By Ekaterina Yurasovskaya
With its highly abstract material and proof-writing that students encounter for the first time, Introduction to Proofs often turns into a gatekeeper to a mathematics major. At the Mathematics Department at Seattle University, we achieved significant results by creating a lecture-free, problem-solving ungraded co-requisite course called Mathematical Communication and Reasoning (MCR). The new course is based on the Berkeley Calculus workshops of Uri Treisman and Math Circles, an East European pedagogy that has been increasingly popular in the US in recent decades. After taking the MCR, students in need of additional support increased their desirable grades (B- to A) by 50%, and cut their non-passing grades (C- to F) in half.
The MCR was designed to offer an informal and friendly space for mathematical practice; to eliminate academic isolation and create a supportive student cohort; to introduce students to opportunities within the department and the university; and to explore the social and psychological aspects of mathematical practice. Students take the MRC concurrently with the Intro to Proofs, and it covers no new material: Typical topics are logic, elements of number theory, set theory, relations and functions, cardinality. All students in the Intro to Proofs are strongly encouraged to take the MCR, and usually about half of the class do so, such that enrollment on the course is 6- 16 students each quarter.
Weekly structure of the MCR course
For example, if the MCR class meets Thursday, then the MCR instructor meets the Proofs instructor on Monday to discuss topics covered in the main course and student performance and difficulties. This meeting is crucial, as it ensures integration of the two courses for both the students and the instructors.
The MCR instructor then creates a set of 6- 8 problems based on the proofs course material covered the preceding week, which students receive on Tuesday. This way students have had the chance to read the textbook and practice some of the Proofs homework before they see the problem set. Students attempt all the problems at home and bring their handwritten drafts to class on Thursday. We have learned over time that the MCR instructor should focus on a single big concept covered in the Intro to Proofs class in the prior week and select problems that explore the concept from multiple angles and illuminate misconceptions. Higher-level math circle literature is an excellent source of nonstandard and creative problems for the weekly problem set. Past exam problems provide a good source as well.
The MCR class session lasts for two hours. In it, students form groups and discuss solutions with each other. Students gain new understanding as they explain their solutions to others, defend their reasoning, and detect faulty arguments. While they work, the instructor walks between groups and answers questions. When a common theme emerges across groups, the instructor may take 5-10 minutes and discuss a particular topic, misconception, subtlety, or a historical reference.
During the last 45 minutes of class, students present solutions to select problems, while their classmates, and sometimes instructor, suggest mathematical and stylistic edits, corrections, and alternatives. An environment of trust and collaboration is particularly important at this point.
After the MCR session, students write up careful solutions to 2-3 problems selected by the instructor. MCR instructors can choose to introduce LaTeX and ask students to submit typewritten work. In this case they gain a skill that will be useful in other classes as well. MCR instructors either provide traditional written feedback or meet one on one with students. The only possible marks are a 4 (minor corrections needed), a 5 (no corrections), or a non-evaluative mark “please rewrite.” If one-on-one sessions are used, they are typically 20 minutes long and student and instructor discuss solutions and any avenues for improvement, as well as any other problems or questions a student may have. Following receipt of feedback, the student incorporates the feedback and corrects the assignment, which then receives additional feedback the following week.
Students also receive reading assignments that emphasize that mathematics is a human endeavor. This might include expository papers on mathematical education and on mathematical progress by the great topologist William Thurston. “Shitty First Drafts” by Anne Lamott, which emphasizes that good writing is an acquired skill rather than an in-born ability given to the select few, encourages a growth mindset. Terence Tao’s mathematical blog provides a wealth of advice on mathematics, career, and writing.
The key ingredients of the MCR course are the rigorous, carefully constructed problem set, an open in-class and meeting environment full of ungraded formative feedback, and many opportunities for students to correct their work at no penalty to the grade. This way, students can focus on understanding mathematical content and view inevitable mistakes as stepping stones towards understanding, rather than threats to their grade. As an instructor, I really enjoy teaching the course because of its relaxed environment of mathematical exploration and lack of pressure to assign evaluative grades. Student comments about the course have been overwhelmingly positive as well. Students reach a firm understanding of essential proofs material, and many push course boundaries and apply proofs course methods to unexpected mathematical settings. The full article in PRIMUS provides more detail on the model and our measures of success. The model may be applicable to institutions of different sizes and to Proofs courses with different content, and I am eager to communicate with other instructors about how we can use it to better serve all of our students.
Yurasovskaya, E. (2024). Integrating the Treisman Model and Math Circles Improves Student Performance in Introduction to Advanced Mathematics. PRIMUtS, 34(10), 1008–1024. https://doi.org/10.1080/10511970.2024.2403102
Ekaterina (Katya) Yurasovskaya is a teaching professor and Sullivan Chair in the Department of Mathematics at Seattle University. She studies effective teaching practices and international perspectives on teaching and learning. She also runs SUM (Seattle University Mathematics) Corps - a departmental mathematical outreach program to neighborhood schools.