Thierry Dana-Picard

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• PhD
• Full professor at Jerusalem College of Technology

This study investigated students' understanding of mathematical functions and strategies to create artwork using GeoGebra. Itwas framed by the principles of constructionism and examined how students use functions in creating artworks. We gathereddata from students' artworks using the Algebra view and the Construction Protocol in the GeoGebra softwa...

We explore offsets of Cayley ovals, by networking with different kinds of software. Using their specific abilities, algebraic, geometric, dynamic, we conjecture interesting properties of the offsets. For a given progenitor (the given plane curve whose offsets are studied), changes in the offset distance induce great changes in the shape and the top...

Envelopes of parameterized families of plane curves is an important topic, both for the mathematics involved and for its applications. Nowadays, it is generally studied in a technology-rich environment, and automated methods are developed and implemented in software. The exploration involves a dialog between a Dynamic Geometry System (used mostly f...

Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the pedal curves of conics. This provides a construction of interesting geometric loci, given at first by parametri...

Hyperbolism of a given curve with respect to a point and a line is an interesting construct, a special kind of geometric locus, not frequent in the literature. While networking between two different kinds of mathematical software, we explore various cases, involving quartics, among them the so-called Kuelp quartic and topologically equivalent curve...

The study of some parametric integrals is presented with a combined approach of analytical development, the usage of a Computed Algebra System (CAS) and of the Online Encyclopedia of Integer Sequences. The methodology for the solution includes a) an analytical investigation for the study of the parametric integral, b) computations with a CAS of the...

This brief history of SEMT provides details of the life of the Symposium on Elementary Mathematics Teaching (SEMT), which, as its name suggests, focuses on the teaching of mathematics to children within the range of 5 to 12 years of age. The first SEMT symposium was in 1991, with the participation of some 39 educators and researchers, and since the...

Locus computation is an essential issue in mathematics education, and a traditional feature of Dynamic Geometry software (DGS). The rising of programs merging DGS and Computer Algebra software (CAS) has fostered a combined approach to locus computation, quite performing in standard examples, but demanding an extended theoretical, and the related al...

As soon as a new technology emerges, the education community explores its affordances and the possibilities to apply it in education. In this article, we analyze sessions with ChatGPT around topics in basic linear algebra. We reflect on the affordances and changes between two versions of ChatGPT since its worldwide publication in our area of intere...

From string art to models of exotic surfaces in space using 3D printing

In this work, we aim to understand how automatic feedback tasks created in GeoGebra contributed to learning how to tell time on analog clocks. Therefore, several tasks were implemented in a fourth-grade class that already had knowledge of analog clocks and hours. These tasks aimed to revise the topic, supporting students with automatic feedback app...

Estrella Solitaria was designed in 1849, 175 years ago. “It has become much more than a visible symbol for the island nation: Modern states and would-be nations around the globe have imitated the elegant design of this flag”. (Chacón: The Global Legacy of Cuba’s Estrella Solitaria, 2017) ▶ Estrella Solitaria contains a regular pentagram that consis...

We started from a question about string art asked by a teacher. This yielded work on envelopes of parametric families of plane curves and to a transition from2D to 3D. Finally, with 3D printing, we create a new register of representation for mathematical objects.

Integrating 3D modelling and printing in STEAM education presents opportunities and challenges for teachers, particularly those in some European countries where its adoption in schools still needs to be improved. This article presents findings from a cross-cultural examination of 3D modelling and printing in STEAM education, showing results from te...

The solar eclipse of 2024 provide an incitement to present simple models of orbits of planets, of objects around planets and around satellites of planets. This involves constructions of geometric loci using automated commands.

Cassini ovals (also called spiric curves or Spirics of Perseus) are an interesting family of bicircular quartic plane curves, appearing in various scientific fields, such as electric fields and delineation of influence zones of wells. They appear also as bisoptic curves of ellipses and, with some additional conditions, of hyperbolas. They provide c...

Relying on the students' cultural background to teach mathematics may be a strong incitement for the learning process. This background can include artistic creations, items from the news, among others. Because of the strong presence of space related news (in particular the almost simultaneous launching of three spacecrafts towards Mars), we present...

We provide constructions of 3 classical curves, using a new approach, based on tools for automated exploration and reasoning, especially for the determination of geometric loci. Dragging and animations are the core features in use. The different ways yield an output based either on numerical computations or on symbolic computations (these use Gröbn...

We present simple models of trajectories in space, both in 2D and in 3D. The first examples, which model bicircular moves in the same direction, are classical curves (epicycloids , etc.). Then, we explore bicircular moves in reverse direction and tricircular moves. The exploration is followed by 3D printing. Students’ activities are organized aroun...

The study of envelopes requires a strong dialog between algebraic computations and graphic representations. Both are available using a Computer Algebra System, and sometimes accompanied by a Dynamic Geometry System. The two kinds of software have different affordances, whence different characteristics of the animations. Visualization raises specifi...

We use GeoGebra in creating a bridge whose components include graphs of mathematical functions. One interesting issue is that if you plot a graph in the 2D view, it appears simultaneously in the 3D view, but it lies on the xy plane. To obtain a realistic view, this plot has to be rotated. We will also show some sample outputs of the students and di...

Octic curves (i.e. algebraic plane curves of degree 8) appearing as geometric loci

A contribution to teaching continuity and discontinuity of a 1-real variable function. Using animations with a Computer Algebra System.

On the one hand, mathematical software is ubiquitous in mathematics education. On the other hand, word problems are an important part of the curriculum, and they often require modelling skills. This is especially true with optimisation and extrema problems proposed to high school and undergraduate students. We propose two activities around extrema...

Analysis of certain problems encountered with AI for teaching Linear Algebra

The emergence of ChatGPT has been rapid, and although it has demonstrated positive impacts in certain domains, its influence is not universally advantageous. Our analysis focuses on ChatGPT's capabilities in Mathematics Education, particularly in teaching basic Linear Algebra. While there are instances where ChatGPT delivers accurate and well-motiv...

We are given a fixed point $A=(a_1,a_2)$ and lines passing through this fixed point to intersect a conic $c:x^2+q y^2=1$ at $C$ and $D$ ($q>0$). The locus $E$, we are interested in finding, is lying on $CD$ and satisfies $\overrightarrow{ED} = s\cdot \overrightarrow{CD}$ where $s$ is a given real number. We study the topology of the obtained parame...

We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations,...

From simple models of trajectories in space, we built classical curves using a DGS. Animation enhance the understanding of moves in space. A further step is the creation of mathematical art, based on these curves.

Talk at a symposium for medicine doctors, Herzliyah, Israel

The study of envelopes and o�sets requires a strong dialog between algebraic computations and graphic representations. Both are available using a Computer Algebra System, and sometimes accompanied by a Dynamic Geometry System. The two kinds of software have di�erent a�ordances, whence di�erent characteristics of the animations.Visualization raises...

This survey paper is an expanded version of an invited keynote at the ThEdu'22 workshop, August 2022, in Haifa (Israel). After a short introduction on the developments of CAS, DGS and other useful technologies, we show implications in Mathematics Education, and in the broader frame of STEAM Education. In particular, we discuss the transformation of...

Envelopes of parametrized families of surfaces in 3D space. A STEAM education approach.

Using items form the news to model orbits of various objects in space and explore curves with automated methods

We explore envelopes and offsets of plane curves using automated methods, based on the usage of Computer Algebra Systems and Dynamical Geometry Systems. Work is performed using both parametric presentations and implicit equations. Implicitization involves well-known algorithms from the theory of Gröbner bases. In particular, irreducibility of the i...

The usage of functions to create art

Exploration of envelopes of families of surfaces, of offsets and of canal surfaces with DGS and CAS

The development of technology has changed the way people communicate in academic contexts as well as working places, for example from print messages to screened messages, and from face-to-face classroom and office meetings to virtual classes and offices. This has prompted the shift from traditional teaching practices to student-centered in which st...

There is a large amount of research that indicates that the use of 3DMP in STEM education improves students’ knowledge, motivation, and participation in the learning process. Nevertheless, despite the existing attempts to market 3DMP in education, its adoption in schools remains low. A number of studies with teachers in secondary schools and colleg...

Convergence in STEAM Education - what happened with the Covid-129 crisis. In particular, the 4 C's of 21st Century Education are addressed.

Visualising abstract concepts such as for example geometrical objects in mathematics can be a valuable support for learners. Visualisation, however, is a process involving several steps that influence each other. Duval (1998) uses steps connecting reasoning by an explanation or proof to a construction step involving tools creating a visualisation t...

For a given angle α, the α-isoptic curve of a parabola is the geometric locus of points through which passes a pair of tangents to the parabola making an angle equal to α. We explore the inner isoptics of parabolas: they are the envelopes of the lines joining the points of contact of the parabola with the tangents through points on a given isoptic....

Models of circular orbits, bicircular and tricircular plane curves, and the creation of mandalas. All this in a STEAM approach.

Polygons are an important topic in geometry. We propose and analyse activities for the exploration of their properties, with extensive usage of a dynamical geometry system. They are based on the analysis of monuments, connecting students’ cultural backgrounds with their mathematical learning. Moreover, two opposite directions for work are presented...

We report on computing and visualizing the offsets of the Cassini ovals. We use the elimination technique to obtain offsets as envelopes of a family of planar curves in a recent experimental version of GeoGebra. Offset curves are obtained in their implicit form as irreducible polynomials of degree 12 and 16.

Exploration of canal surfaces in a technology-rich environemnt, with a STEAM approach.

For a long time, the traditional way to convey mathematics was structured according to the definition-theorem-proof-example scheme. A few decades ago, digital tools appeared: calculators working numerically, then graphical features (also numerically based) were added. With the developments of algorithms for symbolic computations, things changed pro...

Groundwater pollution is a general concern in countries using pumping wells for water consumption. Also, the determination of protected zone around pumping wells is a practical concern. The delineation of this zone is frequently obtained using numerical models such as MODFLOW and or FEFLOW. In this contribution we derived the equation of the plane...

Several non equivalent definitions exist for the envelope of a 1-parameter family of plane curves. Another notion, often considered as related to envelopes, is the offset at a given distance of a plane curve. Using the so-called analytic definition, we study and compare the envelope of a 1-parameter family of circles centered on a parabola and an o...

The non-uniqueness of a rational parametrization of a rational plane curve may influence the process of computing envelopes of 1-parameter families of plane curves. We study envelopes of family of circles centred on a regular trifolium and its offsets, paying attention to different parametrizations. We use implicitization both to show that two rati...

Envelopes of parameterized families of plane curves, of space curves, of surfaces, are an important topic both because of the mathematics involved and because of their applications (e.g. the determination of safety zones around sprinklers, robotic plants, Luna Park attractions , etc.). A drawback of this domain is the small number of its theorems,...

The non-uniqueness of a rational parametrization of a rational plane curve may influence the process of computing envelopes of 1-parameter families of plane curves. We study envelopes of family of circles centred on a regular trifolium and its offsets, paying attention to different parametrizations. We use implicitization both to show that two rati...

Different technologies such as Computer Algebra Systems (CAS) and Dynamical Geometry Systems (DGS) have been developed along the last decades. CAS have high abilities for symbolic computations and also graphical features, and DGS are characterized by their interactivity and exploratory abilities. Due to the implementation of CAS features, DGS are n...

The Golden Section is a mathematical concept that is one of the most famous examples of connections between mathematics and the arts. Despite its widespread references in various areas of nature, art, architecture, literature, music, or aesthetics, discussions of the golden ratio often turn out to be false or misleading. Most of the incorrect state...

In this PhD outline, I will present highlights from my PhD research project on mathematical modelling with real-world information in the classroom, remote teaching and outdoor learning in Luxemburg. Through design-based explanatory studies, I investigated different technology enhanced tasks, learning and teaching settings that could likely engage s...

We present an initial description of an ongoing research project. Students are attracted to learn mathematics, not only for its application but also for their cultural interest. The students’ cultural backgrounds are used, and situations are analyzed and modeled using technology. This first step has been instrumental in making the transition to dis...

For a given curve C and a given angle θ, the θ-isoptic curve of C is the geometric locus of points through which passes a pair of tangents to C making an angle equal to θ. If the curve C is smooth and convex, isoptics exist for any angle, and through every point exterior to the curve, there is exactly one pair of tangents. The isoptics of conics ar...

Polygons and polyhedras - a STEAM education approach with a DGS

STEAM education - 21st Cenrtury skills

Isoptic curves of plane curves are a live domain of study, mostly for closed, smooth, strictly convex curves. A technology-rich environment allows for a two-fold development: dynamical geometry systems enable us to perform experiments and to derive conjectures, and computer algebra systems (CAS) are the appropriate environments for an algebraic app...

We present a study of a classical plane real curve in a technology-rich environment. The interplay between implicit and parametric presentations is enhanced. Cooperation of a computer algebra system and of a dynamical geometry system enables to discover properties that may be hidden in hand-made computations. Moreover, technology does not act only...

Students learning towards a degree in a STEM related domain learn quite early a course in Advanced Calculus, i.e. a course where the main object of study are multivariate functions. It happens that students do not see the connection between the properties of the function such as continuity or differentiability and the 3-dimensional graphical repres...

We study a couple of geometrical elements of a monumental synagogue (Dohany Synagogue in Budapest). Then we describe how undergraduate students from the Jewish orthodox population have performed a technology-based study, using a Dynamical Geometry System, notions, and tools of analytic geometry. These students come without a classical background in...

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 a technology based study of isoptic curves of an astroid, which is a non smo...

The primary objective of our research with the VENS data is the development and validation of natural and cultivated ecosystem functioning models that will be most suitable for Eastern Mediterranean landscapes. This requires spatial and temporal modeling of changes (and respective exchange rates) between dominant ecosystem players: soil, vegetatio...

Jewish Art and Math Education

We present two approaches to symbolically obtain isoptic curves in GeoGebra in an automated, interactive process. Both methods are based on computing implicit locus equations, by using algebraization of the geometric setup and elimination of the intermediate variables. These methods can be considered as automatic discovery. Our first approach uses...

We present two approaches to symbolically obtain isoptic curves in the dynamic geometry software GeoGebra in an automated, interactive process. Both methods are based on computing implicit locus equations, by using algebraization of the geometric setup and elimination of the intermediate variables. These methods can be considered as automatic disco...

Accurate plots of 2-variable functions: the choice of the mesh

Automated determination of isoptic curves

Fractals and tessellations: a STEAM approach in Math Educ

Teaching fractals and tessellations in technology rich environment and a STEAM approach

Accurate mesh for plotting 2-variable functions

Non homogeneous stiff ODEs

The religious population in Israel is heterogeneous; consequently, there are several kinds of high schools catering to this population. We give a schematic description of the population and of the schools, then present trends at the different levels of study. We relate briefly to outstanding and gifted students, using yeshiva high school for gifted...

We describe working session for the study of 1-parameter families of definite integrals in a technology-rich environment. The joint usage of paper-and-pencil work together with a Computer Algebra System and eventually a web based database may lead to closed forms for the integrals, to the derivation of combinatorial identities, and other kinds of o...

The Applications of Computer Algebra (ACA) conference series is devoted to promoting all kinds of computer algebra applications, and encouraging the interaction of developers of computer algebra systems and packages with researchers and users (including scientists, engineers, educators, and mathematicians). Topics include, but are not limited to, c...

The study of a real function of two real variables can be supported by visualization using a Computer Algebra System (CAS). One type of constraints of the system is due to the implemented algorithms, yielding continuous approximations of the given function by interpolation. This masks often discontinuities of the given function and its curvature at...

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...

We study a 1-parameter family of trigonometric definite integrals, showing how the joint usage of Information and Communication Technologies and paper-and-pencil work lead to different outputs, revealing different mathematical meanings and different concrete meanings. This family of integrals is useful for describing a phenomenon in soil mechanics,...

Managing the constraints of technology for an automated study of envelopes of families of plane curves

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 presente...

Nowadays, the interplay between Geometry and Algebra is present almost everywhere. Classical theorems from Geometry may be revived with the usage of automatic theorem proving, based on Computer Algebra Systems and other technologies. Pech's book displays six stories about classical theorems, providing both a computer based proof and a classical one...

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  or 180° − . 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 n...

Jewish Art and Math Educ

Topics in Math Educ in a technology-rich environment