Division Question

[Algot Runeman]

Dividing

As I understand the math conventions, a standard division equation looks like this:

6 ÷ 3 = 2

Yet, that gives me the impression that the original value (6) is diminished by the division, that there is, at the end of the process, a single chunk with the value of two.

I think I prefer a different interpretation, one which sees division as more than a process embodied by the equation.

As shown by the illustration, dividing six by three creates not just a single value of two, but three values, chunks, of two. Now it may be what the simple equation is intended to convey, but I think the fine distinction is lost on those who are asked to memorize the method of division. The simple answer of "two" is in focus, but the fact that there are three twos is essentially ignored or intentionally left fuzzy.

Is this a valid question of process vs. concept?
Is it a failure of my own that I'm thinking about this at age 70 instead of age 10?

What do I have "wrong" here?

[Christian Baune]

When you divide "2i" by "i"  you get "-2".
That means that you have to make ab exception to anything outside |R.

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[Bradford Hansen-Smith]

Algot, good question that is relevant to the times we live. If our whole is 6, it is not diminished by division, it further expands and enriches our understanding of 6 as 3 sets of 2 each. It reveals a generation or level of organization not obvious when thinking of the 6 as a non-differentiated whole. 2 and 3 are simple relational parts of 6

It makes more sense to me if we first divided 6 into equal halves, 2 set of 3 each, then we can figure out that maybe we can also have 3 sets of 2, or since we have found multiplication in the process of division then maybe there is more about addition and subtraction we can discover without ever diminishing the whole of 6.  In this way conceptually there is a principled process in play that reveals much more than a functional processing isolated information. The concept of unity as a comprehensive context of origin for units is missing in math and in most other areas of our fragmented lives. 

This is an important conceptual issue bringing to question unity of the whole which is never diminished by division but increases operational functionality and potential expression. When we separate units from unity is when we get confused and overwhelmed with the diversity of individualized information, thus loosing the proportional relationship between parts and whole, even on this simple level. 

On Wed, Aug 17, 2016 at 10:54 AM, Christian Baune <progr...@gmail.com> wrote:

When you divide "2i" by "i"  you get "-2".
That means that you have to make ab exception to anything outside |R.

On Wed, 17 Aug 2016 17:14 Algot Runeman, <algot....@verizon.net> wrote:

Dividing

As I understand the math conventions, a standard division equation looks like this:

6 ÷ 3 = 2

Yet, that gives me the impression that the original value (6) is diminished by the division, that there is, at the end of the process, a single chunk with the value of two.

I think I prefer a different interpretation, one which sees division as more than a process embodied by the equation.

As shown by the illustration, dividing six by three creates not just a single value of two, but three values, chunks, of two. Now it may be what the simple equation is intended to convey, but I think the fine distinction is lost on those who are asked to memorize the method of division. The simple answer of "two" is in focus, but the fact that there are three twos is essentially ignored or intentionally left fuzzy.

Is this a valid question of process vs. concept?
Is it a failure of my own that I'm thinking about this at age 70 instead of age 10?

What do I have "wrong" here?

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www.wholemovement.com

[michel paul]

On Wed, Aug 17, 2016 at 8:00 AM, Algot Runeman <algot....@verizon.net> wrote:

As I understand the math conventions, a standard division equation looks like this:

6 ÷ 3 = 2

Yet, that gives me the impression that the original value (6) is diminished by the division, that there is, at the end of the process, a single chunk with the value of two.

That's the problem right there: process. 

Put units in and see what happens:  6 meteors per hour over a span of 3 hours gives us an average rate of 2 meteors per hour. 

​The expression for division of naturals does give the impression that the 6 is being 'diminished', but that is because the concept of rate is not being incorporated.

​It is true that 2 in itself < 6 in itself, but when we're dealing with meteors over a sequence of nights, we're talking about quantities and rates, and the 2 is of a different type than the 6.

I think I prefer a different interpretation, one which sees division as more than a process embodied by the equation.

​ ...​

As shown by the illustration, dividing six by three creates not just a single value of two, but three values, chunks, of two.

​Right. Your rate is 2 units per chunk.

You saw 6 meteors over the course of 3 hours.

Each chunk is an hour, and you have the rate of 2 meteors per hour. (That's very low compared to the Perseids recently.)

The 6 actually never went away or got changed. It is there in the structure of the rate.

Now it may be what the simple equation is intended to convey, but I think the fine distinction is lost on those who are asked to memorize the method of division.

​Yep.​

 

The simple answer of "two" is in focus, but the fact that there are three twos is essentially ignored or intentionally left fuzzy.

​Again, a whole lot of the confusion has to do with lack of attention to the concept of rate, and I believe this also has to do with the typical misunderstanding of multiplication.​ 

The expression 2*3 = 6 ignores the fact that we are talking about seeing 2 meteors per hour over a course of 3 hours. Again, the 2 and the 3 are of different types.

Is this a valid question of process vs. concept?

​It is a perfect example of process vs. concept.​

 

Is it a failure of my own that I'm thinking about this at age 70 instead of age 10?

​Nope. It's cultural. Probably most people think the same way. I think that's why Keith Devlin hit such a nerve when he said he wished math teachers wouldn't define multiplication as repeated addition. Wow. The strong reaction to that really got me curious, and digging into it deeply got me to question my own ideas.

What do I have "wrong" here?

​In summary, the focusing on process that ignores units and ratio, and I believe this is a cultural phenomenon. ​

--

​Michel

​

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[Joseph Austin]

On Aug 17, 2016, at 2:47 PM, michel paul <python...@gmail.com> wrote:

Put units in and see what happens:  6 meteors per hour over a span of 3 hours gives us an average rate of 2 meteors per hour. 

Aren't you multiplying instead of dividing?

I'd say 6 meteors per hour  ÷  3 hours  gives 2 meteors per hour per hour, or an acceleration of the rate of meteors,

e.g. 1 the first hour, 3 the second, 5 the third, etc.

Joe 

[michel paul]

On Wed, Aug 17, 2016 at 11:47 AM, michel paul <python...@gmail.com> wrote:

Put units in and see what happens:  6 meteors per hour over a span of 3 hours gives us an average rate of 2 meteors per hour. 

​Oops - need to correct something.​

It is not "6 meteors per over a span of 3 hours..."

​The line should simply read: ​"6 meteors over a span of 3 hours ..."

--

[Joseph Austin]

Interesting.

Does 6 ÷ 3 mean the same as 6 * 1/3?

The first I would interpret as "how many times does 3 fit into 6?"  Answer; 2 times, that is, there are 2 '"threes" in 6.

So I would draw your diagram as two blocks labelled "3" instead of 3 three blocks labelled "2".

The second I would interpret as "what is the third part of 6?"  Answer: 2 (of the six).

In that case, I would would show the answer as 6 blocks of one each, separated into three groups of two,

with one group of two blocks colored red, and the remaining groups colored white, say.

indicating  "two"  units from the original six.

I might actually arrange them in a 2 x 3 array to emphasize that each one of three groups consists of 2 blocks.

So, we may be speaking of 2 "threes" or 2 "sixths"? But in neither case would I say the answer is 3 "twos".

On the other hand, I agree that "division" could be interpreted as "factoring."

This comes from a different interoperation of multiplication.

What is 3 volts times 2 amps?  It's a measure of power, 6 watts.

But 6 watts also equals 6 volts times 1 amp or 2 volts times 3 amps or 1 volt times 6 amps, etc.

So watts ÷ volts = amps, and watts ÷ amps = volts.

In other words, as i like to say, multiplication is not "closed",

that is, the set of units of the product is not the same set as the sets of units of the factors.

In concrete terms, the product of two lengths is not a length, but an area.

So we could interpret "division" as extracting a complementary component, or dimension, of a product,

for a corresponding transformation of coordinates.

That's my take.

But  it may be I am even more "fuzzy"  than you, because I'm a couple years further from school than you are!

Joe Austin

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[michel paul]

On Wed, Aug 17, 2016 at 3:00 PM, Joseph Austin <drtec...@gmail.com> wrote:

​> ​

Aren't you multiplying instead of dividing?

​Instead? No.

We are describing the same structure in two different ways. It's not that we are 'doing' anything in either description.

We can say either:

• 6 meteors / 3 hours = 2 meteors/hour, or
• 6 meteors = 2 meteors/hour * 3 hours.

(Note as I pointed out a moment ago that I need to correct the expression "6 meteors per hour" to simply "6 meteors".)

​> ​

I'd say 6 meteors per hour  ÷  3 hours  gives 2 meteors per hour per hour, or an acceleration of the rate of meteors,

e.g. 1 the first hour, 3 the second, 5 the third, etc.

--

[Joseph Austin]

OK, given your correction, I withdraw my correction.

Of course, I presume you don't really means "Meteors" either, as meteors and hours are incommensurate,

but some time-dimension-measurable event related to meteors, such as meteor transits or sightings or meteor burnouts, i.e. "falling stars". 

In any case, the "answer" must be 2 somethings, not 3 twos.

But my other point is, whatever units we use, the units of the product are necessarily different than the units of the factors;

multiplication, in the physical sense, is not "closed", in the mathematical sense.

Groups and Fields may be convenient concepts for discussing numbers, 

but I think we really need something more to talk about the "products" and "quotients" of physical quantities.

Or perhaps we need a new notion of "relativity" to account for the physical possibility that the same product quantity 

can be factored in multiple ways into individual factor quantities.

In any case, I think Algot's question raises a profound issue.

Joe

[Algot Runeman]

On 08/17/2016 07:47 PM, michel paul wrote:

We can say either:

• 6 meteors / 3 hours = 2 meteors/hour, or
• 6 meteors = 2 meteors/hour * 3 hours.

I think Michel gets onto the thing I was trying to approach. In both of the meteor examples, the significance is that there are three hours in which two meteors were observed during each of the hour periods.

In neither case was an isolated value of two (meteors) the key element. All six meteors remained in play, the division was of the observations spread out into hour long chunks of time.

I'm sure that I would not have absorbed this complexity at age ten or so. However, I do hope that the bigger issue is getting children to understand this side of the discussion even more than they "get" the right answer of 2. The process focus probably does help children with the operation. I do not think that's enough in the long run.

In the mid-night wanderings of my mind, I felt a subtle cheat had left me with the (I think false) feeling that six divided by three is just two. (emphasis important)

Thank you, all-who-have-responded, for your contributions to clarifying this issue.

----Surprise (for me, anyway) second issue----

Just now, as I reread my writing, I am about to throw a monkey wrench to the gears of my happy feeling.

If I go out to look to the sky for three peaceful hours and see three meteors in the first hour and another three during the final hour, the same division remains possible, averaging 2 meteors during an hour's chunk of time, but the averaging is deceptive. The average is two meteors per hour. It is two meteors in an hour in spite of seeing actual meteors only during the first and last hours. I spent the hour in the "middle" seeing no meteors. The rate was two meteors per hour only in the artificial average. It isn't until we combine lots of hours of observed meteors that an average becomes logical. Over a few hundred nights, perhaps the meteor average is two per hour. I feel far more comfortable with that kind of calculation than I do when averaging just a three hour period.

I think that this uncomfortable observation is one of the main reasons I have come to dislike grading. Too often empty middle hours destroy the actual existence of three meteors in each of two hours, a rate higher than the nominal average we wind up calculating.

By focusing on the average of two, we lose the bright shine of performance of the two hours in which we counted the actual meteors. Life is often lived in an uneven pattern. We shine for some of our hours and we merely maintain a pale background glow during many others. Our average cannot accurately represent the brilliance of our highest performing periods. And it may be those peak performances which serve as someone's real legacy.

--Algot

[Christian Baune]

When you have imaginary numbers, this model breaks.

Consider this :

i / i = -1

Kind regards,

Christian

[Andrius Kulikauskas]

Hi Christian!

I thought that i / i = 1.

If i / i = -1, then that would mean i = -1 * i. But that's not true.

Which is to say, I don't understand. :)

Please explain!

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

2016.08.19 17:27, Christian Baune rašė:
> When you have imaginary numbers, this model breaks.
>
> Consider this :
>
> i / i = -1
>
> Kind regards,
> Christian
>
> Le ven. 19 août 2016 à 16:06, Algot Runeman <algot....@verizon.net

> <mailto:algot....@verizon.net>> a écrit :

>
> On 08/17/2016 07:47 PM, michel paul wrote:
>>
>> We can say either:
>>

>> * 6 meteors / 3 hours = 2 meteors/hour, or
>> * 6 meteors = 2 meteors/hour * 3 hours.

>>
> I think Michel gets onto the thing I was trying to approach. In
> both of the meteor examples, the significance is that there are
> three hours in which two meteors were observed during each of the
> hour periods.
>
> In neither case was an isolated value of two (meteors) the key
> element. All six meteors remained in play, the division was of the
> observations spread out into hour long chunks of time.
>
> I'm sure that I would not have absorbed this complexity at age ten
> or so. However, I do hope that the bigger issue is getting
> children to understand this side of the discussion even more than
> they "get" the right answer of 2. The process focus probably does
> help children with the operation. I do not think that's enough in
> the long run.
>
> In the mid-night wanderings of my mind, I felt a subtle cheat had
> left me with the (I think false) feeling that six divided by three

> is *just* two. (emphasis important)

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[Christian Baune]

Hi Andrius,

I made a typo.

i/i is actually "1".

I wanted to write "1/i=-1" :-\

The "i" is an imaginary number as described here : https://en.wikipedia.org/wiki/Imaginary_number

Basically, a super set of Real where "i²=-1".

Kind regards,

Christian

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[Peter Farrell]

Don't you mean 1/i = -i ?

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[michel paul]

On Thu, Aug 18, 2016 at 7:45 PM, Algot Runeman <algot....@verizon.net> wrote:

If I go out to look to the sky for three peaceful hours and see three meteors in the first hour and another three during the final hour, the same division remains possible, averaging 2 meteors during an hour's chunk of time, but the averaging is deceptive.

​Yep. The same average can result from a number of widely varying situations. I think genuinely understanding that fact is a much more important result for education than being able to readily calculate 'the' final answer.

The rate was two meteors per hour only in the artificial average.

​Although it is the case that the 'artificial' average could well be ​an actual occurrence.

It isn't until we combine lots of hours of observed meteors that an average becomes logical.

​Yep. And this is another important part of mathematical understanding. ​

I think that this uncomfortable observation is one of the main reasons I have come to dislike grading. Too often empty middle hours destroy the actual existence of three meteors in each of two hours, a rate higher than the nominal average we wind up calculating.

​I completely agree. During the last few years of teaching, grades as 'averages' struck me as meaningless as well.

By focusing on the average of two, we lose the bright shine of performance of the two hours in which we counted the actual meteors. Life is often lived in an uneven pattern. We shine for some of our hours and we merely maintain a pale background glow during many others. Our average cannot accurately represent the brilliance of our highest performing periods. And it may be those peak performances which serve as someone's real legacy.

​This is beautifully put.​

 

[Peter Farrell]

I absolutely agree! Well put.

[Joseph Austin]

On Aug 18, 2016, at 10:45 PM, Algot Runeman <algot....@verizon.net> wrote:

The rate was two meteors per hour only in the artificial average. It isn't until we combine lots of hours of observed meteors that an average becomes logical. Over a few hundred nights, perhaps the meteor average is two per hour. I feel far more comfortable with that kind of calculation than I do when averaging just a three hour period.

I think that this uncomfortable observation is one of the main reasons I have come to dislike grading. Too often empty middle hours destroy the actual existence of three meteors in each of two hours, a rate higher than the nominal average we wind up calculating.

By focusing on the average of two, we lose the bright shine of performance of the two hours in which we counted the actual meteors. Life is often lived in an uneven pattern. We shine for some of our hours and we merely maintain a pale background glow during many others. Our average cannot accurately represent the brilliance of our highest performing periods. And it may be those peak performances which serve as someone's real legacy.

I would say that "average" is an operation from statistics, technically called "mean,"

not to be confused with the mathematical operation of "division" 

nor with the physical concept of "proportion." 

Unfortunately, once we remove units from consideration and reduce the problem to pure numbers,

we lose the distinction, which leads to confusion.

I would like to see several instances or examples of a quantitative relationship discussed and understood

before I would describe a method for "calculating" it.

For example, why would you want to discuss an "average" number of meteors?

Doesn't "average" imply that there is some variability?  

(E.g, meteors are greatest when the target passes through the trail of a comet)

Otherwise, we would not speak of "average" but "rate".

And it's probably misleading to speak of "average" without also discussing the variability distribution.

As for grading, is an "average" a reasonable measure of a student's competence?

It would seem the highest score would be a better indicator of student ability;

any scores less than that could reflect as much on the teachers as on the student.

Now what if the "average" score for a given test is lower than that on other tests in the course?

Then that would clearly correlate more to the performance of the teacher than of the students!

I would typically throw out the lowest score or two as being more indicative of "extenuating circumstances"

unrelated to the student's overall competence.

In an ideal world, I would have assessments performed by different people than instruction,

to achieve some measure of objectivity and to avoid all hint of conflict of interest.

BTW, I make these observations from the vantage point of no longer being active in teaching;

I was just as guilty as anyone else in abusing statistics when I was actually grading students.

Joe Austin