Science and Mathematics Education Commons^™

Color Models As Tools In Teaching Mathematics, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

In this paper we discuss various situations how color models and patterns can be used to simplify the study of abstract mathematics and serve as tools in understanding mathematical ideas. We illustrate the realization of such models through the development of advanced computer technology. In particular, we present how a computer algebra software such as Mathematica, or a dynamic geometry environment, can be utilized to facilitate the study of transformation geometry and group theory.

Using The Taylor Center To Teach Odes, Alexander Gofen

CODEE Journal

This article introduces a powerful ODE solver called the Taylor Center for PCs (http://taylorcenter.org/Gofen/) as a tool for teaching and performing numeric experiments with ODEs. The Taylor Center is an All-in-One GUI-style application for integrating ODEs by applying the modern Taylor Method (Automatic Differentiation). The Taylor Center also offers dynamic graphics (including 3D stereo vision). After a brief review of the features of the Taylor Center, we consider instructive examples of ODEs in various applications and also several particular examples illustrating intricacies of numeric integration. The article therefore continues the thesis of Borrelli and Coleman (CODEE Journal, http://www.codee.org/ref/CJ09-0157) that awareness …

The Awards Project: Promoting Good Practices In Award Selection, Betty Mayfield, Francis Su

All HMC Faculty Publications and Research

Every year the MAA honors many members of our community with a wide variety of prizes, awards, and certificates for excellence in teaching, writing, scholarship, and service (see maa.org/awards). The winners exemplify our ideals as an association; consequently, they are often viewed as role models and leaders. So it is important to ask: Do these awards, as a whole, reflect the outstanding contributions of the breadth of association membership?

Prime Ideals In Two-Dimensional Noetherian Domains And Fiber Products And Connected Sums, Ela Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis concerns three topics in commutative algebra:

1) The projective line over the integers (Chapter 2),

2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3),

3) Fiber products and connected sums of local rings (Chapter 4),

In the first chapter we introduce basic terminology used in this thesis for all three topics.

In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj(Z[h,k]) over the integers Z, and we interpret this poset as Spec(Z[x]) U Spec(Z[1/x]) with an appropriate identification. …

An Analysis Of Nonlocal Boundary Value Problems Of Fractional And Integer Order, Christopher Steven Goodrich

Department of Mathematics: Dissertations, Theses, and Student Research

In this work we provide an analysis of both fractional- and integer-order boundary value problems, certain of which contain explicit nonlocal terms. In the discrete fractional case we consider several different types of boundary value problems including the well-known right-focal problem. Attendant to our analysis of discrete fractional boundary value problems, we also provide an analysis of the continuity properties of solutions to discrete fractional initial value problems. Finally, we conclude by providing new techniques for analyzing integer-order nonlocal boundary value problems.

Adviser: Lynn Erbe and Allan Peterson

Commutative Rings Graded By Abelian Groups, Brian P. Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

Rings graded by Z and Z^d play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the …

Characterizing And Supporting Change In Algebra Students' Representational Fluency In A Cas/Paper-And-Pencil Environment, Nicole L. Fonger

Dissertations

Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students’ RF in learning environments that utilize both computer algebra systems (CAS) and paper-and-pencil. The purpose of this research was to: (a) characterize change in ninth-grade algebra students’ RF in solving problems involving linear equations, and (b) determine conditions of a CAS and paper-and-pencil learning environment in which those students changed their RF.

Change in RF was measured by comparing results from initial to …

Joanne Growney's Poetry-With-Mathematics Blog -- An Appreciation, Gregory E. Coxson

Journal of Humanistic Mathematics

Now is a good time to work on the boundaries of practice and theory, of art and science. We are seeing a rising tide of interest in these boundaries. Witness the growing Bridges movement, which has been exploring the connections between mathematics and the arts. Similarly, JoAnne Growney's blog, Intersections -- Poetry with Mathematics, explores the connections between mathematics and poetry. Through this review, I aim to give readers a taste of what can be found in Intersections as a way of encouraging others, be they mathematicians, poets, or neither, to visit the blog.

Communicating Quantitative Literacy: An Examination Of Open-Ended Assessment Items In Timss, Nals, Ials, And Pisa, Karl Kosko, Jesse Wilkins

Karl W Kosko

Quantitative Literacy (QL) has been described as the skill set an individual uses when interacting with the world in a quantitative manner. A necessary component of this interaction is communication. To this end, assessments of QL have included open-ended items as a means of including communicative aspects of QL. The present study sought to examine whether such open-ended items typically measured aspects of quantitative communication, as compared to mathematical communication, or mathematical skills. We focused on public-released items and rubrics from four of the most widely referenced assessments: the Third International Mathematics and Science Study (TIMSS-95): the National Adult Literacy …

The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

A linear extension of a partially ordered set is simply a total ordering of the poset that is consistent with the original ordering. The linear extension diameter is a measure of how different two linear extensions could be, that is, the number of pairs of elements that are ordered differently by the two extensions. In this dissertation, we calculate the linear extension diameter of grids. This also gives us a nice characterization of the linear extensions that are the farthest from each other, and allows us to conclude that grids are diametrally reversing.

A linear extension of a poset might …

Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager

Department of Mathematics: Dissertations, Theses, and Student Research

Population dynamics tries to explain in a simple mechanistic way the variations of the size and structure of biological populations. In this dissertation we use mathematical modeling and analysis to study the various aspects of the dynamics of plant populations and their seed banks.

In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for Cirsium canescens. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to …

Une Histoire De La Formation Mathématique En France: Les Réformes Et Les Philosophies De L’Enseignement Primaire Et Secondaire De 1420 Jusqu'À Aujourd’Hui, Rebecca Robinson

Honors Theses

France has produced many illustrious mathematicians who have profoundly impacted mathematics as they are today. While Descartes, Cauchy, and Borel (among others) viewed math as a lifelong pursuit, they began their education in an elementary school classroom with everybody else. In this study, I explore mathematical reforms and governmental documents throughout history to show how the education system has grown to emphasize a strong mathematical curriculum for all students and have consulted many philosophical articles on both the importance of math in a student’s education as well as different views on the manner in which mathematics should be taught to …

K-8 Preservice Teachers’ Inductive Reasoning In The Problem-Solving Contexts, Marta Magiera

Mathematics, Statistics and Computer Science Faculty Research and Publications

This paper reports the results from an exploratory study of K-8 pre-service teachers’ inductive reasoning. The analysis of 130 written solutions to seven tasks and 77 reflective journals completed by 20 pre-service teachers lead to descriptions of inductive reasoning processes, i.e. specializing, conjecturing, generalizing, and justifying, in the problem-solving contexts. The uncovered characterizations of the four inductive reasoning processes were further used to describe pathways of successful generalizations. The results highlight the importance of specializing and justifying in constructing powerful generalizations. Implications for teacher education are discussed.

Team-Teaching Experiences Of A Mathematician And A Mathematics Teacher Educator: An Interpretative Phenomenological Case Study, Sarah K. Bleiler

USF Tampa Graduate Theses and Dissertations

In recent years, experts and organizations involved in mathematics education have emphasized the importance of collaboration between mathematicians and mathematics teacher educators as a means of improving the professional preparation of mathematics teachers. While several such collaborative endeavors have been documented in the extant literature, most research reports have focused on the products, rather than the process, of collaboration. The purpose of this interpretative phenomenological case study is to gain an understanding of the lived experiences of a mathematician and a mathematics teacher educator as they engaged in a team-teaching collaboration within the context of prospective secondary mathematics teacher preparation. …

The Mathematics Portfolio: An Alternative Tool To Evaluate Students’ Progress, Marla A. Sole

Publications and Research

This article describes the need for more thorough and varied forms of assessment to evaluate students’ level of understanding in mathematics. Portfolios are one type of assessment tool that, when added to a teacher’s repertoire can improve students’ comprehension and retention and enable students to monitor their own progress and to take more responsibility for their own learning. Portfolio assignments can also help students and teachers to detect and remedy weaknesses and misunderstandings and can increase students’ self-confidence in mathematics. This article discusses what a portfolio is, gives an example of a unit portfolio used in an undergraduate Finite Mathematics …

Combinatorics Using Computational Methods, Derrick Stolee

Department of Mathematics: Dissertations, Theses, and Student Research

Computational combinatorics involves combining pure mathematics, algorithms, and computational resources to solve problems in pure combinatorics. This thesis provides a theoretical framework for combinatorial search, which is then applied to several problems in combinatorics. Some results in space-bounded computational complexity are also presented.

Undergraduate Students' Self-Reported Use Of Mathematics Textbooks, Aaron Weinberg, Emilie Wiesner, Bret Benesh, Timothy Boester

Mathematics Faculty Publications

Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed, and asked to describe how they used their textbook. The results indicate that students tend to use examples, instead of the expository text, to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself, as well as students' beliefs about reading and the …

What Mathematics Do Elementary Education Teachers Need To Know?, Bret Benesh

Forum Lectures

Almost no one is happy with the state of America's mathematics education. I examine the mathematics textbooks elementary education majors commonly use in college to determine what effect this might be having on their future elementary school students. In this Thursday Forum, I report on what I found in these textbooks---and why I do not like them. I then supply an alternative vision that would better serve our elementary school students.

Parts Of The Whole: Learn More, Learn Better, Dorothy Wallace

Numeracy

Building on previous columns in Numeracy, this column analyzes various teaching techniques in terms of their ability to build cognitive schema, extend existing schema, reinforce learning, move mean understanding of a group of students, and reduce variance in understanding of a group. We offer a pedagogical cycle as an example of how to address multiple learning goals using common teaching methods.

Incorporating Quantitative Reasoning In Common Core Courses: Mathematics For The Ghost Map, John R. Jungck

Numeracy

How can mathematics be integrated into multi-section interdisciplinary courses to enhance thematic understandings and shared common readings? As an example, four forms of quantitative reasoning are used to understand and critique one such common reading: Steven Berlin Johnson’s "The Ghost Map: The Story of London's Most Terrifying Epidemic - and How it Changed Science, Cities and the Modern World" (Riverhead Books, 2006). Geometry, statistics, modeling, and networks are featured in this essay as the means of depicting, understanding, elaborating, and critiquing the public health issues raised in Johnson’s book. Specific pedagogical examples and resources are included to illustrate applications and …

If Only Math Majors Could Write..., Bernard L. Madison

Numeracy

This text of the opening plenary address to the 2011 Summit of the Appalachian College Association and the meeting of the National Numeracy Network makes an argument that quantitative reasoning and writing should be taught together. The argument is set up by noting that humanists have historically banished quantitative issues from their study of the liberal arts and that science, engineering, and mathematics education suffers from lack of approaches to learning that promote complex, deeper understanding, most notably integrative and reflective learning. Therefore, everyone would profit from combining writing and quantitative reasoning. Five more specific reasons are discussed, drawing evidence …

Period Life Tables: A Resource For Quantitative Literacy, Thomas J. Pfaff, Stanley Seltzer

Numeracy

A period life table provides an estimate of the probability that a person will die at a particular age. Using data available online, we examine tables of expected years to live for males and females against age for three populations: the United States in 2007, the U.S. at the turn of the twentieth century, and the Roman Empire. Scatter plots of males and females for each population show how life expectancy increases with age (e.g., U.S. 2007: 50 year-old female > 40 year-old female > 45 year-old male). The three data sets allow historical comparisons (e.g., of gender disparity, larger now; of …

Motivation For Achievement And Attitudes Toward Mathematics Instruction In A Required Calculus Course At The Norwegian University Of Science And Technology, Donna Sundre, Carol Barry, Vidar Gynnild, Erin Tangen Ostgard

Numeracy

This study from the Norwegian University of Science and Technology (NTNU) examines students’ learning goals and attitudes toward mathematics in a first-year calculus course in undergraduate engineering education. Achievement motivation research using the Achievement Goal Questionnaire (AGQ) is advanced from current literature with two additions: (1) a course specific context using introductory college calculus students, and (2) participation of Norwegian students.

Pre- and posttest measures of attitudes indicate that students do change learning goals over time, unfortunately opposite to the instructors’ aspirations. A significant increase in “Mastery Avoidance” and “Work Avoidance” was accompanied with a drop in “Mastery Approach” and …

Quantitative Literacy Interventions At University Of Cape Town: Effects Of Separation From Academic Disciplines, Vera Frith

Numeracy

The aim of the Numeracy Centre at the University of Cape Town is to develop students’ quantitative literacy (QL) in a manner consistent with their programmes of study and intended roles in the community. Our theoretical perspective on the nature of QL is in line with that of the New Literacies Studies and sees academic QL as practices in different academic disciplinary contexts. This means that for us the ideal curriculum structure for developing QL would fully integrate it into the teaching of the disciplines. This is in practice not achievable in most cases, especially since many students do not …

Numeracy, Financial Literacy, And Financial Decision-Making, Annamaria Lusardi

Numeracy

Financial decisions, be they related to asset building or debt management, require the capacity to do calculations, including some complex ones. But how numerate are individuals, in particular when it comes to calculations related to financial decisions? Studies and surveys implemented in both the United States and in other countries that are described in this paper show the level of numeracy among the population to be very low. Moreover, lack of numeracy is not only widespread but is particularly severe among some demographic groups, such as women, the elderly, and those with low educational attainment. This has potential consequences for …

The Joy Of Quantitative Reasoning, Caren Diefenderfer

Numeracy

One of the advantages of focusing on quantitative reasoning is that it spans a wide variety of topics. As incoming president of the National Numeracy Network, I would like to take the opportunity of this editorial to tell my story of intellectual reward from finding common purpose in quantitative reasoning with colleagues from disciplines outside of mathematics. The story starts with an NSF-funded faculty development project (DUE-9952807) to further a QR across-the-curriculum program and the finding from that program that merging authentic context with mathematics brings interaction and collaboration. That joy in learning from and working with colleagues in other …

"Our Research Was Centered On Whether Or Not The Following Changes Made To The Pre-Calculus Algebra Class Positively Affected Student Performance…", Andrea Winchester

Summer Community of Scholars Posters (RCEU and HCR Combined Programs)

No abstract provided.

Mathematical Competitions In Hungary: Promoting A Tradition Of Excellence & Creativity, Julianna Connelly Stockton

Mathematics Faculty Publications

Hungary has long been known for its outstanding production of mathematical talent. Extracurricular programs such as camps and competitions form a strong foundation within the Hungarian tradition. New types of competitions in recent years include team competitions, multiple choice competitions, and some exclusively for students who are not in a special mathematics class. This study explores some of the recent developments in Hungarian mathematics competitions and the potential implications these changes have for the very competition-driven system that currently exists. The founding of so many new competitions reflects a possible shift in the focus and purpose of competitions away from …

Supporting Implementation Of The Common Core State Standards For Mathematics: Recommendations For Professional Development, Paola Sztajn, Karen A. Marrongelle, Peg Smith, Bonnie L. Melton

Mathematics and Statistics Faculty Publications and Presentations

In 2010, the National Governor’s Association and the Council of Chief State School Officers published the Common Core State Standards for Mathematics (CCSSM) and to date, 44 states, the District of Columbia, and the U.S. Virgin Islands have adopted the document. These content and practice standards, which specify what students are expected to understand and be able to do in K-12 mathematics, represent a significant departure from what mathematics is currently taught in most classrooms and how it is taught. Developing teachers’ capacity to enact these new standards in ways that support the intended student learning outcomes will require considerable …

Math Department Newsletter, 2011-2012 (Year In Review), Mathematics Department

Mathematics Newsletter

No abstract provided.