Either a mental chalkboard or carrying out a detailed demonstration without the visuals.
I am absolutely handicapped in this sense. I need pen and paper even for the simplest thing.
I'd like to use a math book without having to write down the math, just do it in my head.
I do mathematics entirely with pens. In class and out of class. For this I have sort of trained myself to think the entire proof before I even write anything.
This will probably stop in a couple of years as I am only a freshman in mathematics and the longest proof I have ever written on a test was for Euclidean Geometry and was just about half a page (face) long.
Still, keeping half a page worth of deductive reasoning stored in your brain at a time should be pretty impressive, I think. However, imagining stuff in euclidean planes is pretty easy.
Serious maths - yes.
Serious physics? Definitely not.
I'm not talking about memorizing proofs, as I used to memorize tens of them (otherwise I'll easily forget the reasoning). I memorized like a hundred proofs and proof like reasonings for a functional analysis test once, it took me a lifetime.
I'm talking about actively doing math in your head. Like, you want to write charts for a sphere in spherical coordinates as an exercise so you sit in a chair, do all your reasoning and once you are finished write down the final answer.
When I did this I used a pencil on a white table and it took me forever, through wrong attempts and useless ones. At the end I did it but I can't imagine something like this done exclusively inside my head. It would have taken weeks.
>I'm not talking about memorizing proofs
I didn't mean this either. I meant figuring out the proof in your head. Doing the actual math in your head.
Ah, ok.
yes but i prefer to do it on windows
Doing math entirely mentally seems like it would actually be counter-productive.
The most important part about math is that one step logically follows another, and anyone can follow along. Constructing a problem and its solution entirely in your head would make it harder to find out if you'd made a mistake in your reasoning.
And even if you don't make a mistake, a third party would have nothing to follow, and might just assert that you're wrong on some other grounds. If you didn't write anything down, it's your word against theirs, but if you did write it all down as you worked it out, they can't possibly disagree (if you're right).
Stephen hawking would do huge calculations in his head by geometrically thinking about them.
It's not like there's much he can do lololol
Yeah I can do proofs in my head.
But writing down forces me to be a bit more rigorous.
If I'm alone doing math I'll "draw" what I'm doing out in front of me with my finger. It makes a fairly big difference in using mental math for me. I remember seeing some human calculator guy do it and thought it might help a little bit and it actually did.
ITT: le John Nash meme
Related question: how can I improve my visualization ability? I hear so much shit about visualization and memory techniques, and I've been garbage at visualization since probably ten or so. I've only gotten it back on draugs and not repeatably. Even when I smoked the devil's lettuce daily, nothing.
Depends on what you consider as serious. I guess the best places to do that is when you're walking to someplace or in the train, etc.
Algebra is easy to do in your head with multiple variable manipulations. I agree with this user But while I can do multiple algebraic manipulations without losing my place, I can't do something simple like 74*60 in my head anymore.
It's the only way I do math. I construct the proofs in my head and convince myself that everything is correct. However I hate writing down the exact details and having to verbalize the shit. It's just tedious work and doesn't add anymore to the satisfaction of having figured out the proof.
Kek
Nope, and I'm not ashamed of it. During one of my first meetings with my adviser she asked me to start working out the beginnings of a new result (i.e. I had not prepared the calculations and only tangentially thought about it) while I was up at the board explaining my progress so far. I got to a point that involved 'brute calculations' (not actual number crunching, as im in pure maths, but not just clever applications of theorems and geometric reasoning either) and I spent probably over 5 minutes standing up there, looking like an idiot trying to do all the maths in my head because I thought that "any student worthy of this position could do it in his head."
It was quite a stupid thing to think, and so I just ended up writing out the steps and it was business as usual. Adviser didn't comment at that pause, but in any case I don't think having to "resort to doing maths by writing it down" is a bad thing in any way, especially when doing research.
for prooving things, I think it's the only way to do it properly and to understand what you're doing.
for doing dull stuff like integration/differentiation excercises, I don't even try
Once you get past sophomore-level math, there's no more problems or exercises.
Literally just you sit in one place, your professor details a bunch of theorems/definitions/proofs on the blackboard, you copy them down and study them, then hope you know the course well enough to pass the final.
Then you fucking suck at arithmetic, bro.
I can perform almost any computation in my head, but my working memory is only about 16 digits, after which I usually need to jot down the rest.
For non-computation, again, I can work with about 8 discrete 'objects' at a time, without needing to reference pen and paper. That's always been the trick for me to work with things in my head, is to categorize what I'm working on to distinct blocks of information and to try and reduce the problem down to 8 or less blocks of info.
visual thinkers are the best creative and innovating thinkers
semantic and syntactic thinkers are far superior in solving problems that have known methods of solving them
its like 2 different types of simulations being used to understand and solve a problem... one constructs more dynamic and large simulation that is harder to parse fast, but easier to dissect and delve into deeper, whilst the other has speed but rigidity
both have their merits, such as breakthrough researcher or rich quantitative "researcher"
maybe you should stop doing drugs you pathetic junky
Yeah, but that is because I can only think in words(that includes numbers and whatnot). That is why things like drawing by memory is impossible for me.
You're not supposed to memorize the proofs. The point of a proof is to demonstrate the sequence in which it takes you to arrive at your conclusion, primarily to show/demonstrate/communicate, to someone else, so they can reproduce it exactly as you did. Think of it like any kind of rule-based sequence that is universally understood. You should be able to produce proofs because you understand the rule that produces the sequence which contains the conclusion you need. Memorization should not be done arbitrarily. Always show the relation between things to remember them. Things do not exist in a vacuum.
Know the rules, you can derive the sequence. Don't know the rules, you're going to be stuck memorizing things with flash cards for eternity. What a torturous existence that would be.
To improve your visualization, you must do a few things.
1. Learn how to describe things so as to conjure images in your mind's eye. Really try to find anything that does this for you, and then try to recreate and nurture the effect as often as you can. Good poetry will do this for me, vividly described fantasy, and other times I can visualize something just by mere repetition, i.e. I've watched a movie hundreds of times and can thus visualize the entire movie in my mind. Do anything you can to nurture this ability. Do not resort to memorization of sounds, or mnemonics. This will lead to a literally blind understanding of the content you are studying.
2. Drawing. Do not be afraid of drawing things. Draw from nature, and then try to draw from your imagination. Drawing from nature is just a true a description (assuming you are accurate) as a symbolic representation. Do not be ashamed of cranking out some drawings and diagrams to explain something. Use all of your resources. NEVER sit and stare at a blank page.
3. Don't TV, movies. create your own visualizations. The more you do it, the more your visualizations will become universally understood.
ran out of characters. want to expand on the last point a little.
Television, movies, and video games are all visual representation of other people's ideas. They aren't necessarily universal. When you mindlessly take in this type of content, depending on how strong your mental constitution is, you might start 'absorbing' their point of view, and thus sacrifice the more universal (natural) view point. Strangely enough, the more accurate a description of nature is, assuming natural laws are universal, the more universal your description is. Much of stories from television, movies and videos games do not describe nature in universal terms, but in local, asymmetric terms, so as to create dramatic effects. Like a casino, stories create situations in which the odds are against you, and that challenge keeps you coming back. Good art will of course be perceived as being closer to nature and thus more universal, so I'm not saying don't consume art, but just be careful about what it is you are consuming. I have often found myself binge-ing on "bad art", and then found my imagination stunted for a disturbingly long length of time. I don't like that, it's limiting. What is good and bad changes from person to person, thus I'm not mentioning specific works of art as having universal effects on humans. You'll have to figure out what has what effect on you.
Essentially, pay attention to what you consume, it can limit your ability to visualize and think clearly. That's all I'm saying. This is from personal experience, and I know of others who have experienced the same thing. I'm not claiming it's universal, but it might be. Be careful.
sorry, one more thing on proofs
Often, I see people get stuck on the linguistic aspects of proofs. Usually this is, in my vastly biased opinion, due to the methods by which they were to taught to remember things, i.e. by sound, through auditory memorization, things they heard but did not see. Auditory memorization is admittedly, less resource intensive, it's clearly easier to remember what you heard than it is to remember what you saw, so I understand why this is the most common method for thinking, but it is limited, again in my opinion (not trying to step on toes here). One's imagination should really be a direct and complete* (not lacking in any of the senses) simulation of your perception, and should thus include all of your senses. Since, of course, mathematics typically does not deal with smells as much as it does things you've seen, I don't recommend describing the 'smell' or 'texture' (although, tactile analogies are quite common, think 'smooth', 'hairy', 'rough', 'spiky', etc) of a theorem, but I also wouldn't stop someone from trying.
So, with that in mind, just remember that while you're trying to describe a sequence that is the product of some rule, in a proof, there are many ways in which you can arrive at that sequence which you are describing, and the best way is often what is best for you, how you explain things to yourself. Eventually, you are going to have to, as said, explain it to someone else, as is the 'point' of a proof, but, if you can't explain it to yourself first, you will probably never really understand it, and be stuck playing the telephone game with the textbook.
If you're interested in other perspectives on proof writing- I highly recommend reading George Polya's "Induction and Analogy", which covers general "mathematical thinking", and not just proofs. And also, of course, always read as many different perspectives as you can!