Transform your approach to maths problems with this simple framework
Solving mathematics problems can be a daunting task for many students. However, with the right framework, even the most complex problems become manageable. One highly effective method for tackling mathematics problems is Polya’s Four-Step Problem-Solving Framework. Developed by mathematician George Polya, this approach offers a clear and structured way to break down problems into manageable steps. In this guide, we’ll explore each step in detail, providing tips and examples to help you build confidence and solve problems more efficiently.
Step 1: Understand the problem
The first and most essential step is to fully understand the problem. While this may seem straightforward, my experience of over a decade marking mathematics exams has shown it is crucial to emphasise. Many students, feeling the pressure of limited time, skim the question and assume they know what is being asked. They immediately dive into solving and fill the page with detailed calculations or even a well-thought-out solution — only to discover it doesn’t address the actual question. No matter how brilliant the solution, if it doesn’t answer the problem as presented, it won’t earn the marks students expect. Taking the time to thoroughly understand the problem is key to success. This approach is particularly important for complex word problems, where information is often presented in a non-mathematical form.
Start by carefully reading the problem to avoid misinterpretation. Dissect the question and identify key information by asking yourself these essential questions:
What is the problem asking for?
What information is provided?
Are there any constraints or conditions to consider?
It may be useful to take the time to restate the problem in your own words or draw a visual representation. Diagrams, tables, or even simple sketches can often make relationships clearer.
It may also be helpful to quickly estimate what your answer should be or in what range you expect it to be. This estimation before solving allow to catch potential calculation errors at the end (see Step 4 below).
Step 2: Devise a plan
Once you fully understand the problem, the next step is to devise a plan — choosing the most appropriate strategy based on the problem’s type and complexity. Problem-solving in mathematics occurs at various levels of difficulty, each requiring increasingly sophisticated decision-making:
Level 1: Initially, you may only be able to solve problems that match familiar patterns. Recognising the problem type immediately triggers a known strategy to apply.
Level 2: As your experience grows, you’ll encounter problems that can be approached in multiple ways. At this stage, selecting the most effective strategy — such as choosing the method that is quickest or simplest — becomes essential for efficiency and accuracy.
Level 3: The highest level involves tackling problems of a type you’ve never seen before. This stage demands creativity and adaptability, as you must analyse the situation and devise a new or modified approach to solve the problem.
Progressing through these levels takes extensive practice and reflection. To navigate this process, ask yourself the following essential questions:
How am I going to solve this problem?
Are there multiple approaches?
Which strategy should I choose, and why?
Mastering the art of selecting and adapting strategies for increasingly complex problems is a critical skill that can significantly enhance your mathematical problem-solving capabilities. Stay tuned for a future blog post on problem solving techniques.
Step 3: Carry out the plan
Now it’s time to execute your plan. This step involves applying your chosen strategy and working through the problem step by step.
Tips for carrying out the plan:
Be precise with calculations and notation.
Double-check intermediate steps to avoid small errors that could compound.
Stay organised and always write down all your steps clearly. Why?
Clear calculations allows you to double-check your reasoning (if time permits) and hence, can help you catch mistakes more easily.
It helps examiners award partial credit, even if the final answer is incorrect.
If your plan doesn’t seem to be working, don’t be afraid to revise it or try a different strategy.
Step 4: Review the solution
The final step is to review your solution. This step is often overlooked, but it’s essential for ensuring accuracy and reinforcing problem-solving habits. Double-check calculations and look for forgotten units. Ask yourself:
Does the solution answer the question?
Did I use all the given information correctly?
Is the answer reasonable given the context of the problem or can I verify the accuracy of my solution?
Does the answer satisfy any restriction or conditions?
Is there an alternative method to solve this problem?
For example, if you’re solving a word problem that asks for the velocity of a car and you calculate a result of -20 km/h, it’s crucial to recognise that this cannot be correct, as velocity cannot be negative in this context. Similarly, if your result is 230 km/h, it should raise a red flag since such a speed is unusually high for most vehicles. In both cases, it’s important to develop the habit of critically evaluating your answers and revisiting your calculations to check for potential errors. This kind of reflective thinking is essential for ensuring accuracy and building confidence in your problem-solving abilities.
Conclusion
Polya’s Four-Step Framework — understanding the problem, devising a plan, carrying out the plan, and reviewing the solution — provides a structured, effective way to approach mathematics problems. By following these steps, students can enhance their problem-solving skills, build confidence, and tackle even the most challenging questions with greater ease. Whether working on basic arithmetic or advanced calculus, this method promotes logical thinking and precision, empowering learners to reach new levels of mathematical success.
