Analúcia D. Schliemann

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.

Logic of meanings and situated cognition

Abstract Inspired by the views of Piaget & Garcia (1991) that logico-mathematical knowledge has its foundations in a logic of meanings, this paper aims at a better understanding of how knowledge develops in everyday contexts, by taking into account situational aspects, individual reasoning processes, and logico-mathematical understanding. We first provide a re-analysis of data on the understanding of proportionality developed outside of school, in buying and selling activities. The set of studies we discuss shows how procedures to solve proportionality problems in specific contexts later develop as general and flexible approaches to solve problems in different contexts. Then, in an interview with an adult with restricted schooling, we show how understanding of graphical representation is achieved through the attribution of meanings to a graph on the basis of personal wishes, opinions, and knowledge about the situation and by a continuous search for logical coherence.

Culture, Arithmetic and Mathematical Models

Three sources of data on Mathematical knowledge are analyzed here in detail, leading to four main conclusions. First, reasoning principles underlying written and oral Mathematics appear to be the same. Second, there are diverse ways of understanding and using mathematical concepts which depend upon the cultural conditions under which Mathematics is practised. Third, schools transmit culturally perfected mathematical tools (such as numeration systems, algorithms, and formulas) and tend to make students good model-users but are perhaps carrying out this function at the expense of meaning. Fourth, Mathematics learning in daily life produces meaningful procedures which may, however, be of restricted applicability.

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.

Constraint and cooperation in algebraic problem solving

Abstract Piaget (1965/1995. Sociological Studies . London: Routledge) proposes that neither the individual nor the set of individuals modify individual mental processes. Instead, what matters are the continuously modified relationships among individuals. But not all types of social interactions are sources of logical reasoning. Social interactions based on constraint only provide the individual with superficial notions, while cooperation, as a system of interpersonal actions governed by the laws of equilibrium and organized into reversible systems, allows for the development of objective and coherent system of operations. With examples of how children try to solve problems involving algebra relations and notation, we will contrast the effect of learning that results from the transmission of rules to the effect of learning that occur in cooperative interview settings. We will argue that Piagets analysis of the role of social life and cooperation constitute a rich theoretical approach for understanding how children come to develop, understand, and use culturally developed mathematical representations.

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.

ANPEPP: Symposia, Research, and Postgraduate Studies from 1988 to 2014

This article briefly describes (a) how ANPEPP was started and its first few years of activities, (b) the planning, implementation, evaluation, and recommendations by participants in the first symposium that led to future symposia format, and (c) ANPEPPs evolving activities from 1988 to 2014, taking one of its working groups (grupos de trabalho [GTs]) as an example. The analysis suggests that ANPEPP is playing a vital role in promoting fruitful interactions among researchers, contributing to a broad research agenda, and building a rich database of psychological knowledge in Brazil.

Capitalizing on Multiple Perspectives

In recent decades the study of cognitive development has increasingly shifted its focus towards sociocultural practices. Much has been gained from this shift. For one thing, development has become viewed much less as a blossoming or unfolding of competences resulting from the interaction between individuals and their physical environments. Modern developmental theories and approaches, many inspired by Vygotsky’s ideas, have made accommodations so that culture and history can be seen to exert their influences on cognitive development through language, practices, and institutions. But this sociocultural shift led to a certain obscuring of the role of individual mental processes. In part this may be due to the growing distrust of general-stage views of cognitive development, notably Piaget’s. Whatever the case, it has been difficult to achieve a balance between, much less a theoretically satisfying account of, sociocultural and individual mechanisms of cognitive development. Children’s Economic Experience attempts to find such a balance through a focus on children’s evolving understanding of economic exchange. The study of how institutions, social interactions, and cultural artifacts contribute to cognitive development requires that children’s perspectives, ways of reasoning, and construction processes be taken into account. That is where Faigenbaum starts: obtaining, through observations and interviews, children’s views about events involving money and financial exchanges that have a meaning in their everyday activities. To interpret the data thus obtained, he draws on insights from history, philosophy, anthropology, economics, and various theoretical perspectives and studies of cognitive development. Along the way, he employs Piaget’s methods and theoretical insights as analytical tools. Significantly, it is not the pre-operational, concrete operations, formal operations stage characterization he employs, but rather the less known work from Piaget’s collaboration with Rolando