# Proof-Based Courses

**MAT 210, 214, 215-217, 216-218**

**MAT 210, 214, 215-217, 216-218**

**Schedule & Organization: **These courses typically meet twice a week for 80-minute sessions on a TTh schedule; 210, 215, and 217 also include a Friday precept. There are weekly readings and problem sets which function as the main learning tool and count for about 30-50% of the course grade. The midterm and final exams count for the remainder of the course grade and these often have a take-home component as well as an in-class component. Collaboration on problem sets is welcome, and substantial support is offered in the form of evening problem sessions run by undergraduate TAs.

**Teaching Style: **The instructor establishes the main definitions and theorems in full mathematical rigor. The textbook is similarly abstract and formal. Students learn to construct formal proofs and counter-examples. By contrast to example-based courses where general principles emerge gradually from hands-on experience with concrete examples, in these courses definitions and techniques are presented and developed with maximal abstraction and generality. Students in 216-218 should already be familiar with this approach and have extensive experience in this style of reasoning about mathematical ideas.

**Work Load: **The weekly readings and problem sets require a substantial time investment outside of class which may well exceed 10 hours per week. These courses are intended as an immersive experience in authentic mathematical thinking for highly motivated students. While most students in MAT214 or MAT215-217 have already demonstrated excellence in example-based mathematics courses that emphasize computation, learning how to construct proofs and to understand mathematics presented in highly abstract and general form is a new approach and a challenging transition for many students. It takes time and sustained effort to learn this new way of doing mathematics and in the meantime it is difficult to assess progress. It is crucial to work with other students in the class and to actively engage in the evening review sessions with the undergraduate course assistants. Take full advantage of time in class and in office hours to discuss your questions with the instructor in order to get the most out of these classes.

*The pace in MAT216-218 is extremely fast, and assumes that much of the material is already familiar from university-level proof-based courses, extracurricular proof-based math programs or (in exceptional cases) substantial reading at the university level. Students without that background will find the workload in this sequence to be overwhelming and will get more out of 215-217 instead.*

**Grading:** The grading scale takes into account that these courses may be academically risky for students because the transition from concrete calculations to abstract mathematical thinking is very challenging. Even a semester may not be enough time to fully master this new approach to mathematics.