Abstract: Starting with a quotation describing mathematical research, this paper presents the beginning of an imaginary dialogue between a practicing mathematician and a didactician about ways of providing students with adequate experiences in mathematical research. The conversation begins with a discussion of the benefits and implications for the students of such experiences, followed by a more detailed discussion of what forms these experiences might take, including two examples. Later, the didactical goals of the experimented situations are elaborated from a more theoretical perspective. The last section includes two proposed hierarchies of knowledge, and concludes with an illustration of how one goes about fostering these hierarchies of knowledge through an experience in mathematical research.

Abstract: This paper presents the early results of an experiment involving a class of elementary student teachers within the context of their mathematics preparation. The motivation of the exercise centred on giving them an experience with mathematical research a their own level and ascertaining its impact on their attitudes and beliefs. The students spent the first month working on open-ended geometrical topics. In the second month, working alone or in groups of up to four, they chose one or more of these topics then worked on a problem of their own design. The students spent the class time developing their ideas using strategies such as generating examples and non-examples, generalising, etc. Reference to books was not accepted as a research tool, but the instruction team monitored student progress and was available for questions.

Abstract: The goal of this paper is to illustrate how octahedra and tetrahedra pack together to fill space, and to identify and visualize the dual to this packing. First, we examine a progression of 2-D and 3-D space-filling packings that relate the tetrahedral-octahedral space-filling packing to the packing of 2-D space by squares. The process will use a combination of stretching, truncation and 2-D to 3-D topological correspondence. Through slicing, we will also relate certain stages of the process back to simple 2-D packings such as the triangular grid and the 3,6,3,6 Archimedean tiling of the plane. Second, we will illustrate the meaning of duality as it relates to polygons, polyhedra and 2-D and 3-D packings. At a later stage, we will reason out the dual packing of the tetrahedral-octahedral packing. Finally, we will demonstrate that it is indeed a 3-D space filler in its own right by showing different construction methods.

Abstract: The present paper reports on children's investigations using the giant equilateral triangles from the Geraldine Project. It took place at the De Zavala Elementary School as the initial stage of a project in mathematics education. The triangles are a part of a modular construction kit made using kite technology. Their size, sturdiness and light weight make them ideal for in-class activities with children of all ages and stages of development.
The school is located in a low socio-economic hispanic neighbourhood consisting of blue-collar families living in apartments and rental houses as well as small businesses and industries. Most of the students at the school are recent immigrants from Mexico or Central America or first generation born to immigrant families. Their parents have little or no education and are forced to work on jobs that entail long hours, frequently into the evening or night. This situation makes it difficult for parents to provide their children with appropriate support as students.
At this stage, the structure of the activities that make up lessons emerged from the response of the children as the activities were tried. This approach, despite its unplanned nature, allowed for the introduction of much mathematical content, and the attention of the children was relatively easy to catch and hold. The activities successfully combined the play aspect of the giant triangles with the mathematical concept explorations that the instructors overlayed. In some cases the children were allowed to build their own shapes, which were then examined with them. The outcome of these trial activities was then used as a basis for lesson planning in later stages of the pilot project.

Abstract: The following paper recounts the stages of a stroll through symmetry relationships between the regular tetrahedron whose faces were subdivided into symmetrical kites and the regular dodecahedron. I will use transformations such as stretching edges and faces and splitting vertices. The simplest non-adjacent regular coloring , which illustrates inherent symmetry properties of regular solids, will help to keep track of the transformations and reveal underlying relationships between the polyhedra. In the conclusion, we will make observations about the handedness of the various stages, and discuss the possibility of applying the process to other regular polyhedra.

Book description: "The book contains papers from the proceedings of the 3rd International Meeting of Origami Science, Math, and Education, sponsored by OrigamiUSA. They cover topics ranging from the mathematics of origami using polygon constructions and geometric projections, applications, and science of origami, and the use of origami in education."

Abstract: Deltahedra are polyhedra with all equilateral triangular faces of the same size. We consider a class of we will call 'regular' deltahedra which possess the icosahedral rotational symmetry group and have either six or five triangles meeting at each vertex. Some, but not all of this class can be generated using operations of subdivision, stellation and truncation on the platonic solids. We develop a method of generating and classifying all deltahedra in this class using the idea of a generating vector on a triangular grid that is made into the net of the deltahedron.
We observed and proved a geometric property of the length of these generating vectors and the surface area of the corresponding deltahedra. A consequence of this is that all deltahedra in our class have an integer multiple of 20 faces, starting with the icosahedron which has the minimum of 20 faces.

Abstract: This is a preliminary report on design features of large,light-weight,modular equilateral triangles and classroom activities developed for using them.They facilitate the fast teaching of three dimensional geometry together with basic math skills,and create a lasting motivational impact on low achievers and their subsequent performance in math and science.
In directed discovery activities,lasting from 20 to 90 minutes,large models of basic polyhedra are made, enabling their properties to be explored.Faces,edges and vertices can all be counted and tabulated,providing opportunities to see number patterns and inter-relationships,to plot graphs,to extract algebraic relationships and to look for proofs of those relationships.These building activities can be kept central,under the teacher ’s control for large classes with limited time,or building can be split out into groups of children where co-operative problem solving skills are also developed.
In interviews,children have stressed the effectiveness of learning by building the shapes themselves.In classroom activities,it is clear to see that these triangles make children excited.Learning by building gives a concrete,active,authentic and personal experience of mathematics to children and teachers enabling them to feel the full excitement of the subject.

Abstract: The following exercise is based on experiments conducted in circular Origami. This type of paper folding allows for a completely different geometry than the square type since it lends itself very easily to the creation of shapes based on 30-60-90 degree angles. This allows for experimentation with shapes made up of equilateral triangles such as deltahedra. The results of this research were used in Annenberg sponsored activities conducted in a progressive middle school in Houston TX, as well as a workshop presented at the 1999 Bridges Conference in Winfield, KS. Not including preparatory and follow up work by the teacher, the activities in Houston were composed of two main parts, the collaborative construction of a three-yard-across, eighty-faced regular deltahedron (the Endo-Pentakis Icosi-dodecahedron) and the following exercise. The barn-raising was presented last year in Winfield, and the paper folding is the topic of this paper.

Extrait du sommaire: Le parallèle entre les structures des mondes 2-D et 3-D est un concept qui a depuis longtemps été pris pour acquis. La question peut se poser, pourtant, sur la façon dont cela se traduit dans le monde pratique, particulièrement dans le domaine de la conception formelle. En effet, quelle forme pourront rendre le ou les objets 3-D régis par les mêmes relations dans l’espace qu’obéit un objet 2-D dans son propre environnement...