Tag: Subject-Specific Education Methods

Many concepts in subjects like math and science can be very technical, requiring lots of background knowledge to understand. This limits the audience of people who can discuss and work with these concepts. But given how much these subjects can influence our society, for example with the modern tech industry and computer science, there is a real need for more people to understand these concepts. Simplified explanations can go a long way towards enabling this. However, it can be hard to write such explanations while also describing the concepts accurately and incorporating scientists’ and engineers’ input on how to present their own work.

I have been realizing recently that many of my math ideas in their “ultimate” forms seem highly suited for formulations in model theory. However, when first trying to study some of the resources I found online for model theory, I didn’t even understand that well what the subject was about or why certain definitions were made, let alone appreciate the overall power of the subject. I think there should be a different approach to teaching model theory, incorporating better motivation and more fully reflecting the incredible power and scope of the subject.

Sometimes, it can be overwhelming to even think about how to approach a subject. Where do you start?

Here, I list some pointers and resources that have helped me most effectively learn math. I also point out some resources which I may not have used myself, but that other people I know have had positive experiences with.

When we learn math, a typical approach is the following. We discuss some motivation, based on what we have already done. This motivation leads, in a possibly informal way, to a formal definition. (It could also be a set of formal axioms, but here we encompass that, as well as any other type of formal “framing,” under the term “definition.”) We then work with our definition using formal logic to produce theorems.

This can be a great approach. However, there are important caveats that we must keep in mind, which I call attention to in this post.

Assume we are developing the curriculum for an extracurricular math program for an advanced student. How could we introduce formal reasoning and the axiomatic method, two cornerstones of the practice of math today, as early as possible? And specifically, what are the implications for this from the philosophical discussion of the formal foundations of mathematics, in my post On the Circularity of Mathematics?