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Sunday 30 June 2019

The Newton-Raphson Method

The Newton-Raphson method is a very efficient algorithm to search for the zero of a real function. Most of the time it converges more quickly than a dichotomy approach. Nevertheless, there are some traps to avoid by studying the function and/or its curve.

It is attributed to the mathematicians Newton and Raphson for their respective contributions in 1685 and 1690. However, it is also the result of a slow maturation during the ages, before and after the Newton era. Its definitive principle as known today is from the mathematician Thomas Simpson in 1740. Before, it is derived from the works of mathematicians as the French François Viète in the 16th centuries, Iranians Sharaf al-Din al-Tusi and Jamshīd al-Kāshī in the 13th and 14th centuries. Finally it is a generalization of the Hero of Alexandria method in the 1st century BC also known as the Babylonian method.

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Monday 3 June 2019

La méthode de Newton-Raphson

La méthode de Newton-Raphson est un procédé très efficace dans la recherche du zéro d’une fonction réelle, sa convergence est généralement bien plus rapide que celle de la méthode dichotomique. Elle comporte cependant des chausse-trappes qu’une étude de la fonction et de sa courbe permettent d’éviter.

Elle est attribuée aux mathématiciens Newton et Raphson pour leur contribution respective en 1685 et 1690. Elle est toutefois le fruit d’une lente maturation au cours des ans, avant et après l’époque de Newton. Son procédé définitif tel qu’utilisé aujourd’hui est attribué au mathématicien Thomas Simpson en 1740. Avant, elle est dérivée des travaux du mathématiciens François Viète et des mathématiciens iraniens Sharaf al-Din al-Tusi et Jamshīd al-Kāshī. Enfin elle est une généralisation de la méthode de Héron ou méthode babylonienne d’extraction d’une racine carrée.

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Saturday 12 January 2019

Dynabook, user interface point of view

In our previous writings, we took the point of view of said innovative teaching approaches. It differs from the children's point of view historically taken by A. Kay. This provided us a set of general features, curriculum neutral, we may want for a teacher centered Dynabook; we summarize it in our "Dynabook and learning models, final words" writing.

Three features came out in several learning models: text processor, question tool and tools to collect and represent data. The first one is linked to the human concept of writing and reading, the last one relates to how to represent information: statically or dynamically. They are related to the representation of knowledge and its transport both in space and time. Appearing as key features in a learning system should not be a surprise, though it may seem to be obvious, it is important not to be overlooked.

To enrich our thinking, we want to change our point of view and look from the curriculum and user interface perspectives. Based on our previous thinking, this will be examined with text and representation of knowledge as the two backbones. Later, we will discuss the third feature, the question tool.

In the following paragraphs, we discuss the user interface point of view.

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Wednesday 18 July 2018

The Dynabook Concept

Dynabook concept, A Personal Computer for Children of All Ages.

A Dynabook technical drawing

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Friday 6 July 2018

Dr. Geo and Dynamic Media

A Dynabook mock up Alan Kay coined the term Dynabook, the expressions Dynamic media and Active Essays. How does Dr. Geo connects to the underneath concepts behind these words? This is what we explain in the following paragraphs.

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Wednesday 4 January 2017

Cercle inscrit

Dans un article précédent, nous montrions comment écrire une description textuelle du cercle circonscrit à un triangle. Cette fois-ci nous expliquons comment écrire une description textuelle du cercle inscrit à un triangle Cette description sera toujours en français, et légèrement plus complexe.

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Tuesday 3 January 2017

Mathematics series

My son came to me with a mathematics series, he need to find its convergence. After the mathematics work is done, we want to confirm by calculus the found limit. Dr. Geo can be of some help here.

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Saturday 31 December 2016

Cercle circonscrit

Comment écrire une description textuelle du cercle circonscrit à un triangle

Avec le logiciel Dr. Geo, il est possible d'écrire un code source en français décrivant n'importe quelle figure. Ecrire une telle description pour le cercle circonscrit d'un triangle est assez simple et court comme expliqué dans les paragraphes suivants.

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Circumscribed circle

How to write a textual description of a circumscribed circle to a triangle?

With Dr. Geo software, you can write source code describing any sketch. Writing a description of the circumscribed circle to a triangle is pretty simple and short as explained in the following paragraphs.

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Saturday 12 March 2016

Tiny, yet so beautiful

 How do you introduce an unknown, alternative programming language to other developers ? One way to do so is by showing lots of small examples. That is what we did in Elegant Pharo Code — Beautiful & Powerful One-liners, Expressions and Snippets.

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Thursday 10 March 2016

Fibonacci spiral

The Fibonacci spiral is based on the Fibonacci sequence to construct a curve forming a spiral. This spiral and some variants appear in the constructions of live entities. In this article we present a computed, interactive version of this spiral with the help of Dr. Geo, an interactive geometry and programming application. Later we discuss on the interest of this approach in mathematics secondary teaching.

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Tuesday 23 February 2016

La spirale de Fibonacci

La spirale de Fibonacci s'appuie sur la suite éponyme pour la construction d'une courbe en forme de spirale, cette spirale et ses variantes apparaissent dans des constructions du vivant. Dans cet article nous proposons une construction calculée d'une forme interactive de cette courbe, à l'aide de Dr. Geo, logiciel de géométrie interactive et de programmation. Nous relaterons ensuite les liens à faire avec les programmes d'enseignement secondaire de mathématiques.

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Saturday 17 October 2015

Programmation recursive

Aborder la récursivité n'est pas aisé pour les néophytes. Voici une approche pas à pas de la construction du triangle de Sierpinski pour appréhender sa dimension auto-répétée.


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Termes d'une suite

On continue avec les suites ou comment afficher les termes d'une suite qui est chaotique pour certaines valeurs de son paramètre.

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Diagramme de suite récurrente

Vous avez sans doute tous ce souvenir de classe de terminale, des suites récurrentes, avec recherche d'un point fixe d'une fonction. Dr. Geo permet d'explorer ce sujet.

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Thursday 15 December 2011

Dr. Geo: hacker spirit in a candy box

When you teach mathematics at secondary schools, you enjoy using interactive geometry software with your students. Dr.Geo is one such free software, it comes with some unique features. With such software, students learn geometry the way it should be learned -- by drawing, but Dr. Geo goes one step further.

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Saturday 12 February 2011

Dr. Geo and block closure

Dr. Geo comes with a few power tools related to programming. Since its first incarnation, Dr. Geo is enriched with open doors to let the user explores new way to use Dr. Geo. The first version of Dr. Geo was developed in C++, with that version the power tools came under the form of a Scheme interpretor embedded in Dr. Geo. This Scheme was GNU Guile and it gave freedom to write script or programmed sktech with the scheme language.

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Thursday 20 January 2011

Paradigm shift

With recent releases, and coming version 11.03, Dr. Geo II is becoming more and more an integrated environment to teach or to learn interactively software programming within a geometry and math context.

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Thursday 8 July 2010

Sierpinski triangle

How to construct with Smalltalk script the Sierpinski triangle.

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Saturday 10 January 2009

Les scripts avec Dr. Geo II

Dans cet article je montrerai comment utiliser les scripts Dr. Geo II pour construire dans une figure la courbe représentative d'une fonction, puis la tangente en tout point de cette courbe. La logique de construction de la figure sera indépendante de la fonction. courbeTangente3.png

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