Mathematics Commons^™

Good Models And Good Representations Are A Support For Learners’ Risk Assessment, Laura Martignon, Sebastian Kuntze

The Mathematics Enthusiast

When learners have to make sense of risky situations, they can use mathematical models and representations which facilitate successful risk assessment. Based on theoretical considerations on the benefits of specific models and specific representations in such contexts, we present empirical findings of a study which examined whether students use such models and representations in their risk assessment. We conclude that the availability of adequate models to learners may help them gain transparency when facing risks and thus foster their decision-making.

Risk And Decision Making: The “Logic” Of Probability, Manfred Borovcnik

The Mathematics Enthusiast

Risk is a hot topic. There is an international trend to use examples of risk or the concept of risk in the early teaching of probability. It enriches the problems, it widens the contexts, and it motivates the students to learn probability. This paper illustrates the notion of risk as a multi-faceted concept. The diverse perceptions start with language where risk is used in very different ways. The overlap of risk and hazard is not restricted to the technical context of safety and reliability; Knight’s seminal work on risk and uncertainty has its definite impact on today’s perception of the …

A Dialectical Invariant For Research In Mathematics Education, Mauro García Pupo, Juan E. Nápoles Valdes

The Mathematics Enthusiast

Many current problems in research in mathematics education emerge from pairs of contradictory dialectical categories. In effect, these pairs characterize the problems. When an epistemological study is made to determine the object of research in which a problem is immersed, it is possible to find essential pairs of dialectical categories that become more profound and thus provide enough elements for the determination of appropriate didactic actions to solve the problem under research.

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 …

Edge Colorings Of Graphs And Their Applications, Daniel Johnston

Dissertations

Edge colorings have appeared in a variety of contexts in graph theory. In this work, we study problems occurring in three separate settings of edge colorings.

For more than a quarter century, edge colorings have been studied that induce vertex colorings in some manner. One research topic we investigate concerns edge colorings belonging to this class of problems. By a twin edge coloring of a graph G is meant a proper edge coloring of G whose colors come from the integers modulo k that induce a proper vertex coloring in which the color of a vertex is the sum of …

Nonlocally Maximal Hyperbolic Sets For Flows, Taylor Michael Petty

Theses and Dissertations

In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.

Topics Pertaining To The Group Matrix: K-Characters And Random Walks, Randall Dean Reese

Theses and Dissertations

Linear characters of finite groups can be extended to take k operands. The basics of such a k-fold extension are detailed. We then examine a proposition by Johnson and Sehgal pertaining to these k-characters and disprove its converse. Probabilistic models can be applied to random walks on the Cayley groups of finite order. We examine random walks on dihedral groups which converge after a finite number of steps to the random walk induced by the uniform distribution. We present both sufficient and necessary conditions for such convergence and analyze aspects of algebraic geometry related to this subject.

Analysis Of Multiple Collision-Based Periodic Orbits In Dimension Higher Than One, Skyler C. Simmons

Theses and Dissertations

We exhibit multiple periodic, collision-based orbits of the Newtonian n-body problem. Many of these orbits feature regularizable collisions between the masses. We demonstrate existence of the periodic orbits after performing the appropriate regularization. Stability, including linear stability, for the orbits is then computed using a technique due to Roberts. We point out other interesting features of the orbits as appropriate. When applicable, the results are extended to a broader family of orbits with similar behavior.

Octahedral Extensions And Proofs Of Two Conjectures Of Wong, Kevin Ronald Childers

Theses and Dissertations

Consider a non-Galois cubic extension K/Q ramified at a single prime p > 3. We show that if K is a subfield of an S_4-extension L/Q ramified only at p, we can determine the Artin conductor of the projective representation associated to L/Q, which is based on whether or not K/Q is totally real. We also show that the number of S_4-extensions of this type with K as a subfield is of the form 2^n - 1 for some n >= 0. If K/Q is totally real, n > 1. This proves two conjectures of Siman Wong.

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), …

Risks Worth Taking? Social Risks And The Mathematics Teacher, Ami Mamolo, Laura Elizabeth Pinto

The Mathematics Enthusiast

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 …

Probability, Justice, And The Risk Of Wrongful Conviction, Jeffrey S. Rosenthal

The Mathematics Enthusiast

We consider the issue of standards of proof in legal decisions from the point of view of probability. We compare ``balance of probabilities'' and ``beyond a reasonable doubt'' to the statistical use of p-values. We point out various fallacies which sometimes arise in legal reasoning. And we provide several examples of legal cases which involved probabilities, including some in which incorrect decisions were made and defendants were wrongfully convicted.

Risk: Mathematical And Otherwise, John Adams

The Mathematics Enthusiast

What role might mathematicians have to play in the management of risk? The idea of turning a risk, a possibility of loss or injury, into a “calculated” risk, a quantified probability of loss or injury, is one that has obvious appeal not just to statisticians and mathematicians – but to large numbers of others who would like to know the probability of failure before pursuing some intended course of action. Conclusion: even when risks can be calculated with great precision, they can only be used to inform judgment, but not substitute for it. And it matters who is making the …

Worth The Risk? Modeling Irrational Gambling Behavior, Matt Lane

The Mathematics Enthusiast

In math class, expected value is often used when deciding whether or not a game is worth playing. A common refrain is that games with negative expected value should be avoided. However, nearly all games of chance have a negative expected value, and a simple expected value analysis fails to explain why these games are so popular. In this article, we consider three psychological factors leading to irrational gambling behavior – the illusion of control, hypersensitivity to reward, and beginner’s luck – and explore how these factors affect an otherwise purely rational model of gambling behavior.

A Cognitive Framework For Normative Reasoning Under Uncertainty, And Reasoning About Risk, And Implications For Educational Practice, Sylvia Kuzmak

The Mathematics Enthusiast

Clarifying what is normative or appropriate reasoning under various circumstances provides a valuable reference for guiding what should be taught, and, in contrast, what should not be. This paper proposes a cognitive framework for viewing normative reasoning and behavior under uncertainty, including the applying of knowledge of probability and statistics in real world situations; and identifies implications for educational practice. Factors relevant to normative reasoning under uncertainty that are addressed within the framework include: risk of misapplying statistics knowledge, involvement of mathematical and non-mathematical reasoning, knowledge of real world domains and situation/application detail, and existence of expert consensus. The cognitive …

Risk As An Explanatory Factor For Researchers’ Inferential Interpretations, Rink Hoekstra

The Mathematics Enthusiast

Logical reasoning is crucial in science, but we know that this is not something that humans are innately good at. It becomes even harder to reason logically about data when there is uncertainty, because there is always a chance of being wrong. Dealing with uncertainty is inevitable, for example, in situations in which the evaluation of sample outcomes with respect to some population is required. Inferential statistics is a structured way of reasoning rationally about such data. One could therefore expect that using well-known statistical techniques protects its users against misinterpretations regarding uncertainty. Unfortunately, this does not seem to be …

Tme Volume 12, Numbers 1, 2, And 3

The Mathematics Enthusiast

No abstract provided.

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 …

The Role Of Probabilistic Reasoning Abilities On Adolescent Risk Taking, Maria Anna Donati, Francesca Chiesi, Caterina Primi

The Mathematics Enthusiast

The aim of this work was to investigate the role of the cognitive system and the affective system on adolescents’ risk taking in gambling tasks characterized as different on the basis of information given to decision makers. In Study 1, we explored the role of probabilistic reasoning and sensation seeking on decision making in a non-risky context (Non-Gambling Task) and a risky context (Gambling Task) in which no preliminary information were given to participants. Results showed that adolescents referred to probabilistic reasoning only in the Non-Gambling Task. In Study 2, we explored the role of probabilistic reasoning and sensation seeking …

Calculated Risks: The Teacher As Big Data Producer And Risk Analyst, Nat Banting

The Mathematics Enthusiast

Teachers’ work is often subjected to data analysis from outside sources in the forms of standardized examinations and media critique. This article uses the literature of risk analysis to play with two important analogies for teachers with regards to the emerging big data culture and the risk decisions therein. The complex context of the classroom facilitates the exploration of teacher as big data producer, while the multi-faceted nature of risk decisions provide the groundwork for the exploration of teacher as risk analyst. Illustrative classroom episodes portray examples of real and virtual risk faced by teachers, and a third category—curricular risk—is …

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 …

Judgment Of Association Between Potential Factors And Associated Risk In 2x2 Tables: A Study With Psychology Students, Carmen Batanero, Gustavo R. Cañadas, Carmen Díaz, Maria M. Gea

The Mathematics Enthusiast

This study was aimed to evaluate the accuracy and strategies used in the estimation of association between potential factors and associated risks when data are presented in 2x2 tables. A sample of 414 undergraduate Psychology students from three different Spanish universities was given three different tasks (direct and inverse association and perfect independence) where they had to estimate such association. Most participants judged association in the task where there was perfect independence, but the data contradicted the students’ previous expectations. The estimation of association was consistent with the perception of association and the accuracy of estimates increased with correct strategies. …

Dividing A Pizza Into Equal Parts – An Easy Job?, Hans Humenberger

The Mathematics Enthusiast

Theoretically seen dividing a pizza equally is not an easy task. For instance, with a normal knife (straight cuts) one has to hit the center so that the cut is a diameter. But there are alternatives (also for dividing equally between more than two persons) which have strong connections to elementary geometry and to integral calculus. This paper deals with these alternatives elucidating the so called “pizza theorem”.

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

School of Mathematical & Statistical Sciences Faculty Publications

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 …

Elliptic Curves, Trinity Mecklenburg

Electronic Theses, Projects, and Dissertations

The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q).

From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form …

Symmetric Presentations And Generation, Dustin J. Grindstaff

Electronic Theses, Projects, and Dissertations

The aim of this thesis is to generate original symmetric presentations for finite non-abelian simple groups. We will discuss many permutation progenitors, including but not limited to 2^*14 : D₂₈, 2^∗9 : 3^•(3²), 3^∗9 : 3^•(3²), 2^∗21 : (7X3) : 2 as well as monomial progenitors, including 7^∗5 :_m A₅, 3^∗5 :_m S₅. We have included their homomorphic images which include the Mathieu group M₁₂, 2^•J_{2 …}

Unique Prime Factorization Of Ideals In The Ring Of Algebraic Integers Of An Imaginary Quadratic Number Field, Nolberto Rezola

Electronic Theses, Projects, and Dissertations

The ring of integers is a very interesting ring, it has the amazing property that each of its elements may be expressed uniquely, up to order, as a product of prime elements. Unfortunately, not every ring possesses this property for its elements. The work of mathematicians like Kummer and Dedekind lead to the study of a special type of ring, which we now call a Dedekind domain, where even though unique prime factorization of elements may fail, the ideals of a Dedekind domain still enjoy the property of unique prime factorization into a product of prime ideals, up to order …

Algebra 1 Students’ Ability To Relate The Definition Of A Function To Its Representations, Sarah A. Thomson

Electronic Theses, Projects, and Dissertations

One hundred high school Algebra students from a southern California school participated in this study to provide information on students’ ability to relate the definition of function to its representations. The goals of the study were (1) to explore the extent to which students are able to distinguish between representations of functions/non-functions; (2) to compare students’ ability to distinguish between familiar/unfamiliar representations of functions/non-functions; (3) to explore the extent to which students are able to apply the definition of function to verify function representations; and (4) to explore the extent to which students are able to provide an adequate definition …

Symmetric Presentations Of Non-Abelian Simple Groups, Leonard B. Lamp

Electronic Theses, Projects, and Dissertations

The goal of this thesis is to show constructions of some of the sporadic groups such as the Mathieu group, M₁₂, J₁, Projective Special Linear groups, PSL(2,8), and PSL(2,11), Unitary group U(3,3) and many other non-abelian simple groups. Our purpose is to find all simple non-abelian groups as homomorphic images of permutation or monomial progenitors, as well grasping a deep understanding of group theory and extension theory to determine groups up to isomorphisms. The progenitor, developed by Robert T. Curtis, is a semi-direct product of the following form: P≅2*ⁿ: N = {πw | π …

Homomorphic Images And Related Topics, Kevin J. Baccari

Electronic Theses, Projects, and Dissertations

We will explore progenitors extensively throughout this project. The progenitor, developed by Robert T Curtis, is a special type of infinite group formed by a semi-direct product of a free group m^*n and a transitive permutation group of degree n. Since progenitors are infinite, we add necessary relations to produce finite homomorphic images. Curtis found that any non-abelian simple group is a homomorphic image of a progenitor of the form 2^*n: N. In particular, we will investigate progenitors that generate two of the Mathieu sporadic groups, M₁₁ and M₁₁, as well as …