What is FAM?
"FAM" is an acronym for an undergraduate course, "Foundations of Abstract Mathematics", offered by the Department of Mathematical Sciences of Stellenbosch University (FAM I and FAM II, respectively) since 2012/2011. The course consists of two year-long modules, FAM I (Mathematics 278) and FAM II (Mathematics 378), offered at the second and the third year, respectively. It is possible to enroll for only one of the two modules. Neither of the modules has any prerequisites, although note that admission to the third-year module is subject to approval by the Department of Mathematical Sciences. The course aims to let the students experience mathematical research, at the level corresponding to students' mathematical skills, and in this process, to uplift those skills.
A bit of history
When teaching a course in calculus at the University of Cape Town in 2008, Zurab Janelidze was approached by a student, Pieter du Toit, with a request to show him mathematics beyond what was taught in the course. This led to a Seminar in Pure Mathematics, which was run by Zurab and attended by Pieter and some of his friends. When Zurab was appointed as a lecturer at Stellenbosch University in 2009, he offered Ingrid Rewitzky and Karin Howell to jointly run a seminar, based on the model of the Seminar in Pure Mathematics, for undergraduate students. The enthusiasm of students attending this seminar led to the conception of the Foundations of Abstract Mathematics modules, based on a framework proposed by Ingrid and Zurab. Thanks to the support of the Head of the Mathematics Division at the time, David Holgate, the third-year module was introduced in 2011, and the second-year module in 2012. The Faculty approved this initiative under the understanding that lecturers involved in teaching these modules would do so in their free time instead of it being part of their official teaching workload. Teaching of these modules became part of the official teaching load by the initiative of the new Division Head, Florian Breuer, a few years after the introduction of the modules. He, Zurab, and Stephan Wagner were the main lecturers in the module for several years, until the resignation of Florian and Stephan (who took up academic positions in Australia and Sweden, respectively). During these years, the module was convened by Zurab. In 2023, the module is jointly convened by Zurab and Sophie Marques. A majority of postgraduate students in Mathematics at Stellenbosch University, who have completed their undergraduate studies at SU, would have followed one or both of the FAM modules during their undergraduate studies. For some of them, these modules played a crucial role in inspiring them to switch their existing career choice (e.g., engineering, computer science, theoretical physics) to mathematics. There are examples where ideas conceived in FAM I and FAM II have led to research topics at the Honors, Masters and PhD levels.
FAM assignments
Assessment in FAM is mainly based on assignments, where students need to apply their creativity to solve problems not discussed in class (or tutorial), compose proofs of theorems, sometimes in a symbolic language (in a formal proof system), come up with examples or counterexamples to a concept/hypothesis, or come up with their own theorems (and their proofs). Occasionally, some of these assignments involve coding proofs using a proof assistant software. Students who show readiness for engaging with a mathematical exploration project (a longer version of the usual assignment), which often comes close to an honors project in terms of its quality and content, are given the opportunity to engage with a longer project in the place of smaller assignments.
The following is a fragment of a conversation among a group of students who were required to append this recording, discussing their semi-joint assignment, to the latter (rewind to play the full video - but its long!). Note that the recording is shared with the permission of all students in the video.
How do students find the experience?
The video above ends with the group members discussing how FAM has impacted them. Here is what some of the other students have said when asked what they have learned in the first quarter of the FAM module in 2021 (once again, shared under permission):
- Reading and understanding equations. We knew about logical operators, but now we know how to use them more effectively to get actual results. This equips one with a toolbox to use in other math modules. Reading and interpreting equations in other modules specifically.
- A philosophical look at mathematics: instead of being given a problem and asked to solve it, now we look at the mechanics of how we can solve the problem and what really encompasses mathematical activity. Comparison with language is fascinating. It is a good life skill to understand logic, which this term contributed to.
- Originally, I thought of this like every other math course: numbers and calculations. Now I view this course more as a course in logic which teaches you how to think. This was very cool, very unlike to what I have done before. Excellent pacing: it was important not to go fast to get a good understanding of what we are working on.
- This term gave me a deeper understanding of mathematics - it was not just about learning a method and solving problems. It was nice that in the beginning more emphasis was placed on effort rather than accuracy. Instead of trying to get it right, one had the opportunity to engage deeper and learn more about the subject, than in other modules where the emphasis is to learn something to get it right. In this module, you learn to understand. The focus was more on understanding concepts rather than grasping the language used to interpret the concept.
- Usually, the student is on the receiving end - now it is the student who was expected to produce a precise mathematical statement that others would be able to interpret correctly.
- The concept of breaking things down and unpacking in proofs. A cool skill to learn. Mathematics is neither invented or discovered. Mathematics is rather something that is within every human being.
- How anything can be turned into math. Mathematics can be made from a normal conversation. How to write down logical reasoning through mathematical steps.
- How mathematics is really so broad around us. I kind of new this, but I did not realise the actual broad extent if this.
- The seminar does not force you to parrot learn - it is much more understanding based. It is a nice thing that the focus is on understanding the work.
- Instead of repetitive information, the lecturer gives us information and lets us build on it while learning from each other. I wish other modules were like that too.
- This is probably the only course that brings thought into it. After the lecture, instead of being happy that the lecture is done, you are still thinking about the lecture. Assessments reflect this too. Putting in extra thought and creativity gives you marks. So assessments allow thought input. You also have the freedom to interpret things in your own way.
- This module teaches you how to formulate your thoughts and structure them in terms of assumptions and conclusions. You must think carefully and understand the process, rather than go through everything step by step or parrot learning, as is often the case in other math modules.
See https://foabma.blogspot.com/ for further feedback.
The Abstract Mathematics stream
A few years ago, FAM has inspired creation of a new focal area (aka stream) within the BSc Mathematical Sciences programme. It features FAM I and II as compulsory modules, among other mathematics modules at the second and third year levels. This stream gives students flexibility to combine a mathematics major with majors in biochemistry, chemistry, physics, genetics, computer science, applied mathematics, and mathematical statistics (only four major combinations are possible in all other focal areas of the programme).