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

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

#51

Post by masklinn »

thaumasiotes wrote: post #34
boomboomsubban wrote: post #22

Earlier quoted context omitted.

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.

> 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.

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

#52

Post by ndsipa_pomu »

boomboomsubban wrote: post #22
lozenge wrote: post #13

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.

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.

> 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.

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

#53

Post by pmontra »

thaumasiotes wrote: post #21
pmontra wrote: post #15

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 numbe…

> 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.

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

#54

Post by ndsipa_pomu »

thaumasiotes wrote: post #33
lozenge wrote: post #13

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…

> 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

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

#55

Post by mostlylurks »

ben_w wrote: post #29
kaetemi wrote: post #10

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?…

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 fa…

> 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").

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

#56

Post by lozenge »

gizajob wrote: post #32
boomboomsubban wrote: post #22

Earlier quoted context omitted.

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.

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.

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

#57

Post by lozenge »

baobabKoodaa wrote: post #19

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 kno…

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.

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

#58

Post by ndsipa_pomu »

masklinn wrote: post #51
thaumasiotes wrote: post #34

Earlier quoted context omitted.

> 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

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

#59

Post by thaumasiotes »

pmontra wrote: post #53
thaumasiotes wrote: post #21

Earlier quoted context omitted.

> 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?

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

#60

Post by gizajob »

lozenge wrote: post #56
gizajob wrote: post #32

Earlier quoted context omitted.

In this context, a line is what you draw with a ruler to get the correct answer.

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”
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