Problem solving Math 123 September 10-12, 2008

Why problem solving?

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

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. http://standards.nctm.org/document/chapter3/prob.htm

Washington State Standards http://www.k12.wa.us/curriculumInstruct/mathematics/RevisedStandards/k8-processes.pdf

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?

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?

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.

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

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.

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.

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

Handshake problem Which strategies did we use?

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

Toothpick problem Which strategies did we use?

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

Age problem Which strategies did we use?

Age problem Guess and test Use variables

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.