MATH

PROBLEM

SOLVING

FOUR STAGES

PROBLEMTRANSLATION

PROBLEMINTEGRATION

Basic Facts PresentedMath Knowledge

Recognize Problem TypeBuild Coherent Representation

SOLUTION PLANNING & MONITORING

Break into Subgoals Step by Step Plan

Where am I in my plan?

SOLUTION EXECUTION

Calculations Fast & Accurate

SAMPLE PROBLEM

Floor tiles are sold in squares 30 cm on each side. How much would it cost to tile a rectangular room 7.2 meters long and 5.4 meters wide if the tile cost $ .72 each?

ONE POSSIBLE SOLUTION PLAN

Step 1: change width & length into number of tiles

540/30 = 18 tiles 720/30 = 24 tiles

Step 2: determine how many square tiles cover the floor

18 times 24 = 432 tiles

Step 3: determine the cost of the 432 tiles

432 times $ .72 = $ 311.04

WHAT DO YOU NEED TO KNOW TO SOLVE THE TILE PROBLEM?

Step 1: Problem Translation

Linguistic Knowledge - need to be able to understand English sentences in order to recognize the facts of the problem (What are the givens? What is the problem goal?)

For the tile problem:

- room is rectangle, 7.2 by 5.4 meters

- each tile costs $ .72

- goal is to find total cost of tiling the room

Translation process also requires factual knowledge about mathematics

- one meter equals 100 cm

WHAT DO YOU NEED TO KNOW TO SOLVE THE TILE PROBLEM?

Step 2: Problem Integration

Schematic Knowledge - need to integrate information into a coherent representation, need to recognize problem type

For the tile problem:

- this is a rectangle problem

- need to use the rectangle area formula to solve the problem

Area = length x width

*** Problem Integration involves more than statement by statement translation

PROBLEM INTEGRATION ENABLES YOU TO RECOGNIZE INCONSISTENCIES

“The number of quarters a man has is seven times the

number of dimes he has. The value of dimes exceeds the

value of the quarters by $2.50. How many has he of each

coin?

WRITE THE EQUATIONS YOU NEED TO SOLVE THE PROBLEM AND THEN SOLVE!

Q = 7 D

D (.10) = 2.50 + Q (.25)

Anything wrong?

PROBLEM INTEGRATION ENABLES YOU TO CONSTRUCT A SITUATION MODEL

Can you give an example of a concrete situation that

corresponds to:

1 3

4

1

2

That is, create a simple word problem that could be solved by the above equation.

Problems of this type were given to elementary school teachers in the U.S. and China

(try to do this yourself, sample answers - next slide)

STUDY ON SITUATION MODELS

U.S. vs. CHINESE TEACHERS

Incorrect model: “If you have one pie and 3/4 of another

pie to be divided equally by two people, how much pie will

each person get?”

Correct model: “If a team of workers construct 1/2

kilometer of road per day, how many days will it take them

to construct a road 1 and 3/4 kilometers long?”

Results: 96% of the U.S. teachers either could not describe an appropriate concrete situation or produced an incorrect model. 90% of the Chinese teachers produced correct models. YIKES!!!!!

WHICH TWO PROBLEMS BELONG TOGETHER?

1. A personnel expert wishes to determine whether experienced typists are able to type faster than inexperienced typists. 20 expert typists (5 yr or more experience) and 20 inexperienced typists (less than 5 yrs) are given a typing test. Each typist’s average number of words per minute is recorded.

2. A personnel expert wished to determine whether typing experience goes with faster typing speeds. 40 typists are asked to report their years of experience as typists and are given a typing test to determine their average number of words per minute.

3. After examining weather data for the last 50 years, a meteorologist claims that the annual precipitation varies with average temperature. For each of 50 years, she notes the annual rainfall and average temperature.

Experienced math problem solvers pick 2 and 3Inexperienced math problem solvers pick 1 and 2.

CAN THESE PROBLEMS BE SOLVED?

ANY IRRELEVANT INFORMATION?

1. A rectangular lawn is 12 meters long and 5 meters wide. Calculate the area of a path 1.75 meters wide around the lawn.

2. The length of a rectangular park is 6 meters more than its width. A walkway 3 meters wide surrounds the park. Find the dimensions of the park if it has an area of 432 square meters.

3. The lengths of the sides of a blackboard are in a 2:3 ratio. What is the perimeter (in meters) of the blackboard?

*** Most high school students make mistakes on more than half of problems like the ones shown above.

IMPLICATIONS FOR INSTRUCTION: TEACHING PROBLEM INTEGRATION SKILLS

• use varied presentation to encourage students to

discriminate among problem types

• encourage students to draw diagrams

• practice sorting problems into categories

• practice identifying relevant and irrelevant information

WHAT DO YOU NEED TO KNOW TO SOLVE THE TILE PROBLEM?

Step 3: Solution Planning and Monitoring

Strategic Knowledge - need general strategies that can be used to devise and monitor a solution plan

For the tile problem:

- draw a picture

- work backwards from goal: goal is to find total cost of tiling floor, so you need to know the # of tiles that cover

the floor

- divide into subgoals: change dimensions into # of tiles, then determine how many tiles cover the floor,

then determine the cost of all the tiles

*** general strategies are italicized

5.4 m

7.2 m

WHAT DO YOU NEED TO KNOW TO SOLVE THE TILE PROBLEM?

Step 4: Solution Execution

Procedural Knowledge - computational procedures from simple procedures (e.g., single digit addition or subtraction) to more complex procedures (e.g., subtraction of multiple digit numbers)

For the tile problem:

540/30 = 18 tiles 720/30 = 24 tiles 18 x 24 = 432 tiles

432 times $ .72 = $ 311.04

*** Key Point: able to do computations with no difficulty, fast and accurate (achieve automaticity, direct retrieval from long-term memory)

FAULTY BELIEFS ABOUT MATH THAT UNDERMINE EFFECTIVE PROBLEM SOLVING

1) Ordinary students cannot expect to understand math, they have to memorize it, and just apply what they have learned mechanically and without understanding.

2) All story problems can be solved by applying operations suggested by key words in the story (in all suggests addition, left suggests subtraction, share suggests division - 3rd graders)

3) Any assigned problem should be solved within five minutes or less. (High school students estimated the typical problem should take about 2 minutes)

4) Math is not particularly useful or sensible. Math is mostly a set of rules and mathematics learning means memorizing the rules (54% of 4th graders and 40% of eighth graders; females’ attitudes toward math more negative).

ATTRIBTUTION STYLE UNDERMINES EFFECTIVE MATH PROBLEM SOLVING

Researchers gave 10 year old children a questionnaire

asking about their likely reactions to hypothetical failures.

They identified two attribution styles:

Mastery-oriented: likely to think they should work

harder in the face of failure/difficulty

Helpless: likely to respond to difficulty with negative

attributions about ability

*** There were no IQ differences between these two groups.

*** Many more girls were categorized as “helpless.”

ATTRIBTUTION STYLE UNDERMINES EFFECTIVE MATH PROBLEM SOLVING - CONTINUED

Researchers next gave the children a series of confusing

math problems (difficult to solve), and then a batch of easy

math problems (that all children should be able to solve).

What happened?

Mastery-oriented children: These children were able to

recoup from the negative experience and solved the easy

problems with ease.

Helpless: These children were thrown by the confusing

problems and didn’t try very hard on the easy problems,

getting many of them wrong.

ATTRIBTUTION STYLE UNDERMINES EFFECTIVE MATH PROBLEM SOLVING - CONTINUED

Researchers wanted to know why girls were more likely to

adopt a “helpless” attribution style.

What happens in the classroom?

Boys and girls receive the same amount of negative

comments. But the nature of these comments differ.

Boys: Criticisms sometimes focus on intellectual quality,

sometimes on neatness, conduct, or effort. Boys and girls both

think teachers like girls better.

Girls: Teacher criticisms focus consistently on the

intellectual quality of the work.

End Result: Boys attribute failure to any number of factors, girls

are left with negative attributions concerning their ability.