Project Summary
Connecting Mathematics for Elementary
Teachers II (CMET II)
The CMET II
project will connect prospective elementary teachers’ learning of mathematics
in mathematics content courses with how children understand and learn
mathematics. The goals are to 1) enhance
pre-service teachers’ understanding of mathematics, 2) improve their teaching
of mathematics, 3) improve their understanding of how children learn and
understand mathematics, 4) help them
connect the mathematics they are learning with the mathematical concepts they
will be teaching, and 5) facilitate their understanding of connections between
the mathematics they will be teaching to the mathematics and technological skills children will learn in middle and high
school. To achieve these goals, supplementary materials and instructor’s guide
will be developed for mathematical content courses for elementary teachers. The
project will emphasize mathematics from children’s perspectives and directly
relate this to mathematics they are learning and will eventually be teaching
children. Formative and summative
evaluations will be conducted at multiple and diverse test sites.
Intellectual Merit
CMET II will
advance knowledge on the integration of research on children’s learning of
mathematics with teachers’ learning, thus providing a mechanism for enhancing
and studying the interweaving of content and pedagogy (Ball & Bass,
2000). This original approach is based
on the NSF proof-of-concept grant, CCLI-EMD DUE 0126882, which successfully
produced pilot materials.
Broader Impacts
This project
has the potential to significantly impact the preparation of future elementary
teachers by providing them with in-depth knowledge of how the subject matter
they are learning relates to their future teaching. National dissemination
activities will include: journal articles, conference presentations, and
commercial publication.
Project Description
Project Overview
The Connecting Mathematics for
Elementary Teachers II (CMET II) project is based on the NSF proof-of-concept
grant, CCLI-EMD DUE 0126882, which successfully produced and piloted
supplementary materials for topics in the typical first mathematical content
course for elementary teachers. The
project attempts to connect prospective elementary teachers’ learning of
mathematics in mathematics content courses with how children understand and
learn mathematics. Specifically, the proposed project will develop
supplementary materials for all mathematical content courses for elementary
teachers. Expanded supplements of
teaching notes, i.e., teacher’s guide, will be developed for the
instructors. The supplementary materials
will emphasize and explore four themes:
1)
How children
understand mathematics
2)
How children
learn mathematics
3)
Mathematics
as taught in the elementary school
4)
The
important connections between the mathematics children will learn in elementary
grades to the mathematics and technological skills they will learn in middle
and secondary school.
The project will emphasize
mathematics from children’s perspectives and relate these perspectives to the
mathematics pre-service teachers are learning and will eventually be teaching
children. The supplementary materials
will be designed so that they can easily be adapted for any elementary level
mathematical content courses.
The CMET II project addresses the fundamental
need to improve the mathematics education of prospective elementary teachers by
making their learning of mathematics more meaningful to their future teaching. This need is evident in that the typical
instructor for mathematical content courses for elementary teachers has very
little or no experience in the teaching of mathematics to elementary school
children. As a result, many instructors
do not understand how children learn mathematics and are often unaware of the
actual mathematics taught in elementary school.
Further, prospective elementary teachers usually have only their own
“rule-oriented” experiences of learning mathematics to draw upon; consequently,
their learning of mathematics is not connected to their future teaching. Systemic reform in mathematics education has
typically not addressed the mathematics education of elementary teachers,
except to occasionally point out that they need more subject matter knowledge
(Askey, 1999; Ball, 1993). CMET II will
advance knowledge on the integration of research on children’s learning of
mathematics with teachers’ mathematical learning, thus providing a mechanism
for enhancing and studying the interweaving of content and pedagogy (Ball &
Bass, 2000).
The
project team principal investigators from Purdue University North Central
include: Dr. David Feikes, Dr. Keith Schwingendorf and Dr. David Pratt. The
P.I.s will work with four key contributors/consultants on the development,
evaluation, and dissemination of the project materials. The contributors/consultants include: Dr. Jeff Gregg, professional and free-lance
writer; Dr. Michelle Stephan, Purdue University Calumet; Dr. Marcela Perlwitz, Wabash College;
and Dr. Mary Jane Eisenhauer, Purdue University North Central. The project team will draw upon the
experience of the distinguished advisory board: Professors Dr. Michael
Battista, Kent State
University; Dr. Paul Cobb, Vanderbilt University;
Dr. David Tall, University of Warwick, the United
Kingdom, and Dr. Grayson Wheatley, Florida State
University.
Quantitative and
qualitative analyses of the project materials will be conducted by internal
evaluator Dr. David Pratt, Purdue University North Central, and external
evaluator Dr. Sarah Hough, California University at Santa
Barbara. Their
comprehensive evaluation will include both summative and formative assessments
so the materials can be revised as needed and so that their national merit can
be assessed. The results of the
assessment activities will be disseminated through journal articles, a project
web site, and presentations at national conferences and professional
meetings. In addition, Addison-Wesley
Publishing has expressed a strong interest in publishing the CMET II
supplementary materials for use with mathematical content textbooks.
Results from Prior NSF Support
The CMET II
project is based on the NSF proof-of-concept grant CCLI-EMD DUE 0126882;
$75,000; January 1, 2002
– June 30, 2003;
Connecting Mathematics for Elementary Teachers, (CMET). The CMET project produced, piloted, and
evaluated supplementary materials for the typical, first mathematical content
course for elementary teachers. The
project team conducted a mixed-method evaluation to determine the students’ and
the instructors’ reactions and perceptions to the project supplementary
materials; and the extent to which the CMET project achieved its goals. A summary of the results for the completed
CMET project is given below. (See
Appendix A for the entire Final Project Report for CCLI-EMD DUE 0126882.)
The
materials were piloted in two sections, Fall 02, on the Purdue University North
Central, PUNC, campus; two sections, Fall 02, and two sections, Spring 03, on
the Purdue University West Lafayette, PUWL, campus. In addition, two sections, Fall 02, at PUWL
served as a control group. The data sources included an end of the semester
questionnaire, a quiz where students were asked their thoughts on the CMET
supplement, student solutions to selected test and final questions, 29 student
interviews, two instructor interviews, the coordinator’s notes given to the
instructors, tests, finals, and written feedback solicited from the
coordinator. Data sources were not applicable or available from all sections.
All interviews were transcribed.
The ongoing
assessment conducted by the project team used varying methods of data
analysis. The qualitative analysis of
data searched for themes and patterns across the data – an adaptation of the
constant comparative method (Glazer & Strauss, 1967). The data analysis was
triangulated by having three project staff members independently look over the
data and draw their own results. A
fourth member of the project team then analyzed the compiled results and made
further revisions.
From
the questionnaire given to two sections at PUNC and two sections at PUWL in the
Fall 02 students believed that the CMET Supplement was beneficial.
Was the CMET
Supplement Beneficial?
The vast majority of students, 95%,
found the supplement Very to Somewhat Beneficial.
In response
to two others questions on the questionnaire,
students believed that the CMET supplement should be required for this
first, mathematical content course and that they would like to have a similar
supplement in the next two content courses.
Should the
CMET Supplement be required for this course?
Would you like a
similar supplement in the next two content courses?
There were
four specific reasons why students found the material valuable (presented in
order of significance). The supplemental
materials…
1.
Helped them understand how children learn math.
2.
Presented a different approach or a new perspective for learning math.
3.
Provided practical examples.
4.
Reinforced what had been covered in class.
Other minor reasons stated for
finding the supplement valuable included that it broke information down into
more understandable pieces and provided a more personal approach.
The
instructors for the pilot sections at PUWL also viewed the use of the CMET
supplement as beneficial. They indicated
they learned about how children thought about mathematics by using the CMET
supplement. Specific examples mentioned
were: children view addition and subtraction as counting, children first learn
multiplication as repeated addition, and the importance of concept of ten. The Fall PUWL instructor believed that the
CMET Supplement was particularly beneficial for her weaker mathematical
students. Further, she believed that students with more educational experience,
juniors and seniors, saw more value in the supplement. The most beneficial aspect, from her point of
view, was that the supplement showed how children think about mathematics and
she felt she was able to connect that to her teaching of mathematics to
preservice teachers. She believed that
the supplement would make her students better prepared to teach children
mathematical and she cited two examples during the semester where her students
had tutored children using ideas from the CMET Supplement.
The crux of
the evaluation focused on assessing the extent to which the CMET Project
achieved its goals. The most significant impact on students’ learning was that
CMET Supplement helped students construct knowledge of how children learn and
understand mathematics. The overriding
comment from students about CMET, on both the interviews and questionnaires,
was that it showed them how children think and how children view and solve
problems in a variety of ways.
The supplement gave
fantastic examples of how children in real situations dealt with problems. I learned that each child is different in how
they think mathematically.
The preliminary evidence suggests
that, by using these materials, prospective elementary teachers did connect the
mathematics they were learning with the mathematics they will be teaching
children.
Some
students did indicate that using CMET improved their understanding of
mathematics; however, most indicated that it did not. More significantly, many students indicated
that using the supplement did improve their understanding of how children see
and understand mathematics. From the
project team’s perspective, understanding how children view mathematics also
required a reconstruction of students’ own mathematical knowledge.
Further,
many students’ indicated that the supplement will influence their future
teaching of mathematics to children. The
three most significant responses concerning their future teaching among the
survey data included:
1.
Learned new methods, including use of hands on materials and ways to present
problems.
2.
Will use the supplement directly as an aid to teaching in the future.
3.
Able to see learning math from a child’s perspective and be more
understanding about how children learn.
A common response from students was
that they were going to keep the supplement and use it when they taught.
I like the CMET book. It describes in detail the steps of how to
teach kids. I want to keep it and use it
when I am a teacher. I have highlighted
things. I just like it how it teaches
different ways and how children will do things differently.
In conclusion, the primary
influence of the CMET supplement most evidenced was on students’ new knowledge
of how children think mathematically.
After taking this class a
few semesters ago, I didn’t have the child’s viewpoint and what and how they
think. Now I do, and with more
strategies to teach them.
Presentations that acknowledged the
NSF Award
“Integrating Knowledge of How
Children Learn and Understand Mathematics into Mathematical Content Courses for
Elementary Teachers”, the Association of
Mathematics Teacher Educators (AMTE) Annual Conference, San Diego, CA
(submitted for presentation January 2004).
“Research into Practice: Connecting Mathematics for Elementary School
Teachers”,
NCTM National Meeting, San Antonio,
TX, April 2003.
“Connecting Mathematics for Elementary Teachers in Pre-service Mathematics
Content Courses”, NCTM Regional Meeting, Indianapolis, IN,
January 2003.
Book Chapters that
acknowledged the NSF Award.
“Connecting the Teaching of
Problem Solving to Children’s Learning of
Problem Solving” in Watanabe, T. & Thompson, D.,
eds., The Work of
Mathematics Teacher Educators:
Exchanging ideas for Effective practice,
AMTE
Monograph. (submitted June 2003)
Goals and Objectives
The overarching
goals of the CMET II project are to:
Enhance pre-service elementary teachers’
understanding of mathematics and consequently improve their teaching of
mathematics to children.
Often, the
mathematics prospective elementary teachers learn in content courses is
disconnected from what they will be teaching.
Experts in any field do not view knowledge as isolated facts but rather
as contexts of applicability (Bransford, Brown, & Cocking, 1999). By learning mathematics in the context of how
they will be teaching mathematics, pre-service teachers will gain a better
understanding of mathematics. The
National Council of Teachers of Mathematics Research Companion states,
“Children’s thinking of mathematics needs to be the center of mathematics
instruction” (p. 49, 2003). The aim is
that these novice teachers will organize mathematical knowledge around the
context of how children learn mathematics and how they will be teaching
mathematics to children. Learners of any
subject are more motivated when they see the usefulness of what they are
learning (Bransford, et. al., 1999).
Research has shown that one factor that influences children’s learning
of mathematics is when teachers base instruction on children’s ways of thinking
(NCTM 2003, Gearhart et. al., 1999).
Increasing prospective elementary teachers’ subject matter knowledge and
pedagogical knowledge of children’s learning of mathematics will improve their
future teaching of mathematics.
Exceptional teachers of mathematics must have both a thorough
understanding of the structure of mathematics and of pedagogical content
knowledge.
Specific
objectives of the CMET project are:
·
Improve
pre-service elementary teachers’ understanding of how children learn and
understand mathematics.
The CMET
project will attempt to achieve this goal through the development of
supplementary materials for mathematical content courses for elementary
teachers with teachers’ notes for the instructors of these courses. The materials will directly focus on how
children learn and understand mathematics.
The supplementary materials will correspond with most major textbooks
used for these courses, providing additional information on how children think
about the mathematics in each section of the content textbooks.
·
Help
prospective elementary teachers connect the mathematics they are learning with
the mathematical concepts they will be teaching children.
The
supplementary materials will describe the mathematical concepts that are
important to emphasize with children at different grades, K-6, and different
developmental levels. The supplementary
materials will reference popular elementary mathematics textbooks illustrating
how the mathematics they are learning is connected to the mathematics children
will need to learn. The materials will
stress the conceptual development of mathematics, illustrating how children
naturally progress in their mathematical thinking.
·
Facilitate
prospective elementary teachers understanding of the connections between the
mathematics they will be teaching in elementary school to the mathematics and
technological skills children will learn in middle and high school.
Often elementary teachers are not aware
of the mathematics and technological skills that their students will need to
learn in middle and high school. The supplementary
materials will also emphasize these connections. By knowing the mathematics children will need
to know in the future, they will be able to emphasize salient concepts and
connections in their future teaching.
Detailed Project Plan
Need
One of the major
problems in the mathematics education of prospective elementary teachers is
that mathematics content courses are typically taught by instructors with very
little or no experience in the teaching of mathematics to elementary school
children. Often these courses are taught
by teaching assistants, graduate students, mathematics professors, and adjunct
instructors with little or no elementary school experience. Some instructors may have secondary school
experience in teaching mathematics; however, instructors rarely have elementary
school experience. Consequently, they do
not understand how children learn mathematics and even are often unaware of the
actual mathematics that these prospective elementary teachers will be
teaching. These instructors, although
qualified to teach mathematics, are unable to make the vital connections to
prospective teachers’ future teaching of mathematics to children. The CMET II
supplements will provide prospective elementary teachers opportunities to
explore diversity in students' mathematical reasoning, accentuating students'
diverse ranges of abilities.
An additional
problem is that prospective elementary teachers’ learning of mathematics is not
connected to their future teaching.
Mathematics will be more meaningful to them if they are able to relate
it to how children learn and think of mathematics. Mathematical content courses for elementary
teachers typically encompass a wide variety of mathematical ideas and concepts. Prospective teachers in these courses have
very few mechanisms in place to help them relate the mathematics they are
learning to the actual mathematics they will be teaching children. Prospective elementary teachers typically
have only their own experiences learning mathematics to relate to the
mathematics they are learning and these are often negative. As a consequence,
mathematics is often learned as disassociated facts and procedures without
meaning. Learning mathematics should be
a sense making activity. Building
relationships and connections with how children learn mathematics is one means
of providing a context of applicability (Bransford, et. al., 1999).
Prospective
elementary teachers are often not aware of the specific mathematics children
will eventually learn in middle and secondary school. It is important for
elementary teachers to know what mathematics that their students will need to
know in the future because it will make them better teachers of
mathematics. In addition, this will
provide motivation to learn higher mathematics where they have taken the
position, “We will never need this because we will never teach it to children.”
Reform in
mathematics education is prevalent at almost every level. Elementary mathematics methods courses and
in-service programs are evolving to reflect the calls for reform and the new
standards (NCTM: 2003, 2000, 1991, 1989).
Many teachers are trying to realize these new forms of practice. Other research suggests elementary teachers
need greater subject matter knowledge in mathematics (Rand Report, 2002;
Conference Board of the Mathematical Sciences, 2001; Askey, 1999; Ball, 1993;
Lampert, 1986). However, the movement to
increase the subject matter knowledge of prospective elementary teachers and
their actual learning of mathematics has not been a main focus of reform. More
emphasis has been placed on improving the teaching of mathematics without
examining how teachers actually learn and use mathematical knowledge in their
teaching. Even when teachers have rich understandings of mathematics they may
not always be able to help children construct similar understandings (Thompson
& Thompson, 1994).
While the
aforementioned research advocates an increased emphasis on subject matter
knowledge for teachers of mathematics, Ball and Bass (2000) have argued for the
integration of subject matter knowledge with prospective teachers’ learning of
mathematics. This view takes into
consideration that knowledge is socially constituted and that increasing
subject matter knowledge alone is not enough.
The interwoven nature of subject matter knowledge and pedagogical
practices must be examined. Teaching is more than knowing mathematics well; it
is knowing usable mathematical knowledge and being able to apply it flexibly in
real classroom settings (Ball and Bass, 2000).
Boaler (2000) has suggested that this is a new type of pedagogical
content knowledge (Shulman, 1986). CMET
II will advance knowledge on the integration of research on children’s learning
of mathematics with teachers’ learning, thus providing a mechanism for
enhancing and studying the interweaving of content and pedagogy.
A weak link
is teachers’ learning of mathematics.
There is a systemic effort to help teachers realize new ways of teaching
mathematics; however, a stronger emphasis needs to be placed on prospective
teachers’ learning of mathematics. In,
“Adding It Up: Helping Children Learn Mathematics”, (Kilpatrick, Swafford,
& Findell, 2001), the authors suggests:
To better prepare
teachers for elementary and middle school math instruction, colleges and
universities should create programs or courses that emphasize thorough
knowledge of mathematics and of the process through which school children come
to understand mathematics.
The intent of the CMET II
project is to develop supplementary materials to address this need.
In summary, this project emphasizes and addresses the
following needs: the notion that a systemic reform in mathematics education
must also focus on the learning of mathematics for teachers; that prospective
elementary teachers do not make the connections with the mathematics they are
learning to the mathematics they will be teaching; and that furthermore, it is not feasible to develop a
new core of instructors for these mathematical content courses since
mathematics educators are currently in high demand. This proposal addresses the aforementioned
problems by providing supplementary materials that will help pre-service
teachers develop connections between mathematics they will be learning in their
mathematical content courses and their future teaching of mathematics in
elementary schools.
Theoretical
Orientation
The
supplementary materials developed in the CMET II project will be based on
research on how children understand and learn mathematics (NCTM, 2003; Steffe,
von Glasersfeld, Richards, & Cobb, 1983; Cobb &Wheatley, 1988; Fuson,
1988; Kamii & Housman, 1999; Clements & Battista, 1992). (See Appendix
B for a more complete list of references and the research basis for the CMET
Pilot.) Significant advances have been made in understanding how children come
to know mathematics; and several projects [Cognitively Guided Instruction
(Carpenter, Fennema, & Franke, 1996) and Purdue Problem-Centered
Mathematics (Cobb, Wood, & Yackel, 1991)] have attempted to use this
knowledge to help in-service teachers realize new forms of practice. These projects have also attempted to use
this approach in mathematics methods courses for elementary teachers.
CMET II differs
from other projects in that it will attempt to connect this knowledge in the
mathematical content courses for elementary teachers. CMET II will not specifically attempt to
change ‘how’ these courses are taught but ‘what’ is taught in these
courses. It is more likely that this
change can be accomplished on a wide-scale basis. CMET II does have the potential to influence
or, more appropriately, provide teachers with opportunities to learn. Research has shown that one of the
significant factors in helping practicing teachers realize new forms of
practice is their increased knowledge of how children understand mathematics
(Feikes, 1992). Thus, this project will
be a first step toward helping instructors of elementary mathematics content
courses realize alternative ways of instruction, especially as they realize how
children and their own students come to understand mathematics (Ball &
Bass, 2000).
|
Course II (continued)
D. Variability/Dispersion
1. Range
2. Standard Deviation
E.
Sampling
F. Standardized Test Scores
1. Normal Distribution
2. Grade Equivalents
3. Normal Curve Equivalents
4.
IQ
XIII. Probability
A. Experimental
and Theoretical Probability
B. Simulations
C. Sample
Spaces and Events
D. Equally
Likely Outcomes
E. Mutually
Exclusive
F. Counting
1. Permutations
2. Combinations
G. Expected
Values
H. Odds
I.
Independent and Dependent Events
XI. Algebraic
Reasoning
A. Variables
B. Patterning
C. Equations
D. Relations and Functions
E. Coordinate Geometry
F. Graphs
Course III
VIII.
Geometry
A. Euclidean
Geometry
1.
Points, Lines, Planes
2.
Line Segments, Rays, Angles
3.
Parallel and Intersecting Lines
B. Other
Geometries
|
Course III (continued)
C. Plane
Figures
1. Polygons
a. Triangles
b. Quadrilaterals
c. Diagonals of Polygons
2. van Hiele Model
3. Angle Measure
4. Circles
D. Three-Dimensional Geometry
1. Lines & Planes in Space
2. Regular Polyhedra
3.
Pyramids and Prisms
4. Euler’s Formula
5. Cylinders Cones and Spheres
6. Conic Sections
E. LOGO
IX. More Geometry
A. Transformations
– Rigid Motions
1. Congruence
2. Tessellations
B. Congruence
1. Congruent Figures
2. Transversals
C. Constructions
D. Proof
and Mathematical Reasoning
E. Symmetry
1. Reflectional Symmetry
2. Rotational Symmetry
F. Similarity
1. Applications
2. Trigonometry
X. Measurement
A. US Customary System
B. Metric System
C. Perimeter and Area
D. Pythagorean
Theorem
E. Surface
Area and Volume
|
Not only will the supplements cover the topics listed in
the three courses above, but they will also address concepts that may not be
covered in elementary mathematics content textbooks, but which are mainstays of
the NCTM Principles and Standards (2000); e.g., two concepts are algebraic
reasoning (Carpenter, Franke, & Levi, 2003; Falkner, Levi, & Carpenter,
1999) and representation (Fennel & Rowan, 2001). CMET II materials will be aligned with the NCTM
Principles and Standards (2003, 2000, 1995, 1991, and 1989). Since the topics addressed in the supplements
were devised by thoroughly analyzing the main pre-service textbooks on the
market and major reform documents, we feel confident that the supplements will
be usable by instructors with a wide variety of course structures and
textbooks.
Proposed Sample Materials
Prospective elementary
teachers have a vague view of the exact mathematics they will be teaching
children. They may believe they will be
directing drill and practice problems for children at different grade levels
with little thought to the actual mathematics they will be teaching. Research on children’s understanding of
shapes (Clements, Sudha, Hannibal, & Sarama, 1999) will be used to guide
the development of segments on geometry.
The following is
an example of the teaching notes for the instructors.
Plane Figures (VIII, B,
1)
Many prospective elementary teachers realize
they will be teaching children to name and recognize shapes along with some
basic properties of shapes in the early elementary grades (K-2). Almost every elementary textbook defines a
square as a quadrilateral with four right angles and four congruent sides. If the shape is held parallel to the
floor, , children and prospective
elementary teachers, refer to the shape as a square. However, if the same shape is rotated 45
degrees, , prospective elementary
teachers refer to it as a rhombus because it fits the definition in the
textbook. Children invariably refer to
the shape as a diamond. For young
children the two shapes are not the same.
For adults and older students the second shape is both a rhombus and a
rotated square. This is an example of
the concept of “invariance of shape”.
Young children see these as two different shapes and do not always see
the relationship between the figures.
The
supplementary materials will emphasize that an important concept that
prospective elementary teachers will be teaching is “invariance of shape”. The intent is that prospective elementary
teachers would think about teaching children about shapes in terms of this
concept and thus, their own learning of the geometry of shapes will be more
meaningful. The notes to the teacher
will continue.
Further, this is a
concept and a concept cannot be taught directly to children. One might erroneously believe that you could
simply tell a child that they are in essence the same shape, but this will not
make sense to a child until the child is ready.
Children learn concepts, like “invariance of shape,” through
continued sensori-motor activity with manipulatives such as tangrams,
pentominoes, pattern blocks, and attribute blocks.
To illustrate
how this example might be connected to the mathematics that elementary students
will learn in middle and high school, the supplement will describe the value of
spatial visualization.
An important connection teachers can make at
this level is to help their students mentally visualize and manipulate
shapes. This is a valuable ability in
geometry and calculus. If students can
learn to rotate and mentally manipulate shapes, such as squares, then they will
be better prepared to visualize and solve calculus tasks such as finding the
surface area of a three-dimensional donut cut in half.
The teaching
notes for instructors will be very important because most of these instructors
are not familiar with how children learn mathematics. The supplements will suggest how the
materials should be integrated with the mathematical content of these courses. In the previous example, the concept of
invariance of shape will be introduced when prospective elementary teachers are
studying shapes and their properties.
The teaching notes will also suggest that the instructor obtain
tangrams, which are discussed in most elementary content mathematics courses, and
physically illustrate the property of invariance of shape with a square. A suggested test question over this concept
could be:
Explain
the concept of invariance of shape.
How would you attempt to help children develop this concept?
See the CMET
materials web site located at http://www.pnc.edu/depts/mp/CMETsupps.html
for the CMET materials, except for worksheets/pages from elementary textbooks,
developed in the proof-of-concept NSF grant CCLI-EMD DUE 0126882.
Web Site
A CMET web site
will be developed and maintained which will be tagged with descriptive
metadata. The web site will describe the
CMET II materials and provide samples.
Specifically, it will contain suggestions for integration and alignment
of the CMET sections with the sections from the most popular mathematics
content textbooks for elementary teachers.
Another goal of the website is to provide contact information and a
collaboration forum in which discussions can take place among users of the
materials. Finally, the web site will be
used to collect data through web based surveys as well as posting general
evaluation findings as they become available.
Time Table
Spring
2004-Summer 2004 Revise Pilot for Math
Course I Supplement
Write Pilot for Math Courses
II Supplement
Develop Web Page
Fall
2004 Pilot
I & II at various sites
Spring
2005-Summer 2005 Write Pilot for Math
Course III Supplement
Revise Supplements I &
II
Fall
2005 Test
I & II at various sites
Pilot
III at various sites
Spring
2006 Test I
& II at various sites
Analyze
Data
Summer
2006 Analyze Data
Revise
Preliminary
Dissemination
Fall
2006 Test
III at various sites
Preliminary
Dissemination
Facilities and Resources Available
Purdue
University North Central (PUNC) is a commuter campus of approximately 3,600
students and is part of the Purdue
University statewide
system. Students may choose from a variety of bachelor's and associate degrees,
as well as courses that prepare them to transfer to Purdue West Lafayette and
other institutions. PUNC’s Education
Department offers a four-year undergraduate degree in elementary education and
master’s degree in education. The
mathematics department offers a three course sequence of mathematical content
courses for elementary teachers where the CMET II materials will be
piloted. The campus computing center has
the capacity and capability to maintain the proposed web site and provide the
needed technical support.
Experience and Capability of the Principal Investigators
David Feikes,
Ph.D., Associate Professor of Mathematics, Purdue University North Central, was
the PI for the proof-of-concept grant CMET, CCLI-EMD DUE 0126882. He will be the PI for CMET II and will be
involved in all phases of the project.
He is in the unique situation in that he teaches the mathematical
content courses for elementary teachers, the mathematical methods course for
elementary teachers, the graduate mathematics education course for practicing
teachers, and he works extensively in local elementary schools. One of the strengths of his teaching of
mathematical content course is his ability to relate the mathematics in these
courses to how children think and learn about mathematics. He is the developer and project director of
Purdue Math, an ongoing innovative elementary mathematics program to enhance
the learning of both teachers and students. In this program he has extensive
experience developing curriculum materials. The Purdue Math program is based on
research into how children learn and understand mathematics (Cobb et. al.,
1991; Steffe et. al., 1983) and on research on teacher learning (Feikes,
1992). One of the main assumptions of
Purdue Math is that teachers learn in the course of their practice. Consequently, Dr. Feikes spends a great deal
of time in elementary classrooms helping teachers realize new forms of
practice. This has given him first-hand
experience with children’s thinking beyond reading research reports. He has directed several statewide Eisenhower
grants. He has conducted both
qualitative and quantitative analyses of the Purdue Math Project. Dr. Feikes’ research interests are in teacher
learning and in realizing reform in mathematics education.
Keith
Schwingendorf, Ph.D., Professor of Mathematics, Purdue University North Central
(PUNC) was Co-PI for the proof-of-concept grant CMET, CCLI-EMD DUE
0126882. He will be a CMET II Co-PI and
will collaborate on the development of the CMET II supplementary materials and
help to oversee the project. He will
also teach math content courses for pre-service elementary teachers on the PUNC
campus. One area of his expertise is on the connection between the mathematics
that prospective elementary teachers are learning to the mathematics that
children will need in middle and high school, and even college. Dr. Schwingendorf has been a Co-PI of several
NSF calculus reform grants (e.g., 9450750-DUE, 9053432-USE) and the PI of an
NSF Instrumentation and Laboratory Improvement grant (9252262-USE) that
established a mathematics educational computing laboratory on the PUNC campus.
David Pratt,
Ph.D., is an Assistant Professor of Education, Purdue North Central and he will
serve as the CMET II internal evaluator and he will develop and maintain a Web
Site devoted to the CMET II project. He
has ten years of elementary and middle school experience in California.
While at the University of California at Santa
Barbara, he worked on the Preparing Tomorrow’s
Teachers to Use Technology (PT3), (P342A000194) grant in which he was in charge
of creating the web site for the project. Dr. Pratt has developed dozens of
sites for various professional projects and has completed specialized web work
for companies outside of education. He
has extensive graduate coursework in both qualitative and quantitative research
methodologies. His research focuses on
developing interviews for gathering responses from teachers and using
established coding procedures for analyzing data.
Other Key Personnel
Jeff Gregg,
Ph.D., was a principal co-author of the supplemental materials the first CMET
proof-of-concept grant, CCLI-EMD DUE 0126882, and he will serve as the primary
consultant for CMET II. He has a
doctorate in mathematics education from Purdue University,
he has been a free-lance writer developing mathematics education materials for
Addison-Wesley Publishers, and he is currently a Visiting Assistant Professor
at Purdue University Calumet. He has taught mathematical content courses for
prospective elementary teachers, participated in classroom teaching experiments
in elementary school mathematics, and conducted clinical interviews with
children to investigate how they think about and understand mathematics. Dr. Gregg will edit and help revise the
supplementary materials. In addition, he will collaborate in the development of
the CMET II evaluation resources. His
expertise in writing and knowledge of mathematics education make him an
invaluable member of the CMET II project team.
Michelle
Stephan, Ph.D., is an Associate Professor of Mathematics, Purdue University
Calumet. Her research has focused on
children’s understanding of measurement.
She will develop materials in her areas of expertise and focus on the
creation of research-based journal articles.
Marcela Perlwitz, Ph.D., is an Assistant Professor, Wabash College,
Crawfordsville, IN.
She has a doctorate in mathematics education and she will assist in the
writing and editing of the CMET II supplements.
Mary Jane Eisenhauer, Ph.D., is an Adjunct Professor, Purdue University
North Central, and she has a doctorate in early childhood education. Her contribution to CMET II will focus on
including the early childhood perspective in the project’s supplements.
Sarah Hough,
Ph.D., the CMET II external evaluator, currently holds a research position in
the Gevirtz Graduate School of Education at the University
of California at Santa Barbara. For the past two years she has led the
evaluation of Project RENEW, an NSF funded teacher retention and renewal
project (#0110995), in which her role includes the evaluation research design,
report writing and speaking at professional conferences and also to local
project participants and stakeholders.
Dr. Hough designed and conducted formative evaluation studies for the Tri-County
Mathematics Project in 1997 and she also lead groups of teachers from project
PRIME (an NSF funded Local Systemic Change Initiative) in their efforts to
research mathematics reform in their classrooms in 1998. Dr. Hough will participate in the Summer 2003
Evaluation Institute at Western
Michigan University. Having been selected as one of the 15
participants to attend, she will be involved in the development of mathematics
professional development evaluation materials.
Dr. Hough is also directing the development and evaluation of on-line
multi-media learning modules for Project CASELINKS, a five-year project funded
by the Office of English Language Acquisition, U.S. Department of
Education. (See Appendix D for the
consultants’ and the contributors’ letters of support.)
Evaluation Plan
A
mixed-method evaluation model will be implemented; one that will design
instrumentation and collect and analyze both quantitative and qualitative data
in order to provide project stakeholders with information necessary to guide
the long-term success of the project, as well as to investigate and report on
how the project is implementing and meeting its goals. This evaluation is
two-tiered and consists of: (1) a summative evaluation component, designed to
produce comprehensive reports of the effectiveness of the developed materials
for project stakeholders and to oversee the overall evaluation effort and (2) a
formative component designed to collect ongoing information and to shape and
improve the design and implementation of the CMET II materials.
Summative
Evaluation Component
The purpose
of the CMET II materials is to impact the preparation of prospective elementary
teachers by developing and testing curriculum materials which use knowledge of
how children learn and understand mathematics.
In order to investigate the effectiveness of the project, a full
curriculum analysis of both the intended and implemented curriculum will be
performed. In addition, a
goals-directed, comparison group evaluation study that draws on both
quantitative and qualitative data will be conducted over diverse test sites.
Summative Research Questions
1) How does CMET II improve prospective teachers’ understanding of how
children learn and understand mathematics?
2) Does CMET
II enhance pre-service teachers’ understanding of mathematics?
3) Does CMET II help prospective teachers
to connect the mathematics they are learning with the mathematical concepts
they will be teaching,
4) How does
CMET II facilitate prospective teachers’ understanding of connections between
the mathematics they will be teaching to the mathematics and technological skills children will learn in middle and high
school?
5) What
evidence is available that CMET II will improve pre-service teachers’ future
teaching of mathematics to children?
|
Question 1: How does CMET
II improve prospective teachers’ understanding of how children learn and
understand mathematics?
|
|
Sub-questions
|
Data Collection
|
Respondents
|
Schedule
|
|
What
opportunities does the supplement offer for participants to better understand
the ways that students think about content?
|
Curriculum
analysis
|
|
As
the materials are developed
|
|
How
do preservice teachers interpret what they are learning about children’s
thinking?
|
Task
based interviews
|
Sample
of students*
|
Midway
and at end of course
|
|
Question 2: Does CMET II enhance pre-service teachers’
understanding of mathematics?
|
|
Sub-questions
|
Data Collection
|
Respondents
|
Schedule
|
|
How
does CMET II affect pre-service teachers’ beliefs/understandings about
mathematics?
|
Questionnaires
|
Students* in test and comparison classrooms
|
Pre/post
course
|
|
How
does CMET II affect pre-service teachers’ achievement in mathematics?
|
Performance
measures
|
Students*
in test and comparison classrooms
|
Pre/post
course
|
|
Question 3: Does CMET II help prospective teachers to
connect the mathematics they are learning with the mathematical concepts they
will be teaching?
|
|
Sub-questions
|
Data Collection
|
Respondents
|
Schedule
|
|
What
opportunities does the supplement offer for pre-service teachers to better
understand the connections between the math they are learning and the
concepts they will be teaching?
|
Curriculum
analysis
|
|
As
the materials are developed
|
|
How
are these opportunities for connecting content in the college setting and the
K-6 classroom interpreted by the pre-service teachers?
|
Task
based interviews
|
Sample
of students*
|
Midway
and at end of course
|
|
Question 4: How does CMET II facilitate prospective
teachers’ understanding of connections between the mathematics they will be
teaching to the mathematics and
technological skills children will learn in middle and high school?
|
|
Sub-question
|
Data Collection
|
Respondents
|
Schedule
|
|
Which
mathematics and technology concepts covered in the supplement, are related to
those concepts covered in middle school and high school?
|
Questionnaire
and
Interviews
|
Instructors
and
Sample
of students*
|
Midway and at end
of course
|
|
Question 5: What evidence is available that CMET II
will improve pre-service teachers’ future teaching of mathematics to
children?
|
|
Sub-questions
|
Data Collection
|
Respondents
|
Schedule
|
|
How
does CMET II facilitate pre-service teachers’ future teaching of math?
|
Questionnaires
|
Students
|
Pre/post
course
|
|
How
do pre-service teachers perceive the activities that they participate in
(using the supplement) as relating to their understandings of pedagogy?
|
Task
based interviews
|
Sample
of students*
|
Midway
and at end of course
|
|
*
Consisting of those students that give consent to participate in the evaluation.
|
|
|
|
|
|
|
Samples of
students will be purposefully selected to represent a range of understandings
of content. The CMET materials will be tested at various and diverse test
sites.
Formative Evaluation Component
The formative
evaluation plan will follow a mixed-methods design with the purpose of
reporting periodically to the project leadership team and other key
stakeholders about how the project is being implemented according to its
timeline and how effective the project activities are in achieving the project
goals.
|
Implementation Evaluation Question
1.
Are project activities occurring according to the timeline?
|
|
Sub-questions
|
Data Collection
|
Respondents
|
Schedule
|
|
What
progress is being made on the material development?
|
Interviews
|
Project
team
|
Periodically
|
|
Which
sites have been selected for piloting?
|
Interviews
|
Project
team
|
Periodically
|
|
Progressive Evaluation
Questions
2.
How are the CMET II materials being implemented?
|
|
Sub-questions
|
Data Collection
|
Respondents
|
Schedule
|
|
How
often are the materials being used, when are they used, what student
assessments are made with them, etc?
|
Interview/
Questionnaire
|
Instructors
|
Once
during each course
|
|
What
is the culture of the classroom and how does this affect the use of the materials?
|
Non-participant
observations
|
Instructors
|
Selected
sites will be visited 3 times
|
|
3.
How are CMET II materials being interpreted by participants?
|
|
Sub-questions
|
Data Collection
|
Respondents
|
Schedule
|
|
How
clearly understandable are the materials to the instructors given their
content and pedagogical backgrounds?
|
Interviews
and
Instructor
Questionnaire
|
Instructors
|
Periodically
|
|
How
clearly understandable are the materials to the students given their content
background and prior experience?
|
Interviews
and
Instructor
Questionnaire
|
Sample
of students*
and
Instructors
|
Periodically
|
|
What
is the culture of the classroom during student investigation time and how
does this affect the use of the materials?
|
Participant
observation, videotape
|
Students
|
Selected
sites will be visited 3 times
|
|
Is
there evidence that the materials help to develop participants’
understandings of mathematics and of children’s thinking?
|
Collections
of graded student work
|
Students
and
Instructors
|
As
needed throughout the course
|
This portion of the evaluation
will draw on an extensive data pool including: student work samples from
classroom mathematics investigations; participant-observation analytic notes;
transcripts from interviews with course instructors and student participants;
questionnaires.
Evaluation Time Line and Analyses
|
Date
|
Project Activity
|
Evaluation Activity
|
Analyses and Product
|
|
Spring
2004 – Summer 2004
|
|
Develop
Evaluation Instruments and analyze the CMET II materials
|
|
|
Fall
2004 - Spring 2005
|
Supplement
used in mathematical content courses at various test sites
|
Formative
assessment conducted
|
Ongoing
data analysis and revision of instruments
|
|
Summer
2005
|
|
|
Continued
analysis and finalization of instrumentation
|
|
Fall
2005 – Spring 2006
|
Supplement
used in mathematics content courses at various test sites
|
Conduct
formative and summative evaluation study
|
Ongoing
analysis of data from summative evaluation study
|
|
Summer
2006
|
Plan
and set up more diverse settings in which CMET II materials will be
implemented
|
|
Finish
data analysis and report conclusions
|
|
Fall
2006
|
Supplements
used in more diverse university settings
|
Conduct
summative evaluations
|
Reports
written up and presented to national audiences
|
Faculty at
the institutions listed below are prepared to pilot the CMET II materials (see
Appendix D for letters of support).
Several others institutions have also indicated an interest.
Purdue University
North Central University of Illinois at Urban-Champaign
Purdue Calumet
University Southern Illinois University
Edwardsville
Kent State
University Ohio Valley College
Indiana University-South Bend Bethel College
Dissemination of Results
Addison-Wesley
Publishing has expressed a strong interest in the publication and dissemination
of the CMET II project materials (see the letter in Appendix D). The project web site will also serve as a
vehicle for disseminating information about the project. The project staff is committed to maintaining
the web site and updating the CMET II materials to keep them current. In addition, the project team will provide
information about the feasibility of these materials and the project’s
innovative approach to enhance pre-service teachers’ understanding of
mathematics through national conferences such as the Mathematical Association
of America , National Council of Teachers of Mathematics, Psychology of Mathematics Education,
Association of Mathematics Teacher Educators, The American Mathematical
Association of Two Year Colleges and American Educational Research
Association. Also, manuscripts will be
prepared for publication in journals of the aforementioned organizations. The CMET II project has the potential to
significantly impact the preparation of future teachers by providing them with
in-depth knowledge of how the subject matter they are learning relates to their
future teaching.
The supplements
will be widely applicable since the contents of the most popular textbooks for
these courses are very similar. An
examination of the most popular textbooks of mathematical content for
elementary teachers reveals the chapters and their organization are almost
identical in content with only slight variations in the organization of
chapters. (See Appendix C for a comparison of chapters in six of the most
popular textbooks.) The materials will
be organized to correspond with the content of these textbooks. Further, the CMET II supplements are designed
to be used in any configuration of mathematical content courses for elementary
teachers (e.g., three, 3-hour; two, 4-hour; or even one course attempting to
cover all topics). The supplementary
materials from this project will accentuate that they may be used with any
mathematical content textbook for elementary teachers.
Future Plans
The CMET II
project team envisions and is committed to future activities based on this
project. We anticipate that we will
submit an additional request to NSF to conduct workshops and dissemination
activities for faculty to use the CMET II materials. The project staff intends to continue with
making presentations and writing journal articles after the duration of this
project. Several team members have a
strong interest in research and anticipate that they will be able to publish
research articles based the project data collected. This project has the potential to make
significant contributions in the integration of pedagogy and subject matter
knowledge (Ball and Bass, 2000). The
project team would like to develop these project materials into support
materials for practicing teachers. In
the evaluation of the site testing for the first CMET project, several students
made this suggestion. In addition,
Heinemann Publishing has expressed an interest in publishing the CMET II
materials as a resource for practicing teachers (see the letter in Appendix
D). The team also envisions that the
material could be adopted into a resource for parents. The CMET II project will continue after the
duration of the proposed grant in the three key areas: development of materials, dissemination, and
research.
References
Askey, R. (1999). Knowing and teaching elementary
mathematics. American Educator, 23(3), 6-9, 12-13Ball, D. &
Bass, H. (2000). Interweaving content
and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics. (pp. 83-104).
Westport, CT: Ablex.
Ball, D. (1993). With an eye on the mathematical
horizon: Dilemmas of teaching elementary school mathematics. Elementary
School Journal 93 373-397.
Boaler, J. (2000).
Introduction: Intricacies of
knowledge, practice, and theory. In
J. Boaler (Ed.), Multiple perspectives on
the teaching and learning of mathematics.
(pp. 1-18). Westport, CT:
Ablex.
Brandsford, J., Brown, A., & Cocking, R.
(1999). How people learn: brain, mind, experience, and school. Washington,
D.C: National Academy
Press.
Carpenter, T. P., Franke, M.L. & Levi, L.
(2003). Thinking mathematically:
Integrating arithmetic and algebra in elementary school. Portsmouth, NH:
Heinemann.
Carpenter, T. P., Fennema, E. & Franke, M.L.
(1996). Cognitively guided instruction:
A knowledge base for reform in primary mathematics instruction. Elementary
School Journal 97 3-20.
Clements, D. & Battista, M. (1992). Geometry and
spatial reasoning. In D. Grows (Ed.), Handbook of research on mathematics
teaching and learning. 420-464. New
York: Macmillan.
Clements, D. Sudha, S., Hannibal, M.A., & Sarama,
J. (1999). Young children’s conception of shape. Journal for Research in
Mathematics Education 30, 192-212.
Cobb, P. & Wheatley, G.
(1988). Children's initial understanding
of ten. Focus on Learning Problems in
Mathematics, 10(3),
1-28.
Cobb, P., Wood, T., & Yackel.
E. (1991). A constructivist approach to second grade mathematics. In E. von
Glasersfeld (Ed.), Constructivism
in Mathematics Education. Dordrecht, Holland: Reidel.
Conference Board of the
Mathematical Sciences (2001). The
mathematical education of teachers. Providence, RI:
American Mathematical Society.
Falkner, K. P., Levi, L., &
Carpenter, T. (1999). Children’s understanding of equality: A foundation for algebra.
Teaching Children Mathematics
6, (4).
Feikes, D. (1992). Beliefs and
belief changes of two elementary teachers in a problem-centered second grade,
mathematics project. Unpublished
doctoral dissertation, Purdue University, West
Lafayette.
Fennel, F. & Rowan, T.
(2001). Representation: an important
process for teaching and learning mathematics.
Teaching Children Mathematics,7(5), 288-292.
Fuson, K. (1988). Children’s counting
and concepts of number. New York: Springer-Verlag.
Gearhart, M., Saxe, G. B.,
Seltzer, M., Schlackman, J., Ching, C. C., Nasir., et al. (1999). Opportunities to learn
fractions in elementary
mathematics classrooms, Journal for Research in Mathematics Education, 30,
286-315.
Glaser
, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research.
Chicago: Aldine.
Kamii, C. & Housman, L.
(1999). Young children reinvent
arithmetic: implications of Piaget’s theory. 2nd ed. New
York:
Teachers College Press.
Kilpatrick, J., Swafford, J.,
& Findell, B. (Eds). (2001). Adding it up: Helping children learn
mathematics.
Washington
DC: National
Academy Press.
Mewborn, D. S. (2003). Teaching,
teachers’ knowledge, and their professional development. In J. Kilpatrick,
G.
Martine, & D. Schifter (Eds.),
A research companion to the
principles and standards for school
mathematics.
Reston, VA: NCTM.
National
Council of Teachers of Mathematics. (1989).
Curriculum and evaluation standards for school mathematics.
Reston, VA:
NCTM.
National Council of Teachers of Mathematics. (1991). Professional
standards for teaching mathematics. Reston, VA:
NCTM.
National Council of Teachers of Mathematics. (1995). Assessment
standards for school mathematics. Reston,
VA:
NCTM.
National Council of Teachers of Mathematics. (2000). Principles
and standards for school mathematics.
Reston,
VA: NCTM.
National Council of Teachers of Mathematics. (2003). A research
companion to the principles and standards for
school mathematics. Reston, VA: NCTM.
Rand Report, (2002). Mathematical proficiency for all
students: Toward a strategic research
and development
program in mathematics education. OERI, U.S.
Department of Education
Thompson, P. W., & Thompson,
A. G. (1994). Talking about rates conceptually, Part 1: A teacher’s struggle.
Journal for Research in
Mathematics Education, 25, 279-303.
Shulman,
L. S. (1987). Knowledge and teaching:
Foundations of the new reform. Harvard Educational Review, 57
(1)
99-113.
Simon, M. (1995). Reconstructing
mathematics pedagogy from a constructivist perspective. Journal for Research
in
Mathematics Education 26
114-45.
Steffe, L. P., von Glasersfeld,
E., Richards, J., & Cobb, P. (1983). Children's counting types: Philosophy, theory and
application. New York: Praeger Scientific.