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.

There are many factors – currently not well understood – which influence how efficiently the educational system builds and cultivates digital literacy of pupils. Among them is the quality of the educational policy docu- ments, which set the overall strategy from the governmental point; the level of digital literacy of the schools’ management and teachers; the level of concern of the pupils and their parents in acquiring competency; the quality and richness of the teaching-learning resources; how dedicated the school is to innovation in general; how is competency implemented in schools – through Informatics or ICT as a subject, or through ICT integrated in all subjects and the quality of collaboration between teachers of different subjects etc. In our department we make a great effort to positively stimulate two of these factors: provide modern university pre-service teacher development and pro- duce attractive and inspiring Informatics textbooks, educational software and learning resources for children, students and teachers. In this paper we present our current series of Informatics textbooks for lower secondary schools. We will analyze in detail the most recent of them: Informatics Around Us. We illus- trate the contents and style of the book and we reflect on how our textbooks may help stimulate a kind of bottom-up transformation of our schools into creative and motivating learning playgrounds.

As we witness a push toward studying spatial reasoning as a principal component of mathematical competency and instruction in the twenty first century, we argue that enactivism, with its strong and explicit foci on the coupling of organism and environment, action as cognition, and sensory motor coordination provides an inclusive, expansive, apt, and fit framework. We illustrate the fit of enactivism as a theory of learning with data from an ongoing research project involving teachers and elementary-aged children’s engagement in the design and assembly of motorized robots. We offer that spatial reasoning with its considerations of physical context, the dynamics of a body moving through space, sensorimotor coordination, and cognition, appears different from other conceptual competencies in mathematics. Specifically, we argue that learner engagements with diverse types of informationally ‘dense’ visuo-spatial interfaces (e.g., blueprints, programming icons, blocks, maps), as in the research study, afford some of the necessary experiences with/in a vast number of cases described by Varela et al. (1991) that enable the development of other mathematical competencies.

Key words: enactivism, sensorimotor coordination, spatial reasoning, robotics

A child is not a little adult, nor a learner a little scientist. Children’s concepts differ in structure and meaning from those of adults. Both Vygotsky (children have different kinds of concepts from adults; they don’t have true concepts until puberty) and Piaget (there are shifts between major periods which can be interpreted as changes in representational formats and processes that operate on them) would argue that there are fundamental changes in mental machinery from childhood to adulthood. As such, how to learn or be taught in a domain is quite different from how to perform or ‘do’ in a domain (i.e., learning science vs. doing science). The epistemology of most sciences, for example, is often based upon experimentation and discovery and, since this is so, experimentation and discovery should be a part of any curriculum aimed at ‘producing’ future scientists. But this does not mean that experimentation and discovery should also be the basis for curriculum organization and learning environment designing. Modern curriculum developers and reformers who often refer to themselves as constructivists tend to confuse the epistemological basis of a domain (i.e., how knowledge is acquired and the accepted validation procedures of that knowledge in a domain) with the psychological and pedagogic bases for teaching in that domain (i.e., strategies of instruction or a style of instruction). In other words, they fail to distinguish between learning and doing and thus overlook the fact that students are not miniature experts practicing something, but rather that they are novices learning about something.

This is a selective review of the published literature on object-choice tasks, where participants use directional cues to find hidden objects. This literature comprises the efforts of researchers to make sense of the sense-making capacities of our nearest living relatives. This chapter is written to highlight some nonsensical conclusions that frequently emerge from this research. The data suggest that, when apes are given approximately the same sense-making opportunities as we provide for our children, they will easily make sense of our social signals. The ubiquity of nonsensical contemporary scientific claims to the effect that humans are essentially – or inherently – more capable than other great apes in the understanding of simple directional cues is, itself, a testament to the power of pre-conceived ideas on human perception.

Radical constructivism is currently a major, if not the dominant, theoretical orientation in the mathematics education community, in relation to children’s learning. There are, however, aspects of children’s learning that are challenges to this perspective, and what appears to be “at least temporary states of intersubjectivity” (Cobb, Wood, & Yackel, 1991, p. 162) in the classroom is one such challenge. In this paper I discuss intersubjectivity and through it offer an examination of the limitations of the radical constructivist perspective. I suggest that the extension of radical constructivism toward a social constructivism, in an attempt to incorporate intersubjectivity, leads to an incoherent theory of learning. A comparison of Piaget’s positioning of the individual in relation to social life with that of Vygotsky and his followers is offered, in support of the claim that radical constructivism does not offer enough as an explanation of children’s learning of mathematics.

Neoconstructivism makes the point that new knowledge is built within a context of existing knowledge. The neuroconstructivist approach argues that this is true of development at all levels of representation, from single cells, to functional brain systems, to children growing up within specific social and physical environments. Context dependence is the key to understanding functional development.

This article presents a novel computational framework for modeling cognitive development. The new modeling paradigm provides a language with which to compare and contrast radically different facets of children’s knowledge. Concepts from the study of machine learning are used to explore the power of connectionist networks that construct their own architectures during learning. These so-called generative algorithms are shown to escape from Fodor’s (1980) critique of Constructivist development. We describe one generative connectionist algorithm (cascade-correlation) in detail. We report on the successful use of the algorithm to model cognitive development on balance scale phenomena; seriation; the integration of velocity, time, and distance cues; prediction of effect sizes from magnitudes of causal potencies and effect resistances; and the acquisition of English personal pronouns. The article demonstrates that computer models are invaluable for illuminating otherwise obscure discussions.

This descriptive paper presents findings from an analysis of fifth graders’ classroom discussions aimed at constructing shared knowledge on biological and ecological topics. A Vygotskian frame of reference was used that assumes reasoning in children is externalized through discussing and reasoning with others. This analysis of peer discourse‐reasoning was developed in a “social constructivist learning community” characterized by collaboration, public sharing, and revision of ideas. The argumentative operations and the epistemic operations activated by the students while reasoning and arguing have been illustrated in detail. These classroom discussions demonstrate how students build up new concepts by renegotiating and sharing meanings and ideas during lively, argumentative exchanges. These discussions also indicate the cognitive procedures activated to make sense of the new knowledge, that is, the main thinking actions needed to be engaged in a deep scientific understanding. The data suggest that collaborative discoursereasoning can act as means to support students in gradually mastering some of the discursive practices characteristic of scientific communities.