If you know the history of education, the "new math" thing is a sad story of a mind-blowing stupidity that keeps hurting math students for decades, because it divided most people into two camps that keep promoting two different wrong ideas.
Theories of education are often based on some psychological theory -- you have a theory how people think in general, and you use it to design a process to teach people.
If we skip the medieval theories, one of the relatively modern ones was called "associationism". The theory was that human mind is basically a set of associations. We are born with zero associations; as we observe the world, we learn to associate this with that; and after many years we have learned to associate things properly and now we are smart adults. For extra nuance, some of us form new associations faster than others, probably for biological reasons; that is what intelligence is. Anyway, associations are all there is.
Building an education theory on associanism is quite easy. You need a teacher who understands the subject (has the correct associations, a lot of them). Then the teacher stands in front of the classroom and keeps talking. The more he talks, the more associations the students can make. That's all there is. -- The order of lessons is not relevant; ultimately, after the teacher mentions everything, all associations will be properly connected into one large network; until then, you have to memorize. You don't wait until the students "understand", that would be a waste of time (there is no such thing as "understanding", you either have the right associations or you don't); the more you keep talking, the more associations the students can make. Of course you can (and should) repeat the facts, that's how the associations are deepened. But after a while, if some students don't get it, they are just hopeless: they had the opportunity to make the right associations, and yet they failed. It is their fate to remain farmers.
This is a bit of a strawman, and yet many teachers follow this method intuitively, even without knowing the underlying theory. And their students complain that they don't get it. And the teachers reply that yeah, some students are just talented and some are not, "the camel has two humps", et cetera.
In psychology, the next step after associationism was Piaget's "genetic epistemology". Where "genetic" is an adjective for "genesis", not the DNA. In modern language, we would probably call it "developmental epistemology", i.e. the study of the ontogenetic origins of understanding. The revolutionary approach was to watch how kids actually learn, rather than trying to shoehorn everything into a simplistic framework. One of the interesting findings was that kids actually do not make linear progress from "zero associations" to "correct understanding", but the process often takes a detour through a phase of magical thinking or some other kind of wrong understanding. Instead of "no opinion -> correct opinion", it is often "no opinion -> wrong opinion -> correct opinion". There are specific examples, not important now. Also, Piaget got some things wrong; this was later improved by Vygotsky. The important thing is the idea that the child is making mental models of the world. It is not just associations floating in a vacuum; the child has a paradigm, and tries to fit the new knowledge in that paradigm, and sometimes it doesn't work and the paradigm changes into a better one.
An educational theory built on this, originally called "constructivism", says that the teacher should not just keep saying random true facts, but also check that the students have the right models. This is achieved on one hand by making the models explicit, saying the facts in proper order, putting them in the right context... and on the other hand by checking the students' models, finding the problems and fixing them. If you find out that many students keep making the same mistake, you should adjust your way of teaching accordingly: make it obvious at the beginning that it is X not Y, maybe change the order of lessons so that making the correct model becomes easier. Keep checking the students' models regularly, because the sooner you find the mistake, the easier it is to fix it. Etc.
This is what many good tutors do intuitively, because if you teach 1:1, there is more interaction, and it is easier to catch the mistakes right when they happen and ask "why did you do this?" You do not wait until the student makes the same mistake hundred times to declare him a failure without talent; you notice when the mistake happens for the first time and keep "debugging" until the mistake is fixed. This is easy to do when tutoring; more difficult to make it scale to a classroom full of kids.
And then... there is another thing, I do not know if it has a proper name in psychology, but "postmodernism" is what some people use (and other people object to this usage)... the "edgy" idea that knowledge transfer is actually impossible, everyone lives in their own different reality, trying to teach something is an oppression, if only we left the kids alone they would reinvent the civilization and make it much better (Rousseau's "Emile"). -- For stupid political reasons ("there are exactly two sides of the story, not more, not less", "the enemy of my enemy is my friend"), these people are typically associated with the constructivists, because they both oppose rote memorization. But although the opponent may be the same, the proposed solutions are quite different ("teaching better" vs "not teaching at all").
To increase the confusion, the educational theory build on this was called "radical constructivism", misleadingly suggesting that this might be "something like the famous Piaget, only much more so", when if fact it is something completely different. The kids taught using this philosophy are left alone to reinvent the math... and fail predictably! Or sometimes they are taught dozen different methods how to do addition (bonus point if the method was used by some indigenous population, because, you know, "noble savage", doesn't matter if the specific method only works for adding 7+8 and 8+9), hoping that this will kickstart their math thinking so now they will develop the rest of the math independently. Predictably, that also never happens.
So, how is this related to the "new math"? The curriculum of the "new math" was based exactly on this "radical constructivist" thought, except the authors did not emphasise the "radical" part enough and often just called it "constructivism". So you had the math curriculum that didn't work at all, and was a complete disaster. And when finally people got angry and returned to the traditional math education, the lesson everyone remembered was that "constructivism has been debunked".
So now, whenever someone proposes to teach math in a way that emphasizes understanding over memorization, the kneejerk reaction is "haha, that sounds like constructivism... yeah, we tried that but that didn't work at all", plus a link to some web page that criticizes "new math". And if you try to explain how this is completely unrelated to Piaget, you are dismissed with "yeah right, the true constructivism has never been tried, comrades, hahaha".
And then, ironically, people keep quoting the Feynman's story about how actually understanding physics is better than mere memorization that "light is waves". And the helpless (Piagetian) construstivist is like "yeah guys, that's exactly what I was trying to tell you all the time", but no one cares, and when it comes back to adding some understanding to the lessons, someone inevitably comes with the condescending "haha, but constructivism has been debunked" and a link to Ten Facts Why New Math Sucks.
Possible solution: perhaps the brand of "constructivism" has been thoroughly poisoned, and we need to reinvent it and call it "Feynmanism". Then you can go and say "nope, I am totally not proposing a constructivist curriculum, that has already been tried and debunked, haha, what I am proposing instead is the Feynmanist curriculum", and then people will go online and say "wow, I hated math at school, but then we got a new teacher who used this new Feynmanist method, and the math finally started make sense and now I love it".
I don't really know much about the history of education, I only read the article I linked. From what I understand from that, it is not so much that constructivism has been debunked, but rather that in order to explain something properly, you need to understand it yourself deeply. Now, many teachers don't actually understand math on that level. So it will be difficult for them to teach it based on constructivism. That seems to me to be the main difficulty. Also, everybody understands things differently, so even if the teacher is capable, how do you scale that for an entire classroom?
> in order to explain something properly, you need to understand it yourself deeply. Now, many teachers don't actually understand math on that level. So it will be difficult for them to teach it based on constructivism.
Yes, this is a sad truth about teachers, many of them actually do not understand deeply the stuff they teach.
It seemed to me unlikely that e.g. a 10 years old kid could ask teacher a math question (related to what they are learning) that the math teacher couldn't answer. Like, any adult, especially one with a university education, should be able to answer any question from the first four grades of elementary school, right? Except... nope. I have friends that try to promote constructivist education at schools, and "what if the kids ask me something I will not be able to answer" is indeed a frequent objection made by actual teachers.
Like, okay, it is definitely possible to make an innocently sounding math question that is actually extremely hard to solve. My favorite example is: "can you cut a square into an odd number of triangles of equal area, but not necessarily the same shape?" (The answer is no, but the proof requires university-level math.) But in real life, this is extremely unlikely to happen; and if it happened to me, I would simply say "sorry, I do not know".
> everybody understands things differently
This is perfectly okay. You let the kids think about it individually first, then discuss their solutions with each other. If you get multiple explanations how things work, that's great -- having things described from multiple perspectives helps you understand it even better.
Theories of education are often based on some psychological theory -- you have a theory how people think in general, and you use it to design a process to teach people.
If we skip the medieval theories, one of the relatively modern ones was called "associationism". The theory was that human mind is basically a set of associations. We are born with zero associations; as we observe the world, we learn to associate this with that; and after many years we have learned to associate things properly and now we are smart adults. For extra nuance, some of us form new associations faster than others, probably for biological reasons; that is what intelligence is. Anyway, associations are all there is.
Building an education theory on associanism is quite easy. You need a teacher who understands the subject (has the correct associations, a lot of them). Then the teacher stands in front of the classroom and keeps talking. The more he talks, the more associations the students can make. That's all there is. -- The order of lessons is not relevant; ultimately, after the teacher mentions everything, all associations will be properly connected into one large network; until then, you have to memorize. You don't wait until the students "understand", that would be a waste of time (there is no such thing as "understanding", you either have the right associations or you don't); the more you keep talking, the more associations the students can make. Of course you can (and should) repeat the facts, that's how the associations are deepened. But after a while, if some students don't get it, they are just hopeless: they had the opportunity to make the right associations, and yet they failed. It is their fate to remain farmers.
This is a bit of a strawman, and yet many teachers follow this method intuitively, even without knowing the underlying theory. And their students complain that they don't get it. And the teachers reply that yeah, some students are just talented and some are not, "the camel has two humps", et cetera.
In psychology, the next step after associationism was Piaget's "genetic epistemology". Where "genetic" is an adjective for "genesis", not the DNA. In modern language, we would probably call it "developmental epistemology", i.e. the study of the ontogenetic origins of understanding. The revolutionary approach was to watch how kids actually learn, rather than trying to shoehorn everything into a simplistic framework. One of the interesting findings was that kids actually do not make linear progress from "zero associations" to "correct understanding", but the process often takes a detour through a phase of magical thinking or some other kind of wrong understanding. Instead of "no opinion -> correct opinion", it is often "no opinion -> wrong opinion -> correct opinion". There are specific examples, not important now. Also, Piaget got some things wrong; this was later improved by Vygotsky. The important thing is the idea that the child is making mental models of the world. It is not just associations floating in a vacuum; the child has a paradigm, and tries to fit the new knowledge in that paradigm, and sometimes it doesn't work and the paradigm changes into a better one.
An educational theory built on this, originally called "constructivism", says that the teacher should not just keep saying random true facts, but also check that the students have the right models. This is achieved on one hand by making the models explicit, saying the facts in proper order, putting them in the right context... and on the other hand by checking the students' models, finding the problems and fixing them. If you find out that many students keep making the same mistake, you should adjust your way of teaching accordingly: make it obvious at the beginning that it is X not Y, maybe change the order of lessons so that making the correct model becomes easier. Keep checking the students' models regularly, because the sooner you find the mistake, the easier it is to fix it. Etc.
This is what many good tutors do intuitively, because if you teach 1:1, there is more interaction, and it is easier to catch the mistakes right when they happen and ask "why did you do this?" You do not wait until the student makes the same mistake hundred times to declare him a failure without talent; you notice when the mistake happens for the first time and keep "debugging" until the mistake is fixed. This is easy to do when tutoring; more difficult to make it scale to a classroom full of kids.
And then... there is another thing, I do not know if it has a proper name in psychology, but "postmodernism" is what some people use (and other people object to this usage)... the "edgy" idea that knowledge transfer is actually impossible, everyone lives in their own different reality, trying to teach something is an oppression, if only we left the kids alone they would reinvent the civilization and make it much better (Rousseau's "Emile"). -- For stupid political reasons ("there are exactly two sides of the story, not more, not less", "the enemy of my enemy is my friend"), these people are typically associated with the constructivists, because they both oppose rote memorization. But although the opponent may be the same, the proposed solutions are quite different ("teaching better" vs "not teaching at all").
To increase the confusion, the educational theory build on this was called "radical constructivism", misleadingly suggesting that this might be "something like the famous Piaget, only much more so", when if fact it is something completely different. The kids taught using this philosophy are left alone to reinvent the math... and fail predictably! Or sometimes they are taught dozen different methods how to do addition (bonus point if the method was used by some indigenous population, because, you know, "noble savage", doesn't matter if the specific method only works for adding 7+8 and 8+9), hoping that this will kickstart their math thinking so now they will develop the rest of the math independently. Predictably, that also never happens.
So, how is this related to the "new math"? The curriculum of the "new math" was based exactly on this "radical constructivist" thought, except the authors did not emphasise the "radical" part enough and often just called it "constructivism". So you had the math curriculum that didn't work at all, and was a complete disaster. And when finally people got angry and returned to the traditional math education, the lesson everyone remembered was that "constructivism has been debunked".
So now, whenever someone proposes to teach math in a way that emphasizes understanding over memorization, the kneejerk reaction is "haha, that sounds like constructivism... yeah, we tried that but that didn't work at all", plus a link to some web page that criticizes "new math". And if you try to explain how this is completely unrelated to Piaget, you are dismissed with "yeah right, the true constructivism has never been tried, comrades, hahaha".
And then, ironically, people keep quoting the Feynman's story about how actually understanding physics is better than mere memorization that "light is waves". And the helpless (Piagetian) construstivist is like "yeah guys, that's exactly what I was trying to tell you all the time", but no one cares, and when it comes back to adding some understanding to the lessons, someone inevitably comes with the condescending "haha, but constructivism has been debunked" and a link to Ten Facts Why New Math Sucks.
Possible solution: perhaps the brand of "constructivism" has been thoroughly poisoned, and we need to reinvent it and call it "Feynmanism". Then you can go and say "nope, I am totally not proposing a constructivist curriculum, that has already been tried and debunked, haha, what I am proposing instead is the Feynmanist curriculum", and then people will go online and say "wow, I hated math at school, but then we got a new teacher who used this new Feynmanist method, and the math finally started make sense and now I love it".