Mathematics Commons^™

Reminiscences Of My Past Life As A High School Math Teacher, Wayne Nirode

Journal of Humanistic Mathematics

In this article, I reflect on my twenty years as a high school math teacher. I describe four distinct phases of my career in five-year increments. I aim to make public both what makes teaching high school students rewarding and what makes it difficult. Also, my writing illuminates how indelible the first five years were to my development as a teacher. I hope that this brief memoir helps early-career teachers see past the tumultuous beginnings while also inspiring veteran teachers to reflect on their own careers.

Creating Art With The Generalized Pythagorean Theorem, Ángel Colín

Journal of Humanistic Mathematics

The Pythagorean Theorem and its generalizations have been widely studied. In this article we focus on one particular generalization that asserts that the area of a polygon with one side the hypotenuse of a right triangle is the sum of the areas of similar polygons whose corresponding sides are the other two sides [4]. Using this theorem together with interactive software or simply pencil and paper, we create some visually enticing artwork. This activity, we believe, can contribute positively to geometry teaching at all academic levels.

Let's Play Uno! Training Mathematical Thinking With An Inclusive Card Game, Thierry Meyrath, Clara-Ioana Mincu, Antonella Perucca

Journal of Humanistic Mathematics

We explore probability exercises based on the game UNO. Excluding the Wild cards, the deck contains precisely 100 cards, which makes it convenient to express probabilities as percentages. We discuss the mathematical relation between cards that can be played after other cards. We also propose an algebraic exercise that relates several quantities, such as the number of cards left in the losing player’s hand. Finally, we argue that UNO is a very inclusive game: it is logistically easy to play, allows for players of different skill levels, and can be adapted to meet special needs. Beyond being playful exercises, our …

Partnering With Students To Create A Positive Image Of Mathematics, Manmohan Kaur

Journal of Humanistic Mathematics

In a 1989 paper, Barbeau states, “Probably no area of human activity is as afflicted as mathematics with a gap between the public perception of its nature and what its practitioners believe it to be.” Decades later, the situation is much the same. With the technical advances in the contemporary world, mathematics affects our day-to-day activities in essential ways such as e-commerce and credit card security. Mathematical problem solving intrinsically requires creative and critical thinking. However, mathematics is often perceived as uninventive, sterile, static, and removed from human needs and culture. This paper describes how mathematicians can own the image …

Why Read The Classics (Of Mathematical Proofs)?, Cosimo Perini Brogi

Journal of Humanistic Mathematics

Inspired by Italo Calvino’s (almost) homonymous writing, this short essay suggests a formal perspective to complement the typical categorisation of classical results in mathematics on aesthetic bases. The traditional proof that √2 is not a rational number provides a simple example of proof mining. When studied through the tools of logic, that proof has also led to designing new systems for formal proving based on cyclic proofs. This example thus supports the central thesis of this essay that reading the classics of mathematical proofs enables us to acquire a taste for mathematical beauty and can open new directions in mathematics …

A Mathematician Expresses Grief, Richard Delaware

Journal of Humanistic Mathematics

A famous Scottish mathematician of the eighteenth century shares a tragic sorrow.

How (And Why) To Start A Math Circle At The County Jail, Aram Bingham

Journal of Humanistic Mathematics

Math circles are a common form of outreach usually run by professional mathematicians as enrichment for K-12 students or teachers. As a setting to stimulate interest and develop positive socialization around mathematical practice, a regular math circle can also be an effective bridge to higher education in carceral settings for both facilitators and participants. This article describes my journey through starting and running math circles in county jails, with a view towards the expansion of mathematics education as another tool for the dismantling of mass incarceration.

Making Mathematical Art Via The Harmonic And Perturbed Harmonic Functions, Mehmet Pakdemirli

Journal of Humanistic Mathematics

Starting from the definition of a harmonic function series, we define a new series which we call the perturbed harmonic function series. We explore the relationship of the new series with the Weierstrass and Riemann fractal functions as well as its convergence and differentiability properties. We then illustrate the potential of the harmonic functions in generating complex figures that sometimes resemble natural objects, with explicit numerical examples.

A Note On Modeling Prejudice, Alan Lockard, Edwin Harcourt

Journal of Humanistic Mathematics

We model prejudice in the context of an iterated prisoner’s dilemma tournament. Prejudice here is defined as the inability to distinguish amongst members of an identifiable group. We run a computer simulation where agents of one of two groups with defined strategies are randomly matched against each other. The agents either cooperate with or defect against the agent they are matched with. Our focus is on agents who play an unprejudiced version of the Tit-for-Tat strategy (cooperate with any player in the first encounter, and then apply the same strategy [defect or cooperate] that the opposing player played against them …

How Did Cauchy Prove His Integral Theorem? A Historical Reconstruction For Education Research, José Gerardo Piña-Aguirre, Rosa María Farfán Márquez

Journal of Humanistic Mathematics

In his article “Mémoire sur les Intégrales Définies, prises entre des limites imaginaires” of 1825, Cauchy proves that the integral of a complex function f defined on complex numbers is independent of the path of integration if the function f remains finite and continuous along the chosen path. In contemporary literature that analyzes the mathematics of Cauchy, the specific mathematical process that allowed Cauchy to prove this result is not explicitly presented. The purpose of this manuscript is to communicate a symbolic-algebraic derivation that justifies the narrative argument used by Cauchy to support his proof, thus hopefully filling this gap. …

Gold And Silver In Balance: Khayyam’S Mathematical Analysis Of Binary Alloys, Yousef Yassi, Reza Yassi

Journal of Humanistic Mathematics

The history of titration dates back to Archimedes who established that an object submerged in a liquid displaces an amount of liquid whose weight is equal to the buoyant force acting on the object. Since then, many scientists and engineers have tried to optimize his approach or devise new instruments for titration purposes. Omar Khayyam (1048–1123), in addition to designing a hydro-static balance, developed mathematical approaches for titration of binary alloys, that is, alloys made up of only two elements. Here we explore the different versions of Khayyam’s work on titration given by various sources. We also compare the accuracy …

A Just Balance And The Teaching And Learning Of Mathematics, Marshall Gordon

Journal of Humanistic Mathematics

Society’s response to the question of what is worth sharing and how it best can be provided as conceived by the educational establishment shapes the education of every individual in school. As the classroom experience is a commitment to the future of society, it is good and just for society to support the educational development of each person in order that all can participate productively. In this way a “just balance” is established. In the particular case of educating students in mathematics, however, there is apparently an unjust balance, for despite the considerable time and effort students and teachers in …

Polygonal Number Similarity, Gunhan Caglayan

Journal of Humanistic Mathematics

This note takes an exploratory approach to define and then visualize the notion of polygonal number similarity between pairs of k-gonal numbers P_k(αn) and P_k(n) , where α is an integer scale factor of 2 or greater.

Translating Trattato Dell'alcibra Amuchabile, Olympia Nicodemi

Journal of Humanistic Mathematics

We present a detailed look at one small section of a premodern algebra manuscript from circa 1365 C.E. The manuscript, Tratatto dell’Alcibra Amuchabile (Anonimo), is written in the Tuscan vernacular, with no symbols, no punctuation, and wildly inconsistent (pre-printing press) spellings. We present some of the hurdles to translating this “rhetorical algebra” followed by an annotated translation of one of its algebra rules, Rule 5, through which we learn the methods and mindset of algebra that preceded and seeded the breakthroughs in algebra of the 16^th century. We also point to some of the significant differences between algebra then …

Mathematics And Ethics: Narratives And Conversations, David Pritchard

Journal of Humanistic Mathematics

We provide an eclectic overview of various narratives that mathematicians use to describe mathematics. We discuss some of the ethical issues and approaches that each narrative tends to emphasise, and how these can lead to productive conversation in the absence of a universally agreed philosophical or ethical framework. We also discuss the dangers of a deficit model of ethical engagement, and other threats to conversation.

A Schur-Fire Way To Make Matrix Products Coincide, Keith Copenhaver, Josh Hiller, Christopher Sciortino

Journal of Humanistic Mathematics

When are the Schur product and the matrix product of two matrices the same? With the help of Morpheus (from The Matrix) and more, we explore this question and find both mathematical answers and math-memes.

Exploring The Dynamics Of Romantic Relationships Through The Lens Of Prem Rog, Umang Jain, Dheeraj Sharma, Pranay Goswami, Kuldeep Malik

Journal of Humanistic Mathematics

Romantic relationships are dynamic events that begin, grow, and frequently remain for a long time in a stagnant or fluctuating state until possibly dissipating. Although they are unquestionably the most significant dynamic events in our lives, dynamic systems theory has only recently included them in its formal framework. Without a mathematical model, it would be impossible to analyze and comprehend the dynamics because, in general, love stories are too brief to allow things to stabilize and are affected by the ups and downs of the surrounding community. In this paper, we set up models made up of four ordinary differential …

The More Things Change..., Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Patient-Specific Lumped-Parameter Model For Quantifying Vessel-Specific Remodeling And Predicting Right Ventricular Function In Pulmonary Hypertension, Christopher G. Lechuga, Amirreza Kachabi, Mitchel J. Colebank, Claudia E. Korcarz, Farhan Raza

Faculty Publications

Purpose: Pulmonary hypertension (PH) is a heterogeneous disease with patient-specific variability and vessel-specific remodeling, which eventually lead to right ventricular (RV) failure. The gold standard for RV assessment—pressure–volume (PV) loop acquisition—is invasive and limited to specialized settings. This study aims to develop a patient-specific lumped-parameter model that quantifies vessel-specific remodeling and simulates RV PV loops across PH phenotypes using routine clinical data.

Methods: A lumped-parameter model was calibrated using right heart catheterization and echocardiography data. Model agreement was assessed by R2 values for pressure and flow goodness-of-fit, and model-derived hemodynamic metrics were comparedwith clinical values. A dimensionality reduction approach was …