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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Thinking about Coding

As I’ve written several times, we are spending the next year developing an integrated math curriculum, the first year of which will be implemented in 2016-2017. One of the things we’ve decided we want to do is include coding in the curriculum. The move to include coding is nothing new of course; in fact, some would say we’re behind the curve since many schools began adding coding to courses a couple years ago. I actually think we’re better off doing it now, because we can learn from the experiences of others and we can benefit from the historical perspective of looking back at what people were doing two or three years ago and what they’re doing now.

coding-future

I will say right up front that I’m not a huge fan of adding coding to our curriculum. It felt like a fad two years ago, and it still feels like one now. Rereading Peter Gow’s insightful piece, “Coding is Just the New Surveying” only reinforced that sense, but it also gave me a good way to think about what the role of coding should, and should not, be. To try to get a sense of what coding the in curriculum might look like, I read some articles on the other side of the issue and also spent some time with the newly published book, Doing Math with PythonBetween this reading a great conversation with colleagues, I’ve come up with an initial framework for thinking about how to bring coding into integrated math.

  1. The role of coding in our curriculum should not be, to borrow from Gow, “pre-vocational”. Our goal should not be to teach coding so everyone knows something about it because they’ll need it in the future.
  2. More to the point, coding should be integrated into the curriculum, not feel like an add-on. If we are working to tie geometry and algebra together, then we should make an equal effort to tie coding to other parts of the curriculum in a natural way.
  3. Whenever possible, the coding work students do should have long-term usefulness, not be a one-off project or assignment. This might mean writing a program you’ll use over and over during the year, or it might mean writing a program you’ll return to each year to modify and enhance.
  4. The purpose of being able to write a program is to get technology to do exactly what you want or need. There is no need to write a program to find the mean of a data set because spreadsheets and calculators can already do this for us. Whatever role coding plays in the curriculum, it should focus on enabling students to do things they can’t otherwise do.
  5. The corollary to this is that in order for students to know what technology can’t do for them, they need to learn what it can do for them. Therefore, familiarity with calculators, spreadsheets, and even websites like Desmos and Wolfram Alpha is also necessary, and should be a prerequisite before learning how to code.

Our first big milestone in the curriculum development is in about three weeks, when we hope to have a broad outline of what topics we think should go in each year of the course, how many levels of each course we should have, etc. It’s been hard to think about what to include in the courses since I’ve never been clear on the role of coding in the course, and this framework gives us a way forward. Next up: so what should we include in each course?