multiplication, past and present

The Common Core Blues

So, you thought you knew how to multiply one number by another. You even thought multiplication was easy. Then your kid comes home with “arrays” or “lattices,” and wants you to help him, but you can’t make sense of the new methods. You’ve got the Common Core blues.

These days, you would be hard put to find an education site that isn’t showcasing an argument for or against the Common Core (CC). I admit that it doesn’t take much to get me on my soap box, railing against some of the inane (in my opinion) methods that are being promoted in North Carolina’s Common Core curriculum. (You should not confuse the Federal standards with the various curricula for teaching those standards. North Carolina and some other states have shelled out lots of $ to consultants to create brand new curricula.)

Okay, I’m stepping away from the box.

Multiplication Past

I thought it might be instructive to look at some of the specific approaches to multiplication that are being taught to elementary school kids. I actually find most of them to be very helpful in teaching kids the meaning of multiplication. However, anything can be taken to extremes. In the interest of full disclosure, I do not have extensive experience in using the new methods with my students because those that attend public school are in the higher grades while my younger students are home schooled. But I do run into the new math while working with a little tutoring group that I lead on weekends, so I had to educate myself on the CC methods.

Okay, let’s get started! As an example, let’s use 11×12. This is a simple problem, and the fact that both numbers have two digits makes it useful for demonstrating various methods.

old school multiplicationIn the “old school math,” there were two ways to do this problem, at least as far as I know. One is to extract the answer from the memory bank (analogous to the RAM in a computer) that resides in each of our heads, because we all memorized the multiplication tables up to 12×12. So we old folks can come up with 11×12=132 without much thought.

The second “old” approach to multiplying 11 by 12 is shown in the image to the right. This approach came in handy when we needed to multiply numbers bigger than 12, since we didn’t have those solutions in our memory banks. It is quick and efficient, but I have to admit that it is possible to just memorize this method without really understanding what is going on. This is the issue the Common Core standards and curricula are trying to address.

Dot, Dot, Dot

Okay, out with the old and in with the new. Let’s start with what the younger kids are learning. It is called the “dot” method. We can start our problem by drawing 11 rows with 12 dots in each row. Of course, it would take a while to count them, not to mention the time it takes to draw them by hand. Photoshop solves the drawing problem (for me), but what to do about the counting problem?

We can break the dots up into groups, creating two or more easier multiplication problems. In the image below I divided them into two groups, one with 10 rows of 12 dots and another with 1 row of 12 dots. Now the problem is easier, as long as we know how to multiply by 10. The problem 11×12 becomes 10×12 + 1×12, or 120 + 12 = 132.

dot method of multiplication

If a child has not yet learned to multiply by 10, he can break these groups into smaller groups but I’ll illustrate that in the next section. My take on the dot method is that it is very useful for younger children because it offers the option of brute force counting, as a starting point, and it helps kids learn that no matter how they group the dots, they’ll get the same answer. This is an important concept. The problem with the dot method is that some teachers assign problems with numbers that are way too large, as in our example. I’ve read stories about kids being brought to tears by having to draw hundreds of dots for homework!

Arrays Make Sense

Let’s move on to the “array” method. This approach can be developed naturally from the dots method. Instead of drawing dots, we draw boxes to hold the dots. So the second of the dots images above (i.e., 10×12 + 1×12) is represented by the first of the array images below. At this point we can break the problem up into as many groups as we like, as you can see in the three other array images. The only rule is that the lines dividing up the array have to extend fully from left to right or top to bottom so that we maintain rows and columns.

The lower right image below shows a more formalized version of the array method, one that represents another progression in the learning process. In this approach, we always divide up the numbers according to place values. So we would divide 12 into 10 + 2 (tens place and ones place), and 11 into 10 + 1 (which we did before). Then we have 100 + 20 = 120 for the upper row, and 10 + 2 = 12 for the lower row. Finally, we add 120 + 12 to get 132.

array method of multiplication

I really like this method too. I find that it offers a natural progression from addition to multiplication, and I’ve started using it with my younger students. But there is one multiplication method that I think is completely inane.

Lattices Do Not

Welcome to the “lattice” method. Most parents absolutely despise this approach–I’ll show you why.

Our example does not make a great demonstration problem for the lattice method, but we’ll muddle through. The image below walks us through the process.

Step 1. Set up a table for which the number of cells is determined by the number of digits in the numbers to be multiplied. In our example, both numbers have two digits, so we create a 2 by 2 table.

Step 2. Write our two numbers across the top and down the side, taking up one cell width or height for each digit, as shown.

Step 3. Divide each square cell into two triangles, using diagonal lines, as shown.

Step 4. Multiply all the possible combinations of digits. For us that means: 1×1 = 1, written as 0 and 1 in the upper left cell. The zero represents the tens place while the 1 represents the ones place. We go on to multiply 1×1 again and write 0 and 1 in the upper right cell. Then we multiply 1×2 and write 0 and 2 in the lower left cell. Finally we multiply 1×2 again and write 0 and 2 in the lower right cell.

I’m guessing that at this point, you are thinking something like, “What the heck is going on?” I feel your pain. But let’s keep going.

Step 5. Add the digits along each diagonal, so we have 0 for the top left, 0+1+0 for the next diagonal down, 1+0+2 for the next, and 2 for the last. And there you have it!

What? You don’t see it? The answer is the concatenation of the numbers that we just wrote around the perimeter of the table: 0132, or leaving off the zero, 132.

lattice method of multiplication

This is what I call a parlor trick. When a little girl in my Saturday group showed it to me, my jaw dropped. As she explained it, she did not refer to ones places or tens places or any other terms that might indicate her understanding of the process. She said, “You draw squares, then you write the numbers across and down, then you draw diagonals, then you …….” She really was performing a parlor trick. Do all these steps and there’s your answer!

Sorry. I’m putting away the soap box again.

I hope my explanations are clear enough to enlighten some of you parents who have run across the new ways of doing multiplication, because I know that you really do want to help your kids with math. I’m sorry if you have to deal with the lattice method. Please be sure that your child also learns the array method, so that he or she will develop an understanding of multiplication, rather than memorizing a series of steps to get an answer.

Calculators are Good … and Bad

Wait … I almost forgot! There is yet another way to multiply numbers–you can use a calculator. I have loved calculators and computers for as long as they’ve been available to the general public, and I spend many of my waking hours using them. Kids love them too–they make life so easy! Recently I was reviewing the previous week’s work of a high school student, and he mentioned that the section on complex numbers had been taught by a substitute teacher. I asked him whether he had understood the material. His answer was along the lines of, “Well, the teacher just showed us how to do the problems on our calculators.” Needless to say, my student and I went back over that section, sans calculator. Calculators are very helpful, but they are not a substitute for mathematical thinking. There is no substitute for that.

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