A number of disciplines require the application of problem-solving skills. This often centers on the use of logic, an exploratory mindset and creativity. You may be expected to apply these skills to find a solution to a given problem.
Problem solving tasks can be presented as part of your coursework (e.g. problem sets), an assignment (e.g. a project), assessment (e.g. an exam), or as an additional study area.
In this resource, you will meet a problem-solving framework that can be applied to many different types of problems. The framework will be applied to some examples throughout the resource so that you can see how it can be used in different settings.
The problem-solving framework will be applied to a pure mathematics problem, a calculus problem and a mathematical modelling problem.
This problem-solving resource has been developed by Vicki Brown (Engineering) and Houry Melkonian (Mathematics & Statistics) from the Faculty of Environment, Science and Economy. The process was supported by Cris Burgess (Psychology) from the Faculty of Health and Life Sciences.
In this section, we will consider the first step in solving a problem – understanding what the question is asking. Below are subsections that address different aspects of that understanding.
There are two main things to consider at first
One method you can use to break down the question is the ‘TAP’ method. For this, you want to identify the Topics, Actions and Parameters that the question gives you. Underline or circle these as you consider the problem. In a mathematical equation, the topics will relate to mathematical topics or techniques identified in the question; the actions will be words such as ‘evaluate’, ‘analyse’, ‘solve’, ‘model’, ‘prove’. Identify these in your problem as these will indicate what you are being asked to do to tackle the problem. Parameters can either be numerical values (information given in the question) that will be used in the course of solving the problem, or may be the constraints on the problem, so that it limits the space in which the problem is solved.
Once you have identified the key information in the question and its main goals, focus on the information you need to complete your solution. If you’re struggling with breaking down the question, consider whether you need to talk to your lecturer or revisit your lecture notes to help you determine a starting point.
If you can understand most of the question, your next step is to look more deeply at what the question is asking. Now is the time to consider what knowledge you need in order to tackle the problem. In particular:
At this point, you should understand the question, what you are trying to accomplish and what knowledge you need to start solving it. Once you have gathered all of this information, you need to bring it together and assess your understanding of the overall picture. At this point, you can identify if there are any other gaps in your knowledge or understanding of the problem.
In this section, we will look at methods for planning your solution. You are likely to need to revisit your plan as you work through the problem, so expect your plan to be a ‘live’ document that you amend and adjust as needed.
You will have some of the knowledge needed to solve the problem already, and now is the time to find any additional information you need.
Once you have gathered the knowledge you need to tackle the problem, you are ready to choose your initial approach to your solution.
Your plan should give you a route to the solution. At this point, you know your starting point and need to identify potential next steps from there. Now you’ve identified a chosen approach, begin to write it out in detail following the steps below. This will help you structure your solution and give you a reference as you work through the problem.
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 your plan, and your approach to the problem, will help you stay on track. That said, this is a great point to review your plan.
You are now ready to begin! At this point you should be feeling prepared and ready to start your solution. We can break this down further, however, and look at different aspects of carrying out the plan and things to consider as you work towards your solution.
As you work towards your solution, it is important to keep your plan in mind. You might find you need to deviate from it, but your plan should include your route to your answer, so you will want to make sure you are continuing to go in the right direction.
Return to the original problem and remind yourself of what you are trying to answer. Confirm that your work is still moving towards a solution.
Setbacks happen, but they can have an emotional impact and affect confidence.
If you find you that attempting to solve a problem is having an impact on how you feel, have a look at the advice from Wellbeing.
Attempt to identify the mistake.
With any solution, it is important to review both your answer and the process you used to get there.
Plausibility of solution
What have I learnt?
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/assignments?
To solve a new problem, try to look for patterns and features shared with other existing problems (e.g., compare the new problem to an old one which you have already solved). Check the assumptions given, check the goal which the problem is claiming and try to identify any similarities and/or differences, and then devise a plan and follow the problem-solving framework. Exploring this will help you to develop a toolkit to solve problems.
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