# Essay: Representation in Algebra: A Problem Solving Approach

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Representation in Algebra: A Problem Solving Approach

The need for a solid background in mathematics for high school and college students in the 21st century is well documented (Katz & Barton 2007). A number of emerging career fields in the Age of Information are directly related to mathematical knowledge. For instance, Conaway and Rennolds emphasize that the "With the onset of the technological age, students must complete as much math as possible during high school. Mathematical skills are essential to gaining access to college and pursuing a career in a math-, science-, or technologically related field. A student's mathematical ability can be a determining factor in choosing a career" (2003, p. 218). Indeed, the ten fastest-growing career fields in the United States today include five fields that are specifically focused on mathematics knowledge, including computer engineers, computer support specialists, systems analysts, database administrators, and desktop publishing specialists (Conaway & Reynolds 2003). Clearly, young learners who continue their pursuit of higher mathematics throughout their high school and college years will enjoy additional career opportunities and a competitive advantage over those who do not (Conaway & Reynolds, 2003). Experienced mathematics teachers, though, can verify that engaging young people in the learning process can be a challenging endeavor under the best circumstances, and the effort involved in acquiring a solid foundation in algebra and other higher mathematics can likewise be a challenging experience for many young learners as well. In some cases, some young people become so discouraged and frustrated with their lack of progress at learning the concepts that are involved that they withdraw from the algebra curriculum entirely, never to return.

There are some steps that teachers can take, though, to assist young algebra learners overcome this frustration by helping them make the mental connection between the representational aspects of algebraic equations and what they mean. For example, Juter (2003) points out that there is a point at which students are able to make the mental leap required to make the logical connections that are needed for a thorough understanding of what is involved rather than just the mechanics. In this regard, Juter notes that teaching methods that have concentrated on plausible reasoning have largely failed to assist students in achieving this mental leap; however, there are some innovative approaches such as conceptual knowledge that have been shown to facilitate this process in young learners. According to Juter, "Conceptual knowledge has an emphasis on relations. The items of a notion are connected through relations and together they form a mental web. A part of conceptual knowledge cannot be thought of as a disjointed piece of information. Conceptual knowledge develops via construction of relations between items" (2003, p. 17). The key to success appears to be directly related to helping students understand the connection between the relational aspects involved in algebra in ways that are relevant to what they already know. For instance, Juter adds that, "The items can be other relations or concepts where the connection can be between two (or more) items that are existing already in the mind or between a new and an existing item. When this connection is created, the result often becomes more than its parts jointly. Parts with no prior relations become connected and suddenly more things fit together" (2003, p. 17). While these mental leaps have already been made by the experienced algebra teacher and are therefore largely taken for granted, helping young people do the same thing is a much more challenging enterprise and these issues are discussed further below.

Findings from Existing Literature

Unfortunately, the research is consistent in showing that there is no "one-size-fits-all" approach that is best suited for facilitating the mental leap needed to understand representation in algebra, and effective teaching methods require techniques that draw on students' individual strengths in problem solving operations. It is also important to encourage young learners to overcome a mere rote memory and functional approach to performing such algebraic problem solving by helping them see the "big picture" that is involved by helping them relate the underlying mathematical concepts to what they have learned in the past. In this regard, Osta and Labban report that, "Existing research on students' abilities to model and solve problems using algebra focused mainly on interpreting symbols, formulating and solving equations, constructing and interpreting graphic representations" (2008, p. 2).

When they are initially confronted with letters instead of numbers, students learning algebra may find the transition from the mathematics they have been learning so different that they cannot achieve the mental leap needed to associate these representational elements with what they already know. As Osta and Labban point out, "Most research conducted on the subject tried to analyze the difficulties that students face in understanding the meanings of the unknown and the equality sign. These difficulties are mostly identified when algebra is taught as an independent isolated course, causing a sudden shift from arithmetic, and when it is seen as a set of formal procedures and rules to be memorized and applied in a rote fashion" (2008, p. 2). Semantics, then, can also play an important role as one teacher's use of descriptors and narrative will influence student perceptions of these concepts (Conway & Reynolds 2003). The results of the study by these researchers indicate that making these vital connections can provide young learners with a more comprehensive understanding of semantic as well as the procedural knowledge required to achieve successful academic outcomes so that more young students will pursue higher mathematics coursework in their college and professional careers (Conway & Reynolds 2003).

According to Tall and Vinner, there are also some fundamental challenges involved in even communicating the definition of a concept when teaching algebra to young people: "We shall regard the concept definition to be a form of words used to specify that concept. It may be learnt by an individual in a rote fashion or more meaningfully learnt and related to a greater or lesser degree to the concept as a whole" (1981, p. 152). Although rote memory and drill has its place in algebra instruction, there is also a need for other problem-solving techniques that allow students to formulate their own conceptualizations of what is required to solve an equation. For example, Tall and Vinner add that, "It may also be a personal reconstruction by the student of a definition. It is then the form of words that the student uses for his own explanation of his (evoked) concept image. Whether the concept definition is given to him or constructed by himself, he may vary it from time to time" (p. 152). Likewise, according to Mason, Graham and Johnson-Wilder (2005), generalization activity in algebra instruction has its foundation in the use of algebraic notation as a tool that can be used for expressing proofs and helping students visualize the representational aspects that are involved in solving problems. It is important to note that these alternative personal constructions may be just as valid and effective as the formal methods being used by the teacher, emphasizing the fact that different students learn in different ways. According to Tall and Vinner, "In this way a personal concept definition can differ from a formal concept definition, the latter being a concept definition which is accepted by the mathematical community at large. For each individual a concept definition generates its own concept image (which might, in a flight of fancy be called the 'concept definition image'). This is, of course, part of the concept image" (p. 152).

According to Williams (1991), "The limit concept has long been considered fundamental to an understanding of calculus and real analysis, but recent studies have confirmed that a complete understanding of the limit concept among students is comparatively rare. Conceptions of limit are often confounded by issues of whether a function can reach its limits, whether a limit is actually a bound, whether limits are dynamic processes or static objects, and whether limits are inherently tied to motion concepts" (p. 219). According to Lauten, Graham and Ferrini-Mundy (1994), "Several researchers in mathematics education have been interested in aspects of secondary college students' understanding of function. Collected research in this area has shown that most secondary and post secondary school students are tied to a definition of a function being represented by a rule of correspondence, unvarying over its entire domain. Piecewise defined functions present great difficulty, particularly in moving from the graphical to algebraic mode of thinking" (p. 227).

Based on their cross-curricular structured-probe task-based clinical interviews of 44 pairs of third-year high-school mathematics students, the majority of whom were high achievers, to identify their problem-solving approaches for various algebraic linear equations (of the form ax ± b = cx ± d), Huntley, Marcus, Kahan and Miller (2007) found that most pairs of students were able to solve the equation resulting in a unique solution using symbol-manipulation algorithms or through the use of a graphical approach that made the algebraic representations that were involved more concrete and apparent. These researchers add that, "On…
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