Teaching and Learning a New Algebra With Understanding - 1

To discuss the teaching and learning of algebra with understanding, we must first look at the algebra students too often encounter in their classrooms. The traditional image of algebra, based in more than a century of school algebra, is one of simplifying algebraic expressions, solving equations, learning the rules for manipulating symbols the algebra that almost everyone, it seems, loves to hate. The algebra behind this image fails in virtually all the dimensions of understanding  having taken as a starting point for reform in the classroom. School algebra has traditionally been taught and learned as a set of procedures disconnected both from other mathematical knowledge and from students' real worlds. 

Construction of relationships and application of newly acquired knowledge are not at the heart of traditional algebra: The "applications" used are notoriously artificial (e.g., "age problems" and "coin problems"), and students are neither given the opportunity to reflect on their experiences nor the support to articulate their knowledge to others. Instead, they memorize procedures that they know only as operations on strings of symbols, solve artificial problems that bear no meaning to their lives, and are graded not on understanding of the mathematical concepts and reasoning involved, but on their ability to produce the right symbol string—answers about which they have no reason to reflect and that they found (or as likely guessed) using strategies

they have no need to articulate. Worst of all, their experiences in algebra too often drive them away from mathematics before they have experienced not only their own ability to construct mathematical knowledge and to make it their own, but, more importantly, to understand its importance–and usefulness–to their own lives.

Although algebra has historically served as a gateway to higher mathematics, the gateway has been closed for many students in the United States, who are shunted into academic and career dead ends as a result. And even for those students who manage to pass through the gateway, algebra has been experienced as an unpleasant, even alienating event - mostly about  manipulating symbols that don't stand for anything. On the other hand, algebraic reasoning in its many forms, and the use of algebraic representations such as graphs, tables, spreadsheets and traditional formulas, are among the most powerful intellectual tools that our civilization has developed. Without some form of symbolic algebra, there could be no higher mathematics and no quantitative science, hence no technology and modern life as we know them. 

Our challenge then is to find ways to make the power of algebra (indeed, all mathematics) available to all students—to find ways of teaching that create classroom environments that allow students to learn with understanding. The broad outlines of the needed changes follow from what we already know about algebra teaching and learning:

· begin early (in part, by building on students’ informal knowledge),

· integrate the learning of algebra with the learning of other subject matter (by extending and applying mathematical knowledge),

· include the several different forms of algebraic thinking (by applying mathematical knowledge),

· build on students' naturally occurring linguistic and cognitive powers (encouraging them at the same time to reflect on what they learn and to articulate what they know), and

· encourage active learning (and the construction of relationships) that puts a premium on sense-making and understanding.

Making these changes, however, will not be easy, especially where the new approaches involve new tools, unprecedented applications, populations of students traditionally not targeted to learn algebra, and K-8 teachers traditionally not educated to teach algebra (neither the old algebra nor some new version). Despite these challenges, this chapter suggests a route to deep, long-term algebra reform that begins not with more new-fangled approaches but with the elementary school teachers and the reform efforts that currently exist. This route involves generalizing and expressing that generality using increasingly formal languages, where the

generalizing begins in arithmetic, in modeling situations, in geometry, and in virtually all the mathematics that can or should appear in the elementary grades. Put bluntly, this route involves infusing algebra throughout the mathematics curriculum from the very beginning of school. Although this article is not designed to show this route to teaching for understanding in greater detail, I have chosen to organize the material around the different forms of algebraic reasoning in order to demonstrate how algebra can infuse and enrich most mathematical activity from the early grades onward. These interrelated forms, form a complex composite.  All forms richly interact conceptually as well as in activity. To understand this algebra is to make a rich web of connections. Together, the forms of reasoning and the classroom examples emphasize where we need to go rather than where we are, or have been. I will discuss them in coming articles.