Maintaining dragging and the pivot invariant in processes of conjecture generation

In this paper, we analyze processes of conjecture generation in the context of open problems proposed in a dynamic geometry environment, when a particular dragging modality, maintaining dragging, is used. This involves dragging points while maintaining certain properties, controlling the movement of the figures. Our results suggest that the pragmatic need of physically controlling the simultaneous movements of the different parts of figures can foster the production of two chains of successive properties, hinged together by an invariant that we will call pivot invariant. Moreover, we show how the production of these chains is tied to the production of conjectures and to the processes of argumentation through which they are generated.Comment: Research report at the 40th PME Conference, Hungar

From conjecture generation by maintaining dragging to proof

In this paper we propose a hypothesis about how different uses of maintaining dragging, either as a physical tool in a dynamic geometry environment or as a psychological tool for generating conjectures can influence subsequent processes of proving. Through two examples we support the hypothesis that using maintaining dragging as a physical tool may foster cognitive rupture between the conjecturing phase and the proof, while using it as a psychological tool may foster cognitive unity between them.Comment: Research report at the 40th PME conference, Hungar

To tell a story, you need a protagonist: how dynamic interactive mediators can fulfil this role and foster explorative participation to mathematical discourse

This paper focuses on students’ mathematical discourse emerging from interactions in the digital environment GeoGebra, in which one can construct virtual objects that realize mathematical signifiers, and then interact with them. These virtual object realizations can become dynamic interactive mediators (DIMs) that influence the development of the learners’ mathematical discourse. In this case study, I analyze in fine detail the discourse developed by two dyads of students in response to an unfamiliar interview question. One dyad came from a class in which GeoGebra was not part of classroom practice and included students who, according to the teacher's evaluation, were standard-to-high achieving. The other dyad was from a generally demotivated and low-achieving class in which GeoGebra had become part of classroom practice. The analyses, focused especially on the low-achieving dyad, are guided by the question of how DIMs shaped these students’ discourse. According to the analysis, these students ended up succeeding where standard-to-high-achieving peers did not. Moreover, the detailed analysis of the ways in which the DIMs supported this dyad's learning showed mechanisms that may be general rather than specific to this one case. This suggests that appropriate integration of DIMs into the teaching and learning of high school algebra can be beneficial for low-achieving students

How I stumbled upon a new (to me) construction of the inverse of a point

While explaining to a friend analyst that a theorem about circle inversion that he used could be proved with synthetic geometry, I stumbled upon a new to me construction of the inverse of a point with respect to a circle. In this snapshot I describe episodes from this discovery process, faithfully to how they played out in time, highlighting the main ways in which I used dynamic geometry as a research tool

Forms of generalization in students experiencing mathematical learning difficulties

We shift the view of a special needs student away from the acknowledged view, that is as a student who requires interventions to restore a currently expected functioning behaviour, introducing a new paradigm to frame special needs studentsâ learning of mathematics. We use the theory of objectification and the new paradigm to look at (and characterize) studentsâ learning experiences in mathematics as generalizing reflexive mediated activity. In particular, from this perspective, we present examples of shifts to higher levels of generalization of a student with mathematical learning difficulties working with Mak-Trace, a Logo-like educational software for the iPad

Formas de generalización en estudiantes con dificultades de aprendizaje en matemáticas

We shift the view of a special needs student away from the acknowledged view, that is as a student who requires interventions to restore a currently expected functioning behaviour, introducing a new paradigm to frame special needs students’ learning of mathematics. We use the theory of objectification and the new paradigm to look at (and characterize) students’ learning experiences in mathematics as generalizing reflexive mediated activity. In particular, from this perspective, we present examples of shifts to higher levels of generalization of a student with mathematical learning difficulties working with Mak-Trace, a Logo-like educational software for the iPad.En este artículo introducimos un nuevo paradigma para enmarcar el aprendizaje de las matemáticas de alumnos con necesidades especiales. Consideramos una visión de los estudiantes con necesidades especiales diferente a la comúnmente aceptada que los considera como estudiantes que requieren intervención para reestablecer el comportamiento actualmente esperado. Utilizamos la teoría de la objetivización y ese nuevo paradigma para observar (y caracterizar) las experiencias de aprendizaje de las matemáticas entendido como actividad reflexiva y mediada de generalización. En particular, desde esta perspectiva proponemos ejemplos de acceso a niveles superiores de generalización de un estudiante con dificultades de aprendizaje de las matemáticas que utiliza Mak-Trace, un software didáctico para iPad parecido a Logo.We wish to thank the student, whom we have called Filippo, for accepting to work with us in this experiment, and the school IIS “E. Majorana” of San Lazzaro, Italy for enthusiastically accommodating collaborations between research and practice. We also dearly thank Anna Sfard for the insightful comments that she framed to address the issues we are concerned with

On different strategies to solve problems involving algebraic expressions

This study is part of a larger project that involves Italian teachers and students in grades 6 and 7, with the objective of designing inclusive mathematical activities, through cycles of implementation and revision of the designed materials, in order to promptly address known difficulties or overcome emerging difficulties in mathematical learning

Reasoning By Contradiction in Dynamic Geometry

This paper addresses contributions that dynamic geometry systems (DGSs) may give in reasoning by contradiction in geometry. We present analyses of three excerpts of students’ work and use the notion of pseudo object, elaborated from previous research, to show some specificities of DGS in constructing proof by contradiction. In particular, we support the claim that a DGS can offer “guidance” in the solver’s development of an indirect argument thanks to the potential it offers of both constructing certain properties robustly, and of helping the solver perceive pseudo objects

Surprise-driven abductions in DGEs

Abductive inferences, which are the only types of inference that produce new ideas, are important in mathematical problem solving. Such inferences, according to Peirce, arise from surprising or unexpected situations. Therefore, one way to improve student problem solving may be to provide them with environments that are designed to evoke surprise. In this paper, we examine the potential of dynamic geometry environments (DGEs) to foster surprise. We conjecture that the ease with which students can explore configurations, along with the immediate feedback, may lead them to encounter surprising situations. We analyse three different examples of student problem solving featuring surprised-provoked abduction, and identify the specific role that the DGE played

Developing an analytical tool of the processes of justificational mediation

Within the Instrumental Approach (IA) the newly developed notion of justificational mediation (JM) describes mediations that aim at establishing truth of mathematical statements in the context of CAS-assisted proofs in textbooks. Here we study JM with the intent to broaden the notion to the context of informal justification processes of early secondary students interacting with GeoGebra. Seeing JM as a process that has the objective of changing the status of a claim, we use Toulmin’s model and combine it with the IA to unravel the structure of the process through an analytical tool. The study is part of a broader project on the interplay between reasoning competency and GeoGebra with lower secondary students