Argumentation and proofs in elementary calculus


Ferdinando Arzarello and Cristina Sabena have written an article entitled Semiotic and theoretic control in argumentation and proof activities. This article was recently published online in Educational Studies in Mathematics. Here is an abstract of their article:

We present a model to analyze the students’ activities of argumentation and proof in the graphical context of Elementary Calculus. The theoretical background is provided by the integration of Toulmin’s structural description of arguments, Peirce’s notions of sign, diagrammatic reasoning and abduction, and Habermas’ model for rational behavior. Based on empirical qualitative analysis we identify three different kinds of semiotic actions featuring the organization of the argumentations, and related to different epistemological status of the arguments. In such semiotic actions, the students’ argumentation and proof activities are managed and guided according to two intertwined modalities of control, which we call semiotic and theoretic control. The former refers to decisions concerning the selection and implementation of semiotic resources; the latter refers to decisions concerning the selection and implementation of a more or less explicit theory or parts of it. The structure of the model allows us to pinpoint a dialectical dynamics between the two.

Transition between different coordinate systems

Mariana Montiel, Biguel R. Wilhelmi, Draga Vidakovic and Iwan Elstak have written an article called Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context. The article was published online in Educational Studies in Mathematics on Saturday. Here is the abstract of their article:

The main objective of this paper is to apply the onto-semiotic approach to analyze the mathematical concept of different coordinate systems, as well as some situations and university students’ actions related to these coordinate systems. The identification of objects that emerge from the mathematical activity and a first intent to describe an epistemic network that relates to this activity were carried out. Multivariate calculus students’ responses to questions involving single and multivariate functions in polar, cylindrical, and spherical coordinates were used to classify semiotic functions that relate the different mathematical objects.

Building intellectual infrastructure

James Kaput wrote an article that was published online in Educational Studies in Mathematics on Friday. The article is entitled: Building intellectual infrastructure to expose and understand ever-increasing complexity. Here is the abstract of the article:

This paper comments on the expanded repertoire of techniques, conceptual frameworks, and perspectives developed to study the phenomena of gesture, bodily action and other modalities as related to thinking, learning, acting, and speaking. Certain broad issues are considered, including (1) the distinction between “contextual” generalization of instances across context (of virtually any kind—numeric, situational, etc.) and the generalization of structured actions on symbols, (2) fundamental distinctions between the use of semiotic means to describe specific situations versus semiosis serving the process of generalization, and (3) the challenges of building generalizable research findings at such an early stage in infrastructure building.

Gestures as semiotic resources

Ferdinando Arzarello, Domingo Paola, Ornella Robutti and Cristina Sabena have written an article called Gestures as semiotic resources in the mathematics classroom. The article was published online in Educational Studies in Mathematics a while ago. Here is the abstract of their paper:

In this paper, we consider gestures as part of the resources activated in the mathematics classroom: speech, inscriptions, artifacts, etc. As such, gestures are seen as one of the semiotic tools used by students and teacher in mathematics teaching–learning. To analyze them, we introduce a suitable model, the semiotic bundle. It allows focusing on the relationships of gestures with the other semiotic resources within a multimodal approach. It also enables framing the mediating action of the teacher in the classroom: in this respect, we introduce the notion of semiotic game where gestures are one of the major ingredients.

Working with artefacts

Michela Maschietto and Maria G. Bartolini Bussi have written an article entitled Working with artefacts: gestures, drawings and speech in the construction of the mathematical meaning of the visual pyramid. The article was published online in Educational Studies in Mathematics two days ago. Here is a copy of the abstract:

This paper reports a part of a study on the construction of mathematical meanings in terms of development of semiotic systems (gestures, speech in oral and written form, drawings) in a Vygotskian framework, where artefacts are used as tools of semiotic mediation. It describes a teaching experiment on perspective drawing at primary school (fourth to fifth grade classes), starting from a concrete experience with a Dürer’s glass to the interpretation of a new artefact. We analyse the long term process of appropriation of the mathematical model of perspective drawing (visual pyramid) through the development of gestures, speech and drawings under the teacher’s guidance.