This dissertation explores bridge n-sections of knotted surfaces in 4-manifolds by defining invariants that measure the complexity of their topology and by giving geometric constructions that determine Lagrangian surfaces in 4-manifolds under certain conditions. First, we present an elegant geometric construction that generates all triple grid…

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…

In high school, unit discs were all circular. Squares and diamonds were added to the list of shapes in a Real Analysis course and although more bizarre shapes appeared in graduate school, the list of possible shapes didn't appear endless. In this note, we'll show that unit discs can have almost any shape imaginable.

This paper presents the ways that reflective questions were used to develop pre-service mathematics teachers' understanding of metric and topological spaces. In particular, the purpose of developing PSTs' understanding of metric and topological spaces is that understanding these mathematical concepts leads to PSTs' understandings on how to teach…

This paper aims at defining and exploring design principles in a distance technological setting for an educational activity for mathematics undergraduate students, devoted to the construction of basic concepts in general topology, the promotion of problem-solving processes, the development of metacognitive aspects, and, in general, the development…

This study aims to specify to what extent students understand topology during the lesson and to determine possible misconceptions. 14 teacher trainees registered at Secondary School Mathematics education department were observed in the topology lessons throughout a semester and data collected at the first topology lesson is presented here.…

Brian Griffiths (1927-2008) was a British mathematician and educator who served as a member of the founding editorial board of "Educational Studies in Mathematics". As a mathematician, Griffiths is remembered through his work on what continue to be known as "Griffiths-type" topological spaces. As a mathematics educator, his…

This brief biography of Riemann is one of a series of biographical sketches in this journal. Riemann was born in Germany in 1826 and studied with Gauss among others. His numerous contributions to mathematics include the invention of a non-euclidean geometry and the founding of topology. (SD)

Suggestions for what and how to teach topological ideas are presented. (DT)

The origin and evolution of the four-color problem are traced and the proof of the problem is outlined. (DT)

Recorded are some biographical data about the late professor Freudenthal along with some indications about his mathematical work. In an appendix, a tiny part of his mathematical work which can be explained in a fairly direct manner is discussed. (Author)

This study investigated the effect an advanced organizer, a post organizer, or knowledge of a behavioral objective has on learning the mathematical concept of network tracing. Self-instructional booklets on network tracing were given to eighth grade students. The booklets were designed so that the instruction and posttest could be completed within…