Part 1 of a three-part article analyzing radical constructivism (as one interpretation of Piaget) and the socio-cultural perspective (as one interpretation of Vygotsky), including major principles, primary contributions to mathematics education, and potential limitations. Introduces an integration of the two theories through a feminist perspective.

Demonstrates that Vygotskian theory can support two opposing interpretations: supporting reform and undermining reform. Discussion is organized by: sociocultural perspectives, Marxist influences on historical analysis and the role of labor, semiotics and psychological tools, dialectic of thought and language, conceptual development, and learning and development.

Presents a theory of intellectual development in which human development depends on environment, self is autonomous and communal, diversity and dissent are anticipated, emotional intelligence is acknowledged, abstraction is reconceptualized and placed in a dialectic, learning is a reciprocal activity, and classrooms are interactions among interactions.

Excerpt: I draw from recent developments in philosophy, biology, ecological thought, phenomenology, and curriculum theory in an effort to re-formulate a response to the question, Why teach mathematics?

The purpose of this article is to describe a model of mathematical learning and mathematical caring relations, where caring is conceived of as work toward balancing the ongoing depletion and stimulation involved in student-teacher mathematical interaction. Acts of mathematical learning are conceived of as modifications or reorganizations in a person’s ways of operating in the context of on-going interactions in her environment. Modifications or reorganizations occur in response to perturbations, or disturbances in the functioning of a person that is brought about by that functioning. Perturbations are a point of connection between learning and caring, because perturbations can be accompanied by an emotional response, such as disappointment or surprise. Implications of holding learning and caring together are explored. Relevance: This article specifically takes the frame of radical constructivism for mathematical learning and explores extensions into emotion, caring, and teacher-student relationships.

In this response to three articles in the immediately previous issue of For the Learning of Mathematics, the author addresses the theme of holding together what is seemingly disparate or even conflicting. First, the author asks whether we can hold together “mathematics” and “care for another” by viewing mathematics as always being someone’s mathematics, and by using the idea of first-order and second-order models of knowing. Second, the author suggests the need to hold together care for students with care for ourselves as teachers and teacher-educators. Third, the author presents a perspective on what is liberating about a radical constructivist perspective on knowing. Relevance: This publication is partly in response to Paul Ernest’s critique of radical constructivism in an article published in the immediately previous issue of For the Learning of Mathematics. In addition, this publication uses a radical constructivist perspective on knowing, and specifically Les Steffe’s first-order and second-order models, to suggest different ways to view tensions in the conduct of research and professional development, in response to an article by McCloskey.

Excerpt: How do people learn mathematics? How is individual learning related to collective knowledge? What part does biology play in learning processes? What is the influence of the social context on the learning of mathematics? My thinking around these issues has been shaped by ideas coming from biology and neuroscience, particularly those related to the enactive approach to cognition. Ideas emerging from these theories emphasise the complexity of the nature of cognition and learning.

Excerpt: Our focus, in this article, on the originary phenomenological sources of mathematical reasoning, moves beyond cognitivist approaches to examining mathematical incomprehension, such as focusing on issues of working memory, semiotic representations, and varied aspects of cognitive function and dysfunction (e.g., Geary, Hoard & Hamson, 1999). We propose to shift the investigative locus of research on mathematical learning to earlier phenomenological events in students’ subjective process of meaning making, just prior to engaging in formal mathematical representation and modeling of psychological content. Our proposition rests on the adoption of a contemplative orientation that promotes a deep focus on somatic and preconceptual realms. In our development of this approach, we introduce contemplative practice as a means to resolve the bottleneck introduced above. Contemplative practices can do this, we suggest, by providing a pre-conceptual or liminal space that bridges the nuanced apprehension of tacit sensorimotor activity and conscious configuring of this ineffable psychological content into expressive forms.

This commentary responds to a criticism of constructivism by Wolff-Michael Roth, published in For the Learning of Mathematics 30(2). At times, Roth oversimplifies and mischaracterizes constructivist perspectives on learning while promoting embodied cognition as an alternative. I argue that a simple transposition of terms largely aligns his description of embodied mathematical objects with the constructivist conception of mathematical objects as interiorized action.