Mathematics Commons^™

Farey Recursion And Hyperbolic Dehn Filling, Jose Ebenezer Martinez

Graduate Student Theses, Dissertations, & Professional Papers

In this work, we present a solution to William Thurston's edge gluing equations for Dehn fillings of hyperbolic 3-manifolds. This is done for triangulations that involve the layered solid torus. Our approach uses Farey recursive functions, and we present a Farey recursive function that provides a solution to the gluing equations for any hyperbolic Dehn filling admitting a triangulation by the layered solid torus. We provide examples that demonstrate our solution for multiple 3-manifolds, and study the roots of the corresponding Farey recursive polynomials. As an additional application of our solution, we provide a formula for the complex length of …

Eating Disorders And Autism: A Network Approach, Lillian C. King

Undergraduate Theses, Professional Papers, and Capstone Artifacts

This paper explores the overlap of ED and ASD symptoms, and evaluates the results of a study that used network analysis to investigate pathway and core ED and ASD comorbidity symptoms. Eating Disorders (EDs) and autism spectrum disorder (ASD) have several overlapping symptoms that may inform our understanding of both disorders. Increased knowledge of the overlap of EDs and ASD can improve the treatment of EDs in those with ASD.

Lasso: Listing All Subset Sums Obediently For Evaluating Unbounded Subset Sums, Christopher N. Burgoyne, Travis J. Wheeler

Graduate Student Theses, Dissertations, & Professional Papers

In this study we present a novel algorithm, LASSO, for solving the unbounded and bounded subset sum problem. The LASSO algorithm was designed to solve the unbounded SSP quickly and to return all subsets summing to a target sum. As speed was the highest priority, we benchmarked the run time performance of LASSO against implementations of some common approaches to the bounded SSP, as well as the only comparable implementation for solving the unbounded SSP that we could find. In solving the bounded SSP, our algorithm had a significantly faster run time than the competing algorithms when the target sum …

On Coloring Oriented Graphs Of Large Girth, Michael Morris

Graduate Student Theses, Dissertations, & Professional Papers

No abstract provided.

Introductory Calculus: Through The Lenses Of Covariation And Approximation, Caleb Huber

Graduate Student Portfolios, Professional Papers, and Capstone Projects

Over the course of a year, I investigated reformative approaches to the teaching of calculus. My research revealed the substantial findings of two educators, Michael Oehrtman and Pat Thompson, and inspired me to design a course based upon two key ideas, covariation and approximation metaphors. Over a period of six weeks, I taught a course tailored around these ideas and documented student responses to both classroom activities and quizzes. Responses were organized intonarratives, covariation, rates of change, limits, and delta notation. Covariation with respect to rates of change was found to be incredibly complex, and students would often see it …

Student's Self Discovery Of Right Triangle Trigonometry., Elizabeth Dereu

Undergraduate Theses, Professional Papers, and Capstone Artifacts

Research has shown that right triangle trigonometry poses a significant learning challenge for high school geometry students. One potential source of this challenge is the tendency for students to experience right triangle trig as an exclusively action-oriented set of strategies (i.e. SOHCAHTOA). This article describes a teaching experiment where students built understanding of right triangle trig using a unique set of manipulatives and direct measurement. Results of the experiment show that the lessons provided opportunities for students to develop their understanding beyond action-oriented strategies.

A Dual State Hierarchical Ensemble Kalman Filter Algorithm, William J. Cook, Jesse Johnson, Marko Maneta, Doug Brinkerhoff

Graduate Student Theses, Dissertations, & Professional Papers

Dynamic models that simulate processes across large geographic locations, such as hydrologic models, are often informed by empirical parameters that are distributed across a geographical area and segmented by geological features such as watersheds. These parameters may be referred to as spatially distributed parameters. Spatially distributed parameters are frequently spatially correlated and any techniques utilized in their calibration ideally incorporate existing spatial hierarchical relationships into their structure. In this paper, a parameter estimation method based on the Dual State Ensemble Kalman Filter called the Dual State Hierarchical Ensemble Kalman Filter (DSHEnKF) is presented. This modified filter is innovative in that …

Dealing With Mathematical Anxiety: Should One Size Fit All?, Jon Warwick

The Mathematics Enthusiast

Many students who have to study mathematics as an enabling subject within higher education experience mathematical anxiety to a greater or lesser extent. This affliction can impact student learning and achievement in mathematics and so a number of strategies have been suggested for alleviating mathematical anxiety or at least moderating its effects. This paper reports on a comparison of the mathematical anxiety experienced by two groups of students each studying a different subject discipline. The results indicate that the groups have quite different levels of anxiety and the differing contributing factors between the groups suggest that approaches to remediation need …

Common Core And Stem Opportunities, Lane H. Walker, Helene J. Sherman

The Mathematics Enthusiast

There is an increasing need for educators at all levels to equip more students with problem-solving skills that better fit our changing work force. Students are largely unaware of many science-, technology-, engineering-, and math-related (STEM) careers. They often do not understand the importance of those careers or what skills are required to pursue them. Students are exposed to some of those skills if they take Career Technical Education (CTE) classes, but rarely do they see the connections in their core math classes. Math teachers have pointed to their dense curricula as making STEM integration impractical. A study of the …

Mathematical Creativity For The Youngest School Children: Kindergarten To Third Grade Teachers’ Interpretations Of What It Is And How To Promote It, Yinjing Shen, Carolyn Pope Edwards

The Mathematics Enthusiast

Creativity is important for young children learning mathematics. However, much literature has claimed creativity in the learning of mathematics for young children is not adequately supported by teachers in the classroom due to such reasons as teachers’ poor college preparation in mathematics content knowledge, teachers’ negativity toward creative students, teachers’ occupational pressure, and low quality curriculum. The purpose of this grounded theory study was to generate a model that describes and explains how a particular group of early childhood teachers make sense of creativity in the learning of mathematics and how they think they can promote or fail to promote …

Reorganizing Algebraic Thinking: An Introduction To Dynamic System Modeling, Diana Fisher

The Mathematics Enthusiast

System Dynamics (SD) modeling is a powerful analytical method used by professional scientists, academics, and governmental officials to study the behavior patterns of complex systems. Specifically through use of the Stella software, it is a method that I and others have used for over two decades with high school, and even middle school, math and science students. In this paper I describe an introduction to SD modeling intended for an algebra class (in either middle or high school). In the body of the paper, a nested sequence of simple bank account examples, increasing in complexity, is used to demonstrate a …

On Mathematics With Distinction, A Learner-Centered Conceptualization Of Challenge And Choice-Based Pedagogies, Boris Koichu

The Mathematics Enthusiast

The main argument of this article is that “challenging mathematics for all” can be more than just a nice slogan, on condition that all students are empowered to make informed choices of: a challenge to be dealt with, a way of dealing with the challenge, a mode of interaction, an extent of collaboration, and an agent to learn from. Pedagogies supporting such choices are called choice-based pedagogies. The article begins from a theoretical discussion of the relationships between the notions ‘mathematics with distinction,’ ‘giftedness’, ‘challenge’ and ‘choice’. As a result, the learner-centered conceptualization of mathematical challenge is proposed. Then two …

Examining The Interaction Of Mathematical Abilities And Mathematical Memory: A Study Of Problem-Solving Activity Of High-Achieving Swedish Upper Secondary Students, Attila Szabo, Paul Andrews

The Mathematics Enthusiast

In this paper we investigate the abilities that six high-achieving Swedish upper secondary students demonstrate when solving challenging, non-routine mathematical problems. Data, which were derived from clinical interviews, were analysed against an adaptation of the framework developed by the Soviet psychologist Vadim Krutetskii (1976). Analyses showed that when solving problems students pass through three phases, here called orientation, processing and checking, during which students exhibited particular forms of ability. In particular, the mathematical memory was principally observed in the orientation phase, playing a crucial role in the ways in which students’ selected their problem-solving methods; where these methods failed to …

Mathematical Modeling Cycles As A Task Design Heuristic, Jennifer A. Czocher

The Mathematics Enthusiast

There are many approaches to task design (Watson & Ohtani, 2015) from a large number of local and global design heuristics. The purpose of this paper is to present how mathematical modeling cycles, a popular way of describing mathematical modeling processes, were used as a task design heuristic.

A Proposed Local Instruction Theory For Teaching Instantaneous Speed In Grade Five, Huub De Beer, Koeno Gravemeijer, Michiel Van Eijck

The Mathematics Enthusiast

In answer to a call for innovative science and technology education in primary education, we started a design research project to explore how to teach instantaneous speed in grade five. In this article we present the results of a series of teaching experiments that were conducted to design, try out, and improve a local instruction theory on teaching instantaneous speed in grade five. In a retrospective analysis, looking for patterns in the whole data set, encompassing all experiments, we identified a set of key learning moment of the students. Based on these patterns, a potentially viable local instruction theory was …

A 21st Century Economic, Educational And Ethical Mathematics Curriculum Policy, Glen S. Aikenhead

The Mathematics Enthusiast

Reformulating a mathematics curriculum is a political act, contextualized by economic, educational and ethical considerations, among others. This article represents a political rendition of a rationale to update the Saskatchewan, Canada, curriculum so it harmonizes with a cultural understanding of school mathematics (Ernest, 1988). By revisiting the 19th century definition of school mathematics that we generally follow today, its economic, educational and ethical consequences seem contrary to the expectations of 21st century Saskatchewan.The article’s point form organization, uninterrupted by the appearance of references, conforms to the genre of submissions to government committees. However, footnotes will clarify information for readers unfamiliar …

Editorial: All Good Things Come In Three, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Problems In Relating Various Tasks And Their Sample Solutions To Bloom’S Taxonomy, Torsten Lindstrom

The Mathematics Enthusiast

In this paper we analyze sample solutions of a number of problems and relate them to their level as prescribed by Bloom’s taxonomy. We relate these solutions to a number of other frameworks, too. Our key message is that it remains insufficient to analyze written forms of these tasks. We emphasize careful observations of how different students approach a solution before finally assessing the level of tasks used.

We take the arithmetic series as our starting point and point out that the objective of the discussion of the examples here in no way is to indicate an optimal way towards …

A Brief Look At The Evolution Of Modeling Hyperbolic Space, Shelby Frazier

The Mathematics Enthusiast

Now considered one of the greatest discoveries of mathematical history, hyperbolic geometry was once laughed at and deemed insignificant. Credited most to Lobachevsky and Bolyai, they independently discovered the subject by proving the negation of Euclid’s Fifth Postulate but neither lived long enough to see their work receive any validation. Several models have been constructed since its acceptance as a subject. I’m going to discuss three graphical models: the Klein Model, the Upper Half Plane Model and the Poincaré Disk Model. I will also discuss and construct three tape and paper models: the Annular Model, the Polyhedral Model and the …

Students’ Reactions To Reform Mathematics Pedagogy In A Postsecondary Remedial Mathematics Course, Luke Smith, W. Gary Martin, Anna Wan, Gilbert Duenas

The Mathematics Enthusiast

The students in this study were enrolled in a remedial mathematics course at a small 4-year university and were taught according to the reform pedagogical principles advocated by NCTM, AMATYC, and MAA. Since most of the students had not been previously exposed to these teaching methods, this study obtained students’ reactions (n = 22) to the course through an anonymous, free-response (not multiple choice) survey at the end of the course. Surveys from students in two equivalent “traditional” lecture courses (n = 44) were also analyzed and served as a baseline by which to gauge students’ responses from the reform …

Rational Number Operations On The Number Line, Christopher Yakes

The Mathematics Enthusiast

The Common Core State Standards make clear that teachers should use the number line to represent integer operations in Grades 6, 7 and 8, continuing the development of number line understanding begun in earlier grades. In this paper, we will investigate the basics of the colored chip and number line models for representing integers and their operations. We will then draw connections between the two models. Finally, we will argue that the number line model is an important one that should be taught alongside or in lieu of the chip model in order to deepen students’ understanding operations with integers, …

A Commentary On Freudenthal’S Didactic Phenomenology Of The Mathematical Structures Associated With The Notion Of Measurement, Omar Hernandez Rodriguez, Jorge Lopez Fernandez

The Mathematics Enthusiast

This paper discusses Freudenthal's didactical phenomenology for the mathematical structures related to measurement. Freudenthal starts with the set G of all objects having the attribute of weight, an attribute that is to be measured. He proposes two operations, the first one among the measurements themselves, called “addition". Addition is interpreted in terms of the mental actions associated with measurement and, in the case of weights, it consists on the process of placing weights on one of the two dishes of a balance in order to balance them with a predetermined gauge, its fractions or multiples thereof, which are placed on …

Linking Pre-Service Teachers’ Questioning And Students’ Strategies In Solving Contextual Problems: A Case Study In Indonesia And The Netherlands, Rahmah Johar, Sitti Maesuri Patahuddin, Wanty Widjaja

The Mathematics Enthusiast

This study examined the relationship between teachers’ questioning techniques and students’ strategies in solving contextual mathematical problems. This case study was undertaken with one pre-service teacher (and 22 Year 4 students) from Indonesia and one pre-service teacher (and 25 Year 4 students) in the Netherlands. Both pre-service teachers assigned the same problems to their students and these problems were novel for the students in both countries. The lessons were observed by the first author and video recorded for data analysis. Qualitative data analysis was undertaken through within-case and cross-case analysis. The findings suggest that the contextual problems, the way pre-service …

On The Definition Of Linear Independence, Yonah Cherniavsky, Artour Mouftakhov

The Mathematics Enthusiast

We discuss a certain very common flaw in the definition of linear independence, which is one of the most important concepts taught in any college or university course of Linear Algebra. This note may be useful to lecturers and students which teach and study Linear Algebra of any level and like the mathematically rigorous approach.

Subtraction Involving Negative Numbers: Connecting To Whole Number Reasoning, Laura Bofferding, Nicole Wessman-Enzinger

The Mathematics Enthusiast

In this article, we explore how students attempt to bridge from their whole number reasoning to integer reasoning as they solve subtraction problems involving negative numbers. Based on interviews with students ranging from first graders to preservice teachers, we identify two overarching strategies: making connections to known problem types and leveraging conceptions of subtraction. Their initial connections suggest that rather than identifying the best instructional models to teach integer concepts, we should focus on identifying integer instructional models that build on the potentially productive connections that students’ already make; we propose an example of one such form of instruction.

Numberlines: Hockey Line Nicknames Based On Jersey Numbers, Egan J. Chernoff

The Mathematics Enthusiast

The purpose of this article, in general, is to expound Chernoff’s (2016) notion of numberlines, that is, hockey line nicknames based on jersey numbers. The article begins with a brief discussion of the history of hockey line nicknames, which allows for the parsing of numberlines and quasi-numberlines (nicknames based on numbers associated with hockey players). Focusing, next, on jersey number restrictions for the National Hockey League (NHL), a repeated calculation of the number of possible numberlines winnows down the number from a theoretical upper bound to a practical upper bound. Moving beyond the numbers, the names of natural numbers – …

Developing Mental Rotation Ability Through Engagement In Assignments That Involve Solids Of Revolution, Atara Shriki, Ruthi Barkai, Dorit Patkin

The Mathematics Enthusiast

Spatial ability is essential for succeeding in the STEM (Sciences, Technology, Engineering and Mathematics) disciplines, especially mental rotation. Research points out that spatial ability is malleable, and therefore calls for developing learners’ ability by engaging them in appropriate assignments, starting from kindergarten. Given this, our paper presents several assignments designed for mathematics prospective teachers with the aim of fostering their mental rotation skills. Specifically, these assignments deal with solids of revolution, three-dimensional shapes formed by revolving a planar shape about a given axis that lies on the same plane.

Studying “Moments” Of The Central Limit Theorem, Benjamin A. Stark

The Mathematics Enthusiast

The central limit theorem ranks high amongst the most important discoveries in the field of mathematics over the last three hundred years. This theorem provided a basis for approximation that turned the question of reaction into the art of prediction. This paper aims to map a course for the history and evolution of the famed theorem from its’ initial origins in 1733, from Abraham de Moivre’s inquiries to the most recent expressions of the theorem. The journey encompassing central limit theorem includes reformations of definition, relaxing of important associated conditions, and numerous types of rigorous proofs.

Critical Examination Of Ways Students Mirror The Teacher’S Classroom Practice: What Does It Mean To Be Successful At Mathematics?, Paula Guerra, Woong Lim

The Mathematics Enthusiast

In this paper, the authors report the mathematical learning experiences of “successful” female students in secondary mathematics classrooms taught by a “successful” teacher with the traditional mathematics’ behaviorist approach. The authors’ claim that the traditional view of mathematics held by the teacher and supported by the school system could not promote rigorous mathematics for girls to understand the importance of mathematical thinking as a foundation for success in mathematics-related professions. The authors recommend future studies creating opportunities for discussion in the field about the teacher’s view on mathematics, classroom practice, and how these resonate with girls’ experiences of learning mathematics.

Coherence And Enrichment Across The Middle And Secondary Levels: Four Mathematically Authentic Learning Experiences, Keith A. Nabb, Jaclyn M. Murawska

The Mathematics Enthusiast

This article discusses four mathematically rich settings with origins in the elementary, middle, and secondary school curricula. Depending on the questions asked and the connections made within each setting, the problem spaces allow the instructor to import tools leading to sophisticated extensions appropriate for college-level study. These topics include the Heaviside function, randomness, symmetry, modular arithmetic, the generalized Pythagorean Theorem, and the theory of groups. Given the potentially extensive ground covered by these settings, they serve to reward those students who are inherently curious while highlighting the coherence in the curriculum as one progresses through the grades. The mathematical experiences …