Borg P., Hewitt D. & Jones I. (2016) Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom. Constructivist Foundations 12(1): 59–69. https://cepa.info/3810

Context: Constructivist teachers who find themselves working within an educational system that adopts a realist epistemology, may find themselves at odds with their own beliefs when they catch themselves paying closer attention to the knowledge authorities intend them to teach rather than the knowledge being constructed by their learners. Method: In the preliminary analysis of the mathematical learning of six low-performing Year 7 boys in a Maltese secondary school, whom one of us taught during the scholastic year 2014-15, we constructed a conceptual framework which would help us analyze the extent to which he managed to be sensitive to constructivism in a typical classroom setting. We describe the development of the framework M-N-L (Mathematics-Negotiation-Learner) as a viable analytical tool to search for significant moments in the lessons in which the teacher appeared to engage in what we define as “constructivist teaching” (CT) during mathematics lessons. The development of M-N-L is part of a research program investigating the way low-performing students make mathematical sense of new notation with the help of the software Grid Algebra. Results: M-N-L was found to be an effective instrument which helped to determine the extent to which the teacher was sensitive to his own constructivist beliefs while trying to negotiate a balance between the mathematical concepts he was expected to teach and the conceptual constructions of his students. Implications: One major implication is that it is indeed possible for mathematics teachers to be sensitive to the individual constructions of their learners without losing sight of the concepts that society, represented by curriculum planners, deems necessary for students to learn. The other is that researchers in the field of education may find M-N-L a helpful tool to analyze CT during typical didactical situations established in classroom settings.

Cobb P., Perlwitz M. & Underwood D. (1994) Construction individuelle, acculturation mathématique et communauté scolaire. Revue des sciences de l’éducation 20(1): 41–61. https://cepa.info/5944

We first distinguish between the school mathematics tradition typically established in textbook-based classrooms and the inquiry mathematics tradition established in classrooms where instruction is compatible with constructivism. We then focus on the inquiry mathematics tradition and consider the theoretical and pragmatic tensions inherent in the view that mathematical learning is both a process of active individual construction and a process of acculturation. Particular attention is given to the ways in which both constructivist and sociocultural theorists address this issue. Finally, we discuss the development of instructional activities for inquiry mathematics classrooms.

Cobb P., Perlwitz M. & Underwood-Gregg D. (1998) Individual construction, mathematical acculturation, and the classroom community. In: Larochelle M., Bednarz N. & Garrison J. (eds.) Constructivism in education. Cambridge University Press, New York NY: 63–80. https://cepa.info/5933

Excerpt: For the past six years we, together with Erna Yackel and Terry Wood, have conducted a classroom-based research and development project in elementary school mathematics.’ In this paper, we draw on our experiences of collaborating with teachers and of analyzing what might be happening in their classrooms to consider three interrelated issues. First, we argue that the teacher and students together create a classroom mathematics tradition or microculture and that this profoundly influences students’ mathematical activity and learning. Sample episodes are used to clarify the distinction between the school mathematics tradition in which the teacher acts as the sole mathe-matical authority and the inquiry mathematics tradition in which the teacher and students together constitute a community of validators. Second, we consider the theoretical and pragmatic tensions inherent in the view that mathematical learning is both a process of individual cognitive construction and a process of acculturation into the mathematical practices of wider society. In the course of the discussion, we contrast constructivist attempts to cope with this tension with approaches proposed by sociocultural theorists. Finally, we use the preceding issues as a backdrop against which to consider the development of instructional activities that might be appropriate for inquiry mathematics classrooms.

Coles A. (2015) On enactivism and language: Towards a methodology for studying talk in mathematics classrooms. ZDM Mathematics Education 47(2): 235–246.

This article is an early step in the development of a methodological approach to the study of language deriving from an enactivist theoretical stance. Language is seen as a co-ordination of co-ordinations of action. Meaning and intention cannot easily be interpreted from the actions and words of others; instead, careful attention can be placed in not going beyond what is observable within the text itself, for example by focusing on patterns in word use. Conversations are highly ritualised affairs and from an enactivist perspective these rituals can be read in terms of pattern. The notion of the ‘structural coupling’ of systems, which will inevitably have taken place in a classroom, means that the history and context of communication needs to be taken into account. The methodological perspective put forward in this article is exemplified with an analysis of two classroom incidents (involving different teachers) in which almost identical words are used by the teachers, but markedly different things happen next. The analysis reveals a complexity within the classroom that, although available to direct observation, only became apparent using an approach to studying language that took account of the context and history of communication in a recursive process of data collection and analysis.

François K. (2014) Convergences between Radical Constructivism and Critical Learning Theory. Constructivist Foundations 9(3): 377–379. https://cepa.info/1098

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: The value of Cifarelli & Sevim’s target article lies in the analysis of how reflective abstraction contributes to the description of mathematical learning through problem solving. The additional value of the article lies in its emphasis of some aspects of the learning process that goes beyond radical constructivist learning theory. I will look for common ground between the humanist philosophy of mathematics and radical constructivism. By doing so, I want to stress two converging elements: (i) the move away from traditionalist ontological positions and (ii) the central role of the students’ activity in the learning process.

Goldenberg E. P. (2019) Problem Posing and Creativity in Elementary-School Mathematics. Constructivist Foundations 14(3): 319–331. https://cepa.info/6045

Context: In 1972, Papert emphasized that “[t]he important difference between the work of a child in an elementary mathematics class and […]a mathematician” is “not in the subject matter […]but in the fact that the mathematician is creatively engaged […]” Along with creative, Papert kept saying children should be engaged in projects rather than problems. A project is not just a large problem, but involves sustained, active engagement, like children’s play. For Papert, in 1972, computer programming suggested a flexible construction medium, ideal for a research-lab/playground tuned to mathematics for children. In 1964, without computers, Sawyer also articulated research-playgrounds for children, rooted in conventional content, in which children would learn to act and think like mathematicians. Problem: This target article addresses the issue of designing a formal curriculum that helps children develop the mathematical habits of mind of creative tinkering, puzzling through, and perseverance. I connect the two mathematicians/educators - Papert and Sawyer - tackling three questions: How do genuine puzzles differ from school problems? What is useful about children creating puzzles? How might puzzles, problem-posing and programming-centric playgrounds enhance mathematical learning? Method: This analysis is based on forty years of curriculum analysis, comparison and construction, and on research with children. Results: In physical playgrounds most children choose challenge. Papert’s ideas tapped that try-something-new and puzzle-it-out-for-yourself spirit, the drive for challenge. Children can learn a lot in such an environment, but what (and how much) they learn is left to chance. Formal educational systems set standards and structures to ensure some common learning and some equity across students. For a curriculum to tap curiosity and the drive for challenge, it needs both the playful looseness that invites exploration and the structure that organizes content. Implications: My aim is to provide support for mathematics teachers and curriculum designers to design or teach in accord with their constructivist thinking. Constructivist content: This article enriches Papert’s constructionism with curricular ideas from Sawyer and from the work that I and my colleagues have done. Key words: Problem posing, puzzles, mathematics, algebra, computer programming.

Hackenberg A. J. (2005) A model of mathematical learning and caring relations. For the Learning of Mathematics 25: 45–51. https://cepa.info/763

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.

Hackenberg A. J. (2007) Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior 26: 27–47. https://cepa.info/764

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.

Hackenberg A. J. (2010) Mathematical caring relations in action. Journal for Research in Mathematics Education 41(3): 236–273.

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.

Hackenberg A. J. (2010) Mathematical caring relations: A challenging case. Mathematics Education Research Journal 22(3): 57–83.

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.