What use is mental math in 2022?

https://www.johndcook.com/blog/2022/02/17/mental-math-2022/

by ColinWright • 4 years ago

101 153 4 years ago

If you're quick at simple calculations early on in your life, it probably has a compounding effect on the other aspects of your life. Not sure how you'd prove it but alot of people who have a Math/Science acumen, do so because they're above average early on and it gets reinforced. Mental math could give one that edge.

sockpuppet69• 4 years ago

People who have more experience in a subject are relatively better at it…. Revolutionary.

cinntaile• 4 years ago

His point was that it creates a snowball effect if you have positive experiences early in life. No need to be snarky.

unfocussed_mike2• 4 years ago

That is not quite what raghavtoshniwal meant now is it?

The key word is "compounding", surely.

For example, I think young children need a few good ways to be taught about managing and understanding very large numbers, about combinational explosion, precision and estimation; essential in today's world.

I think kids from about seven or eight should be taught things about tree structures (decision trees, classification trees, parts diagram enclosures) but also about fractals -- about measuring coastlines, etc.; perhaps even show them Cantor's Comb.

They could be shown the liquid-nitrogen potato experiment to help them understand power series in future.

I feel 2020 taught us that young people need to be taught about the rice-on-chessboard problem, about the Small World experiment, etc.

There are good few kid-friendly ways to teach concepts like this which compound in building understanding that current adults do not have.

Kids in the UK are already taught some quite innovative ways to estimate, multiply and divide; maths has changed a bit since I was a kid. But there's a long way to go.

Workaccount2• 4 years ago

I am excellent at arithmetic (mental math) and can do "beat the calculator" party tricks.

I fall flat on my face with actual math (algebra onwards).

Thankfully, despite the teachings, the engineering I do is heavier on mental math skills. Anything more advanced and we use tools to do it.

sokoloff2• 4 years ago

Another super-power is dimensional analysis. People make sloppy mistakes all the time by ignoring this. ( https://en.wikipedia.org/wiki/Dimensional_analysis )

Between that and simple order of magnitude approximations, it's distressing how often seemingly carefully prepared materials contain errors that can be spotted in a 5 minute reading.

jihadjihad5• 4 years ago

it's distressing how often seemingly carefully prepared materials contain errors that can be spotted in a 5 minute reading.

It really is remarkable how many mistakes like this happen. One I've seen a lot is not knowing how percentages work, particular when it comes to changes between them. Going from 10% of something up to 15% is not a 5% increase--it's a 50% increase.

LordDragonfang• 4 years ago

The issue is that, especially with very small percentages, the absolute percentage change is often more important to the discussion than the relative change, especially when we're talking about risk.

In particular I remember there were a number of breathless articles about covid vaccine side effects where they were talking about a 50% increase in incidence - from 2 in 100,000 to 3 in 100,000*. That's not something the average person needs to factor into their risk model, and headlining it as a 50% increase makes it seem more significant than it is.

*not the actual numbers, but around the right order of magnitude, fit to the example.

rland• 4 years ago

That still trips me up from time to time. I'm not sure why anyone ever uses percentage increase to communicate anything.

Just say the quantity increased "from 0.0015 to 0.002" or something.

It's particularly prevalent in news announcement of scientific studies. "New drug decreases x by 350%!"

gowld3• 4 years ago

More generally, the slightly gauchely named "Street-Fighting Mathematics" http://streetfightingmath.com/

http://web.mit.edu/6.055/

* Divide and conquer * Abstraction * Symmetry and conservation * Proportional reasoning * Dimensional analysis * Easy cases (plugging simple values into complex formulas) * Lumping (discrete approximations of continuous functions) * Probabilistic reasoning * Springs (approximate complex systems as simple oscillators)

lanstin5• 4 years ago

A similar thing that drives me nuts is when people report quantities that only have interest as a flux over some time, like eg tax cuts or investment costs or whatever and they just report the dollars. Is the tax cut 1 trillion USD / year or decade or century? It isn’t just 1 trillion USD that is a meaningless statement. If we switch to AWS we save how much per month? Not just how much.

reactspa• 4 years ago

In Thinking Fast and Slow, Kahneman describes a mental math exercise [1] which can turn on System 2. I love using it to warm up my brain sometimes.

If anyone can share links to mental math resources that can be used instrumentally to improve brain performance, please do.

[1] https://psychology.stackexchange.com/questions/16531/kahnema...

melony• 4 years ago

Interviewing for trading jobs. It's the leetcoding of prop shops

jake_morrison• 4 years ago

In consulting, I have had to train myself to not give specific answers in a sales call. Any number for cost or delivery date that you give a customer will become stuck in their head, and you will have to fight to change it to match reality.

vsareto5• 4 years ago

I don't doubt that estimations are useful, but you don't necessarily need to provide an estimation. Why not just say, "I'll need to look into the impact of that change and I can get back to you". What are you gaining by introducing an opportunity for your head math to be wrong? You can still keep the conversation velocity by committing to providing an estimate later, especially a more accurate one. You might be making a careless error because you're doing the math during a meeting.

Jensson4• 4 years ago

That is good in theory but most people will have forgotten most of the conversation by the time you get back to them with your calculated values. People don't care that much about what others do, in order to grab their attention you need to be well prepared and answer things quickly or you lose them.

MathCodeLove• 4 years ago

It's not quite that binary. There are times where your approach is the best one, likely more times than not. But there are also situations in which it'd be advantageous to provide some concrete figure/estimate in the moment, even if its not completely accurate.

gowld3• 4 years ago

You can still keep the conversation velocity by committing to providing an estimate later,

conversation velocity, yes, but not project velocity, which is much more important.

RcouF1uZ4gsC• 4 years ago

Why not just say, "I'll need to look into the impact of that change and I can get back to you".

When you say that, do you then end the conversation about the change. Or do you keep on taking about the specifics of the change. If you keep on talking, and the change turns out to be infeasible, then you just wasted a bunch of time. However, if you stop talking and the change turns out to be feasible, you just wasted time that you and the other person had set aside.

spaetzleesser6• 4 years ago

I find it infuriating when people can’t come up with a ballpark number. You can and should check the numbers later but don’t stop a discussion.

mkr-hn2• 4 years ago

A lot of people have the unfortunate experience of being pressed for estimates and guesses only to be held to them as though they were 100% certain, and possibly punished for being wrong.

vsareto5• 4 years ago

I'd rather delay for an estimate than give something I'm not confident in. I'm most confident in estimations I've had time to think over, instead of trying to ballpark something. If I can ballpark something, it's usually because I happen to have done it before or it's an otherwise easy problem.

I don't know why one is considered a superpower over the other though (superpower is often a code word for "skills I wish everyone had because I have them"). When your loose estimations propagate to others, there's always a chance of it being taken out of context too.

If it's only you and a single person as your client in the conversation, usually things won't be taken out of context. If it's a larger org, it might become more than just an estimate.

This might even be framed as optimistic estimates vs. pessimistic estimates. One is faster but more error-prone, the other slower but more accurate. Different tools for different situations.

Jtsummers2• 4 years ago

Free resource that has come up a few times on HN in the past: Street-Fighting Mathematics by Sanjoy Mahajan.

https://mitpress.mit.edu/books/street-fighting-mathematics

Click on "open access" to get to the PDF link.

sitkack• 4 years ago

If the number you are calculating will be used as a lower bound, then rounding all the items up to the nearest 10s place and then using a little factoring can get one to the go/nogo solution.

"can't be more than X"

"can't be less than Y"

are powerful devices that allow one to keep thinking, rather than making tasks to be done later. But like the article mentions in the comments, if precision is warranted, then one needs to slow down and double check their work.

As mentioned in the preface to Street Fighting Mathematics, "How to Solve It" by George Polya is another very approachable book. This tl;dr is pretty good [1] (aside, this book should be in the public domain by now)

[1] https://math.ucr.edu/~res/math138A-2012/polya.pdf

There are other books that train folks on how to do mental mathematics like

* The Trachtenberg Speed System of Basic Mathematics https://www.abebooks.com/products/isbn/9780285629165/3011837...

* Secrets of Mental Math https://www.abebooks.com/products/isbn/9780307338402

I found this neat publication from the American Mathematical Society on Back of the Envelope calculation.

https://www.ams.org/publications/TEXT-33-chap-1.pdf

jihadjihad5• 4 years ago

All good points, and the only thing I'll add is that, for me, the utility of mental math (and putting it into practice often) is that it strengthens your mathematical intuition--you start to know, and even feel how numbers work. Having strong mathematical intuition that is reinforced by even basic/rudimentary mental math abilities is indeed a real superpower.

gowld3• 4 years ago

you start to know, and even feel how numbers work.
mathematical intuition ... is indeed a real superpower.

What is this power used for?

thomascgalvin• 4 years ago

Math is all around us, and it's nice to have a gut-level check on whether or not something makes sense.

Say you're at a restaurant, and you and your other both ordered $25-ish dollar entrees, and $10-ish drinks, and split a $15-ish appetizer, and the bill comes closer to $150 than $100, something was probably mixed up.

That kind of quick approximation gives you a nudge to check real numbers.

jihadjihad5• 4 years ago

Good question. For me, in my line of work, it allows me to very quickly and on the spot (similar to John's blog post) confirm or deny if something passes the sniff test. If someone says "but that will cost X!", it is a superpower to easily refute or corroborate that. Along those lines, if someone throws out that doing Y will easily provide Z amount of value, simply being able to work through the assumptions the person made when coming up with Z, and offering up your own Z' if necessary, is very advantageous in a live, synchronous setting like a meeting with a client.

peteradio• 4 years ago

Error spotting during otherwise very quick meetings.

mkr-hn2• 4 years ago

The replies suggest a need to separate mental math from mental arithmetic. I can't do mental arithmetic despite a lifetime of trying to develop the ability, but still have no trouble with the mathematic intuition based on expected costs described in other comments. It's easy enough to round things to 0 or 5 and sum it up for comparison even if doing the actual numbers in my head just won't happen.

thrashh• 4 years ago

Saving you a shit ton of work down the road because you on a quick glance caught a weirdness.

spaetzleesser6• 4 years ago

For deciding if it makes sense to buy a house or finance a car. Or if a high deductible health plan makes sense. It also helps with cooking. Math is all around you.

ajkjk• 4 years ago

Everything, oh my god. (imo, it changes the whole way you think about the world).

ThatGeoGuy• 4 years ago

If you can get used to applying Bayes Theorem in your head as a second nature, it helps greatly as a quick dummy check with reality. Obviously this doesn't mean calculating hundreds of probabilities in your head at once, but it can be very useful as a litmus test for all manner of things.

Knowing roughly what the value will come out to be if your priors are ~10, ~100, or ~1000x apart is very helpful when trying to discern if you're just being led down a wrong path.

Generally speaking doing Monte-Carlo simulations in your head would be nothing more than a stress test on your brain, but the simple multiplications and divisions used in reasoning under uncertainty can take you quite far, and with very little effort. You can even double check your work in the future if you really felt you did a bad job, and you'll be able to point out exactly where the error was (to yourselves or others) because it's more formalized than "I had a gut feeling."

teawrecks• 4 years ago

Once you are familiar with Fourier series expansion, you start seeing everything as a signal that can be broken up and manipulated in interesting ways. From silly stuff like a talking piano (https://youtu.be/-6e2c0v4sBM), to more complicated and useful inventions like jpeg, real time dynamic light rendering (spherical harmonics), and countless others.

watwut• 4 years ago

Basically, you will be able to "feel" when numbers are wrong - without doing calculations or making an effort.

Then you of course have to check before you say something. But without that intuition you would not checked.

klyrs• 4 years ago

When I’m on Zoom with a client, I can’t say “Excuse me a second. Something you said gave me an idea, and I’d like to pull out my calculator app.”

You can't? Have you thought about investing in a separate device for doing calculations? A good scientific calculator is tens of dollars, and would probably blow your calculator app out of the water in terms of usability (mmm, those tactile buttons). They're even solar powered so you never need to worry about charging. Just because phones can replace handheld electronics, doesn't mean they must.

Don't get me wrong, mental math is great, and in my work, I do a lot of off-the-cuff estimation with orders of magnitude and limiting behavior. But a calculator is way more precise, and can handle way more complexity than my poor little head. I tend to use a repl in my favorite language. But it sounds like the author is using their phone for zoom and can't switch windows for some reason.

ekanes• 4 years ago

I used it all the time in business. You're talking to someone, and can know what you want in real-time, without saying, "ok I'll run the numbers and get back to you." Sometimes that doesn't matter, but lots of business is done casually/socially, and finishing the conversation/negoatiation while it's still over drinks at the bar vs on a call later can very much matter.

jasfi• 4 years ago

The main benefit is that I know mental math makes my brain sharp. I can't believe the difference it makes, and that it is not more widely known and practiced. However, it takes discipline to make it a habit, which isn't a very popular thing.

Any practical utility is a distant second.

petercooper• 4 years ago

It's super useful as a way to have confidence in computer/calculator generated values. I have quite often seen people perform calculations and claim their answer is correct but it's out by orders of magnitude because they didn't bother to even think about the basics of what their answer should be.

pyfan878• 4 years ago

It's useful for the estimation part of system design.

corry• 4 years ago

Curious if others have had this experience too - many successful founder CEOs I've known through my career (working closely with them) are excellent at ball-parking / order-of-magnitude / back-of-napkin thinking off the top of their mind.

There's something about it that IMO is an excellent 'tell' for a certain kind of intelligence especially for startups. I imagine part of it is that it's a good sign of 'technical skills' like fluency with math, and fluency with modelling scenarios. The other part is it's a 'demonstration of fitness' to others. Like, wow, our CEO is SMART and thinks FAST - great!

lordnacho• 4 years ago

Having anything in cache allows you to check for errors and requirements.

So for example if you're going to be away for 7 days and you need to spend $100 a day in notes, you might be able to avoid hitting a withdrawal limit, eg if you bank for some reason limits your weekly to $500.

Basically the estimate serves as an input into your requirements for the next step, and tells you whether there's something to worry about.

Same as if you're painting your house and you need to know whether you need to go buy a new bucket of paint or the existing one will do.

ogogmad• 4 years ago

Do people just memorise their times tables up to 100? Is that what it takes?

ColinWright7• 4 years ago

It's a sense of what numbers are. As a decimal, 1/6 is 0.1666.., so that will mean that 6 times 16 is around 100.

How many seconds in a day? It's 60 times 60 times 24, but 24 is around 25, which is 100/4. So it's 60 times 15 times 100, and 6 times 15 is about 6 times 16, which is 100, so it's 10 times 100 times 100, which is 100,000.

But that's too big by about 4% (because we used 25 instead of 24) and another 6% or 7% (from using 16 instead of 15) so it's about 11% smaller.

So seconds in a day is about 88,000.

All small numbers, few facts, but being comfortable in using them quickly with a side order of sanity checking.

tombert2• 4 years ago

I can't speak for anyone else, but for my mental math (which admittedly is only so-so, not spectacular), I make pretty liberal use of factoring. Want to multiply a number by 15? Multiply it by ten, then multiply it by five, and add them together.

Usually that's good enough to get me a rough number, if I need a precise number I whip out my phone.

ColinWright7• 4 years ago

Or if you want to multiply by 15, multiply by ten, then add on half again.

So 34 times 15 is 340 plus half of 340, which is 170, so the answer is around 500 (actually 510).

But if you want 34 (specifically) times 15 there are other ways to go.

Or (34 times 15) is (17 times 30). Six times 16 is around 100, so three times 15 will be half that, or around 50. So 34 times 15 is around 500.

Or 34 is about 100/3, so multiply that by 15 is 100/3 times 15, which is 100 times 15/3, which is about 500.

Being good at mental arithmetic and estimation isn't about being able to do sums quickly and accurately, it's about being happy with rough calculations, retaining a sense of how accurate you are so you can fix it later (if necessary).

shadowofneptune• 4 years ago

Memorizing the tables past 12 is something stressed in older education, but isn't necessary. It really comes down to knowing strategies to simplify the calculation. If you can round the operands to a more easily used number, say 23 becomes 22 or 20, it's much simpler to do the mental math.

chousuke• 4 years ago

Do people actually do that? I never bothered and made up my own method instead.

10x is trivial. 9x is 10x - x, which is also trivial. 5x is also trivial because it's 10x / 2. For the rest I just memorized whatever stuck the easiest and calculate the rest. For example, 7*4 is not automatic for me but it becomes 7*2*2 which is easy.

ALittleLight2• 4 years ago

Once I was in a meeting where someone was presenting from a spreadsheet. As the meeting was insanely boring I was entertaining myself double checking the math in the spreadsheet and spotted a formula error thanks to mental math. I don't recall that it changed anything, but the amount of smugness I felt when I pointed out the error and the presenter confirmed it was so great that I have a smug smile on my face even now, five or six years later, just thinking about it.

conductr8• 4 years ago

It happens. I work in corporate finance and am presenting boring spreadsheet stuff daily. I/My team makes mistakes too and I don’t always catch them before the presentation. Sometimes there’s literally no time it’s like dev on prod without tests.

Anyways, what you did is very common. Many people are actively looking for mistakes in the math instead of focusing on the content of the presentation. And will derail my entire meeting when they find one. Just word of warning , don’t be that guy. Usually it’s some unimportant metric (or someone would have noticed) and directional correctness facilitates the same conversation. For example, oh you’re right overtime is running at 10% instead of 8% is usually not a material fact when your department is 30% over budget and some productivity metric is where the businesses focus needs to be (eg. widgets/hour).

Instead, bring it to my attention after the fact. Or shoot me a email while in the meeting. Only verbalize it if it’s crucial to the conversation or impacts the decision being made. It’s a bit like expecting bug free software, it’s just impossible.

Not saying you handled it wrong perse, most people do this. Just suggesting how I have been on the other side of this and prefer for others to act. I find this more professional as I wouldn’t turn up to your product launch party and just tell everyone in the room about the GitHub issues log. Or, as another example, if I were your manager, I wouldn’t call you out publicly but assume you prefer a private feedback session when you screw something up.

kragen10• 4 years ago

If it isn't important whether the numbers are real or fake, why show them? To intimidate people with a baseless appearance of mathematical rigor? If so, I'd say the people who point out that they're fake are doing a service to the decisionmakers in the meeting, though it's unsurprising that the presenters don't appreciate it.

kragen10• 4 years ago

Ego. It's a hell of a drug.

singularity2001• 4 years ago

Conclusion (also from own experience): Mental math in 2022 can be a career boost and significantly increase your appreciation among peers.

curiousllama3• 4 years ago

Happens all the time! Just be careful with this stuff - often, the small mistakes in unimportant stuff are just a junior futzing up something small. If it derails the meeting, you may be ruining some faceless 23 YO's performance review for an otherwise-innocuous bug.

treeman792• 4 years ago

Was in physics class and the professor put a problem on board to keep the entire class busy for the hour. She had grading or something. I immediately shouted out the answer at first glance. Did that several times.

dotancohen• 4 years ago

Smugness in meetings is fun, but a comment right here on HN demonstrates that even the presumably-intelligent people browsing HN don't have mental math skills.

https://news.ycombinator.com/item?id=30505286

karaterobot• 4 years ago

Since he said "use" and not "practical use" or "lucrative use" or even "productive use", I'll answer for myself: I really enjoy passing time making fermi estimates and back-of-the-envelope calculations (sans envelope).

For example, any time I go to a new restaurant, I try to work out the annual profits based on the time of day, cost of a dish, number of people, staff, location, etc. I don't want to open a restaurant myself, but it's fun to try to think of all the inputs I can, and do some arithmetic in my head to get an answer. It's an enjoyable way to make time pass while waiting for my meal to arrive. For me, it beats reading Twitter.

milansm• 4 years ago

It would be great if you could get a correct (or close) answer every time if you made a guess, and where you made an error. I assume you would get better after a while and would be even more fun.

mimimi31• 4 years ago

I often pass the time while e.g. running or sitting on a train by approximating square roots of and squaring random numbers or calculating the day of the week for random days. I don't think I've ever used these skills for a practical purpose.

jancsika3• 4 years ago

Of course nobody knows exactly how many piano tuners there are in New York, but you could guess about how many piano owners there are, how often a piano needs to be tuned, and how many tuners it would take to service this demand.

You don't want to know how often a piano needs to be tuned. You want to know 1) how often the average piano owner pays to get their piano tuned, and 2) how often the outliers (like Julliard) pay to get theirs tuned. For #1, my own ears doubt there's much if any relationship to how often they need to be tuned.

I feel like this is a common problem with this approach-- something about starting the journey leads one to choose an imagined ideal contingency rather than one that is practical. Almost like the deeper you go, the more likely the thing you're guesstimating will be a fantasy.

Edit: clarification

spaetzleesser6• 4 years ago

“Almost like the deeper you go, the more likely the thing you're guesstimating will be a fantasy.”

Like with every other method you need to know when to stop.

jancsika3• 4 years ago

Seems like people stop. Just rank speculation that the last level they go to before they did was probably not a great choice for a branch.

jodrellblank2• 4 years ago

"Of course nobody knows exactly how many piano tuners there are in New York, but you could guess about how many piano owners there are"

You could guess how many piano tuners there are. Why is a guess at piano owners expected to be any more accurate?

(I know that's in the blog, and original question, not just your comment).

jancsika3• 4 years ago

Because a meaningful guesstimate for tuners needs a workable sample size, and it would generally be too close to zero to tell if any exist.

At the time Fermi lived a metric shitton of abodes housed a piano-- that's almost certainly why he used that example. You could just peruse a block of abodes to get a decent estimate (or simply think of one in your head). But in general there are fewer tuners than there are pianos, so even at the time a rough estimate for a city block could return zero.

Even today, if I walk a city block I'll find six buildings that house a piano. But I'd probably need a 20-mile radius before one of two piano tuners come into range.

Another question occurs to me-- when I peruse the pianos in the block, what is the chance I'll actually see one of the tuners doing the tuning? Smart tuners leave their card inside the piano with a list of dates of tuning-- if I have a chance I'll sneak into one of the houses of worship and see what the frequency is.

I'll bet it's more likely I'll spot some deer meandering on the way.

lacrosse_tannin• 4 years ago

Combine this with memorizing a small amount of facts about the world, like "the population of the united states". If you know the population of new york, you're one step closer to the answer to that piano tuner example question.

spaetzleesser6• 4 years ago

Being able to do quick basic math is also very useful outside of business. A lot of people have no idea how much financing of something like a car will really cost them. Or the decision buy vs rent. Obviously there are a lot of other factors and you will want to check your numbers before making a final decision but I find it very useful to make a quick calculation as a baseline.

frontman1988• 4 years ago

Practising mental maths at a young age probably also helps in improving working memory. Brains of young people are so malleable, it's very important to do the hard work even if it's drudgery cause it helps improve the thinking capabilities.

ejb9992• 4 years ago

To prevent showing the world how truly ignorant a person you are, like this classic: https://www.youtube.com/watch?v=krHkjdnniDE

from an editor of the New York Times, and 'the most trusted named in journalism' Brian Williams, not to mention all the other people along the way in producing this piece who couldn't do basic math, that allowed this 'story' to make it onto the air without anyone sensing that it was off by a factor of about a million.

christiangenco• 4 years ago

This mistake is so interesting to me. Here's a great explanation of what's probably happening: https://youtu.be/6egeUxIEQnM

tl;dr: mentally we take off "millions" as a unit, do the calculation, then add "millions" back on. This works for addition, subtraction, and inequality but fails for multiplication and division.

ejb9992• 4 years ago

the fact that this basic math error made it thru so many people and still made it onto the air is very telling - and scary.

MontyCarloHall2• 4 years ago

That one is especially egregious because you don’t need any mental math to realize the absurdity of the statement (“the $500 million Bloomberg spent on his failed presidential campaign could have given every American over a million dollars each”). Realizing that statement is absurd just requires a basic sense of how the world works.

As an analogy, imagine someone said that “a cow that weighs 500 million milligrams could give every American more than a kilogram of meat each.” You don’t need to know that 500 million milligrams is 500 kilograms and that this clearly would not be enough to give 300 million Americans a kilo of meat apiece — you just have to have an elementary enough understanding of the world that it’s obvious a single cow cannot feed an entire country.

rainonmoon• 4 years ago

The analogy falls a little flat because most people can conceptualise a cow easily enough, but most people have zero intuitive concept of how much 500 million dollars is.

jihadjihad5• 4 years ago

I'm thankful every day that throughout my education it was drilled into me never to circle my answer at the end of a math problem until I first re-read the problem and asked if my answer made any sense.

I feel for the math teachers out there who receive homework where a sink faucet spews 70.4 gallons of water per second.

mtlguitarist• 4 years ago

One funny incident I had on this topic was when I was TAing calculus one student proudly proclaimed that there's no gravity on Mars while doing velocity problems.

This didn't stop them from using the gravitational constant provided in the assignment or from getting the correct answer even, but it was funny regardless.

mmcdermott• 4 years ago

I'm thankful every day that throughout my education it was drilled into me never to circle my answer at the end of a math problem until I first re-read the problem and asked if my answer made any sense.

I had always done this, but I remember hitting a word problem in calculus where the calculation came back with something that made no intuitive sense. I remember staring at it and re-tracing each step before turning it in, certain that I had have made a mistake, but unable to find a problem. It turned out that nothing was wrong, the values just weren't real-world consistent.

booleandilemma• 4 years ago

It always amuses/scares me how much of a separation there is between people who think and people who just parrot others. The people in that video just copied what they saw people commenting on twitter. It’s ok though, they have nice looks and voices and it’s impolite to call them idiots.

It’s how we end up with eggcorns. People just say things without understanding what it is they’re saying.

When your words don’t have to adhere to reality, you can end up saying some ridiculous things, and the other parrots in the room will nod and agree with you.

https://en.m.wikipedia.org/wiki/Eggcorn

exolymph• 4 years ago

Early-stage covid was so revealing of this — who actually thinks, versus downloading updates from NPR and CNN.

gnicholas• 4 years ago

Last week I was overcharged for a large takeout order at a restaurant. They had billed me twice for one of the items, and when the total came up I asked for an itemized receipt. If I weren't at least somewhat confident in my ability to do mental math, I might have just let it go and figured their prices had gone up (as so many have recently.

squidbot• 4 years ago

Mental math is useful (vital) when playing poker or other odds based skill games! You'd be thrown out of any casino if you broke out your phone every hand :D

And, this sort of goes with the article, it's really useful for negotiations/haggling where you may not want to use a calculator.

mathattack• 4 years ago

I find that it is hard to discuss strategy in real time with people who can’t do mental math. They just can’t keep up if they need to go to Excel every time there’s a number.

jmyeet• 4 years ago

We tend to think of the second one as "back of the evenlope" estimation, which is both pervasive and important in software engineering and related fields (eg capacity planning, resource allocation, etc). I wouldn't necessarily attribute this to mental math however. You can still be good at this without mental math (IMHO).

I might rephrase it to something like: mental math is a reflection your ability to use numbers as one would use tools. So, for example, when you learn probability you use lots of examples of games of chance or you boil it does to coin tosses, drawing stones from bags or rolling dice. Part of doing this is being able to (among other things) correctly negate probabilities, which tends to be far more obvious if you have a foundation in set theory.

Example: expected values. If you roll a d6 you expect to roll a 1 approximately once in every six rools. You get an array of possibilites from this based on how you choose to look at it (eg the probability of rolling a 1 after N rolls or the probability distribution of M 1s from N rolls). A common question comes up is "what is the probability I don't roll a 1 in N rolls?" and those naive in probability will often get this wrong. The answer is of course 1-(5/6)^N. And while you do learn that in probability, even if you don't specifically learn it I find that people who are comfortable with numbers as tools (as evidenced by mental math) will tend to figure it out anyway, or at least a good approximation of it.

Edit: corrected "not rolling" to "rolling". My bad.

leephillips3• 4 years ago

To see that your answer for “what is the probability I don't roll a 1 in N rolls?” is incorrect, consider what happens as N gets large. The probability → 1, which should be intuitively wrong.

kragen10• 4 years ago

Probably they meant (1-⅚)ⁿ.

yummypaint• 4 years ago

Mental math can be like an extention of your natural intuition and vise versa. Ideas that wouldn't otherwise arise can spontaneously come to you because you have the feeling in the back of your mind that A<<B, A=B, etc.

Sometimes it seems a problem has dozens of possible solutions. Mental math is great for whittling away the less optimal ones quickly.

It has been said that the best way to have a good idea is to have alot of ideas. Mental math is the natural companion to testing and refining those ideas in an intuitive but quantitative way.

astroalex2• 4 years ago

I have always been terrible at mental math. Numbers are goop to me. I can create systems which process the goop, but when I try to play with the goop directly, it's like the numbers are mentally "slippery" — I can't get traction.

I did okay in math in college (almost completing a math minor in addition to my cs major), but I continue to struggle with doing arithmetic in my head. I also have a hard time memorizing and processing dates and times. I can't even quickly recall the order of the months!

However, once I move up a level of abstraction, I'm totally fine. For example, I successfully implemented some relatively complex date & time logic (with timezone messiness) on a recent work project. I almost think that my lack of intuition around dates and times helped me there, because it forced me to logic my way through each and every part of the system.

Just don't make me touch the goop directly!

lmkg• 4 years ago

Are you aware of "Dyscalculia?" It's similar to dyslexia, but with numbers. Less-studied, so less-well-understood. On the surface, it sounds similar to what you describe: no issue with math per se, but numbers in particular try to run away from you.

astroalex2• 4 years ago

Interesting, I have not heard of that! Thanks for the suggestion, I will look it up.

nofunsir• 4 years ago

New Math - Tom Lehrer (1965)

https://www.youtube.com/watch?v=UIKGV2cTgqA

1e-9• 4 years ago

It makes you look like a genius rather than like this: https://www.youtube.com/watch?v=DHFB40WOMOo

curiousllama3• 4 years ago

Data issues!

I never bother to pull out my calculator to see how much 639/42536 is, but good mental math/intuition lets me thoughtlessly go "oh, that's only 1.5%" rather than "600+ seems like a lot"

ddingus• 4 years ago

Do people not need estimate scale order of magnitude check and any number of other common tasks?

I hold a number of mental models that I used to make all sorts of decisions. These don't need to be precise oh, but I often need them to be fast and or handy.

Sometimes one needs the math just to think about things.

Always needing a machine transfer the data in and out input and all of that is a real pain in the ass sometimes. Who wants that dependency?

whoomp12342• 4 years ago

when all of our calculators are SAAS on the cloud, and then the stock market crashes and all cloud vendors go offline, you will be thankful you memorized your multiplication tables! or atleast thats what I will tell the kids

asdff• 4 years ago

Because you never always have a device on you. Sometimes its dead. Sometimes its charging at a dock across the house. Its ultimately a lot clunkier to pull up a calculator app and tip tap at a faux 4 function calculator than it is to just think.

MandieD• 4 years ago

Mental math can give you a quick idea of whether or not something is worth investigating more closely.

I play a guessing game at the grocery store after I load my cart onto the belt and am waiting for the checker to finish. I’m usually within 5% when there’s a large variety of things and produce is involved, and have been alerted that I needed to check the receipt more carefully than usual.

This habit helps me be more confident at restaurants in a place where I’m a second language speaker - and when discussing pricing with vendors in that second language.

r00tanon• 4 years ago

Good cognitive exercise and fun.

Also if Putin sends the missiles, it may come in handy in 2022.

What use is mental math in 2022?

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