“Looking Back ” by Extending Problems
I like posing and solving problems on-line. Is this reflected in my classroom? Not enough, but yes.
In the fall term of algebra classes I carve out a day here or there, or maybe a few half-periods, to work on extended problem solving. It is generally not on-topic. On-topic would allow the kids to know before they start HOW they should solve the problem, and that would spoil the joy. I usually choose problems with multiple paths to success. And I certainly do not choose problems that have an accessible formula – that would spoil the challenge.
I use an Understand / Plan / Carry out the Plan / Look Back approach with the kiddies, but too often “looking back” for them just means “check.” Over the years I have pushed “find another approach” or “find the relationship between two successful approaches” or “generalize a solution.” But this year I pushed in a new direction.
“Use your work and solution to think of a new, interesting problem.” The idea is not to simply make the problem bigger, or generalize it, but to come up with something related, but new, probably closely related and more complicated, but not necessarily so. And it was quite possible for the new problem posed to be easier than they realized or harder, to yield to a similar approach as the original problem did, or not to yield at all. After all, if they knew there would be a solution of appropriate difficulty, it would mean that there was not original problem posing going on. And after practicing generating ideas on earlier problems, we hit the checkerboard, and I assigned them to extend the problem, gave them time in class and at home, and required them to write up a problem solving “experience”:
- Understand the Checkerboard
- Devise a Plan
- Carry out the Plan
- Look Back (include posing a new problem)
- Devise a Plan
- Carry out the Plan
- Look Back (since many new problems were not solved, this included commentary on obstacles. Where problems were partially solved, we got suggestions for the next team to pick the problem up. Where problems were solved, we got ‘normal’ generalizations, but also suggestions for future work. From 9th graders. )
So, post-checkerboard, what problems got posed? Here’s a few that I recall:
- Solve for an abnormal 8 x 9 checkerboard. Generalize to squares on an m x n checkerboard.
- Solve for a checkerboard with the four corners missing. Try again with the four 2 x 2 corners missing. 3 x 3. Generalize to an n x n checkerboard with four m x m corners missing
- Variation (different group). Solve for a checkerboard with one corner missing. Then a 2 x 2 corner…. Generalize to an n x n checkerboard with a single m x m missing.
- Variation (there was a lot of removing squares going on). Solve for a checkerboard with a 2 x 2 hole in the center. 4 x 4. 6 x 6. Generalize to an n x n board with an m x m hole in the center.
- Solve for an 8 x 4 checkerboard. Account for the difference between two 8 x 4 boards and one 8 x 8 boards (the write up for this was beautiful)
- Solve for rectangles on a checkerboard.
- Leaving the board out of it, count trimonos, tetrominos, pentominos, hexominos. (I think this group got side-tracked into some fascinating but for the moment fruitless discussions of symmetry and handedness. Product? Nah. Discussion – excellent.
So, when you get an answer, are you at the end? For most of the kids the response is still “check, and that’s enough, unless the teacher makes you go on” – but for a substantial minority I think they got used to the idea that mucking around further is a good idea, and potentially fun or interesting.