The building blocks of understanding are memorization and repetition(nautil.us)
nautil.us
The building blocks of understanding are memorization and repetition
http://nautil.us/issue/40/learning/how-i-rewired-my-brain-to-become-fluent-in-math-rp
98 comments
A lot of people connect with concepts via story-telling. Life lessons aren't always learned by just consuming a stream of facts. And facts themselves aren't inherently valuable. A good story helps some readers filter what applies to them and what does not.
I recently picked up the book "How to Win Friends and Influence People". It's full of story telling, with very little fact, and the main thrusts of each chapter are summarized in a sentence. The bulk of the chapters is mostly just stories.
Overall this seems like a really effective strategy for the subject matter. It's easy to intellectually understand the summary sentences, but all the stories in the chapters are illustrative, and the idea is that spending time with the book helps reinforce the mindset that leads to the attitudes it espouses.
So that was a long winded way of saying: Good point.
I will say that a lot of long form journalism pieces these days seem overly focused on their narratives, and often, feel like time wasting to me. But I'm usually in information gathering mode when I read them. I don't want personal stories, I want facts, and links to research when I'm in that mindset. Maybe it just feels to me like they are too often considering the story they're telling as proof enough of their point without doing enough to support it otherwise.
Overall this seems like a really effective strategy for the subject matter. It's easy to intellectually understand the summary sentences, but all the stories in the chapters are illustrative, and the idea is that spending time with the book helps reinforce the mindset that leads to the attitudes it espouses.
So that was a long winded way of saying: Good point.
I will say that a lot of long form journalism pieces these days seem overly focused on their narratives, and often, feel like time wasting to me. But I'm usually in information gathering mode when I read them. I don't want personal stories, I want facts, and links to research when I'm in that mindset. Maybe it just feels to me like they are too often considering the story they're telling as proof enough of their point without doing enough to support it otherwise.
Agreed. I filed this story away as a way to explain math-centric learning to the more literary minded.
Especially in our modern time where our mind spends its time curating zillion of tiny messages to extract meaningful messages.
A few extra points to add:
- expanding on repeating: Enumerate and solve/memorize every permutation of a problem/theorem/etc. (ex: in f = ma, what happens if m is high and a is low and the reverse). Each case becomes a subroutine (or a chunk) in your brain that quickly fires once memorized.
- explaining your understanding of a topic does not translate to the ability of applying that understanding to an actual application
The writing was superfluous and can be summed up in 2-3 points, although enjoyable to read as a story if you're up for that.
I'm a bit disappointed because the title is misleading. The author didn't really rewire her brain. She just spent more time learning and set a different goal (of trying out every permutation instead of simply doing well on tests) than her classmates.
- expanding on repeating: Enumerate and solve/memorize every permutation of a problem/theorem/etc. (ex: in f = ma, what happens if m is high and a is low and the reverse). Each case becomes a subroutine (or a chunk) in your brain that quickly fires once memorized.
- explaining your understanding of a topic does not translate to the ability of applying that understanding to an actual application
The writing was superfluous and can be summed up in 2-3 points, although enjoyable to read as a story if you're up for that.
I'm a bit disappointed because the title is misleading. The author didn't really rewire her brain. She just spent more time learning and set a different goal (of trying out every permutation instead of simply doing well on tests) than her classmates.
Yes, the title is misleading and the author is verbose, but the gist of the story is corroborated :)
Not a big deal but was written by Barbara Oakley (woman) "As a young woman with a yen for learning language and no money or skills to speak of ..."
Her on line course: "Learning how to learn" is the most successful MOOC of all time: https://www.coursera.org/learn/learning-how-to-learn
I really enjoyed following it.
I really enjoyed following it.
She also wrote A Mind for Numbers: How to Excel at Math and Science (Even If
You Flunked Algebra) which is a good read.
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.
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?
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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
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
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 " (which has a link to the PDF).
Fair enough. Unfortunately, I can't edit the comment now.
Anybody experienced the opposite ? first time I was exposed to combinatorics everything felt natural and intuitive. The following lesson everything had become a blur.
QFT was my watershed the point where I really needed to learn, study, concentrate, apply, ... it wasn't at all intuitive to me.
Unfortunately I hadn't learnt to do any of those things; result was scraping a minimum pass.
To me it was like trying to picture n-dimensional hyper-cubes.
Unfortunately I hadn't learnt to do any of those things; result was scraping a minimum pass.
To me it was like trying to picture n-dimensional hyper-cubes.
This was my exact experience learning how to program at 24.
+1 for storytelling. This article is pretty much my life story, and I constantly tell that story to other people, because my own painful personal experience suggests our educational system does not do a good job of delivering the same message. Maybe tucaz doesn't need to hear it. But there are a lot, a lot, of people that do. How many times have you heard someone say "I'm just not good at math" or "I'll never be able to draw" or "I have two left feet, I can't dance"? It's just because they never learned how to learn.
I suggest that author's personal experience does not give any insight into large phenomena affecting many more lives than the author's own that could not otherwise and more succinctly come forward without the addition of the overly personal voice into the prose.
This style of delivering information is called gonzo journalism and has taken over a large part of the media. What people forget is that gonzo journalism was initially a form of entertainment. The kind of prose that serves telling the readers about the Kentucky Derby is not same that serves the conveying of information in a large number of categories. This is one of them.
This style of delivering information is called gonzo journalism and has taken over a large part of the media. What people forget is that gonzo journalism was initially a form of entertainment. The kind of prose that serves telling the readers about the Kentucky Derby is not same that serves the conveying of information in a large number of categories. This is one of them.
So you don't know any people that are mathphobes? You are right that it would be nice to have some hard science to back up the author's claims. Maybe we should read her book. I'm sure it's basically "Fear and Loathing in Calculus".
> TL;DR: Study foundational blocks by repeating then until you understand. Keep repeating as you build knowledge and move to "harder" things.
If I just read that, I'd file that away as "something someone once said on the internet," wouldn't give it much weight, and would forget about. I see dozens of competing one-liners like that every day.
The story made the message much stronger, made me believe it, and will make me remember it.
If I just read that, I'd file that away as "something someone once said on the internet," wouldn't give it much weight, and would forget about. I see dozens of competing one-liners like that every day.
The story made the message much stronger, made me believe it, and will make me remember it.
Thanks for the TL;DR the topic was interesting but I was tired for read more after all day.
My grandma used to say "brain is a muscle".
And I used to see it as an over-simplification, because I was studying, etc...
With the time, I've come to a similar conclusion, a different muscle, and with a micro-services architecture, but at the end of the day, it inherits the attributes, methods and behavior, of the "muscle" class.
My grandma used to say "brain is a muscle".
And I used to see it as an over-simplification, because I was studying, etc...
With the time, I've come to a similar conclusion, a different muscle, and with a micro-services architecture, but at the end of the day, it inherits the attributes, methods and behavior, of the "muscle" class.
> at the end of the day, it inherits the attributes, methods and behavior, of the "muscle" class
Really, it inherits from the general "biological system" class, of which "muscle" is the common subclass referenced, since it's the one most people have familiarity in utilizing.
The differentiation of "muscle" from other "biological system" classes is entirely unrelated to the "use-it-or-lose-it" feature, however, and your subclassing is inappropriate.
While this seems like being pedantic, I'd argue it's important to know when we're using a member to exemplify a class versus when we're talking about the definition of a class.
The same use-it-or-lose-it applies to everything from lung capacity to stomach size to hormone production by organs, and is a good principle to know about how biology works. The emphasis on "muscle" as the class (rather than an exemplifying member) hides this underlying truth about all biological systems -- that they dynamically adapt to efficiency.
Really, it inherits from the general "biological system" class, of which "muscle" is the common subclass referenced, since it's the one most people have familiarity in utilizing.
The differentiation of "muscle" from other "biological system" classes is entirely unrelated to the "use-it-or-lose-it" feature, however, and your subclassing is inappropriate.
While this seems like being pedantic, I'd argue it's important to know when we're using a member to exemplify a class versus when we're talking about the definition of a class.
The same use-it-or-lose-it applies to everything from lung capacity to stomach size to hormone production by organs, and is a good principle to know about how biology works. The emphasis on "muscle" as the class (rather than an exemplifying member) hides this underlying truth about all biological systems -- that they dynamically adapt to efficiency.
Username checks out.
Some people extoll the virtues of long-form writing. But even a site like HN is far too information dense to spend so much time reading one article. It's a different matter when we're learning something of an academic nature, of course. That's why I love summaries such as yours.
There's something to said about attention spans in the digital age. Nautilus isn't a single serving news site; it provides long essays with sufficient background to digest a topic you probably don't really spend much time thinking about otherwise.
They aren't trying to give you a single sentence summary of some result. They are trying to get you to care about some esoteric subject. This one just happens to gel with the ycombinator crowd.
They aren't trying to give you a single sentence summary of some result. They are trying to get you to care about some esoteric subject. This one just happens to gel with the ycombinator crowd.
I completely agree. I have found a speed reading extension to be quite valuable in getting a quick overview of an article. The speed reading time for this article was 5 minutes at 600 wpm (practiced up from 300 wpm). The speed read is good enough to get the TL;DR and a little more depth about the article.
This is the extension I use (it lets you rewind with k and pause with space -- helpful for getting more understanding out):
https://chrome.google.com/webstore/detail/spreed-speed-read-...
This is the extension I use (it lets you rewind with k and pause with space -- helpful for getting more understanding out):
https://chrome.google.com/webstore/detail/spreed-speed-read-...
Have you tried text-to-speech software? The built-in one for Mac OS X is quite good (Alex, the default voice is not annoying).
I set the words per minute to a very high rate, and I get my hacker news while exercising or doing chores around the house. I've completely stopped reading long articles, except when there are intersperced code samples.
I highly recommend.
I set the words per minute to a very high rate, and I get my hacker news while exercising or doing chores around the house. I've completely stopped reading long articles, except when there are intersperced code samples.
I highly recommend.
While I tend to agree with the thesis, personal anecdote does not make a valid conclusion.
We probably have decent research on this at this point. Anybody versed in it weighing it?
We probably have decent research on this at this point. Anybody versed in it weighing it?
The world-class neuroscientist with whom she teaches a free online class as mentioned at the bottom of the piece clearly agrees with her.
Not surprising as she didn't come up with the ideas herself. She stumbled on the underlying principles and did research like anyone else on the terminology scientific experts in learning use to talk about those principles.
Not surprising as she didn't come up with the ideas herself. She stumbled on the underlying principles and did research like anyone else on the terminology scientific experts in learning use to talk about those principles.
> The world-class neuroscientist with whom she teaches a free online class as mentioned at the bottom of the piece clearly agrees with her.
Yet neither of these folks seem to be doing primary research in the particular area they are teaching the course in.
Oakley's peer-reviewed resume on this is quite scarce: http://www.barbaraoakley.com/articlesandreviews.html
Sejnowski is similarly light: http://www.salk.edu/scientist/terrence-sejnowski/publication...
Are we going to wind up with another "10,000 hours" debacle?
Yet neither of these folks seem to be doing primary research in the particular area they are teaching the course in.
Oakley's peer-reviewed resume on this is quite scarce: http://www.barbaraoakley.com/articlesandreviews.html
Sejnowski is similarly light: http://www.salk.edu/scientist/terrence-sejnowski/publication...
Are we going to wind up with another "10,000 hours" debacle?
Being an expert in something doesn't require being an author of a peer-reviewed journal article on the subject.
Journal articles are written to communicate information to those who couldn't perform the experiment/research themselves, either because they have their own research areas (overlapping or not) or because they simply aren't researchers.
No one person is going to have enough depth and breadth to satisfy your requirements. Not even textbook authors would meet them.
Scientists, as part of their training, learn to quickly dissect papers written by others and assimilate the information to the point where they can often make modifications and create experiments that improve on the original or answer a related question not addressed by the first. This is not dissimilar to the way that programmers can take a program written by another and modify/extend it.
You're weirdly focused on this "argument from ultimate authority" instead of simply evaluating the claims and the data that support them. Googling a few of the terms in the article that you're unfamiliar with would be a more useful way to spend your time.
Though first, if you're going to make an informed criticism of a scientist's publication record, you might do well to have enough experience with searching for publications to know that a world-class scientist doesn't always have all of his publications on his website. That was a highlight reel of decades of research [0].
[0] https://www.ncbi.nlm.nih.gov/pubmed?term=Sejnowski%20TJ
Journal articles are written to communicate information to those who couldn't perform the experiment/research themselves, either because they have their own research areas (overlapping or not) or because they simply aren't researchers.
No one person is going to have enough depth and breadth to satisfy your requirements. Not even textbook authors would meet them.
Scientists, as part of their training, learn to quickly dissect papers written by others and assimilate the information to the point where they can often make modifications and create experiments that improve on the original or answer a related question not addressed by the first. This is not dissimilar to the way that programmers can take a program written by another and modify/extend it.
You're weirdly focused on this "argument from ultimate authority" instead of simply evaluating the claims and the data that support them. Googling a few of the terms in the article that you're unfamiliar with would be a more useful way to spend your time.
Though first, if you're going to make an informed criticism of a scientist's publication record, you might do well to have enough experience with searching for publications to know that a world-class scientist doesn't always have all of his publications on his website. That was a highlight reel of decades of research [0].
[0] https://www.ncbi.nlm.nih.gov/pubmed?term=Sejnowski%20TJ
I don't think this is the right metric for weighing these claims. As a mathematician, I do not do primary research in calculus or linear algebra...
...but teaching those classes pays the bills.
...but teaching those classes pays the bills.
But you do not claim to be an expert on teaching or to have "The BEST Way To Learn Linear Algebra".
... the author wrote a book on the topic, and teaches a fairly famous course on it. I'd hope you'd give those credentials at least as much weight as some HN self-proclaimed expert.
[deleted]
> ... the author wrote a book on the topic, and teaches a fairly famous course on it.
Sadly, that doesn't automatically mean anything from a correctness perspective.
Educational methodology, in particular, tends to be absolutely rife with fads more than fact and immediately sets off my alerts.
This is doubly problematic when someone touts how popular their course is instead of how effective it is.
Sadly, that doesn't automatically mean anything from a correctness perspective.
Educational methodology, in particular, tends to be absolutely rife with fads more than fact and immediately sets off my alerts.
This is doubly problematic when someone touts how popular their course is instead of how effective it is.
Context: You disparage someone as simply touting their personal anecdote, and ask if anyone around here is a "real" expert.
If someone had replied "why yes, actually I just wrote a book on the topic, and I teach a huge course on it," would you have accepted that person as being "versed" in the topic?
Can we at least agree that the author is actually more qualified than you seem to give credit for in your original post?
If someone had replied "why yes, actually I just wrote a book on the topic, and I teach a huge course on it," would you have accepted that person as being "versed" in the topic?
Can we at least agree that the author is actually more qualified than you seem to give credit for in your original post?
See also authors of dieting fads. Popularity doesn't make you right [nor wrong].
There is a whole field in cognitive psychology, cognitive learning theory, devoted to this topic. One element of learning theory that may be of interest to HN readers is their notion of a "schema" (not an RDBMS schema). Schemas are the networked data structures in the brain (based ultimately on the underlying physical brain structure--neurons, regions, etc.) There is some interesting work in how schemas are acquired, primarily through imitation and repetition (which is what you would expect when training a neural network).
Yeah, advice I heard from parents and teachers countless times growing up.
"Memorize this." "Don't quit." "Keep trying."
Then I had to repeat verbatim whatever it was I was supposed to memorize, whether a reading selection, scales on an instrument, or multiplication tables. I didn't get to move forward till I got it. And I didn't get to choose what I studied.
Seems like we need more of this today.
"Memorize this." "Don't quit." "Keep trying."
Then I had to repeat verbatim whatever it was I was supposed to memorize, whether a reading selection, scales on an instrument, or multiplication tables. I didn't get to move forward till I got it. And I didn't get to choose what I studied.
Seems like we need more of this today.
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.
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.
Interesting that you should use the phrase "I went to the store"... I took Russian for two years in college, and, while I can get around Moscow alright, ask for directions, and exchange a few sentences of banter with Russian speakers I meet at parties, I'm hardly fluent. But the phrase "I went to the store" could be constructed in a HUGE number of ways in Russian, as there is no word in Russian that simply means "went" or "go" like we have in English.
Did you go to the store and come back? Was it a unidirectional journey? Do you go to the store often? Did you drive to the store? Did you walk to the store? Did you go inside the store? Did you obtain the endpoint of store as a goal?
The phrase "<foo> went to <bar>" can be (and, indeed must be -- no easy outs) conjugated and declined in so many freaking ways in Russian, with subtle meanings encoded in those conjugations and declensions, it's nuts.
I used to be great at the verbs of motion a long time ago, but, because I haven't been religiously practicing and thinking about them, I just avoid them altogether at this point. (The last time I tried saying "Are you driving to Boston?" I accidentally said "Are you wandering around aimlessly in Boston by car?")
My point though, is that I would argue that there is a very good reason to think deeply about the sentence "I went to the store" in Russian, and think deeply about all of the ways you can say it, modify it, and replace its component parts to achieve different meanings, and how those meanings relate to the meanings of other sentences containing the same component parts. I certainly wish I had done that more often!
Did you go to the store and come back? Was it a unidirectional journey? Do you go to the store often? Did you drive to the store? Did you walk to the store? Did you go inside the store? Did you obtain the endpoint of store as a goal?
The phrase "<foo> went to <bar>" can be (and, indeed must be -- no easy outs) conjugated and declined in so many freaking ways in Russian, with subtle meanings encoded in those conjugations and declensions, it's nuts.
I used to be great at the verbs of motion a long time ago, but, because I haven't been religiously practicing and thinking about them, I just avoid them altogether at this point. (The last time I tried saying "Are you driving to Boston?" I accidentally said "Are you wandering around aimlessly in Boston by car?")
My point though, is that I would argue that there is a very good reason to think deeply about the sentence "I went to the store" in Russian, and think deeply about all of the ways you can say it, modify it, and replace its component parts to achieve different meanings, and how those meanings relate to the meanings of other sentences containing the same component parts. I certainly wish I had done that more often!
Didn't this turn out a bit of a straw man? As far as I understand, she advocates an understanding of the concepts complemented by some appropriate type of repetition.
There is also a third concept: That of imagining. She speaks specifically of one equation in classical mechanics, which she played around with in her mind until it became second nature.
Of course, this is based on her experience. It appeals to me based on my own experience, yet it isn't a showcasing of scientific rigour within pedagogics.
There is also a third concept: That of imagining. She speaks specifically of one equation in classical mechanics, which she played around with in her mind until it became second nature.
Of course, this is based on her experience. It appeals to me based on my own experience, yet it isn't a showcasing of scientific rigour within pedagogics.
> yet it isn't a showcasing of scientific rigour within pedagogics.
I think this is exactly the correct criticism of this and most other critiques of mathematics curricula. It's ironic to see a huge conversation about mathematics pedagogy that never goes and looks at the data.
I think this is exactly the correct criticism of this and most other critiques of mathematics curricula. It's ironic to see a huge conversation about mathematics pedagogy that never goes and looks at the data.
> 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".
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".
> 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.
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.
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.
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.
> ... 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.
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.
> 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.
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.
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
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
> Nonsense comparison
Uh-huh. From the original post you're responding to:
I think the error arises from making an overly strong identification between learning a foreign language and learning mathematics
i.e. the parent's criticism is that the author of the article makes exactly the mistake you're claiming parent made...
Uh-huh. From the original post you're responding to:
I think the error arises from making an overly strong identification between learning a foreign language and learning mathematics
i.e. the parent's criticism is that the author of the article makes exactly the mistake you're claiming parent made...
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.
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.
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.
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
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...
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...
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.
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?
Is it the specific mechanics of orthography that you SD as valuable?
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.
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.
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...
>my brain seems to require more repetition to acquire the same knowledge that my daughter picks up near instantly. //
Perhaps you're hitting storage limitations?
Hypothesis: you need more repetitions to tell your brain to store the info in a more accessible manner, like more cache-hits indicating the data should be placed in a higher level cache?
Additional hypothesis: without wiping earlier memories the brain needs to rearrange how things are stored - maybe recategorising, re-"compressing" - in which case new knowledge would need to be for in to a right framework, conceptually, and possibly placed physically to optimise it's storage.
I'm imaging the brain like a spider-diagram - when you start you can easily find a space to draw a new concept and the lines connecting it. As you go on that becomes more difficult.
Now to work on creating an experiment to test my wild hypothesising!
Perhaps you're hitting storage limitations?
Hypothesis: you need more repetitions to tell your brain to store the info in a more accessible manner, like more cache-hits indicating the data should be placed in a higher level cache?
Additional hypothesis: without wiping earlier memories the brain needs to rearrange how things are stored - maybe recategorising, re-"compressing" - in which case new knowledge would need to be for in to a right framework, conceptually, and possibly placed physically to optimise it's storage.
I'm imaging the brain like a spider-diagram - when you start you can easily find a space to draw a new concept and the lines connecting it. As you go on that becomes more difficult.
Now to work on creating an experiment to test my wild hypothesising!
> 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.
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.
>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?
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?
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.
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...
seems many did "pivot" from competitive memorization and repetition to some lovely mathmakers.
Although https://en.wikipedia.org/wiki/List_of_International_Mathemat...
seems many did "pivot" from competitive memorization and repetition to some lovely mathmakers.
You don't get to the IMO by mere memorization.
Pattern matching is huge contributor to solving problems fast or even having the problems reachable.
Just checkout these results 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!
Just checkout these results 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!
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.
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.
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
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
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!)
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!)
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).
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.
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.
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.
Do they excel under the current one?
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.
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.
Rule #5: Repeat to remember. http://www.brainrules.net/short-term-memory
Rule #6: Remember to repeat. 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
Rule #6: Remember to repeat. 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
If this is true, then the spaced repitition learning technique is the best there's for learning.
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.
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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?
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?
You develop new ideas by connecting old facts. The better you're able to recall old facts, the better you're able to connect them to synthesize new ideas.
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?
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.
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.
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?
"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?
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?
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)
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)
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.
And is perhaps a reason mathematics seems to simply and "unreasonably powerfully" explain the world - it's not simple.
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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.
Schools need to be adjusted for this, I think that currently schools are based on the rationalist point of view.
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 happy to chat!
>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 happy to chat!
>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 author's argument is that you build understanding by doing. learning the forms and perfecting your "doing" of them is fundamental to developing insights into what is going on
some students make the leap themselves, some students need it explained to them, either way, the student needs to know the HOW before they can understand the WHY
the author's argument is that you build understanding by doing. learning the forms and perfecting your "doing" of them is fundamental to developing insights into what is going on
some students make the leap themselves, some students need it explained to them, either way, the student needs to know the HOW before they can understand the WHY
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 %.
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 %.
The title is completely false and not even the title of the essay.
But see the author's Learning How to Learn MOOC.
But see the author's Learning How to Learn MOOC.
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), 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.
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), 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.
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.)
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.