Peter Damerow

Mathematics Education and Society

It must be stated at the very outset that the topic of mathematics education and society is not an essential part of the prevailing discourse in education.’ Those who design and plan mathematics education usually do not at the same time give serious thought to the circumstances that determine the domain of the educational efforts. Therefore I do not feel obliged to discuss primarily the “state of research,” to delineate “approaches,” and to cite “investigations” when presenting the facts relevant to the topic. This would presuppose a consistent or, at the very least, a coherent, albeit possibly controversial, educational discourse on this subject.

Hunting the White Elephant: When and How did Galileo Discover the Law of Fall?

we present a number of findings concerning galileos major discoveries which question both the methods and the results of dating his achievements by common historiographic criteria. the dating of galileos discoveries is, however, not our primary concern. this paper is intended to contribute to a critical reexamination of the notion of discovery from the point of view of historical epistemology. we claim that the puzzling course of galileos discoveries is not an exceptional comedy of errors but rather illustrates the normal way in which scientific progress is achieved. we argue that scientific knowledge generally develops not as a sequence of independent discoveries accumulating to a new body of knowledge but rather as a network of interdependent activities which only as a whole makes the individual steps understandable as meaningful “discoveries.”

Representation and Meaning

The concept of cognitive structure is central to those theories of thought that emerged as a result of the critical evaluation of the studies on developmental psychology and epistemology of the Geneva group around Jean Piaget. It denotes developmental stages of logico-mathematical structures of thinking which—based on empirical observations and experiments—were genetically reconstructed in this area of investigation.1 The considerations to be presented here deal with a particularly controversial question: What is the importance of the material representation of cognitive structures with regard to their ontogenetic and phylogenetic development and how should the representation be defined empirically and theoretically.

The Challenging Images of Artillery

Princely families of old were in the habit of engaging historians who were charged with producing tailor-made histories, in which the achievements of these families received due attention. For a long time this, remarkably, was exactly what natural scientists also expected from their historians. As a matter of fact, from the point of view of developed science, the knowledge of a discipline is simply represented by the natural laws that define its object. It was accordingly a matter of course for its historians to concentrate solely on the question of who had discovered which of these laws when and in what manner. In this sense, the history of science is a biographically oriented, heroic history of the great discoverers and their discoveries (fig. 1).

How Can Discontinuities in Evolution Be Conceptualized

Causal effects are different from evolutionary discontinuities. As far as Lock (2000) and Savage-Rumbaugh and Fields (2000) deal with the relation of animal behavior and human activity, they have in common that they argue against an implicit misinterpretation of evolutionary discontinuities as causal effects. As a consequence, it is argued in both papers that it is not the biologically determined animal nature that prevents primates from using human language. The two papers disagree, however, on Bickerton’s distinction of protolanguage and language proper and, therefore, on the nature of ‘ape language’. It is argued in the present commentary that this different judgment results from the fact that evolutionary progress appears continuous on the level of the individual, but discontinuous on the level of a whole population.

Scientific Revolution, History and Sociology of

This contribution analyzes the material, social, and cognitive dimensions of the Scientific Revolution as independent and irreducible dimensions. In discussing the social dimension, the contribution will delineate the impact of the great engineering projects of the Renaissance on the formation of a new social group, dealing both with the practical problems and with the theory of nature. In discussing the material dimension, it will address the intellectual challenges represented by new objects of contemporary technological practices and, in discussing the intellectual dimensions, it will show how mental models of traditional thinking merge with those of practical thinking to form the basis for what eventually will become the conceptual novelties generated by the Scientific Revolution.

Number as a second-order concept

My contribution will focus on a central issue of Yehuda Elkanas anthropology of knowledge — namely, the role of reflectivity in the development of knowledge. Let me therefore start with a quotation from Yehudas paper “Experiment as a Second-Order Concept.”

On the Relationship between Ontogenesis and Historiogenesis of the Number Concept

The present study has as its subject Piaget’s psychogenetic theory of the historical development of cognitive structures. Using the number concept as an example, the connections between the ontogenetic and the historical development of cognitive constructs, between cognition and culture, postulated by Piaget are examined.

The Development of Arithmetical Thinking: On the Role of Calculating Aids in Ancient Egyptian and Babylonian Arithmetic

The early forms of thinking can only be reconstructed on the basis of mere speculation. Up to now the detailed analyses of the ontogenetic development of thinking presented by developmental psychology are without a phylogenetic counterpart of comparable complexity. This situation is unsatisfactory, all the more so because the explanatory models provided by developmental psychology have far-reaching implications with regard to phylogenetic development. Taking a position in the controversy within developmental psychology concerning the existence of cognitive universals is at one and the same time a decision regarding the historical or the unhistorical nature of certain forms of thinking. A choice between competing explanatory models adapted from biology and socialization theory and applied to the development of language and thinking in children ipso facto determines the delimitations between the evolution predating the emergence of human beings and the historical development of the human species. In this situation progress is to be expected only to the extent that the speculative theories are increasingly supplemented by reconstructions of the early forms of thinking guided primarily by the available evidence of the past and corroborated by the latter.

Mental Models as Cognitive Instruments in the Transformation of Knowledge

The chapter is concerned with the epistemic structures of mechanical knowledge in its historical transformations. It describes these structures using the concept of mental models as cognitive instruments, which function as mediators between the realm of practice and experience on the one hand, and conceptual systems on the other. With the help of the concept of mental model, the chapter discusses how mechanical knowledge has emerged from experience in practical contexts and how it was transformed into theoretical and mathematically formalized knowledge systems. Focusing on one particular mental model, which describes the cognitive structure conceptualizing motion as being caused by forces, the chapter then follows its transformations in the long-term history of mechanical thinking. This so-called “motion-implies-force” model is rooted in intuitive, non-written mechanical knowledge. Over the course of history, the model was recruited, complemented, and transformed in the context of the use of mechanical tools and articulated in the work of practitioners dealing with machines, arms, ships, buildings, fortifications, and the like. Eventually, under specific cultural circumstances, this and other mental models were elaborated and integrated into mathematically formalized systems that were used, for example, in the explanation of terrestrial and celestial motions in early modern natural philosophy and the mathematical disciplines of European universities.