As we discussed elsewhere (Schliemann & Carraher, 2002), any analysis of how children develop mathematical knowledge must take into consideration the tools and representations they come to use and understand as they participate in everyday and in formal mathematics instructional activities. Children construct a foundation for logical and mathematical thinking on the basis of their direct experience and individual reflection as they participate in out-of-school activities. However, school classrooms are the privileged place where children are given access to mathematical tools and to experiences that will expand and strengthen their mathematical ideas. In attempting to fully understand the development of mathematical understanding we need analyses of how children learn as they participate in school cultural practices, as they interact with teachers and peers in the classroom, as they become familiar with mathematical symbols and tools, and as they deal with mathematics across a variety of situations.

In this chapter we explore third graders' strategies for dealing with linear functions and constant rates of change, as they participate in activities developed across two 90-minute meetings. As in chapter 4, the examples come from the same exploratory, year-long, third grade teaching experiment we carried out in a classroom of 18 third-grade students at a public elementary school in the Boston area. Here we describe how we gradually adapted mathematical problems involving multiplicative linear functions using function tables and algebraic notation to encourage children to describe functions with increasing clarity.

Our previous work on everyday mathematics suggests that informal mathematical learning and understanding can constitute a solid basis for the development of school mathematics and for the meaningful learning of conventional symbolic systems. However, a student's understanding of mathematics should not be restricted to his or her former everyday experiences. The field of mathematics, although indebted to its origins in farming and commercial activities, cannot be reduced to the circumstances that gave rise to its emergence (Schliemann, Carraher, & Ceci, 1997). This perspective undermines the argument that teachers should bring out-of-school activities to the classroom or that apprenticeship training should replace teaching (Schliemann, 1995; Carraher & Schliemann, 2002).

The contribution of everyday mathematics to the learning of mathematics in school is not a matter of reproducing contexts but rather a recognition that children bring to the classroom ways of understanding and dealing with mathematical problems that should be recognized as legitimate steps towards more advanced mathematical understanding. At the same time, we have to be aware of the differences between everyday approaches constructed as ways to reach everyday goals and the school mathematics goal of exploring multiple properties and representations of mathematical relations. This paper will explore the tension between third-graders' own ways of solving problems and attempts to expand their understanding of proportionality, ratios, and linear functions.

1

1.This
chapter is based on a Plenary Address by the authors at the XXII Meeting of
the Psychology of Mathematics Education, North American Chapter, Tucson, AZ,
October 7 to 10, 2000 and on the video paper "Treating operations as functions"
by Carraher, Schliemann, & Brizuela, to appear in D. W. Carraher, R. Nemirovsky,
& C. DiMattia, (Eds.) Media and Meaning. CD-ROM issue of *Monographs for
the Journal of Research in Mathematics Education*. The
videopaper version of this chapter was assembled by Darrell Earnest.