Excerpt: In a typical study of students’ conceptions and conceptual change, researchers analyze what a student does or says in a classroom or in an interview and recognizes ideas that match or do not match their own understanding of the topic. Attributing the perspective they recognize in the student, those studies support the idea that a conception is the way by means of which an individual intrinsically conceives (of) a given phenomenon. They then hypothesize the existence of some mental structures that can be theoretically and objectively re-constructed based on what is observed in a student’s performance. Thus, researchers studying conceptions commonly assume that the observer and the observed are separate entities. However, even in the most theoretical and hardest of all sciences, physics, the independence of the measured object and the measuring subject is not taken for granted: Light, for example, will present itself as waves or as particles depending on how we examine it. The artificial sense of separation from the object(s) of study found in many accounts on students’ conceptions makes irrelevant the relationship that exists between the observer and the observed: an interdependence and co-emergence of the observer and the observed. This tight relation exists because each participant not only reacts upon what others say but also acts upon the reactions that his/her own actions give rise to. With this situation come epistemological, practical, and ethical implications for those researching in mathematics and science education. Positing or questioning the existence of an objective reality mediates how we accept or reject another human being and the worldviews s/he develops. It provides a rationale that guides our actions. This is especially important when it comes to teaching and learning at a time where the ability to deal with the plurality and diversity of human culture have emerged as significant referents for our social behavior.

I am going to use the occasion of the arrival of the (proper-ish) new millennium to write about aspects of my own work. I hope this won’t seem very self-centred or indulgent. The idea came to me at Christmas: like many, I send relatives and friends a “Year Report.” This year I decided, also, to attempt to explain what I’ve been working at in cybernetics for the last 30 years, or maybe my whole life. I know that what I wrote (and present here in reworked form) is under-argued and it might be relatively easy to pick holes in it. I hope you’ll not want to do that: this is an attempt to set a personal second order cybernetic scene which was originally intended for the completely uninformed, rather than a technical piece. On friend commented (about the Christmas version) that not only was it about a constructed world, but it actually constructed that world as well.

This paper starts by providing a succinct overview of the sensorimotor approach to phenomenal consciousness, describing its two parts: the part that concerns the quality of sensations, and the part that concerns whether or not such qualities are (consciously) experienced. The paper goes on to discuss the explanatory status of the approach, claiming that the approach does not simply “explain away” qualia, but that on the contrary, it provides a way of thinking about qualia that explains why they are the way they are, stimulates scientific paradigms and produces testable predictions. A final part of the paper examines the relation of the theory to radical enactivism, claiming that some kind of “higher order” cognitive mechanism similar to that used in Higher Order Thought theories of consciousness is needed to account for what is usually meant by being conscious of something.

Key words: qualia, phenomenal consciousness, radical enactivism, higher order thought, sensorimotor theory.

Excerpt: How do people learn mathematics? How is individual learning related to collective knowledge? What part does biology play in learning processes? What is the influence of the social context on the learning of mathematics? My thinking around these issues has been shaped by ideas coming from biology and neuroscience, particularly those related to the enactive approach to cognition. Ideas emerging from these theories emphasise the complexity of the nature of cognition and learning.

Ernst von Glasersfeld (1974) used the phrase Radical Constructivism (RC) in the context of clarifying the meaning of Piaget’s genetic epistemology. Initially, much of the work that followed was directly related to educational issues and topics like maths and science. The implications of radical constructivism for social understandings led the present author to study stereotypes. This work emphasised the role of identity in prejudice. Identity as an individual construction is related to one’s interpretations of the social heuristics and world views of one’s culture. Balancing self-perception with acceptable cultural expression is a key to well-being, personal development and one’s social functioning.

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.