Mathematics Commons^™

Investigating The Role Of The Written Curriculum On Lesson Planning For First-Year Elementary Mathematics Teachers, Anna Kron

Honors Program Theses

While studies have previously explored aspects of lesson planning from the perspective of experienced teachers, there is a lack of research investigating how this process is different for brand new educators. The researcher, a preservice elementary teacher studying at the University of Northern Iowa, wanted to learn more about the available resources and how she would utilize them when planning for mathematics instruction in her first elementary classroom. At the completion of this project, the researcher hoped to have a better understanding of the process first-year elementary teachers experience when transforming written textbook materials into a mathematics lesson plan. The …

Complexity Of Linear Summary Statistics, Micah G. Pedrick

HMC Senior Theses

Families of linear functionals on a vector space that are mapped to each other by a group of symmetries of the space have a significant amount of structure. This results in computational redundancies which can be used to make computing the entire family of functionals at once more efficient than applying each in turn. This thesis explores asymptotic complexity results for a few such families: contingency tables and unranked choice data. These are used to explore the framework of Radon transform diagrams, which promise to allow general theorems about linear summary statistics to be stated and proved.

Emergence And Complexity In Music, Zoe Tucker

HMC Senior Theses

How can we apply mathematical notions of complexity and emergence to music, and how can these mathematical ideas then inspire new musical works? Using Steve Reich's Clapping Music as a starting point, we look for emergent patterns in music by considering cases where a piece's complexity is significantly different from the total complexity of each of the individual parts. Definitions of complexity inspired by information theory, data compression, and musical practice are considered. We also consider the number of distinct musical pieces that could be composed in the same manner as Clapping Music. Finally, we present a new musical …

Platonismes, Marco Panza

MPP Published Research

Selon la vulgata philosophique, le platonisme concernant un certain domaine de recherche est la thèse affirmant que ce domaine concerne des objets qui lui sont propres, dont l’existence est indépendante de l’activité cognitive humaine. Souvent, dans la même vulgata on parle aussi de platonisme pour se référer à une thèse un peu différente, d’après laquelle ce qu’on dit concernant ce domaine est vrai ou faux indépendamment de toute justification ou réfutation que l’on puisse apporter. Naturellement, si parmi les énoncées ayant trait à ce demain, il y en a qu’on peut prendre comme particulièrement surs du fait d’en avoir une …

Bifurcation From Infinity And Multiplicity Of Solutions For Nonlinear Periodic Boundary Value Problems, Nsoki Mavinga, M. N. Nkashama

Mathematics & Statistics Faculty Works

We are concerned with multiplicity and bifurcation results for solutions of nonlinear second order differential equations with general linear part and periodic boundary conditions. We impose asymptotic conditions on the nonlinearity and let the parameter vary. We then proceed to establish a priori estimates and prove multiplicity results (for large-norm solutions) when the parameter belongs to a (nontrivial) continuum of real numbers. Our results extend and complement those in the literature. The proofs are based on degree theory, continuation methods, and bifurcation from infinity techniques.

Use Of Sex-Specific Clinical And Exercise Risk Scores To Identify Patients At Increased Risk For All-Cause Mortality, Paul C. Cremer, Yuping Wu, Haitham M. Ahmed, Lee M. Pierson, Danielle M. Brennan, Mouaz H. Al-Mallah, Clinton A. Brawner, Jonathan K. Ehrman, Steven J. Keteyian, Roger S. Blumenthal, Michael J. Blaha, Leslie Cho

Mathematics and Statistics Faculty Publications

Importance Risk assessment tools for exercise treadmill testing may have limited external validity. Cardiovascular mortality has decreased in recent decades, and women have been underrepresented in prior cohorts.

Objectives To determine whether exercise and clinical variables are associated with differential mortality outcomes in men and women and to assess whether sex-specific risk scores better estimate all-cause mortality.

Design, Setting, and Participants This retrospective cohort study included 59 877 patients seen at the Cleveland Clinic Foundation (CCF cohort) from January 1, 2000, through December 31, 2010, and 49 278 patients seen at the Henry Ford Hospital (FIT cohort) from January 1, …

Mind The Gap: An Initial Analysis Of The Transition Of A Second Level Curriculum Reform To Higher Education, Michael Carr, Fiona Faulkner, Cormac Breen, Mark Prendergast

Articles

This article details an initial analysis of the transition of a second level curriculum reform to higher education in Ireland. The reform entitled ‘Project Maths’ involved changes to what second level students learn in mathematics, how they learn it, and how they are assessed. Changes were rolled out nationally on a phased basis in September 2010. Students who were taught and assessed through the new curriculum first entered third level education in September 2012. It is important that third level mathematics lecturers are aware of the changes to the curriculum since certain topics such as vectors and matrices are no …

13: Polar Path, Ruth Dover

Mathematica Notebooks for Pre-Calculus and Calculus

PolarPath.nb allows the user to input any polar function and use a slider to see how the path is created. That is, it will allow the user to see the order in which the petals or loops are created.

08: Sine Transformations, Ruth Dover

Mathematica Notebooks for Pre-Calculus and Calculus

SineTransformations.nb allows you to use sliders to see the effects of parameters on the amplitude, period, phase shift, and vertical shift.

15: Koch Snowflakes, Ruth Dover

Mathematica Notebooks for Pre-Calculus and Calculus

KochSnowflakes.nb contains several animations that show several steps of graphical iteration for Koch Snowflakes and several other patterns.

04: Domain And Range, Ruth Dover

Mathematica Notebooks for Pre-Calculus and Calculus

DomainAndRange.nb allows you to graph various types of functions and shows the domain and/or range along the appropriate axis.

08: Accumulator, Ruth Dover

Mathematica Notebooks for Pre-Calculus and Calculus

Accumulator.nb takes a function ƒ and other information to sketch both the graph of ƒ and the graph of F(x) = Integral of ƒ(t) from a to x. Use the slider for the x-value to see how the accumulated area under ƒ helps to create the graph of F.

11: Parametric Path, Ruth Dover

Mathematica Notebooks for Pre-Calculus and Calculus

ParametricPath.nb allows the user to input parametrically defined curves and a domain for the parameter t. An example is given to show how the curve is traced out.

10: Slope Fields + Solns, Ruth Dover

Mathematica Notebooks for Pre-Calculus and Calculus

SlopeFields+Solns.nb follows similarly from the previous notebook. Here, however, there is a 2D slider that shows a specific solution function as you move the initial point.

A Commentary On Freudenthal’S Didactic Phenomenology Of The Mathematical Structures Associated With The Notion Of Measurement, Omar Hernandez Rodriguez, Jorge Lopez Fernandez

The Mathematics Enthusiast

This paper discusses Freudenthal's didactical phenomenology for the mathematical structures related to measurement. Freudenthal starts with the set G of all objects having the attribute of weight, an attribute that is to be measured. He proposes two operations, the first one among the measurements themselves, called “addition". Addition is interpreted in terms of the mental actions associated with measurement and, in the case of weights, it consists on the process of placing weights on one of the two dishes of a balance in order to balance them with a predetermined gauge, its fractions or multiples thereof, which are placed on …

The Historical Connection Of Fourier Analysis To Music, Shunteal Jessop

The Mathematics Enthusiast

This paper will discuss the relevance between mathematics and music throughout a few periods of history. The paper will first discuss how the Ancient Chinese hired mathematicians in order to “perfect the music” used in the court rooms. Mathematics was typically used in music to develop ratios and intervals that are found in music. This paper will then discuss the history of Fourier analysis, as well as give a brief history of Jean Baptiste Fourier. The Fourier analysis was used to find naturally occurring harmonics, to model sound, and to define sound by breaking it up into pieces. Many examples …

Editorial: All Good Things Come In Three, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Linking Pre-Service Teachers’ Questioning And Students’ Strategies In Solving Contextual Problems: A Case Study In Indonesia And The Netherlands, Rahmah Johar, Sitti Maesuri Patahuddin, Wanty Widjaja

The Mathematics Enthusiast

This study examined the relationship between teachers’ questioning techniques and students’ strategies in solving contextual mathematical problems. This case study was undertaken with one pre-service teacher (and 22 Year 4 students) from Indonesia and one pre-service teacher (and 25 Year 4 students) in the Netherlands. Both pre-service teachers assigned the same problems to their students and these problems were novel for the students in both countries. The lessons were observed by the first author and video recorded for data analysis. Qualitative data analysis was undertaken through within-case and cross-case analysis. The findings suggest that the contextual problems, the way pre-service …

Subtraction Involving Negative Numbers: Connecting To Whole Number Reasoning, Laura Bofferding, Nicole Wessman-Enzinger

The Mathematics Enthusiast

In this article, we explore how students attempt to bridge from their whole number reasoning to integer reasoning as they solve subtraction problems involving negative numbers. Based on interviews with students ranging from first graders to preservice teachers, we identify two overarching strategies: making connections to known problem types and leveraging conceptions of subtraction. Their initial connections suggest that rather than identifying the best instructional models to teach integer concepts, we should focus on identifying integer instructional models that build on the potentially productive connections that students’ already make; we propose an example of one such form of instruction.

Students’ Reactions To Reform Mathematics Pedagogy In A Postsecondary Remedial Mathematics Course, Luke Smith, W. Gary Martin, Anna Wan, Gilbert Duenas

The Mathematics Enthusiast

The students in this study were enrolled in a remedial mathematics course at a small 4-year university and were taught according to the reform pedagogical principles advocated by NCTM, AMATYC, and MAA. Since most of the students had not been previously exposed to these teaching methods, this study obtained students’ reactions (n = 22) to the course through an anonymous, free-response (not multiple choice) survey at the end of the course. Surveys from students in two equivalent “traditional” lecture courses (n = 44) were also analyzed and served as a baseline by which to gauge students’ responses from the reform …

On Mathematics With Distinction, A Learner-Centered Conceptualization Of Challenge And Choice-Based Pedagogies, Boris Koichu

The Mathematics Enthusiast

The main argument of this article is that “challenging mathematics for all” can be more than just a nice slogan, on condition that all students are empowered to make informed choices of: a challenge to be dealt with, a way of dealing with the challenge, a mode of interaction, an extent of collaboration, and an agent to learn from. Pedagogies supporting such choices are called choice-based pedagogies. The article begins from a theoretical discussion of the relationships between the notions ‘mathematics with distinction,’ ‘giftedness’, ‘challenge’ and ‘choice’. As a result, the learner-centered conceptualization of mathematical challenge is proposed. Then two …

A 21st Century Economic, Educational And Ethical Mathematics Curriculum Policy, Glen S. Aikenhead

The Mathematics Enthusiast

Reformulating a mathematics curriculum is a political act, contextualized by economic, educational and ethical considerations, among others. This article represents a political rendition of a rationale to update the Saskatchewan, Canada, curriculum so it harmonizes with a cultural understanding of school mathematics (Ernest, 1988). By revisiting the 19th century definition of school mathematics that we generally follow today, its economic, educational and ethical consequences seem contrary to the expectations of 21st century Saskatchewan.The article’s point form organization, uninterrupted by the appearance of references, conforms to the genre of submissions to government committees. However, footnotes will clarify information for readers unfamiliar …

Long And Short-Range Air Navigation On Spherical Earth, Nihad E. Daidzic

International Journal of Aviation, Aeronautics, and Aerospace

Global range air navigation implies non-stop flight between any two airports on Earth. Such effort would require airplanes with the operational air range of at least 12,500 NM which is about 40-60% longer than anything existing in commercial air transport today. Air transportation economy requires flying shortest distance, which in the case of spherical Earth are Orthodrome arcs. Rhumb-line navigation has little practical use in long-range flights, but has been presented for historical reasons and for comparison. Database of about 50 major international airports from every corner of the world has been designed and used in testing and route validation. …

College Algebra Through Problem Solving, Danielle Cifone, Karan Puri, Debra Maslanko, Ewa Dabkowska

Open Educational Resources

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

Positive Solutions Of A Singular Fractional Boundary Value Problem With A Fractional Boundary Condition, Jeffrey W. Lyons, Jeffrey T. Neugebauer

EKU Faculty and Staff Scholarship

For \(\alpha\in(1,2]\), the singular fractional boundary value problem \[D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,\] satisfying the boundary conditions \(x(0)=D^{\beta}_{0^+}x(1)=0\), where \(\beta\in(0,\alpha-1]\), \(\mu\in(0,\alpha-1]\), and \(D^{\alpha}_{0^+}\), \(D^{\beta}_{0^+}\) and \(D^{\mu}_{0^+}\) are Riemann-Liouville derivatives of order \(\alpha\), \(\beta\) and \(\mu\) respectively, is considered. Here \(f\) satisfies a local Carathéodory condition, and \(f(t,x,y)\) may be singular at the value 0 in its space variable \(x\). Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.

Forcing Optimality And Brandt's Principle, Domenico Napoletani, Marco Panza, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Books and Book Chapters

We argue that many optimization methods can be viewed as representatives of “forcing”, a methodological approach that attempts to bridge the gap between data and mathematics on the basis of an a priori trust in the power of a mathematical technique, even when detailed, credible models of a phenomenon are lacking or do not justify the use of this technique. In particular, we show that forcing is implied in particle swarms optimization methods, and in modeling image processing problems through optimization. From these considerations, we extrapolate a principle for general data analysis methods, what we call ‘Brandt’s principle’, namely the …

Artin Presentations And Closed 4-Manifolds, Jun Li

Honors Papers

In this paper we focus on the study of smooth closed simply-connected 4-manifolds. In particular, we study such manifolds arising from Artin presentations. Artin pre- sentations are intimately related to the pure braid group, and they characterize the fundamental groups of closed orientable 3-manifolds. Each Artin presentation r gives rise to a 4-manifold W4(r), whose boundary is a closed orientable 3-manifold M3(r) with fundamental group presented by r. In case the 4-manifold has bound- ary S3, we may close up W 4(r) by attaching a 4-handle to obtain W 4(r) ¿S3 D4. This closed 4-manifold has quadratic form represented by …

Average Shortest Path Length In A Novel Small-World Network, Andrea J. Allen

Honors Papers

We study a novel model of random graph which exhibits the structural characteristics of the Watts- Strogatz small-world network. The small-world network is characterized by a high level of local clustering while also having a relatively small graph diameter. The same behavior that makes the Watts-Strogatz model behave like this also makes it difficult to analyze. Our model addresses this issue, closely mimicking the same structure experimentally while following a constructive process that makes it easier to analyze mathematically. We present a bound on the average shortest path length in our new model, which we approach by looking at the …

Predicting Risk Of Adverse Outcomes In Knee Replacement Surgery With Reconstructability Analysis, Cecily Corrine Froemke, Martin Zwick

Complex Systems Faculty Publications and Presentations

Reconstructability Analysis (RA) is a data mining method that searches for relations in data, especially non-linear and higher order relations. This study shows that RA can provide useful predictions of complications in knee replacement surgery.

Evolution Equations With A Parameter And Application To Transport-Convection Differential Equations, Emile Franc Doungmo Goufo

Turkish Journal of Mathematics

We deeply investigate the well-posedness of models taking the form $_0^AD^{\beta }_tu(t) = Au(t),\;\; u(0)= \,f,\;\;\;00$ where $_0^AD^{\beta }_t$ is a derivative with the fractional parameter $\beta$ and $A$ is a closed densely defined operator in a Banach space. We show that, unlike other systems, solutions of our models are not governed by Mittag--Leffler functions and their variants. We extend and adapt Peano's idea to our models and establish conditions for existence and uniqueness of solutions. In particular, relations between the two-parameter solution operator, its resolvent, and its generator are provided; the issue of subordination and prolongation principles are addressed; …