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.

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.

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.

Developed from Noddings’s (2002) care theory and von Glasersfeld’s (1995) constructivism, a mathematical caring relation (MCR) is a quality of interaction between a student and a teacher that conjoins affective and cognitive realms in the process of aiming for mathematical learning. In this paper I examine the challenge of establishing an MCR with one mathematically talented 11-year-old student, Deborah, during an 8-month constructivist teaching experiment. This publication is relevant for constructivist approaches because it develops a framework for student-teacher interaction based on constructivism.

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.

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.

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students, with each of three different multiplicative concepts, participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed. Relevance: In this paper the author uses second-order models of students’ multiplicative concepts and fractional knowledge built from radical constructivism to explore relationships between students’ fractional knowledge and algebraic reasoning. The paper is therefore one contribution to the construction of second-order models of students’ algebraic reasoning, which is sorely needed by the field of mathematics education, particularly for students who struggle to learn algebra.

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.

Upshot: In reading the commentaries, we were struck by the fact that all of them were in some capacity related to what we consider a core principle of radical constructivism - interaction. We characterize interaction from a radical constructivist perspective, and then discuss how the authors of the commentaries address one kind of interaction.