Teaching Teaching Mathematics via Mathematics via Cooperative Problem Cooperative Problem SolvingSolving

Dr. Patrick M. KimaniDr. Patrick M. KimaniAssistant ProfessorAssistant ProfessorDepartment of MathematicsDepartment of MathematicsMcCarthy Hall 154McCarthy Hall 154California State University, Fullerton California State University, Fullerton

pkimani@fullerton.edupkimani@fullerton.edu

OverviewOverview

BackgroundBackground StandardsStandards Activity!Activity! DiscussionDiscussion ReflectionReflection

BackgroundBackground Research findings have indicated that Research findings have indicated that

learning occurs when students learning occurs when students are actively are actively involved in the learning process, by involved in the learning process, by assimilating information and constructing assimilating information and constructing their own meaningstheir own meanings..

Learning via problem solving is one way in Learning via problem solving is one way in which students are engaged in the learning which students are engaged in the learning process and have opportunities to make process and have opportunities to make their own interpretations of the their own interpretations of the mathematics they are learning.mathematics they are learning.

BackgroundBackground

Students are actively engaged in Students are actively engaged in learning mathematics via problem learning mathematics via problem solving.solving.

  Learning/teaching (via problem-Learning/teaching (via problem-solving) differs from traditional solving) differs from traditional teacher led approaches with teacher led approaches with regard to both the teacher’s and regard to both the teacher’s and the students’ roles. the students’ roles.

Teaching/Learning via

Traditional Approach

Teaching/Learning via

Problem Solving

Teacher’s Role Lectures Assigns seat work Dispenses knowledge

Guides and facilitates. Poses challenging questions. Helps students share

knowledge.

Students’ Role Works individually Learns passively Forms mainly “weak”

constructions

Works in a group Learns actively Forms mainly “strong”

constructions.

StandardsStandards California (Problem Solving)California (Problem Solving)

High school graduates should be High school graduates should be able to use logical reasoning able to use logical reasoning inherent in mathematics to solve inherent in mathematics to solve practical problems with accuracy. practical problems with accuracy.

California Standards California Standards ((Problem Solving continued) …Problem Solving continued) …

In particular students should be able In particular students should be able to:to:– make decisions about how to make decisions about how to

approach problems.approach problems.– use strategies, skills, and concepts in use strategies, skills, and concepts in

finding solutions.finding solutions.– determine if a solution is complete determine if a solution is complete

and move beyond a particular and move beyond a particular problem to generalizing the result to problem to generalizing the result to other situations.other situations.

California StandardsCalifornia Standards(Number Sense)(Number Sense) By sixth grade students should be By sixth grade students should be

able to determine the least able to determine the least common multiple and the common multiple and the greatest common divisor of whole greatest common divisor of whole numbers; use them to solve numbers; use them to solve problems with fractions (e.g., to problems with fractions (e.g., to find a common denominator to find a common denominator to add two fractions or to find the add two fractions or to find the reduced form for a fraction).reduced form for a fraction).

StandardsStandards NCTM (Problem Solving)NCTM (Problem Solving)Instructional programs from prekindergarten through Instructional programs from prekindergarten through

grade 12 should enable all students to:grade 12 should enable all students to: build new mathematical knowledge through problem build new mathematical knowledge through problem

solving;solving; solve problems that arise in mathematics and in solve problems that arise in mathematics and in

other contexts;other contexts; apply and adapt a variety of appropriate strategies to apply and adapt a variety of appropriate strategies to

solve problems;solve problems; monitor and reflect on the process of mathematical monitor and reflect on the process of mathematical

problem solving.problem solving.

Activity!Activity!

Form a group of 3-4 people to Form a group of 3-4 people to work togetherwork together

Solve the locker problem!Solve the locker problem! Keep Polya’s problem solving Keep Polya’s problem solving

steps in mind! steps in mind!

The Locker ProblemThe Locker ProblemStudents at an elementary school have decided to try an Students at an elementary school have decided to try an experiment. When recess is over, each student will walk into experiment. When recess is over, each student will walk into the school one at a time. The first student will open all the the school one at a time. The first student will open all the first 100 locker doors. The second student will close all of the first 100 locker doors. The second student will close all of the locker doors with even numbers. The third student will locker doors with even numbers. The third student will change all the locker doors with numbers that are multiples change all the locker doors with numbers that are multiples of three. (Change means closing locker doors that are open of three. (Change means closing locker doors that are open and opening locker doors that are closed). The fourth student and opening locker doors that are closed). The fourth student will change the position of all locker doors numbered with will change the position of all locker doors numbered with multiples of four; the fifth student will change the position of multiples of four; the fifth student will change the position of the lockers that are multiples of five. And so on. After 100 the lockers that are multiples of five. And so on. After 100 students have entered the school, which locker doors will be students have entered the school, which locker doors will be open? open?

DiscussionDiscussion

Which locker doors are open?Which locker doors are open? Why are these the open lockers?Why are these the open lockers? Can your solution be extended to 1000 Can your solution be extended to 1000

lockers? 1, 000, 000 lockers? lockers? 1, 000, 000 lockers? nn lockers?lockers?

What mathematical ideas are in this What mathematical ideas are in this problem?problem?

Where do we go from here?Where do we go from here?

Using what you learned from Using what you learned from the locker problem can you the locker problem can you complete this table?complete this table?Number Prime

Factorization

Even/Odd Number of Factors

Exact Number of Factors

What are the Factors?

529 23x23

126 2x3x3x7

441 3x3x7x7

169 13x13

11025 3x3x5x5x7x7

20

X

19

X

18

X

17

X

16

X

15

X

14

X

13

X

12

X

11

X X

F 10

X X

A 9 X X

C 8 X X

T 7 X X X

O 6 X X X

R 5 X X X X

S 4 X X X X X

3 X X X X X X X

2 X X X X X X X 2 X X X

1 X X X X X X X X X X X X X X X X X X X X X X X

1 2 3 4 5 6 7 8 9 10 11

12

13

14

15

16

17

18

19

20

21

22

23

N U M B E R

Number of Factors Numbers

1 1

2 2, 3, 5, 7, 11, 13, 17, 19 …

3 4, 9, 25, 49, 121, …

4 6, 8, 10, 14, 15, 21, 22, 23, 27, …

5 16, 81, 625, …

6 12, 18, 20, 28, 32, …

7 64, …

8 128,

9 256,

Number of Factors Numbers (where p is prime)

1 P0

2 p1

3 p2

4 P3 or p1p2

5 P4

6 P5 or p1.p22

7 P6

8 P7, p1.p23

9 p8 or p21p2

2

Sample Assessment Sample Assessment QuestionsQuestions How many factors does the How many factors does the

number 2number 233.3.355.7.722 have? have? Extension: What are the factors of Extension: What are the factors of

the number 2 the number 233.7.722 Write a number with 12 factors.Write a number with 12 factors.

ReflectionReflection

Unit Objective:Unit Objective:

By the end of the unit students By the end of the unit students should be able to find the number of should be able to find the number of factors any whole number has using factors any whole number has using the fundamental theorem of the fundamental theorem of arithmetic. arithmetic.

Unit Assumption: Unit Assumption:

Students have been exposed to basic Students have been exposed to basic factoring. factoring.

ReferencesReferences

California State Standards.California State Standards. National Council of Teachers of National Council of Teachers of

Mathematics (2000). Mathematics (2000). Principles and Principles and standards of school mathematicsstandards of school mathematics. . Reston, VA: Author. Reston, VA: Author.

Masingila, Lester, & Raymond (2006). Masingila, Lester, & Raymond (2006). Mathematics for Elementary Teachers Mathematics for Elementary Teachers via Problem Solving. via Problem Solving. Tichenor Tichenor Publishing & Printing.Publishing & Printing.