind went to my favorite post of his, "How Do You Turn Something Interesting Into Something Challenging?" I asked myself how Dan pulled that lesson off, and I thought if I could fit this picture into that paradigm. I decided that I could. (E.g. Take different piles of currency and offer them to kids, and ask them to rank them. Then give exchange rates.)

As I was working through Andrew's question, I realized that a lot of my pedagogical knowledge takes the form of lesson exemplars that I hold in my mind. When I face a curricular problem, my first thought is whether I can fit my problem into an of these exemplar approaches.* This is imitation.

* Here's a short list of my exemplars: Dan's liquids, Christopher's hexagons, Paul's exponents, Kate's logarithms and Fawn's slope. More generally, MARS developing metrics lessons. There are more, for sure.

Imitation has been an important part of my development of a teacher so far. My teaching knowledge is structured around certain major landmarks which I use to situate myself in the landscape of teaching. They ground me.

4. Good vs. Great

I may have overstated things, and I'd appreciate pushback in the comments if you think that I did. "Beg, borrow, and steal" is too prominent an adage to be totally false. Of course, everybody needs resources for teaching, and I know that collecting resources can help your teaching get good.

But can you become great through "beg, borrow, and steal"? I'd say that you can't, because becoming great involves finding your own voice as a teacher.*

* Some of you might say: "You can be great at teaching without having your own voice or lesson-style," and I think that we should have a good discussion about this in the comments. I'd say that teaching involves context-specific problems, and you can only solve these problems that are particular to your context with original solutions. Finding original solutions is a creative act, and is impossible without an independent, and ultimately original, style. I also think that original creation is a far more efficient way to solve problems than to seek out resources. Disagreement in the comments, please.

The fundamental issue with "beg, borrow, and steal" is that it emphasizes resources instead of personal development. But resource accumulation has a very low ceiling, compared to personal development, when it comes to improving as a teacher. To illustrate this, consider a story from comedian Patton Oswalt about an up-and-comer who stole jokes. He writes:

Don’t worry – this story has a happy ending. Blaine and I eventually moved west. So did the thief. But when it came time for him to make the transition to television, to movies, to big-time fame and success? He had nothing. And, without going into details, he flamed out, rather spectacularly, on national television. Like, spectacularly.

Maybe "beg, borrow and steal" can help get you through your first year, but will it help you keep growing (and stay interested?) past your fourth?*

* I sometimes wonder whether my general tendency to create my own materials, rather than steal others stuff, was responsible for my particularly rocky first year.

That's my case:  "Beg, borrow, and steal" is fine advice, but it's got a very low ceiling. I don't know how low, but judging from the model of other creative professions, it's no way to get great. If you're interested in reaching the point where teaching is a creative endeavor -- e.g. you're coming up with ideas that are original, and that other teachers might care about -- then I think imitation is a far more reliable model for growth. Personally, I've found imitation particularly helpful, and for me it takes the form of approaching lessons with certain exemplars in mind.

Out with stealing, and on with imitation!

5. Call for Comments

Here are some disagreements that I can anticipate, and while I'd appreciate comments of any kind, I'd especially appreciate comments if you disagree with any of the following:

• Great teachers are always creative, especially when it comes to lesson-planning
• "Beg, borrow and steal" isn't good advice, even if it helps in those first few months, because it instills a false model of teacher growth.
• Artists tend to go through a long(?) period of imitation on their way to creating great things.

As always: thanks for reading and sharing your thoughts. 

[Makeover] Periodic Cycles

Dan's asking us to give this activity a makeover:

In some ways this is a classic pre-Dan Meyer problem. There's a call for prediction. There's a text-heavy scenario with some stock footage that could easily be replaced by a simple video. There's an abstraction of a Ferris Wheel that's too eager to offer the minimum height.

So we could give it the 3act treatment. We'd start by figuring out an image or video that would create the need to know where the Ferris Wheel will be at a certain time, to motivate the prediction.

But when exactly would you need to predict the position of a spot on a Ferris Wheel? All I can come up with are contrived scenarios:

• You want to see something from the very top of the ride. Maybe fireworks or something. But they'll only be there for a split second so you'd better get the timing right.
• There's a swinging blade at the bottom of the ride and if you'd like to keep your legs you'd best make sure you hop on at the right time.
• The spouse is dangling from a burning building at precisely 20 feet up and if you can only get the timing right...

Whatever. This isn't working. And the power to predict isn't what's interesting about this anyway.

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Last spring I started class with this video:

(Truth is, I actually showed two videos, one of a larger, slower Ferris Wheel, and the one above. I asked kids to draw graphs of both.)

I asked the kids to draw a graph showing the height of one of these people over time. We needed an example to get everyone on the same page, but then whiteboards were distributed and the kids were on it.

I walked around, looking for interesting things. And, hey, there was an interesting thing.

Some kids made graphs that were all pointy like this:

Others thought that it would be nice and smooth.

Once I pointed out this disagreement to the kids, we had a nice old-fashioned slug fest. Why ought it be pointy? Why ought it be rounded? What evidence could we bring? Would you play that video again? Of course I will.

What's the difference between a pointy and a curvey graph? What does reality look like according to each? What does it mean for the world to be like one and not the other?

Eventually we used this applet to be especially careful in our observations. The argument was settled. There were winners, there were losers. Folks licked their wounds, figured out where they went wrong. It was fun.

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I've done the Trigonometry thing three times now, and I've started to become convinced that real-world prediction is just harder with periodic functions, because of their periodicity. Things that repeat aren't super-interesting to predict. Plus, the fact that it's periodic means that you don't really need much math to get a decent prediction. Which, I guess, is a lesson of its own sort, but still...

I tend to do a lot of astronomy in trig. We take a look at moon cycles,  we compare sunsets at different latitudes over the course of the year. We can look at tide-charts if that's your thing. The problem is that the power of prediction here is fairly low. The moon cycle repeats roughly every month. Once you know this, there isn't a ton that you don't know, unless you want to know exactly how much moon there is at a given instant in time. And why would you?

(Maybe there's a good werewolf problem hidden in there somewhere. Like, you want to know which days are full moons so you can avoid tearing out your family's guts. Or something.)

Anyway, I think the more interesting thing is the way relatively simple patterns of motion turn out to be periodic in particular ways. That argument that my kids got into was cool. If they don't bring it up next time around, I'll ask them to tell me how they know that it isn't pointy/square/circular or whatever. Maybe we don't need to get into mathematical modeling.

So my makeover is to show a video of a Ferris Wheel and then to ask this question:

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By the way, for completeness, I'll mention that I've also asked kids to predict the position of a pendulum in this stupid video at 1 minute in after 13 seconds. Not the greatest task in a world, but it seemed less contrived than Ferris Wheels and such.

It was sort of fun to do in class, in a nerdy "you've got to be kidding me" sort of way. You might want to turn your sound off before viewing. It's pretty freaking annoying.

Update (8/4/13): I had a quick thought that, if you're interested in upping the difficulty of the graph sketching, you might consider different shapes of Ferris Wheels. Like, what would the graph look like if it had a square wheel? A triangular wheel? Check this out for a visualization and more ideas.

Also, there's got to be a way to ask kids to find the derivative of sine without asking them to find the derivative of sine, right? I can't figure out how to do this, but maybe you can.