I still would argue that method of teaching is perfectly fine.
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.
> Ask why do I need to determine similarity of two vectors
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.
Take a random group of students from the general population and one of those examples (Edit: or any single given example whatsoeve). Turns out 95% are not really interested.
Edit 2: The teacher probably gave some example from biology or something that you didn't care about and therefore forgot about it.
The core skill of the teacher lies in recognizing the interests of the pupil and then working on refining those skills so that the pupil can use those skills for at least their betterment, if not the society.
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.
This is exactly how I would teach SO many subjects in school: take what kids are actually interested about and make them see the connection.
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.
If I'm writing an instruction manual for a lathe, I'm probably not going to run through all the varying things you produce with the lathe. I'm going to talk about correct use of the lathe. It won't help you understand what a lathe is for, but it will help you use the lathe correctly.
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 think more intuitive/holistic ways of teaching would be a lot better. But it's hard to do, especially in dead tree format.
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.
> You cannot simply explain to someone complex stuff
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.
Like I said, I've had many of those moments. It is strange you haven't. You telling me I just wasn't aware of my own grinding is a bit strange. I've literally had examples where person A explains X and I don't get it, then 2 hours later person B explains X again and I get it. There's no grinding in the middle. There's good and bad ways to explain complex things, and the success rate will vary and the amount of "grinding" you need to do I think also varies depending on the quality of the explanations you get. Maybe I'm missing the core of your point.
The only charitable reading of the comment is that the GP is a purely sequential learner - they've dont have epiphanies and gasps of insight like a partially global learner. The learning style stuff is semi-debunked since usually people dont just fit in one category, but they are classically divided as:
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.
What you're describing is well-known to mathematicians. The idea is that if you struggle hard with something (perhaps till you have to give up), your brain feels there is "unfinished business" and keeps working on the problem even when you're not consciously thinking ("grinding" in your words) about it. The second time you engage with the problem, you're primed to understand it - or understanding may just pop into your head, like an unexpected pizza delivery. A couple of examples:
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!]
No, that is different. I'm aware when my background brain does work because like you say next time I arrive at the subject it's easier. This is not that.
I agree and disagree. I've definitely experienced this, but I've come to the conclusion that the problem is bouncing around in my head for those 2 hours, quietly grinding away in the background. Then, when person B explains it, my brain is more receptive to it. At work as a developer, I'll often encounter a difficult problem, and walk away for a few minutes. Then, if still stuck, I'll go to the gym. Usually when I come back I'll have the answer coded in 5 minutes.
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.
Exactly, I don't understand how people are bending backwards to not accept this. I could be explained integrals as "it's a banana", 2 hours later someone explains it properly and I get it, and some of these commenters would tell me my brain was working on "it's a banana" and that's why I understand integrals the second time.
Well I've definitely had those moments but I happen to have an alternative explanation.
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.
I do not argue that guidance is not needed, it also is, but there is no shortcut that let's people skip own hard work by telling someone incantation of words or sentences in special order that will make things click - without even trying to grasp the subject.
You're making a false dichotomy. Learning is a combination of guidance and own hard work.
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).
What I argue is that there is no approach to guidance that will substitute for own hard work.
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.
This is quite an old book. I wish I had an access to or even knew about this book during my school days 40 years ago. When I discovered this book, I was already a middle-aged engineer and was just looking for books for my kids. The first two pages blew my mind. If only I had this book back then...
However, I remember now the several sleepless days and nights in sequence when I tried to make sense of finite fields at university. I could not sleep; I could not rest... And then I understood that you can create your own algebra whenever you want; you just need to follow the rules. This was so mind-blowing, and at the same time, no single teacher even tried to point out this wondrous fact that actually changed my mind in such a significant way. It could literally have saved a couple of years of my life if I had read this book back in high school. I'm not a mathematician; math exists for me only when it is applicable to what I have at hand. But you realize that you missed a HUGE AMOUNT OF TOOLS, too late in your life.
And you'd be dead wrong by all methods of pedagogy that we find useful!
People learn best with stories and meaning, its why the ancients were able to reproduce stories of almost inhuman length via memory.
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.
Knowing why and when to use math is equally as important as knowing it. One of the reasons I lost my love for learning it was this missing information.
What I found very annoying with calculus specifically was the previous 15 years I had been memorizing formulas. Formula to get the area of something remember this thing. Formula to get the volume of something remember this formula. Formula to get the angle of something remember this formula. But if I had known the way of calculus and derivatives I could make those formulas. I now have the ability to have a formula factory instead of devoting tons of mental space to keeping those formulas. I feel I wasted 15 years rote memorizing things instead of understanding the N spaces things live in and how to get the formulas.
But you--or at least most of your classmates--probably weren't in a place to just learn calculus before taking high school physics or even simple geometry. And this happens at a lot of different levels with math, physics, chemistry, etc. There are a lot of inter-relationships and often moving forward requires taking some things on faith (for now).
Illustrative point for the “the value isn’t explained” issue: I’ve spent two years on calculus in school plus some more time on my own and don’t know how or why I’d e.g. use it to derive the formula for the area of an oval (I think that’s the kind of thing you’re getting at?)
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?”
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.