LibGuides: Problem Solving (Numerate & Conceptual): Home

Introduction

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.

Understanding Your Problem Question or Assignment

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.

Breaking your question/assignment down:

There are two main things to consider at first

Identify assumption(s) – these are information given in the question, i.e., what does the question tell you about the problem?
Identify the goal(s) - these are the desired outcome(s), i.e., what are you trying to achieve?

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.

Recall context and basic knowledge needed – ascertain a starting point:

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:

Search for the meaning and the definition of the terms, symbols, and notations used (in the context of your discipline/module);
Look for theoretical and practical aspects relevant to this problem (remember to use the assumption(s) and the goal(s) you have identified when breaking your assignment/question down, as an inspiration).

Extracting pertinent information and data.

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.

Devising a Plan

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.

Ascertain avenues to fill the gaps in your knowledge and/or data – identify how to find the information you need.

Check and honestly assess your knowledge, identifying ‘what you know’ and ‘what you do not know’. This builds on your initial work in assessing what the question is asking.
Make note of your findings and determine how to bridge those gaps in knowledge.

Investigate where to gain knowledge and/or data 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.

Use module lecture notes, online resources, reading lists to form a good understanding of those topics before any further implementation of techniques.
Read and practice relevant and similar examples and exercise problems.
If you still struggle, make note of any questions arising and seek help from your lecturers.
Consider peer discussion as a collective and supportive way of learning.

Considering appropriate approaches to find a solution.

Once you have gathered the knowledge you need to tackle the problem, you are ready to choose your initial approach to your solution.

Develop your approach using the assumptions and the goals you have identified when first assessing the question.
For each approach, revisit the assumptions and definitions given in the question. Do they give you enough information to start this approach? Is the starting point for this approach clear from the information given in the question?
Analyse and check whether an approach is suitable for the problem by making sure that the given problem fulfils its criteria: for example:
- If the chosen technique is only applicable for whole numbers, then it will not be suitable for a problem that contains fractions.
- The choice of the statistical distribution used to study a set of numerical data (measurements) depends on the nature of the random variable, i.e., whether it is a discrete or a continuous random variable.

Identify your chosen course of action.

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.

Begin by looking at your starting point. Establish what you intend to do after that. Does it appear to be moving you closer to your goal? Is it achievable with the information you have?
Once the appropriate technique for the next step is selected, consider applying it to the current point in your solution. Do any issues arise at this point?
Look for similar examples or exercises that have been discussed or outlined in the module material – forming a background understanding of such problems will enhance your ability to identify any patterns; and it will help you develop 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 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.

Once your plan is complete, review it. Identify any points of concern or where extra information may be needed.
Confirm that this appears to be the best way to solve the problem – does any of your plans need revising before you begin?

Carrying Out 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.

Common steps to take

Reread the question and ensure you are confident that you have identified the key points.
Consider your starting point and if there was a particular idea that helped formulate your plan. Are you still happy that this is the right place to start?

Progressing through the plan

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.

Revisit your plan frequently and confirm that you are still following it.
Check if your plan is still leading you to your goal. Assess whether you need to revise your plan.
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.

What to do 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.
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.

How to effectively retrace your steps

Attempt to identify the mistake.

Establish whether this is a correctable error (e.g. a calculation error), or a mis-step towards the solution. If the former, correct and continue. If the latter, revisit your plan for solving the problem. Is it still correct? Does it need revising, with a new route to the solution?
Return to a point where you are confident that your approach is correct and start again from there.

Review, Make Sense of and Inform

With any solution, it is important to review both your answer and the process you used to get there.

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 your answer make sense in terms of the problem?
- Does it appear to be a reasonable answer?

What have I learnt?

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/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.