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: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.Learning is having new questions to ask. If I have learned something, it is because I can ask questions that I previously could not.

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:

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.

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

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.

Hooray! I agree that this is a wonderful and important approach for students to learn to use, to create and explore their own mathematics instead of only solving problems that they're given.

ReplyDeleteOne of my inspirations on this front was a lovely book called The Art of Problem Posing, http://www.amazon.com/The-Problem-Posing-Stephen-Brown/dp/0805849777 . It shows some good ways of making an investigation structured enough for students (at least older students) to have an idea of what to do, while leaving it open enough that they're genuinely exploring their own thing.

Yes, yes, yes. Absolutely more problem-posing and question-asking.

ReplyDeleteOne comment I wanted to make, though... I wonder whether your circle & square image might be a little, for lack of a better word, "scary" for some students. What about just circle-in-square-in-circle? Simpler, and just as easy to jump to your extension questions. Just a thought.

If you haven't done so already, DEFINITELY check out Paul Lockhart's _Measurement_. So much good stuff about math, problem-posing, and wrestling with ideas. As well as plenty of question-inspiring drawings!

>Any thoughts on this all would be appreciated, especially if you disagree.

ReplyDeleteWe respectfully disagree.

Open-ended investigation is one teaching approach, but not one to which we subscribe. Our philosophy is that carefully selected problems be posed because the instructor can ensure that tasks are grade-appropriate and it makes for less directionless time in the limited classroom time that's available.

This ratio problem (the antecedent to yours) could be done in high school, but was actually designed for grades 3 or 4: http://fivetriangles.blogspot.com/2012/08/34-area-ratios.html

This problem http://fivetriangles.blogspot.com/2013/08/87-embedded-squares.html could be done in high school geometry as well, but what if you pose it to a primary school student who has never heard of the Pythagorean theorem?

While extending problems is not without value, the intricacies inside a problem are already limitless.

I certainly agree that if you're just worried about your kids mastering content, then open-ended investigations aren't the way to go. Like you said, it's about efficiency and directionless time just isn't worth it.

DeleteBut I think that investigation is awesome. I want to share it for its own sake, not for the sake of mastering content. So where exactly is your disagreement? Are you saying that I shouldn't value it? Or are you just offering that you don't?

I have two criticisms to -- totally respectfully -- toss your way:

1. I have a really hard time seeing those problems work for 3rd or 4th grade. I know that your work has weathered this criticism before, especially from Dan Meyer, but I want to add my voice to the mix.

2. I have a hard time with the "we" in your responses. How many people are we talking about? Are seven people in a room and all reading this post and disagreeing? Are you one person?

I think that the way that you present yourself matters. By presenting yourself as a "we" you're making it harder to have a conversation. It's always easier to have a conversation with a person -- even an pseudononymous person -- than with a corporation.

If this sounds unfair, it's because I did a poor job communicating in this comment. I sincerely appreciate the criticism, I really appreciate your commitment to sharing your vision of math education. But I also think that work is easier when you offer a bit more of a peek behind the curtain than you guys are currently doing.

Has Meyer weighed in beyond a tweet which called one of our 6th grade construction problems "demon"?

DeleteBut to backtrack, we're not at all interested in the notion of "mastering content". We're actually not that interested in mathematics, which is probably why our viewpoints on "investigation" clash a bit. Our interest is in problem solving. Mathematics is simply the vehicle to acquiring the higher order analytical skills that complex problem solving entails.

Not many share our view: we get that.

About a third of our problems (which we call "applied arithmetic") are solvable in grades 3 or 4, but we agree: they are challenging. (We just call the blog "years 6-8" to mitigate the implied insult.)

As we've written in many places, it's irresponsible to throw difficult problems at students out of context. The proper foundation needs to be laid, so that students experience success at least some of the time.

Much mathematics that is currently offered is on the opposite tack. We just emailed an ed professor named Liljedahl whose website poses a problem that involves telling time, addition and subtraction, grade 3 skills. We asked him what the criteria were for placing this problem in a list for "junior high school".

Finally, we have legitimate reasons for maintaining our anonymity, which we wrote about on our Common Core blog. Sorry if this causes some consternation. We are definitely not a corporation or part of one, but if you dismiss us as hacks, we certainly understand.

But the polygon problem was interesting to you because you thought of it. I think that if you can have kids "go with" a problem that *they* think of, then you have investigation. Otherwise you just have - problems. And not all the kids can articulate, voice or even think of their own problems.

ReplyDeleteA couple of years ago I had kids ask what was the most paper you could get out of a tree - volumetric, density,estimation... I had them "go with it" for about a week, while the rest of the class plowed through bookwork, and then they project group investigated how their estimate was different from the google answer they found. A worthwhile endeavor. But it would not have been if I had posed it.