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.

Excerpt: The first five contributions to this Special Issue on Theories of Mathematical Learning take a cognitive perspective whereas the sixth, that by Voigt, takes an interactionist perspective. The common theme that links the six articles is the focus on students’ inferred experiences as the starting point in the theory-building process. This emphasis on the meanings that objects and events have for students within their experiential realities can be contrasted with approaches in which the goal is to specify cognitive behaviors that yield an input-output match with observed behavior. It is important to note that the term ‘experience’ as it is used in these articles is not restricted to physical or sensory-motor experience. A perusal of the first five articles indicates that it includes reflective experiences that involve reviewing prior activity and anticipating the results of potential activity. In addition, by emphasizing interaction and communication, Voigt’s contribution reminds us that personal experiences do not arise in a vacuum but instead have a social aspect.

Cobb P. (2000) Constructivism in social context. In: Steffe L. P. & Thompson P. (eds.) Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld. Falmer Press, London: 152–178. https://cepa.info/6709

In this chapter, I focus on one of the aspects of constructivist theory that Glasersfeld (Ch. 1) identifies as in need of further development. This aspect of the theory involves locating students’ mathematical development in social and cultural context while simultaneously treating learning as a process of adaptive reorganization. In addressing this issue, I illustrate the approach that I and my colleagues currently take when accounting for the process of students’ mathematical learning as it occurs in the social context of the classroom. In the opening section of the chapter, I clarify why this is a significant issue for us as mathematics educators. I then outline my general theoretical orientation by discussing Glasersfeld’s constructivism and Bauersfeld’s interactionism. Against this background, I develop criteria for classroom analyses that are relevant to our interests as researchers who develop learning environments for students in collaboration with teachers. Next, I illustrate the interpretive framework that I and my colleagues currently use by presenting a sample classroom analysis. Finally, in the concluding sections of the chapter, I reflect on the sample analysis to address four more general issues. These concern the contributions of analyses of the type outlined in the illustrative example, the relationship between instructional design and classroom-based research, the role of symbols and other tools in mathematical learning, and the relation between individual students’ mathematical activity and communal classroom processes.

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 and 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.

D’Ambrosio B. S. (2004) The dilemmas of preparing teachers to teach mathematics within a constructivist framework. In: Fujita H., Hashimoto Y., Hodgson B. R., Lee P. Y., Lerman S. & Sawada T. (eds.) Proceedings of The Ninth International Congress on Mathematical Education. Springer, Dordrecht: 115–117. https://cepa.info/7136

Excerpt: The difficulties of embracing a constructivist perspective towards teaching mathematics are evident when we analyze future teachers’ difficulties in building models of children’s understanding. Our understanding of these difficulties can be further shaped by considering characteristics of future teachers as to their own approaches to learning mathematics and their perceptions of the nature of mathematical knowledge and mathematical learning.

Ferreira M. D. C. R. (2012) A construção do número: os modelos de Klahr & Wallace; Von Glasersfeld e K. Fuson [The number construction: Klahr & Wallace, Von Glasersfeld & K. Fuson models]. Psicologia Escolar e Educacional 16: 197–207. https://cepa.info/7914

In studies on arithmetical thinking, the concept of number development has been a subject of scientific controversy, since the interpretation that mathematical learning occurs mainly through the formation of internal connections, up to the study of cognitive processes underlying arithmetical reasoning. In this case, the influence of constructivism on the genesis of logical structures, extends not only to the work of von Glasersfeld (1988), which postulates that abstract concepts are constructed from everyday life experiences, but it also extends to the social influence on the construction of underlying processes of arithmetical reasoning (Fuson, 1988; Fuson & Burghardt, 2003). Opposing this interpretation, the prospect of innate development (Klahr & Wallace, 1973; Gelman & Gallistel, 1978) assumes that children are born with logical principles required to the construction of elementary arithmetical knowledge.

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

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.