Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks

Do bold shakeups of the learning-teaching agreement work? A commognitive perspective on a LUMOS low lecture innovation

Mathematics undergraduates, and their lecturers, often describe the transition into university mathematics as a process of enculturation into new mathematical practices and new ways of constructing and conveying mathematical meaning (Nardi, 1996). Whatcharacterises the breadth and intensity of this enculturation varies according to factors such as (Artigue, Kent & Batanero, 2007): student background and preparedness for university level studies of mathematics; the aims and scope of each of the courses that thestudents take in the early days of their arrival at university; how distant the pedagogical approaches taken in these courses are from those taken in the secondary schools that the students come from; the students’ affective dispositions towards the subject and their expectations for what role mathematics is expected to play in their professional life. On their part, lecturers’ views on their pedagogical role may also vary according to factors such as (Nardi, 2008): length of teaching experience; type of courses (pure, applied, optional, compulsory etc.) they teach; perceptions of the goals of university mathematics teaching (such as to facilitate access to the widest possible population of participants in mathematics or select those likely to push the frontiers of the discipline); and, crucially, institutional access to innovative practices, e.g. through funded, encouraged and acknowledged research into such practices.In this paper I draw on my experiences as a member of the International Advisory Board of the LUMOS project (Barton & Paterson, 2013) to comment on aspects of aforementioned student enculturation. Here I see this enculturation as the adaptation of different ways to act and communicate mathematically. I take a perspective on these ways to act and communicate as discourses and I treat the changes to the mathematical and pedagogical perspectives of those who act as discursive shifts. To this purpose, I deploythe approach introduced by Anna Sfard (2008) and known as the commognitive approach

Name Strategy: Its Existence and Implications

It is argued that colour name strategy, object name strategy, and chunking strategy in memory are all aspects of the same general phenomena, called stereotyping, and this in turn is an example of a know-how representation. Such representations are argued to have their origin in a principle called the minimum duplication of resources. For most the subsequent discussions existence of colour name strategy suffices. It is pointed out that the Berlin†- Kay† universal partial ordering of colours and the frequency of traffic accidents classified by colour are surprisingly similar; a detailed analysis is not carried out as the specific colours recorded are not identical. Some consequences of the existence of a name strategy for the philosophy of language and mathematics are discussed: specifically it is argued that in accounts of truth and meaning it is necessary throughout to use real numbers as opposed to bi-valent quantities; and also that the concomitant label associated with sentences should not be of unconditional truth, but rather several real-valued quantities associated with visual communication. The implication of real-valued truth quantities is that the Continuum Hypothesis of pure mathematics is side-stepped, because real valued quantities occur ab initio. The existence of name strategy shows that thought/sememes and talk/phonemes can be separate, and this vindicates the assumption of thought occurring before talk used in psycho-linguistic speech production models.

Teaching mathematics with the brain in mind : learning pure mathematics with meaning and understanding

ix, 82 leaves ; 29 cm. --This culminating master's project applies data and information discovered in a content analysis of research documents to create a brain-based pure math teacher resource that will help teachers teach the pure mathematics 20 program with meaning and understanding. The resource includes a rationale, as well as explanations for the brainbased mathematics lesson framework. Teacher friendly daily lessons are laid out in a thematic unit for the algebraic equations, relations and functions section of the curriculum. The resource utilizes a brain-based approach to teaching and learning providing teachers with an easy to understand, practical, everyday guide that can easily be implemented into the pure math classroom. This resource is needed because students continue to feel inadequate and inferior in pure math classrooms across Alberta. Changes needed to resolve this disturbing situation include teachers themselves altering their teaching strategies to help minimize the existing problems in the pure math program, and this project contributes to the knowledge about improving best teaching practices. Research on the multiple intelligence theory reminds us of the different student learning styles and the fact that, more than one type of teaching strategy should be used to deliver the pure math program. Current research on the science of learning has brought to light some very interesting ideas of how a student's brain works and the applications of this work to classroom practice. As teachers, we can translate this information into classroom practice in order to help our students learn pure mathematics with meaning and understanding

Large Deviations for Brownian Intersection Measures

We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable set $A\subset \R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen \cite{Ch09} derived the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\R^d)$ as $t\to\infty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^{-p}\ell_t$ on the set of positive measures on some open bounded set $B\subset\R^d$ as $t\to\infty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $U\subset B$. This extends earlier studies on the intersection measure by K\"onig and M\"orters \cite{KM01,KM05}.Comment: To appear in "Communications on Pure and Applied Mathematics