“Show your work” means “Write a proof”
Sunday December 12, 2021
At any level of mathematics, “showing your work” should not be a chore, but a chance to communicate an argument for correctness: to write a proof. “Work” as “proof” keeps mathematics relevant, authentic, and interesting.
Many school math exercises can be answered with calculators. But how do we know the calculator is correct? The difference between inductive (“the calculator has always been right”) and deductive (“this answer is proven correct”) is important. The answer itself isn't important; it's the demonstration that the answer follows from the question.
High school geometry is sometimes pointed to as the first place students encounter the idea of proofs. This need not be. Arithmetic is a process of proof, using theorems of single-digit operations and place-based algorithms to build new results. In this way, 2+2=4 is used to demonstrate that 22+22=44, and so on. These could be called constructive proofs.
“Work” as “proof” opens up the world of math. “Checking your work” is finding a second proof. For the advanced student who might have been bored with “showing work,” there is an invitation to find further proofs, understand algorithms more deeply, and be creative, rather than being limited to mechanical processes.
Computational fluency is not without value, but math is about much more. We waste opportunities to deliver on teaching mathematical ways of thinking if students don't realize that they're constructing logical arguments in all their math classes.