[0] https://www.freerangekids.com/ [1] https://khanlabschool.org/About-Khan-Lab-School
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 :-)
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 hopefully didn't make mistakes in the several representations of twelve.
You can’t really use the number 0 to make 100.
Would you rather have $100.00 or $0100?
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”
Teacher, are you even paying attention? We've been over this.
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.
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.
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
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).
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.
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.
If you asked most adults what a number is, probably many would come up with a definition that didn't include i.
Ask toddlers if 0 is a round number. :')
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.
Or is that my imagination?
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"?
The teacher is acting as if its existence as a number is obvious.
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."
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.
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.
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."
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.
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").
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.)
The phrase highlights the concept of whole vs. fractional numbers.
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.
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!
Good answer.
