
The Math Myth - forloop
http://econlog.econlib.org/archives/2016/09/the_math_myth.html
======
nostrademons
This largely matches my experience - as a software engineer, I spend probably
< 1% of my time doing math more complex than arithmetic and simple
mathematical logic - but it's missing something crucial:

In a market economy, basically all returns come from _marginal_ gains. The
vast majority of your lifetime income will come from a dozen or fewer
opportunities that you happen to be in a position to take advantage of,
whether it's a new job offer or a high-profile project you volunteer for or a
startup that takes off. You will qualify for those opportunities based on the
skills you have _that other people don 't have_. They will make money for the
organization because of features or insights that your competitors lack. Your
customers will buy it because it lets them do things that they couldn't
otherwise do.

The stuff that you and everyone else spends 99% of your time doing is
economically irrelevant. You probably still need to do it (though if you can
program a computer to do it, you have a huge leg up on competitors), but it
doesn't get you anywhere.

Ironically, this is one of those insights that a good understanding of math
will give you. The common-sense understanding is that we should be teaching
what the majority of people are doing; the data says that we should be
teaching what the majority of people are _not_ doing but desire the results
of.

~~~
NhanH
Wah.

>The vast majority of your lifetime income will come from a dozen or fewer
opportunities that you happen to be in a position to take advantage of,
whether it's a new job offer or a high-profile project you volunteer for or a
startup that takes off

Is this really true? I'm still way too early in my career to know. But I'd
have thought if you stick with the more "traditional" route, your income would
be fairly consistent (as opposed to something similar to the startup lottery,
for example).

~~~
xapata
Even if you stick with the traditional route, having a tiny initial advantage
for your first job will have compounding results over your lifetime. This is
significant enough that people who graduate college during a recession on
average have a 9% lower salary during the first 10 years of working [0].

Many people can tell you about their "big break". That one success that seemed
to cascade into other successes. Whether this is a real phenomenon or simply
an illusion of memory, I don't know. It's quite possible that if that big
event hadn't taken place, there'd have been a similar one just a little later,
taking the person on a different, but equally pleasant path. I suggest you not
trust individual narratives, only systematic analysis.

[0]
[http://www.nber.org/digest/nov06/w12159.html](http://www.nber.org/digest/nov06/w12159.html)

~~~
spdionis
The big break has also an important psychological effect that acts basically
like compound interest.

~~~
xapata
It certainly feels like I have had a series of lucky events in my own life,
but who knows...

------
analog31
At my workplace, we have about 60 scientists and engineers. The author's
observation is accurate, that most people never use math beyond Excel and 8th
grade math. They also never use most of the theory that they learned in their
science (including CS) and engineering educations.

The typical career arc is to get through college, then sit down at a CAD
workstation, or programming terminal, and forget all of your math and theory
within a few years or even months. Time that isn't spent doing CAD, is spent
on testing, troubleshooting, dealing with vendors, and so forth. A few of them
start prepping for management. It's becoming increasingly common for engineers
to start their MBA training as soon as the company agrees to pay for it.

Truth be told, outside of a few life-support-critical applications, most
design is done by trial and error. Very little real engineering gets done.

And the products we make, are designed to provide similar benefits in another
profession. We are told by management: "Our customers don't want products that
require them to think. They want something where a person with an 8th grade
education can push a button and get an answer."

When a math or theory problem arises, they take it to the resident "math
person." That's me. I'm glad that I spent the better part of my youth learning
math and theory, because I'd be as suited to the CAD workstation as I'd have
been to working at a loom, or a lathe, 100 years ago. For the most part, the
people who emerge as "math people" are the ones who were interested in it as
an end unto itself, in the first place. I didn't study math because I expected
it to be necessary for a job. I studied math (and physics, programming,
electronics, etc.) because I was interested in those things. They were for me
an _escape_ from preparing for my career.

~~~
nabla9
All mathematics is applied mathematics. Pure mathematics is just mathematics
applied to mathematics.

This is problematic, because the way mathematics is currently taught only
small number of students actually grok it and make deep connections that
enable them to build up on what they previously learned and learn more. Others
have the constant feeling of things getting progressively harder to understand
and use. I'm sure that most people (including engineers) would benefit from
the ability think and really internalize concepts taught in high school level.

Pedagogical research is almost entirely directed towards small children.
Psychologists have studied children and know the common hangups children have.
What are common misconceptions, how to use them to make children learn. How
they learn to count numbers past ten. Competent teacher can help small
children to learn faster.

I think it's possible to teach most people to think _in math_ but it's much
slower process.

~~~
petra
>> I think it's possible to teach most people to think _in math_

Something like betterexplained.com who tries to develop people's math
intuition ?

~~~
rudedogg
Thanks so much for sharing
[https://betterexplained.com](https://betterexplained.com). I'm amazed I
haven't heard of it before.

------
cs702
I think this essay asks the wrong question, and then reaches doubtful
conclusions from it.

We should not be asking whether most individuals today use higher-level math
in their daily lives, because the answer we get will depend on the degree of
math literacy of the people with whom those individuals must interact every
day. The level of discourse is often dictated by the 'lowest common
denominators' \-- that is, the people with the least math literacy.

For example, freshly minted engineers who are surrounded by math-illiterate
work colleagues quickly learn that they must avoid higher-level math if they
want to interact successfully with others at work. Over time, the level of
discourse of these engineers gradually drops toward that of the work
colleagues with the least math literacy.

A type of "Gresham's Law for math literacy" is at work.[1]

The question we should be asking instead is whether society would be better
off if more people had greater math training and literacy. Would our debates
be more informed and higher-quality? Would our decisions be smarter? Would
there be more technological innovation and wealth creation? Would society as a
whole be better off if more people were trained to think creatively and
critically with the rigor of higher-level mathematics?

I suspect the answer is yes.

[1] Gresham's Law --
[https://en.wikipedia.org/wiki/Gresham%27s_law](https://en.wikipedia.org/wiki/Gresham%27s_law)
\-- states that "bad money drives out good." In this case, unsophisticated
discourse drives out high-level discourse.

~~~
initram
Can't we look at places where that is the case today and see? I always hear
about how in places like Russia, every student learns much harder math than
here in the US, and they learn it better. But I don't see other countries like
Russia producing better engineered products or better science than we do here.
Same with China or Japan, or whoever is supposedly the best this year.

~~~
mistermumble
The good ones come to the US to get better paying jobs. I ran a team of 25
software engineers. I tried to hire the most talented programmers in a tight
job market. All but one of the engineers I hired were born outside the US.

~~~
ap22213
It could be because people who immigrate here are significantly more motivated
than people in general.

I read somewhere that many of the most successful entrepreneurs in the US are
foreign born. That doesn't necessarily mean that foreign born people are
better than American born people (which would be unlikely). It probably just
means that people who have tenacity to immigrate also have tenacity to do
bigger things than average.

------
Animats
I used to write physics engines for animation, back in the 1990s when nobody
had one that worked right. That required reading books on nonlinear
differential equations and getting consulting from experts at Stanford. I had
to learn about quaternions. I had more of a classical computer science
education - number theory, mathematical logic, combinatorics, proof of
correctness - but not enough number crunching.

Before that, I'd worked on automatic theorem proving and proof of correctness.
I still like Boyer-Moore theory. I recently revived the old 1970s-1992 Boyer-
Moore theorem prover and put a working version on Github. It's fun to run that
again; it's a thousand times faster than it was in the early 1980s.

If you do anything serious with graphics, you need to understand 4x4 matrix
transformations throughly. I have the whole shelf of Graphics Gems books, and
they're mostly math. At one point I rewrote many of the C code in C++, and got
rid of their start-at-one arrays. (The original was Graphics Gems in FORTRAN,
and the C version used a horrible hack to make arrays start at 1.)

I didn't know enough filter theory when we were doing the DARPA Grand
Challenge. We had a lot of trouble integrating the GPS and AHRS data into a
good position and orientation. We had about 3 degrees of heading noise, which
kept messing up the map-making function. We really need 3D SLAM, but didn't
know how.

Now I need more math to understand machine learning.

I'm also looking at designing a specialized switching power supply for the
antique Teletypes I restore. You can get enough energy from a USB port to
drive the big selector magnet if you use and store it properly. Fortunately I
can get LTSpice to do most of the number crunching.

I think I've used all the math I was ever taught. And I'm not really into
math.

------
iopq
I think society would be a lot better if BASIC math and statistics would be
better understood.

How many times do you see a study posted here with N=23 and people say "the
sample size is too small" when it's clearly not? How many people ask for a
card deck change to change their luck? How many times do people read a poll
like 49% +/\- 3% vs. 43% +/\- 3% and conclude the two candidates are
statistically tied?

I could probably keep going with just examples from
statistics/probability/combinatorics. But there are other examples of people
misunderstanding math.

I mean I wonder how many people even understand that 0.999... = 1?

~~~
NhanH
To be fair, 0.9999... = 1 is not quite basic. You need to know things like
infinitesimals, the distinction between value and representation of numbers
etc.

~~~
yequalsx
Property of real numbers: between distinct real numbers is at least one other
number. Now try to find a decimal representation of a number bigger than
0.9999999... but less than 1.0. You clearly can't. They must be equal. No need
for infinitesimals.

~~~
ryanmonroe
Okay, so it follows from completeness. Now prove that real numbers are
complete (or provide a construction of the reals that uses completeness as an
axiom) without using concepts foreign or confusing to someone with a middle
school level exposure to math.

~~~
yequalsx
It depends on where you want to start your axioms. We can go the
Whitehead/Russell route or just use this as an axiom.

We convince children that 1+1 =2 without delving into the Peano axioms. It's
ok to not delve too deeply into the axiomatic structure of the reals.

~~~
nzp
You're just proving their point. None of the things you talk about are even
remotely obvious or “natural” to people we're talking about. You want them to
”choose axioms”? Axiom-a-whaaa? 1+1=2 does not need Peano axioms because it's
cognitively fundamentally different than 0.999...=1.0. It is immediately
obvious because dealing with simple arithmetic on natural numbers is in
everybody's experience constantly. In other words we _do not need_ to convince
them. Clever untrained people would be able to get 0.999... thing quickly if
you gave them some explanation and let them think a bit, if they cared, but
properties of real numbers are far from obvious or basic.

------
majos
I think one non-obvious benefit of a good mathematics education is that you
have little choice but to develop a tolerance for and understanding of being
wrong. See Jeremy Kun's blog post [1] for more, but my own experience has been
that in e.g. discussing different ways to solve a problem or prove something
almost every person eventually has an "oh, no, I see, I'm wrong and you're
right" moment. Not that every mathematician is necessarily a font of humility
and grace, but I think math offers more regular and irrefutable demonstrations
of your own fallibility than many other fields, and this is good.

[1] [https://medium.com/@jeremyjkun/habits-of-highly-
mathematical...](https://medium.com/@jeremyjkun/habits-of-highly-mathematical-
people-b719df12d15e#.2ikc1ut7w)

~~~
jdietrich
In my experience, a surprising number of people with a humanities education
simply don't believe in "wrongness", but merely differences of opinion. They
regard truth as peculiar abstraction used by mathematicians and hard
scientists, not a phenomenon that actually exists. It's hard for someone to
admit to being wrong if they don't even believe in the concept.

~~~
mikebelanger
I had a philosophy professor who believed evolution was 'just a theory', and
routinely dismissed it. I don't think she was religious either. Then again, I
had another philosophy teacher who taught logic, which is the most pure use of
right/wrong that I can think of.

~~~
PunchTornado
evolution IS 'just a theory'. like relativity and other scientific theories.
it is just a coherent framework to explain observations. today it is our best
theory, tomorrow it may not be. you cannot say that evolution is Truth. Truth
exists only in mathematics, logics etc.

------
ar0
I don't buy the sports analogy with which he argues that it is "self-serving
nonsense" if people state that mathematics education trains your general
problem solving skills. His argument that soccer players should only play
soccer seems not to be anchored in reality: _Of course_ professional soccer
players spend a lot of time in the weights room or go running to enhance their
general strength and stamina [1]! They do _not_ only train their bodies by
playing soccer...

I do think that learning math does help you to think more clearly and to
analyze problems in a more systematic matter.

Now, he does not define well what he means by "higher mathematics": I agree
that (as with almost all learning) there is diminishing marginal utility in
mathematics education. While I would argue that learning how to work with
percentages and also basic calculus (to get a feeling for the difference
between a change in position and a change in velocity, for example) increase
your general problem solving skills by a lot, if you have been through all
this then learning about Ricci flow will probably not do that much to your
general problem solving anymore.

[1]: [http://well.blogs.nytimes.com/2014/07/16/train-like-a-
german...](http://well.blogs.nytimes.com/2014/07/16/train-like-a-german-
soccer-star/?_r=0)

~~~
rm445
Allegedly (association) football coaches seventy years ago would make players
train without the ball all week, on the grounds that they would be keener to
actually play football come Saturday. Of course they ended up under-skilled.
My point is that one should find the combination of training that gets best
results.

In my engineering career, successfully solving technical problems has
generally consisted of working out what basic techniques solve an
approximation of the problem and leaving it at that. I would say first-year
undergrad level rather than 8th-grade, but definitely not the most complex
mathematics I've ever looked at. Apparently being able to put together any
sort of solution from scratch is relatively rare.

I do think problem-solving could be better taught. And schoolkids should
definitely learn more about finance and statistics. Going on, the OP's stance
seems fairly objectionable, but it's hard to disagree that employers use
success in maths-heavy degrees as a proxy for selecting who may be best at a
technical job. It seems like a fairly good filter, but it probably leads to
injustice in certain cases, and the credential-chasing and learning less-
necessary things may be inefficient.

------
dhd415
The value of studying more advanced mathematics is not tied strictly to what
will be used on a day-to-day basis in one's job. I studied math well beyond
what I use in my day-to-day work as a software engineer, but I've found it
valuable for at least two different reasons. First, it exposed me to ideas and
concepts beyond what is right in front of me every day. If I happen upon the
occasional question about computational theory or cryptography or whatever, I
am at least aware that there's a field of study around it and I know where to
look for solutions to known problems. Second, I don't think I'm entirely
unique in that my mastery of lesser math was improved by studying higher math.
In other words, I'm pretty rusty on things such as partial differential
equations, but because I studied them, I know algebra, trig, basic calculus,
etc., cold and that is beneficial both in my day-to-day work and normal life.

~~~
initram
I agree wholeheartedly! It reminds me of when I was studying music. When the
teacher wanted you to perform a piece for a recital or concert, you'd always
play a piece that was a level or two below where you were at because you could
nail it. You didn't play the piece you were currently working on because it's
already hard enough to play during practice, without the added stress of doing
it in front of an audience.

------
sp527
This has so much more to do with the lack of easily monetizable applications
of complex mathematics. I'm sure a significant number of engineers and STEM
professionals feel (as I do) that they're deliberately eschewing those
subjects not for a lack of interest, but rather as a response to market
demand.

The market of people who are genuinely passionate about complex subjects in
math and science is saturated relative to available opportunities. It makes
more sense for an intelligent person to take the lower overhead and more
achievable approach to becoming a value creator (e.g. full stack engineer with
a strong focus on product development) than waste time competing against the
countless PhDs vacating academia.

I use a similar argument for avoiding the ML/Deep Learning hype train. At a
large corp, that job should be left to people who've spent a lot of time
mastering the subject. And if you're using ML heavily in an early stage
company and don't have a PhD, you may very well be out of your depth
competitively or wasting your time optimizing prematurely.

But even ignoring all of that: anyone who's either spent time on or interacted
with a data science team understands how difficult it is to create value with
ML as well as how intangible the value that's created can often be. I worked
at a fairly well known company that told clients we have a data science team
and could use ML, knowing full well that the team rarely if at all manages to
generate meaningful insights, because dropping buzzwords is an essential
branding tool.

Here's a better approach and the crux of why higher math is often superfluous:
the best way to create value is to specialize in problem-solving first
principles and remain amenable to either adopting new skills ad hoc or hiring
to fill any skill deficiencies.

The caveat: if you're passionate about STEM and that's a higher priority than
'creating value' in a deterministic and practical way (and maybe it is and
that's perfectly fine and even reasonable), then by all means indulge in it.
But it's important to align your expectations about what you want to do with
yourself with the way in which you spend your time. A lot of pain arises in
misconceptions around the question of what we want and the reality of what
we're doing.

~~~
yodsanklai
> It makes more sense for an intelligent person to take the lower overhead and
> more achievable approach to becoming a value creator (e.g. full stack
> engineer with a strong focus on product development)

I would say that the surest way to make money for a mathematicaly-inclined
person is to graduate in maths from a prestigious school and work in finance.

At least, that's how I feel when I look at alumni from my school. People
basically could specialize in finance or CS. Those that went into finance make
consistently much more than the others.

I wish I knew that at the time. I thought banks were boring and unappealing
places. But now I think finance is one of the rare field (if not the only)
where you can earn a lot with a technical, non-managerial position.

~~~
kozikow
Only few people in finance really "make it" \- and it mostly consists of
portfolio managers (quantitative or else). "Superstar economy" analogy
discussed in this thread have very strong effect in finance.

Luck is also a huge factor. I know cases of International Olympiad gold
Medalists, who didn't make it as portfolio managers. Do you really think you
are smarter?

If you are mathematically inclined software engineer, I would avoid finance
unless you are immediately hired into the quantitative role in the front
office. In Silicon Valley you will get similar or better salary, more freedom,
more respect and better culture. I worked on the both sides.

~~~
ethan_g
If you define "really make it" as making millions every year, yes that's rare.
But if it's making 300k+ per year, there are multitudes of math-types doing
that, and not much luck is involved.

~~~
kozikow
> But if it's making 300k+ per year

In Silicon Valley it's not uncommon to see new grads (not even PhD) getting
200K total first year compensation. I was comparing quantitative finance vs
silicon valley as a career for mathematically inclined software engineers. You
won't end up poor either way.

------
TheOtherHobbes
Digital Signal Processing - the kind of programming that makes your phone and
your MP3 player work - is math.

3D rendering and animation and 2D browser transforms are math.

AI and ML have large math components.

Speech recognition is math.

Industrial electrical power distribution engineering is math.

Bridge and other kinds of structural engineering are math.

Analog circuit design is math. Once you get past the op-amp cookbook stage it
can get quite complicated, especially if you need to handle RF issues.

Rocket science and aerospace design is math.

Supply chain process optimisation is math.

Traffic modelling is math.

Quant fintech is math.

Encryption and security are math.

At the absolute minimum these need geometry and trig/complex numbers. Many are
impossible without differential equations/calc.

So this is one of the most idiotic comment pieces I've ever read. But
unfortunately it proves that many people don't understand professional
engineering _at all_ , which makes it very hard for them to value it.

Even if the math is packaged and hidden (CAD etc) _someone still has to write
and check the software._ If math isn't taught properly at school, the number
of people capable of that shrinks.

Because these people are disproportionately valuable, that's a very bad policy
indeed.

------
js8
I can tell from personal experience that I only properly understood simpler
mathematics when I started learning more complicated one. For instance, in
linear algebra, finite dimensional (euclidean) vector spaces became a cakewalk
once we started talking about functional analysis.

So, I think, even if you don't need that particular stuff in your work, it's
still a good training.

Also, there has been a pushback against "rote learning" in the past couple
decades. I believe that our minds need repetition in order to learn patterns
and understand abstractions properly. Yes, you forget most of it, but without
it, you won't learn it properly. I don't think you can be any good in any
field without lot of time spent on boring and repetitive things (AKA "work").

~~~
Davidp00
>I can tell from personal experience that I only properly understood simpler
mathematics when I started learning more complicated one.

That's interesting, I have similar experience. Maybe that just means we didn't
learn the original material well enough?

~~~
randcraw
But the only way to maaster new material is to apply it to something more
advanced, rather than just practice it. Rote takes you only so far; you can
solve only problems you understand already. But learning new kinds of problems
expands your understanding of the _limits_ of the concepts and techniques you
already know.

------
heydenberk
I have relatively little mathematical education and have been seeking to
correct that by self-educating over the last year, so I have a certain bias.
Nonetheless, I disagree with the key points of this article.

The article asserts that most modern professional jobs requires only "Excel"
and 8th grade programming. In my experience, over-reliance on software like
Excel rather than a basic competency in numerical programming is a hindrance
to economic growth. Spreadsheet-based numerical programming is opaque and ill-
suited to interoperation. This leads to subtle errors, duplication of work,
difficulty of replication and silo-ing of meaningful results in the private
sector, the public sector and academia.

I take the second point that the transferability of critical thinking skills
developed by learning mathematics is unproven. Nonetheless, history is flush
with anecdotal evidence of this hypothesis, and in the absence of empirical
evidence, it seems unwise to reject that hypothesis out of hand.

EDIT: removed an assertion that the article was poorly argued.

~~~
rudedogg
What are you using to self-educate? I'm kind of doing the same. I decided to
start with "Mathematics for the Nonmathematician", which so far is good (only
on chapter 3 though).

Someone in the comments shared
[https://betterexplained.com/](https://betterexplained.com/) \- which I hadn't
heard of and looks great.

~~~
heydenberk
I'm cobbling it together — I looked at what the requirements were for various
universities' math degree programs and then found classes on Coursera, MIT
OCW, etc., which matched those as well as possible.

I've also found QuantStart's guide[0] to be a particularly good resource, but
bear in mind that is oriented toward learning quantitative finance (in which I
have no particular interest per se).

[0] [https://www.quantstart.com/articles/How-to-Learn-Advanced-
Ma...](https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-
Without-Heading-to-University-Part-1)

------
moron4hire
I think this reflects more our culture of compartmentalizing specific "math"
and specific "science" topics and putting them on a pedestal.

Linear algebra, computational complexity, type theory, Newtonian physics,
circuit design (it's weird being a computer scientist a room full of
electrical engineers and being the only person who knows Ohm's law off the top
of his head and what it means for the project we're dealing with right now)
all of it has been a constant companion for the last 15 years of my career.
The more I can get my hands on, the better.

I know my colleagues in the past [0] haven't employed knowledge to the same
degree that I have, but they have also typically given up and come to me to
solve even fairly trivial problems in trigonometry or object oriented design.
They don't "need" math because they don't care if the only work they work on
is solved problems with easy copypasta solutions on StackOverflow.

My take away from this is not that math isn't "necessary" for work. To me, it
is necessary because I could not be happy living the kind of mediocre, under
achieving lifestyle that it takes to willfully ignore math. My takeaway from
this is that most people are just bad at their jobs. If you want to be any
good (and being this site is focused on startups, I think that is a fair
assumption), you necessarily have to avoid doing what most everyone else does.

[0] I'm finally out of those sorts of environments.

~~~
hamburglar1
I made an account just to upvote this. I think it also leads to the
possibility that even if the authors simplistic conjecture that ~10% of MIT
graduates actually use math, potentially this number (10%) doesn't vary with
tremendously with the number of people taught. I.e. If we teach 100 people
math, 10 use it but if we teach 10,000, 1,000 will. I'm in a profession where
using higher level math is highly, highly encouraged although avoidable. It is
incredible to see CMU level grads not taking advantage of their education. I
think this leads me to believe that the application of math is more of a
personal choice (do you desire to be helpful and add vale) than skill-based.

------
Tyr42
As a mathematician, I would like to point out that there are a lot of
different areas of math, and higher math isn't just learning more calculus.
Graph Theory and Stats, for example.

I have no idea what he's talking about with including Stats in up to 8th grade
math. I've taken a few university classes on it, and I still don't feel like I
have enough to be confident solving all but the simplest statistical problems.

There's a lot you learn in High School. Functions is a big one that comes to
mind. The idea that f(x) = x^2 + 2 or something, and you can compare it to
another function g(x) is pretty important, but not really covered till the end
of High School. Sure, if you have studied programming too, then you know what
functions are, but that's not quite a good assumption to make for the general
population.

~~~
Ologn
> graph theory

Then once you learn graph theory and about trees and graphs, you can learn
about data structures like self-balance binary trees, dawgs, flow networks
etc., then algorithms that run on those data structures like Dijkstra's
algorithm or the Ford-Fulkerson algorithm.

------
exDM69
I think this gets it all wrong by considering mathematics to be a set of
discrete tricks, like 8th grade arithmetic, algebra and statistics.

Mathematical thinking and problem solving are skills that need to be honed and
kept up to date. You do that by learning new methods and tricks constantly.
There are disciplines that require similar skills and have a positive cross-
over to other skills. Computer science theory is very obvious application.
Cryptography is another. The "tricks" in CS or crypto are not taught in school
for everyone, yet having the background in math will undoubtably help getting
into CS and crypto.

What I wish that mathematics education would get through to students is a
better understanding on how mathematical methods are used in a lot of domains.
I see too much of a divide between "math guys" and "non-math guys", with the
latter group sometimes getting quite anti-intellectual when it comes to math
(even if they seem smart otherwise). Even the author of this article has a
very dismissive tone, if we just teach people how to apply 8th grade math and
Excel, who will be the guys developing Excel and other tools?

Even if math education is learning new methods and tricks, they are not the
skill that should be learned. It's the methodology of what it takes to master
a new method - learning how to learn.

Just to give a counter point: I regularly use math skills, advanced calculus,
arcane series formulations and spherical and hyperbolic trigonometry. A lot of
these methods were _not_ taught to me in formal education, but my education
gave me the tools to tackle these advanced subjects on my own by reading text
books and old research papers.

~~~
johnminter
Yes! This! I am 30+ years post Ph.D. and work in microscopy and image
analysis. I have repeatedly needed to go into areas that I never anticipated
and learn what I needed to solve the problem at hand. Happily, my graduate
advisor taught our group to expect this to happen and to become self-directed
learners and to embrace the process.

I will also note that I found this works best when surrounded by a few like-
minded individuals with complementary skills to serve one another as "a second
pair of eyes" and a sounding board for hypotheses and conclusions.

------
fromwayuphigh
Reading someone call for in-depth study in one sentence and saying they're
already convinced of their own pet theory in the next because of anecdata
(anecdatum?) has me puzzled - I can't decide if it's an indictment of the
author or merely ironic evidence for his thesis.

~~~
hamburglar1
I lean towards indictment

~~~
fromwayuphigh
little from column A, little from column B

------
nzp
Of course, the economy doesn't _depend_ on masses having solid mathematical
education (and knowledge), but the world would be a much, much better place if
all kinds of “advanced” math was common knowledge and skill (and not just
math). I am aware that that is currently a bit of a sci-fi scenario. Anyway,
the author need not worry a thing — wishful thinking aside, as long as we live
in a capitalist society, we're in no danger of large percentages of population
being educated in any advanced subject. Or at all.

~~~
dTal
Indeed; just because a job can be done without mathematics, doesn't mean it
can be done as well, or as fast, or with as much confidence in the solution.

One could interpret a lack of use of mathematics as less of an indication it's
not needed and more of an indication we're not maximising our efficiency.

------
dkarapetyan
My current project is using GLPK to do some basic mixed integer programming to
optimize AWS spot instance allocation. If I had not taken linear algebra,
calculus, and a few courses in linear programming the idea would not even have
crossed my mind that I could use mixed integer programming to solve the spot
allocation problem. That's the first half. The second half can be considered a
problem in control theory because it requires taking the new allocations and
gracefully transitioning from the old set of allocations.

You can go even further and say that the whole thing would be even better if I
understood more about stochastic processes and could potentially model the
spot market and make predictions ahead of time to simplify the control problem
and get ahead of the price fluctuations. Saying all you need is Excel and 8th
grade is in the words of one famous physicist "not even wrong".

If you're in an engineering discipline then the more math you know the better.

------
mathattack
Perhaps the arc of my career is different but I've seen the opposite. I've
been in Finance jobs where people who don't understand more advanced
probability can't figure out how to price things. And even people with
advanced knowledge make mistakes.

I've also been in analytics jobs where college educated people mistake
correlation for causality. (It seemed so profound when I learned the concept
only in how often it's abused)

I've seen people in customer support management make enormous judgment errors
because they think don't comprehend the difference between a 500K account and
a 1K account.

Requiring calculus of everyone may not solve this, but requiring a couple
years of hard (beyond 8th grade) stats could help.

As for the CS/engineering/Math requirement for jobs - I think that's just a
reaction to the weak rigor (on average) of so many other majors.

------
__s
Relevant Carmack tweet:
[https://twitter.com/ID_AA_Carmack/status/767911253763170304](https://twitter.com/ID_AA_Carmack/status/767911253763170304)

~~~
Ologn
But Carmack didn't finish college. He's using what he has. This tweet is him
defending his math ignorance in a particular case.

I consider him a better programmer than me, and he is honest about his
shortcomings (like in this tweet), but I am often very surprised about what
things he says he just learned - things any CS undergrad would know.

It's kind of like stories of programmers who were allowed to feed punchcards
to the mainframe once a week - it's amazing what they accomplished with that
limitation, but one thinks how much more productive they would have been with
a more robust interaction.

------
morgante
His conjecture is correct, but his conclusion is not.

Advanced mathematics are rarely used for _any_ professional position
(including software engineering), but that doesn't mean that technical degrees
are irrelevant. In my experience, such filters (like an MIT degree in CS) are
invaluable for two reasons:

1\. Math _does_ teach you to think logically, which is an invaluable skill in
all careers and essential in some (software engineering, specifically). He
claims that "transference of mathematical skills is unsettled," but in my
experience that's totally untrue: try teaching programming to a bunch of math
majors and a bunch of sociology majors, see how learns more easily. Of course,
math is definitely not the only way to learn this—philosophy is also an
excellent way to learn logical thinking, and I wish that more CS departments
required some basic philosophy courses.

That being said, what would be a better way to teach logic skills directly?
The sports analogy is pretty bogus because athletes typically spend the
majority of their time practicing things _besides_ full games of their sport.

2\. For almost any professional field, having smarter employees is an
advantage. Unfortunately, administering and/or requiring IQ tests is
cumbersome and potentially illegal. A technical degree from a top university
is a convenient proxy.

~~~
Camillo
#2 is spot on. That is also why going to university no longer guarantees a
"good job". A degree used to be a pretty decent proxy for intellectual talent.
For many degrees, that was actually most of the value to potential employers,
rather than the specific skills learned. You could get an English degree and
get a job in an unrelated field not because your English education made you
particularly valuable as an employee, but because the fact that you had made
it through college was proof of the value that you had to begin with.

But now, since more and more people are pushed towards college, the value of a
degree as a proxy for talent has been debased. On top of that, there happens
to be some correlation between the actual economic value of the skills taught
in a degree and its power as a proxy for intellectual talent. For example, a
degree in engineering teaches economically useful skills, and is also less
accessible to those of middling intelligence. But a degree in English, on top
of teaching skills of little economic utility, also lacks a strong filter for
intelligence. So more of its value was in its role as a proxy, and it took a
bigger hit to it.

And that is why you see so many people with English degrees working as
baristas. But the key point is that those are mostly people who would have
been baristas _anyway_. The magical feather was fake; the employability, or
lack thereof, was within them all along.

------
stesch
Last week I needed an arctangent at work. Looked it up on Wikipedia and let
Wolfram Alpha compute the result.

It was partly my fault because I was using Blender instead of a CAD system.
Had to rotate something to align it to the base plane for 3D printing. But
hey, I'm no engineer and it worked. And all for a door stopper with the
company logo.

~~~
ddebernardy
Perhaps you're not aware of your math literacy? You knew, remembered, and
understood what an arctangent is. That makes you tremendously more math
literate than the typical non-STEM educated Joe or Jane.

~~~
moron4hire
And, according to the article, more math literate than most STEM-educated
people, too.

~~~
ddebernardy
Suggests the article, yeah. I honestly don't buy that - citation needed. After
some time STEM-educated people might not always remember exactly how what they
learned works, sure. I certainly don't, for one. But I'd be hard pressed to
believe they don't remember it exists - i.e. or at least remember enough to
google their way into rediscovering it and finding a shortcut to solve their
problem.

------
vladislavp
While I agree with some observations: a) most only use basic maths at their
daily jobs)

b) math and computer science degrees are used as a filtering criteria, by
recruiters hiring for actuary/stats/finance and programming jobs

I disagree with what appear to be a conjecture, and the subsequent conclusion:

    
    
      > Acceptance of the conjecture should have revolutionary 
      > educational implications . 
      > In particular, it undermines the legitimacy of requiring higher mathematics of all students. 
      > Such mathematics is actually needed by only a 
      > minute fraction of the workforce
    

Being able to abstract business-specific/domain specific problems into
something that already has well-researched, validated and implement solution
-- is critical, and gives a business an edge.

This is the type of capability (together with knowing a broad universe of
solved topics), that the graduates with CS and Math degrees should bring in
into the workforce.

I do agree with the author's implication, that there is a 'placebo-style'
filtering that's going on by most of the recruiter.

And it is unfortunate, because it brings into Computer Science, especially, a
huge number of people who have neither the passion, no life-long perseverance
to be current in the subject.

------
merpnderp
How would this same conjecture apply to History, Literature, Biology, Physics,
etc etc?

How much of any advanced learning do most people use in their day to day
lives? All of it in the periphery would be my counter-conjecture.

I was always taught trade schools were for learning a particular skill.
College was to equip you with the knowledge and ability to think logically
required to have a better life.

~~~
sramsay
I had the same thought. Follow the author's conclusions, and you'll end up
rejecting college as such on the basis of its apparent lack of "utility."

But why should everything be tied to the demands of the workforce? At some
level, saying "Don't bother learning calculus because you'll never need it"
seems akin to saying, "Don't bother looking at the Mona Lisa, because you'll
only ever have to read road signs."

Is there no intrinsic value to learning? No need to be connected to the
cultures of the past (or the present)? Nothing to be gained by studying all
those ideas that underlie those CAD programs?

The author traces the myth back to Sputnik. It sounds to me much more like
every kid who's ever wept over their algebra homework and asked "When am I
ever going to use this?"

------
kpil
I think the author is both right and wrong. Maths skills is a litmus test that
reflects the scientific education in general among the population.

Given the anti-intellectual and antiscience trends in US and Europe - where US
seems to lead the way, it's at least one way of monitoring the situation.

As a practical skill, anything beyond 5th grade is rarely used, except if in
rather specialized professions, but learning math most probably gives you
tools for abstract reasoning, and probably changes how you look at the world.

~~~
notahacker
I think the author is mostly attacking a straw man.

The argument that mathematics skills in the US (or many other Western
countries) are inadequate usually isn't a complaint that there aren't enough
graduates familiar with advanced pure mathematical theorems. It's usually a
complaint that after years of compulsory education the masses struggle to do
basic arithmetic and understand basic statistics, and plenty of people whose
jobs do entail working with figures or making calculations from time to time
lack the "eighth grade level" numeracy to spot the figures in the Excel output
table are out by two orders of magnitude because someone screwed up inputting
the formula.

------
munificent
"You don't use X the majority of the time." is only a compelling argument to
not learn X if the minority of the time where you do use it isn't that
important.

Most people will spend very little time giving first aid, controlling a
vehicle in dangerous weather, resolving serious relationship discussions,
negotiating important deals, or doing cost/benefit analysis of large
purchases.

However, in each of those cases, the tiny fraction of time where they do those
is so important, it's still worth preparing for them. It may be that most
people rarely use math, but when they do, they use it on important enough
things to still warrant teaching them.

------
jostmey
Few people have to use calculus or advanced probability or number theory, but
everyone relies on it. The article missed this point.

Need some examples? Public key cryptography, machine learning, physics
simulations of the aerodynamic properties of an airplane. I could go on and
on. Just because a vast majority of the population never has to think about
how this stuff works does not imply that it is somehow useless. We would not
be where we are today without all this mathematics.

~~~
mjfl
I totally agree. I feel like this argument is the equivalent of saying "I'll
never need to learn linear programming, I'll just plug it into lpSolve" "I'll
never need to learn cryptography, I'll just use the Unix libraries" "I'll
never need to know thermodynamics, my car just works!" "Google just works!"

At some point _somebody_ had to be the person that figured out all these
things so the other 99% of people can use it and not think about it ever. And
I'd much rather be the person solving the interesting problems than the person
using the tools without truly understanding them to solve more mundane (but
still probably useful and important) tasks, but that is just my personal
taste.

------
jabrown10
From my experience, people at the forefront of innovation have mathematics
background. Quantitate Portfolio Management has a ton of advanced mathematics
and the people designing those strategies definitely use mathematics in
finance.

If you look at the requirements to be a software engineer for a company that
makes video games these days, the mathematics needed is rigorous in the
geometry aspect.

I'm not sure what kind of actuary this guy was interviewing but all the
actuaries I know in the industry that are respected have used a significant
amount of math in their career before reaching management.

I myself am no expert. I have a MS in applied Mathematics from a regular
school and make over $150k in the Reinsurance industry.... I only have 4 years
of experience. My superiors are definitely making 7 figures.

These days with emergence of predictive analytics which definitely using above
8th grade math, shows the relevance of advanced mathematics.

Because we can program computers, Of course you don't have to write these
formulas/equations etc... Everyday but to initially design these systems,
implement, revise, research and innovate, the skills are needed.

That's why at these top companies at the forefront of the industry have a very
diverse international makeup of countries that excel in mathematics

------
aws_ls
Generalizing observations from the lowest common denominator in any work
place, at its core this article is very cynical and perhaps wishes everyone
would just be happy in their mediocrity.

In most projects, there are minority extra-brilliant people, whose
talent/knowledge reflect on the entire outcome/product. So you always need
people to handle more complexity. And as some others in the discussion have
pointed out, often concepts at a level, become clearer when you grapple the
next level of complexity.

Soviet society did not fail because they were better at Maths. It may have
happened despite it. The right point to infer about that would be, brilliance
in Maths is not a sufficient condition for society as a whole to excel. And
that's a moot point, as there are so many other necessary conditions -
food/shelter/being-alive/etc - leave aside politics.

Also the article ignores probability as a core life concept, by which you can
understand so many things. I use it with my kids all the time.

Another thing which frustrates me recently is my inability to grasp modern
physics. Without the relevant understanding of the complex maths, one can only
get the vaguest idea, of what they(the physicists) are saying. This creates a
huge intellectual gap in society.

Also the knowledge gap has another problem. If it gets too wide, then there
will be a very-very few ultra elites who all know what they are saying
(perhaps that's already the case, unfortunately). And the rest of us, only
take their word on face value. I know one person can't know everything and
this is an era of specialization. But still, I think Maths is a fundamental
thing. And society would only gain when more number of people are proficient
at it.

------
linkmotif
Every little bit of math I've learned has helped me in so many inconceivable
and unexpected ways. Articles like this are sad and make me discouraged about
the future of humanity.

------
ontouchstart
The article only addressed the "operational" aspect of math, there is also a
"communicational" aspect of math that enables people to express, share and
understand complex problems and solutions.

Mathematical communication skills will become more important in the
information age with huge amount of quantitative data.

------
dredmorbius
Biologist Edward O. Wilson makes a case for a similar, though not identical
view, in his _Letters to a Young Scientist_. 2nd essay is "Mathematics".
Distilled:

* A strong mathematical background does not guarantee success in science.

* There's a large amount of foundational theory and work which involves thinking in images and facts, not mathematics.

* Maths phobia deprives science of an immeasurable amount of talent.

* True maths talent is probably at least partially hereditary.

* Maths and conceptual work are complements, not replacements.

[http://www.worldcat.org/title/letters-to-a-young-
scientist/o...](http://www.worldcat.org/title/letters-to-a-young-
scientist/oclc/812254231&referer=brief_resultshttp://www.worldcat.org/title/letters-
to-a-young-scientist/oclc/812254231&referer=brief_results)

------
NhanH
> I find it difficult to find anyone who uses more than Excel and eighth grade
> level mathematics (=arithmetic, and a little bit of algebra, statistics and
> programming)

Statistics and programming is way way higher than eighth grade from what I've
seen.

But taking his premise above, then I think no one is arguing for the general
public to learn more than the aforementioned eighth-grade maths. It's just
that the majority of the population isn't any where near that. Specifically in
statistics, programming, and a bit of logical reasoning, I might add (around
modus tollens).

I might be mistaken here, but I've always thought that when someone talks
about "higher maths" the public should learn, it is capped around calculus I,
or some basic linear algebra. Which is like half a year more study over the
list of the author.

------
heisenbit
There is some truth to it - almost all tasks can be done without higher math
skills in my job as consultant.

On the other hand I tend to believe - possibly misguided - that a lot of my
thinking is influenced by having gone through the math education. I may seldom
need exponential functions but I know what is linear and exponential by heart.
Consultants, engineers, architects and managers work with long levers and
knowing how things scale up and down and when they don't matter. Understanding
linear systems, frequency domain and where nonlinearity starts mattering
informs quite a number of my decisions.

Math as a filter for hiring is questionable as imho. most grades. The skills
that matter every day are mostly not analytical skills. Universities as they
are set up are not well geared towards filling that educational need.

------
ausjke
This is not just Math.

I learnt chemistry, physics, biology and all that from middle to high school.
Now as a software engineer they're totally useless and I have long forgotten
all those details that I spent months and years to memorize and master. Even
reading a science-101 book in one day now will teach me more than what I can
remember. Unless you plan to major in those fields, should we just take some
introduction courses instead?

Also I can testify that I rarely need use any math beyond 8th grade since
graduate school as a software engineer, I mean those calculus, matrix theory,
fuzzy logic, neural network, etc. Well I may pick up some AI stuff now, but
it's more like a start-from-scratch-now as I forgot what I learnt then totally
already.

So yes the education system can be optimized to be more efficient.

~~~
ThrustVectoring
Matching people with the professional roles that are best for each other is
highly valuable - so even if most people forget chemistry, it's often worth
teaching, because the introductory chemistry class is how chemists decided to
become chemists.

~~~
ausjke
yes my point is how deep you need learn, maybe have some introduction courses
at middle school is a good idea.

------
klunger
Hmm, I agree that much of the higher level math is not required for
engineering work that has extended math requirements for the engineering
degree. However, at least for aerospace, it goes a bit beyond 8th grade
algebra and excel.

I used to work as an aerospace engineer, doing trajectory analysis. We used
high school trig and algebra, as well as first semester calculus pretty much
all the time. But, I don't think any of the other required math classes for my
aerospace degree were ever used (3 semesters calc, 2 semesters linear algebra,
diff. eq, IIRC). I _did_ end up using a fair amount of stats, which was not
required for my degree, but really should have been.

------
nhebb
In defense of math:

Statistics was the one ongoing use of math in my former roles as a
manufacturing engineer. Beyond that I haven't used much math directly.
However, I have replied upon my knowledge of engineering core courses to
understand and solve problems. I needed to understand calculus-based math in
order to understand that coursework. So math is important.

There is a saying among teachers that in K-3 you "learn to read" and from then
on you "read to learn". The same principle holds for math. Math itself may not
be the end goal for many degrees, but after you "learn to math", you "math to
learn".

------
Jimmy
100000% accurate article. It frustrates me when I see even professional
programmers perpetuating the "Math Myth", as it's called here. "It's important
for all programmers to have a foundation in CS theory!" No, it's not.

The vast majority of human beings will never do anything intellectually
intensive post-college. Even those in STEM fields. Not that an undergrad
degree is "intellectually intensive" anyway.

EDIT:

>The second argument is the one I always hear around the mathematics
department: mathematics helps you to think clearly. I have a very low opinion
of this self-serving nonsense. In sports there is the concept of the
specificity of skills: if you want to improve your racquetball game, don't
practice squash! I believe the same holds true for intellectual skills.

Dear God, I'm so happy to see this in writing. For a while, I was afraid that
I was the only one who had realized this. This observation has several useful
immediate corollaries. For one, it shows that those "brain training" games
that some people like to play are a waste of time. Also, it shows that if you
ever catch yourself saying "I'm working on my X to help with Y", it probably
means that you're just afraid of the failure that will inevitably come when
you initially begin to practice Y, and that fear can only be ameliorated if
you just dive in and start doing Y.

~~~
mempko
The reason for it is because the vast majority of jobs college people get are
bullshit jobs.

I blame capitalism and it's tendency to create useless positions in
hierarchical organizations.

If you think the gov makes all the useless and pointless paper pushing jobs,
you have never been in a large company.

In fact, market capitalism creates all kinds of pointless soul sucking jobs
like lawyers, police, and insurance brokers. This is because nobody trusts
each other.

You can blame gov regulation, except most regulation is written by lobbyists
for mega corps

~~~
Jimmy
I said "intellectually intensive", not "meaningful".

The majority of people throughout history worked manual labor jobs that
weren't intellectually intensive, but they were often quite necessary.

>If you think the gov makes all the useless and pointless paper pushing jobs,
you have never been in a large company.

I'm currently employed at a large company doing mostly pointless work, so I'm
not sure where you got this assumption from.

~~~
mempko
In Adam Smith's wealth of nations he talks about division of labor and how it
makes people "as stupid as a creature can be". I would argue the work a farmer
and people in tribal societies do is more intellectually stimulating and
varied than in the modern world. Unless for a select few engineering jobs, we
mostly damn people to do the same narrow set of tasks every day.

Also about the gov bit. I was talking to the larger HN readers than just you.
Sounds like you know exactly what I am talking about.

------
SZJX
The problem with the American mass has never ever been the lack of "advanced
math" or whatsoever. They're missing the point. It's the tremendous gap
between elite education and "common" education, as well as the lack of very
basic scientific common sense among the population. It's not required for
people to possess outstanding advanced skills like a PhD, but when many get
even some of the most basic facts wrong, and even believe the earth is 4000
years old for example, then there's a massive problem.

Of course I know it's the elites among the upper echelons of the society who
are more than happy to see and maintain such a situation, and unfortunately
this article might well be another addition, a so-called academic/think-tank
publication that serves their agenda. It can't get more obvious at the end of
the article: "leave elite education to those who 'need' it! Keep the mass
ignorant!" Yeah, sure, so that the children of the elites always stay powerful
and the mass keep remaining ignorant. It doesn't matter for the massive power
wielded by the US, the state terrorism employed by Uncle Sam, but it matters,
a lot, for genuine empowerment of the people and true democracy, which people
including the author here doubtlessly want to stop at all costs.

------
mamcx
I think that in the pursuit of higher levels of understanding, some people
miss what is even more important to know.

Is better to have strong basic skills, than high-level skills.

I was (supposedly) of the bests student of my college. However, terrible at
math? Of course.

My grandmother was able to do arithmetic in his head like _nothing_ , yet I
even have trouble with sum and rest.

She only have _3 years_ of education after kindergarten.

\---

In the first class of calculus in the University, the teacher make us do a
division between a largueish number and a small number, at hand. We was
something like 50 people. I don't remember anyone was able to perform it in
time _and_ give the correct result (or if somebody was able, surely was a very
small number, I don't remember it well).

At that moment I know that the whole point of learn calculus will be a
disaster.

\-----

People not need to learn advanced math. They need to have strong, fluent
understanding of the very basic (imagine if a developer can't perform without
look basic list manipulations).

Like Bruce Lee say:

"I fear not the man who has practiced 10,000 kicks once, but I fear the man
who has practiced one kick 10,000 times."

~~~
morgante
> They need to have strong, fluent understanding of the very basic (imagine if
> a developer can't perform without look basic list manipulations).

Why? Basic arithmetic is almost entirely useless as a skill.

Why should I spend my time learning something which computers will always be
able to do more reliably and faster than me?

~~~
mamcx
Ok, so what could be the true "base-skill" in math?

And is possible to achieve knowledge of it, without a strong foundation of
arithmetic?

Truly, _I don 't know_, as I say I'm bad at math.

However, the point is that without strong basics the rest is a lost cause
(IMHO). What is the basic, I'm open to know, however, I think arithmetic is
part of it...

~~~
nzp
Nope. There are people (I'm one of them) who have sometimes extreme difficulty
doing basic arithmetic manually and in the head. In my case it's not the full-
blown dyscalculia[1] but to this day I haven't been able to learn full
multiplication table for example, so forget about multiplying numbers in my
head. I can do it on paper, but the whole process looks like a computer under
heavy swapping. I even have hard time adding and subtracting numbers mentally.
If I'm expected to produce a numerical result while being watched it all
becomes a catastrophe because then I also get substantial anxiety for not
being able to do what for most people is a simple thing. There's something
about operations with concrete numbers (although I have no problem with
numbers themselves) that turns my mind into mush.

OTOH, I have basically little problem with any other branch of mathematics.
The more advanced the better, actually. In school it got easier as it got more
advanced (although geometry was always easy no matter what level). Getting
calculus in high school felt like I could suddenly breathe with full lungs.
And I studied theoretical physics later at university. Apart from the basic
reality that the math required is _hard_ and needs a lot of work no matter how
smart/gifted/whatever you are, I had no substantial problem with most of it.
Abstract algebra, group theory, vector spaces and manifolds... oh the joys!
Because none of it requires you to do any kind of mental arithmetical
computation. This is not a unique experience, I once saw a TV report on a
young successful astrophysicist with basically the same problem: Calculus on
manifolds, GR field equations? Pfff... All day and every day. But, give her
some numbers on a blackboard to multiply and she gets completely lost.

[1]:
[https://en.wikipedia.org/wiki/Dyscalculia](https://en.wikipedia.org/wiki/Dyscalculia)

------
unabst
Context is the problem. This myth applies to all subjects. But it also applies
to Excel, because it's not the subject, it's the context. What they still
teach at school is the "Excel" of their time. They thought it would be most
useful.

But even with excel spreadsheet, if you leave out the "why" you still end up
with a boring class of formulas and UI work through tutorials that will leave
you questioning relevance just the same. And your school paid MS how much?

If you're developing a game, or designing a building, or analyzing online
sales, or trying to build a web site, you now have the context, but you
probably don't have the education unless you took special courses in higher
education which is practically the only place they teach with context. Maybe
they just need to add more context to lower and general levels as well?

Anecdotally, it's fair to say good students manage to identify context
beforehand and keep things relevant. It really helps with learning when you're
driven by purpose, not obligation.

------
hzhou321
You can't always use results to justify cause. Given the situation that most
adults (even with higher education) are not good at math (possibly due to the
failure of education) -- in particular, most adults are infused with the
perception that math is hard -- you will find them naturally trying to avoid
math as much as they can. So you find that eighty percent of adult rarely ever
use beyond Excel and 8th grade math.

Can you use this result to go back and justify that we don't need math beyond
8th grade?

I beg to differ. I only can provide anecdote. As myself are not bad at math, I
find myself use calculus and linear algebra all the time. In fact, I think
differently. As another anecdote, my son, who I consider is not nearly good at
math, he is in middle school and he uses trigonometry all the time.

You use what you have. Knowledge changes the way we think and work. With
knowledge, you simply see the world differently.

------
tgarma1234
I agree that for an average american child to invest time in studying math as
opposed to learning to play a musical instrument or doing sports or learning
how to cook has very little utility. I think one of the main reasons people
push math at civilizational/educational theory/political level is that it is
basically a meaningless topic and non-controversial and fills up the school
day with material that nobody on earth would find offensive... as opposed to,
for example, history or political science. And I say all of that that even
though I have a math degree from a legitimate university. If you need math
done you can pretty much just pay someone else to do it. You almost never need
math done. If there is some important math to be done, someone else is almost
certainly better qualified to do it than you are, so let them do it. Life is
just too short.

------
doozy
I was a maths minor and dropped out of a MSc in Applied Maths. I took many
courses that would be considered advanced by most, such as Stochastic
Calculus, Partial Differential Equations, Multivariate Statistics, Time Series
Econometrics, etc.

But I left academia over a decade ago, and have never used any of that. I
have, however, used a few things above 8th grade maths, such as linear
algebra, regression analysis, and some basic numerical analysis. I used once a
FFT.

All in all the author is spot on. And I believe an advanced degree in maths is
not a legitimate requirement for anything but a handful of positions, and most
of those aren't particularly desirable.

I still remember fondly my days studying Baby Rudin, though. Definitely one of
the courses that made an impact in my education. But in hindsight, it's been
as useful in my career as my study of Latin.

~~~
vidbina
I use Latin quite frequently to infer the meaning of words in a number of
languages including but not limited to English, French, Spanish, Italian and
Portuguese. Quite oddly, I'm generally not even aware that I'm inferring
meaning of a word by simply studying the roots. In that sense I'm quite
grateful for being taught a limited portion of its fundamentals while in high-
school. All I am saying is that we may use certain bits of knowledge or
certain dormant skills without our awareness to their utilisation.

All of our former experiences help us shape intuition, the same intuition
which we may use to make decisions in seemingly unrelated fields. I have
therefore become somewhat careful in dismissing a skill, domain of knowledge
or lesson for apparent lack of use. The human brain is a fascinating machine;
even while we sleep ideas and solutions take shape and who's to say which
information is used to form those mental artefacts?

------
sriram_malhar
This argument, that "higher math" is only for those few who are interested and
capable, is infuriating.

I detest utilitarian arguments, that something is worth learning only because
it is useful in my day-to-day. I haven't had the faintest use in my daily life
for knowing anything about igneous rocks, sorghum, golgi bodies, Chandragupta
Maurya, black holes, playing hockey. Yet, it would be singularly depressing to
not know it or something to this level of detail.

Second, regarding the argument that only a select few will be interested in
ascending the peak and that the rest are content in the plains. While that is
true, it takes a whole community of people interested in an area for there to
be a star. A Messi or Usain Bolt comes out of having a sporting culture, in
addition to athletic and soccer academies of a high enough standard.

~~~
autokad
knowing about gogi bodies, hockey, and black holes is a personal choice, and
it wouldnt be efficient to teach these topics in depth to everyone.

needing higher level math is absolutely unnecessary for 99% of the working
population, so the argument that our economy will fall to pieces unless we
force high level calculus and trig on every student seems suspect.

however, it is hurting america in an indirect way. we have an elitism going on
that only students from top universities get dibs on the best and important
positions out of college, and our graduates are becoming increasingly foreign,
as american students aren't prepared to get 165+ math GRE scores.

~~~
sriram_malhar
The specifics of hockey and golgi bodies are not important. Practically
everything we learn in high school is unimportant for 99% of the working
population.

My point is that it is important for a culture to aim high. There is a reason
why Israel produces way more innovation than Saudi Arabia, although most of
the population in both countries has no use for calculus or igneous rocks.

------
th0ma5
I saw someone joke on Twitter that their anxiety level lately is the first
derivative of the graph on the 538 2016 election forecast. So to get that I
guess I needed to be able to see that in my head briefly. I think I didn't
pick up that skill until calculus which for me at least was 11th grade.

------
WhitneyLand
The author has a point on the benefit of transference, but he's too extreme in
his conclusions.

Trigonometry is useful in so many ways. It's even useful for projects around
the house, let alone for a lot of careers. Last I checked it comes after 8th
grade.

On the other end of the spectrum he concedes Harvard philosophy undergrads
might want to read "The Road to Reality". Bullshit - No undergrad can
understand all the math in this book and no one is proposing that they should.
Reductio ad absurdum.

And don't forget gaming. Lots of young people these days dream about a career
at a game studio and there are a lot more options if you have good math.

He mentions Sputnik but it's not the 1950's anymore. The number of careers
that benefit from math will only continue to grow.

------
haddr
I think it has something to do with the "theatre paradox": when someone stands
up from her seat, then finally everybody needs to stand up in order to see the
show. If we start having a surplus of people with degree, then everybody
starts to look for "harder" degrees, better universities or just higher
degrees (phd). And you need to have one in order to be successful. Side
effects? Look at Google for instance: "They can hire the very best people — so
_everyone_ is overqualified." [1]

[1] [https://www.quora.com/Working-at-Google-1/What-is-the-
worst-...](https://www.quora.com/Working-at-Google-1/What-is-the-worst-part-
about-working-at-Google?share=1)

------
arilib
the purpose of teaching math is not purely to be applied in the context of
daily work. It is to be able to think through complex problems of different
nature and divide into multiple steps that can be tackled more easily.
Technology has allowed us to easily graph and instantaneously observe math as
it unfolds in daily life. Think bell curves in statistics, regression analysis
and divide and conquer algorithms.

This article misses the point. The purpose of teaching math is not to memorize
equations and solving methods, but to teach to approach problems in different
ways.

As a developer and now a PM I've used complex math at many different times.
I'm not solving in paper, but actually using it to solve real world problems.

------
mrcactu5

        This is a conjecture that desperately needs resolving with solid statistics and in-depth interviews. 
    

This thread is not representative -- include engineers and professionals who
may do math for a living. That's not everybody.

I think more empirical data is needed. If I go on the street or the math is
comparable to 5th grade (at the very best) and in a business setting might
bump up to 8th grade.

Does that preclude there being opportunities to need/use/benefit from math?
No...

I think it just means there are opportunities that nobody is taking advantage
of. Left open and collecting dust.

------
Ologn
I studied undergrad CS including the required math department classes.
Recently for my Android app (
[http://play.google.com/store/apps/details?id=com.unwrappedap...](http://play.google.com/store/apps/details?id=com.unwrappedapps.android.wallpapers)
) I wanted to list the most popular wallpapers. I had a problem though - I was
continually adding new wallpapers. How do I compare a new wallpaper which a
few people took versus a very old wallpaper which hundreds took?

The answer I came up with was N0e^-λt. Exponential decay. Set N0 to 1. I could
set t in various ways, I decided to make it days, so today is 0, yesterday 1,
the day before yesterday 2 etc. The lambda I tunes, right now it is 0.04 (or
-0.04 times t). So the score added for each use decays as it ages, giving new
additions a chance at the top.

Worked real well. Straight out of calculus. I never learned exactly what e was
until college. Who knows what I would have done if I didn't know what
exponential decay, e etc. was. I can't even think of an "eight-grade math"
solution of the type this article mentions.

I had to hash a small list of small numbers once when I had the epiphany -
Goedel numbering! I Google'd that and saw the solution was unoriginal, but I
wouldn't have even saw those pages without knowing what to Google.

I was looking at a large NP hard problem many years ago and thought I could
program a solution. After a complexity class later on, I realized the futility
of that approach, in a direct manner any how.

I am not sure where the line is between math and CS. Graph theory underlies
graphs and trees and the algorithms which run on them. Math functions and
theory of computation underlie functions and methods. Statistics and
probablity underlie ML. Geometry and matrix math and algebra underlie computer
graphics. I don't get people here who say they program without needing post
high school math.

Or seeing the garbage code out there maybe I do. Github is beset with people
who make basic errors in mutual exclusion, critical section violations, lack
of understanding of concurrency etc. I forget and make these mistakes myself
sometimes. I hardly think there is a problem in _over-education_ in these
things. On the contrary, race conditions are spun out all over the software
infrastructure by people writing code who don't have the needed math and CS
understanding of what they're doing. Understanding mutual exclusion and
critical sections and avoiding critical sections is not something picked up in
an hour, a day, or even a week.

~~~
vidbina
I continually study more sources to improve my competence as a software
developer. I have seen too many blunders as elementary as divisions by zero in
code and understand that a decent understanding of math and CS aids in writing
"better" programs. I have also experienced multiple accounts where my math/CS
background, however limited, helped me find the right direction to venture
towards. It helped me develop some sense of intuition that is extremely
valuable in my line of work. So I will keep on studying.

------
losteverything
I never enjoyed math as work but I like to work at Math. See everything in a
math way. For me it's binary, sampling, data presence and relevance.

To convince others I often use examples. One is the discipline issues over the
years. All BS. (1) totally made up. No data.

(1)[http://www.snopes.com/language/document/school.asp](http://www.snopes.com/language/document/school.asp)

------
jeffdavis
Math is abstract, and so are most of its benefits.

------
raarts
I fear that if this becomes reality it will result in even more people without
respect for science or engineering, and thinking that everything is easy.

You need to have experienced that some things are complicated. And we need a
lot of people to respect science and engineering, because they will be the
ones taking decisions, and those decisions need to be good ones.

------
jupiter90000
Seems kind of ironic that the math professor writing this article uses
conjecture as evidence for recommendations to what people should be learning
instead of actual statistics. He didn't even need 8th grade math to make his
argument for fewer people needing to learn higher mathematics (though perhaps
he's correct).

------
dboreham
This is idiotic nonsense.

I use math all the time, especially when I help my kids with their homework
because they don't understand their math assignments properly because the
school can't hire anyone with a decent math understanding to teach because
those people all took high paid jobs elsewhere..

------
brianberns
I'm a software developer trying to grok Machine Learning. I have to understand
trig (e.g. tanh and other sigmoid functions), calculus (e.g. derivatives,
gradients), linear algebra (e.g. vectors, matrices), probabilities, etc. Maybe
it doesn't happen every day, but I need math.

------
iptables
> the former consulting part of the now defunct Arthur Anderson

looks like they [relaunched this
year]([https://en.wikipedia.org/wiki/Arthur_Andersen](https://en.wikipedia.org/wiki/Arthur_Andersen))

------
Double_Cast
How does an actuary get by without having learned trig? Surely, they must
understand statistics at least. Is the job description the same deal as
engineers, where they just look up numbers from a reference table and multiply
them?

------
eli_gottlieb
See, if you tell me this is actually true, then to my ears, it just says that
people who can actually wield real math in anger have a massive advantage over
everyone else.

------
skybrian
This is probably true for many programmers today, but machine learning is hot
and that definitely requires heavy math, so I wouldn't bet on it remaining
true.

~~~
visarga
Maybe, but not necessarily. There is a lot of space for hacking on neural
network frameworks, and even people who don't understand 100% the math
involved can use them to make cool projects. When you understand a powerful
idea, new connections pop up and you see potential applications in your
domain.

------
kragen
Even if this article were correct that math isn't necessary for employment, it
would be wrong that math education is unimportant.

It isn't even correct on the employment front, though, because it is
attempting to unimaginatively extrapolate from the current state of
employment.

John Nagle's comment at
[https://news.ycombinator.com/item?id=12422307](https://news.ycombinator.com/item?id=12422307)
probably expresses this better than I can, but if you're going to make an
advance in a scientific or engineering field — _any_ advance — you need math.
If you're planning to spend your working life as a button-pusher, carrying out
algorithms that other people have designed, or proceeding blindly by trial and
error, you don't need math.

But those button-pusher and blind blunderer jobs will be automated in five,
ten, or maybe twenty years. And the article's comment section suggests that
even today they aren't nearly as common as the article asserts.

There are other categories of work, such as child care, elder care, sex work
(which, defined broadly, includes trophy wives, Hollywood, and a substantial
fraction of secretaries and maids), sales, and family counseling. So there
will probably still be employment that doesn't require math as long as there
are humans, even if it's not the kind of employment the article discusses.

But the bigger question is whether education should be directed at employment.
Is being an employee what you aspire to in your life? It is very good to be
useful to other people. Allowing other people to employ me has benefited me
greatly, and that's true for most people I know. But being used by others is
not the only or even the primary good in life.

Education is what makes us human. Education is a process of personal evolution
from a dumb beast into a human being. Education gives us control over our
impulses and prevents us from being suckered by predatory salespeople,
politicians, lawyers, preachers, and others. At its best, education makes
democracy possible despite such predators, although democracy is rarely
possible because the people is nearly always sufficiently uneducated to vote
it down unintentionally. Education begins before schooling and doesn't end
when schooling ends, but I, like many people, have found that schooling can
speed education up considerably.

And mathematics is fundamental to education in all of these senses. Even if
mathematics isn't necessary for someone else to use you — which is all this
"Math Myth" article tries to show — mathematics is necessary for you to
judiciously choose when and how you will be used, and mathematics is necessary
for citizenship.

~~~
vidbina
The moment we manage to cover the "education gives us control over our
impulses" part for a majority of the population, democracy would truly work to
our benefit. Although some may not actively rely on whatever they may have
learned in a math class, they may still use some of that information to form
intuition and subliminally guide them in their thought processes. Engineers
who studied or practiced the arts in some portion of their lives may
subliminally transfer some of their learnings from that domain into their work
as well. Simply arguing that one doesn't need higher math because they don't
actively use it is therefore unfair.

------
mruniverse
Not confined to math, but it's helped me become aware that problems can be
solved by reasoning.

Also to be measured in how sure I am about something being right.

------
p333347
Its quite well known that engineers need maths only to pass exams and for work
all they need is an appropriate handbook. :-]

------
Chinjut
Huh! I previously submitted this very same article (see
[https://news.ycombinator.com/item?id=10493543](https://news.ycombinator.com/item?id=10493543))
with zero uptick. I wonder why it's managed to do so well on this go-round.
Perhaps the different host? Perhaps just capricious luck.

------
blurge
I used math once. Not for me.

------
insulanian
Excel - on of the best pieces of application software ever made.

------
graycat
The OP is a special case of the old, big question of what to teach.

It is fair to say that there is an old and strong belief that a person who has
studied broadly, and deeply through, say, college, in math, physical,
biological, medical, social, and computer science, and the humanities will
have a significant advantage in much of the rest of life. Lacking a better
name, here I call such study a _broad education_.

To argue this belief in the context of the OP, the OP seems to claim that for
90% or so of people, it is enough for them to stop their math education, and
by extension all their education, after the eighth grade. But in life it is
fairly easy to tell the difference between the OP's eighth grade education and
a _broad education_ as I described it. So, there is a difference. Maybe the
difference is significant and the broad education an advantage and worthwhile.

One point not mentioned very often is that, whatever 90% of the students do,
the broad education was hoping that some of the students would find some
really good uses of some of the education well past the eighth grade. The
educators could have that hope even without knowing just what the good uses
might be.

I studied a lot of math and physics heavily, but not entirely, because I hoped
that they would help me make money. Well, early in my career within 100 miles
of the Washington Monument, that hope was fully correct. I used what I had and
was learning more as fast as I could drinking from a fire hose. Of course that
work was mostly for US national security; there the math and physics were
crucial.

Yes, it does appear that away from the work of US national security, the math
and physics are less commonly used.

Still, in US commercial work, there are significant applications of the math
and physics. Examples:

(A) How to operate an oil refinery. In simple terms, here is a list, with
prices, of crude oil can buy and put into the refinery and a list, with
prices, of refined products get out of the refinery, so a question is what to
buy, produce, and sell to make the most money? First cut, the problem is
linear programming, and for a while there was good money in selling IBM
mainframe computers just for that work. Of course, past the first cut, the
problem is in non-linear optimization.

A practical challenge is: It's a good guess that the first refinery management
that did well seeing and exploiting this opportunity was well paid for their
insight. Since much of the crucial core of that work was some college and/or
grad school applied math and numerical analysis, knowing some math could have
been an advantage for the management trying to understand and make good
decisions.

(B) Take a big hammer and hit the ground and send an acoustic pulse through
the ground. That pulse is commonly partially reflected at the boundaries of
layers of rock, sand, etc. So, the acoustic signal that comes back is a
convolution of the original. Doing a deconvolution, can map the underground
layers and get some good hints of where to drill for oil. The deconvolution is
basically some Fourier theory, and the fast way to do the computations is the
fast Fourier transform (FFT). After Cooley, Tukey, etc. invented the FFT, such
acoustic processing had an explosion that is still active. So, again, oil
prospecting management needed to see, understand, and actively exploit the
FFT. For that, some math was no doubt an advantage.

There are more commercial applications of math and physics. Some of the
applications have been valuable already, and likely some more will be valuable
in the future. So, in looking for what might be valuable in business, some
math and physics stands to be an advantage.

So, in part, with a broad education we are fishing for advantages in the
future. We are not sure just what subjects will lead to what advantages in the
future, but we are quite sure that there will be powerful, valuable new work
where, for successful exploitation, some studies will be important.

Or, the OP is concentrating on what the 90% of the people actually are using
now. Well, in a sense the education wants to concentrate on what is new no one
is doing yet.

~~~
pdm55
As a Math/Science teacher who has to make decisions every day about who and
what to teach, I agree that "with a broad education we are fishing for
advantages in the future". I see a similarity with playing sport. When we are
young, we play various sports. Some will make a career out of a sport they are
good at. Some will continue to play occasionally just for enjoyment it brings.
Most will probably benefit health-wise from the experience. Similarly with
Math/Science education. It may become a career, an occasional interest, or
simply a memory that gives some quantitative insight into what goes on around
us.

------
n00b101
_the percent of such individuals holding engineering as opposed to management,
financial or other positions, and using more than Excel and eighth grade level
mathematics (arithmetic, a little bit of algebra, a little bit of statistics,
and a little bit of programming) is less than 25% and possibly less than 10%._

I would state this differently. Borrowing from the Pareto principle, one could
conjecture that 80% of mathematically advanced work in the economy is
performed by less than 20% of STEM graduates. The remaining 80% of STEM
graduates do not get the economic opportunity to apply the skills which they
trained for and end up doing less prominent work (e.g. middle management).

As the OP and others have pointed out, there is a lot of anecdotal evidence to
support this conjecture.

But it is hardly surprising, and it is not limited to mathematical talent.

Take management, for example. Just because you studied business in school,
does not mean that you will be an executive. I would guess that less than 20%
of MBA graduates manage 80% of economic resources (senior executives, bankers,
consultants, traders, etc) , while the remaining 80% of MBA graduates are left
managing relatively small and inconsequential activities.

Similarly, I would bet that less than 20% of design school graduates do 80% of
the design work in the economy. I bet that less than 20% of classical
musicians perform 80% of orchestral music. Less than 20% of programmers
implement 80% of software used. Less than 20% of athletes win 80% of medals.
Less than 20% of science graduates produce 80% of scientific research. And so
on.

OP's conclusion is that, in light of this dismal reality, students should not
bother learning mathematics after the 8th-grade level (except for "those who
need it"). Well, if we apply the same logic across all disciplines, then the
OP should conclude that all forms of education should stop after the 8th-grade
level for the vast majority of students (and only a minute fraction should
need to pursue higher education). That is exactly what the state of education
looks like in undeveloped feudal economies, and this was also the state of
Western education until relatively recently. I don't think I need to expend a
lot of effort convincing anyone that this a socially, economically and
ethically terrible idea.

I'll also point out that there there are a couple false assumptions implicit
in the OP's original, imprecisely worded conjecture. Firstly, advanced
industrial mathematics is not the exclusive preserve of traditional
engineering. The generalization that "engineering positions" use advanced math
and "management/financial positions" use 8th-grade math, is obviously false.
Many areas in finance require advanced mathematics (derivatives, trading,
fixed income, etc). Much of actuarial science also depends on advanced
mathematics. Marketing, management sciences and operations research are also
steadily moving towards advanced analytics. Secondly, it is a false assumption
that use of Excel implies that the underlying mathematics is limited to an
8th-grade level. For example, in finance, it is easy to find Excel add-ins for
performing highly advanced mathematics (e.g. stochastic differential equation
solvers for derivatives pricing).

~~~
douche
> the OP should conclude that all forms of education should stop after the
> 8th-grade level for the vast majority of students (and only a minute
> fraction should need to pursue higher education). That is exactly what the
> state of education looks like in undeveloped feudal economies

I don't think you have to go back to undeveloped feudal economies. Even a
generation ago, the bulk of people in the US effectively did not receive more
than an elementary education. Ironically, they were often better prepared than
students today to actually enter the workforce after graduating high school,
since vocational education was more in vogue, and so they spent more of the
four years of their high school education learning practical skills, rather
than the vague, college prep holding pattern that is the norm now.

------
oneloop
At the level of individuals, this article is complete bullshit. There is a
WORLD of difference between an average mathematician/physicist/engineer and an
average English literature graduate solving the same problem with the same
tools. Those disciplines teach and/or reinforce A) critical thinking (question
assumptions, look for counter-examples), B) decomposing problems to smaller
problems, C) pattern recognition. The average software engineer would be
somewhere in the middle, he's probably as good as a mathematician at A) and
B), but probably not C). Even amongst reasonably mathematically educated
people (say physicists), you see a difference in C) depending on the depth of
their math education. There is a difference between people who know eg
fractions, and people who know eg fractions AND integrals, and people who know
eg fractions AND integrals AND group theory, in how often these individuals
look at a situation and go "I've seen something like this before". Acquiring
some areas of more advanced maths isn't a quantitative increase like doing
bigger sums. In terms of the patterns that you see around you, it's a
qualitative jump. It's like being able to use your eyes AND having IR goggles:
you will see some aspects of the world very differently.

At the level of societies, maybe. Can a poor society with lots of
mathematicians "beat" a society with lots of wealth and infrastructure and
comfortable niceties but whose individuals can only use Excel? Probably not,
certainly not in < 1 generation. It certainly didn't turn out great for the
soviets.

------
tmptmp
>>The second argument is the one I always hear around the mathematics
department: mathematics helps you to think clearly. I have a very low opinion
of this self-serving nonsense. In sports there is the concept of the
specificity of skills: if you want to improve your racquetball game, don't
practice squash!

The analogy with sports fails miserably and the author seems to not understand
this. Math is a brain skill and we do need to apply brain to understand a
given situation in a better manner, to abstract away some things and focus on
some other things. So, if you expect someone to better understand complex
situations, then you need them to have some knowledge of higher math.

One may ask where do you encounter such situations? Insurance, debates of
fiscal policies, debates about racial biases and social structures, anything
to do with modern finance, language structures, medical decisions. Take your
pick.

So, if you have to do anything complicated in such social areas too, you need
to have some knowledge of higher math.

Skills in sports are not of such versatile nature, hence the analogy fails.

------
hackaflocka
The author in 1885:

"People say that we should train people for factory jobs, but everyone I know
is gainfully employed in agriculture, and we don't have any factories where I
live."

As a college professor, let me assure the author that statistics and
programming is not a standard part of an eighth graders program. In fact, the
ones I teach have passed the 12th grade, and most are woefully unprepared in
algebra, statistics, probability and programming.

For me, understanding slightly advanced math (the type discussed in Taleb's
Fooled By Randomness) helped me realize that Financial academic math is
complete B.S. (in its use of the Gaussian Dist. in non-Gaussian processes).
Yes, that's how learning more math has benefited me: it helped me discover how
math is used to support complete B.S.

------
justinlardinois
I'm not surprised that this article is pretty divisive on Hacker News.

> I find it difficult to find anyone who uses more than Excel and eighth grade
> level mathematics (=arithmetic, and a little bit of algebra, statistics and
> programming)

I think even that's a bit optimistic; I think people whose further studies or
jobs don't require that level of mathematics forget it pretty quickly. For a
base level of "everyday life," you probably only need basic arithmetic
operations.

As anecdotal evidence, look to all those times that a relatively convoluted
expression is posted on Facebook or Reddit and people argue for weeks about
what the proper solution is. Of course there's plenty of people who get it
right, but the wrongs range from a subtle misunderstanding of order of
operations to a complete lack of knowledge about it.

------
andrewclunn
Good, now replace "Math" with "liberal arts." Secondary education for most
fields is a waste.

~~~
Retric
I learned more useful things from my liberal arts class than my CS or Math
classes. Granted, I had been programming for years at that point so a single
public speaking class was more useful than my first 1-2 years of computer
classes. The problem is not the material, the problem is 4 years is just not a
lot of time vs 13 years of prior education.

