David Coffey wrote a post about teachers giving students problems that they themselves don’t know the answers to–doing this deliberately, as a pedagogical strategy. I’d like to share an example from my own classroom, as well as some reasons why I’ve pursued this path.
Before I begin, allow me to say that what follows is a “sometimes” occurrence in my classes. Most times when I share open-ended explorations with my students, I’ve already investigated the problem pretty thoroughly–either with a previous class, or on my own in preparation. And of course, there are plenty of times when my classes are more closed-ended. But on a few occasions I’ve purposefully held myself back from thinking about a problem before sharing it with students.
I think the first time I did this with serious malice and forethought–teaching in the first degree–was at the beginning of last year with my sixth and seventh graders. In addition to beginning our diagnostic and review work with arithmetical topics, I gave out this first sheet of problems that were inspired by / stolen from James Tanton’s Math Without Words:
Not a free-form task–plenty of specificity here–but at the same time, no instructions. I hadn’t done these problems before I handed them out. What I mean is that I had played around with them a little, but hadn’t convinced myself that any were impossible, and certainly didn’t have any kind of general theory. I recognized that there was some mathematical depth here and felt that there were likely many connections and extensions to be made. I wanted to open the year by modeling problem definition, problem creation, and problem solving in a live, off-the-cuff way, and this starting problem seemed like a great way to do it.
The kids dove in–faster, slower, with frustration and accomplishment and bits of insight. The variations they composed on the back were fabulous–different sized grids, different shaped grids, obstacles, fixed end points, three-dimensional versions, and on and on. They shared these with each other and critiqued each other’s work, and we discussed the gradations of variation that had sprung up in these new problems.
Here’s what came next:
Here I was pushing for clear language and problem definition, as well as another go-round at creating variations to the original problem.
For the next assignment, I asked my students to explore the nearby space of the original problem–looking for insight into it by looking at variations, and especially at simpler variations. That’s a problem-solving strategy that worth of the name. I gave them graph paper and asked them to try out this kind of grid-path problem on some smaller and perhaps bigger boards. I did the same.
That was actually a homework assignment. When they came back in, I had them get into small groups to share and compile their findings:
What they uncovered was really awesome and–as best as I can remember–a surprise to me! On the even boards, it seemed like the problem was possible no matter when the starting dot was located. The odd boards, on the other hand, had a very suggestive checkerboard pattern. Conjectures and excitement filled the room.
The following came next–I don’t know if the idea came from me or from a kid:
I won’t spell out the gory details, but there’s so much here–parity, induction, tiling, Hamiltonian circuits. And I didn’t know it was all there when we started, except in my gut. So much depends on having a stock of such problems. Finding them is both hard and easy. Maybe I’ll talk about that some other time.
At this point last year, some kids went on to explore their own variations during free-choice time. (Geofix polyhedra laced with string, anyone?) There were probably loose ends that never got resolved, but that’s not important. What is important is that having this open-ended, let’s-figure-it-out-together exploration at the start of the year really helped to set the tone that I was hoping for. So here’s to say that posing problems that you don’t know the answer to is entirely possible and potentially extraordinarily fruitful. It can happen in large or small doses, but it belongs in the hands of every kid–not just the ones who are “good at math” in conventional ways. Everyone needs the experience of a math classroom where there are no “haves” and “have-nots”–the teacher included.
Right now I’m in the midst of doing something similar with my fifth grade classes. We’re working to figure out how many different combinations of pattern blocks can fit together to make a full turn at a point. I have no idea of what the answer is, or what patterns will crop up. I’ll let you know when we figure it out!
I hope you find something worth chewing on in the above. Wow–it feels so great to finally share some math!
PS With respect to all of this business about problem creation and variations, extensions, and generalizations, let me say that I owe so much Avery Pickford. I had the great fortune of cutting my teeth on this crazy profession while sitting next to Avery in our math department office at Saint Ann’s. Louder voices than mine have sung his praises, but I know firsthand how thoughtful and awesome he is. Just a for-instance: check out this sweet description of the problem cycle and variations, extensions, and generalizations that he wrote and I tweaked. For this and so much–thank you, Avery!
Dang, this is inspirational! I’ve got Tanton’s book around here somewhere, now I need to find it. I love the idea of extensions, the adjacent-possible, problem posing… And I’ll second Avery’s work (which I only know through his blog) to anyone out there. I’ve really enjoyed your thoughts recently. Thanks!
@eddi I’m so glad! Your comment inspires me back, so look for some related posts soon. Thanks for reading!
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A) That is a really awesome series of assignments. Thank you for sharing them; I can’t wait to see what I and my kids do with them! Modelling thinking aloud, and metacognition in general are such great strategies for discovery!
B) I used to teach biology, and used a same “try and find out together” approach when drawing structural formulae. My students and I all knew the rules and conventions, so we’d each try and draw the molecules, and see what variations we came up with that still followed a given chemical formulae. I’d routinely have kids asking me if something was “right” or not, and I loved seeing the look on their faces when I said I didn’t know, and went through the same rules that they did so I could check. Teaching without the answer key shows students the processes of enquiry in action; what could be more important?
C) You posted a comment today on my post on cheering math like it was kindergarten soccer, which is how I got here (hurray!), but I have no clue how you found my blog in the first place, except that I seem to be getting a heap of twitter hits today. Could you fill me in on how you found me, pretty please? If there are more great blogs like this at the end of that trail, I’m very curious to follow it!
A) You’re welcome! Let me know what you guys come up with!
B) Totally. In a related vein, I sometimes talk about the choice I make between figuring something out for myself and looking it up. The choice isn’t always available–on the one hand, I won’t be able to figure it out in a reasonable time or at all, and on the other hand human beings might not yet have figured it out at all! But often there does exist this choice that I have to make as a learning adult. Kids need to be clued in on that.
C) I was linked to your kindy soccer post by this tweet by @republicofmath. Do you tweet?
Actually thinking out loud about how we solve problems, not just in math, but in time management, our own education, and our professional lives? Say it isn’t so! ;)
Wow – that tweet about my post was somewhat scathing, actually. Yikes! I figure that kids have to feel comfortable and interested before they are going to actually want to engage in learning. Guess the guy didn’t notice that I was writing about *kindergarteners*. >;)
Yes, I do tweet: @turkeydoodles ! @republicofmath is on my feed, I must just have missed it. Thanks for filling me in!
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Happy Back to the Future Day!
So I’m nearly a half decade behind on this topic. I discussed this exact topic yesterday with both my geometry honors students and a few colleagues after using the “word proofs” exercises (we’re still looking for a solution to 8). I use the teach-without-a-key approach in a majority of my lessons. I had not completed all of the word proofs yesterday before giving them to my students, and I do this frequently with any exploration or investigation lesson.
The easy reason is cockiness and planning. My limited planning time consists of researching new ways to approach mathematics outside of our bland textbook, not coming up with beautiful answer keys. I guess I’m also confident in my ability to reason my way out of any paper bag. Ha!
More importantly, I don’t like imposing preconceived perceptions/feelings/approaches on my students. I believe their journey to the truth of any given problem is vastly more valuable than my methods.
You guys echoed my sentiments precisely with “modeling”. I want students to hear my thinking, to walk the problem-solving path with me, to make choices that end up in failure and to reassess failed decisions in order to make better ones.
I had a great experience trying to explain “sufficient” and “necessary” last week because, 1. It was an “extension” topic I had not taught before, and 2. It forced me to analyze the definitions provided by the text with the students in order to reword each example. I struggled mightily in my first year of college because I did not know how to study on my own nor read textbook examples for understanding.
You’re also spot on with the teacher not being a “have”. Problem-solving is not something some can do and some can not. All have to wrestle with problems of some kind in all contexts; showing that solutions can be derived when it didn’t seem possible is a wonderful life lesson. Number 8 of those word proofs is a great example of the “unsolvable” problem that creates educational value through the problem-solving wrestling match.