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I'm far from a mathematician, but the field I'm trying to break into (management consulting) requires a fair amount of mental arithmetic. I'm okay, but I'm not even close to as good as I need to be in terms of both speed and accuracy.

I have math apps on my iPhone. I use online mental math tools. I try to do mental when I'm just sitting around. Anyhow, I've been noticing that I'm not getting better. In fact I'm making more mistakes and it seems like I'm struggling more to get answers.

I suspect that I may be pushing it too hard. Right now I'm doing all four major operations in sets of 50 and 100, and taking times tests multiple times per day. After it all sometimes my brain just hurts. Nevertheless I push on because the old adage with math is "practice, practice, practice."

Yet, I'm curious -- is there decreasing returns in speed and accuracy to more mental math practice? If I push through another set of hundred basic questions at 2:00 am before I go to bed -- am I helping myself productivity or hurting it?

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Practice will only help to a point. You might better spend some of your energy looking for better ways to compute or computational shortcuts. There are many many books written about such things. –  Bill Cook Sep 16 '12 at 23:57
Just a comment: I've never heard about a mathematician that cares about being able to do mental arithmetic. It is true that some good mathematicians were able to do calculations very quickly, but then again, there were outstanding mathematicians who couldn't. –  student Sep 17 '12 at 0:04
Mental arithmetic is not a big deal in research mathematics, perhaps, but practitioners and engineers may find it essential... it is important to know what is useful for your purposes, and it sounds like the OP has a good idea of that. That said, I'm not sure if this is a mathematics question as such, and I wonder if it would not be better on some other stackexchange... –  Ben Millwood Sep 17 '12 at 0:21
@Ben, you may be correct. I posted here because my question is certainly related to learning math and I imagined the crowd here would include some educators that may have insights on practicing mental arithmetic. –  George Sep 17 '12 at 0:33
@Drew I am practicing addition, subtraction, multiplication, division of integers ranging from two digits up to 12 digits. I'm also calculating the percents of integers (again two to 12 digits). A link to my primary practice tool is caseinterview.com/math/mathtest.php –  George Sep 17 '12 at 0:46

5 Answers 5

People who are good at mental arithmetic know lots of tricks. We can generally think of three or four different ways of solving any problem. The clever part is seeing these different methods quickly; then picking one and going for it.

It sounds like you have been practising speed a lot, which is good. Now look at the other side of the equation - finding different methods for solving a problem. So, next time you practise, pick a problem, and try to think of three different ways of solving it. Forget about speed, just concentrate on finding different methods. You need to practise doing this, as well as practising speed.

Take the time to read all the books on mental arithmetic, to find out about all the different methods that are available, for solving different arithmetic problems.

Good luck.

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I agree with David. You really should find more ways of computing mentally. –  Euclidean Sep 17 '12 at 4:34

Arthur Benjamin does a whole bunch of mental math stuff. Maybe you should check him out? Most of it is just pattern recognition and shortcuts - things you probably already do anyway.

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Pushing anything at 2:00am sounds like a bad idea. It's true that memory retention is particularly good just before sleep but it sounds like you aren't improving your mental math abilities because you might not be using enough tricks. Fast, efficient mental math is an artform. If you're looking to improve it, check out some books like this on on the subject. The bottom line is your returns on what you learn are not going to be very high if you keep doing mental math problems problems without actually learning a good set of skills for it.

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From experience what I know is to take advantage of daylight hours... Indeed, taking it too long at night can even be counterproductive –  PALEN Sep 17 '12 at 3:56

You're doing arithmetic, not math.

There are lots of ways to train to execute rote tasks quickly with mechanical precision: I doubt mental arithmetic is any different from any other rote task in that regard.

But I think your biggest benefit would be to step back and ask yourself what you are actually trying to do, rather than focus on how to better enact the method you've chosen to try to do it. It's very easy to focus so much on a specific method of doing things that you forget what you were trying to do!

I don't know anything about management consulting, but I am extremely skeptical that feats of mental arithmetic are actually useful at all. Instead, you have probably seen people make use of methods of simple estimation.

In any situation where you have to do arithmetic in your head, nobody will care that it will cost \$24,374.44 to purchase 1638 widgets at a price of \$14.88 each. Even being able to quickly come up with \$24,000 is probably already more than enough precision -- it's probably good enough to simply come up with "it's in the \$20,000 to \$30,000 range".

p.s. I came up with 24,000 as 16 * 15 * 100. As one rounded up and the other rounded down, I figured this to be a good estimate, probably a little low based on how much rounding. I computed 16 * 15 as 15 * 15 + 15. The square of 15 I had memorized, although there is a standard trick for squaring numbers that end in 5.

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You say "You're doing arithmetic, not math." Actually, arithmetic is a branch of mathematics. See, for example, en.wikipedia.org/wiki/Arithmetic. –  Joel Reyes Noche Sep 17 '12 at 4:47
@JoelReyesNoche Of course. Do you recognize the difference between literal and figurative language? –  anon Sep 17 '12 at 5:06
@anon: Now when you say difference, do you mean that literally or figuratively? :-) –  Asaf Karagila Sep 17 '12 at 8:36

Nobody's directly answered your question: the answer is that, yes, it's absolutely possible to exhaust your mental capacity for arithmetic, as it is your willpower or your physical muscles. Take a few days off.

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