Sorry to keep harping on the Education Trust “study” that finds middle school assignments lacking in the “conceptual understanding” department. In the Education Week article on the “study” (and no mention whatsoever in the study of what mathematicians, engineers and/or scientists were consulted in writing it) a commenter agreed wholeheartedly with it and said:

And I COMPLETELY agree that my experience in MS (and HS) classrooms focuses way too much on procedural over conceptual understanding.

Which caused me to wonder: Just what conceptual understanding do people think is missing from middle school math? It isn’t that students are just given problems to solve without explaining what the concepts are. Percentages are explained, as are decimals, as are fractions, and why one uses common denominators, and even the why and how of multiplication and division. Anyone who teachers middle schoolers or even high school students, knows that students gravitate to the “how” rather than the why.

SteveH who comments here frequently notes the following about procedural vs conceptual understanding:

Kids LOVE being good at facts and skills. They are easier to ensure and test, and that success drives engagement and much deeper learning and understanding. Skills come before understanding and engagement, not the other way around. The best musicians are the ones with the most individual private lesson skill instruction. They didn’t get those skills top-down by playing in band or orchestra only. The process is difficult and many don’t like it and drop out. This is true for any real life competitive learning – it is not natural.

Such disputes are usually settled in the same manner as the one about whether inquiry or direct instruction is better, and someone says “You need a ‘balanced’ approach” without defining what that balance is. In the argument about understanding vs procedure, the usual bromide is “they work in tandem”. This tells us absolutely nothing.

Sometimes the conceptual understanding is part and parcel to the procedure like place value and carrying and borrowing (two terms for which I make no apology for using). Other times it is not.

Having understanding is only part of the process. If I may talk about calculus for a moment. In upper level math courses in college, one learns the concepts behind why calculus works–including the delta-epsilon definitions of limits and continuity. A student may be able to recite the definition of continuity and tell you what needs to be done to show continuity at a specific point in a function.

(I.e., a function *f* from R to R is continuous at a point *p* ∈ R if given *ε* > 0 there exists *δ* > 0 such that if |*p* – *x*| < *δ* then |*f* (*p*) – *f* (*x*)| < *ε*.) But having a student prove that the function y = x^2 is continuous at the point x=2 involves a procedural knowledge of how to go about doing that. Just knowing the theorem is not enough.)

So I’d like to know. Do these people who claim they focus too much on procedure think that the majority of middle school students are just operating blindly as “math zombies” as some bloggers like to call it, without any knowledge of what it is they’re actually doing? Really?

Reblogged this on Nonpartisan Education Group.

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“…my experience in MS (and HS) classrooms focuses way too much on procedural over conceptual understanding.”

Conceptual, for the nth time is: “based on an abstract idea; a general notion.”

All proper traditional math classes introduce concepts on a unit basis and an overall math basis. That is not enough. Proper and full math understanding is achieved on a skill or unit level with mastery of p-sets, and full understanding of the bigger picture is obtained from the mastery of skills of problem solving – again, p-sets. This has to be done from the bottom – unit skills to the top – problem solving skills.

I did not properly understand algebra until part way through my Algebra 2 class. I think that is quite normal. It takes a lot of homework. I’m still working on my problem solving skills and understandings after 40+ years of programming complex math and engineering software.

When these people talk about problem solving and understanding, I NEVER hear them talk about knowing specific things like when and how to use explicit, implicit, and parametric forms of equations. I used to teach lectures on how to “see” all equations and factors. But there is no one (small) set of concepts or problems that creates a magic transference ability. It takes a lot of hard work, but the process is made easier by mastery and understanding of basic skills. Bottom-up, not top-down.

Their big lie is that mastery of basic unit skills is rote and that you can swap some of that time for more conceptual (?) work.

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So, what is the problem, lack of “conceptual” understanding or problem solving ability/transference. Will more conceptual understanding fix transference? Is it a matter of just getting really good at Polya’s 4-step process that is so vague and general as to be worthless. In all of my engineering education, I never heard of Polya.

I would pay more attention if educators at least put forward any ideas that made mathematical sense. I don’t hear educators pushing the “Beast Academy”, AoPS, and WOOT. That’s what schools should offer as after-school programs while spending class time pushing and enforcing mastery of the basics. All twelve of the 2017 USAMO winners are WOOT alumni.

However, I’m not a big fan of timed math competitions, and that’s why they should be opt-in after school programs. In-class teaching should push and enforce basic (STEM-level) math skills and understanding. High schools do that in math (AMC prep is opt-in), but not K-6. The lower grades want to flip it around and do meaningless group in-class Polya-type problem solving. Pushing and mastery of the basics is now left up to parents and tutors.

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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.

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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.

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