The One True Way of teaching maths
... is self-evidently a ridiculous notion to me. Different approaches in different contexts are going to be more and less suitable for different people. But in my experience, there seems to be (what i consider to be) an odd resistance to providing background / context / motivation when teaching undergrad mathematics.
As an example, let's consider an introductory unit on mathematical group theory.
In my travels online, i regularly encounter people who are midway through such a unit, or who have successfully passed it, asking “So, okay, what use is this? It has no real-world applications, right?” (where a “real-world application” is usually intended to mean “an application outside of maths itself”). Well, it totally does, e.g.:
To me, it seems a no-brainer for the first lesson/lecture to begin with a brief blurb like:
This unit is about mathematical group theory, which emerged in the late 19th century in response to questions around various geometries and the solvability of polynomials. Since then, group theory has become an important area of mathematics with applications to many non-mathematical areas, including cryptography - for example, to facilitate e-commerce - chemistry, and physics. More details, including references, are included in the course guide / handout.
Why isn't this done, or done so infrequently? i have no idea.
Similarly, there's a common perspective that anything more than various bare combinations of definition / theorem / proof is distracting. And not only “background / context / motivation” stuff; i encounter examples being regarded this way[a]. Subsequently, such things can be absent, and many undergrads get the impression that the mathematics they're being taught just sprang fully-formed from the heads of geniuses - that it wasn't developed as a result of wrestling with some practical issue, but as a result of Minds Forever Voyaging.
This makes sense to me if you're a ‘maths person’ and/or love abstraction; the maths itself is its own reward. But for most people, and for significant numbers of people at the undergrad level, maths is not an end in itself. And people can find it difficult to learn about a subject when it doesn't seem to connect with anything else in their head. “I want to get into machine learning, how is all this linear algebra bullshit relevant?”
i suspect that part of the problem is that much of the time it's ‘maths people’ who end up teaching undergrads (whether via lessons/lectures or textbooks), and who can't relate to those not interested in maths for its own sake. Although i'm certainly someone who can, and does, enjoy studying maths regardless of its applications (or potential for applications), it feels a disservice to many students, and to society in general, to prioritise the preferences of us ‘maths people’. The extent to which students actually understand the mathematics they're being taught can affect their lives and the lives of others once they're in the workforce. What are the costs of not making more of an effort to meet them where they're at?☙
[a] Theoretical physicist / category theorist John Baez once said (though i can't seem to find the source now):
Category theorists are dual to ordinary people: they often get more confused when you surround an abstract concept with a lot of distracting specifics. ‘Could you please not give me an example, to help me understand what you’re saying?’