7 ms·

Meet the New Math, Unlike the Old Math

66 points by M_Grey 10y ago

adamnemecek 10y ago
Unless you remove the concept of a teacher and replace it with a person who helps you with your school work when you get stuck, schools won't be fixed.
- sethammons 10y ago
  I like the idea of reciprocal teaching. You view well made lectures from home, and practice at school, where an expert can help you when you stumble.
ivan_ah 10y ago
> “[Teachers] are woefully unprepared [for] engaging in inquiry driven lessons. [...] instructional changes entailed by NGSS are too big to internalize in isolated chunks of professional development. I think this is currently the main obstacle with the CCSSM and NGSS. The structure and logic of the news standards are good, but they're new so it will take some time for teachers to adjust. We need better course plans, better textbooks for teachers, better textbooks for students, and better exercise frameworks. We can't look to mainstream publishers to provide this, because they're large, slow, and extractive organizations. Therefore, this seismic shift in the structure and material taught represents a tremendous opportunity for OER to come in and eat the big publishers lunch, e.g. [1]. Our company (Minireference Co.) is currently focussed on university-level textbooks so we can't "attack" the high-school market, but we're happy to share technology and insights with anyone out there who is working on OER for primary/middle/high school. [1] State-sponsored content push to create common-core-aligned textbooks for high schools: http://k12oercollaborative.org/ http://k12oercollaborative.org/
- Spivak 10y ago
  > but they're new so it will take some time for teachers to adjust It's actually much worse than this, the only math teachers that actually understand how to teach this material are math teacher teachers and their newly graduated students who have gone through their curriculum. It's going to take a literal lifetime to actually meaningfully execute the new methods even if you manage to get the textbooks in every classroom. I really truly hope the kinds of resources you're developing can help existing teachers transition, because as it currently stands the outlook is bleak.
lowbloodsugar 10y ago
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)
helthanatos 10y ago
Experimenting with education is what broke the system in the first place. We can't very well go back to how anything was in the 1960s because times have changed, but that's not to say those methods are completely broken. Common core is a hideous atrocity that has been proven to hurt young children's minds and cause them to do things in redundant or roundabout ways. I've been experimented on throughout my school career - some of it good most of it bad. Taking education out of its broken state is important, but breaking it more helps no one. Proceed carefully and think things through before possibly ruining the education and, by extension, their lives. Education is seriously important. It takes a lot of work to rewrite your brain after being taught wrong.
- Spooky23 10y ago
  This is why my wife and I decided to make a serious sacrifice and send our kids to private school.
- Conlectus 10y ago
  I agree that education is extremely important. "a hideous atrocity that has been proven to hurt young children's minds" is extremely alarmist. I don't think that helps conversation, but I can understand your passion. Where have you seen that Common Core is harmful? Most sources in my (admittedly limited) research suggest that it is far to early to tell an effect. The most cited report seems to be https://www.brookings.edu/research/a-progress-report-on-the-common-core/#cancel https://www.brookings.edu/research/a-progress-report-on-the-... . However, the data reported here is extremely messy. Looking at table 3-2 the conclusion that CC may be harmful is drawn from the fact that n=1 has an improvement of ~2.25, whereas n=5 has an improvement of only 1.94. However, n=4 is at -0.83, n=3 at 1.8 etc. Results are all over the map. In my opinion there is no clear pattern, let alone a "proven" result. If there is research I have missed I would genuinely like to hear it. I don't have a fixed opinion.
  - helthanatos 10y ago
    My passion comes from being taught some math wrong. The common core algebra books don't cover some things and the books we used for algebra 1 and 2 were pretty much the same. When I moved schools from a charter school that was an early adopter of common core to a public high school (common core is repealed in my state) my new math teacher told me I should have learned a lot more than I did. It's not completely the common core book's fault - my teacher was also pretty lax. My teachers that I would classify as "good" have all had disdain for it. Whether they just don't like change or if they see harm in that change, I prefer an even playing ground for everyone and guessing about what changes are needed can be harmful. I don't have any research about it because I just have personal feelings about it. Some of my relatives can't count without drawing something or multiply because they have to draw a lattice.
platz 10y ago
Tom Lehrer: New Math https://www.youtube.com/watch?v=DfCJgC2zezw https://www.youtube.com/watch?v=DfCJgC2zezw
- slavik81 10y ago
  I enjoy this song, but it is rather dated. The new and confusing way of doing math he describes is the way I learned it in the 90s. It's mostly confusing in the song because it's done quickly and the visualizations added in this version are intentionally obtuse. The song is occasionally used as an argument for going back to the old way, but it seems somewhat contradictory when parents of school-aged children aren't old enough to have experienced the old style of math instruction he advocated for. New math was a trend from the 1960s, so you'd have to be pushing 70 to have experienced elementary school math before it.
  - ghaff 10y ago
    What the "New Math" was about as I remember/understand it--I suppose that's what I went through we also did plenty of arithmetic drill--was a lot of emphasis on simple set theory, doing arithmetic in different bases, etc. As he jokes in the song, the focus was on certain types of mathematical theory as opposed to being able to add, multiple, divide, and subtract. Of course, this sort of tension is present in a lot of academics. Focus too much on the theory and students can't actually do anything useful; and many will be completely lost. Focus too much on the rote learning and the students don't actually understand any underlying principles (and, in the case of arithmetic, can't actually do anything a $5 calculator can't today).
    - WalterBright 10y ago
      If you don't understand arithmetic, a $5 calculator will inevitably produce the wrong answer for you. Sometimes people question the value of teaching math to kids. But I can say anecdotally that the people I've met who did not understand arithmetic correlated strongly with people who were disasters with their personal financial affairs.
hondo77 10y ago
My daughter is a senior in high school. She used to love math. Then they switched to Common Core. Now she hates it. Having looked at her textbooks, I don't blame her. Confusing concepts presented quickly, with no depth, and it's unclear how those concepts solve the problems presented. Say what you will about the old way to solve for "b", at least you learned how to do it.
- mjevans 10y ago
  If she mostly learned the old way, get her old books your self and finish the job?
empath75 10y ago
I have a son now, and I was sort of considering trying to introduce him to abstract algebra and category theory kind of early, because it seems like something that might be easier to understand if you know less about math than if you know more about it.
- throwaway729 10y ago
  Abstract algebra seems doable at a young age for a typical child (say, somewhere b/w late middle school and late high school depending on the kid). I've done bits of group theory with kids in that age group before. It usually gets pretty well, but we never get past the first chapter or two. Basic proofs, but it's a good introduction to what "real" math is like. IMO category theory would be a waste of time without first going through a mathematics undergraduate course of study. Getting past the preface of a reasonable book on category theory requires internalizing the idea of a natural transformation. If you can't get that far, you won't get anything non-superficial out of studying category theory. Understanding what a natural transformation is requires at least one example you can relate to. Understanding why it'd be useful to abstract away from that particular example requires knowing enough about at least one other example. And to do anything even moderately interesting in category theory you need a fair bit of mathematical maturity, which means being able to write and read proofs together with a bit of knowledge about modern logic, analysis and algebra. Without that, a lot of the motivation for category theory is kind of lost (and probably most of the available exercise aren't even comprehensible. And learning math without exercises is close to impossible.).
  - mathgenius 10y ago
    There's quite alot you can do with finite categories. Along the lines of "a group is a category with one object". etc. I'm pretty sure a sharp 12 year-old could get this.
    - throwaway729 10y ago
      Yeah, I suppose that's true. You'd have to pick your topics carefully and probably reformulate a lot of definitions. And you'd need the student to think it was cool for its own sake. But I could see this working out.
jcoffland 10y ago
It seems like a lot of parents these days are appalled when schools actually make their kids do work. Sure it's not fun at first, especially when you're not used to having anyone tell you what to do, but once you get the hang of it, work is fun. It's not about an oppressive system making you in to another drone. Kids should be encouraged to think for themselves but the humans that came before us have a lot of beneficial knlowedge to share. Teach your kids to not take advantage if this and to play all day at their peril.
sporkologist 10y ago
Meanwhile in Finland.....
ch4s3 10y ago
I just read this National Review article [1] criticizing common core, but it all kind of makes sense to me. I get that when you have 6 of soemthing and visually move 4 away, two things remain. IS there something more insidious I'm missing here? Is the idea that 82 is 0 hundreds, 8 tens, and 2 ones all the crazy? I kind of like it. But, I don't have kids and the only thing I ever taught kids was how to tie knots and sharpen an ax at summer camp, so what do I know. [1] http://www.nationalreview.com/article/373840/ten-dumbest-common-core-problems-alec-torres http://www.nationalreview.com/article/373840/ten-dumbest-com...
- maus42 10y ago
  Some of that is just bad material (#3, shapes that should be shaded are not shaded). Others try to explain how at least I do arithmetic mentally (the particular example 7 + 7 feels silly because the answer was drilled to my brain in primary school, but the idea of decomposing number to parts that easier to sum together is sound -- and it feels like nice intuition to algebraic way of thinking). On the other hand, insisting in drawing graphical representations might be boring and feel silly if you already got it. In other words, the Common Core seems to get flak because some of the material is simply bad and sometimes seems not to have good idea how to convey the Common Core ideas.
  - ch4s3 10y ago
    That's my read on it as well. Funny that you mention 7 + 7 which I've always visualized as 5 + 5 + 2 + 2, or reduced to 10+4 in my head. Whenever I add 7 to anything that doesn't end in 3, I quickly add 5 and 2 instead of 7. With respect to base 10 math, 7 is awkward.
sopooneo 10y ago
How about this for a "reform": small, incremental improvement to the system year be year. Because we have one of these "revolutions" in education approximately 1.8 times per decade since 1970. And no matter how noble the ideas, they almost always turn to nonsense by the time the hit the classroom.

66 points by M_Grey 10y ago

Unless you remove the concept of a teacher and replace it with a person who helps you with your school work when you get stuck, schools won't be fixed.

sethammons 10y ago
I like the idea of reciprocal teaching. You view well made lectures from home, and practice at school, where an expert can help you when you stumble.

ivan_ah 10y ago

> “[Teachers] are woefully unprepared [for] engaging in inquiry driven lessons. [...] instructional changes entailed by NGSS are too big to internalize in isolated chunks of professional development. I think this is currently the main obstacle with the CCSSM and NGSS. The structure and logic of the news standards are good, but they're new so it will take some time for teachers to adjust. We need better course plans, better textbooks for teachers, better textbooks for students, and better exercise frameworks. We can't look to mainstream publishers to provide this, because they're large, slow, and extractive organizations. Therefore, this seismic shift in the structure and material taught represents a tremendous opportunity for OER to come in and eat the big publishers lunch, e.g. [1]. Our company (Minireference Co.) is currently focussed on university-level textbooks so we can't "attack" the high-school market, but we're happy to share technology and insights with anyone out there who is working on OER for primary/middle/high school. [1] State-sponsored content push to create common-core-aligned textbooks for high schools: http://k12oercollaborative.org/ http://k12oercollaborative.org/

Spivak 10y ago
> but they're new so it will take some time for teachers to adjust It's actually much worse than this, the only math teachers that actually understand how to teach this material are math teacher teachers and their newly graduated students who have gone through their curriculum. It's going to take a literal lifetime to actually meaningfully execute the new methods even if you manage to get the textbooks in every classroom. I really truly hope the kinds of resources you're developing can help existing teachers transition, because as it currently stands the outlook is bleak.

lowbloodsugar 10y ago

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)

helthanatos 10y ago

Experimenting with education is what broke the system in the first place. We can't very well go back to how anything was in the 1960s because times have changed, but that's not to say those methods are completely broken. Common core is a hideous atrocity that has been proven to hurt young children's minds and cause them to do things in redundant or roundabout ways. I've been experimented on throughout my school career - some of it good most of it bad. Taking education out of its broken state is important, but breaking it more helps no one. Proceed carefully and think things through before possibly ruining the education and, by extension, their lives. Education is seriously important. It takes a lot of work to rewrite your brain after being taught wrong.

Spooky23 10y ago
This is why my wife and I decided to make a serious sacrifice and send our kids to private school.
Conlectus 10y ago
I agree that education is extremely important. "a hideous atrocity that has been proven to hurt young children's minds" is extremely alarmist. I don't think that helps conversation, but I can understand your passion. Where have you seen that Common Core is harmful? Most sources in my (admittedly limited) research suggest that it is far to early to tell an effect. The most cited report seems to be https://www.brookings.edu/research/a-progress-report-on-the-common-core/#cancel https://www.brookings.edu/research/a-progress-report-on-the-... . However, the data reported here is extremely messy. Looking at table 3-2 the conclusion that CC may be harmful is drawn from the fact that n=1 has an improvement of ~2.25, whereas n=5 has an improvement of only 1.94. However, n=4 is at -0.83, n=3 at 1.8 etc. Results are all over the map. In my opinion there is no clear pattern, let alone a "proven" result. If there is research I have missed I would genuinely like to hear it. I don't have a fixed opinion.
- helthanatos 10y ago
  My passion comes from being taught some math wrong. The common core algebra books don't cover some things and the books we used for algebra 1 and 2 were pretty much the same. When I moved schools from a charter school that was an early adopter of common core to a public high school (common core is repealed in my state) my new math teacher told me I should have learned a lot more than I did. It's not completely the common core book's fault - my teacher was also pretty lax. My teachers that I would classify as "good" have all had disdain for it. Whether they just don't like change or if they see harm in that change, I prefer an even playing ground for everyone and guessing about what changes are needed can be harmful. I don't have any research about it because I just have personal feelings about it. Some of my relatives can't count without drawing something or multiply because they have to draw a lattice.

platz 10y ago

Tom Lehrer: New Math https://www.youtube.com/watch?v=DfCJgC2zezw https://www.youtube.com/watch?v=DfCJgC2zezw

slavik81 10y ago
I enjoy this song, but it is rather dated. The new and confusing way of doing math he describes is the way I learned it in the 90s. It's mostly confusing in the song because it's done quickly and the visualizations added in this version are intentionally obtuse. The song is occasionally used as an argument for going back to the old way, but it seems somewhat contradictory when parents of school-aged children aren't old enough to have experienced the old style of math instruction he advocated for. New math was a trend from the 1960s, so you'd have to be pushing 70 to have experienced elementary school math before it.
- ghaff 10y ago
  What the "New Math" was about as I remember/understand it--I suppose that's what I went through we also did plenty of arithmetic drill--was a lot of emphasis on simple set theory, doing arithmetic in different bases, etc. As he jokes in the song, the focus was on certain types of mathematical theory as opposed to being able to add, multiple, divide, and subtract. Of course, this sort of tension is present in a lot of academics. Focus too much on the theory and students can't actually do anything useful; and many will be completely lost. Focus too much on the rote learning and the students don't actually understand any underlying principles (and, in the case of arithmetic, can't actually do anything a $5 calculator can't today).
  - WalterBright 10y ago
    If you don't understand arithmetic, a $5 calculator will inevitably produce the wrong answer for you. Sometimes people question the value of teaching math to kids. But I can say anecdotally that the people I've met who did not understand arithmetic correlated strongly with people who were disasters with their personal financial affairs.

hondo77 10y ago

My daughter is a senior in high school. She used to love math. Then they switched to Common Core. Now she hates it. Having looked at her textbooks, I don't blame her. Confusing concepts presented quickly, with no depth, and it's unclear how those concepts solve the problems presented. Say what you will about the old way to solve for "b", at least you learned how to do it.

mjevans 10y ago
If she mostly learned the old way, get her old books your self and finish the job?

empath75 10y ago

I have a son now, and I was sort of considering trying to introduce him to abstract algebra and category theory kind of early, because it seems like something that might be easier to understand if you know less about math than if you know more about it.

throwaway729 10y ago
Abstract algebra seems doable at a young age for a typical child (say, somewhere b/w late middle school and late high school depending on the kid). I've done bits of group theory with kids in that age group before. It usually gets pretty well, but we never get past the first chapter or two. Basic proofs, but it's a good introduction to what "real" math is like. IMO category theory would be a waste of time without first going through a mathematics undergraduate course of study. Getting past the preface of a reasonable book on category theory requires internalizing the idea of a natural transformation. If you can't get that far, you won't get anything non-superficial out of studying category theory. Understanding what a natural transformation is requires at least one example you can relate to. Understanding why it'd be useful to abstract away from that particular example requires knowing enough about at least one other example. And to do anything even moderately interesting in category theory you need a fair bit of mathematical maturity, which means being able to write and read proofs together with a bit of knowledge about modern logic, analysis and algebra. Without that, a lot of the motivation for category theory is kind of lost (and probably most of the available exercise aren't even comprehensible. And learning math without exercises is close to impossible.).
- mathgenius 10y ago
  There's quite alot you can do with finite categories. Along the lines of "a group is a category with one object". etc. I'm pretty sure a sharp 12 year-old could get this.
  - throwaway729 10y ago
    Yeah, I suppose that's true. You'd have to pick your topics carefully and probably reformulate a lot of definitions. And you'd need the student to think it was cool for its own sake. But I could see this working out.

jcoffland 10y ago

It seems like a lot of parents these days are appalled when schools actually make their kids do work. Sure it's not fun at first, especially when you're not used to having anyone tell you what to do, but once you get the hang of it, work is fun. It's not about an oppressive system making you in to another drone. Kids should be encouraged to think for themselves but the humans that came before us have a lot of beneficial knlowedge to share. Teach your kids to not take advantage if this and to play all day at their peril.

sporkologist 10y ago

Meanwhile in Finland.....

ch4s3 10y ago

I just read this National Review article [1] criticizing common core, but it all kind of makes sense to me. I get that when you have 6 of soemthing and visually move 4 away, two things remain. IS there something more insidious I'm missing here? Is the idea that 82 is 0 hundreds, 8 tens, and 2 ones all the crazy? I kind of like it. But, I don't have kids and the only thing I ever taught kids was how to tie knots and sharpen an ax at summer camp, so what do I know. [1] http://www.nationalreview.com/article/373840/ten-dumbest-common-core-problems-alec-torres http://www.nationalreview.com/article/373840/ten-dumbest-com...

maus42 10y ago
Some of that is just bad material (#3, shapes that should be shaded are not shaded). Others try to explain how at least I do arithmetic mentally (the particular example 7 + 7 feels silly because the answer was drilled to my brain in primary school, but the idea of decomposing number to parts that easier to sum together is sound -- and it feels like nice intuition to algebraic way of thinking). On the other hand, insisting in drawing graphical representations might be boring and feel silly if you already got it. In other words, the Common Core seems to get flak because some of the material is simply bad and sometimes seems not to have good idea how to convey the Common Core ideas.
- ch4s3 10y ago
  That's my read on it as well. Funny that you mention 7 + 7 which I've always visualized as 5 + 5 + 2 + 2, or reduced to 10+4 in my head. Whenever I add 7 to anything that doesn't end in 3, I quickly add 5 and 2 instead of 7. With respect to base 10 math, 7 is awkward.

sopooneo 10y ago

How about this for a "reform": small, incremental improvement to the system year be year. Because we have one of these "revolutions" in education approximately 1.8 times per decade since 1970. And no matter how noble the ideas, they almost always turn to nonsense by the time the hit the classroom.