Problem Solving

Background

Problem solving is a fundamental skill, both within physics and a wider context. There are several stages to solving problems. Recognising the fact that there are a number of stages with their associated strategies and tactics should make problem solving easier.

There are differences between strategies and tactics. A strategy is concerned with the approach taken, while a tactic is concerned with the method of doing. For example, an initial strategy might be to find out what the crux of the problem is, while the tactic to do that might be to read the question carefully and in detail.

The problem solving stages

There are four stages to solving a problem:

  1. Understand what the problem is about.
  2. Plan how to tackle the problem.
  3. Do - try and solve the problem.
  4. Evaluate the solution and the methods by which it was achieved.

1. Understanding the problem

In this stage, we need to decide what the problem is about and what might be required for a solution. The strategies for this stage are:

  • Determine what the nub of the problem is.
  • Divide the problem into manageable sub units.
  • Explore the different ways of looking at the problem.

Some tactics that might be used with these strategies are:

  • Read carefully the problem description.
  • It can be useful to restate the problem, particularly if the description is quite long. The aim is to get a simplified, clear definition of the problem.
  • Note key points and focus on the relationships between different aspects of the problem.
  • Drawing a diagram or graph can help to visualise the problem.
  • Note the explicit information in the description - these can be key points in gaining an overview of the problem.
  • Implicit information can be more difficult to spot - a list of relevant variables might highlight hidden information.
  • Look for the key concepts involved in the problem - what areas of physics do they involve?

2. Planning a solution

In this stage, we need to decide how the problem will be tackled. Strategies for this stage might include:

  • Developing a mental toolbox for solving problems. This is more of a long term solution so that when you meet a problem that you can't immediately solve, there are a set of procedures that can be tried that may lead to a solution.
  • Relate the problem to something else such as a lecture, laboratory material or another problem previously encountered.
  • Look for the structure of the problem.

Tactics for this stage would include:

  • Splitting the problem into more manageable sub-units. Quite often a problem appears complex but consists of several simpler problems that need to be solved in sequence.
  • Step by step. Similar to splitting the problem, but focussing on the mechanics of the solution. Look at the data, assign variables, draw a diagram and trace through the sequence.
  • Look for similarities. Have you seen a similar problem in the past? How similar was it? Are there subtle differences you need to take notice of?
  • Look for alternatives. Sometimes the first method of attacking the problem can lead to difficulties later, so consider alternatives before getting bogged down in lengthy algebra.
  • Simplify. Could sensible approximations be made? For example, could the effect of friction be ignored?
  • List all variables and fixed quantities. Include units so that you can keep track of dimensions.

3. Work the problem

In this stage, we implement the plan. In simpler problems, this stage is jumped to directly, with the first two stages being done almost subconsciously. The strategy for this stage is to solve the problem. Some of the tactics that may be involved are:

  • Generate relevant formulae.
  • Evaluate or 'solve' these formulae.
  • Try trial and error solutions.
  • Iteration. Try some initial guesses and rough calculations, and then modify your approach to home in on a solution through iteration.
  • Proof by contradiction. Prove something to be true by proving the opposite to be false. For example, in a multiple choice question you could identify obviously incorrect answers rather than work through each answer in turn.
  • Dimensional analysis is more often used to check whether a particular solution could be correct rather than to generate a solution. However, it can also be useful to determine whether or not all the relevant variables have been incorporated when obtaining the solution.

4. Evaluate the solution and method

This stage is often forgotten, but is vital. Evaluating the method helps to develop the mental toolbox strategy mentioned in stage two, since you are explicitly assessing how well the methods used worked in practice. Some strategies for this stage are:

  • Test the results.
  • Reflect on the methods used to obtain them.

The tactics would be:

  • Are the results sensible? Does the final solution match that of an order of magnitude estimate? Are the dimensions correct? If you change the initial conditions do you still get a sensible answer?
  • Evaluate procedures. Check that any approximations made were in fact valid. Spend a little time thinking about how this problem was solved and whether a different method might have been easier. The main point is to evaluate your methods so that you learn something from each problem that can help with future problems.

Other resources

Physics specific resources are:

  • Walker, J. (2007). The flying circus of Physics. Hoboken, N.J.: Wiley. [Classmark q QC 32/W]
  • Riley, K.F. (1982). Problems for physics students. Cambridge: Cambridge University Press. [Classmark QC32/R].

A general chapter in problem solving can be found in Brown, G. and Atkins, M. (1994). Effective Teaching in Higher Education. London: Routledge. [Classmark LB 2331/B]

Summary

Stage Strategies Tactics
Understand. Determine.
Divide.
Explore. 		Read carefully.
Restate the problem.
Note key points.
Sketch / graph / diagram.
Note explicit/implicit information.
Note key concepts.
Plan. Develop a mental toolbox.
Relate to other problems.
Look for the structure.
 Break the problem into smaller units.
Take a step-by-step approach.
Look for similarities.
Look for alternatives.
Simplify.
List variables.
Work the problem. Solve. Generate formulae.
Evaluate formulae.
Trial and error.
Iteration.
Proof by contradiction.
Dimensional analysis.
Evaluate. Test the results.
Reflect on the methods. 		Are the results sensible?
Evaluate procedures.