Wikipedia:Reference desk/Archives/Mathematics/2015 November 2

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November 2

Real life example when 3x5 is not 5x3

Both are the same when multiplying items, and it´s easy to explain that by taking 3 times 5 apples or 5 times 3 apples. Sometime es is 3x5 not 5x3, I want some real life examples of this, for educational purposes. — Preceding unsigned comment added by 80.33.29.221 (talk) 19:56, 2 November 2015 (UTC)

In other words, you want real-life examples of operators that are not commutative? One obvious one is exponentiation: 3⁵ = 243 while 5³ = 125. --70.49.170.168 (talk) 20:27, 2 November 2015 (UTC)

(I am 80.33). Yes, like exponentiation, but with more real-life details (apples, holes, km). --Mth htm (talk) 22:17, 2 November 2015 (UTC)

On a spreadsheet, usually row data and column data are different kinds of data. So rows might be something like clients, and column data something like quarterly debts. So a 5x3 table is 5 clients with 3 periods, whereas a 3x5 table is 3 clients over 5 periods. Sławomir
Biały 22:23, 2 November 2015 (UTC)

In the US, men's pants are an example. 32×30 means, I believe, a 32 inch waist and 30 inch inseam. StuRat (talk) 22:39, 2 November 2015 (UTC)

I'm going to take a flyer here and guess that the OP is talking about a recent social-media kerfuffle stemming from a third-grade math quiz where the students were asked to compute 5×3 by "repeated addition". The answer 5+5+5=15 was marked "wrong"; the desired answer was apparently 3+3+3+3+3=15.

This is apparently being used as a piece of evidence by people who don't like the US common core mathematics standards. I don't actually know whether this little piece of overly rigid thinking is part of common core, or just something this particular teacher came up with. --Trovatore (talk) 22:47, 2 November 2015 (UTC)

My brother had a similar case where he had his answered marked wrong. He was asked to draw 5 dots, and drew:

 o o
o o o

rather than what the teacher had taught:

o o
 o
o o

I suspect the teacher's real lesson was "never question authority". Personally, I'd have fired a teacher who does things like that. StuRat (talk) 23:37, 2 November 2015 (UTC)

Context seems important here. If the learning objective is to think of 3x5 as 5 sets of three apples, for instance, then 3+3+3+3+3 is the correct answer per that objective and 5+5+5 is not correct. That seems like a valid instructional objective, and I don't think it is necessarily a symptom of overly rigid thinking. Unfortunately, it's also very likely the case that this instructional objective was not really properly implemented by the teachers. So the problem isn't common core versus whatever else, but rather that American secondary schools need competent teachers. I don't think this problem has really changed much in the past 20+ years. There are certainly reasons to be opposed to Common Core (primarily, that the standards themselves are written by the publishing industry, which also designs all of the textbooks and testing, thus locking out the competition and ensuring a steady revenue stream from those schools that can afford it), but instructional objectives that seem silly on some Facebook soundbite are not high on the list. Sławomir
Biały 23:40, 2 November 2015 (UTC)

One of my biggest frustrations in teaching mathematics to college underclassmen I'm including the women here, of course was always the way they expected everything to be a fixed algorithm, following some exact procedure that they had been taught. This seems to me to be contributing to that sort of mindset. Granted, with third-graders you can't really expect most of them to come away with any really deep understanding, but still, it would be better not to actively promote this way of thinking.

By the way, I have no strong considered opinions on Common Core. The ideas that they say are behind it all sound good; I'm worried about how they're going to be put in practice, but I don't know enough about the actual practice to criticize in detail. So I don't have a chip on my shoulder about Common Core particularly. But I do have one about rigid algorithmic teaching. --Trovatore (talk) 23:50, 2 November 2015 (UTC)

Well, I think we all agree that it's silly as an algorithm, and it would be regardless of how one approaches the problem. But it seems to me like it would be a legitimate workshop exercise to see that 5 bowls of 3 apples is 3+3+3+3+3 apples. But if you're asked "How many apples are in the five bowls?" and you write 5+5+5, arguably you haven't really gotten the point of the lesson. What you've written might be true mathematically, but if anything it actually shows that you're not able to think beyond what you were rigidly taught. As an example, suppose you ask someone who just finished a university calculus course to calculate the limit $\lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}$. Of course, they all use L'Hopital's rule. But then, tell them they aren't allowed to use L'Hopital's rule, and must instead use the standard rules for limits and derivatives to work it out, and they are completely unable to do so. And they whine and complain that they aren't allowed to do it the way they learned. But you've asked it that way because you're testing a different skill. It's possible that $3\times 5=3+3+3+3+3$ is asked in a similar context (notice that there's even a chorus of whiners on Facebook!), in which case I think there is definitely a case to be made that this is not a case of "rigid" thinking, but rather one of trying to show the student some new approach to the concept that synthesizes some other parts of their knowledge. That's the opposite of rigid, algorithmic thinking. Sławomir
Biały 02:39, 3 November 2015 (UTC)

But who says 5×3 means five bowls of three apples? Maybe it means three bowls of five apples. There is no consensus mathematical standard for which factor is the thing repeated and which is the number of repetitions. In fact, in one case where there is such a definite standard, namely ordinal arithmetic, the opposite order is used. I'm against trying to impose extra conditions on students, when they aren't actually used in mathematics. --Trovatore (talk) 03:00, 3 November 2015 (UTC)

"But who says 5×3 means five bowls of three apples?" Well, no one says this. I'm just saying that there is a conceivable learning objective in which 3+3+3+3+3 is the "right" answer and 5+5+5 is the "wrong" answer. Maybe we just showed how to compute 2x7 by counting the number of apples in two baskets of seven for instance, and it's implied that you are expected to follow the model. The goal isn't just to get the right answer, but to visualize the problem in a certain way. The fact that you seem unable to acknowledge that there can be different learning objectives other than purely the outcome 3x5=15 suggests that, perhaps you are the one who is suffering from rigid and algorithmic thinking here. As an example, if you just taught a lesson on the derivatives of trigonometric functions, and set the exercise $\lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}$, do you think it would be reasonable for students to use L'Hopital's rule? I wouldn't have thought so, especially not from someone who purports to be against "algorithms" like L'Hopital's rule. Sławomir
Biały 03:17, 3 November 2015 (UTC)

I don't seem to have gotten my point across. Of course it's legitimate to ask the students to understand that there's a difference between five bowls of three apples and three bowls of five apples, and that the fact that it's the same number of apples all told is an additional fact, not part of the statement.

What I object to is making it part of the interpretation of the notation as taught in class, not just a nonce requirement on an individual problem. Kids internalize that sort of stuff, and it's extremely difficult to un-teach. --Trovatore (talk) 03:24, 3 November 2015 (UTC)

Morris Kline complained about this kind of thing in Why Johnny Can't Add (1973), so more like 40+ years. Certainly colleges and employers should be able to expect some level of mathematical knowledge from a high school graduate, though what that level should be, how it should be measured, and who to blame when the level isn't met are issues for debate. My biggest frustration when I was teaching calculus was that I was trying to explain techniques of integration to people who still made mistakes like (x+2)² = x²+2². To me this was a product of the fact that 70% correct C in algebra was considered sufficient to move on to the next subject; don't worry about what the students got wrong since, to the educational system at least, a C student will never be more that a C student. Lately though I've realized that this is just a symptom of much larger systemic issues. At least we're starting to get away from the "Johnny can't add because he's lazy and/or stupid" mentality, though the "Johnny can't add because his teacher's lazy and/or stupid" it's being replaced with isn't a big improvement. Personally, my biggest problem with Common Core is that you can still graduate high school without knowing who Vi Hart is. --RDBury (talk) 01:13, 3 November 2015 (UTC)

True. A good education system has to start somewhere. Eliminating poverty, providing better social services, a future that doesn't involve prison for the majority of public school students. These seem like a necessary foundation before any "education" can happen. But yes, in many cases the teachers don't know anything about the subject they are required to teach, and that's a big problem. In some of the better schools, teachers have been provided with the resources to teach to the test. But a different set of standards don't really change the fundamentals. It would be better to implement standardized testing of the teachers. Sławomir
Biały 01:39, 3 November 2015 (UTC)

To me some of the saddest commentaries on modern America are the PSAs asking people to go into teaching. It tells me that teaching is such an underpaid career path that schools can't recruit enough capable people and we have to appeal to altruistic motives to fill the jobs. My conclusion is that if we want better teachers we need to increase the pool by offering better pay, better working conditions, etc. to something commensurate with the level of education needed, then we can talk about testing to weed out the people who don't have what it takes. That is, after all, the way it's supposed to work in a capitalist society. But you're right to point out that the social condition of the pupils is just as important. So much of the educational system is the way it is just because it's been that way for the last 100 years and no seems to be questioning whether it's really the way it should be. Perhaps the day spent in teaching how to add fractions would be better spent teaching how to refine a search engine query, or communication skills, or checking if any of the kids have dyslexia, or aren't being supervised at home. If education was a commercial enterprise, what would the product be? Is it adults who can add fractions? jump into the workforce without additional training? understand economics well enough to make an informed decision at the voting booth? invent a cheap energy storage device? And if it is a product, then who is paying for it and what is it worth to them? Who should be paying for it and what is it worth to them. Is the product being made as efficiently as possible or is a person talking in front of a few dozen other people the best technology we can come up with? These are questions that no one will be able to answer easily, but they seem fundamental and no one seems willing to ask them. It's so much easier to pick at the surface symptoms, like pointing out the peeling paint when the foundation is about to collapse. --RDBury (talk) 11:00, 3 November 2015 (UTC)

• @RDBury: what's the problem of not knowing who is Vi Hart? It's not as if she were Martin Gardner. --Mth htm (talk) 03:11, 3 November 2015 (UTC)

My thinking that if a high-schooler somehow gets the idea that math is something interesting to investigate rather than random collection of trivia designed to bring down their GPA, and they do what a modern teenager would normally do which is to look for on-line videos on it, they will quickly come across Vi Hart's channel. All due praise to Martin Gardner but, afaik he wasn't a YouTuber. --RDBury (talk) 08:32, 3 November 2015 (UTC)

For the record: 1. I get the idea that math is something interesting to investigate. 2. Youtube wasn't around when I was a teenager, but I do frequently browse Youtube now. 3. I hadn't heard about Vi Hart. There's a lot of stuff out there, there's no guarantee you'd run into any one in particular. (I have heard about vsauce though). -- Meni Rosenfeld (talk) 13:44, 3 November 2015 (UTC)

The article [Multiplication and repeated addition] refers ta series by Keith Devlin which discusses the business of repeated addition and how it differs from scaling. For the apples and bowls example he says

5 bowls × 3 apples per bowl = 15 apples

Personally I have no problem with thinking of that as either 5 lots of 3 as in I count each the bowls one at a time, or 3 lots of 5 where I put an apple into each bowl and do that three times till there are three per bowl. He also talks about how adding is just wrong for other problems like when you stretch a piece of elastic to three times its length. Dmcq (talk) 10:38, 3 November 2015 (UTC)

If they show those apples arranged in parallel lines instead of putting them in bowls, everyone would immediately see that the same 15 apples are both 3×5 and 5×3, depending on the point of view (a side of the table, where you stand). Children can then catch the commutativity property almost immediately. And they develop no absurd pattern 'this is what you repeat, that is a repetitions number' because they can see at once it doesn't matter. --CiaPan (talk) 14:57, 3 November 2015 (UTC)

The OP's question appears to be being ignored. Perhaps the OP would be interested to look at matrix multiplication or quaternion for examples of multiplication which tick many of the boxes for regular multiplication (such as having a norm) but are nevertheless not commutative. HTH, Robinh (talk) 22:37, 4 November 2015 (UTC)