Mathematics Commons^™

A Models And Modeling Approach To Risk And Uncertainty, Corey Brady, Richard Lesh

The Mathematics Enthusiast

In this article we describe potential contributions of a Models and Modeling Perspective to research focused on learners’ developing conceptions about uncertainty and variation. In particular, we show how a particular class of realistic problem-solving tasks can illuminate how learners develop models to identify, describe, and predict emergent patterns of regularity in the behavior of various types of systems and in the data these systems generate. We begin by situating current design work in this area within a larger project to investigate idea development in the domain of data modeling over extended (course-length) periods. We give design principles and examples …

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. …

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., …

Promoting Risk Taking In Mathematics Classrooms: The Importance Of Creating A Safe Learning Environment, Sashi Sharma

The Mathematics Enthusiast

Students beliefs and attitudes towards risk taking can impact on their mathematics learning and performance. However, at present, risk is not established in the field of mathematics education. The challenge for mathematics teachers in developing their students’ risk taking dispositions is to choose appropriate activities and tools that match this concept and the learning needs of the students. This paper describes some research-based ideas for promoting risk taking behaviours in a mathematics classroom. It presents interactional pedagogical strategies from a design collaborative research conducted at one secondary school. As part of the learning activities, students critically evaluated statistical investigations undertaken …

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 …

Editorial: The Economics Of Risk, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Dihedral-Like Constructions Of Automorphic Loops, Mouna Ramadan Aboras

Electronic Theses and Dissertations

In this dissertation we study dihedral-like constructions of automorphic loops. Automorphic loops are loops in which all inner mappings are automorphisms. We start by describing a generalization of the dihedral construction for groups. Namely, if (G , +) is an abelian group, m > 1 and α ∈2 Aut(G ), let Dih(m, G, α) on Z_m × G be defined by

(i, u )(j, v ) = (i + j , ((-1)^j u + v )α^ij ).

We prove that the resulting loop is automorphic if and only if m = 2 …

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.

Difference Equation For Tracking Perturbations In Systems Of Boolean Nested Canalyzing Functions, Elena S. Dimitrova, Oleg I. Yordanov, Mihaela Teodora Matache

Mathematics Faculty Publications

This paper studies the spread of perturbations through networks composed of Boolean functions with special canalyzing properties. Canalyzing functions have the property that at least for one value of one of the inputs the output is fixed, irrespective of the values of the other inputs. In this paper the focus is on partially nested canalyzing functions, in which multiple, but not all inputs have this property in a cascading fashion. They naturally describe many relationships in real networks. For example, in a gene regulatory network, the statement “if gene A is expressed, then gene B is not expressed regardless of …

Bounded Rationality Alters The Dynamics Of Paediatric Immunization Acceptance, Tamer Oraby, Chris T. Bauch

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Interactions between disease dynamics and vaccinating behavior have been explored in many coupled behavior-disease models. Cognitive effects such as risk perception, framing, and subjective probabilities of adverse events can be important determinants of the vaccinating behaviour, and represent departures from the pure “rational” decision model that are often described as “bounded rationality”. However, the impact of such cognitive effects in the context of paediatric infectious disease vaccines has received relatively little attention. Here, we develop a disease-behavior model that accounts for bounded rationality through prospect theory. We analyze the model and compare its predictions to a reduced model that lacks …

Liftings And Stresses For Planar Periodic Frameworks, Ciprian Borcea, Ileana Streinu

Computer Science: Faculty Publications

We formulate and prove a periodic analog of Maxwell’s theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.

Non-Communicable Diseases And Preventive Health Behaviors: A Comparison Of Hispanics Nationally And Those Living Along The Us-Mexico Border, Belinda M. Reininger, Jing Wang, Susan P. Fisher-Hoch, Alycia Boutte, Kristina Vatcheva, Joseph B. Mccormick

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Background: Non-communicable diseases (NCDs) are rising among US Hispanics, but few studies have examined the preventive health behaviors for these NCDs among Hispanics. This study compared the preventive health behaviors of smoke-free living, physical activity, fruit and vegetable consumption, and avoidance of heavy alcohol use in Hispanics in the United States and Hispanics living along the US-Mexico border.

Methods: Two weighted data sets with information on Hispanic populations were analyzed: 1) the national Behavioral Risk Factor Surveillance Survey (n = 29,942) from 2009; and 2) the Cameron County Hispanic Cohort (n = 1,439) recruited from the US-Mexico border between 2008–2011. …

Optimal Defensive Strategies In One-Dimensional Risk, Darren B. Glass, Todd W. Neller

Math Faculty Publications

We consider a one-dimensional version of the board game RISK and discuss the problem of how a defending player might choose to distribute his armies along a chain of territories in order to maximize the probability of survival. In particular, we analyze a Markov chain model of this situation and run computer simulations in order to make conjectures as to the optimal strategies. The latter sections of the paper analyze this strategy rigorously and use results on recurrence relations and probability theory in order to prove a related result.

Comparison Of Robotics, Functional Electrical Stimulation, And Motor Learning Methods For Treatment Of Persistent Upper Extremity Dysfunction After Stroke: A Randomized Controlled Trial, Jessica Mccabe, Michelle Monkiewicz, John P. Holcomb, Svetlana Pundik, Janis J. Daly

Mathematics and Statistics Faculty Publications

Objective

To compare response to upper-limb treatment using robotics plus motor learning (ML) versus functional electrical stimulation (FES) plus ML versus ML alone, according to a measure of complex functional everyday tasks for chronic, severely impaired stroke survivors.

Design

Single-blind, randomized trial.

Setting

Medical center.

Participants

Enrolled subjects (N=39) were >1 year post single stroke (attrition rate=10%; 35 completed the study).

Interventions

All groups received treatment 5d/wk for 5h/d (60 sessions), with unique treatment as follows: ML alone (n=11) (5h/d partial- and whole-task practice of complex functional tasks), robotics plus ML (n=12) (3.5h/d of ML and 1.5h/d of shoulder/elbow robotics), …

Modeling Of Humoral Immune Response To Repeated Influenza A Virus Infections, Abbiana Arenas, Safiyah Muhammad, Ly Nguyen, Samita Andreansky, Evan Haskell

Mathematics Faculty Proceedings, Presentations, Speeches, Lectures

Seasonal infections by Influenza A virus (IAV) causes hundreds of thousands of deaths worldwide each year, with most individuals being infected multiple times throughout their lifetimes. The relative impact of the components of the host immune system in controlling the severity and duration of repeated challenges from an IAV infection remains unclear. In particular, the differential contribution of the humoral immune response in primary and secondary challenges from IAV are relatively little explored. We develop a parsimonious mathematical model of the humoral immune response to IAV infection with biologically meaningful and identifiable parameters. We show the relative sensitivity of the …

On Spectra Of Composition Operators, Valentin Matache

Mathematics Faculty Publications

In this paper we consider composition operators Cφ on the Hilbert Hardy space over the unit disc, induced by analytic selfmaps φ. We use the fact that the operator C∗φCφ is asymptotically Toeplitz to obtain information on the essential spectrum and spectrum of Cϕ, which we are able to describe in select cases (including the case of some hypercyclic composition operators or that of composition operators with the property that the asymptotic symbol of C∗φCφ is constant a.e.). One of our tools is the Nikodym derivative of the pull-back measure induced by φ. An alternative formula for the essential norm …

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.

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 …