Becerra G. (2016) Los usos del constructivismo en las publicaciones científicas de Latinoamérica [Uses of constructivism among Latin-American scientific publications]. Mad 35: 38–59. https://cepa.info/4529

Constructivism is a heterogeneous intellectual movement that spans across different fields of knowledge. Within constructivism there is a variety of discussions that deal with their own questions and particular references, and that appear clustered in the journals and publications of different scientific areas. Attempting to clarify this communication, the present paper explores scientific publications from Latin America that include the term “constructivism” among their descriptors, as listed on CLASE, PERIODICA and SCIELO databases. These publications have been segmented into 3 very general groups, according to the way in which constructivism is used: (1) those that seek to “apply” constructivism to the problems of their area; (2) those that take “constructivism” as their object of study or criticism; (3) those that adopt constructivism as a “framework” for notional or conceptual analysis. Some data about those publication groups is described and compared in an attempt to show how scientific communication about constructivism organizes in Latin America (publication area, subjects, keywords, main authors).

Cariani P. (2012) Infinity and the Observer: Radical Constructivism and the Foundations of Mathematics. Constructivist Foundations 7(2): 116–125. https://cepa.info/254

Problem: There is currently a great deal of mysticism, uncritical hype, and blind adulation of imaginary mathematical and physical entities in popular culture. We seek to explore what a radical constructivist perspective on mathematical entities might entail, and to draw out the implications of this perspective for how we think about the nature of mathematical entities. Method: Conceptual analysis. Results: If we want to avoid the introduction of entities that are ill-defined and inaccessible to verification, then formal systems need to avoid introduction of potential and actual infinities. If decidability and consistency are desired, keep formal systems finite. Infinity is a useful heuristic concept, but has no place in proof theory. Implications: We attempt to debunk many of the mysticisms and uncritical adulations of Gödelian arguments and to ground mathematical foundations in intersubjectively verifiable operations of limited observers. We hope that these insights will be useful to anyone trying to make sense of claims about the nature of formal systems. If we return to the notion of formal systems as concrete, finite systems, then we can be clear about the nature of computations that can be physically realized. In practical terms, the answer is not to proscribe notions of the infinite, but to recognize that these concepts have a different status with respect to their verifiability. We need to demarcate clearly the realm of free creation and imagination, where platonic entities are useful heuristic devices, and the realm of verification, testing, and proof, where infinities introduce ill-defined entities that create ambiguities and undecidable, ill-posed sets of propositions. Constructivist content: The paper attempts to extend the scope of radical constructivist perspective to mathematical systems, and to discuss the relationships between radical constructivism and other allied, yet distinct perspectives in the debate over the foundations of mathematics, such as psychological constructivism and mathematical constructivism.

Castillo-Garsow C. W. (2014) Mathematical Modeling and the Nature of Problem Solving. Constructivist Foundations 9(3): 373–375. https://constructivist.info/9/3/373

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: Problem solving is an enormous field of study, where so-called “problems” can end up having very little in common. One of the least studied categories of problems is open-ended mathematical modeling research. Cifarelli and Sevim’s framework - although not developed for this purpose - may be a useful lens for studying the development of mathematical modelers and researchers in applied mathematics.

Cifarelli V. V. & Sevim V. (2014) Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis. Constructivist Foundations 9(3): 360–369. https://constructivist.info/9/3/360

Context: The paper utilizes a conceptual analysis to examine the development of abstract conceptual structures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC influenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? Problem: The paper documents how Ernst von Glasersfeld’s view of mental representation can be illustrated in the context of mathematical problem solving and used to explain the development of conceptual structure in mathematical problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internalization and interiorization of solution activity. Method: A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particularly her emerging and developing reflections about her solution activity. The interview was videotaped and written transcripts of the solver’s verbal responses were prepared. Results: The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly abstract levels of conceptual knowledge demonstrated by the solver. Implications: The results suggest a framework for an explanation of problem solving that is activity-based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed.

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.

François K. (2014) Convergences between Radical Constructivism and Critical Learning Theory. Constructivist Foundations 9(3): 377–379. https://constructivist.info/9/3/377

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: The value of Cifarelli & Sevim’s target article lies in the analysis of how reflective abstraction contributes to the description of mathematical learning through problem solving. The additional value of the article lies in its emphasis of some aspects of the learning process that goes beyond radical constructivist learning theory. I will look for common ground between the humanist philosophy of mathematics and radical constructivism. By doing so, I want to stress two converging elements: (i) the move away from traditionalist ontological positions and (ii) the central role of the students’ activity in the learning process.

Füllsack M. (2012) Communication Emerging? On Simulating Structural Coupling in Multiple Contingency. Constructivist Foundations 8(1): 103-110. https://constructivist.info/8/1/103

Problem: Can communication emerge from the interaction of “self-referentially closed systems,” conceived as operating solely on the base of the “internal” output of their onboard means? Or in terms of philosophical conceptions: can communication emerge without (“outward” directed) “intention” or “will to be understood”? Method: Multi-agent simulation based on a conceptual analysis of the theory of social systems as suggested by Niklas Luhmann. Results: Agents that co-evolutionarily aggregate probabilities on how to cope with their environment can structurally couple and generate a form of “eigenbehavior” that retrospectively (i.e., by an observer) might be interpreted as communication. Implications: The “intention” or the “will to be understood,” as prominently claimed to be indispensable in communication by theoreticians such as Jürgen Habermas, can be seen as a retrospective ascription to an emergent property of complex interaction. Constructivist content: The paper attempts to base constructivist reasoning on data generated in simulations.

Füllsack M. (2014) The Circular Conditions of Second-order Science Sporadically Illustrated with Agent-based Experiments at the Roots of Observation. Constructivist Foundations 10(1): 46–54. https://cepa.info/1160

Problem: The inclusion of the observer into scientific observation entails a vicious circle of having to observe the observer as dependent on observation. Second-order science has to clarify how its underlying circularity can be scientifically conceived. Method: Essayistic and conceptual analysis, sporadically illustrated with agent-based experiments. Results: Second-order science – implying science in general – is fundamentally and ineluctably circular. Implications: The circularity of second-order science asks for analytical methods able to cope with phenomena of complex causation and “synchronous asynchrony,” such as tools for analyzing non-linearly interacting dynamics, decentralized, clustered networks and in general, systems of complex interacting components.

Gash H. (2011) Moving Forward from Radical or Social Constructivism to a Higher Level Synthesis. Constructivist Foundations 7(1): 20–21. https://constructivist.info/7/1/020

Open peer commentary on the target article “From Objects to Processes: A Proposal to Rewrite Radical Constructivism” by Siegfried J. Schmidt. Upshot: Siegfried J. Schmidt’s timely article offers a fresh look at radical constructivism with an emphasis on contextually and culturally located action as an expression of knowing. Perhaps it remains cautious in making connections with neighbouring philosophical approaches. Two areas that are largely unmentioned are the issue of viability and the conceptual analysis, which remained largely on the sidelines in von Glasersfeld’s later work.

Glasersfeld E. von (1998) Die Radikal-Konstruktivistische Wissenstheorie. Ethik und Sozialwissenschaften 9(4): 503–511. https://cepa.info/1500

Die epistemologische Stellungnahme, die ich in meinen Büchern ausgeführt habe, wird hier kurz zusammengefaßt. Die Herkunft der konstruktivistischen Wissenstheorie aus vier Quellen – die Tradition des Skeptizismus, Jean Piagets Genetische Epistemologie, Ideen der Kybernetik und operationale Analyse der sprachlichen Kommunikation – werden erläutert und die konstruktivistische Orientierung im Aufbau grundlegender Begriffe wird an Hand von einigen Beispielen gezeigt. English: The paper is a brief exposition of the epistemological position I have presented in a number of books. The four, sources of the constructivist theory of knowing are explained: The tradition of scepticism, Jean Piaget’s Genetic Epistemology, cybernetical ideas, and the operational analysis of linguistic communication. The constructivist method of conceptual analysis is demonstrated with some basic examples.