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October 3, 2014 6:24 AM Subscribe

Time after time, professors in mathematics and the sciences have told me that building well-ingrained chunks of expertise through practice and repetition was absolutely vital to their succes Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

Nice post (though I'm not sure what the point of the random Russian word-forms in the title is); the general idea is true, and of course I loved the language examples. This is a great anecdote:posted by languagehat at 7:17 AM on October 3, 2014 [4 favorites]

And this is a summary of a number of Vasily Aksyonov stories from the '60s:
Much of what Oakley says is right, but her claim that understanding what's going on is simply going to happen by itself given enough practice is just wrong. Wrong for math, wrong for sports, wrong for language. She could have memorized an entire Russian-English dictionary back to front, known all the conjugations perfectly, and she wouldn't have known Russian -- not without actually spending time talking to Russians, reading Russian books -- understanding the language as an actual thing in the world, not as a set of rules. Math is much the same.

The idea of the Common Core, and all other post-1960 math curricula, isn't to get rid of computational fluency training and replace it with some kind of wispy "understanding." It's that we shouldn't

posted by escabeche at 7:36 AM on October 3, 2014 [6 favorites]

The idea of the Common Core, and all other post-1960 math curricula, isn't to get rid of computational fluency training and replace it with some kind of wispy "understanding." It's that we shouldn't

*settle*for fluency training, any more than we should settle for people taking "Spanish" whose knowledge consists of translating words one for one. Go to any elementary school classroom right now and you'll see students being tested on lists of addition and multiplication facts, being trained to do these problems quickly and correctly. And you'll also see exercises aimed at teaching them how to get information off a graph. Some people think it's too ambitious to ask our students to do it all. But how can we not try?posted by escabeche at 7:36 AM on October 3, 2014 [6 favorites]

There's a complex feedback relationship. It's oversimplifying even to reduce things to algorithm /drill/repetition on one hand vs. conceptual understanding on the other, but we'll start with that because it is an important and useful opposition. I've paid a lot of attention to how the two interact in my own learning and in my students' learning (and I'm a little unusual in that I have spent large amounts of time teaching math to university students as well as elementary school students).

I'm going to boil my thoughts down as much as possible: Neither one can come entirely before the other. Algorithms (such as, for example, multiplication of multidigit numbers or division of fractions) learned without any deeper understanding of what's represented and why the steps are undertaken in the way and order that they are will be retained poorly and used in a brittle, inflexible way, and students in this situation will have difficulty recognizing when an algorithm should be used unless it is explicitly spelled out for them.

On the other hand, students who have, say, a good conceptual understanding of multiplication (they can explain it in terms of groupings or repeated addition, represent it graphically, etc) but do not have fluency in computation will get bogged down in long problems - their lack of speed simply gives them too much time to get lost and lose attention during a multistep problem. "Long" problems might mean two steps. By the time Jessica has finished counting out the answer to the first step on her fingers, she has completely forgotten what she was going to use it for or what the second step in the problem was.

Experienced mathematicians working on problems flip back and forth a lot between "blind" calculation - just letting the algorithms take over for a while until you get to the next conceptual step - and planning/strategizing/conceptual thinking. Learners also have to go through these phases on a larger scale. Algorithms facilitate problem-solving. Algorithms can't be retained well, applied flexibly, or selected appropriately without conceptual understanding. The mere repetition of algorithms is

posted by Wolfdog at 7:40 AM on October 3, 2014 [18 favorites]

I'm going to boil my thoughts down as much as possible: Neither one can come entirely before the other. Algorithms (such as, for example, multiplication of multidigit numbers or division of fractions) learned without any deeper understanding of what's represented and why the steps are undertaken in the way and order that they are will be retained poorly and used in a brittle, inflexible way, and students in this situation will have difficulty recognizing when an algorithm should be used unless it is explicitly spelled out for them.

On the other hand, students who have, say, a good conceptual understanding of multiplication (they can explain it in terms of groupings or repeated addition, represent it graphically, etc) but do not have fluency in computation will get bogged down in long problems - their lack of speed simply gives them too much time to get lost and lose attention during a multistep problem. "Long" problems might mean two steps. By the time Jessica has finished counting out the answer to the first step on her fingers, she has completely forgotten what she was going to use it for or what the second step in the problem was.

Experienced mathematicians working on problems flip back and forth a lot between "blind" calculation - just letting the algorithms take over for a while until you get to the next conceptual step - and planning/strategizing/conceptual thinking. Learners also have to go through these phases on a larger scale. Algorithms facilitate problem-solving. Algorithms can't be retained well, applied flexibly, or selected appropriately without conceptual understanding. The mere repetition of algorithms is

*not*sufficient to add that understanding, and so there has to be a constant effort to pause and reflect. It's not a first thing and second thing. It's a constantly alternating flow.posted by Wolfdog at 7:40 AM on October 3, 2014 [18 favorites]

*Algorithms (such as, for example, multiplication of multidigit numbers or division of fractions) learned without any deeper understanding of what's represented and why the steps are undertaken in the way and order that they are will be retained poorly and used in a brittle, inflexible way, and students in this situation will have difficulty recognizing when an algorithm should be used unless it is explicitly spelled out for them.*

And my point there was, that remains true no matter how many times the process is repeated and no matter how quickly the student is able to carry it out.

posted by Wolfdog at 7:48 AM on October 3, 2014 [6 favorites]

Yep, I'm 100% with Wolfdog here. And what's frustrating is that this common-sense take is seldom presented in the media because it's not "punchy" and it doesn't "take a side."

posted by escabeche at 8:02 AM on October 3, 2014 [2 favorites]

posted by escabeche at 8:02 AM on October 3, 2014 [2 favorites]

I am available to punch people, if it will help.

posted by Wolfdog at 8:04 AM on October 3, 2014 [7 favorites]

posted by Wolfdog at 8:04 AM on October 3, 2014 [7 favorites]

I think the article refers to complex feedback relationship in the "chunking" part? I would love to be in a room with you guys and the author and hear it all. I'm pretty excited about the ideas in the article and above.

posted by drowsy at 8:24 AM on October 3, 2014 [1 favorite]

posted by drowsy at 8:24 AM on October 3, 2014 [1 favorite]

from comments at TFA:

Memorizing a programming language using spaced repetition software

Tips For Mastering A Programming Language Using Spaced Repetition

* rubs hands together *

posted by drowsy at 8:31 AM on October 3, 2014 [5 favorites]

In algorithmic terms, rote learning is just very thorough indexing. What the author is describing is closer to an exhaustive search or even heuristic optimization.That seems right. So I googled relevant to my interests and found these:

Memorizing a programming language using spaced repetition software

Tips For Mastering A Programming Language Using Spaced Repetition

* rubs hands together *

posted by drowsy at 8:31 AM on October 3, 2014 [5 favorites]

There is a third element missing, and that is imagination. One imagines how to do something before doing it. The concept of how to do the thing is influenced by experience/fluency ("how did I do this before") and by understanding of the core concepts ("how does this work in general"). This may seem common sense, but too often I have tried to do something complex without spending enough time "thinking it through" beforehand. And so in a rush to complete the thing, I end up spending more time and energy making and fixing mistakes in the implementation than if I had just thought it through first. This is why "day dreaming" is key to work (of a creative nature). It's a skill of its own that interplays with fluency and understanding.

posted by stbalbach at 8:41 AM on October 3, 2014 [4 favorites]

posted by stbalbach at 8:41 AM on October 3, 2014 [4 favorites]

Two things:

1. In a lot of higher math situations, there needs to be a light bulb moment where a lot of lower level knowledge suddenly merges into a unified core and you see how all of these disparate pieces fit together. Complex Analysis IMHO is a good example of a course where I think this happens. But, not every student can do this. When they can't, I think a lot of people spend time coming up with complicated explanations of why, pedagogically, it didn't happen when the simple answer is not everyone can learn higher level math.

2. I am always suspicious of the degree to which we really know what our students learned. I am shocked every semester by things that students apparently learned that I did not teach ( and which aren't true.) Students do a lot of "learning and purging", that is, learning it for the test then immediately forgetting it if they can. I hope my tests give me some idea of what my students learned from me, but I'm sure I'd be shocked at their grades on the same tests if they took them a year from now.

posted by wittgenstein at 9:01 AM on October 3, 2014 [4 favorites]

1. In a lot of higher math situations, there needs to be a light bulb moment where a lot of lower level knowledge suddenly merges into a unified core and you see how all of these disparate pieces fit together. Complex Analysis IMHO is a good example of a course where I think this happens. But, not every student can do this. When they can't, I think a lot of people spend time coming up with complicated explanations of why, pedagogically, it didn't happen when the simple answer is not everyone can learn higher level math.

2. I am always suspicious of the degree to which we really know what our students learned. I am shocked every semester by things that students apparently learned that I did not teach ( and which aren't true.) Students do a lot of "learning and purging", that is, learning it for the test then immediately forgetting it if they can. I hope my tests give me some idea of what my students learned from me, but I'm sure I'd be shocked at their grades on the same tests if they took them a year from now.

posted by wittgenstein at 9:01 AM on October 3, 2014 [4 favorites]

*Algorithms (such as, for example, multiplication of multidigit numbers or division of fractions) learned without any deeper understanding of what's represented and why the steps are undertaken in the way and order that they are will be retained poorly and used in a brittle, inflexible way, and students in this situation will have difficulty recognizing when an algorithm should be used unless it is explicitly spelled out for them.*

And yet I've seen generations of university calculus students unable to go through an algorithm that takes more than 2 steps. I'm totally on board with understanding mathematics, but as a mathematician and a teacher I've seen student after student coming into university calculus unable to solve basic problems because they perform algebraic calculations like a 1st year Russian student speaks Russian. And, if you drill down into it, the reason they can't do algebra is because they can't manipulate fractions, and they can't manipulate fractions because their ability to do arithmetic is entirely rote.

However, the problem isn't that they don't understand arithmetic. No amount of manipulating number cubes is going to help them get multiplication. The problem is that they have been taught, all the way, by people who don't have any comfort, much less understanding of basic mathematics. As a result, they've only done arithmetic one way. I've seen the results of so many pedagogical fads on student homework it gets sickening.

There's no royal road to mathematics. Understanding isn't something you can teach. Fluency comes from actually doing mathematics (even arithmetic), but if your teacher isn't fluent, no amount of central curriculum planning is going to change things.

*Yep, I'm 100% with Wolfdog here. And what's frustrating is that this common-sense take is seldom presented in the media because it's not "punchy" and it doesn't "take a side."*

"Common sense" is one of things, like "no true Scotsman" which should be banned in any discussion. Of course the thing you support is common sense.

posted by ennui.bz at 9:34 AM on October 3, 2014 [4 favorites]

Also, if you've taught mathematics outside of higher education, you'll notice that teachers who've been around awhile have this curious 'Maoist' paranoia when you talk to them about math. There have been so many curriculums, pedagogies, programs, re-education sessions, that they feel like they are always looking over their shoulder for the next ideological line. 5 years ago it was manipulatives, last year, back to basics, this year "Common Core." To maintain your position you have to be prepared to teach in what ever way is being pushed down, from above, this year.

Fundamentally, whether it's Common Core or something else, the people who actually run the US education system tend to think of teachers as "educators." That is, whatever curriculum you put in front of them, a good teacher will be able to implement it successfully. In this environment, paradoxically, feeling like you "understand" mathematics just gets in the way, because your "understanding" might not fit into the latest program.

posted by ennui.bz at 10:06 AM on October 3, 2014 [3 favorites]

Fundamentally, whether it's Common Core or something else, the people who actually run the US education system tend to think of teachers as "educators." That is, whatever curriculum you put in front of them, a good teacher will be able to implement it successfully. In this environment, paradoxically, feeling like you "understand" mathematics just gets in the way, because your "understanding" might not fit into the latest program.

posted by ennui.bz at 10:06 AM on October 3, 2014 [3 favorites]

*Of course the thing you support is common sense.*

Yes, good point. So delete that I said it was a "common sense" stance, and just keep the part where I say it's the stance I think is correct.

posted by escabeche at 10:34 AM on October 3, 2014 [1 favorite]

*2. I am always suspicious of the degree to which we really know what our students learned...*

posted by wittgenstein at 12:01 PM on October 3

posted by wittgenstein at 12:01 PM on October 3

eponysterical

posted by MisantropicPainforest at 10:53 AM on October 3, 2014 [2 favorites]

*Fluency comes from actually doing mathematics (even arithmetic), but if your teacher isn't fluent, no amount of central curriculum planning is going to change things.*

Our kids are now being taught by teachers whose own teachers grew up with television.

The results are predictable and were predicted.

posted by flabdablet at 11:14 AM on October 3, 2014 [2 favorites]

*In algorithmic terms, rote learning is just very thorough indexing. What the author is describing is closer to an exhaustive search or even heuristic optimization.*

Rule automation enhances one's ability to use the rules learned in novel ways.

posted by Jpfed at 11:57 AM on October 3, 2014 [1 favorite]

>

Why is it teachers always assume you didn't study? Spend the whole class learning one pattern of problem, and then the test has those damn tricky questions. As though insight is supposed to 'erupt' from repetition fully formed. Its like they're really testing if you have one of those 'lucky' brain mutation which allows you to remember everything you ever saw, but which also makes you hysterical around schnauzers.

Whatever. I was enjoying the article, despite its academic barbarisms, until it became a nostalgia trip or 'this is my life' piece. I prescanned the length of the article, so when she got to

Meh. I lost patience at,

Who is this article for?

posted by xtian at 4:09 PM on October 3, 2014

*"... not developed any kind of procedural fluency or ability to apply what he thought he understood."*Why is it teachers always assume you didn't study? Spend the whole class learning one pattern of problem, and then the test has those damn tricky questions. As though insight is supposed to 'erupt' from repetition fully formed. Its like they're really testing if you have one of those 'lucky' brain mutation which allows you to remember everything you ever saw, but which also makes you hysterical around schnauzers.

Whatever. I was enjoying the article, despite its academic barbarisms, until it became a nostalgia trip or 'this is my life' piece. I prescanned the length of the article, so when she got to

*"Continually focusing on understanding itself actually gets in the way."*, I thought the rest would explain the solution to adult learning. If its there I can't find it.Meh. I lost patience at,

*"I became a translator for the Russians on Soviet trawlers [...] a decent salary while getting drunk all the time."*Who is this article for?

posted by xtian at 4:09 PM on October 3, 2014

wittgenstein:

This is true of other disciplines too. My favourite part of university was one to four weeks after the end of the school year, my brain would start synthesizing all those bits and pieces into various revelations about what we'd been studying.

Yes, there's a whole bunch of other stuff involved, like reading, discussion, writing exams -- handwriting exams is a great way to force your mind to pull everything together and deal with it, even if it's ugly and stressful -- and time.

posted by sneebler at 8:54 AM on October 4, 2014

*In a lot of higher math situations, there needs to be a light bulb moment where a lot of lower level knowledge suddenly merges into a unified core and you see how all of these disparate pieces fit together.*This is true of other disciplines too. My favourite part of university was one to four weeks after the end of the school year, my brain would start synthesizing all those bits and pieces into various revelations about what we'd been studying.

*As though insight is supposed to 'erupt' from repetition fully formed.*Yes, there's a whole bunch of other stuff involved, like reading, discussion, writing exams -- handwriting exams is a great way to force your mind to pull everything together and deal with it, even if it's ugly and stressful -- and time.

posted by sneebler at 8:54 AM on October 4, 2014

I'm fumbling with trying to untangle three themes put forth by the author: a tale of the author's cognitive evolution, her broad thoughts on how we learn, and some general thoughts on how we ought to teach STEM disciplines.

I would love some more anecdotes from Russian trawlers, and the front lines of learning math as an adult. She outlines an interesting career path, both culturally and intellectually. Maybe a couple of "look at me!" style stories, always a bit incongruous in a piece when the author is trying to make a larger point, but I can look past that.

From those tales she brings out a "how we learn" thesis: fluency over understanding. Repetition and memorization are critical. More-or-less, I can agree with this.

Beyond that, I sense more than a little push towards saying that is the best approach in the classroom. Nope. I think it's been said far better by others in this thread: figuring out when to focus on repetition vs. big-picture concepts is an incredibly fluid process. In the classroom, you need to walk that knife-edge every day, every class period, to find the right spot on the "drill vs. discussion" spectrum to communicate this knowledge to these students.

I struggle to comprehend that folks think it is preferable -- or even possible! -- to codify the approach: "repeat this exercise X times, then move on to concept B, then practice the next exercise," etc etc.

Learning, educating: they are fluid processes. Methodologies, policies: they aren't. Can there be a push to highlight the fact that "education policy" is an oxymoron? (especially when policies are created by folks with more political allegiances than educational ones....)

posted by Theophrastus Johnson at 11:07 AM on October 4, 2014 [1 favorite]

I would love some more anecdotes from Russian trawlers, and the front lines of learning math as an adult. She outlines an interesting career path, both culturally and intellectually. Maybe a couple of "look at me!" style stories, always a bit incongruous in a piece when the author is trying to make a larger point, but I can look past that.

From those tales she brings out a "how we learn" thesis: fluency over understanding. Repetition and memorization are critical. More-or-less, I can agree with this.

Beyond that, I sense more than a little push towards saying that is the best approach in the classroom. Nope. I think it's been said far better by others in this thread: figuring out when to focus on repetition vs. big-picture concepts is an incredibly fluid process. In the classroom, you need to walk that knife-edge every day, every class period, to find the right spot on the "drill vs. discussion" spectrum to communicate this knowledge to these students.

I struggle to comprehend that folks think it is preferable -- or even possible! -- to codify the approach: "repeat this exercise X times, then move on to concept B, then practice the next exercise," etc etc.

Learning, educating: they are fluid processes. Methodologies, policies: they aren't. Can there be a push to highlight the fact that "education policy" is an oxymoron? (especially when policies are created by folks with more political allegiances than educational ones....)

posted by Theophrastus Johnson at 11:07 AM on October 4, 2014 [1 favorite]

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