Last November, the online Atlantic published an article that Katherine Beals and I wrote on the current trend to have students explain their work in math.
Our main point was that doing so was in many cases superfluous to the mathematical process. That is, showing one’s work in solving a problem can be the explanation itself. Asking for more tends to be make-work and for students in lower grades whose articulation skills are still developing, a waste of time.
There was quite a flurry of both agreement and protest about this article. (Despite all the complaints, however, it will resurface in the annual anthology “Best Math Writing of 2016” which will be released around December.)
There is a back-story to the article. In the article there is a sample of a student’s explanation of a particular problem. I had solicited such explanations while I was assisting math teachers at a middle school. I had given students some instruction on how to solve particular types of percent problems, and then gave them a problem, asking them to explain their answer. I had them use a template that the school was recommending: “Need/Know/Do”.
There was only one student who got the answer right; and that was the one I used and which appears in the article. It turns out that I really should have also used an example of a wrong answer–complete with incorrect explanation, which might have better made the point. (Though I doubt that there would have been any less criticisms of the article).
I say this because of a meta-study conducted by psychologist Bethany Rittle-Johnson of Vanderbilt University. Rittle-Johnson examined 85 studies and one of her conclusions was “If kids are just off explaining their own thinking without guidance, then they can be spending their time essentially justifying stuff that’s wrong.”
traditional math 
General explaining (not proof explaining) works only for one problem. If you do an algebra problem by giving the rule or identity justification for each step (which I had to do way back in the 1960’s), that’s better, but it’s still for one problem. Success on one problem doesn’t mean that you better understand the next problem that comes along. That’s why good math classes use proper textbooks with large individual homework sets. Each set starts with simple problems that focus only on mechanics followed by more advanced variations and approaches. Explaining is not some magic way to avoid doing lots of problems and doing homework sets is not a rote process. All of this talk of explaining comes from the stupid idea that words show better understanding than doing the whole set of homework problems and getting good grades on tests that just have you do the work without extraneous words..
In my son’s old Glencoe Algebra 1 textbook, each unit has homework sections like this: “Check Your Understanding”, “Exercises”, applied problems (such as baseball, geometry, reasoning, open ended, and challenge), “Writing in Math”, and “Standardized Test Practice.” Only a few questions require explaining, and that’s the way it should be. There are different explaining needs for different types of problems. For basic algebra problems, one expects to see the rule or identity that is applied. I had to do that long ago, but it became unnecessary after our first algebra class. General problems require different explanations, but not at the silly, wordy level expected by most ed school pedagogues. In most cases, the math does the explaining. Back when I taught math, I required the rule or identity explanation on homework, but not on tests. It takes too much time and it’s better to use that time to give more problem variations. Teachers are not potted plants in the classroom. They should be able to figure out whether the students they see every day really know the material. This new idea of excess explaining is just a rote ed school meme. It’s worse when it comes to yearly tests when you can’t identify any specific problem like properly manipulating fractions. Yearly tests only offer feedback for a school, not individuals, where any feedback is a year late and many tutoring dollars short.
It’s amazing how there is still a whole world of high school math that uses proper textbooks and traditional methods and how integrated math has lost the battle. These are the classes that create all of the STEM-ready students. They just had to get help at home or with tutors to survive the silliness of K-8 math.
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Exactly, Steve! What the fuzzy math folks mean by “explain” often means go back to a “first principles” which does NOT mean the same as first principles in math — it means explain how you “see” this “intuitively”. That generally does not represent transferrable learning, particularly when these are ad-hoc explanations built on students’ own convoluted thought processes. Humankind advances in knowledge by effective, efficient coding of the best knowledge and understanding of the past into forms that can be passed along to the next generation. Students should see excellent models of the best thoughts of great minds of the past, and they will thus be equipped to advance further with their own original minds. To go back to intuition is like (metaphorically) bombing their education back into the dark ages and demanding that they rebuild the rubble using hand tools. And much as it might “work” in some sense on simple addition and subtraction problems involving a small number of “pieces” it does not provide an appropriate foundation for learning advanced math. And what understanding is “conferred” by this approach does not appear to be transferrable — it is limited use insight, akin to a disposable lighter. Use it, then toss it.
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