The goal of “algebra for all” has been in place in this country for more than a decade, driven by the need for quantitatively literate citizens and a recognition that algebra is a gatekeeper to more advanced mathematics and opportunities (Silver, 1997; Dudley, 1997). To accomplish this goal, many states, including California, have established algebra as its grade level course for eighth graders (California Board of Education, 1997) . Unfortunately, the data clearly show that all students are not succeeding in algebra in the eighth grade. For example, in 2006 only 22% of California’s eighth graders demonstrated proficiency in a course equivalent to algebra or higher (Kriegler & Lee, 2007). The implication is clear: elementary and middle school mathematics instruction must focus greater attention on preparing all students for challenging middle and high school mathematics programs that include algebra (Chambers, 1994; Silver, 1997). Thus, “algebraic thinking” has become a catch-all phrase for the mathematics teaching and learning that will prepare students with the critical thinking skills needed to fully participate in our democratic society and for successful experiences in algebra.
In this article, algebraic thinking is organized into two major components: the development of mathematical thinking tools and the study of fundamental algebraic ideas. Mathematical thinking tools are analytical habits of mind. They are organized around three topics: problem- solving skills, representation skills, and quantitative reasoning skills. Fundamental algebraic ideas represent the content domain in which mathematical thinking tools develop. They are explored through three lenses: algebra as generalized arithmetic, algebra as a language, and algebra as a tool for functions and mathematical modeling.
Within the algebraic thinking framework outlined here, it is easy to understand why lively discussions occur within the mathematics community regarding what mathematics should be taught and how. Those who argue that the study of mathematics is important because it helps to develop logical processes may consider mathematical thinking tools as the more critical component of mathematics instruction. On the other hand, those who express concern about the lack of content and rigor within the discipline itself may be focusing greater emphasis on the algebraic ideas themselves. In reality, both are important. One can hardly imagine thinking logically (mathematical thinking tools) with nothing to think about (algebraic ideas). On the other hand, algebra skills that are not understood or connected in logical ways by the learner remain “factoids” of information that are unlikely to increase true mathematical competence.
Mathematical thinking tools are organized here into three general categories: problem-solving skills, representation skills, and reasoning skills. These thinking tools are essential in many subject areas, including mathematics; and quantitatively literate citizens utilize them on a regular basis in the workplace and as part of daily living.
Problem-solving requires having the mathematical tools to figure out what to do when one does not know what to do. Students who have a toolkit of problem-solving strategies (e.g., guess and check, make a list, work backwards, use a model, solve a simpler problem, etc.) are better able to get started on a problem, attack the problem, and figure out what to do with it. Furthermore, since the real world does not include an answer key, exploring mathematical problems using multiple approaches or devising mathematical problems that have multiple solutions gives students opportunities to develop good problem-solving skills and experience the utility of mathematics.
The ability to make connections among multiple representations of mathematical information gives students quantitative communication tools. Mathematical relationships can be displayed in many forms including visually (i.e. diagrams, pictures, or graphs), numerically (i.e. tables, lists, with computations), symbolically, and verbally1. Often a good mathematical explanation includes several of these representations because each one contributes to the understanding of the ideas presented. The ability to create, interpret, and translate among representations gives students powerful tools for mathematical thinking.
The quantitative reasoning is fundamental to success in mathematics, and algebraic thinking helps develop quantitative reasoning within an algebraic framework (Kieran and Chalouh, 1993). Since applications of mathematics rarely require making calculations on “naked numbers,” analyzing problems to extract and quantify relevant information is an essential reasoning skill. Inductive reasoning involves examining particular cases, identifying patterns and relationships among those cases, and extending the patterns and relationships. Deductive reasoning involves drawing conclusions by examining a problem’s structure. Quantitatively literate citizens routinely utilize these types of reasoning on a regular basis.
The line between the study of informal algebraic ideas and formal algebra is often blurred, and the algebra ideas identified here are intended to be studied in concrete or familiar contexts so that students will develop a strong conceptual base for later abstract study of mathematics. In this framework, algebraic ideas are viewed in three ways: algebra as generalized arithmetic, algebra as a language, and algebra as a tool for functions and mathematical modeling.
Algebra is sometimes referred to as generalized arithmetic; therefore, it is essential that instruction give students opportunities to make sense of general procedures performed on numbers and quantities (Battista and Van Auken Borrow, 1998; Vance, 1998). According to Battista, thinking about numerical procedures should begin in the elementary grades and continue until students can eventually express and reflect on procedures using algebraic symbol manipulation (1998). By routinely encouraging conceptual approaches when studying arithmetic procedures, students will develop a network of mathematical structures to draw upon when they begin their study of formal algebra. Here are three examples:
• Elementary school children typically learn to multiply whole numbers using the “U.S. Standard Algorithm.” This procedure is efficient, but the algorithm easily obscures important mathematical connections, such as the role of the distributive property in multiplication or how area and multiplication are connected. These require attention as well.
• The “means-extremes” procedure for solving proportions provides middle school students with an easy-to-learn rule, but does little to help them understand the role of the multiplication property of equality in solving equations or develop sense-making notions about proportionality. These ideas are essential to the study of algebra, and attention to their conceptual development will ease the transition to a more formal study of the subject.
• The widely accepted distance from the earth to the sun is estimated at 93 million miles, but establishing some referants for the meaning of the magnitude of 93,000,000 requires manipulation of ratios and rates and a well-developed generalized number sense.
Algebra is a language (Usiskin, 1997). To comprehend this language, one must understand the concept of a variable and variable expressions, and the meanings of solutions. It involves appropriate use of the properties of the number system. It requires the ability to read, write, and manipulate both numbers and symbolic representations in formulas, expressions, equations, and inequalities. In short, being fluent in the language of algebra requires understanding the meaning of its vocabulary (i.e. symbols and variables) and flexibility to use its grammar rules (i.e. mathematical properties and conventions). Historically, beginning algebra courses have emphasized this view of algebra. Here are two examples:
• How to interpret symbols or numbers that are written next to each other can be problematic for students. In our number system, the symbol “149” means “one hundred forty-nine.” However, in the language of algebra, the expression “14x” means “multiply fourteen by ‘x.’” Furthermore, x14 = 14x, but “14x” is the preferred expression because, by convention, we write the numeral or “coefficient” first.
• The variables used in algebra take on different meanings, depending on context. For example, in the equation 3+x = 7, “x” is an unknown, and “4” is the solution to the equation. But in the statement A(x+y) = Ax+Ay, the “x” is being used to generalize a pattern.
Finally, algebra can be viewed as a tool for functions and mathematical modeling. Through this lens, algebraic thinking shows students the real-life uses and relevance of algebra (Herbert and Brown, 1997). Seeking, expressing, and generalizing patterns and rules in real world contexts; representing mathematical ideas using equations, tables, and graphs; working with input and output patterns; and developing coordinate graphing techniques are mathematical activities that build algebra-related skills. Functions and mathematical modeling represent contexts for the application of these algebraic ideas.
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