Over at the First Excited State, the quasi-anonymous proprietor laments the tendency of basketball replays to focus on the shot rather than the play that set up the shot, and compares this to a maddening student habit:
Students in introductory physics classes inevitably place too much focus on the final numerical answer of the problem, which in reality is the least important part. I graded a quiz last week where I spent way too much time trying to decipher the numbers the students wrote down, because they placed the numbers in their equations rather than writing them clearly with the symbols representing the quantities in question. Grading problems written in this way is like trying to analyze a basketball play from the replay that only shows the shot. It can be done, but it requires more effort on the part of the grader. It’s also a bigger risk for the student, because if they don’t get everything right, it’s harder for the grader to assign partial credit when there’s no symbols to show exactly what they’re doing.
This is something that I think most physics teachers try to drive home: the setup and the problem-solving process is more important than the final answer. Basketball coaches drive home the idea that focus on execution of the play is paramount and that the scoring will take care of itself. But the fans always focus on the shot, and the students always want you to tell them if they got the right answer.
I’m once again teaching introductory mechanics (out of a new textbook, though, which is both good and bad), and I’m once again going to be forced to grapple with this problem. Anybody who teaches a class in the physical sciences will– students have an amazing insistence on plugging numbers in as soon as possible, and then manipulating six-digit decimal numbers for the rest of the problem, which makes it all but impossible to tell when they’ve transposed two digits in the second line of a fourteen-line problem, leading to an error that’s off by 15% for no obvious reason.
As it happens, I do know a cure for this problem. It may be worse than the disease, though.
The only way I know of to get students not to plug numbers in from the very first step is to give them problems that don’t have numbers in them. Rather than a 1000 kg car moving at 21.5 m/s, it’s a car of mass M, moving at a speed v, and the end result is some algebraic expression.
This is the only method I know of for forcing students to do algebra. Even that isn’t foolproof, as I’ve had students make up completely arbitrary numbers to plug in just so they could manipulate numbers through the whole problem. The ingeniousness of students looking for ways to get problems wrong knows no bounds.
The problem with this method, of course, is that students hate it with the burning passion of a million white-hot suns. If you think they get unhappy when they don’t have the exact numerical answers to work toward, just wait until you see their reaction to no numbers at all.
My compromise is to work in one or two questions without numbers every test, and to always work example problems and questions asked in class algebraically. I also try to say at every opportunity that the exact numbers aren’t important until the very end, but that just gets lost.
An interesting alternative approach, and one that I’ve tried with mixed success as a way of making lectures more participatory, is to assign problems where the students have to supply the necessary numbers themselves. A lot of newer textbooks include questions that ask students to estimate values for something based on their own experience. That way, there really isn’t a numerical answer that they can check in the back of the book, but there’s still the option of working with numbers all the way through, for the deeply symbol-phobic.
This is really only a problem for the intro classes, though. As soon as we get past the courses required by the engineering majors (say, the sophomore modern physics class I taught last term), I flip the balance in the other direction– only one or two questions per test have numerical values all the way through, while the bulk of the credit is for questions with algebraic solutions. I figure, if the engineering students want to continue punching 47 things into their calculators all the time, that’s somebody else’s problem. Anybody who’s going to major in physics, though, will need to know how to work in purely symbolic terms, and the sooner they get used to that (or change majors) the better.