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Dittman, Marki; Soto-Johnson, Hortensia; Dickinson, Scott; Harr, Tim – PRIMUS, 2017

In this paper, we describe how we integrated complex analysis into the second semester of a geometry course designed for preservice secondary mathematics teachers. As part of this inquiry-based course, the preservice teachers incorporated their geometric understanding of the arithmetic of complex numbers and complex-valued functions to create a…

Descriptors: Secondary School Teachers, Secondary School Mathematics, Geometry, Preservice Teachers

D'Angelo, John P. – PRIMUS, 2017

We offer many specific detailed examples, several of which are new, that instructors can use (in lecture or as student projects) to revitalize the role of complex variables throughout the curriculum. We conclude with three primary recommendations: revise the syllabus of Calculus II to allow early introductions of complex numbers and linear…

Descriptors: Mathematics Instruction, College Mathematics, Undergraduate Study, Calculus

Garcia, Stephan Ramon – PRIMUS, 2017

A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by…

Descriptors: Algebra, Mathematics Instruction, Numbers, College Mathematics

Howell, Russell W.; Schrohe, Elmar – PRIMUS, 2017

Rouché's Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The "winding number" provides a geometric interpretation relating to the…

Descriptors: Mathematics Instruction, College Mathematics, Undergraduate Study, Mathematical Logic

Newton, Paul K. – PRIMUS, 2017

The subject of fluid mechanics is a rich, vibrant, and rapidly developing branch of applied mathematics. Historically, it has developed hand-in-hand with the elegant subject of complex variable theory. The Westmont College NSF-sponsored workshop on the revitalization of complex variable theory in the undergraduate curriculum focused partly on…

Descriptors: Mathematics Instruction, Undergraduate Study, College Mathematics, Mechanics (Physics)

Bolt, Michael – PRIMUS, 2017

The sheet resistance of a conducting material of uniform thickness is analogous to the resistivity of a solid material and provides a measure of electrical resistance. In 1958, L. J. van der Pauw found an effective method for computing sheet resistance that requires taking two electrical measurements from four points on the edge of a simply…

Descriptors: Mathematics Instruction, College Mathematics, Undergraduate Study, Physics

Brilleslyper, Michael A.; Schaubroeck, Beth – PRIMUS, 2017

The Gauss-Lucas Theorem is a classical complex analysis result that states the critical points of a single-variable complex polynomial lie inside the closed convex hull of the zeros of the polynomial. Although the result is well-known, it is not typically presented in a first course in complex analysis. The ease with which modern technology allows…

Descriptors: Graphs, Physics, Geometry, Mathematics Instruction

Garcia, Stephan Ramon; Ross, William T. – PRIMUS, 2017

We hope to initiate a discussion about various methods for introducing Cauchy's Theorem. Although Cauchy's Theorem is the fundamental theorem upon which complex analysis is based, there is no "standard approach." The appropriate choice depends upon the prerequisites for the course and the level of rigor intended. Common methods include…

Descriptors: Mathematics Instruction, College Mathematics, Mathematical Logic, Undergraduate Study

Farris, Frank A. – PRIMUS, 2017

The "domain-coloring algorithm" allows us to visualize complex-valued functions on the plane in a single image--an alternative to before-and-after mapping diagrams. It helps us see when a function is analytic and aids in understanding contour integrals. The culmination of this article is a visual discovery and subsequent proof of the…

Descriptors: Color, Mathematical Concepts, Mathematical Logic, Plane Geometry

Kinney, William M. – PRIMUS, 2017

Educational modules can play an important part in revitalizing the teaching and learning of complex analysis. At the Westmont College workshop on the subject in June 2014, time was spent generating ideas and creating structures for module proposals. Sharing some of those ideas and giving a few example modules is the main purpose of this paper. The…

Descriptors: Learning Modules, Teaching Methods, Mathematical Concepts, Mathematical Formulas

Sachs, Robert – PRIMUS, 2017

A new transition course centered on complex topics would help in revitalizing complex analysis in two ways: first, provide early exposure to complex functions, sparking greater interest in the complex analysis course; second, create extra time in the complex analysis course by eliminating the "complex precalculus" part of the course. In…

Descriptors: Mathematics Instruction, Undergraduate Study, Validity, Mathematical Logic

Cooper, Thomas; Bailey, Brad; Briggs, Karen; Holliday, John – PRIMUS, 2017

The authors have completed a 2-year quasi-experimental study on the use of inquiry-based learning (IBL) in precalculus. This study included six traditional lecture-style courses and seven modified Moore method courses taught by three instructors. Both quantitative and qualitative analyses were used to investigate the attitudes and beliefs of the…

Descriptors: Longitudinal Studies, Quasiexperimental Design, Inquiry, Teaching Methods

White, Jonathan J. – PRIMUS, 2017

A problem sequence is presented developing the basic properties of the set of natural numbers (including associativity and commutativity of addition and multiplication, among others) from the Peano axioms, with the last portion using von Neumann's construction to provide a model satisfying these axioms. This sequence is appropriate for…

Descriptors: Numbers, Sequential Learning, Active Learning, Inquiry

Greene, M.; von Renesse, C. – PRIMUS, 2017

This paper aims to illustrate a design cycle of inquiry-based mathematics activities. We highlight a series of questions that we use when creating inquiry-based materials, testing and evaluating those materials, and revising the materials following this evaluation. These questions highlight the many decisions necessary to find just the right tasks…

Descriptors: Mathematics Instruction, Learning Activities, Mathematics Activities, Inquiry

Toews, Carl – PRIMUS, 2017

Inquiry-based pedagogies have a strong presence in proof-based undergraduate mathematics courses, but can be difficult to implement in courses that are large, procedural, or highly computational. An introductory course in statistics would thus seem an unlikely candidate for an inquiry-based approach, as these courses typically steer well clear of…

Descriptors: Computation, Inquiry, Introductory Courses, Statistics