
How I Rewired My Brain to Become Fluent in Math - nautilus
http://nautil.us/issue/17/big-bangs/how-i-rewired-my-brain-to-become-fluent-in-math-rd
======
ivan_ah
My observation from years of experience teaching calculus and mechanics is
that every student has a different learning style. In particular, there are
"theory people" who need to understand the reasoning behind the concepts first
and "practice people" who best understand concepts by looking at worked
examples. There may be other subdivisions, but these are the main types.

I had two cases when my students showed no signs of progress after hours of
theory-style training, and then became Einsteins after solving some practice
problems. I was like... why did we waste so much time with the theory???

I don't necessarily agree with the OP that rote memorization is for everyone.
Ultimately both the theoretical/symbolic understanding and the
"doing"-experience are necessary, but the order in which students acquire
these skills is up to them.

Here's a nice quote:

    
    
        不闻不若闻之，      Not having heard is not as good as having heard,
        闻之不若见之，      having heard is not as good as having seen,
        见之不若知之，      having seen is not as good as mentally knowing,
        知之不若行之；      mentally knowing is not as good as putting into action;
        学至于行之而止矣    true learning is complete only when action has been put forth
               -- 荀子                                    -- Xunzi

~~~
xsace
不: not

闻: hear

若: good

见: see

之: have

知: know

looks easy :)

~~~
dmytrish
That's not as easy as it seems.

若 literally means [some kind of] "if", 之 is not "have" but rather "the" [?].
The last line is completely obscure for my very beginner Chinese skills.

I'd be glad if someone explained this.

~~~
ivan_ah
My chinese is probably more rudimentary than yours, but I often use a firefox
plugin that helps you get _some_ idea of what the sentence means:
[https://addons.mozilla.org/en-US/firefox/addon/perapera-
kun-...](https://addons.mozilla.org/en-US/firefox/addon/perapera-kun-chinese-
popup-tra/) (note it recognizes expressions, instead of working only character
by character ... )

------
eric_bullington
I come from a very similar background as the writer. As a young man, I showed
high potential for languages. So when I went to college, I loaded up on
languages, graduating with 6 different languages at at least the 200-level and
fluency in 3 of those languages. Unlike the writer, I chose neither military
nor government work upon graduation, but instead worked as an ATA-certified
translator for 10+ years, primarily translating pharmaceutical documentation
and medical studies.

I've since moved into professional software development and am now rusty in
the spoken languages I used to be fluent in. I work with a number of different
programming languages, primarily doing web development since that's where the
money is in remote work (although I also have a strong interest in embedded
programming). In between, I also finished graduate school in public health
(don't ask!), taking multiple upper-level biostats and epidemiology courses.

Anyway, people were always interested in how I learned so many languages. As
it turned out, I used similar techniques as what the writer describes to not
only learn languages, but also learn graduate-level quantitative coursework,
and now programming languages, algorithms, and data structures (an ongoing
task!). People are taking exception to her description of the technique as
"rote learning". Perhaps a better emphasis would be on repetition, drilling,
and fluency. But for me at least, this did entail things like flash cards --
Quizlet is a godsend -- and repeated, focused practice of short problems.

I have found that the technique doesn't seem to work as well for complex
algorithms as it did for math and languages. Maybe because it's difficult to
break down some algorithms into small parts, perhaps because I just haven't
figured out how to do so yet, or maybe I'm just getting older.

But I am in total agreement with the writer that modern education does a grave
disservice to young learners by discarding rote learning and repetition.

~~~
stdbrouw
> But I am in total agreement with the writer that modern education does a
> grave disservice to young learners by discarding rote learning and
> repetition.

I've come to appreciate the importance of rote learning over the past couple
of years... but despite its simplicity, I would call it a very advanced
technique.

You don't need rote learning to get to understand something. You don't even
need rote learning to become very good at something. You only ever need it to
be able to think faster about certain advanced topics or if you'd like to
become fully fluent in a language.

These are not things a high school student should necessarily spend much time
on. Especially considering that rote learning, if not self-motivated, can be
incredibly demotivating and have the opposite effect of what was intended.

~~~
jdbernard
I disagree. We are expecting students to learn more than ever before. Math is
a great example. Their ability to handle even a simple concept like
multiplication requires fluency in addition. The ability to learn algebra
requires fluency in arithmetic. The ability to learn calculus requires fluency
in algebra. Tons of kids pass one grade but emerge unprepared for the next.
The author hits the nail on the head: they understand the material at the
time, but never develop fluency. Then next year when they try to build on that
foundation it has evaporated. Rote learning should begin in elementary. I am
thankful that I was required to learn my multiplication tables.

Yes, we need to tailor our teaching to the attention span and abilities of our
students, but that doesn't mean we shouldn't be using this powerful tool of
fluency through rote learning.

~~~
Retric
It really depends on the goal. From a mathematician's perspective Engineering
is still part of the kiddie pool of math. The kind of things some people did
in high school mostly though route memorization.

However, for ~98% of the population it's by far the most useful parts of math.
Then again you can also say the same thing about just Arithmetic, basic
Algebra, Logic and Statistics. So, it's really a question of what your goals
are.

There is actually a lot of Math that's been dropped from K-12 education. EX:
Understanding logarithms is really fundamental for using a slide rule or
understanding floating point arithmetic, but it's not really that useful for
most people.

------
nautilus
As a thank-you present to our readers on HN, Nautilus wanted to give you
exclusive full access to this Prelude article from our latest print Quarterly
online "preview". Subscriptions to our print edition are how we make all our
online content free. We thought the HN audience would like this piece so in
appreciation for your traffic and your comments, here it is.

~~~
otoburb
I love Nautilus! Gorgeous layouts (print & online) with engaging and
opinionated content.

I notice that you only sell print subscriptions. Do you plan on selling any
digital-only subscriptions? I would sign up in a heartbeat because your online
layout is Really That Good (TM).

~~~
nautilus
We are planning to have ebooks of each issue available next month. Thanks

------
wwweston
I can recognize some of my experience in what she's saying -- as I've been
told several times in math/stats "you don't understand this, you just get used
to it."

That's a bit of troubling statement at first, but I think it's sortof a
shorthand way of saying that much of the understanding available comes through
the process of repeated manipulation and observing outcomes.

I've also noticed that a decade or two after going through a math undergrad,
the limited material I've retained best was the stuff I learned my freshman
year and had to repeatedly re-apply in other coursework.

Still, I think she may be overstating the case about the limits of conceptual
understanding, though (or perhaps bringing an engineer's perspective to the
discussion rather than a mathematician's. ;)

Practice is crucial, but when I need to dredge old mostly-forgotten material
up out of my brain, the interconnections formed by the mapping of conceptual
space we call "proofs" often turn out to be pretty helpful. There's plenty of
things I can't remember that I can derive from what I can recall.

~~~
shostack
I got tired of not having an intuitive understanding of particular
mathematical concepts. The rote memorization was very limiting for be because
I wasn't as easily able to grasp when it might apply to certain concepts.

Pretty sure Kalid is a commenter here, but I have to give him a huge thanks
and plug for BetterExplained.com--really helped me grasp some of these higher-
level concepts in a more intuitive manner, even if I didn't walk away with a
full understanding of the intricacies.

His Cheatsheet is a great starting point if you are interested in a particular
topic:
[http://betterexplained.com/cheatsheet/](http://betterexplained.com/cheatsheet/)

~~~
kalid
Wow, thanks for the plug and the kind words! Glad to hear the site is helping
-- most of the time I miss the intricacies too, and end up adding things years
later [after a re-read] or in response to a comment from the article.

As a more general reply to the article, I was fortunate enough to work with
Prof. Oakley on her Coursera Class
([https://class.coursera.org/learning-001/lecture](https://class.coursera.org/learning-001/lecture),
I'm the last guest interview at the bottom) and I really, really like her
learning strategy.

Her article didn't use the exact phrase "deliberate practice" but I think that
captures the essence of what she means by repetition. You need enough
conceptual understanding to make sure you're following the path correctly, but
then you want to practice -- at the edge of your comfort zone, with feedback,
etc. -- to make sure it's really clicking.

It's really easy to fool yourself into thinking "I've got this" when it's
untested. In my own case, I realized I didn't "get" imaginary numbers and
exponents when I couldn't estimate a^b [a raised to the bth power] in my head
with equal fluency for all numbers a and b. I had never really tested every
type of number in every type of exponent position (base and power).

For example,

3^4 => this should be a positive real number greater than 1

3^(-4) => this should be a positive real number, very close to 0.

3^i => Hrm.

i^i => Uh oh.

I knew that unless I had fluency with all of these scenarios, I didn't truly
"get" exponents or complex numbers. Sure, maybe I had a baby version where I
could use them in well-defined ways, but I had a subconscious fear of i
appearing as a base and/or exponent. I had to challenge myself and practice
thinking through the various permutations before I recognized the gaps. Then I
had to deepen my conceptual understanding, and practice again.

(For the previous questions, 3^i should be a complex number on the unit
circle, maybe around 50 or 60 degrees, but less than 90, and i^i should be a
real number, greater than 0 but less than 1. I can estimate these without
calculating them, see [http://betterexplained.com/articles/intuitive-
understanding-...](http://betterexplained.com/articles/intuitive-
understanding-of-eulers-formula/) for more details.)

Again, thanks for the mention!

~~~
kev6168
The entries on complex number, exponential functions & e, and many other
subjects at betterexplained.com are simply amazing! It helped to clear my
decade-long confusions or fears when dealing with them, Thank you for the
great work.

In my opinion the three essential characteristics of a great teacher are
expert level in the subject at hand, great communication skill and enthusiasm
for teaching. Possessing two of them already makes a good teacher. You have
all three.

~~~
kalid
That is really humbling and gratifying to hear, thank you.

------
oldbuzzard
I found the article rambling and mostly pointless...

However, if you have a weak background in math and want to get up to speed
before going into calculus and beyond, I have 2 suggestions.

1) Lial's Basic College Math[1] is adequate and will get you up to speed. 2)
Serge Lang's "Basic Mathematics" is great and will cover all you need to go
into a rigorous theory based college math class.

[1] [http://www.amazon.com/s?ie=UTF8&field-
keywords=lials%20basic...](http://www.amazon.com/s?ie=UTF8&field-
keywords=lials%20basic%20college%20math&index=blended&link_code=qs&sourceid=Mozilla-
search&tag=mozilla-20) The editions basically the same... pick the cheapest

[2] [http://www.amazon.com/Basic-Mathematics-Serge-
Lang/dp/038796...](http://www.amazon.com/Basic-Mathematics-Serge-
Lang/dp/0387967877/ref=sr_1_1?ie=UTF8&qid=1412284774&sr=8-1&keywords=serge+langs+basic+mathematics)

~~~
ivan_ah
Since we are on the topic of math textbooks, I will suggest the _No bullshit
guide to math and physics_ which is a math textbook written specifically for
adult learners. See [http://minireference.com/](http://minireference.com/) for
more info.

<discl>I'm the author</discl>

~~~
clarry
It's an interesting looking book, and I'm somewhat inclined to buy it.
Bookmarked!

I agree that you don't need to read thousands of pages to learn calculus.
However, I don't want to stop at calculus. Basically, what I'd really love to
have is a "mother of all maths textbook" \-- a thick and heavy tome that
compacts information from all the other thick and expensive books (which I'm
never going to read) and different fields of mathematics. With enough detail
that you can actually learn from it -- so it shouldn't be just for review and
looking up formulas you couldn't memorize. I'd like to call it a reference
book I can forever keep in my bookshelf (under my bed) and always look in it
if I'm unsure about something...

If someone has book suggestions, I'm all eyes.

~~~
ivan_ah
I've looked at a bunch of these math compendiums while researching what to
include in my book, and this one seemed the best so far:
[http://www.amazon.com/Mathematics-Content-Methods-Meaning-
Do...](http://www.amazon.com/Mathematics-Content-Methods-Meaning-
Dover/dp/0486409163) The writing isn't very hand-holdy, but it covers a lot of
important topics, and without too much fluff.

For a more "math for general culture" I'd recommend this one:
[http://www.amazon.ca/Mathematics-1001-Absolutely-
Everything-...](http://www.amazon.ca/Mathematics-1001-Absolutely-Everything-
Explanations/dp/1554077192) which covers a lot of fundamental topics in an
intuitive manner.

I have both books on the shelf, but not finished reading through all of them
so I can't give my full endorsement, but from what I've seen so far, they're
good stuff.

------
Lambdanaut
She never really gave a definition of "understanding". Personally, I see
"understanding" as exactly the way she described her method of learning. When
I learned f=ma I did it the same way she did, but I would've called it
"understanding".

In fact, "rote memorization" is the opposite of how I'd describe her method of
learning. She says she learns language by using it in many different
situations and learning the situations when it's usable and when it's not to
be used. That sounds exactly like a search for intuitive understanding to me.

Rote memorization is more like using flash cards or scanning over a list of
words and memorizing their definitions.

Contrast this with her method. She's playing with the subject matter.

I find that playing with a subject is the best way to understand it and to be
able to apply it.

------
simoneau
A similar comment Joel Spolsky from years ago always stuck with me.

"... Even though I understand all the little bits, I can’t understand them
fast enough to get the big picture. And the same thing happens in programming.
If the basic concepts aren’t so easy that you don’t even have to think about
them, you’re not going to get the big concepts."

[http://www.joelonsoftware.com/articles/GuerrillaInterviewing...](http://www.joelonsoftware.com/articles/GuerrillaInterviewing3.html):

------
was_hellbanned
As someone who graduated with a math minor and promptly forgot the vast
majority of what he learned, something that really messed me up were gaps in
my education.

For example, I took a graduate level course in the foundations of mathematics
(proving natural numbers and arithmetic). I grasped the lectures, but one day
I was completely stumped by a step in the proof where algebra was performed
across an inequality.

I had never in my life seen algebra across an inequality! As an undergrad
senior in a 500-level course.

I grew up moving a few times, and even within single schools I found the
teachers weren't on the same page in terms of how math would be taught. For
this alone, the Common Core sounds like a good idea to me.

~~~
Chinjut
What do you mean by "algebra across an inequality"? (Does this mean things
like "x < y is equivalent to 5 + x < 5 + y"?)

~~~
peterfirefly
Or perhaps more likely, multiplying with a negative number which involves
changing the ordering relation in the inequality.

------
mfrankel
Take a look at Daniel Willingham's material and his book "Why Don't Students
Like School"

Here's a review of the book: [http://ed-policy.blogspot.com/2009/04/one-of-
handfull-one-of...](http://ed-policy.blogspot.com/2009/04/one-of-handfull-one-
of-most-important.html)

Here's Dr Willingham's web site with a lot of articles worth reading:
[http://www.danielwillingham.com/articles.html](http://www.danielwillingham.com/articles.html)

Here's his credentials: Earned his B.A. from Duke University in 1983 and his
Ph.D. in Cognitive Psychology from Harvard University in 1990. He is currently
Professor of Psychology at the University of Virginia, where he has taught
since 1992.

------
aeharding
What an incredibly short-sighted sentence: "Sorry, education reformers, it’s
still memorization and repetition we need.".

I could not disagree more with this article. Memorization helps, but MAKING
CONNECTIONS between concepts is _HOW YOU MEMORIZE EFFECTIVELY_.

Memorizing is great for learning different languages. NOT MATH.

Sure, maybe students are bullshitting in understanding concepts, and forgot
quickly since they didn't learn anything. However, just memorizing shit
doesn't do anything. So many students in other cultures are being forced to
memorize everything, and since they don't understand the connections, they
cannot come up with the answer on their own. There was a piece on HN about a
math teacher that was amazed by how little students knew: They memorized how
to solve a particular problem instead of actually knowing _why_ something was
true, and therefore couldn't work through basic proofs and stupidly basic,
foundational stuff.

A terrible article!

~~~
dubcanada
This is why the education system will never ever change. Too many people learn
too many different ways. Some people can read a formula and instantly memorize
it. Other's need a connection to that formula. Some need to figure out how
that formula was derived in order to understand and memorize it. And some know
the formula before you even show it too them.

Not to rehash what you hashed, but it depends who you are. If you are really
good at memorizing, you may not need to make connections. You may just be able
to memorize everything. Some people can.

~~~
MilnerRoute
This is very true. A corporate trainer once told me there's specific kinds of
learners. They've actually identified each type, and which specific
methodologies you need to reach each one. The best teachers find a way to
synthesize it all into an effective single presentation. (This may also be why
tutors are so helpful -- they can customize their material to the audience...)

~~~
Dylan16807
Although there is ample evidence that individuals express preferences for how
they prefer to receive information, few studies have found any validity in
using learning styles in education.[2] Critics say there is no evidence that
identifying an individual student's learning style produces better outcomes.
There is evidence of empirical and pedagogical problems related to the use of
learning tasks to "correspond to differences in a one-to-one fashion".[3]
Well-designed studies contradict the widespread "meshing hypothesis", that a
student will learn best if taught in a method deemed appropriate for the
student's learning style.[2]

~~~
MilnerRoute
Heh. Okay, that's good to know. These comments prompted me to look up
"Learning Styles" on Wikipedia, which confirms that scientific studies have
not confirmed the validity of the "different learning styles" theory.

[http://en.wikipedia.org/wiki/Learning_styles#Criticism](http://en.wikipedia.org/wiki/Learning_styles#Criticism)

That probably says something about corporate trainers. I still remember the HR
department at one company where I worked that insisted on giving the Myers-
Briggs personality test to every employee. So maybe this also says something
about junk science and the way it lingers on in our workplaces...

------
kev6168
I am interested in this kind of topic, but came away disappointed. Not only is
it long-winded, but also she never gave clear definitions of 'fluency' and
'understanding', which are the key ideas in the article.

I think the reason she has succeeded is quite simple: she is an avid learner.
She has put in huge amount of effort into learning. It's not about some magic
methods she discovered.

Carrying the formula f=ma in head all day long, thinking about it, practicing
its various forms in different situations, that is not rote learning or simple
repetition (as the author claims), it is working one's a$$ off to understand
something.

------
g9yuayon
It's always the combination of conceptual learning and memorization when it
comes to learn math. It's like performing a complex computation job once, and
then caching the result for fast retrieval. Memorization is for efficiency: it
clears the way for our brains to focus on higher-level thinking. The great
Euler "memorized not only the first 100 prime numbers but also all of their
squares, their cubes, and their fourth, fifth, and sixth powers. While others
are digging through tables or pulling out pencils and paper, Euler could
simply recite from memory...". Besides, math is all about making invisible
visible, about discovery patterns, or about connecting dots. That means we
need to have dots to connect, and need to have patterns to work with. If we
don't remember them, what the hell can we use for?

On the other hand, there's no need for rote memory. Just practice by solving
interesting problems. There are plenty of opportunities to use math every day.
There is also a very effective way of learning: work on slightly harder than
usual problems. When learning calculus, I started to work on Demidovich's
Problems in Mathematical Analysis, and I thought it was hard. Then, I stumbled
upon solution book for college math competitions. Man, that was a huge help.
After working through the problems, a lot of concepts became clear to me, and
Demidovich's problems became reasonably easy too. It turned out the hard
problems were hard because they required me to make non-obvious connections,
which nudged me to really understand, from different angles, the concepts that
I learned in the classroom.

By the way, when did arithmetic become so hard? It seems kids nowadays are
being spoiled by their parents...

------
ilitirit
He highlights one of the problems I had with mathematics when I reached
University. I was in the top 2% of students in the first few months. I
considered it a trivial subject. "Here's a class of problems, here's how you
solve them." Easy peasy. Then we started integrating and differentiating forms
of problems we weren't familiar with. All of a sudden, everything we were
taught didn't seem to apply any more.

Our lecturer told us: "I can see that some of you are struggling and are
confused. You want to know how you become skilled at solving these problems?
Forget your social life. Solve every problem in that 900 page textbook of
yours. When you've done that, come to me and I'll give you another 900 page
text book and finish that as well."

In one fell swoop Mathematics had lost all its appeal. I switched to Applied
Math and aced it.

On the other hand, my brother actually followed this advice. For an entire
year he spent a few hours every night doing Calculus till he recognized every
different form they could conceivably throw at him. I asked him how he knew
how to solve some problems. He said he didn't. It was all just memorization
and recognition.

~~~
chestervonwinch
>> I switched to Applied Math and aced it.

What sort of integration and differentiation were you doing originally that is
not included in an applied math curriculum?

~~~
ilitirit
My Applied Math curriculum (roughly):

\- Logic of Compound/Quantitative statements.

\- Elementary Number Theory and Proof Methods

\- Sequences and Mathematical Induction

\- Graphs/Tree/Set theory

\- Algorithms

\- Functions and Application

------
B-Con
> memorized the equation so I could carry it around with me in my head and
> play with it. If m and a were big numbers, what did that do to f when I
> pushed it through the equation? If f was big and a was small, what did that
> do to m? How did the units match on each side? Playing with the equation was
> like conjugating a verb. I was beginning to intuit that the sparse outlines
> of the equation were like a metaphorical poem, with all sorts of beautiful
> symbolic representations embedded within it.

I cannot emphasize how useful this approach is. What's more, I am very
surprised by how many students lack the basic ability to sit and "play around"
with math concepts.

I used to tutor algebra, trig, and pre-calc. I was surprised by how few
students had the ability to look at something and break it into pieces. If you
gave them piece A which they knew, piece B which they knew, and put them
together, the result was something new the student couldn't understand and
they'd sit and wait for an explanation.

This was sad because this is largely what math learning is. Given pieces you
do understand, you put together bigger pieces and then build an understanding
of those bigger pieces.

I have always advocated something similar to what what the author said here:

> What I had done in learning Russian was to emphasize not just understanding
> of the language, but fluency. Fluency of something whole like a language
> requires a kind of familiarity that only repeated and varied interaction
> with the parts can develop.

I love this point. I am borderline convinced it it is near impossible to learn
math concepts more than one or two steps beyond the point that you are
_fluent_ in. Beyond that, it's just memorization, guessing, and poor
heuristics that just get you to skate past the test and do little or nothing
for understanding or retention.

The bigger picture here is that the author has learned how to learn. It amazes
me how significant a divide there is between people who know how to learn and
people who don't. Forget IQ or test scores, I think knowing how to learn is
the biggest indicator of how far someone will go in life. So perhaps most of
all I love the fact that they began this journey relatively late (most
students going through a similar math path would have done in their late teens
or very early 20s what the author was doing in their late 20s).

------
x1798DE
Case studies in education as applied to non-atypical brains are essentially
worthless. You'd think that someone with a newfound scientific disposition
might realize that you probably shouldn't make sweeping prescriptions based on
personal anecdotes (though honestly, based on the quality of the educational
literature I've seen, it's not uncommon even for professional education
researchers).

------
kenjackson
I think those of us who aren't a fan of "memorize/repeat" is that it is
extremely inefficient. And you lose as many people as you gain.

The classic example is long division. In elementary school I did a thousand
long division problems, but I never understood it. I just knew the pattern. I
didn't really know the math. It wasn't until much later did I learn how and
why it worked. Having done those thousand problems didn't really help my
understanding at all. And honestly, it's not a skill I use today.

Rote learning is useful when what you memorize is useful in itself. Learning
what 2+2 is useful, but because adding 2 to another small number, including
itself, is something you do a lot.

Addition/times tables are great because the scope of numbers is something you
will run across a lot of day to day. A thousand long division (or
multiplication) problems is just rote work for little benefit. It's better to
teach the concepts, and then build on them.

~~~
Verdex
I can relate to your long division example. In grade school my mom helped[1]
me put in a ridiculous amount of effort in order to get through my school's
spelling tests. I have come to believe that the reason that I couldn't spell
was because at the time I did not understand the rules behind how the letters
form together to get words. Simply memorizing how words are spelled took a lot
of painful practice.

A more recent example is when I was first learning lambda calculus I wanted to
understand how church encoding worked to allow basic integer operations with
just function abstraction and application. Most of the operators are easy
except for predecessor, which is absolutely nightmarish to get on your own. I
tried memorizing the definition I found on wikipedia for over a year to no
avail, but once I understood how it was working ...

    
    
      pred = λ n . λ f . λ v. 
               n ( λ a . λ b . b (a f) )
               λ k . v
               λ i . i
    

I haven't yet forgotten what it is.

[1] - well, basically dragged me kicking and screaming

EDIT: formatting

~~~
eric_bullington
> I couldn't spell was because at the time I did not understand the rules
> behind how the letters form together to get words.

One of the main goals of this kind of learning (at least for me) is to
recognize patterns and rules on your own. When I repeat something often
enough, patterns and rules began to jump out at me.

------
ahmadss
Just saw that the author, Barbara Oakley, has a (free) Coursera course titled
"Learning How to Learn" starting Oct 3. Sign-up here -
[https://www.coursera.org/course/learning](https://www.coursera.org/course/learning)

------
mw67
For those interested in this learning approach, I recommend the book "The
Talent Code" by Daniel Coyle [1] which covers 3 areas for mastering new
skills.

Here the author only mentions about the importance of repetition (which she
refers to as fluency). In the Talent code book Repetition is the first step,
but we can learn that the brain is wired to master new skills by taking
advantage of 2 others areas as well: A/ Ignition (or passion and motivation),
and B/ Discipline and long term commitment.

[1] [http://www.amazon.com/The-Talent-Code-Greatness-
Grown/dp/055...](http://www.amazon.com/The-Talent-Code-Greatness-
Grown/dp/055380684X)

------
hyp0
Understanding is "facile" [superficial], but rote learning is the real
way....?

Seems reversed. An explanation is she's in Electrical Engineering - my ugrad
experience of EE was a emphasis on _using_ tools (formula etc) as opposed to
understanding them (that's for Science). Engineers, after all, are paid to get
stuff done, not just sit there and grok it. "Fluency" with tools works well
for EE.

OTOH, all discplines have "tools" \- even pure maths has algebraic
manipulation. If you're not fluent, it will slow you down (fortunately, you'll
have years of algebraic practice from school).

OAH, most professional mathematicians don't _think_ symbolically (a survey
found, IIRC, about 70% visual, 25% kinesthetic, 5% linguistic) - notation not
so much a tool of thought as a serial representation (serialization/data
format), for recording/communicating. So practice in thinking is what's
helpful.

I think fluency with "standard" modes of thought is a double-edged sword. Yes,
you become expert with those tools, fast and able. But that very expertise
biases your perception and reasoning in terms of them. Thus, it's hard for you
to see another way; you'll tend to build on top of them instead. Fortunately,
since our tools are pretty good, this approach works well. It's just that
you're less likely to see fundamentally new approaches (though to be fair,
that's pretty damn unlikely anyway).

------
kamaal
>>student may have thought he’d understood it at the time, and perhaps he did,
but he’d never practiced using the concept to truly internalize it.

I'm changing jobs and I've interviewed at quite a few places off late. Having
faced with so many Algorithm and Data Structure questions, I took it upon
myself to read upon a good deal of material before I started interviewing. So
I did the usual approach one takes while studying math, understand all the
basic concepts and look for places where you could apply them, rather than
studying individual problems.

Recently I interviewed at one place, where I guess a guy who was just out of
college asked a few questions. My general approach is to look for how the
problem would fit in the concepts I've learned. I gradually derived the
solution out of the concepts I knew. And to my surprise, the solution was
simply unacceptable to the interviewer, he just clung to his position that I
was wrong. He then proceeded to write the answer, he even wrote the test
cases.

Now I go back home and check it out of the internet and it was hardly
surprising for me to see what I had suspected. The guy didn't want the answer
to the question. He wanted exactly the same answer he had memorized. The guy
had even memorized the variable names, even the exact input values.

Most people interviewing you on your Algorithm and Data structure skills. They
are simply testing your algorithm and data structure rote memorization skills.
The same goes in exams.

~~~
pbhjpbhj
And isn't that completely bogus too - you don't want programmers who can
regurgitate algos you want ones that can take an overview and know which algos
to apply and how to implement them in a timely way using reference material if
needs be ... perhaps it's too hard to test for that ability in interview?

It's like upthread someone talked about engineering exams - seemingly you
don't need working engineers to run through equation derivations and do
arithmetic. You want them to understand the basis for the derivations and to
have a mental model that will spot if the calculations are way out, for sure,
but really you want them to be able to take an overview: know which factors
are important, use the engineering tools they have. A practical engineer is
going to be using computers to perform calculations (be that chemical
processes or electronics or civil structures or whatever).

In a way I think we're on the cusp of a change in how we need to approach job
focused learning. We're getting to the point where we've stood on the
shoulders of so many giants that if you want to operate at the top you can't
see the floor any more. By which I mean that maybe the human mind can't cope
with the full vertical stack in some fields, that we need to have people
either focus at the bottom - understanding the basis on which particular
knowledge is built. Or understand the top - be able to work with the tools for
implementation.

Maybe.

------
blisterpeanuts
I have a crappy memory. I tried med school for a year. In all my pre-med and
medical courses, they constantly warned us to understand, not memorize. I
gladly took their advice, tried to understand rather than rote-memorize, and I
ended up flunking 5 exams, which finished my medical career.

The fact is, in many fields of human endeavor, you need to memorize before you
can think. Would we have better physicians if they weren't required to
memorize and spit out so much rote information? Not sure about that. I think a
good doc not only has great personal and intuitive skills, he or she needs
good analytical skills and a "think outside the box" problem solving mentality
similar to that of a really good computer programmer (which I thought I had,
hence crossing over into medicine from computers).

But, a doc also needs to _remember_ stuff. There's so much data and you need
to remember signs and symptoms, case histories, and see the similarities and
differences with previous cases you've had.

The OP was about math, and I'm definitely in the memorize-the-formula camp;
I've never really grasped the theoretical niceties of mathematics and am more
of a connect-the-dots kind of guy. But definitely there are similarities
between learning math and learning the physical and biological sciences. Some
of it is intuitive, but some of it is just wacky stuff out of left field and
you have to just take it on faith.

I think we need to emphasize critical thinking and analysis more, but we also
need to teach the ancient Greek and Roman methods of memorizing, e.g. the
memory palace method -- see _Moonwalking with Einstein_ for more details.

------
snarfy
It reminds me a lot of the Pimsleur method.

[http://en.wikipedia.org/wiki/Pimsleur_method](http://en.wikipedia.org/wiki/Pimsleur_method)

It's rote memorization and repetition, but it also incorporates interaction
and active thinking. Additionally, the repetition is non linear. As you learn
new terms they throw old terms at you once in a while to make sure you
remember them, and the spacing is set out for maximum retention.

------
EGreg
I believe that math is something a lot of people can learn and become
effective at using, way more than Americans currently believe about
themselves.

I think it all comes down to making sure people understand the actual
abstractions and concepts rather than formulas and rote learning. The
advantage of math is that you can figure stuff out using what you already
know, without having to learn every little detail - while that's impossible in
other subjects, such as history.

The missing piece is, in fact, granularity and feedback. And real world
studies support my position:

[http://opinionator.blogs.nytimes.com/2011/04/18/a-better-
way...](http://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-
math/)

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

If you want to read how I'd reform education and improve the economy, these
are my thoughts:

[http://magarshak.com/blog/?p=158](http://magarshak.com/blog/?p=158)

------
rubicon33
Assuming the author's convictions are true, how would one apply a 'rote
learning' technique to programming?

I believe in repetition, but, personally haven't been able to find a useful
way of implementing it to become a top-notch programmer.

Why? Well, take calculus. Buy a book, pound problems. Do 1,000 derivatives.
Next chapter. 1,000 integrals.

Is there a similar "exercise" for programming?

~~~
notduncansmith
Depending on what you want to learn, sure. Googling for "[someLanguage]
practice problems" appears to turn up good results for most languages.

For data structures, write some unit tests and then make them pass.

I personally wouldn't recommend you bother learning the implementation details
of algorithms: just learn the names and use-cases. Make some Quizlet flash
cards and you're good. If you're having trouble finding some to learn, try
Googling "sorting algorithms" or "sorting algorithm for [someDataStructure]".
Go from there. I'd recommend a similar approach for design patterns.

If you want to get better at building applications, practice by reimplementing
apps you use every day in your favorite language (or just come up with your
own). "Building applications" encompasses a wide variety of things not
mentioned above that you'll only be exposed to by building stuff, so do that.

~~~
Scuds
> "[someLanguage] practice problems" It might also be fall under "Études"

such as 'Études for Elixir'
[http://chimera.labs.oreilly.com/books/1234000001642](http://chimera.labs.oreilly.com/books/1234000001642)

which is a collection of exercises that slowly introduce you to a programming
language.

I can read stuff out of a book, but I won't have that 'muscle memory' until
I've spent that time in Vim and the compiler/REPL. So that when something new
gets thrown my way, I'm not having to think back to the basics.

------
nitrobeast
"Sorry, education reformers, it’s still memorization and repetition we need."
This is very true. For example, doing the advanced engineering calculus can
often boil down to manipulating polynomial, which are still arithmetic. A
person with "understanding" but no fluency in basic arithmetic is setup to
fail.

~~~
cbd1984
If your work seriously involves manipulating polynomials, install some
software.

Done, and done better and faster than you could possibly do it.

------
jcagalawan
I thought thus would be about the guy on HN who used a psychedelic trip to
gett over math anxiety. Still a good read.

------
casebash
I think that it is strange that she is saying that there is too much focus on
understanding at school. From what I could see, most people learned maths by
just memorising the process. Except for a few who invested an exceptional
amount of effort, they scored poorly. In contrast, everyone who I knew who
focused on gaining a deep understanding was able to pick up maths quickly.

There is one area, though, where there is probably too much focus on
understanding at the cost of memorisation and that is in the Olympiads. I
remember that many of us held the rather uncharitable attitude that the people
who tried to succeed in maths by memorising it were 'stupid'. But clearly,
understanding combined with targeted memorisation of the key building blocks
will lead to the most success.

------
conistonwater
How is "rewiring your brain to become fluent in maths" different from
"learning maths"? Can someone explain this? I read the article, and I don't
understand it. To me it sounds like she learned maths, in much the same way
one learns anything else.

~~~
hudibras
Authors usually don't get to pick the titles of their articles. That's the
editor's job and they try to pick headlines which will attract readers.

------
graycat
It's not so hard:

(1) Pick a topic at about the right level.

(2) Get one main, maybe plus a few extra, good books on the topic.

(3) In a good chair in a quiet room, in the good book, read the material on
the topic and try to understand it well enough to make sense. Then work the
exercises.

If something doesn't make sense, look in one or more of the other books.

(4) At times, review the last several topics and try to get more
understanding, e.g., what else is true, what is not true, what makes the true
statements true and the false statements false.

Maybe take a class that covers the main book and check work and understanding
with the professor.

Rinse, repeat with other topics.

Done.

Worked for me.

------
edwinespinosa
I liked her point of repetition being the main problem. It makes sense as we
strengthen the new synopsis that are made. Josh Foer author of moonwalking
with Einstein proposed an interesting theory that, essentially fucked up
imagery of any kind helps increase and strength synaptic connections.

Interesting watch (first 5mins gives good context)
[http://books.google.com/books/about/Moonwalking_with_Einstei...](http://books.google.com/books/about/Moonwalking_with_Einstein.html?id=f_0XXXjJk3gC)

------
lordnacho
I don't understand why there seems to be a languages/math dichotomy. I
remember in school teachers used to bucket people in one or the other.

But from what I can tell, and also from the article, they have a lot of common
ground. Learning either requires understanding of structure, but also a fair
bit of memorization. Learning conjugations by heart is not all the different
to learning trig relations. You also need both on the fly when you're using
your language/calculus.

------
MisterBastahrd
Memorization is not learning. It is a part of learning. Being able to memorize
things and understand their relationships in context is learning.

~~~
robaato
My German teacher at school did an accelerated course from scratch at
university (no previous knowledge) - a full language degree. It involved rote
memorisation of German texts. It worked! She said that the advantage was that
you later recognised phrases and grammar and were able to analyse what was
previously just rote memorisation.

I spent a couple of years in Italy in my early 20s and learnt fluent Italian
in that time. I took a couple of classes a week in the evenings, but spent the
day programming with a team of English colleagues and perforce spoke English.
I made an effort to socialise with Italians outside of work.

I went out with groups of Italians many a time, and experienced many a boring
evening where if people didn't speak to me specifically I couldn't understand
a thing.

I realised that after 9 months I could hold a good 1-to-1 conversation but
that it was another 9 months before I was comfortable in a group situation -
in your native tongue you can keep tabs on the various threads of a group
conversation with ease, whereas if you are having to explicitly process it,
it's so easy to fall behind and then you're lost.

~~~
e12e
From experience with exchange students, staying with families and going to
regular local secondary school (ages 15-18, look at www.afs.org for more
information) -- that sounds typical for an adult in a non-native work setting,
making an effort.

Generally students starting at zero will use 3/5/7 months to approach fluency,
approaching native level at the end of a stay. The difference being the "jump
distance" \-- so German to Norwegian or Spanish to Italian is 3 -- Arabic to
Norwegian might be 5 -- and Japanese to Norwegian might be closer to 7.

Coming to Norway, sometimes students with poor English skills will do better -
not being tempted to fall back to English, breaking the immersion - and not
having such a hard time "forcing" class mates to speak Norwegian.

Having spent a year in Japan myself, it's fascinating how one can go from
seemingly "nothing" to fluent after those months of no apparent progress. But
language learning tends to be like a staircase -- you can feel stuck at one
level for a long time (and being in a foreign setting it can be incredibly
frustrating to be constrained to a preschool level of speech) -- only to seem
to jump up a level. And then you'll be stuck there... etc

------
facepalm
I wonder if part of the effect is simply studying a lot. It sounds as if she
devoted a lot of time to studying Russian and then maths. Perhaps the actual
way she spent the time is not that relevant? Not saying it is irrelevant, but
for example maybe simply doing problems in maths for the same amount of time
would have been just as effective?

------
larssorenson
Reading this reminds me of how much I really want to be better at math, and
especially the quick fire recall and understanding. I'm taking courses with a
lot of math that I've learned before, and encountering similar struggles this
time around as last time. Maybe I'll try a new approach, one such as this, to
insure my success.

------
Tycho
I get this sense that mathematicians tend to focus on abstracting
relationships, whereas programmers tend to focus on abstracting
objects/entities. Hard to explain though. Like programmers are constantly
building models of the world, whereas mathematicians are constantly building
new functions and general transformative techniques.

~~~
emillon
This is the general idea behind category theory: it describes relations and
not objects themselves.

That's quite a shift from set theory, where the core idea is that objects can
be described with what's inside of them (and building the whole idea of
relation as a set of tuples).

------
lohankin
Author makes generalizations based on her own individual experience - the
process which she doesn't understand. Rot-learning in music (mentioned as
example of good practice) produces millions of classically-trained musicians
that don't understand anything about music, and can't play even Christmas song
by ear.

------
liox
so many aspects of this article echo my own experiences, though i was likely a
far worse example of a student than the author of this article. i excelled at
literature (and took this talent for granted) and i had a vicious disdain for
math. my time in high school was spent anticipating a “mathless” career in the
NHL that never quite came to fruition… sometimes you just don’t grow any
taller after your freshmen year :)

here’s how a gift can actually be a curse: i’ve always had an extremely strong
visual memory and it enabled me to get by (and sometimes even excel) at
courses like algebra and pre-calc. i’d simply memorize the exact structure of
a solution, and “plug and play” from memory when teacher asked identical
questions on exams with different numbers. it wasn’t until i was an undergrad
and i decided to do my own “rewiring” experiment that i realized i hadn’t ever
learned a thing about math my entire academic career. it made for quite the
rude awakening, being that i was a college sophomore who’d just switched from
pre-med biology to electrical engineering!

i struggled with basic problems that required even a little bit of
mathematical intuition because i’d never developed an intuitive understanding
of anything–to paraphrase bart simpson, it was like i was cheating on tests by
using my own mind, since i could remember my notes with such vivid detail.

how’d i fix it? i became a voracious reader. i started reading “popular”
science books about math that actually interested me, and i realized that
textbooks were only part of the puzzle. once i was able to grasp the
theoretical concepts (aided by the “pop” books) i could return to the
textbooks without being bored to tears with the dry, abstract concepts–i could
actually relate the concepts to reality! i also took a keen interest in
physics because it connected calculus to the real world. when i realized that
a parabola could actually be the path of a cannon ball, it changed my entire
outlook on math. things started to click.

so how’d it work out? getting through b.s. degrees in both electrical and
biomedical engineering (i was a masochist and picked up a second degree) was
nonetheless a struggle, due to the fact i was essentially several years behind
my peers in terms of my mathematical ability. also, i constantly had to fight
the urge to rely on my memory. old habits die hard or simply just go into
remission, but i made it!

where am i now? as an MBA student (please bear with me here) i was fortunate
enough to be at a university that recently embraced data science. i fell in
love with the field because data can tell a story, and my (now) strong
mathematical background enabled me to grasp the underlying math & algorithms
that yield the wonderful insights you can uncover if you look in the right
places.

was it worth it? absolutely. i feel as though i am a different person that’s
capable of using both sides of their brain! although i must admit i’m still a
natural “right brainer” and i need to use a calculator more frequently than
i’d like to admit. i’m an engineering manager at a 9-5, i work as a data
science lead for a startup, and i dabble with inventing products. i’m
convinced that had i not put myself through the wonderful torture that was
rewiring my brain, i’d likely not have achieved a single thing in my career.

if you’re really curious to see how someone like me thinks i’ve done some
writing here: www.takeiteasythursday.com/liox

------
icantthinkofone
I remember struggling with Calculus in high school. I sat in the library next
to the smartest guy in the class and explained to him that I couldn't figure
out why a certain formula was to be used to solve a problem. It went something
like:

Steve: "You have to use that formula to solve this."

Me: "I understand that but why do we use that formula?"

Steve: "Cause that's the only formula that works."

Me: "OK but how did they come up with that one formula?"

Steve: "Cause it just works!!"

He didn't know. That's when I first realized a lot of his knowledge was just
memorization while I needed to know the why. It wasn't till my first year in
college that I finally found a math instructor who always shows us the reasons
why that I was able to take off and grasp math by connecting the dots.

So, no, I do not think memorization is the key but, like most zen things,
understanding is.

~~~
jxm262
I've constantly struggled with something along these lines too. I've always
had a difficult time in Chemistry because i wanted to know the why's of things
as opposed to just memorizing things. Even in programming, my intro Java class
everyone learned the basic public static void main(String[] args){} and we
were expected to take it for granted. The first Hello World assignment took
most students a few minutes to complete, but sent me researching the Oracle
docs, which led me to the java trail, which made my assignment like a week
late lol.

If there's any life hacks on how to minimize this mindset, I'd love to hear
about it. I'm constantly struggling to find this balance of memorizing
abstractions vs digging in the details.

~~~
spion
YMMW, but I use "I can always fill in that gap later if I have the time" and
move on. Later, if I do indeed have the time or I find out that I'm struggling
because of missing details, I'll get back to it.

Its like making a temporary mess in a project while trying to understand the
problem you are solving and making a promise that you will refactor it later,
once you get to something that works.

~~~
RankingMember
This is where OCD really kicks my ass. I want to understand everything that
I'm typing, and it's a major hurdle to have to accept that, no, I just can't
have all the answers.

------
thenerdfiles
Theorems are muscles of the mind.

------
torkable
Nice!

~~~
corysama
FYI: This comment will get downvotes. Not because people here are senselessly
mean. But, because the HN community takes a strict approach to comments that
add noise without contributing signal to the conversation. Downvotes on "Nice
article!" comments frustrate many new users. So, here's your explanation.
Welcome to Hacker News!

~~~
torkable
I was simply replying to the explanation that the article is available for
free and I care nothing for points or votes, thanks anyway

------
Dewie
I could hardly stomach the self-congratulatory tone of the introduction. Jeez.

