High School Level—Algebra I

Taught by Monica Neagoy
National Science Foundation
Ph.D., University of Maryland

The teaching of algebra in most of today’s classrooms is not significantly different from what it was 50 years ago. Certainly, there have been some attempts to change algebra instruction, such as the "new math" reform movement of the 1960s. But the changes that persist in today’s algebra curricula, as a result of that movement, are more superficial than substantial.

On the other hand, mathematics and its applications have changed spectacularly in the past 50 years. The advent of technology, for example, in both applied and pure mathematics, has changed the way mathematicians, scientists, and social scientists do and use mathematics.

It is high time for the classroom instruction of algebra to reflect some of these changes. This is why, in some exciting, interactive, hands-on algebra classes, we are beginning to see changes in what, how, and in what order mathematics is being taught.

Our Algebra I course is on this cutting edge of mathematics teaching and learning for the many reasons stated below.

The Approach to Algebra in a Technological World

"In a technological world, variables actually vary and functions describe real-world phenomena… Not only does technology suggest an increased 'front stage' role for functions, but it also allows for the dynamic study of families of functions" (National Council of Teachers of Mathematics, 1995, Algebra in a Technological World).

Inspired and informed by research on the teaching and learning of algebra, Dr. Neagoy gives functions a "front stage" role in her Algebra I course design.

After a historical overview of the evolution of algebra, she explores the various families of functions, in a logical and ascending order from linear to quadratic to rational and, finally, to the family of exponential functions.

Each family serves as the building blocks for understanding the following and more advanced family of functions.

The penultimate section addresses systems of equations and inequalities—taken from among the families studied in previous sections—and the final section looks to the future by giving a taste of fascinating fractals and captivating chaos.

Algebra the "Gatekeeper"

Algebra I has a well-established reputation as one of the primary gatekeepers for access to college, and in particular for access to competitive public and private institutions. One of the main reasons for algebra’s longstanding reputation as a gatekeeper is the way in which it functions as a prerequisite to all other college-required mathematics courses. For this reason, it is very important that students acquire a rich foundation for both a conceptual understanding and procedural fluidity that will serve them not only along their journey through algebra, but also beyond.

You will be amazed by Dr. Neagoy’s ability to help her audience tackle complex concepts, deep questions, and rich problems with ease and joy. For example, after taking this course, "functions" will no longer represent merely abstract objects that "pass the vertical line test" for instance, but rather meaningful and powerful tools that can be used in all subsequent mathematics classes as well. Furthermore, "applications of algebra" will no longer be synonymous with those meaningless "age," "coin," "mixture," and "distance-rate-time" word problems but rather with real-world problems that will expand horizons, ease understanding, and stimulate curiosity to learn more.

Moving beyond the abstract, Dr. Neagoy uses historical anecdotes, stories, and myths about mathematicians to humanize their work and the problems they were trying to solve. Concrete models such as prisms, cubes, and disks are employed to help students connect algebraic expressions with the shapes and quantities they describe. Finally, the latest graphing calculators are used throughout this course to illustrate key concepts and enhance student understanding.

The Language of Representations

Many studies have shown that when students are exposed to multiple representations of the concept or topic studied, the resulting understanding is deeper, for rich connections are made among these various forms of representations, thus solidifying a multifaceted mental construct.

Dr. Neagoy constantly travels back and forth among the various "universes" of algebra, which she calls (1) Words, (2) Equations, (3) Numbers, and (4) Graphs:

• Words are used to formulate questions and pose problems. If the use of language is not clear, it can be an impediment to the transition from words to algebraic symbols.
• Equations are the realm of algebraic symbols, often called "number models" in pre-algebra courses. The language of symbols has its own semantics and grammar. In this course much attention is paid to the correct use and rich understanding of symbolic algebra.
• Tables of Numbers in which one column contains the x-values and another the y-values can be used to represent algebraic relationships between two variables. With the use of modern technology, Dr. Neagoy effortlessly illustrates the fluid and beautiful transition from numbers to graphs, and back.
• Two-dimensional Graphs are the visual representations in the Cartesian plane of algebraic relationships between x- and y-variables. With the use of dynamic technology, she shows the students how to "trace" a graph and watch how the change in x affects the change in y, and vice versa.

Dr. Neagoy sometimes even goes beyond these four worlds and uses pictorial or concrete representations to render the problem investigated more "hands-on" and realistic. In her "mathematics laboratory," she uses a variety of 1-, 2-, and 3-D concrete models including string, square tiles, cylinders, cubes, and other polyhedra to bring problems alive. The use of mathematics manipulatives enhances her teaching and makes the learning more exciting.

In short, it is important to note that Dr. Neagoy focuses on the meaningful and related multiple representations of functions, variables, and relationships rather than focusing on the narrow acquisition of skills in manipulating dry symbols stripped of any meaning.

Multiple Audiences

While Algebra I was originally designed to target high school students, many "returning adults" have purchased this course for their own edification. We have received numerous letters and e-mail messages from such customers praising not only the content and approach of the course but also the excitement, passion, and expertise with which Dr. Neagoy infects, injects, and infuses her audience.

So if you are a "returning adult" and feel as if you’ve never really understood algebra—or appreciated its power or utility for that matter and would like to give it another chance, trust the word of your peers and embark on the algebra journey with Dr. Neagoy.

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