All good questions. My gut response is that these exact problems—which level to memorize vs. construct mental models—are exactly what I was attempting to communicate. You’re always telling something, even when you’re showing.
A more concrete response is that every level is constructed atop the one beneath it; multiplication is repeated addition, quadratic equations require multiplication, etc.
So for children, pick something near the bottom, something relevant to their actual, real, Newtonian lives (so addition, because adding things happens all the time, and not set theory, which is arguably more fundamental but further removed from the normal life of a child) and have them memorize it. Then construct the thing on top of it.
Once they’ve constructed the thing on top of whatever they’ve memorized, have them memorize that and construct the next layer. And then keep going.
A basic math course might then look like this:
Memorize addition, then
Construct multiplication on top of addition, then
Memorize multiplication, then
Construct exponentiation on top of multiplication, then
Memorize exponentiation, then
Construct quadratic equations on top of exponentiations, then
...
So the repeated pattern is:
Memorize the lower level skill, then
Construct the higher level skill, then
Repeat, but with the higher level skill taking the place of the lower level skill
This makes me think of climbing a ladder, where first you reach for a rung with your hands, then eventually step on the same rung as you climb higher.
This can also work going “down” as well to get to more fundamental levels:
Memorize Newtonian physics, then
Construct relativity (by exploring where Newtonian physics fails, or where the simplification/approximation reaches its limits), then
Memorize relativity, then
Construct quantum mechanics (similar to above, but with different limits/failures), then
Memorize quantum mechanics, then
Construct string theory or something...?
In the end, you’ve got to start somewhere, with something, knowing that you won’t be able to really explain it because doing so requires knowledge the student doesn’t yet possess (by definition). So pick something simple and get started up the ladder, knowing that at some point you’ll want to come back around and construct the original thing that was taught.
You’re always telling something, even when you’re showing.
I find myself thinking “yes, obviously”, but at the same time there seems to be something we disagree about, so what it is exactly? I think it might be about what exactly the teacher is telling; to put it simply, at the extreme, there are two teaching styles:
build the models carefully, check that the students have the right models;
just say a bunch of true facts in random order without feedback, the smart kids will sort it out, and the rest of them… I guess they were not talented enough, “the camel has two humps”, etc.
And “constructivism” as I use it, kinda means: do more of the former and less of the latter. But there is of course more to it, like provide enough time for the kids to build and debug those models, show how the existing models relate to the new things, etc.
To get more specific, I believe that kids should definitely have a model of what addition is, on the level of “adding five and eight is like having five apples in one hand and eight apples in the other hand, how many total apples do you have?”. This model is enough to derive e.g. that 5+8 = 8+5; you just need to notice this first on a few specific cases, and then you can justify it using the model, like “if I am already holding the apples in my hands, and then I like switch my hands, the number of apples remains the same”.
Without a model, only having memorized lots of additions, if you ask whether addition is commutative, the answer will be like “well, I don’t remember any example to the contrary, so… maybe?”
Maybe what I am trying to say could be put like: of course you are always telling something, that is inevitable, but it is better to increase the parts where the kids themselves can connect the dots. And then we can go to specific techniques how to prepare the dots so that they are easy to connect.
not set theory
100% agree. I suspect that set theory was just high-status at some moment and people couldn’t resist “hey, if we can make the kids use the word ‘set’, we will make them high-status mathematicians”. Of course it does not work that way; the things that kids do with “sets” at elementary school have nothing in common with the set theory as usual. So, drop the “sets” entirely, IMHO.
More general, we need to distinguish between something being simple in the… vulgar sense, such as addition being simpler (less work, easier to understand for a child) than multiplication, and being more fundamental from some perspective, like quarks are more simple (easier to define for a scientist) than apples. The former is a good heuristic for elementary schools, the latter is not.
So the repeated pattern is:
Memorize the lower level skill, then
Construct the higher level skill, then
Repeat, but with the higher level skill taking the place of the lower level skill
Doesn’t this means that all skills—except for the first one—are explained first and memorized later? Then why make the exception for the first one?
Technically, addition of integers is also a sequence of skills. There is a difference between 2+3 (counting on fingers or memorizing), 8+5 (thinking how it wraps across ten: 8+2+3), and 7416+2872 (arranging vertically, then adding the digits starting from the last one).
(By coincidence, I recently made this tool to teach/train addition and subtraction; don’t mind the language, just click on the bullet points in order, it is self-explanatory. Needs JavaScript enabled.)
Maybe I should say it explicitly that I am not opposed to training/memorization, that I agree that it is super necessary, and I think maybe we should use computers at school and just spend the first 5 minutes of the lesson doing a quick “spaced repetition” exercise at the beginning of each lesson. I just think that the proper moment for memorization is after the things were properly understood.
...at some moment in future I would like to write a few articles on this, including screenshots from an actual constructivist math textbook that I already have at home, so that we don’t have to discuss this in abstract. The reason is that there were many shitty “constructivist” textbooks published in USA, and I want to make it clear that I am definitely not defending those ones, not even the ideas they were built upon, which to me feel like a strawman of the original idea. I suspect that if I showed you the actual textbook, you would see nothing wrong with it (at least compared to the usual textbooks). But I always procrastinate a lot with writing articles.
In the end, you’ve got to start somewhere, with something,
That “something” is the knowledge kids already have (hopefully) when they go to school. Mathematics is not a separate magisterium; it is an abstraction built upon things from everyday life. The primitive pre-mathematical knowledge is like “realizing that two apples are more than one”; you build on that.
at the same time there seems to be something we disagree about, so what it is exactly?
As to the disagreement you mention, I think I’m starting from the position of the child being taught being a complete blank slate, which is obviously inaccurate when taken literally.
A blank slate would have to be told things first, before any models could be built, because you need a nonzero amount of knowledge about the world to build any models at all.
More realistically (and as you said), a child should be expected to be a functional human child by the time they get to a teacher (and math is not a separate magisterium), so starting with a model isn’t impossible (because the child already possesses facts to build the model with). I do believe that the memorize → model → memorize → model loop is how learning happens; the question is where in that loop the teacher meets the student.
I suspect that set theory was just high-status at some moment
Completely agree with you on Set Theory.
Doesn’t this means that all skills—except for the first one—are explained first and memorized later? Then why make the exception for the first one?
The exception is made because I assumed a human was starting with a complete blank slate, which is not literally true as I agreed above.
Perhaps a better example than addition would be the first time someone tries to learn a foreign language; I would argue that some vocabulary has to be memorized first, because that’s the foundation upon which everything else rests (you can’t start by trying to teach grammar, for instance).
...at some moment in future I would like to write a few articles on this
I look forward to reading them!
(By coincidence, I recently made this tool to teach/train addition and subtraction; don’t mind the language, just click on the bullet points in order, it is self-explanatory. Needs JavaScript enabled.)
I tried it out. Simple but nice! One thing I noticed was that for the answers that were “10”, you could just leave them as “1″ without it being marked as either right or wrong (green or red). Not sure if that’s a feature or a bug.
Wow, glad we came to an agreement, I actually didn’t expect that.
Not sure if that’s a feature or a bug.
That’s on purpose, glad you noticed! Green = the correct answer. Yellow = not the correct answer, but a prefix of it (that includes an empty string). Red = neither the correct answer, nor a prefix of it.
Like, if the correct answer is “42”, then “4″ is yellow, because for all I know maybe you are halfway to writing the correct answer, so I don’t want to scare you needlessly. (Though maybe I should later update it to red when you leave the text field… and update back to yellow when you return? Nah, sounds like too much work.)
Seems to happen to me here a lot more often than IRL.
Like, if the correct answer is “42”, then “4″ is yellow, because for all I know maybe you are halfway to writing the correct answer, so I don’t want to scare you needlessly. (Though maybe I should later update it to red when you leave the text field… and update back to yellow when you return? Nah, sounds like too much work.)
The only problem with the box remaining yellow that I see is that it conveys partial information, because it turns red if the digit is wrong.
In other words, if a student wanted to fill out the boxes by brute force, without actually doing any math, just by trying numbers, they’d be able to get to multi-digit answers by trying out 1-9 until they found the number that didn’t cause the box to turn red, then moving on to the next digit.
Off the top of my head, the simple way to fix it would be to do the correctness check after focus leaves the box (triggered by leaving the box, as it were); that can apply to every box and ensures the student can’t brute-force the answer as above.
For some reason I procrastinate for months when trying to write articles, but can write an insanely long comment whenever I get angry about something. So here is a story about “constructivism” in education, as a Hacker News comment.
Give me a few more months, and I will probably rewrite it to a LW article, and then it will get like 5 karma total, heh.
I mean, what other response is possible when someone is wrong on the internet?
Either way, I’m looking forward to it.
Edit: after reading the comment, I feel like I have a better understanding of how we might’ve been talking past each other a bit. I do agree with your position.
Also, the history of education is a terrifying and depressing subject, in my experience.
All good questions. My gut response is that these exact problems—which level to memorize vs. construct mental models—are exactly what I was attempting to communicate. You’re always telling something, even when you’re showing.
A more concrete response is that every level is constructed atop the one beneath it; multiplication is repeated addition, quadratic equations require multiplication, etc.
So for children, pick something near the bottom, something relevant to their actual, real, Newtonian lives (so addition, because adding things happens all the time, and not set theory, which is arguably more fundamental but further removed from the normal life of a child) and have them memorize it. Then construct the thing on top of it.
Once they’ve constructed the thing on top of whatever they’ve memorized, have them memorize that and construct the next layer. And then keep going.
A basic math course might then look like this:
Memorize addition, then
Construct multiplication on top of addition, then
Memorize multiplication, then
Construct exponentiation on top of multiplication, then
Memorize exponentiation, then
Construct quadratic equations on top of exponentiations, then
...
So the repeated pattern is:
Memorize the lower level skill, then
Construct the higher level skill, then
Repeat, but with the higher level skill taking the place of the lower level skill
This makes me think of climbing a ladder, where first you reach for a rung with your hands, then eventually step on the same rung as you climb higher.
This can also work going “down” as well to get to more fundamental levels:
Memorize Newtonian physics, then
Construct relativity (by exploring where Newtonian physics fails, or where the simplification/approximation reaches its limits), then
Memorize relativity, then
Construct quantum mechanics (similar to above, but with different limits/failures), then
Memorize quantum mechanics, then
Construct string theory or something...?
In the end, you’ve got to start somewhere, with something, knowing that you won’t be able to really explain it because doing so requires knowledge the student doesn’t yet possess (by definition). So pick something simple and get started up the ladder, knowing that at some point you’ll want to come back around and construct the original thing that was taught.
I find myself thinking “yes, obviously”, but at the same time there seems to be something we disagree about, so what it is exactly? I think it might be about what exactly the teacher is telling; to put it simply, at the extreme, there are two teaching styles:
build the models carefully, check that the students have the right models;
just say a bunch of true facts in random order without feedback, the smart kids will sort it out, and the rest of them… I guess they were not talented enough, “the camel has two humps”, etc.
And “constructivism” as I use it, kinda means: do more of the former and less of the latter. But there is of course more to it, like provide enough time for the kids to build and debug those models, show how the existing models relate to the new things, etc.
To get more specific, I believe that kids should definitely have a model of what addition is, on the level of “adding five and eight is like having five apples in one hand and eight apples in the other hand, how many total apples do you have?”. This model is enough to derive e.g. that 5+8 = 8+5; you just need to notice this first on a few specific cases, and then you can justify it using the model, like “if I am already holding the apples in my hands, and then I like switch my hands, the number of apples remains the same”.
Without a model, only having memorized lots of additions, if you ask whether addition is commutative, the answer will be like “well, I don’t remember any example to the contrary, so… maybe?”
Maybe what I am trying to say could be put like: of course you are always telling something, that is inevitable, but it is better to increase the parts where the kids themselves can connect the dots. And then we can go to specific techniques how to prepare the dots so that they are easy to connect.
100% agree. I suspect that set theory was just high-status at some moment and people couldn’t resist “hey, if we can make the kids use the word ‘set’, we will make them high-status mathematicians”. Of course it does not work that way; the things that kids do with “sets” at elementary school have nothing in common with the set theory as usual. So, drop the “sets” entirely, IMHO.
More general, we need to distinguish between something being simple in the… vulgar sense, such as addition being simpler (less work, easier to understand for a child) than multiplication, and being more fundamental from some perspective, like quarks are more simple (easier to define for a scientist) than apples. The former is a good heuristic for elementary schools, the latter is not.
Doesn’t this means that all skills—except for the first one—are explained first and memorized later? Then why make the exception for the first one?
Technically, addition of integers is also a sequence of skills. There is a difference between 2+3 (counting on fingers or memorizing), 8+5 (thinking how it wraps across ten: 8+2+3), and 7416+2872 (arranging vertically, then adding the digits starting from the last one).
(By coincidence, I recently made this tool to teach/train addition and subtraction; don’t mind the language, just click on the bullet points in order, it is self-explanatory. Needs JavaScript enabled.)
Maybe I should say it explicitly that I am not opposed to training/memorization, that I agree that it is super necessary, and I think maybe we should use computers at school and just spend the first 5 minutes of the lesson doing a quick “spaced repetition” exercise at the beginning of each lesson. I just think that the proper moment for memorization is after the things were properly understood.
...at some moment in future I would like to write a few articles on this, including screenshots from an actual constructivist math textbook that I already have at home, so that we don’t have to discuss this in abstract. The reason is that there were many shitty “constructivist” textbooks published in USA, and I want to make it clear that I am definitely not defending those ones, not even the ideas they were built upon, which to me feel like a strawman of the original idea. I suspect that if I showed you the actual textbook, you would see nothing wrong with it (at least compared to the usual textbooks). But I always procrastinate a lot with writing articles.
That “something” is the knowledge kids already have (hopefully) when they go to school. Mathematics is not a separate magisterium; it is an abstraction built upon things from everyday life. The primitive pre-mathematical knowledge is like “realizing that two apples are more than one”; you build on that.
As to the disagreement you mention, I think I’m starting from the position of the child being taught being a complete blank slate, which is obviously inaccurate when taken literally.
A blank slate would have to be told things first, before any models could be built, because you need a nonzero amount of knowledge about the world to build any models at all.
More realistically (and as you said), a child should be expected to be a functional human child by the time they get to a teacher (and math is not a separate magisterium), so starting with a model isn’t impossible (because the child already possesses facts to build the model with). I do believe that the memorize → model → memorize → model loop is how learning happens; the question is where in that loop the teacher meets the student.
Completely agree with you on Set Theory.
The exception is made because I assumed a human was starting with a complete blank slate, which is not literally true as I agreed above.
Perhaps a better example than addition would be the first time someone tries to learn a foreign language; I would argue that some vocabulary has to be memorized first, because that’s the foundation upon which everything else rests (you can’t start by trying to teach grammar, for instance).
I look forward to reading them!
I tried it out. Simple but nice! One thing I noticed was that for the answers that were “10”, you could just leave them as “1″ without it being marked as either right or wrong (green or red). Not sure if that’s a feature or a bug.
Wow, glad we came to an agreement, I actually didn’t expect that.
That’s on purpose, glad you noticed! Green = the correct answer. Yellow = not the correct answer, but a prefix of it (that includes an empty string). Red = neither the correct answer, nor a prefix of it.
Like, if the correct answer is “42”, then “4″ is yellow, because for all I know maybe you are halfway to writing the correct answer, so I don’t want to scare you needlessly. (Though maybe I should later update it to red when you leave the text field… and update back to yellow when you return? Nah, sounds like too much work.)
I like coming to agreements too!
Seems to happen to me here a lot more often than IRL.
The only problem with the box remaining yellow that I see is that it conveys partial information, because it turns red if the digit is wrong.
In other words, if a student wanted to fill out the boxes by brute force, without actually doing any math, just by trying numbers, they’d be able to get to multi-digit answers by trying out 1-9 until they found the number that didn’t cause the box to turn red, then moving on to the next digit.
Off the top of my head, the simple way to fix it would be to do the correctness check after focus leaves the box (triggered by leaving the box, as it were); that can apply to every box and ensures the student can’t brute-force the answer as above.
For some reason I procrastinate for months when trying to write articles, but can write an insanely long comment whenever I get angry about something. So here is a story about “constructivism” in education, as a Hacker News comment.
Give me a few more months, and I will probably rewrite it to a LW article, and then it will get like 5 karma total, heh.
I mean, what other response is possible when someone is wrong on the internet?
Either way, I’m looking forward to it.
Edit: after reading the comment, I feel like I have a better understanding of how we might’ve been talking past each other a bit. I do agree with your position.
Also, the history of education is a terrifying and depressing subject, in my experience.