Maybe, but it's interesting because I had quite a few discussions with my kids about infinity. It came up when playing the "how much do I love you" game. I introduced infinity, and from there I was able to teach them a little about it. They were very young and I am still impressed with how easily they grasped some of the concepts.
Do you think it's possible that they accepted your ideas of infinity in a dogmatic sort of way, similar to how many young children accept god? I remember first having discussions with my father about infinity as young as 6, but I didn't really start exploring infinity for myself until my mid teens. It would be interesting to see how your kids thoughts change as they mature.
I do think they accepted it in a dogmatic way. Incidentally, I am raising our children Catholic so it's an idea that fits pretty well with the idea of God, the Holy Trinity, and all that. Three in One, yet one is the Father and the other is the Son? Yes, dear. Oh, ok. <grin>
Please don't post snarky dismissals to HN. They're corrosive of the higher-quality discussion we're hoping for here, especially because they generate replies that end up taking the thread in a less interesting direction.
Interesting read. Though Socratic method much? The writer uses leading questions[1] as a primary expository method, directing the kid towards conclusions that "should" be made within the proposed system, rather than merely exposing a small set of rules and letting the child freely explore their consequences. All the talk about the different infinities felt that way.
This leading style may be good for teaching the history of mathematics, i.e. introducing concepts that have been relevant to solve problems in the past. But for teaching math skills I prefer methods where a problem is stated, and the student is left to build their own abstractions about it; the IOUs and thermometer examples were more in this line. This competence will prove more useful when solving new problems than merely following the trodden path of already invented mathematics.
I think it's noteworthy that the article you point to pertains to common law and testimony, rather than to philosophy, pedagogy, or education. When a student freely explores a concept, that clearly demonstrates a higher ability or skill in mathematics. But "leading questions" are scaffold to show the student how to explore and build abstractions, and most will have to learn how to do that from example.
The Wikipedia article may be primarily written about legal cases, but leading questions appear in many other contexts. In psychology and interviews they're often advised against, as they induce a precise answer and thus make it hard to learn about the subject of study, instead reinforcing the interviewer preconceptions; in user interface design they can produce the wrong conclusions about how easy an interface is to use.
Leading questions have their use in education, but as the other commentator below points out, the child might be giving the answer that the questioner wants to hear, rather than showing a true understanding of the topic. Having a stated problem to solve may prevent that, as the child can instead answer the question in a way that solves the problem.
I know this isn't the point of the article, but did anyone else LOL at the bottom of page 3 when the difference of 10^100 - (ninety nine 9's) was given as 1? The author seems proud to attest "There really are ninety nine 9s!" The problem though is that this doesn't equal 1. You would need to subtract a number with one hundred 9's in a row to get an answer of 1.
I would like to see fMRI studies on how human experience the infinite or perhaps near infinite. There is an instinctual appeal of vastness (near infinite, perceptually infinite?) - mountains are climbed, sunsets into far away horizons over seemingly infinite oceans, skyscrapers are built and their height and subsequent view are valued,... - These are times when humans first experience vastness which may trigger pathways in the brain related to comprehending / thinking about the infinite. I would like to see is the parts of the brain associated with self-awareness are active during these experiences.
Mathematical concepts like nothing, straight lines, circles, are also abstractions that have no complete reference in nature - is it just infinity that you consider needs self-awareness? I'm not quite sure what you're hinting at?
I was just musing, no concrete connections at the moment. I don't know if infinity needs self-awareness, but I was thinking that it could prompt one to call into question his/her self relative to the infinite that is being experienced, and in that perhaps adjust one's concept of self through comparison. Or even just thinking about how one can conceptualize such a thing based upon all life experiences where almost everything encountered is finite.
In my experience, children (and most people) just conceptualize infinity as a very large set, just as they perceive a tire a "circle", and think a square drawn on paper is made up of "lines".
It requires years of thinking about these mathematical entities before people get a first glimpse of what they really mean, beyond a simplistic qualitative experience.
To most people, children included, infinity is conceptually no different from the amount of sand on a beach. They just allow that it is simply not quantifiable. That turns off the brain.
The more interesting thing to test, I would argue, would be the human mind conceptualizing a very, very large but ultimately finite set of something. That requires that we make a distinction between the discreteness of something and our ability to quantify it.
That is a much more difficult concept to comprehend, and it must be understood before we can even begin to think about teaching a child about mathematical entities like infinity, circles, or lines.
I vaguely recall having a "night terror" centered around infinity when I was very young. I think I was trying to truly grasp how tiny and insignificant I am and how immensely huge everything else is and it just freaked me out.
I think I know what you mean. When I was a kid, I got a feeling of complete emptiness lasting maybe couple of seconds when I was thinking about what if there wouldn't be anything.
I still fall into a rabbit hole every-time i imagine infinity. it started when i was 5 or so. brain goes into a loop and imagines the infinite. for me it was and is connected to space. you have to hit a wall at some point, but then there is something behind the wall.. there can never be something that isn't there. the limited donut universe theory framed the space imagination. but the donut has to be surrounded by something. here goes the infinite loop again.
Just a few days ago I had an interesting conversation with my 4 year old son that went something like this:
"Daddy, what's the biggest number?"
"There is no such thing as the biggest number; numbers keep getting bigger and bigger forever. This is a concept called infinity."
"Which number is the biggest?"
And so on; he's only 4, so I don't expect him to grasp the concept of infinity, but I was impressed that he's already asking about it.
My son is in his first year of Montessori education, and has learned to count into two digit numbers already, and understands zero, but doesn't yet understand negative numbers. Montessori is fairly interesting because they teach math in a different way, by learning powers of 10 and using a bead system similar to an abacus, so Montessori educated kids can actually do simple arithmetic with large numbers (in the thousands) by around age 5.
Tony Benn (UK politician) relates his grandson doing this in his diaries, but I've heard it related many times and suspect it doesn't come from that single source.
I'm not sure if this is normal or expected, but my two-year-old can count to 20. I'm not trying to brag-- maybe just seeing if maybe I should start bragging? ;)
Not really. We lived in an apartment from the time our youngest daughter was born until she was 2 or so. There was no elevator so we would count the stairs when we went up. I don't recall how many stairs there were, but she could count higher than 20 because of that. Of course she was a bit precocious. We were at university at the time and the only people she interacted with were other university students (no other kids around). By the time she was 2 she was fully fluent and could carry on lengthy conversation with any adult as if she was a college student. It was quite amusing.
Had a near exact conversation with my then 5 year old. He got really frustrated during the discussion as he wanted a straight answer but later understood what I meant.
Sounds like a child was picking up on OPs intonation when he asked the questions. That's how children answer. They pick on body language about what the correct answer is and basically tell what the one who asks wants them to tell. This was made especially easy by the binary format of questions. Yes/no answers are very easy to guess based on the way you ask. And people tend to ask extra suggestively when talking with children.
I'm not sure my son was even 6 yet (he may have been five) when he learned about subtraction. We started by talking about things: to illustrate 4 minus 2, you start with four marbles and remove 2 of them, leaving 2. For a couple of weeks he would ask me to subtract numbers, then he moved on to asking me to ask him numbers (and writing down simply arithmetic problems to solve).
Then, one day, out of the blue, he asked with a look of incredulity in his eye: "Daddy. What is one minus TWO???"
He clearly realized that he had come up with a question which had no representation in the marble-based construct.
I was so proud.
I sat down with him and drew a number line...and then showed him how we could extend the number line to the other side of zero.
He does know about infinity now (he's 7) -- and his most recent revelation that there is no such thing as an infinite number of any physical object. Even germs. Even atoms -- no matter how tiny, an infinite number of them can't exist in the universe (which he certainly pictures as finite).
I had a similar experience when my oldest was still young. The conversation was a little flipped. We live in a cold climate and she had started taking note of the thermometer. I had asked her "what is 5 - 5?" > "zero!"; "what is that minus 5?" > "you can't do that!" > "What happens if it is 0 degrees out and it gets 5 degrees colder?" > * eyes open wide * "there are all the same numbers below zero!?"
Some time later, when she was learning algebra, I told her "0.999... == 1, can you tell me why?". I was expecting an algebra lesson to show her how that was true. She surprised me when, a second or two later, she looked up and said, "1/9 is 0.111... so 9/9 is 0.999... and 9/9 is 1". I was very proud.
So this resonates with me. In general I don't have a particularly detailed memory of my early childhood. However, I do recall one day when I was 4 years old sitting at the end of the driveway waiting for my dad to get home from work.
I had picked up 2 digit counting from my listening to my mom work with my older sister. As I am sitting there by myself counting and then I got to 99. At that moment something partly clicked and I realized that 3 digits let me continue counting and I could just carry on the pattern. I remember being pretty excited at this and counted for awhile longer ... maybe another 20 or 30 digits and then in a flash I realized that the same thing works when I run out of 3 digits and I intuited the fact that there is really no end to counting. And, even though I didn't have a name for it I remember being kind of awestruck at something that could just go on and on without end.
It makes me smile today at 54 to be reminded of that rush of discovery.
I was born in communist Poland and back then everybody was catholic there, and so was I. So everyone had to go to the church, so I did too. I listened what the preast had to say about the heaven and the hell, but particularly that he said one would end up there for eternety.
That sunday lying in my bed I couldn't sleep. I was thinking about this infinity thing. I thought to my self, going to hell is bad because you get tortured and stuff, but going to heaven for eternity isn't really much better either, because it never ends! This was so frightening to me I couldn't sleep.
I was ok with going to hell for some really long amount of time, say 10k years, as long as I knew it would end some day. But being in heaven for ever? I still get a bad feeling when I remember my thoughts from back then, that would be unbearable, you couldn't even commit suicide, nothing would change anything, you'd be trapped forever with no hope of escape.
That was when I was seven or something, when I was eleven we moved to Germany where not everybody was catholic, half of the people were protestants. Then later when I was 27, so 10 years ago, I moved to Sweden where most people are atheists, and with time I also liberated myself from this scary religion.
Now I feel really happy about the fact that there most probably is no god and thus no inifinite amount of time in heaven either.
By the conservation of matter, matter cannot be created or destroyed. Thus in a certain sense you were not created but have always existed to infinity in the past (as far as we know the singularity which came before the big bang had no beginning). So is it a strange thing to think of yourself as continuing to exist in some form to infinity in the future?
For the same reason I have found the notion of Christian God to be very scary. A father who does not die and keeps judging his children and is always powerful.
40 comments
[ 3.0 ms ] story [ 71.3 ms ] threadThis leading style may be good for teaching the history of mathematics, i.e. introducing concepts that have been relevant to solve problems in the past. But for teaching math skills I prefer methods where a problem is stated, and the student is left to build their own abstractions about it; the IOUs and thermometer examples were more in this line. This competence will prove more useful when solving new problems than merely following the trodden path of already invented mathematics.
[1] https://en.wikipedia.org/wiki/Leading_question
Leading questions have their use in education, but as the other commentator below points out, the child might be giving the answer that the questioner wants to hear, rather than showing a true understanding of the topic. Having a stated problem to solve may prevent that, as the child can instead answer the question in a way that solves the problem.
It requires years of thinking about these mathematical entities before people get a first glimpse of what they really mean, beyond a simplistic qualitative experience.
To most people, children included, infinity is conceptually no different from the amount of sand on a beach. They just allow that it is simply not quantifiable. That turns off the brain.
The more interesting thing to test, I would argue, would be the human mind conceptualizing a very, very large but ultimately finite set of something. That requires that we make a distinction between the discreteness of something and our ability to quantify it.
That is a much more difficult concept to comprehend, and it must be understood before we can even begin to think about teaching a child about mathematical entities like infinity, circles, or lines.
"Daddy, what's the biggest number?"
"There is no such thing as the biggest number; numbers keep getting bigger and bigger forever. This is a concept called infinity."
"Which number is the biggest?"
And so on; he's only 4, so I don't expect him to grasp the concept of infinity, but I was impressed that he's already asking about it.
My son is in his first year of Montessori education, and has learned to count into two digit numbers already, and understands zero, but doesn't yet understand negative numbers. Montessori is fairly interesting because they teach math in a different way, by learning powers of 10 and using a bead system similar to an abacus, so Montessori educated kids can actually do simple arithmetic with large numbers (in the thousands) by around age 5.
Son: the biggest number is 1081
Dad: what about 1082
Son: well, I was close
Then, one day, out of the blue, he asked with a look of incredulity in his eye: "Daddy. What is one minus TWO???"
He clearly realized that he had come up with a question which had no representation in the marble-based construct.
I was so proud.
I sat down with him and drew a number line...and then showed him how we could extend the number line to the other side of zero.
He does know about infinity now (he's 7) -- and his most recent revelation that there is no such thing as an infinite number of any physical object. Even germs. Even atoms -- no matter how tiny, an infinite number of them can't exist in the universe (which he certainly pictures as finite).
Some time later, when she was learning algebra, I told her "0.999... == 1, can you tell me why?". I was expecting an algebra lesson to show her how that was true. She surprised me when, a second or two later, she looked up and said, "1/9 is 0.111... so 9/9 is 0.999... and 9/9 is 1". I was very proud.
I had picked up 2 digit counting from my listening to my mom work with my older sister. As I am sitting there by myself counting and then I got to 99. At that moment something partly clicked and I realized that 3 digits let me continue counting and I could just carry on the pattern. I remember being pretty excited at this and counted for awhile longer ... maybe another 20 or 30 digits and then in a flash I realized that the same thing works when I run out of 3 digits and I intuited the fact that there is really no end to counting. And, even though I didn't have a name for it I remember being kind of awestruck at something that could just go on and on without end.
It makes me smile today at 54 to be reminded of that rush of discovery.
That sunday lying in my bed I couldn't sleep. I was thinking about this infinity thing. I thought to my self, going to hell is bad because you get tortured and stuff, but going to heaven for eternity isn't really much better either, because it never ends! This was so frightening to me I couldn't sleep.
I was ok with going to hell for some really long amount of time, say 10k years, as long as I knew it would end some day. But being in heaven for ever? I still get a bad feeling when I remember my thoughts from back then, that would be unbearable, you couldn't even commit suicide, nothing would change anything, you'd be trapped forever with no hope of escape.
That was when I was seven or something, when I was eleven we moved to Germany where not everybody was catholic, half of the people were protestants. Then later when I was 27, so 10 years ago, I moved to Sweden where most people are atheists, and with time I also liberated myself from this scary religion.
Now I feel really happy about the fact that there most probably is no god and thus no inifinite amount of time in heaven either.