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This short article (1996) explains how the textbook Algebra: Themes, Tools, Concepts (by Anita Wah and Henri Picciotto) and the Lab Gear (an algebra manipulative designed by Henri Picciotto) attempt to make the algebra course both more accessible and more mathematically rich.
Functions at the Core
Geometric Connections

Algebra for All?

by Henri Picciotto

Over the past several years, many in the math reform movement have pointed out that algebra plays the role of a "filter". The class is difficult to get into, and difficult to pass. Even students who pass the class often show minimum understanding of the ideas involved. (If you don't believe this, just think of the number of adults you know who got A's and B's in algebra and yet do not remember any of it.)

Since facility with algebra is essential to further work in mathematics, as well as in all the sciences, the algebra course serves a gatekeeper to many college preparatory courses, and therefore to college admissions. The College Board even claims that success in algebra and geometry is the best predictor of graduation from college.

Because of this substantial social responsibility, many educators have argued that we should aim to teach algebra to all students. In fact, the legislatures in a number of states have made passing an algebra course a requirement for high school graduation. Unfortunately, this is easier said than done: attempts to teach the traditional course in the traditional way to a broader population have often led to record numbers of D's and F's.

On the other hand, there are a number of promising ideas and approaches in algebra education, which may yet result in much broader mathematical literacy in our country. By rethinking what concepts should be included in our algebra courses, what themes we can use to increase student motivation, and what tools are available to make the concepts accessible, we are starting to see the possibility for fundamental change. Here are some thoughts on two key components of this reform, and how Algebra: Themes, Tools, Concepts and the Lab Gear fit in.


Functions at the Core

ATTC starts out with many lessons on functions: 1.1-1.2, 1.4, 2.5-2.11, 3.1, 3.4, 3.8, 3.10, and just about all of Chapter 4. This is a lot to do so early on, when you think about the fact that functions were traditionally a topic that was saved for Precalculus, or at least Algebra 2.

Why did we give functions such prominence right at the beginning of the course, and maintained this central role throughout? Because by putting functions at the core of the algebra course, we are reaching many more students, without sacrificing mathematical rigor.

We approach functions in six ways: (a) real world situations; (b) tables of values; (c) Cartesian graphs; (d) symbolic notation; (e) function diagrams; and (f) Lab Gear. The multiplicity of approaches serves two purposes: access, and depth.

Access: Because people learn differently, a given student may be more comfortable with one or another representation, which therefore provides him or her an entry-way into this central concept.

Depth: Understanding how the idea of function appears in all these forms represents a much greater depth of understanding than the mastery of just one or two of them (symbols and Cartesian graphs were the only ones that the traditional course really supported.) In fact, some educational researchers identify the ability to switch representations as a crucial sign of understanding.

Moreover, the use of functions lays the foundation for a different sort of understanding of equations, inequalities, and identities, which can be thought of as statements about functions. For example, (x+1)2=x2+2x+1 can be seen as a statement that these two functions have the same table of values, or the same graph. Or solving (x+1)2=25 can be thought of as finding the values of x that equalize the two functions. This sort of thinking, of course, is well supported by graphing technology -- whether on computers or graphing calculators.


Geometric Connections

The separation of algebra from geometry is an artificial feature of the traditional curriculum, one that does not serve the students well. For one thing, one year is too short a time to learn anything important, whether algebra or geometry. For another, the connections between algebra and geometry enrich both fields, and form the foundation of trigonometry and calculus.

In ATTC, we make many such connections with the help of the geoboard, which we use to throw light on concepts of equivalent fractions, proportional thinking, slope, square root, distance, the manipulation of radicals, and the Pythagorean Theorem. (A listing of all the geoboard lessons appears on page 534 of the Teachers' Edition.)

Other geometric connections are made through topics from recreational math (polyominoes, supertangrams -- see the index), fanciful stories (the flat scientist's jarring discoveries in 4.8) and even word puzzles (1.11).

But the main geometric vehicle is the Lab Gear, which we use to introduce integer arithmetic, the distributive law, and equations and inequalities. With the Lab Gear, variables do not vary, but are manipulated as objects, while we temporarily ignore what the object stands for. This is entirely legitimate, as these two uses of variables coexist in mathematics. In many Lab Gear and geoboard lessons, we are concentrating on issues of mathematical structure, a necessary component of understanding mathematical systems.

Once again, the geometry in ATTC (and the Lab Gear) is intended to provide both access and depth.

Access: visuals and manipulatives increase student participation and interest. They offer support in understanding concepts which were beyond the reach of most students when our only tools were repetition and drill in symbol manipulation.

Depth: students who cannot "see" what is going on with the visual representation of ideas have only a shallow understanding of the symbols, even if they are competent at their manipulation.

To take once again the example of (x+1)2, many students believe this expression is equal to x2+1. With Lab Gear users, it is often enough to mention the Lab Gear representation for them to be able to visualize the correct result of multiplying the binomials. But even the students who find it easy to "get this right" gain by having the visual image, which ties in to powerful and interesting ideas about (for example) dimension.

Of course, the geometric connections do not make the subject matter easy. What they do is provide environments ("microworlds") in which discussion of important ideas is facilitated, both among students, and between student and teachers. As the French say, "de la discussion jaillit la lumière" --from discussion, light emerges. It is our job as teachers to make sure that such discussions occur, and that through interaction with those visual domains, our students become better able to think mathematically.


Other Changes

This article concentrated on two features of a reformed algebra curriculum: putting functions at the core, and making geometric connections. For algebra reform to be most effective, other changes are also needed: we should start earlier and continue longer; we should learn a wide range of instructional and assessment techniques; we should become adept at using technology; and so on. But the function-centered and geometry-rich algebra curriculum can be a big help in the movement to make algebra accessible to all -- without sacrificing mathematical rigor and challenge.

More about Algebra
Visit Henri Picciotto's Math Education Page. Send me e-mail .