Middle School Students’ Experiences with Symbolism
University of Georgia
Introduction to the study
Whenever we think something and express or communicate about it, we need some tools. In doing algebra, symbols provide one such tool with which we can think of and communicate about our thoughts and ideas. Not only are symbols a tool for representation, they have also played a critical role in developing algebra. If we consider generality as what makes algebra most different from arithmetic, then the beginning of algebra is historically traced back to ancient Mesopotamia and Egypt. In spite of almost four thousand years of history of algebra, the history of symbols had not begun until the 16th century. It was Vieta who used symbols purposefully and systematically after some mathematical symbols (e.g. , , ) were introduced with letters used for unknowns (Kline, 1972). Before then, algebraic ideas were stated rhetorically, and special words, abbreviations, and number symbols were used as notations. Since Vieta, algebra has rapidly developed from a science of generalized numerical computations, to a science of universal computations and then into a science of abstract structures thanks to symbolism (Sfard, 1995).
Even though symbolism made it possible to study abstract structures in algebra by expressing complicated mathematical ideas succinctly, symbolism is one of the major difficulties for young students in learning algebra. Hiebert et al. (1997) explained that the difficulties in dealing with symbols as a learning tool were attributed to the fact that “meaning is not inherent” in symbols (p. 55). They insisted that meaning is not attached to symbols automatically and without meaning symbols could not be used effectively. So students should construct meaning for and with symbols as they actively use them. The National Council of Teachers of Mathematics’ (2000) Algebra Standard also encouraged using symbols as a tool to represent and analyze mathematical situations and structures in all grade levels. In particular, students in Grades 6 – 8 are recommended to have
… extensive experience in interpreting relationships among quantities in a variety of problem contexts before they can work meaningfully with variables and symbolic expressions. An understanding of the meanings and uses of variables develops gradually as students create and use symbolic expressions and relate them to verbal, tabular, and graphical representations. Relationships among quantities can often be expressed symbolically in more than one way, providing opportunities for students to examine the equivalence of various algebraic expressions (p. 225-226).
In this recommendation, NCTM put emphasis on using problem contexts to help students develop meaning for symbols and appreciate quantitative relationships.
In line with the issues mentioned above, the present study is intended to provide insight into students’ experiences with symbolism. In particular, the educational purpose of this study is to inform mathematics educators of how students construct meaning for algebraic symbols and learn mathematical concepts with symbols so that mathematics educators can enhance students’ learning of mathematics with symbols.
In the abstract development of algebra with systematic symbolism, Wheeler (1989) argued that abstract algebra sacrificed the implicit meanings for its applicability unlike rhetorical and syncopated algebra. For instance, Diophantus, as a syncopated algebraist, created numeral expressions like 10 – x and 10 + x and multiplied them to get 100 – x2as if they were numbers like 2 and 3 to solve his word problem. As he denoted the letter x as an unknown but fixed value in the context of the problem, he could keep the meaning of x for its applicability explicitly. However, he might have not obtained the notion of variables, which abstract algebra achieved (Sfard & Linchevski, 1994). As Kieran (1992) elaborated, symbolic language made algebra more powerful and applicable by eliminating “many of the distinctions that the vernacular preserves” and inducing the essences (p. 394). However, the powerful yet decontextualized language brought difficulties for young students who were beginning to learn algebra:
Thus, the cognitive demands placed on algebra students included, on the one hand, treating symbolic representations, which have little or no semantic content, as mathematical objects and operating upon these objects with processes that usually do not yield numerical solutions, and, on the other hand, modifying their former interpretations of certain symbols and beginning to represent the relationships of word-problem situations with operations that are often the inverse of those that they used almost automatically for solving similar problems in arithmetic (Kieran, 1992, p. 394).
In fact, some researchers (see Kieran, 1992) have studied students’ difficulties in manipulating symbols as mathematical objects and modifying their interpretations of symbols. Also some studies (e.g. Stacey & MacGregor, 1997) were conducted to investigate how meaning for symbols could be developed. Hiebert and Carpenter (1992) reviewed such literature and summarized that making meaning for symbols could develop in two ways: through connections between symbols and other representational forms or through connections within the system of symbols. Then they analyzed that the former way mainly served a public function of symbols as “recording what is already known” for communication and the latter served a private function as “organizing and manipulating ideas” (p. 73-74). This analysis drew my attention and provoked my curiosity about how early algebra students begin to make sense of symbols, which we as adults with mathematical knowledge take for granted.
My initial curiosity included sporadic questions like: how do young students interpret mathematical symbols?; in what ways do they use symbols?; do they feel the need for symbols?; what do they want to represent with symbols?; in what ways do their understanding of symbols affect learning mathematical concepts?; and so on. Inspired by these questions, the present study will investigate how middle school students develop algebraic reasoning with algebraic symbols while doing mathematical activities. The following questions will guide this study:
How do students make sense of algebraic notations in relation to other forms of representation throughout mathematical activities?
How do students’ mathematical concepts form and develop as they use the algebraic notations throughout mathematical activities?
I presume that students’ prevalent experiences with algebraic symbolism occur in classroom learning situations where the learning experience includes the teacher’s lectures, reading mathematics books, doing hands-on activities, observing how the teacher and other students use symbols, and discussions with other students. So I will mainly focus on mathematical activities in the classroom setting in the present study. In addition, it is most likely that the learning experiences begin with introducing algebraic symbols and students try to make sense of them throughout the following activities. So when students are introduced to algebraic symbols for the first time, they do not yet have their own meanings for the symbols. In order to differentiate symbols before and after students develop their meanings for them, I will use the term, ‘algebraic notations’, before their meaning development.
Thus the first research question is about how algebraic notations become symbols to students as students do mathematical activities in the classroom setting. Specifically, I will investigate it through how they relate algebraic notations to other forms of representation such as narrative, tabular and graphical representations. Moreover, as students have experiences about symbolism, their new mathematical concepts will be created or previous ones may change or develop. With the second question, I will investigate the development of students’ mathematical concepts throughout mathematical activities. I will explain the research questions in detail in the next section.