# Is division overrated?

Discussion in 'Mathematics' started by Eureka!, Mar 21, 2007.

Division is really multiplication in disguise. "Division' into equal pieces might be an everday rewquirement of the real world, but is there any need to make the concept so pervasive in mathematics? Wouldn't maths be easier to teach and understand if the division line were abolished?

By using the reciprocal R(x), definition R(x)x=1, all problems can be solved, and equations manipulated. (Please don't say reciprocation is division in diguise, it is not. If anything the opposite is true. The process of long division is essentially the process of finding the reciprocal).

Without the fussiness of 'division', the action of multiplication and its representation on the number line can be focused on. For a diagram see

http://www.infet.co.uk/forums/viewtopic.php?t=3230&postda...

Division is really multiplication in disguise. "Division' into equal pieces might be an everday rewquirement of the real world, but is there any need to make the concept so pervasive in mathematics? Wouldn't maths be easier to teach and understand if the division line were abolished?

By using the reciprocal R(x), definition R(x)x=1, all problems can be solved, and equations manipulated. (Please don't say reciprocation is division in diguise, it is not. If anything the opposite is true. The process of long division is essentially the process of finding the reciprocal).

Without the fussiness of 'division', the action of multiplication and its representation on the number line can be focused on. For a diagram see

http://www.infet.co.uk/forums/viewtopic.php?t=3230&postda...

3. ### Polecat

In the early days of computers, division was done by repeated subtraction until the remainder was less than the divisor.
I agree, however, that reciprocation is an underrated operation - of course in higher maths reciprocation corresponds to the very important concept of the inverse.
I wouldn't bother with the number line when discussing multiplication and division.

4. ### ragpicker

do you think young children will come to that understanding? Or do you accept that their arithmetical thinking is different and will make sense of division by seeing it as being about sharing and grouping ie partitive & measurement division.

Arguably division and multiplication are underated and introduced too late within the primary curriculum. Developing an understanding of measurement division can build and support place value understanding in young children.

You might be interested in the work of Tom Carpenter & Elizabeth Fennema on this. It is based on research of children's arithmetical thinking and really blows your suggestion out the water.

5. ### goldbug

Ragpicker,

I'm interested!
Have you got more details?

goldbug

You can explore the concept of ACTUAL DIVISION just as well using reciprocals. Probably better.

A half is R(2). A quarter is R(4).

Somehow I feel division has gatecrtashed maths. Just because a half is obtained by the process of dividing 1 by 2, there is no need to write 1/2 to represent a half all the time.

Kids get confused between process and value.

7. ### coyoteNew commenter

In a similar fashion, we do not need subtraction. Instead, we just add negative numbers.

When to teach this? To me these ideas are abstractions that are simply not understood until we have a concrete idea of the processes of addition, subtraction, multiplication and division as "real life" operations.

But I know there is research out there that challenges this model of maths education...

8. ### maths126New commenter

While we are thinking about too many operations, we might as well reject multiplication too.
Multiplication was only ever intended to "fold together" many additions into a single step: the French word "Plier" = "To fold" as in 2-ply, 3-ply, combined with "many" gives us "manifold" or "multiplier".

As mentioned above, subtraction is just disguised addition, so everything just comes down to addition.

Counting is just a convenient notation for a notched stick or bone.

That in turn, was just a matching device to support a one-to-one correspondence.

So we reduce it all down to equality.

Perhaps I made a few oversimplifications....

9. ### ragpicker

Goldbug - Im not sure what in particular you are interested in? I have posted frequently on the work of Tom Carpeneter and others in the area of children's arithmetical thinking. If you trawl through my posts you will find examples of practice as well as references to the literature. The practice that they developed comes form their research and there is a significant body of this going back over twenty years. I studied with these researchers and teachers in the States and have been developing these ideas here.

coyote sums up very well the importance of working from children's conceptulisations and both coyote and maths126 highlight how ludicrous Eureka's recommendation is. It's interesting that someone who argues against prescriptive curricula should promote ideas that are completely alien to learning with understanding.

10. ### HB1

If you want to make things as simple as possible - why not just go with the set theoretic definition of the natural numbers. 0 is defined as the empty set, all other numbers are defined in terms of sets of sets of the empty set, greater than becomes 'is a member of' and all mathematical functions are defined as sets of ordered pairs.

No need for anything other than the set membership relation and the empty set. And you can then define rational, irrational and complex numbers once you've done that.

11. ### mecky

I agree with Ragpicker. We can introduce the idea of multiplying / dividing to 5year olds by giving them a sheet folded into 3x4 squares. I'm fascinated by the Maths126's post.

12. ### Maths_MikeNew commenter

If you share 6 sweets between three children they get 2 each. How exactly do you explain that with reciprocation. What a load of academic nonsense. Might be great for your masters or whatever else you are doing but for thoe of us in the real world its a waste of time

13. ### ragpicker

Matthew has 24 cakes and some plates. He puts 6 cakes on each plate. How many plates will he need?

Matthew has 24 cakes and 4 plates. He puts the same number of cakes on each plate. How many cakes are on each plate?

Two different types of DIVISION problems that very young children can solve without instruction by modelling out the action of the problems.

Eureka please explain the relevance of reciprocation to the above examples and also why you consider it useful to disregard children's arithmetical understanding in place of arcane procedures that need to be directly taught?

14. ### jtw521

Division is a Joy

Ragpicker - can you elborate about research that "blows me out of the water"?

HB1 & set.... gulp! derrrrr.... lol

Mmike said "If you share 6 sweets between three children they get 2 each.

Ragpicker said : "Matthew has 24 cakes and some plates. He puts 6 cakes on each plate. How many plates will he need?

Matthew has 24 cakes and 4 plates. He puts the same number of cakes on each plate. How many cakes are on each plate? "

These are all examples of finding the reciprocal. When you "do division", you are actually searching for the reciprical. |Long division" is actually long reciprocation.

I repeat - division is a real world concept, but when it comes to the maths "division" does not exist in one's mental processes when performing the calculation. If kids were trained to hunt the reciprical in real world "division", then the unecessary mathematical middleman called division could be eliminated

Understanding multiplication and reciprocation on the number line is a valuable tool in problem solving. See my diagram from post 1. It is simple and straighforward enough is it not? Putting "divsion" in as well would be awkward and pointless.

Let's talk specifics here. To anyone who wants to shoot me down... fine, but try aiming first!

16. ### emilyisobel

.----------------
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|----Eureka------|
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|_______|________|

KABOOM!

17. ### emilyisobel

That was supposed to be Eureka in the gunsight, but it didn't really work. I really need to learn to preview posts first.

Love you really Eur ;0)

Actually, I was wrong to say that 6 divided by 2 involves finding the reciprocal. It involves finding the factor. The process is very similar to using the reciprocal in problem solving, where the reciprocal is simply a factor. But the main point is that there IS NO SUCH THING IN ARITHMETIC AS DIVISION. One is finding the factor in a MULTIPLICATION. That is why the reciprocal is important - because you can keep everything within the concept of multiplication.

For example, scale a recipe for 4 up to 10. 10 x R(4) x 4 will take 4 to 10.

Lol ! Emily ... luv u 2!

20. ### ragpicker

Eureka - i may have got the wrong end of the stick here.
I'm interested in children's arithmetical thinking and how they make sense of real life problems. I don't know what you mean by 'traniing children to hunt the reciprocal'.

the problems I gave require no training. Young children will solve them without prior instruction, whether you choose to call this division or reciprocation is up to you. 5 year olds wouldn't call it either of those.

What they would do is solve each of those problems in very different ways buy directly modelling the language of the problem.
For the first problem they would make a pile of 24, then group sets of 6 from that pile. The second would involve sharing out. Teachers can then help children build on these informal strategies and eventually connect them to more formal algorithms. At what stage do you propose telling them that about reciprocation?