Anybody who has taught introductory physics has noticed the tendency, particuarly among weaker students, to plug numbers into equations at the first opportunity, and spend the rest of the problem manipulating nine-digit decimal numbers (because, of course, you want to copy down all the digits the calculator gives you. Many faculty, myself included, find this kind of maddening, as it’s pretty much the opposite of what professional physicists do– we tend to work primarily with equations in abstract, symbolic form, and plug numbers in only at the very end of the problem.
Thus, very few people will be surprised by the conclusion of a recent preprint by Eugene Torigo, How Numbers Help Students Solve Physics Problems, which finds more or less what you’d expect:
Previous research has found that introductory physics students perform far better on numeric problems than on otherwise equivalent symbolic problems. This paper describes the results from a series of interviews with introductory physics students as they worked on analogous numeric and symbolic problems. This analysis revealed important differences between numeric and symbolic problem solving. In almost every respect the inclusion of numbers makes information more transparent throughout the problem solving process.
The basic idea seems to be that when they’re given numbers to work with, students are readily able to manipulate them with a calculator, because they don’t have to worry about what the numbers mean. When they have to work the same problem symbolically, though, they get tripped up because they lose track of what symbols mean what things, and will, for example, put in the final velocity where the initial velocity ought to go, and end up with the wrong final formula.
This is the culmination of a series of papers on this subject by Torigoe, and is sort of interesting as an explanation of why students go so wrong when asked to solve problems symbolically rather than numerically. The problem is, it’s kind of weak on the prescriptive end of things.
This comes out in posts by John Burk at Quantum Progress, and in a series of posts (one, two, three) at Gas Station Without Pumps (I can’t find a name to associate with these, so I will henceforth refer to the author as GSWP, because I’m lazy). GSWP read all of Torigoe’s papers (the second post above), and sums them up fairly but somewhat harshly:
Bottom line: Torigoe may have identified some structural characteristics of problems that give the bottom Â¼ of large, introductory, calculus-based physics classes particular difficulty. More and better experiments are needed to see whether his analysis captures the phenomenon correctly or is the result of some confounding variable. Most of his analysis is of no relevance for the top Â¼ of the class, where the future physics and engineering majors should be concentrated.
(Some of the methodological concerns are addressed by Torigoe in comments to that post and others in the series.)
The easy and obvious take-away from this would be to stick to numerical rather than symbolic problems in introductory classes. We already mostly do this– all of the free-response problems on our exams will give numerical values, and can be solved by putting numbers in and chugging away– but that’s problematic, in that we eventually need to move students away from plug-and-chug and toward the more physicist-like practice of solving problems algebraically. We could push this back past the intro-level classes that are mostly populated by future engineers who will never take another physics course. As satisfying as it might be to dump the algebra problem back into the engineers’ laps, though, it would just force us to grapple with the same issue later in the curriculum, and some of those courses are kind of overloaded as it is.
Another somewhat cynical approach would be to say, as GSWP does, that this is mostly a problem afflicting the bottom 25% of the class, and just write those students off. That’s awfully crass, though, and while Steve Hsu might be happy to claim that these problems reflect an innate lack of mathematical ability, meaning that these students will never be able to make it in technical fields, I’m a little more open to the idea that if we could get some of these students past this particular roadblock, they might do better than we think.
Which leaves finding some way to help teach students how to work with symbols. Torigoe offers a few suggestions, but I don’t find them very convincing– he advocates using subscripts to make different variables clear, but anecdotally, at least, I find that students who are confused by algebra are bewildered by additional notation. GSWP advocates for computer-program-style variable names (“hare_final_velocity” or some such), but I think that’s hopelessly messy for anything beyond really simple equations. Torigoe also suggests adding some explicit notation to distinguish known and unknown quantities– circling or underlining symbols, perhaps– which might work, but again, strikes me as kind of cumbersome for anything beyond the really basic problems.
As GSWP notes in that final post, though, prompted by a math teacher’s similar anecdotes, this is a general problem for all mathematical subjects. And it seems like the sort of thing that somebody in math education would’ve looked at. As near as I can tell, though, the traditional approach there as in science and engineering has been basically option two– demand that students do symbolic manipulation, and keep doing so until those who can’t get the hang of it stop taking your classes.
So, anyway, I don’t have a good idea where to go with this, so I’ll just trow it out there: anybody know a good way to lead students who aren’t already comfortable manipulating equations symbolically into doing so?