
Math education: It’s not about numbers, it’s about learning how to think - CarolineW
http://www.nwaonline.com/news/2017/jul/03/slaying-the-math-monster-20170703/
======
d3ckard
Maybe I'm wrong, but I have always believed that if you want people to be good
at math, it's their first years of education which are important, not the last
ones. In other worlds, push for STEM should be present in kindergartens and
elementary schools. By the time people go to high school it is to late.

I never had any problems with math until I went to university, so I was merely
a passive observer of everyday struggle for some people. I honestly believe
that foundations are the key. Either you're taught to think critically, see
patterns and focus on the train of thought, or you focus on numbers and
memorization.

The latter obviously fails at some point, in many cases sufficiently late to
make it really hard to go back and relearn everything.

Math is extremely hierarchical and I believe schools do not do enough to make
sure students are on the same page. If we want to fix teaching math, I would
start there, instead of working on motivation and general attitude. Those are
consequences, not the reasons.

~~~
Merovius
> Maybe I'm wrong, but I have always believed that if you want people to be
> good at math, it's their first years of education which are important, not
> the last ones. In other worlds, push for STEM should be present in
> kindergartens and elementary schools. By the time people go to high school
> it is to late.

I indeed believe you are wrong. I started getting interested in STEM when I
was 16 and now have a masters in math on the one hand and work for one of the
most prestigious Tech firms in existence on the other. Meanwhile, when I _was_
young, I was encouraged to enter creative fields (mostly writing in my case)
which I did have enthusiasm for but then dropped as a teenager. So I switched
at least twice, what I wanted to do with my life and still turned out fine.

Honestly, the whole "you need to get them when they are young" idea never
convinced me very much. It's never too late to learn a thing and often you
need a certain age to actually appreciate something. You don't have to be a
childhood prodigy to be good at something.

I rather find the idea pretty harming; I've met a lot of people who did not
pursue the career they wanted, because they bought into the myth, that you
need to start programming at 6 to get a job at a company like Google or
Facebook.

Let kids be kids for a while. Don't worry, they can always figure out what
they love later and it's never too late to reconsider.

~~~
Balgair
Anecdote time: I was a late bloomer in math too. From about 6th to 10th grade,
I was terrible at math. I think it was because of inapplication problems.

Yes, you get a word problem about Jack and Kwanzeeah trying to fill up a pail
or something, it was nonsense to me. Why are they filling up a pail? Why are
they using a bucket that is 173/9th's the size of the hose? Why do they have
to not let it overflow? Why do I have to decipher this inanity, why can't they
just ask me to do this stupid problem of stupid fractions? Why the cloak and
dagger?

Then you get to 'real' algebra and you start to find roots of quadratic
equations and other useless information. Dear Lord! When has anyone ever done
this for any reason? Yes, something something eigenvalues something something
computational modeling something runge-kutte. But for real though, quadratic
equations are just nonsense gibberish to weed out the poor kids from the rich
ones. Trig, yes, it's useful when you are building a shed that has to be
totally perfect and crisp otherwise the rich white lady will not pay you.
Geometry was a bit 'fun' actually, but in the same way that learning that
Custer's Men took a poop at this road-side stop on the way to Little-Big Horn.

It wasn't until Chem in my 10th grade year that I actually 'got' algebra. We
were doing moles to molarity calculations to get the solution to the right pH
and make it all turn pink or something. It was then that I realized that all
this mathy stuff was actually useful to me, that I could use it to make things
easier and better for myself and my family. Like, I could 'know' what to do
then. Before that it was just drill and kill and bad grades and shame. I
remember staying after class and into the next one in the room, just sitting
in the back doing these calculations over and over. I got kicked out for doing
math! It was such a relief! I finally 'got it' when I needed it for something
real to me personally.

For me, it was the issue of 'math' being this blunt-shame-thing that made no
sense to use ever. But once I _needed_ to use math, it was trivially easy.
Learning math for each of us is individual, we each have our own motivations
that are unique that need to be met. I know that is not easy to
institutionalize, but if you want to do it, you have to find the 'correct
motivation for each kid.

~~~
brightball
I'd agree with this completely. It's less about teaching what, how and when
than it is teaching WHY you need to know this.

In high school, if you can convince a single person that they need to know
calculus, more power to you. 99% of people don't and it's literally a waste of
time. Teach people finance though...suddenly we have math that people will
know is important.

~~~
hinkley
I was so pissed in college when I found out that all of those equations we had
to memorize in physics were just the integrals.

Why did I have to memorize all that shit if there's a simple way to derive the
formula? Jesus.

~~~
analog31
At my college, there were "calculus based" and "non calculus based" physics
classes. It seemed like the calculus based class was a lot easier for just the
reason that you state.

------
gusmd
I studied Mechanical Engineering, and it was my experience that several
professors are only interested in having the students learn how to solve
problems (which in the end boil down to math and applying equations), instead
of actually learning the interesting and important concepts behind them.

My wife went to school for Architecture, where she learned "basic" structural
mechanics, and some Calculus, but still cannot explain to me in simple words
what an integral or a derivative is. Not her fault at all: her Calculus
professor had them calculate polynomial derivatives for 3 months, without ever
making them understand the concept of "rate or change", or what
"infinitesimal" means.

For me that's a big failure of our current "science" education system: too
much focus on stupid application of equations and formulas, and too little
focus on actually comprehending the abstract concepts behind them.

~~~
acjohnson55
Nailed it.

I remember "hitting the wall" in Honors Algebra II, when I just couldn't keep
up with all the new things I was expected to compute: radicals, synthetic
division, logarithms, etc. It was a major dent in my math confidence, which up
until that point had be reinforced by being told I was "good at math".

I didn't start regaining that confidence until Chemistry, when I learned what
pH actually meant. That sparked the epiphany that _scale_ was deeply important
in nature, and with that motivation, logarithms seemed much simpler.

We so quickly forget that most math wasn't discovered as some abstract
exercise. But we still teach it that way, with the concepts before the
applications.

~~~
kaitai
This is compounded in the weird American class called "precalculus," which is
just a random grab bag of math crap that might maybe needed before calculus
but might not. In most precalc classes I've encountered, not only is there no
explanation of why the concepts will be used in calculus, but there is little
attempt to relate the concepts covered in the class. Let's go from linear
equations to complex numbers to polar coordinates! And let's not mention that
complex numbers and polar coordinates are related!

To top it all off, a lot of young college students taking precalculus are
taking it as a terminal math class -- they'll never take calculus! It's just
for a credit! So you've got some poor kid who hit that wall in high school and
is _repeating_ precalc and going through some unit on polar coordinates who
will never take another math class again and will somewhat justifiably think
this whole thing is stupid.

In theory, I strongly support "math for liberal arts"-type classes instead in
college. When done well they might cover voting systems and how you can prove
there isn't a perfect one, gerrymandering and how you can use math to get
several solutions, perspective drawing and its relation to projective geometry
and Renaissance art, a bit of number theory and basic cryptography....

~~~
dsfyu404ed
>"math for liberal arts"-type classes

I took a class that met in a room used by a class like that the period prior
and got to listen to the better part of it. In the one lecture the prof coaxed
them into creating an algorithm for the number of moves to solve a rubix cube
based on the number of cells who's configuration is known and where those
cells are located on the cube. In a different lecture they were doing the math
necessary to cut hypoid gears on a manual milling machine. They didn't even
know what they had did until he tied it back to the underlying concept.

The quality of those courses depends on the prof.

~~~
kaitai
> The quality of those courses depends on the prof.

Entirely. In theory they are great :)

The one you listened in on sounds interesting!

------
Koshkin
Learning "how to think" is just one part of it. The other part - the one that
makes it _much_ more difficult for many, if not most, people to learn math -
especially the more abstract branches of it - is learning to think about math
_specifically_. The reason is that mathematics creates its own universe of
concepts and ideas, and this universe, all these notions are so different from
what we have to deal with every day that learning them takes a lot of
training, _years_ of intensive experience dealing with mathematical structures
of one kind or another, so it should come as no surprise that people have
difficulty learning math.

~~~
jdmichal
Is that really true before calculus? In my recollection, even in pre-calculus
it was pretty easy to accurately tie constructs to physical representations.
Limits is the first time I remember learning something that could not be
generalized to a physical form, because it specifically dealt with what
happened when a function "broke".

~~~
hasenj
I find this sentiment somewhat surprising.

Limits is incredibly easy to link to real physical things: movement, speed,
acceleration.

If speed is distance over time, what's the speed of a moving object at a very
specific instant?

~~~
jdmichal
Are you thinking of derivatives? I agree that those are relatively easy --
it's the rate of change for a function. Or, in geometric terms, it's the slope
of the tangent line of the function's curve.

Like I said, limits are used when a function "breaks", so they're naturally
hard to visualize. I'll use my typical example for when I discuss teaching
math by solving problems: Approximating the area under a curve using
rectangles. The thinner your rectangles are, the more accurate your
approximation is. So you narrow the rectangles, getting more and more
accurate... Until you hit zero width, where the value of the function suddenly
drops to zero... A discontinuity. A limit lets you find what the value _would
have been_ had that discontinuity not happened. So, there's a clear
visualization of what the limit is letting you do _in this particular case_
\-- pretend like the rectangles of zero width don't have zero area -- but
there's no clear generalization on how to visualize a limit.

~~~
hasenj
Will, speed breaks when you consider a specific instant in thing because there
is no time span nor a distance the be traveled.

~~~
jdmichal
Ah, good point. I forgot about that problem with instantaneous speed.

------
spodek
> _it 's about learning how to think_

It's about learning a set of thinking skills, not _how to think_. Many people
who know no math can think and function very well in their domains and many
people who know lots of math function and think poorly outside of math.

~~~
randcraw
Math skills (esp proofs) shape the way you approach problems, giving the
problem structure and providing you the tools to reduce it into components,
relations, and dependencies. To make mysteries unmysterious, such tools are
indispensable.

I'd go so far to disagree with you that people do well without learning math
skill (those tools of thinking). If you can deconstruct a problem, then you
certainly learned that skill somewhere, just not in a math class.

Abstraction and reductionism are unusual capabilities to acquire on your own.
As I recall, they're almost nonexistent in pre-linguistic societies like
hunter-gatherers. If you have no words for abstract concepts, your thinking
will be strictly concrete.

~~~
FLUX-YOU
I feel like there's at least three ways of thinking about problems that are
practical:

\- systematic (math, rule-based)

\- social (history, emotions, people-based)

\- creative (art, invention)

These all overlap and feed into each other because we're a complex species,
but we really only ever hear about the necessity of math to "learn to think".

------
J_Sherz
My problem with Math education was always that speed was an enormous factor in
testing. You can methodically go through each question aiming for 100%
accuracy and not finish the test paper, while other students can comfortably
breeze through all the questions and get 80% accuracy but ultimately score
higher on the test. This kind of penalizing for a lack of speed can lead to
younger kids who are maximizing for grades to move away from Math for the
wrong reasons.

Source: I'm slow but good at Math and ended up dropping it as soon as I could
because it would not get me the grades I needed to enter a top tier
university.

~~~
MengerSponge
You couldn't finish your tests in high school, but you're good at math? Have
you ever been evaluated for test anxiety?

[https://www.adaa.org/living-with-anxiety/children/test-
anxie...](https://www.adaa.org/living-with-anxiety/children/test-anxiety)

There's no shame in it (although high schoolers can be assholes), and it can
help your school accommodate your needs eg extra time to take your exams.

~~~
J_Sherz
I've never been evaluated but I don't think test anxiety fits in my case - the
speed issue was only ever in math / physics.

My accuracy on the questions I got to was very high, I just couldn't go fast
enough to complete enough questions. Same deal on SAT type math papers too.

~~~
alimw
Where I did my undergraduate degree they squared the marks scored for each
question before adding them up! Slow but accurate was an advantage.

~~~
xfer
But, most exams at that level gives enough time to make them accurate.

------
BrandiATMuhkuh
Disclaimer: I'm CTO of [https://www.amy.ac](https://www.amy.ac) an online math
tutor.

From our experience most people struggle with math since they forgot/missed a
curtain math skill they might have learned a year or two before. But most
teaching methods only tell the students to practise more of the same. When
looking at good tutors, we could see that a tutor observes a student and then
teaches them the missing skill before they actually go to the problem the
student wanted help with. That seems to be a usefull/working approach.

------
Nihilartikel
This is something I've been pondering quite a bit recently. It is my firm
belief that mathematical skill and general numeracy are actually a small
subset of abstract thought. Am I wrong in thinking that school math is the
closest to deliberate training in abstract reasoning that one would find in
public education?

Abstract reasoning, intuition, and creativity, to me, represent the
underpinnings of software engineering, and really, most engineering and
science, but are taught more by osmosis along side the unintuitive often
boring mechanics of subjects. The difference between a good engineer of any
sort and one that 'just knows the formulas' is the ability to fluently
manipulate and reason with symbols and effects that don't necessarily have any
relation or simple metaphor in the tangible world. And taking it further,
creativity and intuition beyond dull calculation are the crucial art behind
choosing the right hypothesis to investigate. Essentially, learning to 'see'
in this non-spacial space of relations. When I'm doing system engineering
work, I don't think in terms of X Gb/s throughput and Y FLOPS... (until later
at least) but in my mind I have a model of the information and data structures
clicking and buzzing, like watching the gears of a clock, and I sort of
visualize working with this, playing with changes. It wouldn't surprise me of
most knowledge workers arrive have similar mental models of their own. But
what I have observed is that people who have trouble with mathematics or
coding aren't primed at all to 'see' abstractions in their minds eye. This
skill takes years to cultivate, but, it seems that its cultivation is left
entirely to chance by orthodox STEM education.

I was just thinking that this sort of thing could be approached a lot more
deliberately and could yield very broad positive results in STEM teaching.

------
jeffdavis
My theory is that math anxiety is really anxiety about a cold assessment.

In other subjects you can rationalize to yourself in various ways: the teacher
doesn't like me, or I got unlucky and they only asked the history questions I
didn't know.

But with math, no rationalization is possible. There's no hope the teacher
will go easy on you, or be happy that you got the gist of the solution.

Failure in math is often (but not always) a sign that education has failed in
general. Teachers can be lazy or too nice and give good grades in art or
history or reading to any student. But when the standardized math test comes
around, there's no hiding from it (teacher or student).

~~~
mbizzle88
Marking a math test is only objective when an answer is entirely correct. What
mark do you give a student who makes a minor arithmetic error on a single step
of a multi-step problem?

~~~
jeffdavis
Objective tests for math ability are pretty good. Arguing that they are only
99% objective doesn't really change my point.

People aren't afraid of making tiny mistakes. They are afraid of looking at a
problem similar to a dozen they saw in lecture; and having no idea what to do,
and then being told under no incertain terms that they did not succeed.

------
monic_binomial
I was a math teacher for 10 years. I had to give it up when I came to realize
that "how to think" is about 90% biological and strongly correlated to what we
measure with IQ tests.

This may be grave heresy in the Temple of Tabula Rasa where most education
policy is concocted, but nonetheless every teacher I ever knew was ultimately
forced to chose between teaching real math class with a ~30% pass rate or a
watered-down math Kabuki show with a pass rate just high enough to keep their
admins' complaints to a low grumble.

In the end we teachers would all go about loudly professing to each other that
"It's not about numbers, it's about learning how to think" in a desperate bid
to quash our private suspicions that there's actually precious little that can
be done to teach "how to think."

~~~
jacobolus
Unfortunately your realization is empirically unsound, and the problems of
“every teacher you know” have as much to do with lack of available expert
teacher attention per student and standard expected teaching styles and school
structure with substantial friction/latency in feedback and huge amounts of
wasted time as with inherent student ability.

Start here [http://web.mit.edu/5.95/readings/bloom-two-
sigma.pdf](http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf) but then
there is a vast literature (thousands of studies and other research papers)
exploring the general topic of math pedagogy. Some approaches and some
teachers are radically more successful than others, given comparable students.

If you hand me one of your 20th percentile students who is struggling but
willing to learn, and give us the same amount of time which would otherwise be
spent in a classroom for a year of one-on-one face-to-face time, with the
student also spending a typical amount of time working independently, I can
have them outperforming your 80th percentile students by the end of the year,
without issue. The difficulty of course is that direct mentoring by an expert
tutor is too expensive for society to be willing to pay for at scale.

The main reasons that your teacher friends are stuck is because (a) many if
not most of their students are unprepared before they arrive in any particular
course, which is largely down to structural social factors and school
scheduling inflexibility, and (b) the combination of lectures and independent
work with inadequate feedback are for the average student a terribly
inefficient and ineffective way to learn. Neither of those has all too much to
do with biological predestination or whatever.

~~~
monic_binomial
Heh. You can cite all the old wives' tales, superstitious folklore, and
education research papers you want. None of that makes it more likely that a
95 IQ student will ever understand precalculus.

~~~
jacobolus
I actually have direct experience getting a (I would guess) 95 IQ student to
understand precalculus, tutoring for a few months back when I was a high
school student ~15 years ago. She got an A, after having had a string of Cs in
every prior high school math course, and not being properly fluent with 9th
grade algebra at the beginning of the year. [I don’t say this to brag; I
expect any competent tutor can achieve the same thing with a typical below-
average student if the student is motivated and they get lots of 1-on-1 time.]

However, I can well believe that a 95 IQ student who was dramatically
unprepared at the beginning of the year and believes himself to be an
irredeemable failure will have a lot of trouble in a standard lecture +
homework drills format high school precalculus course taught by a teacher with
no time to help him catch up on remedial material at his current level of
understanding, give him any special attention, or make the class material
engaging enough to give him reason to care. In that context, failure (or
scraping by with an undeserved C) is the obvious outcome, but can’t fairly be
blamed on the student’s effort during that year or some kind of mental
deficiency.

(Personally, I skipped half of precalculus and somewhat wish I had skipped the
other half; I found it to be an uninspired grab-bag of disconnected topics,
with much too rote a focus.)

------
gxs
Late to the party but wanted to share my experience.

I was an Applied Math major at Berkely. Why?

When I was in 7th grade, I had an old school Russian math teacher. She was
tough, not one for niceties, but extremely fair.

One day, being the typical smart ass that I was, I said, why the hell do I
need to do this, I have 0 interest in Geometry.

Her answer completely changed my outlook and eventually was the reason why I
took extensive math in HS and majored in math in college.

Instead of dismissing me, instead of just telling me to shut up and sit down,
she explained things to me very calmly.

She said doing math beyond improving your math skills improves your reasoning
ability. It's a workout for your brain and helps develop your logical
thinking. Studying it now at a young age will help it become part of your
intuition so that in the future you can reason about complex topics that
require more than a moment's thoughts.

She really reached me on that day, took me a while to realize it. Wish I could
have said thank you.

Wherever you are Ms. Zavesova, thank you.

Other beneits: doing hard math really builds up your tolerance for building
hard problems. Reasoning through long problems, trying and failing, really
requires a certain kind of stamina. My major definitely gave me this. I am a
product manager now and while I don't code, I have an extremely easy time
working with engineers to get stuff done.

------
ouid
When people talk about the failure of mathematics education, we often talk
about it in terms of the students inability to "think mathematically".

It's impossible to tell if students are capable of thinking mathematically,
however, because I have not met a single (non-mathlete) student who could give
me the mathematical definition of... anything. How can we evaluate student's
mathematical reasoning ability if they have zero mathematical objects about
which to reason?

------
mindcrime
This part really resonates with me as well:

 _" You read all the time, right? We constantly have to read. If you're not
someone who picks up a book, you have to read menus, you've got to read
traffic signs, you've got to read instructions, you've got to read subtitles
-- all sorts of things. But how often do you have to do any sort of
complicated problem-solving with mathematics? The average person, not too
often."_

 _From this, two deductions:_

 _• Having trouble remembering the quadratic equation formula doesn 't mean
you're not a "numbers-person."_

 _• To remember your math skills, use them more often._

What I remember from high-school and college was this: I'd take a given math
class (say, Algebra I) and learn it reasonably well. Then, summer vacation
hits. Next term, taking Algebra II, all the Algebra I stuff is forgotten
because, well, who uses Algebra I over their summer vacation? Now, Algebra II
is harder than it should be because it builds on the previous stuff. Lather,
rinse, repeat.

This is one reason I love Khan Academy so much. You can just pop over there
anytime and spend a few minutes going back over stuff at any level, from basic
freaking fractions, up through Calculus and Linear Algebra.

~~~
sdrothrock
I wanted to quote this passage for another reason -- it seems like you're
agreeing with it, but I completely disagree with it, especially this:

> But how often do you have to do any sort of complicated problem-solving with
> mathematics?

There are a LOT of chances to do that kind of complicated problem solving,
especially if you're shopping or comparison shopping on your own. It's not
that people don't have the chances, it's that people avoid the work involved
in doing those kinds of comparisons.

~~~
mindcrime
I mean, in a certain sense I agree with you. There are _chances_ to do that
stuff, yes. To a point. Basic algebra could come up in a shopping scenario for
example. But even then, people aren't going to be doing that stuff every day,
or even close to it. And when you get even slightly more esoteric, like, say,
exponents... how often are people (especially kids) really going to be using
exponents, or logarithms, or the quadratic formula, etc. in their daily lives?

Sure, if you go out of your way to actively _look_ for reasons to find that
stuff, you can find them. But my point is that when you're a kid learning
math, it's summer vacation and you're busy playing with your friends, you
_aren 't_ out actively looking for reasons to apply the quadratic formula,
etc. At least not for most people, from what I've seen.

------
jtreagan
You say "it's not about numbers, it's about learning how to think," but the
truth is it's about both. Without the number skills and the memorization of
all those number facts and formulas, a person is handicapped both in learning
other subjects and skills and in succeeding and progressing in their work and
daily life. The two concepts -- number skills and thinking skills -- go hand
in hand. Thinking skills can't grow if the number skills aren't there as a
foundation. That's what's wrong with the Common Core and all the other fads
that are driving math education these days. They push thinking skills and
shove a calculator at you for the number skills -- and you stall, crash and
burn.

The article brings out a good point about math anxiety. I have had to deal
with it a lot in my years of teaching math. Sometimes my classroom has seemed
so full of math anxiety that you could cut it with a butter knife. I read one
comment that advocated starting our children out even earlier on learning
these skills, but the truth is the root of math anxiety in most people lies in
being forced to try to learn it at too early an age. Most children's brains
are not cognitively developed enough in the early grades to learn the concepts
we are pushing at them, so when a child finds failure at being asked to do
something he/she is not capable of doing, anxiety results and eventually
becomes habit, a part of their basic self-concept and personality. What we
should instead do is delay starting school until age 8 or even 9. Some people
don't develop cognitively until 12. Sweden recently raised their mandatory
school age to 7 because of what the research has been telling us about this.

------
g9yuayon
Is this a US thing? Why would people still think that math is about numbers?
Math is about patterns, which got drilled into us by our teachers in primary
school. I really don't understand how US education system can fuck up so badly
on fundamental subject like math.

~~~
Jtsummers
It is a US thing. And it's because until middle school (age 11-13) your
teacher is likely not that familiar with math (they have taken math courses,
but likely little formal math beyond geometry and trigonometry, perhaps some
courses on math pedagogy). So for them, math is all about the numbers, they
barely got to the more abstract concepts.

In middle school in the US you start getting teachers specialized in their
teaching subject matter, but often tangentially. Any one with enough college
math credits can teach middle school math (with some other certifications,
usually). High school you may, finally, get a math teacher who is a
mathematician. If you're lucky.

~~~
mamon
What kind of education is required from teachers in US? In Poland it is
Bachelor's degree in relevant topic (for elementary and middle school) or
Master's degree (for high schools) plus a course in pedagogy. Most teachers
however get Master's degree even if not required (they get salary increase for
that). It is hard for me to imagine math teacher not good in match themselves.

~~~
kaitai
The sibling commenter is right re: bachelors and certifications, but due to
shortages in the US often teachers with no certification in a subject are
teaching. See Table 2 in [1] and find that 70% of high school math teachers
have a math major and 81% are certified, while 10% of public high school math
teachers have neither a math major nor a certification in math.

[1]
[https://nces.ed.gov/pubs2015/2015814.pdf](https://nces.ed.gov/pubs2015/2015814.pdf)

------
quantum_state
Wow ... this blows me away ... in a few short hours, so many people chimed in
sharing thoughts ... It is great ... Would like to share mine as well.
Fundamentally, math to me is like a language. It's meant to help us to
describe things a bit more quantitatively and to reason a bit more abstractly
and consistently ... if it can be made mechanical and reduce the burden on
one's brain, it would be ideal. Since it's like a language, as long as one
knows the basics, such as some basic things of set theory, function, etc., one
should be ready to explore the world with it. Math is often perceived as a set
of concepts, theorems, rules, etc. But if one gets behind the scene to get to
know some of the original stories of the things, it would become very nature.
At some point, one would have one's mind liberated and start to use math or
create math like we usually do with day to day languages such as English.

------
Tommyixi
For me, math has always been a source of unplugging. I'd sit at my kitchen
table, put in some headphones, and just get lost in endless math problems.

Interestingly, now as a masters student in a statistics graduate program, I've
learned that I don't like "doing" math but get enjoyment from teaching it. I
really like it when students challenge me when I'm at the chalkboard and I'll
do anything for those "ah-ha!" moments. The best is at the end of the semester
hearing students say "I thought this class was going to suck but I worked hard
and am proud of the work I did." I'm hoping that on some small scale I'm
shaping their views on math. Or at least give them the confidence to say, "I
don't get this, but I'm not afraid to learn it."

------
lordnacho
I think a major issue with math problems in school is that they're obvious.

By that I don't mean it's easy. But when you're grappling with some problem,
whatever it is, eg find some angle or integrate some function, if you don't
find the answer, someone will show you, and you'll think "OMG why didn't I
think of that?"

And you won't have any excuses for why you didn't think of it. Because math is
a bunch of little logical steps. If you'd followed them, you'd have gotten
everything right.

Which is a good reason to feel stupid.

But don't worry. There are things that mathematicians, real ones with PhDs,
will discover in the future. By taking a number of little logical steps that
haven't been taken yet. They could have gone that way towards the next big
theorem, but they haven't done it yet for whatever reason (eg there's a LOT of
connections to be made).

~~~
throwawayjava
Reminds me of this quote:

"The view that machines cannot give rise to surprises is due, I believe, to a
fallacy to which philosophers and mathematicians are particularly subject.
This is the assumption that as soon as a fact is presented to a mind all
consequences of that fact spring into the mind simultaneously with it. It is a
very useful assumption under many circumstances, but one too easily forgets
that it is false. A natural consequence of doing so is that one then assumes
that there is no virtue in the mere working out of consequences from data and
general principles."

Alan Turing

~~~
CarolineW
You might want to edit that to fix the name. It's "Turing", not "Turning".

~~~
throwawayjava
Thanks. That's what I get for posting on my phone :)

------
yellowapple
I wish school curricula would embrace that "learning how to think" bit.

With the sole exception of Geometry, every single math class I took in middle
and high school was an absolutely miserable time of rote memorization and
soul-crushing "do this same problem 100 times" busy work. Geometry, meanwhile,
taught me about proofs and theorems v. postulates and actually using logical
reasoning. Unsurprisingly, Geometry was the one and only math class I ever
actually enjoyed.

------
taneq
As my old boss once said, "never confuse _mathematics_ with mere
_arithmetic_."

~~~
dahart
I've always liked that line, but it's clearly being cheeky and there are times
it's applicable and times it's not.

All arithmetic is math, but not all math is arithmetic.

Except that if you dig deep enough, all math is arithmetic under the hood.
Algebra is doing the arithmetic on variables before they're numbers. Calculus
is just useful shorthand for compound arithmetic; the derivative is a divide
and the integral is a sum. Exponentiation is repeated multiplication, which is
repeated addition. Linear algebra is arithmetic on blocks of numbers.

So we abstract over the individual arithmetic operations for our own
convenience, there are too many of them. But it's all still fancy arithmetic.

Maybe we should call math "fancy arithmetic" to help all the people who are
intimidated by math...

~~~
bjl
Actually, the opposite is true. If you dig deep enough, it turns out that
arithmetic is far more complicated than most other areas.

~~~
dahart
Then maybe instead of calling math fancy arithmetic, we should call arithmetic
"hard math"? :P

Seems like you agree more than disagree; math and arithmetic aren't easily
separable, if you're being serious instead of cheeky?

~~~
bjl
Well, I was trying to make two points.

1) Arithmetic (number theory) is far harder than most people realise, to the
point that its considered one of the most complicated mathematical
disciplines.

2) Most subject areas in math bear little to no resemblance to arithmetic, so
it doesn't make sense to call all math 'Fancy Arithmetic'.

~~~
dahart
Sure, understood and I agree. I was also trying to make one point: all math is
built on arithmetic operations if you dig down to first principles, so it does
make some sense to call all math 'fancy arithmetic'. There is no math at all
without the concept of addition sitting somewhere under it.

But I'm not actually proposing that, just sharing a cheeky point of view to
contrast @taneq's boss, so don't take me _too_ seriously. ;)

~~~
julian_1
> I was also trying to make one point: all math is built on arithmetic
> operations if you dig down to first principles,

I am a newb at this stuff, but I believe (basic) number theory actually builds
on recursive structures, and then uses induction to make statements about
behavior.

[https://en.wikipedia.org/wiki/Peano_axioms#Addition](https://en.wikipedia.org/wiki/Peano_axioms#Addition)

What is interesting is how the same approach works for analogous structures
like lists - because of the monoid abstraction.

~~~
dahart
This is interesting! I'm not sure, but I think this may reinforce what I said,
not contradict it. I'm suggesting that math operations we know all build on
top of addition, not that addition is necessarily the atom. Addition can build
on top of something else, and still sit under all higher level math.

But that said, I didn't learn addition in terms of monoids, and I don't know
anyone else who did. This might be a grand unified theory of math under which
addition fits, but physical addition has no recursive monoid analogue. You can
add two weights together to get a measurable sum, and it does not depend on a
recursive structure that uses induction. You just put two separate things side
by side on the scale.

Just because we can explain addition using monoids and induction does not mean
that addition _is_ made of monoids and induction.

Also, I probably wouldn't use Peano theory as the starting place to teach
addition. As a pedagogical tool, this is advanced math, and I guess would be a
bigger impediment to learning than Greek symbols, for beginners. Right?

------
alexandercrohde
Enough "I" statements already. It's ironic how many people seem to think their
personal experience is somehow relevant on a post about "critical thinking."

The _ONLY_ sane way to answer these questions: \- Does math increase critical
thinking? \- Does critical thinking lead to more career
earnings/happiness/etc? \- When does math education increase critical thinking
most? \- What kind of math education increases critical thinking?

Is with a large-scale research study that defines an objective way to measure
critical thinking and controls for relevant variables.

Meaning you don't get an anecdotal opinion on the matter on your study-of-1
no-control-group no-objective-measure personal experience.

------
brendan_a_b
My mind was blown when I came across this Github repo that demonstrates
mathematical notation by showing comparisons with JavaScript code
[https://github.com/Jam3/math-as-code](https://github.com/Jam3/math-as-code)

I think I often struggled or was intimidated by the syntax of math. I started
web development after years of thinking I just wasn't a math person. When
looking at this repo, I was surprised at how much more easily and naturally I
was able to grasp concepts in code compared to being introduced to them in
math classes.

~~~
sethammons
I found programming (started my first year in college) to be easily
accessible, and quite a natural next step due to my success in math at the
time. It is interesting how folks develop a mental model of a problem space
and scaffold knowledge onto previous models to understand problems at hand.
What do you feel made programming accessible when you started?

------
jrells
I often worry that mathematics education is strongly supported on the grounds
that it is about "learning how to think", yet the way it is executed rarely
prioritizes this goal. What would it look like if math curriculum were
redesigned to be super focused on "learning how to think"? Different, for
sure.

------
alistproducer2
I can't agree more. Math is about intuition of what the symbols are doing. In
the case of functions, intuition about how the symbols are transforming the
input. I've always thought I was "bad at math." It wasn't until my late 20's
when I took it upon myself to get better at calculus and I used "Calculus
Success in 20 Minute a Day[0]" did I finally realize why I was "bad" at it; I
never understood what I was doing.

That series of book really put intuition at the forefront. I began to realize
that the crazy symbols and formulas were stand-in for living, breathing
dynamic systems: number transformers. Each formula and symbol represented an
action. Once I understood Math as a way to encode useful number
transformation, it all clicked. Those rules and functions were encoded after a
person came up with something they wanted to do. The formula or function is
merely a compact way of describing this dynamic system to other people.

The irony was I always thought math was boring. In retrospect it was because
it was taught as if it had no purpose other than to provide useless mental
exercise. Once I started realizing that derivatives are used all around me to
do cool shit, I was inspired to learn how they worked because I wanted to use
them to do cool shit too. I went through several years of math courses and
none of them even attempted to tell me that math was just a way to represent
cool real world things. It took a $10 used book from amazon to do that. Ain't
life grand?

[0]:[https://www.amazon.com/Calculus-Success-20-Minutes-
Day/dp/15...](https://www.amazon.com/Calculus-Success-20-Minutes-
Day/dp/1576858898)

~~~
andrepd
But math _isn 't_ about intuition, much to the contrary, it's about rigorous
reasoning. In math "intuition" and "common sense" are worth nothing. Only
objective proof matters. It's a valuable lesson too, learning not to take
anything on a hunch and only take as true what what you carefully and
logically prove.

~~~
ice109
you use intuition to come up with the argument. then you use a proof to
convince others.

~~~
andrepd
No, you use intuition to come up with an idea the use that idea to write a
proof _for yourself_. Intuition is invaluable as a starting point, but it's
often misleading.

~~~
ice109
you're very naive. i think you're probably still an undergrad or maybe have
never done serious math.

[https://terrytao.wordpress.com/career-
advice/there%E2%80%99s...](https://terrytao.wordpress.com/career-
advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/)

~~~
andrepd
But the article agrees with me:

> The point of rigour is not to destroy all intuition; instead, it should be
> used to destroy bad intuition while clarifying and elevating good intuition.

In mathematics, intuition is the starting point and guiding principle, but
never worth anything by itself without rigour.

------
dbcurtis
Permit me to make a tangentially related comment of interest to parents
reading this thread: This camp for 11-14 y/o kids:
[http://www.mathpath.org/](http://www.mathpath.org/) is absolutely excellent.
My kid loved it so much they attended three years. Great faculty... John
Conway, Francis Su, many others. If you have a math-loving kid of middle-
school age, I encourage you to check it out.

------
simias
I completely agree. I think we start all wrong too, the first memories I have
of maths at school was learning how to compute an addition, a subtraction and
later a multiplication and division. Then we had to memorize by heart the
multiplication tables.

That can be useful of course (especially back then when we didn't carry
computers in our pockets at all times) but I think it sends some pupils on a
bad path with regards to mathematics.

Maths shouldn't be mainly about memorizing tables and "dumbly" applying
algorithms without understanding what they _mean_. That's how you end up with
kids who can answer "what's 36 divided by 4" but not "you have 36 candies that
you want to split equally with 3 other people, how many candies do you end up
with?"

And that goes beyond pure maths too. In physics if you pay attention to the
relationship between the various units you probably won't have to memorize
many equations, it'll just make sense. You'll also be much more likely to spot
errors. "Wait, I want to compute a speed and I'm multiplying amperes and
moles, does that really make sense?".

~~~
kungito
A friend's kid did not know how to do subtraction with just numbers but when
you said it like he had some currency in a shop then he understood it. That
was peculiar although the kid was very young (5-6 I guess) so it's interesting
how kids develop ability to do abstract thinking as they age

~~~
steverb
That's the way my kids were able to grasp basic math. For fractions we used
pizzas. Being able to hang the abstract concept on something that they were
used to dealing with made it work for them.

------
dahart
I wonder if a large part of our math problem is our legacy fixation on Greek
letters. Would math be more approachable to English speakers if we just used
English?

I like to think about math as language, rather than thought or logic or
formulas or numbers. The Greek letters are part of that language, and part of
why learning math is learning a completely foreign language, even though so
many people who say they can't do math practice mathematical concepts without
Greek letters. All of the math we do on computers, symbolic and numeric,
analytic and approximations, can be done using a Turing machine that starts
with only symbols and no built-in concept of a number.

~~~
d8421l01vv4r
If the use of greek letters is a major hurdle to learning maths, I'm guessing
that the learner has misunderstood something fundamental about symbols in
maths. It shouldn't make any significant difference if the symbols are Latin,
Greek or even made up by the lecturer (as long as they are sane).

~~~
yequalsx
In theory it shouldn't matter but it does. We brainwash students to think in
terms of y being a function of x.

y = f(x)

If you give a problem with x = f(y) it really confuses people. If I label the
horizontal axis y and the vertical axis x in the cartesian plane then
confusion ensues.

t = b(a) where t is the velocity and a is time

That's confusing. I suppose it's analogous to me saying

"The male, fluffy, white, large sheep."

There is a certain amount of brainwashing that occurs with the symbols that we
use. Using symbols other than the ones that have been standardized on is very
difficult cognitively. What you say is true in theory but I think not in
practice.

~~~
eeZah7Ux
If this is enough to "really confuse" someone than there's a serious problem
with understanding VS rota learning.

~~~
yequalsx
I'm certain you would be confused by non-standard usage of notation. It's not
just symbols and that any symbol can be used. Try reading or speaking English
with the adjectives in an order that is not expected. It's confusing as hell
but grammatically correct. Our brains get trained to see things in a certain
way and when the symbols get jumbled up so do we.

(x1(x2+x3) - x1(x2))/x3

is simply not as understandable to a first year calculus student as

(f(x+h) - f(x))/h

I've been teaching university level mathematics for 25 years and I know mixing
up symbols confuses people. Try reading Newton's Principia. It's really hard
to know what he's talking about.

EDIT: Try this equation. What famous one does it represent?

(sex(sxe) - sex(xes))/(sxe - xes) = xse sex/xse sxe

------
WheelsAtLarge
True, Math is ultimately about how to think but students need to memorize and
grasp the basics in addition to making sure that new material is truly
understood. That's where things fall apart. We are bombarded with new concepts
before we ultimately know how to use what we learned. How many people use
imaginary numbers in their daily life? Need I say more?

We don't communicate in Math jargon every day so it's ultimate a losing
battle. We learn new concepts but we lose them since we don't use them.
Additionally a large number of students get lost and frustrated and finally
give up. Which ultimately makes math a poor method to teach thinking since
only a few students can attain the ultimate benefits.

Yes, Math is important, and needs to be taught, but if we want to use it as
away to learn how to think there are better methods. Programming is a great
way. Students can learn it in one semester and can use it for life and can
also expand on what they already know.

Also, exploring literature and discussing what the author tries to convey is a
great way to learn how to think. All those hours in English class trying to
interpret what the author meant was more about exploring your mind and your
peer's thoughts than what the author actually meant. The author lost his
sphere of influence once the book was publish. It's up to the readers of every
generation to interpret the work. So literature is a very strong way to teach
students how to think.

------
lucidguppy
Why aren't people taught how to think explicitly? The Greeks and the Romans
thought it was a good idea.

------
tnone
Is there any other subject that is given as much leeway for its abysmal
pedagogical failures?

"Economics, it's not about learning how money and markets work, it's about
learning how to think."

"Art, it's not about learning about aesthetics, style, or technique, it's
about learning how to think."

"French, it's not about learning how to speak another language, it's..."

Math has a problem, and it's because the math curriculum is a pile of dull,
abstract cart-before-the-horse idiocy posing as discipline.

------
listentojohan
The true eye-opener for me was reading Number - The Language of Science by
Tobias Dantzig. The philosophy part of math as an abstraction layer for what
is observed or deducted was a nice touch.

------
yequalsx
I teach math at a community college. I've tried many times to teach my courses
in such a way that understanding the concepts and thinking were the goals.
Perhaps I'm jaded by the failures I encountered but students do not want to
think. They want to see a set of problem types that need to be mimicked.

In our lowest level course we teach beginning algebra. Almost everyone has an
intuition that 2x + 3x should be 5x. It's very difficult to get them to
understand that there is a rule for this that makes sense. And that it is the
application of this rule that allows you to conclude that 2x + 3x is 5x.
Furthermore, and here is the difficulty, that same rule is why 3x + a x is
(3+a)x.

I believe that for most people mathematics is just brainwashing via
familiarity. Most people end up understanding math by collecting knowledge
about problem types, tricks, and becoming situationally aware. Very few people
actually discover a problem type on their own. Very few people are willing, or
have been trained to be willing, to really contemplate a new problem type or
situation.

Math education in its practice has nothing to do with learning how to think.
At least in my experience and as I understand what it means to learn how to
think.

~~~
joe_the_user
I've had the same experience - teaching math for a short time at a State
University. I think the basic situation is that by the time students have
reached the university level, the habit of looking for a set algorithm for
completing a class is completely ingrained and moreover, the average student
has little-to-no free time so they feel a panic concerning anything that will
take an _unpredictable_ amount of time.

------
0xFFC
Exactly, as ordinary hacker i was always afraid of math. But after taking
mathematical Analysis I realized how wonderful math is. These day i am in love
with pure mathematics. It literally corrected my brain pipeline in so many
ways and it continues to do it further and further.

I have thought about changing my major to pure mathematics too.

------
katdev
You know what helps kids (and adults) learn math? The abacus/soroban. Yes,
automaticity with math facts/basic math is important but what's really
important is being able to represent the base-10 system mentally.

The abacus is an amazing tool that's been successful in creating math savants
- here's the world champion adding 10 four-digit numbers in 1.7 seconds using
mental math [https://www.theguardian.com/science/alexs-adventures-in-
numb...](https://www.theguardian.com/science/alexs-adventures-in-
numberland/2012/oct/29/mathematics)

Students are actually taught how to think of numbers in groups of tens, fives,
ones in Common Core math -- however, most are not given the abacus as a
tool/manipulative.

------
jmml97
I'm studying math right now and I have that problem. We're just being vomited
theorems and propositions in class instead of making us think. There's not a
single subject dedicated to learning the process of thinking in maths. So I
think we're learning the wrong (the hard) way.

~~~
bikenaga
Many undergrad programs (in the U.S.) now have a course in math proof, which
is intended to introduce students to proof techniques. It seems helpful to
learn how to write proofs without the pressure of learning the content of a
subject (like linear algebra, abstract algebra, and so on) at the same time.

------
Mz
Well, I actually liked math and took kind of a lot of it in K-12. I was in my
30s before I knew there were actual applications for some of the things I
memorized my way through without really understanding.

When I homeschooled my sons, I knew this approach would not work. My oldest
has trouble with numbers, but he got a solid education in the concepts. He has
a better grasp of things like GIGO than most folks. We also pursued a stats
track (at their choice) rather than an algebra-geometry-trig track.

Stats is much more relevant to life for most people most of the time and there
are very user-friendly books on the topic, like "How to lie with statistics."
If you are struggling with this stuff, I highly recommend pursuing something
like that.

------
keymone
i always found munging numbers and memorizing formulas discouraging. i think
physics classes teach kids more math than math classes and in more interesting
ways (or at least have potential to).

~~~
throwawayjava
Interestingly, my experience was that your two sentences are totally
contradictory. Physics and engineering was all about using memorized formulas
to solve word problems, and mathematics was where you learned how those
equations systematically fall out of some obvious assumptions.

I think the real take-away is that both physics and mathematics can be taught
poorly, and both can be taught well.

~~~
keymone
i get a feeling you're not talking about elementary school. but i agree - both
can be taught good or bad. bad is dumb memorization, good is understanding
real world phenomena.

------
crb002
Programming needs to be taught alongside Algebra I. Especially in a language
like Haskell or Scheme where algebraic refactoring of type signatures looks
like normal algebra notation.

~~~
sanderjd
Learning multiple different highly abstract concepts at the same time does not
seem like a solution to the difficulty of learning a single highly abstract
concept.

~~~
Jtsummers
My experience with this comes from college, but I disagree. I added a math
major late into my education (I was one course from a minor, then 4 more from
a major so why not?). I ended up in several math courses at once that were
often taken separately, but specifically abstract algebra and a higher level
(4xxx) linear algebra course.

I pretty much aced both, though I probably would have gotten an A in them
taken separately. The reason I did better, though, was that they were both
algebra. They were both expressing the same concepts, but on different _types_
of objects. The moment that that clicked in my head, my Monday lesson in
linear became the clarified by my Tuesday lesson in abstract, which was
further clarified by my Wednesday linear.

If you structure it correctly (particularly, with programming, by selecting
appropriate problem domains for programming exercises), I'm quite confident
you could create a better math and programming education experience by
properly pairing the two.

~~~
sanderjd
Maybe! I just don't think programming (even in Haskell!) is similar enough to
algebra for this to be effective, no matter how much we all want it to be.
Programming still requires a mental model of how machines compute, which is a
good thing to learn about but not (in my opinion!) a strong complement to
learning math.

Don't get me wrong though, I learned programming and calculus at the same
time, I'm not saying you can't learn both at once, I just don't think one
helps much in learning the other, for a novice in both.

------
JoshTriplett
One of the most critical skills I see differentiating people around me (co-
workers and otherwise) who succeed and those who don't is an analytical,
pattern-recognizing and pattern-applying mindset. Math itself is quite useful,
but I really like the way this particular article highlights the mental blocks
and misconceptions that seem to particularly crop up around mathematics; those
same blocks and misconceptions tend to get applied to other topics as well,
just less overtly.

------
andyjohnson0
A couple of years ago I did the _Introduction to Mathematical Thinking_ course
on Coursera [1]. Even though I found it hard, I enjoyed it and learned a lot,
and I feel I got some insight into mathematical though processes. Recommended.

[1] [https://www.coursera.org/learn/mathematical-
thinking](https://www.coursera.org/learn/mathematical-thinking)

------
cosinetau
As a someone with a degree in applied mathematics, I feel the problem with
learning mathematics is more often than not a problem or a fault of the
instructor of mathematics.

Many instructors approach the subject with a very broad understanding of the
subject, and it's very difficult (more difficult than math) to shake that
understanding and abstract it to understandable chunks of knowledge or
reasoning.

------
archeantus
If we want to teach people how to think, I propose that math isn't the best
way to do it. I can't tell you how many times I complained about how senseless
math was. The real-world application is very limited, for the most part.

Contrast that to if I had learned programming instead. Programming definitely
teaches you how to think, but it also has immense value and definite real-
world application.

------
djohnston
Anecdotally, I was a pretty average math student growing up and a pretty good
math student in university. One of the reasons I studied math in college was
to improve what was objectively my weakest area intellectually, but I found
that once we were working with much more abstract models and theories, I was
more competent.

------
k__
I always had the feeling I failed to grasp math because I never got good at
mid level things.

It took me reeeally long to grasp things like linear algebra and calculus and
I never was any good at it.

It was a struggle to get my CS degree.

Funny thing is, I'm really good at the low level elementary school stuff so
most people think I'm good at math...

------
EGreg
There just needs to be faster feedback than _once a test_.

[https://opinionator.blogs.nytimes.com/2011/04/21/teaching-
ma...](https://opinionator.blogs.nytimes.com/2011/04/21/teaching-math-
advanced-discussion/)

------
bojo
When I first saw it I thought the sign in the mentioned tweet may have been
because the deli was next to a mathematics department and the
professors/students would stand around and hold up the line while discussing
math.

Overactive imagination I guess.

------
kchr
Dammit, I glanced at the domain and expected a statement from the gangsta rap
group... School is cool, kids!

------
CoolNickname
School is not about learning but learning how to think. The way it is now it's
more about showing off than it is about anything actually useful. They don't
reward effort, they reward talent.

------
GarvielLoken
tl;dr A couple of numbers-nerds are sad and offended that math is not as
recognized as reading and literature, where there are great works that speaks
of the human condition and illustrates life.

Also they have the mandatory "everything is really math! ™". "LeGrand notes
that dancing and music are mathematics in motion. So ... dance, play an
instrument."

Just because i can describe history through the perspective of capitalism or
Marx theories, does not make history the same thing as either of those.

------
humbleMouse
On a somewhat related tangent, I think about programming the same way.

I always tell people programming and syntax are easy - it's learning to think
in a systems and design mindset that is the hard part.

------
calebm
I agree, but have a small caveat: math does typically strongly involve
numbers, so in a way, it is about numbers, though it's definitely not about
just memorizing things or blindly applying formulas.

It just bugs me sometimes when people make hyperbolic statements like that. I
remember coworkers saying things like "software consulting isn't about
programming". Yes it is! The primary skill involved is programming, even
programming is not the ONLY required skill.

------
dorianm
Maths problems are cool too, like counting apples and oranges :) (Or gold and
rubies)

------
pklausler
How do you "learn to think" _without_ numbers?

Depressing.

------
bitwize
Only really a problem in the USA. In civilized countries, there's no
particular aversion to math or to disciplined thinking in general.

