Problem-solving is a key transferable skill, developing the use of strategic planning, logic, curiosity, an exploratory mindset and creativity across many disciplines. Problem-solving tasks can take the form of coursework, such as problem sets; an assignment; assessment or as an additional study area.
Here, we’ll present a problem-solving framework that can be applied to many different types of problems. The framework uses Polya’s four-step model of problem-solving and elaborates on each of its stages.
Stage 1: Understanding the problem question or assignment
Stage 2: Devising a plan
Stage 3: Carrying out the plan
Stage 4: Review, make sense of and inform
It also provides a list of questions that learners and educators can use to support each part of the model.
Stage 1: understanding the problem question or assignment
Let’s consider the first step in solving a problem – understanding what the question is asking. Show your students how to break your question or assignment down:
- Identify any assumptions: what does the question tell you about the problem?
- Identify the goals: what are you trying to achieve?
Recall context and basic knowledge needed – ascertain a starting point:
- Search for the meaning and the definition of the terms and notations used
- Look for theoretical and practical aspects relevant to this problem. Remember to use the assumptions and the goals you have identified.
Extracting pertinent information and data
At this point, your student should understand the question, what they are trying to accomplish and what knowledge they need to start solving it. Once they have gathered all of this information, they need to bring it together and assess their understanding of the overall picture. Help them identify if there are any other gaps in their knowledge or understanding of the problem.
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Stage 2: devising a plan
Here, we’ll look at methods for planning the solution to a problem-solving project. Your students will likely need to revisit their plan as they work through the problem, so tell them to expect it to be a live document that they amend and adjust as needed.
Ascertain avenues to fill the gaps in students’ knowledge and/or data
Help them check and honestly assess their knowledge, identifying what they know and what they do not know.
Encourage them to make note of their findings and determine how to bridge those gaps in knowledge.
Investigate where to gain knowledge and/or data needed
Your students will have some of the knowledge needed to solve the problem already, and now is the time to find any additional information they need.
Point them towards module notes and other resources to form a good understanding of those topics before any further implementation of techniques.
Read and practise relevant and similar examples and exercise problems.
If they still struggle, make it clear this is the point to seek help, perhaps from you or their other lecturers, and consider peer discussion as a supportive way of learning.
Consider appropriate approaches to find a solution
Help your students develop their approach using the assumptions and the goals they have identified.
For each approach, encourage them to revisit the assumptions and definitions given in the question. Do they give enough information to start this approach? Is the starting point for this approach clear from the information given in the question?
Show them how to analyse and check whether an approach is suitable for the problem. For example, the choice of the statistical distribution used to study a set of numerical data (measurements) depends on the nature of the random variable – ie, whether it is a discrete or a continuous random variable.
Identify a chosen course of action
Now they have identified a chosen approach, they can begin to write it out in detail following the steps below.
- Begin by looking at the starting point. They should establish what they intend to do after that. Does it appear to be moving them closer to the goal? Is it achievable with the information they have?
- Once the appropriate technique for the next step is selected, prompt them to consider applying it to the current point in their solution. Do any issues arise at this point?
- Look for similar examples or exercises that have been discussed or outlined. Forming a background understanding of such problems will enhance their ability to identify any patterns, and it will help develop their analytical skills.
Sense-check
This is one of the most important steps in problem-solving and it does not just happen at the end of a section. Regularly reviewing a plan, and the approach to the problem, will help your students stay on track.
Once their plan is complete, help them review it. Identify any points of concern or where extra information may be needed.
Do any plans need revising before starting?
Stage 3: carrying out the plan
You are now ready to begin! At this point your students should be feeling prepared and ready to start their solution.
Useful steps for students to take
Reread the question and ensure they are confident that they have identified the key points.
Consider their starting point and if there was a particular idea that helped formulate their plan. Are they still happy that this is the right place to start?
When progressing through the plan, as they work towards their solution, it is important to keep the plan in mind.
Revisit the plan frequently and confirm that they are still following it.
Assess whether they need to revise the plan.
Confirm that their work is still moving towards a solution.
What to tell your students when things go wrong
Setbacks happen, but they can have an emotional impact and affect confidence.
Take a deep breath! Mistakes happen and it’s important to see what this one can tell you.
Struggling to understand problems like this is all part of the learning process and perfectly natural. If it was always easy, you wouldn’t learn from the experience and understand your approach the next time you try to solve a similar problem.
How to effectively retrace your steps
Help students identify their mistake. Establish whether this is a correctable error, such as a calculation error, or a misstep towards the solution. If the former, prompt them to correct and continue. If the latter, help them revisit their plan for solving the problem. Is it still correct? Does it need revising?
Return to a point where they are confident that their approach is correct and start again from there.
Stage 4: review, make sense of and inform
With any solution, it is important your students review both their answer and the process they used to get there. Here’s how they can do it.
Plausibility of solution
- Consider whether you can check the answer in some way
- Can you estimate the answer with another method?
- Can you put your solution into the original problem?
- Can you challenge your answer with new data?
- Does it appear to be a reasonable answer?
What have you learned?
- Reflect on the problem and the solution you have found.
- Review the methods you used to solve the problem.
- Evaluate how you approached any mistakes.
- Study skills – are you happy with your time management in solving the problem?
- What can you take from this problem to help you approach problem solving in the future?
Evaluating your approach – how might you apply this approach in other settings or assignments? To solve a new problem:
- look for patterns shared with other problems.
- check assumptions, and the goal which the problem is claiming.
- identify any similarities and/or differences, and then devise a plan.
- follow the problem-solving framework. Exploring this will help you to develop a toolkit to solve problems.
Houry Melkonian is a senior lecturer in mathematics and the director of STEM Foundation Year Programme, Vicki Brown is a lecturer in engineering and Cris Burgess is a senior lecturer in psychology, all at the University of Exeter.
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