I know no good math teachers. Most math teachers pre-college don't know the material themselves. I still don't know why something is "efficient with" (a coefficient). And every time I try to get into any math there's 79 unspecified prerequisites I supposedly need to understand all the jargon. 3blue1brown is preaching to the choir AFAIK. I get lost trying to watch his videos as they also expect you already know the material.
> I still don't know why something is "efficient with" (a coefficient).
That's just a name. You don't need to know the etymology to understand the topic at hand. But if you want to know, you can think of co-efficient as "together-effector". In 3x, 3 is a coefficient of x, as it's something that "acts together" with x. There isn't really much more to it. You don't understand concepts by studying their names in much detail.
Another thing to get used to is that the same word may be used for different concepts in different contexts (or different words for the same thing). Often there is very little connection between them. You have to accept this as a fact of history. You can of course try to look for connections, but don't get frustrated because of not finding any. And some concepts are just badly named. Naming things is hard. Don't try to deduce things based on the names of things. Learning to understand names as just labels is an important step in acquiring good abstraction skills necessary for math.
I think this may be an issue for certain people that they approach math from just the wrong angle. The jargon and terminology is of course important for understanding of the content, but the prerequisites are not really lingo-understanding, but concept-understanding. Unlike in other fields, you don't learn math by pouring over the text and reading page after page. You read something and try to understand it. You play with it, you stand up, take a walk and try to work out why something is the way it is, think of examples, try to clear up what you don't quite get. You cannot use the same learning tactics that you'd use for, say, accounting or humanities. The pages read per hour is pretty low in math.
> you don't learn math by pouring over the text and reading page after page. You read something and try to understand it. You play with it, you stand up, take a walk and try to work out why something is the way it is, think of examples, try to clear up what you don't quite get.
Absolutely this. Math is a not a spectator sport, you can't acquire the skill and knowledge by reading. You must engage with it, fence with it, wrestle with it, make it your friend and spend time with it.
I remember not so long ago playing with some equations and accidentally proving Pythagoras' Theorem. It wasn't new, it wasn't deep, but it was satisfying, in much the same way as getting code to perform as you want, accomplishing some task in an elegant way.
But if you don't spend the time, you'll never get the insight(s).
Yes, I think some people falsely learn early on that stumbling upon something they don't quite understand is a sign that they are failing.
I sometimes even think (though I'm not so sure) that having too good textbooks can be an impediment for above-average students as they have less to wrestle with and things are laid out and pre-disgested too much. I may be wrong though, because you can then just get to the next levels until you hit the wall.
But I feel like anecdotally I got good value from deriving some basic things and reorganizing the somewhat badly presented materials in some of my classes. Later on I found very good American textbooks (although widely out of the reach of my wallet at the time), from MIT etc. which were crystal clear in some of the intuition that I had to sweat out for myself. Maybe I could have spent that energy at higher levels, though.
In my country (Hungary, but I saw similar attitude in Germany as well) its not unusual to learn entirely from the lecture material (own notes or provided script and assignments) without any textbook. Some people do get a book from the library, but it's not really expected from you.
my original comment before editing said book/teacher. I use them interchangeably, but I hope the message is still there. Namely: Is it a difficult subject or do you need a better teacher?
More people should play with math like they play with new programming languages or tools. I've learned a lot by just trying to describe in geometric terms the relationships among shapes I see around me or use in my design work.
I learned more about trig working out what angle I needed to input in some design software to make a line cross two exact points than I ever did in high school.
I'm very much a visual learner so I find geometry in particular very satisfying since I work out relationships visually (and then mathematically to prove it).
I find that I learn immensely more in even tiny little tasks where I care about finding the result, compared to any assignment or exercise. Suddenly you look at the resources in a totally different light, critically evaluating what they are saying. Like, I'd pull some lecture slides or papers from the internet and read them in a very different way, compared to actual lectures. The roles are changed, I suddenly see the math (or whatever the topic is), and evaluate what it can do for me in the particular case. Not like what I need to do for the math to conquer it and learn it to the satisfaction of a teacher.
This also applies to computers. Just get your hands dirty, try to build something and you'll find yourself learning about so many things and tools incidentally while trying to fix something.
I can learn much better this way than going chapter by chapter in a book.
But interestingly, even graded, project-style assignments don't accomplish the same effect, only if I'm really onboard with the topic. I seem to need a spark of excitement of "this is not exactly the way you're supposed to be doing things" and just breaking things and playing.
Couldn’t have said it any better. Whenever someone is asking me if I have been introduced to a tool needed to do a job, I tell them “no, but only because I never had a reason to learn it. I’ll pick it up.”
I can pick something up very quickly when I have an actual reason to know it, and getting a good grade in a course is not a strong enough reason compared to “Learning this will help me save the world”. I’m not going to be passionate about it or inquisitive or asking myself the extra questions I normally ask myself when I’m learning something in order to apply it to a problem. And real problems are better than made up problems.
I know no good math teachers. Most math teachers pre-college don't know the material themselves
I’m hoping to change this, not for all high school students, but for those I meet. I’m majoring in pure mathematics with a teaching option right now. My current plan is to teach high school algebra and calculus, as well as physics.
The other comments are right about terminology. I have no idea where the word coefficient comes from, yet I use them every day and understand them pretty well. At the moment, I’m studying ring theory, ideals, quotient rings, and ring homomorphisms and isomorphisms. Trying to figure out where these abstract terms come from may be interesting but it doesn’t help me in my proofs at all.
If you want to understand anything in math, the best approach is to read the definition, read some basic theorems, look at a few examples, and then start playing around with these things to get a feel for them. Try to prove some of the basic properties of the object you’re working with in terms of the axioms. Keep going, proving larger and more complicated theorems, and try some exercises that apply this knowledge to solve counting problems or measurement problems. If you’re learning about isomorphisms then look at some examples and try to understand how classes of object relate to one another, for example by having the same structure in their operations.
If you’re not able to understand 3blue1brown’s videos then you might benefit from using Khan academy to build up your understanding of the fundamentals. It has material stretching from grade 1 to college. Solving problems feels like playing a game, so it can be fun to sit there for hours just doing the problem sets.
Math is a deep, deep subject but it doesn’t require memorization. It just requires lots of effort and struggle in order to develop mastery.
You've probably seen this lecture but I'll pass it along anyway as I found it to be very insightful on the topic of teaching math and physics by using computation.
As someone with a mechanical engineering degree with minors in math and physics I would have loved to have been exposed to this while I was learning the stuff.
> I still don't know why something is "efficient with" (a coefficient).
I can't parse this - would you care to expand on it?
> 3blue1brown is preaching to the choir AFAIK. I get lost trying to watch his videos as they also expect you already know the material.
If you care, you can always track back to earlier material and build your knowledge. Most people I know in a similar situation to yours don't actually care, and won't actually put in any effort.
And that's a reasonable choice. There are places where math is hard, needs work, and most people have come to a point where they feel that the effort won't be worth it, that the ROI isn't there. But if that's your decision, then own it.
If that's not your decision, and you actually want to know more, there are things you can do about it.
FWIW, I know a lot of really good math teachers, but I agree that there are also some really poor teachers. If yours were bad then I feel bad about that, although there's nothing I can do about your experience in the education sector. What you do about it now, if anything, is up to you.
This is the realisation I had (20 years late). Especially beyond high school, maths is hard. You have to put the effort in, and you have to do the example problems.
I think its expressiveness and conciseness makes it more dispiriting to study: one page of a maths textbook can contain so much information, and it feels like you're just stupid when it takes an hour to understand that page fully.
> I think its expressiveness and conciseness makes it more dispiriting to study: one page of a maths textbook can contain so much information, and it feels like you're just stupid when it takes an hour to understand that page fully.
This is very important. Apparently some people get frustrated by this because they are used to studying lower-information-density textbooks. Maybe there would be value in teaching these meta-skills, like how to approach learning each subject. People may apply techniques, like rote learning or brute-force cramming, reading the same page for hours on end, expecting to magically grok it, where these are ineffective of even counterproductive. Not sure how well this is teachable though.
- illustration — whether drawings, gestures, animations, using objects, whatever works, but have a visual equivalent to 'translate' notation; the typical example would be fractions which are as simple as it gets with drawings and hard as hell using only numbers, for most people discovering them. The same is true for most common functions and transformations (which, when you reduce it to the maximum, are the entire body of applied math).
- real-world application (typically the best candidates are simple physics or statistics in high school). The point here is to give a point of reference, a "feeling" of the mechanism or logic or paradigm. The deeper problem is that this application must feel 'interesting', like a puzzle, and not everyone is interested by the same things. That's where, imho, narrow AI could help craft customized courses with ad hoc examples that fit people's "world view" and "approach" to problems, and show animations for every variable, equation, let people play with those and see the results.
Both probably fit in the "faster feedback" approach, i.e. have the teacher/teaching material give feedback as fast and as often as possible, to guide the learning mind on the right path. This is an extremely important discovery of UX in the last 50 years and I believe education has much to learn from these very applied insights into human cognition.
illustration — whether drawings, gestures, animations, using objects, whatever works, but have a visual equivalent to 'translate' notation
This approach breaks down when you go into higher math. You end up working with these abstract objects that have no visual representation. You may be able to find an equivalent object from geometry or graph theory, but it may be so complicated you could never draw it. Heck, even basic shapes like the Platonic solids are difficult to visualize, let alone draw, correctly.
I solved a problem recently requiring me to count the number of vertex colourings of an icosahedron, up to rotational symmetry. Trying to draw one of these things and visualize all of the possible rotations was basically impossible for me. I ended up loading Blender and playing around with one in 3D, using the coordinate planes to see all the axes of symmetry.
If I had been given a higher dimensional object like a tesseract then all bets would have been off. I would have had to find another approach entirely, likely algebraic.
> This approach breaks down when you go into higher math.
It doesn't have to be literally visualization. Just some "feeling" for the objects. Some conception of them as actors, actees, things interacting, or having emotions or whatever. Again, whatever works. Maybe some people can work with concepts just based on strictly memorizing the definitions and drilling through proofs, but others need to conceptualize it all into a narrative. I don't mean dumbing it down, but making it easier to "grab" mentally. Hard to describe.
> There are places where math is hard, needs work, and most people have come to a point where they feel that the effort won't be worth it
The problem with math is that it's seems much harder than it actually is. Every time I have to look into some specific math I haven't worked with before I spend a lot of time decoding what they actually mean. Usually what is happening is not that difficult at all, mathematicians just insist on writing it down in the most convoluted and incomprehensible way imaginable.
It's like they took every bad coding practice and applied it all to math. Why have descriptive variable names if you can just use random greek letters ? Why give a function or operation a name at all if you could make up some weird symbol instead ? And of course you want those symbols to have different meanings depending on context. Imagine if programmers worked like that and wrote everything as obfuscated C++ code, because fuck you.
> Why have descriptive variable names if you can just use random greek letters?
This is a common complaint I see from programmers all the time. However, I've seen people trying to do real maths with descriptive names, and it fails very, very quickly.
This is by no means comprehensive, but as a (probably thought) experiment, try writing reasonable complex code without variable name completion. You write the same variable name over and over and over, and after a while you realise that since this variable is only use in this screenful or two of code, it's easy enough just to use a short name and know what it is, because you're only thinking about it here, in this context.
I'm not going to try to defend mathematical notation in general because there are enormous inconsistencies, many of which are historical, accidental, and indefensible.
But when you're doing the maths it becomes tolerable, then usable, then actively helpful. It's like the parentheses in Lisp - all newbies complain about them, and those versed in the art know that after a while they not only don't matter, they are a genuine positive.
But unless you take the time to do the math, that won't happen for you. That makes it sound like a deliberate barrier, but it's not, it really isn't.
> This is by no means comprehensive, but as a (probably thought) experiment, try writing reasonable complex code without variable name completion.
So why don't math tools have variable name completion ? and/or why insist on still doing things on paper in 2019 ?
The other things about variable names is that they force you to think about what a variable actually contains. This in itself is very helpful not only for others reading your work, but also for yourself when trying to grasp a problem.
The research mathematicians I work with just stare in incomprehension at the idea of doing their work on any computer-mediated system. Yes, there are things that can be done, and yes, computer proof-assistants have made huge strides, and yes, there are always people at the cutting edge doing amazing work.
But your everyday research mathematician will just stare in disbelief.
I don't know your background, your profile is empty, but it sounds like you are someone who genuinely has no idea of how research in math works, and therefore feel that you really must have a better way of doing things. And maybe you have. But speaking as someone who has a PhD in pure math, and who has worked in safety critical software, I can only say that so far everything you're suggesting just really doesn't make sense.
The reason that for centuries mathematicians use single letter glyphs to represent the things they're dealing with is because it is, for the purpose of doing the work, the most effective thing to use.
> The research mathematicians I work with just stare in incomprehension at the idea of doing their work on any computer-mediated system. Yes, there are things that can be done, and yes, computer proof-assistants have made huge strides, and yes, there are always people at the cutting edge doing amazing work.
I'm not talking about computer assisted proofs or anything like that. Just using a readable syntax and the mathematical equivalent of a word processor would be an enormous step forward. No one is writing books in cursive with a fountain pen anymore either, which is basically analogue to what mathematicians are still doing.
> I'm not talking about computer assisted proofs or anything like that.
No, I'm not talking about proof assistants either, I'm talking about actually doing math using any kind of computer system.
> Just using a readable syntax and the mathematical equivalent of a word processor would be an enormous step forward.
Do you have any idea of how to do that? I've done research in math, and I've written software for safety critical systems. I don't know how to create a system like a word processor for math that would let me actually do the math.
Do you know how to do that? If so, please, let me know.
Why have descriptive variable names if you can just use random greek letters
Because math is abstract. The variables and constants don’t mean anything in most cases and are only given different symbols to distinguish one from another. Descriptive variable names or even descriptive subscripts get OLD very quickly when you’re working through a page of calculations to solve a problem.
The beauty of math, if you’re using it to solve some real world problem, is that you can forget about what the symbols mean and focus on solving the equations, evaluating the integral or derivative, or whatever other calculation you’re doing.
Once you’ve completed all of the calculations and gotten your answer simplified as much as possible, then you can go back to the problem you were trying to solve and interpret your result in context. Since you’ve done all of your calculations purely symbolically, you can see the relationships between the quantities you’re working with. In many cases, some quantities you thought were important actually cancel out and don’t appear in your final answer at all. This tells you something fundamental about the independence of your quantities. Perhaps the most famous case of this comes from physics: the acceleration of an object in free fall is independent of its mass. You would not see this so clearly if you substituted numerical quantities as soon as possible.
> The variables and constants don’t mean anything in most cases
Don't they really mean anything or are mathematicians just unable to grasp what they mean ? To me this seems like a symptom of an underlying bigger problem in our understanding of math rather than those variables not having a meaning.
True. Many times the notation gets in the way, but math is very conservative in this regard. Often I'd just prefer seeing a formula in SymPy (symbolic math library for Python) code.
