Intuitive and Formal Approaches

Andrew Heckler, a physics education researcher at the Ohio State University, wrote an interesting paper about the consequences of prompting students to draw free-body diagrams. Heckler found that asking novice college students to draw free-body diagrams as part of physics problems has some interesting (negative) consequences for their problem-solving. It’s worth reading yourself, but here are some notes.

One problem involved asking students to simply identify the forces exerted on a basketball rolling across a ‘frictionless’ floor. (correction: the problem states that friction is small enough to be ignored)* Students who were asked to draw and then identify were more likely to identify a “force of motion” than students who were not asked to draw a diagram. Why might this be so? I’d speculate that it’s simply compelling to draw an arrow in the direction of motion as part of a sketch. Then, upon seeing what one has drawn, one is persuaded into thinking that it must be a force. My argument is that it isn’t so much that students have a force of motion misconception, but that there is a dynamic between what one draws, what ones sees, and how one responds. Drawing an arrow in the direction of motion is part of the dynamic by which students engage in thinking that there must be a force in that direction.

Another problem from the study involved students having to figure out the minimum mass needed to get a box initially moving where in the problem the box is being pulled on by both sides with known but different forces and there is friction. Once again, some students were just asked to solve the problem, and other students were first asked to draw a FBD and then solve the problem. With this problem (as with the others), students were more successful in solving the problem when they weren’t asked to draw FBD.

Many of the students who were successful used intuitive approaches that were not taught. One of these approaches Heckler calls the two-step method, in which students first simply subtract the two pulling forces, and then set them equal to the friction force. Some students even went so far to draw 2 different diagrams, one with only the pulling forces opposing each other. And then a new one with the combined pulling forces opposing the friction. In contrast, students are taught to draw 1 FBD that shows all the forces, and then they are taught to write out a complete ΣF statement. The students’ intuitive approach has several benefits. First, it has a divide and conquer strategy–if you can’t figure everything out, start with what you know and work from there. Second, it allows you to figure out the direction of the friction force along the way, instead of having to guess and then adjust at the end if you find you’ve gotten a friction with a negative sign. Third, since the strategy makes sense to the students, they have ways of spotting errors and correcting mistakes along the way. When students take the expert approach, they are more likely to make mistakes and less likely to correct mistakes.

Overall, Heckler found that students did not typically see the FBD as a way to help organize the problem or to check for consistency. Rather, FBDs were more of just something an instructor was asking you to do. In fact, many successful students would draw a wrong FBD, and then proceed to ignore it, so that they could solve the problem correctly using an intuitive approach. And many students who drew incomplete or wrong FBDs often still solved the problem correctly using an intuitive approach. Still, overall, students who weren’t prompted to draw diagrams did better than students who were.

Intuitive Approaches in Energy in my Classroom

Speaking, of intuitive approaches. Last week, I showed students how to draw energy pie charts instead of starting with equations for energy conservation. This led students to use some intuitive approaches that were successful, but quite different than the formal approaches. In one problem, a roller coaster started a height of 85 cm and then goes around a loop with a radius of 17 cm. Students were asked to find the speed at the top of the loop. The formal approach would have students write

PEi + KEi = PEf + KEf

mgH + 0 = mg2R + 1/2 mv²

but several student groups noted that 34 cm was 40% of 85 cm, which meant than the potential energy on the loop was 40% of the original , leaving 60% of the initial energy for Kinetic.

They then wrote this equation

KEf = .6 PEi

1/2 mv² = .6 mgH

I let students go down this path, knowing that this approach might not be easy to implement all the time. Instead  of steering them away from it in the moment, I let them continue. In order to make sure they had an opportunity to make contact with the formal approach they would be expected to use on the exam, I then had them explain their solution to another group, and that other group share their approach, which was more closely aligned with the formal approach.

The Big Picture

I’m pretty convinced that students have a wealth of problem-solving strategies and reasoning skills that go untapped when we teach formal methods to soon. It leaves these formal approaches disconnected from the the good things students have to bring to the table. Of course, I know that students’ intuitive approaches will need to be formalized at some point, and that many intuitive approaches will run into problems later. But I feel that teaching students to use formal approaches without helping them anchor it to their own sensibilities and ideas is much more problematic. I’d rather help them to refine and objectify their own approaches, and introduce formality as authentic need arises.

* Note that many physicists will initially have a problem with this. They’ll say, “Rolling on a frictionless floor? That’s impossible. This question is flawed!” Remind them that students often believe that force is required to maintain motion, and that this is a misconception. Then ask them if they have a similar misconception that rolling (or spinning) objects must be maintained by a torque.

4 thoughts on “Intuitive and Formal Approaches

Add yours

  1. This is an interesting paper; thanks for writing about it. Of course, it’s no surprise to anyone who teaches this subject that students will draw wrong f-b diagrams and then ignore them and solve the problem some other way.

    I’d like to see this study reproduced with problems that aren’t strictly one-dimensional (e.g. adding blocks on inclines). I wonder how the two groups would compare on those sorts of problems. Many of the “intuitive” students will probably still simply add and subtract force magnitudes, which “works” in 1D but fails in more complex situations.

    I also have to disagree with your description of the basketball problem and accompanying footnote. The problem doesn’t say the floor is frictionless, it says “The friction between the basketball and the ground is so small it may be ignored.” That’s not quite the same thing when considering rolling objects. You can roll with very low friction on a surface with a non-zero coefficient of friction only by maintaining the no-slip condition (v = omega r). On a truly frictionless surface, v and omega are completely decoupled, and I would say that describing such motion as “rolling” would in fact be flawed.

  2. Hey Chris,

    Great point about 2D. I think that’s part of my point. Eventually, we will need to help students formalize their approaches because their intuitive approaches will run into problems later. The question is, “When and how do we do this?” I’d argue that we often do this too soon, so that instead of anchoring formal approaches to a sensible starting place, we just teach them formal approaches (that are very generalizable to experts) but non-sense to students. Of course, maybe it’s not a bad thing for students to get worse before they get better, but then we should recognize that this is the path of learning, and teach and assess with that knowledge in mind.

    I also hear what you are saying about the basketball problem, and the physics. I didn’t mean to misconstrue the problem. I was writing most of this post from memory of having read the paper about 2 years ago. Either way, I’m just not convinced that “rolling” is a technical term… if I saw a spherical object spinning and translating across an icy surface, I’d still say it was rolling. My point is to get things rolling you need friction, but if something rolls across a rough surface over to an icy pond, nothing would change. Conservation of momentum and angular momentum would suffice to keep v = omega r. I think you’d say that once the ball goes over to the icy pond, it isn’t “rolling” anymore. It is just “spinning and translating” in such a way that the meets the constraints of rolling-without-slipping, but it is not in fact rolling. Did I get that right?

    1. One thing I didn’t put in my original reply is that I think it’s dangerous to use “Does better on traditional problems” as a measure of success. Isn’t one of the lessons of PER that the ability to solve traditional problems can be unrelated (to some extent) to actual understanding? So I wouldn’t necessarily characterize the f-b-diagram-drawing students as getting worse. It’s tricky. Or as my once-upon-a-time guitar teacher told me, to improve you have to play through your mistakes.

      Of course I try to build on student intuition as much as possible (if for no other reason than I don’t want them to think that they are natively bad at physics), but this is tricky too. I try to get them to analyze their own intuition. When learning Newton’s laws and more formal problem solving, students often ask me “Can’t I just subtract a from b and divide by m” (or similar) and I always say, “I don’t know, is that equivalent to Newton’s second law?” or “You can do it, but unless it’s equivalent to Newton’s second law it’ll give you the wrong answer.” I think there’s a lot of value in getting students to think about how their intuition matches up with more formal techniques. I’m also very wary of students just using naive numerology or pattern matching. Many problems are inadvertently written so that such techniques actually work.

      I had an interesting situation related to this on the Monday warm-up exercises this week. Two questions about vertical circular motion with gravity. First was just a conceptual free-response question about why a string could go slack as object went over the top if the rotation rate was too low. Almost everyone answered correctly. Second asked the students to calculate the tension in the string as the object went over the top at a certain speed. About 1/4 correct, with the most common wrong answer being the naive result of setting tension equal to the centripetal force. I don’t know what to make of that, except naive numerology seems very strong when students are faced with a traditional quantitative question.

      Regarding rolling, yes, I think you summarize my thinking correctly. I don’t know if rolling is a technical term, but I think it has certain expectations built in. If you were driving and hit an icy patch and started skidding, would you say the car is still rolling? Saying that something is rolling implies a certain relationship between v and omega, but this relationship need not be true on a frictionless surface. Personally, I would only use the term rolling when the no-slip condition is met and when that is enforced by static friction.

  3. Another by-product of this conflict between intuitive and formal, I think, are the many elegant mathematical shortcuts used by the experts. FBDs are one of them – a non-intuitive method that, when used properly, make the problem easier to solve.

    But why do we use (or take) shortcuts? Because we know the “usual” way us longer and harder. That extra anguish is very good motivation for learning the shortcut.

    My point is, if we *only* teach the elegant shortcut, why should students have any motivation to use it? Because it’s good for them? Yeah, we know how well that works! I support letting students struggle with the problem, for a measured amount of time, and maybe even solving it the hard way. Then introduce the expert’s approach to solving the problem. The students will have a better understanding of the problem being solved and will, I feel, better appreciate (and hence use) the more advanced technique.


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