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# Eva Knoll

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Eva Knoll is an Assistant Professor in the Faculty of Education. Eva Knoll's current academic work consists of research into the mathematical elements of craft, design and visual art practices, with an emphasis on the potential application in mathematics education.
In particular, she is currently leading a research group composed of a mathematician, a textile scholar, two schoolteachers, an archivist and herself. Their work is focusing on an exploration of the mathematical thinking inherent in the design, creation and appreciation of textiles. Topics include:

Other Research and Teaching Interests

- Geometry: symmetries and transformations, groups in geometry, topology
- Number sense and patterns in numbers
- Diagrams and their effective creation and use for communication both in mathematics and art making

Other Research and Teaching Interests

- Mathematical Research Situations in the Classroom (RSC)
- Research experience in pure mathematics as a learning context
- Affect in mathematics learning
- Creativity in and outside education
- Interdisciplinary/integrated teaching
- Mathematics in visual art and in crafts
- Arts in mathematics

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- ItemBarn-Raising an Endo-Pentakis-Icosi-Dodecahedron(Bridges: Mathematical Connections in Art, Music, and Science, 1999) Knoll, Eva; Morgan, Simon
Show more The 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.Show more - ItemBuilding a Möbius Bracelet Using Safety Pins: A Problem of Modular Arithmetic and Staggered Positions(2008) Knoll, Eva
Show more This article reports on the resolution of a mathematical problem that emerged when two ideas were brought together. The first idea consists of a method for constructing a decorated bracelet made with safety pins that are strung together at both ends, creating a band. The other is suggested by the word band: why not introduce a twist and make the bracelet a Möbius band? As Isaksen and Petrofsky demonstrated in their paper [1] discussing the knitting of a Möbius band, the endless nature of the band’s single face and edge introduces an additional design constraint, particularly if the connection is to appear seamless. To make the creation appear seamless, the decoration applied to the design must itself be regular, as this helps the eye travel along the endless length. The paper discusses the mathematical and practical constraints of this result for a design that uses a repeating pattern throughout the band, first in the standard design, then in the Möbius bracelet. This resolution involves some simple modular arithmetic and an unusual way to lay out the pins in preparation for their being strung together.Show more - ItemCircular Origami: a Survey of Recent Results(A.K Peters, 2001) Knoll, Eva
Show more - ItemDecomposing Deltahedra(International Society of the Arts, Mathematics, and Architecture, 2000) Knoll, Eva
Show more 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.Show more - ItemDeveloping a Procedure to Transfer Geometrical Constraints from the Plane into Space(International Society for Geometry and Graphics, 1998) Knoll, Eva
Show more Topology 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.Show more - ItemDiscussing Beauty in Mathematics and in Art(Taylor & Francis Ltd., 2009-09-04) Knoll, Eva
Show more - ItemDiscussing the challenge of categorizising mathematical knowledge in mathematics research situations(Bridges: Mathematical Connections in Art, Music, and Science, 2005) Knoll, Eva; Ouvrier-Buffet, Cécile
Show more Starting 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.Show more - ItemElementary Student Teachers Practising Mathematical Enquiry at their Level: Experience and Affect(2013-02-28) Knoll, Eva
Show more From the time of publication of Polya’s “How to Solve It” (1954), many researchers and policy makers in mathematics education have advocated an integration of more problem solving activities into the mathematics classroom. In contemporary mathematics education, this development is sometimes taken further, through programmes involving students in mathematics research projects. The activities promoted by some of these programmes differ from more traditional classroom activities, particularly with regards to the pedagogic aim. Several of the programmes which can claim to belong to this trend are designed to promote a less static view of the discipline of mathematics, and to encourage a stronger engagement in the community of practice that creates it. The question remains, however, about what such an experience can bring the students who engage in it, particularly given the de-emphasis on the acquisition of notional knowledge. In the study described in this thesis, I investigate possible experiential and affective outcomes of such a programme in the context of a mathematics course targeted at elementary student teachers. The study is composed of three main parts. Firstly, the theoretical foundations of the teaching approach are laid down, with the expressed purpose of creating a module that would embody these foundations. The teaching approach is applied in an elementary teacher education context and the experience of the participating students, as well as its affective outcomes, are examined both from the point of view of authenticity with respect to the exemplar experience, and for the expected–and unexpected–affective outcomes. Both of these examinations are based on the establishment of a theoretical framework which emerges from an investigation of mathematicians’ experience of their research work, as well as the literature on affective issues in mathematics education.Show more - 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, Paul
Show more 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 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.Show more - ItemAn Exploration of Froebel’s Gift Number 14 leads to Monolinear, Re-entrant, Dichromic Mono-Polyomino Weavings(Bridges Coimbra Conference Proceedings, 2011) Knoll, Eva; Landry, Wendy
Show more When Froebel, the inventor of the Kindergarten [1] designed the “Gifts” and “Occupations” given to the children, he deliberately selected materials that provided a haptic dimension to their explorations. This physicality in the interaction with the gifts can create a significant potential learning focus making full use of concepts of spatial reasoning (front-back, over-under, etc.). For adults playing with the materials for the first time, and incorporating a reflective component in their doing and their thinking, the Gifts can provide a novel perspective on other, deeper mathematical concepts. The following paper and its accompanying workshop present some activities possible with Gift # 14, which involves the Occupation of paper weaving, and explore ideas in modular arithmetic, combinatorial geometry, ethnomathematics and more.Show more

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