SteveH, a frequent commenter at this blog, has made some cogent observations about the transferability of problem solving skills, and about “understanding” in general. Like me and others I know, he feels that far too much importance is placed on understanding in math in K-6 than is necessary. He also posits that mastery of procedural skills in the early grades and even high school, doth not a “math zombie” maketh. I’ve taken two of his recent comments and placed them here for general interest.
I. Problem solving and transference
I want to say more about problem solving and transference. Polya is worthless, so what problem solving skills are transferable? You can draw pictures, label variables, and write equations to look for m=n. Everybody does that, but still, it’s difficult. You can study governing equations and their variations. This is classic homework p-set work. However, does D=RT problems and variations easily transfer to work problems? Yes and no. Both are amount = rate * time sorts of things, but there are lots of odd variations. Look at AMC test problems. You could know a lot about logarithms, but the testers are really good at finding odd problem variations. I look at some of their questions and feel really stupid. Success on the AMC is not really about general transference, but very specific and detailed preparation of problem classes for a timed test. You don’t study general problem solving skills. You study in detail every past test question you can get your hands on. Only that reduces transference distance.
How about general problem solving transference? First, it’s not a timed test issue. The question is whether you can do the problem eventually. I’ve seen cases where the smart kids in a class take longer to figure out the trick or angle. Knowing and working with lots of governing equation variations is probably the best foundation. How about this problem?
You are sitting in a row boat in a big tank of water that has a scale showing the level of the water in the tank. You take out the anchor and drop it into the water. Does the level of water in the tank go up, down, or stay the same?
What general problem solving skills help with this if you don’t know the governing equation? Do you have to do a JIT (thanks Barry) discovery of Archimedes? There really is no such thing as general magical transference and problem solving outside of m=n, and that’s not a big help even with a LOT of p-set practice. If you figured out this problem quickly, does that mean that you will do the same for any other problem? I doubt it. Even Feynman used to study up on trick problems so that he could impress people when he pretended to figure them out off the top of his head. Colleagues called him a faker.
Feynman’s true understanding brilliance was based on not only on knowing many governing equations in physics, but making sure he had a real physical sense and connection of those equations to reality. However, even that ability has limited trasference. You have to develop that sense for every different governing equation. Feynman struggled a lot. I still struggle a lot with problem solving. Epiphanies of understanding come to me, but only after I struggle and work on problems for a long time. Then again, there is a huge gap of hard work and discovery between insight and any sort of final solution.
II. Understanding and challenging assignments
This is from my son’s old Glencoe Algebra 1 textbook on factoring differences of squares – Exercises. (page 451)
Section 1 – Basic skill problems like this – Factor or prime?
256g^4 – 1
Section 2 – Factor and solve
9y^2 = 64
Section 3-6 various word problem applications
Section 7 – Open Ended
“Create a binomial that is the difference of two squares. Then factor your binomial”
Section 8 – Challenge
Section 9 – Find the error
Section 10 – Reasoning – find the “flaw”
a = b
a^2 = ab
a^2 – b^2 = ab – b^2
(a-b)(a+b) = b(a-b)
a+b = b
a+a = a (remember that a=b)
2a = a
2 = 1
Section 11 – Writing in Math
Section 12 – Standardized Test Practice
Section 13 – Spiral REview questions
Section 14 – Reading Math
Learn to use a two-column proof for algebraic manipulation
So what’s the problem here? Is it just that teachers only assign the problems in the first two sections? Clearly, those are the most important sections and they are not dumb, rote, busy work just for speed. You have to understand the basic concepts and see the variations. However, there are many layers of understanding and nobody can fully understand the implications with just a few problem variations. That’s why it’s common for many to not really “understand” algebra until sometime during Algebra 2.
You can get good grades in math, but still feel like you are struggling. It’s not a “zombie” issue. It’s normal. You can get poor grades and not understand, but that’s another issue. If you get good grades and really don’t understand, then probably your grades are OK, but dropping. Some honors (proper math) classes say that you can only enter if your grade is 80+ or some such thing. Gaps and weaknesses in understanding will eventually cause you to fall off the cliff. Math is tough in high school and college. However, it’s NOT that in K-6, but too many kids struggle. It’s NOT an understanding issue.