Understanding irrational numbers by means of their representation as non-repeating decimals

International audienceResearch study on students' conceptions of irrational numbers upon entering university is of importance towards the transition to university. In this paper, we analyze students' conceptions of irrational numbers using their representation as non-repeating infinite decimals. The majority of students in the study identify the set of all decimals (finite and infinite) with the set of rational numbers. In spite of the fact that around 80% of the students claimed that they had learned about irrational numbers, only a small percentage of students (19%) showed awareness of the existence of non-repeating infinite decimals.</p

The epistemological dimension revisited

International audienceEpistemology and networking was discussed in the last CERME working group on theory. This paper aims to continue the discussion. I reflect on epistemological analysis and the cultural dimension of knowing and present examples which demonstrate how the changes in the cultural context influence the epistemological analysis. Then, I reconsider the epistemological dimension and the networking of theories. In some cases, the epistemological dimension permits the networking. In other cases, we notice how by means of networking, strong epistemological concerns in one theory might be integrated in another theory in a way that reinforces the underlying assumptions of this other theory. I end the paper with an example of networking that demonstrates how the social dimension might influence the epistemological analysis

Comparing a priori analyses

International audienceThe present research study is a theoretical reflection on the notion of a priori analysis in two different theories: the Theory of Didactical Situations (TDS) and the theory of Abstraction in Context (AiC). For both theories the epistemological perspective is of importance. The a priori analyses offered by the two theories are different. This difference reflects the different priorities of focus of analysis of both theories. The study demonstrates that, in their effort of networking theories, researchers of both theories will benefit of comparing their a priori analyses

Comparing a priori analyses

International audienceThe present research study is a theoretical reflection on the notion of a priori analysis in two different theories: the Theory of Didactical Situations (TDS) and the theory of Abstraction in Context (AiC). For both theories the epistemological perspective is of importance. The a priori analyses offered by the two theories are different. This difference reflects the different priorities of focus of analysis of both theories. The study demonstrates that, in their effort of networking theories, researchers of both theories will benefit of comparing their a priori analyses

The roles of visualization and symbolism in the potential and actual infinity of the limit process

A teaching experiment—using Mathematica to investigate the convergence of sequence of functions visually as a sequence of objects (graphs) converging onto a fixed object (the graph of the limit function)—is here used to analyze how the approach can support the dynamic blending of visual and symbolic representations that has the potential to lead to the formal definition of the concept of limit. The study is placed in a broad context that links the historical development with cognitive development and has implications in the use of technology to blend dynamic perception and symbolic operation as a natural basis for formal mathematical reasoning. The approach offered in this study stimulated explicit discussion not only of the relationship between the potential infinity of the process and the actual infinity of the limit but also of the transition from the Taylor polynomials as approximations to a desired accuracy towards the formal definition of limit. At the end of the study, a wide spectrum of conceptions remained. Some students only allowed finite computations as approximations and denied actual infinity, but for half of the students involved in the study, the infinite sum of functions was perceived as a legitimate “object” and was not perceived as a dynamic “process” that passes through a potentially infinite number of terms. For some students, the legitimate object was vague or generic, but we also observed other students developing a sense of the formal limit concept

Preface: CERME11 in lovely Utrecht historic sites

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THE ROLES OF EMBODIMENT AND SYMBOLISM IN THE POTENTIAL AND ACTUAL INFINITY OF THE LIMIT PROCESS

A teaching experiment—using Mathematica to investigate the convergence of sequence of functions visually as a sequence of objects (graphs) converging onto a fixed object (the graph of the limit function)—is here used to analyse how the approach can support the dynamic blending of visual and symbolic representations that has the potential to lead to the formal definition of the concept of limit. The study is placed in a broad context that links the historical development with cognitive development and has implications in the use of technology to blend dynamic perception and symbolic operation as a natural basis for formal mathematical reasoning. 1

Preface: CERME11 in lovely Utrecht historic sites

International audienc