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?