Problems occur in all fields of study, not just quantitative ones like mathematics.

What is a “problem”?

A problem can’t be solved by just simply remembering the solution or following a straightforward, well rehearsed procedure (algorithm). It may also be that you don’t even fully understand the problem to start with and aren’t sure what is and is not relevant information for solving it. Consequently, the solution and method of solution to some extent need to be figured out or discovered through exploration.

What does it take to be a good problem solver?

A rich, well organised body of relevant domain knowledge

It’s a bit hard to diagnose what’s wrong with a medical patient if you have no medical knowledge! Your knowledge also needs to be well organised or it’s hard to “find” what you need when you need it!

A repertoire of generic problem solving strategies

Since a vague set of symptoms like stomach pains could have a large number of causes, it is not enough for a doctor to just “know” a lot about the possible causes, they also have to be able to figure out (“diagnose”) which cause is the most likely one.

Determination

If at first you don’t succeed, try, try and try again! (But try different things if your initial ideas don’t work!)

Many people fail to solve problems because they give up too quickly, often because of a variety of common barriers to successful problem solving. (In contrast, consider Einstein’s struggle to develop his general theory of relativity.)

What are some generic problem solving strategies worth remembering?

Review relevant knowledge / identify what you need to know to solve the problem

Many people fail to solve problems, not because they don’t know enough to solve the problem, but because they have trouble activating the knowledge they need to solve the problem.

In quantitative courses like physics, engineering and finance, building up a formula sheet as a ready reference guide when doing problems can be a big help.

When tackling an essay or report, first reviewing the concepts or theories from lectures which might help with the analysis can also be a very useful early step.

Identify the factors which need to be taken into account in coming to an answer

List possibilities and use a process of elimination

Consider parents trying to determine why their baby is crying. First they consider the possibilities – the baby’s nappy could be dirty, the baby could be hungry / tired / too hot or cold / bored / ... – and then test each idea in turn until the baby stops crying.

Use a process of trial and error and where possible

See if what doesn’t work gives you clues about what might

This is how a lot of people learn how to operate a new piece of electronic equipment when they can’t understand the instruction manual, and the process cooks follow when developing new recipes.

In an academic context, a lot of writing proceeds like this: you try ideas for how to organise and explain your thinking and then evaluate these. If you aren’t happy with the result you try to figure out why and modify the writing accordingly.

Organise the information you have into a table or diagram

Doing this might help overcome information overload and can also greatly help with the process of analysis.

Diagrams are often very useful in physics and engineering problem solving, while computer programmers often use flowcharts to help them with software development.

Not making progress?

Check to see if you’re making a constraining assumption which might not be valid

In the classic “9 dots problem” where you are asked to join 9 dots in a square array by four straight lines without lifting your pen, most people fail to make any progress because they sub-consciously assume that the lines have to remain within the confines of the square. However, this problem can only be solved by extending the lines beyond the square – you have to “think outside the box”.

Problem seems unsolvable? Try reframing it

E.g. Perkins (2000) relates the example of a building manager who received numerous complaints from people working in the building that the lift was too slow. The building manager couldn’t solve this problem by putting in a faster lift because that would be too expensive, so the manager reframed the problem. Rather than think that the problem was that the lift was too slow, the manager considered the problem to be that workers were getting impatient/bored while waiting for the lift, and that was a solvable problem – he put in a large ornamental mirror so the workers could admire themselves while they were waiting!

Further Reading: D. Perkins (2000), The Eureka Effect: The Art and Logic of Breakthrough Thinking, New York: W W Norton and Co.

Downloadable resources

Tips and observations about problem solving

Graphic organisers (PDF 3877kb)