Chapter 1: Mathematical Processes

1.3 Mathematics as Problem Solving – p. 35

1.3.1. The Role of Problem Solving

1.3.1.1. Central to the development and application of mathematics

1.3.1.2. Used extensively in all branches of mathematics

1.3.1.3. The Meaning of a Problem p. 36

1.3.1.3.1. Description of a problem: A problem is a situation for which the following conditions exist

· It involves a question that represents a challenge for the individual

· The question cannot be answered immediately by some routine procedures known to the individual

· The individual accepts the challenge

1.3.1.3.2. Mini-Investigation 1.7 – p. 36

1.3.1.4. The Meaning of Problem Solving p. 37

1.3.1.4.1. Description of Problem Solving: Problem solving is a process by which an individual uses previously learned concepts, facts, and relationships, along with various reasoning skills and strategies, to answer a question about a situation

1.3.1.4.2. Algorithms are known steps used for solving different types of equations – the problem solving process CANNOT be made into an algorithm

1.3.1.4.3. Answer vs. Solution

1.3.1.4.3.1. Answer – final result

1.3.1.4.3.2. Solution – process used to find the answer

1.3.1.4.4. Example 1.10: Pizza problem – see solutions p. 37-38

1.3.1.4.5. Your Turn p. 39 Solve in groups

1.3.2. A Problem-Solving Model

1.3.2.1. George Polya’s (1945) model p. 39

1.3.2.1.1. Problem posing

1.3.2.1.2. Understanding the problem

1.3.2.1.3. Making a plan

1.3.2.1.4. Carrying out the plan

1.3.2.1.5. Looking back

1.3.2.2. Example 1.11: Telephone Lines p. 39-41

1.3.2.3. Your Turn p. 41 Solve in groups. Use the example on p. 40-41 as a guide

1.3.2.4. Estimation is the process of determining an answer that is reasonably close to the exact answer used in different stages of problem solving

1.3.3. Problem-Solving Strategies

· Make a model

· Act it out

· Choose an operation

· Write an equation

· Draw a diagram

· Guess – check – revise

· Simplify the problem

· Make a list

· Look for a pattern

· Make a table

· Use a specific case

· Work backward

· Use reasoning

1.3.3.1. Learning when and how to use problem solving strategies is an important problem solving skill

1.3.3.2. Example 1.12: The Bicycle-built-for-two rides p. 42

1.3.3.3. Your Turn p. 42-43 Solve in groups

1.3.4. Importance of Problem Solving

1.3.4.1. Mathematics is primarily used to solve problems in mathematics and in the real world

1.3.4.2. Learning to solve problems is the principal reason for studying mathematics

1.3.4.3. Mathematics is MUCH more than algorithms

1.3.4.4. Problem solving applies to all aspects of our lives, NOT just mathematics