On reforming practice in mathematics education

Based on principles of constructivism, an analysis is made of how practice in mathematical education might be reformed towards a professional practice. In addition to the widespread recommendations that mathematical teaching be based on interactive communication and that mathematical learning be active, we argue that conventional school mathematics be replaced by a constructivist school mathematics. A constructivist school mathematics is based on children's use of their schemes of action and operation in learning situations, and whatever accommodation the children make in these schemes as they use them. Through examples of our learning of the numerical schemes of five year old children we illustrate what we mean by a constructivist school mathematics. In our examples, we characterize the schemes of action and operation that we attribute to children as our interpretations of the children's activities. For this reason, we define a constructivist school mathematics to be the results of the observer's experiential abstractions in the context of interacting with children mathematically. A professional teacher is cast as one with the intellectual autonomy and power to produce a constructivist school mathematics, including the involved situations of learning and interactive mathematical communication.     

A pedagogy for mathematical microworlds

In this paper we attempt to map out some relationships between pedagogy and student behaviour in a mathematical microworld. We illustrate our analysis by a series of episodes which occurred in a Logo-based microworld constructed around the notions of ratio and proportion. We explore the patterns of pedagogy associated with on and off-computer activities and suggest how the teacher has significant roles in both settings; particularly in helping pupils to bridge the discursive disjuncture between the practices of Logo- and schoolmathematics.     

Social,environmental, and developmental issues and creativity

Parents and teachers alike have been led to believe that the motivation that stems from wanting to earn a good grade or receive a gold star or do better than the person sitting at the next desk is a positive force. However, in the case of open-ended, creative tasks—tasks for which there is more than one possible solution—extrinsic constraints have frequently proven harmful. A social-psychological approach to the study of creativity is especially useful for answering the question of how best to foster creative behavior in the classroom. Rather than focus on the (largely innate or at least immalleable) differences between creative and uncreative persons, social psychologists concentrate on creative situations (i.e., the particular social and environmental conditions that can positively or negatively impact the creativity of most individuals). The present article explored this research tradition and proposed a theoretical explanation for the impact of motivation on creativity that is based upon children's conceptions of work and play.     

Dolls in the early childhood classroom

Dolls are a part of every culture. From the beginning of time, children have played with dolls, using them as friends and confidants, as surrogates while the child recreates domestic situations, and as props for acting out fantasies. Dolls help children practice the nurturing role they see their parents performing. Dolls can be loved or punished, kissed or spanked, rocked to sleep or sent to bed with no supper, providing an acceptable outlet for a child's range of emotions. Children have the same kind of control over their dolls' lives that parents have over the lives of their children. Eden Ross Lipson, children's book editor ofThe New York Times, notes the universality of children's play with dolls. She suggests, The will to create dolls, to give them personalities and family relationships, to imagine their emotions and act out fantasies in which they figure, is one of the characteristics that makes us all more alike than different as human beings (Lipson, 1986).Blakely Fetridge Bundy is a writer, consultant, and teacher at Willow Wood Pre-School in Winnetka, IL.     

Using Intuition From Everyday Life in 'Filling' the gap in Children's Extension of Their Number Concept to Include the Negative Numbers

We report here an instructional method designed to address the cognitive gaps in children's mathematical development where operational conceptions give rise to structural conceptions (such as when the subtraction process leads to the negative number concept). The method involves the linking of process and object conceptions through semiotic activity with models which first record processes in situations outside mathematics and subsequently mediate activity with the signs of mathematics. We describe two experiments in teaching integers, an interesting case in which previous literature has focused on the dichotomy between the algebraic approach and the modelling approach to instruction. We conceptualise modelling as the transformation of outside-school knowledge into school mathematics, and discuss the opportunities and difficulties involved. This revised version was published online in July 2006 with corrections to the Cover Date.     

Integrating Tinbergen's inquiries: Mimicry and play in humans and other social mammals

Visual signals convey emotions and intentions between individuals. Darwin underlined that human facial expressions represent a shared heritage between our species and many other social mammals. Social play is a fertile field to examine the role and the potential communicative function of facial expressions. The relaxed open-mouth (or play face) is a context-specific playful expression, which is widespread in human and non-human mammals. Here, we focus on playful communication by applying Tinbergen’s four areas of inquiry: proximate causation, ontogeny, function, and evolution. First of all we explore mimicry by focusing on its neural substrates and factors of modulation within playful and non-playful context (proximate causation). Play face is one of the earliest facial expressions to appear and be mimicked in neonates. The motor resonance between infants and their caregivers is essential later in life when individuals begin to engage in increasingly complex social interactions, including play (ontogeny). The success of a playful session can be evaluated by its duration in time. Mirroring facial expressions prolongs the session by favoring individuals to fine-tune their own motor sequences accordingly (function). Finally, through a comparative approach we also demonstrate that the elements constituting play communication and mimicry are sensitive to the quality of interindividual relationships of a species, thus reflecting the nature of its social network and style (evolution). In conclusion, our goal is to integrate Tinbergen’s four areas of ethological inquiry to provide a broader framework regarding the importance of communication and mimicry in the play domain of humans and other social mammals.     

Knowledge construction and diverging thinking in elementary & advanced mathematics

This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action upon it and reflection on both perception and action. We see an emphasis on one or more of these activities leading not only to different kinds of mathematics, but also to a spectrum of success and failure depending on the nature of the focus in the individual activity. For instance, geometry builds from the fundamental perception of figures and their shape, supported by action and reflection to move from practical measurement to theoretical deduction and euclidean proof. Arithmetic, on the other hand, initially focuses on the action of counting and later changes focus to the use of symbols for both the process of counting and the concept of number. The evidence that we draw together from a number of studies on children's arithmetic shows a divergence in performance. The less successful seem to focus more on perceptions of their physical activities than on the flexible use of symbol as process and concept appropriate for a conceptual development in arithmetic and algebra. Advanced mathematical thinking introduces a new feature in which concept definitions are formulated and formal concepts are constructed by deduction. We show how students cope with the transition to advanced mathematical thinking in different ways leading once more to a diverging spectrum of success.This revised version was published online in September 2005 with corrections to the Cover Date.     

More than just fun: a place for games in playful learning / Más que diversión: el lugar de los juegos reglados en el aprendizaje lúdico

Games play a significant role in childhood, fuelling hours of engagement and social interaction, and probably much learning as well. Board games, card games and outdoor games (such as Tag) first come to mind, but more recently, games have also gone digital. In this piece, we offer a new perspective by placing games within the established construct of ‘playful learning’. We review key elements of playful learning and the two subtypes: free play and guided play. We suggest that games promote learning in ways similar to other playful learning situations. Games involve fun, a sense of curiosity and an inhibition of reality, in active, engaging, meaningful and socially interactive contexts. We argue that games therefore belong alongside free play and guided play to form a trio of playful learning experiences. This perspective adds to a growing understanding of the role of games in supporting children’s learning and development.     

Every unsuccessful problem solver is unsuccessful in his or her own way: affective and cognitive factors in proving

It is widely recognized that purely cognitive behavior is extremely rare in performing mathematical activity: other factors, such as the affective ones, play a crucial role. In light of this observation, we present a reflection on the presence of affective and cognitive factors in the process of proving. Proof is considered as a special case of problem solving and the proving process is studied adopting a perspective according to which both affective and cognitive factors influence it. To carry out our study, we set up a framework where theoretical tools coming from research on problem solving, proof and affect are present. The study is performed within a university course in mathematics education, where students were given a statement in elementary number theory to be proved and were asked to write down their proving process and the thoughts that accompanied this process. We scrutinize the written protocols of two unsuccessful students, with the aim of disentangling the intertwining between affect and cognition. In particular, we seize the moments in which beliefs about self and beliefs about mathematical activity shape the performance of our students.

Mathematics learning and practice in and out of school: A framework for connecting these experiences

A variety of researchers in the last fifteen years have described how people learn and use mathematics in out-of-school situations. These researchers have found that mathematics learning and practice in and out of school differ in a number of ways. In this paper we examine and discuss these differences while maintaining the position that while some differences may be inherent, many differences can be narrowed so that mathematics learning and practice in school and out of school can build on each other and be connected. Before discussing a framework that we think sheds some light on connecting these experiences, we present some research from several of our studies that illustrates some of the differences between in-school and out-of-school mathematics practice and lays the groundwork for the discussion of the framework.We then discuss Saxe's (1991) research framework for gaining insight into the interplay between sociocultural and cognitive development processes through the analysis of practice participation (p. 13). Although Saxe's framework is a method for studying the interplay between sociocultural and cognitive development processes, we propose that it may be helpful in working towards connecting in-school and out-of-school mathematics learning and practice. Thus, we discuss the framework with illustrations from our own research, and then elaborate on ways to make this interplay between in-school and out-of-school contexts more deliberate.     

PERSONALITY-CAREER FIT AND FRESHMAN MEDICAL CAREER ASPIRATIONS: A Test of Holland's Theory

Employing Holland's personality theory ofoccupational decision making, this study explores howpersonality-career fit influences initial medical careeraspirations among college freshmen, and the extent to which such fit is associated with themaintenance or abandonment of those aspirations.Personality types thought to be common among physicians(Holland's investigative, social, or artistic types) are predictive of aspiring to a medicalcareer-findings that validate the importance offit between an individual's personalityand career choice generally, and the choice of medicinespecifically. Moreover, the personality characteristics ofstudents are related to the careers they later choose asalternatives to medicine. Overall, these findingsprovide additional support for Holland's occupational decision-making theory of personality-careerfit, and illustrate how personality characteristics aresystematically associated with changes in career choiceduring college.     

Designing and using open-ended software to promote conceptual change

We describe a recent project that explored the use of interactive computer software for teaching Einstein's special theory of relativity to secondary school studients. Our approach couples results from recent cognitive science research with modern techniques for using computers to help students visualize and experiment with otherwise inaccessible phenomena. One the products of this research is RelLab, a computer-based exploratory tool for constructing gedanken, or thought experiments involving physical systems in relative motion. We will describe our efforts in designing and testing this software for affecting change in students' concepts of space and time. Relativity is ideally suited for such a study because understanding it requires a radical reconceptualization of these quantities.Advanced Physics from an Elementary Viewpoint, NSF grant MDR-9016417.     

`Creative mathematics' – real or rhetoric?

The notion of creativity has its natural home in the fine arts, where the artist literally creates something that can be perceived by the senses. The products of mathematical activity are clearly not of this kind, yet some distinguished mathematicians have claimed that mathematics offers considerable scope for creativity. The title of the book under review, and some claims to be found in it, suggest that creativity can indeed be associated with mathematics, and that young children may experience it in the classroom. We suggest that the word creative is being used in rather different senses in these different contexts, yet the meanings associated with the arts, say, are in danger of being applied to mathematical situations for rhetorical purposes.This revised version was published online in September 2005 with corrections to the Cover Date.     

The road to solving a combinatorial problem is strewn with pitfalls

In this paper we examine typical difficulties involved in solving a combinatorial problem, and demonstrate them throughout the solution of the Problem of the misaddressed letters. We focus on difficulties of two sorts: Those which stem from the fact that a combinatorial problem is usually an infinite set of problems, and others which are involved in finding a systematic approach to their solution.     

Interactive development of subject matter in the mathematics classroom

There are considerable differences among mathematics teachers with regard to the quatlity of their way of developing mathematical knowledge in the classroom. Such differences are analysed. To develop mathematical meaning requires both a consistent presentation of the mathematical symbols and of the referential meaning of these symbols with respect to the given task. On the basis of this conception we assume that the quality of teaching will differ according to how teachers cope with the relation between these two sides of meaning. From a sample of 26 teachers, an expert teacher and a non-expert teacher were selected by means of classroom observation with scales of instructional quality variables. For each of these two teachers, two lessons introducing probability (sixth grade) are analysed. For this purpose, teacher and student contributions are coded. For the expert teacher, graphic visualizations of the development of mathematical concepts across time show soft transitions between the different aspects of mathematical meaning. These transitions are made possible by a consistent explication of the relation between formal symbols and the given mathematical task. In the case of the other teacher, explication of the relationship between the object side and the symbol side of mathematical meaning is much rarer, and there are sudden switches from one aspect of meaning to another. Further differences concern the handling of student contributions.We gratefully acknowledge the help of Wolfgang Barz, Regina Dietrich and Claudia Krüger with recording, transcribing or coding lessons. For their comments on draft versions of the paper the authors thank Deborah Ball, Jere Brophy, Willibald Dörfler, Alexander Gruza and two anonymous reviewers.     

Interference and Inhibition in Cognition and Behavior: Unifying Themes for Educational Psychology

In this article, we review modern work on interference and inhibition in cognition and behavior. We begin by briefly reviewing the historical and conceptual roots of this new body of work. Next we discuss five new theoretical frameworks that use the concepts of interference and inhibition to explain cognitive and behavioral phenomena. In the section that follows, we review evidence of individual and developmental differences in inhibition and susceptibility to interference in at-risk students. We then consider the implications of this new body of work for research on educational psychology by discussing six selected areas of research: intelligence, strategies, reading comprehension, logical and mathematical reasoning, self-regulated learning, and retention. In the next section, we explore four critical issues that pose significant challenges to research in inhibition and interference. Finally, we discuss the relevance of this work to educational practice by focusing on its implications for the curriculum and instruction. Throughout, our principal goal is to bring this new body of work to the attention of the readers of this journal and to show how it might be useful in guiding research and theory in educational psychology.     

Better communication = better relationships: Tips for caregivers

We all live in a social environment with others. Indeed, ... we desire and seek out relationships with others, and we have personal needs that can be satisfied only through interacting with other humans (Johnson, 1986). It is not possible to think of living without relating to other people, nor is it possible to develop satisfying relationships without some sort of communication. Communication skills are the tools, the means, the mechanism through which we develop our relationships. Just as a carpenter would not expect to produce a fine piece of furniture with a blunt saw, neither should we assume that we can develop productive relationships with other people, such as children and their parents, colleagues or professional associates, using inadequate or ineffective communication skills. Johnson (1986) suggested that interpersonal skills are learned just as any other skills are learned. However, simply having learned communication skills is not enough! We need to continually practice these skills and occasionally be reminded of their value for interacting with and developing relationships with others. This, therefore, is a small reminder that all of us, caregivers and parents alike, can do with sharpening up our style of communicating with others and learn to create better and more satisfying human relationships!Jillian Rodd is a psychologist who lectures at the School of Early Childhood Studies, University of Melbourne, Australia. She acts as a consultant in parent education and in-services child care staff and for preschool teachers.     

From behaviourism to cognitive behaviourism to cognitive development: Steps in the evolution of instructional design

The recent history of instructional technology is traced, starting with the work of Skinner, moving on to the task analytic approach of Gagné, and following through to contemporary efforts associated with the cognitive revolution. It is suggested that an understanding of the process of cognitive development may enable us to build on and improve earlier approaches, by adapting them more directly to students' current levels of cognitive development, and by ensuring that we do not overtax their information processing capabilities. To illustrate and support this claim, a number of recent instructional studies are cited, some of which have utilized classic developmental tasks, and some of which have utilized conventional classroom material.Presented at the conference for Educational Technology in the 80's Caracas, Venezuela, June 14–18, 1982.     

Psychological bases of programmed instruction

This paper puts forward an argument for using a cybernetic approach to programmed instruction in order to make the control of the teaching process more efficient. It argues the need for a micro-approach to program construction which takes into account not only the correctness of response but also the psychological activity of the learner.The author presents a summary of Prof. P. Galperin's theory of stage by stage formation of mental actions as well as presenting a method for putting the theory into practice in the construction of teaching programs.Finally, there is a presentation of the methodology and results of various teaching programs in which control over the cognitive activity of the learner was carried out.This article is based on Professor Talyzina's work up to 1971. Ed.