Appendix A:
Excerpts From Different Standards

            The diverse ways that different organizations express expectations for mathematics illustrate a variety of approaches to setting standards. The excerpts that follow illustrate this variety in the particular case of algebra, the core of high school mathematics.

From the National Council of Teachers of Mathematics (1989):
            In grades 9-12, the mathematics curriculum should include the continued study of algebraic concepts and methods so that all students can

• represent situations that involve variable quantities with expressions, equations, inequalities, and matrices;
• use tables and graphs as tools to interpret expressions, equations, and inequalities;
• operate on expressions and matrices, and solve equations and inequalities;
• appreciate the power of mathematical abstraction and symbolism;
• and so that, in addition, college-intending students can
• use matrices to solve linear systems;
• demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations.

From the California Academic Standards Commission (1997):
By the end of Grade 10, all students should be able to:

• Solve linear equations and inequalities with rational coefficients; use the slope-intercept equation of a line (y = mx + b) to model a linear situation and represent the problem in terms of a graph.
• Describe, graph, and solve problems using linear, quadratic, power, exponential, absolute value, polynomial, and rational functions; identify key characteristics of functions (domain, range, intercepts, asymptotes).
• Derive and use the quadratic formula to solve any quadratic equation with real coefficients; graph equations of the conic sections (parabola, ellipse, circle, hyperbola), identifying key features such as intercepts and axes.
• Describe, extend, and find the nth term of arithmetic, geometric, and other regular series.

And in Grades 11-12, mathematics students should learn about:

• Piece-wise defined functions; logarithm function and as inverse of exponential; polar coordinates; parametric equations; recursive formulas, binomial theorem, mathematical induction; trigonometric functions, graphs, identities, key values, and applications; vector decomposition.

From the American Mathematical Association of Two-Year Colleges (1995):
            The study of algebra must focus on modeling real phenomena via mathematical relationships. Students should explore the relationship between abstract variables and concrete applications and develop an intuitive sense of mathematical functions. Within this context, students should develop an understanding of the abstract versions of basic number properties and learn how to apply these properties. Students should develop reasonable facility in simplifying the most common and useful types of algebraic expressions, recognizing equivalent expressions and equations, and understanding and applying principles for solving simple equations.
            Rote algebraic manipulations and step-by-step algorithms, which have received central attention in traditional algebra courses, are not the main focus. Topics such as specialized factoring techniques and complicated operations with rational and radical expressions should be eliminated. The inclusion of such topics has been justified on the basis that they would be needed later in calculus. This argument lacks validity in view of the reforms taking place in calculus and the mathematics being used in the workplace.

From the Secretary's Commission on Achieving Necessary Skills (1991):
            Mathematics. Approaches practical problems by choosing appropriately from a variety of mathematical techniques; uses quantitative data to construct logical explanations for real world situations; expresses mathematical ideas and concepts orally and in writing; and understands the role of chance in the occurrence and prediction of events.
            Reasoning. Discovers a rule or principle underlying the relationship between two or more objects and applies it in solving a problem. For example, uses logic to draw conclusions from available information, extracts rules or principles from a set of objects or written text, applies rules and principles to a new situation, or determines which conclusions are correct when given a set of facts and a set of conclusions.