Teaching Mathematics
via Cooperative Problem
Solving
Dr. Patrick M. Kimani
Assistant Professor
Department of Mathematics
McCarthy Hall 154
California State University, Fullerton
pkimani@fullerton.edu
Overview

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Background
Standards
Activity!
Discussion
Reflection
Background

Research findings have indicated that
learning occurs when students are
actively involved in the learning process,
by assimilating information and
constructing their own meanings.

Learning via problem solving is one way
in which students are engaged in the
learning process and have opportunities
to make their own interpretations of the
mathematics they are learning.
Background


Students are actively engaged in
learning mathematics via problem
solving.
Learning/teaching (via problemsolving) differs from traditional teacher
led approaches with regard to both
the teacher’s and 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.
Standards

California (Problem Solving)
High school graduates should be able
to use logical reasoning inherent in
mathematics to solve practical
problems with accuracy.
California Standards
(Problem Solving continued) …
In particular students should be able to:
– make decisions about how to approach
problems.
– use strategies, skills, and concepts in
finding solutions.
– determine if a solution is complete and
move beyond a particular problem to
generalizing the result to other situations.
California Standards
(Number Sense)

By sixth grade students should be able
to determine the least common
multiple and the greatest common
divisor of whole numbers; use them to
solve problems with fractions (e.g., to
find a common denominator to add
two fractions or to find the reduced
form for a fraction).
Standards

NCTM (Problem Solving)
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.
Activity!
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Form a group of 3-4 people to work
together
Solve the locker problem!
Keep Polya’s problem solving steps in
mind!
The Locker Problem
Students at an elementary school have decided to try an
experiment. When recess is over, each student will walk
into 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 locker doors with even numbers. The third student
will change all the locker doors with numbers that are
multiples of three. (Change means closing locker doors that
are open and opening locker doors that are closed). The
fourth student will change the position of all locker doors
numbered with multiples of four; the fifth student will
change the position of the lockers that are multiples of five.
And so on. After 100 students have entered the school,
which locker doors will be open?
Discussion
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Which locker doors are open?
Why are these the open lockers?
Can your solution be extended to 1000
lockers? 1, 000, 000 lockers? n lockers?
What mathematical ideas are in this
problem?
Where do we go from here?
Using what you learned from the
locker problem can you complete
this table?
Number
Prime
Even/Odd Exact
What are
Factorization Number of Number of the
Factors
Factors
Factors?
529
23x23
126
2x3x3x7
441
3x3x7x7
169
13x13
11025
3x3x5x5x7x7
16
X
15
X
14
X
13
X
12
X
11
F
10
A
9
C
8
T
7
O
6
R
5
S
4
X
X
X
X
X
X
X
X
X
X
2
X
X
X
X
X
X
X
X
X
X
X
X
3
1
X
X
X
X
X
X
X
X
X
X
X
X
X
X
2
X
X
X
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
2
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 p21p22
Sample Assessment
Questions
How many factors does the number
23.35.72 have?
 Extension: What are the factors of
the number 23.72
 Write a number with 12 factors.
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Reflection
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Unit Objective:
By the end of the unit students should
be able to find the number of factors
any whole number has using the
fundamental theorem of arithmetic.
Unit Assumption:
Students have been exposed to basic
factoring.
References
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California State Standards.
National Council of Teachers of Mathematics
(2000). Principles and standards of school
mathematics. Reston, VA: Author.
Masingila, Lester, & Raymond (2006).
Mathematics for Elementary Teachers via
Problem Solving. Tichenor Publishing &
Printing.