Tanzania, Mathematics, and Me: Reflections from my work with Tanzanian Teachers

In June 2006 I had the privilege of participating in a four-day teacher training workshop in Mumba, Tanzania. In this paper I will discuss the challenges and triumphs of working with Tanzanian Secondary Mathematics teachers. We will discuss the educational environment, teaching strategies, and curricular issues that affect mathematics teachers in rural areas of Tanzania and contrast that with the American educational experience. We will also discuss some of the goals of the Teacher Training workshop that my colleagues and I led and look at some of the specific mathematical ideas and applications that I shared with the Mathematics teachers in attendance

Almost perfect nonlinear functions and related combinatorial structures

A map f(x) from the finite field Fpn to itself is said to be differentially k-uniform if k is the maximum number of solutions of the equation f(x + a) - f(x) = b where a, b [is in] Fpn , a ≠ 0. In particular, 2-uniform maps over F2n are called almost perfect nonlinear (APN) maps. These maps are of interest in cryptography because they offer optimum resistance to linear and differential attacks on certain cryptosystems. They can also be used to construct several combinatorial structures of interest.;In this dissertation, we characterize and classify all known power maps f(x) = xd over F2n , which are APN or of low uniformity. We discuss some basic properties of APN maps, collect all known APN power maps, and give a classification of APN power maps up to equivalence. We also give some insight regarding efforts to find other APN functions or prove that others do not exist and classify all power maps according to their degree of uniformity for n up to 13.;In the latter part of this dissertation, through the introduction of an incidence structure, we study how these functions can be used to construct semi-biplanes utilizing the method of Robert S. Coulter and Marie Henderson. We then consider a particular class of APN functions, from which we construct symmetric association schemes of class two and three. Using the result of E. R. van Dam and D. Fon-Der-Flaass, we can see that the relation graphs of some of these association schemes are distance-regular graphs. We discuss the local structure of these distance-regular graphs and characterize them

Reading Assignments and Assessments: Are Your Students Reading Math Texts Before Class, After Class, Both, or Neither?

In his recent book What the Best College Students Do [Bain, 2012], Ken Bain defines a number of different types of students including “surface learners,” “strategic learners,” “routine experts,” and finally, “deep learners.” In our mathematics courses at Trinity, we have found examples of all of these student types. A major determinant of their preferred approach to learning appears to be the ways and degrees to which mathematical texts and other written materials are read prior to class sessions. Each full-time member of the department both assigns and assesses the reading of mathematical materials prior to class sessions. Assessment methods, as well as the corresponding pedagogical choices, vary significantly. We also discuss the results of a related survey of over 100 Trinity undergraduates enrolled in mathematics courses during fall 2012

Start a Math Teacher Circle: Connect K-12 Teachers with Engaging, Approachable, and Meaningful Mahtematical Problems

Many K-12 math teachers are not ready to teach from a conceptual and inquiry-oriented pe

Start a Math Teacher Circle: Connect K-12 Teachers with Engaging, Approachable, and Meaningful Mathematical Problems

Many K-12 math teachers are not ready to teach from a conceptual and inquiry-oriented perspective because they have an algorithmic understanding of mathematics. One solution is to create a math teacher circle (MTC), which provides conceptual and inquiry-based learning activities and builds professionalism among the teachers. In this paper, we describe the origins of two such MTCs, highlighting the process of identifying leadership team members, submitting the grant proposal for seed money, and hosting launch events, intensive summer workshops, and monthly meetings during the academic year. We also share opportunities for professional development for college and university faculty, including research linked to shifts in in-service teacher attitudes. We finish the paper with several of this year’s best activities used at our MTC meetings, including fair division, extensions and generalizations of numerical and algebraic patterns, and applications in cryptography

The P0-matrix completion problem

In this paper the P0-matrix completion problem is considered. It is established that every asymmetric partial P0-matrix has P0-completion. All 4 × 4 patterns that include all diagonal positions are classified as either having P0-completion or not having P0-completion. It is shown that any positionally symmetric pattern whose graph is an n-cycle with n ≥ 5 has P0-completion

Almost perfect nonlinear functions and related combinatorial structures

A map f(x) from the finite field Fpn to itself is said to be differentially k-uniform if k is the maximum number of solutions of the equation f(x + a) - f(x) = b where a, b [is in] Fpn , a ≠ 0. In particular, 2-uniform maps over F2n are called almost perfect nonlinear (APN) maps. These maps are of interest in cryptography because they offer optimum resistance to linear and differential attacks on certain cryptosystems. They can also be used to construct several combinatorial structures of interest.;In this dissertation, we characterize and classify all known power maps f(x) = xd over F2n , which are APN or of low uniformity. We discuss some basic properties of APN maps, collect all known APN power maps, and give a classification of APN power maps up to equivalence. We also give some insight regarding efforts to find other APN functions or prove that others do not exist and classify all power maps according to their degree of uniformity for n up to 13.;In the latter part of this dissertation, through the introduction of an incidence structure, we study how these functions can be used to construct semi-biplanes utilizing the method of Robert S. Coulter and Marie Henderson. We then consider a particular class of APN functions, from which we construct symmetric association schemes of class two and three. Using the result of E. R. van Dam and D. Fon-Der-Flaass, we can see that the relation graphs of some of these association schemes are distance-regular graphs. We discuss the local structure of these distance-regular graphs and characterize them.</p

Boletín del Clero del Obispado de León: Año XLII Tomo XLII Número 25 - 1894 junio 21

Copia digital. Madrid : Ministerio de Cultura. Subdirección General de Coordinación Bibliotecaria, 200

The P0-matrix completion problem

In this paper the P0-matrix completion problem is considered. It is established that every asymmetric partial P0-matrix has P0-completion. All 4 × 4 patterns that include all diagonal positions are classified as either having P0-completion or not having P0-completion. It is shown that any positionally symmetric pattern whose graph is an n-cycle with n ≥ 5 has P0-completion.This article is published as Choi, Ji Young, Luz Maria DeAlba, Leslie Hogben, Mandi S. Maxwell, and Amy Wangsness. "The P0-matrix completion problem." The Electronic Journal of Linear Algebra 9 (2002): 1-20. DOI: 10.13001/1081-3810.1068. Posted with permission.</p