mathematicians didn’t (and still mostly don’t) understand survivorship bias.
Yes. To use the terms from the article, mathematicians have the content knowledge, but most of them miss the pedagogical knowledge. Also, a frequent objection to constructivist education of math in Czechia today is some high-status mathematician saying “I didn’t need this at school, so what’s the point?” And I feel like: “Dude, this is not about the few lucky people like you and me who got it (often because we had some kind of math education outside the school system, e.g. from parents) but about the 90% of kids who don’t get math and then hate it.”
Cannot comment on math ed researchers; haven’t met any of them in person, other than the group of activists that promote the constructivist-math-as-I-know-it, which of course is an extremely biased sample.
Politicians, yeah. One Czech member of parliament keeps saying things like “experimenting on kids is dangerous” and “constructivism causes kids to have mental disorders”; and then the next day he goes “constructivism actually says nothing new, it’s merely a summary of things that all good teachers already know”. (Does he even realize that his opinions, taken together, imply that good teachers cause mental disorders?) His qualifications? He used to be a sales manager, and now is a politician… so the newspapers are interested in whatever he says, and then the (Czech) Wikipedia page quotes him in the “criticism” section, so the people who make their research by reading Wikipedia (yes, we were all told we shouldn’t, but...) can hear both sides of the controversy.
E.g., one reform effort experimented with removing drilling times tables, and parents were in revolt about this. Why? Because they learned via times tables, so why wasn’t that good enough for their kids??
I’m going to nitpick here. There is a difference between learning and practice. In my opinion, one should learn multiplication without the tables (by repeated addition), then explore it (e.g. notice that “10n” is just appending a zero, “9n” can be calculated as “10n—n”, multiplication is commutative, etc.), but at some moment after the understanding is solid (maybe a year later, not sure) the kids should practice until they memorize it. The “tables” are merely a tool; one might use a spaced repetition software instead or whatever else, I don’t care; the important part is that at some moment you hear “8×9” and respond “72″ by instinct, otherwise you will have a problem focusing on complex tasks that require “8×9” somewhere in the middle. Put the understand first, sure. Removing the practice entirely is taking it ad absurdum.
Also, let’s empathise with the parents a little. They don’t get the math, because of how they were taught decades ago. Yet, they will have to answer their kids’ questions at home. What if the child gets sick and misses a week or two at school? Tutoring may be too expensive for some. The problem is real, we can’t just dismiss it. (A simple solution that took me 5 seconds to think of, would be to make a YouTube channel for each reform, containing videos that explain all the changed parts: the motivation, and how to teach it. The channel would be visibly mentioned in the textbook.)
I fully agree about the set theory; what kids do at elementary school has almost nothing in common with the set theory qua set theory (ZFC, etc.), they were just happy to use a high-status keyword.
The thing to understand here is, CGI is mostly inquiry-based. You come from such a rich foundation of PCK that you know what problems to present to your students to guide them to notice and sometimes reinvent the mathematical tools they need for their next stage of mathematical growth. If they get stuck, you either have the PCK to know what hint to give them, or you seek PCK in real time to see why they’re stuck and what hint will actually help them think instead of just get the right answer. This doesn’t always follow the linear progression of topics that’s normal in a lecture-based math course, especially as conversation in the classroom meanders student attention and interest into a different direction.
Haven’t read the textbook yet, but my first impression is that this is a weakness (which according to your story turned out to be fatal). Realistically, teachers need a textbook with a linear progression of topics, with the right problems already provided for them. It’s great if they can improvise, but they shouldn’t be required to, if the textbook can do this for them. “The method needs to work for average humans” is a sensible constraint.
I hope to have kids one day. If I do, I will move mountains and seas to make sure they’re educated at home. The only benefit of public schools anymore, from what I can tell, is that very wise and patient parents can use it to support their children in mastering Defense Against the Dark Arts. Well, that and getting to play with other kids. Which is still pretty cool.
Actually, homeschooling is even better for “playing with other kids”; you just need to have other non-crazy homeschooling families in your neighborhood. Then the kids can play all day long (and also get huge discounts at various private playgrounds which are mostly empty in the morning).
My daughter just finished her first grade. So far, she is quite happy at school, because everything is super easy for her… because she already knew it before she went there. Well, if she is happy, then I am happy, too. No need to homeschool now. But of course, this may change later.
*
Now, one thing that is missing from your story, and which seems very relevant to me. The Wikipedia page on constructivism mentions something called “radical constructivism”, which allegedly “rejects an objective reality independent of human perception or reason” and “looks to break with the conception of knowledge as a correspondence between a knower’s understanding of their experience and the world beyond that experience”. They claim to draw on the work of Piaget, but also call his version of constructivism “trivial”.
Wikipedia says they were “influential in educational research and the philosophy of science”. Their online journal is called “Constructivist Foundations”.
This seems to contradict your framing, which—if I may simplify it so—says that the theory was good, and everyone tried to be a good Piagetian at first, but then… arrogant mathematicians, incompetent teachers, limited budgets, Goodharting.
But this information from Wikipedia suggests that the bad (from my perspective, anti-rational) ideas were already out there, and presented themselves as the true constructivism (as opposed to silly “trivial” Piaget who naively believed that there is such a thing as reality, ha ha). I admit I have never heard of them before, but Wikipedia says that they were influential, and… some stupid ideas used in the math-wars textbooks actually make sense from their perspective. Like, what is the point of actually teaching proper math, if even the reality itself is not real… just let the kids do whatever comes to their minds, and signal that we are more constructivist than Piaget by carefully avoiding anything that might resemble actual teaching.
So to me it seems like the teachers / politicians / school administrators were told to do “constructivism”, but then there came two groups of people claiming to be experts, who gave them completely contradictory advice on what “constructivism” actually is. And the bad guys had greater political clout, or maybe just their ideas seemed easier and cheaper to implement, so most people went with their version… and then it turned out that “constructivism” is a bad idea.
(BTW I didn’t “×-vote” your comment. Saying it explicitly, because now it’s just the two of us debating.)
but at some moment after the understanding is solid (maybe a year later, not sure) the kids should practice until they memorize it.
I totally disagree. This alienates kids from their developing inner agency. “You’ll now learn this because we said so.” As opposed to learning what they need because they can see & feel their own need for it.
Also, let’s empathise with the parents a little.
Obviously.
I’m not going to filter my speech to be appropriate across all dimensions. I don’t regret what I said or how I said it.
But obviously I agree with you.
(I mean, it’s kind of the main point of the PCK stuff!)
Realistically, teachers need a textbook with a linear progression of topics, with the right problems already provided for them.
Only if you’re teaching computation classes instead of mathematics.
Notice this isn’t an issue in English literature, or music.
But I agree that given the current bonkers goals — both overt and implicit — for math classes, this kind of linearity is key.
I just also think those goals are bollocks and in some cases downright nasty.
Now, one thing that is missing from your story, and which seems very relevant to me. The Wikipedia page on constructivism mentions something called “radical constructivism”…
Oh jeez. I honestly just forgot about those guys.
The heyday of radical constructivism was before my time. But my impression was that it was mostly philosophy that math ed researchers used to confused each other. Kind of like how modern art is some bizarre combo of oneupmanship amongst artists plus a front for tax evasion. Radical constructivism was part of the wave of postmodern philosophy justifying grants for all kinds of weird research projects. It offered context for endless intelligent-sounding debate.
It had incoherent suggestions about what to do for education. It was kind of the behaviorism of education theory. It sounded technically right and quite compelling if you didn’t look too closely. Once you started trying to figure out what to actually do with it though… :-P
But I don’t recall it claiming there’s no objective reality. I remember it rejecting the ability to know about any objective reality, so talking about it directly is a kind of absurdity in their view. But like all postmodern philosophies, it runs into a basic paradox: Is that claim absolutely true, meaning you can access some parts of objective reality, thereby invalidating the point? Or is it only sometimes true, which also implies there are some objective truths you can access somewhere sometimes?
(I could totally believe some radical constructivists thought about this and tried to get around it by claiming there’s no objective reality. This is exactly the same postmodern move as when radical social relativism insists that the laws of physics are just a bunch of ideas imposed by the racist patriarchy to support the established social structures of power. It tries to sidestep the logical challenge by shattering the mind’s ability to follow reason at all.)
I don’t recall the radical constructivists having an oversized effect on the math wars. They affected how math ed researchers thought and talked, but roughly the kind of way behaviorism affected how psychologists thought. I think it’d be pretty weird to list behaviorism as an influence on the math wars (even though it actually did play a pretty big role for context-setting in decades prior).
But hey, I could be mistaken. Maybe the postmodern philosophers of math ed research shoved themselves to the front and started babbling gibberish at teachers until everything fell apart. And maybe if we could own that and stop it from happening again, we could get sane policies into schools.
I’m not holding my breath for that one though.
So to me it seems like the teachers / politicians / school administrators were told to do “constructivism”, but then there came two groups of people claiming to be experts…
I honestly wasn’t aware that the word “constructivism” was a hot topic. I wouldn’t have thought there’d be much point in talking about “constructivism” explicitly when proposing education policy. That’d be like a plumber trying to explain viscosity and Bernoulli’s principle to house owners.
But given that math ed researchers did, I think the cause is messier than what you’re describing.
Basically anyone who was inspired by Piaget called their thing “constructivism”. There are lots of variations. The “radical” vs. “trivial” split isn’t the only one. Like I hinted at before, mathematicians created a “constructivist” approach based on constructing (!) mathematical objects and operations in the minds of children following the same rigorous pathway the mathematicians used to derive their tools from ZMF.
I faintly remember encountering at least one paper that claimed that it was constructivist because it was looking at levels of development. Because Piaget inspired the idea of levels of development, you see, and that’s based on constructing understanding, and so with “Piaget” and “construction” in the same idea cluster surely that makes you a constructivist, right?
I remember one of my teachers was basically a Piaget scholar and encouraged us to refer to Piagetian Constructivism, and to dissect various papers for where their implicit philosophy or explicit methodology differed from key points of Piaget’s approach (as she interpreted it). Frankly there weren’t many papers that survived that kind of analysis.
It’s possible the word “constructivism” also underwent something kind of like “nanotechnology” did: If funding appeared for “constructivist” approaches, then anyone who wanted funding could get it by finding a way to describe their pet project as “constructivist”, thus diluting the term.
For the most part I never want to use that word in educational contexts. It’s a messy ball of confusion just begging for pointless arguments. Some of the reason I honed in on Gears is precisely because it promises to cut through… gosh, I don’t know, but my gut claims it’s something like 2⁄3 of the babbling nonsense that shows up in education spheres once someone brings up “constructivism” (or “understanding”).
(BTW I didn’t “×-vote” your comment. Saying it explicitly, because now it’s just the two of us debating.)
Okay. I wasn’t worried about it. But thanks for mentioning.
And… honestly I really dislike framing this as a debate. I guess it is now? I wasn’t debating you before this. Just letting you know my impressions. I thought we were just comparing notes.
This alienates kids from their developing inner agency.
If you have 20-30 kids in the classroom, and an externally given list of goals to achieve, this puts a constraint on agency.
Also, some kids have an aversion against practicing stuff. Often the smart ones—they sometimes identify as “intelligent”, and it is a part of their self-image that they are supposed to learn things by mere understanding; anything that resembles work means for them that they have failed, because they were supposed to learn it without working hard. I knew very smart kids who just couldn’t learn a foreign language, because the idea of “memorizing by repetition” horrified them, and… nothing else worked. Their less smart classmates already learned the languages by practicing.
There are schools that try to maximize agency. And there is also unschooling, with the same goal. I suspect that kids who learn this way, will usually miss all the stuff that has very long inferential distances—because to get there, you need to walk a long way, and not each step is intrinsically exciting. (Reminds me of people in Mensa who can spend endless hours debating relativity or quantum physics, but never find time to read a textbook and fix their elementary misconceptions.)
So… yeah, I would seek some compromise between agency and knowledge. I might be convinced otherwise by some research that would show that average unschooled kids are more successful along some dimension than average school kids. It seems to me that unschooling is more enjoyable, but does not typically translate into following one’s own educational goals or projects.
Basically anyone who was inspired by Piaget called their thing “constructivism”.
If the label is diluted to uselessness, we need some new way to talk about the useful parts. One possibility is to just list the useful parts individually, without having an umbrella term. Not sure how well this would work… I guess I would need to compile the list first.
So… yeah, I would seek some compromise between agency and knowledge.
To each their own. I don’t value any knowledge so dearly that it’s worth sacrificing chunks of children’s agency to make sure they have said knowledge. The willingness to make that trade is key to the lifecycle of that which would create unFriendly AI.
If the label is diluted to uselessness, we need some new way to talk about the useful parts. One possibility is to just list the useful parts individually, without having an umbrella term.
Well… if you look above, you’ll see that you were the one who introduced the label!
As I said, I gave up using “constructivism” to describe things in this space years ago.
Yes. To use the terms from the article, mathematicians have the content knowledge, but most of them miss the pedagogical knowledge. Also, a frequent objection to constructivist education of math in Czechia today is some high-status mathematician saying “I didn’t need this at school, so what’s the point?” And I feel like: “Dude, this is not about the few lucky people like you and me who got it (often because we had some kind of math education outside the school system, e.g. from parents) but about the 90% of kids who don’t get math and then hate it.”
Cannot comment on math ed researchers; haven’t met any of them in person, other than the group of activists that promote the constructivist-math-as-I-know-it, which of course is an extremely biased sample.
Politicians, yeah. One Czech member of parliament keeps saying things like “experimenting on kids is dangerous” and “constructivism causes kids to have mental disorders”; and then the next day he goes “constructivism actually says nothing new, it’s merely a summary of things that all good teachers already know”. (Does he even realize that his opinions, taken together, imply that good teachers cause mental disorders?) His qualifications? He used to be a sales manager, and now is a politician… so the newspapers are interested in whatever he says, and then the (Czech) Wikipedia page quotes him in the “criticism” section, so the people who make their research by reading Wikipedia (yes, we were all told we shouldn’t, but...) can hear both sides of the controversy.
I’m going to nitpick here. There is a difference between learning and practice. In my opinion, one should learn multiplication without the tables (by repeated addition), then explore it (e.g. notice that “10n” is just appending a zero, “9n” can be calculated as “10n—n”, multiplication is commutative, etc.), but at some moment after the understanding is solid (maybe a year later, not sure) the kids should practice until they memorize it. The “tables” are merely a tool; one might use a spaced repetition software instead or whatever else, I don’t care; the important part is that at some moment you hear “8×9” and respond “72″ by instinct, otherwise you will have a problem focusing on complex tasks that require “8×9” somewhere in the middle. Put the understand first, sure. Removing the practice entirely is taking it ad absurdum.
Also, let’s empathise with the parents a little. They don’t get the math, because of how they were taught decades ago. Yet, they will have to answer their kids’ questions at home. What if the child gets sick and misses a week or two at school? Tutoring may be too expensive for some. The problem is real, we can’t just dismiss it. (A simple solution that took me 5 seconds to think of, would be to make a YouTube channel for each reform, containing videos that explain all the changed parts: the motivation, and how to teach it. The channel would be visibly mentioned in the textbook.)
I fully agree about the set theory; what kids do at elementary school has almost nothing in common with the set theory qua set theory (ZFC, etc.), they were just happy to use a high-status keyword.
Haven’t read the textbook yet, but my first impression is that this is a weakness (which according to your story turned out to be fatal). Realistically, teachers need a textbook with a linear progression of topics, with the right problems already provided for them. It’s great if they can improvise, but they shouldn’t be required to, if the textbook can do this for them. “The method needs to work for average humans” is a sensible constraint.
Actually, homeschooling is even better for “playing with other kids”; you just need to have other non-crazy homeschooling families in your neighborhood. Then the kids can play all day long (and also get huge discounts at various private playgrounds which are mostly empty in the morning).
My daughter just finished her first grade. So far, she is quite happy at school, because everything is super easy for her… because she already knew it before she went there. Well, if she is happy, then I am happy, too. No need to homeschool now. But of course, this may change later.
*
Now, one thing that is missing from your story, and which seems very relevant to me. The Wikipedia page on constructivism mentions something called “radical constructivism”, which allegedly “rejects an objective reality independent of human perception or reason” and “looks to break with the conception of knowledge as a correspondence between a knower’s understanding of their experience and the world beyond that experience”. They claim to draw on the work of Piaget, but also call his version of constructivism “trivial”.
Wikipedia says they were “influential in educational research and the philosophy of science”. Their online journal is called “Constructivist Foundations”.
This seems to contradict your framing, which—if I may simplify it so—says that the theory was good, and everyone tried to be a good Piagetian at first, but then… arrogant mathematicians, incompetent teachers, limited budgets, Goodharting.
But this information from Wikipedia suggests that the bad (from my perspective, anti-rational) ideas were already out there, and presented themselves as the true constructivism (as opposed to silly “trivial” Piaget who naively believed that there is such a thing as reality, ha ha). I admit I have never heard of them before, but Wikipedia says that they were influential, and… some stupid ideas used in the math-wars textbooks actually make sense from their perspective. Like, what is the point of actually teaching proper math, if even the reality itself is not real… just let the kids do whatever comes to their minds, and signal that we are more constructivist than Piaget by carefully avoiding anything that might resemble actual teaching.
So to me it seems like the teachers / politicians / school administrators were told to do “constructivism”, but then there came two groups of people claiming to be experts, who gave them completely contradictory advice on what “constructivism” actually is. And the bad guys had greater political clout, or maybe just their ideas seemed easier and cheaper to implement, so most people went with their version… and then it turned out that “constructivism” is a bad idea.
(BTW I didn’t “×-vote” your comment. Saying it explicitly, because now it’s just the two of us debating.)
Cool. Have fun.
I totally disagree. This alienates kids from their developing inner agency. “You’ll now learn this because we said so.” As opposed to learning what they need because they can see & feel their own need for it.
Obviously.
I’m not going to filter my speech to be appropriate across all dimensions. I don’t regret what I said or how I said it.
But obviously I agree with you.
(I mean, it’s kind of the main point of the PCK stuff!)
Only if you’re teaching computation classes instead of mathematics.
Notice this isn’t an issue in English literature, or music.
But I agree that given the current bonkers goals — both overt and implicit — for math classes, this kind of linearity is key.
I just also think those goals are bollocks and in some cases downright nasty.
Oh jeez. I honestly just forgot about those guys.
The heyday of radical constructivism was before my time. But my impression was that it was mostly philosophy that math ed researchers used to confused each other. Kind of like how modern art is some bizarre combo of oneupmanship amongst artists plus a front for tax evasion. Radical constructivism was part of the wave of postmodern philosophy justifying grants for all kinds of weird research projects. It offered context for endless intelligent-sounding debate.
It had incoherent suggestions about what to do for education. It was kind of the behaviorism of education theory. It sounded technically right and quite compelling if you didn’t look too closely. Once you started trying to figure out what to actually do with it though… :-P
But I don’t recall it claiming there’s no objective reality. I remember it rejecting the ability to know about any objective reality, so talking about it directly is a kind of absurdity in their view. But like all postmodern philosophies, it runs into a basic paradox: Is that claim absolutely true, meaning you can access some parts of objective reality, thereby invalidating the point? Or is it only sometimes true, which also implies there are some objective truths you can access somewhere sometimes?
(I could totally believe some radical constructivists thought about this and tried to get around it by claiming there’s no objective reality. This is exactly the same postmodern move as when radical social relativism insists that the laws of physics are just a bunch of ideas imposed by the racist patriarchy to support the established social structures of power. It tries to sidestep the logical challenge by shattering the mind’s ability to follow reason at all.)
I don’t recall the radical constructivists having an oversized effect on the math wars. They affected how math ed researchers thought and talked, but roughly the kind of way behaviorism affected how psychologists thought. I think it’d be pretty weird to list behaviorism as an influence on the math wars (even though it actually did play a pretty big role for context-setting in decades prior).
But hey, I could be mistaken. Maybe the postmodern philosophers of math ed research shoved themselves to the front and started babbling gibberish at teachers until everything fell apart. And maybe if we could own that and stop it from happening again, we could get sane policies into schools.
I’m not holding my breath for that one though.
I honestly wasn’t aware that the word “constructivism” was a hot topic. I wouldn’t have thought there’d be much point in talking about “constructivism” explicitly when proposing education policy. That’d be like a plumber trying to explain viscosity and Bernoulli’s principle to house owners.
But given that math ed researchers did, I think the cause is messier than what you’re describing.
Basically anyone who was inspired by Piaget called their thing “constructivism”. There are lots of variations. The “radical” vs. “trivial” split isn’t the only one. Like I hinted at before, mathematicians created a “constructivist” approach based on constructing (!) mathematical objects and operations in the minds of children following the same rigorous pathway the mathematicians used to derive their tools from ZMF.
I faintly remember encountering at least one paper that claimed that it was constructivist because it was looking at levels of development. Because Piaget inspired the idea of levels of development, you see, and that’s based on constructing understanding, and so with “Piaget” and “construction” in the same idea cluster surely that makes you a constructivist, right?
I remember one of my teachers was basically a Piaget scholar and encouraged us to refer to Piagetian Constructivism, and to dissect various papers for where their implicit philosophy or explicit methodology differed from key points of Piaget’s approach (as she interpreted it). Frankly there weren’t many papers that survived that kind of analysis.
It’s possible the word “constructivism” also underwent something kind of like “nanotechnology” did: If funding appeared for “constructivist” approaches, then anyone who wanted funding could get it by finding a way to describe their pet project as “constructivist”, thus diluting the term.
For the most part I never want to use that word in educational contexts. It’s a messy ball of confusion just begging for pointless arguments. Some of the reason I honed in on Gears is precisely because it promises to cut through… gosh, I don’t know, but my gut claims it’s something like 2⁄3 of the babbling nonsense that shows up in education spheres once someone brings up “constructivism” (or “understanding”).
Okay. I wasn’t worried about it. But thanks for mentioning.
And… honestly I really dislike framing this as a debate. I guess it is now? I wasn’t debating you before this. Just letting you know my impressions. I thought we were just comparing notes.
If you have 20-30 kids in the classroom, and an externally given list of goals to achieve, this puts a constraint on agency.
Also, some kids have an aversion against practicing stuff. Often the smart ones—they sometimes identify as “intelligent”, and it is a part of their self-image that they are supposed to learn things by mere understanding; anything that resembles work means for them that they have failed, because they were supposed to learn it without working hard. I knew very smart kids who just couldn’t learn a foreign language, because the idea of “memorizing by repetition” horrified them, and… nothing else worked. Their less smart classmates already learned the languages by practicing.
There are schools that try to maximize agency. And there is also unschooling, with the same goal. I suspect that kids who learn this way, will usually miss all the stuff that has very long inferential distances—because to get there, you need to walk a long way, and not each step is intrinsically exciting. (Reminds me of people in Mensa who can spend endless hours debating relativity or quantum physics, but never find time to read a textbook and fix their elementary misconceptions.)
So… yeah, I would seek some compromise between agency and knowledge. I might be convinced otherwise by some research that would show that average unschooled kids are more successful along some dimension than average school kids. It seems to me that unschooling is more enjoyable, but does not typically translate into following one’s own educational goals or projects.
If the label is diluted to uselessness, we need some new way to talk about the useful parts. One possibility is to just list the useful parts individually, without having an umbrella term. Not sure how well this would work… I guess I would need to compile the list first.
To each their own. I don’t value any knowledge so dearly that it’s worth sacrificing chunks of children’s agency to make sure they have said knowledge. The willingness to make that trade is key to the lifecycle of that which would create unFriendly AI.
Well… if you look above, you’ll see that you were the one who introduced the label!
As I said, I gave up using “constructivism” to describe things in this space years ago.