continued from the previous post – “Explaining Math Terminology” (Please read that first in order to understand the points made here).
Since I wrote the last post, I’ve come across more examples of math terms that are incompletely, inadequately or misleadingly explained. The next one comes from a source I never thought I’d find anything like that on. It just goes to show that even the most well-informed and well-intentioned sources can goof up occasionally. That is why we have to make more of an effort to be aware that if even the great teachers can occasionally slip up, we can expect the hacks who write “curriculum” and “standards” to be total screw-ups. And it should go without saying that yours truly, dilettante that I am, have a black belt in screwing up.
On page 22 of the very excellent “The Book of Numbers,”in the second paragraph, the authors write:
“The familiar whole numbers, 1,2,3…”
Let me say now that this book is otherwise awesome, and the authors (John H. Conway and Richard K. Guy) are amazing. John Conway is legendary. So when, three pages later, they say:
“…the ordinary whole numbers, 0, 1, 2, 3, …”
anyone could get confused. And get confused we do. I believe that millions of children each year, as well as almost that many adults, are not clear about what the whole numbers are. It’s not just the whole numbers, either. Fractions, decimals, algebra, even math itself. What are they? Do you have a good working definition? Or do you maybe think there is only one “right” definition for any of them?
(See the thoughtful comment from a reader below. It will help clarify those last questions.)
This is not really a big issue. As the very authoritative wolfram.mathworld.com observes, ’0 is sometimes included in the list of “whole” numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement.’
Actually, it’s a teachable moment, especially since we often use zero-indexed sequences and we often use one-indexed sequences. I don’t think it counts as a “slip up” on Conway’s part; it’s just an alternative definition. Like most Americans, I happen to prefer the term “whole numbers” to include zero, and “counting numbers” or “natural numbers” to exclude it. But Conway isn’t American. I believe that in Britain it’s more common to exclude zero from the whole numbers.
Larry,
You are absolutely right. This does not contradict my point, though. It strengthens it. I suppose I should have made more clear that this post is a continuation of the last, in which I made the point that there is usually more than one way to look at many points of math (or life in general, for that matter).
The confusion comes when we are told that “So and so is such and such,” without the caveat that it is only such and such in certain cases, and that there is more to the story.
So we are told that the whole numbers are 1, 2, 3… Fine. We walk away though, reasonably thinking that that is the case. We weren’t told that there was more. People who are not versed in mathematics should not be expected to understand that the definition is nuanced, unless we give them a clue when we first tell them about it.
This reminds me of a pet peeve of mine – in school we are usually told that “multiplication is repeated addition.” Well, of course it’s not. It acts like repeated addition in the case of whole numbers, so it might be reasonable to think that it is repeated addition. Then fractions come along, and screw the whole thing up.
No wonder kids hate math. They feel like the rug is pulled out from under them. We teach them one thing, then we use it in a different way that we gave them no hint was coming along.
The problem isn’t different definitions. A mature mind accepts that there can be different meanings for different situations. The problem is that too often we expect the minds of the people we are teaching to already understand what we are teaching without giving them a reasonable chance to mature into it.
And the worst thing is, some damned school “standards-weenie” is going to test them on it. The weenie doesn’t even understand that there can be two takes on whole numbers, but the book we use in class from the “Cosmo-Demonic Textbook Company” (from Texas, of course) taught that the whole numbers include zero. But on the quiz, if little Johnny says that the whole numbers are 1, 2, 3.. , little Johnny gets a raspberry
So my beef is not that there are more than one way to see something (heaven forbid! See my post on The Parallax View), it’s that nowhere was it mentioned in those pages of “The Book of Numbers” that we sometimes use 0, and sometimes not, when defining whole numbers. If they explained it as you had, that would have been a fine lesson – it would have taken advantage of the teachable moment.
Unfortunately, it was like telling a child that two plus two equals 4, and then later telling him that two plus two equals 11. It does, in base three, but you have to tell the child about base three, first, or it’s just plain negligent.
I’m sure you know what I mean. I should have begun the post by telling readers that they should read the previous one to get my point. I’ll go correct that now.
Thanks for your point, and thanks for the info about why there is a discrepancy about the usage of 0 in the whole numbers.
I never thought about it before having a kid in elementary school, but now I’m finding terminology to be a huge issue. For instance, in 4th grade math they had to look at data distributions and determine a “typical number” but we had nothing at all that defined what was meant by “typical number”. I asked was it the “mode” maybe? or “mean”? It turns out the teacher couldn’t provide me with a definition. With fractions, they were doing some method in which “least common denominator” was not supposed to be used, but we had no substitute term. When long division started, they had a “hangman” method that I’d never heard of and again, no explanation was provided. For parents who are trying to help, it’s really difficult to even know how given the differences in terminology (method?). I’m all for methods that are more helpful than the ones I grew up with, but it would be helpful for parents to know what the terms mean.
this is great:)