Gobbo F. & Benin M. (2011) Constructive adpositional grammars: Foundations of constructive linguistics. Cambridge Scholar Publishing, Newcastle upon Tyne.

This book presents a new paradigm of natural language grammar analysis, based on adposition as the key concept, considered a general connection between two morphemes or group of morphemes. The adpositional paradigm considers the morpheme as the basic unit to represent morphosyntax, taken as a whole, in terms of constructions, while semantics and pragmatics are treated accordingly. All linguistic observations within the book can be described through the methods and tools of Constructive Mathematics, so that the modelling becomes formally feasible. A lot of examples taken from natural languages belonging to different typological areas are offered throughout the volume in order to explain and validate the modeling with special attention given to ergativity. Finally, an application of the paradigm is given, i.e., conversational analysis of the transcript of therapeutic settings in terms of constructive speech acts. The main goal of this book is to broaden the scope of Linguistics by including Constructive Mathematics in order to deal with known topics such as grammaticalization, children’s speech, language comparison, dependency and valency from a different perspective. It primarily concerns advanced students and researchers in the field of Theoretical and Mathematical Linguistics but the audience can also include scholars interested in applications of Topos Theory in Linguistics. Relevance: The book is relevant for constructivism in linguistics, derived from cognitivism.

Goldenberg E. P. (2019) Author’s Response: Constructionist Curriculum Construction, Nutritional Supplements, and Language. Constructivist Foundations 14(3): 337–341. Fulltext at https://cepa.info/6048

Abstract: Crafting constructionist supplements to enrich curriculum is not easy; crafting a full set of constructionist-designed materials for day-to-day use by students and teachers is downright hard; both are possible. If one chooses to build in programming, decisions about what computer language has the “ideal” characteristics may depend on the specific subject matter or purpose to which that language will be applied. Mathematics, even for young children, imposes demands on that programming language - among them, the ability to create and compose functions - that other expressive purposes may not.

Goldenberg E. P. (2019) Problem Posing and Creativity in Elementary-School Mathematics. Constructivist Foundations 14(3): 319–331. Fulltext at https://cepa.info/6045

Context: In 1972, Papert emphasized that “[t]he important difference between the work of a child in an elementary mathematics class and […]a mathematician” is “not in the subject matter […]but in the fact that the mathematician is creatively engaged […]” Along with creative, Papert kept saying children should be engaged in projects rather than problems. A project is not just a large problem, but involves sustained, active engagement, like children’s play. For Papert, in 1972, computer programming suggested a flexible construction medium, ideal for a research-lab/playground tuned to mathematics for children. In 1964, without computers, Sawyer also articulated research-playgrounds for children, rooted in conventional content, in which children would learn to act and think like mathematicians. Problem: This target article addresses the issue of designing a formal curriculum that helps children develop the mathematical habits of mind of creative tinkering, puzzling through, and perseverance. I connect the two mathematicians/educators - Papert and Sawyer - tackling three questions: How do genuine puzzles differ from school problems? What is useful about children creating puzzles? How might puzzles, problem-posing and programming-centric playgrounds enhance mathematical learning? Method: This analysis is based on forty years of curriculum analysis, comparison and construction, and on research with children. Results: In physical playgrounds most children choose challenge. Papert’s ideas tapped that try-something-new and puzzle-it-out-for-yourself spirit, the drive for challenge. Children can learn a lot in such an environment, but what (and how much) they learn is left to chance. Formal educational systems set standards and structures to ensure some common learning and some equity across students. For a curriculum to tap curiosity and the drive for challenge, it needs both the playful looseness that invites exploration and the structure that organizes content. Implications: My aim is to provide support for mathematics teachers and curriculum designers to design or teach in accord with their constructivist thinking. Constructivist content: This article enriches Papert’s constructionism with curricular ideas from Sawyer and from the work that I and my colleagues have done. Key words: Problem posing, puzzles, mathematics, algebra, computer programming.

Goldin G. A. (1990) Epistemology, constructivism, and discovery learning of mathematics. In: Davis R. B., Maher C. A. & Noddings N. (eds.) Constructivist views on the teaching and learning of mathematics. National Council of Teachers of Mathematics, Reston VA: 31–47. Fulltext at https://cepa.info/2976

Excerpt: What is the best way to characterize the body of knowledge that we call mathematics? How do children and adults learn mathematics most effectively? How can we best study their learning processes, and assess the outcomes of learning? Can meaningful learning be consistently distinguished from nonmeaningful or rote learning? What constitutes effective mathematics teaching, and how can elementary and secondary school teachers be enabled to provide it?

Gopnik A. & Wellman H. M. (2012) Reconstructing constructivism: Causal models, Bayesian learning mechanisms, and the theory theory. Psychological Bulletin 138(6): 1085–1108. Fulltext at https://cepa.info/4909

We propose a new version of the “theory theory” grounded in the computational framework of probabilistic causal models and Bayesian learning. Probabilistic models allow a constructivist but rigorous and detailed approach to cognitive development. They also explain the learning of both more specific causal hypotheses and more abstract framework theories. We outline the new theoretical ideas, explain the computational framework in an intuitive and nontechnical way, and review an extensive but relatively recent body of empirical results that supports these ideas. These include new studies of the mechanisms of learning. Children infer causal structure from statistical information, through their own actions on the world and through observations of the actions of others. Studies demonstrate these learning mechanisms in children from 16 months to 4 years old and include research on causal statistical learning, informal experimentation through play, and imitation and informal pedagogy. They also include studies of the variability and progressive character of intuitive theory change, particularly theory of mind. These studies investigate both the physical and the psychological and social domains. We conclude with suggestions for further collaborative projects between developmental and computational cognitive scientists.

Greenstein S. (2014) Making sense of qualitative geometry: The case of amanda. Journal of Mathematical Behavior 36: 73–94. Fulltext at https://cepa.info/1195

This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry environment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as “qualitative geometry,” that have implications for the teaching and learning of geometry and for research into students’ mathematical reasoning. Relevance: This article describes a study for which I used Steffe & Thompson’s teaching experiment methodology to produce a learning trajectory (Steffe 2003, 2004) resulting from the actual teaching of children. In order to perform the conceptual analysis, the theoretical framework draws on von Glasersfeld’s scheme theory, which is an interpretation of Piaget’s theory of cognitive development.

Gremmo M. J. & Riley P. (1995) Autonomy, self-direction and self access in language teaching and learning: The history of an idea. System 23(2): 151–164. Fulltext at https://cepa.info/4814

The terms “autonomy” and “self-direction” are being used more and more frequently in educational discussion. This article identifies and examines the ideas and historical contingencies which form the background to these developments, including minority rights movements, shifts in educational philosophy, reactions against behaviourism, linguistic pragmatism, wider access to education, increased internationalism, the commercialization of language provision and easier availability of educational technology. A number of objections to “autonomy” (it could not work with children or adults of low educational attainment, nor for “difficult” languages, or in examination-led syllabuses) have largely been overcome. Research into a wide range of educational topics, such as learning styles and strategies, resource centres and counsellor and learner training has directly contributed to present practice. Much remains to be done, however, particularly if cultural variation in learning attitudes, roles and activities is to be taken into account and if “autonomy” and “self-direction” are to be situated and understood within the workings of the social knowledge system.

Hackenberg A. J. (2007) Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior 26: 27–47. Fulltext at https://cepa.info/764

This article communicates findings from a year-long constructivist teaching experiment about the relationship between four sixth grade students’ multiplicative structures and their construction of improper fractions. Students’ multiplicative structures are the units coordinations that they can take as given prior to activity – i.e., the units coordinations that they have interiorized. This research indicates that the construction of improper fractions requires having interiorized three levels of units. Students who have interiorized only two levels of units may operate with fractions greater than one, but they don’t produce improper fractions. These findings call for a revision in Steffe’s hypothesis (Steffe L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior 20: 267–307) that upon the construction of the splitting operation, students’ fractional schemes can be regarded as essentially including improper fractions. While the splitting operation seems crucial in the construction of improper fractions, it is not necessarily accompanied by the interiorization of three levels of units. Relevance: This article takes a radical constructivist approach to mathematical learning and develops local theory about how students’ units coordinations are related to the fraction schemes they can construct.

Hjorth A. (2015) Body Syntonicity in Multi-Point Rotation? Constructivist Foundations 10(3): 351–352. Fulltext at https://cepa.info/2149

Open peer commentary on the article “Elementary Students’ Construction of Geometric Transformation Reasoning in a Dynamic Animation Environment” by Alan Maloney. Upshot: Parnorkou and Maloney’s article presents an interesting, well-structured and clearly described study of children’s reasoning about mental rotations. Specifically, Parnorkou and Maloney deploy the microworld Graphs ’n Glyphs, and use it as a “window on thinking-in-change” as they observe and interview children who use it. Reading the article raised a few questions for me about the role of body syntonicity in learning about rotation of geometric shapes, and I wonder where (or if) the authors feel these foundational concepts fit in with their research.

Hromkovič J. & Staub J. (2019) Constructing Computational Thinking using CS Unplugged. Constructivist Foundations 14(3): 353–355. Fulltext at https://cepa.info/6051

Open peer commentary on the article “Constructing Computational Thinking Without Using Computers” by Tim Bell & Michael Lodi. Abstract: With CS Unplugged activities, children all across the world can learn fundamental CS concepts without the requirement of learning how to program first. In the target article, the authors show the connection between CS Unplugged, on the one hand, and constructionism and computational thinking, on the other hand. They do so by providing a mapping between three CS Unplugged activities, Papert’s constructionism, and six elements characterizing computational thinking. Here we reflect on what computational thinking is, how it is related to CS Unplugged, and how the CS Unplugged approach can be deepened in order to explore the full potential of constructionism.