9 ms·

"Is Zero a Number?": Interviews with a Whole Class of Kindergartners

MrRolleyes 3y ago
[dead]
lionkor 3y ago
What a fascinating article, thank you! I see why we have exams in schools. Some of the kids just seemed to entirely miss what they were taught! The way kids approach these questions are very telling on how to teach them, and that teaching math in particular is about understanding, not memorization.
- ndsipa_pomu 3y ago
  Personally, I think exams are a horrible way to determine what people have actually understood. Most school exams are just looking for specific "magic" words that are learnt by rote. Questions might be set up to ask why water appears on the outside of a cold glass and the answer would be considered correct if the answer used the word "condensation" without needing any concept of humidity.
  - lionkor 3y ago
    I believe this is an issue of how exams are done. There are certainly exams I've written, in Math, in English, in Philosophy, that tested my understanding, and there are certainly some that didnt.
    - ndsipa_pomu 3y ago
      Definitely agree. As a kid, it really bugged me when there were poorly worded questions that were often ambiguous if you thought about it. I found that exam success often relied on trying to think what the questions were looking for rather than what they were actually asking - basically considering the questions in the context of the course material.
- datavirtue 3y ago
  They may have missed what the teacher was trying to convey but this serves as an example, to me, of how some people just think differently. The girl who was fixated on shape, for instance. She saw two zeros making an 8. I think we need to be able to recognize early on that some people think differently than the "typical" person, and that they should be understood and perhaps taken down an educational path that fits their mind instead of just trying to pound the correct answers into their head. Sure, teach them the concept of zero as a number, but also recognize that these people are not typical. I used to assume that all humans had an internal dialog. This is not the case. Although most do, some people do not or cannot talk to themselves silently in their own head. This blew my mind. I viewed this as the core of being human. I had been interviewing for jobs and found that my internal dialog would kick in and just wreck my shit during interviews. After learning that some people don't have this I considered that I can just shut mine off if I need to. Basically telling myself to shut up. This did wonders for me, and helped me stay on track during interviews and helped to stop negative thoughts that I might have at other times. The internal dialog has value, but I have learned to control it because I no longer consider it essential to being human. This had just never occurred to me before. The point is, there is neurodivergance in humans. We do not all think the same and trying to mold all children into a thinking the same "correct" way is rather barbaric. Failing to acknowledge that people think differently on a fundamental level is also hurting everyone in some way. Some people can't see images in their head. Again, I had always assumed that all humans could picture anything they wanted, not true. And often these people are visual artists!
Rapzid 3y ago
Haha, this could pass as the results from asking 9th graders.
deleted 3y ago
[deleted]
bawolff 3y ago
I mean philosophically what is a "number" is a hard concept. I don't think any of the children saying 0 is not were really wrong so much as had a different definition of number. If you asked most adults what a number is, probably many would come up with a definition that didn't include i.
- ndsipa_pomu 3y ago
  Probably the simplest concept is that they're for counting things, but that doesn't lead to a clear answer as to whether you can count zero things. When you count things, zero has a physical meaning (if you have one apple and give it away, how many apples do you have?), so maybe they should be considering it a number, but not negative numbers until they get a more abstract understanding.
  - asddubs 3y ago
    maybe measuring is more accurate to say than counting. that way you include things like units of speed, and negative numbers (you can't really count -3)
    - LudwigNagasena 3y ago
      That means "huge" and "small" are numbers.
      - MrRolleyes 3y ago
        [dead]
    - ndsipa_pomu 3y ago
      Yes, measuring is a more advanced concept than counting as it introduces fractions. I don't think simple measuring would include negative numbers though as that would include a measuring direction concept (i.e. vectors) as rulers don't have negative numbers on them and you can't have a speed of -5kph.
Almondsetat 3y ago
People seem to forget that it took us tens of thousands of years to develop maths and a couple thousands more to conceptualize zero (and some time more for negative quantities and infinities). So in this light this result is absolutely not surprising. In fact, it’s funny to see that the reason the answers changed is basically “because the school told me so”
deleted 3y ago
[deleted]
kaetemi 3y ago
Try asking if B is a letter or a word. (Pronounce it as 'b', not 'bee', or you'll just make it more confusing.) Seems a lot of these basic classifications are an early stumbling block. Even if they can count or rattle the whole alphabet, they haven't learned the meaning of the words "word", "letter", or "number" as terminology. Initially sentences/words/letters/numbers are all just 'sound'. "What's the sound of this?" is equivalent to "Read this."
- morsch 3y ago
  Once you get into academic linguistics, determining what's a letter and what's a word gets very murky again. And you mostly just don't use those categories and talk about phonemes and utterances instead, ie. sound.
- hnlmorg 3y ago
  My wife is a primary school teacher and I assure you, UK kids are taught terminologies. I don’t think they were when I was a kid but curriculums change.
  - kaetemi 3y ago
    It is taught in primary school, yeah. I remember learning pseudo terminology like "do-words" for "verbs" in first and second grade. Pre-school they try to teach "reading" letters nowadays, but then they get drilled in that "A is for apple" and now the shape A could mean the sound apple, and they have even less of a clue on what letters are. (Even though in the 90s research already pointed out that reading should be taught with complete lowercase words instead of by letter picking.)
    - hnlmorg 3y ago
      No, they get taught terms like verb, digraph, as well as the difference in sounds between an and A. They don’t even teach uppercase letters until long after kids have mastered lower case. The whole “Annie Apple” thing went out of favour years ago. UK primary schools are very jargon heavy these days and phonics is all about teaching lower case letters. My wife is a primary school teacher, a few of my friends are teachers, and I have two kids in primary school. So this is a topic I’m very familiar with. What people forget is that the curriculum changes. It changes frequently. When I was at school the curriculum was very different to what it is like now. For example I actually learned the term “digraph” from my kids, not my own schooling. I mean, I always knew what digraphs and split digraphs were, but I never knew they were called digraphs.
otteromkram 3y ago
I like how one of the students noted that his dad said some kids can be zero years old. The phrase highlights the concept of whole vs. fractional numbers.
- gizajob 3y ago
  Yeah cool dad award. I wonder how many hours of questioning came from the back of the car before he explained it.
lozenge 3y ago
Good article. At age 11, I went downstairs to announce I was ready for tomorrow's maths exam, where we would need to plot equations like y = x - 3. My dad said, "well, here's my question - what is a line?" This angered me greatly as I couldn't answer.
- boomboomsubban 3y ago
  That doesn't really make sense. How did they explain what plotting was without defining a line? I guess they could have taught how to plot a line segment then explained they go on forever.
  - gizajob 3y ago
    In this context, a line is what you draw with a ruler to get the correct answer.
    - boomboomsubban 3y ago
      My math teachers would have marked it incorrect if I didn't indicate it went on forever, unless the problem stipulated bounds.
    - lozenge 3y ago
      This is it. He also asked me what the width of the line was and how could I say a line doesn't have a width.
      - gizajob 3y ago
        “Because Euclid says so in the Elements”
  - thaumasiotes 3y ago
    > How did they explain what plotting was without defining a line? Most plots aren't lines, so there's no reason this task would require defining a line. In fact, the shape formed by a graph is called a "curve" for just this reason.
pmontra 3y ago
The gist of it seems to be: intuitively zero is nothing and not a number but eventually you discover that it's useful to think about it and use it as a number so you change your mind, maybe without even noticing. By the way, some kids misunderstood the digit zero for the number zero. > “Yes, you already asked me! It’s a number for nothing. You can use it to make 100.” > “Yes, you already asked this! It’s in the number 10, so it must be a number. We don’t mix shapes like this.” Imagine the surprise if told that A, B, C etc are also numbers :-)
- thaumasiotes 3y ago
  > By the way, some kids misunderstood the digit zero for the number zero. That's not a misunderstanding. Place-value notation requires that digits are numbers. The fact that 10 is a number doesn't mean that the 0 within the 10 is not also a number; the kindergartener is correct in saying that that location means the 0 is a number.
  - gizajob 3y ago
    They’re also correct in saying it represents pure, unadulterated nothingness though. The teacher is confusing number theory with reality by limiting them in the way she does.
    - thaumasiotes 3y ago
      > They’re also correct in saying it represents pure, unadulterated nothingness though. This makes me hesitate. For example, if I ask you how many quiches there currently are in your refrigerator, it does not follow that nothing is currently in your refrigerator. The zero is bound to the unit it measures.
      - gizajob 3y ago
        Not in an empty universe.
    - c22 3y ago
dkbrk 3y ago
> Yes, because you can add it to other things. Good answer.
baobabKoodaa 3y ago
I was genuinely impressed how good the "before" answers were. Many adults refuse to accept that they don't know something and will rather invent a fictitious answer, start believing in it and proclaim it as fact - but many of these kindergarteners actually said "I don't know". Stunning emotional maturity! Also, the ones who answered yes or no were mostly able to craft arguments supporting their position. I didn't know kids are that smart.
- hiAndrewQuinn 3y ago
  Kids these days are indeed very willing to admit when they don't know things! I do suspect the openness is downstream of the slow cultural realization that there is just too much to know for any one human being to truly know it all. Smart parents are more willing than ever on the margin to say they don't know things, and the children learn from that example that this is okay to admit.
- OmarShehata 3y ago
  "I didn't know kids are that smart." Yes!! I wish more people recognized this. There's a "Free Range Kids"[0] movement growing out of this fact. A lot of people have ended up in a place where unstructured/unsupervised time is seen as a "waste" for kids, but I think this is the wrong direction. I think Khan Academy's new experimental high school[1] is also built around this fact (that kids are smart & just need access to resources & meaningful work) [0] https://www.freerangekids.com/ https://www.freerangekids.com/ [1] https://khanlabschool.org/About-Khan-Lab-School https://khanlabschool.org/About-Khan-Lab-School
- lozenge 3y ago
  At that age, isn't not knowing the default, and experienced many times every day? I wonder when we swap to assuming we know or can deduce everything.
  - Staple_Diet 3y ago
    [dead]
  - ksaj 3y ago
gizajob 3y ago
I feel like these children have been enlightened into number theory, but their original answers were nevertheless more correct than the later answers.
- darkerside 3y ago
  Education is a serious of increasingly smaller lies
yodsanklai 3y ago
It's a philosophical question which is always fun to think about. As far as teaching maths is concerned, I don't know if it's needed to think too much about it. Could it even be more confusing for kids?
- gizajob 3y ago
  I think giving them a clear, limited definition when their original answers were more intuitive is the part that introduces the confusion. Toddlers and kindergarteners are always bang on the money and don’t mess around where it comes to confusion, they have other things going on. That is to say, I’ve found that asking under sevens questions in ethics is always a brilliant process - where a professor could spend thousands of words on an article on something like the trolley problem, preschoolers just pull the lever.
vehementi 3y ago
> “Yes, you asked this, and I said my dad told me and he said some babies are zero.” Teacher, are you even paying attention? We've been over this.
yodsanklai 3y ago
I see natural numbers as an abstraction for counting things. Like "3" is what "3 apples", "3 cars", "3 stars" have in common. In that sense, "0" doesn't have a different status. I find it more difficult to explain "-1". Probably using a position along an axis with an origin, which gives you all the real numbers.
cubefox 3y ago
Zero is interesting because it is the only natural number that is a cardinal number but not an ordinal number. An array can have zero (no) elements, but it can't have a "zeroth" element. Despite some programming languages suggesting otherwise.
- ndsipa_pomu 3y ago
  Just imagine how perfect the world would be if we started counting at zero, though
mgaunard 3y ago
0 as a symbol didn't exist for millenia, and people were still counting and doing complex math. The teacher is acting as if its existence as a number is obvious.
a_c 3y ago
It seems to me after the lessons, more students are in the "because you said so" camp whereas before the lesson they have more reasoning. I wonder, as parent, how much "indoctrination" and how much "reasoning" shall we impart to our children.
- ndsipa_pomu 3y ago
  My experience of education in the UK is that reasoning and questioning were actively discouraged by some teachers. They were pressured to go through course material quickly, and having students presenting counter-arguments would slow things down.
devit 3y ago
The simplest definition of natural numbers is as the free algebraic structure generated by a nullary and unary operation (conventionally named "zero" and "succ"), which makes it natural to associate zero as the value of the nullary operation, so that one can be it's successor and addition by one works as expected. Not having zero and associating one as the result of the nullary operation works as well, but then the natural definition of addition will result in x + y - 1 rather than x + y and it seems likely extension to rational numbers would not give an intuitive result.
_8j50 3y ago
If zero is a number, atheism is a religion and asexuality is a sexuality. I don't disagree with how zero is used but the english language is the debate here not math. Zero is a legitimate concept in math, if a term describes certain things then is the concept of the lack of those things by default considered a member of those things? Sounds like a linguistic convention in part. Why not have a separate term to describe zero and infinity as part of the same set, "anumerics"?
gls2ro 3y ago
(My math is probably old and maybe I forgot some things) I think zero is kinda strange in a way: it is intuitive mostly when thinking about quantities but when dealing with other concepts I find it strange to think about it. Here are some situations where I know how to answer but reasoning about it is still in a way hard: N^0 or 0^N or 0! or or more simpler when thinking about limits and zero try to see what happens when you try to reach 0 while going like this 0.9^0.9, 0.8^0.8, … 0.001^0.001, 0.0001^0.0001 In most cases I think we are defining the answer. Maybe there are some complex proof for these answers but they only prove that zero is kinda hard to grasp. Or it might be that I forgot a lot of math and I am wrong with these examples.
ksaj 3y ago
In my younger days, I was a Beaver leader. Beavers are Boy Scouts in the 5-6 age group. They are not empty headed. They are extremely inquisitive and easily understand things that are surprising to adults. I would even say on average, their observational skills are better than those seen in adulthood. They're still young enough to have not had every question they ever hear from an adult as something they'll be graded on yet. Pass or Fail. So when they don't know, they tell you and hope that you'll tell them or guide them to the right answer. I personally believe this pressure in older kids and adults alike, to not FAIL (ie: not know the expected answer) is because of the way school tends to turn everything into a binary test. We are literally taught to avoid "I don't know."

[dead]

What a fascinating article, thank you! I see why we have exams in schools. Some of the kids just seemed to entirely miss what they were taught! The way kids approach these questions are very telling on how to teach them, and that teaching math in particular is about understanding, not memorization.

ndsipa_pomu 3y ago
Personally, I think exams are a horrible way to determine what people have actually understood. Most school exams are just looking for specific "magic" words that are learnt by rote. Questions might be set up to ask why water appears on the outside of a cold glass and the answer would be considered correct if the answer used the word "condensation" without needing any concept of humidity.
- lionkor 3y ago
  I believe this is an issue of how exams are done. There are certainly exams I've written, in Math, in English, in Philosophy, that tested my understanding, and there are certainly some that didnt.
  - ndsipa_pomu 3y ago
    Definitely agree. As a kid, it really bugged me when there were poorly worded questions that were often ambiguous if you thought about it. I found that exam success often relied on trying to think what the questions were looking for rather than what they were actually asking - basically considering the questions in the context of the course material.
datavirtue 3y ago
They may have missed what the teacher was trying to convey but this serves as an example, to me, of how some people just think differently. The girl who was fixated on shape, for instance. She saw two zeros making an 8. I think we need to be able to recognize early on that some people think differently than the "typical" person, and that they should be understood and perhaps taken down an educational path that fits their mind instead of just trying to pound the correct answers into their head. Sure, teach them the concept of zero as a number, but also recognize that these people are not typical. I used to assume that all humans had an internal dialog. This is not the case. Although most do, some people do not or cannot talk to themselves silently in their own head. This blew my mind. I viewed this as the core of being human. I had been interviewing for jobs and found that my internal dialog would kick in and just wreck my shit during interviews. After learning that some people don't have this I considered that I can just shut mine off if I need to. Basically telling myself to shut up. This did wonders for me, and helped me stay on track during interviews and helped to stop negative thoughts that I might have at other times. The internal dialog has value, but I have learned to control it because I no longer consider it essential to being human. This had just never occurred to me before. The point is, there is neurodivergance in humans. We do not all think the same and trying to mold all children into a thinking the same "correct" way is rather barbaric. Failing to acknowledge that people think differently on a fundamental level is also hurting everyone in some way. Some people can't see images in their head. Again, I had always assumed that all humans could picture anything they wanted, not true. And often these people are visual artists!

Rapzid 3y ago

Haha, this could pass as the results from asking 9th graders.

deleted 3y ago

[deleted]

bawolff 3y ago

I mean philosophically what is a "number" is a hard concept. I don't think any of the children saying 0 is not were really wrong so much as had a different definition of number. If you asked most adults what a number is, probably many would come up with a definition that didn't include i.

ndsipa_pomu 3y ago
Probably the simplest concept is that they're for counting things, but that doesn't lead to a clear answer as to whether you can count zero things. When you count things, zero has a physical meaning (if you have one apple and give it away, how many apples do you have?), so maybe they should be considering it a number, but not negative numbers until they get a more abstract understanding.
- asddubs 3y ago
  maybe measuring is more accurate to say than counting. that way you include things like units of speed, and negative numbers (you can't really count -3)
  - LudwigNagasena 3y ago
    That means "huge" and "small" are numbers.
    - MrRolleyes 3y ago
      [dead]
  - ndsipa_pomu 3y ago
    Yes, measuring is a more advanced concept than counting as it introduces fractions. I don't think simple measuring would include negative numbers though as that would include a measuring direction concept (i.e. vectors) as rulers don't have negative numbers on them and you can't have a speed of -5kph.

Almondsetat 3y ago

People seem to forget that it took us tens of thousands of years to develop maths and a couple thousands more to conceptualize zero (and some time more for negative quantities and infinities). So in this light this result is absolutely not surprising. In fact, it’s funny to see that the reason the answers changed is basically “because the school told me so”

deleted 3y ago

[deleted]

kaetemi 3y ago

Try asking if B is a letter or a word. (Pronounce it as 'b', not 'bee', or you'll just make it more confusing.) Seems a lot of these basic classifications are an early stumbling block. Even if they can count or rattle the whole alphabet, they haven't learned the meaning of the words "word", "letter", or "number" as terminology. Initially sentences/words/letters/numbers are all just 'sound'. "What's the sound of this?" is equivalent to "Read this."

morsch 3y ago
Once you get into academic linguistics, determining what's a letter and what's a word gets very murky again. And you mostly just don't use those categories and talk about phonemes and utterances instead, ie. sound.
hnlmorg 3y ago
My wife is a primary school teacher and I assure you, UK kids are taught terminologies. I don’t think they were when I was a kid but curriculums change.
- kaetemi 3y ago
  It is taught in primary school, yeah. I remember learning pseudo terminology like "do-words" for "verbs" in first and second grade. Pre-school they try to teach "reading" letters nowadays, but then they get drilled in that "A is for apple" and now the shape A could mean the sound apple, and they have even less of a clue on what letters are. (Even though in the 90s research already pointed out that reading should be taught with complete lowercase words instead of by letter picking.)
  - hnlmorg 3y ago
    No, they get taught terms like verb, digraph, as well as the difference in sounds between an and A. They don’t even teach uppercase letters until long after kids have mastered lower case. The whole “Annie Apple” thing went out of favour years ago. UK primary schools are very jargon heavy these days and phonics is all about teaching lower case letters. My wife is a primary school teacher, a few of my friends are teachers, and I have two kids in primary school. So this is a topic I’m very familiar with. What people forget is that the curriculum changes. It changes frequently. When I was at school the curriculum was very different to what it is like now. For example I actually learned the term “digraph” from my kids, not my own schooling. I mean, I always knew what digraphs and split digraphs were, but I never knew they were called digraphs.

otteromkram 3y ago

I like how one of the students noted that his dad said some kids can be zero years old. The phrase highlights the concept of whole vs. fractional numbers.

gizajob 3y ago
Yeah cool dad award. I wonder how many hours of questioning came from the back of the car before he explained it.

lozenge 3y ago

Good article. At age 11, I went downstairs to announce I was ready for tomorrow's maths exam, where we would need to plot equations like y = x - 3. My dad said, "well, here's my question - what is a line?" This angered me greatly as I couldn't answer.

boomboomsubban 3y ago
That doesn't really make sense. How did they explain what plotting was without defining a line? I guess they could have taught how to plot a line segment then explained they go on forever.
- gizajob 3y ago
  In this context, a line is what you draw with a ruler to get the correct answer.
  - boomboomsubban 3y ago
    My math teachers would have marked it incorrect if I didn't indicate it went on forever, unless the problem stipulated bounds.
  - lozenge 3y ago
    This is it. He also asked me what the width of the line was and how could I say a line doesn't have a width.
    - gizajob 3y ago
      “Because Euclid says so in the Elements”
- thaumasiotes 3y ago
  > How did they explain what plotting was without defining a line? Most plots aren't lines, so there's no reason this task would require defining a line. In fact, the shape formed by a graph is called a "curve" for just this reason.

pmontra 3y ago

The gist of it seems to be: intuitively zero is nothing and not a number but eventually you discover that it's useful to think about it and use it as a number so you change your mind, maybe without even noticing. By the way, some kids misunderstood the digit zero for the number zero. > “Yes, you already asked me! It’s a number for nothing. You can use it to make 100.” > “Yes, you already asked this! It’s in the number 10, so it must be a number. We don’t mix shapes like this.” Imagine the surprise if told that A, B, C etc are also numbers :-)

thaumasiotes 3y ago
> By the way, some kids misunderstood the digit zero for the number zero. That's not a misunderstanding. Place-value notation requires that digits are numbers. The fact that 10 is a number doesn't mean that the 0 within the 10 is not also a number; the kindergartener is correct in saying that that location means the 0 is a number.
- gizajob 3y ago
  They’re also correct in saying it represents pure, unadulterated nothingness though. The teacher is confusing number theory with reality by limiting them in the way she does.
  - thaumasiotes 3y ago
    > They’re also correct in saying it represents pure, unadulterated nothingness though. This makes me hesitate. For example, if I ask you how many quiches there currently are in your refrigerator, it does not follow that nothing is currently in your refrigerator. The zero is bound to the unit it measures.
    - gizajob 3y ago
      Not in an empty universe.
  - c22 3y ago

dkbrk 3y ago

> Yes, because you can add it to other things. Good answer.

baobabKoodaa 3y ago

I was genuinely impressed how good the "before" answers were. Many adults refuse to accept that they don't know something and will rather invent a fictitious answer, start believing in it and proclaim it as fact - but many of these kindergarteners actually said "I don't know". Stunning emotional maturity! Also, the ones who answered yes or no were mostly able to craft arguments supporting their position. I didn't know kids are that smart.

hiAndrewQuinn 3y ago
Kids these days are indeed very willing to admit when they don't know things! I do suspect the openness is downstream of the slow cultural realization that there is just too much to know for any one human being to truly know it all. Smart parents are more willing than ever on the margin to say they don't know things, and the children learn from that example that this is okay to admit.
OmarShehata 3y ago
"I didn't know kids are that smart." Yes!! I wish more people recognized this. There's a "Free Range Kids"[0] movement growing out of this fact. A lot of people have ended up in a place where unstructured/unsupervised time is seen as a "waste" for kids, but I think this is the wrong direction. I think Khan Academy's new experimental high school[1] is also built around this fact (that kids are smart & just need access to resources & meaningful work) [0] https://www.freerangekids.com/ https://www.freerangekids.com/ [1] https://khanlabschool.org/About-Khan-Lab-School https://khanlabschool.org/About-Khan-Lab-School
lozenge 3y ago
At that age, isn't not knowing the default, and experienced many times every day? I wonder when we swap to assuming we know or can deduce everything.
- Staple_Diet 3y ago
  [dead]
- ksaj 3y ago

gizajob 3y ago

I feel like these children have been enlightened into number theory, but their original answers were nevertheless more correct than the later answers.

darkerside 3y ago
Education is a serious of increasingly smaller lies

yodsanklai 3y ago

It's a philosophical question which is always fun to think about. As far as teaching maths is concerned, I don't know if it's needed to think too much about it. Could it even be more confusing for kids?

gizajob 3y ago
I think giving them a clear, limited definition when their original answers were more intuitive is the part that introduces the confusion. Toddlers and kindergarteners are always bang on the money and don’t mess around where it comes to confusion, they have other things going on. That is to say, I’ve found that asking under sevens questions in ethics is always a brilliant process - where a professor could spend thousands of words on an article on something like the trolley problem, preschoolers just pull the lever.

vehementi 3y ago

> “Yes, you asked this, and I said my dad told me and he said some babies are zero.” Teacher, are you even paying attention? We've been over this.

yodsanklai 3y ago

I see natural numbers as an abstraction for counting things. Like "3" is what "3 apples", "3 cars", "3 stars" have in common. In that sense, "0" doesn't have a different status. I find it more difficult to explain "-1". Probably using a position along an axis with an origin, which gives you all the real numbers.

cubefox 3y ago

Zero is interesting because it is the only natural number that is a cardinal number but not an ordinal number. An array can have zero (no) elements, but it can't have a "zeroth" element. Despite some programming languages suggesting otherwise.

ndsipa_pomu 3y ago
Just imagine how perfect the world would be if we started counting at zero, though

mgaunard 3y ago

0 as a symbol didn't exist for millenia, and people were still counting and doing complex math. The teacher is acting as if its existence as a number is obvious.

a_c 3y ago

It seems to me after the lessons, more students are in the "because you said so" camp whereas before the lesson they have more reasoning. I wonder, as parent, how much "indoctrination" and how much "reasoning" shall we impart to our children.

ndsipa_pomu 3y ago
My experience of education in the UK is that reasoning and questioning were actively discouraged by some teachers. They were pressured to go through course material quickly, and having students presenting counter-arguments would slow things down.

devit 3y ago

The simplest definition of natural numbers is as the free algebraic structure generated by a nullary and unary operation (conventionally named "zero" and "succ"), which makes it natural to associate zero as the value of the nullary operation, so that one can be it's successor and addition by one works as expected. Not having zero and associating one as the result of the nullary operation works as well, but then the natural definition of addition will result in x + y - 1 rather than x + y and it seems likely extension to rational numbers would not give an intuitive result.

_8j50 3y ago

If zero is a number, atheism is a religion and asexuality is a sexuality. I don't disagree with how zero is used but the english language is the debate here not math. Zero is a legitimate concept in math, if a term describes certain things then is the concept of the lack of those things by default considered a member of those things? Sounds like a linguistic convention in part. Why not have a separate term to describe zero and infinity as part of the same set, "anumerics"?

gls2ro 3y ago

(My math is probably old and maybe I forgot some things) I think zero is kinda strange in a way: it is intuitive mostly when thinking about quantities but when dealing with other concepts I find it strange to think about it. Here are some situations where I know how to answer but reasoning about it is still in a way hard: N^0 or 0^N or 0! or or more simpler when thinking about limits and zero try to see what happens when you try to reach 0 while going like this 0.9^0.9, 0.8^0.8, … 0.001^0.001, 0.0001^0.0001 In most cases I think we are defining the answer. Maybe there are some complex proof for these answers but they only prove that zero is kinda hard to grasp. Or it might be that I forgot a lot of math and I am wrong with these examples.

ksaj 3y ago

In my younger days, I was a Beaver leader. Beavers are Boy Scouts in the 5-6 age group. They are not empty headed. They are extremely inquisitive and easily understand things that are surprising to adults. I would even say on average, their observational skills are better than those seen in adulthood. They're still young enough to have not had every question they ever hear from an adult as something they'll be graded on yet. Pass or Fail. So when they don't know, they tell you and hope that you'll tell them or guide them to the right answer. I personally believe this pressure in older kids and adults alike, to not FAIL (ie: not know the expected answer) is because of the way school tends to turn everything into a binary test. We are literally taught to avoid "I don't know."