Why Are Word Problems So Hard?

One of the courses I teach is an interdisciplinary course in AP Physics 1 and Precalculus Honors. In a future post, I’ll describe the course, the things we’ve tried to do (successfully and unsuccessfully) in it, and what we’ve learned in the first two years it’s been offered. For now, suffice it to say that the course is a lot of fun to teach, and it’s also an incredible challenge. Wednesday was a good example of both those things.

One of the reasons to create the course was the natural overlap that exists in content between typical physics and precalculus courses. Among other things, both courses cover vectors and parametric equations; rely heavily on trigonometry; require students to solve equations; and spend time on word problems. It seems redundant to cover the same material in both classes, and it implies that the math they use in physics is somehow different than the math they learn in precalculus. Why not just talk about these topcs once, and use the time saved to go into more depth about something else? This made perfect sense when we created the course, and for the most part it’s worked pretty well. But…it hasn’t always worked well, and when it hasn’t it’s told us a lot about where our students are relative to where we think they are.

Our students have been working on one-dimensional motion for a few days now, and they’re having trouble with it. The ideas are hard, which we expected, but earlier this week we realized that their struggles were not really with the physics, they were with the problem-solving process itself. Despite the fact that they’ve been drawing pictures, stating knowns and unknowns, determining equations, and solving for two or three years at this point, they couldn’t seem to do it well enough to allow them to solve the problems. So, we came up with a way to diagnose their issues and help them through their struggles.

The problem-solving process we’re using in class follows the one laid out in our textbook, Eugenia Etkina’s College Physics:

  1. Sketch and Translate – draw a picture of the situation and put all the information from the problem on the picture
  2. Simplify and Diagram – model the objects in the problem as points (if appropriate), and draw the appropriate vector diagrams and/or graphs of the situation to help visualize the relationships. To this step we’ve also asked the students to explicitly state what they Want To Find (WTF).
  3. Represent Mathematically –  state the equation(s) that relate the knowns and unknowns in the situation
  4. Solve and Evaluate – solve the equations and decide if the answers make sense

Our students were getting bogged down in the third step – they couldn’t seem to figure out how to create equations. As we (my co-teacher and I) thought it about, we realized that wasn’t really where their issue was, though. Their real issue was in setting up the problem well enough to clearly understand the WTF, which in turn enables one to identify the appropriate equation. To help them understand the importance of the first two steps of the process, we threw out what we had planned to do in class and did the following instead.

We have groups of students in class, with three of four students in each group. Each group was given a different problem. Together, on a whiteboard, they had to complete the first two steps of the problem solving process, ending by stating the WTF. The whiteboards were then switched, but the problems were not. Each group had to try to find the WTF using only what was on the board. If the problem had been sketched, translated, simplified, and diagrammed well, then the second group should be able to finish the problem off. Our students quickly realized this because, well, the problems were not all sketched, translated, simplified, and diagrammed well.

When a group got stuck, we invited them to give feedback to the group that had set up the problem. For example, one group stated that they wanted to find the time when two objects’ paths intersected, but their diagram showed the objects moving away from each other. Another group correctly showed two objects as moving towards each other, but they stated both velocities as positive instead of changing the sign on one to indicate movement in the negative direction. With these and other mistakes, the students were able to see the importance of setting a problem up well. At the same time, we were able to see what concepts they had misconceptions about, and we were able to address those misconceptions on the spot.

This exercise went very well, and in theory it helped resolve a lot of the issues the students were having. Did it actually help? We’ll find out this afternoon when we ask them how their homework went.

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