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Tips to Improve Your Mental Math Ability (gizmodo.com)
153 points by artsandsci on Feb 23, 2017 | hide | past | favorite | 50 comments

I don't care too much about mental arithmetic, but I care a lot about my "mathematical literacy."

I find that reading anything "mathy" can quickly become hard to follow. Sometimes it seems the authors are being very obtuse by introducing variables and not specifying what they represent (also doesn't help that they are all greek letters which I forget the names of). But regardless, I'd love to be able to read pages of mathematics as easily as I can read pages of source code.

Anyone got tips for that? Is there some sort of bootcamp for learning the language of maths?

I feel the language part can be disentangled from the problem solving part. The same way one doesn't need to be able to come up with brand new algorithms to read and understand the algorithms used in computer science. I don't want math superpowers, just math fluency...

The 'Princeton Companion to Mathematics' has the best introduction I've seen to understanding what mathematics is, and what the most important, orientation-giving concepts are. I think your problem is missing implicit context; once you begin understanding what mathematicians value and how various branches interrelate, and how parts of it developed historically—it becomes much easier to follow all kinds of mathematical writing.

The intro is about 150 pages, and somewhat dense, but with work definitely accessible to someone with little knowledge of mathematics. I wish they would sell it separately from the rest of the book because it's that good, but unfortunately it comes attached to another thousand pages of material you probably won't have a use for unless you get into mathematics research.

I tutor on the side, and it's often for this very reason.

I am not sure of any particular source that makes this easy, but the majority of technical things written are written by people who don't really have a reason to have especially strong communication skills, especially communication skills for people that might not be familiar with some of the basics.

Since this is HN --- I actually think this is a business opportunity. People would pay plenty for the best educational resources, at varying times in their life, and having good explanations of things is a valuable resource.

I have actually thought about getting a tutor for this reason. Do you think tutoring is a good solution for this particular goal?

Yes, but only if you are putting time into it on your own outside of tutoring. Otherwise, sessions will either seemingly not go very far, or you spend a lot of wasted time in the tutoring sessions (just blank time when you are reading and re-reading things that you could have done on your own). A lot of this kind of learning still happens independently even though a tutor can (and usually does!) accelerate the learning.

In history mathematics was not for everyone[0], not in the same sense as isn't today. IMHO, having chance to learn mathematics is a privilege. Most people not even have chances to train "mental arithmetic", "mental survival" is more likely a thing. Sorry for not answering your question.

[0]: https://blogs.scientificamerican.com/guest-blog/mathematicia...

I've found a good basis in abstract math makes a really wide range of mathy stuff accessible. The first two chunks I would tackle from a high school math + college calculus foundation would be set theory and group theory, and though it's an oldie, I'd honestly recommend Russel and Whitehead's "Principles of Mathematics" for the former as a start.

I was proud that my PhD thesis was about 1/3 as long as one in the humanities might be, counting by pages. Then I realized that it probably would take just as long to read out loud. :)

Mathematics is dense. Period.

Try "What is Mathematics?" by Courant et al.

However, you might be mistaken about language and problem solving being independent.

Are you just reading mathematics or are you also doing mathematics?

Just reading.

That's probably the biggest problem, then.

You can't understand mathematics without doing mathematics anymore than you can understand a complicated program without having a commensurate amount of experience in programming.

There was a great MITx course called Street-Fighting Math. Most of the techniques and ideas taught apply well to mental math. The course materials are freely readable after logging into edX and enrolling. https://www.edx.org/course/street-fighting-math-mitx-6-sfmx

EDIT: Apparently the similarly-titled MIT Press book is freely available as PDF. Now that the course isn't currently running, I'd probably prefer the book. https://mitpress.mit.edu/books/street-fighting-mathematics

this is great - thanks!

While these are simple tricks almost everyone here knows, there are great books on advanced techniques. I heard about this one from a math blog and got it. Goes into gory detail about to manually calculate logs and roots, reciprocals and transcendentals. It was a fascinating read, I expected it to help me sleep at night and it did the opposite.

"Dead Reckoning: Calculating Without Instruments"


How would you respond to the negative Amazon reviews that say the book is very difficult, not well explained, and has no exercises?

All those things are true. ;) And the author admits them in the introduction, which you should be able to read from the amazon page. It is still an amazing book. A lot of the reviews clearly expected this to be a way to learn quick tricks to impress their friends. There are lots of books like that, but this isn't one of them, this one goes deep into the theory and long lost methods for doing difficult problems without a computer. For this audience, the difficulty level will be mixed and not necessarily all hard. Anyone who's done some programming will find plenty of material that is both accessible and super interesting.

Lately I've been picking up the soroban[1], the Japanese abacus, and it's been tons of fun. It feels a little like solving a Rubik's cube with arithmetic, or maybe like working with a finite state machine. There are different algorithms to apply depending on the state of the soroban, and applying the right sequence of these algorithms will get you to the right result.

I find it to be a little addictive, and sometimes find it a bit hard to stop. I always feel like wanting to improve my skills a little more, become a little faster at it, and increase the number of digits and terms I can handle without making a mistake.

The soroban is a great tool for developing concentration, a memory for numbers, a facility for performing a relatively complex series of steps in a certain sequence, and eventually for lightning fast mental arithmetic.

In Japan, soroban use is taught to young kids[2], who after a while develop enough proficiency not to need the physical device any longer and can perform the calculations on an imaginary soroban, and eventually can achieve some really amazing feats of mental arithmetic, such as this example from their national competitions: [3]

Forward to the 1'45" mark in the following video for another really impressive display of soroban-trained mental math prowess: [4]

Finally, there's a great subforum for mental math in the Art of Memory Forum: [5]

[1] - https://en.wikipedia.org/wiki/Soroban

[2] - https://www.youtube.com/watch?v=Px_hvzYS3_Y

[3] - https://www.youtube.com/watch?v=7ktpme4xcoQ

[4] - https://www.youtube.com/watch?v=rPTKZ4PLkMc

[5] - http://mt.artofmemory.com/forums/mathematics-and-mental-calc...

Agree. I taught myself in order to teach a young relative, and it really improved my ability to handle mental calculations at the simple end of the mathematical operational spectrum.

What was really weird was doing the abacus work, moving beads, all in chunks of 5 or 10, and then looking at the final results without any preconceived notion of what the results were. Totally fresh eyes as I counted the final set of beads.

The manual movement was one thing, but the motion didn't cause me to visualize "numbers" as I understand them. Completely separate pathway to the same answer.

> For instance, you can multiply by 8 by doubling three times. So instead of trying to figure out 12x8, just double 12 three times: 24, 48, 96

This can be more generally thought of as using exponentiation.

12 * 8 = 12 * 2^3

Using the more general rule, again.

12 * 9 = 12 * 3^2. Or 12 * 3 * 3.

I find both of these easier and faster using other methods. Maybe just because I've been doing them since childhood.

12 * 8 = 10 * 8 + 2 * 8 = 96

12 * 9 = 12 * 10 - 12 = 108

That said, most of the things in the article are things I've picked up more or less automatically. If you could train yourself to do them, I think it could help, but I think most people who do them just figured them out on their own.

I disagree entirely with "Memorizing building blocks", mostly because I see the huge dichotomy among students between those who try to memorize and those who attempt to learn concepts. Those who memorize do worse on tests, because if they forget, they can't just apply their knowledge. They also seem to forget the material faster after the class. Those who learn the concepts are usually okay, even if they forget a detail they can often work it out. I also think that someone who needs to memorize that 3/5 = 0.6 is unclear on some deeper concept somewhere.

Edit: formating

> I find both of these easier and faster using other methods. Maybe just because I've been doing them since childhood.

Sure. It's just another tool.

> I disagree entirely with "Memorizing building blocks", mostly because I see the huge dichotomy among students between those who try to memorize and those who attempt to learn concepts.

Do you find the best students are fluent with the techniques and have an understanding?

It would seem to me, spending time pondering meaning also has the benefits of remembering 'building blocks'; because they're the primitives involved in the struggle for understanding.

There are three components: facts, techniques, concepts.

Being fluent with techniques is not strictly necessary, but is generally important and can aid in helping to understand concepts by demonstrating relationships. But a student who prioritizes concepts is probably better off than one who only understands techniques.[0]

My objection to the article is that it is advocating memorizing facts. Figuring out that 3/5 == 0.6 is trivial if you already understand the concepts and knowing that 3/5 == 0.6 is pointless if you don't know what it means.

[0] - Anecdotally, I breezed through calc 1 in high school. First done with every test, always aced those tests, never studied. Had no idea what I was doing, because I was just memorizing and applying rules. Knowing just the techniques was equivalent to knowing nothing.

> Had no idea what I was doing, because I was just memorizing and applying rules. Knowing just the techniques was equivalent to knowing nothing.

I suppose so. If you wrote some code to carry out the rules, you wouldn't say the computer knows anything.

If you were always first done and aced your tests, but think you had no idea what you were doing, then think how everyone else must have felt.


1111128 = 1111118+8=888896

Those who grown up in asian schools would see these as super easy tricks. Not intend to make superiority sound here but just the well-known fact. There was a trend in my high school when math teacher ask to solve (expect elaborate steps clearly) a problem on blackboard. Ones who can make a shortest steps possible is cool. I always love the moment that I can skip 5 steps with a magic line of equation and go straight to the answer. And then the teacher will remember you, not to appreciate but to revenge you with semester exam!

Sorry for being irrelevant.

Interesting indeed! In the west maths is widely not taught/practiced in such "fun" ways sadly, and by and large mostly formulaically and by rote. Unsurprisingly, it instils mostly a yawn reflex in a majority of people for the rest of their lives. Not a good outcome.

Asia is a big continent, what kinda countries or areas are you talking of?

I am in Thailand where their education system is crappy. IMO, west math is better when it comes to systematic thinking, less hacky, more stable. Asian is good at mess thing up in hacky ways (or in the black side, cheating!), system designed is not stable in general.

Chinese and korean are more serious about teaching in my perspective. It's probably not that bad that Thailand's education system is crappy, it's more relax.

There's a widespread effort in USA to teach calculation using multiple algorithms (thanks to Common Core), and the unsurprising result is that parents are howling mad about the "nonsense" their kids are being taught.

An amusing way to keep your mental math skills working is to compute the day of the week for calendar dates. (There's a few ways to do this that are published on the Web, e.g. the "Doomsday" method, but one can derive much faster methods with a little analysis.)

You can quickly gain a kind of "Rain Man" reputation if somebody mentions a date and you instantly mutter "Saturday". But it's just a matter of adding three numbers and taking the remainder modulo 7.

Taking a number of any size mod 7 can also be done in your head, from left to right! First time I saw this was in intro to theoretical CS, where the exercise is create a state machine for mod 7. It was a big surprise for me that it's super simple and you just feed it digits til you run out. Almost trivial to do mentally on very large numbers as fast as someone can speak the number, with just a little practice.

Can you provide a link to a method that is faster than the doomsday algorithm? If I know the doomsday for a given year, I can get the day of week for a date in about 3 seconds. I'd love to get that even faster(or have a faster method when I don't know the doomsday)

Memorize this table of offsets:

  0 Oct, Jan
  1 May
  2 Aug
  3 Feb, Mar, Nov
  4 Jun
  5 Sep, Dec
  6 Apr, Jul
Precalculate the offset for the year (2017 is 117 + trunc(117/4) == 146 == 6 mod 7). Then the day of the week for, say, today, is 6 (for the year) + 3 (for Feb) + 23 (day of the month) == 32 == 4 mod 7, which is Thursday .

Other examples: D-Day, 6 June 1944, was (44+11 == 55 == 6 mod 7 for the year) + 4 (Jun) + 6 == 16 == 2 mod 7 == Tuesday.

The only twist is needing to subtract 1 from the result for dates in Jan and Feb on leap years, since those dates precede the leap day.

I don't understand how you got the "6 mod 7" in the two examples above. Everything else makes sense, though. Thanks.

> Add and Subtract From Left to Right

This one's my favorite, and I like to combine it with doing all borrows/carries in a separate pass, which means intermediate "digits" are sometimes no longer 0..9.

For example: 78 + 89 = 7+8|8+9 = 15|17 = 1|6|7 = 167

90 + 78 - 1 is faster (to me).

Actually, I believe it just depends on ways which either fits your brain or you gets used to it.

For this example, totally. What I'm talking about is more useful with larger numbers, when working on paper, and simply as an alternative way to think. I'm not making any claims about what's "faster", that's up to individual experience and preference. That said, rounding tricks get hard to use on 4+ digit numbers.

78+89 = 80+90-3=167

I'm a bit surprised - I do almost all these things in my head anyway, never thought them as tricks just that I'm weird and cannot model school methods like others do. (Not trying to brag)

But, how did you learn them? I didn't do many of these things naturally, but I've read about them over the years and have learned to do them from reading so now I do almost all these things in my head too.

Laziness? I didn't like remembering to carry when multiplying 45x16 so I did 40x16 + 5x16 - easy to add two numbers in your head, hard to remember two shifted partial results (the way it's done on paper). Different media require different methods. 45x 16 on paper: 270+ 45 It's easy to see here but in your head you have to remember that 45 is shifted left when added to 270. Visually it makes sense but otherwise hard. Breaking things into multiplications by 10, five or other easy substeps is recursive and easy to trace back up.

Truth is I don't know how I came up with it, I think I was just lazy.

I thought pretty much the same thing as I was reading this.

I don't think of these as "tricks", just "how I do it in my head".



This is awesome book, it teach you all sort of tricks from 3 digits addition to 3 digits multiplication.

I know its written for school teens, but still pretty helpful life skills.

reminds me of one of my favorite books from when I was a kid - https://www.amazon.com/Math-Magic-Everyday-Problems-Revised/...

One of my favorites from eighth grade was Isaac Asimov's Realm of Algebra - I read that before any of his other books.

There's a great segment in one of Richard Feynman's books where he talks about picking up tricks like this, and ends up beating an abacus salesman in a cube-root-finding race. I think it involved memorizing log tables.

The story of "Feynman vs The Abacus" can be read here: [1]

While it's a great story and is very entertainingly written, as is typical of Feynman, I don't think he was being very fair to the merits of the two approaches to mental math -- his own vs that of the soroban.

Feynman clearly hadn't done any research on the merits of the soroban method, he just had this single run in with a random soroban salesman. From this one experience, and without very much thought on the subject, he just kind of dismisses the method almost out of hand. Such offhand dismissals are unfortunately also pretty typical of Feynman regarding subjects he is mostly ignorant of and has little experience with (such as his dismissals of philosophy in other talks).

In the story, the random soroban salesman, who clearly hasn't even very much proficiency with the soroban (as evidenced by his need for a physical soroban, vs soroban-trained practitioners who no longer need the physical device but can apply the soroban methods in their head) is matched against one of the greatest physicists in history, a Physics Nobel Prize winner who has advanced math training. That's really not a fair match.

It would have been great to see how Feynman would have fared against someone like this guy: [2] or against one of the national soroban competition winners, or against a top mathematician or physicist who was soroban-trained. That would be a fairer match and the results more indicative of the merits of the two approaches.

[1] - http://www.ee.ryerson.ca/~elf/abacus/feynman.html

[2] - https://www.youtube.com/watch?v=rPTKZ4PLkMc

You miss the point of the story. The story is about "calculating" vs "thinking", and how refusing to think holds one back.

Worth a read: "Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks" by Arthur Benjamin, one of my wife's college professors.

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