Solve It Science and Mathematics

Polya defines the four phases of problem-solving as understanding the problem, devising a plan, carrying out the plan, and looking back. Understanding the problem involves comprehending the problem statement and desired outcome. This phase is crucial as it sets the foundation for the rest of the process.

The second phase, devising a plan, is where the problem solver creates a strategy to achieve the solution. This phase is significant because it outlines the steps needed to solve the problem, ensuring that the problem solver has a clear direction.

Carrying out the plan is the execution phase, where the problem solver follows the steps outlined in the previous phase. This phase is essential as it is where the problem solver applies their knowledge and skills to solve the problem.

Finally, looking back involves reviewing the solution to ensure its correctness and understanding. This phase is significant because it allows the problem solver to learn from their experience, improve their problem-solving skills, and apply the knowledge gained to future problems.

The book "How to Solve It" by George Polya outlines several key heuristic techniques and strategies for effective problem-solving:

1. Understanding the Problem: This involves identifying the unknown, data, and conditions, and determining if the conditions are sufficient or redundant. Drawing a figure and introducing suitable notation can also aid in understanding.

2. Devising a Plan: This step requires finding connections between data and the unknown, considering auxiliary problems if needed. It involves looking for related problems, analogies, and familiar problems with similar unknowns.

3. Carrying Out the Plan: This is the execution of the solution plan, with each step checked for correctness.

4. Looking Back: This involves examining the solution, checking the result and argument, and considering alternative methods or applications of the result.

5. Variation of the Problem: This technique includes decomposing and recombining elements, going back to definitions, and using generalization, specialization, and analogy.

6. Heuristic Reasoning: This involves using plausible reasoning and not mixing it with rigorous proof. It's important for discovering solutions but must be examined for correctness.

7. Progress and Achievement: Recognizing progress and achievements in problem-solving is crucial for maintaining motivation and focus.

These techniques and strategies contribute to effective problem-solving by guiding the solver through a structured process, encouraging creativity and critical thinking, and helping to identify and overcome obstacles. They also promote active learning and the development of problem-solving skills that can be applied to a wide range of problems.

Polya's approach to problem-solving emphasizes understanding the problem, devising a plan, carrying out the plan, and looking back to evaluate the solution. This method is applicable to both mathematical and practical problems, but the implications differ slightly.

For mathematical problems, the focus is on logical reasoning and abstract thinking. Polya's approach encourages identifying the unknown, data, and conditions, and using general principles to derive a solution. This approach is crucial for teaching mathematics as it fosters critical thinking and the development of problem-solving skills.

For practical problems, the emphasis is on applying mathematical concepts to real-world situations. Polya's approach still applies, but it also requires understanding the context, considering various perspectives, and making trade-offs. This approach is essential for teaching practical subjects, as it encourages students to apply their knowledge to solve real-world challenges.

The implications for teaching and learning are that educators should encourage students to develop both mathematical and practical problem-solving skills. This can be achieved by incorporating real-world examples into mathematical lessons and using mathematical principles to solve practical problems. By doing so, students can develop a deeper understanding of both mathematical and practical concepts and become better prepared for future challenges.

In Polya's problem-solving process, heuristic reasoning plays a crucial role as a guiding tool for discovering and understanding mathematical truths. It serves as a provisional, plausible approach to problem-solving that allows for exploration and hypothesis formation before rigorous proof is established. Heuristic reasoning is often based on induction, analogy, and other inductive methods, which help to generate initial ideas and solutions.

This approach contributes to the discovery and understanding of mathematical truths by:

1. Generating Ideas: Heuristic reasoning encourages the exploration of various perspectives and connections, which can lead to the discovery of new ideas and solutions that might not be immediately apparent.

2. Facilitating Learning: By using heuristic reasoning, students can develop a deeper understanding of mathematical concepts and their interrelations, as they learn to think flexibly and creatively.

3. Building Intuition: Through heuristic reasoning, problem-solvers can develop a sense of intuition about mathematical problems, which can help them to anticipate potential solutions and avoid common pitfalls.

4. Guiding Proof: While heuristic reasoning is not a substitute for rigorous proof, it can guide the development of a proof by suggesting potential paths and strategies for investigation.

5. Encouraging Exploration: Heuristic reasoning encourages exploration of different problem-solving methods, which can lead to the discovery of new mathematical truths and the refinement of existing ones.

In summary, heuristic reasoning is a valuable tool in Polya's problem-solving process, as it fosters creativity, intuition, and exploration, which are essential for the discovery and understanding of mathematical truths.