The introduction to general topology represents a challenging transition for students of advanced mathematics. It requires the generalization of their previous understanding of ideas from fields like geometry, linear algebra, and real or complex analysis to fit within a more abstract conceptual system. Students must adopt a new lexicon of…

Concepts of Mathematics (21-127 at CMU) is a course designed to introduce students to the world of abstract mathematics, guiding them from more calculation-based math (that one learns in high school) to higher mathematics, which focuses more on abstract thinking, problem solving, and writing "proofs." This transition tends to be a shock:…

Proof is a foundational mathematical activity that has been underrepresented in school mathematics. The recently adopted Common Core State Standards in Mathematics includes eight process standards, several of which promote the inclusion of reasoning and proof across all grades, courses, and students. If students are to reach the expectations…