1. Problem solving
Math 123
September 10-12, 2008

2. Why problem solving?

3. Why problem solving? Essential for mathematics
According to NCTM, one of the processes through which mathematics should be taught
Fun way to begin semester.

4. NCTM Standards
Instructional programs from prekindergarten through grade 12 should enable all students to--
build new mathematical knowledge through problem solving;
solve problems that arise in mathematics and in other contexts;•
apply and adapt a variety of appropriate strategies to solve problems;
monitor and reflect on the process of mathematical problem solving.

5. Washington State Standards

6. Problem 1
There are 31 people at the party. If everyone shakes hands with each of the other party-goers exactly once, how many handshakes will there be?

7. Problem 2
If 14 toothpicks are arranged to form a triangle so none of the toothpicks are broken and all 14 toothpicks are used, how many different-shaped triangles can there be?

8. Problem 3
In three years, Chad will be three times my present age. I will then be half as old as he. How old am I now? 3.

9. Polya’s four steps -- page 4 in the book
Understand the problem
Devise a plan
Carry out the plan
Look back

10. Polya’s ten commandments for teachers -- page 1 in the book
Be interested in your subject.
Know your subject.
Try to read the faces of your students; try to see their expectations and difficulties; put yourself in their place.
Realize that the best way to learn anything is to discover it by yourself.
Give your students not only information, but also know-how, mental attitudes, the habit of methodical work.

11. 6. Let them learn guessing.
7. Let them learn proving.
8. Look out for such features of the problem at hand as may be useful in solving the problem to come -- try to disclose general pattern that lies behind the present concrete situation.
9. Do not give away your whole secret at once -- let the students guess before you tell it -- let them find out by themselves as much as is feasible.
10. Suggest, do not force information down their throats.

12. Problem solving strategies
Guess and test (i.e. guess and check)
Draw a picture
Use a variable
Look for a pattern
Make a list
Solve a simpler problem

13. Handshake problem
Which strategies did we use?

14. Handshake problem Draw a picture Look for patterns
Solve a simpler problem

15. Toothpick problem
Which strategies did we use?

16. Toothpick problem
Draw a picture
Make a list
Solve a simpler problem

17. Age problem
Which strategies did we use?

18. Age problem
Guess and test
Use variables

19. Important!
Make sure to read this chapter because it solves sample problems for each strategy and teaches you how to recognize which strategy to use.
Note that you will hardly ever be using one strategy in isolation. It is not as important to recognize strategies as it is to solve problems. This comes with practice.