Digital Commons Network^™

Stepping Stone Problem On Graphs, Rory O'Dwyer

The Mathematics Enthusiast

This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N, which assign a fixed number of 1’s to some vertices and assign higher numbers to other vertices by adding up their neighbors’ assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient …

A General Expression For Hermite Expansions With Applications, Tom P. Davis

The Mathematics Enthusiast

Hermite polynomials arise when dealing with functions of normally distributed variables, and are commonly thought of as the analog of the simple polynomials on functions of regular variables. Therefore the Hermite expansion should be an analog of the Taylor expansion. Indeed there is a strong connection between the two – the general coefficient in the Hermite expansion is the weighted integral of the nth derivative, as compared to the nth derivative evaluated at zero in the case of Taylor. This fact can be used to derive the Hermite expansion for the integral and the derivative of a function. Furthermore, it …

The Tradition Of Large Integers In Historical Arithmetical Textbooks, Franka Miriam Bruckler, Vladimir Stilinovic

The Mathematics Enthusiast

After the Hindu-Arabic decimal positional system was introduced in Europe, throughout many centuries textbooks on elementary arithmetic, intended for beginners, had a more or less fixed organization of content, usually starting with chapters on numeration. These chapters, as a rule, contained one or more examples of large integers the purpose of which was simply to be named (read out loud), sometimes also vice versa. This tradition apparently began with the two first texts that significantly contributed to the spread of the decimal system in Europe—the Latin translations of al- Khwarizmi’s treatise on decimal arithmetic, and Leonardo’s Liber Abaci, containing examples …

Will They Automatically Work Together? Cooperation Among Non-Fools In Hobbes´S Leviathan, Karim Pluma

The Mathematics Enthusiast

Thomas Hobbes´s State of War is commonly imagined as a harrowing condition where hostile interactions are the rule and non-hostile encounters are the rare exception. However, while it is generally true that Hobbes purposely outlined his famed condition of anarchy as a condition of perennial conflict, it is also equally true that cooperative behavior was not uncommon. In fact, by only taking into account cooperative behavior, as presented in Leviathan, the anarchic humans leave the absolute uncertainties of the State of War and create the Commonwealth for their safety and well-being. Over the past fifty years or so, several exits …

An Examination Of The Factors And Characteristics That Contribute To The Success Of Putnam Fellows, Robert A. J. Stroud, Thomas C. Defranco

The Mathematics Enthusiast

The William Lowell Putnam Mathematical Competition is an intercollegiate mathematics competition for students in the United States and Canada and is regarded as the most prestigious and challenging mathematics competition in North America (Alexanderson, 2004; AMS, 2020; Grossman, 2002; Reznick, 1994; Schoenfeld, 1985). Students who earn the five highest scores on the examination are named Putnam Fellows. Since its inception in 1938, only 306 individuals have won the competition and a select few have won multiple times. Clearly, being named a Putnam Fellow is a remarkable achievement and therefore, understanding the factors and characteristics that contribute to their success is …

The Construction Of Figured Numbers In Geogebra Software Using Algebraic Properties, Francisco Evamar Barros, Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino, Renata Teofilo De Sousa

The Mathematics Enthusiast

Figured numbers are sequences of natural numbers that, when represented by dots, form geometric configurations. The immediate sequence of figured numbers presents in its configuration the equilateral triangles, being the basis for finding several others, as long as a relationship between them is discovered. This article shows the construction of polygonal, stellar and three-dimensional figured numbers that can be constructed using the GeoGebra software as a resource to visualize the properties that relate these numbers to the triangular numbers. It is a way of observing the relationship that the figures shown have to each other, how we can understand the …

Tagging Opportunities To Learn: A Coding Scheme For Student Tasks, Janet Heine Barnett, Cihan Can, Daniel E. Otero

The Mathematics Enthusiast

This article describes the development of a coding scheme for analyzing mathematical tasks in Primary Source Projects (PSPs), curriculum materials based on primary historical sources designed for teaching standard topics from today’s undergraduate mathematics curriculum. Our scheme attends to social-cultural aspects of mathematical learning while focusing on the student actions expected as they work tasks. We exemplify our scheme with tasks drawn from a diverse set of PSPs and report results from its application to these projects. We conclude with comments on how our work can assist instructors, curriculum developers, and researchers, including those who are interested in other types …

On The Even Distribution Of Odd Primes: An On-Ramp To Mathematical Research, James Quinlan, Michael Todd Edwards

The Mathematics Enthusiast

The authors consider a conjecture by Chebyshev in 1853 on the distribution of odd primes among those that are one more than a multiple of four and those three more than a multiple of four—and use technology to explore the cardinality of these subsets. Generalizations are presented for student exploration along with several sources for more in-depth research.

The Chain Rule Does Not Have To Be A Pain Rule, Elizabeth Beeman, Jennifer Runge, Lauren Holden, Devon Maxwell, Yi-Yin (Winnie) Ko

The Mathematics Enthusiast

Relational and instrumental understanding of the Chain Rule can help teachers provide students a deeper and more meaningful calculus experience.

The Monty Brawl And Other Unexpected Consequences Of Mathematical Instruction, Francesco Beccuti

The Mathematics Enthusiast

No abstract provided.

Using Gardner's Three-Squares Problem For A Group Project In A Mathematical Problem Solving Module, Jonathan Hoseana

The Mathematics Enthusiast

Consider a 1 x 3 grid whose top-left vertex is connected to the bottom-right vertex of each of the unit squares. What is the sum of the three acute angles formed by the connecting segments with the unit squares' bases? This is the so-called three-squares problem, often attributed to Gardner. In a recent academic year, the author used a video on this problem, produced by the YouTube channel Numberphile, for a group project in a first-semester undergraduate module: Mathematical Problem Solving. The project involved collaborative writing on the problem and individual completion of a peer-assessment form. We report the outcomes …

Going Off On A Tangent: An Inclusion-Exclusion Identity, Aloysius Bathi Kasturiarachi

The Mathematics Enthusiast

Undergraduate research in tertiary education offers mathematics students a pathway to engage in high-impact practices. Using a simple sum equals product identity from number theory as a motivator, we build a series of inclusion-exclusion identities for convex polygons using the symmetry inherent in the tangent function. The techniques used are simple and accessible, illuminating and generalizable, in a manner that rejects a singular line of inquiry in favor of a plurality of mathematical ideas.

Investigating Problems Posed By Pre-Service Mathematics Teachers For The Four Operations In Fractions, Yasemin Kiymaz

The Mathematics Enthusiast

This study investigates 40 preservice middle school teachers’ problems posed for two given fractions. Pre-service teachers were asked to pose three problems (farea, length and set model) for each of the operations of addition, subtraction, multiplication and division with these fractions. 12 problems posed by each pre-service teacher were examined in terms of their suitability for the given operation, the targeted model and real life relevance. Errors in the posed problems were also examined. As a result of the analysis, it was observed that the model in which pre-service teachers were the most unsuccessful in posing problems was the set …

On Convincing Power Of Counterexamples, Orly Buchbinder, Rina Zazkis

The Mathematics Enthusiast

Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students’ arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, …

A Role For Affect In The Future Of Mathematics Education (With Thoughts On Intelligence), Marcelo Almora Rios

The Mathematics Enthusiast

A Review of:

1. Alan Schoenfeld, Heather Fink, Alyssa Sayavedra, Anna Weltman, & Sandra Zuñiga-Ruiz’s (2023) Mathematics Teaching on Target, Routledge, 164pp. ISBN (HB): 978-1-0324-4167-2;

2. Alan Schoenfeld, Heather Fink, Sandra Zuñiga-Ruiz, Siqi Huang, Xinyu Wei, & Brantina Chirinda’s (2023) Helping Students Become Powerful Mathematics Thinkers, Routledge, 272pp. ISBN (HB): 978-1-0324-4168-9.

Nike's Product Recommendation System And Incorporation Of Ai, Quinton Truman Lamers

Undergraduate Theses, Professional Papers, and Capstone Artifacts

No abstract provided.

A Road Map For Place Based Collaboration For Conflict Reduction, Joseph L. Zecher

Undergraduate Theses, Professional Papers, and Capstone Artifacts

No abstract provided.

Creation Of A Digital Storage System For Genome Sequencing Metadata, Jacquelin W. Olexa

Undergraduate Theses, Professional Papers, and Capstone Artifacts

As the field of computational genomics continues to expand in both potential and application, it is now more imperative than ever to ensure that massive genetic sequencing datasets are properly stored in an accessible manner. This project sought to establish a practical, user-friendly, secure system for a genomics research lab (the Good Lab; thegoodlab.org) at the University of Montana. A MySQL database and connected web application was ruled the best configuration to maximize utility and accessibility for the lab’s researchers. Building the logical framework for the database, creating the server, and sourcing data occurred over several months. The dataset ranged …

Effects Of El Nino And Sunspot Cycles On Global And Reginal Climate, August Nathan Tolzman

Undergraduate Theses, Professional Papers, and Capstone Artifacts

Sunspots and El Nino affect the interannual variability of temperature on earth. Separately there is a positive relationship between higher sunspots and warmer temperatures or more intense El Nino and warmer temperatures globally and in Missoula. This is with the bias of climate change removed. Putting these two processes together it is expected that when intense El Ninos and high sunspots happen in the same year there will be warmer temperatures. This was not the case, there was no data to support that putting these to processes together makes them more or less intense.

Water Quality Contaminants Long-Term Spatial Analysis Along The Upper Clark Fork River Superfund Site Between Warm Springs And Garrison, Montana, Raegan Michele Osentowski

Undergraduate Theses, Professional Papers, and Capstone Artifacts

Long-term data of heavy metal contaminants, cadmium, copper, lead, arsenic, and zinc were compiled from the Montana Department of Environmental Quality, DEQ, and the United States Geological Survey, USGS, to investigate the effects of remediation on water quality along the upper Clark Fork River superfund site. Long-term data sets were analyzed at CFR-03A, CFR-27H, CFR-34, and SS-25 along the upper Clark Fork River and Silver Bow Creek. CFR-03A and CFR-27H had full remediation in the immediate vicinity, SS-25 had on-going remediation in the immediate vicinity, and CFR-34 had no remediation in the immediate vicinity. Analysis of heavy metal contaminants is …

Influence Of Cross-Country Skiing & Off-Leash Dogs On Wildlife Habitat Use In Lubrecht Experimental Forest, Sophie Ann Morris

Graduate Student Theses, Dissertations, & Professional Papers

Outdoor recreation activities are seen as a type of anthropogenic stressor that can significantly impact wildlife. Despite winter recreational activities having a more negative effect on wildlife compared to summer or aquatic recreation, they are the least studied in terms of impact (Larson et al., 2016). Wildlife species can be influenced both temporally and spatially by the development of winter recreation trails and snow-based recreation (Braunisch et al., 2011; Olsen et al., 2018). This research combines natural and social science methods to assess how cross-country skiers affect wildlife activity patterns, guiding decisions about managing recreation. During two winter data collections, …

A Social-Ecological Approach To Evaluating Short-Term Water Leasing For Streamflow Conservation In Montana, Usa, Logan A. Brauer

Graduate Student Theses, Dissertations, & Professional Papers

Streams throughout Montana are increasingly caught between increasing temperatures, decreasing summer precipitation, and heavy water demand for irrigated agriculture. Amid rapidly growing urban populations and uncertain climatic futures, many watersheds in Montana may face water shortages that impact aquatic species and habitat in the upcoming years. Conservation efforts around the state are working to mitigate water stress in these streams especially during the late summer season, when streamflow is naturally lower, and/or in times of drought. One policy solution to address low flows is a concept known as temporary leasing or split season leasing programs (SSLP), whereby an entity (e.g., …

Assessing Algebra Readiness: A Three-Year Study Of The Orleans-Hanna Prognosis Test And Middle School Math Placements, Tabitha Nicole Ringler

Graduate Student Theses, Dissertations, & Professional Papers

This study investigates the viability of the Orleans-Hanna Algebra Prognosis Test (Hanna and Orleans, 1982) as a modern predictor of algebra readiness among eighth grade students. At over 40 years old, the Orleans-Hanna Algebra Prognosis Test (1982) is still commonly utilized by schools as the main source of mathematical ability grouping. The guiding question for this research is whether the test is outdated and should be retired, or is it still a valid predictor of algebra readiness.

Utilizing a longitudinal study spanning three years, the research examines the performance of math students placed in an advanced class versus students who …

How Phonological And Syntactic Overlap Impact Cognate Processing Speeds In Bilinguals, Ella Marie Peterson

Undergraduate Theses, Professional Papers, and Capstone Artifacts

This paper investigates individuals who are proficient in two languages (bilinguals) and the speeds at which bilinguals process cognates (words with comparable form and meaning across languages). This paper cites two ongoing experiments: The Language Identification (LID) task and the Self-Paced Listening (SPL) task. Findings from the LID suggest that phonological overlap in cognates facilitates bilingual language processing speeds, when cognates are presented in isolation. Findings from the SPL suggest that syntactic overlap in cognates also facilitates bilingual language processing speeds, when cognates are presented in sentences. These findings are significant in that the tasks present cognates to participants in …

Effects Of Timber Harvest On Understory Vegetation Succession: A Time Series Study, Reese A. Poindexter, Trevor Weeks, Chad Bishop, Akasha Faist

Undergraduate Theses, Professional Papers, and Capstone Artifacts

Timber harvest, as a necessary and common global practice, can significantly affect the ecosystems being harvested through opening the canopy, disturbing the understory, and resetting successional trajectories. This study examined how timber harvest influenced understory plant functional group dominance, understory species richness, and understory species dominance along a 40-year, or older, harvest timeline across two forest types (mesic and xeric) in Northwestern Montana, U.S.A. In our two forest types, vegetation sampling was completed at 208 sites with 1,040 quadrats across six timber-harvest timelines. Harvest timelines varied from the most recent at 1-5 years post-harvest to over 40 years as the …

Moving Forward, Looking Back: Narratives Of Indigenous Identity In Montana, 2024, University Of Montana--Missoula. School Of Journalism. Native News Honors Project

Native News Honors Project, 1992-2026

An annual publication that is reported, photographed, edited, and designed by students in the University of Montana’s School of Journalism.

Hydrogeomorphic Response To Flooding In Northern Yellowstone National Park, Zachary P. Deluca

Graduate Student Theses, Dissertations, & Professional Papers

Understanding and predicting flood-induced geomorphic change, and the relative influences of fluvial forces and valley-bottom geometry on system response, are persistent quandaries in geomorphic process studies. We combine field surveys, remote sensing, and hydraulic modeling to assess the hydrogeomorphic effects of historic flooding in northern Yellowstone National Park (YNP) in a variety of channel configurations. We compare impulse, a metric that incorporates a flow duration threshold based on threshold channel theory, grain size, channel-bed slope, and flood depth and stream power estimates with hydrogeomorphic response. Measurements of pre- and post-flood active-channel width change in aerial photos captured geomorphic response associated …

A Comparison Of Concurrent Training And Moderate Intensity Continuous Training In Patients With Coronary Artery Disease, Amanda Alfaro-Chaverri, John Quindry, Charles Dumke, Brian Loyd

Graduate Student Theses, Dissertations, & Professional Papers

Moderate intensity continuous training (MICT) is the most studied method in cardiac rehabilitation (CR) for patients with coronary artery disease (CAD). Traditionally, resistance training has been utilized as a secondary role in CR. Recent interest in the topic suggests that MICT and resistance training should be applied collectively, an approach termed concurrent training (CT). The collective understanding of the potential of CT in CR patients is currently lacking.

Purpose: The aim of this professional paper is to analyze scientific evidence regarding the efficacy and safety of CT versus MICT. The analysis will focus on five primary measurements: exercise capacity, cardiovascular …

Regulation Of Er To Golgi Transport By Alg-2 And Peflin In Response To Calcium Signaling, And A Review Of Relevant Genetically-Encoded Fluorescent Tools, Danette M. Kowal Seiler

Graduate Student Theses, Dissertations, & Professional Papers

The endoplasmic reticulum performs multiple functions in the eurkaryotic cell: biomolecule synthesis, protein folding, cargo sorting and secretion, and helping to maintain cellular calcium homeostasis. Though much has been discovered about these processes and the myriad proteins, enzymes, transcription factors, ions and other factors that play their own parts, much is still left to learn. To facilitate continued study of ER-to-Golgi transport, an array of genetically-encoded fluorescent tools are available to researchers. The featured review in Chapter 1 details the history of these tools and highlights some of the newest, which also incorporate information from the fields of synthetic chemistry …

Coalescence: A Carnivore Coexistence Curriculum That Braids Indigenous & Western Ecological Knowledge Into A Relevant And Experiential Learning Opportunity For Youth, Stephanie Anne Barron

Graduate Student Theses, Dissertations, & Professional Papers

As grizzly bears (Ursus arctos horriblis) begin to reoccupy more of their historic range, and as humans and large carnivore populations continue to increase, incidences of human carnivore conflict are on the rise. A decolonial curriculum designed in collaboration with the Confederated Salish and Kootenai Tribe’s wildlife biologists stands to increase awareness of Indigenous ecological knowledge and teach youth about the importance of coexistence with carnivores. Additionally, this project could greatly influence youth perceptions of grizzly bears and other large carnivores. This research project examines the development and implementation of a carnivore coexistence curriculum for youth that is guided by …