That is the main struggle for many math teachers, they try to do all these fun and interesting explanations, and the kids just ignore it and go directly for the formulas and forgets everything else. the problem seems easy to solve until you have experienced trying to apply it yourself to a real class of kids needing the material for real grades. It can be done, but a teacher who could do it could make way more money in entertainment etc, since that is what is required to get kids to pay attention.
That was novel. That is how research is done…you start with a problem and figure out a way forward.
That is not how textbooks are written. It’s like, why waste our valuable paper to print the wrong way to do something even if it helps people learn? They only print the right way to do it. The student loses the ability to participate in the discovery process and just becomes a dumb initiate who is forced to believe whatever is written down. It’s more like a degenerate religion you’re forced to memorize without the inspiring examples of all the saints and martyrs who showed others the way before you.
I'm all but certain mine didn't. For sure there was the odd contextualization and examples of real world applications here and there but definitely not enough and definitely not interesting or fascinating ones.
Eg if there was one for trigonometry it would have been something lame like "Alice is standing in the field some distance from Bob and Billy. Calculate the distance based on <trigon blabla>"
Hardly riveting stuff :P
They don't have the time, attention spans or memory to be able to take in both the contextualization and to understand the formulas. After all it's only the formulas' that really matter in the end when performing the calculations in test/exams when you have to show your working out. I also find that I can understand the contextual quite easily but it's another thing to then apply it through formulas.
This is coming from a non-teacher but past student.
At University I had more time to think about the contextuals during my degree, but not in high school jumping from subject to subject.
I did not do well in either set of subjects in K-12 -- even to the point that my graduation from high school was threatened. In college I forced by way back through it all by sheer force of will and got my A's.
There's something fundamentally broken in math/science pedagogy as these subjects aren't really all that difficult. There's far too much time spent memorizing things that are trivial to look up and way too little time understanding how to use them.
An analogy might be learning to cook, and spending all of your time remembering precisely how many spoons, bowls, cups, and cloves of garlic or other ingredients you have. And doing that kind of thing for years, and maybe seeing a demo once of pouring water into a cup. And tests might contain problems like "a party of 5 is coming over for dinner, are you able to set places for all attendees for a 7 course meal?"
The real message being sent is this: "Sorry kids, actually cooking from recipes is only for academics, and to get your PhD and be allowed into the hallowed halls of these academic cooks you must come up with one original recipe (edibility will be determined by peer review)".
In college I retook everything from Algebra up and found the math pedagogy focused more on symbolic manipulation and getting used to how that works in each subject rather than drilling arithmetic in various guises. Tests that required various pre-derived formula were usually just an open book problem. And what mattered was how one went about solving the problem, not the rightness or wrongness of it. Calculators were absolutely expected so you didn't waste time fighting with trivial mistakes.
The sciences usually had a mandatory lab portion that forced application of math to the problem space. Because the labs typically had you collecting your own measurements, it forced you to work through the calculations yourself anyways since there was nowhere else to look up the answer. Again the methods and approaches were where the grade came from, not the slavery to memorization.
Still, while I think the approach I encountered in college was much better than grade school, it still wasn't as good as it could be.
A couple of years ago, randomly browsing YouTube, I came across this home made video asking how they figured out the distance to the moon before modern technology. The host starts out small scale showing he can calculate the distance to things in his back yard using trigonometry and then scales it up to the moon.
My mind was blown, because no one ever told me that. It was simple, anyone could understand it. When I was in school, all I was told was to memorize abstract formulae like calculating the length of sides of triangles based on angles and known length of one side. It was never contextualized to any actual, let alone interesting or fascinating, applications.