As summer vacation enters its second half, it’s probably a good idea to have kids do some of the basic how-tos. In the last decade or two, there’s been a shift in how basic arithmetic is taught. One of those is how to do multidigit multiplication. Consider, for example, $$34 \times 56$$

The Traditional Way

For many adults, even if they have been out of touch with mathematics, the procedure they were taught as kids is still ingrained in reflex. They can dutifully go through the motions and arrive at \(1904\) through a process that looks like this

$$
\begin{array}{r r}
&34\\
\times & 56\\
\hline
&204\\
&170*\\
\hline
&1904
\end{array}$$

Sometimes the “* is replaced by a zero, sometimes it is left as an empty space.

A criticism of this approach is that it’s just a process and most people (adults) can’t explain what they’re doing or why it works — they just know that it does work. The inability to explain the method or the steps involved is partially a consequence of how it’s often taught (as simply a method without explanation). And even if the why is taught, as is often the case with math education, the why is forgotten and the how remains. As such, another reason for an inability to explain the method is also partially a consequence of the opaqueness of the method.

Finally, as a process, it requires a strict adherence to proper alignment. When the “*” is left as white space, students often misalign the resultant arithmetic and wind up computing something like \(204 + 170\) for a result of \(374\). From here, determining if that answer is correct or not either requires the student to already have had some intuition about what the answer should be or it requires the student to “debug” the procedure. The debugging process rarely goes well since, and especially for young students, it is often difficult to see one’s own math mistake. As for intuition, this all depends on the student’s prior math education at home and in school — both of which tend to be wanting for more.

Box Method

As a response to the perceived and documented pedagogical shortcomings of the traditional way, and maybe also because things just have to change, in comes a new approach which aims to be more transparent and conducive to understanding the why. [As an aside, I don’t consider the box method to be “new”, but rather that it is more fashionable / in vogue.]

The box method works as follows for \(34 \times 56\). First, construct a \(3 \times 3\) box with the top row containing \(30\) and \(4\) and the left column containing \(50\) and \(6\). The top left cell can be empty or can be a placeholder for the multiplication symbol as a reminder that this is a multiplication problem. The inner cells are the result of the individual multiplications. Doing so, gives us this

$$
\begin{array}{c | c c}
\times & 30 & 4\\
\hline
50 & 1500 & 200\\
6 & 180 & 24\\
\end{array}$$

The second, and final step, is to add the numbers in the inner box. $$1500 + 200 + 180 + 24 = 1904$$

This is clean and transparent. It also allows for some natural exploration. For example, \(1500 + 200 = 1700\) and \(180 + 24 = 204\) which then allow for \(1700 + 204 = 1904\). Alternatively, adding the rows, gives \(1500 + 180 = 1680\) and \(200 + 24 = 224\), yielding \(1680 + 224 = 1904\). We can also add the diagonals if we wanted to.

Compare / Contrast

I think both methods are good ways to do multiplication by hand. How good the method is depends on how it is taught. It’s just as easy to teach the box method as a process to memorize as it has been to teach the traditional method. Further, what gets lost sometimes in the push for new methods is how extendable they are.

Consider, \(345 \times 678\). The stacked multiplication way gives

$$
\begin{array}{r r r r r r r}
& & & & 3 & 4 & 5\\
\times & & & & 6 & 7 & 8\\
\hline
& & & 2 & 7 & 6 & 0\\
& & 2 & 4 & 1 & 5 & \\
& 2 & 0 & 7 & 0 & &\\
\hline
& 2 & 3 & 3 & 9 & 1 & 0
\end{array}$$

And the box method with an added line for columnar sum is

$$
\begin{array}{r | r r r}
\times & 300 & 40 & 5\\
\hline
600 & 180000 & 24000 & 3000\\
70 & 21000 & 2800 & 350\\
8 & 2400 & 320 & 40\\
\hline
\mbox{sum} & 203400 & 27120 & 3390
\end{array}$$

Finally, \(203400 + 27120 + 3390 = 233910\), which is consistent with stacked multiplication.

From a labor standpoint (not algorithmic complexity), the box method is going to take more time than stacked multiplication. The stacked multiplication way is also more compact, while the box method takes up space on the order of \(n \times m\) where \(n\) and \(m\) are the number of digits in the multiplicand and multiplier as well as any side work that has to be done for the final additions.

On the other hand, the box method is certainly more easy to audit since all the component calculations have been broken down. As such, it is arguably easier to teach what is going on with multiplication than with the stacked method. Plus, it provides for a nice segue into some mechanics students see in Algebra (eg, \((a + b)(c + d)\)) — this is no small comment either. It is extremely helpful for students to see the same concept used in many different ways.

Another thing that’s forgotten or glossed over is that both methods are identical if we sum in a particular way.

With the box method, if we sum the rows instead of the columns, we have
$$
\begin{array}{r | r r r | r}
\times & 300 & 40 & 5 & \mbox{sum} \\
\hline
600 & 180000 & 24000 & 3000 & 207000\\
70 & 21000 & 2800 & 350 & 24150\\
8 & 2400 & 320 & 40 & 2760\\
\hline
\mbox{sum} & 203400 & 27120 & 3390
\end{array}$$

Notice that the right column sum is exactly the rows in the calculation for stacked multiplication. Neither of these ways are materially different. It’s just a matter of how things are exposed.

If anything, stacked multiplication combines the row-sum of the box method on a digit by digit basis, which leaves the final addition as a stacked addition.

I will say this, though. If the reason we’re teaching stacked multiplication is solely for its speed over other methods, I’d urge us to consider a different approach. Perhaps something like boxed-stacked multiplication. It could look like

$$
\begin{array}{r r r r}
&3&4\\
\times & 5&6 &\\
\hline
& 180 & + 24 & = 204\\
& 1500 & + 200 & = 1700\\
\hline
& & & 1904
\end{array}$$

And from here, we can treat the traditional stacked multiplication as a nice shortcut once we’ve become proficient with how the numbers breakdown [place value!].

For these reasons, I don’t think it is meaningful to make comparisons about which method is “better”. Rather, in my book, either method is good for learning how to multiply. What matters, no matter what method is used, is why the method would work and how it relates to place value.

Place Value

Stacked multiplication is powerful in this way since it exposes and utilizes the place value system. However, lost in the shuffle of doing is exactly the meaning of the place value system.

If we really, truly, can teach multiplication correctly, one standard would be to see if students can understand how to multiply \(345 \times 678\) in base \(x \geq 9\). Both methods should work just the same, but do students really understand the place value system?

It’s this question that goes unanswered. Students simply do not understand the place value system other than the mechanics in base 10. The moment we switch base, it’s as if all math is unknown.

To me, that should be a major focus when we teach multiplication methods. The mechanics are good and all, regardless of the method, but there’s no reason that a change in base throws everything off. We work in mixed bases ALL the time.

Three weeks plus four days is how many total days?

I have a carton that holds twelve eggs and I have a crate that holds twelve cartons. I have 6 full crates, 2 extra cartons, and three extra eggs. How many total eggs do I have?

HH:MM:SS \(\rightarrow\) If the time is 11:43:59 and one second passes, what is the new time? We effortlessly know that this is 11:44:00.

So the summer excursion? Try \(345 \times 678\) in bases 9, 11, 12, 13, 14, 15, 16 and get a firmer grasp on the place-value system!