Ricardo Nemirovsky

When the Classroom Floor Becomes the Complex Plane: Addition and Multiplication as Ways of Bodily Navigation

In this article we contribute a perspective on mathematical embodied cognition consistent with a phenomenological understanding of perception and body motion. It is based on the analysis of 4 selected episodes in 1 session of an undergraduate mathematics class. The theme of this particular class session was the geometric interpretation of the addition and multiplication of complex numbers. On the basis of these episodes, the article examines 2 conjectures: (a) The mathematical insights developed by an individual or a group are expressed in and constituted by perceptuo-motor activity, and (b) the learning of mathematical ideas is shaped in nondeterministic ways by the setting or learning environment.

Experiencing Change: The Mathematics of Change in Multiple Environments

ed from their work. Instead, the boys engaged in a process of making each environment into a lived-in space for themselves and of developing family resemblances across their trips in these lived-in spaces, to create a family of trips that included the similarities among trips as well as the distinct identity of each trip. Of course it is sometimes important for a teacher and her or his students to step back from a diverse set of activities and ask what they have in common and to reflect on the general mathematical principles that describe the activities. However, we argue that these general principles become meaningful and relevant only to the extent that they are rooted in an ongoing background of experiences.

Introduction to the Special Issue: Modalities of Body Engagement in Mathematical Activity and Learning

How is the active human body—through gesture production, manipulation of tools, mobility in the local environment, and interaction with others—involved in mathematical thinking and learning? This special issue of the Journal of the Learning Sciences presents three articles, each taking a different approach to this question. Two commentaries by accomplished scholars in the cognitive and learning sciences provide a critical perspective on the articles and pose far-reaching questions about studies of embodiment in human thinking and learning. These articles and commentaries appear amid a diverse set of theoretical proposals for how cognition is necessarily embodied, supported by a growing collection of empirical findings regarding the body, concepts, and cognition that come from a range of scientific disciplines. Not surprisingly, these articles and the larger interdisciplinary field reflect different commitments to the nature of the mind, to modes of scientific investigation in the learning sciences, and to what should count as an adequate or a productive explanation in our field. In this introduction, we start with a brief review of proposals for embodied cognition and what they can tell us about mathematical activity in particular. What is the nature of mathematical knowledge, what is the role of the body in

On Mathematical Visualization and the Place Where We Live

This paper is a case study of how a high school student, whom we call Karen, used a computer-based tool, the Contour Analyzer, to create graphs of height vs. distance and slope vs. distance for a flat board that she positioned with different slants and orientations. With the Contour Analyzer one can generate, on a computer screen, graphs representing functions of height and slope vs. distance corresponding to a line traced along the surface of a real object. Karen was interviewed for three one-hour sessions in an individual teaching experiment. In this paper, our focus is on how Karen came to recognize by visual inspection the mathematical behavior of the slope vs. distance function corresponding to contours traced on a flat board. Karen strove to organize her visual experience by distinguishing which aspects of the board are to be noticed and which ones are to be ignored, as well as by determining the point of view that one should adopt in order to ‘see’ the variation of slope along an object. We have found it inspiring to use Winnicotts (1971) ideas about transitional objects to examine the role of the graphing instrument for Karen. This theoretical background helped us to articulate a perspective on mathematical visualization that goes beyond the dualism between internal and external representations frequently assumed in the literature, and focuses on the lived-in space that Karen experienced which encompassed at once physical attributes of the tool and human possibilities of action.

Episodic Feelings and Transfer of Learning

The goal of this article is to develop a new perspective on transfer of learning integrating cognition, emotion, and bodily experience. It is based on a case study with a 10-year-old girl as she explored the use of a motion detector, allowing for the simultaneous graphing of the position versus time of 2 moving points. The paper elaborates on the notion of episodic feeling and illustrates a phenomenological approach for the study of transfer of learning whose point is not ascertaining mechanisms of transfer but elucidating within the infinite landscape of human experiences certain ones that seem amenable to characterization as transfer of learning.

From private intuitions to public symbol systems: An examination of the creative process in georg cantor and sigmund freud

Case studies of major creative figures who were active in different domains can help to indicate commonalities and distinctive features in the creative process. With this goal in mind, a comparison is made between the mathematician Georg Cantors study of various orders of infinity and the psychologist Sigmund Freuds exploration of the operation of the unconscious. In both cases, similar processes can be discerned: (a) articulations of a new intuition; (b) construction of local coherences; (c) the reworking of standard symbol systems, giving way to the creation of a new, more adequate symbolic system; and (d) the articulation of a new thema (Holton, 1988). The study also describes a number of contrasts, among them the criteria by which formulations are judged in the two domains, the contrasting cosmological stances assumed by the investigators toward their projects, and the differing needs for a formal symbol system.

Facilitating Grounded Online Interactions in Video-Case-Based Teacher Professional Development.

The use of interactive video cases for teacher professional development is an emergent medium inspired by case study methods used extensively in law, management, and medicine, and by the advent of multimedia technology available to support online discussions. This paper focuses on Web-based “grounded” discussions—in which the participants base their contributions on specific events portrayed in the case—and the role facilitators play in these online interactions. This paper analyzes the online exchange of messages in one school district that participated in a video-case-based program of teacher professional development and derives principles that will help facilitators lead grounded online interactions.

Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics?

Children creating ways to represent changing situations: On the development of homogeneous spaces

This paper focuses on children creating representations on paper for situations that change over time. We articulate the distinction between homogeneous and heterogeneous spaces and reflect on childrens tendency to create hybrids between them. Through classroom and interview examples we discuss two families of tasks that seem to facilitate childrens development of homogeneous spaces: 1) Making selected features directly visible, instead of requiring intermediate steps and calculations; for example, to be able to directly compare different sets of data combined in a single graph, and 2) Exploring well-defined figural components that can be used in graphing, such as line segments or sequencing from left to right, that are introduced as a resource.

On Guessing the Essential Thing

This paper relates to the ongoing discussion in the domain of psychology of (mathematics) education about the consequences of the ‘situated cognition’ paradigm for the issues of generalization and transfer of learning. This chapter focuses on the nature on generalizing. A first issue that is examined is the relationship between principles, laws, and definitions, on the one hand, and the circumstances in which they are validly applied. In this respect, a distinction is made between formal and situated generalizations across the following three areas: (a) where the generalization takes place, (b) values associated with generalizing, and (c) how generalizations relate to particular instances. A second part to be published elsewhere elaborates on transfer of learning. These papers articulate the claims through an in-depth analysis of an interview with Clio, an 11-year old girl working with problems involving the graphical representation of motion.