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.