11
Jeonglim Chae
Prospectus
Middle School Students’ Experiences with Symbolism
Introduction to the study
Background
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 and represent 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, 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. Before then, algebraic ideas were stated rhetorically, and special words, abbreviations, and number symbols were used as notations. It was Vieta who used symbols purposefully and systematically after some mathematical symbols (e.g. +, -, =) were introduced with letters used for unknowns (Kline, 1972). 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).
However, symbolism is one of the major difficulties for young students in learning algebra even though symbolism made it possible to study abstract structures in algebra by expressing complicated mathematical ideas succinctly. 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. Specifically, 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 of symbols and learn mathematical concepts with symbols so that mathematics educators can enhance students’ learning of mathematics with symbols.
Research questions
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 (cited in Kieran, 1992). For instance, Diophantus, as a syncopated algebraist, created numeral expressions like 10 – x and 10 + x and multiplied them to get 100 – x2 as 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, many researchers (e.g. 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 connections between symbols and other representation forms. In analyzing two primary functions of symbols of a public function and a private function, they insisted that connections between symbols were required for a public function, involved in representing something known already for communication, and connections with other representation forms were necessary for a private function, involved in organizing and manipulating ideas as objects (p. 73-74). This analysis led to my curiosity about early algebraic students’ experiences with symbolism. 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 of 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 symbols while doing mathematical activities. The following questions will guide this study:
1. How do students make sense of symbols used in mathematical activities?
2. How do students’ mathematical concepts develop and evolve as they use symbols throughout mathematical activities?
I presume that students’ prevalent experiences with symbolism occur in classroom learning situations and the learning experience includes teacher’s lecture, reading mathematics books, doing hands-on activities, observing how teacher and other students use symbols, and discussion with other students. Under these circumstances, we can never simply assume that students understand symbols in the way that each activity means to provide. So I would like to investigate how students make sense of symbols used in classroom activities through the first question. The second question focuses on students’ understandings of mathematical concepts that certain symbols are intended to represent. For instance, y = mx + b is a symbol as a whole or a symbolic representation that describes linear relationship between two variables of x and y. Also m and b are symbols for the slope and the y-intercept in the linear relationship, respectively. Embedded mathematical ideas in the symbol or symbolic representation are what students need to learn ultimately. Since I believe that students’ understanding of mathematical concepts cannot occur once and for all, I expect students’ understanding of certain mathematical ideas to evolve throughout various activities. So I would like to investigate how students’ mathematical concepts develop as they engage in mathematical activities.
Theoretical framework
This section includes theoretical perspectives with which the present study will be guided. The first part is allotted to my perspectives on learning mathematics in general, and the second part is to show briefly what theoretical perspectives on symbolism were considered that eventually lead to the adoption of the procept model (Tall et al., 2001).
Perspectives on learning
Perspectives on learning concern statements of subjectivity, that will inform and affect all the activities of the present study in general, rather than establish a theoretical framework. In particular, this section mainly includes my current personal beliefs formed through learning and teaching mathematics and studying mathematics education as a graduate student. Since the beliefs will provide lenses and constraints with which I design the present study and interpret all possible phenomena the study will bring, I believe that it is worthwhile to state here. My perspectives on learning are quite parallel to what radical constructivists assume as underlying principles: (1) knowledge is not passively received but built up by the cognizing subject, and (2) the function of cognition is adaptive and serves the organization of the experiential world, not the discovery of ontological reality (von Glasersfeld, 1995, p. 18). Although the way I interpret the principles might not be as identical as most radical constructivists presume, these inform my view of mathematics learning. What I believe about learning mathematics are:
· A learner does not receive mathematical knowledge passively.
· A learner constructs his own knowledge.
· Learning occurs through experiences.
· Knowledge is primarily personal.
As an elaboration, I believe that a learner does not receive mathematical knowledge passively. Even without taking a constructivist perspective that a learner actively constructs mathematical knowledge, I have seen evidences of my belief. For example, I had learned mathematics via lectures and fortunately most of my mathematics teachers helped me understand mathematical concepts. When I discussed or worked on problems with my classmates who had shown similar mathematical abilities and performances, I could find differences in how we understood a certain topic and strategies we used. If we were receivers of knowledge, we should barely find differences in knowledge or understanding of it. I think the differences came from what we did in our own mind. My second belief is that a learner constructs his own knowledge. What I mean by “construct” is not necessarily the same as constructivist perspective. Construction means neither invention like what professional mathematicians do nor isolated construction without help. Back in my example aforementioned, I believe that my friends and I constructed our own mathematical knowledge mainly with the teacher’s help. We could not absorb what the teacher told us, but instead each of us tried to make sense of mathematics around us. In other words, we made sense of all mathematics from previous learning, textbook, everyday life experiences, communication with classmates, and teacher’s explanation in order to fit all of them together. Third, I believe that learning occurs through experiences. What I mean by experiences is mostly the learner’s interaction with his/her environment, and examples of experiences related to learning include learning activities, reading books, communicating with others, listening to teachers, reflecting and so on. Finally, I believe that knowledge is primarily personal. In fact, mathematical knowledge definitely had social and cultural aspects not only because mathematics has been built historically and culturally throughout a long period of time but also because most of the mathematical experiences we have occur in school settings with a teacher and a class of students. Although we develop mathematical knowledge or meaning as a community in classroom, it is ultimately the learner who makes choices of whether the developed knowledge will be included meaningfully into his or her knowledge structure and whether the knowledge will be actively used for mathematical activities like solving problems.
Perspectives on symbolism
In order to study students’ experiences with symbolism, the present study needs to decide which symbols will be focused on and how to define them. Cobb (2000) seemed to define the term symbol broadly such as:
…to denote any situation in which a concrete entity such as a mark on paper, an icon on a computer screen, or an arrangement of physical materials is interpreted as standing for or signifying something else (p. 17).
Following his definition, I could use two distinct erasers pretending they were cars in order to explain a traffic accident that I experienced. By moving the two erasers, I could explain how my car was hit by the other. Here the erasers were symbols since they were concrete entities in the accident situation and stood for cars. In addition, Janvier et al. (1992) differentiated the erasers and the car that the erasers stood for as “signifier or referent” and “signified or referenced”, respectively. Unlike symbols, the signified or referenced (what symbols stand for) is not limited to a concrete thing. It could be an action, an idea or a concept, depending on context.
Although accepting Cobb’s (2000) notion of symbols and the concepts of signifier (or referent) and signified (referenced) by Janvier et al. (1992), the present study will focus on standard mathematical written symbols since mathematical activities considered in this study will happen in classroom contexts. Unlike the previous example of everyday-life symbols, it is not always clear to tell what mathematical written symbols stand for. For example, a fraction 3/4 is a mathematical symbol, or signifier, but its signified is not as clear as that of the erasers in the previous example. In various contexts, 3/4 could stand for a fair share of sharing 3 apples among 4 children, a ratio of 3 out of 4, or an operator as in 3/4 of something. Even when removing a specific context, 3/4 still could refer to the abstract concept of fraction. This example shows two faces of symbols as process and concept. It seems the frequent case that symbols as a vehicle to signify process are prioritized, and after the process is familiarized enough they can be used as objects carrying concepts. Mason’s (1980) spiral model also explained the shift of symbol uses as:
from confidently manipulable objects/symbols,
through their use to gain a ‘sense of’ some idea involving a full range of imagery but at an inarticulate level,
through a symbolic record of that sense,
to a confidently manipulable use of the new symbols,
and so on in a continuing spiral (cited in Mason, 1987, p. 74-75).
In particular, Mason (1980) not only explained the shift of symbol uses between as process and as concept but also showed the continuous acquisitions of new symbol uses with confidently usable symbols.
However, both the dichotomous uses of symbols and the acquisition of symbols as concept after process seem artificial since I believe process and concept are mingled together so that the demarcation is not clear and they grow together. Thus, I believe, building on Mason’s (1980) spiral model, Gray and Tall (1991, cited in Tall et al., 2001, p5.) introduced the notion of procept. Tall et al. (2001) elaborated as: