Representation in Algebra: A Problem Essay

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2007, p. 115). Likewise, a study by Wyndhamm and Saljo found that young algebra learners were more successful in their problem-solving efforts when collaborating in a group environment. According to these researchers, "An experiment involving 14 small groups of Swedish students (usually 3 per group) aged 10, 11, and 12 years shows that these students acting in groups and creating shared contextualizations were able to solve mathematics word problems calling for real-world knowledge. Research has shown students acting alone to have difficulty with the same types of problems" (Wyndhamm & Saljo 1997, p. 361). Other teachers report that algebra story problems can help make learning more relevant to young people's lives. For instance, according to Homann and Lulay, "Algebra story problems are an important practical application of mathematics since real-world problems usually do not arise in terms of equations but as verbal or pictorial representations. The problems are solved by understanding, abstraction, and transformation of these representations into symbolic equational forms which can be solved by algebraic algorithms" (1996, p. 1). Likewise, Laughbaum makes the point that, "Our students see relationships in their lives, but do not know that the study of functions is the tool for analyzing and understanding them. What our students must be taught is to recognize and understand these mathematical relationships in the world they live in now, and will live in as adults" (2003, p. 64). Even here, though, there are some constraints to learning. For example, Dillon and Sternberg emphasize that, "Problem solving involves building a representation of the words of the problem and finding the solution of the problem using the rules of algebra. A major difficulty in students' performance on word problems seems to involve representation of the problem, i.e., moving from the words in the problem to a coherent mental representation of the problem. One major subcomponent in the representation process for word problems in the translation of each sentence" (1986, p. 145).

Critical Evaluation from Own Experience

The argument has been made that some subjects, such as Shakespeare, should not be taught until students reach college because they do not possess the requisite maturity, life experience and interest that are needed to pursue them. The same argument can be made for teaching algebra at the secondary level, of course, but these arguments are misguided and do young learners a disservice. According to Stacey and MacGregor, "Algebra is hard to teach and hard to learn. [However], with commitment it is possible to teach a large proportion of the school population" (1999, p. 58). Therefore, when teachers take the time to explain the fundamentals that are involved in representation in algebra, most students are able to overcome their initial fear of the unknown and make the mental leap that is needed to understand how linear equations operate. In this regard, Staszkow suggests that teachers should seek to eliminate the mystery involved and just explain to students that, "To understand what algebra is all about, you must realize that, in algebra, letters are used to stand for numbers. Just as you operated with numbers in arithmetic, in algebra you simply replace those numbers with letters and work with them" (1986, p. 327). These types of elementary explanations that introduce the fundamental representational concepts that are involved in algebra will likely go a long way in reducing the initial anxiety that can result from being introduced to algebraic concepts that may appear to be so much arcane and unattainable mumbo-jumbo to young learners (Russell & O'Dwyer 2009). As Stacey and MacGregor point out, "Outside the algebra sections of their textbooks, students rarely see algebraic letters used except in formulas or as labels indicating the quantity to be found in diagrams or formulas. Their exercises almost always have numerical (rather than algebraic) answers" (1999, p. 58).

Indeed, some students appear to mirror the adverse reaction to being presented with learning algebra as being a form of severe punishment in the same fashion that humorist Dave Barry did when Sputnik was launched by the Soviet Union in 1957 and his mathematics teacher told his class that, "We would have to learn a LOT more math, as if it was our fault" (1989, p. 139). By helping young learners understand that algebra is not in fact a type of "punishment" and that the rules involved in solving algebraic problems are readily accessible and understandable with some effort, the first step to achieving the mental leap needed to successfully recognize the representational elements involved in algebra will have been made.

Certainly, while it is important to stress the "what's-in-it-for-me" aspects of learning algebra to students, this importance may not be readily appreciated by young learners who may not care a whit about learning algebra just because an adult says it is important for them to do so. A number of valuable goals and outcomes have been advanced in recent years in support of teaching algebra, including the following:

1.

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To develop student skills in the solution of equations, finding numbers that meet specified conditions;

2. To teach students to use symbols to help solve real problems, such as mixture problems, rate problems, and so forth;

3. To prepare students to follow derivations in other subjects, for example, in physics and engineering; and,

4. To enable students to become sufficiently at ease with algebraic formulas that they can read popular scientific literature intelligently (Wagner & Kieran 1999, p. 12).

Therefore, by making the instructional material relevant to their lives and by drawing on what they already know, though, algebra teachers at all levels of instruction can facilitate the learning process even if students do not appreciate how important the subject matter may be to them in their later lives and professional career pursuits. For example, Stacey and MacGregor report that, "Ideas essential for learning algebra have a place in the primary curriculum, but only in secondary school do students begin formal algebra, which for us is signified. This late introduction reflects the special role of algebra as a gateway to higher mathematics. Algebra is the language of higher mathematics and is also a set of methods to solve problems encountered in professional, rather than everyday, life" (1999, p. 58). This point is also made by Wagner and Kieran who emphasize that, "All mathematics instruction and algebra instruction in particular, should be designed to promote understanding of concepts and to encourage thinking. Drill and practice should be required whenever necessary to reinforce and automatize essential skills. but, whenever drill and practice are required, students should always have a clear understanding of why the particular skill is so important that its mastery is required" (1999, p. 12).This is not to say, of course, that algebra teachers must resort to "tricking" students to learn, but it does mean that different students will learn in different ways and there is a need to provide an individualized approach to teaching the representational aspects of algebra.

Most classroom teachers can readily testify that they are able to identify the point at which students achieve the "a-ha" moment in learning, where they make the mental connection between the curricular offering and comprehension. In this regard, Tall and Vinner (1981) advise that the mental leap described above can be conceptualized in terms of the "evoked concept image" which will vary for different students: "At different times, seemingly conflicting images may be evoked. Only when conflicting aspects are evoked simultaneously need there be any actual sense of conflict or confusion. Children doing mathematics often use different processes according to the context, making different errors depending on the specific problem under consideration" (Tall & Vinner 1981, p. 152). Moreover, different students can achieve successful academic outcomes by using different problem-solving methods, including those preferred by the teacher. In this regard, Tall and Vinner emphasize that, "For instance adding 1/2 + 1/4 may be performed correctly but when confronted by 1/2 + 1/3 an erroneous method may be used. Such a child need see no conflict in the different methods, he simply utilizes the method he considers appropriate on each occasion" (1981 p. 152).

Once the initial mental leap regarding these representational aspects is achieved, teachers can apply a more standardized approach to the entire classroom, but helping individual learners get started is an essential requirement for success -- even if this means taking the time to tutor struggling students or arrange for peer mentors to help them in the process. In this regard, it is educationally axiomatic that, "If students aren't learning the way I'm teaching, then I must teach the way they learn." Unfortunately, some parents lack the basic background in algebra needed to help their children in this area, making the classroom the only place where young learners can acquire this important knowledge. Therefore, it is incumbent upon classroom….....

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