Mathematics Commons^™

To The Mathematical Beach, Francis Su

All HMC Faculty Publications and Research

What context am I missing that hinders my connection with my students? How often do I take the time to get to know their backgrounds? What are the primary experiences that shaped them, and do those present obstacles or opportunities for learning? And in what ways does the mathematical beach say “open to all” but still feel restricted?

These questions appear unrelated to mathematics, but if we ignore their effects, some of our students will not flourish.

Guest Editorial: Risk – Mathematical Or Otherwise, Egan J. Chernoff

The Mathematics Enthusiast

No abstract provided.

What Can Education Learn From Real-World Communication Of Risk And Uncertainty?, David Spiegelhalter, Jenny Gage

The Mathematics Enthusiast

Probability is a difficult topic to teach, not least because it is rather unclear what it actually means. Modern risk communication has tackled general public incomprehension of probability statements by using the metaphor of ‘expected frequencies’ – for example, “of 100 people like you, we would expect 10 to have a heart attack or stroke in the next 10 years.” We show how these ideas can be taken into the classroom as the basis for teaching probability, using frequency tree diagrams as the fundamental representation. Empirical frequency trees can be used to summarise a series of classroom experiments, and then …

Developing Strategic And Mathematical Thinking Via Game Play: Programming To Investigate A Risky Strategy For Quarto, Peter Rowlett

The Mathematics Enthusiast

The Maths Arcade is an extracurricular club for undergraduate students to play and analyse strategy board games, aimed at building a mathematical community of staff and students as well as improving strategic and mathematical thinking. This educational initiative, used at several universities in the U.K., will be described. Quarto is an impartial game played at the Maths Arcade, in that there is one set of common pieces used by both players, and one where stalemates are a common outcome. While some students play without apparent direction until a winning opportunity appears, others adopt a more risky strategy of building the …

Risky Research Business: Mathematics Education Research On The Margins, Erika C. Bullock

The Mathematics Enthusiast

Although we would like to believe that decisions about what research to conduct and how to conduct it are based solely on researcher interest and societal need, the reality is that external political and disciplinary factors do play a role. Scientifically based research (SBR) is one example of external political pressures that shape researcher choice both directly and indirectly. Additionally, disciplines like mathematics education operate under hidden curricula that have the potential to marginalize particular research foci. The purpose of this paper is to consider the implications of such a narrow focus on a young mathematics education researcher’s choices about …

Teaching Risk In School, Andreas Eichler, Markus Vogel

The Mathematics Enthusiast

Although risk is an important topic for society it is seldom addressed when teaching statistics and probability. In this paper we refer to this discrepancy identifying three obstacles for teaching risk in school regarding the mathematical and the situational aspect of risk. Based on two educational constructs, i.e. probability literacy and modelling, we discuss existing approaches for teaching risk in school and propose two strategies for promoting risk as a valuable issue for students based again on the distinction of the mathematical and situational aspect of risk.

Risk—A Fundamental Condition Of Doing Mathematics, Wolff-Michael Roth, Jean-François Maheux

The Mathematics Enthusiast

The theme of this special issue is risk. But risk is not a common topic of investigation in mathematics education, lest it be an occasional interest in “at risk” students, generally defined as those who likely will fail at school. In this study, we are not interested in this rather limited use of the risk concept. Instead, we show that risk not only is a condition of human life generally, but also a necessity for teaching and learning mathematics. To show this, we develop the concept of risk with materials from a second-grade mathematics unit on geometry. Implications are drawn …

Making Decisions About Gambling: The Influence Of Risk On Children’S Arguments, Annie Savard

The Mathematics Enthusiast

This article presents results from a study on decision-making towards eventual participation to gambling activities by grade 4 students. For this study, six learning situations were proposed in a fourth grade classroom. The researcher, who was also the teacher, proposed some extra activities in order to define gambling. Students learned about probability and developed, at the same time, the ability to think critically about gambling. She then proposed three fictional situations of gambling to the students, and asked them if and why they would (or would not) participate in the situation. By studying the arguments that students provided, she explored …

Students’ Language Repertoires For Prediction, David Wagner, Joseph Dicks, Paula Kristmanson

The Mathematics Enthusiast

Communication about prediction is complex in a number of ways. First, language is by nature recursive — language is an indicator of meaning as well as a force that shapes meaning. Second, the same language used to communicate prediction in uncertain environments is used for other purposes. In this article, we describe how the recursive nature of language impacted the choices we made in a cross-sectional longitudinal study aimed at gaining insight into children’s language repertoires relating to conjecture. We then explore some Grade 6 students’ communication about prediction to develop insight into their meaning and meaning-making with prediction language. …

Levels Of Reasoning Of Middle School Students About Data Dispersion In Risk Contexts, Ernesto Sánchez, Antonio Orta

The Mathematics Enthusiast

The aim of this research study is to explore students’ reasoning concerning variation when they compare groups and have to interpret dispersion in terms of risk. In particular, we analyze in this paper the responses to two problems from a questionnaire administered to 82 ninth-grade students. The problems consist of choosing between two and three groups of data by comparing them. The first one composed of losses and winnings coming from a hypothetical game; the second is about medical treatments. The results show the difficulty students had in interpreting variation in a risk context. Although they identify the data group …

Pedagogy Of Risk: Why And How Should We Teach Risk In High School Math Classes?, Nenad Radakovic

The Mathematics Enthusiast

Risk is everywhere yet the concept of risk is seldom investigated in high school mathematics. After presenting arguments for teaching risk in the context of high school mathematics, the article describes a case study of teaching risk in two grade 11 classes in Canada- an all-boy independent school (23 boys) and a publicly funded religious school (19 girls and 4 boys). The findings suggest that the students possessed intuitive knowledge that risk of an event should be assessed by both its likelihood and its impact. Following and amending pedagogic model of risk (Levinson, R., Kent, P., Pratt, D., Kapadia, R., …

Risk Intuitions And Perceptions: A Case Study Of Four Year 13 (Grade 12) Students, Stephanie Budgett, Lorraine O'Carroll, Maxine Pfannkuch

The Mathematics Enthusiast

In the New Zealand school statistics curriculum, year 12 students (aged 16-17) are required to solve problems that involve interpreting risk and relative risk within a range of meaningful contexts. In a small exploratory study we investigate the risk conceptions of four year 13 students who performed at the excellence level in their year 12 externally-assessed examination on this topic. Through questionnaires and interviews we investigate the ways in which these students perceive and express risks associated with a variety of everyday activities and also how they compare the risks of several adverse outcomes. We also explore the strategies they …

Adults’ Perceptions Of Risk In The Big Data Era, Theodosia Prodromou

The Mathematics Enthusiast

The present digital era has seen rapid growth in the availability of big data; we were curious about whether such availability of data changes perceptions and assessments of risk. In this paper, we investigate adults’ (35-63 years old) perceptions of risk in the big-data era and how it figures in their everyday life. We developed decision-making scenarios for socio-economic, environmental and health topics that involve modelling with personal value systems alongside Gapminder word map data. Going beyond the idea of risk in statistical theory, we attempt to gain an understanding of the processes by which adults assess risks.

A Conversation With Herbert Tate: Mathematics Educator And Builder, Christian Genest

The Mathematics Enthusiast

Herbert Tate was a Professor of Mathematics at McGill University (Montréal, Canada) from 1921 to 1964. As the author of four textbooks, and in his capacity as Chairman of the Department of Mathematics from 1948 to 1960, he played a key role in structuring the institution’s research and study programs in mathematics during an important period of growth. McGill’s current position as a hub of mathematical research owes much to him. In this interview given shortly after his retirement, Herbert Tate describes his career and shares some of his views about mathematics and related topics. Beyond its archival value, this …

Mathematical Creativity: The Unexpected Links, Amine El-Sahili, Nour Al-Sharif, Sahar Khanafer

The Mathematics Enthusiast

Creativity in mathematics is identified in many forms or we can say is made up of many components. One of these components is The Unexpected Links where one tries to solve a mathematical problem in a nontraditional manner that requires the formation of hidden bridges between distinct mathematical domains or even between seemingly far ideas within the same domain. In this article, we design problems that express unexpected links in mathematics and suit students of intermediate and secondary levels. We prove their feasibility through teachers’ testimonies and through introducing them in classrooms and collecting students’ attitudes with respect to understanding …

Risk Measures On Orlicz Heart Spaces, Coenraad Labuschagne, Habib Ouerdiane, Imen Salhi

Communications on Stochastic Analysis

No abstract provided.

Regular Transformation Groups Based On Fourier-Gauss Transforms, Maximilian Bock, Wolfgang Bock

Communications on Stochastic Analysis

No abstract provided.

Fractional Hida-Malliavan Derivatives And Series Representations Of Fractional Conditional Expectations, Sixian Jin, Qidi Peng, Henry Schellhorn

Communications on Stochastic Analysis

No abstract provided.

On P-Confidence Intervals For G-Expectations, Leo Shen, Rodney C Wolff

Communications on Stochastic Analysis

No abstract provided.

Structure And Decompositions Of The Linear Span Of Generalized Stochastic Matrices, Andreas Boukas, Philip Feinsilver, Anargyros Fellouris

Communications on Stochastic Analysis

No abstract provided.

On Approximation Rates For Boundary Crossing Probabilities For The Multivariate Brownian Motion Process, Shaun Mckinlay, Konstantin Borovkov

Communications on Stochastic Analysis

No abstract provided.

On The Fundamental Theorem Of Asset Pricing, Abhay G Bhatt, Rajeeva L Karandikar

Communications on Stochastic Analysis

No abstract provided.

Periodic Solutions Of Discrete Generalized Schrödinger Equations On Cayley Trees, Fumio Hiroshima, József Lörinczi, Utkir Rozikov

Communications on Stochastic Analysis

No abstract provided.

Almost-Sure Explosive Solutions Of Some Nonlinear Parabolic Itô Equations, Pao-Liu Chow, Rafail Khasminskii

Communications on Stochastic Analysis

No abstract provided.

Math Department Newsletter, 2014-2015, Mathematics Department

Mathematics Newsletter

No abstract provided.

Extremal H-Colorings Of Graphs With Fixed Minimum Degree, John Engbers

Mathematics, Statistics and Computer Science Faculty Research and Publications

For graphs G and H, a homomorphism from G to H, or H-coloring of G, is a map from the vertices of G to the vertices of H that preserves adjacency. When H is composed of an edge with one looped endvertex, an H-coloring of G corresponds to an independent set in G. Galvin showed that, for sufficiently large n, the complete bipartite graph Κ is the n-vertex graph with minimum degree δ that has the largest number of independent sets. In this article, we begin the project of generalizing this result …

Free Split Bands, Francis Pastijn, Justin Albert

Mathematics, Statistics and Computer Science Faculty Research and Publications

We solve the word problem for the free objects in the variety consisting of bands with a semilattice transversal. It follows that every free band can be embedded into a band with a semilattice transversal.

Editorial: The Economics Of Risk, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Understanding Risk Through Board Games, Joshua T. Hertel

The Mathematics Enthusiast

In this article, I describe a potential avenue for investigating individual’s understanding of and reactions to risk using the medium of board games. I first discuss some challenges that researchers face in studying risk situations. Connecting to the existing probabilistic reasoning literature, I then present a rationale for using board games to model these situations. Following this, I draw upon intuition and dual-process theory to outline an integrated theoretical perspective for such investigations. The article concludes with two vignettes demonstrating how this perspective might be used to analyze thinking about risk in a board game setting.

Risk Education: A Worldview Analysis Of What Is Present And Could Be, Gale L. Russell

The Mathematics Enthusiast

Risk, risk analysis, risk management and risk-based decision-making are ubiquitous ideas in the modern world. Consequently, risk education is emerging as a new field of research. However, just as the defining of risk and what it entails is a contested topic, so too is the field of risk education research open to many possible approaches. In this paper notions of risk, particularly as they play out in research on risk education, are analyzed (within an ethical space) using a theoretical framework based on the Traditional Western and an Indigenous worldview. Through this analysis, along with the identification of the kinds …