I was trying to recall if I recalled thinking of this age as being so focused on enquiry. In general, the impression that I have of children this age is of an intelligent but somewhat alien creature. He does not yet really follow how our language works, though he seems to have some of his own and he has moments of communication breakthrough when he understands what we say or communicates his desires or impressions. He does not really see why it is that we encounter things the way that we do, nor why he should follow our methods if his own seem more interesting. Why do the grown-ups insist on showing him that his little wooden train can be rolled, when it's clearly much more interesting to decide exactly how it tastes? Why do the adults seem so fixated on the eating of applesauce, when they have put no real exploration into its use as a hair styling aid?
Though it also struck me that having a child this age during the lockdown of March-April this year when the piece seemed to have been written must have been a very unique experience. As I was casting memories back to when our eldest was that age, I realized it was during a period when Cat and I had moved in with my dad's 93 year old mother to help provide her with live in care, and we were already pregnant with our second, and I had a commute which took me nearly two hours each way across the LA basin. No wonder my memories of that era were a bit vague.
And while, of course, each baby goes through this age, it's a different experience for a baby with other small children in the family as well. I think it was our third, at this age, who would toddle after the older two calling out one of her first words, "Guys! Guys!" There was a small tribe of persons who looked at the world in a <5 kind of way, and in some ways they formed their own child-view world which we interacted with as outsiders.
I certainly hope I will one day be outnumbered by a little-child society of siblings. That's a fascinating point about large families having such a strong shift in the average perspective in the house.
I do like baby led weaning (our approach) for giving the baby a lot of room to be curious about food.
OK, in imitation of classical UY discourse I'm associating this with something vaguely math-y while ignoring most of the poetical and cute aspects:
A lot of this is actually about graph traversal! Things you ask/explore have follow-up relations. For example, Fox is starting out with a lot of things to pull on. Some of these lead to reactions and then he could either next explore something that changed or appeared (follow-up) or he could just pull on something different and see what happens there. So basically possible explorations are nodes of a graph connected by the follow-up relation as edges. Fox wants to find a big and interesting number of explorations fast, or in math to rapidly traverse a big and interesting part of the graph.
Graph traversal is also a big part of what computer science is about. CS101 offers two extremist strategies: In depth-first-traversal you always explore the first follow-up and then the first follow-up to that and so on. The problem with that is, if the graph is infinitely deep like this one you never finish your first exploration. The other extreme strategy is what Fox is doing now: In breadth-first-traversal you start with your initially visible explorations and do them all before doing any of the follow-ups. Only after finishing the original explorations you do the same with your original follow-ups as next first-level explorations &c. That way you get to every exploration eventually, better then depth-first! But! In the long run it may not be optimal either, because you take very long getting not very deep and doing a lot of low-interest explorations such as yellow play-douh tastes just like blue play-doh. So eventually you probably adopt some sort of compromise strategy. The exact compromise varies from person to person, people who like more depth than most are called nerds, often x-nerds where x is a name for the region of the tree they are recursing in. But almost everybody eventually sacrifices a lot of breadth for depth, doing so less than most is sometimes called ADD.
But all such compromise strategies depend on special knowledge of how the graph looks like. In Martyn's imagery, you need to do a lot of minotaur thought to build your mental labyrinth in the first place. So for the moment Fox is doing optimal computer science!
I chuckled when I read "imitation of classical UY discourse" and glanced up to see if it would be any name I'd recognize, and all I can say is "GILBERT!" Hi! I remember you! (Specifically, I remember that one brief comment you made to help someone else was incredibly helpful to me at the time. I even remember what it was about.)
So... about the breadth-first traversal of The Graph Of Experiments: it also has the value of making it easier for little ones to be able to track patterns ...like what things are in common to a series of experiments on knocking different objects over.
Or... one example that's mostly not about knocking things over. I remember when one little one was in the "two eyes!" phase, enthusiastically pointing out when a given animal in a book had the "has exactly two eyes" property. And then one day he wobbled excitedly into the kitchen, catching his balance on the handle of the stove and proclaimed to me, "He haves two eyes!" "Who has two eyes?" I asked, pretty sure I knew the answer. "Brudder!"
Back to the topic of knocking things over, and therefore gravity... my husband tutors physics. He agonizes over the way that many students come to him functionally having, in his words, "the pagan worldview of a chaotic universe... where they sincerely don't expect the universe to be governed by consistent laws." (Ideally, the breadth-first Tree-o-Experiments would help counter that!)
Could you point me toward one of Montessori's writings you particularly love? I've read more _about_ Montessori education but little _by_ Maria Montessori herself!
It's been a while since I had a child at this stage. It's fun to remember how delightful it was to ponder and imagine what was going on in there, what it was like to be that age and to try to understand how exactly early learning is happening.
I loved reading books about the topic of early learning What's Going on in There was one, The Scientist in the Crib. But I stopped reading as much about it or thinking about it now that I no longer have any babies or toddlers to observe and wonder about. I rather miss that stage of parenting...
Thanks for this month's read. I'm especially looking forward to your conversation with Chana Messinger since I found parts of this article connecting with questions I have about numerical understanding in young children. I've recently been spending more time with my three-year old godson and the paragraph in the piece about whether Fox associates the word "Bingley" with their specific dog or dogs in general reminds me of an experience I had during my last meeting with him.
He has a large toy car in the shape of a lion (i.e. large enough to sit on) and as a mane, the lion has the numbers 1 to 9 written, each one in a separate yellow circle. He now knows how to point to each circle in order and count relatively quickly from 1 to 9. However, when I tried to extend him further by asking him to go backwards from 9, he couldn't do it. I tried pointing at the 9, to which he accurately responded "nine", then pointed at the 8, to which he accurately said "eight", and so on with 7. However, when I asked him to keep going independently, he said "seven eight nine". I then tried pointing somewhere in the middle (say 5), and it took him a while before he was able to identify the number. Once we got that down and I returned to trying to get him to go in reverse order, he would make it a few numbers in, before seemingly uncontrollably heading back in the positive direction ("five... four... fivesixseveneightnine").
I found this fascinating and it made me wonder - what is actually going on in his mind? Is he connecting each of the symbols in front of him to specific sounds, or connecting the physical motion (pointing and moving his hand in a circle) to specific sounds? Or rather, is he simply memorising a sequence of noises and actions that he can produce on command to elicit praise (like Fox picks up his toys)? It's probably worth adding that English is not his first language.
I happen to also be a high school math teacher (another reason I'm looking forward to your conversation with Chana!) and I am aware through educational research and personal experience how difficult it is to correct incorrect models students have - say for example using "moving" to do algebra - models which work for simple problems, but make it challenging for them to do anything meaningfully complex. Obviously minds as young as Fox's and my godson's are more flexible, but it is interesting to note that there are in fact consequences to developing these wrong models in the first place.
" I am aware through educational research and personal experience how difficult it is to correct incorrect models students have - say for example using "moving" to do algebra - models which work for simple problems, but make it challenging for them to do anything meaningfully complex. Obviously minds as young as Fox's and my godson's are more flexible, but it is interesting to note that there are in fact consequences to developing these wrong models in the first place."
I think I've witnessed that in action and a child who had to totally remake her model when the old one failed to be sufficient to her needs. I homeschool so I've had the delight of teaching early math five times with only my own previous math experiences (and a lot of books, of course) to go on. It's been fascinating to watch my kids enter into the world of numeracy.
My oldest daughter seems to have synesthesia and early on numbers clearly had an association with colors for her, but also I think she had some kind of mental map of them, they seemed to live in houses. But only the numbers 1 through 10. When she started to have to do math that regularly required adding and subtracting larger numbers we went thought a period of intense frustration for her and after many conversations with her I became convinced that her mental model for numbers had become inadequate and she had to build a new one from the ground up. And again and again as she progressed through math, she'd hit a wall, get incredibly frustrated over and over again and eventually seem to have figured out a different way to visualize/conceptualize the world of numbers.
I should talk to her about this again, we haven't talked about it in a few years.
Yes! I have a stronger sense of the *personality* of small numbers, but didn't carry that lightly synesthetic sense over to big ones. The movement through models reminds me of my experience in HS chemistry, where we kept moving through different models of what an atom was (from a solar system of nuclei and orbiting electrons to a probability field of orbits!). It felt like being let in on a secret.
Struck by the presumed interrogative character of Fox's learning behaviors.
Possibly a distinction without a difference, but is that what's going on? It certainly seems that way when watching a child put things in her mouth, drop objects, look underneath things.
Which comes first? Trial and error observation or questioning orientation? Or do they emerge from and with each other?
This was an interesting read.
I was trying to recall if I recalled thinking of this age as being so focused on enquiry. In general, the impression that I have of children this age is of an intelligent but somewhat alien creature. He does not yet really follow how our language works, though he seems to have some of his own and he has moments of communication breakthrough when he understands what we say or communicates his desires or impressions. He does not really see why it is that we encounter things the way that we do, nor why he should follow our methods if his own seem more interesting. Why do the grown-ups insist on showing him that his little wooden train can be rolled, when it's clearly much more interesting to decide exactly how it tastes? Why do the adults seem so fixated on the eating of applesauce, when they have put no real exploration into its use as a hair styling aid?
Though it also struck me that having a child this age during the lockdown of March-April this year when the piece seemed to have been written must have been a very unique experience. As I was casting memories back to when our eldest was that age, I realized it was during a period when Cat and I had moved in with my dad's 93 year old mother to help provide her with live in care, and we were already pregnant with our second, and I had a commute which took me nearly two hours each way across the LA basin. No wonder my memories of that era were a bit vague.
And while, of course, each baby goes through this age, it's a different experience for a baby with other small children in the family as well. I think it was our third, at this age, who would toddle after the older two calling out one of her first words, "Guys! Guys!" There was a small tribe of persons who looked at the world in a <5 kind of way, and in some ways they formed their own child-view world which we interacted with as outsiders.
I certainly hope I will one day be outnumbered by a little-child society of siblings. That's a fascinating point about large families having such a strong shift in the average perspective in the house.
I do like baby led weaning (our approach) for giving the baby a lot of room to be curious about food.
OK, in imitation of classical UY discourse I'm associating this with something vaguely math-y while ignoring most of the poetical and cute aspects:
A lot of this is actually about graph traversal! Things you ask/explore have follow-up relations. For example, Fox is starting out with a lot of things to pull on. Some of these lead to reactions and then he could either next explore something that changed or appeared (follow-up) or he could just pull on something different and see what happens there. So basically possible explorations are nodes of a graph connected by the follow-up relation as edges. Fox wants to find a big and interesting number of explorations fast, or in math to rapidly traverse a big and interesting part of the graph.
Graph traversal is also a big part of what computer science is about. CS101 offers two extremist strategies: In depth-first-traversal you always explore the first follow-up and then the first follow-up to that and so on. The problem with that is, if the graph is infinitely deep like this one you never finish your first exploration. The other extreme strategy is what Fox is doing now: In breadth-first-traversal you start with your initially visible explorations and do them all before doing any of the follow-ups. Only after finishing the original explorations you do the same with your original follow-ups as next first-level explorations &c. That way you get to every exploration eventually, better then depth-first! But! In the long run it may not be optimal either, because you take very long getting not very deep and doing a lot of low-interest explorations such as yellow play-douh tastes just like blue play-doh. So eventually you probably adopt some sort of compromise strategy. The exact compromise varies from person to person, people who like more depth than most are called nerds, often x-nerds where x is a name for the region of the tree they are recursing in. But almost everybody eventually sacrifices a lot of breadth for depth, doing so less than most is sometimes called ADD.
But all such compromise strategies depend on special knowledge of how the graph looks like. In Martyn's imagery, you need to do a lot of minotaur thought to build your mental labyrinth in the first place. So for the moment Fox is doing optimal computer science!
I chuckled when I read "imitation of classical UY discourse" and glanced up to see if it would be any name I'd recognize, and all I can say is "GILBERT!" Hi! I remember you! (Specifically, I remember that one brief comment you made to help someone else was incredibly helpful to me at the time. I even remember what it was about.)
So... about the breadth-first traversal of The Graph Of Experiments: it also has the value of making it easier for little ones to be able to track patterns ...like what things are in common to a series of experiments on knocking different objects over.
Or... one example that's mostly not about knocking things over. I remember when one little one was in the "two eyes!" phase, enthusiastically pointing out when a given animal in a book had the "has exactly two eyes" property. And then one day he wobbled excitedly into the kitchen, catching his balance on the handle of the stove and proclaimed to me, "He haves two eyes!" "Who has two eyes?" I asked, pretty sure I knew the answer. "Brudder!"
Back to the topic of knocking things over, and therefore gravity... my husband tutors physics. He agonizes over the way that many students come to him functionally having, in his words, "the pagan worldview of a chaotic universe... where they sincerely don't expect the universe to be governed by consistent laws." (Ideally, the breadth-first Tree-o-Experiments would help counter that!)
I adore Montessori's writing on this topic, and Martyn's writing reminds me a bit of hers in flow and form. It's a beautiful short essay!
Could you point me toward one of Montessori's writings you particularly love? I've read more _about_ Montessori education but little _by_ Maria Montessori herself!
It's been a while since I had a child at this stage. It's fun to remember how delightful it was to ponder and imagine what was going on in there, what it was like to be that age and to try to understand how exactly early learning is happening.
I loved reading books about the topic of early learning What's Going on in There was one, The Scientist in the Crib. But I stopped reading as much about it or thinking about it now that I no longer have any babies or toddlers to observe and wonder about. I rather miss that stage of parenting...
Oooh, I've placed a library hold on The Scientist in the Crib
Thanks for this month's read. I'm especially looking forward to your conversation with Chana Messinger since I found parts of this article connecting with questions I have about numerical understanding in young children. I've recently been spending more time with my three-year old godson and the paragraph in the piece about whether Fox associates the word "Bingley" with their specific dog or dogs in general reminds me of an experience I had during my last meeting with him.
He has a large toy car in the shape of a lion (i.e. large enough to sit on) and as a mane, the lion has the numbers 1 to 9 written, each one in a separate yellow circle. He now knows how to point to each circle in order and count relatively quickly from 1 to 9. However, when I tried to extend him further by asking him to go backwards from 9, he couldn't do it. I tried pointing at the 9, to which he accurately responded "nine", then pointed at the 8, to which he accurately said "eight", and so on with 7. However, when I asked him to keep going independently, he said "seven eight nine". I then tried pointing somewhere in the middle (say 5), and it took him a while before he was able to identify the number. Once we got that down and I returned to trying to get him to go in reverse order, he would make it a few numbers in, before seemingly uncontrollably heading back in the positive direction ("five... four... fivesixseveneightnine").
I found this fascinating and it made me wonder - what is actually going on in his mind? Is he connecting each of the symbols in front of him to specific sounds, or connecting the physical motion (pointing and moving his hand in a circle) to specific sounds? Or rather, is he simply memorising a sequence of noises and actions that he can produce on command to elicit praise (like Fox picks up his toys)? It's probably worth adding that English is not his first language.
I happen to also be a high school math teacher (another reason I'm looking forward to your conversation with Chana!) and I am aware through educational research and personal experience how difficult it is to correct incorrect models students have - say for example using "moving" to do algebra - models which work for simple problems, but make it challenging for them to do anything meaningfully complex. Obviously minds as young as Fox's and my godson's are more flexible, but it is interesting to note that there are in fact consequences to developing these wrong models in the first place.
" I am aware through educational research and personal experience how difficult it is to correct incorrect models students have - say for example using "moving" to do algebra - models which work for simple problems, but make it challenging for them to do anything meaningfully complex. Obviously minds as young as Fox's and my godson's are more flexible, but it is interesting to note that there are in fact consequences to developing these wrong models in the first place."
I think I've witnessed that in action and a child who had to totally remake her model when the old one failed to be sufficient to her needs. I homeschool so I've had the delight of teaching early math five times with only my own previous math experiences (and a lot of books, of course) to go on. It's been fascinating to watch my kids enter into the world of numeracy.
My oldest daughter seems to have synesthesia and early on numbers clearly had an association with colors for her, but also I think she had some kind of mental map of them, they seemed to live in houses. But only the numbers 1 through 10. When she started to have to do math that regularly required adding and subtracting larger numbers we went thought a period of intense frustration for her and after many conversations with her I became convinced that her mental model for numbers had become inadequate and she had to build a new one from the ground up. And again and again as she progressed through math, she'd hit a wall, get incredibly frustrated over and over again and eventually seem to have figured out a different way to visualize/conceptualize the world of numbers.
I should talk to her about this again, we haven't talked about it in a few years.
Yes! I have a stronger sense of the *personality* of small numbers, but didn't carry that lightly synesthetic sense over to big ones. The movement through models reminds me of my experience in HS chemistry, where we kept moving through different models of what an atom was (from a solar system of nuclei and orbiting electrons to a probability field of orbits!). It felt like being let in on a secret.
Struck by the presumed interrogative character of Fox's learning behaviors.
Possibly a distinction without a difference, but is that what's going on? It certainly seems that way when watching a child put things in her mouth, drop objects, look underneath things.
Which comes first? Trial and error observation or questioning orientation? Or do they emerge from and with each other?
It's hard to tell! Certainly, at first everything is equally surprising. It takes time to build up expectations that might be contradicted.