Meaningful Mathematical Representations and Early Algebraic Reasoning
Lubinski, Cheryl A., Otto, Albert D., Teaching Children Mathematics
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.
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 additional obstacles in the more formal study of equations that occurs later in the curriculum. Furthermore, students' desire to find a sum for such an expression creates a cognitive obstacle when they first encounter such representations as 3x + 5 or [a.sup.2] + [a.sup.3]; they believe they need to complete the calculation to further simplify the expression.
When young students recognize that the equals sign describes the relationship of equality between two quantities, we found that they begin to focus their reasoning on the quantities and operations, not just the numbers. This increased level of abstraction is instrumental in developing prealgebraic thinking.
A Worthwhile Task to Establish Meaning for Mathematical Representations
We chose How Many Snails? (Giganti 1988) to use in discussing representations in a first-grade class of twenty-eight students. This counting book can also help students realize that numbers and arithmetic expressions serve as representations. One of the authors of this article, an experienced classroom teacher, led the discussion with the first-grade students, and the other observed and offered suggestions to contribute to the teacher-student discourse. The session was videotaped and transcribed, providing an accurate record of the dialogue, which has been condensed for this article, and enabling us to easily identify major points in the discussions. …