# 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.

## Problem Solving Process

• 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.

## Key Techniques in Problem Solving

• 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.

## Common Challenges in Problem Solving

• 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.

## Importance of Problem Solving

• 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.