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What You Can Learn from How to Solve It

A How to Solve It Guide by the University of Dating

『How to Solve It』 is a book written by the renowned mathematician George Pólya. This book explains the techniques of problem-solving, primarily focusing on mathematics, and discusses methodologies for problem-solving in mathematics education. Pólya's approach serves as a very useful guide for students to systematically learn how to approach mathematical problems and reach solutions.

Author: George Pólya

George Pólya

George Pólya (1887–1985) was a Hungarian-born mathematician widely known for his research in mathematics education, particularly in problem-solving methods. After teaching at institutions such as the University of Zurich in Switzerland, he spent many years as a professor at Stanford University in the United States. Pólya made substantial contributions to teaching mathematical thinking and the process of problem-solving.

As part of his contributions to mathematics education, his approach to problem-solving has had a long-lasting impact on the field. How to Solve It is his most significant work and has become an essential book for educators and mathematicians.

Content and Purpose of the Book

How to Solve It aims to explain how to solve mathematical problems and the process behind it. Pólya presents a practical and succinct methodology for problem-solving and proposes the following four main stages:

  1. Understanding the Problem: Understanding the content of the problem accurately.
  2. Making a Plan: Thinking of strategies and methods to solve the problem.
  3. Carrying Out the Plan: Solving the problem based on the plan.
  4. Looking Back: Checking the solution and considering whether other methods could have been used.

Additionally, Pólya discusses potential obstacles and ways of thinking students may face during the problem-solving process and provides advice for effective learning.

Audience of the Book

How to Solve It is primarily aimed at students and educators interested in mathematics. It has been strongly recommended, especially for teachers involved in mathematics education and students wishing to learn problem-solving techniques. Pólya’s approach is applicable to a wide range of learners, from beginners to advanced students, and his methodologies have been highly regarded as techniques for problem-solving that can be applied in other academic fields and life situations.

Who Has Supported the Book

How to Solve It has garnered support from many mathematicians and educators interested in teaching methods and problem-solving. Pólya’s methodology aims to help learners develop their own ability to solve problems, and as a result, many mathematics educators have incorporated this book into their teaching materials and continue to use it in educational settings.

Furthermore, Pólya's approach is valuable not only in mathematics but also in science, technology, and general problem-solving. As a universal guide for developing problem-solving skills, many educators continue to recommend this book, and its influence persists to this day.

Impact and Evaluation

How to Solve It has had a revolutionary impact on mathematics education, particularly in the approach to problem-solving, and is considered a classic for those who want to learn how to solve problems. The "Four Phases of Problem Solving" proposed by Pólya have been applied across many disciplines beyond mathematics, and today, this approach continues to be widely used in educational settings.

How to Solve It is a book that systematically explains methods for solving mathematical problems and delves deeply into approaches to mathematics education and problem-solving. Below, we provide an overview of the contents of each chapter.


Introduction

  • The introduction emphasizes that effective problem-solving requires not only intuition and experience but also logical thinking and creativity. Understanding "solution methods" in mathematics is presented as a key element for the reader’s growth.

PART I. IN THE CLASSROOM

Purpose This section introduces effective methods for problem-solving between teachers and students.

  1. Helping the Student This section discusses how teachers can support students when they are engaged in problem-solving. A key point is asking questions that prompt students to think for themselves.

  2. Questions, Recommendations, Mental Operations Proper questions are crucial for problem-solving, and this section explains how guiding thought through questions can help students.

  3. Generality Using "general methods" in problem-solving is essential as these methods can be applied to other problems as well.

  4. Common Sense The importance of using common sense is emphasized, and it suggests starting with basic problem-solving methods.

  5. Teacher and Student. Imitation and Practice This section explains the importance of both the teacher’s guidance and the student’s practice in learning how to solve problems.

  6. Main Divisions, Main Questions Understanding the major divisions of problem-solving and the fundamental questions related to each stage is essential.


The Four Phases of Problem Solving

This chapter breaks down the problem-solving process into four stages. These are a series of steps: understanding the problem, making a plan, carrying it out, and reflecting on the solution.

  1. Understanding the Problem To solve a problem, it is necessary to first understand what the problem is asking.

  2. Devising a Plan This involves thinking of strategies and approaches to solve the problem. Creating a plan helps clarify the specific method to be used.

  3. Carrying Out the Plan When executing the plan, it is important to follow the strategy, but also to be prepared to adapt as needed.

  4. Looking Back After solving the problem, reflect on the solution and consider if there were other methods that could have been applied. This reflection will be helpful for future problems.


PART II. HOW TO SOLVE IT

This section explains specific methodologies for solving problems in a dialogue format. Readers can learn techniques and approaches that are directly applicable to problem-solving.


PART III. SHORT DICTIONARY OF HEURISTIC

  • Heuristic A brief definition of intuitive methods and approaches used to solve problems. Heuristics provide strategies for solving problems efficiently, even when a complete proof is not available.

  • Analogy A method of finding solutions based on similar problems or situations.

  • Auxiliary Elements This section discusses additional elements needed to solve a problem.

  • Induction and Mathematical Induction Induction is a method of deriving general rules from specific examples, while mathematical induction is its formal application.


PART IV. PROBLEMS, HINTS, SOLUTIONS

In the final section, actual problems are presented along with hints and solutions. By following the thought process for each problem, readers can gain a deeper understanding of problem-solving techniques.


How to Solve It - Cheat Sheet

Here’s a quick review of the four steps for solving problems. These are based on the following four fundamental steps.


1. Understand the Problem

  • Reading the problem carefully: Carefully read the problem statement.
  • Clarifying the question: Clearly identify what is being asked.
  • Organizing known information: List the information given in the problem.
  • Identifying missing information: Identify what information is needed to find the solution.
  • Confirming conditions: Check the constraints or conditions related to the problem.

2. Devise a Plan

  • Selecting a solution method: Think of mathematical methods or strategies that can be used to solve the problem.

    • Example: Using formulas, drawing diagrams, working backwards, or applying induction or deduction.
  • Applying previous experience: If you’ve solved similar problems before, use that experience to guide you.

  • Considering multiple approaches: It is important not to focus on just one method; explore various possible solutions.


3. Carry Out the Plan

  • Follow the plan: Solve the problem according to the plan you devised.
  • Detailed calculations: Carry out calculations and steps precisely. If something is unclear, double-check at each stage.
  • Check intermediate results: During the problem-solving process, check intermediate results to ensure accuracy.

4. Look Back

  • Confirming the answer: Check if the final answer fits within the context of the problem and meets all conditions.
  • Comparing with other solutions: If possible, try solving the problem using different methods.
  • Applying to other problems: Consider how the solution process can be applied to other problems.

Helpful Tips

  • Visualization: Use diagrams or graphs to help deepen your understanding of the problem.
  • Simplification: If the problem feels difficult, break it down into simpler parts.
  • Think backwards: Try working backwards or using a reverse approach.
  • Check the validity: Always verify that the final answer is correct, possibly through alternative methods.

Basic Strategies for Problem Solving

  • Trial and Error: Try multiple solutions to find the correct one.
  • Pattern Recognition: Identify common patterns in problems.
  • Inductive Reasoning: Derive general laws from specific examples.
  • Deductive Reasoning: Derive specific conclusions from general laws.

By using this cheat sheet, you can make your problem-solving process more efficient and improve your approach to mathematics and other challenges!


Through this book, you can learn the fundamental approaches and techniques for solving mathematical problems and understand how to organize your thinking and efficiently tackle problems.

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