Helena Rocha

This chapter aims to make more explicit the grounded or ‘pragmatic theories’ that inform the design of mathematics teachers’ professional development (PD) to exploit technological affordances. It uses aspects of some representative projects that took place in four countries (Colombia, Italy, Mexico, and Portugal) to illustrate lessons learned (e.g., similarities and differences, barriers and opportunities) and provide important insights to inform future PD implementations. To do this, we have identified a set of aspects (and sub-aspects) that emerged in relation to five major themes and reveal our ‘pragmatic theories’ alongside a consideration of the interconnections between these aspects. Our contribution offers a methodological frame to support future PD designs for teachers of mathematics concerning digital technology uses.

For most teachers, the use of games is not a common practice. This limited professional experience forces the teacher to reflect on the use of games, on how to integrate them into class work, and on how to enhance the development of learning based on them. Our objective is to analyse a set of game scenarios and study aspects that characterise each of them, relating these characteristics to aspects of the professional knowledge to teach with technology. The study adopts an interpretative and qualitative methodology that seeks to understand the relationship between teachers' choices and their professional knowledge. The main findings suggest that the different characteristics of the game scenarios imply the creation of different working environments in the classroom; enhance the emergence of different student difficulties, requiring different ways to support their work; allowing/promoting different conceptual approaches; and the work with different representations. These differences between the game scenarios point to different emphases on the teaching and learning knowledge and the chemistry knowledge, suggesting different mobilizations of the teacher's professional knowledge.

Through writing, students express many of their processes and ways of thinking. Since at high school level some of the activities are carried out with the graphing calculator, we intend to investigate the contribution of this resource to promote the mathematical writing in the learning of continuous nonlinear models at 11th grade. Adopting a qualitative methodology, we collected and analyzed the students’ writing productions. What they write when using the calculator gives evidence about the information valued (when they sketch graphics without any justification); about the strategies used (when they define the viewing window and relate different menus on the graphing calculator); and about the reasoning developed (when they justify the information given by the calculator and the formulation of generalizations and conjectures validation).

Deductive reasoning, being central in mathematics, is also usually a source of difficulties for students, more used to the empirical approaches. In this study we focus on mathematical proof and we try to give attention to how this kind of reasoning is envisaged by the students, to the options they assume when asked to develop a deductive reasoning and to the factors affecting the implementation of this kind of reasoning. The study follows a qualitative and interpretative methodological approach, including the completion of two case studies of students of the 10th grade. Data were collected in work sessions and through interviews. The main findings point to a devaluation of mathematical proof and a strong preference for empirical approaches. Yet students show ability to develop different approaches. The preference for the mathematical subject and the attention given in class to the deduction work, appears to be relevant factors when considering the students' ability to develop a deductive reasoning when involved on a mathematical proof.

This article focuses the choice and use of examples by two students of the 11th grade to prove or refute a set of statements. The use of representations of sequences and functions is also considered. The study adopts a qualitative approach and data were collected by interviews and documental gathering. The conclusions suggest most of the examples used were well-known sequences or functions. However, the students sought different purposes for the use of examples, such as understanding the conjecture, demonstrate the falsity or truthfulness of the statement and conveying a general argument. The students made a satisfactory articulation between the various types of representations but relied mostly in the cartesian graph.

The integration of the graphing calculator in mathematical activity encourages students to express many of their processes and ways of thinking. Since some of the activities at the high school level are carried out with the graphing calculator, we intend to investigate the contribution of this resource to promote the learning of nonlinear continuous models in the 11th grade. By adopting a qualitative methodology, we collected and analysed the students‟ writing productions. At first, students used to present the information given by the calculator with no justification. As they acquire skills in the use of this resource, they usually set up the viewing window in order to visualize the graphical representations of functions that model the problem situation they are working on and also relate the different existing menus in the study of those functions characteristics. Such procedures make students to present the data collected in the calculator with a justification of their arguments and a validation of their conjectures.