Dividing instead of Multiplying--why??

[Otis]

… what number when multiplied by a given number results in a second given number, the general method of solution is division …

Does "general method" depend upon the given numbers or grade level?

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[JeffM]

Does "general method" depend upon the given numbers or grade level?

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Neither

As Halls said earlier in this thread, we do not expect students at any level to know the hundred and threes table

 

[Otis]

Neither …

Please correct me, if I'm missing something, but it seems like you're saying that, if a third-grader asked for help with, "what number when multiplied by 2 results in 4", for example, then division is the general approach to teach.

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[Dr.Peterson]

But division is not a method; it is an operation. It is not even an "approach"; in a sense, it is just a name for a kind of question.

There are many ways to carry out division, one of which is "inspection" -- that is, to observe that you know the other factor by virtue of having memorized multiplication facts. Another is long division, which you would use for hard cases. Another is short division. Another is to convert to binary and do some shifting. Another is to use a calculator.

So why not just point that what we are doing when we try to figure out "what number when multiplied by 2 results in 4" is called division, and then talk about how you can recognize the answer? So you're talking about something general, and then doing something specific.

I don't think there's any need to argue about this; you may all really be thinking the same thing, but emphasizing different parts.

 

[Otis]

… I don't think there's any need to argue about this …

I hope that we may participate in a discussion. I'm interested in different viewpoints.

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[JeffM]

Please correct me, if I'm missing something, but it seems like you're saying that, if a third-grader asked for help with, "what number when multiplied by 2 results in 4", for example, then division is the general approach to teach.

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Otis

First, this thread was cut off from a thread that involved an algebraic fraction in an equation. So, we have gone a bit afield to be talking about third graders.

However, it is a legitimate question to ask whether finding "what number when multiplied by 2 results in 4" involves the operation of multiplication or the operation of division. I would certainly not tell even a third grader that such problems are solved using the operation of multiplication. They are solved using the operation of division. If we want to say that very easy problems requiring the operation of division can be solved using only the times table, I have no objection. In fact, that is a better way than trial and error. And before Napier, the only efficient way to solve division problems was by using the times tables (although in a more sophisticated way than mere rote memorization).

I am genuinely perplexed by how much effort should go into teaching and learning the times tables in an age with cheap, reliable, and sophisticated hand calculators. I feel that we should and could be devoting less time and effort to teaching and learning brute mechanics and more on when to use what techniques in solving problems. I am not sure, however, what the balance should be. But I certainly would not be stressing the times tables to the extent that they were stressed when you and I were in grade school. And I would be expecting even grade school students to day, let alone algebra students, to have calculators readily available.

The very first line in this thread is

I am not a big fan of going from 9x=90 to x=10 by dividing by 9.

I say in rebuttal that whatever method is used is simply a way of doing division.

And I do not see that it is simpler or quicker or avoids division to go

[MATH]9x = 90 \implies 9x = 9 * 10 \implies \dfrac{9x}{9} = \dfrac{9 * 10}{9} \implies x = 9[/MATH]
rather than go

[MATH]9x = 90 \implies \dfrac{9x}{9} = \dfrac{90}{9} \implies x = 90.[/MATH]
EDIT: I spent time last spring teaching a girl how to do long division with two and three digit divisors. I kept asking myself why. There is little practical use in the modern world to doing so, and it is boring work that turns people off mathematics. I cannot see the point of fetishizing the times table.

 

[krimzondeleeuw]

Yeah, well it never made sense to me why teachers taught what they wanted instead of what’s true.

 

[JeffM]

Yeah, well it never made sense to me why teachers taught what they wanted instead of what’s true.

Mmm. Assuming that was a response to my latest post, I'd like to clarify that I was not saying that the following are false:

[MATH]10 * 0 = 0.[/MATH]
[MATH]10 * 1 = 10.[/MATH]
[MATH]10 * 2 = 20.[/MATH]
[MATH]10 * 3 = 30.[/MATH]
[MATH]10 * 4 = 40.[/MATH]
[MATH]10 * 5 = 50.[/MATH]
[MATH]10 * 6 = 60.[/MATH]
[MATH]10 * 7 = 70.[/MATH]
[MATH]10 * 8 = 80.[/MATH]
[MATH]10 * 9 = 90.[/MATH]
Would you care to explain why you believe that the times tables are lies?

 

[firemath]

The earth rumbles..........

 

[Otis]

… why teachers taught what they wanted instead of what’s true.

Hi krimzondeleeuw. If by 'true' you mean the 'bigger picture', maybe they didn't have time for that. (I'm thinking of experiences in grade school.)

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[Otis]

… we have gone a bit afield …

That's the new normal, here.

… I would certainly not tell even a third grader that such problems are solved using the operation of multiplication …

Thanks for the confirmation, Jeff. I know that you've worked with children. Whether tutoring or in a classroom, I've worked only with adults.

Your comments got me thinking about experiences in my childhood. In grade school, multiplication was never explained in terms of repeated addition (that I recall). I was taught that × means what's on the multiplication table, and we had to memorize that, to handle the steps when × appears between bigger numbers. Likewise, division was never related to subtraction (only symbols and setups) or having any relationship to multiplication; division was another set of repeated steps to get answers.

I was exposed to different ways of viewing operations only after I attended community college (in my early 30s). I wonder whether I could have grasped any deeper meaning, had those grade schools (I attended four of them) presented multiplication and division together, with more demonstration about their practical meaning and why they're related.

I'd like to think that grade schoolers today could handle an approach more comprehensive than I got. Maybe they already do?

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[Dr.Peterson]

I'd like to think that grade schoolers today could handle an approach more comprehensive than I got. Maybe they already do?

In principle, they do: http://www.corestandards.org/Math/Content/3/OA/

The relationships among operations, and multiple meanings of each operation, are scattered throughout these third-grade standards.

Whether that's all done effectively by most teachers, or in most curricula, is another issue, but there are certainly those who are trying.

 

[JeffM]

That's the new normal, here.


Thanks for the confirmation, Jeff. I know that you've worked with children. Whether tutoring or in a classroom, I've worked only with adults.

Your comments got me thinking about experiences in my childhood. In grade school, multiplication was never explained in terms of repeated addition (that I recall). I was taught that × means what's on the multiplication table, and we had to memorize that, to handle the steps when × appears between bigger numbers. Likewise, division was never related to subtraction (only symbols and setups) or having any relationship to multiplication; division was another set of repeated steps to get answers.

I was exposed to different ways of viewing operations only after I attended community college (in my early 30s). I wonder whether I could have grasped any deeper meaning, had those grade schools (I attended four of them) presented multiplication and division together, with more demonstration about their practical meaning and why they're related.

I'd like to think that grade schoolers today could handle an approach more comprehensive than I got. Maybe they already do?

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I cannot remember where I had intellectual trouble in arithmetic (other than being bored witless) whereas I do remember where I had intellectual troubles in algebra, calculus, and statistics.

I am also unsure about what the right balance is between teaching the mechanics of arithmetic like the times tables and teaching how to recognize which arithmetical tool to use.

For those two reasons, I seldom teach grade-school children.

The admittedly little I have seen of the current teaching of math in grade schools has certainly not persuaded me that the changes in methods and topics over the last several decades are resulting overall in a higher level of mathematical skill or understanding.

EDIT: While I was writing, Dr. P. posted. I am absolutely sure that he is much better informed about what is supposed to be taught and how, but he does not seem to be willing to be very sanguine about the results, at least so far.

 

[hoosie]

Coming from the perspective of someone who has taught High School Math for many years I always encourage students when solving a problem like 9 x = 90 to use their mental arithmetic to fill in the box. Some students would use their tables to answer it directly while others would do a division sum in their head. Both approaches are equally valid - the important thing is that the students understand the inverse relationship between the two. When I asked students what was the question asking I often got two different replies. Some students said the question was asking “what number do I multiply by 9 to get 90?” Others said how many 9’s are in 90?” I always gave the response that you are both correct because a sum like that can be written in three different ways.
9 x = 90 or 90 ÷ 9 = or 90 ÷ = 9

To the students I would say “sometimes we are asked to solve an equation like
6 x = 45 Can we use a direct approach here using our tables?”
Some students would say no because 45 is not in the 6 times table. Others would say that because 45 is half way between 7 x 6 and 8 x 6 then
7½ x 6 = 48. The direct multiplication approach was always encouraged in my classes but there are times when a second approach is the better option.
Consider: 9 x = 51 (either approach)
Multiplication approach 5 x 9 = 45 5⅓ x 9 = 48 5⅔ x 9 = 51
6 x 9 = 54 Answer: 5⅔
Division approach 51 ÷ 9 = 51/9 = 17/3 = 5⅔ =
Consider 8 x = 43 (second approach)
Division approach 43 ÷ 8 = 40/8 + 3/8 = 5⅜ =

The second approach is the better option in algebra when transposing equations to make another letter the subject.

Consider rearranging a = bc to make c the subject.
Multiplication: What do we multiply by b to give a ? b(a/b) = a c = a/b
Division: a ÷ b = c (simply writing sum in reverse)
(Forwards: 6 x 3 = 18
Reverse to isolate 6: 18 ÷ 3 = 6
Reverse to isolate 3: 18 ÷ 6 = 3)