Symbolizing and Communicating in Mathematics Classrooms: Perspectives and Discourse, Tools and Instructional Design

Journal of Literacy Research, Spring 2003 by Draper, Roni Jo

Symbolizing and Communicating in Mathematics Classrooms: Perspectives on Discourse, Tools, and Instructional Design

Symbolizing and Communicating in Mathematics Classrooms: Perspectives on Discourse, Tools, and Instructional Design. Paul Cobb, Erna Yackel, & Kay McClain (Editors), 2000. Mahwah, NJ: Lawrence Erlbaum & Associates (10 Industrial Avenue, Mahwah, NJ 07430). Softcover, 411 pages.

Symbolizing and Communicating in Mathematics Classrooms: Perspectives on Discourse, Tools, and Instructional Design, is a collection of chapters by various researchers and theorists interested in mathematics education and specifically in thinking about how students become mathematical symbol users. Cobb, Yackel, and McClain, the editors of the book, assembled the various chapter authors together to participate in a symposium on discourse, tools, and instructional design as related to mathematics learning and teaching. Yackel explains in the introduction that,

The discussions [at the symposium] were permeated by questions such as: How do symbols acquire meaning? How is meaning achieved? What do we mean by "meaning"? What role do symbols play in the [mathematician's] development of mathematical concepts? What role do symbols play in students' mathematical learning? (p. 1)

As a literacy educator interested in issues related to content area literacy, I saw these questions as literacy questions or at least as parallel to the kinds of questions that permeate discussions about literacy and literacy education.

I began my professional teaching as a mathematics teacher and became attracted to literacy because I was interested in the very questions Yackel poses in the introduction. I came to see literacy theories and methods as providing answers or at least opportunities to answer these questions. However, I have found little reference to these kinds of questions in the literacy research related to thinking and learning in mathematics classrooms. Generally, the literature related to literacy in mathematics classrooms focuses on word problems and the use of texts (i.e., mathematics textbooks, children's literature, history of mathematics) in the mathematics classroom (Rudnitsky, Etheredge, Freeman, & Gilbert, 1995; Siegel & Fonzi, 1995; Tanner & Casados, 1998). Although this literature may address print literacy issues related to learning mathematics, it does not address the issues raised by the contributors of Symbolizing and Communicating in Mathematics Classrooms related to how students attribute meaning to mathematical symbols and how students become mathematical symbol users. These issues expand the questions literacy educators and researchers should consider as they explore the role of literacy instruction in mathematics classrooms.

The book is divided into two sections. The contributing authors of the first section of the book offer their ideas and theories surrounding symbolizing, communicating, and mathematizing grounded in research done in mathematics classrooms. The contributors of the second section of the book focus on instructional design implications for mathematics classrooms that support symbolizing, and communicating mathematics.

Theoretical Considerations

The authors use social semiotics and socio-cultural theories of learning to provide a foundation for describing how students become symbol users in mathematics classrooms. At the heart of the semiotic theory used to frame the discussions in the book is the idea that people do not have the ability to convey meaning directly to other people. Instead, human beings endow objects with meaning, and these socially negotiated and agreed upon meanings assist people in making sense of their world and in communicating with one another. The meanings that individuals create and attach to objects (signs, symbols, texts, and so forth) are socially mediated and agreed upon by the individuals who create and use the objects for communicative purposes.

To introduce the book and to frame the discussion of semiotics, Yackel defines symbolizing as, "reasoning with physical materials, pictures, diagrams, computer graphics, and verbal expressions, as well as with conventional and nonstandard written symbols" (p. 15). This symbolization seems to have two purposes: (a) to assist the thinker by providing a place and means of recording thought, and (b) to provide the thinker with a means of communicating thought to another thinker. For the first purpose, any kind of symbolizing that makes sense to the individual will work, because the symbols need not have a shared meaning. Cobb points out that the first use of symbolizing is more than an aid for reasoning, but rather an "integral aspect of [people's] mathematical reasoning" (p. 19). However, when symbolizing is used for communicative purposes the individual must rely on socially agreed upon uses or conventional symbols to help other thinkers recreate the meanings of the symbols.

In the first chapter of the theoretical section of the book, Sfard considers reasoning, meaning, and the role of symbolizing in the creation and communication of meaning. She argues that symbolizing is not simply the result of thinking but plays a role in the creation of meaning itself; her thesis is that users of mathematic symbols symbolize meaning into existence. Sfard explains that the emergence of the notion of function in the 16th century was preceded by mathematical expressions (i.e., f(x)). Despite the use of the expressions, widespread doubts remained among mathematicians about the exact meaning of the expressions and the validity of their use. Sfard uses this example, and others like it from the history of mathematics and from research done in schools with learners of mathematics, to point out that the meanings associated with the mathematical symbols and expressions are born not simply from the meanings created prior to the first use of the symbols but rather the meanings continue to evolve as thinkers use and act with symbols. In making this claim, Sfard argues that meaning construction related to symbols and other mathematics texts is a discursive and ongoing process that is predicated on the background knowledge, experience, expectations, and verifications of the symbol user.