Earlier approaches to sense-making in mathematics have looked at the ways students comprehend a mathematical concept. Recent research suggests that some students make sense not only of mathematical objects that have a being, but also of objects that have yet to become. In such cases, learning mathematics is not just an act of comprehending a given…

The aim of this study is to explore the possibility of introducing the general notions of function and inverse function through a mathematical activity on linear functions, focusing on the quantitative meaning associated to the connection between a relation and its inverse. We present a genetic decomposition, that is, a viable cognitive path for…

This paper introduces a model built upon the Method of Varying Inquiry, offering a didactical approach to problem posing and solving activities that stimulates inquiry-based learning in mathematics classrooms. The model combines the inquiry-based framework with the variation theory and with specific didactical and theoretical elements (the…

In this paper I analyze a problem solving event in a secondary mathematics classroom. As the event unfolds, the teachers, including me, understand the event as involving interactions that were not related to learning. By adopting an expansive view of learning, I advance a different interpretation, that the interactions tell a story of student…

Certain aspects of theoretical frameworks in mathematics education can remain unexplained in the literature, hence unnoticed by the readers. It is thus interesting to participate in a dialogue where they can be discussed and compared in terms of the aims, objects studied, results and their relation to teaching. This study is focused on how APOS…

In this article, a case is made that affect is central in determining students' experience of learning or not learning mathematics. I show how reversal theory (Apter, 2001), and particularly its taxonomy of motivations and emotions, provides a basis for a thick description of students' experiences of learning in a mathematics classroom. Using data…

This is the first of a two-part article that presents a theory of unit construction and coordination that underlies radical constructivist empirical studies of student learning ranging from young students' counting strategies to high school students' algebraic reasoning. My explanation starts with the formation of arithmetical units, which presage…

Motivated by the question, "What exactly about a mathematical concept should students discover, when they study it via discovery learning?", I present and demonstrate an interpretation of discovery pedagogy that attempts to address its criticism. My approach hinges on decoupling the solution process from its resultant product. Whereas theories of…

Presents and criticizes current theories of multiplication in order to pave the way for an alternative foundation for the concept of multiplication. Formulates a uniform concept of multiplication that recognizes anew the importance of the roles of multiplier and multiplicand. (Contains 22 references.) (ASK)

Discusses Brousseau's work on the theory of didactical situations in mathematics and its applications in mathematics classrooms which do not address the collection of obvious components, but instead discuss phenomena by analyzing knowledge in given situations. Meanings emerge in situations engineered by analyzing a situation and developing an…

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…

Introduces a Vygotskian approach to learning with an emphasis on mediated activity within a particular socio-historical context. Addresses the relationship between mathematical learner and different sign systems (multiple representations) afforded by the graphic calculator. Contains 32 references. (ASK)

Proposes an entirely different perspective and challenges both the current para-dogmatic character of constructivism in some quarters of mathematics education and the dismissive attitude towards 'humanist non-sense' in others. Discusses the linguistic aspects of mathematical knowledge through a language acquisition device and a language…

Discusses the role of play in the learning of mathematics. Describes a play for the third grade mathematics classroom that focuses on negative numbers and number concepts, and revisits Vygotsky from this point of view. Concludes that mathematically imaginative play can contribute greatly to mathematics learning in elementary school. (Contains 20…

The role of mental discipline theory in the history of mathematics education is considered. The author argues that we now remember mental discipline theory only as a caricature of what it was; moreover, it persists among both teachers and researchers. (MNS)