The Calculator as an Instrument of Validation of Mathematical Knowledge: A Case Study (original) (raw)

TITE Problem Solving: Integrating Computing and Proving in Secondary Mathematics Teacher Education

2017

The paper reflects on an earlier research on the use of technology in secondary mathematics teacher education through the lenses of newer digital tools (Wolfram Alpha, Maple), most recent standards for teaching mathematics, and recommendations for the preparation of schoolteachers. New ideas of technology integration into mathematics education courses have been proposed. The goal of the paper is to show how one can connect mathematical understanding, conceptual knowledge, procedural skills, and the modern day technological competence.

1 Beyond verification: Proof can teach new methods

2015

The fact that proofs can convey new mathematical techniques to students effectively, as shown in recent literature, is an important advantage of the classroom use of proof, but it is one that mathematics educators seem to have overlooked to a large extent. The paper argues that teachers should make use of the potential of proof for presenting new techniques and demonstrating their value, and that mathematics educators in general should accord this potential its due importance among the many reasons for teaching proof. The teaching of proof in schools has been the topic of extensive investigations over the last two decades in the scholarly literature on mathematics education and in particular in the proceedings of the International Group for the Psychology of Mathematics. In her survey of research on proof in mathematics education, Mariotti (2006) found that most of the investigations on this topic have dealt primarily with the logical aspects of proof and with the cognitive problems...

Proof in school mathematics curriculum: A historical perspective

This paper is a case study of the place of proof in the United States school mathematics curriculum during the last century. The changes in the place of proof that took place in the United States school mathematics curriculum are indicative of similar trends in the school mathematics curricula of other countries. The paper addresses the following question: What story can be said about the place of proof in the United States school mathematics curriculum during the last century, as this is reflected (primarily) in available scholarly commentaries and curriculum frameworks? The examination of this question is intended to enhance mathematics educators' understanding of contemporary research and policy calls in many countries for a widespread presence of proof in the school mathematics curriculum, by helping them situate these calls in the 'standards era.' This era is the last in a series of four historical eras discussed in the paper: the unified and applied mathematics era, the modern mathematics era, the post-modern mathematics era, and the standards era.

Calculators in mathematics education: A rapid evolution of tools, with differential effects

The Didactical Challenge of Symbolic Calculators Turning a Computational Device into a Mathematical Instrument, 9-40, 2005

The appearance of more and more complex tools in mathematics classes is not a response to an institutional need of school. It is, rather, the expression within this institution of a huge social phenomenon (the increase in the number of screens and machines) arising from the utilization of computerized tools by certain branches of mathematics and science.

A model for the educational role of calculators

2014

Calculators can be used effectively for mathematics education in a number of ways, although frequently they are regarded merely as devices for undertaking computations. In this analytic paper, we describe and illustrate a four-part model to understand more fully the potential role of calculators for learning mathematics. The four elements of the model include representation, computation, exploration and affirmation. Effective use of a calculator by students learning mathematics will often involve more than one of these four components. The model has been derived from analysis of educational materials developed to support rich calculator use.

PROVING AND PROOF AS AN EDUCATIONAL TASK

Situated in the context of current research, my talk presents a point of view on proof as an educational issue. This point of view intends to combine three perspectives: epistemological, what we mean by proof; cognitive, which are the main difficulties that students meet; and didactical, what kind of didactic interventions can be proposed, among those that have been experimented and showed their effectiveness. The examples that will be presented are drawn from the current literature, but also from research studies which I have been directly involved in recently. In particular, I will discuss some results coming from long-term teaching experiments related to the use of computer-based artefacts as tools of semiotic mediation that teachers can exploit to develop the mathematical meaning of proof.

VALIDATING IN THE MATHEMATICS CLASSROOM

The focus of this report is on the process of resolution of a task; specifically, on the validation of the mathematical model proposed by a group of students and the numerical result that is constructed within this model. Habermas' construct of rational behavior is used to describe validity conditions that emerge and are used by the students as means for validation. We take a classroom episode from a design experiment to examine how the emergence of these conditions points to a socially constituted mathematical epistemology in the secondary school mathematics classroom, to shared and tacit principles of the didactic contract concerning the knowledge there, and to non-mathematical references that are taken for granted.