Life is not a word problem
Our tutoring consultant continues to gain insights from his students.
Our consultant writes:
Having worked now with students from Middle School all the way through undergraduate level, I have seen quite a few different ways of learning. It struck me recently that, by the time I see a student, he or she has already been in school for many years and has developed strategies for dealing with its various aspects. These range from a simple method to do a math problem on a test all the way up to the general approach to school (which is much more than just the sum of all the classwork). And (this is important) all the strategies have worked. Even the method of multiplying or dividing every number in a word problem in turn until you get one of the multiple-choice choices will yield the right answer sometimes, often enough for some students to retain it. It’s no way to learn the class material; in fact it’s a way to avoid learning it; but it can work for the student’s immediate purpose.
Well, schoolwork gets progressively more complicated and sophisticated as one goes along. At some point a strategy that has been perfectly fine stops producing the right answer often enough–or at all. Then it becomes my role to suggest, as tactfully as possible, than a different approach is called for. It’s hard for the student to change “the way you do this kind of problem” when it always (or often enough) worked before. It means unlearning as well as re-learning, and that can be painful.
It’s harder the further along you are. In fact, for a student at the undergraduate level it can be quite as traumatic as (say) the replacing of Aristotelian physics by the Newtonian version in the seventeenth century, when a familiar and well-understood way of thinking about the universe had to be discarded and an entirely new one developed. The one saving grace I can see is that students are still young and flexible, so in fact it happens often and without ill effects.
The question for the student is how to tell the coping strategy from actual understanding. It’s not too hard when the strategy is as simple as multiplying numbers together until something familiar shows up. But a student may actually believe that matching a certain pattern in physics problems is the point of the course, rather than developing a thorough grasp of F=ma.
The question for me–and the one I put to you–is: which of our ways of thinking are actually coping strategies, rather than understanding? And how do you tell?
Edward Celarier
January 4, 2017 at 7:55 pm“… all the strategies have worked. ”
Not to put too fine a point on it, but this begs the question, What do you mean by “work?” If the metric is “student passed a multiple-choice exam,” fine. But should that be the point of education? I mean, more than it appears to have now become?
Certainly, life is more like a word problem than it is like a multiple-choice question, no? As you say, there is a distinction between a coping strategy and actual understanding. I would say this even goes for multiplying numbers.
Somewhere between the old math and the new math (using the Lehrer summary) and something that the new math morphed into since, there may be a point where the purpose of learning is explicitly for the student to have a kernel of reasoning and understanding. Odds are against it.
One could perhaps even wish for the student to appreciate the fact that, at any point in one’s formation, one does not actually understand fully. What, after all, *is* a “thorough grasp of F=ma?” You can use it repeatedly and reliably to solve a variety of problems–even ones that happen to appear on a multiple-choice exam–but that doesn’t mean you understand it in any deeper sense of the word. I’m not arguing that we should teach the deeper meaning, but we can always tease the student by asking, “Why *should* F=ma?” not because we expect him to answer, but because we want him to learn that there is something underneath the expression to think about. Should one not at least understand how little one understands? The alternative is to foster a delusion that one’s education in some area, at some time, is somehow complete.
I’ll leave you with the thought that, when I was in university, I had some professors who would write multiple-choice answers that could all be got from the data, but with different common mistakes (e.g. incorrect sequences of operations) having been made. Stumbling upon one of the listed answers in no way ensured that one had stumbled upon the correct answer.
fivecolorssandt@icloud.com
January 5, 2017 at 10:17 amSuch a thoughtful comment deserves more than a short response, so perhaps another blog post is in order. But for the moment:
I meant the strategies were successful from the immediate perspective of the student, in that they have allowed him or her to pass tests and be promoted to the next grade. I didn’t mean to imply that such “success” was desirable or something teachers should work for; in fact, rather the opposite. My point was that it’s hard, sometimes, to convince a student that something that is apparently successful should be discarded.
I have been working at a far less sophisticated level than I think you’re contemplating. By “a thorough grasp of F=ma” I meant only being able to take the step from the free-body diagram to writing down an equation, as opposed to looking through the class notes and finding gsin(theta) minus mu cos(theta) from a different problem and using that.
And, yes, the standardized tests are constructed so that alternative (wrong) answers are those produced by common mistakes, so many good-until-now strategies fall foul of them. That is one useful feature of these tests (and the preparation material we have for them).
The question of what sort of understanding we should be trying to produce in our students is much bigger, and I won’t try to address it now. It is certainly worth pondering, though.