Thanks for the article. Your comment on common core math being developed by absconding with the smart kids' shortcuts and trying to force those shortcuts on everyone--regardless of innate ability or learning style--makes a lot of sense.
The more examples I see of common core math, the more thankful I am that I graduated before that became a thing. I love math, and I've always been good at it (including at a university level; I have a STEMmy degree that required calculus, differential equations, and linear algebra). I can see what common core math is trying to do, but I expect it would have felt confusing and arbitrary if I was taught that method with none of my own experience or intuition to fall back on. I don't know if this is a personal quirk or a general rule, but I've always found that rote memorizing the basics (be it multiplication tables, phonics, or foreign alphabets) is crucial to understanding the more difficult areas of each subject.
I'm about the opposite from you - I can learn math if it's necessary to do something, but I don't enjoy it, and I usually forget what it is as soon as I don't need it anymore (the example I use with my wife is that I've forgotten what a standard deviation is about ninety times in ten years).
I agree with you on rote being more valuable than people make it out to be. One thing that I don't think I did a good job of explaining here is why I think rote is a high standard for common core to overcome in the first place. Rote has a huge advantage I don't actually really believe can be overcome; basically, it teaches people who can't be advanced mechanical ways to "cheat" to the right answer, and with the same method teaches people who can be advanced that "cheat" as well as getting them close enough to "number sense" that they find it themselves.
My big worry on common core is that what seems to be happening is happening - i.e. that we aren't getting the slower kids rote capabilities as well, and that the number sense thing isn't teachable so much as it is a result of being smart. If that's the case it's worst-of-all-worlds. I hope I'm wrong, actually, because that's pretty bad if true.
I've always personally preferred the teaching style where the teacher explains the theory first, then works through lots of example problems to teach the rote solution. I feel like that gives a good balance: the kids who naturally "get it" understand why the rote solution works (and can develop shortcuts at their leisure) and the kids who don't naturally "get it" will at least know a practical solution to the problem if they ever need it. My teachers did this for everything from basic addition/subtraction to algebra to 3 dimensional integration. One of the few things they didn't do this on is logs, and my understanding of what logs actually ARE still suffers for it.
Another potential problem with common core math: different kids will come up with different shortcuts. How are they determining which shortcuts to teach? That Apple Hill problem has me scratching my head. To me, the obvious shortcut is to subtract 300 and then add 7. They're making it needlessly complicated and I don't understand why.
Rote learning has another advantage - the repetition ingrains it to the student. It's like rote repetition builds a foundation of granite to start building a fancy house on. Sure, you may understand how simplifying polynomials or representing them differently works, but if you always refer to reference books or cheatsheets to do it, you'll likely be lost when solving a harder problem while someone who's done a lot of rote practice has a practiced ease to it, and they can focus on the big thing. Rote practice moves things out of System 2 and more to System 1 -ish processing, and keeping the mental stack clean is essential to high performance. Musicians practice by rote a shitton and the result is they attain a fluidity with their tool and can make it sing effortlessly. They don't do the math or write the notes, things just feel right and sure enough, they sound nice.
Even on the European side of the pond, I find that teachers stress understanding the principles above all else far, far too much. It's sexier than old fashioned rote learning, but creates reference material addled, clumsy practicioners. Rote creates a foundation for actual mastery. Doing things by rote just sucks, sadly.
I think the problem comes back to the fact that we've never really addressed the question of 'why is studying math valuable'. And while opinions of this will obviously vary, my own is that math is important because of its ability to train your brain to approach problems in a very specific way, and that way is transferable to just about every discipline/pursuit there is.
What does math teach you? I think it boils down to two things.
1) It teaches you how to train the entirety of your attention on a single problem for > 30 seconds.
2) It teaches you that becoming frustrated will destroy your ability to solve a problem.
The ability to avoid getting frustrated, and to focus on a single problem for more than 30 seconds, are just enormously valuable skills that are worth cultivating-- and they apply to everything from analyzing stock values to glazing windows. Moreover, you can cultivate those skills via math regardless of what 'method' your using to solve the problem. If you believe that math is important because of how it trains your mind, and not because of the earning potential of the people who master it, then a lot of these questions about 'how to teach it' start looking irrelevant.
As an academic myself, I feel like that the last quote from the paper is the standard "our data might be incorrect so take the findings with a grain of salt" disclaimer that has to appear in every paper nowadays, (otherwise reviewers are quick to point out that the data might be incorrect!), rather than an attempt to spin results in their own way.
My premise there was based on them saying "These are reason why our common-core-is-bad-for-kids results might not be true" and then listing "this gets worse the longer common core exists" as one of those reasons - it's a conclusion counter to their own argument.
I do acknowledge that this is basically their limitations section, though, so besides that heading there's not really a problem. I'll even do you one better - it could be they were just being tactical, not because of the limitations of their study but because they were afraid to publish the papers in the first place and needed to soften it as much as possible. I'm suspicious there are other papers just like this one sitting in desk drawers for that reason - there is suspiciously little published on this subject for how important it is.
Thanks for that! Regarding 3, my wife said about the same thing - I wasn't stressed, but I think both of you are right that I need to slow down the pace.
Air Traffic Control is another domain where "not every skill can be taught to every kind of person, and sometimes it’s harmful to try" applies.
Lots of people get 80 to 90% of the way, but can never make it over the last hurdle. All the extra training and studying in the world doesn't help at this point. And continuing training beyond a certain point results in stress, exhaustion, added costs, etc.
I've always wondered though if changing the training structure would yield a higher number of controllers (for example, say 3 in 10 versus 1 in 10). In other words, even though some people just can't be taught the skills needed, some people could probably still make it but don't because the training program falls short.
I hesitate to name one skill since that oversimplifies too much, but maybe it can best be described as "predicting conflicts and taking effective action". "Spacial ability" is part of it, but it doesn't capture the component of making plans and executing plans under tight time constraints. People can do this up to a certain level of busyness, but fail once their capacity is exceeded.
Common core focused on reading too, though, which kinda undermines your argument about STEM. It wasn't only a math thing, the math thing simply became a culture war rallying cry for the right.
I didn't really address the reading side in this article - as it says directly after the intro section, the article is specifically about common core math. Common core reading might be great - I haven't really looked into it as heavily. But even if the reading half of it was sent down from Olympias as a gift to man, I'm not sure that effects the math half much.
The STEM argument is similarly purely about the math half - it seems to me they constructed CC math in such a way that if it worked how they wanted, they'd be able to convert math-hating kids into data analyst/physicist/engineer types. I actually think that's fine, provided it works. Or even fine if it doesn't work, so long as it doesn't then hurt their ability to do simple math. The worry is that it does hurt that - the current data I can find indicates scores are dropping rather than improving; the hope I hold is that I'm wrong and that pattern doesn't continue.
For a contrary example of a new approach leading to better results, teaching foreign languages by mostly feeding people massive amounts of comprehensible input and letting the brain's pattern-matching machinery do the heavy lifting works much better than traditional language classes.
Thanks for the article and podcast. I found it super useful and thought you did an excellent job! Practical feedback would be: the backing music isn't needed for the entire podcast - I listen to podcasts on 1.8x speed and the music can be quite distracting and doesn't add to what you're saying; when describing pictures I feel that if you're better cutting your losses and just saying look at the post to understand as "badly" describing it is probably worse than not describing it at all (not saying you were bad this time, it's just something that could happen); when describing an external link (just saying 'link in post' automatically might help maintain the flow); linking directly to the post in the episode description would be handy. Otherwise I hope you keep it up, definitely would mean I would engage with more of your stuff if you kept this up.
I'm mostly with you, with a few quibbles. First, thank you for the exposition. The weird-ass subtraction example actually seemed really analogous to long division -- call it long subtraction. The mildly racist poster you called "long division" is actually <i>short</i> division, the difference being that in short division you always guess right about what the next digit is, while in long division you can try 6 and subtract out whatever that gives you and then try 2 and then if you have to even 1, and add it all up at the end. So maybe the theory is that "long subtraction" is actually easier for a slow student than learning the shortest-time algorithm with borrowing and all that jazz, not that the manipulation is supposed to make them comfortable with numbers.
Dunno, I've never read any of the educational theory behind common core, or any educational theory for that matter.
One of the other problems with common core is that parents aren't able to tutor their children in math, because they themselves haven't learned the new techniques. This has the effect of cementing teachers as the gatekeepers of educational success. We should be making it easier for parents to take an interest in their child's education, and making parental tutoring more streamlined and normal.
I enjoyed this, but would have been interested for more discussion of the theory behind common core math from the perspective of its proponents. I agree with your critique of the one line of pro-common core thinking you discussed, but am curious how much of the argument for common core is driven by that line of thinking.
(Of course, that's a lot to ask from a blog post - this was already good!)
Of course the narration is useful; I don’t have to be sitting in front of and staring at the text every second in order to be consuming the article.
BUT why are you using “tick” in this sense up there in the text...isn’t it really a Britishism? (The Internet-Age proliferation of Britishisms among Americans is my pet peeve. And why, you ask. It may have something to do with me being a crank.)
It’s in the sentence “the logical solution is to find boxes successful student A ticked that successful student B didn’t, and make them tick the same boxes.”
Thanks for the article. Your comment on common core math being developed by absconding with the smart kids' shortcuts and trying to force those shortcuts on everyone--regardless of innate ability or learning style--makes a lot of sense.
The more examples I see of common core math, the more thankful I am that I graduated before that became a thing. I love math, and I've always been good at it (including at a university level; I have a STEMmy degree that required calculus, differential equations, and linear algebra). I can see what common core math is trying to do, but I expect it would have felt confusing and arbitrary if I was taught that method with none of my own experience or intuition to fall back on. I don't know if this is a personal quirk or a general rule, but I've always found that rote memorizing the basics (be it multiplication tables, phonics, or foreign alphabets) is crucial to understanding the more difficult areas of each subject.
I'm about the opposite from you - I can learn math if it's necessary to do something, but I don't enjoy it, and I usually forget what it is as soon as I don't need it anymore (the example I use with my wife is that I've forgotten what a standard deviation is about ninety times in ten years).
I agree with you on rote being more valuable than people make it out to be. One thing that I don't think I did a good job of explaining here is why I think rote is a high standard for common core to overcome in the first place. Rote has a huge advantage I don't actually really believe can be overcome; basically, it teaches people who can't be advanced mechanical ways to "cheat" to the right answer, and with the same method teaches people who can be advanced that "cheat" as well as getting them close enough to "number sense" that they find it themselves.
My big worry on common core is that what seems to be happening is happening - i.e. that we aren't getting the slower kids rote capabilities as well, and that the number sense thing isn't teachable so much as it is a result of being smart. If that's the case it's worst-of-all-worlds. I hope I'm wrong, actually, because that's pretty bad if true.
I've always personally preferred the teaching style where the teacher explains the theory first, then works through lots of example problems to teach the rote solution. I feel like that gives a good balance: the kids who naturally "get it" understand why the rote solution works (and can develop shortcuts at their leisure) and the kids who don't naturally "get it" will at least know a practical solution to the problem if they ever need it. My teachers did this for everything from basic addition/subtraction to algebra to 3 dimensional integration. One of the few things they didn't do this on is logs, and my understanding of what logs actually ARE still suffers for it.
Another potential problem with common core math: different kids will come up with different shortcuts. How are they determining which shortcuts to teach? That Apple Hill problem has me scratching my head. To me, the obvious shortcut is to subtract 300 and then add 7. They're making it needlessly complicated and I don't understand why.
I think all decent people can agree that the apple hill problem is bullshit.
Answer 1: Maybe not; see my comment about "long subtraction".
Answer 2: Yes, of course. I mean it's subtraction; surely that's within reach of nearly every student.
Rote learning has another advantage - the repetition ingrains it to the student. It's like rote repetition builds a foundation of granite to start building a fancy house on. Sure, you may understand how simplifying polynomials or representing them differently works, but if you always refer to reference books or cheatsheets to do it, you'll likely be lost when solving a harder problem while someone who's done a lot of rote practice has a practiced ease to it, and they can focus on the big thing. Rote practice moves things out of System 2 and more to System 1 -ish processing, and keeping the mental stack clean is essential to high performance. Musicians practice by rote a shitton and the result is they attain a fluidity with their tool and can make it sing effortlessly. They don't do the math or write the notes, things just feel right and sure enough, they sound nice.
Even on the European side of the pond, I find that teachers stress understanding the principles above all else far, far too much. It's sexier than old fashioned rote learning, but creates reference material addled, clumsy practicioners. Rote creates a foundation for actual mastery. Doing things by rote just sucks, sadly.
I think the problem comes back to the fact that we've never really addressed the question of 'why is studying math valuable'. And while opinions of this will obviously vary, my own is that math is important because of its ability to train your brain to approach problems in a very specific way, and that way is transferable to just about every discipline/pursuit there is.
What does math teach you? I think it boils down to two things.
1) It teaches you how to train the entirety of your attention on a single problem for > 30 seconds.
2) It teaches you that becoming frustrated will destroy your ability to solve a problem.
The ability to avoid getting frustrated, and to focus on a single problem for more than 30 seconds, are just enormously valuable skills that are worth cultivating-- and they apply to everything from analyzing stock values to glazing windows. Moreover, you can cultivate those skills via math regardless of what 'method' your using to solve the problem. If you believe that math is important because of how it trains your mind, and not because of the earning potential of the people who master it, then a lot of these questions about 'how to teach it' start looking irrelevant.
As an academic myself, I feel like that the last quote from the paper is the standard "our data might be incorrect so take the findings with a grain of salt" disclaimer that has to appear in every paper nowadays, (otherwise reviewers are quick to point out that the data might be incorrect!), rather than an attempt to spin results in their own way.
My premise there was based on them saying "These are reason why our common-core-is-bad-for-kids results might not be true" and then listing "this gets worse the longer common core exists" as one of those reasons - it's a conclusion counter to their own argument.
I do acknowledge that this is basically their limitations section, though, so besides that heading there's not really a problem. I'll even do you one better - it could be they were just being tactical, not because of the limitations of their study but because they were afraid to publish the papers in the first place and needed to soften it as much as possible. I'm suspicious there are other papers just like this one sitting in desk drawers for that reason - there is suspiciously little published on this subject for how important it is.
Great article!
I have a few comments about the audio narration, since you asked for feedback:
1. You have a nicely crisp and distinct voice.
2. I gave up on listening once you got to the memes and I realized I could just go take a look at them for myself.
3. A more relaxed approach to reading the article might be better: you sounded pretty stressed out and I don't think that's what you were going for.
Same, I stopped to look at the meme and then finished reading.
Well, that's final: no more fun memes and videos!
Thanks for that! Regarding 3, my wife said about the same thing - I wasn't stressed, but I think both of you are right that I need to slow down the pace.
Air Traffic Control is another domain where "not every skill can be taught to every kind of person, and sometimes it’s harmful to try" applies.
Lots of people get 80 to 90% of the way, but can never make it over the last hurdle. All the extra training and studying in the world doesn't help at this point. And continuing training beyond a certain point results in stress, exhaustion, added costs, etc.
I've always wondered though if changing the training structure would yield a higher number of controllers (for example, say 3 in 10 versus 1 in 10). In other words, even though some people just can't be taught the skills needed, some people could probably still make it but don't because the training program falls short.
What skill is it they can't get? Is it just a stress management thing?
I hesitate to name one skill since that oversimplifies too much, but maybe it can best be described as "predicting conflicts and taking effective action". "Spacial ability" is part of it, but it doesn't capture the component of making plans and executing plans under tight time constraints. People can do this up to a certain level of busyness, but fail once their capacity is exceeded.
That's interesting, thank you. I'll add "air traffic control" to my list of jobs I absolutely shouldn't be allowed to do.
ha! you never know
The narration was well done and I appreciate the speed being adjustable. I think the narration is a great addition.
Common core focused on reading too, though, which kinda undermines your argument about STEM. It wasn't only a math thing, the math thing simply became a culture war rallying cry for the right.
I didn't really address the reading side in this article - as it says directly after the intro section, the article is specifically about common core math. Common core reading might be great - I haven't really looked into it as heavily. But even if the reading half of it was sent down from Olympias as a gift to man, I'm not sure that effects the math half much.
The STEM argument is similarly purely about the math half - it seems to me they constructed CC math in such a way that if it worked how they wanted, they'd be able to convert math-hating kids into data analyst/physicist/engineer types. I actually think that's fine, provided it works. Or even fine if it doesn't work, so long as it doesn't then hurt their ability to do simple math. The worry is that it does hurt that - the current data I can find indicates scores are dropping rather than improving; the hope I hold is that I'm wrong and that pattern doesn't continue.
For a contrary example of a new approach leading to better results, teaching foreign languages by mostly feeding people massive amounts of comprehensible input and letting the brain's pattern-matching machinery do the heavy lifting works much better than traditional language classes.
Thanks for the article and podcast. I found it super useful and thought you did an excellent job! Practical feedback would be: the backing music isn't needed for the entire podcast - I listen to podcasts on 1.8x speed and the music can be quite distracting and doesn't add to what you're saying; when describing pictures I feel that if you're better cutting your losses and just saying look at the post to understand as "badly" describing it is probably worse than not describing it at all (not saying you were bad this time, it's just something that could happen); when describing an external link (just saying 'link in post' automatically might help maintain the flow); linking directly to the post in the episode description would be handy. Otherwise I hope you keep it up, definitely would mean I would engage with more of your stuff if you kept this up.
my min setting for any audio is 1.3x, but usually faster so I agree about the music
I'm mostly with you, with a few quibbles. First, thank you for the exposition. The weird-ass subtraction example actually seemed really analogous to long division -- call it long subtraction. The mildly racist poster you called "long division" is actually <i>short</i> division, the difference being that in short division you always guess right about what the next digit is, while in long division you can try 6 and subtract out whatever that gives you and then try 2 and then if you have to even 1, and add it all up at the end. So maybe the theory is that "long subtraction" is actually easier for a slow student than learning the shortest-time algorithm with borrowing and all that jazz, not that the manipulation is supposed to make them comfortable with numbers.
Dunno, I've never read any of the educational theory behind common core, or any educational theory for that matter.
One of the other problems with common core is that parents aren't able to tutor their children in math, because they themselves haven't learned the new techniques. This has the effect of cementing teachers as the gatekeepers of educational success. We should be making it easier for parents to take an interest in their child's education, and making parental tutoring more streamlined and normal.
I enjoyed this, but would have been interested for more discussion of the theory behind common core math from the perspective of its proponents. I agree with your critique of the one line of pro-common core thinking you discussed, but am curious how much of the argument for common core is driven by that line of thinking.
(Of course, that's a lot to ask from a blog post - this was already good!)
Thanks for the audio! I would read if it didn't exist, but I'll probably subscribe via podcast rss if you keep doing it.
I think enough people are indicating they like it that it will probably continue to be a thing I do at this point. I'm glad to find it's useful!
Of course the narration is useful; I don’t have to be sitting in front of and staring at the text every second in order to be consuming the article.
BUT why are you using “tick” in this sense up there in the text...isn’t it really a Britishism? (The Internet-Age proliferation of Britishisms among Americans is my pet peeve. And why, you ask. It may have something to do with me being a crank.)
I'm not sure I follow - did I use tick? I can't find it in the text with a ctrl-f and I don't remember using it. I did say check a couple times.
It’s in the sentence “the logical solution is to find boxes successful student A ticked that successful student B didn’t, and make them tick the same boxes.”
Oh, so it is! Yeah, I guess to the extent that's British I was being Britishish. I wasn't aware of it as a British thing, if that helps.