An Example of Contextual Understanding

Or "concepts" as they are sometimes called

I’ve been writing lately about “conceptual understanding” really being “contextual understanding” in the early grades, and even in high school. I thought an example might be appropriate, so I’ve taken a figure from a 1948 textbook for fifth grade, called “Study Arithmetics”, showing an alternate method for adding two digit numbers.

In the cartoon, the little girl knows that 27 + 24 is the same as 20+7 + 20+4, which of course is the same as 20 + 20 + 7 + 4. She then adds tens, ones, and then adds the sub-totals.

Some will say "But does she understand the concept?" What concept are we talking about then? That 27 is 20 + 7 and so forth? That 20+7+20+4 can be rearranged via the commutative rule to 20 + 7 + 20 + 4? The concept at play here is ultimately understood in the context of a collection of specific procedures.

Later on in algebra, students learn that a two digit number can be represented as 10x + y, and the sum of two such numbers can be represented as 10x + y + 10a + b, which is equivalent to 10(x+a) + y + b. Is that a concept? Again, the concept is contained within the context of a collection of algebraic procedures.

In learning math, we understand concepts initially in the context of procedures and build upon such foundations. But don't tell anyone I told you.