What's strange to me is that the Russian Math discussion of factors and multiples, which I've had huge trouble separating & defining in my mind, worked.....while the Parker & Baldridge discussion of quotitive/partitive does not work (for me).

I can't tell why.

-- CatherineJohnson - 11 Jul 2005

What I didn't know was whether equally ranked operations are performed left to right or right to left, and that's probably because I have great trouble with left and right. Always have, always will.

Yeah, that may be a left-to-right thing.

Left-to-right is 'natural' and 'instinctive' to me.

Probably left-to-right is overlearned for me. I remember having a trick for telling the two apart when I was a child, which was to remember which hand I put over my heart when I said the pledge of allegiance.

-- CatherineJohnson - 11 Jul 2005

How much is in the cup, versus how many cups.are the two types of division. 32 oz of water poured into 4 cups gets you how much water in each cup?

How many 8 oz cups can be filled with 32 oz of water?

These two things sound like the exact same thing to me.

I can't tell whether that's because it's too late to disaggregate, or because I'm not specifically talented in math.

-- CatherineJohnson - 11 Jul 2005

How much is in the cup, versus how many cups.are the two types of division. 32 oz of water poured into 4 cups gets you how much water in each cup?

How many 8 oz cups can be filled with 32 oz of water?

Yes, definitely check out Wu's chapter, though Parker and Baldridge's discussion which you quoted is also pretty good.

-- BarryGarelick - 11 Jul 2005

_Thirty years after the last time I took a look at the rules, I didn't even have to review them to teach them to Christopher.

And then I find out Carolyn barely even knows them!_

Whoops! Not quite accurate. I know that exponents come before mult/div come before add/subtract,

What I didn't know was whether equally ranked operations are performed left to right or right to left, and that's probably because I have great trouble with left and right. Always have, always will.

-- CarolynJohnston - 11 Jul 2005

Why distinguish partitive and measurement division? This distinction is not something students need to know, but it is something that can make you a better teacher. A key point of this section [on partitive & quotitive division] is that students must come to associate the operation of division with completely two different types of word problems. That difference is not apparent to adults, who long ago learned to instantly associate both with division; it requires effort to become conscious of that automatic association. But teachers who learn to distinguish partitive and measurement division are better able to understand their students' thinking They are in a better position to insure that their students see an appropriate mix of division problems, and are better prepared for making up word problems for division.

I've got to go corral materials for Andrew's teacher (he finally has a summer program, 25 miles away, headed up by a high school resource room teacher who apparently has never taught severely autistic elementary school-age kids before.

Just got THE CALL:

Andrew is lying on the floor of the room screaming, all the other students are 'upset,' they had to be sent out of the class to the computer room, she is 'very concerned' about Andrew, etc.

It never ends.

-- CatherineJohnson - 11 Jul 2005

Where I completely and totally lost it on quotitive & partitive was in the division of fractions section in Liping Ma's book.

[pause]

Oh my gosh.

I'm losing my 'student confusion.'

When I first read the chapter on division I could barely understand a word--especially the section on division of a fraction by a fraction.

Now I'm having trouble remembering what I couldn't understand.

This is upsetting.

-- CatherineJohnson - 11 Jul 2005

This explanation hasn't 'worked' for me, and I think I can explain why, for once.

You say that:

Partitive problems ask you to divide number of objects by number of groups, and get number of objects as an answer.

then you give as a sample problem:

I have a board of length 16 inches, and I need to make 10 shorter boards of equal length out of it. How long can each board be? (16 objects, 10 groups)

To me, the word 'board' implies one single object.

To you, it's not one single object; it's 16 objects, because it's 16 inches and inches are the units you're using.

But by the time I get to that distinction, I'm lost.

-- CatherineJohnson - 11 Jul 2005

I don't know how respected teaching is in China. As I say, teachers have a high school education.

In Singapore, becoming a teacher is a huge, big deal. Lots of respect, lots of competition, and I believe the pay is considered desireable & enviable (though I don't know about this last).

-- CatherineJohnson - 11 Jul 2005

The funny thing about teaching not being a respected profession is that the entire origins of constructivism were an effort to professionalize the job by stripping away all power from lay people, including school boards & parents.

-- CatherineJohnson - 11 Jul 2005

I keep planning to get to Hu's chapters, and I just HAVE NOT DONE SO YET!

-- CatherineJohnson - 11 Jul 2005

The interesting thing about the Chinese teachers is that they are barely educated (radically not the case in Singapore).

They have the equivalent, IIRC, of a high school degree.

They acquire most of their knowledge of mathematics on the job:

• they study the textbooks intensively

• they meet with colleagues in study groups (YES! WE NEED TEACHER RELEASE TIME!)

-- CatherineJohnson - 11 Jul 2005

The other hilarious example of MATHEMATICAL KNOWLEDGE THAT IS USELESS TO MATHEMATICIANS appears to be order of operations.

I have ZERO problem grasping & reading & using the standard rules concerning order of operations.

Thirty years after the last time I took a look at the rules, I didn't even have to review them to teach them to Christopher.

And then I find out Carolyn barely even knows them!

I wonder if there's something about those rules that makes them more easily learned & retained by a nonmathematician.

-- CatherineJohnson - 11 Jul 2005

But that's another distinction that's meaningless in terms of mathematical properties and principles (thanks to the commutative property).

I've had a huge amount of trouble dealing with 3 x 4 versus 4 x 3.

I can't explain why.

Once again, I hit the limits of words.

Not only can I not explain what the sensation of having one's mathematical knowledge 'grow' or 'deepen' or 'improve' or whatever, I also can't explain what it is I 'don't understand' about 3 x 4 versus 4 x 3.

And only an extremely good teacher could figure it out.

If a mathematician asks me what is 3 x 4 versus 4 x 3, of course I'll say it's 12, and I'll believe it.

That's not where the problem is.

The RUSSIAN MATH book is probably, finally, letting me 'get' this.

-- CatherineJohnson - 11 Jul 2005

My friend Kris said to me one day, 'My husband thinks 3 x 4 is the same thing as 4 x 3. But it's not.

I can't remember the example we came up with, though.

-- CatherineJohnson - 11 Jul 2005

I love that line about the difference that is no longer apparent to adults.

I actually find it beautiful, in an emporer's new clothes kind of way: children can still see a thing we no longer can (in this case the huge gaping distinction between partitive and quotitive division).

-- CatherineJohnson - 11 Jul 2005

Here's what Parker & Baldridge have to say:

Why distinguish partitive and measurement division? This distinction is not something students need to know, but it is something that can make you a better teacher. A key point of this section [on partitive & quotitive division] is that students must come to associate the operation of division with completely two different types of word problems. That difference is not apparent to adults, who long ago learned to instantly associate both with division; it requires effort to become conscious of that automatic association. But teachers who learn to distinguish partitive and measurement division are better able to understand their students' thinking They are in a better position to insure that their students see an appropriate mix of division problems, and are better prepared for making up word problems for division.</blockquote.

-- CatherineJohnson - 11 Jul 2005

I am going to have to WORK to understand this distinction even as Bernie has explained it!

ALTHOUGH: one thing I've learned, I THINK, is that the best way to finally grasp a mathematical distinction is to do lots of problems that exemplify that distinction.

Even better is to do problem sets like the ones in RUSSIAN MATH that somehow, and brilliantly, lead the student to the distinction problem by problem.

That book is stunning.

-- CatherineJohnson - 11 Jul 2005

Hung-Hsi Wu, a mathematician at UC Berkeley, also discusses these two types of division problems, though refers to quotitive as "measurement". It is discussed in his draft chapter 1 of his textbook for future math teachers; the chapter is on whole numbers. It can be found at http://math.berkeley.edu/~wu/EMI1c.pdf. I've talked to Carolyn about his chapter on fractions (chapter 2) which is also available on his web site. I enjoy both chapters and think they're both worth reading. As I pointed out to Carolyn, these are drafts and he is welcome to comments from people, so I encourage you to contact him. He is a long veteran of the math wars and worked with Milgram and others in writing the California standards. He has devoted much of his time away from math research to work with teachers in California to help them understand and teach math, and he has also been very supportive and helpful to me in my endeavors. Ralph Raimi, another mathematician, carefully reviewed these draft chapters and provided comments to Wu during the writing. They are excellently written and worth while reading.

Liping Ma, by the way, also makes the point in her book that the problem with math teaching in the U.S. is that teachers have learned math imperfectly, and therefore teach it imperfectly, whereas in China, the teachers have what she terms the PUFM: Profound Understanding of Fundamaental Mathematics. Wu gets at this too, but perhaps too rigorously for some.

Barry

-- BarryGarelick - 11 Jul 2005

partitive and quotitive pedagogy: a primer and an opinion

Catherine mentioned in one of her comments that she always finds it amusing when a mathematician encounters the notion of partitive vs. quotitive division:

I absolutely think there's all kinds of elementary math knowledge real mathematicians don't have, or did have but forgot, etc.

I always crack up when i see or read real mathematicians reacting to the 'partitive'-'quotitive' distinction in division.

They think it's ridiculous!

(And btw, I STILL can't explain the difference, so I'm not even going to bother to try....)

She's absolutely right. When I first encountered the notion of partitive vs. quotitive division (Liping Ma goes into a lot of detail about it in her book) I thought it was unnecessary obfuscation.

I know I never learned it myself. I don't know if my teachers knew it, but I know they never taught it to me (although Liping Ma says they didn't need to). And I don't know whether I need to know it in order to teach young children the full meaning of division, although Liping Ma says I do.

But as it happens, I do know what the difference is: my husband explained it to me in brilliantly simple terms (having learned it at the same time I did, and distilled its meaning more efficiently than I did). Here it is:

Partitive problems ask you to divide number of objects by number of groups, and get number of objects as an answer.

the partitive type of word problem asks this question: if I have x objects, and I want to split them into y groups, how many objects will be in each group?

Examples of partitive problems:

I have a board of length 16 inches, and I need to make 10 shorter boards of equal length out of it. How long can each board be? (16 objects, 10 groups)

I have a batch of 128 cookies. I need to split it into 8 equal bags of cookies. How many cookies will there be in each bag? (128 objects, 8 groups)

I have 12 cans of pears, and I need to serve 24 kids at lunch. How many cans of pears will each kid get? (12 objects, 24 groups)

It is somewhat difficult to frame word problems involving division by fractions as partitive problems, because you are dividing by the number of groups you want. Generally, you don't want a fractional number of groups. Note that in the problems I gave as examples of partitive division, the divisors are always whole numbers.

But here is a partitive word problem that uses a fractional divisor:

I have two cans of dog food that I need to split into 1-1/2 servings for my big and small dog. How many cans will be in a single serving? (2 objects, 1-1/2 groups -- awkward!)

Quotitive problems ask you to divide number of objects by number of objects, and get number of groups as an answer.

the quotitive word problem asks: If I have x objects, how many groups of y objects can I make from them?

Examples of quotitive problems:

I have a board of length 16 inches, and I need boards of length 1-3/4 inches. How many short boards can I cut from the longer board?(16 objects, 1-3/4 objects)

I have a batch of 128 cookies. I need to split it into bags of 12 cookies to give to children at school. How many such bags can I give away? (128 objects, 12 objects)

I have 12 cans of pears, and I need to serve a half can of pears to every kid at lunch. How many kids can I serve? (12 objects, 1/2 objects)

Problems involving division by fractions are easier to frame as quotitive word problems. Note that in the first and third sample problem, the divisor is a fraction; I didn't have to gin up an awkward problem involving big and small dogs in order to give you an example of quotitive division by a fraction.

Liping Ma's only point vis a vis quotitive and partitive division is that teachers should know the difference. It doesn't have to be explicitly laid out for the kids. But teachers need to know about it because they need to give a mix of types of word problems. She says that it may be obvious to us that numerically they are the same problems (in fact it is SO obvious that we miss the distinction!), but to the kids it may not be.

I'm not sure that's true, but I'm willing to give her the benefit of the doubt.

Liping Ma actually gave a set of US and a set of Chinese elementary school teachers the following problem: frame a word problem for 1-3/4 divided by 1/2.

The best of the Chinese teachers gave examples of both partitive and quotitive word problems; they were all able to give at least one word problem for the division. But some of the US teachers couldn't do the calculation.

The difference: in China, elementary math teachers are respected for what they do, and given time to consult with each other in order to improve their pedagogical knowledge. Elementary Chinese math teachers are specialists in math education.

Catherine has studied the Liping Ma book very carefully. I think Catherine concluded that the fundamental problem in the US is that teachers need release time to consult with each other and improve their knowledge.

I believe that the fundamental problem is that teaching is not a respected profession in the U.S., and that the other problems -- lack of release time, and mathematical weakness in the teachers themselves -- all follow from this.

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• Posted by: CarolynJohnston
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