## Finding Common Ground, Indianapolis, March 2006 Algebra Group Report

Working Group
David Bressoud, Macalester College
John Carter, Adlai E. Stevenson High School, IL
Susan Forman, Bronx Community College
Ira Papick, University of Missouri
Alan Tucker, SUNY Stony Brook
Hung-Hsi Wu, University of California-Berkeley

#### Preface

The universal value of algebra lies in its use as a tool for problem solving, strengthening reasoning skills, and interpreting functional relationships. A good foundation in algebra is a prerequisite to success in the study of mathematics, science, engineering, and the social sciences. Mathematics becomes more meaningful to students when they see the connection to their lives, prior learning experiences, or future learning.

#### Mathematics Learning Prior to Algebra

It is important that elementary and middle school students be appropriately prepared to take Algebra I in Grade 8 or Grade 9. Experience with symbols and generalizations should increase  in the years preceding Algebra I. (There was disagreement as to when students should take Algebra I.)

#### Elementary and Middle School Mathematics

• Reinforce the understanding of whole number arithmetic through mathematical explanations of algorithms, i.e., use of the commutative, associative, and distributive properties.
• Gradually incorporate the use of generality and symbols. One natural way to do this is in the development of rational numbers and operations on them.
• Introduce linear relations, linear functions, and basic graphing techniques.
• Provide students with problems that exemplify the power of generalization. For example, 18 is divisible by 3 and 27 is divisible by 3 is (18 + 27) divisible by 3 Evaluate (43 x 63) + (57 x 63).
• Use well-structured patterns to help build students’ capacity for generalization. Students should be able to explain the process they used to arrive at a generalization. (In process.)

#### Algebra I

Algebra begins as generalized arithmetic. In algebra classes, students learn to handle the concept of generality and to use symbols to translate verbal information into symbolic language. In our search for common ground, there was agreement that Algebra I should contain: solving linear equations, linear functions and their graphs, systems of linear equations, quadratic equations and their graphs, solving general quadratic equations, operations on polynomials, and an introduction to exponential growth. Other topics include rate of change and linear inequalities.

#### Algebra II

Students should complete the equivalent of Algebra I and Algebra II by the end of 12th grade. There was agreement that Algebra II primarily comprises an extension of topics in Algebra I, complex numbers, in-depth study of functions including composition and inverses, rational functions and the asymptotes of their graphs, radical functions, logarithmic and exponential functions and their graphs, and polynomial functions.

#### Alternate Algebra II

The group considered an alternative to the Algebra II course described above. Suggested topics for inclusion in this alternative course are in-depth study of functions, logarithmic and exponential functions, linear programming, data analysis and modeling, and topics in discrete mathematics such as recurrence relations, permutations and combinations, and matrices.

#### Finally…

Due to the fact that algebra requires students to think at a higher level of abstraction and generality, due attention should be paid to precise definitions, logical reasoning, and mathematical closure.