In traditional mathematics classes, to set written equations to solve problems and to use the syntactic rules of algebra to solve them constitute the core of algebra learning. Even though this vision of algebra is too restrictive and does not consider the many other features of algebra (see Kaput, 1995, Usiskin, 1988), we believe that algebra notation plays an important role in learning algebra and in solving algebra problems. As such, throughout our interview and classroom studies, analysis of children’s notation has been a central focus of our analysis.

The interview data we analyzed in chapter 3 exemplify some of our first explorations of children's understanding and use of algebraic notation, and, in more general terms, of children's ways of representing their algebraic understandings and the algebraic problems they are presented with. It is clear that:

- Some children feel that they need to make notations
for problems with numerical information but not for problems that deal with
logical relationships; they prefer to solve problems with logical relationships
among quantities “in their heads”.

- Notation can do more than register the data
in a problem and the operations to be performed.

- Notation can serve multiple roles such as to
register and guide students’ thinking, to keep track of different parts
of a problem, and to help them find an answer to a problem.

- Children can combine both idiosyncratic and
conventional notation in their representations.

- Conventional and algebraic notation can be used
by children to further their understanding of problems and algebraic concepts.

Our classroom studies described in chapters 4 and 5 introduced children to conventional algebraic notation to represent variables. Brito Lima (1996; see also Brito Lima & da Rocha Falcão, 1997) and Bodanskii (1991) have already shown that written algebraic notation is within reach of elementary school children. In our teaching experiments reported in Chapters 4 and 5 we observed among our students the gradual way in which their notations become more and more context independent. At the beginning of the school year, the notations that children created to represent and solve algebraic problems were imbued with features peculiar to the problem at hand. For example, in representing a problem in which 17 fish had been reduced to 11 fish, children drew fish, with eyes, tails, and fins. While these notations served well the purpose of representing the problem at hand, they would probably not serve well the task of representing problems with a similar underlying arithmetical structure, such as representing how a bank balance of 17 dollars fell to 11 dollars. As the weeks went by, however, the children’s notations became ever more schematic and general, focusing on the logical relationships among quantities instead of the physical properties of the quantities themselves. In the two previous chapters we also documented how the use of a letter to represent any number helped children in considering variables and functions.

In this chapter we further explore children’s notations as they participate in early algebra classroom activities and consider the role that written notation may play in children’s thinking about different problems. In the back of our minds was constantly a question posed by Kaput (1991): “How do material notations and mental constructions interact to produce new constructions?” (p. 55).

In his work regarding cultural tools and mathematical learning, Cobb (1995) highlights two opposing perspectives—the sociocultural and the constructivist—in the analysis of children’s notations. One could argue from a sociocultural perspective that children internalize the algebraic notations used by the mathematical community. The other would argue, presumably from a constructivist perspective, that conceptual development will occur independently of the cultural tools--such as algebraic notation--that learners make use of. Our position is midway between these dichotomous views. Therefore, our task is to explore and document how the assimilation of conventional algebraic notation interacts with both children’s conceptual development regarding algebraic relations and their incipient and spontaneous ways of representing.

Our examples are taken from the same third-grade classroom we described in Chapters 4 and 5. The specific examples we would like to focus on refer to Sara. Sara exemplifies, through her actions and her words, how notations can represent not only what was done while solving a problem and what happened in the context of the problem (as we saw in chapter 3 with Eliza, Maggie, and Melanie), but also how notations can become tools for thinking and reflecting about the relationships between quantities in the problem (as happened with Charles, also in chapter 3). In this way, we can begin to think about children’s notations not only as tools for learners to represent their understanding and thinking about algebraic relations or as precursors of conventional algebra representation, but also as tools to further those understandings and that thinking. As Sara explained to one of us during an interview, referring to the pie chart she had drawn to represent the fractions she was thinking about, “Well, I don’t…when I draw this [the pie chart] it’s just to help me think of something, so it doesn’t really matter [if the pieces of the pie chart are different sizes].”

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1.This chapter is based on previous presentations by B.M. Brizuela, D.W. Carraher, & A.D. Schliemann at the Research Presession, 2000 Meeting of the National Council of Teachers of Mathematics, Chicago, IL, and at the Ninth International Congress of Mathematical Education, Tokyo-Makuhari, Japan, August 2000. The videopaper version of this chapter was assembled by Darrell Earnest.