Answer: Common Core Math: Multiply & Divide 2-Digit Numbers Intuitively

[EM_Guy]

47 x 62 = ?

(40+7) x (60 + 2)

40 x 60 + 40 x 2 + 7 x 60 + 7 x 2

2400 + 80 + 420 + 14

2480 + 434

2914

Consider a rectangle with dimensions 47 units by 62 units. The rectangle can be subdivided into four smaller rectangles: 40 x 60, 40 x 2, 7 x 60, 7 x 2.

So when kids multiply two digit numbers together, they don't need to split up partial products and carry. Such an algorithm works, but is not very intuitive and doesn't get the concept of the multiplication operation across to the average elementary age kid. Instead, kids can find partial products, and then add them together. This also has a geometric interpretation as I just demonstrated.

How about division?

3000 / 62

Well, 62 goes into 3000 at least 10 times, so we write down 10.

10 x 62 = 620. 3000-620 = 2380 remainder.

62 goes into 2380 at least 10 times, so we write down another 10.

10 x 62 = 620. 2380-620 = 1760 remainder.

62 goes into 1760 at least 20 times, so we write down 20.

20 x 62 = 1240. 1760-1240 = 520 remainder.

62 goes into 520 at least 8 times, so we write down 8.

8 x 62 = 496. 520-496 = 24 remainder.

So, 10 + 10 + 20 + 8 is 48. The answer is 48 R24, or 48 + 24/62 = 48 + 12/31.

Again, this is a more intuitive way to perform division. The traditional algorithm for performing long division I think is a lot less intuitive. Most people, especially kids, don't understand why each step in the traditional algorithm works. So they are really learning simply how to follow a procedure. There is value to practicing standard operating procedures; however, mathematical education should focus on teaching concepts and helping children develop intuition. Kids should learn the structure of numbers and the concepts of operations performed on those numbers. Geometric interpretations are also useful to help kids make sense of what they are doing.

Common Core gets very political, and I'm sure that there are all kinds of political arguments that can be made for or against Common Core. But for the moment, I think that the Common Core methods of performing arithmetic have a lot of merit.

Any thoughts about this?

 

[jerromyjon]

I still have yet to find a comprehensive description of common core math and fail to see where it provides any additional insight compared to "traditional" math. I've always been good at doing math in my head so perhaps I am just missing something simple.

 

[symbolipoint]

47 x 62 = ?

(40+7) x (60 + 2)

40 x 60 + 40 x 2 + 7 x 60 + 7 x 2

2400 + 80 + 420 + 14

2480 + 434

2914

Consider a rectangle with dimensions 47 units by 62 units. The rectangle can be subdivided into four smaller rectangles: 40 x 60, 40 x 2, 7 x 60, 7 x 2.

So when kids multiply two digit numbers together, they don't need to split up partial products and carry. Such an algorithm works, but is not very intuitive and doesn't get the concept of the multiplication operation across to the average elementary age kid. Instead, kids can find partial products, and then add them together. This also has a geometric interpretation as I just demonstrated.

How about division?

3000 / 62

Well, 62 goes into 3000 at least 10 times, so we write down 10.

10 x 62 = 620. 3000-620 = 2380 remainder.

62 goes into 2380 at least 10 times, so we write down another 10.

10 x 62 = 620. 2380-620 = 1760 remainder.

62 goes into 1760 at least 20 times, so we write down 20.

20 x 62 = 1240. 1760-1240 = 520 remainder.

62 goes into 520 at least 8 times, so we write down 8.

8 x 62 = 496. 520-496 = 24 remainder.

So, 10 + 10 + 20 + 8 is 48. The answer is 48 R24, or 48 + 24/62 = 48 + 12/31.

Again, this is a more intuitive way to perform division. The traditional algorithm for performing long division I think is a lot less intuitive. Most people, especially kids, don't understand why each step in the traditional algorithm works. So they are really learning simply how to follow a procedure. There is value to practicing standard operating procedures; however, mathematical education should focus on teaching concepts and helping children develop intuition. Kids should learn the structure of numbers and the concepts of operations performed on those numbers. Geometric interpretations are also useful to help kids make sense of what they are doing.

Common Core gets very political, and I'm sure that there are all kinds of political arguments that can be made for or against Common Core. But for the moment, I think that the Common Core methods of performing arithmetic have a lot of merit.

Any thoughts about this?

"Common Core"? Not sure.
The most part of your discussion is nothing new. It makes sense. Trying to teach children to blindly follow a procedure does not work for every child. Showing how the structure works can help. Since that also becomes detailed, repeated teaching is needed - depending how fast an individual child can learn and how much detail he can handle for learning just one set of parts for a procedure.

 

[EM_Guy]

I found this video very convincing:

Obviously the old, traditional algorithms work. But the point is this: Is teaching the old, traditional algorithms the best pedagogical way to teach the concepts that we are trying to get children to learn? Or are these Common Core methods better (from a pedagogical perspective)? I think they are.

 

[EM_Guy]

I think this video is good too - for subtraction.



The point is not that the old methods are bad, but that thinking about these problems in a variety of ways is helpful.

 

[Greg Bernhardt]

I found this video very convincing

The video is good and I agree. I was terrible at math in school. I think much of it was because I wasn't shown any application or reason for anything I was learning. It was all mechanical.

 

[jerromyjon]

I think I understand now, but I don't think you can teach intelligence. These methods rely on the ability of the child to associate relationships mentally. It seems to me the logic is that smaller, simpler pieces are easier to put together than larger, more complicated pieces, therefore more children should understand math better. These methods seem to contradict the way my brain works, and I am very good at math.

 

[EM_Guy]

I think I understand now, but I don't think you can teach intelligence. These methods rely on the ability of the child to associate relationships mentally. It seems to me the logic is that smaller, simpler pieces are easier to put together than larger, more complicated pieces, therefore more children should understand math better. These methods seem to contradict the way my brain works, and I am very good at math.

I don't think you can teach intelligence. I don't think you can teach people to have talent or to be born with gifts. But the idea here is to help children learn how to find unknowns (the answer to a problem for example) from what is known in a logical and intuitive manner. The traditional algorithms are logical, but I don't think they are as intuitive as these other methods. Think about it: In multiplication, why are we splitting up a partial product and then "carrying." What exactly is going on there? And what is with the "place holder zero"? Perhaps your brain doesn't think this way simply because you don't have much practice with these methods. The traditional algorithms have been drilled into you, so they are familiar. Asking you to consider these other methods is somewhat like asking the United States to go onto the metric system. We think in terms of inches, feet, yards, miles, pounds, etc. That's what we are used to. Changing that is difficult. But children have not learned the traditional algorithms.

Don't get me wrong; I'm not anti-traditional algorithms. I think that there is benefit to solving problems several different ways.

 

[jerromyjon]

I'm certainly all for teaching a variety methods and allow the students to decide which method suits their minds. My having been neglected and misinformed throughout most of my "education" certainly biases my opinion, fortunately I decided to self-educate myself at an early age.

 

[SteamKing]

If you want to teach Common Core division to illustrate (or supplement) the traditional long division or multiplication algorithms, fine.

If you want the Common Core method to replace the traditional algorithms, that's where I have a problem.

The CC math methods seem incredibly tedious and drawn out. I don't see a room full of fidgety kids being very patient with learning these methods.
The CC division algorithm suggests how the IRS might teach how to do long division.

 

[JorisL]

I found this video very convincing: [video removed]

Obviously the old, traditional algorithms work. But the point is this: Is teaching the old, traditional algorithms the best pedagogical way to teach the concepts that we are trying to get children to learn? Or are these Common Core methods better (from a pedagogical perspective)? I think they are.

I don't think this always works.
The main problem I see is that the kids have to remember that in the example of 45 times 24, the 4 and the 2 actually represent 40 and 20.
This will work well for those good with maths. For students that struggle with the old method, I'm not really convinced that they can remember this.
In fact it is similar to the old way as in we need to teach them the "rules" of the calculations. (We did learn tricks like 45*24 = (40+5)*(20+4) for doing such calculations without paper)

What is good, is the graphical support. This helps at all levels of education.

After teaching this, shorthand is what we need (like SteamKing said). This means going back to the old algorithm.

I think this video is good too - for subtraction.

[video removed]

The point is not that the old methods are bad, but that thinking about these problems in a variety of ways is helpful.

This is a very nice video. The example of change is awesome (didn't immediately make the link to this).
We have again the need for shorthand.

Now the main question I have, I imagine this is a repetition game for the students just as much as before. That is what you need for actually calculating stuff.
So will it benefit the students later on? Are there good statistics about "mathematical abilities" to compare the methods?

I am certain there is not a unique method that's the best for every single student (without considering learning disorders).

 

[MidgetDwarf]

I remember as a student I was having problems learning how to multiply the product of two double digit real numbers. My teacher showed me a method that involved diagonals, which was more like adding then multiplyijg.

He later eased me into the normal algorithm method.

I think being able to attack a concept from different angles is key to student success, sadly, a lot of teachers are not qualified to teach mathematics. Have you seen the books kids use now adays?

 

[symbolipoint]

I remember as a student I was having problems learning how to multiply the product of two double digit real numbers. My teacher showed me a method that involved diagonals, which was more like adding then multiplyijg.

He later eased me into the normal algorithm method.

I think being able to attack a concept from different angles is key to student success, sadly, a lot of teachers are not qualified to teach mathematics. Have you seen the books kids use now adays?

Lattice Method - works well because it gives partial multiples based on place values, just as does the normal multidigit multiplication way, but can be easier to handle. Student multiplies each partial product and puts it into a cell. After all the partial products are found, only THEN does the student do the addition step.

One way to bring some students to the regular multi digit multiplication method is to break each factor into its expanded form, and teach the students about multiplying by powers of ten; but again, some students handle this well, and some get confused with all the number-writing. It worked well for me, but by that time, I was in high school.

 

[EM_Guy]

The main problem I see is that the kids have to remember that in the example of 45 times 24, the 4 and the 2 actually represent 40 and 20.
This will work well for those good with math. For students that struggle with the old method, I'm not really convinced that they can remember this.

But whatever method is taught, in my mind, it is absolutely essential that students understand that the digit 4 in the number 45 represents 40 (4 groups of 10), and the digit 2 in the number 24 represents 20 (2 groups of 10). If students don't get this, then they are just learning how to follow a procedure; they are not actually learning math. If we want kids to learn how to follow procedures, why not just give them a calculator and provide instruction on how to multiply numbers together?

I think that this principle of teaching concepts instead of operational procedures should apply all the way up. When I took linear algebra in college, one of the first things we were taught was how to find the determinant of a matrix. Then, based on that, we learned things about eigenvalues, eigenvectors, etc., etc. I somehow managed to pass that undergraduate class without actually learning much about linear algebra concepts. Finding the determinant of a matrix is a non-intuitive procedure; it is a mathematical trick. Learning this procedure doesn’t actually help you get an intuitive understanding of the concepts of linear algebra that matter. It wasn’t until I was learning numerical methods in a course on fast computational electromagnetics in graduate school that I actually began to learn the concepts of linear algebra, and the subject became alive to me. I have since acquired a book called “Linear Algebra Done Right” in which the author puts determinants in the last chapter of the book – almost like an optional appendix. He develops the concepts of linear algebra intuitively, concept by concept – not just teaching procedures, but teaching students how to think about vector spaces, linear independence, span, orthogonality, basis functions, etc.

 

[SteamKing]

But whatever method is taught, in my mind, it is absolutely essential that students understand that the digit 4 in the number 45 represents 40 (4 groups of 10), and the digit 2 in the number 24 represents 20 (2 groups of 10). If students don't get this, then they are just learning how to follow a procedure; they are not actually learning math. If we want kids to learn how to follow procedures, why not just give them a calculator and provide instruction on how to multiply numbers together?

Learning place value notation and what it means is essential to doing all decimal arithmetic and should be thoroughly mastered before starting addition and subtraction. If this crucial step is not made, then the rest of a student's mathematics education is built on a shaky foundation indeed.

It doesn't matter if you are teaching traditional algorithms or CC, if you don't know what the numbers mean, how can you make sense out of any calculations with them?

 

[Incand]

This is very similar to the method taught the last 20 years in the swedish school (the same is also used for addition/substraktion doing it term by term called "räkning med mellanled" and then it was extended for multiplication and subtraction). There been a lot about it in the media lately in sweden that it had no research evidence behind it and it may be the reason for the decreased performance in international studys (PISA for example).

I think it's a method that actually really hurt the student and while it works nicely for 2 digit numbers once you 3-4-5 it gets progressively harder to do while the traditional algorithm always is fast and less prone to mistakes. Actually I really regret never having learning to do it that way myself.

Edit: I like to add that it's true that the algorithm may be harder to understand but I believe understanding comes from doing it over again and following a reliable algorithm is the solid way to do it. But if the student asks why the algorithm works there may be a problem and a "I don't know" isn't a good answer. For example proving that long division works is rather complicated and I'm doubtful most teachers teaching it even know why it works. Still there's no research supporting that it's a worse method to teach afaik and it got a lot of advantages.

 

[Greg Bernhardt]

This is very similar to the method taught the last 20 years in the swedish school

Yeah, I showed this to my wife who is a teacher and she said New Zealand has been teaching like this for about as long too. The US is so behind.

 

[EM_Guy]

I think it's a method that actually really hurt the student and while it works nicely for 2 digit numbers once you 3-4-5 it gets progressively harder to do while the traditional algorithm always is fast and less prone to mistakes.

This is a good point. Doing the CC method, the number of partial products needed to find the product of two numbers is the product of the number of digits of each number. For example, if you multiply two 2-digit numbers together, you have 4 partial products. If you multiply 2 3-digit numbers together, you have 9 partial products. If you multiply a 7-digit number by a 5-digit number, you have 35 partial products! Obviously, we should not abandon the traditional algorithm!

However, I still think it is beneficial (maybe even fun) for kids to perform multiplication several ways. I just multiplied 365 x 472. That product is the sum of nine partial products, but not all partial products are equal (or even on the same order of magnitude). For instance, in 365 x 472, you can tell right away that the answer will be between 120,000 (300 x 400) and 150,000 (300 x 500). 300 x 400 is one of the nine partial products (and it is obviously the largest of the partial products). As a side note, there is value to teaching kids how to estimate well. Again, drawing a 365 x 472 rectangle, the width is 300 + 65, and the length is 400 + 72. So, again, even this multiplication problem can be broken down into four partial products: 300 x 400, 300 x 72, 65 x 400, and 65 x 72. Well, 300 x 400 is easy and immediately gets us "in the ballpark" of the answer. 300 x 72 and 65 x 400 are also fairly easy. If you can do 72 x 3, you can do 72 x 300. But if a kid has a hard time doing 72 x 300, he can divide this into partial products too (70 x 300 + 2 x 300). Again, 70 x 300 = 21,000 gets us in the ballpark of the 72 x 300 partial product. Finally, we have 65 x 72, which is itself the sum of four partial products (60 x 70, 5 x 70, 60 x 2, and 5 x 2). Again, rectangles can be drawn to give a geometric interpretation. This obviously isn't the most efficient way to compute 365 x 472, but if we are going for efficiency alone, we can just give kids calculators.

The traditional algorithm gets the problem done fast. The CC method teaches the concept of partial products. The geometric interpretation (rectangle areas) really reinforces this partial products concept.

Also, it is worth noting that in the traditional method, you do compute partial products. 365 x 472 = 365 x 2 + 365 x 70 + 365 x 400 = 730 + 25,550 + 146,000. Each of these partial products are themselves the sums of three CC partial products. 365 x 2 = 730 = 10 + 120 + 600. Etc., etc. Teaching kids to take apart and put back together these numbers in a whole variety of ways I think can only be beneficial - and hopefully fun. And obviously, learning a multiple of methods enables students to check their work. If they made a mistake, they can catch it. If they did it right, they get the satisfaction of getting to the right answer by two different methods.

 

[SteamKing]

This is a good point. Doing the CC method, the number of partial products needed to find the product of two numbers is the product of the number of digits of each number. For example, if you multiply two 2-digit numbers together, you have 4 partial products. If you multiply 2 3-digit numbers together, you have 9 partial products. If you multiply a 7-digit number by a 5-digit number, you have 35 partial products! Obviously, we should not abandon the traditional algorithm!

That's why I said doing CC math is like learning to do arithmetic the IRS way: you fill out reams of paper to calculate one number: a product, a quotient, whatever.

Back in the day, when the "New Math" first appeared, multiplication was taught using a heavy emphasis on the distributive property, with some place value lessons thrown in for good measure:

To wit: a * (B + C) = a*B + a*C

Say the example wanted to find 9 * 87, then you would write 9 * 87 = 9 * (8 * 10 + 7) = (72 * 10 + 63) = (720 + 63) = 783

Of course, as the example multiplications got more complex, then the "New Math" procedure got more complex and lengthier. Fortunately, most of my elementary school teachers were of an age where their training had relied on teaching the traditional algorithms, so we got parallel instruction in the old and new math simultaneously.

This was all before electronic calculators, so math instruction involved heavy doses of not only working out scads of multiplication problems using pencil and paper, but also drills with flash cards, to practice quick recall of the basic facts contained in the multiplication table.

 

[QuantumCurt]

A lot of people seem to be under the impression that CC math is something new, but it really isn't. These methods have been around for basically as long as arithmetic has been around. People seem to believe that the conventional subtraction algorithm (for example) is much easier...but it's really not. It's actually significantly more abstract and difficult, and provides no insight into WHY the difference of two numbers is a given value. When we look at a subtraction problem in terms of a number line, we can intuitively understand this.

There have been a lot of deliberately poor examples without explanations posted online. I can certainly see how that would look bizarre to some people, but really all they're doing is counting back change.

If I buy something for $13.86 and I pay with a $20.00 I get back

13.86+0.14=14.00
14.00+6.00=20.00

6.00+0.14=6.14

It's really as simple as that. This is reinforcing the methods of mental arithmetic that many people use. I have been using many common core methods for my entire life, and I never learned them in school. They come directly from number intuition, and this is a very good thing.