On Doubts and Multiplication

I wanted to write a simple text about teaching multiplying large numbers, but as it is the case with all the teaching, many questions immediately popped out. Consequently, what supposed to be a logical, clear, linear chain of thoughts , became an entangled knot of half doubts. Those doubts happened to be as, if not more, important than the multiplication itself. Yet, to appreciate the importance of  doubts I have to talk about issues Robert and I encountered while teaching/learning multiplication.

Robert mastered times table in the first half of 2006.

He did not have any problem with multiplication of two or three digit number by a one digit when there was no need to regroup.  For instance 52 times 3.

When Robert was learning to multiply two digit numbers by one digit with regrouping I asked him to always write the result of first multiplication (ones by ones) on the side and then rewrite it in a split way.  The ones under ones but tens above tens.  For instance when counting   43 times 4 , Robert first writes 12 on the side and then he rewrites it by placing 2 under 3 and 1 above 4.  And so on.

Of course there are other multiplication algorithms and some of them are much more intuitive, but I chose this one as it is the most frequently used.

I have to say, that it did help a lot that Robert became good at mental addition finding the sums  similar to those: 17+4, 56+8  as this skill is utilized a lot during multiplication.

Well, not exactly.  When Robert started multiplying larger numbers, he was not able to do all the required additions mentally.  I often asked him to write on the side, and he dutifully followed.  But as he continued to do that I often stopped him just after he wrote the numbers under each other.  I let him look at them , but didn’t let him write the answer below.  I wanted him to say the sum just by looking at addenda and do regrouping in his mind. I think that was an important step to develop ability to add in his mind.  It was during multiplying when Robert mastered mental addition (in the narrow range required to find products).

When Robert started multiplying by two digit numbers I used (as many curricula suggest) vertical and horizontal lines to help organize partial products in proper spaces.

Robert learned to multiply three and four digit factors  by two digit factors.  He made , however, mistakes of forgetting to multiply by ten’s digit.  The horizontal lines let him remember.  Still, it was a problem I addressed by making worksheets in which I mixed up multiplying by one digit with multiplying by two digits. Interspersing problems this way demanded that Robert pays attention.  This problem presented itself  many times, specially after periods of time  when we did not count products.

Multiplying by three digit factor, is still a problem.   Robert can be completely confused and often cannot find even partial product.  When, around 2008 I dealt with this problem for the first time, I tried to hide under tiny stickers two out of the three digits, so Robert concentrated on only one.  (That is what we did initially while multiplying by two digit number).  But even so, it was still very baffling.  Robert  wanted to multiply the hundred’s digit only by hundred’s digit.  Since it was so hard for Robert to follow with those numbers and for me to find a way to make it easier, I decided to stop working intensively on that skill. But over the next four years, while Robert was learning other things, I kept returning  to the task.  I wanted Robert to learn, but mainly I wanted to understand HOW HE LEARNS.

Clearly, the hundred’s digit was a problem. Somehow Robert was not able to connect it with one’s and ten’s digit of the other factor.

I drew three arrows from hundred’s digit to all the digits of the factor above, as the way of visualizing the process.  Each arrow had a tiny number attached to it: 1, 2,3.   That seemed to help a little.  But not much.

Today, I wrote just 4 multiplication problems.  Two on each piece of paper.  With vertical lines and horizontal lines. I asked Robert to cross out each digit in the bottom number he had already multiplied by.  With some prompting, he kept crossing one’s, then ten’s digit until he got to hundred’s.  He was still a little uneasy about connecting hundred’s with one’s and ten’s of the other factor, but did not make mistake.  Crossing off seemed to make a difference.  But we are not of the woods yet.

Now I pose the question, which probably any person reading this blog would like to ask , “Why am I teaching Robert multiplication he probably will not have a chance to use in any of the adult programs he might be included in the future? ”   Why don’t I just give him simple one digit problems (If I really need to do something with him.) he can solve easily and feel good about himselfThat is after all,  what Robert’s school is doing almost every day this year.

First , Robert doesn’t feel proud when he fills easy worksheets.  To the contrary, he does feel very proud in those magical moments when he grasps something new, something he could not do before. 

Second, had Robert had language and was able to communicate with me sufficiently, I might find other things more appropriate to teach. But when Robert learns multiplication, he learns to follow a complicated algorithm, he learns to follow steps, to pay attention, to redirect his attention as he moves from one digit to another. 

Third, the fact that Robert had so many problems with multiplying by three digit numbers was a REASON to teach that skill, as a way of maybe (just maybe) addressing the deficit which caused those problems in the first place. 

Ability to multiply is not a goal in itself.  I am teaching multiplication for the same reason I am trying to teach everything else. To discover and possibly address  the learning (thinking?) difficulties  Robert has.  I don’t know any better method of understanding  Robert and helping him to understand the world. 

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