Mathematics Commons^™

Three Strikes, Tom O'Brien, Marianne Smith

The Mathematics Enthusiast

The first part of the paper describes and comments upon three aspects of the back-to-basics movement: the make-up and mindset of the National Mathematics Advisory Panel (NMP), the movement’s history in California, and recent “grassroots” activities of the movement in the state of Washington. The second part reports and comments on the principal findings of the NMP report.

Tme Volume 5, Numbers 2 And 3

The Mathematics Enthusiast

No abstract provided.

Chess And Problem Solving Involving Patterns, Dores Ferreira, Pedro Palhares

The Mathematics Enthusiast

In this paper we present the context and results from a study, with 3rd to 6th grades children, about the relationship between chess and problem solving involving geometric and numeric patterns. The main result of this study is the existence of a relation between strength of play and patterns involving problem solving. We have included in the beginning an analysis of chess as a context for elementary mathematics problems, also showing its richness historically.

Matlab Report, Nancy Hale

Cornerstone 3 Reports : Interdisciplinary Informatics

The objective of this effort is to create a multidisciplinary approach to problem solving using technology. Through Thinkfinity, the Seidenberg School of Computer Science and Information Systems is collaborating with the science and quantitative faculty to create lessons that can be used by high school and first -year college students to work interactively with data to solve problems. MatLab is the tool of choice. It gives the user the ability to create visual models of large data sets and is used in the science and finance areas.

The key features of MatLab:

• Interactive tools for iterative exploration, design, and …

A Case Study Using Soft Systems Methodology In The Evolution Of A Mathematics Module, Jon Warwick

The Mathematics Enthusiast

This paper describes the application of Soft Systems Methodology as a tool for facilitating the review of a taught mathematics module so that the views of those engaged with the module could be captured and conflicting expectations and views highlighted. Checkland’s Soft Systems Methodology is used since it enables the capture of stakeholder views and addresses both ‘hard’ and ‘soft’ aspects of the learning experience. Stages in the application of Soft Systems Methodology are illustrated including the development of a rich picture and conceptual models and the work was conducted using a stakeholder group that included students taking the module …

History Of Mathematics In Mathematics Education: A Saussurean Perspective, Michael Fried

The Mathematics Enthusiast

It is not only because of a certain eclecticism in mathematics education research that semiotic ideas have begun to take root there: it is also because of the dawning recognition that key areas of interest in mathematics education genuinely have a semiotic nature. For this, one need only point to research focused on meaning, communication, language, and culture (e.g., Presmeg, 1997; Ernest,1997; Radford, 2001; Brown, 2001). The ways in which semiotics informs cultural aspects of mathematics education, particularly those connected with the history of mathematics, were highlighted in a discussion session at the 2003 meeting of the International Group for …

Comparison Of Geometric Figures, Spyros Glenis

The Mathematics Enthusiast

Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein’s view for the Euclidean Geometry, we defined a relation “≤ ” as: S1 ≤ S2 whenever there is a Euclidean isometry f :R² →R² , so that . This relation is not an order because there are …

Incoherence Of A Concept Image And Erroneous Conclusions In The Case Of Differentiability, Antti Viholainen

The Mathematics Enthusiast

The level of the coherence of a concept image conveys how well the cognitive structure concerning the concept is organized. This study considers the relationship between deficiencies in the coherence of the concept image and erroneous conclusions in the case of differentiability. The study is based on an interview where the student made conclusions contradictory to the formal theory of mathematics. He used an erroneous method to study the differentiability of piecewise defined functions. This method became the key factor which maintained the internal coherence of the concept image. It made it possible to build a cognitive structure whose basis …

Generically There Is But One Self Homeomorphism Of The Cantor Set, Ethan Akin, Eli Glasner, Benjamin Weiss

Mathematics and Statistics Faculty Research & Creative Works

We describe a self-homeomorphism R of the Cantor set X and then show that its conjugacy class in the Polish group H(X) of all homeomorphisms of X forms a dense Gδ subset of H(X). We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class. © 2008 American Mathematical Society.

A Review Of Fibroblast Populated Collagen Lattices, J. C. Dallon, Paul H. Ehrlich

Faculty Publications

Bellaes introduction of the fibroblast populated collagen lattice (FPCL) (1) has facilitated the study of collagen-cell interactions. As a result of the numerous modifications of the casting of FPCL's, the in vivo applications of these in vitro findings has been confusing. Here experimental FPCL contraction findings are viewed in regard to three proposed mechanisms responsible for lattice contraction. The cellular mechanisms responsible for generating FPCL contraction are: cell contraction, cell tractional forces related to cell locomotion, and initial cell elongation and spreading.

Enumerating Constrained Non-Crossing Minimally Rigid Frameworks, David Avis, Naoki Katoh, Makoto Ohsaki, Ileana Streinu, Shin-Ichi Tanigawa

Computer Science: Faculty Publications

In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints, which we call constrained non-crossing Laman frameworks, on a given set of n points in the plane. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In particular, we obtain that the set of all the constrained non-crossing Laman …

Unfolding Manhattan Towers, Mirela Damian, Robin Flatland, Joseph O'Rourke

Computer Science: Faculty Publications

We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4×5×1 refinement of the vertex grid.

Criteria For Components Of A Function Space To Be Homotopy Equivalent, Gregory Lupton, Samuel Bruce Smith

Mathematics and Statistics Faculty Publications

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

Editorial: 2 To 3- An Omnibus Of Features & Policy Issues In Mathematics Education, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Inverses - Why We Teach And Why We Need Talk More About It More Often!, Woong Lim

The Mathematics Enthusiast

This article examines the key role that the notion of inverses plays in numerous mathematical concepts.

Chopping Logs: A Look At The History And Uses Of Logarithms, Rafael Villarreal-Calderon

The Mathematics Enthusiast

Logarithms are an integral part of many forms of technology, and their history and development help to see their importance and relevance. This paper surveys the origins of logarithms and their usefulness both in ancient and modern times.

Mathematics And The World: What Do Teachers Recognize As Mathematics In Real World Practice?, Barbara Garii, Lillian Okumu

The Mathematics Enthusiast

Elementary school teachers are encouraged to better integrate appropriate mathematics pedagogy with deeper, more relevant mathematics content. However, many teach a mathematics they do not fully understand to students who see, recognize, and use less mathematics than ever before. Both teachers and students struggle to articulate the role mathematics plays in society as mathematics becomes more embedded into our technology. In this study, we asked teachers to record the mathematics they used on a daily basis during a 1-week period. Their responses indicate that they do not recognize that mathematics plays any important role in technological and professional practices. This …

Associative Operations On A Three-Element Set, Friðrik Diego, Kristín Halla Jónsdóttir

The Mathematics Enthusiast

A set with a binary operation is a fundamental concept in algebra and one of the most fundamental properties of a binary operation is associativity. In this paper the authors discuss binary operations on a three-element set and show, by an inclusion-exclusion argument, that exactly 113 operations out of the 19,683 existing operations on the set are associative. Moreover these 113 associative operations are accounted for by means of their operation tables.

Magic Math Cards, Steve Humble

The Mathematics Enthusiast

Starting lessons with a good story, captures students imagination. The story then leads nicely into a magic card experiment which can be used to help teach the topic. I have used magic with a variety of age groups and found that they ask different questions of the task, and search for different levels of answers.

Mathematical Cognition And The Final Report Of The National Mathematics Advisory Panel: A Critical, Cultural-Historical Activity Theoretic Analysis, Wolff-Michael Roth

The Mathematics Enthusiast

In its Final Report, the National Mathematics Advisory Panel has depicted a stark image of mathematical competencies and achievement among U.S. students. The Advisory Panel notes the lack of research with “truly scientific” rigor and, mentioning Vygotsky’s cultural-historical activity theory in passing, suggests its utility to be untested. In reading the report, I noted the limited understanding of the mathematics education literature it articulates and a complete failure to draw on established, “tried and proven,” theory and practice in mathematics education founded upon an encompassing culturalhistorical activity theory. This theory is comprehensive and encompassing, because it retains activity in its …

E(Race)Ing Race From A National Conversation On Mathematics Teaching And Learning: The National Mathematics Advisory Panel As White Institutional Space, Danny Bernard Martin

The Mathematics Enthusiast

In this paper, I wish to argue that several factors support a characterization of the National Mathematics Advisory Panel as an instantiation of the white institutional space (Moore, 2008) that characterizes mathematics education research and policy contexts more generally. In particular, my analysis highlights that mathematics education research and policy contexts such as the National Mathematics Advisory Panel are not immune to the structural and institutional racism that characterize many other areas of U.S. society.

Guest Editorial: Reaction To The Final Report Of The National Mathematics Advisory Panel, Brian Greer

The Mathematics Enthusiast

In 2006, President Bush appointed the National Mathematics Advisory Panel (NMAP henceforth) which issued its Final Report on March 13, 2008. The Final Report (summary), plus reports of three subcommittees and five Task Groups, together with two appendices containing the Presidential Executive Order setting up the panel, and details of the panel membership and other personnel, can be downloaded from www.ed.gov/about/bdscomm/list/mathpanel/index.html

Algebra For All?, Brian Greer

The Mathematics Enthusiast

In 1957, the director of one of the major math curriculum projects in the UK was quoted in a newspaper as saying: "Up went Sputnik and down came all the pure mathematicians saying we must do sets and be saved". The corresponding message from the National Math Panel Final Report is "Up went the economic performance of China and India and down came all the pure mathematicians saying we must do algebra and be saved".

The Political Context Of The National Mathematics Advisory Panel, Eric Rico Gutstein

The Mathematics Enthusiast

The National Mathematics Advisory Panel needs to be situated in its broader political context to more fully understand it. Who created it, for what purpose, and who will (and will not) benefit from it are key questions I address in this article. My argument is that the NMAP, as part of a larger initiative undertaken by the Bush Administration and US financial/corporate elites, serves capital’s efforts to shore up the US’s weakening economic global position and does not benefit the majority of the US people—particularly marginalized and excluded students of color and low-income students.

Meet The Authors

The Mathematics Enthusiast

No abstract provided.

New Titles In Mathematics Education

The Mathematics Enthusiast

No abstract provided.

Positive Solutions Of A Third-Order Three-Point Boundary-Value Problem, Bo Yang

Faculty Articles

We obtain upper and lower estimates for positive solutions of a third-order three-point boundary-value problem. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are also obtained. Then to illustrate our results, we include an example.

Nonbinary Quantum Error-Correcting Codes From Algebraic Curves, Jon-Lark Kim, Judy L. Walker

Department of Mathematics: Faculty Publications

We give a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia- Stichtenoth tower of function fields which are constructible in polynomial time.

Binary quantum error-correcting codes have been constructed in several ways. One interesting construction uses algebraic-geometry codes [2], [6], [7], [12], with the main idea being to apply the binary CSS construction [4], [5], [16] to the asymptotically good algebraic-geometry codes arising from the Garcia-Stichtenoth [11] tower of function fields over F_q2 (where q is a …

Integer Ratio Logarithm Based Edge Detection For Runway Hazard Detection, Sri Satya Vardhan Gogineni

Electrical & Computer Engineering Theses & Dissertations

As air traffic is growing in leaps and bounds every day, flying safe has been a major concern in day to day life. Aviation safety is always dependent on the performance of the pilot and air traffic controllers. The very involvement of human discretion introduces judgment as a component of a safe takeoff and landing. A convenient and immediate alternative would be the use of technology to alleviate the problem. Several algorithms have been developed to solve this problem. However, most computer vision algorithms leave the task of interpreting the end data unaddressed. Juggling the myriad tasks during landing and …

Fusion Of Character Tables And Schur Rings Of Dihedral Groups, Long Pham Bao Nguyen

Theses and Dissertations

A finite group H is said to fuse to a finite group G if the class algebra of G is isomorphic to an S-ring over H which is a subalgebra of the class algebra of H. We will also say that G fuses from H. In this case, the classes and characters of H can fuse to give the character table of G. We investigate the case where H is the dihedral group. In many cases, G can be completely determined. In general, G can be proven to have many interesting properties. The theory is developed in terms of S-ring …