Transcript Using Children’s Mathematical Thinking to Promote

Knowledge of Children’s
Learning of Mathematics:
A Common Denominator
in Preservice, Teacher, and
Parent Education
David Feikes
David Pratt
Purdue North Central
Purdue North Central
Sarah Hough
University California
Santa Barbara
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.
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!
Methods Courses
 Graduate Courses
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CMET Materials:
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Descriptions, written for prospective elementary teachers, on how
children think about, misunderstand, and come to understand
mathematics.
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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.
Preservice Teachers
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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.
Teachers and Parents
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Methods – Journaling, Interviews, Reviews
Participants were asked to describe:
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what they learned by using the materials
times when they considered or used information from
the project materials in their teaching or with their
children
how they used this information,
when they did something differently or tried something
new based on an idea from the text
any instance where a child’s mathematical thinking was
like or unlike that described.
Indicators of:
 Verification of CMET
 Learning
 Influence Practice
Teachers - Verification
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[On children’s understanding of graphs] I can
hear my class talk about “being the winner” if
their choice is picked by most. It’s kind of like
being the first one to arrive in the classroom.
[Probability] At the third grade level, personal
experiences will even influence how a child
answers reading comprehension questions.
They will choose the answer they would like it
to be, rather than the correct answer.
Teachers - Learning
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One specific thing that I learned had to do with the
concept of ten. Until reading about this in your
materials, I had assumed that the concept of ten as
a unit was pretty easy for students. I’d seen many
suggestions for using manipulatives (base-ten
blocks or an equivalent), but I’d never seen anything
suggesting that even with the manipulatives it’s a big
step for children to think of ten as a unit.
Teachers - Learning
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Children focus on “filling in the blanks” when using
pre-partitioned shapes instead of looking at the
meaning of the fraction.
Students need the mental images in order to
understand geometry.
I was helpful to see how the children progress
through the stages of measurement. I also liked the
different concepts of measurement; iteration was a
new term for me.
I learned about things not to do, like stringing
problems together, which misrepresents the facts.
Teachers - Learning
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I found it interesting that one of the problems
children have with decimals is that they try to
understand them by relating to their prior
understanding of whole numbers.
I also enjoyed reading about invariance of
shape. Every year I draw a “downward”
triangle on the board and ask my students
what “this” is. Each year they say, “An upside
down triangle”.
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.
Teachers – Influence Practice
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Many of my kindergarten students counted
one, two, three, four, five, seven and I didn’t
understand why. [Later she wrote] One-toone correspondence is how young children
count.
I also forgot how important it is to let children
count on their fingers. In my fifth grade
classroom, some children are still …[counting
on their fingers] and the try to hide their
hands.
Teachers – Influence Practice
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The paragraph about multiplication thinking strategies was very
helpful to me. I began using this in class just recently. My
students are eager to share their thinking strategies. I believe
once they can express their thought process in words, it helps
solidify the sense making aspect of math.
It is hard for teachers to try to think like students. Many times,
you are teaching the children and they are not getting it. And,
you keep persisting and cannot figure out why they do not get it.
And, cannot see what or why they are thinking what they do.
This part [of CMET] explains that.
I worked with a student one night that was having problems with
integers. We were working on the rules for multiplying and
dividing negative numbers. I was able to use this chapter to help
explain.
Teachers – Influence Practice
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I liked the section called Key Concepts of Measurement. That
section was very detailed. I liked the detailed pictures. The
explanations of actual children doing the activities were great. I
have tried several of the activities with the students that I work
with at Sylvan.
I also found the area problem where you cover up or erase part
of the area to be beneficial. I tried this with my students to see
who really could visualize and understand the concept of area.
The time section was very helpful. I now understand why time is
so confusing. They can’t see it!! They can’t touch it!!
Conservation is a hard skill, so it makes sense why elapsed time
can be so hard and confusing for students.
Teachers – Influence Practice
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This chapter has allowed me to accept a student’s
self-generated algorithm for solving a computation
problem. If the child is able to explain their process,
and it correctly leads to the answer, I will encourage
this child to continue to use this algorithm. One of
my students recently created his own algorithm for
subtracting two-digit numbers that required “trading.”
In the problem 64 – 29, Austin came up with a
“partial differences” algorithm. He subtracted 60 –
20 = 40, and then subtracted 9 – 4 = 5, he then took
40 – 5 = 35. This algorithm worked for him, and he
taught it to a friend who was struggling with “tradefirst” subtraction.
Teachers – Influence Practice
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I didn’t feel that my students really understood the
concept of congruence, so I took a suggestion from
CMET, and we physically cut out shapes to compare
congruency. Students are now able to determine
congruency without having to physically manipulate
the shapes. I have been making a conscious effort
to incorporate geometry into all areas of my
teaching.
I had an occasion to discuss the problem: 8 + 4 =
___ + 5 with my second grade students. We have
since been working several problems like this during
calendar, and they really seem to get the concept of
equality.
Parent - Verification
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As a parent of an almost 4 year old, I tried the
measure activity with his footsteps. It is so
neat to see he did exactly what you illustrated
was typical at that developmental level.
Parent - Verification
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I thought you’re example of time and speed
conservation was a nice way to present the
problem that children face when trying to
understand time. I did a similar activity with
my pre-schooler. He too thought that the
sand moved faster when he moved faster. It
is very interesting to put these examples into
action.
Parent - Verification
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I found myself actually having my 4 year old
collect data with a mini clipboard. I had him
go around our house when we had company
lately and see who was wearing jeans, pants,
or skirts. He drew pictures and reported his
“data” to us at dinnertime. It was so neat to
see him do this. I have since used this with
my friends, mothers as well, and they too
have been pleasantly surprised by their little
statisticians!
Parent - Learning
[A parent talking on her daughter’s
understanding of place value] I think she
understands place value to a certain degree,
but I know she doesn’t totally get it. She
knows that she must trade or borrow, but I
don’t think she truly understands why. I
would like to get a set of unifix cubes for
home for when she does her homework to
help her better understand this concept.
Parent – Influence Practice
A parent of a sixth grader
 In her journaling, she bulleted the following
changes:
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restraining myself to not "jump in" and allow her to
experience problem solving for herself
becoming more open to using other
resources/tools to assist her, as "my way" of
thinking/teaching, may be more harmful than
helpful to her
Parent – Influence Practice
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CMET changed how she helped her daughter
with her mathematics homework.
She found herself more willing to let her
daughter experience mathematics rather than
just telling her how to do it
She began to realize that her way of helping
her daughter may not be the best approach
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
 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 need letters of commitments from
faculty to attend a faculty workshop or use
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.