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FTA: "Traditional math teaches an algorithm. Common Core teaches the concept of 'proportionality' [over a period of six years]" My 11 and 13yr olds are dealing

FTA: "Traditional math teaches an algorithm. Common Core teaches the concept of 'proportionality' [over a period of six years]" My 11 and 13yr olds are dealing with this Common Core bullshit right now. This article might give an insight into why CC is so bad, yet they think its so good. Like when we demo a software product that we've all been working on for six months, and discover that users can't figure out how to make it work: it only works for us because we've been using it continuously for six months. Maybe Common Core can teach Proportionality if you sit down with a kid and give them all six years of it over two days. What happens in reality, is that my kids (a) don't understand the confusing ways its presented and (b) don't make the connection between the different expressions of the same underlying concept. Its not useful. Its actually immensely frustrating for everyone. I think they have it exactly backwards. Teach the algorithm. Rote practice. Then teach the concept. I'm reminded of my first year thermodynamics lecture at Imperial: "We're going to teach this to you every year. The first year you wont understand it. The second year you'll think you understand it. The final year you'll realize you never really understood it, but at least you'll be able to use it."

mjevans 10y ago
That might work /for you/ what works for me is being shown the algorithm (the path to the true answer) and ALSO being shown WHY at the same time. And yes, the rote practice is what makes it sink in. I just also need the WHY so that it makes 'sense' what I'm doing and to help me understand why I am doing it.
- haberman 10y ago
  Agree. When I took linear algebra I calculated plenty of eigenvalues/eigenvectors without having any idea what they actually were or why I was doing it. Then one day years later I saw the Mona Lisa examples on this page: https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors I was simultaneously thrilled that these concepts actually had meaning and annoyed that no one explained this meaning to me at the time. Math is filled with definitions and proofs, but none of that has any point unless the result actually corresponds to something interesting! (btw I don't have any experience of learning from common core, so I don't know if I would like it or not)
  - ivan_ah 10y ago
    On the topic of eigenvalues, this video is a very nice introduction: https://www.youtube.com/watch?v=PFDu9oVAE-g&index=14&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/watch?v=PFDu9oVAE-g&index=14&list=PL... (the whole playlist kicks ass; highly recommended for anyone interested in "seeing" what linear algebra is all about)
  - M_Grey 10y ago
    I'm like this too; I need a framework of meaning to fit the rote facts into, or they don't stick.
  - seanp2k2 10y ago
    Yep, same. I also independently discovered existing concepts which were way above what we were being taught which really lifted the veil on some things. I'm guessing that these concepts would have also been helpful for other kids to understand, or at least be aware of.
seanmcdirmid 10y ago
It is increasingly important for people to be able to formulate, and therefore understand intuitively, algorithms rather than just to execute them quickly and accurately. Rote is just getting less useful year after year, computers are already great at execution.
- swsieber 10y ago
  What I took away from their comment is that "rote execution -> concept" is faster than "concept -> execution"
  - ThrustVectoring 10y ago
    That's not quite what I got: "rote execution -> concept" is a faster way to get the concept than trying to teach the concept itself.
- candiodari 10y ago
  That's only useful in cases where human intuition matches reality. In a lot of cases it does, in many more cases it doesn't. Where it doesn't rote is the only thing that may work and intuition without rote will actually prevent you from understanding, and lead you to believe falsehoods. In statistics, this is quite common. You should never use intuition, always use the math, always work it out, even when "it's obvious". By the way, I have a friend who's a TA in statistics and this is a big point of frustration for them. Undergrads are starting to do just what you said : unquestioningly follow intuition. It's a disaster. Not that there wasn't always 50% of the class doing this, it's just that it's now 90%.
  - seanmcdirmid 10y ago
    I don't think we disagree that much. My problem is simply training humans to become better pattern marchers via rote practice is simply less useful now that computers are getting better (much much better) at that anyways. There is not much point for a person to do model training by hand, for example. So what is left? Abstract reasoning, problem solving rather than number crunching. Being able to understand the math so that it becomes intuition. Many classes are still stuck in teaching students to pattern match rather than really understand what they are doing, and why. Practice on a bunch problem sets without does nothing to fix that. Rote learning is just not useful for the jobs that are humans are going to do next, where computers are going to be able to pattern match much better than us anyways.
wvenable 10y ago
My daughter went through this in Canada; same basic concept and she did not like it and neither of us could make sense of a lot of it. My theory is that this version of Math is designed by people who hate Math (English teachers). It's all about estimation and fuzzy thinking. Everything that I personally liked about Math as a kid seems to have been stripped away. In my opinion, children who want to become computer programmers, scientists, and engineers are probably not well serviced by this type of teaching. I'm not saying the old method was perfect either; mostly because if you're just teaching algorithms then there isn't much understanding about why they actually work.
- candiodari 10y ago
  One of the things machine learning has taught me that for many complex algorithms, I don't know how they work, and ... no-one does, and here's the great insight : it's not important, it is in fact a very good thing that I don't know what they're doing. It wouldn't make any difference if I did, and it would in fact be more limiting than not knowing. Having the ability to execute and design them is all that's important. The issue is that machine learning, at least in the modelling domain, is exceeding the maximum complexity the human mind can contain. Attempting to analyze neural networks is fun, and it's what you do as a researcher. But no amount of study is going to tell you what more than 10 neurons are doing to any reasonable amount of detail, no matter how smart you are. Machine learning algorithms suck at explaining what they're doing mostly because what they're doing is far more complex than what we do. And yes, I can build toy examples like a 100 bit binary adder where I do know, I'm talking about non-obvious cases. Therefore: rote is all that matters. You should be able to execute these things, forwards and backwards and take it's code apart and put it back together again. That's all there is to it. And if machine learning does indeed conquer the world, that's all there will be to anything.
  - johncolanduoni 10y ago
    That may apply to machine learning, but not to anything you'd get in a grade school math education. Or a graduate one for that matter. Moving from rote implementation of something else came up with to figuring out how to approach problems where the steps aren't already laid out for you is a big speed bump in university education for mathematics majors. Also, unless our current machine learning algorithms just need bigger computers to take over the world, somebody is going to have to develop more. If some kind of guiding intuition isn't a part of that process, the universe will hit heat death before we build an AGI.
WalterBright 10y ago
novice: follows the rules because he's told to master: follows the rules because he understands them guru: transcends the rules because he understands where they don't apply
msie 10y ago
This is recommended by Barbara Oakley, one of the authors of the "Learning How to Learn" course at Coursera. http://nautil.us/issue/40/learning/how-i-rewired-my-brain-to-become-fluent-in-math-rp http://nautil.us/issue/40/learning/how-i-rewired-my-brain-to... The stigma against rote practice has damaged my past learning efforts. I thought I was learning things the wrong way. :-(
- johncolanduoni 10y ago
  The problem is that virtually every grade school math program will teach you rote practice... and then nothing else. Seeing examples and doing exercises is still key even if you're teaching category theory to graduate math students, but nobody would consider someone who had only mastered that part to understand the subject. The efforts to de-emphasize it seem to be in the hope that it'll magically cause people to learn the higher concepts as long as you use the rest of the time to read passages out of a book about them. They've clearly been unsuccessful.
GavinMcG 10y ago
Consider also that she might simply have a teacher who is not very good at teaching that material.
rdtsc 10y ago
> What happens in reality, is that my kids (a) don't understand the confusing ways its presented and (b) don't make the connection between the different expressions of the same underlying concept. Its not useful. Its actually immensely frustrating for everyone. I agree. I remember how I learned: lots rote exercises and word problems. Clearly recall having insights and making discoveries on my own, but those made sense based on how I visualized things. And because of discovering them on my own, they were very memorable and fun. The funny thing, repetition is how kids learn. They copy and repeat things since day 0 without understanding why or how anything works. Insights and understanding the meaning of things come later. Reversing that process is counter-productive and goes against all the evolutionary mechanisms optimized up to this point. That kind of build-up from first principle will be useful, but that's closer to highschool or college level.
throwaway729 10y ago
> Common Core teaches the concept of 'proportionality' [over a period of six years] It sounds like you're poo-pooing the centrality and depth of notions of proportionality in Mathematics. Just in case that was your intention: expecting a student's understanding of proportionality to fully mature over 6 years is ridiculously optimistic -- often a big chunk of the first year of a graduate mathematics program is devoted to introducing measure theory. > Maybe Common Core can teach Proportionality if you sit down with a kid and give them all six years of it over two days Hell no it can't. How in god's name are you going to teach about fractions and the Riemann integral in two days? To be clear, proportionality is extremely important in both cases -- I'm not sure what it would even mean to successfully teach either of those things without your students noticing something essential about proportionality. Obviously, a competent teacher will notice that some analogies and connections could be useful to point out and elucidate, while some are still a decade away from being useful to the student. > Teach the algorithm. Rote practice. Then teach the concept. This isn't at all contradictory with Common Core... what makes you think it is? For me, "State and motivate the concept. Teach the algorithm. Rote practice. Then teach the concept. Rote practice with the concept as opposed to the algorithm." is far more effective, but anyways, the practice and algorithm is still there. > What happens in reality, is that my kids (a) don't understand the confusing ways its presented Oh right, shitty execution. I'd be unsurprised if the teacher is as confused as everyone else. You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME. I know why I'm not a teacher... > and (b) don't make the connection between the different expressions of the same underlying concept... Its actually immensely frustrating for everyone. Well, yeah, learning hard stuff usually is.
- johncolanduoni 10y ago
  > Just in case that was your intention: expecting a student's understanding of proportionality to fully mature over 6 years is ridiculously optimistic -- often a big chunk of the first year of a graduate mathematics program is devoted to introducing measure theory. First off, I'm not getting the proportionality => measure theory link. Proportionality figures in much more prominently in other areas (e.g. projective spaces, geometry in general). Where's the time you allot for studying PU(n)? Second, you forgot to demand that high school science teach kids how to derive hyperfine structure so they can really understand what a hydrogen atom is. You should probably give up on teaching them about any of the other elements at all. > You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME. Except what you'll get is as many opinions on what to spend the 6 years on as there are mathematicians. And any discussion is likely to be dominated by group theorists thinking group theory is the most important thing mankind has come up with, analysts thinking the same about calculus, etc. Doesn't sound helpful to me at all.

mjevans 10y ago

That might work /for you/ what works for me is being shown the algorithm (the path to the true answer) and ALSO being shown WHY at the same time. And yes, the rote practice is what makes it sink in. I just also need the WHY so that it makes 'sense' what I'm doing and to help me understand why I am doing it.

haberman 10y ago
Agree. When I took linear algebra I calculated plenty of eigenvalues/eigenvectors without having any idea what they actually were or why I was doing it. Then one day years later I saw the Mona Lisa examples on this page: https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors I was simultaneously thrilled that these concepts actually had meaning and annoyed that no one explained this meaning to me at the time. Math is filled with definitions and proofs, but none of that has any point unless the result actually corresponds to something interesting! (btw I don't have any experience of learning from common core, so I don't know if I would like it or not)
- ivan_ah 10y ago
  On the topic of eigenvalues, this video is a very nice introduction: https://www.youtube.com/watch?v=PFDu9oVAE-g&index=14&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/watch?v=PFDu9oVAE-g&index=14&list=PL... (the whole playlist kicks ass; highly recommended for anyone interested in "seeing" what linear algebra is all about)
- M_Grey 10y ago
  I'm like this too; I need a framework of meaning to fit the rote facts into, or they don't stick.
- seanp2k2 10y ago
  Yep, same. I also independently discovered existing concepts which were way above what we were being taught which really lifted the veil on some things. I'm guessing that these concepts would have also been helpful for other kids to understand, or at least be aware of.

seanmcdirmid 10y ago

It is increasingly important for people to be able to formulate, and therefore understand intuitively, algorithms rather than just to execute them quickly and accurately. Rote is just getting less useful year after year, computers are already great at execution.

swsieber 10y ago
What I took away from their comment is that "rote execution -> concept" is faster than "concept -> execution"
- ThrustVectoring 10y ago
  That's not quite what I got: "rote execution -> concept" is a faster way to get the concept than trying to teach the concept itself.
candiodari 10y ago
That's only useful in cases where human intuition matches reality. In a lot of cases it does, in many more cases it doesn't. Where it doesn't rote is the only thing that may work and intuition without rote will actually prevent you from understanding, and lead you to believe falsehoods. In statistics, this is quite common. You should never use intuition, always use the math, always work it out, even when "it's obvious". By the way, I have a friend who's a TA in statistics and this is a big point of frustration for them. Undergrads are starting to do just what you said : unquestioningly follow intuition. It's a disaster. Not that there wasn't always 50% of the class doing this, it's just that it's now 90%.
- seanmcdirmid 10y ago
  I don't think we disagree that much. My problem is simply training humans to become better pattern marchers via rote practice is simply less useful now that computers are getting better (much much better) at that anyways. There is not much point for a person to do model training by hand, for example. So what is left? Abstract reasoning, problem solving rather than number crunching. Being able to understand the math so that it becomes intuition. Many classes are still stuck in teaching students to pattern match rather than really understand what they are doing, and why. Practice on a bunch problem sets without does nothing to fix that. Rote learning is just not useful for the jobs that are humans are going to do next, where computers are going to be able to pattern match much better than us anyways.

wvenable 10y ago

My daughter went through this in Canada; same basic concept and she did not like it and neither of us could make sense of a lot of it. My theory is that this version of Math is designed by people who hate Math (English teachers). It's all about estimation and fuzzy thinking. Everything that I personally liked about Math as a kid seems to have been stripped away. In my opinion, children who want to become computer programmers, scientists, and engineers are probably not well serviced by this type of teaching. I'm not saying the old method was perfect either; mostly because if you're just teaching algorithms then there isn't much understanding about why they actually work.

candiodari 10y ago
One of the things machine learning has taught me that for many complex algorithms, I don't know how they work, and ... no-one does, and here's the great insight : it's not important, it is in fact a very good thing that I don't know what they're doing. It wouldn't make any difference if I did, and it would in fact be more limiting than not knowing. Having the ability to execute and design them is all that's important. The issue is that machine learning, at least in the modelling domain, is exceeding the maximum complexity the human mind can contain. Attempting to analyze neural networks is fun, and it's what you do as a researcher. But no amount of study is going to tell you what more than 10 neurons are doing to any reasonable amount of detail, no matter how smart you are. Machine learning algorithms suck at explaining what they're doing mostly because what they're doing is far more complex than what we do. And yes, I can build toy examples like a 100 bit binary adder where I do know, I'm talking about non-obvious cases. Therefore: rote is all that matters. You should be able to execute these things, forwards and backwards and take it's code apart and put it back together again. That's all there is to it. And if machine learning does indeed conquer the world, that's all there will be to anything.
- johncolanduoni 10y ago
  That may apply to machine learning, but not to anything you'd get in a grade school math education. Or a graduate one for that matter. Moving from rote implementation of something else came up with to figuring out how to approach problems where the steps aren't already laid out for you is a big speed bump in university education for mathematics majors. Also, unless our current machine learning algorithms just need bigger computers to take over the world, somebody is going to have to develop more. If some kind of guiding intuition isn't a part of that process, the universe will hit heat death before we build an AGI.

WalterBright 10y ago

novice: follows the rules because he's told to master: follows the rules because he understands them guru: transcends the rules because he understands where they don't apply

msie 10y ago

This is recommended by Barbara Oakley, one of the authors of the "Learning How to Learn" course at Coursera. http://nautil.us/issue/40/learning/how-i-rewired-my-brain-to-become-fluent-in-math-rp http://nautil.us/issue/40/learning/how-i-rewired-my-brain-to... The stigma against rote practice has damaged my past learning efforts. I thought I was learning things the wrong way. :-(

johncolanduoni 10y ago
The problem is that virtually every grade school math program will teach you rote practice... and then nothing else. Seeing examples and doing exercises is still key even if you're teaching category theory to graduate math students, but nobody would consider someone who had only mastered that part to understand the subject. The efforts to de-emphasize it seem to be in the hope that it'll magically cause people to learn the higher concepts as long as you use the rest of the time to read passages out of a book about them. They've clearly been unsuccessful.

GavinMcG 10y ago

Consider also that she might simply have a teacher who is not very good at teaching that material.

rdtsc 10y ago

> What happens in reality, is that my kids (a) don't understand the confusing ways its presented and (b) don't make the connection between the different expressions of the same underlying concept. Its not useful. Its actually immensely frustrating for everyone. I agree. I remember how I learned: lots rote exercises and word problems. Clearly recall having insights and making discoveries on my own, but those made sense based on how I visualized things. And because of discovering them on my own, they were very memorable and fun. The funny thing, repetition is how kids learn. They copy and repeat things since day 0 without understanding why or how anything works. Insights and understanding the meaning of things come later. Reversing that process is counter-productive and goes against all the evolutionary mechanisms optimized up to this point. That kind of build-up from first principle will be useful, but that's closer to highschool or college level.

throwaway729 10y ago

> Common Core teaches the concept of 'proportionality' [over a period of six years] It sounds like you're poo-pooing the centrality and depth of notions of proportionality in Mathematics. Just in case that was your intention: expecting a student's understanding of proportionality to fully mature over 6 years is ridiculously optimistic -- often a big chunk of the first year of a graduate mathematics program is devoted to introducing measure theory. > Maybe Common Core can teach Proportionality if you sit down with a kid and give them all six years of it over two days Hell no it can't. How in god's name are you going to teach about fractions and the Riemann integral in two days? To be clear, proportionality is extremely important in both cases -- I'm not sure what it would even mean to successfully teach either of those things without your students noticing something essential about proportionality. Obviously, a competent teacher will notice that some analogies and connections could be useful to point out and elucidate, while some are still a decade away from being useful to the student. > Teach the algorithm. Rote practice. Then teach the concept. This isn't at all contradictory with Common Core... what makes you think it is? For me, "State and motivate the concept. Teach the algorithm. Rote practice. Then teach the concept. Rote practice with the concept as opposed to the algorithm." is far more effective, but anyways, the practice and algorithm is still there. > What happens in reality, is that my kids (a) don't understand the confusing ways its presented Oh right, shitty execution. I'd be unsurprised if the teacher is as confused as everyone else. You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME. I know why I'm not a teacher... > and (b) don't make the connection between the different expressions of the same underlying concept... Its actually immensely frustrating for everyone. Well, yeah, learning hard stuff usually is.

johncolanduoni 10y ago
> Just in case that was your intention: expecting a student's understanding of proportionality to fully mature over 6 years is ridiculously optimistic -- often a big chunk of the first year of a graduate mathematics program is devoted to introducing measure theory. First off, I'm not getting the proportionality => measure theory link. Proportionality figures in much more prominently in other areas (e.g. projective spaces, geometry in general). Where's the time you allot for studying PU(n)? Second, you forgot to demand that high school science teach kids how to derive hyperfine structure so they can really understand what a hydrogen atom is. You should probably give up on teaching them about any of the other elements at all. > You know what would be great? If excellent (or even mediocre) mathematicians who know what it takes to learn math well became invested teachers. When that happens, the outcomes are amazing. Pretty much without exception, IME. Except what you'll get is as many opinions on what to spend the 6 years on as there are mathematicians. And any discussion is likely to be dominated by group theorists thinking group theory is the most important thing mankind has come up with, analysts thinking the same about calculus, etc. Doesn't sound helpful to me at all.