Catalysts for Change

•Principles and standards for school mathematics (NCTM, 2000)

•Before It’s Too Late: Glenn Commission Report, (DOE, 2000)

•Mathematics Education of Teachers Report (CBMS/MAA, 2001)

Before It’s Too Late: Glenn Commission Report

Primary message: “America’s students must improve their performance in mathematics andscience if they are to succeed in today’s world and if the United States is to stay competitive in an integrated global economy.”

Second message: “The most direct route to improving mathematics and science achievementfor all students is better mathematics and scienceteaching.”

Teachers need a deep understanding of the mathematics they will teach.

Teachers need to learn how fundamental mathematical principles underlie classroom practice.

Teachers need to develop a mastery of the mathematics in several grades beyond (and below) the grade level they expect to teach.

CBMS Mathematical Education of Teachers Report: General

Recommendations

Courses for teachers should involve: + foundational mathematics + careful reasoning + connections of principles / practice + mathematical ‘common sense’ + habits of mind / inquiry + utility, power and elegance + multiple ways to engage students + technology-computation /exploration

CBMS Mathematical Education of Teachers Report: General

Recommendations

Teacher education must be recognized as an important part of mathematics departments’ mission at institutions that educate teachers.

Mathematics departments should devote commensurate resources to designing and offering courses for teachers. They should value and properly reward the faculty members that are heavily involved in teacher education.

The mathematical education of teachers should be seen as a partnership

between mathematics faculty and mathematics education faculty.

CBMS Mathematical Education of Teachers Report: General

Recommendations

Project Outcomes

Developed 4 mathematics courses with accompanying text books for pre-service / in-service middle grade mathematics teachers

Developed recruitment models for attracting middle grade mathematics teachers

Developed options for post-bac certification (middle / high school mathematics)

The Curriculum:

Making the Connections

•Algebra Connections

•Geometry Connections

•Data Analysis and Probability Connections

•Calculus Connections

The text books utilize middle school curricular materials in

multiple ways

∑ As a springboard to college level mathematics

∑ To expose future (or present) teachers to current middle grade curricular materials

∑ To provide strong motivation to learn more and better mathematics

∑ To support curriculum dissection—critically analyzing middle school curriculum content—developing improved middle grade lessons through lesson study approach

∑ To use college content to gain new perspectives on middle grade content and vice versa

∑ To apply middle grade instructional strategies and multiple forms of assessment to the college classroom

Instructional Components:Classroom Connections, Classroom Discussions, and Classroom Problems.

Classroom Connections: middle grade investigations that serve as launch pads to the college level Classroom Discussions, Classroom Problems, and other related collegiate mathematics.

Classroom Discussions: detailed mathematical conversations between college teacher and pre-service middle grade teachers, and are used to introduce and explore a variety of important concepts during class periods.

Classroom Problems: a collection of problems with complete or partially complete solutions and are meant to illustrate and engage pre-service teachers in various problem solving techniques and strategies.

For desk copies contact:Michael Bell

Project Manager, Statistics & Service MathematicsPrentice Hall

Michael_Bell@Prenhall.com

The Middle Grade Connection Series

Algebra Connections-Contents

Chapter 1. Patterns

1.1 Classroom connections: representing patterns

1.2 Reflections on classroom connections: representing patterns

1.3 Arithmetic sequences

1.4 Classroom connections: a quadratic sequence

1.5 Reflections on classroom connections: a quadratic sequence

1.6 Finite arithmetic sequences

1.7 Geometric sequences

1.8 Mathematical induction

1.9 Classroom connections: counting tools

1.10 The Binomial Theorem1.11 The Fibonacci sequence

Algebra Connections-Contents

Chapter 2. Arithmetic and Algebra of the Integers2.1 A few mathematical questions concerning the periodical cicadas

2.2 Classroom connections: multiples and divisors

2.3 Reflections on classroom connections: multiples and divisors

2.4. Multiples and divisors

2.5 Least common multiple and greatest common divisor

2.6 The Fundamental Theorem of Arithmetic

2.7 Revisiting the lcm and gcd

2.8 Relations and results concerning lcm and gcd

Algebra Connections-Contents

Chapter 3. The Division Algorithm and the Euclidean Algorithm

3.1 Measuring integer lengths and the Division Algorithm

3.2 The Euclidean Algorithm

3.3 Applications of the representation gcd(a, b) = ax + by

3.4 Place value

3.5 Prime thoughts

Algebra Connections-Contents

Chapter 4. Arithmetic and Algebra of the Integers Modulo n4.1 Classroom connections: divisibility tests4.2 Reflections on classroom connections: justifying the divisibility tests4.3 Clock addition4.4 Modular arithmetic4.5 Comparing arithmetic properties of Z and Zn

4.6 Multiplicative inverses in Zn

4.7 Elementary applications of modular arithmetic4.8 Fermat’s Theorem and Wilson’s Theorem4.9 Linear equations defined over Zn

4.10 Extended studies: the Chinese Remainder Theorem4.11 Extended studies: quadratic equations defined over Zn

Algebra Connections-Contents

Chapter 5. Algebraic Modeling in Geometry: The Pythagorean Theorem and More 5.1 The significance of Daryl’s measurements and related geometry 5.2 Classroom connections: The Pythagorean Theorem and its converse5.3 Reflections on classroom connections: The Pythagorean Theorem and its

converse5.4 Computing distance in 2-dimensional and 3-dimensional Euclidean space: the

distance formula5.5 An extension of the Pythagorean Theorem: the law of cosines5.6 Integer distances in the plane5.7 Pythagorean triples: positive integer solutions to

x2 + y2 = z2

5.8 Extended studies: further investigations into integer distance point sets - a Theorem of Erdos

5.9 Extended studies: additional questions concerningPythagorean triples

Algebra Connections-Contents

Chapter 6. Arithmetic and Algebra of Matrices6.1 Classroom Connections: systems of linear equations

6.2 Reflections on classroom connections: systems of linear equations

6.3 Rational and irrational numbers

6.4 Systems of linear equations

6.5 Polynomial curve fitting: an application of systems of linear equations

6.6 Matrix arithmetic and matrix algebra

6.7 Multiplicative inverses: solving the matrix equation AX = B

6.8 Coding with matrices

QED