"Is Zero a Number?": Interviews with a Whole Class of Kindergartners
jennalaib.wordpress.comI 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.
"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/ [1] https://khanlabschool.org/About-Khan-Lab-School
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.
When they started adding the possibility of a "FAIL" to their tests.
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.
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 :-)
> 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.
I concede that my opinion is worth very little to nothing and I didn't even check what's the mathematical consensus on this matter, but when one sees that 1100, 110, 30, 22, 20, 15, 14, 13, 12, 11, 10, A are all the same number one starts thinking about the difference between symbols as numbers and symbols as digit.
I hopefully didn't make mistakes in the several representations of twelve.
> I hopefully didn't make mistakes in the several representations of twelve.
That depends! What is the numeric value of A?
And then: how can you answer that question without admitting that A is a number?
You probably meant "C" as the final one, instead of "A": https://www.wolframalpha.com/input?i=12+in+base+13
Yeah, a made a mistake on the easiest base
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.
I agree, I thought a lot of the "before" answers were excellent and insightful. The teacher did these children no favors by foisting such a meaninglessly pedantic distinction on them at this young age.
> “When we ask ‘how many,’ the answer is a number.
How about a few is a few a number?
Fun way to look at it. In Common Lisp, NULL is not a number. But in Assembly language, it absolutely is.
Is "no" a number? As in "I have no apples." It is strange that we can't say "I have yes apples" otherwise.
> It is strange that we can't say "I have yes apples" otherwise.
You do see ungrammatical "yes" applied sometimes when people are stumped as to how to indicate the opposite of a negative sentence.
It is not strange in general that certain things are, for no apparent reason, impossible to say. That occurs in every language. English modal auxiliaries, for example, do not have non-finite forms, so in many sentences it is mandatory to replace modal can with lexical to be able [to...]
> 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.
Not in an empty universe.
Digit is a symbol, number is a concept. It’s a map-territory distinction. While a single digit can be used to represent a numerical value, it isn’t a number per se.
You can’t really use the number 0 to make 100.
That's the argument in a nutshell. Is 10 a single number, or two numbers? In context, it is one number. But I digress because typing it out suggests it is two.
Would you rather have $100.00 or $0100?
Imagine the surprise learning that anything other than combinations of 0's and 1's are not numbers. For the same reason, but more extreme and in the other direction.
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”
> “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.
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.
The simple answer is "it's the set of all points in a plane that are equidistant from two particular distinct fixed points", but that just gives you a clean definition that won't capture any of the properties that make lines interesting.
You want something like "it's the shortest path between two points, if you extend it to be infinitely long", which glosses over how you know how to do that correctly, or "it's a 180 degree angle" [same problem].
What did your dad say?
This question reminds me that one of Euclid's postulates was "all right angles are equal". We don't consider that an axiom today, and we don't understand why, in Greek mathematics, this was considered to require noting, or what exactly it meant to them. They obviously had something in mind when they said this, but we don't know what that was.
Thinking of a line as being defined by the 180 degree angle it forms at every point of itself feels like a similar sort of thing to whatever it was that the Greeks were thinking about right angles.
> it's the set of all points in a plane that are equidistant from two particular distinct fixed points
I don't like that definition as how do you determine the distance from a fixed point without using a line?
> it's the shortest path between two points, if you extend it to be infinitely long
That's probably a better definition
You use the metric to the measure the distance. the Euclidian metric is the standard sum of squares.
for any pair of points P you can construct a pair of spheres centered on P1 and P2 and take the intersection to find points on the line.
> This question reminds me that one of Euclid's postulates
The question is one of Euclid's postulates (or the combination of two, 1 which defines a straight line segment, and 2 which extends the straight line segment to a straight line).
Postulate 1 doesn't define a straight line segment. It tells you that it's possible to connect two points with a straight line segment. A definition would tell you whether or not something qualified as a straight line segment.
The definition appears to be given, appropriately enough, in the "definitions" section that precedes the postulates, which says "a straight line is one which lies evenly with the points on itself". (As translated in https://farside.ph.utexas.edu/books/Euclid/Elements.pdf )
But although that is indeed how people recognize lines, it's not so easy to define what it means.
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.
In this context, a line is what you draw with a ruler to get the correct answer.
My math teachers would have marked it incorrect if I didn't indicate it went on forever, unless the problem stipulated bounds.
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.
“Because Euclid says so in the Elements”
> How did they explain what plotting was without defining a line?
Plotting is simply putting a point for a specific value. Drawing the line is inferring the un-plotted points from the plotted ones which may or may not be correct though for y = x - 3, it's easy enough to prove that a straight line covers all the solutions.
If "plotting" is only defined in terms of lines, then a student could attempt to plot y = x ^ 2 using only two points and a straight line.
> 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.
Interestingly in Elements that is a line: a line was defined only by having no breadth (= being 1-dimensional). What is now generally called a line would have to be qualified as straight.
Does that mean that a point is also a line as a point has no breadth
No because a line does have a length. In Elements a point is dimension-less.
Okay, so "a line was defined only by having no breadth" should be "a line was defined only by having length and no breadth"
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.
Terminology questions are very very ambiguous in general without academic context bias.
Ask toddlers if 0 is a round number. :')
"Round number" is specifically an ambiguous terminology though. You could use "integer" instead, but it's really more to do with the concept of significant figures. e.g. 129 is an integer, but 130 is a "rounder number" than 129 and 100 is rounder still. To make it more confusing, decimal numbers ending in 5 (before the zeros) are often considered round, 150 is rounder than 130.
However, mathematically there's a totally different definition: A round number is a number that is the product of a considerable number of comparatively small factors. That means that 24 is rounder than 25.
8 is twice as round as 0.
Source: parent of 3.
Technically 3 is equally round as 0, but half as round as 8. It's just split and flipped.
Precisely. With ambiguous terminology, that seems perfectly logical to me.
0 is round because it's a circle.
Except that it isn't
People use the euclidean definition of circle in common parlance? If I can have dark circles under my eyes, then my zeroes can be circles too.
Beautiful pedantry. :D
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.
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)
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.
you can measure -5°c though
Yeah, but that's just an artefact of that particular scale. You'd have trouble measuring -5K
you can measure negative pressure, negative acceleration
Negative pressure is usually a comparison with e.g. atmospheric pressure, so I wouldn't consider it a simple measurement. An actual negative pressure would presumably be less empty than a vacuum e.g. the Casimir effect.
Negative acceleration has a directional component, so again, it's like a negative velocity and is a vector.
I've read articles about negative Kelvin where they surmise it would get hotter. Keeping in mind what heat actually is, the atoms would just be out of phase.
Yes, temperatures below absolute zero are possible and have been achieved (something to do with flipping the high and low energy states), but I stand by my comment that it isn't easy to measure and definitely a far more advanced concept for kids than negative numbers.
That means "huge" and "small" are numbers.
It's a number in Base 18.
Or is that my imagination?
I feel like these children have been enlightened into number theory, but their original answers were nevertheless more correct than the later answers.
Education is a serious of increasingly smaller lies
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.
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.
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?
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.
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"?
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.
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."
(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.
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.
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."
Indeed, but even the standard definitions are flawed.
When I fist was interested in NLP, I thought "word" was easily defined as "sequence of letters bounded by non-letters". But we don't normally pronounce spaces, do we? AndyoucanwritewithoutspacestoindicatespeakinglikeYahtzeeCroshawonZeroPunctuation. And when we've got a contraction, is that "we've" two words or one? Compound words in general, for which German is famous ("Antibabypille") but that's also present in English ("antidisestablishmentarianism", or less severely "internet" and "smartphone")? What about single nouns with spaces in the middle like "remote control" or "mobile phone")? And as my original definition was letters, what about the apostrophe in "we've" or "O'Neil", or the hyphen in older posters saying "to-day"? If all names are words, what about numbers in product names such as "iPhone 15"?
And all those pedants complaining about "to boldly go" in Star Trek because of the split infinitive; I'm told that's because infinitives in Latin were single words and therefore un-splittable, but should admit that I don't know Latin and I don't care to learn it.
> antidisestablishmentarianism
This is not actually a compound word, merely a word which stacks several layers of derivation on top of a single stem.
> What about single nouns with spaces in the middle like "remote control" or "mobile phone")?
These are compound words, identical in their nature to the stereotypical long German compound words, regardless of the fact that they're written with a space in the middle. The difference is merely orthographical.
> When I fist was interested in NLP, I thought "word" was easily defined as "sequence of letters bounded by non-letters".
There's an additional problem with this view which you didn't mention, which is that written language is merely one possible encoding of the actual underlying language, which is not merely a theoretical concern, but also a practical one. After all, even in English the encoding is far from 1-to-1, since you have multiple differing spellings for what would generally be considered to be the same word (e.g. "color" and "colour"), as well as identical sequences of letters which would generally be considered to be different words (e.g. "lead" and "lead").
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.
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.
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.)
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.
UK curriculum sounds good, then.
Where I live a lot of the pre-school kids are excessively exposed to YouTube, which is primarily still drilling on the uppercase alphabet songs.
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.
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.
Yeah cool dad award. I wonder how many hours of questioning came from the back of the car before he explained it.
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.
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!
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.
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.
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.
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.
Just imagine how perfect the world would be if we started counting at zero, though
Haha, this could pass as the results from asking 9th graders.
> Yes, because you can add it to other things.
Good answer.