The paradox of scientific expertise is that the growth of science leads to a fragmentation of scientific expertise. To resolve this paradox, this paper probes three hypotheses: 1) All scientific knowledge is perspectival. 2) The perspectival structure of science leads to specific forms of knowledge asymmetries. 3) Such perspectival knowledge asymmetries must be handled through second order perspectives. We substantiate these hypotheses on the basis of a perspectivist philosophy of science grounded in Peircean semiotics and autopoietic systems theory. Perspectivism is an important elaboration of constructivist approaches to help overcome problems in cross-disciplinary collaboration and use of science, and thereby make society better able to solve complex, real-world problems.

Context: The problems that are most in need of interdisciplinary collaboration are “wicked problems,” such as food crises, climate change mitigation, and sustainable development, with many relevant aspects, disagreement on what the problem is, and contradicting solutions. Such complex problems both require and challenge interdisciplinarity. Problem: The conventional methods of interdisciplinary research fall short in the case of wicked problems because they remain first-order science. Our aim is to present workable methods and research designs for doing second-order science in domains where there are many different scientific knowledges on any complex problem. Method: We synthesize and elaborate a framework for second-order science in interdisciplinary research based on a number of earlier publications, experiences from large interdisciplinary research projects, and a perspectivist theory of science. Results: The second-order polyocular framework for interdisciplinary research is characterized by five principles. Second-order science of interdisciplinary research must: 1. draw on the observations of first-order perspectives, 2. address a shared dynamical object, 3. establish a shared problem, 4. rely on first-order perspectives to see themselves as perspectives, and 5. be based on other rules than first-order research. Implications: The perspectivist insights of second-order science provide a new way of understanding interdisciplinary research that leads to new polyocular methods and research designs. It also points to more reflexive ways of dealing with scientific expertise in democratic processes. The main challenge is that this is a paradigmatic shift, which demands that the involved disciplines, at least to some degree, subscribe to a perspectivist view. Constructivist content: Our perspectivist approach to science is based on the second-order cybernetics and systems theories of von Foerster, Maruyama, Maturana & Varela, and Luhmann, coupled with embodied theories of cognition and semiotics as a general theory of meaning from von Uexküll and Peirce.

Key words: Perspectivism, semiotics, complex phenomena, social systems theory, differentiation of science, perspectival knowledge asymmetries.

In the Developmental Teacher Education (DTE) program at the University of California at Berkeley, Piagetian developmental theory and research is used as core knowledge for preparing elementary school teachers. Developmental-constructivist principles of knowledge acquisition are particularly well-suited for this purpose because they have implications for what and how children are taught, how progress toward expertise in teaching is conceptualized, and how teachers are educated. The authors describe the 2-year, postgraduate program, whose features include small cohorts of students, course work organized to address key topics repeatedly and hierarchically, multiple student teaching placements in diverse settings, and a master’s project on a teaching-learning issue. They also comment on teaching practices of program graduates and the development of their understanding of children, learning, and teaching during and after graduation from the program.

Abstract In this article, we present a re-framing of tea- cher development that derives from our convictions regarding the enactive approach to cognition and the bio- logical basis of being. We firstly set out our enactivist stance and then distinguish our approach to teacher development from others in the mathematics education literature. We show how a way of working that develops expertise runs through all mathematics education courses at the University of Bristol, and distil key principles for running collaborative groups of teachers. We exemplify these principles further through analysis of one group that met over 2 years as part of a research project focused on the work of Gattegno. We provide evidence for the effec- tiveness of the group in terms of teacher development. We conclude by arguing that the way of working in this group cannot be separated from the history of interaction of participants.

Key words: mathematics teacher development, awareness, purposes, collaborative group, enactivism, expertise.

Purpose: This study aims to explore, illuminate and hence evoke further reflections on the implications of creating and conserving distinctions that inherently act as simplifications and limit appropriate action. Design/methodology/approach – The approach used was reflective regarding the chosen concept of designing and learning from the perspective of a constitutive epistemology. These were investigated as circularities and as distinctions in language. The variety of intended meanings and hence implicit entailments was examined from the perspective of implicit domains. Findings: A tendency to focus on the results of designing and learning rather than the processes was attributed to several factors including cultural relevance, tangibility, durability and observability. Further, it was found that result and process are arbitrary distinctions in a circular system. It was noted that lack of awareness of multiple domains encourages reification, and that distinctions inherently obscure what happens in the non-articulated aspects of living. However, expertise embraces an ability to attend to such “betweens”. This applies to expertise in the assessment of learning and designing. Originality/value – The most obvious value of the findings is for the field of education. The insights gained indicate that the path of individualized learning with an emphasis on attention to the processes, inclusive of those that are not distinguished and named but can, with reflective experience, be sensed and acted on, has deep epistemological roots. A further implication is that educators require expertise to effectively work with learners, and that effective assessment depends on recurrent conversational interactions between the educator and learner.

Key words: recursion, language, learning, education, epistemology.

Context: There seem to be relatively few sustained implementations of microworlds in mathematics instruction. Problem: We explore the roles of and demands on university instructors to create an environment that supports students’ constructionist learning experiences as they design, program, and use interactive environments (i.e., microworlds) for doing mathematics. Method: We draw on the experiences of instructors in programming-based courses implemented since 2001 at Brock University, Canada, as a case study, and use Ruthven’s model on the professional adaptation of classroom practice with technology to guide our analysis of these experiences. Results: We describe how, in adapting to a design of empowering students to engage in programming for authentic mathematical explorations, instructors adopt characteristics of constructionist teaching that, nevertheless, demand expertise, a shift in traditional roles, and time from instructors. Implications: The results contribute to our understanding of roles of and demands on “ordinary” instructors in classrooms, who aim to create rich environments for supporting students’ constructionist learning experiences of computational thinking for mathematics. Constructivist content: The teaching approach aligns with Papert’s constructionism: a constructivist learning theory, but also a pedagogical paradigm. However, the approach presented has two salient characteristics: it is a university-level constructionist implementation, and it is a sustained long-term authentic classroom implementation. The focus is on the roles of and demands on instructors in that kind of implementation. Through the analysis using Ruthven’s work, we enrich our understanding of constructionist teaching features.

Key words: Computational thinking, constructionism, mathematics, programming, microworlds, university, teachers.

Open peer commentary on the article “Eigenform and Reflexivity” by Louis H. Kauffman. Upshot: Kauffman proposes to understand scientific thinking as including not only observations but also the act that enables their intentional use. This provides a constructivist opportunity: extending scientific thinking to gaining personal expertise.

Existential phenomenologists hold that the two most basic forms of intelligent behavior, learning, and skillful action, can be described and explained without recourse to mind or brain representations. This claim is expressed in two central notions in Merleau-Ponty’s Phenomenology of Perception: the intentional arc and the tendency to achieve a maximal grip. The intentional arc names the tight connection between body and world, such that, as the active body acquires skills, those skills are “stored”, not as representations in the mind, but as dispositions to respond to the solicitations of situations in the world. A phenomenology of skill acquisition confirms that, as one acquires expertise, the acquired know-how is experienced as finer and finer discriminations of situations paired with the appropriate response to each. Maximal grip names the body’s tendency to refine its responses so as to bring the current situation closer to an optimal gestalt. Thus, successful learning and action do not require propositional mental representations. They do not require semantically interpretable brain representations either. Simulated neural networks exhibit crucial structural features of the intentional arc, and Walter Freeman’s account of the brain dynamics underlying perception and action is structurally isomorphic with Merleau-Ponty’s account of the way a skilled agent is led by the situation to move towards obtaining a maximal grip.

Key words: mental representation, successful learning, maximal grip, skillful action, tight connection.

Existential phenomenologists hold that the two most basic forms of intelligent behavior, learning, and skillful action, can be described and explained without recourse to mind or brain representations. This claim is expressed in two central notions in Merleau-Ponty’s Phenomenology of Perception: the intentional arc and the tendency to achieve a maximal grip. The intentional arc names the tight connection between body and world, such that, as the active body acquires skills, those skills are “stored,” not as representations in the mind, but as dispositions to respond to the solicitations of situations in the world. A phenomenology of skill acquisition confirms that, as one acquires expertise, the acquired know-how is experienced as finer and finer discriminations of situations paired with the appropriate response to each. Maximal grip names the body’s tendency to refine its responses so as to bring the current situation closer to an optimal gestalt. Thus, successful learning and action do not require propositional mental representations. They do not require semantically interpretable brain representations either. Simulated neural networks exhibit crucial structural features of the intentional arc, and Walter Freeman’s account of the brain dynamics underlying perception and action is structurally isomorphic with Merleau-Ponty’s account of the way a skilled agent is led by the situation to move towards obtaining a maximal grip.

Key words: Mental representation, successful learning, maximal grip, skillful action, tight connection.

This article offers an account and defence of constructionism, both as a metaphilosophical approach and as a philosophical methodology, with references to the so-called maker’s knowledge tradition. Its main thesis is that Plato’s ‘‘user’s knowledge’’ tradition should be complemented, if not replaced, by a constructionist approach to philosophical problems in general and to knowledge in particular. Epistemic agents know something when they are able to build (reproduce, simulate, model, construct, etc.) that something and plug the obtained information into the correct network of relations that account for it. Their epistemic expertise increases with the scope and depth of the questions that they are able to ask and answer. Thus, constructionism deprioritises mimetic, passive, and declarative knowledge that something is the case, in favour of poietic, interactive, and practical knowledge of something being the case. Metaphilosophically, constructionism suggests adding conceptual engineering to conceptual analysis as a fundamental method.

Key words: constructionism, constructivism, epistemology, levels of abstraction, minimalism, philosophy of information.