Chapter 4: Addition Operations
David Carraher, TERC
Analúcia D. Schliemann and Bárbara M. Brizuela
We believe, as others do, that many of the difficulties students have with algebra arise from the fact that algebra has been kept out of the early mathematics curriculum (see, for example, Booth, 1988 and Kaput, 1995). But it would be naïve to hope that the situation could be improved simply by adding algebraic content and methods to current curricula. As we already stressed in this book, the real challenge consists in finding opportunities for bringing out the algebraic character of elementary mathematics. Arithmetic is a part of algebra. As such, arithmetic topics should be approached as instances of more abstract ideas and concepts. This will not only enrich children's understanding of arithmetic, but will build the foundations for the meaningful learning of more advanced algebra in later years. This shift towards an algebrafied arithmetic could include various, possibly overlapping, approaches, such as: generalizing arithmetic; moving from particular to generalized numbers; focusing on mathematical structures common to sets of algorithms; introducing variables and covariation in word problems; reorganizing curricula around abstract concepts that offer the possibility of uniting a range of topics that currently stand isolated in the curriculum, such as the concept of function. As we showed in chapter 1, there is some degree of consensus that algebraic reasoning belongs to the early mathematics curriculum. The question for many educators and curriculum developers becomes: what does this mean for the topics already in the curriculum? Presuming that many of the time-honored topics of early mathematics (number sense, addition, subtraction, multiplication, division) will stay around, what will they look like with their new, algebrafied personalities? What shifts in classroom activities are likely to occur as a result of elevating the importance of making generalizations? What opportunities for exploring the potentially algebraic character of standard school mathematics have been overlooked in the past? And how might they now be seized upon so that young students can cultivate ways of reasoning that will prove useful not only now but also much later, when dealing with even more advanced mathematical ideas? Since changing the nature of the early mathematics curriculum only makes sense if it effectively helps students learn mathematics better and more deeply, we need research that looks into how students (and teachers) deal with standard topics presented in new ways.
Here we will focus on a remarkably simple idea: that the operations of arithmetic can be treated as functions in early mathematics education. Admittedly, some prominent mathematics educators have argued for the importance of functions in the mathematics curriculum, though only well after arithmetical operations have been taught (e.g., Schwartz & Yerushalmy, 1992). Others have defended the general goal of "algebrafying" early mathematics (Kaput, 1995). However, the field has largely failed to note that addition, subtraction, multiplication, and division operations can be treated from the start as functions, and that this idea could provide an important foothold for early algebra reform in current curricula.
In this and the next two chapters we will describe some results from a 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. The study aimed at understanding and documenting issues of learning and teaching in an "algebrafied" arithmetical setting (see Carraher, Schliemann, & Brizuela, 1999 and Kaput, 1995). The school where the classroom is set serves a diverse multiethnic and multiracial community, reflected well in the class composition, which included children from South America, Asia, Europe, and North America. Our activities in the classroom consisted of teaching a two-hour "math class" on a bi-weekly basis. The topics for the class sessions evolved from a combination of the curriculum content, the teacher's main goals for each semester, the questions we brought to the table regarding our research, and the interests, revelations, and confusions put forth by the children in our classrooms. We were well aware of the relative difficulty of additive comparison problems and seized upon additive comparisons as a special opportunity for approaching arithmetic from an algebraic standpoint in the first six lessons we developed and implemented in the Fall term. The seven lessons implemented in the Spring term focused on additive and multiplicative operations as functions.
In this chapter, we will present examples of students taking part in activities regarding additive operations in the classroom and in an end of the term interview. The examples were chosen to exemplify instances of children's algebraic reasoning as well as the challenges they face as they are asked to focus on functional relationships and the use of letters to represent functional relations. Special attention will be given to the tension we identified in children's development as they deal with number relations and contextual constraints that are inherent to problem solving activities.
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.