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- ItemPolyhedra, Learning by Building: Design and Use of a Math-Ed. Tool.(Bridges: Mathematical Connections in Art, Music, and Science, 2000) Knoll, EvaThis 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.
- ItemExperiencing research practice in pure mathematics in a teacher training context(International group for the Psychology of Mathematics Education, 2004) Knoll, Eva; Morgan, Simon; Ernest, PaulThis 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 at 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 nonexamples, generalising, etc. Reference to books was not accepted as a research tool, but the instruction team monitored student progress and was available for questions.
- ItemDiscussing the challenge of categorizising mathematical knowledge in mathematics research situations(Bridges: Mathematical Connections in Art, Music, and Science, 2005) Knoll, Eva; Ouvrier-Buffet, CécileStarting with a quotation describing mathematical research, this paper presents ways of providing students with comparable experiences in mathematical research, in the classroom. The paper focuses on the benefits and implications for the students of such experiences. “Real mathematics research-situations” are defined, and the didactical goals of these situations, as they are experienced are elaborated on. These elements are presented through examples, looking at similar situations (research situations) in two contexts and using different theoretical frameworks.
- ItemDeveloping a Procedure to Transfer Geometrical Constraints from the Plane into Space(International Society for Geometry and Graphics, 1998) Knoll, EvaTopology teaches us that the two dimensional plane and three dimensional space have a comparable structure. In fact, this apparent parallel is deeply rooted in our consciousness and is applied in many domains, including various fields in the design industry, through the use of such tools as descriptive geometry and perspective drawing. From the particular point of view of the designer, however, this parallel in structure has often been simplified to plans, sections and elevations i.e. 2-D slices through a 3-D object. It has therefore not been an integral part of the design process, but rather a tool of representation of the design process. In the following paper, the relationship between plane and space will be explored as a design element. The question will be answered whether it is possible, starting with a 2-dimensional system of design parameters, to construct a 3-dimensional object based on the spatial equivalents of the initial parameters. To illustrate this process, the painting Opus 84 of Hans Hinterreiter (1902-1992), a Swiss Concrete painter, will be re-interpreted in space.
- ItemBarn-Raising an Endo-Pentakis-Icosi-Dodecahedron(Bridges: Mathematical Connections in Art, Music, and Science, 1999) Knoll, Eva; Morgan, SimonThe workshop is planned as the raising of an endo-pentakis-icosi-dodecahedron with a 1 meter edge length. This collective experience will give the participants new insights about polyhedra in general, and deltahedra in particular. The specific method of construction applied here, using kite technology and the snowflake layout allows for a perspective entirely different from that found in the construction of hand-held models or the observation of computer animations. In the present case, the participants will be able to pace the area of the flat shape and physically enter the space defined by the polyhedron.