Mathematics Commons^™

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 …

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”.

Practical Applications Of An Integrally Christian Approach To Teaching Mathematics, Valorie L. Zonnefeld

Faculty Work Comprehensive List

Descriptions of various frameworks and approaches to integrating Christian faith in the mathematics classroom are explored, as well as examples and techniques. In particular, a subject-centered approach is advocated in contrast to the traditional teacher-centered approach or, more recently, the student-centered approach.

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.

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.

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.