
Tips to Improve Your Mental Math Ability - artsandsci
http://gizmodo.com/10-tips-to-improve-your-mental-math-ability-1792597814
======
Tycho
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...

~~~
asafira
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.

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

~~~
asafira
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.

------
dancek
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](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](https://mitpress.mit.edu/books/street-fighting-mathematics)

~~~
imranq
this is great - thanks!

------
dahart
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"

[https://www.amazon.com/gp/product/0884150879/](https://www.amazon.com/gp/product/0884150879/)

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

~~~
dahart
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.

------
pmoriarty
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](https://en.wikipedia.org/wiki/Soroban)

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

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

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

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

~~~
rabboRubble
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.

------
discombobulate
> 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.

~~~
cgriswald
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

~~~
discombobulate
> 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.

~~~
cgriswald
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.

~~~
discombobulate
> 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.

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Existenceblinks
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.

~~~
dualogy
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?

~~~
Existenceblinks
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.

------
pklausler
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.

~~~
oh_sigh
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)

~~~
pklausler
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.

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

------
dahart
> 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

~~~
Existenceblinks
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.

~~~
dahart
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.

------
george_ciobanu
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)

~~~
irrational
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.

~~~
george_ciobanu
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.

------
vitohuang
'Mathemagics'

[https://www.amazon.co.uk/Mathemagics-Genius-without-
Really-T...](https://www.amazon.co.uk/Mathemagics-Genius-without-Really-
Trying/dp/0737300086)

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.

------
jflowers45
reminds me of one of my favorite books from when I was a kid -
[https://www.amazon.com/Math-Magic-Everyday-Problems-
Revised/...](https://www.amazon.com/Math-Magic-Everyday-Problems-
Revised/dp/0060726350)

~~~
dwe3000
One of my favorites from eighth grade was Isaac Asimov's Realm of Algebra - I
read that before any of his other books.

------
Pete_D
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.

~~~
pmoriarty
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](http://www.ee.ryerson.ca/~elf/abacus/feynman.html)

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

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

------
penglish1
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.

