Mathematics Education

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Materials authored by Dr. Eva Knoll pertaining to Mathematics Education.

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Discussing the challenge of categorizising mathematical knowledge in mathematics research situations
(Bridges: Mathematical Connections in Art, Music, and Science, 2005) Knoll, Eva; Ouvrier-Buffet, Cécile
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.
Elementary Student Teachers Practising Mathematical Enquiry at their Level: Experience and Affect
(2013-02-28) Knoll, Eva
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.
Experiencing 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
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.
Investigating ‘Epistemologically Correct’ Experiences of Mathematical Learning Expériences d’apprentissage mathématique « épistémologiquement correctes » : une investigation
(Canadian Mathematics Education Study Group / Groupe Canadien d'Étude en Didactique des Mathématiques, 2008-05-23) Knoll, Eva; Knoll, Eva
Noticing and engaging the mathematicians in our classroom
(Canadian Mathematics Education Study Group / Groupe Canadien d'Étude en Didactique des Mathématiques, 2010-05) Knoll, Eva; Chernoff, Egan; Mamolo, Ami
Many characteristics describe the work of a mathematician. These characteristics just as readily apply to the work of “professional” mathematicians (e.g. people who “do math” as a career, researching and publishing in the field) as they do to “amateur” mathematicians (e.g. people who “do math” (without funding), be it students, teachers, or teacher educators). The focus of this working group was to explore different ways in which teachers, mathematics educators, and (professional) mathematicians come to appreciate themselves and their students as mathematicians. Through engagement with mathematical tasks, our working group attempted to establish a sense of what it means to “be a mathematician.” This developed from a shared vision of fundamental aspects of “doing math” that were exemplified in the tasks, discussions and experiences of our group members. Specific questions we designed to help shape our discussions included: How is it that teachers/teacher educators/mathematicians come to notice and foster mathematical thinking in primary, secondary, and tertiary classrooms? This question is motivated by Wheeler’s concern that “the majority of teachers [do] not encourage their students to ‘function like a mathematician’” (Wheeler, 1982, p. 46). How can we as teachers engage students as mathematicians and what types of tasks model what it is that mathematicians “do”? This question is motivated by a recognized disconnect between how students experience mathematics in the classroom and how professional mathematicians experience mathematics in research (e.g. Boaler, 2008; Lockhart, 2009).
Polyhedra, Learning by Building: Design and Use of a Math-Ed. Tool.
(Bridges: Mathematical Connections in Art, Music, and Science, 2000) Knoll, Eva
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.
Preliminary Field Explorations in K-6 Math-Ed: the Giant Triangles as Classroom Manipulatives
(Bridges: Mathematical Connections in Art, Music, and Science, 2002) Knoll, Eva; Morgan, Simon
The present paper reports on children’s investigations using the giant equilateral triangles from the Geraldine Project2. 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.
Research-Situations for the Classroom and for Teacher Education
(2006) Knoll, Eva; Cartier, Léa; Godot, Karine; Ouvrier- Buffet, Cécile

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