Transcript Problem solving - Pacific Lutheran University

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/m
athematics/RevisedStandards/k8processes.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
3. I will then be half as
my present age.
old as he. How old am I now?
Polya’s four steps -- page 4 in the
book
1.
2.
3.
4.
Understand the problem
Devise a plan
Carry out the plan
Look back
Polya’s ten commandments for
teachers -- page 1 in the book
1. Be interested in your subject.
2. Know your subject.
3. Try to read the faces of your students; try to see
their expectations and difficulties; put yourself
in their place.
4. Realize that the best way to learn anything is to
discover it by yourself.
5. 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
1.
2.
3.
4.
5.
6.
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.