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?

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.

Key words: Cognitive development, Bayesian inference, theory of mind, causal knowledge, intuitive theories

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.

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.

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.

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.

In this article, we integrate two constructionist approaches – the theory of constructed emotion and rational constructivism – to introduce several novel hypotheses for understanding emotional development. We first discuss the hypothesis that emotion categories are abstract and conceptual, whose instances share a goal-based function in a particular context but are highly variable in their affective, physical, and perceptual features. Next, we discuss the possibility that emotional development is the process of developing emotion concepts, and that emotion words may be a critical part of this process. We hypothesize that infants and children learn emotion categories the way they learn other abstract conceptual categories – by observing others use the same emotion word to label highly variable events. Finally, we hypothesize that emotional development can be understood as a concept construction problem: a child becomes capable of experiencing and perceiving emotion only when her brain develops the capacity to assemble ad hoc, situated emotion concepts for the purposes of guiding behavior and giving meaning to sensory inputs. Specifically, we offer a predictive processing account of emotional development.

Key words: prediction, construction, variation, social learning, culture

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.

In this study, we challenge the deficit perspective on mathematical knowing and learning for children labeled as LD, focusing on their struggles not as a within student attribute, but rather as within teacher-learner interactions. We present two cases of fifth-grade students labeled LD as they interacted with a researcher-teacher during two constructivist-oriented teaching experiments designed to foster a concept of unit fraction. Data analysis revealed three main types of interactions, and how they changed over time, which seemed to support the students’ learning: Assess, Cause and Effect Reflection, and Comparison/Prediction Reflection. We thus argue for an intervention in interaction that occurs in the instructional process for students with LD, which should replace attempts to “fix” ‘deficiencies’ that we claim to contribute to disabling such students.

Key words: constructivism, student-teacher interaction, mathematics, intervention, learning difference

Open peer commentary on the article “Building Bridges to Algebra through a Constructionist Learning Environment” by Eirini Geraniou & Manolis Mavrikis. Upshot: In their article, Geraniou and Mavrikis describe an environment to help children explore algebraic relationships through pattern building. They report on transfer of learning from the computer to paper, but also implicit is transfer from concrete to abstract contexts. I make the case that transfer from abstract to concrete contexts should complement such approaches.