Improving math calculations ability

What do you think about the possibility of improving your calculations speed in mathematics? Do you think that it is possible, or do you think that it is an inborn skill such as intelligence.

I can understand math quite well, but I am horrible at doing calculations. Whenever I attempt to do any sort of calculation at all, my mind almost shuts off and it is very mentally taxing for me to do them. Since about middle school, I have always used a calculator because of my laziness towards/inability to efficiently do problems in my head.

I also have a friend who can preform math problems in his head very easily. He multiplies and divides 3 digit numbers in his head faster than I can type them on a calculator. According to him, it is easy and does not require any more effort than normal thought.

As I move into higher-level math, I am becoming really interested in improving my speed of calculations to a higher level, but I am unsure if it is possible. When I search it on google, I only get results about mental math tricks, but I want to be able to do the actual operations faster and without becoming mentally tired afterwards.

I think it's mostly an innate skill, but I wouldn't worry too much about it. I'm quite good at mental arithmetic, but the times where I need to use it are fairly limited, especially when I started doing more advanced maths.

It just takes practice, the more math you do the faster you will become. When your friend was young he probably practiced a lot and used flash cards, and memorized his times table and stuff. You did not. As you said you were lazy and used your calculator to much and now your slow. I am like you, I never really practiced much in math, and I don't do much home work(didn't need to). Hence I am very slow with most math. However, I am pretty smart and I grasp concepts easily, so I learn math easily and I can do most problems, I am just slow some times.

I think that may be a trap people fall in, when doing easy low level math you don't practice enough, because it seems easy. Well the practice isn't to learn the math, it is to master it so you can do it very quickly. So even though you understand it all you didn't master the speed factor that only comes from extensive practice.

So you just need a lot of practice. There are a ton of math games on the internet, I would try to find one that isn't to childish and play that. Or just get out some flash cards and actually study like you would for something really difficult.

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Originally Posted by sheogorath

or do you think that it is an inborn skill such as intelligence

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Everything is malleable, including intelligence.

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Originally Posted by Alric

It just takes practice, the more math you do the faster you will become. When your friend was young he probably practiced a lot and used flash cards, and memorized his times table and stuff. You did not. As you said you were lazy and used your calculator to much and now your slow. I am like you, I never really practiced much in math, and I don't do much home work(didn't need to). Hence I am very slow with most math. However, I am pretty smart and I grasp concepts easily, so I learn math easily and I can do most problems, I am just slow some times.

I think that may be a trap people fall in, when doing easy low level math you don't practice enough, because it seems easy. Well the practice isn't to learn the math, it is to master it so you can do it very quickly. So even though you understand it all you didn't master the speed factor that only comes from extensive practice.

So you just need a lot of practice. There are a ton of math games on the internet, I would try to find one that isn't to childish and play that. Or just get out some flash cards and actually study like you would for something really difficult.

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I wish we had a dislike button for this answer...

When I was a kid, we used to have a test every week where we had to fill in a product table for the integers 1-10 in a certain amount of time. I sucked at it. My step dad literally made me practice for 1 hour a day. I dreaded that hour. I never got any better. I still can't memorize them. I can figure them out though as I need them. And lol at flash cards being how one studies for something "really difficult".

Also, this isn't really math. It's just arithmetic. It's practically useless. It's important to know how to multiply by 1 and how to add 0. In higher math, you're going to want to get really good at doing those two things in creative ways. Think rationalizing the denominator for an elementary example of multiplying by 1 in a creative way. As far as adding zero in a creative way, think factoring a difference of squares for an elementary example:

a² - b² = a² - ab + ab - b² = a(a - b) + b(a - b) = (a + b)(a - b)

Beyond that, I wouldn't sweat arithmetic. That's what calculators and computers are for. One of the greatest stories in math is that of "Grothendiek's Prime": 57 = 3*19. Suffice it to say that Grothendiek sparked off a golden age in algebraic geometry in the 1960s. He was brilliant with abstract mathematics. He was demonstrating a procedure he had invented involving a prime number to another mathematician. The other mathematician says "How about we try this out on an actual prime so I can get a feel for it." Grothendiek says "p is taken to be an actual prime, that's what we've been doing." The other mathematician says "No. Like a real prime." Grothendiek says "You mean like an actual number?" The other mathematician says "Yes, an actual number." Grothendiek thinks for a minute and says "Okay. Take p=57."

So if you want to study higher math, I would focus on properties that can be used in proofs and not arithmetic.

E.G.

if p is prime and divides mn then p divides m or p divides n.
if 0 = mn then m = 0 or n = 0. If one of them is taken to be arbitrary, then the other one is 0.
If you can integrate a function over an arbitrary interval and it's 0, then the function is 0.

Things like that are important.

If you are just trying to memorize something, it is the way to go. I don't really believe you when you say you can't memorize something, because anyone could memorize something if they do it enough. If you dreaded it so much, you probably were just going through the motions without really trying very hard, which is why you failed to memorize them. Now don't get me wrong, I understand trying to memorize some things is like trying to bang your head into the wall until you learn in and isn't all that effective in a lot of cases, but it does work. Things like using the little mental tricks to help you, are actually very useful for it as well.

My main point though, as you need a lot of practice, and a lot of practice is helpful in higher level math as well. Though I would focus more on doing it accurately every time first, before worrying about speed. That said I do agree with what Philospher said being a lot more important than basic arithmetic. Realistically your never going to be doing difficult math without a calculator, so I wouldn't be ashamed of using one, as long as you understand what the calculator is doing.

I actually find flash cards really helpful. I'm a visual learner, so what I do is write down a topic on a card, draw something that's related and going to stick in my head on the back of it, and go through them. It's how I remember all the stuff in sciences. Saying that though, maths is my worst subject. But I guess if I had put my mind to it more, it would have been better. I'm too lazy.

But really, as people have said, for bigger calculations you'll get a calculator. So no big deal right? Unless it's a challenge you wanna set yourself ;)

The problem with flash cards (as I see it) is that I don't see how they can do anything but help you memorize stuff. I don't see how they can help you understand stuff. There will always be a higher information compression ratio with understanding than with memorization. If I understand a principle, I can "decompress" it into an infinite amount of facts. If I remember a fact, then I merely remember a fact. That's it. So I'd need infinite memory to memorize enough facts to match understanding one concept. What a waste of time.

Here's how you study math.

Do problems.

When, in the course of doing a problem, you run across the need for a formula or theorem that you don't remember, derive or prove it. If you can't derive it, go to where the book derived it (if it didn't, get a better book) and follow along. I mean act like you're reading it for the first time and fill in any missing details and write the derivation out in full. Do this every single time you don't know how to derive something.

Soon you'll be complaining about how you're slow and stupid just like me.

It doesn't help you understand stuff, and it isn't suppose to help you to understand thing. Flashcards are purely for memorization. Obviously you need to understand what you are doing, but some times memorizing is important.

If you memorize a formula and all its variations, then whatever you need it or see it, you can quickly use it. On the other hand if you do not memorize the formula you will have to derive or prove it every time you see it, and you will have to do it for all its variations as well. Which is far slower. Which is why if you are looking for speed it takes memorizing stuff as well.

I literally have no idea how I would go about straight out memorising a formula or method. It's a terrible idea.

So instead of memorizing A squared plus B squared equals C squared for a right triangle, you regularly spend fifteen minutes measuring the sides and angles and doing trig to figure it out? There are a ton of formulas you end up memorizing, and you either do it because you used it so often that you memorizes it, or you practice memorizing it. Though it is often a mix of both, of you repeating it in your head a bunch of times, and then also practicing it in actual problems.

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Originally Posted by Alric

you regularly spend fifteen minutes measuring the sides and angles and doing trig to figure it out?

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...what does this even mean?

Anyway, you've chosen an extremely basic example. Pythagoras's theorem is so well known it's pop culture, probably only E = mc^2 tops it. Let's try some proper maths; the stuff that you need to learn for exams. How about sin^2 + cos^2 = 1. Do you seriously remember that, symbol by symbol, without reference to anything else? You think that's sensible? Do you also memorise tan^2 + 1 = sec^2, and 1 + cot^2 = csc^2, separately? 1/2*a*b*sinC? sinA/a = sinB/b = sinC/c? All of these are extremely trivial facts if you understand the basis of them.

Even on the most basic level this doesn't work. Exponentiation; do you learn

x^a*x^b = x^a+b,
x^a/x^b = x^a-b,
(x^a)^b = x^a*b
x^-a = 1/x^a

et al? Or do you simply learn that x^a means x*x*...*x, a times? The rest follows easily with a good understanding of this.

Or fractions, even. a/b + c/d. Did you learn that you calculate it via a*d + b*c / bd ? Or do you think it's a better idea to just understand the basics of arithmetic; if you want to add up two things you just have to make them of the same type by changing the bottoms, and you can do that easily by timesing top and bottom by the same number.

Learning reasons rather than by rote is infinitely better: it is self-checking; the knowledge lasts a lifetime rather than a few days; having learnt the reasons behind them, things become far easier to recall anyway; it makes you vastly more confident and fluent; it means can adapt things to new situations with ease.

The entire process of learning in the first place becomes easier, too. Maths is fundamentally a highly interconnected structure. If you understand what went before, learning something new is like driving to a house and placing down some stepping stones up to it; if you don't, however, it's like ignoring the existing infrastructure and building a whole new road just to get home. And then doing this for every single house in the country.

And if those reasons aren't enough, the method is tried and tested by yours truly, and it's working pretty damn well so far. As for all the guys at school who used to spend revision time writing out hundreds of little squares of paper... well, they make good coffee.

Actually yes, when I was learning stuff like sin^2+cos^2=1, I memorized it in that form. Same with the others. It makes spotting them in equations, or using them far easier. I think you are misunderstanding, in that I said you are suppose to learn how it works first. So your not memorizing anything you don't already understand. You are memorizing stuff you already know, so you can do it a lot faster, and more accurately.

In the originally post, he already said he knew how to do it, he just wanted to increase his speed. In all the examples you give, if you have to figure the stuff out it is going to take multiple steps, and if their multiples of them in a problem it will take you forever. If your adding an extra 30 steps in a problem, it doesn't matter how easy those steps are, its going to make it take a lot longer to do.

You want to be reducing the steps you need to take, not adding steps. By memorizing the formulas and understanding how they work you reduce your time by a great deal. I am actually surprised that you think you can do high level math without knowing any formulas at all. My guess is that you probably do really know them.

Alric's post + Xei's post = the same thing.

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Originally Posted by Alric

You want to be reducing the steps you need to take, not adding steps. By memorizing the formulas and understanding how they work you reduce your time by a great deal. I am actually surprised that you think you can do high level math without knowing any formulas at all. My guess is that you probably do really know them.

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You don't get permanently faster by sitting down and memorizing some formulas. You get faster by solving a ton of exercises so that it sticks to memory by itself.

Isn't that what I already said? You need a ton of practice.

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Originally Posted by Alric

Isn't that what I already said? You need a ton of practice.

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Well, kinda. But that was a long time ago :p

Hey this question goes out to anyone who is relatively good at maths. So I want to start teaching myself maths but I don't really know where to start, a rubric would be great like "learn X then you will be great at Y and then you can start Z". Should I just start out with algebra and work my way through geometry, trigonometry to calculus?

The best approach is an algebraic one. Learn all of the rules of arithmetic (including things like exponential identities); learn about polynomials (the factor theorem, the fundamental theorem of algebra, et al), and get to grips with solving general equations and manipulating math. Algebra is basically the language you will do all maths in, at least nowadays. Also learn about arithmetic and geometric series and the binomial theorem. You then want to learn about some of the basic theory of functions; learn some useful functions such as sin, cos, log and the relations between them. I barely know any geometry; the most important things to learn by far are the trig identities, and some other little things like the pythag theorem, radians, and simple stuff about circles, along with a basic understanding of coordinates. At this point I'd imagine you can start to learn calculus; the meaning of differentiation and how to differentiate any function you can write; various applications of differentiation. Then learn about definite integration and Riemann sums, and learn how it relates to anti differentiation. Then there's a whole host of integration techniques.

At this point you should have a good understanding of the basics, and where to go. You should also get to grips with vectors at some point; scalar and vector products; the equations of planes, etc. You'll find the stuff you can potentially learn increases very quickly due to the high degree of interrelation; for instance, a very fundamental and basic function is the exponential, but you need the basics of calculus before you can understand it.

Other places to go: differential equations; matrices; Taylor series; complex numbers; induction; conic sections; multivariable calculus.

Awesome thanks. I have a basic understanding of algebra so I'm on the right track!