How to Solve It: A New Aspect of Mathematical Method

Polya emphasizes the importance of mental operations and heuristic reasoning in problem-solving by highlighting their role in discovering solutions and fostering independent thinking. He argues that understanding and applying these mental operations is more valuable than rote learning, as it encourages active engagement with problems.

Key heuristic strategies Polya suggests include:

1. Understanding the Problem: Clearly defining the unknown, data, and conditions to grasp the problem's essence.
2. Devising a Plan: Using heuristic reasoning to generate a plausible solution strategy.
3. Carrying Out the Plan: Rigorously checking each step for correctness.
4. Looking Back: Reviewing the solution to ensure its validity and identify areas for improvement.
5. Variation of the Problem: Generalizing, specializing, analogizing, and decomposing/recombining elements to explore different perspectives.
6. Heuristic Reasoning: Employing plausible, provisional reasoning to guide the search for a solution, while maintaining a critical eye for its limitations.
7. Checking Your Guess: Examining assumptions and intermediate results to ensure they align with the problem's requirements.
8. Using Auxiliary Elements and Problems: Introducing additional elements or related problems to facilitate the solution process.

According to Polya, the teacher plays a crucial role in fostering problem-solving skills by guiding students through the process of understanding, planning, executing, and reviewing solutions. The teacher should:

1. Challenge curiosity: Present problems that are proportionate to the students' knowledge, stimulating their interest and encouraging independent thinking.
2. Facilitate understanding: Help students understand the problem by asking questions like "What is the unknown?" and "What are the data?" and by encouraging them to draw diagrams and introduce notation.
3. Guide planning: Assist students in finding connections between data and the unknown, suggesting related problems or theorems, and helping them devise a plan for solving the problem.
4. Support execution: Encourage students to check each step of their plan and ensure they understand the reasoning behind it.
5. Encourage reflection: After solving the problem, prompt students to verify their solution, consider alternative methods, and apply their findings to other problems.

To encourage students to develop their own problem-solving abilities, teachers should:

1. Model problem-solving: Demonstrate their own problem-solving process, showing how they ask questions, make connections, and evaluate solutions.
2. Provide opportunities for practice: Offer a variety of problems and encourage students to try solving them independently.
3. Encourage questioning: Allow students to ask questions and explore different approaches to problems.
4. Praise effort and progress: Acknowledge students' efforts and progress, even if they don't find the solution, to foster a growth mindset.

Polya distinguishes understanding a problem from mere memorization by emphasizing the importance of comprehending the problem's structure and the reasoning behind the solution. He suggests that understanding involves a deep engagement with the problem, including identifying the unknown, the data, and the conditions, and understanding their relationships. This contrasts with memorization, which involves rote learning of solutions without understanding the underlying concepts or the process of deriving those solutions.

For students, understanding a problem is crucial for developing problem-solving skills and critical thinking. It allows them to apply their knowledge flexibly and creatively to new situations. For educators, fostering understanding rather than mere memorization is essential for effective teaching. It encourages active learning, promotes deeper engagement with the subject matter, and prepares students for real-world challenges where rote learning is insufficient.

George Polya identifies several common pitfalls in problem-solving, which can be avoided through effective strategies:

1. Not Understanding the Problem: Misunderstanding the problem is a frequent pitfall. To avoid this, carefully analyze the problem, identify the unknown, data, and conditions, and ensure clarity in understanding.

2. Overlooking Auxiliary Problems: Sometimes, the main problem is too complex. Polya suggests considering auxiliary problems that are simpler or more general to gain insights.

3. Ignoring the Use of All Data: Not using all available data can lead to incorrect solutions. Always check if you've utilized all the given information and conditions.

4. Failing to Check Each Step: Skipping steps or not verifying them can lead to errors. It's crucial to check each step for correctness and completeness.

5. Not Revisiting Definitions: Definitions are the foundation of mathematical problems. Revisiting definitions can clarify misunderstandings and ensure a solid understanding of the problem.

6. Being Pedantic: Applying rules rigidly without considering the context can lead to mistakes. Use your judgment and avoid being overly pedantic.

7. Neglecting the Solution's Evaluation: After solving a problem, it's important to review the solution, consider alternative methods, and check its applicability to other problems.

By following Polya's problem-solving strategies, such as understanding the problem, devising a plan, carrying out the plan, and looking back, one can avoid these pitfalls and improve problem-solving skills.