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.
"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
#32Good 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.
In this context, a line is what you draw with a ruler to get the correct answer.
Re: "Is Zero a Number?": Interviews with a Whole Class of Kindergartners
#33Good 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.
Re: "Is Zero a Number?": Interviews with a Whole Class of Kindergartners
#34Good 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?
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.
Re: "Is Zero a Number?": Interviews with a Whole Class of Kindergartners
#35I 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.
Re: "Is Zero a Number?": Interviews with a Whole Class of Kindergartners
#36Zero 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.
Re: "Is Zero a Number?": Interviews with a Whole Class of Kindergartners
#370 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.
Re: "Is Zero a Number?": Interviews with a Whole Class of Kindergartners
#38I 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.
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.
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Re: "Is Zero a Number?": Interviews with a Whole Class of Kindergartners
#39Re: "Is Zero a Number?": Interviews with a Whole Class of Kindergartners
#40What 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!