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.
traditional math 
Reblogged this on Nonpartisan Education Group.
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For those interested, I want to talk about problem solving on a large scale. I have two master’s degrees in engineering (one in computer engineering) and I’ve written over a million lines of mathematical and engineering code over the last 45 years. I taught college math and computer science full time for about 6 years in the 1980’s. One of the courses I taught was called Systems Analysis, which is about problem solving on a very large scale. The textbook I used introduced about 50 different analysis and design diagramming techniques. Over the years, I’ve used everything from flowcharts in the 1970s (I still have my template) through pseudo-code, data flow diagrams, and now into UML (Unified Modeling Language), where I can identify users, use cases, and draw lots of pretty pictures. There is no one great solution or diagramming technique. And don’t get me started on “Agile Software Development” or SCRUM. Note that in systems analysis, many of the tools end up being useful for documentation, not analysis. In one project I was on, management really believed in data flow bubble charts for analysis, but we programmers always drew them after we got the code done. Too many tools fall into the documentation, not analysis category.
In a data structure class I once taught, I wanted to show the students how to program just one routine to do an insertion into a B-Tree. I could spend time to do it all at home and then present and explain the clean final version, but they wouldn’t see the thought process I went through. I decided to not to prepare at home, but do it on the board from scratch while they watched. It failed. They couldn’t follow my thought process. I was too far ahead of them and I couldn’t simulate their level. I could have used pseudo-code, but pseudo-code only helps on a conceptual level because there is a lot of hand waving – just like when words are used to describe math problem solving. There is a huge (non rote) distance between pseudo-code and the solution, so my goal in class was to do the exact routine. Since that didn’t work, how do you teach problem solving? It’s not words or concepts, but experience and practice.
This doesn’t mean that no structure works, but there are limits. Traditional systems analysis is based on a strict top-down approach: Problem Statement, Analysis, Design, Code, and Test. Set up your project time line in advance (when you have no clue) and then try to keep to the schedule. The big problem is that you can’t know what you are doing before you actually do it. This leads to what many call “analysis paralysis”, where you always come up with new ideas or you want to get it right the first time to avoid changes later in the process. This is a difficult trade-off between paralysis and screw-up.
The opposite of Top-Down is Bottom-Up where you just dive right in and learn while you are doing a project. This is a prototype approach where you begin with a stub of a program that doesn’t do anything, and then you start adding pieces, testing and getting feedback as you go. The problem is that you can invest too much on a clueless wrong path that can be very difficult to correct.
When my son and I went to MIT for an open house tour, they really pushed their “hacking” meme and history (it was rather annoying), showing us when students turned the Green Building into a very large Tetris game. However, they slid right over the fact that these students were taking traditional courses with huge amounts of p-sets. It’s one thing to hack or prototype solutions on top of good fundamental learning of separate subjects and skills, and quite another to use Project Based Learning to learn the basics. This is where educational pedagogues get it completely wrong. PBL = hacking learning = vocational learning = not a proper education. You might get to the ‘T’ in STEM, but that’s it.
I love prototyping for solutions, not education, but it has to be built on top of mastery of base knowledge and skills. And general hacking is not enough even on top of base knowledge. I use what I call an Outside-In approach to systems analysis, where you start with limited top-down analysis followed by defining general low level tools that can be combined in many different ways. Then you start from the bottom up to define and test those general tools to create a prototype system. My latest big project, started in 2014, is built that way and I’m in the process of finishing a third working prototype after doing a lot of learning and discovery. It will soon be offered as open source software. Even now I have new insights that clearly show that a top-down solution would never have worked, but a dive right in bottom-up hacking approach wouldn’t work either. PBL is a nice idea, but engagement will evaporate quickly when it runs into too many JIT learning roadblocks. You can spend a LOT of time on problems that have simple known solutions – solutions learned when you develop separate subject and unit skills. Engagement will never overcome that. Just try to learn and implement a B-Tree structure from scratch while trying to get a project done. A hacked solution might show that something can be done, but it will be light years away from a proper solution. You end up with clueless engagement success or deep anti-engagement failure.
The problem with PBL, as I saw with the First Lego League, is that time-based projects induce hacking with no time allocated for proper learning. The best organizations were the ones with parents or coaches who established low level base skills and tools and ensured that students built solutions on top. PBL never works from scratch in a new area or subject. The successes become robotics clubs that just happen to be involved with the competition. In education, this means skills and knowledge in class and projects or competitions after school. PBL in class is guaranteed to fail.
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