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I just finished reading a fascinating preprint, How numbers help students solve physics problems, by Eugene Torigoe.

Through interviews and task analysis, the paper found that students consistently fared better on problems that called for numeric rather than symbolic computation. Here’s an example:

A jet airliner on a runway can accelerate at a constant rat from 0 to a speed $v_1$ in $t$ seconds. How far has the jet traveled, $L$ when it reaches its minimum takeoff speed of $\left(v_1/2\right)$? (Assume constant acceleration?

A jet airliner on a runway can accelerate at a constant rat from 0 to a speed $80\;\frac{\textrm{m}}{\textrm{s}}$ in $90\;\textrm{s}$ seconds. How far has the jet traveled, $L$ when it reaches its minimum takeoff speed of $40\;\frac{\textrm{m}}{\textrm{s}}$? (Assume constant acceleration?

When making symbolic computations, Torigoe found students had great difficulty distinguishing between knowns and unknowns, and different quantities represented by similar symbols (such as the takeoff speed and half the takeoff speed). When students were presented with the numeric version of the problem, they did not make any of these errors and were able to solve the problem completely.

In the past few years, I’ve tried to take my class further away from equation manipulation, but Torigoe’s findings differ from what I would have expected. I find my students have lots of difficulty manipulating numbers and especially keeping track of units through a calculation, and as a partial remedy to that, I’ve encouraged students to always try to solve problems symbolically, and not use numbers, even if they are given, until the very last step of their work, when they arrived at a symbolic expression for the solution.

Torigoe recognizes that this is a common practice and encourages the following additional steps:

1. Encourage the use of subscripts—this helps students avoid errors like confusing the velocity of two different objects, and very few students use subscripts on their own.
2. Encourage students to distinguish known an unknown quantities. This is critical. Often in the middle of algebra for a problem, I’ll stop and take a “symbol census” and ask students what each of the symbols mean and whether each is known or unknown.
3. Consider an explicit notation to distinguish unknowns, such as underlining or circling the symbols of unknown quantities.

All of these seem like reasonable steps. I would also add that in many cases, adding another representation, usually a graphical representation, can really help students to ascribe greater meaning to symbols and expressions like $\frac{1}{2}at^2$

The paper is short and very readable, so I encourage you to check it out if you’re interested in a deeper analysis of the kinds of errors that students make in symbolic computation of physics problems.

1. December 19, 2011 3:42 pm

Like everything else how much you encourage/require symbolic manipulation depends on the student’s abilities as they come into the course. When students are weak in algebra, then I’ve found that finding an symbolic solution before substituting numbers increases confusion about the problem. Specifically, without numbers students are not sure what the next algebraic step is, while with numbers they are more likely to get it. For example, let’s say students are solving a linear completely inelastic collision problem and trying to find the mass of one of the objects. Most of my kids can set up m1v1+m2v2=(m1+m2)v. However, most regular students will have more trouble solving for m1 in an algebraic form than if they plugged in numbers for all of the measured quantities.

I’m glad you posted on this topic, because I’ve been looking for more ideas about how to get kids better at solving symbolic problems. Symbolic problem solving is a pretty important part of College Board’s AP Physics C exam, and I’ve been trying to figure out how best to get our kids ready for this.

2. December 21, 2011 4:37 pm

I wonder if certain students (poor math background, math-phobic, bad test-taker) tend to make certain mistakes as listed in the paper (poor strategies, variable confusion, symbol property isolation, confusion of known and unknown).

3. December 22, 2011 10:36 pm

December 23, 2011 3:24 pm

Thanks for taking the time to read and comment on my paper. If you are interested, this is a follow-up on earlier work published in the American Journal of Physics earlier this year.

http://ajp.aapt.org/resource/1/ajpias/v79/i1/p133_s1?isAuthorized=no

If you are not a member of the AAPT, then you can download it directly from my personal website.

This paper deals with comparisons of numeric and symbolic physics problems with a large population of students. It also discusses numeric question structures like simultaneous equations, which similar to symbolic problems place an emphasis on the understanding of equations.

• January 1, 2012 11:11 pm

Eugene,
Thanks for the comment and links. I’ll check these out.

5. December 31, 2011 6:31 pm

Hm, I’ve never really thought of it as trouble with numbers vs non-numbers. I find students have trouble with quantitative vs qualitative. To me, using letters instead of numbers doesn’t switch it from quantitative to qualitative, but increases the difficulty from tangible quantitative to abstract quantitative haha. One might argue that symbols and equations represent qualitative relationships, but when most of your students are struggling with algebra, fractions, ratios, decimals, etc., I don’t depend on them seeing the connection between an equation and a relationship (even though that’s like… what much of physics is haha-… 😦 ). It’s not easy to teach either (even though that’s one of the purposes of modeling… but that doesn’t help when your students are that far behind…)

6. December 31, 2011 6:43 pm

Oh, and I’m a bit surprised that someone found out “numbers help students solve physics problems”. I admit I didn’t look at the article at all. 😛 I suppose it depends on the students, but at least for mine, they almost always plug in the numbers first. Symbols are more abstract and I’m guessing most physics classes boil down to plug-and-chug problem solving. If that’s what students are trained to do, I would guess that’s what they expect. It’s weird to see something not-so-surprising as “students have trouble with physics problems stated as variables only” be rephrased to “numbers help students solve physics problems”. Like someone doing research and finds out “people have trouble walking on ice” and phrases it “friction helps people walk”.
Anyway, I’m not trying to devalue the research at all. It’s great that there’s actually some solid evidence on this now, so thanks for sharing!
As for solving problems symbolically, I’ve seen another teacher model the problem-solving using three different colors (which goes with that idea of circling/underlining). One color for known variables, one color for the unknown, and one for everything else. It looked very cumbersome writing it. Does make it easier to read though.

• January 1, 2012 11:14 pm

Frank,
I agree that students default to plugging in numbers and prefer this, but I think I’d always felt that solving problems completely symbolically avoids a bunch of messy tracking of numbers (and sometimes units) through a problem would be bigger handicap than it is. I’ve also tried to sell doing things symbolically as an easier route, when this research would seem to say it isn’t the case.