Conceptual Understanding vs Vague, Transitory Notions
“Conceptual understanding” is one of those terms, that despite many definitions that have been provided, often remains more as Andrew Old once described it: “What people consider to be an important concept to be understood, is often just a notion that is vague or transitory.”
I view conceptual understanding, in lower grades, to be more of a “contextual understanding”. Students are given the context for which a procedure takes place, so that they see what it is that the procedure is doing mathematically. If the concept is “place value”, this is demonstrated by actual examples. Case in point is the long division algorithm, now frequently derided as “unuseful” because everyone uses calculators anyway. But ignoring that last criticism, long division (like other procedures) can indeed be taught with the concepts accompanying the procedure.
That's how I learned it. I have attached a picture illustrating how it was taught in the the arithmetic book I used in 4th grade (Arithmetic We Need, 1955, by Buswell, Brownell and Sauble). But even when taught using correct place value language, and students are made to say “We divide 400 by 4 to get 100, which we write as 1 placed over the 4…” kids ultimately glom onto the procedure. The context is illustrated in the picture and it does work to explain what is being done mathematically. But chances are fairly good to excellent that not much time passes before they do what is advised in the illustration; that is, they think “4 divided by 4 is 1, subtract, bring down, and divide the next number.” (How many people who took calculus and maybe even those who have PhD’s in math for that matter take the derivative of a quotient by saying “Hi-dee ho minus ho-dee hi, over ho-ho”?)
Over time, those students who may not have fully understood the concept when first presented with the procedure, may come to better understand the underlying concepts of place value in long division. And still some others may never fully understand (cut to teacher saying “We won’t move on till you really understand”). But they will likely know what division represents, what type of problems are solved with it, and when and how to use the procedure when necessary, (i.e., when a calculator is not handy) because of the contexts in which they originally learned it.
