Justin Skycak @justinskycak 2024-08-08
How to get from high school math to cutting-edge ML/AI: a detailed 4-stage roadmap with links to the best learning resources that Iâm aware of.
I recently talked to a number of people who work in software and want to get to the point where they can read serious ML/AI papers like
Ram @ramsetty123 2024-08-08
Good post.
In my opinion, as a former math major, the single biggest hinderance to learning and doing math, is proofs.
Archaic, generally useless to learn, and they get in the way of actually doing math while devouring hours of your time.
Alex Smith @ninja_maths 2024-08-10
I found your comment really interesting. Iâm an applied mathematician, but I also very much enjoy pure math and always have. I recently completed our Methods of Proof course, which places me at the center of this discussion.
Here are some of my immediate thoughts:
Every undergraduate math student should take an Introduction to Proofs course. Proofs form the very fabric of mathematics, and you can only truly call yourself a mathematician if you understand how they are constructed. A foundational understanding of sets, logic, and methods of proofâsuch as direct proof, induction, contrapositive, and contradictionâis essential knowledge that all students should possess.
However, a Methods of Proof course doesnât need to be arduous. It should be one of the most enjoyable parts of learning math.
To help students transition to Methods of Proof with as little friction as possible, I employed the same principles
I use when designing any course:
A highly scaffolded curriculum: Mastery-based, highly granular, strong emphasis on worked examples.
Master concrete examples first: This helps students develop an understanding and intuition of the objects and the relationship between them first before proving general statements.
Tight feedback loop: Feedback should be provided after every problem. This helps students quickly identify mistakes and misconceptions and take corrective action before the next problem.
We launched our Methods of Proof course in June, and itâs been a great success. Students are completing the entire course in less than six weeks!
In a university settingâwhere most students encounter proof for the first timeâmany find introductory proof courses incredibly tough. But why? Hereâs my analysis:
Combined subject matter: About 20% of Methods of Proof courses are taught in a âtopic onlyâ setting. In other words, students are expected to learn methods of proof while simultaneously studying challenging subjects like real analysis! This is a tough ask, even for the most capable students.
Unguided, inquiry-based learning: An unguided, inquiry-based approach is often the norm even for institutions offering a dedicated Methods of Proof course. This isnât to say that inquiry-based learning should be entirely avoided in this type of course, but it should be on the periphery and not the central underpinning of every problem set.
Limited feedback: In traditional university settings, providing the immediate and frequent feedback students need to succeed is impossible, and active learning is also virtually impossible.
After an introductory Methods of Proof course, students should take at least one or two more proof-based courses, such as Real Analysis or Abstract Algebra. The topics covered in a Methods of Proof course are pretty elementary and disparate. A second course allows students to see how these methods can be used to develop an entire mathematical theory from the ground up, which is a central aspect of math that every student should see.
Beyond that, students should be able to pursue non-proof-based mathematics if that aligns with their interests. I studied math at the University of Manchester in the UK, and we had this option. I quickly gravitated towards applied mathematicsânot because I didnât enjoy pure math (I did, especially number theory), but because I felt applied math was more relevant for solving real-world problems.
I get incredibly frustrated with some of the resources I read. Undergraduate texts on graph theory are my go-to example here. Many of these texts follow a strict theorem-proof-theorem-proof structure. But this is Graph Theory! Why not let students first gain practical experience by running Dijkstraâs Algorithm before diving into proofs about its properties?