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The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. McCarthy 2016 Washington University in St Louis

The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

We prove an implicit function theorem for non-commutative functions. We use this to show that if p ( X;Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of p ( X;Y ) = 0 will commute with X


Generating Functions And Wilf Equivalence For Generalized Interval Embeddings, Russell Chamberlain, Garner Cochran, Sam Ginsburg, Brian K. Miceli, Manda Riehl, Chi Zhang 2016 Trinity University

Generating Functions And Wilf Equivalence For Generalized Interval Embeddings, Russell Chamberlain, Garner Cochran, Sam Ginsburg, Brian K. Miceli, Manda Riehl, Chi Zhang

Mathematics Faculty Research

In 1999 in [J. Difference Equ. Appl. 5, 355–377], Noonan and Zeilberger extended the Goulden-Jackson Cluster Method to find generating functions of word factors. Then in 2009 in [Electron. J. Combin. 16(2), RZZ], Kitaev, Liese, Remmel and Sagan found generating functions for word embeddings and proved several results on Wilf-equivalence in that setting. In this article, the authors focus on generalized interval embeddings, which encapsulate both factors and embeddings, as well as the “space between” these two ideas. The authors present some results in the most general case of interval embeddings. Two special cases of interval embeddings are ...


Tme Volume 13, Numbers 1 And 2, 2016 University of Montana

Tme Volume 13, Numbers 1 And 2

The Mathematics Enthusiast

No abstract provided.


Guest Editorial: Mathematical Knowledge For Teaching: Developing Measures And Measuring Development, Reidar Mosvold, Mark Hoover 2016 University of Montana

Guest Editorial: Mathematical Knowledge For Teaching: Developing Measures And Measuring Development, Reidar Mosvold, Mark Hoover

The Mathematics Enthusiast

No abstract provided.


Interview Prompts To Uncover Mathematical Knowledge For Teaching: Focus On Providing Written Feedback, Yeon Kim 2016 University of Montana

Interview Prompts To Uncover Mathematical Knowledge For Teaching: Focus On Providing Written Feedback, Yeon Kim

The Mathematics Enthusiast

One area of study that has been gathering enthusiastic attention and interest is mathematical knowledge for teaching (MKT). How to research MKT, however, is still unsettled despite the plethora of unexamined areas of practice. As one of ways to unearth and measure MKT, this study uses interview prompts designed to providing written feedback, as a target area of practice. This study specifies in what ways the interview prompts are used in order to provide a comprehensive method to researching MKT. From interviews across professional communities with different kinds of mathematical expertise, the author develops a conceptual model based on the ...


Why Defining The Construct Matters: An Examination Of Teacher Knowledge Using Different Lenses On One Assessment, Chandra H. Orrill, Allan S. Cohen 2016 University of Montana

Why Defining The Construct Matters: An Examination Of Teacher Knowledge Using Different Lenses On One Assessment, Chandra H. Orrill, Allan S. Cohen

The Mathematics Enthusiast

What does it mean to align an assessment to the domain of interest? In this paper, we analyze teachers’ performance on the Learning Mathematics for Teaching assessment of Proportional Reasoning. Using a mixture Rasch model, we analyze their performance on the entire assessment, then on two different subsets of items from the original assessment. We consider the affordances of different conceptualizations of the domain and consider the implications of the domain definition on the claims we can make about teacher performance. We use a single assessment to illustrate the differences in results that can arise based on the ways in ...


Knowledge For Equitable Mathematics Teaching: The Case Of Latino Ells In U.S. Schools, Aaron T. Wilson 2016 University of Montana

Knowledge For Equitable Mathematics Teaching: The Case Of Latino Ells In U.S. Schools, Aaron T. Wilson

The Mathematics Enthusiast

This paper reports the exploration of an aspect of knowledge needed for equitable mathematics teaching. Pedagogical Content Knowledge for Teaching Mathematics to English Language Learners (PCK-MELL) was proposed as a theoretical knowledge construct, a subdomain of MKT, and the construct was investigated through a process of survey instrument development and administration. The survey contained items intended to measure teachers’ knowledge of the obstacles encountered by ELLs in math classes, of the resources that ELLs draw upon, and of instructional strategies for teaching ELLs. Analysis of middle school mathematics teachers’ responses (N = 42) offered insights into how to improve the reliability ...


In-Service Teachers' Reasoning About Scenarios Of Teaching Mathematics To English Language Learners, Sultan Turkan 2016 University of Montana

In-Service Teachers' Reasoning About Scenarios Of Teaching Mathematics To English Language Learners, Sultan Turkan

The Mathematics Enthusiast

The student population in the U.S. and worldwide is becoming increasingly diverse, creating a need to support all learners, especially linguistically and culturally diverse subpopulations such as English language learners (ELLs). From a social equity standpoint, the need to support these learners is critical especially in mathematics classrooms. In the U.S, the demand for mathematics teachers who are adequately prepared to teach ELLs has in fact risen. Yet, little is known about what knowledge base is essential to teach mathematics to ELLs. Driven by the need to explore this knowledge base, in this paper I explore what is ...


Teachers And Their Educators - Views On Contents And Their Development Needs In Mathematics Teacher Education, Mika Koponen, Mervi A. Asikainen, Antti Viholainen, Pekka E. Hirvonen 2016 University of Montana

Teachers And Their Educators - Views On Contents And Their Development Needs In Mathematics Teacher Education, Mika Koponen, Mervi A. Asikainen, Antti Viholainen, Pekka E. Hirvonen

The Mathematics Enthusiast

Finland has scored well in international assessments (e.g. PISA, TIMSS), and the pressure to attain excellent scores has activated a drive toward even more effective mathematics teacher education. This article presents the results of a qualitative assessment of the mathematics teacher education provided by the University of Eastern Finland. In this study, the views held by practicing teachers (N=101) and teacher educators (N=19) are compared so that the outstanding development needs of mathematics teacher education in terms of their contents can be revealed. The data was gathered via an electronic survey and was mainly analyzed using data-driven ...


Making Progress On Mathematical Knowledge For Teaching, Mark Hoover, Reidar Mosvold, Deborah L. Ball, Yvonne Lai 2016 University of Montana

Making Progress On Mathematical Knowledge For Teaching, Mark Hoover, Reidar Mosvold, Deborah L. Ball, Yvonne Lai

The Mathematics Enthusiast

Although the field lacks a theoretically grounded, well-defined, and shared conception of mathematical knowledge required for teaching, there appears to be broad agreement that a specialized body of knowledge is vital to improvement. Further, such a construct serves as the foundation for different kinds of studies with different agendas. This article reviews what is known and needs to be known to advance research on mathematical knowledge for teaching. It argues for three priorities: (i) finding common ground for engaging in complementary studies that together advance the field; (ii) innovating and reflecting on method; and (iii) addressing the relationship of such ...


What Does It Take To Develop Assessments Of Mathematical Knowledge For Teaching?: Unpacking The Mathematical Work Of Teaching, Sarah Kate Selling, Nicole Garcia, Deborah L. Ball 2016 University of Montana

What Does It Take To Develop Assessments Of Mathematical Knowledge For Teaching?: Unpacking The Mathematical Work Of Teaching, Sarah Kate Selling, Nicole Garcia, Deborah L. Ball

The Mathematics Enthusiast

In the context of the increased mathematical demands of the Common Core State Standards and data showing that many elementary school teachers lack strong mathematical knowledge for teaching, there is an urgent need to grow teachers’ MKT. With this goal in mind, it is crucial to have research and assessment tools that are able to measure and track aspects of teachers’ MKT at scale. Building on the concept of “mathematical tasks of teaching” (Ball et al., 2008), we report on a new framework that unpacks the mathematical work of teaching that could serve as a scaffold for item writers who ...


Assessing Mathematical Knowledge For Teaching: The Role Of Teaching Context, Geoffrey Phelps, Heather Howell 2016 University of Montana

Assessing Mathematical Knowledge For Teaching: The Role Of Teaching Context, Geoffrey Phelps, Heather Howell

The Mathematics Enthusiast

Assessments of mathematical knowledge for teaching (MKT), which are often designed to measure specialized types of mathematical knowledge, typically include a representation of teaching practice in the assessment task. This analysis makes use of an existing, validated set of 10 assessment tasks to both describe and explore the function of the teaching contexts represented. We found that teaching context serves a variety of functions, some more critical than others. These context features play an important role in both the design of assessments of MKT and the types of mathematical knowledge assessed.


Use Of Mathematical Tasks Of Teaching And The Corresponding Lmt Meaures In The Malawi Context, Mercy Kazima, Arne Jakobsen, Dun N. Kasoka 2016 University of Montana

Use Of Mathematical Tasks Of Teaching And The Corresponding Lmt Meaures In The Malawi Context, Mercy Kazima, Arne Jakobsen, Dun N. Kasoka

The Mathematics Enthusiast

We discuss the adaptation and piloting of the previously developed U.S.-specific measures of mathematical knowledge for teaching to the Malawi context. The purpose is to produce measures that can be used to evaluate changes in mathematical knowledge for teaching gained through primary teacher education, thus informing teacher educators on the most effective evidence-based practices. By interviewing 14 teachers, we first examine whether the 16 recurrent mathematical tasks of teaching tasks identified in the U.S. are applicable to the Malawi context. This is followed by the discussion of the adaptability of the U.S. developed number concept and ...


Cyclic Critical Groups Of Graphs, Ryan P. Becker, Darren B. Glass 2016 Gettysburg College

Cyclic Critical Groups Of Graphs, Ryan P. Becker, Darren B. Glass

Math Faculty Publications

In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the group exactly.


Pythagoras Plays His Lyre, Sarah Glaz 2016 University of Connecticut

Pythagoras Plays His Lyre, Sarah Glaz

Journal of Humanistic Mathematics

A poem about the Pythagoreans' beliefs and way of life.


Pension Building, Washington Dc, E Laura Golberg 2016 Claremont Colleges

Pension Building, Washington Dc, E Laura Golberg

Journal of Humanistic Mathematics

No abstract provided.


Jay Leno And Abstract Algebra, Adam Glesser, Martin Bonsangue 2016 California State University, Fullerton

Jay Leno And Abstract Algebra, Adam Glesser, Martin Bonsangue

Journal of Humanistic Mathematics

The Jay Leno skit Jaywalking, showing ordinary people struggling to answer basic questions, is both entertaining and applicable to teaching. This article describes how an instructor can strengthen students' conceptual understanding by creating an element of confusion, or "cognitive dissonance," in the students' minds using Jaywalking-style interactions in the classroom.


Dramathizing Functions: Building Connections Between Mathematics And Arts, Gunhan Caglayan 2016 Florida International University

Dramathizing Functions: Building Connections Between Mathematics And Arts, Gunhan Caglayan

Journal of Humanistic Mathematics

This article focuses on connections between mathematics and performance arts (drama). More specifically we offer an exposition of a segment of college algebra mathematics (an introduction to functions), with an approach primarily emphasizing the aesthetic aspects of mathematical learning, teaching, and performing.


A Meeting Of Minds: An Alternate Humor For Teaching Mathematics To Non-Stem Majors, Paul H. Grawe 2016 Winona State University and Institute for Travesty, Comedy, and Humor Studies

A Meeting Of Minds: An Alternate Humor For Teaching Mathematics To Non-Stem Majors, Paul H. Grawe

Journal of Humanistic Mathematics

John Allen Paulos argued essentially for three forms of humor dear to mathematics: Incongruity, Gotcha, and Word Play. Unfortunately, these three are often combative forms and easily drive non-STEM majors out of mathematics and statistics.

William Dunham in The Mathematical Universe shows how a fine mathematician can use humor to draw non-specialists in. Central to Dunham’s success is his use of Sympathetic Pain humor, which creates softer synthetic Reconciler, Consoler, or Bridgebuilder humor styles.


What If?: Mathematics, Creative Writing, And Play, Emily Clader 2016 ETH Zurich, Institute for Theoretical Studies

What If?: Mathematics, Creative Writing, And Play, Emily Clader

Journal of Humanistic Mathematics

Mathematics can inform creative writing by suggesting structures for it to follow, as well as by providing the imaginative impetus for common rules to be broken. In a workshop co-taught by the author, a class of sixth-grade students explored this interplay as they produced fractal-inspired poetry and geometry-inspired fiction. This article describes the form and results of the workshop in the context of a broader discussion of the influence of mathematics upon literature.


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