Exploring argumentation, objectivity, and bias: The case of mathematical infinity

This paper presents an overview of several years of my research into individuals’ reasoning, argumentation, and bias when addressing problems, scenarios, and symbols related to mathematical infinity. There is a long history of debate around what constitutes “objective truth” in the realm of mathematical infinity, dating back to ancient Greece (e.g., Dubinsky et al., 2005). Modes of argumentation, hindrances, and intuitions have been largely consistent over the years and across levels of expertise (e.g., Brown et al., 2010; Fischbein et al., 1979, Tsamir, 1999). This presentation examines the interrelated complexities of notions of objectivity, bias, and argumentation as manifested in different presentations and normative interpretations or resolutions of well-known paradoxes of infinity. Paradoxes have been described as occasioning major epistemological reconstructions (e.g., Quine, 1966), and I highlight such occasions as they emerged for both novices and experts with connection to current conceptualisations of objectivity (e.g., Daston, 1992). Of interest is the perception that one single objective truth about “actual” mathematical infinity exists – indeed, this is brought to question at an axiomatic level with both theoretical and empirical research implications

Polysemy of symbols: Signs of ambiguity

This article explores instances of symbol polysemy within mathematics as it manifests in different areas within the mathematics register. In particular, it illustrates how even basic symbols, such as ‘+’ and ‘1’, may carry with them meaning in ‘new’ contexts that is inconsistent with their use in ‘familiar’ contexts. This article illustrates that knowledge of mathematics includes learning a meaning of a symbol, learning more than one meaning, and learning how to choose the contextually supported meaning of that symbol

Risks Worth Taking? Social Risks and the Mathematics Teacher

In this article, we explore notions of risk as perceived or experienced by individuals involved in mathematical education. We present this exploration in the form of vignettes, each illustrating a form of risk: a parent’s reaction to classroom “propaganda”; a teacher trying to do justice by her students; a teacher confronted by his administration; and a college professor who believes university policy to be unjust. Each vignette sheds light on areas in which teacher education may offer additional support in fostering the mathematical knowledge, pedagogical sensitivity, and social awareness required to foster, what are in our view, much needed risks in the mathematical (and otherwise) education of pupils. Following the vignettes, we offer a discussion of factors that contributed to the risks perceived or experienced by teachers: neoliberal discourses, and the powerful cultural scripts that leave teachers feeling that they must hold all control, authority, and knowledge

Full STEAM Ahead: Building Preservice Teachers’ Capacity in Makerspace Pedagogies

This paper explores teacher candidates’ understandings of 1) makerspace/constructionist pedagogies; 2) the issue of bullying; and, 3) working with at-risk youth, as they evolved over the course of a six-month partnership. The partnership included researchers and teacher candidates at a Faculty of Education and the teacher librarian at a local elementary school who were participating in a larger Social Sciences and Humanities Research Council of Canada (SSHRC)- funded project that focuses on building, implementing and evaluating an effective model for a school improvement program that increases teachers’ capacity, experience and specific fluency and expertise with technologies supporting STEAM learning and digital literacies. In this paper, we discuss qualitative ethnographic case study research, which examines in depth the experiences of five teacher candidates as they worked with 20 students in a grade 6 class in a high needs school on makerspace activities related to bullying prevention in their school community. Qualitative research documentation includes digital video and audio recordings, on the-ground field notes and observational notes, pre and post interviews with participants and focus group sessions. Results from this study contribute new knowledge in the areas of preservice teacher development and digitally-enhanced learning environments for K-6 learners

INFINITE MAGNITUDE VS INFINITE REPRESENTATION: INTUITIONS OF “INFINITE NUMBERS”

Abstract. This report explores students ’ naïve conceptions of infinity as they compared the number of points on line segments of different lengths. Their innovative resolutions to tensions that arose between intuitions and properties of infinity are addressed. Attempting to make sense of such properties, students reduced the level of abstraction of tasks by analysing a single number rather than infinitely many. In particular, confusion between the infinite magnitude of points and the infinite amount of digits in the decimal representation of numbers was observed. Furthermore, students struggled to draw a connection between real numbers and their representation on a number line. The research presented in this paper is part of a broader study that investigates changes in students ’ conceptions of infinity as personal reflection, instruction, and intuitions are combined. It strives to uncover naïve interpretations of a concept that has puzzled and intrigued minds for centuries. The counter-intuitive and abstract nature of infinity provides a particularly interesting avenue for investigation. Moreover, infinity is a concept steeped in personal convictions that may stem from religion or philosophy. Consequently mathematical arguments might not be sufficient to convince an individual of properties that even Cantor saw but could not believe. It has been well established that intuitions are persistent, especially when dealing with counter-intuitive concepts (Fischbein, 1987). As a learner’s mathematical background increases, so too does his or her reliance on systematic, logical schemes (Fischbein, Tirosh, and Hess, 1979). Thus an individual might feel ill equipped to deal with the mathematical anomalies that arise with infinity. This study presented undergraduate students with a geometric representation of infinity, and observed how those students responded to contradictory or inconsistent results that they themselves discovered. A benefit of a geometric approach is that it provided a context for investigating infinity without the necessity of introducing unfamiliar symbolic representations or terminology, such as in a set theoretic approach. Students were able to reflect on and develop their ideas by considering familiar and accessible objects, and with minimal instruction

Glimpses of infinity: intuitions, paradoxes, and cognitive leaps

This dissertation examines undergraduate and graduate university students’ emergent conceptions of mathematical infinity. In particular, my research focuses on identifying the cognitive leaps required to overcome epistemological obstacles related to the idea of actual infinity. Extending on prior research regarding intuitive approaches to set comparison tasks, my research offers a refined analysis of the tacit conceptions and philosophies which influence learners’ emergent understanding of mathematical infinity, as manifested through their engagement with geometric tasks and two well known paradoxes – Hilbert’s Grand Hotel paradox and the Ping-Pong Ball Conundrum. In addition, my research sheds new light on specific features involved in accommodating the idea of actual infinity. The results of my research indicate that accommodating the idea of actual infinity requires a leap of imagination away from ‘realistic’ considerations and philosophical beliefs towards the ‘realm of mathematics’. The abilities to clarify a separation between an intuitive and a formal understanding of infinity, and to conceive of ‘infinite’ as an answer to the question ‘how many?’ are also recognised as fundamental features in developing a normative understanding of actual infinity. Further, in order to accommodate the idea of actual infinity it is necessary to understand specific properties of transfinite arithmetic, in particular the indeterminacy of transfinite subtraction