You cannot simply explain to someone complex stuff - best way is to let people grind through to build their own understanding.
Parent poster wrote that "it’s useful for determining the similarity of two vectors" - now I would ask why do I need to determine similarity of two vectors as it does not mean much to me - if I would be grinding through math problems I would most likely find out why, but there is no way I could understand and retain it when someone would just tell me.
Simple:
Start with
a) Suppose you are making a video game..
b) Suppose you are determining ballistic trajectory of your missile system based on model rockets
c) Suppose you are running a fighter robot group..
Or any of the stuff children are supposed to *actually* do and then take these classes with determination to do the actual creative things that they wanna do all life.
There is an aspect of jest in the above comment, but it also contains some likely truth. Children love doing stuff, and these are the things that may enable them.
Edit 2: The teacher probably gave some example from biology or something that you didn't care about and therefore forgot about it.
And that is one of the toughest things to get right. Children are extremely curious, that's how they learn and master absolutely anything including arts, dancing, music, history, skating, catching insects, street smarts etc. It's on us as teachers to not let that curiosity wither into nothingness.
The worst thing was to learn something just because the teacher said so. If I hadn't had the motivation not to fail, I would definitely not have gone this far in my studies and in life.
I think the same goes for mathematics instruction. The thinking is that you will soon take courses that will make specific application of these tools that have broad application.
I am not from the west, but I watched Band of Brothers. From most of the ww2 media there is always a term being used. "Why we fight?"
To get someone understand something holistically, as in link to their previous knowledge base, requires knowledge of what their knowledge base is. Traditionally this has been done with structuring the teaching with prerequisites etc and hoping it works.
I struggle with this quite a bit when I teach students with heterogeneous background. To be effective, one has to first probe what the students already knows to be able to relate the new stuff to that, and this requires interaction. Hypertext is/would be helpful for self-learning, but it's sadly very underutilized. LLMs may be better. But probably even those can't at least in the current form replace interactive human teaching as they don't really form/retain a model of what the user knows.
This is absolutely not true. Many times I've experienced the moment of something complex "clicking" after hearing or reading an appropriate explanation for a phenomenon - finally seeing the right visual or appropriate example or comparison. I find it hard to believe you've never experienced this other than through grinding problem sets.
I believe something "clicked" only because you were grinding or already quite familiar with the topic.
There is no way to simply explain complex topic so someone would just get it or someone would "click" on after reading one book on the topic.
Sequential learners prefer to organize information in a linear, orderly fashion. They learn in logically sequenced steps and work with information in an organized and systematic way.
Global learners prefer to organize information more holistically and in a seemingly random manner without seeing connections. They often appear scattered and disorganised in their thinking yet often arrive at a creative or correct end product.
The great mathematician Henri Poincare was struggling with a problem on fuchsian functions. He made some progress, and then: "Just at this time, I left Caen, where I was living, to go on a geological exursion under the auspices of the Schools of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it ..." [1]
"Incubation is the work of the subconscious during the waiting time, which may be several years. Illumination, which can happen in a fraction of a second, is the emergence of the creative idea into the conscious. This almost always occurs when the mind is in a state of relaxation, and engaged lightly with ordinary matters. Helholtz's ideas usually came to him when he was walking in hilly country. ... Incidentally, the relaxed activity of shaving can be a fruitful source of minor idea; I used to postpone it, when possible, till after a period of work." [2]
[1] Jacques Hadamard, "The Psychology of Invention in the Mathematical Field" (p. 12--13)
[2] J. E. Littlewood, "Littlewood's Miscellany" (p. 192) [A wonderful book, by the way!]
However, I've definitely had the case where person B just explains it better (for me). I still can't completely discount that my brain was primed for it by person A.
Person B explains X and I don't get it, then 2 hours later person A explains X again and I get it. There's no grinding in the middle. I just needed both perspectives to make sense of X.
Maybe you prefer to figure out everything yourself, but you have just one lifetime, and having access to guidance while grinding will allow you to learn things faster (and thus more).
I do not argue that guidance is not needed, it also is, but there is no shortcut that lets people skip own hard work by telling someone incantation of words or sentences in special order that will make things click.
Grinding through to build your own understanding when someone can just give you useful meaning and context to connect to your other parts of learning is a core teaching skill, and anyone avoiding that because its "too hard" is doing a deep disservice to their students.
Actually, I can count the times I’ve applied math from later than 6th or 7th grade on one hand. I’m almost 40 and have been writing code for pay since I was 15. I struggle with this with my own kids and dread their reaching those later classes because I have no compelling answer for “why do I have to learn this boring shit?”
Otherwise at the end you get people dissatisfied they cannot get job but they have a degree.