What use is mental math in 2022?
johndcook.com
johndcook.com
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.
If someone asked you "If it isn't important whether the software works or not, why ship it?", what would your reaction be?
Note: > Directional correctness facilitates the same conversation
There’s such a wide spectrum of situations it’s hard to be too rigid with this. My software analogy was just pandering to this audience. Probably not a perfect analogy.
You might say that the same can be true of calculations; after all, calculations about the contingent universe (as opposed to, say, number theory) are always based on data incorporating some uncertainty. But there are two key differences:
① I don't put up a slide showing all the intermediate values of the variables in my program in order to make an argument for something. Instead, I try to show my chain of reasoning in enough detail to be convincing and to expose any relevant potentially incorrect assumptions or reasoning steps I've made so that others can find my errors, without including trivial or irrelevant details. Including trivial and irrelevant details makes it harder, not easier, to find relevant incorrect assumptions or reasoning steps.
That's why I question the motivation of people who include those in their slides: it sounds like they're attempting to make their argument so complex that there are no obvious errors, rather than so simple that there are obviously no errors.
② A 10% uncertainty in an input datum can be traced through the calculation from beginning to end and will often result in a 10% or 21% uncertainty in the result, or even less. (And where that's not the case the sensitivity should be called out.) A calculation that's simply incorrect—for example, treating millions as thousands, or forgetting to divide by the relevant denominator—commonly produces results that are off by multiple orders of magnitude.
I also think there is a difference between software and spreadsheets. At least, in the role I was in, software let us do things and spreadsheets let us reason about the business.
If there was a bug in software, say, 5% of requests to save a configuration failed, and that failure triggered an automatic retry, and the result of this bug was elevated configuration save latency at the 95+ percentile, we might decide to ship that bug (depending on other factors). On discovering the bug we would try to understand its impact and then we would reason about the impact on the software's goals. For example, we might say that people rarely change configurations, and an additional 100ms of latency for 5% of customers is something we can live with until such time as we solve the underlying cause.
What we would not do, in any software org that I've been apart of, is say "Hey, we notice some of our operations are failing sometimes, anyway - let's ship it!" To me, that seems analogous to what you are saying. You know there are errors in the spreadsheets but you ignore them on the assumption they are inconsequential, reasoning that if they were consequential you or your team would have noticed them.
There are ways for bugs to exist in software that don't block what the customer is trying to do. Maybe the bug makes it harder to accomplish a workflow, or makes it take longer, or looks silly, or something like that - and those are bugs it may be okay to ship with (depending on context).
If values are wrong in a spreadsheet, then the only way I could think that doesn't matter is if the values are unimportant. That is, either the values are important and it matters if they are wrong, or it doesn't matter if the values are wrong because they are unimportant. If the values are important, let's correct them. If they are unimportant, let's not discuss them.
This was for CEO of a fortune 100
People start aimlessly hypothesizing if you don’t show them the nonissues.
Also, management is conditioned to look at certain key things in combination because they are in fact interconnected.
So for example, is the core issue is profit is lower than budgeted. I will show them sales are on plan. I will show them non labor is on plan. Then I will say now labor is the problem. Overtime is immaterial , but our hours/widget KPI seem way high. We call the plant manager and find out an automation process is down. Because it’s so costly, we fly in a support tech that night to fix it. Seems simple enough but in an enterprise this could involve a dozen people and move slowly unless I craft the story of how much this issue is costing us. And since I’m not in the plant, I didn’t even know the problem that I was pointing out to begin with. I probably don’t even know who the plant manager or his boss is. So it’s not like I can just pick up the phone and figure out the whole story (but generally would try to).
There are many times when the math is in fact correct and the numbers are just implausible. That’s usually the meat of the meeting and something to dig deeper on. Hopefully we noticed it, double checked it, and it’s one of my talking points to address in the meeting. Either I want to point it out and explain why or point it out and ask why depending on who I’m talking to.
I wonder if the other teams who present in the same meetings are doing the same kind of careful behind-the-scenes double-checking, motivated by the concern that someone might find an error in their numbers during the meeting. That conscientiousness seems like it might be beneficial to you, even though it's an unwanted cost when it's your team that's saddled with it?
> mathematical intuition ... is indeed a real superpower.
What is this power used for?
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.
On the other side of the spectrum, many people don't even understand percent. So if an article says "1% of corona patients died", they don't understand what that actually means. They understand some people died, but not really what the probabilities are or if they should be worried. They don't really understand much at all about the world as even the simplest of analytical articles are beyond their understanding. They can pick up the emotions in the article and understand that 1% dying is bad, but if you told them that 1% dying isn't a big deal they would trust you as well, as the number doesn't tell them anything its just how you say it.
A worryingly (to me) large number of people can't tell you what 3% of 200 is.
Should we care? Should people have a clue about this? Rates of interest are quoted in percentages. Deaths from diseases are quoted in percentages. Growth rates of economies are quoted in percentages.
If you have no idea about them, how can you make properly informed decisions?
And yet people survive, so maybe it really doesn't matter.
They survive thanks to people who knows math. We can't run modern society without a lot of people who knows math, we would regress centuries, every natural science requires people who are good at math. There is a reason why the people who are decent at math are valued in every modern society, even soviet understood that part.
I think we should care if kids can't do it because they learned it recently, but who cares if an adult does it in their head vs. uses a calculator correctly?
> "being able to do quick approximations in mid-conversation is a superpower."
So here's something to think about:
* People who can do quick, rough estimations in their heads say that they find it useful, and that the experience is that they have a sort of superpower.
* People who can't do it say that there's nothing special about it, and using a calculator is just as good.
Perhaps people in the second group don't have the information needed to make an accurate assessment.
The math doesn't add up for me, and I'm hesitant to believe anyone who says it feels like a superpower because that sounds like an exaggeration. What does that comparatively make people who use calculators to do estimates? Simple primates using an iPhone?
Plus the article's reasoning is really bogus to me too: "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.”"
This feels like a forced problem to justify his conclusions. It goes away if the other party is willing to wait for you.
But we're repeating ourselves. I'm saying I agree with the submitted post, that it's everywhere, and in everything, and feels like a superpower. You're saying you don't understand, therefore it can't be that important. That's fine, the post (and my replies here) have tried to give you an insight into this experience ... you don't have to believe it.
Back in a previous job we'd be in a meeting and people would be discussing strategies. The ability to sketch out options, choices, rough costs, and expected benefits on the fly meant that it all flowed without pause, without hesitation. Pulling out calculators and working out the numbers would have made things horribly staccato ... I've been in meetings where that happened.
> What does that comparatively make people who use calculators to do estimates? Simple primates using an iPhone?
It makes them ordinary people (in this sense) without this superpower. They may have other skills that people find extraordinary and don't understand, they might not.
I think the issue that we're leaving out is how right were those calculations? You can throw out a number to move the conversation on, but if you have to double back because you got something wrong, you lost time there maybe some further conclusions were also wrong. I'd rather just try to get it right the first time, and I know I'm flawed as human, which is why I rely on devices rather than gray matter. But I feel like this is optimizing for the wrong thing: flow of conversation vs. accuracy of calculations. That really starts to seem like the same tradeoffs as experience vs. raw speed.
Part of this entire thing is to know when a number is "right enough". Anyone can choose to just throw out random numbers ... we're talking about being able to come up with numbers that are right enough.
Otherwise the whole thing is genuinely pointless.
I don’t know if the calculations are essential (tho I guess they are) but having the feeling for the numbers and the ways things become inevitable because of numeric truths and relationships is essential for understanding reality.
x% of y = x/100 * y
= y/100 * x
= y% of x
For instance, a job offer. One job offers you $28/hr, the other offers you $55,000 a year with 2 weeks paid vacation time
Simple math shows that $28/hr, with 2 weeks unpaid vacation time, is $56,000/year. ($28*2000), and can be ~$2.3k more if you don't take vacation.
Understanding of the world and yourself may make you realize that earning $1k-~$3.3k less a year would be worth it to have a set income that doesn't change even when you take vacations, but it's nice to be able to quickly compare the stark math so that you don't immediately think "big number better".
That's not the best allegory to explain the utility of math, but the basic concept has rung true in many situations anecdotally in my life.
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."
Then you of course have to check before you say something. But without that intuition you would not checked.
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.
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.
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.
- One person could have enough to give every American $1M without first taking $1M from every American. Did you and everyone you know buy $1,000,000 of Washington Post or Wallstreet Journal subscriptions, for example?
- One person could spend $1M-per-American without anything very interesting or noteworthy happening to the people taking that money in payment.
- One person could spend enough for $1M/American on a Presidential run and lose; did Hillary and Trump and the other candidates also spend that much (WOW!) or did they spend nothing by comparison and still beat Bloomberg? (also wow!).
- One Presidential run could cost so much; $1M/American is a lot more than Americans earn every year and therefore pay in tax every year, and handwaving debt away it's therefore many times more than government annual spending on everything; infrastructure, military, social services, schooling, etc. Suspiciously high.
- A million per American is in the region of 340 Trillion, several times the world's GDP.
"There are 10^11 stars in the galaxy. That used to be a huge number. But it's only a hundred billion. It's less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers." - Richard Feynman
Once net worth is that high, the majority of it seems to be attributable to stock market and other similarly intangible $, rather than $ someone has literally directly taken/earned from others. It seems unfair to expect humans to contemplate big numbers (something we're known to be bad at) by also thinking about imaginary $ as though they had a 1-for-1 tangible source.
That being said, you certainly aren't wrong to think of it this way. For example, Bezos' net worth of ~$110B / ~333M Americans = ~$333/American , which sounds like a reasonable ballpark approximation of how much _profit_ the average person has spent on Amazon + Audible + AWS + WaPo.
Thanks for giving examples of logical ways to process large numbers. These are good ways to double-check quick maths.
I feel for the math teachers out there who receive homework where a sink faucet spews 70.4 gallons of water per second.
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.
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.
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.
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.
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.
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.
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%!"
Obviously this can and is constantly abused, but it is undeniably useful.
Is x% of y the same as y% of x?
So 14% of 50 becomes 1.4 * 5 == 7
Or to simplify even more…
1 * 5 = 5
4 * 5 = 20 -> remove the “0”
5 + 2 = 7
One significant underlying reason is that different authors make different value judgments and want to tell different narratives.
For a given set of values and narrative, there are many questions to consider, such as:
- Should a change be expressed summarized as one scalar?
- As a difference? As a division? As a percentage change? (Read https://en.wikipedia.org/wiki/Relative_change_and_difference for many ways to do it; you might be surprised at the number of ways)
- Should the initial and final quantities be specifically mentioned?
- But what are the appropriate "initial" and "final" quantities? These are tied into the goals (above).
Is a relative change best expressed as
* 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)
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
Like with every other method you need to know when to stop.
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).
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.
conversation velocity, yes, but not project velocity, which is much more important.
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.
Ballpark estimates for software tasks are useful in situations like https://xkcd.com/1425/: before ImageNet, determining whether a photo contained a bird really would have been a multi-year research project with uncertain success, while determining if EXIF places a photo within a national park was a relatively simple GIS query. But this is not obvious to muggle stakeholders.
Of course, for things like bug fixes, scientific discoveries, and market research, often even estimation of that level of precision is unavailable. It's true that that can be infuriating, but that's just the way it is. Will this new model sell to a thousand customers or a billion? You can't tell in advance. Will this bug take five minutes or five months of effort to fix? You can't tell until you find the cause. When will we discover the next new class of antibiotics? It may already have happened or it may take a century.
Very often you can know if it has a chance to be in the thousands or billions just by looking at the market size. Or you can estimate a possible size of installed base by looking at how many companies may be using your stuff.
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.
I am not so sure about that. From my experience quick guesses are often as good as detailed estimates that are based on assumptions that will change.
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.
Just to add when I mean fractals at 7 and 8, I don't mean the hard maths or the programming, for sure.
But some ways to imagine the context of the hard maths.
Like how to recognise fractals (or just self-similarity) in nature, or explore drawing some by hand, that is a really crucial thing that a seven or eight year old should be introduced to. Why are leaves like trees? Why is broccoli simpler than it looks? River deltas, snowflakes, you know...
Kids are shown lots of these things -- and they often notice self-similarity without prompting -- without being told that there is a unifying theory. The unifying theory is amazing in and of itself, and can be demonstrated with a Logo turtle.
(Which brings me onto another thing... where the heck are all the logo turtles)
There's a long and broad tradition in the UK of the "Christmas Lecture" now (started with the Royal Society which is televised; it's what a TED talk is, but better). All-ages family learning, made fun and accessible.
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.
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?
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.
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!
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.
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!
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.
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.
https://mitpress.mit.edu/books/street-fighting-mathematics
Click on "open access" to get to the PDF link.
"can't be more than X"
or
"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.
Any practical utility is a distant second.
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"
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.
Usually that's good enough to get me a rough number, if I need a precise number I whip out my phone.
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).
We who knows our powers of two do
34 ~= 2^5
15 ~= 2^4
2^5 * 2^4 = 2^9 = 51210x 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.
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.
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.
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...
And, this sort of goes with the article, it's really useful for negotiations/haggling where you may not want to use a calculator.
Also if Putin sends the missiles, it may come in handy in 2022.