Abstracts for talks

AMS-MAA-MER Special Sessions on Mathematics and Education Reform

Joint Annual Mathematics Meetings, Atlanta, Georgia, January, 2005

Undergraduate Geometry in the Twenty-First Century
Tom Banchoff

How can we introduce geometric ideas into the undergraduate curriculum in ways that reflect present-day problems and applications, in calculus and linear algebra as well as in more traditional geometry courses? Where do we find good problems? What is the role of the computer in investigations? How do these ideas fit in with the CUPM Guide 2004? Some of the new themes to be explored are scientific visualization in three- and four-dimensional space, catastrophe theory in engineering and in art, critical point theory and linkages, and dynamic non-Euclidean geometry.

Symbolic Logic in a Proofs Course: Finding the Right Balance
Connie M Campbell, Millsaps College

Symbolic logic plays a critical role in a first course in proof writing (a bridge course) by providing students with some of the fundamental skills necessary for developing logical arguments. However, when teaching a bridge course, the goal is not only to teach students how to read and write mathematical proofs, but also to move them from algorithmic thinking to critical, and even creative, thinking. It has been the experience of the author that this transition can be greatly hindered by either an overemphasis,
or an underemphasis, on symbolic logic. She will discuss problems that she has experienced from bridge courses, which erred on either of these two extremes, as well as discuss what she has found to be the right balance for her classroom. A course outline and classroom activities will be discussed as well.

Mathematical Logic in the Undergraduate Curriculum
Martin D. Davis, Professor Emeritus, Courant NYU; Visiting Scholar, UC Berkeley

Experts in mathematical logic can be found in three different academic departments: mathematics, philosophy, and computer science. Some knowledge of technical logic is important for majors in each of these subjects. This creates problems and opportunities. Which topics are important for majors in these different subjects? Can courses be devised to serve the needs of all three? These questions will be addressed in this talk.

Chaos in the Classroom: Exciting Students about Mathematics
Robert L. Devaney, Boston University

We give several examples of how ideas from chaotic dynamics may be used in classes such as calculus and differential equations (and even in pre-calculus and HS algebraclasses) to show students that all is not known in mathematics.

Chicago Algebra Initiative: A Multi-University Collaboration for Middle School Teachers
Naomi Fisher, University of Illinois at Chicago, David Jabon, DePaul University, Lynn Narasimhan, DePaul University, and Paul Sally, University of Chicago

In 2002, faculty from three Chicago universities and staff from the Chicago Public Schools (CPS) met regularly to organize the CPS Algebra Initiative with the goal of increasing the number of CPS middle grade teachers who are prepared to teach algebra in the eighth grade. The Algebra Initiative was launched in 2003-04 with programs at DePaul University, the University of Chicago, and the University of Illinois at Chicago. The programs ran for a full academic-year, and included a common certifying examination jointly administered at the completion of the programs. Over 100 teachers participated in the three programs. In this session, faculty from the three programs will describe the process of organizing the initiative and ongoing plans to continue and expand the initiative. The presenters will describe the needs of the CPS system and its teachers, the evolution of the multi-university effort, the curricula that were developed for the initiative, some early assessment results, and a summary of lessons learned and recommendations.

Teaching Logic to Prospective Elementary and Middle School Mathematics Teachers
Gregory D. Foley, Appalachian State University

According to renowned mathematics educator Bob Davis, mathematics is what you have left over after you have invented ways to solve problems and reflected on those inventions. With this as his motto, the presenter taught an undergraduate course in the Introduction to the Logic and Structure of Mathematics during Spring Semester, 2004 to a group of prospective teachers of Grades K-8, using a modified R. L. Moore method. Much time was devoted to students presenting their work to the class. The course focused on problem solving, exploring, conjecturing, reasoning, and communicating using number systems and algebraic systems as the content. The talk will focus on how this course helped the pre-service teachers enrolled in the class to learn about logic and mathematical reasoning.

Gender, Graduate Education, and Academic Careers in Science
Mary Frank Fox, Georgia Institute of Technology

In improving participation and performance of women as an under-represented group in science, it is important to comprehend the social complexity, or features, of the environments of graduate education, as they may vary for women and men students.

Drawing from a national survey of 3800 women and men doctoral students in five science and engineering fields, and site visits to 22 of the departments in which these students were located, this presentation concentrates upon characteristics and practices of departments, research teams, and advisement in doctoral education--and students reported experiences within them. The findings encompass matters of inclusion, nuances of
training and advising, and evaluative practices as they operate for women and men students. They point to implications of different opportunities to participate in research groups, to collaborate, and to gain significant roles in the scientific enterprise.

Household group theory
Joseph A. Gallian, University of Minnesota Duluth

In this talk we provide examples that an instructor in an abstract algebra class can use to connect groups with commonplace things. The connection between groups and everyday
objects is two-way. Sometimes the groups provide interesting information about the objects and sometimes the objects provide an interesting way to think about the group.

Logic as Compass
Iraj Kalantari, Western Illinois University

For most early students of mathematics, deep understanding of intricate mathematical concepts is a challenging charge. Absorbing and reproducing proofs, as well as authoring one's first set of proofs, because it is a combination of subject and logic, is a major discouragement for many capable but inexperienced students.

At Western Illinois University, where we have about 100 mathematics majors half of whom are pursuing a career in teaching, we meet the students' needs by requiring every mathematics major to take, post Calculus II and before proof embedded courses, a course in logic.

I will describe the history, the nature, the success, and the structure of our course arguing for existence of a similar course in all institutions of higher education including those whose students generally absorb most of the material of such a course implicitly while taking advance mathematics topics.

Linking Introductory Calculus and Statistics with Multivariate Modeling.
Daniel T. Kaplan

We have devised and deployed a new approach to teaching calculus and introductory statistics that is particularly attractive to students from biology and the social sciences. These students need to be able to reason about functions of multiple variables and relate them to data but rarely go beyond Calc I or II.

The year-long course is a common starting point for entering students, regardless of whether they have studied calculus in high school. The first half emphasizes modeling skills and differential calculus in several variables including the basic geometrical concepts of linear algebra.

The second half introduces statistical modeling in several variables. Multivariate regression and multiway analysis of variance and covariance are taught in a geometrical way that uses the formalism gained from linear algebra to build on the students' intuition. Computation is used intensively, including simulation, bootstrapping and logistic regression.

Response from the client disciplines has been strongly positive. We will describe the course's organization and present the approaches that allow introductory students with mixed backgrounds to use and understand concepts and techniques in statistical modeling that have traditionally been considered advanced.

Understanding under-representation in mathematics using research and concepts from social science: An introduction
Cathy Kessel, Mathematics Education Consultant

Mathematicians such as Lipman Bers and mathematics departments such as Potsdam, the University of Nebraska, and the University of Maryland have been remarkably successful in creating productive environments for members of groups often under-represented in mathematics.

Although descriptions of these successes exist, they do not appear to have been used to achieve success elsewhere. This may be because essential features have not been described, have not been described in enough detail to be replicated, or it may not be clear what features are essential. Social science offers methods to describe a range of phenomena from individual beliefs and behaviors to institutional practices. It also offers methods of providing evidence that certain actions promote or discourage participation of particular groups.

This session is intended as an introduction to some concepts and methods, which ameliorate or illuminate the climate for under-represented groups.

This talk is intended as an introduction for the rest of the session. I will provide:

> Statistics that show representation at various levels in mathematics, from undergraduates to faculty.
> Brief examples of social science concepts and how they may be used.
> Caveats for mathematicians about the use of social science.

Logic for undergraduates at Notre Dame
Julia F. Knight, university of Notre Dame

Notre Dame has a universal requirement of two semesters of mathematics. Students intending to major in science, engineering, or business need at least two semesters of calculus. Arts and Letters students have some other options. One is a logic course.

The goal is to give students some formal tools for analyzing arguments. There is an emphasis on formal proofs, and we cover both propositional logic and predicate logic. Such a course is standard in philosophy departments. However, in our course, homework is done in groups, and students also write a group paper, on some topic in logic that is not
covered in the lectures.

For mathematics majors at Notre Dame, there are several junior/senior electives. One of these is on various kinds of automata, and what they can and cannot do. Such a course is standard in computer science programs. However, there are challenging problems that encourage mathematical thinking, and the mathematics majors who choose the course seem to respond well to the material.

Center for the Study of Mathematics Curriculum: What does it do and why should it interest mathematicians?
Glenda Lappan, Michigan State University, and Robert Reys, University of Missouri

The Center for the Study of Mathematics Curriculum is a Learning and Teaching Center supported by the National Science Foundation. The CSMC is a collaboration among Michigan State University, University of Missouri and Western Michigan University with additional partners including the University of Chicago, Horizons Inc. and school districts. It serves the K-12 educational community by focusing scholarly inquiry around issues of mathematics curriculum. A central mission of the CSMC is to increase the number of doctoral students in mathematics education who will have expertise in curriculum research and in matters relating to curriculum: policy, history and status, design and analysis, adoption and enactment, teachers knowledge, and student learning.

This session will highlight efforts to recruit doctoral students and the infrastructure being established to support the professional growth of doctoral students in mathematics curriculum related areas. Outreach activities of the CSMC include monographs under development and forthcoming conferences. Current research activities will be highlighted, including the role of research associates and opportunities for others to collaborate in mathematics curriculum related research.

Developing a Capstone for Pre-service Secondary Teachers: A PMET Initiative
Cameron Sawyer, Southwestern University

The goal of this presentation is to describe the standards-based mathematics capstone course which is designed to develop a deeper conceptual understanding of grades 8-12 mathematics. This capstone is intended for senior math majors in order to strengthen their preparation for becoming secondary teachers. The objectives of the capstone course are to connect students' mathematical knowledge with the mathematics they will be teaching, to develop their knowledge of mathematics and mathematics-specific pedagogy, and to provide students with multiple perspectives through which to understand how mathematics is learned. The project is supported by the MAA's PMET (Preparing Mathematicians to Educate Teachers) program. We will explain how we involved mathematics department faculty and alumni as well as ideas from the education department in the formation of this new course.

Helping future elementary teachers "Reconceptualize Mathematics"
Judith T. Sowder, San Diego State University

Prospective elementary school teachers often need to reconceptualize mathematics rather than to review it. Instructional materials developed at San Diego State University (to be published by Freeman Press) provide prospective and practicing elementary teachers opportunities to develop a deep understanding of mathematics, as recommended in The Mathematical Education of Teachers. We have selected and developed topics and problems that exemplify mathematical reasoning and problem solving. "Sense-making" is the dominant theme, with regular expectations including explanations and justifications. The course content is sufficient for three semester courses but can be tailored to two courses. State frameworks and credentialing tests were considered while refining these materials. Course materials have been piloted numerous time at various institutions. We also provide instructors with assistance in developing the interactive teaching strategies recommended in MET and we will provide a CD with videos and instructions for a seminar course for instructors of courses for teachers. I will provide examples of the materials and of instructor assistance we offer. The courses also provide a foundation for additional coursework for middle school teachers.

Ideas to Enliven Liberal Arts Students
Michael Starbird, The University of Texas at Austin

Mathematics is a living, growing enterprise, full of creativity and imagination. Students can be convinced of this reality if their mathematical education includes contemporary topics. Modern abstract topics such as chaos, fractals, and aperiodic tilings, and modern applied topics such as public key cryptography, election paradoxes, and intriguing applications of statistics and probability all provide eye-opening ideas for students. Students who do not have technical majors can bring mathematics into their own lives by understanding new ideas and by incorporating the analytical strategies of mathematics into their own thinking habits.

Gender Schemas and Mathematics Performance
Virginia V. Valian, Hunter College

How do gender schemas affect students' attitudes to mathematics and performance on mathematics problems? Cognitive and social cognitive studies demonstrate the consequences for girls and women of parents', teachers', peers', and their own beliefs about sex differences. Information about gender schemas is available in four tutorials with voice-over narration at www.hunter.cuny.edu/gendertutorial; documents on gender
equity are available at www.hunter.cuny.edu/genderequity.

Discovery learning in logic education
Walker M White, University of Dallas

Mathematical logic is often seen as a technical discipline with little relevance to how "mathematics is done". But this is a complaint about logic education --- which often focuses on formal syntax --- and not the field itself. With discovery learning in logic education, students can learn how logic and mathematics relate to their everyday use of language.

In this talk, we show how to introduce discovery learning by focusing on natural language, and de-emphasizing formal syntax. Students learn to use natural language to prove theorems and construct models for axiom systems. They also learn how to use nonstandard models as counterexamples, insights into a proof, or the motivation of new axioms.

By itself, however, this is not enough for discovery learning. The structure of the course must motivate and reward this type of learning. We will also discuss how we institute grading, and how this can encourage students to attempt challenging and unfamiliar problems.

Throughout the talk, we will examine the opportunities and challenges that have been encountered in these types of courses over the past 30 years. We show how it has been successfully adopted at all levels of the curriculum, from bridge courses for majors to liberal-arts mathematics classes.

Faculty Resources for Improving the Mathematical Education of Teachers from the Center for Proficiency in Teaching Mathematics
Patricia S. Wilson and Jeremy Kilpatrick, University of Georgia

The principal aim of the Center for Proficiency in Mathematics Teaching (CPTM) is to build the capacity of the system of professional education for preservice and practicing teachers of mathematics. We focus on the improvement of teachers' opportunities to learn mathematics for teaching and to learn to use mathematical knowledge effectively in practice. CPTM is a National Science Foundation (NSF) -funded Center for Learning and Teaching involving the University of Georgia and the University of Michigan.

CPTM supports and investigates a variety of approaches for the education of professionals who prepare teachers of mathematics. This includes mathematicians, doctoral students in mathematics and mathematics education, post-doctoral fellows seeking to develop specialization in the mathematics education of teachers, mathematics educators, mathematics teacher leaders, local curriculum and professional education specialists. Our session will report and discuss its activities for those who teach mathematics for teachers at all grade levels. Our activities include doctoral programs, postdoctoral programs, certificate programs for doctoral students, summer institutes, study groups, research and materials preparation for professional development.

The Mid-Atlantic Center for Mathematics Teaching and Learning: Resources and Coming Together to Improve the Mathematical Education of Teachers
Rose Mary Zbiek, Pennsylvania State University

The Mid-Atlantic Center for Mathematics Teaching and Learning (MAC-MTL) involves mathematics educators, mathematicians, K-12 partners and others in a research agenda that examines both how prospective teachers learn mathematics and how their mathematics arises in their emerging classroom practice. The Center's products include a range of resources that can be used to enhance the mathematical preparation of teachers.

Most obvious among those products are course descriptions and materials addressing mathematical topics that transcend and augment typical content courses, including mathematical modeling, data analysis, and function. The Center also offers a growing base of research-based work to inform everything from development of mathematical tasks or assessments to interaction between campus and classroom components of teacher preparation.

More importantly, MAC-MTL is a group of faculty and emerging scholars in mathematics education with strong backgrounds and intense interest in mathematics and statistics. There are opportunities for collaboration and projects that further both the challenges of educating teachers and the continued development of a research base. This presentation elaborates on both the products and the collaborative opportunities.

MER Forum

Last update: October 14, 2004

Please address questions and comments to mer@math.uic.edu