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