Digital Commons Network^™

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 …

Toward Greater Reproducibility Of Undergraduate Behavioral Science Research, Bruce Evan Blaine

Mathematical and Computing Sciences Faculty/Staff Publications

Reproducibility crises have arisen in psychology and other behavioral sciences, spurring efforts to ensure research findings are credible and replicable. Although reforms are occurring at professional levels in terms of new publication parameters and open science initiatives, the credibility and reproducibility of undergraduate research deserves attention. Undergraduate behavioral science research projects that rely on small convenience samples of participants, overuse hypothesis testing for drawing meaning from data, and engage in opaque statistical computing are vulnerable to producing nonreproducible findings. These vulnerabilities are reviewed, and practical recommendations for improving the credibility and reproducibility of undergraduate behavioral science research are offered.

Historical Study Of The Relationship Between The Federal Funds Rate And The Inflation Rate, Aaron Wilkins

Undergraduate External Publications

It is believed that in order to control high inflation rates, the Federal Reserve Bank (“the Fed”) increases the federal funds rate and when the inflation rate gets low, the Fed takes the opposite approach. (The federal funds rate is a rate of interest that banks charge each other to lend funds and stay above the reserve requirement, set by the government.) This project examines the relationship between the federal funds rate and the inflation rate. Sixty-five years of historical inflation rates and federal funds rates were used as the basis for this exploration. Because a time lag between the …

Temporal Network Analysis Of Small Group Discourse, Bernard P. Ricca, Michelle Jordan

Mathematical and Computing Sciences Faculty/Staff Publications

The analysis of school-age children engaged in engineering projects has proceeded by examining the conversations that take place among those children. The analysis of classroom discourse often considers a conversational turn to be the unit of analysis. In this study, small-group conversations among students engaged in a robotics project are analyzed by forming a dynamic network with the students as nodes and the utterances of each turn as edges. The data collected for this project contained more than 1000 turns for each group, with each group consisting of 4 students (and the occasional inclusion of a teacher or other interloper). …

State Space Analysis And Its Connection To The Classroom, Bernard P. Ricca, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

Discrete dynamical systems have been used to theoretically model the complex dynamics of classrooms. While time-series analyses of these models has yielded some insights, state space analyses can yield additional insights; this paper will explore state space analyses and their application to classroom situations. One benefit of state space analysis is that it allows simultaneous exploration of multiple time-series, and so can more easily provide information about divergence and convergence of paths. Additionally, state space analysis, more easily than time-series analysis, can provide information about the existence of multiple paths leading toward a desired state. Further, state space analysis can …

Graph Theoretic Methods For The Analysis Of Data In Developing Systems, Kris H. Green, Bernard P. Ricca

Mathematical and Computing Sciences Faculty/Staff Publications

A full examination of learning or developing systems requires data analysis approaches beyond the commonplace pre-/post-testing. Drawing on graph theory, three particular approaches to the analysis of data—based on adjacency matrices, affiliation networks, and edit distances—can provide additional insight into data; these methods are applied to student performance in a Calculus course. Data analysis methods based on adjacency matrices demonstrate that learning is not unidimensional, that learning progressions do not always progress monotonically toward desired understandings and also provide insight into the connection between instruction and student learning. The use of affiliation networks supports the concept development theory of Lev …

Asymptotic Spectral Properties Of The Schrödinger Operator With Thue-Morse Potential, William Clark, Rachael Kline, Michaela Stone

Undergraduate External Publications

The open problem

Study the Thue-Morse trace map; in particular, find the asymptotics of the Hausdorff dimension of the spectrum as the coupling constant tends to zero or infinity.

Our research

• We studied the dynamics of the Thue-Morse trace map.

• We developed MATLAB code to help approximate the box-counting dimension, thickness, and Hausdorff measure of the Schrödinger operator with Thue-Morse potential.

• We analysed the resulting data, which confirmed conjectures about the asymptotic behavior of the fractal dimensions of the operator. Asymptotic Spectral Properties of the Schr¨odinger Operator with Thue-

What's In A Name? The Matrix As An Introduction To Mathematics, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

In my classes on the nature of scientific thought, I have often used the movie The Matrix (1999) to illustrate how evidence shapes the reality we perceive (or think we perceive). As a mathematician and self-confessed science fiction fan, I usually field questions related to the movie whenever the subject of linear algebra arises, since this field is the study of matrices and their properties. So it is natural to ask, why does the movie title reference a mathematical object?

Of course, there are many possible explanations for this, each of which probably contributed a little to the naming decision. …

Coming Out Of The Dungeon: Mathematics And Role-Playing Games, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

After hiding it for many years, I have a confession to make.

Throughout middle school and high school my friends and I would gather almost every weekend, spending hours using numbers, probability, and optimization to build models that we could use to simulate almost anything.

That’s right. My big secret is simple. I was a high school mathematical modeler.

Of course, our weekend mathematical models didn’t bear any direct relationship to the models we explored in our mathematics and science classes. You would probably not even recognize our regular gatherings as mathematical exercises. If you looked into the room, you’d …

Matching Functions And Graphs At Multiple Levels Of Bloom’S Revised Taxonomy, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

This paper illustrates the power of Bloom's revised taxonomy for teaching, learning and assessing [3] in aligning our curriculum expectations and our assessment tools in multivariable calculus. The particular assessment tool considered involves a common matching problem to evaluate students' abilities to think about functions from graphical and formulaic representations. Through this analysis we gain additional understanding of why students may have difficulty in performing well on certain activities.

If Mowat And Davis Are Correct, Then Teaching Is Hard: A Response To Elizabeth Mowat & Brent Davis, Kris H. Green, Bernard P. Ricca

Mathematical and Computing Sciences Faculty/Staff Publications

In lieu of an abstract, below is the article's first paragraph.

Mowat & Davis (this issue) present a model of learning mathematics that relies heavily on ideas from network (or graph) theory. The important questions (to us, at least) concern the dynamics of the nodes and links. Answers – even tentative ones such as we present here – to these questions lead to a second set of questions concerning the implications of these answers to teachers and researchers.

Using Spreadsheets To Discover Meaning For Parameters In Nonlinear Models, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

Using spreadsheets one can develop an exploratory environment where mathematics students can develop their own understanding of the relationship between the parameters of commonly encountered families of functions (linear, logarithmic, exponential and power) and a natural interpretation of “rate of change” for those functions. The key to this understanding involves expanding the concept of rate of change to include percent changes. Through the use of the spreadsheet model, students can explore and easily determine which type of change is most natural for a given family of functions. This, in turn, provides a mechanism for interpreting the parameters of the function …

What's In A Name? The Matrix As An Introduction To Mathematics, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

In lieu of an abstract, here is the article's first paragraph:

In my classes on the nature of scientific thought, I have often used the movie The Matrix to illustrate the nature of evidence and how it shapes the reality we perceive (or think we perceive). As a mathematician, I usually field questions related to the movie whenever the subject of linear algebra arises, since this field is the study of matrices and their properties. So it is natural to ask, why does the movie title reference a mathematical object?

Reorganizing Freshman Business Mathematics Ii: Authentic Assessment In Mathematics Through Professional Memos, Kris H. Green, W. Allen Emerson

Mathematical and Computing Sciences Faculty/Staff Publications

Part I of this paper described the development of a new Freshman Business Mathematics (FBM) course at our college. In this second part of the paper, we discuss our assessment tool, the business memo, as a venue for students to apply mathematical skills, via mathematical modeling, to realistic business problems. These memos have proven a crucial step in turning our FBM course around from a dreaded course with little connection to students’ intended careers into a course where students experience the power of mathematics for solving problems and informing decisions. Comments from students in the course throughout its six-year history …

Promoting Mathematical Communication And Community Via Blackboard, Kris H. Green, Erica L. Johnson

Mathematical and Computing Sciences Faculty/Staff Publications

Major changes in mathematics pedagogy include writing as pedagogy and the role of community in learning. The classroom community is naturally extended by the use of online discussion boards. In this paper several models for student use of online discussion boards that have been successfully used to promote mathematical discourse are presented. Structured and unstructured examples that are easily adaptable and transportable to a variety of mathematics classroom settings are offered. These assignments facilitate student engagement and interaction outside of the classroom. Assessment, utility, and transferability are offered. Although the authors use the discussion boards provided by Blackboard, this particular …

A New Framework For Grading, Kris H. Green, W. Allen Emerson

Mathematical and Computing Sciences Faculty/Staff Publications

Grading is one of the least liked, least understood and least considered aspects of teaching. After years of work, we have developed a grading system that is quite different from traditional and reformed approaches to grading and which meaningfully incorporates and integrates the collection of evidence, the evaluation of evidence, and the reporting of judgments about that evidence. This system satisfies the requirements of good grading system and answers many of the problems faced by more traditional methods by substantially changing the way in which grade information is aggregated, resulting in a final course grade that aligns qualitative evaluation with …

A Solution To Einstein’S Field Equations For A Tachyonic Gas: Possible Astrophysical Applications, Kris H. Green, W. John Cocke

Mathematical and Computing Sciences Faculty/Staff Publications

In this paper we show that a change in the signs of some of the metric components of the solution of the field equations for the classical cosmic string results in a solution which we interpret as a time-dependent wall composed of tachyons. We show that the walls have the property of focusing the paths of particles which pass through them. As an illustration of this focusing, we demonstrate the results of a simple simulation of the interaction between one such tachyon wall and a rotating disk of point masses. This interaction leads to the temporary formation of spiral structures. …

Creating Successful Calculus Writing Assignments, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

I discuss three different writing assignments that I have used in my calculus courses. These assignments are introduced with a discussion of purpose and audience. Defining these qualities of an assignment will ensure that your writing assignments are more successful. The assignments discussed and explored here represent three different purposes: personal, informational and a blend of the two. The audiences for these assignments are diverse and force the students to incorporate particular modes of writing that demonstrate much of their thinking. Assessment of student learning as a result of these assignments is discussed. A fourth writing assignment is developed from …