In February 1994, NCTM's Board of Directors passed a position statement regarding Algebra for Everyone. In this statement it was indicated that all secondary school students should have the opportunity to learn the basic ideas and methods of algebra. It also indicated that the assumption that the algebraic skills formerly taught to a narrow range of students would serve as a passport to jobs and further educational opportunities in todays world, was simply not true. Many first year algebra courses in their present form suffer from several major weaknesses:

• they advance only a narrow range of by-hand skills for transforming and simplifying symbolic expressions and solving equations and inequalities;
• they are often removed from natural context, thus the algebraic understandings that are cultivated are far removed from the needs of the learner and employer;
• as a separate course, they effectively isolate the concepts and methods of algebra from other major strands of school mathematics;
• they neither acknowledge nor encourage the development of informal understanding of algebraic ideas in grades K - 8.

In particular, the NCTM Standards indicate that ALL students need to develop reasoning and problem-solving skills built upon exploration, modeling, describing/conjecturing, explaining and generalizing. All of these, we feel are basic to algebraic reasoning. These concepts are the knowledge-base upon which many future employers base decisions to hire and more advanced mathematics courses depend for studying the formal structures of their theoretical bases.

Thus, the first step toward algebra for everyone is to reconceptualize the current vision of algebra and how and when these reconceptualized concepts and skills can be internalized by the students. The responsibility of formal implementation of this reconceptualization begins with the kindergarten teacher and extends past the identified algebra teacher. Analysis of the Standards reveals that early development of algebraic thinking is the study of patterns and relationships which represent major components of the revised K - 4 curriculum. This theme is carried on in a more formal manner in the 5 - 8 recommendations of the Standards, and finally reaches maturity in the 9 - 12 standards of Algebra and Functions. This analysis reveals two major shifts in emphasis as we reconceptualize algebra: 1) a shift from algebra as a collection of manipulative skills to algebra as a means of representation; and 2) a shift from algebra as a separate course or courses to a content strand that is strengthened each year by braiding it with statistics, geometry and discrete mathematics.

As a means of representation, algebra provides a language that uses verbal, tabular, graphical, and symbolic forms to model, and help answer questions about quantitative patterns and relationships. These patterns and relationships often arise in contexts involving collection and analysis of data, shape of data, change of data and optimization which are the real substance of the workplace. It is dealing with these real applications that employers suggest will provide employment advantages for the upcoming workforce.

The emergence of hand-held graphing calculators, symbolic manipulators, and computers has provided powerful new visual, numeric, and symbolic approaches to traditional tasks of algebra. These tools must be an integral part of the learning opportunities afforded students. Access to and utilization of these technologies for graphically representing quantitative relationships have shifted the main purpose of algebra from fine-tuning techniques for by-hand symbolic manipulation. This change was cited in Algebra in a Technological World from NCTM, Although some of the attention now paid to symbolic representations will be rededicated to developing symbol sense, most class time will be spent in helping students develop a sense for how algebra can be used to explain the world around them.1

This publication also indicated the following:

• The language of technology quite naturally depends on the concepts of variable and function. But the concepts of variable and function in a technological world are much richer than those found in current school textbooks or in the minds of todays students. The search for variable values that satisfy equations need no longer be the unquestioned and primary goal of beginning algebra. Functions are no longer merely abstract objects that pass the vertical line test. In a technological world, variables actually vary and functions describe real-world phenomena. Variables represent quantities that change, and algebra is the study of relationships among these changing quantities. What was the search for fixed values that fit statistically defined relationships is now the dynamic exploration of mathematical relationships.
• Technology allows students to study algebra as meaningful and related representations of functions, variables, and relations rather than as the acquisition of skills in manipulating symbolic representations stripped of other meaning.1
• Thus, we recommend that the reconceptualization of the algebra strand in the school curriculum should follow three general principles regarding goals and teaching approaches:
• When algebraic concepts are reserved for an Algebra I course students are often frightened and bewildered by the sudden appearance of symbolism, obscurity, and abstractness. Threading the concepts of algebraic reasoning appropriately throughout the K - 12 program softens the impact of uniqueness and strengthens the understanding of algebra as a mechanism for modeling and solving real applications.
• The primary role of algebra at the school level is to develop confidence and facility in using variables and functions to model numerical patterns and quantitative relations and to analyze data in order to solve problems both within pure mathematics and in a broad range of settings in which numerical data are important.
• The use of graphing calculators and computers makes the focus on modeling and functions attractive and accessible for students across a broad range of interests, aptitudes and prior achievement. The use of these calculating tools offer students a variety of powerful new learning and problem-solving strategies.2

These principles will permit the algebra strand to concentrate on developing students ability to recognize, represent, and solve problems involving the type of variables and functions that occur most often in quantitative models.

The heart of NCTM's vision of a quality mathematics program is for all students to experience learning which embrace five general student oriented goals: (1) help students learn to value mathematics; (2) increase students confidence in their ability to do mathematics; (3) help students develop problem solving abilities; (4) increase students abilities to communicate mathematically; and (5) improve students abilities to reason mathematically.3 The learning opportunities presented for the students during their involvement with the algebra strand should contribute toward each students development of mathematical power as defined by NCTM. This requires that each student grow throughout each school year in the ability to perform effectively and efficiently in a variety of real-world settings. These settings should be embedded in a context that is both realistic and significant.

In submitting this report, the committee acknowledges the local autonomy of school districts in course/program development. We also suggest that simply mandating a conventional algebra course for all students is not the solution to the challenge. The position stated in this document is founded upon numerous recent publications, information gained through conferences, meeting with representatives of business and industry, and the professional experiences of the committee members.

Structuring the Program

It is recommended that students who complete the algebra program available in basic education should demonstrate functional knowledge of the following standards:

Primary Level

• Use concrete objects and symbols to model the concepts of variables, expressions, equations, and inequalities.
• Substitute a missing addend in a number sentence.
• Create a story to match a given combination of symbols and numbers.
• Use concrete objects, trial and error, and technology to solve number sentences and check if solutions are sensible and accurate.
• Explain the meaning of solutions and symbols.
• Recognize, describe, extend, create and replicate a variety of patterns including attribute, activity, number and geometric patterns.
• Gather information and display it in the form of a table or a chart.
• Describe and interpret the data shown in tables and charts.
• Demonstrate simple function rules using input and output machines.
• Analyze simple functions and relationships and locate points on a simple grid.

Intermediate Level

• Use concrete objects (manipulatives) and combinations of symbols and numbers to create expressions that model mathematical situations
• Explain the meaning of combinations of symbols and numbers in expressions, equations, and inequalities.
• Write a story to match given equations or inequalities.
• Select and use appropriate strategies to solve number sentences and explain the method of solution.
• Recognize, reproduce, extend, create and describe patterns, sequences and relationships verbally, numerically, symbolically, and graphically, using a variety of materials.
• Connect patterns to geometric relations and basic number skills.
• Form rules based on patterns.
• Generate functions from tables of data and relate data to corresponding graphs and functions.
• Locate and identify points on a coordinate system.

Middle Level

• Create expressions, equations, or inequalities that model problem situations
• Interpret expressions, equations and inequalities
• Use concrete objects to model algebraic concepts
• Select and use a strategy to solve an equation or inequality, explain the solution, and check the solution for accuracy.
• Solve and graph equations and inequalities using scientific and graphing calculators and computer spreadsheets.
• Discover, describe, and generalize patterns, including linear, exponential, and simple quadratic relationships.
• Connect simple algebraic patterns to basic number theory and to spatial relations.
• Use patterns and function concepts to solve routine and non-routine problems.
• Represent relationships with tables, graphs in the coordinate plane, and verbal or symbolic rules.
• Graph a linear function from a rule or table
• Generate a table or graph from a function and use graphing calculators and computer spreadsheets to graph and analyze functions.
• Show that an equality relationship between two quantities remains the same as long as the same change is made to both quantities; and explain how a change in one quantity determines another quantity in a functional relationship.

Secondary Level

• Formulate expressions, equations, inequalities, systems of equations, systems of inequalities, and matrices to model routine and non-routine problem situations
• Analyze and explain systems of equations, systems of inequalities and matrices
• Select and use an appropriate strategy to solve systems of equations and inequalities using graphing calculators, symbol manipulators, spreadsheets, and other software
• Use matrices to organize and manipulate data, including matrix addition, subtraction, multiplication, and scalar multiplication.
• Identify whether systems of equations and inequalities are consistent or inconsistent.
• Use equations to represent curves such as lines, circles, ellipses, parabolas, and hyperbolas.
• Demonstrate the connection between algebraic equations and inequalities and the geometry of relations in the coordinate plane.
• Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically.
• Give examples of patterns that occur in data from other disciplines.
• Use patterns, sequences, and series to solve routine and nonroutine problems
• Create and interpret functional models
• Analyze properties and relationships of functions (linear, polynomial, rational, trigonometric, exponential, and logarithmic).
• Analyze and categorize functions by their characteristics
• Solve linear, quadratic, and exponential equations both symbolically and graphically.
• Determine the domain and range of a relation, given a graph or a set of ordered pairs.
• Analyze a relation to determine whether a direct or inverse variation exists and represent it algebraically and graphically.
• Represent functional relationships in tables, charts, and graphs.
• Select, justify, and apply an appropriate technique to graph a linear function in two variables, including slope-intercept, x- and y- intercepts, graphing by transformations, and the use of a graphing calculator.
• Write an equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.
• Given a set of data points, write an equation for a line of best fit.

Implementation of an Algebra program for ALL students requires grappling with the many dimensions of this complex problem. Simplistic quick-fix solutions must be avoided to ensure that quality experiences are made available to these students. It is essential that students learn algebra as a way of thinking as well as a set of competencies involving the representation of quantitative relationships. Connections among mathematical strands of the total school curriculum permit insights developed in one strand to infuse into others. Multiple strands with strong interconnections developed throughout the students educational program open gateways for true mathematical power and heightened enthusiasm in the students. Programs based upon the standards cited, reflecting the philosophy outlined and effectively utilizing the available technology will be the corridors to these gateways. There are many individuals contributing to a quality algebra experience, but TEACHERS and CURRICULUM COORDINATORS must take a leadership role.

1 Heid, M. Kathleen. Algebra in a Technological World. Reston, VA.: National Council of Teacher of Mathematics, 1995.

2 NCTM. News Bulletin. Reston, VA.: National Council of Teacher of Mathematics. May, 1995.

3 National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: The Council, 1989.

4 Steen, L. A. Does Everybody Need to Study Algebra? The Mathematics Teacher. 85 (April 1992): 258 - 260.

5 Academic Standards, Commonwealth of Pennsylvania, 1997.

6 The National Council of Teachers of Mathematics publications:

• Mathematics Teacher, February 1997
• Mathematics Teaching in the Middle School, February 1997
• Teaching Children Mathematics, February 1997