Problem Solving

Introduction

Problem solving is a crucial aspect of numerical methods, requiring clear understanding of mathematical concepts and techniques.
Effective problem solving involves a mix of analytical, numerical and graphical methods, as well as appropriate use of technology.

Start by understanding the problem: Identify what’s known, what needs to be found, and any constraints or conditions.
Formulate a mathematical model: This could involve formulating equations, setting up calculation sequences, or devising appropriate diagrams or visualisations.
Solve the problem: Apply relevant methods or techniques, considering the context and the nature of the problem.
Interpret the solution: Make sure the solution makes sense in the context of the problem and check if it’s reasonable in terms of magnitude, direction, etc.
Always reflect on the problem-solving process: Identify effective strategies and consider alternative approaches.

Dialogue and Discussion: Tackling complex problems often requires discussion with peers or teachers to generate ideas and clarify understanding.
Sketching and Diagrams: Help to visualise a problem, making it easier to grasp the underlying concepts.
Approaches to Equations: Knowing when to apply specific techniques to solve equations, like iterative methods, simplification, substitution or rearrangement, is vital.
Hypothesis Testing: Useful in checking the validity of a solution or predicting its behaviour.
Using Technology: Tools like graphic calculators, spreadsheet software or specific mathematical software allow for manipulation and visualisation of data and equations.

Ambiguity or vagueness: In real-world problems, information may be incomplete or unclear, and part of the challenge is to cope with and make sense of such uncertainty.
Complexity and multidimensionality: Problems may involve several unknowns or require a combination of different methods or techniques.
Computational errors: Watch out for mistakes in calculations. Use tools like calculators or software to avoid such errors.
Misinterpretation of solution: Always reflect on whether the solution is reasonable and makes sense in the context of the problem.

Problem solving is key to the application of mathematics in real-world situations and is integral to advanced studies in mathematics and related disciplines.
It enhances critical thinking, analytical skills, creativity and adaptability, important attributes in further studying and careers in STEM fields.