Mathematics Learning

ALGEBRA
Mitchell J. Nathan

COMPLEX PROBLEM SOLVING
Alan H. Schoenfeld

GEOMETRY
Vera Kemeny

LEARNING TOOLS
Susanne P. Lajoie
Nancy C. Lavigne

MYTHS, MYSTERIES, AND REALITIES
Michael T. Battista

NUMBER SENSE
Chris Lowber
Teruni Lamberg

NUMERACY AND CULTURE
Yukari Okamoto
Mary E. Brenner
Reagan Curtis

WORD-PROBLEM SOLVING
Lieven Verschaffel
Brian Greer
Erik De Corte

ALGEBRA

Algebraic reasoning is a major development (circa 800), both culturally and individually. Culturally, the invention of algebraic representations in graphical and symbolic form is viewed as central for the advancement of mathematics and science. Algebra provides a succinct notation for recording mathematical relationships and describing computational algorithms and scientific laws. Algebraic reasoning is viewed as a major conceptual advancement beyond arithmetic thinking. For individuals, algebra can serve as a thinking tool, or, more aptly, a toolkit, for describing relationships in terms of unknown quantities and modeling complex and dynamic situations. For many, algebra instruction is the sole source for the formal study of abstract representations and problem solving.

Algebra has also been identified as a societal gatekeeper for further development of mathematical and scientific instruction, and for wide-ranging economic opportunities. Consequently, at the beginning of the twenty-first century, there is a major shift in the United States to move algebra education into the middle and primary grades, and to reconceptualize instruction appropriately for these age groups. The utility of algebra is boundless. However, learning and teaching algebra, particularly algebra word problems, is often viewed as the bane of mathematics education.

What Is Algebra?

Abstract algebra refers to the use of formal mathematical structures and symbols, such as F (X), to represent relations between terms or objects. It includes the operations that operate on those structures, such as the inverse, F-1 (X), and the identity, I (X). An especially important structure is the function, which specifies a one-to-many relationship (or mapping) between an independent variable (the input) and a set of dependent variables (the output).

School-based algebra is most commonly viewed as the generalization of arithmetic to include the use of literal symbols (such as the letters X and Y) and arbitrarily complex symbolic expressions. School algebra also includes the study of functions as well as the construction of abstract formalisms that inductively describe a pattern of instances, predict future instances, and characterize the general form of the pattern (e.g., linear).

Challenges of Learning Algebra

To encompass the multifaceted nature of school-based algebra, new concepts arise that contribute to its learning difficulties. The variable is expanded from being a place-holder (or box) in arithmetic, to representing an unknown value or set of values that stand in relation to (and may covary with) other values and expressions. Symbols that represent variables in algebra must denote the same thing everywhere in a problem, but generally take on new meanings with each new problem.

Detailed analyses of problem solving show that result-unknown problems, such as 25x4+8=?, are solvable by direct application of the arithmetic operators, or by using counting objects to physically model the number sentence; whereas start-unknown problems, such as 25Y+8=108, defy modeling and are considered to be algebraic. Both children and adults exhibit lower levels of performance with start-unknown than with result-unknown problems.

In algebra, the equal sign (=) takes on a relational or structural role, as when two sides of an equation are compared, such as 25Y=108. This is in addition to its operational role in arithmetic where the equal sign signals one to perform a computation, as with 25x4=?. Facility with and among multiple representations, including symbolic, tabular, graphical, and verbal formats, is an important aspect of algebraic reasoning, as is an understanding of the relative utility of each. Each representational format has unique advantages. However, in practice, equations receive far more attention that other representations, such as graphs.

The prototypical activities for school-based algebra are solving symbolic equations and word problems. Equation solving most typically involves applying legal rules of symbolic manipulation to isolate an unknown value (see Figure 1, part a). Legal rules typically entail performing the same symbolic calculations to both sides (such as subtracting 10 from both sides of Equation 1 in step 1) to maintain the relations specified in the original problem.

Novices often perform actions that violate the syntactic, hierarchically nested relationships contained in equations, and so inappropriately change their original meanings. For example, in Figure 1 it is algebraically illegal to combine 10 and 5 in step 1 because the original relation is no longer preserved. However, preconceptions from reading and arithmetic, where processing is done from left to right, can lead students to misstep.

Word-problem solving is the next most common activity in algebra education. Word problems can be in the form of a story (Figure 1, part b), or a word equation (Figure 1, part c). Although based on the same quantitative relations as Equation 1, students perform very differently on these three tasks. A misconception generally held by high school mathematics teachers is that high school students solve Equation 1 more easily than a matched story problem or word equation. Teachers justify this prediction by noting that a student must first write a symbolic equation that models the verbal statement, and that this invites other types of errors. While translation from words to mathematical expressions is error-ridden for novices, high school students typically circumvent this step when permitted. Instead, they use highly reliable informal methods, such as guess-and-test and working backwards, which produce higher levels of performance than equation solving.

FIGURE 1

Curricular and Technological Advances in Algebra Education

Like many algebra teachers, traditional algebra textbooks take a largely symbol precedence view of the development of algebraic reasoning, introducing algebraic concepts through symbolic problem solving, and later applying them to verbal reasoning activities. In contrast, several alternative curricula have recently emerged that begin by eliciting students' invented strategies and representations for describing patterns and data, and developing from these inventions algebraic equations and graphs through a process called progressive formalization. These reform-based curricula typically draw on problem-based learning (PBL), which emphasizes complex, multi-day, collaborative problem solving. Three of these approaches, Mathematics in Context, Connected Mathematics, and The Adventures of Jasper Woodbury, have produced commercially available curricula that cover the major topics in middle grade mathematics, such as geometry and algebra.

Technology has also been effectively wedded to innovative curriculum designs. Graphing calculators have had a profound effect on the teaching of algebra using graphical, tabular, and programming forms. The Algebra Sketchbook supports the relationship between verbal descriptions and graphics. The Animate system helps students to construct situation-based meaning for equations. Jasper uses multimedia to present rich problem contexts and encourage production of Smart Tools, representations that support modeling, analysis, and comparison. The Pump Algebra Tutor provides individualized computer-based instruction by relying on adaptive cognitive models of individual students.