Physical Sciences and Mathematics Commons^™

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.

Numerical Issues For A Non-Autonomous Logistic Model, Marina Mancuso, Kaitlyn M. Martinez, Carrie Manore, Fabio Milner

CODEE Journal

The user-friendly aspects of standardized, built-in numerical solvers in
computational software aid in the simulations of many problems solved using
differential equations. The tendency to trust output from built-in numerical
solvers may stem from their ease-of-use or the user’s unfamiliarity with the
inner workings of the numerical methods. Here, we show a case where the
most frequently used and trusted built-in numerical methods in Python’s
SciPy library produce incorrect, inconsistent, and even unstable approxima-
tions for a the non-autonomous logistic equation, which is used to model
biological phenomena across a variety of disciplines. Some of the most com-
monly used …

Linear Ode Systems Having A Fundamental Matrix Of The Form F(Mt), Kevin L. Shirley, Vicky W. Klima

CODEE Journal

We interweave scaffolded problem statements with exposition and examples to support the reader as they explore specific linear systems of differential equations with variable coefficients, $\vec{x}'(t)=A(t)\vec{x}(t)$ and initial value $\vec{x}(0)$. We begin with a constant $n\times n$ matrix $M$ and a real or complex-valued function $f$, analytic at the eigenvalues of $M$ with $f(0) = 1$, and construct a linear system of differential equations with solutions $x(t)=f(Mt)\vec{x}(0)$, where $t$ is a parameter in some interval including zero. In general, the solutions to the resulting non-autonomous system are more difficult to analyze than solutions to the constant coefficient case. However, some …

Engaging Students In Partial Differential Equations Through Modeling Fourier's Law Of Conduction, Justin G. Trulen, Kayla Keller, John Sinclair, Lauren Meagher

CODEE Journal

In an effort to make active learning exercises in a partial differential equations course, we present a student activity modeling Fourier's law of conduction under the framework of the heat equation. An overview of the heat equation, including several avenues of study, is provided. Then we give an intuitive way of constructing the heat equation from a couple of fundamental properties of physics including Fourier's law of conduction. We outline an experiment that can be run to collect their own data to model Fourier's law of conduction as well as provide data we collected. We conclude with a student activity …

Differential Equations For Modeling Pathways Leading To Diabetes Onset, Viktoria Savatorova, Aleksei Talonov

CODEE Journal

This paper presents a mathematical model that explains potential pathways leading to diabetes onset. By utilizing a system of nonlinear differential equations to describe the dynamics of the glucose regulatory system, the model can serve as a pedagogical tool for teaching and learning differential equations, dynamical systems, mathematical modeling, and introduction to biomathematics. Within this framework, students can analyze equilibrium solutions, investigate stability, assess parameter sensitivity, and explore the potential for bifurcations. Theoretical analysis is complemented by illustrative numerical examples. Instructors have the flexibility to adapt and incorporate suggested activities according to their teaching preferences and objectives.

Mpt And Capm Mismeasure Risk, Gary N. Smith

Pomona Economics

Mean-variance analysis and the capital asset pricing model provide many useful insights for investors who want to measure and manage risk. However, their focus on short-term returns is of limited use and potentially misleading for investors with long horizons. A value investing approach suggests that risk might be better measured by long-run uncertainty about asset income than by short-run uncertainty about asset prices.

Automorphisms: New Cars For 2024, Jane P. Sheldon

Journal of Humanistic Mathematics

This review of new cars for 2024 demonstrates total variation.

Mathematical Relationships, David Sheskin

Journal of Humanistic Mathematics

A digital collage depicting the mathematics of interpersonal relationships.

Hypatia's Math: A Play, Daniel S. Helman

Journal of Humanistic Mathematics

Hypatia of Alexandria was the most famous woman mathematician of antiquity, and her story is as relevant now as it was in her day. This play tells her story. While the events in the play are based closely on extant historical texts, inspiring excerpts from classical works by Porphyry, Plotinus, Aristotle and Plato are woven into the action to further the plot and add to the classical mood. Dance and music, including an Homeric hymn, add to the artistic texture.

The Free Numbers, Jessica Greenbaum

Journal of Humanistic Mathematics

No abstract provided.

Persistence: A Romance In Many Dimensions, Tony Bedenikovic

Journal of Humanistic Mathematics

This is a short poem which is inspired by Edwin Abbott's Flatland. The poem follows the narrator through a typical day with a focus on the individual's persistence in the face of nature's brawn and magnitude.

The Point At Infinity, Josh Hiller

Journal of Humanistic Mathematics

A poem about the longing for parallel lines to meet.

The Conviction Of Miss Prediction, Dane C. Joseph

Journal of Humanistic Mathematics

Miss Prediction is questioned in a court of law over her involvement in the mischaracterization of linear models when they were inappropriate.

Pied Beauty, Pamela Vale

Journal of Humanistic Mathematics

This is a shape poem about the beauty of mathematics formed with the lines containing the number of words corresponding to the digits in pi (to 26 places).

Variations On A Meme: Thoughts And Responses To Isaac Elishakoff’S “Differential Equations Of Love And Love Of Differential Equations", Jonathan Kinsman

Journal of Humanistic Mathematics

This folder contains reflections and three poems written in response to Isaac Elishakoff’s JHM article “Differential Equations of Love and Love of Differential Equations" (July 2019).

A Personal And Multicultural Journey Through The World Of Games (With A Little Math) Or Book Review: Around The World In Eighty Games, By Marcus Du Sautoy, Karl-Dieter Crisman

Journal of Humanistic Mathematics

Author and number theorist Marcus Du Sautoy has written a book about games, including a light dose of mathematics. In it, he journeys through far more than (his choice of) eighty of “the world’s greatest games”, laying out a charming voyage through many lands. Despite some reservations, this reviewer finds that Around the World in Eighty Games is well-written and quite fascinating, largely due to its idiosyncratic, personal nature.

Quik Church, Route 3.141592, Sarah Voss

Journal of Humanistic Mathematics

The following set of poems are from one of ten sections in a collection of poetry called Quik Church: Short Poems that Travel Far. Each section illustrates one of many “streets” which individuals often take on their spiritual journey through life, e.g., the Old Gods Path, Nature Trail, Memory Skyway, Mystic Avenue, Pastoral Lane, and so on. This one, Route 3.141592, is the route of mathematics and the science that depends on mathematics.

Hockey Card Statistics Are Stagnant And Stale, Egan J. Chernoff

Journal of Humanistic Mathematics

The purchase of a coffee at a Canadian institution, Tim Hortons, turned into an informal investigation into hockey card statistics. Turns out, hockey card statistics are stagnant and stale. This was disappointing to see because the game of hockey has changed, the statistics used to keep track of the game have changed. Even the cards have changed. Well, not the back of the cards, which do not well enough paint a statistical picture of the hockey player photographed on the front of the card.

The Limits Of Data Science, David E. Drew

Journal of Humanistic Mathematics

Data science can contribute valuable predictions in diverse fields. But I write to express some concerns and red flags. I suggest that data science is being oversold. This article contains three questions that I believe data science must address as this new discipline matures. Is data science significantly different from statistics? This is a question that has haunted the field since the term first was introduced. By creating algorithms based on current societal decision rules that may be biased, even bigoted, does data science lock in and exacerbate inequality? Scholars have identified a continuum from data to information to knowledge …

Reflections On Teaching Mathematics In Prisons, Matthew Junge

Journal of Humanistic Mathematics

I started teaching math in prison ten years ago during my PhD studies. Since then, I have taught for three different college-in-prison programs across the country. The goal of this piece is to communicate my experiences with the hopes of encouraging more mathematicians to get involved.

Polygon Quadrature And Dodecagonal Tessellation With Pattern Blocks, Gunhan Caglayan, Ben Kamau

Journal of Humanistic Mathematics

The age-old challenge of polygon quadrature involves converting a polygon into a square of equal area. In this educational resource, we utilize pattern blocks, commonly employed instructional aids in K-12 education across the United States, to visually demonstrate the transformation of different equilateral and regular pattern block polygons into squares. This is achieved through the application of the area conservation principle and geometric congruence/similarity reasoning.

Benefits Of Risk-Taking In Teaching Mathematics, Mehmet Kirmizi, Abigail Quansah, Zafer Buber

Journal of Humanistic Mathematics

In this paper, we, a group of graduate students in mathematics education, discuss some of the metacognitive benefits of the non-traditional teaching methods we observed employed by one of our professors. This professor’s methods challenge the common belief that well-managed class time is key for positive learning outcomes. Instead, he orients his teaching to share the exploration and sense-making phases of doing mathematics. The goal of his teaching is to share the idea that learning mathematics is a process of “refining our mathematical thinking”. We argue that this approach to teaching helps students see that mathematics is a human endeavor, …

Does Chatgpt Know Calculus?, Kris H. Green

Journal of Humanistic Mathematics

Academics and educators across the world are grappling with how OpenAI’s new software, ChatGPT, will impact teaching and learning. This essay explores ChatGPT’s response to a typical calculus problem as a way of illustrating its functionality and limitations.

The Frankensteinian Nature Of Mathematics, Ali Barahmand

Journal of Humanistic Mathematics

Frankenstein is a story about a scientist who creates a sapient creature that gets out of control and horrifies its creator by its unexpected behavior. In this note, we show that this type of undesirable behavior can reflect a part of the nature of mathematics, and that its origin is related to the ontological question of whether mathematics is invented or discovered. Based on a review of the relationship be- tween discovery and invention, we demonstrate that mathematics has similarities and differences with both discovery and invention. In the natural sciences, new instruments have to be invented to discover new …

Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett

Journal of Humanistic Mathematics

Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques.

Logical Or Epistemological? A Study On Kepler’S 3m Cosmological Model, Po-Hung Liu

Journal of Humanistic Mathematics

Kepler published Mysterium Cosmographicum in 1597 constructing his cosmo- logical model based on the five regular polyhedra. Such a creative but weird idea was almost consistent with empirical evidence. Furthermore, following the Pythagorean belief about the connection between music and astronomy, Kepler delved into looking for what things having to do with the planetary movements have the harmonic consonances. This article claims that Kepler’s model is a 3M (mathematical, musical, and metaphysical) model and demonstrates how it had been constructed. Furthermore, I explore the reasons behind Kepler’s departure from the 3M model and his subsequent consideration of a non-circular orbit, …

On Parallels Between Words And Music, Will Turner

Journal of Humanistic Mathematics

A generalised song is a means of drawing parallels between words and music. The parallels are encoded in a mathematical structure, which is interpreted in a verbal structure and a musical structure. Here we develop a number of new techniques for drawing such parallels, in giving two examples of generalised songs, `Relation', and `Merge/Split'.

The first five partials of a note played on a piano are roughly 0,12,19,24,28 semitones above the fundamental.`Relation' is a generalised song, whose musical part is played on a piano, constructed from the mathematical relation 4.28 = 3.12 + 4.19.

`Merge/Split' is a generalised song whose …