"How is a teacher, who uses formative assessments, supposed to know what a child truly understands or is grappling to understand, if they don’t have a window into their thinking?"
"By giving them truly challenging problems and seeing if they consistently get the right answers!"From a recent exchange I had on an Education Next thread on whether students should be required to explain their answers.
It strikes me that there are two things about Reform Math assignments that cause today’s educators to be so insistent that kids explain their answers:
1. Reform Math problems, in comparison with traditional math (and overseas math, including Singapore Math) are often so easy that there’s no work to show. The calculations are relatively simple, and the word problems usually don’t involve more than one or two steps. Many students, unless you instruct them otherwise, have no reason to write down anything beyond the bare answers. Teachers, meanwhile, remembering how important it was to show their work back when they were in school, may conflate showing your work with explaining your answers.
2. Reform math assignments often involve no more than a half dozen problems, again with relatively easy calculations, where traditional (and overseas) assignments involve dozens of problems with much more complex calculations. It’s therefore conceivable for a student to get every problem on a Reform Math sheet correct by chance rather than by understanding. Getting all correct answers for several dozen complex calculations without understanding what you’re doing, in contrast, is so unlikely that, when students accomplish this, then it’s reasonable to assume that (unless they were cheating) they know what they are doing.
Two additional factors underlie Reform math’s insistence on explained answers. One is the notion that, regardless of how much your math skills exceed your verbal skills, if you can’t communicate your mathematical thinking then that thinking must be deficient. The other is that “mere calculation” has little to do with “deep understanding.”
These notions, of course, also help justify the educational malpractice of restricting students to problem sets in which the calculations are a fraction of the frequency and difficulty they once were and still are nearly everywhere else in the developed world.
I recently showed the Singapore Math curriculum to a student visiting from Mainland China, and the first thing she observed was that the 4th grade Singapore Math problems were problems that she and her classmates routinely did in first grade.
"You must have gone to one of the top elementary schools," I said.
"Not one of the top schools; just a pretty good school" was her reply.