“we have to start by being just a little less helpful. It doesn’t help if the problem is past the point they can solve without us swooping back in to save them. They have to be able to figure it out on their own with just a quick dip into the disorienting waters of independent thought. Then, the experience of satisfaction will give them a little more tolerance for that discomfort the next time. And it builds incrementally.”

Sam

I second Kate in that I don’t think it’s like this radical dichotomy “teach the curriculum lockstep” or “totally open-ended creative fun problem solving.” (Although I agree that there is an essential tension between the time pressure of the curriculum and the sense of expansiveness about time needed to really settle in and start cultivating resourceful and creative thinking in our students. I think the only thing to do about this is to fight for the curricula being streamlined and the big important priorities clarified. Everyone knows US state frameworks are too broad and not deep enough and yet they continue to be…) Here’s how I see it:

We all agree that over-directing our students’ efforts prevents math class from being a place to learn critical thought, creativity and resourcefulness. The flipside is that being totally disoriented is also disempowering. So, giving them a POW for which they don’t have any way in and watching them flounder for a few minutes and then give up is not the solution. (This I know from personal experience and imagine many in the math education blogosphere do too.) Students that are not used to being asked to think for themselves in math class find the experience uncomfortable. Tolerance of this discomfort, and an appreciation of the value of it, only comes from successfully solving a problem in this state. Solving a problem no one told you how to solve is infinitely more rewarding, as everybody on here knows from personal experience, but a student used to just following orders is never going to solve such a problem if just dropped into a totally disorienting situation. It has to happen in an incremental, Vygotsky sort of a way. To paraphrase Dan Meyer, we have to start by being just a little less helpful. It doesn’t help if the problem is past the point they can solve without us swooping back in to save them. They have to be able to figure it out on their own with just a quick dip into the disorienting waters of independent thought. Then, the experience of satisfaction will give them a little more tolerance for that discomfort the next time. And it builds incrementally.

I’ve had a lot of confidence with independent problem solving for a long time, but I myself had a developmental experience of this kind in the last 2 years, during which I picked up the serious study of math again. I’m now taking grad classes but when I started I was totally on my own. I was tending to get frustrated if I couldn’t solve a problem for two or three days. The book (Michael Artin’s Algebra) had starred certain problems as particularly hard and I solved two of the starred ones and felt very accomplished. The next starred problem I came to, I really struggled with, but because I had solved the last two I stuck with it. It took two weeks before I found a solution; but once I found it, I was never going to give up in a day or two again. About 5 months later, the memory of this experience of a problem yielding after 2 weeks of frustration carried me through 2 months of unsuccessful work on a problem. Now 2 months feels fairly normal. The point is, the experience of success after time spent in the frustrated, disoriented state makes it much easier to stay involved in a problem through this frustration later. So the willingness to stay engaged, and so the opportunities for creativity and resourcefulness, grow incrementally. So the problems need to grow less helpful, but they need to do it gradually.

Most groups choose one solution, a decimal solution, and complex number solution. I pointed out that one of the complex number solutions was a decimal, so those solutions could be put into two categories, which was against the rules. One group got the expected categories and shared their thinking with the rest of class. (This group did a lot of whining about how their heads hurt thinking so hard. :))

While not brain science, it got folks thinking on their own, even if reluctantly. I find the biggest challenge in these situations is motivating students to take the problem on, rather than give up and wait for the answer. Since this was a small problem that didn’t take too long, they humored me. Any advice on that front would be appreciated.

I hope this is helpful.

Joan

I have had the best luck by just turning the lesson on its head. Instead of
TOPIC
Definition/Derivation/Formula
Example example example
Practice

Just be like:
PROBLEM.

OK so today in Algebra 2, it was Composition of Functions with Equations Day. Like, if f(x) = x-3 and g(x) is x^2, what are f(g(x)) and g(f(x)). Last year I did a very traditional lesson, and the last example was this texty problem about how if you have a % off coupon and a $ off coupon and you can use them both, which one should you use first, and how much would you save. But it walked you through in parts and it was very boring. And it came at the end, when everyone is half asleep. So today I started the lesson by projecting images of a 20% off coupon and a $10 off coupon from Old Navy (which I have dozens in my Inbox), and said sometimes Old Navy lets you use them both on the same purchase, so What Question Am I About To Ask You? (a trick I got from Dan) And they spent the first 5 minutes arguing about which order was better, how much you would save that way, whether the price of your purchase mattered. They didn’t get to .8x – 10 vs .8(x-10) on their own, but at least they were curious, and it was something they could hook the lesson on to. I had to leave out some of the annoying-gotcha-algebra-pain examples, but still. Definitely better than last year.

I hope that helps. And looking forward to seeing what you produce with your project.

I’m sorry if this feels more like a rant than a productive thought–I was just wondering if you might have suggestions for how to incorporate some type of problem solving into Saxon program (if you’re familiar with it).

Thanks!

My own experience with students suggests that, if they have a strong grounding in all of the relevant basic concepts, they are better able to handle novel situations. I did a lab with my physics students in which I asked them to determine the acceleration of an object on inclines of various angles. Because they had already done cookbook labs relating to measuring motion and because we had spent some time working with the kinematics equations, most of the groups were able to devise a method for determining acceleration without much help. Granted, this particular scenario wasn’t intensely novel for them, but I think they successfully applied some problem solving.

Thanks for taking the time to write this entry. If you happen to figure anything out, I’d like to hear about it. I just started following you on Twitter.