About a month ago, there were 2 posts on “problem-solving paralysis”. First one by Frank Noschese Problem-solving Paralysis, then a followup by Mylène: Problem-solving paralysis and the game layer. Frank noticed that when students were presented by a puzzle (he used an example of an Android game), they dove right in and explored possibilities, doing some sort of backtracking search to find a solution, but when presented with a physics problem that they did not immediately know how to do, they froze, rather than doing the same sort of puzzle-like problem solving.

Mylène asked *“What if part of the difference is students’ reliance on brute force approaches?”* She pointed out that games and puzzles usually have a fairly small number of possible moves (often the same from game to game), and that the sort of easy puzzles that most students tackle reward fairly random exploration. She goes on *“I suspect that some students experience a physics experiment as an infinite playing field with infinite moves, of which every point must be explored.”* The important skill in Mylène’s view is recognizing value even in initially incorrect approaches:

I don’t experience the game space as an infinite playing field of which each point must be explored. I experience it as an infinite playing field where it’s (almost) always possible to play “warmer-colder.” I mine my failures for information about whether I’m getting closer to or farther away from the solution. I’m comfortable with the idea that I will spend my time getting less wrong.

I’m not sure that is a correct view of the situation. Certainly, partial information and heuristics can be very useful in exploring a large space, but the space of physics problems at the high school level is not that large. There are only a few general principles that need to be applied (conservation of momentum, conservation of energy, … ), and only a few variants to try (constant velocity, constant acceleration, computing net force, using a relativity correction, … ). The space of physics problems is a lot more game-like than Mylène gives it credit for.

So why don’t students see it that way? I think part of the problem has been the relentless pursuit of special cases in their math and science classes. Very rarely have students been taught to lay out their tools and explore using them in any sort of systematic way. Random flailing is all they have, because they’ve never seen anything other than magic solutions with no dead ends that appear out of nowhere. They expect solving a math or physics problem to be a simple matter of writing down one step after another in a straight line from problem statement to solution, just like the examples in their books and in class. When they don’t see an obvious path from problem to solution, then they don’t see any way forward at all.

I highly recommend that teachers read Pólya‘s book *How to Solve It* (still in print, but also available on the web in PDF files). Those who have good problem solving skills will often regard Pólya’s advice as trivial—as it is for those who have already mastered problem solving. But if you find that your students have not—if they are suffering from problem-solving paralysis, then Pólya provides general questions that you can use to guide the student without giving them the answers by magic. But the book should not be treated as a remedial technique for students who are lost—it should be taught to the students from the beginning, so that they take a puzzle-solving approach to their physics and math problems.

Some students do not need to be taught—they’ve already made the generalization from various forms of game or puzzle solving to the academic contexts of math and physics. But many have not made that connection. The rationale for requiring math and science of everyone is that it is supposed to help them learn how to think, and so we should spend the time teaching them how to think about math and physics, even if it takes time away from “content”.

A few years ago, I would have recommended learning computer programming as a good way to develop problem-solving skills. But in recent years I’ve seen more and more students trying to debug a program into existence—randomly flailing until error messages go away and some sort of output appears. Now I think that the problem-solving skills need to be taught explicitly. Computer programming still provides a richer set of problems to learn on than most high-school level math and physics classes, in part because it is easier to devise exercises that reward partial correctness and provide precise feedback about what is right and what still needs to be corrected.

Have you ever successfully used Polya to transform a clueless problem-solver into a proficient one?

I’ve read “How to Solve It” after receiving rave reviews from others. What went through my mind while reading it was that if you took a high school student who was used to an algorithm-approach to mathematics (“Here’s what you do for step 1, then this what you do for step 2 . . . “) they would ask themselves all of Polya’s suggested questions. And still have nary a clue what to do to solve the problem.

One problem-solving skills builder textbook for high school and college students that DID impress me is “Crossing the River with Dogs”:

http://www.amazon.com/Problem-Solving-Strategies-Mathematical-Adventures/dp/1559533706/ref=sr_1_2?ie=UTF8&qid=1319166751&sr=8-2

They teach problem-solving strategies with lots of practice and analysis. (Your son might enjoy it, btw.)

Paul Hawking

Blog: The Challenge of Teaching Math

Comment by Paul Hawking — 2011 October 20 @ 20:20 |

I don’t know that any book can successfully transform a clueless problem-solver into a proficient one. That takes practice and (in most cases) careful coaching. Polya’s book is more a coach’s manual than textbook.

Right now, my son has sufficient problem-solving practice in the Art of Problem Solving class on calculus and debugging his Python programs. I don’t believe that he is currently wanting or needing another source of problem-solving advice, but I’ll look into the book.

Comment by gasstationwithoutpumps — 2011 October 21 @ 04:43 |