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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Kathy Can Stay

Oh, how embarrassed I felt on those Tuesday’s evenings in 1996, when my daughter’s ballet teacher, Kathy, was bringing Amanda home.   At that time it was rather difficult to  pick up Amanda after her dance classes. Waiting until she changes her clothes was not easy  with Robert, who couldn’t stay still even for a fraction of the second. To help us, Kathy drove Amanda home after every  lesson.  I always asked Kathy to stop for a few minutes, and  had a bowl of her favorite, home-made chicken soup ready for her. But Robert did not want Kathy to come in. He let Amanda in, but tried to prevent Kathy from crossing through the door.  The little four years old creature, looking  like two years old toddler, was pushing her outside with his arms and… head.   The first time this happened, Kathy wanted to leave as not to cause any problems.  I begged her not too.  Holding  Robert  I explained that Robert had to learn to tolerate people coming in. ” I know it would be hard for you, but please stay at least for 5 minutes.”  The most important thing was, in my opinion, for Kathy to sit down and pretend Robert’s objections did not bother her.  When Robert was not protesting too strongly, Kathy and I ate the soup together at our kitchen table and  chatted.  When Robert was protesting vehemently, I calmly and slowly filled a plastic container and gave it to Kathy, so she could take the soup home and eat in a calm atmosphere.

Every time Kathy came, she said in her calm, friendly way, “Hi Robert, How are you?”  Maybe sometimes she changed a word or two, but her voice was always friendly and NEVER artificially sweet.  Robert did not respond.  At that time, he knew maybe 5 expressive words and not even one receptive label. I am not sure what he understood.  The best he could do was to turn around, ran to the living room, and let us be.    After generously leaving us alone for 2-3 minutes, he came back and handled Kathy her purse, or (later) her jacket. It was a hint, I asked Kathy to ignore.  She agreed to get up when Robert was NOT “asking” her out.  When in the door, she never forgot to say, “Bye Robert.”

One Tuesday, two or three months later, as soon as he heard the door bell, Robert rushed to the door, which I was just unlocking.   I wanted to pick him up, terrified that the old habit would return and Robert would push Kathy, when I noticed that he was bouncing happily in place, and pulled the door to open it…. wider.  He was happy!  He wanted both Kathy and Amanda to come in.

When they entered, Robert shut the door and  stretching  his arm toward the latch at the top of the door he made clear that he wanted to lock it.  After I closed the latch, Robert bounced happily off to the living room.   Kathy, Amanda, and I ate chicken soup (Robert did not eat that stuff.) and even some ice cream. We chatted  and laughed realizing that Robert demonstrated the same attitudes toward Kathy, he previously expressed toward Louis, the Cat.  Did Robert accepted Kathy and Louis  as extended family members, close family members, or just friends?

We did not know.  Robert did not explain himself.

I am not really sure how many times Kathy came before Robert considered her a very good friend but I know that he certainly appreciated that she ALWAYS addressed him at the beginning and the end of each visit and did not mind that he did not answer. 

The Functions of the Latch

In the fall of 1996, a day after Robert’s opened the door and aiming for the playground ran through the narrow streets and the treacherous parking lot of our apartment complex , my husband installed a latch on top of the door.  Robert quickly understood its function  and sincerely detested it. It was there to prevent  him from going out.  Robert wanted out, out, out.  So a few times a day, he stretched himself along the door’s surface, his head tilted backwards while short, little hands extended dramatically toward the latch.  Sometimes it worked, sometimes it did not.

There were many reasons Robert wanted out.  He wanted to go to the playground, to the swimming pool, to the grocery store to buy more bubbles, and he wanted to admire Louis.

Louis was a black and white cat, living across the street (and the parking lot), who at some point started to spent a lot of time on the steps leading to our townhouse.  It is possible that it was because in the afternoons, the sun warmed up our side of the street.  It is possible. However,  I am convinced that Louis kept coming because he loved Robert and Amanda. He waited for each of them to return from their schools, but he hid between the bushes and the wall of the townhouse  when other children tried to approach him.  Louis  clearly felt pampered and entertained  by Amanda who, pretending to be Isadora Duncan, danced for him with thin shawls twirling around. Louis also accepted gracefully Robert’s dance of appreciation, quick bouncing in place and flapping arms.  So he kept coming to our stairs, and Robert kept going outside.

It was unavoidable that at some point Louis would enter our apartment. And he did.  As soon as Louis entered, Robert shut the door, stretched his arms as high up as possible, and kept repeating,”Close, close, close”.

Suddenly, the latch from a despicable obstruction changed into Robert’s ally. Robert wanted Louis to stay in our home, and the only thing capable of assuring that was, in Robert’s mind,  the latch.

So Louis stayed for the whole afternoon until his owner, Ann, came to pick him up.

Take It to your Brain

I experienced a lot of chaos while teaching Robert to round up or down natural numbers or those decimals that relate to dollars and cents.
I started to teach rounding to the nearest ten approximately eight years ago. I simply used a segment of a number line, for instance, from 20 to 30.  I asked Robert to find 32 and then decide if it was closer to 30 or 40.   I soon stopped because I I begun to teach  Robert to divide with a reminder.  In a process, I  used a number line in a slightly different way.  I did not want Robert to be confused as to what was expected of him.

To make the matter even more baffling , in a school year 2005/2006, the teacher in a collaborative program was teaching rounding prices to the next dollar.  So for $25.99 and for $ 25.01 the proper answer was $26.  The authors of that part of the curriculum rightly believed that  when the price is  $25.30, you have to handle $26 and not $25 to the cashier.  It made a lot of sense from a practical point of view, but not from mathematical understanding of the values of numbers.

In the school year 2006/2007 Robert’s teacher’s aide introduced rounding up by a set of instructions, identical to those I received in my childhood. I described it in the comment below.

I tried to continue in this manner, but Robert had a lot of problems, I was not sure how to deal with.  So I stopped and waited until the teacher at school would move on to another topic.  Then…. I forgot as I moved to other things.

From time to time, the  need to round the numbers to estimate the result of math operation surfaced.  We returned to the old approach.  It seemed more visual.  To round 47 to the nearest ten, we drew a line segment which had 40 and 50 written at its ends and 45 placed in the middle.  Robert had to place 47 on the proper side of 45.

He had the same difficulties as in the past. I realized that he was looking for EXACT point he should place the number. That was confusing.   I remedied that by drawing one circle above each half of the line segment. Now there was a space to place the number,  Everything became  simpler and more straightforward.

As we worked today on such problems, I realized what I am aiming for while teaching Robert.

In the past I just wanted Robert to be able to turn verbal direction into proper math operations. After reading “find average” , to follow by  adding and dividing.  It was a worthy goal.

That was also my first goal with rounding to the nearest ten or hundred. I wanted Robert to almost automatically follow the steps as I described them above with number 47.

Now, I consider it only a partial achievement, not the goal by itself.

The real goal  for Robert is to draw a line segment with two numbers at the ends and one in the middle… in his head.  Make a mental picture and use it to solve a problem.

I only vaguely understand why I consider this last step so important for Robert that I made it a goal worthy reaching.  Is it because I want Robert to pick up the problem from a piece of paper and take it to his brain or is it because I want a proof that he can do it?

When Schools Teach and When They Don’t

I rarely write about skills Robert acquired at one of his schools.  There are two reasons for that.

1. When the school did a good job of teaching, my part was mostly supportive.  I just either followed the suggestions given to me  during home visits by one of Robert’s teachers (when he attended private, ABA driven school) or I provided extra practice based on  the worksheets Robert brought from his collaborative program.  I did not have to think, design a program, or adjust it when it didn’t work.

2.There were long periods, extending into months when Robert was not taught anything new, when so-called “progress reports” although very wordy did not indicate one concrete thing Robert learned at school. So there was nothing to write about the school teaching.

I remember most vividly those attempts to teach Robert which aimed at the skills he was most resistant to learn. The longer I tried, the more approaches I experimented with, the stronger my memory. If I remember how  Robert finally memorized the addition  facts it is because for six years, his school and I tried, tried, and failed. So the final success felt like a miracle, although it was the result of finding a proper approach, as I documented in Looking for Variables.

Sadly, I do not recall as clearly the ways Robert learned those skills that were mainly taught at schools.  I cannot describe, for instance, how Robert was toilet trained.  I did not contribute much time, effort, or thought to that developmental achievement.  It was all the doing of wonderful, although always changing, teachers in Private School, Robert was attending at that time.  I believe that in the process many hours were spent in the toilet, but I am not sure even of that.    This school’s young and dedicated teachers also managed, to my disbelief,  to teach Robert to tie his shoelaces.  I suspect that they wrote a good task analysis and used either forward or backward chaining.  Thanks to them we did not have to search a store after a  store for Robert’s size shoes with Velcro straps.

Most importantly at that school Robert learned to stop when asked to do so.  It was a life saving skill.  Many times Robert managed to wiggle out of my grasps  and run without looking or stopping to the point of getting out of my site and placing himself in danger.  I couldn’t keep up with him and consequently I could take him to park, playground, or store only together  with his sister.  She was five, six, or seven, but she knew it was her role to stop Robert. Two teachers taught Robert to stop when he heard the word “Stop” or his name.  One walked with Robert in a maze of hallways, the other one called Robert from behind.

From what I know now about other programs, collaborative or public, I am sure that neither of them would teach Robert those important life skills.  I doubt if they would even try. That possibility still sends chill through my spine.

Similarly,  I  cannot say, how Robert learned multiplication facts because teaching and learning took place at school and I only supported what the teachers were doing in the collaborative program which Robert attended at that time.

I also don’t know how the occupational therapist in the public school managed to teach Robert to manipulate his  combination padlock to open his locker.

I value greatly those times when Robert learned something with only minimal support from me or without my support at all.  I felt that the world was full of people able to support Robert.  That was a source of great relief and hope.

On the other hand, the fact that I write so much on these pages about Robert’s learning is a very unfortunate one.  I became the main force  and sometimes the only force in Robert’s learning only because I had to fill the terrible void caused by lack of any appropriate teaching at schools. And this is quite depressing.

What About Reminder?

Robert memorized multiplication facts during his 2005/2006 year in Collaborative program.  I  cannot describe how Robert learned multiplication facts because my role was only marginal.  It, probably, helped  that in 2004, Robert knew already how to count by five, two, and ten or that I practiced with him counting by  other numbers.   I made worksheets to practice changing repetitive addition into multiplication and  worksheets  in which a new multiplication fact was scattered  among the facts that Robert already mastered.  The Collaborative Program, however, led the way.  I think that the worksheets the teachers used relied a lot on patterns. I recall a worksheet that introduced multiplying nine by consecutive numbers by emphasizing a pattern of results:  increasing ten’s digit and simultaneously decreasing one’s digit. I am not sure  if at that time Robert and I worked on those lessons from Saxon Math that presented multiplication as an array of objects.  Maybe not yet.  I recall vividly, that Robert learned quickly and almost effortlessly as it was always the case when the school assumed responsibility for teaching.  Unfortunately, not much later, Robert was forced to leave this program.

For the next four months I was teaching Robert at home much more intensively than before.  We practiced multiplication facts and family of facts  as a way to almost mechanically tie multiplication to division.  (That is what we had done before when Robert had been learning to memorize subtraction facts). That allowed Robert to memorize division facts relatively quickly.  The problem arose when reminders had to be taken into account. It is one thing to remember that 32 :8=4 and another to remember what is 34, 35, or 38  divided by 8.

We started by learning to  divide by 2.  On top of each page, I wrote  the multiples of two on a number line in large print and the odd numbers in small.  Robert had to place each dividend on the number line and then look to the left to find the closest multiple.  For 17, he pointed to 16, for 9, he found 8 and so on.  In the next step, we skipped number line, but Robert still had to write all the multiples of two with empty spaces between them and  locate the odd number somewhere between the multiples.  For 11, for instance, he had to point to empty space between 10 and 12,   then move his finger to 10, and proceed from here.
Finally, when Robert practiced division by two, he was prompted to help himself by listing the multiples of 2 in order but only when he was unsure or confused.

We repeated these steps for all the divisors from 3 to 9.  Of course while learning to divide by 3 with a reminder, Robert also practiced, on a separate page, dividing by 2.  Then I mixed two kinds of problems on the same page.

I have to say, however, that we didn’t go  past dividing by more than 12.

Learning to divide with a reminder helps with long division and with changing improper fractions to mixed numbers.  It is clear that  it would be beneficial  to practice dividing by divisors larger than 12.  That, however, is still a problem, and when we have to do that, I just ask Robert to find a few multiples of the divisor (by multiplying).  Cumbersome process that still requires hand over hand (prompt over prompt) guidance.  We do that only rarely, when Robert wants to finish the problem which is a little over his (or mine) head.

The algorithm of dividing by numbers larger than 12 depends on  ability to round numbers up or down.  This is a very hard skill to teach, and Robert doesn’t have it.

For school this is NOT a functional skills, so instead Robert is practicing counting values of nickles: 5cents, 10 cents, 15 cents…

Didn’t he do that 10 years ago?