Common Core-era rules that force kids to diagram their thought processes can make the equations a lot more confusing than they need to be.
One way is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, or where a mathematical rule comes from.
The underlying assumption here is that if a student understands something, he or she can explain it—and that deficient explanation signals deficient understanding. But this raises yet another question: What constitutes a satisfactory explanation?
x = original cost of coat in dollars
100% – 20% = 80%
0.8x = $160
x = $200
Clearly, the student knows the mathematical procedure necessary to solve the problem. In fact, for years students were told not to explain their answers, but to show their work, and if presented in a clear and organized manner, the math contained in this work was considered to be its own explanation. But the above demonstration might, through the prism of the Common Core standards, be considered an inadequate explanation. That is, inspired by what the standards say about understanding, one could ask “Does the student know why the subtraction operation is done to obtain the 80 percent used in the equation or is he doing it as a mechanical procedure—i.e., without understanding?”
For problems at this level, the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious. As the above example shows, the explanations may not offer the “why” of a particular procedure.
Advocates for math reform are reluctant to accept that delays in understanding are normal and do not signal a failure of the teaching method. Students learn to do, they learn to apply what they’ve mastered, they learn to do more, they begin to see why and eventually the light comes on. Furthermore, math reformers often fail to understand that conceptual understanding works in tandem with procedural fluency. Doing a procedure devoid of any understanding of what is being done is actually hard to accomplish with elementary math because the very learning of procedures is, itself, informative of meaning, and the repetitious use of them conveys understanding to the user.
Most exemplary are children on the autism spectrum. As the autism researcher Tony Attwood has observed, mathematics has special appeal to individuals with autism: It is, often, the school subject that best matches their cognitive strengths. Indeed, writing about Asperger’s Syndrome (a high-functioning subtype of autism), Attwood in his 2007 book The Complete Guide to Asperger’s Syndromenotes that “the personalities of some of the great mathematicians include many of the characteristics of Asperger’s syndrome.”
Measuring understanding, or learning in general, isn’t easy. What testing does is measure “markers” or byproducts of learning and understanding. Explaining answers is but one possible marker.
At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.
It is a profoundly erroneous truism … that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.