2. 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)

3. Before It’s Too Late: Glenn Commission Report Primary message: “America’s students must improve their performance in mathematics and science 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 achievement for all students is better mathematics and science teaching.”

4. o Teachers need a deep understanding of the mathematics they will teach. o Teachers need to learn how fundamental mathematical principles underlie classroom practice. o 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

5. o 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

6. o Teacher education must be recognized as an important part of mathematics departments’ mission at institutions that educate teachers. o Mathematics departments should devote commensurate resources to designing and offering courses for teachers. o They should value and properly reward the faculty members that are heavily involved in teacher education. o 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

7. 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)

8. The Curriculum: Making the Connections Algebra Connections Geometry Connections Data Analysis and Probability Connections Calculus Connections

9. 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

10. 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.

11. For desk copies contact: Michael Bell Project Manager, Statistics & Service Mathematics Prentice Hall Michael_Bell@Prenhall.com The Middle Grade Connection Series

13. Algebra Connections-Contents Chapter 1.Patterns 1.1Classroom connections: representing patterns 1.2Reflections on classroom connections: representing patterns 1.3 Arithmetic sequences 1.4Classroom connections: a quadratic sequence 1.5Reflections on classroom connections: a quadratic sequence 1.6Finite arithmetic sequences 1.7Geometric sequences 1.8Mathematical induction 1.9Classroom connections: counting tools 1.10The Binomial Theorem 1.11The Fibonacci sequence

14. Algebra Connections-Contents Chapter 2.Arithmetic and Algebra of the Integers 2.1A few mathematical questions concerning the periodical cicadas 2.2Classroom connections: multiples and divisors 2.3Reflections on classroom connections: multiples and divisors 2.4.Multiples and divisors 2.5Least common multiple and greatest common divisor 2.6 The Fundamental Theorem of Arithmetic 2.7Revisiting the lcm and gcd 2.8 Relations and results concerning lcm and gcd

15. Algebra Connections-Contents Chapter 3.The Division Algorithm and the Euclidean Algorithm 3.1Measuring integer lengths and the Division Algorithm 3.2The Euclidean Algorithm 3.3 Applications of the representation gcd(a, b) = ax + by 3.4Place value 3.5Prime thoughts

16. Algebra Connections-Contents Chapter 4.Arithmetic and Algebra of the Integers Modulo n 4.1Classroom connections: divisibility tests 4.2Reflections on classroom connections: justifying the divisibility tests 4.3Clock addition 4.4Modular arithmetic 4.5Comparing arithmetic properties of Z and Z n 4.6Multiplicative inverses in Z n 4.7 Elementary applications of modular arithmetic 4.8Fermat’s Theorem and Wilson’s Theorem 4.9Linear equations defined over Z n 4.10Extended studies: the Chinese Remainder Theorem 4.11Extended studies: quadratic equations defined over Z n

17. Algebra Connections-Contents Chapter 5. Algebraic Modeling in Geometry: The Pythagorean Theorem and More 5.1The significance of Daryl’s measurements and related geometry 5.2Classroom connections: The Pythagorean Theorem and its converse 5.3 Reflections on classroom connections: The Pythagorean Theorem and its converse 5.4Computing distance in 2-dimensional and 3-dimensional Euclidean space: the distance formula 5.5 An extension of the Pythagorean Theorem: the law of cosines 5.6 Integer distances in the plane 5.7 Pythagorean triples: positive integer solutions to x 2 + y 2 = z 2 5.8Extended studies: further investigations into integer distance point sets - a Theorem of Erdos 5.9 Extended studies: additional questions concerning Pythagorean triples

18. Algebra Connections-Contents Chapter 6.Arithmetic and Algebra of Matrices 6.1Classroom Connections: systems of linear equations 6.2Reflections on classroom connections: systems of linear equations 6.3 Rational and irrational numbers 6.4Systems of linear equations 6.5Polynomial curve fitting: an application of systems of linear equations 6.6Matrix arithmetic and matrix algebra 6.7Multiplicative inverses: solving the matrix equation AX = B 6.8Coding with matrices

19. QED