Nurit Zehavi

Bisoptic curves of hyperbolas

Given a hyperbola, we study its bisoptic curves, i.e. the geometric locus of points through which passes a pair of tangents making a fixed angle θ or 180° − θ. This question has been addressed in a previous paper for parabolas and for ellipses, showing hyperbolas and spiric curves, respectively. Here the requested geometric locus can be empty. If not, it is a punctured spiric curve, and two cases occur: the curve can have either one loop or two loops. Finally, we reconstruct explicitly the spiric curve as the intersection of a plane with a self-intersecting torus.

Revival of a classical topic in differential geometry: the exploration of envelopes in a computerized environment

Learning mathematics in a technology-rich environment enables us to revive classical topics which have been removed from the curriculum a long time ago. Both theoretical issues and applications can be studied with an experimental process. We present how envelopes of 1-parameter families of plane curves and some of their applications can be presented early in the curriculum either for pre-service teachers or for in-service teachers. This approach may be useful for students in an engineering curriculum. Working with technology yields important effects, such as reviving classical topics, broadening perspectives on already known topics, and enhancing the learners experimental skills, where conversion between various registers of representation is an important issue.

The Impact of Computers on Student and Teacher Commitment to Learning and Teaching

During the past 15 years, the slow penetration of computers into schools, as well as their limited impact there, gives credence to the claim that computers have had a modest impact on the average student. We identify four possible sources of this limited impact of computers on school learning: (1) lack of a critical mass of computers, (2) inappropriate goals and expectations, (3) insufficient interaction of computers with the content of school learning, and (4) insufficient professional development and related support. What might be the impact of computers and information technology to learning and teaching if serious efforts were undertaken to offset these four limitations?

Qualitative evaluation of mathematical activity and its relation to effective guidance

A model of qualities of mathematical activity was developed within a framework of learning goals. In this paper we describe a ‘clinical’ application of the model, from which a hierarchy of effective guidance was established, permitting diagnostic evaluation and progressive development of mathematical activity.

Problem solving: linear functions and magic circles

Problem‐solving experience should be an integral part of a students mathematical experience at whatever level he may be. We describe such a situation which requires little more than an elementary knowledge of linear functions, and can be developed, with suitable prompting by the teacher, towards ideas which usually come much later. Most of the description is, for the sake of brevity, mathematical; the didactical strategy should, however, be clear by implication.

Automated Study of Envelopes of One-Parameter Families of Surfaces

Learning mathematics in a technology-rich environment enables to revive classical topics which have been removed from the curriculum a long time ago. Theoretical issues and their applications can be studied within an experimental process, using automated proofs. We present how envelopes of one-parameter families of surfaces in 3D space and some of their properties can be presented using technology. This approach may be useful for students in an engineering curriculum and for in-service/pre-service teachers. Working with technology and taking advantage of both algebraic symbolic features, such as algorithms computing Grobner bases, and visualization tools, educational and professional profit is obtained such as reviving classical topics from differential geometry, broadening horizons, introducing new topics. The purpose is also to enhance the learners experimental skills. In such a framework, conversion between various registers of representation is an important issue.

The Use of ITS Techniques in Research on Pairing Representations

AbstractPairing external representations of the same concept is ubiquitous in educational software. A study integrating cognitive research and development of an Intelligent Tutoring System (ITS) was undertaken to evaluate the effects of pairing representations of mathematical functions on high school students. The study consisted of (a) the administration of a questionnaire (N = 200) for assessing knowledge of properties of functions displayed separately in graphical and algebraic representations; and (b) a clinical study showing the effectiveness of pairing as an instructional action; (c) the development of an ITS, the Function Characteristics Tutor (FCT), for learning function characteristics by comparing (a special case of pairing) representations; (d) an experiment (N = 60) in which students solved tasks with FCT. Overall results show that comparing graphical and algebraic representations is efficient in learning important function characteristics, not only when the two representations refer to the sa...