Mathematics Commons^™

Got Books? A Story Of Creative Mathematics Children's Book Writing, Maria G. Fung, Pamela Hollander

Journal of Humanistic Mathematics

In this report, we describe a joint project between a mathematics content course and literacy methods course for pre-service elementary teachers that resulted in our students writing children's mathematically themed books. Out report then focuses first on the creative process of combining mathematical knowledge and ideas for teaching with the craft of writing for children, and second on the creative connection between mathematics and literacy education at the elementary school level.

What Would The Nautilus Say? Unleashing Creativity In Mathematics!, Megan E. Selbach-Allen, Cathy A. Williams, Jo Boaler

Journal of Humanistic Mathematics

While the nautilus shell is often represented in popular culture as an example of a golden spiral, according to many mathematicians it is not. In this paper we examine multiple arguments for and against considering the nautilus as a “golden” spiral and offer a semi-structured task that is accessible to middle school students and beyond to begin their own investigation. Our hope is that asking, what would the nautilus say, can serve as a starting point for children and adults alike to push against the walls we so often draw around mathematical questions and begin to see where their creativity …

Tactivities: Fostering Creativity Through Tactile Learning Activities, Angie Hodge-Zickerman, Eric Stade, Cindy S. York, Janice Rech

Journal of Humanistic Mathematics

As mathematics teachers, we hope our students will approach problems with a spirit of creativity. One way to both model and encourage this spirit – and, at the same time, to keep ourselves from getting bored – is through creative approaches to problem design. In this paper, we discuss ``TACTivities'' – mathematical activities with a tactile component – as a creative outlet for those of us who teach mathematics, and as a resource for stimulating creative thinking in our students. We use examples, such as our ``derivative fridge magnets'' TACTivity, to illustrate the main ideas. We emphasize that TACTivities can …

Going Beyond Promoting: Preparing Students To Creatively Solve Future Problems, Kristin M. Arney, Kayla K. Blyman, Jennifer D. Cepeda, Scott A. Lynch, Michael J. Prokos, Scott Warnke

Journal of Humanistic Mathematics

While we cannot know what problems the future will bring, we can be almost certain that solving them will require creativity. In this article we describe how our course, a first-year undergraduate mathematics course, supports creative problem solving. Creative problem solving cannot be learned through a single experience, so we provide our students with a blend of experiences. We discuss how the course structure enables creative problem solving through class instruction, during class activities, during out of class assessments, and during in class assessments. We believe this course structure increases student comfort with solving open-ended and ill-defined problems similar to …

Fostering Student Discovery And Conjecture In Multivariable Calculus, Aaron Wangberg

Journal of Humanistic Mathematics

Who owns the mathematical ideas in the undergraduate classroom? A traditional mathematics classroom and curriculum imposes several barriers that prevent students from discovering and engaging with mathematical concepts. Definitions, notations, and theorems require mastery before students can work meaningfully with the underlying mathematical concepts. Raising Calculus to the Surface utilizes a different approach by providing students multiple entry points to engage meaningfully with mathematics ideas and allows students to promote meaningful ideas and conjectures into the classroom discourse to formalize their explorations. In this paper, we describe several characteristics built into the project materials, including a rubric designed to encourage …

Inspiring Mathematical Creativity Through Juggling, Ceire Monahan, Mika Munakata, Ashwin Vaidya, Sean Gandini

Journal of Humanistic Mathematics

The goal of the Creativity in Mathematics and Science project, funded by the National Science Foundation’s [NSF’s] Improving Undergraduate STEM Educa- tion program, is to reconsider how we teach mathematics at the collegiate level. Over the last three years, we have developed interdisciplinary modules that seek to encourage students, including non-STEM majors, to see mathematics in unexpected places, make connections to their own interests and disciplines, and explore creativity in mathematics. Relying on traits of creativity such as the ability to connect ideas, be inquisitive, question norms, and have flexibility [1], we encouraged students to participate and understand mathematics in …

Innovative Induction And Mathematical Code Switching, Benjamin Dickman, Erik Nauman

Journal of Humanistic Mathematics

In the first part of this paper, we provide an example of a project designed to foster mathematical creativity among students at an independent, all-girls school in the Northeast United States. The mathematical motivator for the project is a polyomino proof by induction first formulated by Solomon Golomb. We explain how the project has been implemented over the past two years at the school’s Innovation Lab in collaborative work between a mathematics instructor and an educational technologist, provide instructions and background information to facilitate the implementation of this project at other learning sites, and show examples of student work along …

Virtual Temari: Artistically Inspired Mathematics, Carl Giuffre, Lee Stemkoski

Journal of Humanistic Mathematics

Technology can be a significant aide in understanding and appreciating geometry, beyond theoretical considerations. Both fiber art and technology have been employed as a significant aide and an inspiring vessel in education to explore geometry. The Japanese craft known as temari, or "hand-balls", combines important artistic, spiritual, and familial values, and provides one such approach to exploring geometry. Mathematically, the artwork of temari may be classified based on whether they are inspired by polyhedra and discrete patterns or by periodic functional curves. The resulting designs of these categories provide an ancient vantage for displaying spherical patterns. We illustrate a …

The Surname Impossibility Theorem, Adam Graham-Squire

Journal of Humanistic Mathematics

The Surname Impossibility Theorem offers solace to anyone who has struggled in the quagmire of choosing a surname for a child. I posit that it is impossible to find a method for giving a child a surname that satisfies the important criteria of being traditional, aesthetically pleasing, ancestor-respecting, non-sexist, gender-neutral and non-heterosexist. My mathematical approach defines what those criteria would mean and analyzes different naming systems to conclude that no method could satisfy all criteria. In the same way that Arrow's Impossibility Theorem proved that no voting method can satisfy all criteria for a fair election, I prove the impossibility …

The Emergence Of Creativity: Insights From Carnatic Raaga Improvisation And Mathematical Proof Generation, Srividhya Balaji, Sean Chorney

Journal of Humanistic Mathematics

Creativity is a broad phenomenon that scholars have interpreted in a multitude of ways. We notice that a majority of the views describe creativity as something innate. This paper aims to verge from this perspective and explore creativity in terms of the constant mutual interaction of a person and their environment. Using the theoretical framework, enactivism, and the notion of emergence, we investigate the creative processes involved in musical improvisations of south Indian classical or Carnatic music and mathematical proof generation. Interview excerpts from professional Carnatic musicians and research mathematicians on their respective creative processes during musical improvisation and proof …

Does Your Course Effectively Promote Creativity? Introducing The Mathematical Problem Solving Creativity Rubric, Kayla K. Blyman, Kristin M. Arney, Bryan Adams, Tara A. Hudson

Journal of Humanistic Mathematics

As believers in the power of blending the creative with the quantitative, we design our courses with an eye towards developing creative problem solvers. However, when it comes time to evaluate our course's success in developing creative problem solvers we come away with a plethora of qualitative evidence and yet we are left hungry for the quantitative evidence we desire as mathematicians.

In this article we describe the development of the Mathematical Problem Solving Creativity Rubric and its pilot use in a freshman-level Mathematical Modeling and Introduction to Calculus course at the United States Military Academy. We not only come …

A Study Of Problem Posing As A Means To Help Mathematics Teachers Foster Creativity, Deborah Moore-Russo, Amanda A. Simmons, Michael J.D. Tulino

Journal of Humanistic Mathematics

Research suggests that mathematical creativity often results from extended periods of mathematical activity and reflection based on the use of deep and flexible content knowledge [14, 15]. This implies that instruction can influence creativity. However, for teaching to foster creativity in mathematics, there should be purposefully designed instructional tasks. It is doubtful that routine, mechanical exercises would foster creativity. Moreover, mathematical creativity may neither be explicitly promoted, nor fully appreciated, by students when a learning space involves only problem solving, even if the problems are challenging and engaging. For students to get an authentic sense of mathematics and to develop …

Fostering Mathematical Creativity While Impacting Beliefs And Anxiety In Mathematics, James M. Fetterly

Journal of Humanistic Mathematics

This quantitative study examined the notion of mathematical creativity and its relationship to epistemological beliefs of the nature of mathematics and mathematical anxiety. A counterbalanced design was employed, randomizing a class of elementary pre-service teachers into two groups and giving a pre- and post-test to determine if significant differences exist in the participants who are exposed to problem posing, divergent thought, and invented strategies, that is, a punctuated, intentional experience with mathematical creativity. This difference in mathematical anxiety, beliefs, and creativity was also gauged using repeated measures during the study. Furthermore, beliefs and anxiety were correlated with mathematical creativity employing …

Everyday Examples In Linear Algebra: Individual And Collective Creativity, Aditya P. Adiredja, Michelle Zandieh

Journal of Humanistic Mathematics

This paper investigates creativity in students’ constructions of everyday examples about basis in Linear Algebra. We analyze semi-structured interview data with 18 students from the U.S. and Germany with diverse academic and social backgrounds. Our analysis of creativity in students’ everyday examples is organized into two parts. First, we analyze the range of students’ creative products by investigating the mathematical variability in the more commonly mentioned examples. Second, we unpack some of the collective processes in the construction of students’ examples. We examine how creativity was distributed through the interactions among the student, the interviewers, and other artifacts and ideas. …

Special Issue -- Creativity In Mathematics: Foreword, Emily Cilli-Turner, Houssein El Turkey, Gulden Karakok, Milos Savic, Gail Tang

Journal of Humanistic Mathematics

No abstract provided.

A World Of Creativity, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Integrating Data Science Ethics Into An Undergraduate Major, Benjamin Baumer, Randi L. Garcia, Albert Y. Kim, Katherine M. Kinnaird, Miles Q. Ott

Statistical and Data Sciences: Faculty Publications

We present a programmatic approach to incorporating ethics into an undergraduate major in statistical and data sciences. We discuss departmental-level initiatives designed to meet the National Academy of Sciences recommendation for weaving ethics into the curriculum from top-to-bottom as our majors progress from our introductory courses to our senior capstone course, as well as from side-to-side through co-curricular programming. We also provide six examples of data science ethics modules used in five different courses at our liberal arts college, each focusing on a different ethical consideration. The modules are designed to be portable such that they can be flexibly incorporated …

De Branges Spaces On Compact Riemann Surfaces And A Beurling-Lax Type Theorem, Daniel Alpay, Ariel Pinhas, Victor Vinnikov

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the notion of commutative operator vessels, this work investigates de Branges-Rovnyak spaces whose elements are sections of a line bundle of multiplicative half-order differentials on a compact real Riemann surface. As a special case, we obtain a Beurling-Lax type theorem in the setting of the corresponding Hardy space on a finite bordered Riemann surface.

Uniform Random Variate Generation With The Linear Congruential Method, Joseph Free

PANDION: The Osprey Journal of Research and Ideas

This report considers the issue of using a specific linear congruential generator (LCG) to create random variates from the uniform(0,1) distribution. The LCG is used to generate multiple samples of pseudo-random numbers and statistical computation techniques are used to assess whether those samples could have resulted from a uniform(0,1) distribution. Source code is included with this report in the appendix along with annotations.

Joint Lattice Of Reconstructability Analysis And Bayesian Network General Graphs, Marcus Harris, Martin Zwick

Complex Systems Faculty Publications and Presentations

This paper integrates the structures considered in Reconstructability Analysis (RA) and those considered in Bayesian Networks (BN) into a joint lattice of probabilistic graphical models. This integration and associated lattice visualizations are done in this paper for four variables, but the approach can easily be expanded to more variables. The work builds on the RA work of Klir (1985), Krippendorff (1986), and Zwick (2001), and the BN work of Pearl (1985, 1987, 1988, 2000), Verma (1990), Heckerman (1994), Chickering (1995), Andersson (1997), and others. The RA four variable lattice and the BN four variable lattice partially overlap: there are ten …

Reconstructability Analysis And Its Occam Implementation, Martin Zwick

Complex Systems Faculty Publications and Presentations

This talk will describe Reconstructability Analysis (RA), a probabilistic graphical modeling methodology deriving from the 1960s work of Ross Ashby and developed in the systems community in the 1980s and afterwards. RA, based on information theory and graph theory, resembles and partially overlaps Bayesian networks (BN) and log-linear techniques, but also has some unique capabilities. (A paper explaining the relationship between RA and BN will be given in this special session.) RA is designed for exploratory modeling although it can also be used for confirmatory hypothesis testing. In RA modeling, one either predicts some DV from a set of IVs …

Decomposition Of Certain Complete Graphs And Complete Multipartite Graphs Into Almost-Bipartite Graphs And Bipartite Graphs, G. Sethuraman, M. Sujasree

Theory & Applications of Graphs

In his classical paper [14], Rosa introduced a hierarchical series of labelings called ρ, σ, β and α labeling as a tool to settle Ringel’s Conjecture which states that if T is any tree with m edges then the complete graph K2m+1 can be decomposed into 2m + 1 copies of T . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decomposition using graph labeling. In this direction, in 2004, Blinco et al. [6] introduced γ-labeling as a stronger version of ρ-labeling. A function g defined on the vertex set …

Image Dehazing From The Perspective Of Environmental Illumination., Sanchayan Santra Dr.

Doctoral Theses

Haze and fog are atmospheric phenomena where the particles suspended in the air obscure visibility by scattering the light propagating through the atmosphere. As a result only a part of the reflected light reaches the observer. So, the apparent intensity of the objects get reduced. Apart from that, the in-scatter of the atmospheric light creates a translucent veil, which is a common sight during haze. Image dehazing methods try to recover a haze-free version of a given image by removing the effects of haze.Although attempts have been made to accurately estimate the scene transmittance, the estimation of environmental illumination has …

Lattice Of Maximal-Primary Ideals In Quadratic Orders, Ryan Bridges

Mathematics & Statistics ETDs

An order is a subring of the ring of integers of an algebraic extension, Peruginelli and Zanardo classified the lattices of orders with prime index inside te ring of integers of quadratic extensions of the rational numbers. The lattices are quite striking and have different layered structure depending on whether the prime is inert, split, or ramified. This thesis considers the orders which have prime power index inside the Gaussian integers. This is a nice generalization of the work of Peruginelli and Zanardo, and succeeds in a few classifications of specific instances of orders derived from inert primes.

Maintaining Rich Dialogic Interactions In The Transition To Synchronous Online Learning, Hyunyi Jung, Corey Brady

Mathematical and Statistical Science Faculty Research and Publications

A central premise across a variety of educational research and policy documents is that students learn with greater understanding in classrooms where they engage in exploring, reasoning, and communicating about their thinking (Hiebert and Wearne, 1993; National Council of Teachers of English, 2016; National Council of Teachers of Mathematics, 2000). With the recent emergency transition to remote online instruction in higher education, opportunities for rich synchronous learning have been diminished in many courses. Most instructors have had to adapt rapidly from in-person classroom settings to online environments without sufficient time and training. Accordingly, college students have shared concerns about the …

Expected Resurgence Of Ideals Defining Gorenstein Rings, Eloísa Grifo, Craig Huneke, Vivek Mukundan

Department of Mathematics: Faculty Publications

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is noetherian.

Guided Reinvention As A Context For Investigating Students' Thinking About Mathematical Language And For Supporting Students In Gaining Fluency, Kristen Vroom

Dissertations and Theses

Fluency with mathematical language is important for students' engagement in many disciplinary practices such as defining, conjecturing, and proving; yet, there is growing evidence that mathematical language is challenging for undergraduate students. This dissertation study draws on two design experiments with pairs of students who were supported to encode their mathematical meanings with more formal language. I aimed to investigate the teaching and learning of mathematical language, and particularly the language in statements with multiple quantifiers, by engaging students in this type of activity. In the first paper, I investigated the complex ways in which the students in my study …

Exact And Strongly Exact Filters, M. A. Moshier, A. Pultr, A. L. Suarez

Mathematics, Physics, and Computer Science Faculty Articles and Research

A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame FiltE(L) somewhat surprisingly isomorphic to the frame of joins of closed sublocales. In this paper we present a characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters FiltsE(L).

On Pseudo-Spectral Factorization Over The Complex Numbers And Quaternions, Daniel Alpay, Fabrizio Colombo, Izchak Lewkowicz, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

This paper is a continuation of the research of our previous work[5] and considers quaternionic generalized Carathéodory functions and the related family of generalized positive functions. It is addressed to a wide audience which includes researchers in complex and hypercomplex analysis, in the theory of linear systems, but also electric engineers. For this reason it includes some results on generalized Carathéodory functions and their factorization in the classic complex case which might be of independent interest. An important new result is a pseudo-spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions.