Miroslav Lovric

Transition from secondary to tertiary mathematics: McMaster University experience

The transition (‘gap’) between secondary and tertiary education in mathematics is a complex phenomenon covering a vast array of problems and issues. The aim of this paper is to present the ways in which the issues of mathematics transition have been dealt with at McMaster University. Roughly, the process of transition has been broken into three stages: students’ voluntary preparation for university mathematics courses facilitated by the Mathematics Review Manual; administration of Mathematics Background Survey; and redesign of the first-year Calculus (and, subsequently, other mathematics courses).

Mathematics Textbooks and Their Potential Role in Supporting Misconceptions.

As a fundamental resource, textbooks shape the way we teach and learn mathematics. Based on examination of secondary school and university textbooks, we describe to what extent, and how, the presentation of mathematics material–in our case study, the concept of the line tangent to the graph of a function–could contribute to creation and strengthening of students’ misconceptions. Our findings, roughly classified in several categories, raise awareness of non-obvious problems that need to be addressed when teaching calculus.

Suggestion for a Theoretical Model for Secondary- Tertiary Transition in Mathematics

One of most notable features of existing body of research in transition seems to be the absence of a theoretical model. The suggestion we present in this paper—to view and understand the high school to university transition in mathematics as a modern-day rite of passage—is an attempt at defining such framework. Although dominantly reflecting North-American reality, we believe that the model could be found useful in other countries as well. Let us emphasize that our model is not new in the sense that it recognizes the transition as such. In this paper, we try to determine whether (and, if so, how) the notion of a rite of passage—which is a well-understood concept in anthropology, as well as in some other disciplines (e.g. culture shock in cultural studies)—can help us understand mathematics transition issues better. Can it help us systematize existing body of research, and enhance our understanding of transition in mathematics; does it point at something new? We believe so, and by elaborating some traditional aspects of rites of passage, we hope to provide a useful lens through which we can examine the process of transition in mathematics, and make suggestions for improved management of some transitional issues.

Understanding secondary–tertiary transition in mathematics

In Clark and Lovric (Suggestion for a theoretical model for secondary–tertiary transition in mathematics, Math. Educ. Res. J. 20(2) (2008), pp. 25–37) we began developing a model for the secondary–tertiary transition in mathematics, based on the anthropological notion of a rite of passage. We articulated several reasons why we believe that the educational transition from school to university mathematics should be viewed (and is) a rite of passage, and then examined certain aspects of the process of transition. Present article is a continuation of our study, resulting in an enhanced version of the model. In order to properly address all aspects of transition (such as a number of cognitive and pedagogical issues) we enrich our model with the notions of cognitive conflict (conceptual change) and culture shock (although defined and used in contexts that differ from the transition context, nevertheless, we found these notions highly relevant). After providing further justification for the application of our model to transition in mathematics, we discuss its many implications in detail. By critically examining current practices, we enhance our understanding of the many issues involved in the transition. The core section ‘Messages and implications of the model’ is divided into subsections that were determined by the model (role of community, discontinuity of the transition process, shock of the new, role of time in transition, universality of transition, expectations and responsibilities, transition as a real event). Before making final conclusions, we examine certain aspects of remedial efforts.

In Vivo-to-In Silico Iterations to Investigate Aeroallergen- Host Interactions

Background Allergic asthma is a complex process arising out of the interaction between the immune system and aeroallergens. Yet, the relationship between aeroallergen exposure, allergic sensitization and disease remains unclear. This knowledge is essential to gain further insight into the origin and evolution of allergic diseases. The objective of this research is to develop a computational view of the interaction between aeroallergens and the host by investigating the impact of dose and length of aeroallergen exposure on allergic sensitization and allergic disease outcomes, mainly airway inflammation and to a lesser extent lung dysfunction and airway remodeling. Methods and Principal Findings BALB/C mice were exposed intranasally to a range of concentrations of the most pervasive aeroallergen worldwide, house dust mite (HDM), for up to a quarter of their lifespan (20 weeks). Actual biological data delineating the kinetics, nature and extent of responses for local (airway inflammation) and systemic (HDM-specific immunoglobulins) events were obtained. Mathematical equations for each outcome were developed, evaluated, refined through several iterations involving in vivo experimentation, and validated. The models accurately predicted the original biological data and simulated an extensive array of previously unknown responses, eliciting two- and three-dimensional models. Our data demonstrate the non-linearity of the relationship between aeroallergen exposure and either allergic sensitization or airway inflammation, identify thresholds, behaviours and maximal responsiveness for each outcome, and examine inter-variable relationships. Conclusions This research provides a novel way to visualize allergic responses in vivo and establishes a basic experimental platform upon which additional variables and perturbations can be incorporated into the system.

Recruitment and retention of mathematics students in Canadian universities

Data from Statistics Canada shows that while the number of mathematics degrees at the undergraduate and graduate levels remained relatively constant between 1992 and 2005, the total number of mathematics degrees as a percentage of all degrees awarded has slightly decreased over the same time period. To understand this situation better, we investigate present trends at Canadian universities–in particular, as they relate to the recruitment and retention of students into/within mathematics programs. Using data available from Statistics Canada, results of our own survey, as well as written and electronic references, we produce a snapshot of the situation at Canadian universities, and attempt to identify good practices that might be able to reverse the downward trend. Our survey shows that recruitment and retention are not at the top of the agenda in many mathematics departments across the country. However, we identified activities organized at every university that was represented in our survey that could be interpreted as efforts aimed at increasing numbers of mathematics students. In order to understand variables that affect recruitment and retention we take a look beyond a typical mathematics department, and discuss issues such as: careers in mathematics and the ways information about it is presented to students, promotion of mathematics and science as important areas of human endeavour, students’ and parents’ beliefs about mathematics and its role in ones life, self-selection out of mathematics, and emergence of new fields of applications in mathematics, such as biological sciences.

Understanding and supporting teacher horizon knowledge around limits: a framework for evaluating textbooks for teachers

ABSTRACT As part of recent scrutiny of teacher capacity, the question of teachers’ content knowledge of higher level mathematics emerges as important to the field of mathematics education. Elementary teachers in North America and some other countries tend to be subject generalists, yet it appears that some higher level mathematics background may be appropriate for teachers. An initial examination of a small sample of textbooks for teachers suggested the existence of a wide array of treatments and depth and quality of mathematics coverage. Based on the literature, a new framework was created to assess the mathematical quality of treatments for both specialized knowledge and horizon knowledge in mathematics textbooks for teachers. The framework was tested on a sample topic of the circle area formula derivation, chosen because it draws heavily on both specialized and horizon knowledge. The framework may contribute to similar analyses of other topics in a broader range of resources, in the overall quest to better describe the details of what constitutes appropriate mathematics horizon knowledge for teachers.

Curvature pinching based on integral norms of the curvature

A compact Riemannian manifold (M, g) of dimension 3 or higher admits a metric of constant (positive or negative) sectional curvature if the following conditions hold: the diameter is bounded from above, the part of the Ricci curvature which lies below some fixed negative number is bounded in LP norm for p > n/2, and the metric is almost spherical or almost hyperbolic in the LP sense. The idea of the proof is to obtain stronger (i.e. L°°) pinching by deforming the initial metric using the Ricci flow, thus reducing the problem to the theorems of Gromov in the case r < 0 and of Grove, Karcher and Ruh in the case r > 0. The reduced curvature tensor changes along the flow according to the heat equation, which implies a weak nonlinear parabolic inequality for its norm. The iteration method of De Giorgi, Nash and Moser is applied to obtain the estimate for the maximum norm of the reduced curvature tensor. The crucial step in the iteration consists of controlling the Sobolev constant of the appropriate imbedding (which also changes along the flow, but behaves well) by the isoperimetric constant, which, in turn, can be bounded in terms independent of the particular manifold.

Predicting Morphological Disparities In Sea Urchin Skeleton Growth And Form

Sea urchins exhibit among their many species remarkable diversity in skeleton form (e.g., from spheroid to discoid shapes). However, we still do not understand how some related species show distinct morphologies despite inherent similarities at the genetic level. For this, we use theoretical morphology to disentangle the ontogenic processes that play a role in skeletal growth and form. We developed a model that simulates these processes involved and predicted trajectory obtaining 94% and 77% accuracies. We then use the model to understand how morphologies evolved by exploring the individual effects of three structures (ambulacral column, plate number, and polar regions). These structures have changed over evolutionary time and trends indicate they may influence skeleton shape, specifically height–to-diameter ratio, h:d. Our simulations confirm the trend observed but also show how changes in the attributes affect shape; we show that widening the ambulacral column or increasing plate number in columns produces a decrease in h:d (flattening); whereas increasing apical system radius to column length ratio produces an increase in h:d (gloublar shape). Computer simulated h:d matched h:d measured from real specimens. Our findings provide the first explanation of how small changes in these structures can create the diversity in skeletal morphologies.