Covariation between variables in a modelling process: The ACODESA (collaborative learning, scientific debate and self-reflection) method

Semiotic representations have been an important topic of study in mathematics education. Previous research implicitly placed more importance on the development of institutional representations of mathematical concepts in students rather than other types of representations. In the context of an extensive research project, in progress since 2005, related to modelling mathematical situations in Québec secondary schools (grades 8 and 9), we have addressed the problem of constructing a specific mathematical concept: covariation between variables as a prerequisite for the concept of function and its graphical representation. However, our research differs from previous studies as we attempt to take into consideration, in a cultural semiotic perspective, the spontaneous non-institutional representations that students produce when solving a problem situation in mathematics. We report our results with a group of students in grade 9, discussing the evolution of the representations the students produced to solve a problem situation, and the key role that the concept of covariation seems to play in helping students grasp the graphical representation of functions. We also discuss the different stages of the teaching method used, based upon collaborative learning, scientific debate and self-reflection (the ACODESA method of teaching) which aims to help the students acquire a cultural semiotic system.Conseil de Recherche en Sciences Humaines du Canada (No. 410-2008-1836, CID 130 252)

Sobre la comprensión en estudiantes de matemáticas del concepto de integral impropia : algunas dificultades, obstáculos y errores

En el presente trabajo mostramos algunas de las dificultades, obstáculos y errores que los alumnos universitarios encuentran al aprender los conceptos relativos a la integración impropia; algunos de ellos parecen inherentes al propio concepto de integral impropia y otros vienen relacionados con ausencia de significado o con otros conceptos del cálculo. Con el objetivo de analizar estas dificultades, obstáculos y errores construimos un marco teórico basado, principalmente, en la teoría de Duval sobre los registros semióticos de representación y construimos un modelo de competencia para evaluar la comprensión de nuestros alumnos.In this paper we present some difficulties, obstacles and errors university students face when learning the concepts relative to improper integration. Some of them seem to be inherent to the improper integral concept itself and some others are related to lack of meaning or to other concepts of calculus. With the aim of analysing these difficulties, obstacles and errors we built up a theoretical framework based, fundamentally, on Duval's theory of semiotic registers of representation and we designed a competence model to assess our students' comprehension

The introduction of real numbers in secondary education. An institutional analysis of textbooks

In this paper we analyse the introduction of irrational and real numbers in secondary textbooks, and specifically the propositions on how these should be taught, in a sample of Brazilian textbooks used in state schools and approved by the Ministry of Education. The analyses discussed in this paper follow an institutional perspective (using Chevallard's Anthropological Theory of Didactics). Our results indicate that the notion of irrational number is generally introduced on the basis of the decimal representation of numbers, and that the mathematical need for the construction of the field of real numbers remains unclear in the textbooks. It seems that textbooks used in secondary teaching institutions develop mathematical organisations which focus on the practical block

Didactic Situations and Didactical Engineering in university mathematics: cases from the study of Calculus and proof

This paper discusses the use of the Theory of Didactic Situations (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of Didactical Engineering (DE), particularly adapted at the tertiary level. We then illustrate the potency of the TDS-DE approach in three university level Research Cases, two related to Calculus, and one related to proof. These studies deploy constructs such as didactic contract, milieu, didactic variables, and epistemological analyses, among others, to design Situations at university level. We conclude with a few thoughts on how the TDS-DE approach relates to other approaches, most notably the Anthropological Theory of the Didactic

Growth mode transition involving a potential-dependent isotropic to anisotropic surface atom diffusion change. Gold electrodeposition on HOPG followed by STM

The electrodeposition of gold on highly oriented pyrolytic graphite (HOPG) from acid aqueous solutions was studied by using electrochemical techniques complemented with ex-situ scanning tunneling microscopy (STM). The kinetics of gold electrodeposition is consistent with a nucleation and three-dimensional growth process under diffusion control from the solution side. As the applied potential moves in the negative direction, the gold crystal density increases, and the crystal shape changes from a Euclidean to a dendritic fractal morphology. This transition can be assigned to the anisotropic surface diffusion of gold adatoms induced by the applied electric potential. A model including a potential-dependent energy barrier at step edges accounts for the morphology transition for gold electrodeposition on HOPG.Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA)Facultad de Ciencias Exacta

Introduction to the papers of TWG14: University mathematics education

Research on university level mathematics education is a fast developing field as evident in the growth of the CERME University Mathematics Education (hereafter UME) Thematic Working Group. TWG14 was launched in CERME7 (Nardi, González-Martín, Gueudet, Iannone & Winsløw, 2011). After CERME8 (Nardi, Biza, González-Martín, Gueudet, & Winsløw, 2013), its leader team – in collaboration with TWG14 participants and others – worked towards a Research in Mathematics Education Special Issue on Institutional, sociocultural and discursive approaches to research in university mathematics education (Nardi, Biza, González-Martín, Gueudet & Winsløw, 2014) which focused on research that is conducted in the spirit of the following theoretical frameworks: Anthropological Theory of the Didactic, Theory of Didactic Situations, Instrumental and Documentational Approaches, Communities of Practice and Inquiry and Theory of Commognition

Introduction to the papers of TWG14: University mathematics education

International audienc

From resource to document: Scaffolding content and organising student learning in teachers’ documentation work on the teaching of series

We examine teachers’ use of resources as they prepare to teach the topic of numerical series of real numbers in order to identify how their personal relationship with mathematical content—and its teaching—interacts with their use of a commonly used textbook. We describe this interplay between textbook and personal relationship, a term coined in the Anthropological Theory of the Didactic (ATD, Chevallard, 2003), in the terms of documentation work (resources, aims, rules of action, operational invariants), a key construct from the documentational approach (DA, Gueudet & Trouche, 2009). We do so in the case of five post-secondary teachers who use the same textbook as a main resource to teach the topic. Documentational analysis of interviews with the teachers led to the identification of their aims and rules of action (the what and how of their resource use as they organise their teaching of the topic) as well as the operational invariants (the why for this organisation of their teaching). We describe the teachers’ documentation work in two sets of aims/rules of action: scaffolding mathematical content (series as a stepping stone to learning about Taylor polynomials and Maclaurin series) and organising student learning about series through drill exercises, visualisation, examples, and applications. Our bridging (networking) of theoretical constructs originating in one theoretical framework (personal relationship, ATD) with the constructs of a different, yet compatible, framework (documentation work, DA) aims to enrich the latter (teachers’ documentation work) with the individual agency (teachers’ personal relationship with the topic) provided by the former

Growth mode transition involving a potential-dependent isotropic to anisotropic surface atom diffusion change. Gold electrodeposition on HOPG followed by STM

The electrodeposition of gold on highly oriented pyrolytic graphite (HOPG) from acid aqueous solutions was studied by using electrochemical techniques complemented with ex-situ scanning tunneling microscopy (STM). The kinetics of gold electrodeposition is consistent with a nucleation and three-dimensional growth process under diffusion control from the solution side. As the applied potential moves in the negative direction, the gold crystal density increases, and the crystal shape changes from a Euclidean to a dendritic fractal morphology. This transition can be assigned to the anisotropic surface diffusion of gold adatoms induced by the applied electric potential. A model including a potential-dependent energy barrier at step edges accounts for the morphology transition for gold electrodeposition on HOPG.Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA)Facultad de Ciencias Exacta