Excerpt: In summary, I have agreed that science should include the creation of hypotheses; that if it does, a dialectic mode of argument (a special case of a conversation) must be considered as able to accommodate the abduction or eductive components of induction; and that, if so, the logic of conversation, including analogy, is of a certain type and that its truth valuation schemes are different from standard schemes. You may or may not call a theory constructed on this basis, tested by means proper to this basis, a scientific theory but it does have a predictive, generative, explicative but not strictlyspeaking explanatory or proof-theoretic power and considerable generality. Participants may be scientists, their points of view, social organisations or systems of belief like Lakatos “programmes of scientific research”: to science as a totality. The hypotheses of Section 1 are invented and agreed by the participants; so, also, are the test instruments used to select evidence for datahood, to determine measured values affirming or denying the data. These are all, ultimately, consensual, and their invention, to be agreed (or not) is not simply mechanical in the sense of section 1. The lack of standard objectivity is the price paid for proper scrutiny of the phenomena of concern and interest when it comes to problem formulation and even, maybe, to problem solving. If you accept my argument, the epistemology, logic and ontology of science must be enlarged to encompass the old but as a subject of the emerging enterprise.

Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition might best serve in articulating implications of Enactivism for mathematics education. We offer a blend of Dynamical Systems Theory and Sociocultural Theory as an analytic lens on micro-processes of action-to-concept evolution. We also illustrate the methodological utility of design-research as an approach to such theory development. Building on constructs from ecological psychology, cultural anthropology, studies of motor-skill acquisition, and somatic awareness practices, we develop the notion of an “instrumented field of promoted action”. Children operating in this field first develop environmentally coupled motor-action coordinations. Next, we introduce into the field new artifacts. The children adopt the artifacts as frames of action and reference, yet in so doing they shift into disciplinary semiotic systems. We exemplify our thesis with two selected excerpts from our videography of Grade 4–6 volunteers participating in task-based clinical interviews centered on the Mathematical Imagery Trainer for Proportion. In particular, we present and analyze cases of either smooth or abrupt transformation in learners’ operatory schemes. We situate our design framework vis-à-vis seminal contributions to mathematics education research.

Key words: mathematics, education, motor learning, conceptual learning, motor problem, conceptual metaphor

In recent years constructivism has been increasingly confronted with severe criticism concerning a number of weaknesses in its argumentation: \\– constructivism draws upon the reliability of empirical neurobiological findings\\– it is fixated on individuals and thus disregards the relevance of language, culture and society for reasoning and action\\– the clarification of the concept of construction is deficient\\– constructivism is fixated on cognition and thus underestimates the relevance of (joint) action\\– it disregards the relevance of emotions in processes of cognition and action\\– its epistemology is still dualistic as it maintains the distinction between a constructed reality and a reality inaccessible to cognition (the German words “Wirklichkeit” and “Realität” reflect this distinction) and the differentiation of language and language-independent reality. \\Such weaknesses should henceforth be avoided by a consistent consideration of the observer resulting in a strict non-dualistic process-orientation in describing cognition, communication and action in a framework of culture, narratives and discourses.

In recent years, cognitive science has witnessed the emergence of various ideas that challenge standard conceptions of perception, cognition, and action by emphasizing the cognizing subject’s active role in the construction of experiential world and questioning the subject-world relationship. However familiar these insights might sound to a constructivist, they are rarely linked to constructivist philosophy. This chapter aims at exploring the possibility of relating constructivist epistemology to recent empirical work in cognitive science by focusing on two examples: the predictive processing approach and the interface theory of perception. We suggest that constructivist epistemology could importantly contribute to these developing lines of research. Conversely, these new approaches might offer a suitable framework for the operationalization of constructivist concepts and their further empirical and computational examination.

The author examines education from a constructivist perspective, assuming that its goal is independent thinking rather than the maintenance of the status quo. He distinguishes training from teaching, stressing that only the latter is concerned with understanding. Concepts and conceptual structures are seen as the material of understanding, and their dependence on language and subjective interpretations is presented. The belief that words refer to observer-independent things rather than to speakers’ and listeners’ experiences is seen as one of the age-old obstacles to conceptual education. After a brief mention of important steps in the history of epistemology, the naïve realism of popular writings on science and science textbooks is shown to be incompatible with the ideas of the great scientists of the last century, whose views were remarkably close to constructivism. The construction of concepts is illustrated by an example from visual perception, the conceptual formation of a well-known constellation, and the construction of pluralities. An examination of these constructions yields principles that may be of interest to educators. These principles are then illustrated by means of a didactic experiment in physics teaching.

Key words: education, philosophy

The current frequent use of the words ‘information’, ‘communication’, ‘representation’, and ‘meaning’ introduces misleading metaphors into some of our most important social interactions. The paper presents an outline of the constructivist approach to the underlying phenomena as a more adequate and productive way of thinking – it shows linguistic ‘meaning’ to be the result of individuals’ reflection upon their experience.

Key words: language, semiotics, communication

Students’ views of the nature of scientific knowledge have been recognized as an important component of science learning environments. In this study, we analyze an extensive data base consisting of 23 students’ written and oral discourse about ontology, epistemology, and sociology of scientific knowledge collected over a 15-month period in the context of two consecutive junior- and senior-level physics courses. Over a 2-month period at the beginning of the second year, students read, reflected on, and talked about a text which discusses epistemology in the context of physics. Our study shows that students drew on nine types of discursive resources to support their ontological, epistemological, and sociological claims. Toward the end of the study, the range and number of supportive statements had increased. Simultaneously, few students changed their ontological and sociological claims, but a considerable number changed their epistemological claims. Two case studies illustrate the development of student discourse in the course of the study.

An experimental analysis of the claim that individual constructive activity is social was carried out using two categories of interactions, subjective-environment interactions and interactions of operations within the individual. The primary question of the analysis concerns the contribution of these two kinds of interaction to the individual construction of a mathematical scheme called the equi-partitioning scheme. It was found that interactions of operations within the individual were the principal interactions contributing to the construction of the scheme. It was further found that these interactions were occasioned by subject-environment interactions. But the subject-environment interactions could not be used to explain the individual construction of the scheme.