On the Helpfulness of Numbers

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 ¼ 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 ¼ 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?

16 thoughts on “On the Helpfulness of Numbers

  1. I wonder if part of the problem of algebraic notations is that students lose track of whether an answer makes sense or not. I doubt that’s a real answer — I’m sure there are tons of wildly wrong numeric answers — but I can imagine that things seem fuzzy very quickly in algebraic notation, and kids can feel unmoored and adrift.

    Don’t know if this is at all relevant, but… I remember coasting through HS physics with a feeling that it was cheating to do algebra on units. I was secretly derisive of my teacher’s efforts to keep track of m/s^2 or whatever. (I was, admittedly, partly derisive because he was obviously struggling, as he was a first-year teacher with more of a chemistry background.) When I got to college, I quickly realized that was never going to fly: you had to keep track of units, to make sure you hadn’t screwed up the algebra somewhere. For some bizarre reason, it felt like cheating to me… but when I gave in, things got immediately better.

    My point being, is focusing more intensely on how the units line up possibly a good tactic to help students think things through more symbolically? I don’t know how much of that is done already though, so perhaps I’m covering well-trod ground.

  2. Representing two different quantities by the same variable is just begging for trouble. It’s a terrible idea. Which means you either need to use subscripts, or you need to lose the mnemonics of assigning variables based on names, so f=ma becomes x=bc. I don’t like the latter, as you’re moving towards strictly symbolic manipulation without any obvious connection to what it all means. So I think you have to get the students to accept subscripts, even if you need to repeat the idea over and over, and take off points for reusing variables. Subscripts are part of the basic math of physics, there’s really no way around it.

    Reading the linked paper, it sounds like the biggest problem is confusions with the identity of variables, be it canceling known quantities, conflating variables of the same type, or whatever. For an introductory class, I’d recommend requiring the students to list and identify the variables to start with, adding a question mark to unknown quantities. Like so:

    v_0 = train’s starting velocity
    v_e = train’s end velocity?
    t = elapsed time
    a = train’s acceleration

    One advantage is they can look up if the symbols start to confuse them and reground the meaning of a variable. It also provides some quick red flags when they’re going wrong. Are you solving for a variable that doesn’t have a question mark? If you have numbers in the problem, are they all accounted for? Have you used a variable like v or f multiple times to mean different things? Because you’ve written out the variable identities to start, you can do all the symbolic manipulation without being bogged down by named variables or special annotations. As the student grows comfortable with physics they can just drop the explicit list and they’re now doing regular algebra. I imagine they’ll grumble about the extra work of having to write it all out, but that’s the point: they need to stop and take the time to understand exactly what each variable they’re using means.

  3. Way, way, back in the day, when I was first learning physics in high school, my instructor did pretty much what Paul (@2) suggests. Almost all the problems were numerical, but there was a *required* format for working them: 1) Draw a labeled diagram, 2) list and identify known quantities, 3) list and identify the unknown quantities to be calculated, 4) write down the equation(s) to be used, and only then 5) do the math to solve the problem.

    Looking back, it seems pretty tedious, and doesn’t really allow students to think for themselves too much, BUT I can’t deny that following that required format taught me to see the connections between all the parts of the problem, and helped me learn to think symbolically, rather than numerically.

  4. The last time I was a physics student (about two years ago, at the age of 43) I noticed some things about my classmates. You have the students who are just looking for a passing grade and aren’t really interested in a deep conceptual understanding of the material. “Just let me get my degree(s), get a job, and get on with my life.” Plug-and-chug was their friend and their loyalty to it ran very deep. For serious students (my grades not withstanding, I was a serious student) I noticed that there were two areas that seemed to get inadequate attention:

    Proofs – I never received a specific lesson on the nature of mathematical proofs. We studied them all the time, we were expected to know several, and they seemed terribly important, but we never spent any time explicitly discussing “What is a proof? How are all these relationships and manipulations possible? What’s with all this logic an’ stuff?” I suppose it’s fairly intuitive for an exceptional student, but for an average student, it’s a bit like studying Shakespeare without knowing anything about Elizabethan England — you can do it, but you’re bound to misinterpret some things.

    Units – cisko is quite right: It took me a while to figure it out, but a lot of physics problems are much more easily understood by following the units — especially when it came to constants. It’s what makes physics easier to understand than abstract mathematics, the symbols actually mean something. We only spent a little time on dimensional analysis in chemistry class, but it was probably the most important science lesson I got: always keep track of your units. I had one physics teacher who, very wisely, would not allow us to say “x” or “omega” when reading an equation out loud… he forced us to say the quantity it represented in the equation so we always knew the formulas weren’t just recipes, they were descriptors that told us about relationships between quantities.

    So, assuming you’re starting with a properly motivated student who is free of math anxiety, that’s my advice: Talk about proofs, teach them to follow the units.

  5. Even before I scrolled down to the comments, I had the same thought as cisko. Make them write down f*kg*m/s^2=m*kg*a*m/s^2. I sometimes do this when I want to convert units with certainty; for the trivial example of 300*meters/seconds=300*meters/seconds*1*miles/(1609.344*meters)*3600*seconds/hour, for example, fed into Maple, we get “671.0808877*miles/hour”, with all the unit cancellations checked by computer algebra. Sorry about the spurious accuracy. But my slightly more abstract thought is that it’s just that any willingness to be systematic should be encouraged until concepts are very well set. I suspect that this is best said hundreds of times, so that when, one day in the distant future, they notice that they just solved a problem by being systematic, they can be moved to say “so that’s what Prof. Orzel meant”. It’s ultimately not much different from learning how to do long division by guesswork or by lining up the columns. Up to a point, it applies to research too, “if you don’t know what you’re doing, do it systematically”.

  6. I used the Paul and Pam (#2 and #3) ideas during 35 years of math/science teaching in a suburban high school.
    Subscripts got a little messy. Instead of Fsub x and F sub y we would use F sideways and F up. Torques were worse. We used these so everyone would know what we were talking about before we tried to write or solve any equations.
    I would get annoyed when students would start to plug in numbers before doing any algebra, as so many things can cancel. Gravitation and centripetal force equations get very simple after the cancelations.
    I also tried to make it very clear that 1/ fraction is reciprocal of fraction.

    My students did pretty well, overall. One thing that always bothered the kids, was that we seemed to get the answers to part a), b) c) d) in the reverse order. Whatever was asked last, we found first. I never figured out why.

    I started teaching physics after six years of only math, so I was all over the kids to do algebra. As I got older I was more geared towards the good diagram, then what are you trying to find, then what is needed to find this…. until I didn’t have to describe what to do. “Every problem is not a new adventure!”

  7. In my first worklife incarnation I was an electrical engineer, and I first learned my craft so long ago that Fortran VII was still a viable language. So, when I had to figure things out algebraically or with calculus, I tended to start writing, say, startv instead of V sub 0 and endv instead of V sub e. It actually made a huge difference in being able to do a page of symbol manipulation accurately! (Perhaps, as much as for any other reason, as I could accurately read my handwriting when I used symbols without subscripts.:-) )

  8. @6: “Every problem is not a new adventure!”

    That should be the theme song for first semester physics.

    Helpful observation first. You can work on this difficulty by (a) assigning problems with unspecified quantities, like not giving them the mass when it isn’t actually needed or (b) leaving one variable unspecified and asking for the answer for something like the tension in a cable as a function of some unknown mass and/or the unknown position of the mass.

    You write “the more physicist-like practice of solving problems algebraically”. Why are you assuming that engineers don’t want problems done the “physicist way”? My informants (former students now in engineering classes) tell me their profs are quite strict about writing down every variable, assigning the value (with units) of each that is known [this is graded], writing down the starting equation symbolically [this is also graded separately], doing algebra, substituting in the final expression, and doing one calculation to get the answer. They also have to use a straightedge when drawing free-body diagrams. Few physics majors would pass such a class, but kids in Pam’s HS class would do fine!

    My observation about the origin of the problem is called “graphing calculators starting in middle school”. One consequence of this is entire math courses where the unknown is always x and functions are always y(x). The sudden appearance of f(x) or g(x) or — heaven help us — x(t) in physics can be mind blowing for students who can only solve an equation by graphing y -vs- x on their calculator.

    It is no surprise to me that the bottom quartile in a freshman physics or calculus class can’t do algebra very well and, in particular, might never have used a variable with subscripts. Many are also unfamiliar with the idea that capital and lower case letters might be different symbols. [I love putting all of the different E symbols used in the first weeks of second semester physics, from energy to epsilon, on the board.]

  9. I suspect a good method for getting students to become comfortable with symbols is to start out freshman courses teaching a numerical solution followed by a symbolic solution for each problem. In essence, solve the problem both ways and the students will soon learn to translate symbolic thinking. It isn’t much different than saying a word in English and then Spanish when teaching Spanish. As a Spanish course progresses, less English is used.

  10. Paul wrote @2:
    > Subscripts are part of the basic math of physics, there’s really no way around it.

    At least this presents the convention on notation for distinguishing

    – on the one hand the general type (or operator) of some particular quantity — namely as a particular letter without subscript; or, if it can’t be helped, at least with a “hat” symbol above the letter) and

    – on the other hand specific instances or values of this particular quantity — namely the same letter with subscripts, or with the specific arguments listed in brackets; e.g.

    mA, rAB, φABC, vA[ K ], vA[ Kinitial ], and so on.

    > Representing two different quantities by the same variable is just begging for trouble

    Especially troublesome I find it if no distinction is made between time (symbol letter “t”) and duration (symbol letter “τ).
    Those who are careless in expressing this distinction in their notation sure can’t hope to teach the conceptional distinction to their students.

  11. Peter Morgan wrote @5:
    > 300*meters/seconds =
    > 300*meters/seconds*1*miles/(1609.344*meters)*3600*seconds/hour, for example

    This excellent technique (of inserting suitable “active unit factors” into a product) is known as

    As a related technique, suitable “active zero summands” may be inserted into a given sum expression, allowing some cancellations with terms in the given sum expression and thus leading to an overall simplification.

    Does anyone know how this technique is called? (Perhaps “inserting an active zero”? Wikipedia doesn’t seem to know …)

  12. Numeric calculations are useful to gain a sense of scale in a problem. By the time symbolic manipulations are done, it may not be apparent, for example, that the horizontal component of a 10 N force at 30° is not -9.9 N. (Oops, I took the sine of 30 radians, instead of 30°.) So it has some value, but it is entirely overused.

    I think by the time you’ve got students in a college physics course, it’s too late for that bottom 25%, who just don’t have the natural ability to think abstractly about things. Manipulation of units ought to be taught in science class the same day that the multiplicative identity property is taught in math class, which seems like a 6th grade subject. Algebra classes should be emphasizing symbolic manipulation in their word problems. Of course, that’s not much help to you.

    I wonder if there’s a way to force them to do a symbolic calculation. Maybe on an in-class quiz, you do it in 2 steps. Set aside 10 minutes. Give them the problem with all symbolic values. Require them to solve the problem symbolically and check dimensions, and submit an answer in the first 7 minutes. Then reveal the values, have them plug and chug, and give a numeric answer and justify whether that is a reasonable value in the next 3 minutes. This could be done electronically with the proper infrastructure, or you could maybe get some carbon copy paper out of the back of the supply closet so they could turn in their symbolic solution and still have it to refer to for their numeric solution. Note that checking dimensions in the first step, and evaluating the reasonableness of their answer in the second step, are key parts here.

  13. Some others already touched on this, but I want to explicitly note the helpfulness of considering physical units (MLT, and the particular forms such as force, area etc.) It helps to work through a lot of physical problems, look at the power of “dimensional analysis” which reveals what form of equation can and cannot be right. Seems like magic, but it has to be that way in a consistent universe.

    Not only that, the equivalent consistency helps even with pure math: e.g., derivatives need to be one power lower each time or else units for time derivatives wouldn’t turn out right. Anyone looked into that?

  14. Oh, how I do agree and have noticed the student desire to have numbers instead of symbols. It almost becomes a battle of ‘just do it my way because I say so/that’s where I’m giving the marks’ until they are so used to doing it by the required steps that it becomes reflex. Our series of science texts call it the GRASP method: list Given and Responding, Analyze the necessary formula(e) and rearrangements, Solve, and Present with appropriate sig. digits and units.

    About half if not more of my grade 11 physics course consists of dynamics. Of the 32 lessons into which my course is divided, #14-23 and #32 are all based on two very simple ideas: Fnet = ma, and “you can’t push on a rope” (i.e. circular motion requires an inward force).

    Falling objects, objects sliding on surfaces, objects on ramps, rockets, elevators, horizontal circles, vertical circles, planetary orbits – Same. Damn. Steps. Every. Time.
    1) FBD
    2) list every force and its components, if necessary (the teaching of which, of course, is practically the other half of the course).
    3) organize a table in x and y of the force components
    4) Fnet = ma for each of x and y, and Fnet gets replaced by the sum of everything that has a non-zero value in the appropriate x or y column.
    5) substitution of one of the circular forms of a, if required.

    By the time you get to that point, the variable you’re trying to find should practically be slapping you in the face and demanding to have algebraic expressions combined and reduced. And still there are so many students who will insist that they have to have all the numerical force values calculated first before they can proceed. *aargh*

    Tom and Neil, I like your ideas of having a purely algebraic set of steps to solution before they are given any values, and of encouraging dimensional analysis. Oh, but they’re so hard done by if I require such a thing! I’d be tempted to subtract marks if they ended up using a numerical value in a calculation that would have been eliminated in a symbolic approach…

    joemac53, I sure hear you on the 1/(a/b) = b/a business – I call it the ‘kiss and flip’. When dividing by a fraction, you kiss (multiply => “X”) and you flip the fraction

    ‘factor-label’ is heavily emphasized in our Chemistry program for calculation, so I at least have some expectation that they’ve seen it when I demand they show me exactly how they determined their units or did metric conversions.

    I do try to emphasize all of the above as techniques for the students’ own sake, to help them catch errors, but I do wonder sometimes just how much I’m shouting into the wind.

    Sigh. Two grade 11 physics courses starting in February. Maybe this time they’ll listen…

  15. I encountered this problem with my daughter and her friends in high school. They were taking introductory physics, and they’d get together at my house for the free tutoring. They would plug away with their calculators and get very frustrated at how many times they had to work the problem over again before they got consistent answers. When I saw what they were doing, I was horrified, and I told them that there was actually a reason they took algebra other than for the pure torture of it. I showed them how to do it symbolically, the paul and pam method of keeping track of how they were thinking through the problem, and dimensional analysis as a sanity check. For some of them it never worked, but I convinced my daughter and one of her friends to stick with it, and after a couple of weeks of working symbolically, they were sold. Why? Because it really did end up taking them less time, and the answers were more reliable. But this was undoing damage that had already been done (in algebra class mostly, in my opinion, where they never conveyed any sense of what algebra is good for, apparently), and it only worked because they trusted me at least as much as they had their teachers.

  16. Using numbers allows you to consolidate items which would otherwise be seemingly separate. Yes, 9.6712910 can seem simpler than (a+b).

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