Using Knowledge of How
Children Learn
Mathematics in
Mathematics Content
Courses
David Feikes
Purdue North Central
Keith Schwingendorf
Purdue North Central
Copyright © 2007 Purdue University North Central
Connecting Mathematics for Elementary
Teachers (CMET)
NSF CCLI Grants
DUE-0341217 & DUE-0126882
The views expressed in this paper are those of the
authors and do not necessary reflect those of
NSF.
Table of Contents
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Problem Solving
Sets
Whole Numbers
Number Theory
Integers
Rational Numbers - Fractions
Decimals, Percents, and Real Numbers
Geometry
More Geometry
Measurement
Statistics/Data Analysis
Probability
Algebraic Reasoning
Mathematical Content Courses for Elementary
Teachers
Focus on How Children Learn Mathematics!
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Methods Courses
Graduate Courses
CMET Materials:

Descriptions, written for prospective elementary teachers, on how
children think about, misunderstand, and come to understand
mathematics.

These descriptions are based on current research and include:
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how children come to know number
addition as a counting activity
how manipulatives may embody (Tall, 2004) mathematical activity
concept image (Tall & Vinner, 1981) in understanding geometry
In addition to these descriptions the CMET materials contain:
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problems and performance data from the National Assessment of Educational
Progress (NAEP)
Problems and performance data from the Third International Mathematics and
Science Study (TIMSS)
our own data from problems given to elementary school children
questions for discussion.
The Concept of Measurement
In elementary school, measurement has
traditionally been presented as procedures
and skills. However, a more careful analysis
indicates that measurement is a concept.
Teaching measurement is more than teaching
the procedures for measuring, it is also
helping children understand the concept of
measurement
Over seventy-five percent of the
fourth grade children missed this
question. Most children who
missed this question answered 8 or
6. Why 6?
Iteration/Repeating a unit
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One of the most important underlying concepts
of measurement is the building-up activity of
iteration or repeating a unit (e.g., paperclip,
inch, or centimeter).
Measurement involves learning to repeat a unit
and the mental ability to place the unit end to
end to measure or represent the length of the
object being measured (unit iteration). The unit
can be reused!
Nancy measured her pencil and got 6 inches.
Why do you think she started measuring from
1?
Partitioning/Subdividing
A second key concept is the
breaking-down activity of partitioning
or subdividing.
Partitioning is the mental activity of
“slicing up” an object into the samesized units. Children frequently
struggle creating units of equal size
(Miller, 1984).
Mary says the wall is 10 steps long.
Sara says the wall is 13 steps long.
Preservice Teachers

Preservice elementary teachers’ mathematical
knowledge, beliefs, and efficacy about
teaching and the learning of mathematics can
be developed by focusing on how children
learn and think about mathematics in content
courses.
Methods

The following analysis compares the data
from the treatment courses, which used
CMET and focused on knowledge of
children’s mathematical thinking with
control courses, which were taught
without using the CMET materials or an
emphasis on children mathematical
thinking.
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The Likert items possible responses ranged from
Strongly Agree to Strongly Disagree, responses
were given numerical values accordingly from 5
to 1.
The negatively worded questions are in
bold.
Positively worded questions with higher scores
up to 5 and negatively worded questions with
scores closer to 1 are the closest to our
theoretical position or an indication of an
adherence to the beliefs and knowledge we
believe are most important.
Knowledge of Children’s Mathematical Thinking
Control
Mean
SD
(n)
Treatment
Mean
SD
Initially addition is a
counting activity for
children
4.07
.741
(55)
4.66
.479
(58)
Children who can
count, say the
numbers in order,
understand the
concept of number.
2.84
1.058
(56)
2.27
.997
(59)
.680
(56)
4.17
.699
(59)
The concept of ten is the 3.79
basis for place value.
(n)
Knowledge of Children’s Solution Methods
Children are likely to
cross multiply to
solve ratio and
proportion problems.
Control
Mean
SD
(n)
Treatment
Mean
SD
3.88
(56)
2.88 1.076 (24)
.715
(n)
Knowledge ofControl
Children’s Geometric
Thinking
Treatment
Mean
SD
(n)
Mean
SD
(n)
Children first understand
shapes as a ‘concept
image’, i.e., a shape is a
rectangle because it
looks like a door.
4.14
.787
(37)
4.57
.502
(37)
Children frequently
look at the lengths of
the rays or the
distance between the
arrows to determine
which angle is larger
or smaller.
4.08
.894
(37)
4.57
.502
(37)
Mental imagery is
essential to learning
geometry.
3.78
.947
(37)
4.43
.728
(37)
Teachers - Learning
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I liked the part where it includes the exact
verbiage that a child said in regards to the
decimals. It gives teachers an opportunity to be
inside the head of a child.
This chapter refreshed why we count the
number of decimal factors and why we move the
number of decimal places in the divisor to make
a whole number. I had forgotten why. It was
just how it is suppose to be. It is just the rule.
It reminded me so that I can explain it to the
children. When there is an explanation to give
to the children, they believe it and trust it.
Concluding Comments by a
Teacher
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…you’re not teaching in the CMET how or when
to teach concepts, but rather giving some insight
as to how children think about and learn these.
This chapter made me realize that this is the
first text I have ever read that aims to help the
reader understand what children have been and
will be going through when attempting to learn
math. This is such a neat idea because not only
will pre-service teachers and parents understand
their children’s mathematical thinking better, but
they will also have more empathy for them.
Summary
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The CMET Project is premised on using children’s
mathematical thinking to promote mathematical
understanding for preservice teachers.
We believe this approach will enhance teacher’s learning
of mathematics.
We are also linking research to practice.
As part of our evaluation, we are attempting to both
describe and assess the mathematical knowledge
necessary for teaching.
Our intent is to improve the teaching of mathematics to
children!
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Focusing on knowledge of children’s
mathematical thinking raises prospective
teachers’ efficacy to understand and teach
mathematics as well as having an impact on
their beliefs about mathematics and its teaching.
Initial qualitative evidence suggests that using
this approach also influences teachers and
parents in their teaching and work with children.
Looking to the Future
We are looking for faculty who might be
interested in attending a faculty workshop or
using CMET in either your content or methods
courses for our next grant!
We are seeking a funding source to support the
development of parent resources and parent
research.