Owen Biesel

A frequent reader of this blog sent me a link to this video on Facebook, of a side-by-side comparison between a traditional method for doing multidigit multiplication (which is over very quickly) and a new “grid” method (which takes a long time to explain):

She wanted to know why anyone would use the long, drawn-out method on the left when it takes so hilariously much longer. Here’s what I wrote in reply:

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What a funny video! I love the choice of activities for what to do with the extra time: cooking ramen and playing video games. Hah! I appreciate your curiosity about the new method, though, because I think it gets at the heart of what “new math” is trying to do that “old math” wasn’t doing a great job at.

First, it looks like the video’s intended takeaway is that the new method is vastly inferior to the old method, on the basis of the new method taking so long. However, you may have noticed that it compares someone giving an explanation of how to do the new method, including mnemonics and reasoning, with someone merely using the old method. A more apples-to-apples comparison would have been two videos side-by-side just working through the problem using the two methods. It would be a lot closer in the amount of time it takes, and a lot easier to see the commonalities between what the two methods are doing. I think the old method would probably still be faster, though since there’s less writing and rewriting the intermediate steps on the way to the final answer.

If the goal were really speed, though, there should be a third video, of someone typing 35 * 12 into a calculator and hitting equals. A few extra seconds to play computer games! So why don’t we just teach that to students? It’s faster and less error-prone than doing complicated computations by hand, and for this reason I don’t mind if my students use calculators for their arithmetic if I’m teaching something where that’s not the primary focus. But I can think of a couple reasons why you wouldn’t just want to teach elementary school kids to use calculators when they first start learning multidigit multiplication:

1. You want your students to understand what they’re doing, not just mindlessly follow an opaque procedure the way the calculator does.
2. You want your students to develop a general number sense, so that they know roughly what the answer should be and can notice if a calculator answer is wildly wrong (because of mistyping the input, for example).

On both of these counts, the old method is better than just using a calculator, but it seems to me that the new method is even better than the old:

1. The grid process for multiplying the two two-digit numbers is a way of visualizing the distributive law in action: break apart 35 and 12 into sums of simpler pieces, multiply those pieces by each other, and then add up the results. If the video showing the old method also included an explanation, I think it would need to spend quite a while explaining why each digit in the subtotals goes where it does.
2. Notice that in the old method, the first digit of the answer to be calculated is the ones digit, and you don’t get a sense of how big the final answer is until you’ve multiplied the last pair of digits and counted up how far to the left it is. In the new method, the very first thing you do is multiply 30 and 10 to get 300, which while less than the final answer of 420, gives you at least a sense of what size number to expect in the end.

Something the video doesn’t address is whether the new method is trying to teach something else, besides multiplying two-digit numbers. Is it paving the way for a more conceptual way to approach problems like 499 * 2999 as (500 – 1) * (3000 – 1), which is much faster to expand out and add than to compute with the old digit-by-digit method? Is it laying the groundwork for multiplying algebraic expressions like (x + y) * (3x – 2)?

From a broader perspective, is it trying to communicate that even large problems can be solved by breaking them down into their component pieces and working on them one by one? Is it perhaps trying to illuminate that knowing how an algorithm works is just as important as getting the right answer? That you first need to be able to break down a complex goal into meaningful discrete steps if you want to be able to write a computer program like the person on the right is enjoying?

Or perhaps, the goal is to keep clear at every stage that the 3*1 part is much more important to the final answer than the 5*2 part, and that mistakenly adding them together would be the same kind of mistake as comparing two videos created for very different purposes. I don’t know yet whether kids raised on New Math are less likely to fall for this kind of visual rhetoric, but I’m happy to keep trying to drive the lesson home whenever I can. 　

Thanks again for asking, and I hope this gives you some appreciation for the power of these new methods, as well as compassion for all those parents out there who are being asked to learn a new perspective on something they thought they already knew!

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Addendum: I recently saw this picture on twitter illustrating that the box method can apply to products of more than just multidigit numbers, in contrast to the several different algorithms I learned in school to handle the various cases:

I’m not sure who “One boxy boi” is but it looks like you’re doing a good job multiplying! I especially appreciate that the color-coded letters in “F.O.I.L” match the colors of the squares in the boxes on the right — that’s a nice touch!