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, 2002June 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).


Description of Proposed Materials

For reference in this section, the five goals of the CMET II project are:

1.      Improve pre-service elementary teachers’ understanding of how children learn and understand mathematics.

2.      Enhance pre-service elementary teachers’ understanding of mathematics.

3.      Help prospective elementary teachers connect the mathematics they are learning with the mathematical concepts they will be teaching children.

4.      Facilitate prospective elementary teachers’ understanding of the connection between the mathematics they will be teaching in elementary school and the mathematics children will learn in middle and high school.

5.      Improve pre-service elementary teachers’ teaching of mathematics.

 

 

The following contains a brief description of the content of the CMET II student supplement. 

Each statement is cross-referenced, in parentheses, with the project goal(s) that it addresses.

§         Descriptions and examples of children’s mathematical thinking (1)

§         Discussions of how children learn mathematics (1)

§         Activities for prospective elementary teachers—these activities are intended to illustrate children’s mathematical thinking and especially how children think differently than adults (1,  2, & 3)

§         Mathematical problems for prospective elementary teachers to solve (1 & 2)

 

§         Discussions of specific terminology or notation that is typically confusing to prospective elementary teachers (2)

§         Discussions of the “why” in mathematics (2)

 

§         Discussions of how the “why” is important in teaching children mathematics (3)

§         Rationale for teaching mathematics for understanding (3)

§         Discussions of manipulatives and their role and uses in teaching mathematics (3)

§         Examples from elementary school mathematics textbooks (3)

§         Analysis of, and questions about, the textbook examples (3)

§         Historical, mathematical and educational references (3)

§         Descriptions of the connection between the mathematics children learn in elementary school and the mathematics they will learn in middle and high school (4)

 

§         Discussions of how one might teach mathematical concepts they are learning (5)

§         Key points that a teacher might emphasize when teaching specific mathematical concepts (5)

§         Alternative ways of representing and presenting  (teaching) mathematical concepts and

procedures (2 & 5)

 

§         Questions for discussion (1, 2, 3, 4, & 5)

 

The CMET II Instructor’s Supplement will contain:

§         More detailed discussions

§         Points of emphasis for instructors

§         Further examples and descriptions

§         Test Questions

§         Answers and Solutions

§         Rationale

§         Discussions of other approaches to teaching mathematical concepts

§         References

 

Outline of the Proposed Content

From the analysis of six of the most popular mathematical content textbooks for elementary teachers (see Appendix C) the project staff has identified the major topics to be addressed in the supplementary materials.  The topics are organized by a likely delineation into courses.  However, the material could be organized into any configuration of possible mathematical content courses for elementary teachers. 

 

 

 

 

 

 

Course I

 

I.             Problem Solving

A.    Problems

B.    Patterns

C.    Strategies

D.    Polya’s Four Steps

E.       Types of Problems

F.       Inductive and Deductive Reasoning

II.            Sets

A.    History

B.    Purpose

C.    One-to-one Correspondence

D.    Operations on Sets

E.    Venn Diagrams

III.          Whole Numbers

A.    Numeration Systems

1.       History

2.       Place Value

3.       Expanded Notation

4.       Manipulatives

a.  Unifix Cubes

  b. Base Ten Blocks

B.    Addition and Subtraction

        1.     Closure Property

        2.     Addition

        3.     Subtraction

        4.     Classifying Problems

C.    Multiplication and Division

        1.     Multiplication

        2.     Division

        3.     Closure

        4.     Classifying Problems

        5.     Division by Zero

D.    Properties and Algorithms

        1.     Commutative

        2.     Associative

        3.     Distributive

        4.     Identity

E.    Mental Computation and Estimation

        1.     Mental Computation

        2.     Rounding

3.       Real-life Applications

IV.             Number Theory

                        A.    Factors

        B.    Divisibility

        C.    Prime and Composite

                1.     Prime Number Test

                2.     Prime Factorization

                3.     Fundamental Theorem of    Arithmetic

4.       Famous Problems

 

 

Course I (continued)

 

D.    Common Factors and Multiples

                1.     GCF Factor-list Method

                2.     Relatively Prime

                3.     LCM Multiple-list Method

                                4.     Prime Factorization

V.             Integers

        A.    Addition and Subtraction

                1.     Models

        B.    Multiplication and Division

                1.     Models

        C.    Integer Properties

VI.          Rational Numbers

        A.    Fractions

                1.     Types

                2.     Equivalent Fractions

                3.     Simplifying Fractions

        B.    Addition and Subtraction of Rational     Numbers

                1.     Models

                2.     Unlike Denominators

C.   Multiplication and Division of Rational Numbers

                1.     Models

                2.     Rules

        D.    Properties of Rational Numbers

                1.     Inverses

2.       Denseness

 

Course II

 

VII.         Decimal, Percents and Real Numbers

A.      Decimals

        1.   Place Value

        2.   Models

        3.   Scientific Notation

B.      Operations with Decimals

1.       Estimation

2.       Mental Computation

C.      Ratio and Proportion

D.      Percents, Fractions, and Decimals

E.       Irrational and Real Numbers

XII.          Statistics

A.      Graphs

1.    Real

2.    Picture

3.    Bar

4.    Line

5.    Circle

B.      Deceptions

C.      Mean, Mode and Median

                         



 

 

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?

Summary of Summative Evaluation to Evaluate Project Effects on the Participants

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[P1] 

 

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

[P2] Students* in test and [P3] 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.

[P4] 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...

Formative Research Questions

1.                Are project activities occurring according to the timeline?

2.                How are the CMET II materials being implemented?

3.                How are CMET II materials being interpreted by participants?

 

Summary of Formative Evaluation to Evaluate Project Effects on the Participants

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?[P5] 

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.


 [P1]What does this entail?

 [P2]Human Subjects mandate that  we cannot require all students to participate.

 [P3]We tried a control group in our pilot and we were not happy with the results.   Due to fairness issues we could not put test questions to both groups.  We also found that there were major differences in the instructors, in terms of background and educational experience.  I could not posit any results to CMET due to the instructor factor.

 [P4]What about your question 5 which I changed to 4?  How will that be assessed?

 [P5]I have asked 15 sites to pilot the materials.   Many institutions offer three courses, see proposal, and our intent is for the formatitive evaluation to ask 3-5 to pilot each course supplement.  We would like your input on how we will test for the summative evaluation.