what is the difference between addition and multiplication?

[michel paul]

A few years ago Keith Devlin sparked a lot of controversy in secondary math land by adamantly asserting that multiplication is NOT repeated addition, not even in the case of whole numbers. I was intrigued by this, as I think many of us were in fact taught that it is. If this has been already thoroughly discussed here, I apologize, don't want to reignite any battles, but I recently had reason to think about it again, and it occurred to me - the reason it is not good to define multiplication as equivalent to repeated addition is that it makes it seem that there is essentially no difference between them.

This really is an interesting question to explore - what is the difference between addition and multiplication? 

If we define multiplication as repeated addition, then there really is no difference. And this is just an intuition, haven't completely developed it yet, but I think this might have something to do with students not understanding what ratio is? Because they don't actually understand what multiplication is, they also don't actually understand its inverse? I think the stretching analogy is really to the point - scaling a quantity to two times its original value is definitely not the same idea as adding the original value to itself. The procedures do result in the same value, but they are in fact two different ideas. If you want to enlarge a picture to twice its size, you do not do so by appending a copy beside it.

We usually use rectangular area as a visual model of multiplication, but I think that can promote the limitation of the repeated addition idea. Maybe a better visual model for multiplication would be a line through the origin? Here y is always the product of x and the RATE of change! Bingo. I think it would do a lot of good to emphasize that model. It is essentially connected to ratio in a way that repeated addition is not.

Here are some good examples of products that are not reducible to repeated additions: 'blue circle', 'red triangle', 'even prime', square fibonacci', etc. A deep mathematical understanding of product as the intersection of two sets seems strange to us when we study set theory if we equate multiplication with a kind of addition.

Having said all this, I will admit that I have quite frequently presented multiplication as though it were repeated addition. As a programming problem I have had kids define the basic arithmetic operations using only the concepts of zero, successor, predecessor, equality, and conditional reasoning. For example, what do we mean by the statement ' a is greater than b' in the set of natural numbers? Well, if a is zero, then the statement is false. However, if b is zero (and a is not), then the statement is true. Otherwise, the statement has the same truth value as 'the predecessor of a is greater than the predecessor of b'. 

That is why I was so interested in the Devlin discussion. One of the things you notice in this problem is that defining division in these terms is not only gnarly, there's an inherent limitation to the kind of division you can express. You can't nicely get to 'ratio'.

And so I'm wondering - isn't this where our students tend to be stuck in their understanding as well regarding what division is? They frequently never get to ratio in their actual understanding, though they 'study' it and pass tests regarding it, and perhaps it all starts with an inherently additive idea about what multiplication is?

A possible problem with modeling multiplication with a line through the origin - what if the kids haven't had linear equations or slope? Well, consider a right triangle with a base of 1 and a height of 2, let's say. Now stretch the base and the hypotenuse to create similar triangles ... we're multiplying! The new height will always be twice the new base. The same sort of thing will happen given any initial height with a base of 1. It is very easy to create an interactive model of this in GeoGebra. Experimenting with that model could be a way to introduce 'rate of change' in the first place? Prior to any formulas. Prior to any ridiculous discussions of, "Which comes first when we subtract? Is it y_1 or y_2?"

 
Now watch what happens when we frame the question this way - "What is the difference between a sum and a product?" 

This is very interesting. A 'sum' is the value of a collection of objects of some type. The objects could be collected in a list or in some other way, but there's nothing special about how a sum needs to be structured. However, a 'product' IS a structure. Always. This is true on a lot of levels, even in a very general pre-arithmetic sense, a product is not just a collection of parts. The parts are organized. Again, we typically use an array-like structure to represent 'product', but we could also use similar triangles, and this might be pedagogically valuable as it in no way suggests that multiplication is simply a shorthand for addition. It overcomes a limitation in our curriculum that we math teachers are usually unaware of.

What's so cool is that in addressing this limitation in our curriculum (and apparently collective secondary math understanding from what I've seen) we improve both mathematical rigor and technological literacy simultaneously, as thinking in terms of 'sums' and 'products' vs. 'adding' and 'multiplying' gets back to functional vs. procedural thinking styles. I have gained insight on many occasions by translating my mathematical ideas from procedural terms into functional terms and vice versa - and this is what we want the students to be able to do as well. And note that this is not necessarily 'programming'. This is prior to any particular programming language. However, it's something that the activity of programming can make you conscious of.

In teaching kids that sums and products are different types of things we wouldn't have to resort to superficialities like PEMDAS. When the only difference between multiplication and addition in a student's mind is its rank in PEMDAS, it's really no wonder that we're in the mess we're in.

-- Michel

 

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

- Richard Feynman

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

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

[Oleg]

Dear Michel,

I believe that introducing multiplication as repeated addition is not the best way to teach it. Multiplication, first and foremost, computes area and should be introduced as such. If interested, please see a short PDF file attached to this e-mail. To add to it, this approach to multiplication visualizes repeated multiplication as volume, naturally leads to dimensions higher than three, makes associativity visual, too, etc., etc.

Very Truly Yours,

Oleg Gleizer.

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[Dmitri Droujkov]

Michel,

This reminds me of research Maria Droujkova and myself did a few years back which was summarized in the attached poster image. It's a collection of 12 multiplication models with combinations and fractals being my favorite. I think they land themselves very well to programming problems for students. These models are also less similar to repeated addition than others.

I don't think any one model is the best, but all of them should be introduced to kids with invitation to create their own. Some of the models have very low level of prerequisites and are accessible to very young kids.

A paper copy of the poster is available here http://www.amazon.com/Multiplication-Models-Natural-Math-24X36/dp/0977693910

Thanks,

--Dmitri

<tmp.pdf>

[Darius Clarke]

Dear Oleg, 

I just curious ... where's the area seen in 2^4.

Very Truly Yours,

Darius Clarke

[Sue VanHattum]

This is the (hyper)volume of a hyper-cube with side length 2. Right?

---

From: soci...@gmail.com
Date: Sat, 8 Dec 2012 20:05:26 -0800
Subject: Re: [Math 2.0] what is the difference between addition and multiplication?
To: mathf...@googlegroups.com

[Oleg]

Dear Sue,

Yes, of course. 2^4 is the volume of a tesseract with the side length two, 2^5 is the volume of the 5D cube with the same side length, etc.

Very Truly Yours,

Oleg Gleizer

[Oleg]

Dear Darius,

2^4 is the volume of a tesseract with the side length two.

Very Truly Yours,

Oleg Gleizer

[michel paul]

Hi Oleg,

Thanks for this. I always appreciate your perspective.

On Sat, Dec 8, 2012 at 12:24 PM, Oleg <oleg...@gmail.com> wrote:

> Multiplication, first and foremost, computes area and should be introduced as such. 

I was wondering if you consider an area model more fundamental than a scaling model? 

I would agree that the rectangular area model is what I've primarily used myself in my life, though I do remember it being explained as repeated addition when I was a kid, and I never really questioned that model until Keith Devlin's discussion. He suggests a scaling model, not exclusively of course, but it intrigued me, and as I thought about what he was saying it hit me that a fundamental difference between scaling and repeated addition is the concept of ratio. Ratio is intrinsic to scaling, front and center, in a way that it is not in repeated addition. 

Ratio also shows up nicely in the area model when we draw a diagonal. The area ab necessarily contains the ratio b/a.

So either way, it seems that ratio is an essential component of what multiplication is all about that we do not effectively communicate to students using a repeated addition model.

Of course, that's assuming we're talking about multiplication up through the reals. Once we get into matrix multiplication, then it's another story again.

Which suggests a question - do you have a good way to express to students what matrix multiplication IS? It's one of those things in our curriculum that gets equated with a procedure but with no underlying unifying idea. I have been able to condense the process to the concept of dot products of corresponding row and column vectors, and this becomes a very nice programming exercise for students. What initially seems like an impossible task, programming matrix multiplication, can be organized quite efficiently. I also have them visualize the matrices written diagonally from each other, the right above the left, and this organizes the dot products nicely. But I still don't have a good way to immediately say what this product IS. 

-- Michel

[michel paul]

Hi Dmitri,

Thanks very much for this! I love the poster. It's good to reflect on how all these different models relate to each other. It's good for a lot of levels. I think I will have the students first try to come up with as many models of multiplication as they can and then have them look at the poster. 

The fractal model could also serve as a model of exponentiation. And something interesting in Devlin's discussion is that he also states that exponentiation is not the same thing as repeated multiplication. Just as multiplication is something different from repeated addition, he also says exponentiation is something different from repeated multiplication. So ... what is it?

So far I haven't had a good answer, but the fractal example made me wonder. Could self-similarity have something to do with it? Exponential growth is self-similar in a way that scaling is not?

- Michel

[Christian Baune]

The difference between addition and multiplication is 45º!

[Alexander Bogomolny]

I wonder what is it that people have in mind talking of "repeated addition model" - emphasis on MODEL. Is there just one?

Counting, i.e., the ability to count, the association of magnitude with a group of objects, "numerosity" I believe it's called, precedes arithmetic operations. The ability to count directly implies basic properties of addition; for, the same group of objects could be counted in various ways. 

Now, this said, what is exactly "repeated addition model"? I may be counting six elements by twos. Arithmetically, I can express that as 2+2+2; is that a model? The reason I doubt that is because there are at least two geometric representations of this operation:

1) 

** ** **

2)

**

**

**

The latter is nothing but the "box" model. The same six items can be placed in many ways, in particular, in a line or in 2x3 or 3x2 arrays. That the latter two are the same is also a consequence of inherent numerosity, supported by the usual consideration of the same rectangle on one side or the other.

The bottom line is that there is something to learn from the "repeated addition" and in any way it may be helpful to have several viewpoints available for discussion. Counting is naturally a case of repeated addition of 1. Counting by 2s and 3s, etc., may come as a natural extension, and then all it takes is to arrange objects in a way suggestive of the next step.

Alex

[michel paul]

I agree. There is definitely something one can learn from studying different structures of repeated addition. This has been a great discussion. At this point I would say repeated addition can be subsumed under scaling, and it's this understanding of scaling that is often missing from explanations of multiplication. 

I'm still curious whether there might be a connection between student misunderstanding of ratio and a limited view of multiplication, especially given that understanding of ratio correlates with success in algebra.

- Michel

[kirby urner]

On Sun, Dec 9, 2012 at 4:24 PM, michel paul <python...@gmail.com> wrote:

I agree. There is definitely something one can learn from studying different structures of repeated addition. This has been a great discussion. At this point I would say repeated addition can be subsumed under scaling, and it's this understanding of scaling that is often missing from explanations of multiplication. 

I'm still curious whether there might be a connection between student misunderstanding of ratio and a limited view of multiplication, especially given that understanding of ratio correlates with success in algebra.

- Michel

When it comes to scaling, I think what is too often dropped out is the respective 1st, 2nd, 3rd powering of linear, areal and volumetric measures.

To shrink every linear distance to 1/8 of what it was i.e. to multiply ever length by 0.125, is to shrink the surface area of this item by 0.125 * 0.125, and to shrink volume to (1/8) to the 3rd power.  Start with any shape, how about a steam engine (since model trains exist, are prevalent).

An object to focus on in that respect might be the classroom globe (if you're in a classroom).  If the linear dimensions are all doubled, the volume goes up by eight times.

Integrating these three "growth curves" is not done often enough, especially in relation to simpler polyhedrons (simpler than steam engines).

Kirby

[Christian Baune]

Tetration is repeated exponentiation, exponentiation is repeated multiplication, product is repeated sum and sum is repeated increment.

increment is an atom.

You can now define "f^k" as being an operator of order "k".

When k=0, you got "succ", when k=1, you got addition. When k=1, you get product, When k=3, you get power, When k=4 you get tetration.

As such, you can define f^k as k iterations of f^(k-1).

It does perfectly fit the bill and real numbers aren't an issue at all. For irrational numbers : sqrt(3)*sqrt(5) is only equal to itself while any couple of real number can be shifted as such we got 2 whole numbers, even an infinite number of time.

The fact is that "multiplication as repeated addition is a common pattern bringing confusion in bright minds per its simplicity and the unknown of its extends, seen as too simple and childish it often hide a whole area of overall generalization and complexity."

Kind regards,

Christian

[Bradford Hansen-Smith]

Alexander, you stated, "Now, this said, what is exactly "repeated addition model"? I may be counting six elements by twos. Arithmetically, I can express that as 2+2+2; is that a model? The reason I doubt that is because there are at least two geometric representations of this operation:

1) 

** ** **

2)

**

**

**

The latter is nothing but the "box" model. The same six items can be placed in many ways, in particular, in a line or in 2x3 or 3x2 arrays. That the latter two are the same is also a consequence of inherent numerosity, supported by the usual consideration of the same rectangle on one side or the other."

I just wanted to suggest another 3 sets of 2 opposite edges at right angles to each other; 6 edges of a tetrahedron. I only bring this up because of the fundamental nature of the tetrahedron in 3-D geometry that is degenerated in 2-D geometry.
Brad

--
Bradford Hansen-Smith
www.wholemovement.com

[michel paul]

This is from the very beginning of his Geometry:

"(Multiplication is) taking one line which I shall call unity in order to relate it as closely as possible to numbers, and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity."

- Descartes

The impression we might form from school texts about what Descartes was doing is a little different from what we see when we look at his actual work. 

I also found this relevant:

Descartes professed that the abstract quantity a² could represent length as well as an area. This was in opposition to the teachings of mathematicians, such as Vieta, who argued that it could represent only area.

Very interesting to see how math evolves! These discussions were happening only about 400 years ago. 

[David Chandler]

On Sat, Jan 5, 2013 at 8:59 AM, michel paul <python...@gmail.com> wrote:

"(Multiplication is) taking one line which I shall call unity in order to relate it as closely as possible to numbers, and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity."

- Descartes

Descartes description reminds me of a geometric construction that enacts multiplication:  Draw a triangle and draw a line parallel to one side cutting the other two sides.  If you set one part of one side equal to unity, you can arrange the proportions among the other three lengths either as one being the ratio of the other two, or one being the product of the other two. 

--David Chandler

[mok...@earthtreasury.org]

If you take as your measure of area a rectangle with unit side, then the
area of the rectangle and the length of the second side are in a constant
proportion that makes it evident that either can be the measure of the
product. Later on, it could be explained that the two measures represented
domains that were algebraically isomorphic to each other and to real
numbers. (I skipped over adding in negative numbers, which are not
necessary to the argument.) The ideas of isomorphism, homeomorphism, and
more generally, much later, morphism are among the most powerful in
mathematics.

There was an equally strong argument over powers higher than the third,
given the prejudice against more than four dimensions. Interpreting higher
powers as lengths and thus as real numbers was a huge extension of
algebra, leading eventually, along with other ideas, to acceptance of
higher-dimensional geometry.

On Sat, January 5, 2013 11:59 am, michel paul wrote:
> This is from the very beginning of his

> Geometry<http://books.google.com/books?id=MB7F32p0y5MC&pg=PA2&lpg=PA13&dq=%22Geometry+of+Ren%C3%A9+Descartes%22>
> :

>
> "(Multiplication is) taking one line which I shall call unity in order to
> relate it as closely as possible to numbers, and which can in general be
> chosen arbitrarily, and having given two other lines, to find a fourth
> line
> which shall be to one of the given lines as the other is to unity."
>
> - Descartes
> The impression we might form from school texts about what Descartes was
> doing is a little different from what we see when we look at his actual
> work.
>
> I also found

> this<http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes#Mathematical_legacy>relevant:
>
> Descartes professed that the abstract quantity *a2* could represent length

>> as well as an area. This was in opposition to the teachings of
>> mathematicians, such as

>> Vieta<http://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te>,

>> who argued that it could represent only area.
>
>
> Very interesting to see how math evolves! These discussions were happening
> only about 400 years ago.
>
>
>
> -- Michel
>
> ===================================
> "What I cannot create, I do not understand."
>
> - Richard Feynman
> ===================================
> "Computer science is the new mathematics."
>
> - Dr. Christos Papadimitriou
> ===================================
>

> --
> You received this message because you are subscribed to the Google Groups
> "MathFuture" group.
> To post to this group, send email to mathf...@googlegroups.com.
> To unsubscribe from this group, send email to
> mathfuture+...@googlegroups.com.
> For more options, visit this group at
> http://groups.google.com/group/mathfuture?hl=en.
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>


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[michel paul]

On Sat, Jan 5, 2013 at 9:34 AM, David Chandler <david...@gmail.com> wrote:

Descartes description reminds me of a geometric construction that enacts multiplication:  Draw a triangle and draw a line parallel to one side cutting the other two sides.  If you set one part of one side equal to unity, you can arrange the proportions among the other three lengths either as one being the ratio of the other two, or one being the product of the other two.  

Yes, it's very similar. The resulting structure is identical, but Descartes finds the product, whereas this construction begins with the product already in place. 

Again, what I find interesting is that the way in which Descartes united algebra and geometry isn't quite the same as the caricature 'Cartesian coordinate system' we present in class.

Here's a nice quick Geogebra construction that captures multiplication as proportion:

1. A = Point[xAxis]
2. B = Point[yAxis]
3. a = Segment[(1,0), B]
4. b = Segment[A, Intersect[Line[A, a], yAxis]]

As you drag points A and B, segments a and b remain parallel, and segment b always points to the product x(A)*y(B).

It very nicely illustrates how negative * negative = positive.

-- Michel

[kirby urner]

Likewise if you take to edges a, b and draw these length at 60 degrees, then close off the gap between their tips with a third side c, you may call this a x b (the area). 

Changing either a or b changes the triangle's area in proportion. 

The same may be done using a 90 degree triangle, in which case the resulting area is 1/2 the rectangular area.

The Pythagorean theorem is unchanged.  Draw two equilateral triangles on legs a, b and their areas will be that of the hypotenuse's equilateral triangle.  Actually any similar proportionate figures, scaled by the legs and hypotenuse, will do the job (not just squares).

Kirby

[kirby urner]

On Sat, Dec 8, 2012 at 11:18 AM, michel paul <python...@gmail.com> wrote:

A few years ago Keith Devlin sparked a lot of controversy in secondary math land by adamantly asserting that multiplication is NOT repeated addition, not even in the case of whole numbers. I was intrigued by this, as I think many of us were in fact taught that it is. If this has been already thoroughly discussed here, I apologize, don't want to reignite any battles, but I recently had reason to think about it again, and it occurred to me - the reason it is not good to define multiplication as equivalent to repeated addition is that it makes it seem that there is essentially no difference between them.

Yeah, that was big.

I'd say English the language has its own grammatical models.  When we do something "ten times" that means we repeat the thing ten times.  Reiteration.  Then we use "times" as a synonym for "multiply" e.g. "ten times ten".  The reiterative looping meaning is always there.  Running off to play with other symbols is all fine and good, but English doesn't let go of its speakers that easily.

Out in symbols world, we have logical beasts that multiply but don't sensibly add (matrices) or entities that multiply only with beasts of a different type, like vectors with scalars.

My recommendation is to drop into a computer language that permits operator overriding so that students get a chance to imaginatively define __add__ and __mul__ however they like, but then point out similarities where they do exist i.e. both have an "identity" where a + id = a or a * id = a (the "no change" element). 

Both also tend to have inverses i.e. polar opposites that, when combined, produce the same null particle id. 

Each set-with-binary-op (+ and *) has those, where the set may be really simple, like integers modulo some prime (or just use a set of strangers i.e. totatives to a composite).

Operator overloading with an abstract algebra spin is a cool new advantage for those schools over the hump of allowing computer languages in math class (I'd say any school that takes STEM seriously).

Kirby