David W. Carraher

The Transfer Dilemma.

In this article we provide an overview of research on transfer, highlighting its main tenets. Then we look at interviews of two 5th-grade students learning about mathematical concepts regarding operations on positive and negative quantities. We attempt to focus on how their learning is influenced by their prior knowledge and experience. We take the position that transfer is a theory of learning and we attempt to show that it cannot provide a solid foundation for explaining such examples of learning.

Signed Numbers and Algebraic Thinking

Signed numbers are positive and negative numbers. They refer to integers, notmerely the natural numbers or positive integers. They refer to rational numbers,not merely the non-negative rational numbers. They refer to real numbers, notmerely non-negative real numbers. Students may learn to accept negative num-bers in the co-domain (output of computations) before accepting them in thedomain (input of computations). When they are comfortable with both, we saythey have learned not only that signed numbers exist, but they can serve asthe input for addition, subtraction, multiplication, and division as well as otherfunctions.

Lines of thought: A ratio and operator model of rational number

A model of rational number is described that exploits the fact that a pair of line segments can embody a ratio of numbers and that actions upon segments can embody arithmetical operations. Tasks are described for bringing out diverse meanings of rational numbers. In the software under development algebraic equations are used for formulating descriptions of the relations among segments and for carrying out operations intended to display geometrically the meaning of the equations. Incorrect hypotheses produce arithmetic inequalities, represented in the model as resultant segments of unequal lengths; correct hypotheses produce resultant segments of equal lengths. None of the tasks discussed can be solved through rote numerical computation but require reflecting upon the relations of quantities. This aspect may be useful in drawing out the relational meaning of rational numbers.

Chapter 8: Is Everyday Mathematics Truly Relevant to Mathematics Education?

Early research in everyday mathematics lent support to diverse and often contradictory interpretations of the roles of schools in mathematics education. As research has progressed, we have begun to get a clearer view of the scope and possible contributions of learning out of school to learning in school. In order to appreciate this view it is necessary to carefully scrutinize concepts of real (as in “real life”), utility (or usefulness), context, as well as the distinction between concrete and abstract. These concepts are crucial for determining the relevance of everyday mathematics to mathematics education; yet each concept is deeply problematic. The tension between knowledge and experience acquired in and out of school is not a topic of mathematics. But it deserves to be a fundamental topic in mathematics education. Everyday and Academic Mathematics 239 Introduction Over the years we have witnessed a wide range of reactions to everyday mathematics research. We have seen teachers awed by street vendors solving problems. On the other hand, we have seen mathematics educators question whether serious mathematics was being learned out of school and whether semi-skilled trades encourage the development of advanced mathematical concepts. For some the results suggest that mathematics does not require formal instruction; for others the results merely reaffirm the importance of schooling; for still others, the results call for educational reform focusing on how people think, communicate, and learn out of school. When people interpret the same data in such diverse ways, are they simply projecting their prior views onto inkblots of data? When they agree, are they simply celebrating common beliefs? Can we make headway in understanding the relevance of everyday mathematics for education? Or will our conclusions ultimately reflect our pre-existing educational agenda, our socio-political persuasions, and our views of learning? We share with the other contributors to this monograph a number of premises about the nature of mathematical learning. We view mathematics, for example, as both a cultural and personal enterprise. It is cultural because it draws upon traditions, symbol systems, ideas, and techniques that evolved over the course of centuries, having originated in human activities such as surveying, astronomy, building, commerce, and navigation and eventually becoming a partially autonomous field of endeavor with its own subject matter, purposes, tools, and concerns. It is personal Everyday and Academic Mathematics 240 insofar as it demands from learners constructive processes and creative rediscovery even when they are apparently engaged in mere assimilation of facts and conventions. We also share a number of ideas and beliefs about the purposes and spirit of mathematics education. We tend to distrust rote, unquestioning learning. We favor situations in which students themselves must decide how to frame and represent problems, but we realize that, left to their own devices and inventiveness, students are not going to discover many important concepts of elementary mathematics. We value problems that promote engaging discussions among students. We recommend that students and teachers pursue multiple paths of reasoning when solving problems. We encourage educators to make frequent comparisons between everyday language and symbolism and the formal language and symbolism of mathematics. These value premises are rooted in our personal backgrounds, in the economic and political Zeitgeist, and in our theoretical lenses as well as in the legacies of Plato’s Meno, Rousseau’s Émile and Piaget’s constructivism (even though they may at times push and pull in somewhat opposed directions). Some hallmarks of these influences include the belief that mathematical concepts are “out there” to be discovered but require the careful questioning and guidance of the more learned (Plato), faith in the capabilities of children to learn provided we are sensitive to their views and motivations (Rousseau), and the importance of painstakingly documenting children’s reasoning and handling of mathematical invariants as well as the need to place (mathematical) learning in a longterm developmental perspective (Piaget). Although discussions about everyday mathematics take place in value-laden contexts, there is still plenty of room for scientific work. As the number of studies of Everyday and Academic Mathematics 241 everyday mathematics has grown, we now are beginning to understand the range of mathematical concepts likely to develop (or not) in out-of-school environments. Studies of classroom contexts are providing much needed data on how out-of-school knowledge plays a role in classroom problem-solving. This chapter represents an attempt to stand back from the data and try to make sense of what we have observed and learned. In doing so, we will try to make clear how we have had to adjust our beliefs about the relevance of everyday mathematics as we learned more from research. The present chapters in this monograph provide a rich set of empirical data and discussions that will help to refine and clarify educators’ views and beliefs on the educational relevance of everyday mathematics. What Does Research on Everyday Mathematics Suggest? Our views of the relevance of everyday mathematics for mathematics education developed over the years as a result of our research on the characteristics of everyday mathematical understanding as well as students’ understanding of school mathematics (see reviews by D. W. Carraher, 1991; Nunes, Schliemann, & Carraher, 1993; Schliemann, 1995; Schliemann, Carraher, & Ceci, 1997). Studies such as the ones in this monograph are crucial to the discussion and refinement of those views. What lessons do they offer us about how mathematics is learned or ought to be learned? What do they say about the relevance of informal mathematics to mathematics education? In view of the discussions and data provided in this chapter, we will try to reflect upon what we consider to be the main issues raised by them trying, at the same time, to enrich our own views concerning everyday mathematics and mathematics education. Everyday and Academic Mathematics 242 Learning In and Out of School Everyday mathematics research has repeatedly produced evidence that people learn mathematics outside of school settings. Specific cultural activities such as buying and selling promote the development of mathematical ideas that were previously thought to be only acquired through formal instruction. Our research shows that groups with restricted schooling master arithmetical operations, properties of integers and of the decimal system, and proportional relations (Nunes, Schliemann, & Carraher, 1993). Besides arithmetic (T. N. Carraher, Carraher, & Schliemann, 1982, 1985; Saxe, 1991; Lave, 1977, 1988), concepts and procedures related to measurement (T. N. Carraher, 1986; Gay & Cole, 1967; Saraswathi, 1988, 1989; Saxe & Moylan, 1982; Ueno & Saito, 1994), geometry (Abreu & Carraher, 1989; Acioly, 1993; Gerdes, 1986, 1988; Harris, 1987, 1988; Millroy, 1992; Schliemann, 1985; Zaslavsky, 1973), and probability (Schliemann & Acioly, 1989) are used in everyday settings by children or adults with little access to school instruction. In retrospect, the observation that mathematical learning occurs out of school may seem obvious. Indeed one might wonder how anyone could have ever thought otherwise! After all, commerce and crafts requiring rudimentary measurement skills have often flourished in societies where schooling has been infrequent or even nonexistent. Furthermore, developmental psychological studies, particularly those of the Piagetian tradition, have long since documented that young children discover, for example, the commutative nature of addition before entering school. Some remarks can help clarify how out-of-school mathematical learning could evoke surprise from researchers. Firstly, and this is particularly true of our own Everyday and Academic Mathematics 243 research on mathematics of street vendors in Brazil, we were surprised by the fact that the very same people presumed to be unskilled in mathematics could solve problems in out-of-school settings. When we presented the same street vendors with (what we thought were) the same problems in a school-like setting, they tended to find them confusing and gave answers suggesting they were having trouble with the basic sense of the problems. For example, they would sometimes claim that the amount of change to be returned to a customer after a purchase would be greater than the amount of money originally handed to the seller! Looking back, it seems that we were not asking street vendors to solve “the same” problems at all. By indirectly suggesting in the school-like setting that they should write out their answers, we induced them to use poorly understood computation routines that involved features, such as borrowing from neighboring columns, that they did not employ in their work as vendors. But how were they solving problems in their own ways? This brings up a second source of our surprise. If they were not following school-prescribed routines, but nonetheless producing correct answers, they must have alternative ways of representing and systematically solving problems. Much of our work in everyday mathematics pursued this question. It now seems fairly clear that many of the street vendors did not use a place value notational system when mentally solving problems. Furthermore, they seemed to operate on measured quantities (such as 3 coconuts, 35 Brazilian cruzeiros) as opposed to pure numbers (3, 35). In this manner they did not have to perform calculations on numbers and introduce the result of the computation Everyday and Academic Mathematics 244 back into the meaningful problem context. Rather, they would always be working directly with countable quantities. As we came to document more and more instances of

Learning to Quantify Relationships Among Weight, Size, and Kind of Material

ABSTRACT In a 1-hour teaching interview, 20 children (aged 7 to 11) discovered how to tell whether objects might be made of the same material by using ratios of measures of weight and size. We examine progress in the childrens reasoning about measurement and proportional relations, as well as design features of instruments, materials, and tasks crafted to enable such progress. Findings are discussed in light of theories and research about cognitive development and learning.

Cultivating Early Algebraic Thinking

This chapter describes a functions approach to early algebraic thinking developed in the context of classroom research with young students. We outline our approach, examine examples of students’ reasoning in the classroom, and present interview and written assessment evidence of student learning. We also describe and evaluate a related program aimed at preparing teachers to promote algebraic thinking across the curriculum. Throughout, we attempt to identify conditions favorable to the cultivation of algebraic thinking in mathematics education.