Education Commons^™

A Generalized Method Of Undetermined Coefficients, James S. Cook, William J. Cook

CODEE Journal

The method of undetermined coefficients is used to solve constant coefficient nonhomogeneous differential equations whose forcing function is itself the solution of a homogeneous constant coefficient differential equation. In this paper, we show that the classical methods for tackling constant coefficient equations, including the method of undetermined coefficients, generalize to much wider class linear differential equations which, for example, include Cauchy-Euler type equations. This general method includes an explicit construction of the fundamental solution sets of such equations. We also briefly consider where this method can be applied by producing the most general second and third order differential equations that …

“Math Talks Are Like An Alarm Clock Waking You Up”: Language’S Crucial Role In Mathematics, Gabriella M. Wasser

Journal of Practitioner Research

Whole group math talks, or number talks, are a common practice to get students talking about their own understanding of mathematical concepts. The purpose of this study was to implement math talks in small group settings to see what would happen, specifically to students’ conceptual understanding as well their general perceptions of math talks. This study took place in a fourth-grade math classroom, and math talks were implemented with the whole class for a week and then moved to small groups for the remaining three weeks of the study. During the study, a pre-and post-assessment was given, field notes were …

Collaboration In Mathematics Teacher Education: The What, How, And Why Of Mathematical Modeling, Aubrey Neihaus, Amy Bennett

The Advocate

In this paper, we share our collaboration across the disciplines of mathematics and mathematics education to develop and implement a mathematical modeling task for prospective secondary mathematics teachers. Through this collaboration, we identified three key components of mathematical modeling: the what, how, and why. In this paper, we outline these components from the literature and how each framed our development and implementation of the Sprinkler Task in our mathematics content and mathematics methods courses for secondary teachers. These three components show that mathematical modeling is a particularly fruitful space for collaboration between the disciplines of mathematics and …

Mathematics Education As Dystopia: A Future Beyond, Peter Appelbaum, Charoula Stathopoulou, Constantinos Xenofontos

Journal of Humanistic Mathematics

We argue that scholars and practitioners of mathematics education need to find new directions through recognition of its dystopic characteristics, and embrace these characteristics as both the source of challenges and method of response. This contrasts with the generally utopic approach of most scholarship in the field. We offer critical ethnomathematics education as a model, since it has its own origins in lingering dystopic legacies. A perpetual hopelessness and disempowerment is one implicit curriculum of contemporary mathematics education, where the mathematics one learns might help to describe things, yet hardly assists in transforming the reification of power and agency in …

Calculus Iii: Under The Influence Of Peer Instruction, Alan Von Herrmann, L. Jeneva Clark

Journal of Humanistic Mathematics

In peer Instruction, students engage with core course concepts and then explain those concepts to one another in small groups. Unlike in lecture format, peer instruction involves every student in the class. In Spring 2019, the first authot began using a modified version of peer instruction in Calculus III classes. He started each class by discussing important Calculus III concepts from three standpoints (the formula, the geometry behind the formula, and the physics behind the formula). During the last 20 minutes of each 50-minute class session, he polled the students using questions in the “Goldilocks Zone” – not too hard …

Higher Meanings: A Speaker Series Connecting Mathematics And Religion, Lawrence M. Lesser, Patricia S. Barrientos, Ben Zeidman

Journal of Humanistic Mathematics

An innovative grant-funded general adult audiences international speaker series on connections between mathematics and religion yielded six 2021 (now archived) presentations. We share reflections and lessons learned, informed by two sets of surveys.

Navigating Mathematics Teacher Preparation During A Time Of Crisis, Zareen G. Rahman, Rani Satyam, Younggon Bae

Journal of Humanistic Mathematics

In this paper we highlight the experience of a mathematics teacher educator (MTE) and their prospective teachers (PTs) in a middle school mathematics methods course during the 2020 shift to online instruction due to the COVID-19 pandemic. We believe it is valuable to report how the MTE reflected on their instructional decision-making in response to this massive transition to remote instruction. We also report that PTs needed support and guidance to employ new teaching practices they had learned in the methods course instead of reverting to familiar teaching methods.

Stewardship Of Mathematics: Essential Training For Contributors To, And Users Of, The Practice Of Mathematics, Rochelle E. Tractenberg

Journal of Humanistic Mathematics

A steward of the discipline was originally defined as an individual to whom “we can entrust the vigor, quality, and integrity of the field”, and more specifically, as “someone who will creatively generate new knowledge, critically conserve valuable and useful ideas, and responsibly transform those understandings through writing, teaching, and application” [8]. Originally articulated for doctoral education, in 2019 the construct of stewardship was expanded so that it can also be applied to non-academic practitioners in any field, and can be initiated earlier than doctoral education [18]. In this paper, we apply this construct to the context of mathematics, and …

Towards Pedagogy Supporting Ethics In Modelling, Marie Oldfield

Journal of Humanistic Mathematics

Education for concepts such as ethics and societal responsibility that are critical in building robust and applicable mathematical and statistical models do currently exist in isolation but have not been incorporated into the mainstream curricula at the school or university level. This is partially due to the split between fields (such as mathematics, statistics, and computer science) in an educational setting but also the speed with which education is able to keep up with industry and its requirements. I argue that principles and frameworks of socially responsible modelling should begin at school level and that this would mean that ethics …

Benny, Barbara, And The Ethics Of Edtech, Geillan Aly

Journal of Humanistic Mathematics

Erlwanger (1973) shook the mathematics education world when he introduced Benny, a student who successfully worked through a behavioristic curriculum. Erlwanger showed how far removed Benny’s understanding of mathematics was from expectations. Erlwanger’s legacy is the basis for this comparative case study which explores students’ actions in the modern, in-class computer-centered emporium classroom. Many striking similarities are found between Pearson’s MyMathLabs (MML) and Benny’s Individually Prescribed Instruction curriculum. In this case study we meet Barbara, a student who succeeds in MML but shows little understanding of mathematical concepts and demonstrates that the legacy of Benny is his continued appearance in …

Violence In Mathematics Teaching. Reflections Inspired By Levinas’ Totality And Infinity, Adriano Demattè

Journal of Humanistic Mathematics

In mathematics class, violence is carried out in some usually not recognized situations. In this article, I share some reflections on the topic inspired by some passages of Totality and Infinity, work of the French-Lithuanian philosopher Emmanuel Levinas (1906-1995). I find violence by the teacher in not promoting students’ understanding of mathematics, in a distorted use of rhetoric, and in interrupting an ethical relation. This article analyses situations taken from class activities, focusing on the teacher’s presentation of mathematical content and students’ interventions. I also propose that the improvement of interventions in mathematics education is possible and suggest theoretical …

The Formal Presentation Language Of Mathematics And Communication Ethics, Marshall Gordon

Journal of Humanistic Mathematics

Mathematics employs a formal language where symbols, verbs, and nouns serve to express terms, concepts, and rules that concatenate to definitions, problem-solving procedures, and proofs. Taken together these constitute the expository language of mathematics found in journals, textbooks, and demonstrations. As a communication given to informing, there are epistemological and ethical considerations that deserve examination. For in keeping with the commitment to an aesthetic of concision promulgated by tradition, the formal presentation language and style of mathematics, while valuable in furthering the body of knowledge, provides only the conclusion of an inquiry, completely excluding the language of investigation that informed …

Theoretical Features Of Foreign Systems Of Preschool Education, Ayjamal Bazarbaeva

Karakalpak Scientific Journal

This article analyzes the features of preschool education systems in foreign countries. The positive experience and features of some systems are highlighted. Comparison of domestic and foreign systems of preschool education is carried out.

How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli

The Review: A Journal of Undergraduate Student Research

The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊n/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with n walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph …

Learning Mathematics From The Master: A Collection Of Euler-Based Primary Source Projects For Today’S Students, Part I, Janet Heine Barnett, Dominic Klyve, Kenneth M. Monks, Adam E. Parker

Euleriana

This article and its sequel will together highlight a set of nine classroom ready projects that draw on the remarkable writing of Leonhard Euler (1707–1783) as a means to help students develop an understanding of standard topics from today’s undergraduate mathematics curriculum. Part of a larger collection of primary source projects intended for use in a wide range of undergraduate mathematics courses, these projects are freely available to students and their instructors. We provide a general description of the pedagogical design underlying these projects, more detailed descriptions of the individual projects themselves, and instructions for obtaining downloadable copies for classroom …

High School Student Perspective: My Njit Stem For Success Internship Experience, Michael Mora

STEM Month

During the 2020-2021 school year, I was a senior at the Academy for Mathematics, Science, and Engineering (AMSE) in Rockaway, NJ. At AMSE, a STEM-focused four-year magnet high school program hosted at Morris Hills High School, participating in an extended internship senior year is a cornerstone of the learning process. Required to complete a STEM-related internship to graduate, Academy students are encouraged to seek out an internship they’re passionate about in a field of their choice. The internship, which must be conducted under the mentorship of an industry professional, must meet the New Jersey-approved standards for a work-based learning experience …

Analysis Of A Mathematical Model Of Real-Time Competitive Binding On A Microarray, Frank H. Lynch

CODEE Journal

A mathematical model of competitive binding on a microarray in real-time yields a planar system of nonlinear ordinary differential equations. This model can be used to explore dimensionless formulation, linear approximation, and reduction. Real-time competitive binding is proposed as an uncommon approach to advance the study of planar systems of differential equations.

Revisiting Prejudiced Polygons: Adapting A Familiar Activity During A Time Of Unknowns, Anne M. Ho, Jaime J. Mccauley, Tara T. Craig

Journal of Math Circles

This article describes the design process behind various iterations of Prejudiced Polygons, a Math Circles activity about segregation. In particular, we frame our discussion around two guiding principles from User Experience (UX) Design in thinking about the interconnected components of a Math Circles session, which includes all the people, the physical or virtual setting, the technology, and the world context. Additionally, we describe how we think about developing a “low floor" and “high ceiling" for math content, social issues content, as well as technology and access.

An Urgent Plea For More Graduate Programs In Statistics Education, David Eli Drew, Sam Behseta, Cherie L. Ichinose

Journal of Humanistic Mathematics

Lately, much has been written about the importance of amplifying statistics-related content in the K-12 curricula. This can be viewed in parallel or as an addendum to the existing mathematics curricula in the United States. Nevertheless, a key component of this debate is the lack of robust and cutting-edge academic programs in statistics education. In this piece, we emphasize the urgent need for investing in creating strong statistics education programs, which would significantly contribute to nurturing quantitative literacy as well as preparing a more informed citizenry in the 21^st century.

A Reflection On Growth Mindset And Meritocracy, Rachel L. Petrik, Julianne Vega, Andrés R. Vindas-Meléndez

Journal of Humanistic Mathematics

As mathematicians working in higher education we reflect on meritocracy and growth mindset with a focus on the relationship between the two. We also note the subtle differences between growth mindset and grit. Our reflection ends with suggestions for how to move forward in the math classroom and throughout the collegiate level.

The Problem Of Words: Learning To Teach Mathematics When Numbers And Languages Mix, Gladys Krause

Journal of Humanistic Mathematics

In this piece I propose a perspective shift, from a simplified view of mathematics story problems to a more academically rigorous perspective that integrates mathematical proficiency and language practices crucial for educating bilingual students. The data presented in this article provide a window into what preparing bilingual pre-service teachers to teach mathematics might involve. I discuss issues that arise in the context of preparing Spanish-English bilingual pre-service teachers in a way that can inform their practice in linguistically and culturally diverse classrooms.

Lessons Learned From The Disorder Of Operations, Egan J. Chernoff

Journal of Humanistic Mathematics

The purpose of this article, in general, is to explore certain possible outcomes associated with an underaged gambler attempting to collect his rightful winnings. More specifically, this article is a thought experiment investigating the union of (1) skill testing questions, (2) the equation that recently broke/divided the internet, and (3) how different outcomes render different elements of the thought experiment moot. For example, when the final arbiter has total dominion over a particular outcome, the mathematics of a skill testing question is rendered moot. The article concludes with a discussion revealing how disorder of operations could be considered the teaching …

In Search Of Star Clusters: An Introduction To The K-Means Algorithm, Marcio Nascimento

Journal of Humanistic Mathematics

This article is a gentle introduction to K-means, a mathematical technique of processing data for further classification. We begin with a brief historical introduction, where we find connections with Plato’s Timæus, von Linné’s binomial classification, and the star clustering concept of Mary Sommerville and collaborators. Artificial intelligence algorithms use K-means as a classification methodology to learn about data in a very accurate way, because it is a quantitative procedure based on similarities.

Navigating A Calculus Course During A Pandemic: A Usma Perspective, Shane K. Smith, Tyson H. Walsh, Lee Evans

Journal of Humanistic Mathematics

In this article we analyze publications written about different teaching modalities and evaluate how each applies to a calculus class during the on-going COVID-19 pandemic. We focus on the positives and negatives of teaching and learning in a virtual, classroom, or HyFlex environment. Although arguments could be made for each environment, especially given different institutional objectives, this work aims to explain why we eventually preferred teaching our Fall 2020 multivariable calculus course in a face-to-face classroom setting at the United States Military Academy at West Point. We also offer measures of performance to compare the current COVID-19 semester with previous …

Happiness In Mathematics Education: The Experiences Of Preservice Elementary Teachers, Jeffrey Pair, Kent Dinh

Journal of Humanistic Mathematics

In this paper, we discuss preservice elementary teachers (PSTs) self-report of their happiness or unhappiness. Several times throughout a mathematics content capstone course, PSTs responded to prompts in which they described times from their past schooling experiences or during the course in which they experienced happiness or unhappiness in learning mathematics. Through thematic analysis, we examined their common experiences related to happiness and their mathematics learning. We found that PSTs’ happiness is related to expectations of themselves, their teachers, their peers, and mathematics itself. The study illuminates PST beliefs about mathematics teaching, collaborative group work, and the nature of mathematical …

Middle School Students Generating Mathematical Problems From A Real-Life Situation, David Coffland, Ying Xie

Journal of Humanistic Mathematics

In this study, we examined the effect of different presentation formats of a realistic situation on students’ mathematical problem-posing behavior. We divided thirty-six middle school students into two groups, gave them a pretest, and then showed them a realistic, problem-posing situation in Artifact or Video format. We used Silver’s core dimensions of creativity, namely fluency, flexibility, and originality, to measure participants’ problem-posing activity. The results for the fluency measures showed that the Artifact group wrote more questions than the Video group but the same number of mathematics problems. The Video group posed problems in more mathematical domains than the Artifact …

Plane Figurate Number Proofs Without Words Explained With Pattern Blocks, Gunhan Caglayan

Journal of Humanistic Mathematics

This article focuses on an artistic interpretation of pattern block designs with primary focus on the connection between pattern blocks and plane figurate numbers. Through this interpretation, it tells the story behind a handful of proofs without words (PWWs) that are inspired by such pattern block designs.

There From The Beginning: The Women Of Los Alamos National Laboratory Supporting National And International Nuclear Security, Olga Martin, Laura Mcclellan, Octavio Ramos, Heather Quinn

International Journal of Nuclear Security

From the beginning of the Manhattan Project in the early 1940s, the women of what would become Los Alamos National Laboratory (LANL) worked in technical positions alongside their male counterparts, played a key role as computers, and worked in administrative jobs as secretaries, phone operators, bookkeepers, and on behalf of the U.S. Army in the Women’s Army Corps.

Throughout the history of the Laboratory, women experts at LANL helped establish and lead important national and international security programs, with careers in science, technology, engineering, and mathematics. Over time, the women of Los Alamos have come together under various Employee Resource …