# The Back and Forth Between Procedural Knowledge and Conceptual Understanding

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“Procedural skill” and “conceptual understanding” are typically set off each other as polar opposites. We have a perceived dichotomy in the mathematics education between the two. Generally, one might believe in the priority of one over the other. Or, rather, that is the public perception, and the perception of some educators. Swing too far in either direction, toward either pole, and you might feel like you are not teaching the right way.

To be clear on my own stance on teaching K-12 mathematics, I believe there are very few, if any mathematical ideas that shouldn’t be presented in a powerful visual and conceptual way. Put it this way: don’t show me the Pythagorean formula, let me play with triangles, and then show me the Pythagorean formula. In that one hour block of time, I will then have a visual anchor for the formula, and I will be ready to use the formula. The geometric representation takes priority over the algebraic abstraction in this case. We move from the representational to the abstract, and we can then understand an algebraic relationship that is always true for all right-angled triangles.

But is it wrong to just write “a squared + b squared=c squared” and then explore that? I would argue it’s a lot less interesting, but perhaps not wrong. You can get to the same place, in the end. You can play with the algebraic formula, take it apart, and see how it works. You can then see that relationship is always true for all right-angled triangles.

The swinging back and forth between the procedural and conceptual conditions is my understanding of the main point of the Rittle-Johnson, Siegler, and Alibali paper, which all math teachers should read. Their experimental design deals with decimal fractions on a number line, but I believe their findings have implications for all mathematics teaching and learning.

This paper could be an olive branch held out between “procedures first” and “concepts first” adherents. I see it as a uniting text, and would welcome further work in this area, using different concepts in mathematics.

It’s very important to note that their experimental design involved placing decimal fractions on a number line. The number line is itself a representation of space, and the numbers in the spatial intervals given (the “ticks”).

This experimental design is probably testable and replicable, over and over again, across many different topics in mathematics. Multiplication springs to mind, and although any experimental design wouldn’t be so simple, consider this for a moment:

- you could “just” teach the standard algorithm, and then play with different conceptions of multiplication (scaling, for example).
- you could teach the standard algorithm using partial products (the area model), and consolidate it into the standard algorithm.
- you could spend some time exploring mental multiplication, skip counting, the area model, grouping, and so on, and finally, a while later, get to the standard algorithm.
- you could deploy the standard algorithm as one strategy among many, and play with different strategies.

For me the big promise of the authors’ iterative model is how tight the iterations could be, in time. If they are right, any given mathematics period (let’s say a 60 minute chunk of time) could feature a number of back and forth “moves” that are taken between procedural and conceptual understanding. The orchestrating of these moves: that’s the art of pedagogy. The art of pedagogy is the minute by minute decision making that teachers do, responsive to the thinking of their students. It’s what we are best at. It’s what we are trained to do, and what we practice every single day.

Conceptual and procedural knowledge did not develop in an all-or-none fashion, with acquisition of one type of knowledge preceding the other.

My takeaway is it’s probably best to not downplay either type of knowledge, but to acknowledge the complex bidirectional relationship between them. It’s never going to be an “either/or” situation. One will not strictly precede the other at all times. Our pedagogical moves should reflect this complex interplay.

There are times in mathematics to play with big open ideas, and there are times to apply procedures. Our ultimate goal is better problem representation- having our students in the state called “understanding” where they can bring all their powerful knowledge to bear on the interesting problems and tasks we give them.