Mathematical Foundations of the Common Core (Grades 7-10)

A LaMSTI On-Ramp Course, Fall 2011

In 2010, the Louisiana Board of Elementary and Secondary Education adopted the Common Core State Standards (http://www.corestandards.org/) in English Language Arts and Mathematics for adoption in Louisiana in 2014-2015.

In Fall 2011, the Louisiana Math and Science Teacher Institute (LaMSTI) at Louisiana State University will offer a new course for secondary math teachers that is intended to provide an in-depth view of the algebra and geometry that the Common Core State Standards recommend for Grades 7 and 8 and the first two years of high-school (Algebra and Geometry).  Topics include: ratio, proportion and similarity,  expressions and equations, linear, quadratic, polynomial and exponential functions, coordinates and analytic geometry, geometric transformations, Euclidean geometry, and mathematical modeling.  The instructor will guide the participants in unpacking these topics and the relationships between them to build a coherent understanding of the mathematical vision of the Common Core.  Teachers in the course will work together as a professional learning community to prepare to teach middle- and high-school math as the Common Core State Standards intend.

Mathematical Foundations of the Common Core will also serve and an “On Ramp” for the LSU Masters of Natural Sciences Degree Program.  Teachers who prove themselves in this course will be given priority for admission to the LaMSTI-MNS Degree Program.

Scheduling: This course will meet on Tuesday evenings from 5:00 to 7:30 starting on August 23.  The initial meeting will be in Room 132 Prescott Hall on the LSU Campus.  In subsequent weeks, the location will be 203 Prescott Hall.

This course may be taken for graduate credit as a non-matriculating student.  Apply for non-matriculating status at the LSU Graduate School Web site.

For more information, contact:

Dr. James J. Madden, Director of LaMSTI
213 Prescott Hall, LSU
madden@math.lsu.edu
225-978-3525 (mobile)

I. Course Content

I.A. Sample lecture notes

• The number line
• The distributive law
• Order of Operations

Other sample materials: Measurement | Expressions | Zetetica (Algebra) | Equations of Lines | Seminar notes (June 3, 2011)

I.B. Algebra topics (selected from the following)

• Arithmetic and Algebra
• Number systems.  Natural numbers, integers, rational numbers. The number line.
• Decimal notation. Algorithms for addition and multiplication. Repeating decimals.  Approximation and real numbers.
• The rules of arithmetic.  Rings and fields.
• Advanced topic:  Euclidean algorithm and continued fractions  
• Expressions and Equations.
• Arithmetic expressions.  Structure. Transformations.
• Variables and expressions with variables.
• Equations and solving equations.
• Word problems.
• Advanced topic: Descartes’ Geometry.
• Coordinates and Graphs
• Coordinate systems on a line.  Solving problems with coordinates. Coordinate transforms.
• Coordinate systems in the plane.  Graphing equations.
• Lines in the plane.
• Linear maps from the plane to the plane; matrices.
• Logic.
• Connectives (“and”, “or” and “not”) and logically complex propositions.
• Existential quantifiers (“for all” and “there exists”).
• Sets defined by logical formulae.  Boolean algebra. Products and projections.
• Advanced topic: Semi-linear and semi-algebraic sets.
• Functions
• Definitions and examples.  Functions between sets.
• Linear functions and groups of linear functions.
• Composition and inversion
• Exponent and Logarithm.
• Sequences and recursion.
• Polynomials and rational functions
• Computing with polynomials and rational functions.
• Quadratics.
• Factoring and the Remainder Theorem.
•  Polynomials of several variables.
• Conic sections.
• Advanced topic: Algebraic curves.
• Word problems and modeling.

I.C. Geometry Topics (selected from the following)

• Euclid’s Elements
• Informal deductive rigor: definitions, postulates and propositions
• Book I: Triangles and congruence
• Book I: The Parallel Postulate and its consequences
• Book I: Area by dissection
• Book V: Measurement, ratio and proportion
• Book VI: Similarity
• Descartes’ Geometry
• Introducing a unit.  Multiplying and finding roots by geometry
• Dynamic figures analyzed using similarity
• Coordinates
• Transformations
• Isometries and similarity transforms: classification
• Transformation groups
• Advanced topic: Inversion (in a circle) and hyperbolic geometry

II. Research Base

Educational research (see articles below) demonstrates that mathematics teachers need:

• a deep understanding of the structure, content and goals of the curriculum,
• a large repertoire of fully analyzed mathematical examples that may be incorporated in lessons and tests, and
• ability to conceptualize and assess the mathematical knowledge of others and select appropriate actions in response.

This course is designed to develop these competencies.

1. Ball, D. L., Lubenski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge.  In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp. 433-456).  Washington, DC: American Educational Research Association.
2. Baumert, J. Kunter, M., Blum, W., Brunner, M, Voss, T., Jordan, A., Klusman, U.,  Krauss, S., Neubrand, M., & Tsai, Y. (2010).  Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress.  American Educational Research Journal 47, 133-180.
3. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States.  Hillsdale, NJ: Erlbaum.

III. Content References

High School Reference Texts
We will use the CME Project texts: Algebra I and Geometry.  The CME Project is a four-year, NSF-funded, comprehensive high school mathematics program that is problem-based, student centered, and organized around the traditional high-school course sequence.  The series was developed by the Center for Mathematics Education at Education Development Center, Inc. (EDC) in Newton, Massachusetts, and is published by Pearson Education, Inc.

Main Primary Sources
Euclid.  Euclid’s Elements, Thomas L. Heath, translator.  Green Lion Press, 2002.
Viete, F.  The Analytic Art.  Dover 2006.
Descartes, R.  The Geometry of Rene Descartes.  Dover, 1954.
Newton, I.  The Mathematical Papers of Isaac Newton, Volume I, edited by D. T. Whiteside.  Cambridge Univ. Press, 1967.

Other Important Primary Sources
Archimedes.  The Works of Archimedes, Thomas L. Heath, translator.  Dover 2002.
Apollonius.  Conic Sections, Catesby Taliaferro, translator.  Green Lion Press, 1998.
Fibonacci.  Fibonacci’s Liber Abaci, Lawrence Sigler, translator.  Springer 2002.
Kepler, J.  Selections from Kepler’s Astronomia Nova.  Green Lion Press, 2004.
Newton, I.  The Principia: Mathematical Principles of Natural Philosophy, I.B.Cohen, trans..  Univ. California Press, 1999.

Secondary Sources
Bashmakova, I. G. & Smirnova, G. S. The Beginnings and Evolution of Algebra (Dolciani Mathematical Expositions).  MAA, 2000.
Bashmakova, I. G.  Diophantus and Diophantine Equations (Dolciani Mathematical Expositions). MAA, 1998.
Klein, F., Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis.  Dover, 2004.
Klein, F., Elementary mathematics from an advanced standpoint: Geometry.  Dover, 2004.
Lebesgue, H.  “Measure of magnitudes,” in: Measure and the Integral by Henri Lebesgue, Kenneth O. May, editor, Holden-Day, 1966.
Meng, S.K.  New Elementary Mathematics, Syllabus D: books 1, 2, 3A, 3B.  Singapore: Panpac, 2006.
Moise, E.  Elementary geometry from an advanced standpoint, second edition. Addison-Wesley, 1974.
Tarski, A., Introduction to logic and the methodology of the deductive sciences.  Dover, 1995.