Today I had an interesting experience that revealed a large gap between how my students and I think about the problems we've been doing. This difference corresponds to how you learn the algorithm for a specific skill vs how you tackle a problem you've never seen before.
If you're learning a specific skill (say, how to solve a quadratic equation), the process generally involves recognizing some cue that tells you what to do next, over and over again. The operative question is "What's the next step?"
If you have to solve a problem you've never seen before, you can't ask yourself "What's the next step?", because the whole point is that you don't know; maybe no one knows. So instead you have to ask yourself "What do I want to find out?", "What do I know already?", and "How can I start to connect the two?"
It's astonishing how few of the problems students get in school require them to ask themselves these three questions. Even if a problem starts out requiring these genuine problem solving questions, it's extremely easy for it to become "proceduralized"--either when I (or other students) give too many hints and reduce it to following-the-steps, or if we do the problem so many times that it becomes a "type" of problem the students recognize "the steps" for.
Students (usually) do a pretty good job listing what they know, and what they want to figure out; the whole trick is how to connect them together. You could "work forwards" by asking what else you can figure out with what you know. It might not seem directly relevant, but if you keep figuring out more and more things, eventually you may see a connection to what you want to know. You can also work backwards by identifying what key piece of information would let you solve your problem. This sets a new sub-goal to aim at.
My students seem to get this idea in general, but have trouble applying it. The specific phrasing of the questions I ask seem to make a big difference. To me, all these questions are equivalent:
For struggling students, it was difficult to get them to think of more than 1 thing they could try. I'd given them a diagram with two perpendicular lines, the equation of one of the lines, and a point on the other. I was hoping they'd suggest things to try such as finding the equation of the other line, finding the point of intersection, finding the x or y intercepts of either line, etc. None of these seemed to present themselves as salient pieces of information. I tried to prompt them by asking what kinds of questions they'd been asked before about situations like this, or what skills they remembered from algebra I or II about lines, points, or perpendicular lines, but without specific queues to trigger their memory, they weren't able to brain-storm a list effectively. At least not individually.
For my last block of the day we started as an entire class. I wrote the diagram on the board, and started a list with the two pieces of information we had (the equation of one line, and the coordinates of the point on the other), as well as the piece of information we were trying to find in the problem. Then we played a game where I would hand the pen to a student who would silently add a new "fact" about the diagram to our list of facts to figure out. They could add labels or additional points or lines to the diagram if they wanted. If they were really stuck, they could hand the pen to another student.
This seemed to work. Students were able to get a good list, extending or modifying ideas that were already in the list. It was interesting to notice that sometimes students would add a potentially relevant addition to the drawing (such as a line that made a promising-looking right triangle with existing lines), while a few would add a seemingly random line or point just for the sake of extending the list.
We had a discussion afterwards about which facts seemed more or less "helpful" in solving the problem. Again, as a class we were able to formulate a series of sub-goals to lead from the givens to the solution.
I think this general approach could work for small group problem solving. The main obstacle, I think, is actually getting the students to follow the process. As a whole class, they were happy to do it, but in small groups they tend to focus directly in on the problem, and seem to regard this sort of activity as long and circuitous.
Some ideas I'd like to try are these:
* Give them a situation and tell them to list, and find, as many things about the situation as possible (so there is no specific "problem" to solve).
* Scaffold the process by providing some of the steps. E.g. on the first problem, I provide the list of facts and they figure out how they connect together. Next time, we generate the list together first. After that, I prompt them to do it on their own. After that, I just give them the problem with no prompting to think about all the things they might try.
* Have the work on half-sized poster paper. Each sub-goal they formulate goes on a 3x5 card. They're reponsible (as part of their graded work) for creating and sequencing the cards on the paper (to show their problem-solving plan). After they have their plan, they can actually show the calculations on the paper.
With respect to how each student understands the specific self-questions involved in working forwards or working backwards, I think it might be best to have students say the idea in their own words, and we can just create a small collection of student phrasings. In my prior experience, this works much better than me trying to figure out the clearest way to express it, since student phrasings tend to make sense to other students. Having a diversity of phrasings will also increase the probability that one of them will catch.
Anyway, a lot of interesting issues to think about. More updates on the teaching of basic problem-solving later!
Students (usually) do a pretty good job listing what they know, and what they want to figure out; the whole trick is how to connect them together. You could "work forwards" by asking what else you can figure out with what you know. It might not seem directly relevant, but if you keep figuring out more and more things, eventually you may see a connection to what you want to know. You can also work backwards by identifying what key piece of information would let you solve your problem. This sets a new sub-goal to aim at.
My students seem to get this idea in general, but have trouble applying it. The specific phrasing of the questions I ask seem to make a big difference. To me, all these questions are equivalent:
- What could you figure out from here?
- What could you try to do next?
- What could you try to figure out next?
- What facts could you try and figure out about this situation?
- What information could you try and figure out next?
For struggling students, it was difficult to get them to think of more than 1 thing they could try. I'd given them a diagram with two perpendicular lines, the equation of one of the lines, and a point on the other. I was hoping they'd suggest things to try such as finding the equation of the other line, finding the point of intersection, finding the x or y intercepts of either line, etc. None of these seemed to present themselves as salient pieces of information. I tried to prompt them by asking what kinds of questions they'd been asked before about situations like this, or what skills they remembered from algebra I or II about lines, points, or perpendicular lines, but without specific queues to trigger their memory, they weren't able to brain-storm a list effectively. At least not individually.
For my last block of the day we started as an entire class. I wrote the diagram on the board, and started a list with the two pieces of information we had (the equation of one line, and the coordinates of the point on the other), as well as the piece of information we were trying to find in the problem. Then we played a game where I would hand the pen to a student who would silently add a new "fact" about the diagram to our list of facts to figure out. They could add labels or additional points or lines to the diagram if they wanted. If they were really stuck, they could hand the pen to another student.
This seemed to work. Students were able to get a good list, extending or modifying ideas that were already in the list. It was interesting to notice that sometimes students would add a potentially relevant addition to the drawing (such as a line that made a promising-looking right triangle with existing lines), while a few would add a seemingly random line or point just for the sake of extending the list.
We had a discussion afterwards about which facts seemed more or less "helpful" in solving the problem. Again, as a class we were able to formulate a series of sub-goals to lead from the givens to the solution.
I think this general approach could work for small group problem solving. The main obstacle, I think, is actually getting the students to follow the process. As a whole class, they were happy to do it, but in small groups they tend to focus directly in on the problem, and seem to regard this sort of activity as long and circuitous.
Some ideas I'd like to try are these:
* Give them a situation and tell them to list, and find, as many things about the situation as possible (so there is no specific "problem" to solve).
* Scaffold the process by providing some of the steps. E.g. on the first problem, I provide the list of facts and they figure out how they connect together. Next time, we generate the list together first. After that, I prompt them to do it on their own. After that, I just give them the problem with no prompting to think about all the things they might try.
* Have the work on half-sized poster paper. Each sub-goal they formulate goes on a 3x5 card. They're reponsible (as part of their graded work) for creating and sequencing the cards on the paper (to show their problem-solving plan). After they have their plan, they can actually show the calculations on the paper.
With respect to how each student understands the specific self-questions involved in working forwards or working backwards, I think it might be best to have students say the idea in their own words, and we can just create a small collection of student phrasings. In my prior experience, this works much better than me trying to figure out the clearest way to express it, since student phrasings tend to make sense to other students. Having a diversity of phrasings will also increase the probability that one of them will catch.
Anyway, a lot of interesting issues to think about. More updates on the teaching of basic problem-solving later!
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