Liping Ma book...interesting

[choirfarm]

I am currently reading Kinowing and Teaching Elementary Mathematics by Liping Ma. I've only made it through 2 chapters. The second chapter is about long multiplication. I hope the formatting stays for this as it should line up correctly.

 

123

x645

615

492

738

79335

 

They had one example lined up incorrectly and one correctly. Ok..I have NEVER seen it like that without the zeros. Most of the US teachers said it was a problem of place value if the student lined it up incorrectly BUT they had no concept of what they were doing. Some of them talked about putting in the imaginary zeroes or sometime even other object so they could keep their place. The only thing the zeros were good for were keeping their place??????? Now the Chinese left out the zeros as well many times but they had a much better grasp of the place value...

 

The way I learned to do it was that you multiplied 5 x 123. But then you were multiplying 40 times 123. You had to have the zero there not to keep the place but because that is what you were doing. You were NOT multiplying 4 x 123. Then you are multiplying 600 x123 so once again the zeros must go there because you are NOT multiplying it by 6. So my problem looks like

123

x645

615

4920

73800

79335

 

To me the zeros are absolutely necessary. The Chinese teachers taught this and then had them erase the zeros since they were not necessary. I guess I do not understand how they were not necessary...to me, they are otherwise the number means something different. The American teachers thought the zeros were confusing to the students...

 

So then I did an experiment. I asked my husband to solve the problem. HE DID IT WITHOUT THE ZEROS!!! I asked him why and he said that is because that is the way you do it. Why don't you put zeros there? Why would you need them?? he asked. He couldn't explain why he lined them up the way he did. This is a Phi Beta Kappa guy that took several calculus classes in college... I explained a little bit but he argued that he could do it as quickly as I could and maybe faster without the zeros...

 

So then I gave it to my oldest. He did it with the zeros. I asked him why and he said it had to do with place value. He said dad's way would certainly work.

 

I gave it to my middle boy and he did it with the zeros. I asked him why and he explained like I did because you are multiplying by 40 and by 600. ( Probably because I am the one teaching and dragging him through math!)

Then he noticed that his answer differed from dad's answer..GREAT.. he said but then he realized that dad made a carrying mistake and his answer was correct. (There was great rejoicing that he got something correct.)

 

I just thought it was funny as I have never seen or heard of lining it up that way but it is obviously taught that way. I realize that lining it up my husband's way will work, but to me it just doesn't make logical sense. It is like the numbers are just hanging out in space. Without the zeros it would be so easy to slide them over to where they are not lined up correctly.

 

The formatting didn't work I see and I don't know how to get it to work out correctly so I hope you can get what I am trying to say. There were just spaces where zeros should be and they lined up where they SHOULD be. My example with zeros isn't lining up either.

[kiana]

Yes. I noticed this when I read it.

 

To me it seems like when you teach it without the zeroes it really obfuscates understanding of what's going on. I can understand using it as shorthand later on. Similarly with polynomial long division/synthetic division.

 

I also found it interesting to see some of the Chinese teachers saying "Well, I have never ever seen a student having difficulty with 3 digit by 3 digit multiplication because we would have still been on 2 digit by 2 digit multiplication until they understood that and then 3 digit by 3 digit is easy."

 

(Or something along those lines, my book's at my office at the moment.)

[Nan in Mass]

I was taught to leave out the zeros. It was never a problem for me. I knew why I was doing what I was doing, I think because we were taught that it was a shorthand for:

 

123x654=(600x123)+(50x123)+(4x123)

and

600x123=(600x100)+(600x20)+(600x3)

etc.

 

They talked about the associate property. I wouldn't have been able to tell you the name, but I knew how it worked. (This was new math. My grandmother was completely baffled by the set theory we were taught.)

 

By the time we did it vertically, we had done it horizontally for long enough that we understood that this was just a short way of writing it. Or at least, I did. I saw this with my children, too, using Singapore math. By the time they got to this algoritm, they had been doing the problems horizontally long enough that it wasn't a problem. I think this is why the book made such a big deal about order of operations, even for little children.

 

Nan

[Candid]

Honestly I think the leaving out or keeping zeros is not important (except for a practical reason I'll come back to). Instead what is problematic is knowing what either the zeros or nothing represents: "why do you do that?" Your explanation is dead on the money (and I'm pretty sure it is what Singapore teaches, but it's been a long time since I had anyone at this level, so I don't know for sure).

 

BUT there is a practical reason to put the zeros in: to keep the student from getting his numbers in the wrong place or misaligned due to sloppiness and handwriting vagaries. I don't know what hand writing in your house is like, but in mine it is pretty awful, and to add to the mix some numbers are fat and some are thin. Even in my own case, if I had to do pages of long multiplication, I know I'd make some errors because when I came to the last step and had to add up the numbers, I'd have some numbers just enough out of line due to sloppiness that I'd add them in the wrong column. I suspect that with younger children this is even more likely. So, personally, I'll keep using the zeros to limit this error.

[MyThreeSons]

I know the typesetting on the forum doesn't always look like what we intended, but the way I'm seeing what you've written would be totally confusing to me.

 

When you say they "leave out the zeros", do you mean that they just don't write in a zero, but they still leave a space where one would write a zero? If so, that's what I've done for years. I'm pretty sure that when I learned it as a kid, we wrote the zeros in.

[Cosmos]

I was taught to do it without the zeroes. When I taught my ds, we did lots of horizontal multiplication before we used the shorthand vertical system (with zeroes intact). I told him that many people like to go quickly and just leave out the zeroes, but I think he usually does them with the zeroes.

 

I'm reading Knowing and Teaching right now myself. When I got to that section I showed the scenario to my son. He immediately suggested the teachers should show the students the missing zeroes. He was shocked when I read him some of the U.S. teachers' responses who thought the zeroes were "artificial".

[Nan in Mass]

We used graph paper, and I had my son who tended to misread things write in the zeros for awhile.

Nan

[Candid]

We used graph paper, and I had my son who tended to misread things write in the zeros for awhile.

Nan

 

This is an excellent suggestion that helps with the whole handwriting misalignment thing. I can remember when I was a student drawing my own lines down to keep my columns lined up.

[kiana]

This is an excellent suggestion that helps with the whole handwriting misalignment thing. I can remember when I was a student drawing my own lines down to keep my columns lined up.

 

Another workaround if you're broke and graph paper is expensive is to rotate notebook paper so that you use the lines for columns instead. Of course, then you have to worry about level alignment, but for some reason that seems easier to keep up with.

[choirfarm]

Ok..part of this is missing the point. Even IF you do it with graph paper or whatever tool you use and you leave out the zeros.. To me, it just seems like floating numbers. They may be under the correct digit but if you write 738 looks like 738 to me not 73,800 even if you supposedly put it under the correct digit. To me, it just doesn't logically make sense. You are not adding

615,492, and 738. So why would you write it like that?? It is probably just my problem with my brain.

[Cosmos]

Ok..part of this is missing the point. Even IF you do it with graph paper or whatever tool you use and you leave out the zeros.. To me, it just seems like floating numbers. They may be under the correct digit but if you write 738 looks like 738 to me not 73,800 even if you supposedly put it under the correct digit. To me, it just doesn't logically make sense. You are not adding

615,492, and 738. So why would you write it like that?? It is probably just my problem with my brain.

 

Shorthand. It's just to save time. Many people do something similar when doing long division. Instead of writing the entire dividend on each line, they add only the next digit, in other words only the minimum necessary for performing the calculation steps.

 

I agree, though, that these types of shorthands should be used only if the student has a rock-solid understanding of *what* they are doing in the calculation and *why*. Otherwise you end up with people like the teachers in the book who don't even know why those columns of numbers are moved over!

[kiana]

Shorthand. It's just to save time. Many people do something similar when doing long division. Instead of writing the entire dividend on each line, they add only the next digit, in other words only the minimum necessary for performing the calculation steps.

 

I agree, though, that these types of shorthands should be used only if the student has a rock-solid understanding of *what* they are doing in the calculation and *why*. Otherwise you end up with people like the teachers in the book who don't even know why those columns of numbers are moved over!

 

Also, if you don't like the shorthand, you should ABSOLUTELY not use it. It's not necessary at all to use it. I don't use it, for the same reasons as choirfarm. But other people using it doesn't bother me.

[Nan in Mass]

Perhaps you could think of it this way:

Numbers are a representation of an idea of a concrete thing. When you are doing the long multiplication algorithm, it is a shorthand way of jotting down a procedure, just like numbers are a shorthand way of jotting down a quantity. It is just a way of keeping track of the work that you are doing in your head. There is nothing special about how it is written down. After a long period of doing 2 digit by 2 digit multiplication in his head in the car using me to help him remember his in-between quantities (the ones you have to add), my son devised his own way of jotting down the in-between quatities. My older two were taught some sort of grid algorithm. The standard one is a very efficient way of doing it, so everyone is encouraged to do it, but it is only a short-hand way of doing what I wrote out horizontally, and since it is only shorthand to help you keep track of what you are doing in your head, not real, precise written math language, there is no reason why you need to write any more of it down than is necessary. If I do this for myself for work, I skip writing the x sign and the bar and even sometimes the original numbers themselves, and I write the answer off to the side where I can see it more easily.

 

HTH

Nan

[Alexinmemphis]

Just jumping in since I didn't see much defense of the way we do it here. I was taught not to use 0's and I don't let my children use them either. I want to make sure they understand the principal that you are multiplying by 6 tens, not 60 ones, for example. So 24 x 68 is 24 x 6 tens added to 24 x 8 ones. It's mostly a semantic thing, I guess, but I somehow feel it's important for emphasizing units. This shows up in algebra and in science when they are used to multiplying by 6 -whats?-. They know those words are coming out of my mouth. I annoy them with it regularly.

Best,

Alex

[Nan in Mass]

Hmm... Now that you mention it, I seem to remember saying "six whats?" a lot myself. I began very early on by making a distinction between "fingers" and "person's hands" and then sort of kept on doing it. I probably over emphasized that multiplication means "of" and that multiplication is a rectangle.

Nan