math experts: multi-digit multiplication algorithm

[wathe]

9yo DS is learning multidigit multiplication.  He is balking at learning the standard algorithm where one carries/regroups as one goes along.  He finds that switching between multiplying and adding (when one carries) as one goes along "mixes me up".   Instead, for each digit multiplied, he likes to write his product on a new line, then do all the adding and regrouping at the end. 

 

For example he would solve 34x56 this way:

 

  34

x56

------

0024

0180

0200

1500

------

1904

 

He is using Math U See gamma, and education unboxed videos.  He understands how to view the problem as solving the area of a rectangle, made of smaller rectangles representing units x units, units x tens, tens x units, tens x tens etc.  He understands place value.  He makes very few mistakes.

 

I am thinking giving up on the traditional algorithm and having him continue along as he is doing.   Maybe try to teach the traditional algorithm next year.

 

Other than taking up an awful lot of paper (especially for 3 or 4 digit problems), can any of you mathy folks see a problem with this? 

 

 

Edited April 22, 2017 by wathe

[maize]

No problem, he's showing better conceptual understanding than most kids who follow the algorithm. His method is fine.

[Julie Smith]

I see no problem with it.

[EKS]

If I had to find a problem, it might be that his procedure is more time consuming.  I mean, it probably isn't for him now because he's not fluent with the standard algorithm, but I suspect that if he were equally fluent with both, the standard algorithm would be faster.

 

Otherwise, I'd say it's fine.

[Syllieann]

Math mammoth actually teaches that as an intermediate step in developing the algorithm. After a couple days of doing it that way it shows them the shortcut of carrying/regrouping as you go.

[OneStepAtATime]

He seems to understand quite well.  I say let him do it his way.

 

On a side note, if he wants a faster process that does not require constantly switching between the operations he might be interested in Lattice Method. DD HATED the standard multi-digit algorithm.  Once a teacher showed her lattice and she had a bit of practice with it under her belt it was like night and day.  She can whip through problems very quickly.  Frankly, it is how I now do multi-digit multiplication and I wish someone had showed me this version when I was still in school.  It just makes so much more sense to me.  Now, the first time I saw it it seemed like gobbledygook.  I seriously had no clue what DD was doing.  It also seemed pretty labor intense.  Once I tried it out a few times I realized I could move through the math faster and had a better understanding conceptually doing it this way.  I love it.  I linked some youtube videos that explain it.  Most basic video is the first one and each one after that increases in difficulty a bit.  The last one is multiplying by decimals.

 

 

 

 

[Guest]

The "standard algorithm" is merely shorthand for what he's doing. He's showing good conceptual understanding of what's going on, so I'd leave it alone.

[Targhee]

This is how BA originally covers multi digit multiplication, and it was how my DS choose to do it for a long while (even after learning the short hand). He chose to give it up when he saw, as I worked problems alongside him, how much quicker I finished.

 

FWIW, I remember asking my dad in 4th grade WHY I needed to skip a place (or put an X or a zero there) because I had only learned the shorthand and not the conceptual premise. He couldn't answer me, and neither could my mother, although both are well-educated. I was so mad. Your DS will never have that problem ;-)

[MistyMountain]

I actually like that way to start out with so they know conceptually what is happening with the numbers rather then just starting with carrying but not really thinking about it. I think it is actually a good way to learn it and I will teach it this way at first.

Edited April 22, 2017 by MistyMountain

[wathe]

Thanks all, very reassuring.  I think will have him carry on the way he likes and save the traditional algorithm for later.  I'm thinking that eventually he will get frustrated with how much more space and writing his way takes and maybe be more motivated to learn the traditional way.

[Paige]

I agree that it's fine. I think he'll likely switch to the normal algorithm on his own as he matures because it is faster. If he doesn't naturally switch, I would seriously encourage it later, but for a beginner it is just fine. The reason I'd encourage switching eventually is because it takes longer to do it his way and you do more writing, and I find the more steps you take and the more numbers you write, the more chances you have to make a mistake. When he gets older and into more complicated math, the speed issue with his way will start to take a toll. I would probably continue to model the normal algorithm when I did problems side by side with him; partly to passively encourage him to switch and get him accustomed to it, and partly because I hate doing it the slow, long way. 

Edited April 22, 2017 by Paige

[idnib]

Sounds fine to me. It sounds like he understands place value better than a lot of people.  :)

[Jackie]

I was also going to point out that Beast Academy actually teaches it the way he is doing it, in their third grade books. They move to teaching the standard algorithm the following year, in fourth grade. I thought it was brilliant when I saw the way they taught multiplication and division because it really emphasized why it works.

[kiana]

I honestly would prefer it if more students did the long way that emphasizes what's really going on until they were at a point to make the connection for themselves on how to speed it up. 

 

I feel the same way in algebra about teaching binomial multiplication with "FOIL". If they don't understand what's going on, the mnemonic will produce an illusion of competency which will break down when they have to multiply a binomial by a trinomial. If they do understand what's going on, the mnemonic is unnecessary. 

[Kathy in Richmond]

That's how my daughter taught herself to do multiplication, too. Like everyone else said, there's no problem at all! In fact, he's showing wonderful conceptual understanding. I can't remember how long dd stuck with writing it all out, but eventually she moved to the standard algorithm on her own.

[Butter]

That's how Math-U-See teaches it.  Particularly when you have multiple carried numbers, that way is more likely to keep it straight and end up with a correct answer.  Just because we learned it with the "standard" algorithm does not mean it is the only right way or better.  The best way to solve a math problem is the way that you understand, makes sense to you, and results in the correct answer.  People who do multi-digit multiplication in their head generally do it the way your son is doing it.

[kiana]

People who do multi-digit multiplication in their head generally do it the way your son is doing it.

 

Yep. Except my partial sums are in the reverse order. So for 34x56 it'd be 1500, add 200 to get 1700, add 180 to get 1880, add 24 to get 1904. It's easier to keep track of this in my head for some reason. 

[Kuovonne]

It is fine for now, but he does need to eventually switch to the standard algorithm before tackling long division.

 

Long division will also throw division and subtraction into the mix, so he will have to be able to switch between all four operations, not just two. Long division without the standard algorithm for multiplication would also take up a lot of paper and require a lot of copying.

 

Can you ease him into the standard algorithm with a single digit times a multi-digit number that will still have him switch between multiplying and adding? For example

 

67

x 9

----

 

then

 

476

x 8

-----

 

Also check to make sure that he has both addition and multiplication facts down cold. If he struggles with remembering the facts, that might make him reluctant to switch gears so much.

[forty-two]

I had my dd do multiplication this way for a while, because there was too much going on at a time for her with the standard algorithm - she'd always make at least one computation mistake per problem, and it was demoralizing.  Figured it would do her good to separate out some of the moving parts (so to speak).  After a week or so, she went back to the standard algorithm and did fine.

 

I feel the same way in algebra about teaching binomial multiplication with "FOIL". If they don't understand what's going on, the mnemonic will produce an illusion of competency which will break down when they have to multiply a binomial by a trinomial. If they do understand what's going on, the mnemonic is unnecessary. 

 

I was absent the day my class learned FOIL, and it felt like I'd missed being initiated into the secrets of math or something.  I didn't know what the teacher meant when she said to FOIL something, and when I'd ask a fellow student, they'd say (in accents of horror), "You don't know how to *foil*!?!  Why, you can't do *anything* if you don't know how to *foil*!"  But for some reason they never followed that up with an explanation of what FOIL was :-/.  Very frustrating.  Eventually I got clued in, and after all that build-up it was *so* anti-climactic.  "It's just multiplying terms???  That's *it*?!?  What kind of fundamental math secret is this!?!"
 

[OneStepAtATime]

It is fine for now, but he does need to eventually switch to the standard algorithm before tackling long division.

 

Long division will also throw division and subtraction into the mix, so he will have to be able to switch between all four operations, not just two. Long division without the standard algorithm for multiplication would also take up a lot of paper and require a lot of copying.

 

Can you ease him into the standard algorithm with a single digit times a multi-digit number that will still have him switch between multiplying and adding? For example

 

67

x 9

----

 

then

 

476

x 8

-----

 

Also check to make sure that he has both addition and multiplication facts down cold. If he struggles with remembering the facts, that might make him reluctant to switch gears so much.

Actually, there are other ways to do long division, too.  Such as Partial Quotient division.  

[Kuovonne]

Actually, there are other ways to do long division, too. Such as Partial Quotient division.

Even in partial quotient division you have to switching among multiple operations. The only method for multi-digit division that wouldn't require switching between operations is repeated subtraction.

[MistyMountain]

Education Unboxed which has been mentioned teaches long division concretely in a way that you would not need to know that standard algorithm. Once you get it down it can be applied to bigger numbers too and there are sources out there on how to teach it with examples bigger then they used.

Edited April 24, 2017 by MistyMountain