Thirteen Ways of Looking at Multiplication (with apologies to Wallace Stevens)

Well, no, I’ve lied: not thirteen, but merely four ways of looking at (and doing) multiplication are contained herein. And I hope to convince you that they are essentially the same mathematically, which is very much my point. There are other ways to calculate the product of two positive integers, but I must leave those for another conversation.

Today’s Standard Algorithm

When I was in primary school starting in 1955, there was only one way presented for doing multi-digit multiplication (though it had two variants that differed only cosmetically). I’d like to explore that approach naively (which is to say without looking at the underlying mathematics that justifies the procedure), then look at two other models/methods for doing the same problem before returning to what is generally thought of today as the “standard algorithm” for finding the answers to multi-digit multiplication problems. I will conclude with a visual approach that some students find helpful.

Let’s find the product of 27 x 53. Here is how I was taught to do problems like this:

If you’ve somehow forgotten the procedure (the availability of software calculators on smartphones, tablets, and computers makes handheld-calculators seem almost quaint, and it’s not inconceivable that using them for arithmetic would lead some people to grow rusty with paper-and-pencil algorithms), it works like this: multiply 7 by 3; that’s 21. You write the “1” below the first line under the 3, then either write 2 above the “2” in “27” or mentally hold onto it (either way, this is called “carrying”). Multiply the “2” in “27 by 3 to get 6, then add that carried “2” to get 8, which you write down to the immediate left of the “1” below the first line.

At this point, you multiply 7 by 5 and get 35. You write “5” under the 8 and “carry” the “3” above the “2” in 27 or hold it mentally. Then multiply 2 by 5, yielding 10. Add that carried “3” and write down the resulting “13” to left of the “5” under the “8.” Now you are done multiplying and must add the two lines (known as “partial products” you just created. Doing so column by column, you get 1431 (I’ll assume you still know how to add two multi-digit numbers).

I can’t speak for you, but when I learned the above procedure, I don’t recall the teacher going into an explanation of why it worked. We were just to assume that it did, and it was up to us to do enough examples that we “mastered” the algorithm. If we knew our one-digit multiplication and addition facts and didn’t get lost or wonder about that left-shift business when writing the second partial product, we were “good.” Many teachers, mine included, avoided that issue entirely by telling students to write a “0” under the right-most digit of the first partial product and then writing down the digits of the second partial product to the left of that “0” as they were computed. At some point, students were either told in later grades or deduced for themselves that as long as they weren’t careless, it was not strictly necessary to write that “0” or multiple “0’s” if the calculation entailed numbers with more than two digits. Personally, I varied between being very cautious and less so until the need to write in trailing “0’s” simply extinguished itself, but I’ve seen adults who will write every last one of them in. Given my sloppiness and obsession with being “first done” in elementary school, I probably would have benefited from more caution. I’m sure my teachers would have appreciated clearer writing of all kinds from me. Or perhaps we should have been taught the lattice method. (See below).

You may have also had a teacher who explained what was going on, but in coaching upper-grade elementary mathematics teachers in Michigan in 2004-2006, I observed many teachers who were not big on why things they taught made sense mathematically. And for now, I’m going to be one of those teachers, though not for too long.

Lattice Multiplication

It turns out that the above “standard algorithm” has not always been standard at all. In fact, before the widespread use in Europe of the printing press with moveable type after its invention by Johannes Gutenberg around 1440, the most common way to do multi-digit multiplication was the lattice (aka, “gelosia” – related to the English word “jalousy”) method, first introduced to Europe around 1200 by Leonardo of Pisa, better known as Fibonacci). Let’s see how the lattice method works on our previously solved example, 27 x 53:

What’s going on here? The two factors, 27 and 53, are written – one digit per column or row – above or to the right of the lattice, respectively. Starting in the upper right-hand corner, 7 is multiplied by 5 to obtain the two-digit product, 35. These digits are then put into the upper and lower halves of the corresponding cell. The process is repeated in each cell: 7 x 3 for the lower right cell, 2 x 5 for the upper left cell; 2 x 3 for the lower left cell. Note that in this last case, where the product is only a single digit, 0 is put in the upper half of the cell as a place-holder.

At this juncture, the multiplication process is complete, and the outer digits of the factors do not figure further in the calculation. Instead, we add diagonally, starting in the lower right of the lattice, where we simply have “1.” The next diagonal yields the sum “5 + 2 + 6,” and the first digit of the resulting “13” goes at the bottom, while the “1” is carried into the next diagonal where it is added to “3” to give the sum, “4.” Finally, the “1” in the upper left diagonal is brought down and we read from there down and then right to get the complete product, 1431.

When I first saw this approach demonstrated by a fifth-grade teacher I was coaching in Pontiac, MI, he offered no explanation of the process, just a statement of each step. I assumed this was because in the EVERYDAY MATH curriculum the students were using, they’d been taught lattice multiplication in fourth grade and were now revisiting it for products involving decimals. I checked the result with the standard algorithm and convinced myself that it was correct, but did not get why it worked mathematically. I did another example to be sure I could do the procedure accurately, then approached this teacher after class to ask him about the underlying mathematical justification for the method. Apparently thinking that I didn’t follow the steps, he started to do another example, but I assured him that my question was not about the procedure but about why it worked. He looked at me as if that were a strange thing to want to know.

That night, I sat down and quickly saw that my confusion was simply with the diagonal addition. The arrangement of the lattice was designed to keep each digit of a partial product aligned with others that had the same place value. During the addition process, single digits with the correct place value were added and any sums greater than 9 were “carried” into the next diagonal. In other words, this “new” method was in every respect the same as the standard algorithm, but with diagonals replacing columns and the splitting of cells keeping digits absolutely aligned properly, unlike sometimes was the case with students or adults who wrote the partial products carelessly and then introduced addition errors due to misalignment. So the diagonals starting in the lower right represented the ones, tens, hundreds and thousands digits of the partial products, and naturally the final addition gave the same result as did the standard algorithm. This is not a coincidence.

Let’s turn to another way to look at our problem, in a sort of “expanded notation.“[ii] I will rewrite 27 x 53 as      (20 + 7) (50 + 3), noting that this is a perfectly valid way to regroup one or more multi-digit numbers for any purpose, though it is certainly not the only way to do so). Now, taking advantage of the distributive property of multiplication over addition for the integers, I can multiply these factors as follows: 20*50 + 20*3 + 7*50 + 7*3. This equals 1000 + 60 + 350 + 21, which then yields the sum, 1431. That is comforting since we have already established two other ways that the product we seek is 1431.

Now we can look more productively at both the standard algorithm and the lattice method. Recall that the standard approach looks like this:

Revisiting the Standard Algorithm

Examining the process by which we arrived at this result a little more carefully, we might note that there is a good deal of “lying” going on in our heads when we do the algorithm. After the first step (“multiply 7 x 3”) nearly everything we say to ourselves is a lie. For example, the next step is “2 times 3.” But is it really? That “2” is in the tens place, and hence it would be more accurate to say, “twenty times three is sixty, and we had twenty more than one in our first multiplication of 7 x 3, so adding that twenty to sixty gives us eighty, not eight.” Of course, it would be cumbersome to go through that much chatter to do elementary arithmetic, so we don’t. But we don’t for another reason: we’re never told to think about what we’re actually doing. Everything is reduced, time and time again, to single-digit multiplication. Place value is effectively pushed under the rug throughout the process.

We next say to ourselves, “seven times five is thirty-five.” But we don’t write that five under the one in the first partial product. Why not? Place value. Without really being told why, we move the second partial product one place left, which effectively multiplies that entire line by 10. Were there more digits resulting in more partial product lines, each would be shifted left to account for changes in place value. So like the second multiplication in the first partial product, we have a units digit (“7”) multiplied by a tens digit (“5” which stands for “50”) and our actual result there is 350, not 35.

The final multiplication isn’t 2 x 5 but 20 x 50, which is 1000, not 10. Fortunately, our mindless left-shifting takes care of that and the 1 representing one-thousand winds up conveniently in the thousands column for the addition steps.

“Expanded” Notation

Breaking down each multiplication without this framework, but taking into account the actual place value, we get just what we got with the so-called expanded notation process: 1000 + 350 + 60 + 21 = 1431 (or if you prefer, in the order of actual calculation, 21 + 60 + 350 + 1000 = 1431).

Why don’t we see any of those numbers when we use the standard algorithm?

Because we combine pairs of steps through addition, leaving us with only two numbers to add at the end instead of four. In our desire for “efficiency,” we lose transparency in the standard “long multiplication” algorithm. Indeed, as I will explore in a future piece, something similar happens in the standard long division algorithm, where an even greater opacity for many people results. With multiplication, what gets hidden may not be fatally confusing for a majority of young students, but for those who “don’t get it,” the struggle to memorize a series of poorly understood steps, perhaps exacerbated by a less-than-firm mastery of single-digit multiplication and/or addition facts, can easily lead to confusion, frustration, and inaccuracy. Losing clarity of reasoning for a gain in speed isn’t a problem if you are good at a given procedure and/or don’t care whether you understand why it works (as apparently was the case for that fifth-grade teacher and the lattice method). For a parent, teacher, or learner to see mathematics as strictly a matter of barely grasped, tenuously memorized rules and procedures, there is a great risk of creating a nation in which a vast number of citizens “can’t do math,” which is to say that they find what they believe math to be – calculation – a tedious, intimidating, frustrating, and ultimately pointless enterprise.

Looking again at the lattice method, you might notice that reading the inner diagonals from top left to bottom right, you can see them as representing 1000 + 21, 60, & 350 (appending the necessary zeros). Or taking each cell individually and in terms of place value, we have 1000, 350, 60, and 21. Those who doubt that the lattice method “makes sense” should really consider that it is no LESS transparent than the standard one, and possibly a bit more clear. As for some other objections (takes too long to create the lattice, uses too much room/paper), I’ll leave those for possible comments and for a planned interview with mathematician James Tanton in the context of solving quadratic equations and using lattices.

Long Multiplication Visually: An Area Model

Let me offer one last model/method for doing long multiplication. Before doing so, I need to make clear to readers that this is intended strictly to promote understanding and to appeal to learners and teachers who enjoy a visual flavor to their mathematical comprehension. It is generally known as the area model. Unsurprisingly, I’m going to again employ 27 x 53 to illustrate the model, shown below:

In this model, each large square is 10 x 10 and thus has 100 small unit squares. Across the top, students will mark a line along the horizontal one full block for every 10, and one smaller line along the top of each unit square, then similarly down the left border for the second factor. Here, I’ve marked two “10’s” and seven “1’s” along the top, and five 10’s and three 1’s down the left side to indicate 27 and 53, respectively. I used green to color in each complete hundred-square, red to color in each ten “stick,” and orange for each unit square. Since we have two by five 10’s, the green area covers 1000 unit squares; vertically we have seven by five sticks for 350; horizontally, there are three by two sticks for 60. Finally, there are 3 by 7 unit squares for 21. Adding up the areas gives 1431, of course.

In my experience with elementary students, having them model a couple of multi-digit multiplication problems this way is enjoyable for most and particularly clarifying for some who do not gain a solid understanding from more symbolic models alone, regardless of notation or algorithm. But it is clear to most students without being told that doing an area model drawing is going to be far too tedious and time-consuming for an every-day method of computing products. I’ve never had to tell a student that this isn’t intended as a go-to approach, though I do try to make that crystal-clear to teachers and parents, particularly those who are suspicious of anything but the standard algorithm. Number-lines, algebra tiles, and countless other hands-on, software, and other models and tools are useful for promoting understanding: few are intended to be replacements for or alternatives to the standard algorithms. There are some alternative algorithms, as we’ve seen, such as lattice multiplication, that arguably are a matter of personal preference, particularly for students who struggle to do paper-and-pencil arithmetic the standard way. As a teacher, I don’t see it as my job to dictate which mathematically sensible calculation method an individual settles upon, as long as s/he’s making a reasonably informed choice. Refusal to even consider more than one way to do something mathematically is often the sign of someone who is not confident about mathematics.

I hope that readers take away from this piece the realization that any mathematical procedure has to make sense, that the ones that work can always be justified, and that many methods are intimately related if you look more deeply. In the near future, I hope to unpack long division in ways that will help it make more sense and less of a procedure to be imitated without understanding.

[i] For those unfamiliar with 20th-century American poetry: https://bit.ly/2nWvCYv

[ii] This is not the usual definition of expanded notation in arithmetic, but it is close enough for my purpose here.

Michael Paul Goldenberg