Meaningful Mathematical Representations and Early Algebraic Reasoning

Beginning of article

Both oral and written communication play an important role in teaching and learning mathematics (NCTM 2000). Students and teachers exchange ideas about their understanding of, and thinking about, mathematics by communicating with one another. An important part of this communication process is the choice of symbols used to represent that thinking (Hiebert 1989). The process of representing mathematical ideas using symbols and expressions should begin at the earliest stages of mathematics instruction and appear in the context of ideas to which young students can relate (Carey 1992).

This article illustrates one way that a children's book can be used to develop young students' abilities to make sense of mathematical representations. We selected this mathematical task to establish early in the primary school curriculum a better understanding of what the equals symbol represents. This experience should give students a richer understanding of equations that appear later in the curriculum and act as a precursor to the development of algebraic thinking.

We hope that in reading this article, teachers will be motivated to reflect on the ways that mathematics is used in their communities and how these uses are culturally influenced. With this understanding, teachers can then create connections for students between the mathematical concepts to be taught and the uses of these concepts in the world outside the classroom.

Background

Thompson and others (1994) emphasize the importance of a teacher's having a "conceptual orientation." An outcome of this orientation is found in students' explanations of their solution strategies when they attach meaning to numbers and arithmetic expressions and when they look at relationships in the context of a problem. This result is in contrast with a "calculational orientation," which is seen when students focus on procedures to obtain an answer.

Herscovics and Kieran (1980) note that students have a strong tendency to use the equals sign in a very narrow sense, namely, to indicate only the results of an operation. For these students, the equals sign means "perform an operation and show the result." For example, these students would maintain that 3 + 4 should be followed by an equals sign to indicate the need to add to arrive at 7. These students seem unwilling to accept 3 + 4 as representing something other than 7. Students who think about the equals sign in this manner do not fully appreciate that this sign represents a relationship of equality between two quantities. As Herscovics and Kieran (1980) point out, this type of understanding creates …