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,
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,
Results from Prior NSF Support
The CMET II
project is based on the NSF proof-of-concept grant CCLI-EMD DUE 0126882;
$75,000;
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
“Connecting Mathematics for Elementary Teachers in Pre-service Mathematics
Content Courses”, NCTM Regional Meeting, Indianapolis , IN
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).