What teacher — especially teachers of gifted children or “visual spatial” learners — has not become exasperated with a student’s obstinate refusal to Write Out That Math Problem?
I can’t deny that I am often impressed with what people can accomplish “in their heads.” I am fine at math — well above average, if you trust my GRE score, which is rather dated by now — but I can only skip so many steps before the numbers get all tangled up.
We are doing one last “review” before Violet goes to camp. To put it all out there: in math I consider 80% correct answers to be sufficient “mastery” before moving to a higher level. When Violet doesn’t hit this mark, much of the time it is her habit of doing things in her head and relying on intuition that trips her up. In one way I can’t fault her — if it works almost 80% of the time, wouldn’t you trust it too?
I have many toss off lines I use to cajole “writing it out.” I have stopped short of requiring it, and she has actually had periods where she has willingly written everything out on her own because she felt that it would help. But if not I might say,
“You would have gotten those 3 problems right if you had just written them out,” or
“You can’t check your work if you don’t write it out,” or, as I said tonight,
“It may not seem like a big deal to miss a quiz problem because you didn’t write it out, but imagine if you were a scientist or architect. Then a little miscalculation could cause a big problem!”
Apparently Violet could immediately see how pathetic this reasoning was, and quickly responded,
“I guess that’s why I don’t really have any plans to be a scientist or an architect.”
I assume it happens with all math students that someday, in some subdiscipline of mathematics, they start writing it out. And for now I still don’t plan to force it. If we get to a place where she could get partial credit for getting some of a problem right, that might inspire her. Actually, I think we’re at that place now . . . wait . . . has writing it out just helped me figure out this problem? Ah ha!
I don’t want my kids to be motivated by “points,” but I don’t mind giving a little carrot to encourage them to experience the benefits of something they are not doing on their own. We don’t study math because our children might grow up to be scientists or architects — though one reason we study math is so that door remains open regardless of short-term preferences. One reason I insist on regularly studying math is that it is a great way of training the mind in logic and in language, as Elizabeth Foss suggests in Real Learning. I don’t know a lot about math as a discipline, but I have always been impressed by the suggestion that it is a particularly elegant (in the scientific/mathematical sense) form of language. Apart from acquiring the mathematical literacy that might save them from sub-prime lenders and Sam’s Club bulk faux-bargains, I imagine that studying math helps develop precision of thought in general.
A side note: did you know that a study of law students determined that the most successful (as measured primarily by grades and earning a place on the law review) had majored in one of three subject: English (duh), classics (interesting, but sort of related to English), and math. What do these things have in common? Maybe it’s an interest in expressing ideas effectively. (Or maybe they just attract smart people who can’t figure out what to do after earning a BA . . . )
There are at least superficial similarities between math and English: inexperienced students often have to be reminded to rely on the text. Does the diagram *say* the lines are parallel? How do you *know* that’s a right angle? What words in the text lend credence to the suggestion that Hamlet and Laertes had a little something going on the side? We don’t go with our gut on these things. In both mathematical equations and written expressions, we simplify, simplify, simplify (unless you’re blogging). The final answer is 1/2, not 24/48ths, and “thing used dig food” is generally considered less elegant than “spoon.” But we don’t oversimplify either: if a measurement requires using Pi to 5 places, so be it, and if “eschatological” is really what you mean, then say it.
But I digress. And I think digressions are not very mathematical, though a hallowed literary tradition.
My guess is that the effort to give math a proper place in our homeschool lives will be never-ending. How much time is too much? How challenging is too challenging? Why insist on it at all? We’re not content to do it just because we ought to do it, yet it may well be true that for one or both of my kids, the “usefulness” of math study will often seem indirect, and the fun of it may never compare to the fun of reading a dictionary (new Chinese dictionary came home today!), playing the piano, or studying history — let alone making comic books and rehearsing for plays. Maybe now’s the time to accept trial and error and a little intuition about the necessity of math for reasons I can’t quite write out as the MO for the next 10 years or so.