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A 8th grade student I was working with recently was faced with squaring 37 in a problem that they were working on. The student stared down the numbers for some time, and then came up with some slightly incorrect answer.

I put my own pencil to the margins of the page and did the multiplication by hand, and asked why the student had attempted such a large multiplication in their head rather than making some scratch work of it. The answer surprised me:

We're not allowed to do multiplication by 'stacking'.

Is this practice wide spread? What motivates it?

edit: by 'multiplication by stacking', I mean this:

enter image description here

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    $\begingroup$ "Stacking" is not a term I am familiar with in this context. Perhaps the teacher in question is trying to foster intuitive numeracy by making kids work it out in their heads? $\endgroup$ – Ben I. Jun 11 '17 at 18:48
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    $\begingroup$ $37*37 = (30+7)*(30+7) = 900+420+49 = 1369$ but, maybe $40*37 = 1200+280 = 1480$ then subtract three copies of $37$ is easier. Of course, the "stacking algorithm" works as well. $\endgroup$ – James S. Cook Jun 11 '17 at 19:56
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    $\begingroup$ Skeptical. I've never heard of such a prohibition (northeast U.S. here). Clearly the student knows the standard multiplication algorithm; so, where did they learn it (I assume in class)? Perhaps they misunderstand a one-time exercise at some point (e.g.: "Explain this multiplication in another way other than the standard algorithm"). $\endgroup$ – Daniel R. Collins Jun 12 '17 at 1:12
  • $\begingroup$ This is in Canada. $\endgroup$ – NiloCK Jun 12 '17 at 2:22
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Some context is missing here. But I'll go ahead and speculate scenarios for why a math teacher would (temporarily?) prohibit multiplication by stacking [and in writing? --- nothing stops the student from doing stacked multiplication in their head]. In no particular order:

  • Sometimes, there is too much 'reflex' and not enough thinking. For example, $15 \times 200$ can be done by stacking, but it might be overkill. [cf Ben I's and James S. Cook's comments]
  • Sometimes, a course textbook or an educational standard emphasizes different methods and the teacher may want to motivate other methods by first asking students to multiply numbers but removing the [perhaps] one and only method they know how.
  • Since the question gives an example involving multiplying squares, I can imagine some Algebra related build-ups via $(a + b)^2$ [again, James Cook's comment]
  • Specifically with perfect squares and the previous bullet point, consider this mental method. $(35 + 2)^2 = 1225 + 140 + 4$ ... the "mental" part is recognizing a little 'trick' with multiplying perfect squares that end in $5$ --- namely, and by example, $35\times35$ is the concatenation of $3 \times 4$ and $25$. Another example, $105^2$ is the concatenation of $10\times 11$ and $25$ --- $11025$. So you can take any $x$ and to find $x^2$ it's a matter of computing $((x-k) + k)^2$ where $k$ is the smallest non-negative integer such that $x-k$ is divisible by $5$. For example, $89^2 = (85 + 4)^2 = 7225 + 680 + 16$ --- we can argue that $(80 + 9)^2$ is easier and so be it. I'm speculating about why a certain method is restricted. [Of course, $83^2$ might be more easily handled as $(80 + 3)^2$ rather than $(85 - 2)^2$]
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    $\begingroup$ By personal experience (my daughters) some teachers are doing this in Italy. Their motivation is to develop some number sense in kids, by forcing them to use some ad hoc procedure like the one you mention for squares. It is usually done much earlier than 8th grade (I'd rather sat 5th-6th grade) and expressed in the form of "compute without using pen and paper" since at that age it is very exceptional than someone is able to do stacked multiplication in their head. $\endgroup$ – Nicola Ciccoli Jun 13 '17 at 8:11
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    $\begingroup$ If I were squaring $89$, I'd probably think of it as $(90-1)^2=8100-180+1=7921$. Every integer is within $2$ units of a multiple of $5$, and I can see no reason to favor the odd multiples. The ones that are also multiples of $10$ are generally easier to work with. +1 for good answer, though. $\endgroup$ – G Tony Jacobs Jun 13 '17 at 13:17
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    $\begingroup$ @GTonyJacobs I can see why $(90-1)^2$ would be easier, but I can imagine that one can also argue that working only with addition, a la $(80 + 9)^2$ would be easier than dealing with subtraction. I think it's a matter of personal preference. $\endgroup$ – Math Misery Jun 13 '17 at 15:21
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    $\begingroup$ Yeah, my own take, pedagogically, would be to let students know that $(80+9)^2$, $(85+4)^2$ and $(90-1)^2$ are all options, that they should try all three for practice, make sure they get the same answer each way, and decide which one they like best. For that matter, $88\times 90 +1$ isn't a bad way to do it, either. $88\times 90 = ((8\times 9)\times 11)\times 10$, and those steps are all very quick mental math. $\endgroup$ – G Tony Jacobs Jun 13 '17 at 15:59
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    $\begingroup$ @GTonyJacobs agreed! $\endgroup$ – Math Misery Jun 13 '17 at 16:00
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Maybe instead of the "stacking" method we learned in the last century, they are supposed to use the "diagonal" method like this

enter image description here

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  • $\begingroup$ A number sense approach to $36\times 27$ would be to start with $36$, and then triple it three times. Thus, $108$, $324$, $972$. $\endgroup$ – G Tony Jacobs Jun 12 '17 at 22:09
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Although anecdotes are usually badly reported samples of size one, I'll answer with one. Trying to diagnose his problems with multiplication, I tried giving my grade school son variants of the same problem, written in "stacking" form, e.g. $123 \times 456$, $456 \times 123$, $1230 \times 456$, $4560 \times 123$, $1230 \times 4560$, etc. My hope was that after the first he would realize that rest could be deduced from its result with minimal computation. Instead I discovered that he had not learned to multiply well by "stacking" (he doesn't write neatly, and his columns are like the one in Pisa, so to obtain the final sum he adds digits from different columns; usually each row of the answer is correct, but the final sum is wrong). He did in fact perceive what I had hoped - he recognized that his five answers (four of them incorrect) were inconsistent, e.g. that the first two had to be the same and his answers weren't, and so forth, but this recognition didn't help him decide which answer was correct or where his mistakes were! The moral (for me at any rate) is simple (although perhaps this is just confirmation bias in operation) - for a student who doesn't have operational mastery of the basic rote operations it's silly to worry much about interpreting the operations being performed. Approaches that emphasize such interpretation are important and useful, but they can't substitute for learning some effective and efficient algorithm for reliably performing multiplications correctly. Rather they should accompany and follow the exercise/practice/drill that is their necessary precursor.

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I've never heard of 'stacking', the method shown appears the standard way to multiply digit by digit.

37 squared. For those who wish to do this in their head, I'd go up to 40, and the same 3 down to 34. 34 doubles to 68, and again to 136, then x10 to 1360. Now add back the square of 3, the 9 we lost by taking (a+b)(a-b).

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