Problem-Involves a Situation in Which The

Problem Types

Problem-involves a situation in which the

solution route is not immediately obvious

Exercise-a situation in which the solution

route is obvious

Routine problem-the application of a

mathematical procedure in the same

way it was learned

Nonroutine problem-the choice of

mathematical procedures is not obvious

Which terms are synonyms?

Master 6-1: Problem Types

Suppose students have been multiplying whole numbers.

567748287793

x2x4x5 x7 x8

Are these problems?

15 rows of stamps.24 packs of baseball cards.

8 stamps in each row.8 cards in a package.

How many stamps?How many baseball cards?

Try this. Is this a problem for you?

Use the numerals 0,1,2,3,4,5,6,7,8 to form a 3 by 3

square. The sum of the numbers in every row is 12.

The sum of the numbers in every column is 12.

______

Try this. Is this a problem for you?

Use 1,2,3,4,5,6,7,8,9

______

+______

______

___+___+___= 18

Master 6-2: Is this a problem?

Signposts for Teaching Mathematics Through Problem Solving

Signpost 1:Allow Mathematics to Be

Problematic for Students

Signpost 2: Focus on the Methods Used

to Solve Problems

Signpost 3:Tell the Right Things at

the Right Time

Hiebert, 2003

Master 6-3: Signposts for Teaching Mathematics Through Problem Solving

Factors for Success in Problem Solving

• Instruction should build on what children already know.

• Engaging children in problem solving should not be

postponed until after they have “mastered” computational

skills.

• Children should be taught a variety of problem-solving

strategies to draw from.

• Children’s problem-solving achievements are related to

their developmental level. Thus, they need problems at

appropriate levels of difficulty.

• Factors which contribute to children’s difficulties with

problem solving include knowledge, beliefs and affects,

control, and sociocultural factors.

Master 6-4: Factors for Success in Problem Solving

Choosing Appropriate Problems

Consider including problems that:

• ask students to represent a mathematical idea in various

ways.

• ask students to investigate a numeric or geometric

concept.

• require students to estimate, or to decide on the degree

of accuracy required, or to apply mathematics to

practicalsituations.

• ask students to conceptualize very large or

very small numbers.

• ask students to use logic, to reason, to test conjectures,

or to gauge the reasonableness of information.

• ask students to perform multiple steps, or use more than

onestrategy.

• are open ended.

Master 6-5: Choosing Appropriate Problems

Polya’s Model of Problem Solving

1. Understand the problem.

2. Devise a plan for solving it.

3. Carry out your plan.

4. Look back to examine your solution.

Problem Solving Strategies

Act It Out

Make a Drawing or Diagram

Look for a Pattern

Construct a Table

Guess and Check

Work Backward

Solve a Simpler or Similar Problem

Master 6-6: Problem Solving Strategies

The Horse Problem

A man buys a horse for

$60, sells it for $70, buys

it back for $80, and sells

it for $90. How much

does the man make or

lose in the horse trading

business?

Master 6-7: The Horse Problem

Interviewer, "Paco had 13 cookies. He ate 6 of them. How many

cookies does Paco have left?"

Meredith: Fifth Month of Kindergarten

Master 6-8: Problem Solving Sample

Interviewer, "Carla has 7 candies. How many more candies does she

need so that she will have 11 candies to share with her friends?"

Allan: Fifth Month of First Grade

Master 6-9: Problem Solving Sample

Interviewer, "Nineteen children are going to the circus. Five children

can ride in each car. How many cars will be needed to get all 19

children to the circus?"

Clint: Fifth Month of Second Grade

Master 6-10: Problem Solving Sample

Interviewer, "Robin has 3 packages of gum. There are 6 pieces of

gum in each package. How many pieces of gum does Robin have

altogether?"

Bill: Third Month of Third Grade

Master 6-11: Problem Solving Sample

Interviewer, "Tad had 15 guppies. He put 3 guppies in each jar.

How many jars did Tad put guppies in?"

Darla: Third Month of Third Grade

Master 6-12: Problem Solving Sample

Interviewer, "Tad had 15 guppies. He put 3 guppies in each jar.

How many jars did Tad put guppies in?"

Ellen: Third Month of Third Grade

Master 6-13: Problem Solving Sample

Interviewer, "Nineteen children are taking a mini-bus to the zoo. They

will have to sit either 2 or 3 to a seat. The bus has 7 seats. How

many children will have to sit three to a seat, and how many can sit

two to a seat?"

Allison: Fifth Month of Fourth Grade

Master 6-14: Problem Solving Sample