
The building blocks of understanding are memorization and repetition - rgun
http://nautil.us/issue/40/learning/how-i-rewired-my-brain-to-become-fluent-in-math-rp
======
tucaz
TL;DR: Study foundational blocks by repeating then until you understand. Keep
repeating as you build knowledge and move to "harder" things.

I like the message, but not the messenger.

Articles like this, in my point of view, fail to delivery their messages in a
better way because they are simply too long and with too much noise for my
taste.

The point he is trying to get across could be explained in a simpler way
without the necessity of explaining how he went to the army, what he did there
and all his personal history.

All this information is completely irrelevant to the main goal of the article
and turn it into a boring story instead of useful piece of practical advice.

~~~
noobermin
I remember first learning Quantum Field Theory. One week, my head would ache
from what I was reading but by the next week, that bit would be incorporated
into new parts, so the "confusing stuff" become foundation for the next week's
confusing stuff. Eventually, it became intuitive.

~~~
sevenless
How _do_ you learn quantum field theory? I took a tiny amount of QM in school
- basically, solving the Schrodinger equation for a few simple systems. What's
the next step?

~~~
noobermin
I read Shrednicki[0]'s book, a pre-print is freely available online (or was 4
or so years ago). A class in Grad school solidified it for me, specifically
because they related it back to physics, like the standard model.

If you're going to start learning QFT, I suggest you try to understand time-
dependent perturbation theory as much as possible. Also, be sure you have a
handle on special relativity and scattering theory. QFT that most people use
is really those things combined.

[0] [http://web.physics.ucsb.edu/~mark/ms-qft-
DRAFT.pdf](http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf)

~~~
JadeNB
For what it's worth, that PDF says "Please DO NOT DISTRIBUTE this document.
Instead, link to
[http://www.physics.ucsb.edu/∼mark/qft.html](http://www.physics.ucsb.edu/∼mark/qft.html)
" (which has a link to the PDF).

~~~
noobermin
Fair enough. Unfortunately, I can't edit the comment now.

------
westoncb
The author is arguing that 'the latest wave of educational reform in
mathematics' (in the U.S.) is overemphasizing conceptual understanding, and
that doing so will be damaging to students.

The author advocates instead using a memorization and repetition based
approach.

I can see where she's coming from, but the argument she is making is flawed.
She describes a process for developing fluency both in foreign language and
mathematics, wherein she experiments and plays with the constructs being
learned—this is a great suggestion, but it's not in _opposition_ to
understanding: the two complement each other.

I think the error arises from making an overly strong identification between
learning a foreign language and learning mathematics. If you take the
representation of Euler's equation, for instance, and consider how much
conceptual depth underlies it, versus a string of 20 Cyrillic characters
meaning "I went to the store," or whatever—you can see the difference. You'd
be wasting your time trying to get a deep understanding of the Russian phrase,
but there's a reason to do it with the mathematical phrase.

~~~
SomeStupidPoint
> You'd be wasting your time trying to get a deep understanding of the Russian
> phrase, but there's a reason to do it with the mathematical phrase.

I think you're simply wrong here.

You're not talking about the amount of conceptual depth underlying it, you're
talking about the relative delta in the conceptual structures between
languages you already know and the language you're learning.

On the whole, the Russian phrase has greater conceptual depth; however,
Russian is closer to English than mathematics is, and thus you get greater
conceptual transfer between your knowledge of English and Russian than your
knowledge of English and mathematics, so more of your past experience remains
relevant.

In essence, I have something like 36-72k hours considering semantically
equivalent statements to the Russian one in English, while I have something
like 5-10k hours of studying mathematical statements at all.

Your comment just amounts to saying that having 6x as much practice at task A
as task B makes task A relatively easier, which, while true, isn't perhaps as
profound as you'd meant it.

Consider this: the entire field of academic inquiry containing Euler's
identity is fundamentally an effort to fully understand "I went to the store".

~~~
westoncb
> you're talking about the relative delta in the conceptual structures between
> languages you already know and the language you're learning

I don't think so. In both of my examples, the one from mathematics and the one
from Russian, language is used as a way of _representing_ some concepts—now,
the _language itself_ also has concepts underlying it, but these are about
grammatical structures, lexicon, etc.—not about the subject the language is
used to talk about.

To be clear, I'll call the first category of concepts 'subject concepts' and
the second 'language concepts.'

When you learn a new foreign language you must learn new language concepts,
but not new subject concepts. You aren't re-learning what it means for one to
visit a store, you are only learning a new scheme for representing that
concept, which isn't nearly as deep as the initial concept itself.

In learning mathematics you're having to pick up the language concepts _and_
the subject concepts, and the subject concepts (as in the example I gave) can
be quite deep.

~~~
SomeStupidPoint
You've outlined my point exactly: you're comparing a case of learning language
concepts to a case of learning language and subject concepts. Of course one of
those is easier, but it doesn't mean that the two processes are fundamentally
dissimilar, which was your initial point.

Your comment is no more insightful than to say that it's easier to learn set
theory knowing category theory than it is to learn English while knowing no
languages, since it's purely acquisition of language rather than subject
concepts. (This is actually untrue -- which is why it's easier to say, switch
to romance languages than going to an Asian one from English; there's a bit of
subject conceptualization in the nature of the language concepts.)

It's not comparing apples to apples, which sort of reduces the point about the
relative complexity of statements and the way that you learn the underlying
concepts -- both language and subject. You're comparing the complexity
differential of two encodings on the one hand and the total complexity
involved in the other. Nonsense comparison.

If you want to talk about learning Russian while knowing English, why not
contrast it with learning set theory while knowing category theory?

Because the language differential between spoken languages (eg, you _do_ learn
new subject concepts if you learn Japanese versus English) is comparable to
the difference in mathematical underpinnings, eg, the switch from set theory
to category theory.

~~~
westoncb
> ... it doesn't mean that the two processes are fundamentally dissimilar,
> which was your initial point.

That was not my initial point. You've read the 'fundamentally' part into it.

> If you want to talk about learning Russian while knowing English, why not
> contrast it with learning set theory while knowing category theory?

That is what I was doing. I'm not saying learning math versus foreign language
is fundamentally different, I'm talking about _practical_ differences in
actually learning one or the other: who in the audience here doesn't already
speak one natural language? So, with any foreign language they will be in the
position of somebody knowing category theory and attempting to learn set
theory. That is by definition not, however, the case for someone trying to
break into mathematics for the first time. So you always have language
concepts only for a foreign language and language concepts + subject concepts
for mathematics—practically speaking.

~~~
SomeStupidPoint
> I think the error arises from making an overly strong identification between
> learning a foreign language and learning mathematics.

You explicitly begin by mentioning that you think it's the similarity between
the two topics that you think others are mistaken about -- that learning
language and learning mathematics shouldn't be identified (as strongly). You
now say that it's okay to (strongly) identify those two processes (as being
the same), we need only talk about the relative amounts of subject versus
language concepts needed, given an expected background for a student.

I wish you'd admit your phrasing could be misconstrued (at the very least).

> If you take the representation of Euler's equation, for instance, and
> consider how much conceptual depth underlies it, versus a string of 20
> Cyrillic characters meaning "I went to the store," or whatever—you can see
> the difference. You'd be wasting your time trying to get a deep
> understanding of the Russian phrase, but there's a reason to do it with the
> mathematical phrase.

This statement, of course, only makes sense if you're comparing either just
the language concepts of both or else the language and subject concepts of
both.

Instead, your point here, as you last stated it, isn't that there's no reason
to get a deep understanding of the Russian, but rather, that getting a deep
understanding of the Russian is easier than getting a deep understanding of
the mathematics because the deep understanding of the Russian is easier than
the mathematical one to derive from your pre-existing understanding of
English.

Okay, but is that really surprising? Or what you think you conveyed in that
quote?

> So you always have language concepts only for a foreign language and
> language concepts + subject concepts for mathematics—practically speaking.

You actually always have both, but the bulk of learning a foreign language is
language concepts, while in mathematics, you're exposed to more new subject
concepts, which I think is what your point is.

Again, okay. But I think the language you used to describe that initially
draws a needless (and ultimately, meaningless) distinction, and that we get a
lot more utility out of talking about them as the same process for students,
for which they've already done part in the general language case (and general
arithmetic case!), since this automatically gives us a framework where we can
meaningful discuss how learning Japanese, Russian, English, Category Theory,
Set Theory, etc interrelate, the relative load of different concepts to learn
in each case of learning a language, the process to go through, etc in very
general terms.

So I don't think we actually disagree, except on whether your initial post
accurately conveys what you seem to think.

~~~
westoncb
Without getting as precise as using formal mathematical notation about all of
this, there is going to be sufficient ambiguity that, if you desire, you can
construe it to be taking one or another subtly different stance on several
points.

Interestingly, those ambiguities are pretty readily solved without the formal
notation if both parties are actually interested in communicating, rather than
demonstrating that an initial statement said 'X' rather than 'Y'.

You are clearly in the second camp here, so the ambiguities weren't easily
resolved. _shrug_

------
Falkon1313
I liked this article because it was in direct opposition to my beliefs (based
on my own experience) and yet it gave me a different point of view and
convinced me that there was some truth and value there.

I grew up in an era when we wasted a lot of time on mindless rote repetition
of meaningless things (meaningless because we were not taught to understand
them, only to repeat them). It was boring and useless. I learned by
understanding and then applying that understanding. If you understood it, then
there was no need to memorize something like f = ma because it was then
obvious. But this is a whole new way of looking at 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?

That's not thoughtless memorization, that's gaining understanding. But the
author did it by active exploration via repetitively examining something from
different angles. Not just rote repetition (doing the multiplication over and
over again), but seeking understanding. So in a sense, the author is arguing
in favor of teaching by emphasizing understanding, but explaining how to use
directed repetition to do so. That makes a lot of sense. The key is that it is
not mindless rote repetition, but directed, inquisitive repetition.

~~~
emanuelev
Agreed. I think that's the key, you learn by first grasping a basic
understanding of a concept and then reinforcing it through application
repetition. But the initial understanding is absolutely vital, otherwise
chances are you end-up in the mindless rote repetition you mentioned.

------
dhawalhs
This article was written by Barbara Oakley who also teaches the most popular
MOOC in the world and also my favourite MOOC called Learning How To Learn:
[https://www.coursera.org/learn/learning-how-to-
learn](https://www.coursera.org/learn/learning-how-to-learn)

She also a upcoming book titled 'Mindshift: Break Through Obstacles to
Learning and Discover Your Hidden Potential':
[https://www.amazon.com/Mindshift-Obstacles-Learning-
Discover...](https://www.amazon.com/Mindshift-Obstacles-Learning-Discover-
Potential/dp/1101982853/)

------
fitzwatermellow
My latest hack is to return to paper and pencil. After going paperless about a
decade ago, re-wiring these particular neural pathways is quite an obstacle in
itself. But in terms of retention and creative flow it is unparalleled. I
bring it up because I think the foundation for the memorization and repetition
feedback cycle begins with elementary school lessons in handwriting, cursive,
the alphabet and spelling. And then onto basic arithmetic and mathematics. The
idea that "digital native" students would eschew hours of rote hand writing
practice altogether puts humanity in danger of losing something elemental.

~~~
pbhjpbhj
Such digital natives would be doing hours of input practice for other media
though?

Is it the specific mechanics of orthography that you SD as valuable?

------
sireat
Author makes the point that repetition of a solid foundation is a necessary
condition to reach unconscious mastery of some craft/skill.

I am just not sure it is sufficient. Age/talent/nature do play a big factor.

Also I suspect starting to learn serious math at age of 26 she is an outlier.

Russian is my 2nd language (English is my 3rd) and both I learned through
repetition via reading prodigious amounts in each language.

My Spanish is horrible because I only studied it in high school and never did
read any serious books in Spanish.

Now in my 40s I find that learning German is a formidable task even to read
Brothers Grimm much less some philosophical works that I had hoped to.

Part of the problem is lack of time (6 months of Duolingo is not enough that's
for sure) but also my brain seems to require more repetition to acquire the
same knowledge that my daughter picks up near instantly.

~~~
projektfu
I learned to read French with the book French for Reading. Perhaps the German
version would help you. [https://smile.amazon.com/German-Reading-Second-John-
Wendel/d...](https://smile.amazon.com/German-Reading-Second-John-
Wendel/dp/158510745X/ref=sr_1_1?ie=UTF8&qid=1473976898&sr=8-1&keywords=german+for+reading)

------
danso
> _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. Where my language classmates had often been
> content to concentrate on simply understanding Russian they heard or read, I
> instead tried to gain an internalized, deep-rooted fluency with the words
> and language structure. I wouldn’t just be satisfied to know that понимать
> meant “to understand.” I’d practice with the verb—putting it through its
> paces by conjugating it repeatedly with all sorts of tenses, and then moving
> on to putting it into sentences, and then finally to understanding not only
> when to use this form of the verb, but also when not to use it._

Wow, what great timing. I remember reading this article when it was originally
published (apparently in Sept. 2014) and I believe I read it because it made
the HN front page back then. It's changed the way I think about teaching,
especially technical skills to non-technical students.

The best students I've had so far are ones who are pretty smart and hard-
working already, but by and large, they are also the ones who follow my advice
to type out code by hand, run it, change it, repeat, break it, several times.
And to memorize a few essential keyboard shortcuts (Tab for auto complete,
Cmd-Tab for window switching, etc) so that the work of retyping and debugging
code itself is much more frictionless.

Unfortunately, and understandably, most college students aren't thrilled with
the idea of repetitive practice makes perfect (or at least, makes learning the
important concepts much easier). This year I'm going to do a lot more testing
involving writing code with pen and paper, on the theory that if you can
actually write out code by hand, then you probably know the fundamental
patterns (I'm talking _fundamental_ , as in a common for-loop) essential for
higher programming.

edit: One thing I should point out; this doesn't make me right but I do
dogfood this approach myself when learning any new programming language. I'll
write out a tutorial. Then write it the way I think it should work. If I'm
half distracted, I'll write it backwards. The thing is, I'm an experienced
enough programmer to know that taking the extra time to know how things work
is always worth it in programming, because of how insignificant the work of
_physically_ writing code is compared to actual programming. Non-programmers
do not realize this and approach it as if they were asked to write 20 pages
about Hamlet, and then 20 pages about Hamlet using different adjectives.

~~~
pbhjpbhj
>if you can actually write out code by hand, then you probably know the
fundamental patterns (I'm talking fundamental, as in a common for-loop)
essential for higher programming.//

What if you can write the pseudo-code but not the specific syntax without a
reference it auto-fill out your IDE? Is that enough?

------
piedradura
In maths you learn to solve problem by expanding the abstractions by adding
new elements. There is a crucial difference between problems easy to solve by
repetition and problems that require some insight.

~~~
lokiboi
Exactly, otherwise all those IOI and IMO contestants would be pure geniuses,
probably they are, but they practiced a lot to be good at a single task -
problem solving in programming or mathematics for competitions. That's a good
foundation built on insane amounts of repetition and memorization, but still,
not many turn out to be Terence Tao or Peter Shor.

Although
[https://en.wikipedia.org/wiki/List_of_International_Mathemat...](https://en.wikipedia.org/wiki/List_of_International_Mathematical_Olympiad_participants#Notable_participants)

seems many did "pivot" from competitive memorization and repetition to some
lovely mathmakers.

~~~
tamana
You don't get to the IMO by mere memorization.

~~~
lokiboi
Pattern matching is huge contributor to solving problems fast or even having
the problems reachable.

Just checkout these results
[https://icpc.baylor.edu/scoreboard/](https://icpc.baylor.edu/scoreboard/)

They practically solve a very hard problem (it might take days for me to solve
it) in 5 minutes. If that is not some supreme pattern matching then I don't
know what it is.

Bunch of proof strategies used in math competitions are the same.

You have to have a giant knowledge of algorithms: fast fourier transform, bfs,
dfs, iterative deepening, dijkstra, a*, sweep line algs, practically have to
master dynamic programming (there's DP on trees, hidden markov model like DPs
etc.), flood fill, topological sort, bipartite graph checks, kruskal, prim,
edmonds karp, bunch of combinatorics, geometry algorithms.

data structures like segment trees, fenwick tree, suffix trees, etc.

you see where I'm going, there's a lot of memorization, and a lot of
repetition (which improves memorization).

same thing goes for IMO, you have to do a lot of proofs, have to memorize and
repeat a bunch of proof strategies in a huge number of mathematical areas.

memorization and repetition!

------
cmillard
Everyone has their own roadblocks when it comes to learning. Looking at my
understanding of programming I realize I have put too much an effort on self-
study (rote memorization) and not enough emphasis on interaction (especially
with people).

As a child I was always able to memorize something- whether it be classical
music, or formulas. However, the deeper connections in learning that allow you
to proceed outside of your box come from something else. True understanding is
greater than the sum of it's parts.

I like the adage- see one, do one, teach one. Memorization is a requisite for
all of them, but true understanding and mastery comes when you're able to take
abstract concepts and impart them to other people.

No matter where you are the easiest way to benchmark your competency is to
share it with someone else. They don't even need to be an expert, after all-
it doesn't take a air traffic controller to see that a plane's landing gear
isn't down.

------
daveloyall
The author attended DLI[0]?

Sounds like the way to learn math late in life is to be _really_ smart. Nobody
that attends DLI isn't.

0:
[https://en.wikipedia.org/wiki/Defense_Language_Institute](https://en.wikipedia.org/wiki/Defense_Language_Institute)

------
tsumnia
It's nice to see people agreeing with Oakley, considering I had developers
ready to tar and feather me for suggesting code snippets should be images to
prevent copy/paste.

By no means is memorization the end all be all. Oakley even writes "In the
United States, the emphasis on understanding sometimes seems to have replaced
rather than complemented older teaching methods". The techniques should be
used hand and hand.

RangerScience said it best when he said "perfect practice makes perfect", I
like to reword it so it reads "practice makes permanent". Obviously,
practicing something wrong isn't going to help you advance.

Memorization should be "step one" for introducing a concept. I can explain how
a for loop looks and show it to you, but forcing the student's hand to make a
few loops before trying to implement it in a homework assignment can help
build the neurological pathways (or motor engrams) so that they're no longer
thinking about the syntax, just the chunk. "I need to type
for(something;something;something)..." transitions to "I need a loop to go
through this thing". The syntax is secondary as anyone who knows more than 1
programming language will tell you. Hell, I've build snippets into Sublime
Text so I don't have to waste time with syntax either!

Some things just come from putting in the time and effort. No one gets their
black belt after one class (otherwise I've been seriously doing it wrong for
10 years!)

~~~
BirdieNZ
Having seen the results of a system that focuses entirely on memorization
without comprehension, a balance definitely needs to be struck between
understanding and memorization. Pure memorization is useless, but so is
conceptual teaching without practical application (which helps to reinforce
the concepts and ensure real understanding).

------
ramblenode
I've found a lot of observations in this article to align with my own personal
learning and teaching experiences. Memorization, even in analytical fields, is
important because it allows us to chunk smaller bits of information together
and skip evaluating every detail--it decreases the size of the "mental stack",
if you will. Because recalling a fact from memory tends to be less demanding
than applying rules to derive the fact, more cognitive reservoir remains for
the novel parts of the larger problem.

------
apalmer
I agree with a lot of what she is saying, but two points: 1) The basic
solution outlined is 'do a lot of work'. Which is kind of a no brainer, you
will learn if you do a lot of work. The tension is how do you get the most
productive learning out of the time that the student has

2) The US only recently moved from memorization and rote learning to focus on
understanding for a reason. The reason ultimately is the students werent
excelling under that regime.

~~~
bordercases
Do they excel under the current one?

------
sfink
Bleh. It was a good life story, but you could write out the conclusions either
way, and the article is mostly successfully defeating a strawman. She makes a
good point: if you define "understanding" narrowly, such as "whatever it takes
to score well at Common Core", then it is important to realize that it is
inadequate. Fluency is missing, and important.

But it's really just playing with definitions. I'll attack the contrapositive
strawman: rote repetition and practice is useless if you don't first
understand. You can memorize multiplication tables and the algorithm for
multiplying multiple digit numbers, and yet make orders of magnitude mistakes
without batting an eye. 40x25=100, right?

I would say that you need to understand the basics, then gain fluency, and
from that gain an understanding of the depths and nuances. And I think her
life story could back that description up just as well as it backs up hers.

------
smartbit
Rule #5: _Repeat to remember_. [http://www.brainrules.net/short-term-
memory](http://www.brainrules.net/short-term-memory)

Rule #6: _Remember to repeat_. [http://www.brainrules.net/long-term-
memory](http://www.brainrules.net/long-term-memory)

From John Medina's book Brainrules. Highly recommended, translates scientific
research into an accessible style. The videos are very humorous eg _Whenever I
feel like exercise, I lie down until the feeling passes_ :-D (1:40)
[http://brainrules.net/brain-rules-video](http://brainrules.net/brain-rules-
video)

------
truth_sentinell
If this is true, then the spaced repitition learning technique is the best
there's for learning.

------
utefan001
I talk to a lot of young adults about their challenges in school. Many are
interested in computer science but don't have a strong foundation to ace a C++
course. I wish I could convince them that failing the course is not as big a
deal as it seems. Repeating the class is the right choice. Giving up is not
the answer. Sometimes repetition means taking a class 3 times.

<advertisement>

To help younger kids with repetitive learning, consider smartmadre.com (beta).
Once configured, it will deactivate your child's access to time wasting
websites. In order to get internet access back, the child has to spend a few
minutes earning points on readtheory.org, quizlet.com, khanacademy.org or
typingclub.com etc..

</advertisement>

------
perliosse
Important to note that repetitions are never exactly the same: new connections
are found each time.

Also, choosing _what_ to repeat seems important. Drilling in what is already
understood may help you become a good teacher but how does one develop new and
_original_ ideas?

~~~
WillPostForFood
Someone might be better equipped to write something new and original if they
knew 10,000 words vs 1,000. Or to bring it to programming, do you think a
programmer who has complete mastery of the syntax of a language might be
better equipped to develop innovative software than a developer who had to
refer to stack overflow to lookup whether == was assignment or equivalence?

------
exmuslim
I absolutely agree that memorization and repetition trump conceptual
understanding when dealing with intermediate level of mathematics (I am taking
a course on probability and integration and even though I have a good grasp of
the basic concepts, if I don't keep up with the nitty gritty details and
properties, I completely go astray when the teacher introduces a new concept
built using those details.)

I.e. you need to master your tools (theorems, formulas etc) by repeating them
so that you can use them to build something. Even if you knew what a hammer is
good for (conceptual understanding) but didn't know how to hold it, it would
still be pointless.

------
paublyrne
For someone who wasn't interested in Maths in school, but would like to learn
the basics now (quadratic equations, trigonometry, etc), can any one recommend
a good all round text?

------
RangerScience
"Building blocks" sounds correct, but misleading.

It sounds like this is a lot of "perfect practice makes perfect" combined with
"play begets understanding".

Memorization is a pre-requisite for perfect practice; and, if the core of what
you're playing with isn't immediately at hand, how can you play with it? (If
you have to look up every function every time, you'll have too many
interruptions to grok much more).

I memorize _in order_ to practive, and I repeatedly practice _in order_ to
understand.

Does that sound right?

------
kafkaesq
_Number Twelve: What was the Treaty of Adrianople?_

 _Number Six (looking puzzled, but answering automatically): September...
1829._

 _Number Twelve: Wrong. I said "What," not "When." You need some special
coaching._

[https://en.wikipedia.org/wiki/The_General_(The_Prisoner)](https://en.wikipedia.org/wiki/The_General_\(The_Prisoner\))

------
hyperpallium
Reminds me of "mathematical maturity", that you can't understand mathematics
on its own, but need much background. In this way, maths is more like an Arts
than Science subject, and evaluated similarly.

And is perhaps a reason mathematics seems to simply and "unreasonably
powerfully" explain the world - it's _not_ simple.

------
adamnemecek
This mirrors my experience and this fundamentally proves that empiricists were
right and rationalists were wrong. This also implies that your thinking isn't
exactly logical.

Schools need to be adjusted for this, I think that currently schools are based
on the rationalist point of view.

------
posterboy
I'm starting to think that any head line starting with question words fall
under the law that I forgot the name of. Even without a question mark it
suggest that the author is still looking for an answer, except for idioms like
_how to_.

~~~
throwanem
Betteridge's law.

------
anonymid
TLDR: This seems to be a strawman, at least in how it presents common core.
However, I have to give a concession to the criticism that superficial
understanding is often what common core implementations look like. Still, in
my opinion fluency comes from understanding, not the other way around.

>The problem with focusing relentlessly on understanding is that math and
science students can often grasp essentials of an important idea, but this
understanding can quickly slip away without consolidation through practice and
repetition. Worse, students often believe they understand something when, in
fact, they don’t. By championing the importance of understanding, teachers can
inadvertently set their students up for failure as those students blunder in
illusions of competence. As one (failing) engineering student recently told
me: “I just don’t see how I could have done so poorly. I understood it when
you taught it in class.” My 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. He had not developed any kind of procedural fluency or ability
to apply what he thought he understood.

Teaching for understanding means that teachers are responsible for ensuring
that students are understanding. If a student is mistaken about understanding
something, but the teacher doesn't probe their understanding to expose their
misconceptions, that's not "teaching for understanding".

Common core encourages repetition through its focus on multiple
representations. One might study linear growth as repeated adding, as a table,
as a graph, and in applications to various real-life phenomena. Common core
places emphasis on the student being fluent (as the author states, common core
has fluency as one of its three major focal points) with all of these
representations, and also in seeing the connections between them. This
repeated exposure brings out misconceptions, builds understanding, and (over
time) results in fluency.

I really don't see why the author has a bone to pick with common core since
the sort of practice she describes would fit perfectly into a common core
curriculum:

>I 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?

Common core (and contemporary education movements) are against "rote" or
"procedural" learning. They would be against making up a song to memorize
f=ma, and merely using that song to plug-and-chug through a small collection
of problem types.

One recent example I saw (a colleague works on coaching teachers in common
core) was a class of elementary students who could correctly multiply 4/7 *
5/9, but couldn't shade in 1/4 of a square. They memorized and rehearsed the
procedure for multiplication, but never built understanding of what they were
doing.

The unfortunate thing is that they are able to demonstrate fluency in this
skill - and they will likely score well on standardized tests as a consequence
of this fluency. This skill, however, is shallow - and will be easily
forgotten without continued practice. Furthermore, when the time comes to
learn proportional reasoning, or rates of growth, or any other thing that has
to do with fractions, they will have nothing to build their understanding on.

I have to make a concession to the author, however. It is easy to get this
impression of common core from the sidelines. Most teachers, departments, and
schools were dumped into the core (which is merely a set of standards) without
much support or training. Implementing the core requires a major shift in how
one approaches teaching, and whether it is due to a lack of understanding, a
lack of will, or most likely - a lack of resources, many classrooms are merely
cargo-culting the sorts of things that common core demands.

My favorite introductory book to the subject is
[https://amzn.com/0325052875](https://amzn.com/0325052875) happy to chat!

~~~
WillPostForFood
_Common core encourages repetition through its focus on multiple
representations._

Not sure if I'd say Common Core itself encourages any particular method,
nonetheless many modern math curricula take this approach. I have an
elementary age kid at a school using Everyday Math which is like this, and it
is pretty terrible. The homework is, like you say, a few problems of one type,
then jump to something else, then jump to something else. There is never
mastery of anything. Not basic operations, not coloring squares, not anything.
I'm not opposed to multiple representations, but there isn't enough focus or
repetition, so there ends up being no mastery.

While most of the kids could color 1/4 of a square and understand that
conceptually, if I said color 40/160 of a square, they'd freeze because they
don't work with numbers or basic operations enough to figure out that it is
the same %.

------
edtechdev
The title is completely false and not even the title of the essay.

But see the author's Learning How to Learn MOOC.

------
dredmorbius
Reading this, I'm finding a few elements matching my own learning process
critically missing.

Yes. Memorisation and repetition are useful. But of and by themselves, _they
are not enough_ , at least not for a deep understanding. That Ms. Oakley is a
linguist makes the omission particularly glaring to me: it's finding a
_systemic understanding_ , that is, _understanding the knowledge 's grammar_.

I've picked up a few skills over my life. Some music (poorly). A little bit of
foreign language -- not much, but enough in a couple to get by as a tourist.
Sport. A great deal of spatial knowledge. Some physics and economics, at uni.
Programming and systems administration, some data management and analysis.
More recently, synthesising numerous elements looking at questions of
sustainability, collapse, or various modes of splitting the difference.

Some knowledge is almost mechanical. Music, sport, spelling, multiplication
tables. X comes in, Y goes out. But simple repetition isn't fully sufficient
-- this is what a good coach, in maths, music, or sport, offers. They know
what you _should_ be doing, see what you _are_ doing, and then offer _the
cues_ necessary to get you to where you ought to be. The cues might themselves
not make much sense overtly, but are the adjustment necessary to reach the
desired result.

Practice _without_ that intervention, and focusing on the right cues, only
drills in the bad practices. And _unlearning_ non-useful patterns is
_exceptionally_ difficult.

I'm tempted to say that easy learners are all alike. At the very least, none
of them encounter the limits or barriers to learning (though it's possible
each has some particular fast track to results). It's when learning comes
_hard_ that it's crucial to identify where and what the fault is, and to
either correct it or bypass it.

In my current studies -- economics, political theory, ecology, energy,
systems, and more -- what I'm finding most useful is to cover a great deal of
ground, much of which is essentially circling a central problem sphere, but
giving views on it from different directions. I'm quite literally finding
myself re-acquainting myself with concepts, lessons, materials, and more, from
the past 40+ years, and both dis-integrating and re-integrating them. I've
described it as "refactoring my worldview" (mentioned on HN in a comment
recently, also at
[https://dredmorbius.reddit.com](https://dredmorbius.reddit.com)), more to
describe what the experience is like.

But the crucial element is _not_ simply to repeatedly encounter facts until
they're memorised, _it is to create the structure into which they naturally
fall._ Or at least that's what I've found.

Some "systems" are less systemic than others. Virtually all have at least some
path dependency, so history, law, politics, and literature will, in aggregate,
at least follow some sort of path of low energy, if not an entirely logical
route. In maths, logic, physics, chemistry, and electical engineering, the
structure is more overt.

Again: Oakley approaches this concept, but never quite gets there. I found
that disappointing.

~~~
sfink
Yes, this resonates with me. I thought the article lumped together multiple
different types of understanding, and attacked an "understanding first"
approach by using a very limited definition. From the specific examples in the
article, you can see that practice is useful to gaining a systemic
understanding. But there's a bit of a claim that _repetition_ leads to useful
understanding, which is simplistic. Juggling the terms of F=ma is not what I
would call repetition. It is _practice_ , certainly, but to me repetition
pretty strongly implies the _absence_ of tweaking things according to the
understanding you have so far. And that, I would assert, is a very slow and
ineffective way of building up knowledge and deep understanding. (It will
works, but only because our brains will get bored and sense patterns and start
inferring some level of understanding no matter what. Although I'd gamble that
a disinterested student who only needs to make it through all 50 problems on a
worksheet will be pretty successful in avoiding any learning.)

