Physical Sciences and Mathematics Commons^™

Mathematical Graffiti: Bridges 2023 Clerihew Collection

Journal of Humanistic Mathematics

Clerihews are poems of a form invented by Edmund Clerihew Bentley around the turn of the 19th-20th century. The poems are typically biographical, humorous, and are made up of two couplets. The rhyming pattern is always aabb, but the meter of the two couplets is usually not the same. The first line is simply the name of the person, the other three lines relate to the subject, often in an absurd way. If the rhyme is slightly off, or the rhythm irregular or awkward, or the facts a bit confused, so much the better. The present collection of clerihews, written …

The Value Of Adding Nothing: A Call For Reform-Oriented Polynomial Division, Jonathan Clark, Jeneva Clark

Journal of Humanistic Mathematics

The call to implement reform practices in schools reflects the historical turn away from the behaviorist theory of learning in education. Yet the praxis of this turn remains a significant challenge, particularly within mathematics classrooms where procedural memorization is emphasized. In this article, we show one means of how to advance our pursuit of meaningful mathematics into polynomial division. Building on the literature for reform-based division methods, an alternative to the long division algorithm will be explored that relies solely on adding zero and fundamental algebraic principles.

Book Review: How To Expect The Unexpected: The Science Of Making Predictions -- And The Art Of Knowing When Not To By Kit Yates, Mark Huber

Journal of Humanistic Mathematics

Humans think about the future all the time. Prediction is a part of how we prepare for the coming of both good and bad events in our lives. Kit Yates' book, How to expect the unexpected, concentrates primarily on the question of why prediction is difficult, and what mental shortcuts people take in prediction that can lead to incorrect results. Unfortunately, a lack of concern for details and several omissions undermine the quality of the book.

Geometric Shapes That Sing And Move: An Interdisciplinary Lesson With Pre-Service Teachers, Gladys Krause, Gustavo Velandia

Journal of Humanistic Mathematics

Our work shares a practical example of an interdisciplinary lesson in which two teacher educators collaborated to integrate mathematics and music in an elementary mathematics methods course. This paper describes the process of collaboration in designing the lesson and shares original instructional resources to be used in the classroom. We also discuss what the pre-service teachers participating in the lesson shared about their learning experience, and what we, the teacher educators, learned from this experience. In presenting this work we aim to promote the opening of spaces in teacher preparation programs that allow pre-service teachers to develop their own instructional …

States Of Matter, Todd Sformo

Journal of Humanistic Mathematics

This is a non-fiction essay overtly about three methods used in overwintering physiology, set in the context of my first learning them, along with associated thoughts and ideas as I began working on my PhD. The mathematics shows up mainly in the final section of the essay whose subtitle plays on a poem by Wallace Stevens called “Anecdote of the Jar”. This section is fable-like in its explanation of protein purification and begins with an impossible statement that is slowly adjusted to make sense by words and math.

Fibonacci-Inspired Spiral Quilts, Kathleen Offenholley, Sk Collins, David Radcliffe

Journal of Humanistic Mathematics

This article provides insight into the mathematics and designs of quilts inspired by Fibonacci and logarithmic spirals. We introduce the history and development of the Fibonacci number sequence and how to hand-draw a Fibonacci spiral. Further, we explain the relationship between the Fibonacci spiral and logarithmic spirals, the advantages of using logarithmic spirals to create designs, and how to produce digital spiral designs using Desmos, a free web-based graphing calculator. Finally, we discuss methods for designing spiral quilts or other triangle and spiral designs (such as collage or other media) and derive a formula for calculating the apex angles of …

Badass Women, Richard Delaware

Journal of Humanistic Mathematics

In this true story, one mathematics major supports another in an unexpected way.

Intuitive Explanations In Mathematical Education, Jerzy Pogonowski

Journal of Humanistic Mathematics

I discuss the role of intuitive explanations in the learning, teaching, and popularization of mathematics. Several examples of such explanations are presented, related to linguistic explanations, perception, empirical models, and internal explanations inside mathematics itself. I emphasize the fact that intuitive explanations in a sense transgress mere mathematical arguments. I also discuss in brief the role of paradox resolution in mathematical education.

Bootstraps And Scaffolds: What A Cognitive-Historical Analysis Of The Complex Number System Reveals About Numerical Cognition, Charles R. Card, Gary G. Miller

Journal of Humanistic Mathematics

The following investigation is a cognitive-historical analysis of the conceptual development of complex numbers. The history of this development spans nearly two millennia, from the earliest appearance of the square root of a negative quantity in the calculations of Heron of Alexandra (1st Century CE) to the full flowering of complex numbers in the first half of the 19th Century. The approach used for this analysis is Nersessian's, including her formulations of model-based reasoning and mental models. Additional aspects of the analysis feature the prominent roles played by process representations, including object-process complementarities, and by core numerical systems. Our analysis …

Building Communities Of Care For Equity, Justice, And Culturally Responsive Practice In Mathematics Education, Nicole Fletcher, B Waid

Journal of Humanistic Mathematics

Teaching is widely considered one of the “caring professions,” but conceptualizations of care and how care is put into practice in education are not universal. In this article, we draw from a range of perspectives on care that integrate supportive interpersonal relationships, high expectations, and culturally relevant theories of critical care, as well as Queer Theory and Disability Justice, to explore the application of these ideas in mathematics education. We identify key elements for building communities of care in mathematics education contexts: co-constructing community agreements, redefining participation, shifting traditional power structures, collaborative problem solving, and building networks of care beyond …

The Braids On Your Blanket, Michelle Cheng, Robert Uwe Laugwitz

Journal of Humanistic Mathematics

In this expository essay, we introduce some elements of the study of groups by analysing the braid pattern on a knitted blanket. We determine that the blanket features pure braids with a minimal number of crossings. Moreover, we determine polynomial invariants associated to the links obtained by closing the braid patterns of the blanket.

Mathematical Models And Pedagogy Of Marxist Political Economy, Christopher Perez

Journal of Humanistic Mathematics

How can we teach people about the economics of labor and exploitation in mathematics courses? We define a mathematical model for describing the relationships embodied by commodities and labor. We then use this model to illustrate the exploitative nature of profit and the tendency for catastrophic chain-reactions that lead to market crashes. Lastly, we discuss applications to pedagogy in mathematics courses using a simplified version of the model.

What Is An Imaginary Number? The Plane And Beyond, Andrew W. Powell

Journal of Humanistic Mathematics

In this article I argue that i is a quantity associated with the two-dimensional real number plane, whether as a vector, a bi-vector, a point or a transformation (rotation). This position provides a foundation for the complex numbers and accounts for complex numbers in some equations of applied mathematics and physics. I also argue that complex numbers are fundamentally geometrical and can be described by geometric algebra, and that moreover the meaning of complex numbers in physics varies with dimension and geometry of the manifold.

Language Analysis Via The Run And Flattened Statistics On Permutations, Jennifer Elder, Pamela E. Harris, Anthony Simpson

Journal of Humanistic Mathematics

A permutation π in Sn can be decomposed into its runs π = τ₁τ₂ . . . τ_k, where a run of π is a maximal contiguous subsequence whose elements are in increasing order. If the first values of each run are in increasing order, then π is said to be flattened. Motivated by the study of flattened permutations, we study the words in the Danish, German, English, Spanish, French, Italian, Dutch, and Norwegian languages. In each language considered, our work provides the following: a list of the longest flattened words, histograms for the proportion …

Millions, Billions, Or Trillions: How To Partition Large Numbers Into Friendly Figures, Eryn Michelle Maher, Ha Nguyen, Cynthia Sanchez Tapia

Journal of Humanistic Mathematics

Communicating and making sense of large numbers — millions, billions, and trillions — is a persistent struggle in our society. Using large numbers is a learning requirement for elementary school children, but even adults struggle with it. Hence supporting future teachers in developing their own understanding of the concepts is valuable. To construct, enact, and revise an educational experience for preservice teachers, we apply three frameworks of teaching mathematics for social justice tasks, high cognitive demand tasks, and productive mathematics discussion. The context uses United States educational and defense spending, the national budget, and the gross domestic product. Preservice teachers …

The Modern Geometrician: Euclidean Construction For Digital Paper, Deborah A. Kent, David J. Muraki

Journal of Humanistic Mathematics

The emphasis on traditional hand-drawn compass and straight-edge geometrical constructions has been reduced in the core narrative of most current curricula. In response to this trend, this paper presents a virtual toolkit for producing precision geometrical figures within the popular note-taking app, Notability. These graphical procedures employ the app's stylus-based input and shape tools (for lines, circles and squares) to offer a modern take on classical geometrical construction. These procedures are adaptations of familiar textbook methods, necessary because the app's circle-drawing tool behaves differently from a standard compass. Beyond the familiar canon of elementary Euclidean constructions, such as angle bisectors …

Sociomathematical Norms And Automated Proof Checking In Mathematical Education: Reflections And Experiences, Merlin Carl

Journal of Humanistic Mathematics

According to a widely held view, mathematical proofs are essentially (indications of) formal derivations, and thus in principle mechanically checkable (this view is defended, for example, by Azzouni [3]). This should in particular hold for the kind of simple proof exercises typically given to students of mathematics learning to write proofs. If that is so, then automated proof checking should be an attractive option for math education at the undergraduate level. An opposing view would be that mathematical proofs are social objects and that what constitutes a mathematical proof can thus not be separated from the social context in which …

My Own Private World Of Non-Ordinary Associative Arithmetics, Marion D. Cohen

Journal of Humanistic Mathematics

A binary operation # on Z+ is said to be an associative arithmetic if both # and its iteration — the binary operation ∗ defined recursively by: x∗1 = x and x∗y = [x ∗ (y − 1)]#x — are associative. E. Rosinger [6] showed that under reasonable conditions an associative arithmetic must be ordinary addition. However, in the general case, there are associative arithmetics that are not ordinary addition. This paper gives examples of these as well as results towards a structure theorem for associative arithmetics. The paper also describes the role that this particular math problem has played …

Picturing Mathematics (Education) In New Ways, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Nilpotent Global Centers Of Generalized Polynomial Kukles System With Degree Three, Hebai Chen, Zhaosheng Feng, Rui Zahg

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we study and characterize the nilpotent global centers of a generalized polynomial Kukles system with degree three. A sufficient and necessary condition of global centers is established under certain parametric conditions.

Short-Time Fourier Transform And Superoscillations, Daniel Alpay, Antonino De Martino, Kamal Diki, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the …

Quantitative Reasoning: What’S Math Got To Do With It?, Pamela Burdman

Numeracy

This keynote address explores the history and role of college math requirements with a focus on ensuring math courses serve to expand students’ horizons, rather than serve as gatekeepers. It discusses the advent of general education math courses, which brought more students into math departments, which ultimately contributed to broadening the scope of the courses to align with more students’ interests and majors, since their purpose was to advance quantitative reasoning, not mathematics skill per se. It also examines several practices to address calculus’ gatekeeping role: revising placement practices and prerequisites, redesigning courses, and updating instruction and assessment practices. Lastly, …

Stochastic Solutions For Hyperbolic Pde, Abdol-Reza Mansouri, Zachary Selk

Journal of Stochastic Analysis

No abstract provided.

The Arbitrariness Of Symmetry In Mathematical Proofs, Melisa Vivanco

Philosophy Faculty Publications and Presentations

Symmetry is not an inherent characteristic of mathematical proofs; instead, it is a property that arbitrarily manifests in different modes of presentation. This arbitrariness leads to the conclusion that symmetry cannot be part of the defining or essential properties that characterize proofs. Consequently, contrary to some authors’ claims, symmetry does not significantly contribute to the validity, accuracy, or soundness of mathematical proofs. What is more, it does not even play any critical role in heuristic aspects such as explanatory power. The examples developed in this paper constitute compelling evidence supporting these claims.

Uniformly Distributing Points On A Sphere, Flavio Arrigoni

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we are going to present and discuss different procedures for distributing points on a sphere's surface. Furthermore, we will assess their quality with three different distribution tests. The MATHEMATICA package that we created for testing and plotting the points is publicly available.

Modeling Virus Diffusion On Social Media Networks With The Smirq Model, Justin Browning, Arnav Mazumder, Gowri Nanda

Rose-Hulman Undergraduate Mathematics Journal

As social networking services become more complex and widespread, users become increasingly susceptible to becoming infected with malware and risk their data being compromised. In the United States, it costs the government billions of dollars annually to handle malware attacks. Additionally, computer viruses can be spread through schools, businesses, and individuals’ personal devices and accounts. Malware affecting larger groups of people causes problems with privacy, personal files, and financial security. Thus, we developed the probabilistic SMIRQ (pSMIRQ) model that shows how a virus spreads through a generated network as a way to track and prevent future viruses. Our model is …

Monotone Functions On General Measure Spaces, Alejandro Santacruz Hidalgo

Electronic Thesis and Dissertation Repository

Given a measure space and a totally ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to generalize monotone functions to general measure spaces. The least core decreasing majorant construction, the level function construction, and the greatest core decreasing minorant, already known for functions on the real line, are extended to this general setting. A functional description of these constructions is provided and is shown to be closely related to the pre-order relation of functions induced by integrals over the ordered core.

For an ordered core, the down space …

Resonant Solutions Of The Non-Linear Schrödinger Equation With Periodic Potential, Arein Duaibes, Yulia Karpeshina

Mathematics Faculty Publications

The goal is construction of stationary solutions close to non-trivial combinations of two plane waves at high energies for a periodic non-linear Schrödinger Equation in dimension two. The corresponding isoenergetic surface is described for any sufficiently large energy k2. It is shown that the isoenergetic surface corresponding to k2 is essentially different from that for the zero potential even for small potentials. We use a combination of the perturbative results obtained earlier for the linear case and a method of successive approximation.

Analytic Properties Of Quantum States On Manifolds, Manimugdha Saikia

Electronic Thesis and Dissertation Repository

The principal objective of this study is to investigate how the Kahler geometry of a classical phase space influences the quantum information aspects of the quantum Hilbert space obtained from geometric quantization and vice versa. We associated states with subsets of a product of two integral Kahler manifolds using a quantum line bundle in a particular manner. We proved that the states associated this way are separable when the subset is a finite union of products. We presented an asymptotic result for the average entropy over all the pure states on the Hilbert space H⁰(M₁,L_{1 …}