Interpreting Multiplication

[Shoes+Ships+SealingWax]

My son is using Book A (2nd Ed.) of RightStart Math. I have been very impressed so far with the depth & accuracy of the information given, however Lesson 59 - Introducing Multiplication has me concerned. It states:

 

"Sometimes 6x3 is thought of as '6 groups of 3'. However, consistency with other operations requires a second look. When adding 6+3 we start with 6 and transform it by adding 3. When subtracting 6-3 we start with 6 and transform it by removing 3. When dividing 6/3 we start with 6 and transform it by dividing it into 3 groups. Likewise, 6x3 means we begin with 6 and transform it by multiplying it 3 times."

 

According to every other mathematical text I can find, this is completely incorrect - 6 x 3 IS "6 times (the number) 3"

 

I find the disagreement especially concerning because it has to do with the fundamental understanding of one of the "Big Four" operations.

 

Thoughts? Sources? Solutions?

[regentrude]

Since multiplication is commutative, this makes no sense. 6x3 is six groups of three (six times whatever three objects make the group) - which is the same thing as three groups of 6. Which, for a student, can be very easily visualized drawing arrays.

I see no benefit in creating an artificial confusion that typically is none to the student, since from working with arrays the commutative property of multiplication is very obvious.

In fact, when you think ahead to algebra, it is complete nonsense what these authors say. 2y is twice the quantity y, the multiplication sign is even omitted completely. Nobody thinks of it as "y groups of 2".

 

I would be inclined to ignore this discussion and not confuse my student by creating a seeming difference between 6 groups of 3 and 3 groups of 6.

Edited July 9, 2016 by regentrude

[wapiti]

Since multiplication is commutative, I would ignore this misstatement and move on.  There is a school of thought on teaching the way that is described, but I think it's dumb.

 

ETA, what Regentrude said.  FWIW, there was a thread discussing this not long ago based on some guy's article but I can't find it.

Edited July 9, 2016 by wapiti

[Shoes+Ships+SealingWax]

Since multiplication is commutative, this makes no sense. 6x3 is six groups of three (six times whatever three objects make the group) - which is the same thing as three groups of 6. Which, for a student, can be very easily visualized drawing arrays.

I see no benefit in creating an artificial confusion that typically is none to the student, since from working with arrays the commutative property of multiplication is very obvious.

In fact, when you think ahead to algebra, it is complete nonsense what these authors say. 2y is twice the quantity y, the multiplication sign is even omitted completely. Nobody thinks of it as "y groups of 2".

 

I would be inclined to ignore this discussion and not confuse my student by creating a seeming difference between 6 groups of 3 and 3 groups of 6.

6x3 and 3x6 give the same RESULT (commutative property) but the equations are not themselves interchangeable.

 

If you have 6 friends, then 4 boxes of 6 chocolates a piece do not meet your needs. Sure, 6x4 & 4x6 both result in 24 chocolates... but wrapping will be an issue!

 

2y may not state (2 x y), but it does give you sequence - it is 2 groups of y, rather than y groups of 2. It seems minor at this level, but is increasingly important in higher-level mathematics. Multiplying matrices comes to mind.

[Kiara.I]

Solution? Ignore it and move on. It really hasn't made much difference, and by level D it's just multiplication and the student can flip it in any order that makes sense to him.

 

Honestly, I find the fact that they even ADDRESS it reassuring, since it means the author has enough understanding of arithmetic to have an opinion about it, and to believe that it should be addressed, instead of just saying, "this is how one does multiplication." Even if you disagree with her position, at least she makes it clear there is a position to be discussed!

 

As to it mattering more later on, yes. It does. Which is precisely why you can ignore it right now, because I guarantee that your son will not get stuck in the paradigm of which is the groups and which the how many, but will instead treat them as completely commutative. So in other branches of arithmetic where it does matter, it will be fine to teach the order that matters then.

 

Sent from my Nexus 5 using Tapatalk

[Cake and Pi]

That discussion is in the parent notes (right?  I only have 1st edition on hand), and not intended to be shared with the student (and if it were I'd just not, because that would be confusing).  I'd guess this is their justification to the parent for teaching it differently than is mainstream.  They probably think their way will be more intuitive to the child.  I don't think it would matter to most kids though.  

 

I think it's fine to teach either way.  In fact, I do teach it both ways.  We start with RS's 6x3= "six taken three times" or 3 groups of 6.  After multiplication is solidly solid I discuss it being fine to call it "six of three" and think of it as 6 groups of 3.  Yes 6 groups of 3 is different than 3 groups of 6, but the semantics of the story problem need not be imposed on the equation.  It doesn't matter which way you read the equation as long as you execute it properly.  Mathematics is art; an equation is just a model, and in math the ends justify the means.

 

Yes, 6 boxes of 3 chocolates is quite different from 3 boxes of 6 chocolates, but both instances can be written 6x3=18 (or 3x6=18).  Likewise, if I have 18 cookies and want to divide them among my three friends and find out how many each friend would get,  it would be very, very different from if I had 18 cookies that I wanted to divide into groups of 3 for my friends so I can find out how many friends I can share with.  But both of these scenarios are written 18/3=6.  In one I broke the 18 into 3 groups of 6 and in the other I divided the 18 into 6 groups of 3.  Point being that which number you name as the group and which you call the number in each group is interchangeable when writing the equation, even though they are not interchangeable in the application of the equation.

[Cake and Pi]

And as far as higher level math is concerned, each topic will just have to be learned separately because there really isn't some far reaching, all encompassing naming convention that you see across the board.  I've seen the argument with matrices, that [ 1  4  -2 ] is a 1x3 matrix, and therefore it's more accurate to think of 1x3 as a one row (as on the abacus) of 3.  In matrices we'd name an entry in the same way, row then column, so the first number is how many vertically and the second number is how many horizontally -- which is completely opposite of how you'd name a point with xy coordinates (where the first number is horizontal and the second number is vertical).  Students just have to memorize the conventions for different subjects.  The order in which they multiply in grade school isn't going to matter for these later maths years down the road (IMO!).

[kiwik]

I had a similar discussion with ds7 the other day. We agreed that while it doesn't matter which way round the numbers go for calculations there will be some real life cases where it matters a lot. He is just learning multiplication so that is enough for now.

[regentrude]

2y may not state (2 x y), but it does give you sequence - it is 2 groups of y, rather than y groups of 2. It seems minor at this level, but is increasingly important in higher-level mathematics. Multiplying matrices comes to mind.

 

But he is exactly arguing the opposite: he claims that it is wrong to see 6x3 as "6 groups of 3"

 

 

 

"Sometimes 6x3 is thought of as '6 groups of 3'. ... Likewise, 6x3 means we begin with 6 and transform it by multiplying it 3 times."

 

I would not teach multiplication to an elementary age kid with an eye on matrix multiplication which is not commutative.

Edited July 9, 2016 by regentrude

[wapiti]

6x3 and 3x6 give the same RESULT (commutative property) but the equations are not themselves interchangeable.

 

If you have 6 friends, then 4 boxes of 6 chocolates a piece do not meet your needs. Sure, 6x4 & 4x6 both result in 24 chocolates... but wrapping will be an issue!

 

The fact that they give the same result means that they are interchangeable mathematically.  If we attach the context of a word problem, that may add a complexity, perhaps a sequence of computations (e.g. how many boxes do we need to buy if there are 5 in a box, we want four chocolates each for six kids means we would need to add another step after we compute 6x4 or 4x6), but it does not change how we can write a multiplication portion of an expression nor does it change the manner in which that multiplication portion is computed.

 

2y may not state (2 x y), but it does give you sequence - it is 2 groups of y, rather than y groups of 2. It seems minor at this level, but is increasingly important in higher-level mathematics. Multiplying matrices comes to mind.

 

My understanding is that the coefficient in front of the variable is simply a convention.  We would never write y2 even if we were computing y groups of 2 in the context of some word problem.

 

You may be interested in this thread on this topic and an article arguing that the order may matter - I find the extensive comments to that article particularly interesting and well worth reading through.  In my opinion, the bottom line is that to say order matters in non-matrix multiplication is to say the commutative property of multiplication does not exist, whereas, as we know, the commutative property is a crucial concept to understand in elementary arithmetic.

 

There's an interesting idea in daijobu's post on that thread that really shows the fun of exploring to further develop number sense by taking counters and arranging the same number of counters in different combinations of rows/columns/groups:

 

I'm going to weigh in on the "please be kind" part.  You have asked an excellent question, and I want to encourage you to keep asking questions of this sort on these boards.  You are doing your students a great service by being completely 100% clear on what is going on with even these basic operations.  Good for you!   If you encounter another topic along these lines, please keep asking; the pp's and others will provide lots of great ideas.  

 

I'm going to add another bit that may or may not be helpful.  In addition to arranging your counters into a 4x7 array, you can take those same counters and arrange them into a 2x14 array.  Or a 1x28 array.  Because all those operations provide the same result.  

 

Have fun and go deep whenever possible!

 

The bolded is very important  :)

Edited July 9, 2016 by wapiti

[kiwik]

To go back to the initial point I would say 6*3 is 6 groups of 3 and 3*6 is 3 groups of 6 and too bad if that isn't consistent with addition.