In this post, I suggest that many of the strategies we need to solve problems are also the key to success in solving Mechanics exam questions - perhaps more so than other A-level modules. Although there are only a few concepts to grasp, and relatively simple mathematical techniques involved, Mechanics questions are often non-standard, difficult to interpret, involve a lot of data, and it is often not clear to students how to progress. In this sense, I think that Mechanics questions should be approached using the same strategies we would use to approach a mathematical problem.
Here is an example of a recent M1 exam question - what is your first impression on reading it?
It may not be immediately clear to students how to progress, and some students may feel lost. This is meat and drink to a good problem solver - but this can make many non-problem solvers feel anxious. With a few simple strategies, we should be able to increase our confidence and get a better idea about how to solve the problem.
What is the question asking? Encouraging students to reinterpret the question by drawing a picture is essential - this will get them started, give them some confidence, and also some time to think about what is happening.
Here we can see that we will need to resolve forces parallel and perpendicular to the slope. The lifeboat is accelerating down the slope, but by how much? We can use the information about u, v and s in the equations of motion to find a. And if there are forces and acceleration involved, we will probably need F = m.a at some point.
Now we have some ideas to work with - which one will lead us to the solution? Intuition and experience tells me that resolving forces is the way to go, so let's try that. Resolving parallel to the slope (using F = m.a) gives:
800.g.sin15 - mu.R = 800.a
Suddenly how to progress becomes more clear. We can find R by resolving perpendicular to the slope and we can find a using the equations of motion, which will leave us with one unknown, mu - and the problem is all but solved.
In the book How to Solve It, George Polya describes progress and achievement in solving problems. He outlines several stages, which I paraphrase here:
We search our memory and select the relevant concepts involved (mobilization), then combine these facts and adapt them to the problem at hand (organization). Sometimes we ask: what is missing - have we used all the data? As we work on a problem, our conception of the problem becomes more full. We view the problem from different viewpoints, restate the problem in our own words. As we progress toward our final goal we see more of it and can see more clearly what needs to be done. We do not foresee such things with certainty, only with a degree of plausibility. Without considerations which are plausible and provisional, we could never find the solution, which is certain and final.
This matches exactly with my experience of solving the exam question above. After using all the data and restating the problem, we selected the concepts that may have been involved and combined them to solve the problem at hand. We were not certain that these concepts would give us our solution, we were following plausible paths (made more plausible by experience), and only as we progressed did it become more clear how to proceed.
Considering there are only a few concepts in (say) M1, and the maths is not particularly demanding, I would argue that it is this complex problem solving process that students find difficult. Of course, we need to master the basic mathematical techniques and concepts involved. But ,in addition, I am going to ensure that my students are comfortable with the problem solving process. We need to:
- Get something on paper by restating the question - usually with with a diagram - and include all the relevant data.
- Select the relevant concepts - there are only a few in M1 to choose from!
- Use our intuition and experience - it may only lead to a partial solution (or no solution at all) - but it's a start.
- Enjoy the feeling of being lost - this is part of problem solving and what makes finding the solution rewarding!