We Must Teach the Concepts, Dept.
517th edition
For those of you have read Traditional Math, or even just glanced at the lesson re-creations in the 7th and 8th grade parts of the book, "concepts" are taught in tandem with the procedure. That is, I provide a context for what the procedure is doing mathematically; e.g., a diagram of how many 3/4 inch pieces of ribbon go into a 12 inch piece, to illustrate why this is equivalent to 12 x 4/3. Or I show that the equation 2/3 = 4/6 when multiplied by a common denominator of 18 is equivalent to 6 x 2 = 4 x 3, which is also equivalent to what one obtains when cross-multiplying. (See pp 213-214 of Traditional Math.)
“Conceptual understanding” in early stages of learning is more like “contextual understanding” which provides the context for what the procedure is doing mathematically. Such explanations then lead naturally into the procedure. I don't obsess over the conceptual part, knowing that the students are thinking “OK, just show me where this is going and what I have to do to solve the problems for homework”. Most teachers do this, though there are plenty who feel obligated to spend a lot of time on teaching the concepts, and when students ask “We get it; can we move on?” they respond “Yes, but do you really get it?”, before spending even more time on it.
The textbooks they are given to use will have students drawing pictures and using convoluted ways to solve problems without using “cross multiplication” or whatever standard algorithm/procedure would work best.
What concepts are in the early grades, when students do not have a lot of information at hand, is the acquisition of procedures that over time can coalesce into a broader and more general and abstract understanding. For example, the concept of division is summarized as a/b = a*1/b where b does not equal 0, and a,b are in R (as in reals). Students can understand that probably by algebra 2 (although it is presented in Dolciani’s Algebra 1 book).
Students at that point are more capable of understanding this abstract explanation of division and how it explains why we invert and multiply when dividing fractions. This is because by then, they have mastered the procedures of fraction multiplication which leads to the understanding that a/b * b/a = 1, which then leads to the understanding that 1/(a/b) = b/a. The understanding comes with experience, practice and instruction of the procedures just mentioned that lead up to it.
It is not as if we are teaching procedures in a vacuum, saying “Just do this when you want to solve problems like these.” When I explain what I just did above to those steeped in the math reform edict that “students must be taught the concepts or they will die”, I often hear: "Of course! We're both saying the same things."
Except that we’re not, but that gets into “nuance” which I don’t want to talk about.
No concessions should be made to reformers—conceptual understanding has no place in the discipline. The key difference between traditional and progressive approaches is that the former is fact-based, while the latter is concept-based. Facts provide the solid foundation for understanding.
An analogy: think of knowledge as a raster image, where pixels represent facts. If the image is low-resolution or incomplete, understanding comes from acquiring more pixels, not by guessing the missing parts. Guesswork, like in whole language reading approaches, is not a valid path to knowledge.