"If a student can consistently solve a variety of problems, that student likely has some level of mathematical understanding. Teachers can assess this more deeply by looking at the solutions and any work shown and asking some spontaneous follow-up questions tailored to the child’s verbal abilities. But it’s far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way. At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone."

This has produced quite a bit of chatter across the blogosphere. I initially had a strong instinctive reaction. Personally, I find these written explanations to be tedious and not very useful so its not a technique I would use with others either. However, I acknowledge that my own feelings are often in the minority and more interestingly I tend to be more progressive in practice when dealing with a room of the kids than I am in private when imagining these type of pedagogy questions.

Thinking about this some more, I think one can disregard the straw man position where these type questions are overused, or not handled effectively. One can find bad teaching any where that misapplies a good tool. So let's imagine, a teacher is only asking these questions occasionally to truly confirm understanding, is generous in reading the answers, provides appropriate feedback and adjusts based on the information discovered and doesn't mandate a particular format. Reluctantly I think there is probably a place for such a technique. The key is to use it carefully. I also think that a lot of the same effects can often be had just through regular classroom interactions where good questions are asked back and forth and where you listen carefully to what the students are saying. Likewise, where ever possible I think understanding should be demonstrated through problems. Usually these discussion are framed around fairly simple problems that are mostly straight procedure like:

"A coat has been reduced by 20 percent to sell for $160. What was the original price of the coat?”

I'd much rather see conceptual understanding of percentages pushed through a series of more difficult and interesting problems prior to requiring any written queries. Likewise, any written explanations ought to be directed first towards proof building skills. Its a lot more valuable to have students explain how they made a break through, than to parrot back a procedure they've already learned (and this is subject to blind memorization just as much as anything else).

Bringing this back to my experiences with the math club because my first priority is making a fun exploratory environment, I will never require such explanations. But I do find myself having many conversations about why a technique works. For example, our recent dive into why the divisibility for nines works which involved me asking a series of questions while working through the basic idea i.e. what does it mean to be a multiple, if you add a multiple of 9 to another multiple of 9 why is it also a multiple of 9 etc.?

I'm always looking at the kids work and questions for evidence of what areas they don't quite understand yet and thinking about how to circle back if necessary. Having said this, of course, I have the immense luxury of not needing to hit any curriculum and if I miss something I can rely on their regular teacher's to fill the gaps.

I also have no illusions that I've mastered the technique of lecturing yet. I tend to minimize the amount of work I do on the whiteboard in front of the kids as a result. Every time I engage in a longer talk I think about it afterwards and worry about whether the kids followed along with me. Asking questions during a talk helps and having kids demo parts of the problem on the board also helps but neither is perfect. In sum, I don't have the perfect answer yet here and its something I will worry about and continue to think about how to improve.

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