Euler

Infusing Connections into Core Courses for Secondary Teachers

Contemporary College Algebra: A Refocused College Algebra Course

Fair Division: From Cake-Cutting to Dispute Resolution

Combinatorially Thinking

Teaching a Proof-Based Course as the Gateway to the Mathematics Major

MINICOURSE #1

William W. Dunham, Muhlenberg College
Edward C. Sandifer, Western Connecticut State University

Part 1: Thursday, August 10, 1:00 pm – 3:00 pm

Part 2:  Friday, August 11, 1:00 pm – 3:00 pm

Euler wrote and published over 850 books and papers. They form the basis for huge segments of modern mathematics. We will survey his many contributions and take a close look at a few of them. We will demonstrate how to use Euler’s 18th century mathematics in a 21st century environment, and we will show by example why Laplace was giving good advice when he said, “Read Euler, read Euler.  He is the master of us all.”

Steve R. Benson, Education Development Center and University of New Hampshire
Karen J. Graham, University of New Hampshire

Part 1: Thursday, August 10, 1:00 pm – 3:00 pm

Part 2:  Friday, August 11, 1:00 pm – 3:00 pm

National recommendations call for content courses for prospective teachers that make explicit connections between the mathematics that teachers learn and the mathematics they will use as teachers. Most content courses for preservice secondary teachers are core courses for the mathematics major and texts for these courses do not typically address these connections. Minicourse participants will work with materials that contain the mathematical rigor of an upper division course and help prospective teachers build connections to secondary mathematics, discuss implementation issues with colleagues who have used such materials, and begin to adapt these materials for the courses they teach.

Laurette Foster, Prairie View A&M University
Alex Fluellen, Clark Atlanta University
Don Small, U.S. Military Academy

Part 1: Thursday, August 10, 3:30 pm – 5:30 pm

Part 2: Saturday, August 12, 1:00 pm – 3:00 pm

This minicourse will take participants on a typical journey through a refocused college algebra program. The trip will include small-group project presentations, assignments requiring the use of a graphing calculator, writing assignments, and assessment techniques. Participants will receive a collection of existing small-group projects and will create at least one new one during the minicourse. Familiarity with a graphing calculator will be helpful but is not a prerequisite.

Steven J. Brams, New York University

Part 1: Thursday, August 10, 3:30 pm – 5:30 pm

Part 2: Saturday, August 12, 1:00 pm – 3:00 pm

Cutting a cake, dividing up property in an estate, determining the borders in an international dispute – such problems of fair division are ubiquitous. Rigorous procedures for allocating goods (or “bads,” like chores), or deciding who wins on what issues in disputes, will be analyzed, starting with the well-known cake-cutting procedure of “I cut, you choose.” Particular attention will be given to procedures that produce “envy-free” allocations, in which everybody thinks he or she received the largest portion and hence does not envy anybody else. Results obtained in the last five years will be highlighted.  Applications to real-life conflicts, from interpersonal to international, will be discussed.

Arthur T. Benjamin, Harvey Mudd College
Jennifer J. Quinn, Association for Women in Mathematics

Part 1: Friday, August 11, 3:30 pm – 5:30 pm

Part 2: Saturday, August 12, 3:30 pm – 5:30 pm

Faced with an identity, how do you create a combinatorial proof?  This hands-on minicourse will provide you with some useful combinatorial interpretations, well-selected examples, and the challenge of finding your own combinatorial proofs.  Along with numbers that are defined through counting (binomial coefficients, Stirling numbers, Catalan numbers), you will acquire a combinatorial appreciation for quantities like harmonic numbers, continued fractions, determinants, Fibonacci numbers, and the golden ratio.  An extensive list of identities – some with known interpretations and others without – will serve as the basis for your exploration.  Of course, you are welcome to bring along your personal favorites to add to the excitement.

James Sandefur, Georgetown University

Part 1: Friday, August 11, 3:30 pm – 5:30 pm

Part 2: Saturday, August 12, 3:30 pm – 5:30 pm

Many colleges and universities have a gateway course to help mathematics students make the transition to more theoretical courses, with a goal of helping students learn how to understand and construct proofs. The organizer of this course, guided by 5 years of videotaping his students doing their homework for a proof-based course, will lead participants in an exploration of effective approaches to teaching “proof.” We will discuss appropriate types of problems, the wording of problems, effective hints and prompts, and a variety of pedagogical approaches. Suggestions and questions from participants will be encouraged.