That's a very good approach, which I try to follow in all my writings. Basically, I imagine a reader that is generally uninterested in the material, so the first thing to do is to "pitch" the mathematical concept using a simple example, or just say why the concept is useful. In the remainder of the lesson, I assume the reader might lose interest and stop reading at any point, so this is why I put the most valuable content first (definitions and formulas), followed by explanations, and finally a general discussion about how the material relates to other concepts.
I am always so disappointed when I hear people express this. Being able to place an effort in a broader context is so helpful in being able to approach the work well. A few teachers in the Literature department in high school had the same attitude and it was incredibly demoralizing and left me kinda directionless in their classes. I wish things like https://youtube.com/watch?v=suf0Jdt2Hpo had existed back then to give me some idea of what useful and interesting literary analysis looked like and could do.
I used to know a math lecturer, and his attitude was very much that it was his job to throw proofs at his students, and the bright ones would put the rest together for themselves.
He wasn't even remotely interested in the less bright ones, and certainly not in presenting the material in a way that made it easier for them to follow.
Digital has real potential here, because you can build animations and virtual math labs to explore concepts and give them a context, and suddenly math becomes practical and not just an excuse for wrangling abstract symbols for the sake of it.
It'd be cool if, once you start your major, there was basically an overview class explaining why each of your courses is important and what concepts (at a high level) you should be grokking with each course and semester; basically some context to frame your learnings.
I learned trig and geometry from my shop and programming courses... Trying to do graphics in QBasic and determine lengths and angles in carpentry gives concrete examples. It's not as though most math sprang spontaneously from pure thought-stuff - at some point, architects, inventors, astronomers and others in concrete endeavors discovered these rules.
It's not difficult to explain the first few topics on calculus in terms of distance, speed and acceleration. Other examples I remember were washing lines (a catenary), the path traced by a stream train's wheels and rocketry.
My teacher in England always had a real world example, but students with the other teacher didn't.
But sometimes there is no real life example. Fundamentally, math is abstract. Yes, mercifully, its models often have analogues in nature, making its purpose utilitarian and intuitive. But sometimes no analogue exists. As in much of modern physics, in math, often you have only abstraction.
I think that's why math is difficult to learn. Without compelling illustrations based in the physical world the student must follow the concepts and proofs using the rigor of math's legal transformations, fortified only by the faith that these formalisms will sustain truth. But too often the practitioner must remain oblivious to the utility and implications of both the end and the means.
We did a bunch of linear regression(with maple?) i believe. And some manual differential equations. I was one of the best in that class, but when I asked what we need this stuff for the other people in the class looked at me and said the following:
"if you ask this question you're in the wrong class"
I went on to study engineering, and i guarantee you the other two still don't know what that stuff is good for. They learned the formulas by heart and then went on with their lives.
I mean it was fun for me, it was basically a coding exercise. And I was happy because I was faster than everyone else, but I didn't get why we were doing it.
The same actually bothered my about the studies. Just one example: folding is a fairly easy concept. But for some reason they first want to drill it into your head, you learn a bunch of techniques and then if you're lucky and you stick with it, eventually it clicks and you'll realize why in a completely unrelated class.
Why can't we just provide a simple real life example first and then go on explaining the details?
Aren't things easier to grasp when you can have a real understanding/connection to it? Isn't that precisely why people that learn coding at home tend to be better than those that just studied it, because they were told it's a solid profession to study?
I think with a good example and visual representation you can probably teach most of the stuff that's taught in Uni to young kids. But then you would be forced to admit that you wasted a lot of time during your own life, who'd want to admit such a thing.