Wednesday, June 22, 2011

The virtues of explaining your answers

Throughout this blog I've complained about requirements that students explain their answers to easy Reform Math problems. What is gained by making children put into words that which they can easily do in their heads? Why should those who repeatedly get the right answers lose points for not explaining how they got them?

Does that mean that there's no virtue to explaining your solutions? Not when it comes to harder problems, e.g. multi-step problems in algebra and beyond, where there's actually some work to show. Furthermore, although I believe it's quite possible to have a perfectly adequate mathematical understanding without being able to verbalize what you're doing (consider all those language-impaired, mathematically-gifted children on the autistic spectrum for example), I also believe that there are different levels of understanding, and that the ability to verbalize things indicates something about how deeply you get it. When it comes to math in particular, it strikes me that there are at least two major dimensions to understanding: one is how abstract you can go; another is how well you can explain it to others (a third might be how well you can visualize it).

Throughout my middle school and high school years, the resident student math genius would constantly pester me about why I bothered taking math classes when it was clear to him that I didn't really understand what was going on. He couldn't appreciate that I understood things at a level that was meaningful to me, if not to him.

Weak though I am compared to math buffs, I think my relative strength is in being able to explain how things work to others (again, not in that inane "explain your answer" sense, but more in terms of verbalizing math concepts; of verbalizing what's going on logically, as I try to do when I teach math to my kids). Driving this home in a very odd way have been math classes I've taken with classmates who outperformed me on tests, but would turn to me for verbal explanations of what was going on. I helped them out, and then did ultimately did worse than them overall.

Given a choice, I actually prefer to have strengths in the ability to verbalize rather than in the ability to solve really hard problems. I like being conscious of what's going on--and, as a linguist, I've come to believe that, for those of us who aren't language impaired, there's a very strong connection between verbalizing things and being fully conscious of them.

I wonder whether those who can solve really hard problems but can't put into words what they're doing are truly conscious of what's going on. Again, I mean something different here from the idea that if you can't explain your answer you don't know what you're doing. If you can solve hard problems, at some level you must know what you're doing; you just may not be fully conscious of it, and that lack of full consciousness doesn't really matter unless you happen to be a teacher and it impairs your ability to teach others.


Brian Rude said...

I think it was in about fifth grade when one day the teacher told us, "If you can't say it, you don't know it." That always stayed with me, not because I believed it, but because I questioned it. I wondered if it were really true. It's totally understandable that a teacher would say that, but that doesn't mean it's really true. Over my lifetime I have given the idea some thought and concluded that it's not a simple matter. There are times when explicitly and precisely verbalizing what we are learning is very important. There are other times when verbalizing what we are learning is practically impossible and certainly not worth the time trying. Language is obviously a very powerful tool that we ought to develop as much as we can. But it is definitely not true that thought is only linguistic. With a little reflection one can come up with plenty of examples of nonverbal thought

I have a few years experience teaching college math. Many times a student would come to my office for help. A common pattern was repeated many times. After getting oriented to the students difficulty (not always a quick or easy task) and selecting an appropriate problem to work on, I would be struggling to find just the right words to explain something, when the student would suddenly say, "Oh, I get it!". That would be my cue to shut up. Language is a wonderful tool, but a lot of thought consists of assembling ideas together in certain ways. When helping students I use language identify the mathematical ideas needed and assemble them in a way that will apply to the problem at hand. But the actual assembling of ideas itself is not linguistic so much as conceptual.

Most of us are quite adept at using a computer word processing program for writing. That involves a lot of learning. Were we forced to verbalize each step along the way when we learned? Can you verbalize everything you know about writing with a computer? Would it be beneficial to try to verbalize all that? What about driving. Have you ever tried to verbalize everything you know about driving? Would that be beneficial?

All this is not to say that language is not important. Obviously it is. But you can over do a good thing. I have been aware of the "explain your answer" fad in math education, but have always considered it just that, a fad. In a math class I think "explain your answer" should be translated into "show your work". And writing in a math class should simply mean "show your work".

So I am very much in agreement with what you say in this post.

Katharine Beals said...

"Have you ever tried to verbalize everything you know about driving? Would that be beneficial? "

Only if I'm trying to teach my 17-year-old how to drive, which, fortunately, I'm not yet having to do.

Even worse (here I'm thinking of my younger son): explaining how to carry on a normal conversation.

The hardest stuff to verbalize is that which comes most naturally--at least to neurotypicals. Unfortunately, some of this doesn't come naturally to people on the AS spectrum, and those who try to teach these skills to them are highly handicapped by this very phenomenon.

kcab said...

I think the very best math students are able to explain their answers as well as work the problems. In reading a couple of books recently, "Count Down" and "Perfect Rigor", it seemed to me that the systems described for developing top math contest competitors had a lot of emphasis on teaching kids (not necessarily neurotypical) to explain their thought processes. My own mathy child is extremely good at explaining work, particularly aloud to others (the process of writing it seems tedious to him at times).

I think there is something to the idea that being able to explain one's work requires understanding a problem in a different way. I wonder though, whether anything is lost in the translation to words - is it necessary sometimes to hold off on the verbalization for a bit so that the problem can be completely seen? Wasn't there something about verbal descriptions of a memory altering and weakening the memory itself? I wonder if other non-verbal information is similarly affected.

ChemProf said...

In General Chemistry, I teach basic quantum mechanics. I occasionally get a very verbal student who wants me to explain why atoms act the way they do at a more fundamental level. But at that fundamental level, you are really talking about mathematics (or a description of mathematics -- saying "a superposition of basis sets" to a student who is in Calc I is not too meaningful). A few have been insistent that I must be able to explain in words, since "if you can't say it, you don't know it," and I've had to shut them down with a little bra-ket notation! So, there are a few things that aren't easily explained in words. And the people I know who really internalize quantum (I am not one of them) seem to do so in some non-verbal way.