The higraph, a general kind of diagramming object, forms a visual formalism of topological nature. Higraphs are suited for a wide array of applications to databases, knowledge representation, and, most notably, the behavioral specification of complex concurrent systems using the higraph-based language of statecharts.

On visual formalisms

Computational synergetics and mathematical innovation

"in mathematics ... we find two tendencies present. On the one hand, the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.

Editors' introduction: What is mathematical visualization?

Our main goal in this study is to exemplify that a meticulous design can lead pre-service teachers to engage in productive unguided peer argumentation. By productivity, we mean here a shift from reasoning based on intuitions to reasoning moved by logical necessity. As a subsidiary goal, we aimed at identifying the kinds of reasoning processes (visual, inquiry-based, and deductive) pre-service teacher's students adopt, and how these reasoning processes are interwoven in peer-unguided argumentation. We report on a case study in which one dyad participating in a pre-service teachers program solved a mathematical task. We relied on three principles to design an activity: (a) creating a situation of conflict, (b) creating a collaborative situation, and (c) providing a device for checking hypotheses/conjectures. We show how the design afforded productive argumentation. We show that the design of the task entailed argumentation which first relied on intuition, then intertwined the activities of conjecturing and checking conjectures by means of various hypotheses-testing devices (measurement, manipulations, and dynamic change of figures with Dynamic Geometry software), leading to a conflict between conjectures and the outcome of the manipulation of DG software. Peer argumentation then shifted to abductive and deductive considerations towards the solution of the mathematical task. These beneficial outcomes resulted from collaborative rather than adversarial interactions as the students tried to accommodate their divergent views through the co-elaboration of new explanations.

From visual reasoning to logical necessity through argumentative design

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 Winnicott's (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.

On Mathematical Visualization and the Place Where We Live

In this paper we make a distinction between the practice of mathematics as it is usually presented—a logical chain of abstract, analytical reasoning from premises to conclusions—and how mathematics seems to be done in actuality—as a series of nonverbal, analog, often kinesthetic or visual insights Mathematics in recent years has created a hierarchy with highly abstract,, logical and symbolic material at the peak and with more geometrical, visual, and analog material held to be of lesser worth We argue that humans are known to vary widely in their approaches to cognition and that the areas of the human brain specifically related to language and logical analysis seem to comprise only a part of the machinery of our intellect We suggest that it would be wise for the practitioners of mathematics, and perhaps especially the students of mathematics to be aware of the very important nonverbal elements in mathematics We feel that excessive emphasis on the abstract, analytic aspects of thought may have had dele

James A. Anderson

Papers

Nonanalytic Aspects of Mathematics and Their Implication for Research and Education