Chapter 1: an Introduction to Problem Solving

Chapter 1: An Introduction to Problem Solving

1.2 Mathematics and Problem Solving

1.2.1. Four-Step Problem Solving Process

1.2.1.1. George Polya’s (1887-1985) 4 steps

1.2.1.1.1. Understand the problem

1.2.1.1.2. Make a plan

1.2.1.1.3. Carry out the plan

1.2.1.1.4. Look back

1.2.1.2. Applies to everything, not just mathematics

1.2.1.3. Every discipline requires mathematics. Why? Every discipline needs problem solvers

1.2.2. Avoiding Mind Sets

1.2.2.1. One way or the high way doesn’t work

1.2.2.2. The more options or strategies you know the better off you are

1.2.2.3. Sometimes better to let a problem get “cold” and then try again later

1.2.2.4. Stick-to-itiveness is a big key

1.2.3. Strategies for Problem Solving

· Several strategies discussed here

· Many more strategies exist

· Don’t be afraid to let your thinking get “outside of the box”

1.2.3.1. Strategy: Look for a Pattern

1.2.3.1.1. sometimes patterns are geometric shapes

1.2.3.1.2. sometimes patterns are arithmetic - +, -, x, / with numbers

1.2.3.1.3. sometimes patterns are algebraic

1.2.3.1.4. patterns are often seen when the problem is organized in different ways

1.2.3.1.4.1. Gauss’s problem p. 19

1.2.3.1.5. sometimes we use other strategies to help us see the pattern

1.2.3.2. Strategy: Make a Table

1.2.3.2.1. summarizes data

1.2.3.2.2. helps to see a pattern

1.2.3.2.3. helps to consider all possible cases in a problem

1.2.3.2.4. Your turn: Checkerboard Worksheet

1.2.3.3. Strategy: Examine a Simpler Case

1.2.3.3.1. make a simpler problem out of a complex problem

1.2.3.3.2. gain insight from simpler cases

1.2.3.3.3. synthesize simpler cases into complex model

1.2.3.4. Strategy: Identify a Subgoal

1.2.3.4.1. identify a more familiar related problem

1.2.3.4.2. use the insight of the related problem to solve the original problem

1.2.3.5. Strategy: Examine a related Problem

1.2.3.5.1. look for relationships to previously solved problems

1.2.3.5.2. try applying a similar approach to the new problem

1.2.3.6. Strategy: Write an Equation

1.2.3.6.1. Sometimes the best way to summarize a problem is by writing an equation

1.2.3.6.2. Your turn: Revisit Checkerboard Worksheet

1.2.3.6.3. Your turn: Bean Boxes

1.2.3.7. Strategy: Draw a Diagram

1.2.3.7.1. one of the best ways for gaining insight into the solution of a problem

1.2.3.7.2. doing it “in your head” is not helpful and often leads to overlooking options that diagrams suggest

1.2.3.7.3. A picture simplifies a problem and often makes the solution obvious

1.2.3.8. Strategy: Guess, Check, and Revise

1.2.3.8.1. probably one of the most popular problem solving methods

1.2.3.8.2. important that you understand that guess does not imply “wild” guess

1.2.3.8.3. the check and revise portions of this strategy are critical aspects implying thought and consideration will go into subsequent guesses

1.2.3.9. Strategy: Work Backward

1.2.3.9.1. Sometimes going from the answer to the question is a good way of figuring out how an answer was obtained

1.2.3.9.2. this is a critical skill for a teacher to have because it is a big part of determining error patterns in students’ thinking

1.2.3.9.3. This is an important strategy to be comfortable with before the student takes courses with proofs in them like geometry or trigonometry

1.2.3.10. Strategy: Use Indirect Reasoning

1.2.3.10.1. restate the problem as an absolute

1.2.3.10.2. restate the converse of the problem as an absolute

1.2.3.10.3. look for a contradiction

1.2.3.11. Strategy: Use Direct Reasoning

1.2.3.11.1. problem statement is already an absolute

1.2.3.11.2. identify cases

1.2.3.11.3. look for contradictions to cases

1.2.3.12. Summary of Strategies

1.2.3.12.1. strategies are not used independently

1.2.3.12.2. strategies are used in concert

1.2.3.12.3. not every strategy applies to every problem

1.2.3.12.4. every strategy has not been explored here

1.2.3.12.5. whatever works for you is a good strategy

1.2.3.12.6. be open to trying new strategies

1.2.4. Ongoing Assessment p. 32

1.2.4.1. Home work: 1a, 2, 14, 18, 29, 34

1.2.4.2. All solutions should be written using Polya’s 4 steps