In an 8-month teaching experiment, the author aimed to establish mathematical caring relations (MCRs) with 4 6th-grade students. From a teacher’s perspective, establishing MCRs involves holding the work of orchestrating mathematical learning for students together with an orientation to respond to energetic fluctuations that may accompany student″teacher interactions. From a student’s perspective, participating in an MCR involves some openness to the teacher’s interventions in the student’s mathematical activity and some willingness to pursue questions of interest. Analysis revealed that student″teacher interactions can be viewed as a linked chain of perturbations; in MCRs, the linked chain tends toward perturbations that are bearable for both students and teachers. This publication is relevant for constructivist approaches because it examines how attention to affective responses (specifically, emotion and vital energy) can be included in a radical constructivist approach to knowing and learning.

In an 8-month teaching experiment 4 sixth-grade students reasoned with reversible multiplicative relationships. One problem involved a known quantity that was a multiple of an unknown quantity; students were to determine the unknown. All four students constructed schemes to solve such problems and more complex versions where the known was a fraction of the unknown. Two students could not foresee the results of their schemes in thought. The other two could; their schemes were anticipatory. The study shows that constructing anticipatory schemes requires coordinating three levels of units prior to activity. The paper presents a detailed framework of radical constructivism.

The actions of the mathematics teacher are bound up in ethical decisions that impact the learner and teacher, both within and external to the formal school curriculum. This paper argues that the principles of a radical constructivist theory of knowing underlie a model for an ethics of liberation. A learner’s active construction of their experiential reality, including the construction of the independent existence of the other and resulting social implications, frame guidelines for actions that are liberatory. To demonstrate this point, the paper develops the ideas of responsibility of the self, unique directions of learning, and socially-generated disequilibrium. When teachers and their students act according to such guidelines, they are freed to know mathematics and hence themselves in ways that allow them to work toward social justice and democratic ideals.

Three iterative, after school design experiments with small groups of middle school students were conducted to investigate how students represent fractional relationships between two unknowns and whether they construct reciprocal reasoning with unknowns. Of the 22 students who participated, 9 had constructed fractions as multiples of unit fractions. Seven of these 9 students constructed reciprocal reasoning with unknowns. They did so by constructing a multiplicative relationship between a unit fraction of one unknown and the other unknown. The other 2 students did not make this construction. Instead, they represented relationships between unknowns with whole number multiplication and division, and by adding unknowns and fractional parts of unknowns. The students who constructed reciprocal reasoning demonstrated a link between fractions as measures and fractions as operators, one benefit of working on rational number knowledge and algebraic reasoning together. Implications for teaching include recommendations for supporting students to construct reciprocal reasoning with unknowns.

This article reports on students’ construction of fraction composition schemes. Students’ whole number multiplicative concepts were found to be critical constructive resources for students’ fraction composition schemes. Specifically, the interiorization of two levels of units, a particular multiplicative concept, was found to be necessary for the construction of a unit fraction composition scheme, while the interiorization of three levels of units was necessary for the construction of a general fraction composition scheme.

In this paper, we engage in a thought experiment about how students might notate their reasoning for taking a fraction of a fraction and determining its size in relation to the whole. We situate this discussion within a radical constructivist framework for learning in order to articulate how developing systems of notation with students can contribute to their learning. In particular, we posit that developing systems of notation with students is likely to contribute to what Piaget called reflected abstractions – a retroactive thematization of one’s reasoning.

Context: This paper outlines how radical constructivist theory has led to a particular methodological technique, developing second-order models of student thinking, that has helped mathematics educators to be more effective teachers of their students. Problem: The paper addresses the problem of how radical constructivist theory has been used to explain and engender more viable adaptations to the complexities of teaching and learning. Method: The paper presents empirical data from teaching experiments that illustrate the process of second-order model building. Results: The result of the paper is an illustration of how second-order models are developed and how this process, as it progresses, supports teachers to be more effective. Implications: This paper has the implication that radical constructivism has the potential to impact practice.

Key words: Second-order model, radical constructivism, teaching, mathematics education