
I Rewired My Brain to Become Fluent in Math (2014) - paublyrne
http://nautil.us/issue/17/big-bangs/how-i-rewired-my-brain-to-become-fluent-in-math
======
nicolas_t
This is related to the Curse of the Gifted. When I was a kid, I could grasp
concepts easily and I understood maths very quickly so I didn't really bother
doing things like homework or actually studying.

I then went to a decent university (a French Grande Ecole) and for the first
year, I was still doing okish but I went from being a student who always had
good results to doing average. Then, the second year came, and just
understanding the concepts without cultivating fluency by practicing wasn't
enough anymore and I failed hard. I hadn't learned how to learn and while I
understood the concepts and could easily follow when the teacher solved a
problem together with us, I had difficulties doing it on my own. I worked
doing exercises and was able to pass but time and time again in my
professional life, I've had the same issue where I tend to rely too much on my
intuitive understanding until it fails.

I'm not sure how to help kids not to go down this path. As a kid, I would
avoid any rote memorization and would probably not do any non graded exercise
since I knew that I understood (and also since as a geek wanting to fit in, I
tried to be more accepted by my peers by purposefully not doing my homework).
In the end, I think the key is play and giving maybe challenging exercises
that forces the kid to use his newly found understanding would be a way to get
them used to not just rely on their intuition understanding.

~~~
drakonka
This is exactly what happened to me as a kid, but in my case it wasn't curse
of the "gifted" but curse of the pushed. In my younger years growing up in
Ukraine my parents and granparents pushed us hard to learn math, and so did my
school. Endless repetition, hours of homework, Saturday classes, etc. I didn't
have any sort of gift for math, this is just how children were taught there.

Moving to the US and going into 5th grade, the first several years were just
repetition of things I'd already learned in Ukraine. They were so easy. My
grandmother wasn't there anymore and to my parents it looked like I was doing
_great_ so they also loosened up and didn't really push me further.

When we finally got to substantially new material I started to struggle. I got
too used to everything being so easy and built on concepts I'd already had
drilled into me. I failed a math test once - even my teacher was surprised. I
ended up falling from perfect score As to Cs, then scraping back up to Bs by
the end of high school.

Now I regret not taking math as seriously as I should have after leaving
Ukraine, and am now taking time at home to learn on my own. Thankfully I'm
used to self-learning in other areas, so I think it's going pretty OK.

~~~
arvinsim
I have the same experience. For context, I live in a third world country.

I was lucky in the sense that I was enrolled in a prestigious school from
kindergarten to grade 4. The school has a great curriculum that doesn't dumb
down math and science while also emphasizing the arts. Arts and Sciences were
equally interesting to me. I made no distinction with my enjoyment of solving
arithmetic with creating poems or painting.

Unfortunately, my family went into hard times and I had to transfer to our
public school. In contrast to my previous school, teachers can't personally
help each student because there were 50+ students per class. I also found out
that the lessons they taught are already covered in my earlier years. I
coasted through Grades 5 and 6 using my stock knowledge.

When I got to high school, I was used to leaning back on what I learned. This
is where I first tasted difficulty in learning the advanced math subjects.
Sure, I got good grades in algebra but when it came to calculus, I was
stumped. I could not care less about it because "math should be about
calculation, not this symbolic mumbo jumbo". My boredom and inability to deal
with the challenges resulted in my delinquency. I often cut classes and
plagiarism homework. I still can't remember how I manage to graduate from high
school.

Now I deeply regret that I didn't apply myself back then.I am now a software
developer and a lot of interesting stuff are closed to me because I can't
understand the math behind them. I resolved to return studying high school
algebra last year. Armed with experience and a new perspective, I realized how
much I have missed and how useful math is.

~~~
drakonka
I'm in the same position as you now, I think - a coder who wants to explore
interesting subjects that rely on math knowledge I don't have. How are your
studies going now? I have recently (a couple of weeks ago) started going
through a textbook on discrete mathematics, spending 1-2 hours a day. So far
it feels like I'm going through concepts that I intuitively already know (the
first chapter is on logical thinking), but it has been very interesting to
learn how to put names and definitions to those concepts, and to prove them.
After this I hope to study algorithms and statistics.

------
meesterdude
> Gradually, neuroscientists came to realize that experts such as chess grand
> masters are experts because they have stored thousands of chunks of
> knowledge about their area of expertise in their long-term memory. Chess
> masters, for example, can recall tens of thousands of different chess
> patterns. Whatever the discipline, experts can call up to consciousness one
> or several of these well-knit-together, chunked neural subroutines to
> analyze and react to a new learning situation. This level of true
> understanding, and ability to use that understanding in new situations,
> comes only with the kind of rigor and familiarity that repetition,
> memorization, and practice can foster.

This made a lot of sense to me. A pianist plays through these chunks of
knowledge - and learns through them as well. These "chunked neural
subroutines" can be built, and clearly are.

I think a lot of resistance for me comes from learned helplessness (aka baby
elephant syndrome). For me, growing up my inner narrative was one of failure
and rejection. But as i've grown and changed this narrative to be more
nurturing, I do see more and more, my capacity for a great many things. Am I
crazy? A genius? An idiot? Nope. I just figured out that with a lot of
curiosity, play and patience, you can become proficient or even a domain
expert. That's easy to see, at a certain level.

But I think for a lot of us, we have a concept of who we are, what we are and
what we are not. And this, at least for me, this definition of I, has held me
back more than anything. Change the definition, challenge assumptions, push
boundaries. See who you are.

~~~
justifier
i agree, and i like your 'see who who are'

i say the same when people cite the taxi cab number(o) story and suggest some
kind of intuition or magic

ramanujan simply had worked with cubes enough to recognise the number, still
very impressive, but anyone could do it with the kind of interest ramanujan
had for number theory

(o)
[https://en.m.wikipedia.org/wiki/Taxicab_number](https://en.m.wikipedia.org/wiki/Taxicab_number)

~~~
number-sequence
a rather interesting aspect of the taxicab number story is that ramanujan had
been looking at "near misses" to the fermat equation, and this was the first
values of one of the infinite families of near misses he had constructed. The
story is often dramatized to make it sound like Ramanujan just knew about all
the cubes and their sums and he pulled this out of thin air on the spot, but
the truth is probably that he had worked enough with these equations to
recognize the number when Hardy mentioned it.

check it out, super neat number thy stuff!
[https://arxiv.org/abs/1510.00735](https://arxiv.org/abs/1510.00735)

------
imjk
I've realized lately that I often fool myself into thinking I'm proficient in
some subject by picking up on concepts quickly. I think I "get" it because I
can understand something on a conceptual level, but when it comes to practice,
my deficiencies become apparent. I guess the tricky part is to push myself
past those initial concepts to delve deeper into the subjects with practice. I
need to slow down and remind myself that I don't really "get" it just because
I can understand the concepts in a very broad sense.

~~~
Waterluvian
Binary trees was that for me. I listened to lectures and understood them very
well from a conceptual point of view. But once I started writing them from
scratch as an exercise did I learn that understanding the theory alone was
woefully insufficient.

~~~
trentmb
Maybe you just didn't understand the "theory" as well as you thought you did.

I fell into the "it makes sense when we go over it in class, but not when I do
it at home" trap often.

------
Rainymood
Can highly recommend the book the Coursera course is based on [1]: A mind for
numbers [2]. Have read it multiple times back to from and front to back.

[1]- [https://www.coursera.org/learn/learning-how-to-
learn](https://www.coursera.org/learn/learning-how-to-learn)

[2] [https://www.amazon.com/Mind-Numbers-Science-Flunked-
Algebra/...](https://www.amazon.com/Mind-Numbers-Science-Flunked-
Algebra/dp/039916524X)

------
chrismealy
The author is not wrong to emphasize the fundamental necessity of drilling,
but in her article it sounds like she, having memorized basic facts (in
Russian and math), ultimately achieved fluency through play.

~~~
meesterdude
That was my takeaway as well. The author played with the equation f=ma, and
understood it inside and out in various scenarios.

------
theyoungestgun
This is mentioned in her bio at the end, but worth noting here, I think:
Barbara runs a coursera course on learning how to learn
([https://www.coursera.org/learn/learning-how-to-
learn](https://www.coursera.org/learn/learning-how-to-learn)).

It is interesting in that I find that Coursera courses highlight the flawed
learning processes she mentions quite well. I often find myself watching the
videos, thinking I get it, buzzing through the usually basic follow-up
questions, and moving on. Likely that material won't last in my brain for very
long in a quickly usable fashion.

~~~
fny
Coursera simply mirrors collegiate pedagogy: a professor lectures you for an
hour, you take a quiz here and there, and then there are some larger
assessments that prove mastery. Study habits and methods are entirely left up
to the student. You could employ Dr. Oakley's methods, whatever works for you,
or just breeze by without truly internalizing anything.

Coursera's missing one powerful dynamic of a traditional university, however:
incentives to remember beyond a class.

Say you coast your freshman year without internalizing: you'll pay the price
the following year or when you take some cumulative assessment like the MCAT.

With Coursera everything still feels very disjointed. Even in the
specializations, knowledge doesn't need to compound for success. You can
easily succeed in edutainment mode. Why take notes when you can use your hands
for popcorn?

------
pmoriarty
Lately I've been picking up the soroban[1], the Japanese abacus, and it's been
tons of fun. It feels a little like solving a Rubik's cube with arithmetic, or
maybe like working with a finite state machine. There are different algorithms
to apply depending on the state of the soroban, and applying the right
sequence of these algorithms will get you to the right result.

I find it to be a little addictive, and sometimes find it a bit hard to stop.
I always feel like wanting to improve my skills a little more, become a little
faster at it, and increase the number of digits I can handle without making a
mistake.

The soroban is a great tool for developing concentration, a memory for
numbers, a facility for performing a relatively complex series of steps in a
certain sequence, and eventually for lightning fast mental arithmetic.

In Japan, soroban use is taught to young kids[2], who after a while develop
enough proficiency not to need the physical device any longer and can perform
the calculations on an imaginary soroban, and eventually can achieve some
really amazing feats of mental arithmetic, such as this example from their
national competitions: [3]

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

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

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

------
closed
> By interleaving my learning—in other words, practicing so that I knew not
> only when to use that word, but when not to use it, or to use a different
> variant of it—I was actually using the same approaches that expert
> practitioners use to learn in math and science.

This is huge! Even if you learn something so that you can use it without fail
today, if you interleave your practice of using it (today) with other things,
you'll do much better a week from now.

~~~
Jugurtha
Interleaving has served me very well during my second year of Engineering. I
hadn't attended that much and had a grade of 3/20\. There were the final exams
where the materials of the whole year was fair game.

I buckled down for a month with a friend at my sister's house which was empty.
I eliminated the modules with diminishing returns that I had passed or where I
was close (easier to go from 0/20 to 12/20 than it is to go from 8/20 to 20/20
for the same amount of points).

I was left with five modules I hadn't attended: Numerical Analysis (ANAI),
Rational Mechanics(MECA), Strength of Materials(RDM), Vibrations-Waves-and-
Propagation(VOP), and Atomic and Nuclear Physics(PAN).

I drew a pentagon and organized the modules. Starting at the top, going
counter-clockwise: VOP, ANAI, PAN, MECA, RDM.

Each day, I'd do two modules:

Day1: VOP-ANAI

Day2: PAN-MECA

Day3: RDM-VOP

Day4: ANAI-PAN

Day5: MECA-RDM

Day6: Restart cycle.

Many benefits:

\- You only do a module for half a day. Intensely. Then switch to another
module and you sort of hustle your brain for a fresh start. It's not tired
because you're doing something else now. "It's not like you've been studying
all day" is the impression.

\- Mixing modules gives new insights. Especially in second year, there's a
bootstrapping phenomenon: to understand a module of Physics, you had to
understand a module in Maths.

\- You study a module hard. You don't see it the next day, but the day after.
Not too soon to be sick of it and burn out, but not too far in the future not
to remember any of it.

The problem with the methods most other students followed was that it violated
their brains and common sense: they'd do one module exclusively for a week
(all chapters, all exercises). Then go on to the next one and do the same. By
the time the exam comes: they're sick of the modules, and they remember
nothing for the most part because it's been 3 weeks since they've last done
the first module they started with.

I, on the other hand, have seen any given module at most 3 days before.

This allowed me to study 13 hours per day during a month without burnout
(reading the course material for the first time, going over the exercises and
exams, etc). The key was keeping a schedule.

Up at 0500.

0500 - 0700: Study.

0700 - 0800: Breakfast.

0800 - 1200: Study.

1200 - 1300: Lunch and nap.

1300 - 1700: Study.

1700 - 1800: Afternoon snack and chill.

1800 - 2100: Study.

2100 - 2200: Dinner.

2200 lights out, going to sleep.

------
equalunique
``At some point, self-consciously “understanding” why you do what you do just
slows you down and interrupts flow, resulting in worse decisions.``

I experienced this phenomena while learning the dvorak keyboard layout. The
GNU Typist dvorak lesson can be completed in just one day. A mental map of
where all the keys are in dvorak was developed very quickly. The problem I was
faced with was the relatively slow thought process of envisioning the key
layout, moving my typing finger to where it needed to be, and then continuing
this thought process as I went on to type full words. Knowing I could type
much faster in regular Qwerty layout, this was frustrating. After a month, I
was thankful many words no longer required much thinking to write. Months
later, I now very much prefer to turn on Dvorak layout on whatever computer
I'm logged into.

Great article. I'm also keen on brushing up on my math. Kahn Academy is an
amazing learning resource and math is their biggest offering. Not taking
advantage of it seems like a sin.

------
sigi45
I read her book, watched her videos and i like her.

But while i do already know a shit ton of how to learn, my level of knowledge
is not caped by learning issues or by my iq.

It is caped because my stamina is where it is. I have enough stamina to learn
and understand a shit ton of stuff but not to sit down day/every second day
after day to learn. To Exercise. To do it on a regular base.

~~~
dualogy
Well that's an entirely different problem to attack.

Exercise: I used to do all kinds of programs incl "45 mins a day, 6 days a
week" but found "sufficiently enjoyable results" while _keeping_ the process
enjoyable and in full balance with the rest of my life with just 3 reps a day,
no break days, cycling through 8 exercises. Good enough for both health and
looks if no Mr. Olympia goals.

For learning, the right balance is a lot harder to strike IME. Going too hard
on oneself _or_ too soft is a real danger. The motivating goals for learning
something or other will have to be consistently and sustainedly present to
settle into the right balance over time. Whether (perceived) low stamina is
merely due to "the undertrained stamina muscle", mismatch of expectations and
results, or some deeper real physiological/psychological factor is also the
the likeliest found out by yourself. Not reason not to go meta on this
roadblock!

------
scribu
Previous discussion:
[https://news.ycombinator.com/item?id=8402859](https://news.ycombinator.com/item?id=8402859)

------
hive_mind
Barbara Oakley's MOOC is great. I taught me to take Pomodoro seriously. Also
taught me that taking breaks is important to help the brain integrate
material.

------
Chris2048
> your mind constructed the patterns of meaning. Continually focusing on
> understanding itself actually gets in the way

For an engineer, who only needs to _use_ the math, maybe. But what if you need
an understanding too? Once you have "intuition" it's easy to stick with that,
rather than challenge your understanding.

------
JamilD
I've realized exactly the same thing. A starting intuition is necessary, but
not sufficient, for understanding.

I can watch as many lectures as I want, but nothing beats sitting down with a
pen and a piece of paper, playing around with equations and developing a true,
intimate understanding.

~~~
pm90
This is why a lot of the best mathematics books won't just state theorems, but
will actually heavily encourage the reader to solve problems as well, as that
is considered an indispensable part of the learning process.

------
notvaluable
I seem to recall that this post was posted here previously, I don't remember
when. Well googling 15
seconds,[https://news.ycombinator.com/item?id=12508776](https://news.ycombinator.com/item?id=12508776)

------
EternalData
Every time I see Barbara Oakley I feel the need to post her "Learning How to
Learn Talk".

[https://www.youtube.com/watch?v=O96fE1E-rf8](https://www.youtube.com/watch?v=O96fE1E-rf8)

------
gyrgtyn
Check out the book Make It Stick. It's the recent research, summarized.

