In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and…

Topology is considered an advanced field in mathematics, and it might seem off-putting to people with no previous experience in mathematics. The classification theorem, which lies within the field of algebraic topology, is fascinating, but understanding it requires extensive mathematical knowledge. In this manuscript, we present a modular object…

As mathematics educators we want our students to develop a natural curiosity that will lead them on the path toward solving problems in a changing world, in fields that perhaps do not even exist today. Here we present student projects, adaptable for several mid- and upper-level mathematics courses, that require students to formulate their own…

A pair of elementary exercises, one from topology, the other from group theory are such that if one replaces three words in the topology problem, you get the group theory problem and vice-versa. This suggests connections between the two that are explored here.

Symmetry is found in the visual arts, architecture and design of artefacts since the earliest time. Many natural objects, both organic and inorganic, display symmetry: from microscopic crystals and sub-atomic particles to macro-cosmic galaxies. Today it features strongly in higher mathematics such as Linear and Abstract Algebra, Projective and…

Automobile sunshades always fold into an "odd" number of loops. The explanation why involves elementary topology (braid theory and linking number, both explained in detail here with definitions and examples), and an elementary fact from algebra about symmetric group.

The history of the study of dynamical systems is traced, examples are given, and the contribution of Birkhoff and Smale are described. (DT)

Ten recent results in pure mathematics are described, covering the continuum hypothesis, Diophantine equations, simple groups, resolution of singularities, Weil conjectures, Lie groups, Poincare conjecture, exotic spheres, differential equations, and the index theorem. Proofs are omitted, but references are provided. (DT)

This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)

Part of the report by the Committee on Support of Research in the Mathematical Sciences is presented. It deals with an overview of the mathematical sciences from past to present. (JG)

Describes the effect, method, and mathematics of the following magic tricks which can be used in introducing mathematics lessons: the Ninth Card, Fibonacci Revealed, the Case of the Missing Area, I've Got Your Numbers, and the Card That Turns Inside Out. (MKR)

Quantitative algebraic and qualitative topological models can be used in the quest for simple explanations in the field of teaching and learning. The examples described cover such diverse topics as teacher anxiety and classroom control, process-product studies, aptitude-treatment interactions, the pace of instruction, the size of classes, and…

This is a collection of 24 papers presented or prepared by participants of the Comprehensive School Mathematics Program (CSMP) International Conference in 1970 covering a wide range of topics in mathematics and geometry education. Conference recommendations are summarized, including the use of geometry to lead into other branches of mathematics…

The Czechoslovakian Academy of Sciences is sponsoring an experimental approach to the modernization of the geometry curriculum. Geometry is viewed as ancillary to other parts of the curriculum and is taught as appropriate to other subjects (e.g., algebra). Combinatorial geometry is taught formally. (SD)