Comments

3, 6, 9, 12...

Posted by: mjfrombuffalo at October 19, 2006 6:24 AMBTW, my answer is 3,6,9,12,15... and so on. But don't let that sway anyone; if they have a different first inclination I want to hear about it. (I cite as my source "Schoolhouse Rock.")

I will explain later.

Posted by: James at October 19, 2006 6:45 AMDitto on the 3, 6, 9, 12... are you wondering if we should include 3?

Posted by: Mike at October 19, 2006 7:44 AM0, 3, 6, 9, 12, etc. I would include 0 time 3. I suppose using that logic I should also include -3, -6, -9, ... but I won't. That's my answer and I'm sticking with it.

Posted by: B.O.B. (bob) at October 19, 2006 8:33 AMWell, it depends on the audience. If this is a homework question for one of the kids, I doubt very much they've gotten into negative numbers yet and they may not have gotten to multiplication with zero (I distinctly remember when we got to that in elementary school and I felt almost as if I had been lied to before that about zero as it was never discussed in earlier grades).

Presuming this is the source of the queston, and you don't mean "multple" as in "multiply the number by itself," I will say:

3, 6, 9, 12, and so on

Posted by: Patti M. at October 19, 2006 8:57 AMAs a bonus, people also might want to consider if "multiple" has the same meaning here as in the term "least common multiple."

Posted by: James at October 19, 2006 9:09 AMSo, anyhow, here is why I'm asking the question. Derek asked Ryan and myself this question yesterday, and we both answered 3,6,9... (Just like on *Schoolhouse Rock*)

There was some disagreement over an assignment regarding whether the 0 should be included. But the point of the assignment was not to know the multiples of three -- the multiples were only a peripheral fact.

In my research on this, it appears to me that there is some ambiguity to what people understand to be "multiples" of a number. If you ask for the "multiples of 3" I seem to find a lot of lists that begin with 3. The definition of multiple, however, certainly allows for zero, as B.O.B. points out. So you could include it.

In any case, if a definition like this is somewhat ambiguous, it seems to me to be silly to mark something wrong when a student has a different understanding of what "the multiples of 3" means.

While I understand the inclusion of zero, by the definition of "multiple," it leaves me to wonder why LCM isn't always zero. Since it would then technically be the Least Common Multiple of any two numbers.

I don't think there is a right answer, frankly. But when there is wiggle room people ought not to act like something is set in stone.

For the longest time I thought the natural numbers were "the counting numbers" (1,2,3,4,5...) and that the addition of zero gave you "the whole numbers." Now I learn that some people (especially computer scientists) include zero as a natural number.

Do I go with my upbringing, or switch over to the computer science definition and stand with my peeps?

And then what do I call the counting numbers? "The counting numbers?"

Dilemma!

Posted by: James at October 19, 2006 10:10 AM*For the longest time I thought the natural numbers were "the counting numbers" (1,2,3,4,5...) and that the addition of zero gave you "the whole numbers." Now I learn that some people (especially computer scientists) include zero as a natural number.*

This is *precisely* what I was getting at in my post, and I had forgotten about those terms!

I can see very clearly the laminated strip of paper we had on our desks in the depression for the pencils. It had the "natural numbers." When we got to "whole numbers," I remember feeling confused and kind of ripped off that we hadn't learned about zero right from the get go.

Was the delay because teachers didn't think we could handle an abstract construct such as "nothingness"? For pity's sake, it's counting, not philosophy.

Include zero!

Posted by: Patti M. at October 19, 2006 10:36 AMI wouldn't have included 0, because I think of multiples as something that can be factored back to the original number. Is 3 considered a factor of 0? Is this a meaningful concept? (I don't know. I'm asking.)

Of course, -3 can be factored back to 3 as well, but if you're asking for "the first few multiples of 3" and allow negative numbers, that means, like, negative infinity +3 or something. That might be more difficult to factor.

Posted by: Julie at October 19, 2006 11:17 AMChildren do learn about multiplying by zero these days, Patti. Multiplication is in grade 3 (same as when I was in school), and they learn multiplication by zero. Just FYI. I like the approach they take to math. M, in fourth grade, is doing pre-algebra stuff and I've gotten her to take a word problem and turn it into equations (e.g. Mark ran four more laps today than he ran yesterday. The total number of laps he ran is 20. How many laps did he run each day?). In school they solve that with "guess and check," but I got her to write the equations because it helps you understand what you're looking for and put it in a table to make guesses with, especially when there are three variables. K, in sixth grade, is getting a little deeper into pre-algebra. They definitely take an approach that is less computational with more of an emphasis on understanding the relationships between the numbers than what I remember, although I went to private school and I think my education had more of an emphasis on understanding and less on memorizing/computing/following algorithms.

Posted by: Maggie at October 19, 2006 12:07 PMOh good lord no. No more word problems. Those are the stuff of nightmares. I hated those things.

Word problems are like poetry--they beat around the bush. Just say what you mean already.

Posted by: Patti M. at October 19, 2006 1:14 PM3,6,9,12,15,18,21,...

And I suppose you could put 0 at the front of the list, but most people don't in my experience. Now to read the other comments...

Posted by: Chuck S. at October 19, 2006 1:26 PMHmmm. Looks like it was all about 0 after all.

0 is a valid multiple of ALL numbers, and thus it isn't particularly interesting to include when listing multiples of three. When we list multiples it's typically to assist in memorization of multiples to facilitate speedier arithmetic, or simply to "count by N's" in some practical pursuit, like counting pennies 3 at a time. In either situation you *could* start from zero, but it seems pointless to do so.

In the former since Nx0 = 0 for all N, you hardly need to memorize that 3x0 = 0. If there was such a need, we'd be memorizing our multiples of zero too... and who would seriously ask "list the first few multiples of zero".

In the second, nobody starts counting pennies by saying "I have zero pennies." It's a wasted step. Obviously you've counted zero pennies thus far, that's why you're grabbing the first three and saying "three" to yourself. (Total aside: I count pennies in fives by taking alternating groups of 3's and 2's. I'm weird.)

In my experience computer scientists like starting from zero because it makes their life easier. An index is ultimately an offset from the start of a list. How far do you have to travel from the beginning of a list to get the first item? Duh, zero. If it is the first item it is already AT the beginning of the list. In other words, Item 1 is found at position 0. This is confusing for people who aren't software developers, but for compsci types it's easier. Less work is involved to navigate arrays, strings, and lists.

Posted by: Chuck S. at October 19, 2006 1:43 PMThe plus side of word problems are that if done correctly they show you real life uses for math. The problem is historically they aren't usually done that way.

Posted by: B.O.B. (bob) at October 19, 2006 1:47 PMAnd there are all those European buildings that are zero-indexed for the benefit of us computer types. (Ground floor is G, floor above that is 1, etc.) Thinking of the ground floor as 1 makes my life easier but then again I'm an American.

Posted by: Mike at October 19, 2006 1:59 PM0 is a valid multiple of ALL numbers, and thus it isn't particularly interesting to include when listing multiples of three.

I don't see where interestingness ought really figure in the question. If "2" is a boring prime number, it's still a prime and you'd still list it because it conforms to the definition.

I understand what you're saying, it just seems like either is a multiple by definition and gets listed or isn't a multiple by definition and shouldn't get listed.

Or, there is disagreement about whether it should or shouldn't be listed, and an understanding about that disagreement so that if you have an assignment that relies on "multiples of 3" you either state your definition of what that means, or you're willing to accept more than one reasonable interpretation.

I think not including zero is certainly reasonable.

Poor zero. The "You Forgot Poland" of multiples.

Posted by: James at October 19, 2006 4:17 PMIt all goes back to the fact that you initial question did not contain enough info. Hell every number is a multiple of three, you just need to use fractions to get there (2/3 * 3 = 2)right?

keri I think the term you're looking for is geeks.

Posted by: B.O.B. (bob) at October 19, 2006 4:39 PMThat's part of the issue. Is "multiple" enough info?

In number theory, the meanings of "multiple" and "factor" only apply to natural numbers, or whole numbers. For other purposes, only integers are considered.

I don't think my question was missing information. However, you might say it was missing context.

But the place from which I plucked the question also lacked enough context to answer the issues we're raising, and that's precisely why I posted it here. To see what context people would apply to the question.

Posted by: James at October 19, 2006 4:48 PMSchool assignments lack context. When I give an exam, I realize that students have to understand "where I'm coming from" when they take it. And they get graded accordingly. And I try to be fair, but sometimes it's not fair. And that sucks, but it's just school.

(I know from where James' question arises.)

Posted by: Maggie at October 19, 2006 5:02 PMIf I say "draw a cow" and you draw a brown cow, but I was thinking of a black and white cow, you shouldn't get marked down unless there was a previous agreement as to what "cow" specifically meant. That's my point.

Especially in an art class where you weren't studying cows, you were just using them as something to draw.

Posted by: James at October 19, 2006 5:10 PMWow did I tell you that story (the cow one) or did you come up with it on your own. In kindergarden I was apparently marked down for drawing a black instead of brown (or vice versa I don't even rember the event but it left quite an impression with my mother). Not only marked down but apparently the teacher made a big deal about it in front of the entire class. Of course I was using those stupid fat crayons that had no labels on them. Oh yeah and I'm color blind (OK deficient).

Posted by: B.O.B. (bob) at October 19, 2006 6:07 PMNo - I don't think you've ever told me that story. However, something similar happened to me in kindergarten with a picture of the Sun. I was asked to draw an orange sun and I colored it yellow because the Sun is a yellow star. I was reprimanded.

Whatever. In my case I wasn't following directions, but I thought the directions were stupid.

Posted by: James at October 19, 2006 7:30 PMExactly. It's difficult in school because you're being evaluated, and the evaluation is important. But sometimes stuff is going to be semi-arbitrary and you're going to get marked down unfairly, and that's just life. It's better, of course, if you figure it out in college rather than preschool, because if you figure it out in preschool you've got a long road of oppression ahead.

But why should school be any more fair than anything else? Most teachers really do strive to be fair. But it's a completely artificial little world.

I think, in the case of the assignment given, you have to look at every number and ask "is it a multiple of three?", and if the answer is yes, then you have to give the appropriate output. In that instance, I do think it's reasonable to assume 0, -3, etc. It's a test - "multiple of three?" and the answer can't be no, because zero is a multiple of three, even if it isn't one that comes immediately to mind when listing multiples of three.

Since when does democracy give us answers in math?

Posted by: Maggie at October 20, 2006 7:06 AMWell, that's a good point.

But beyond "fairness" I would be embarassed to mark someone down who obviously understood the material, but misinterpreted something that was both peripheral and ambiguous.

'Course, I've never been a TA.

Posted by: James at October 20, 2006 7:54 AMI would like to say that I am very happy that I am no longer in school! And, if I _do_ go back--not for another degree, but simply to take a class for the pleasure of it, I will audit and not do any of the testing/writing/etc.

I agree, Patti, school is the stuff ulcers are made of. But then so are bad managers, bad landlords, bad presidents... argh. Anybody evil who's in control. I'm not defending the TA's grading, I don't know how much he marked off, and I can't stand these little geek tests like "did you count 0," but let's face it, if you're going to get a geek test, CS is the place to get it.

Posted by: Maggie at October 20, 2006 12:11 PMMaybe some clarification on the assignment:

Generically create a machine (NFA/DFA) that has a string with size of n, and it accepts a string of size k, where k is a multiple of n;

My machine would accept string size k, and return to the second state, always giving a multiple of n, for the common definition (so we can use LCM), where it was a number multipled by all counting numbers.

Posted by: Derek at October 20, 2006 12:20 PMMy brain has just gone "click."

Spot the English major!

Posted by: Patti M. at October 20, 2006 12:42 PMCopyright © 1999-2007 James P. Burke. All Rights Reserved