What We Know

Because algebra includes many concepts, it may be easier to define as a set of thinking abilities. Driscoll (1999) proposes three critical thinking habits: doing-undoing (reversibility and working backwards), building rules to represent functions (recognizing patterns and organizing data) and abstracting from computation (computation separate from particular numbers).

Initially, it is the concept of a variable that allows students to transition from arithmetic to algebra (Schoenfield & Arcavi, 1988). Studies have shown, though, that students have difficulty with variables (Kieran, 1983; Wagner & Parker, 1993 in Principles and Standards for School Mathematics, 2000). To develop this understanding, students must first have a strong sense of numbers and operations. Students must have insight into how operations work in order to make decisions that are independent of particular numbers (Driscoll, 1999). Students need to understand different ways that variables can be used [e.g., it can represent a number (2x + 5 = 9), a formula (A = lw) or covariation (a = 2b) (See Principles and Standards for School Mathematics, 2000, p. 225].

In addition, many students lack understanding of the structural aspects of algebra. Educators need to develop students' ability to move between procedural and structural conceptions of problems and to be able to identify the perspective most useful for the task at hand (Kieran, 1992).

Students must also learn algebra as a way of thinking that involves the formalization of patterns, functions and generalizations (Silver, 1997). Working with patterns can help students to understand functional relationships (Schwartz & Yerushalmy, 1992). It is important that students work with varied representations of function, including numeric, graphic and symbolic representations (Leinhardt, Zaslavsky, & Stein, 1990; Moschkovich, Schoenfeld, & Arcavi, 1993; NRC, 1998 in Principles and Standards for School Mathematics, 2000).

One challenge for high school algebra teachers is meeting the needs of students whose abilities fall below course expectancies. Choike (2000) suggests these strategies:

• emphasize the "big ideas" or conceptual understandings of algebra;
• start by using "friendly numbers" such as multiples of 5 and 10 that will be easier for students to manipulate;
• analyze word problems for appropriate reading level and content;
• use multiple representations (words, tables, graphs and symbols) to help students see the relationships between these representations;
• use and reuse familiar settings and high-interest contexts for problems.