Math and Physics Problems

Papers: "Student Views of Similarity between Math and Physics Problems", McBride, PERC2011;  "Comparing Physics and Math Problems", Jones & Roseman, PERC2012; "Students’ Views of Math and Physics Problems: Structure vs. Content", Jones, PERC2013

Presenter: Shane Frewen

How do students perceive Math vs. Physics problems?

McBride 2011: Students were asked to match physics problems to the mathematical representatives of similar problems. Students were largely able to do this, though there was some confusion over definite vs. indefinite integrals. There wasn't anything incredibly enlightening that came out of the papers, although they did a solid investigation into an interesting topic.

What they might have missed is that physics variables have meaning to physicists, whereas all variables are interchangable to a mathematician. For instance, presented with the problem:

A(x,y) = x^2 + y^2

What is A(r,theta)?

A mathematician would say that A(r,theta) = a^2 + theta^2 because you are just changing out the variables. A physicist would recognize r as a radial distance, and would say that x^2 + y^2 = r^2. So physicists apply a filter to math problems to give them meaning and a potential hint at a solution.

Jones & Roseman 2012: Students are asked to pair math and physics problems based on similarity while working in groups. Students (novices) tend to focus on surface features of the problem rather than trying to put the problems into context. The physics problems were much longer, and students spent longer reading the context.

Jones 2013: Similar to the other two studies, but they focused on the words students used when deciding on a solution and/or approach.

For the math problems, the word cloud created by the students' answers looked like this:

For the physics problems, the word cloud created by the students' answers looked like this:

How useful is an investigation like this?

Did the researchers take into account the words that were actually presented as part of the problem? Also, they started with a math problem, which got the students in the mindset to apply mathematical approaches. It would have been more useful to have a control group.

Bassok and Holyoak 1989 discussed the differences between students who started learning math before physics and those that learned physics before math. A background in math was found to be helpful in approaching physics problems.

Connection between context and motivation:

Would it be easier to solve the physics problems because they have some context? Does context provide motivation for the students to even approach the problem? Which type of problem (math or physics) teaches better problem solving skills?

In the 2011 paper, one of the students involved in the study flat-out refused to complete a math problem because he/she felt it was "pointless and not useful" for solving the physics problem. How do we get students over this hump?

Should we focus more on teaching students to identify the important information in a word problem? Maybe we could ask students what the important variables are and why before asking them to actually solve the problem. There is some opposition to this approach among students because they think they know where to go and don't want to take the time. Maybe the problems need to be more difficult. Then again, they are trying to be efficient to save more time. About half of students in an intro level physics class took the time to fill out that step of the problem in order to get some partial credit for a question. Students shouldn't get full credit unless they have to write out every step.

Giving partial credit by identifying students' intention in long multi-faceted word problems is also difficult.

Teasing out the problem solving techniques that students are using makes assessment/grading a nightmare. Where are the ideas from educational researchers for assessing 150+ students with 1 graduate TA or an undergraduate grader? This is a problem that has not really been addressed.

Problem Solving in Groups

Jones et al. commented that, when solving physics and math problems, "the cognitive load was at times overwhelming for students without peer support." This is why discussion sections are so important. But if you try something they don't like, they won't show up for discussion section in the following week. How do you force students to work effectively in groups? You could make discussion section mandatory (then the students won't attend lecture).

Students do not like being asked questions. They will basically complain when asked to participate actively in class. They also do not like working in groups in general, and would rather go their own way. Once group work becomes part of the culture of the class, it is easier to convince them to work in groups. This is even more true when the solution to a group work problem shows up again on an exam.

It has been seen that students who attend discussion section and work actively in class see an increase in their grades. How do we convince the rest of the class that this is the case? Jeff Bennnet (On Teaching Science) points out that students from today's generation are not spending as much time on their courses as students did 20-30 years ago. There has been a direct correlation between the time spent on classes and the final grade in the class. This should be made absolutely clear to students at the beginning of the course.

Students don't have a good conceptual understanding of what learning actually means. Should they be informed about some background educational psychology to motivate them to put the work in? Would students appreciate the extra material?

Resources for students:

MOOC - Learning how to Learn (online course)

How to excel at math and science (book)

Make it Stick - The Science of Successful Learning (book)

Cold-calling students to get them involved

Give a 1% grade boost to students who volunteer to be on a "cold calling list"? You don't have to get the answer right, and you can say "I don't know." But you have to be willing to participate. Students might complain or drop out, but the ones who do stay WILL do better.

Taking responsibility for learning

These students are in college and need to take responsibility for their own learning. Students should understand by this point that putting hard work in will result in better grades. Students also need to learn to "embrace befuddlement" and appreciate that struggle is a part of the learning process.