Moving Forward, Turning Back

September 9, 2015

A few days ago, Robert completed Geometry section from Level F Momentum Math.  That was the only part of the curriculum we skipped previously mainly because I wasn’t sure if I was capable of teaching Robert the  Pythagorean Theorem.  I probably wasn’t.  Nonetheless, I had to try. All other sections were much easier to teach.  It helped that over the past few years, Robert had a few other opportunities to learn about polygons and circles. He also practiced calculating their perimeters (or circumferences) and areas by mechanically applying proper formulas.

So I didn’t present Robert with something entirely new.  I mainly opted for refreshing former skills, better organizing the facts, and applying formula in different contexts.  So Robert was finding areas of rectangles, triangles, parallelograms, and circles.  There was no formula for area of trapezoid. Instead, Robert had ample opportunities to practice counting areas of shapes made of a few simple polygons or circles.  With my help he noticed that the trapezoid was made of parallelogram and a triangle thus he could add the area of both shape to come with an answer.

He followed the algorithm

  1. Divide the shape into two or more polygons (or circles).
  2. Write a formula for area of each of the simple shape.
  3. Plug in the measurements into formula and do all the arithmetic operations.
  4. Add all the areas.

Although, Robert still hesitates while following those directions he seems to grasp the idea.  Now it is time to go back to the previous chapter and relearn Pythagorean Theorem.




Hard Task of Teaching “Easy” Things

September 2, 2015

What could be easier to teach and to learn than the fact that the length of the diameter is twice the length of the radius and thus the length of the radius equals half of the length of the diameter. You could see this relation clearly in the picture of a circle.  To switch from one to another you could simply multiply or divide by 2.  Nothing to it.  So obvious. And yet, the link between radius and diameter seems to be the source of great confusion for Robert.  And thus for me.  It is very hard to teach obvious facts and  apparent connections.  There is really nothing to explain and not much to memorize.

It is much easier to teach and to learn other, more complex formulas. How to calculate areas of rectangles, triangles, and even trapezoids, not to mention areas of circles.

Today, Robert was dividing circles into six, eight, or twelve congruent sectors, cutting them out, and using them to build figures that resembled rectangles.

Judging by his sly smile, Robert noticed the fact that as the parts of the circle decreased in size but increased in the number, they could be arranged in a shape that more and more looked like a rectangle. (To illustrate the concept you can look at )

I am not sure if he just guessed or sort of understood that the length of the rectangle was approaching half of the circumference while the width was moving toward  the length of the radius.  Nonetheless, we both arrived to the formula for area of the circle, which Robert later applied a few times.

Robert almost automatically calculated areas of the circles when he was given their radii.

When, however, the problem demanded that Robert find the area of the circle with known diameter, Robert hesitated for quite a while, then closed his workbook and said, “Tomorrow.”

Just to put this post in the context of our studying together, I need to add, that we devoted most of the time today not to areas of the circles, but to the clearer pronunciation of CVC words.  That is a real struggle.



Different Kind of Change

June 16, 2015

Over the  last five or more years, Robert had many opportunities  to count the amount of change received after purchasing something. Unfortunately, all those opportunities were reduced to math workbooks – mainly Saxon Math Grade 4. Robert practiced with pencil and paper how much he should pay and how much change he should get back. The only school program  that addressed buying in practice was The Collaborative that  Robert attended when he was 13 and 14 years old.  Every week he went with is classmates on a trip to a store.  With money and short shopping list provided by parents, Robert,under the supervision of his teachers, purchased the items , paid for them, and got the change. None of his other programs followed this approach. Sadly, I wasn’t either.

Despite easily solving math problems that required Robert to count first the total amount paid for two or three items and then the change from $10 dollars, Robert didn’t quite understand the idea of paying with money and getting change back.  How could he if he could so  conveniently paid for everything with his debit card.?  Our trips to grocery stores usually ended with Robert pulling his plastic and paying with it.

I realized that there was a problem, almost a year ago,  when Robert wanted to pay with$1 bill for his lunch which cost $10.  He had $10 dollars in his wallet but in one dollars and five dollar bills. He didn’t have any idea that he should count dollars up to 10.  For him all the bills meant the same thing in practice. He gave one bill strongly believing that it should suffice.


This Friday, another issue came to light.  Robert went to a bowling alley with Pam, his skill instructor. He wanted to play, but he didn’t want to give his only ten-dollar bill to pay for shoes and games.  Finally, he was persuaded to do so.  However, he didn’t want to accept $6 in change.  He tried to give it back over and over. It took Pam some convincing before Robert accepted the change and began bowling.  But he didn’t forget those six dollars.  When he finished playing he wanted to give them back to the attendant.

I am not entirely sure what Robert was thinking.  Did he try to give back  $6 believing that he would get his $10 back?  Or did he thought that $10 dollars was the amount he should pay for the right to bowl and the $6 belonged to the attendant. It is clear, however that Robert didn’t understand the process of paying and receiving change.  That process interfered with Robert extremely strong conviction that the things should stay in the place they were in the beginning. Ten dollars in Robert’s wallet.  Six dollars in the attendant’s cash register.  His efforts to return the money (and maybe even get his money back) were a result of that belief. Robert’s insistence on returning the money was also a consequence of the lack of opportunity for Robert to practice in real life situations what he learned at the table.  For Robert solving money math problems is not the same thing as paying with money at the bowling alley. One might say, Robert didn’t generalize the skill to a different settings.  He didn’t because most of his teachers, and that include me, didn’t realize that as many people with autism, Robert needed to practice the same skill across different settings.

Sadly, I realized that , but didn’t do anything to help Robert connect his academic abilities with real life needs.


When Less is More

June 4, 2015

For the last couple months, Robert and I were rushing through units in Level F Momentum Math. Why shouldn’t we?  After all, we were ONLY reviewing what Robert already knew or what I ASSUMED he knew. So each day, we did one whole unit – nine pages of definitions, examples, and problems.  Then one day, Robert was lost when the tasks required placing fractions on number lines. Since I believed he knew how to do it, I tried to rush him through that unit as well. With every problem Robert became more and more bewildered, but I still pushed forward thinking that the next problem would clarify the whole concept. Instead of stopping and reworking the problem again so Robert could better understand the issues involved, and so I could understand the nature of Robert’s confusion, I presented the next task as if it would provide a better  opportunity to learn.  It didn’t.

It couldn’t as each problem became more complex and thus more difficult.

No wonder, Robert grew tense.

I had to rethink the strategies.

Every day, I presented Robert with one page of 4-5 easy exercises of placing halves, thirds, fourths, or fives on the number lines. 1/2,  2/3,  1/4, or 3/5.

I noticed that instead of counting segments into which one unit was divided, Robert was counting marks on number line starting with the first. The remedy was simple, Robert was asked to draw and count small arches connecting ends of the segments.

The second errors Robert kept making was not to count all the parts in one whole unit, but only up to the first letter representing a fraction (part of the unit). So we went back and I only drew one unit at a time for instance from 3 to 4 divided into a few parts.  This way, the end was clearly visible. Then I extended the number line to include next (or previous) whole number.

For the last three days, Robert and I worked on placing fractions and decimals on number lines. We went slowly, very slowly.  For every example in the book, I prepared a few similar ones. Before any example or problem in the book, we reviewed changing fractions to decimal and vice versa.

We didn’t hurry. Robert solved very few problems from this chapter, and yet he learned something.  That “something”  meant ” a lot more.

Learning More, Understanding Less

May 19, 2015

I am for teaching Robert as many subjects and topics as I am capable to teach.  At the same time, not once, instead of teaching I  confused Robert with too many words when I attempted to explain the subject completely. It might be because Robert processes words in a way that I don’t fully understand. It is not that he grasps the meaning of my directions/explanations slowly. I suspect that he grasps the meaning of just a few last words (or a few first words.  I am not sure even of that.) and then he replaces words he didn’t catch with other ones. I don’t know where those “other” words, unspoken by me, come from.  The less words I use to explain something, the more effects they have.  Adding words, that might show another side of the subject calls for extra caution as they instead might cloud the image already formed in Robert’s mind.

I don’t think this problem is typical of only children with special needs.  Imagine asking for directions and having someone giving you all the information about the way you should take – directions that would consist not only of names of the streets, but  also  descriptions of the all  buildings or  trees.

As I noticed before, Robert  had a strong tendency to compartmentalize his life and refused to accept that some elements moved  from one part of his life to another.  He pushed me out of his classroom when he was five, and he kept pushing his teachers when they arrived for home visits. Only respite worker could take him to McDonald but not his parents. Each of us belonged to specific places, and we shouldn’t encroach on other people terrains.   I do believe that it might be that similar separate structures exist in Robert’s mind that don’t allow him to see the same subject in a different light.

When I was teaching Robert algorithm for multiplying large numbers I was smart enough to ignore the method of partial products.  It is a great method which really demonstrate clearly to the student what is the basis for the multiplication algorithm. But for Robert it was important to associate only one method with one task.  Only when Robert became very good at multiplying large number I dared to present Robert with method of partial products.  We even completed a few examples.  At that point, Robert’s skill was strong enough to withstand the attack of the new method.  To the contrary, he seemed pleased almost as if he understood the idea behind the multiplication algorithm a little better.

Not so much luck with subtracting those fractions that needed regrouping. I made a mistake of switching between the two methods – changing mixed fractions into improper fractions or regrouping by changing just  1 into a fraction of the proper denominator.  I tried to follow Momentum Math  to the fault.

Of course, I did that because I believed that Robert knew already one of the methods and had the prerequisite skills for the second method.  But I also knew that Robert didn’t master any of the methods yet.  He  was still prone to making errors as he had tendency to lose track of what he was doing specially when subtraction demanded not only regrouping but also finding common denominator.

As I said, I followed Momentum Math curriculum without really taking into account Robert’s level of understanding of all steps needed for subtraction. As we read the problems, Robert attempted to solve the problem using the required method.  This way, problem after problem,  he grew more and more bewildered until he didn’t know any more what to do.

So we will go to the beginning following all the steps he had already mastered in the past and those that are still puzzling.

I don’t give up, however, on teaching another algorithm at some point.  If not for the sake of improving Robert’s arithmetical abilities than to give him much more important lesson:  different methods can help to achieve the same goals just like different people can bring you to the same places.

And although Robert learns best with as succinct instruction as possible, adding non important words, phrases, or sentences could better prepare Robert for flexibly adjusting to our noisy, imperfect  world.

Loosing Momentum and Catching It

May 13, 2015

For the last two weeks, Robert and I were working with Momentum Math 6th Grade. It was the third time we used this curriculum.  Since the last time, Robert had many opportunities to practice similar skills with the help of different programs and with the added support of a few worksheets I designed for him to address areas of weaknesses.  Moreover, the first 10+ units didn’t seem to present any new challenges for Robert but to the contrary allowed him to have a better grasp on general ideas and their connections. So, I expected smooth sailing through the chapter addressing placing fractions on number lines.  Many times Robert placed fractions on the number lines while working with Saxon Math grade 4th.

He knew…. Well, he was supposed to know that the  algorithm to complete such tasks began with counting into how many parts one unit (for instance between 0 and 1 ) was divided. He was supposed to know that, at least I thought so. But he didn’t know.  He was frustrated and I was frustrated as well. I had a feeling of being a failure as a teacher not because I didn’t teach Robert, but because I didn’t know why Robert didn’t know what I assumed he should have known.

Although frustration is not a good addition to the lesson, nonetheless it does happen.  The worst thing teacher can do in such situation is to continue subject pupil to similar exercises over and over without taking time to understand the roots of the problem.

That is what I did.  We finished the whole section, which meant that  Robert wrote all the right answers but he didn’t know what he was doing. He didn’t learn anything.  How is that possible?  Yes, it is possible. The student follows the teacher and is able to write the correct answers, but doesn’t  have a slightest idea of what he or she did and why.  It is much more common that any educator would like to admit.  Of course, I felt like a failure. And that is not a feeling that supports learning.

What should I have done instead?  Take a break and give Robert a break. During the break,  analyze what had happened, and understand the nature of Robert’s confusion.  I could quickly made a few easy pages requiring Robert to place only halves, thirds,  and fourths  on the number lines.

I should make the task  clearer, almost self explanatory  with small denominators. I should.  I haven’t done that yet. Instead, in the following days,  I asked Robert to work with me on problems in the following three chapters- adding and subtracting fractions. I knew that he could solve many tasks independently and a few with my minimal support. I just wanted him to regain momentum.  And he did.

Progress, Ever So Evasive

April 26, 2015

In the last few days, Robert and I returned to the workbooks we have completed twice before.

1. Cut, Paste, & Color Logic from Remedia Publication

As I  watched Robert attempting to place four pictures in four spaces by following three or two cues, I wondered how to asses his progress.  It was clear, that he would not have completed this activity if I had not been sitting next to him.  I don’t know however, how exactly did he benefit from my presence.  Did he observe (as he did so cleverly in the past ) my reactions to his manipulations and deduced the proper response based on what he noticed in the movement of my eyes or lips?  Was it possible that my very presence did the trick by giving him psychological support as he tried to find a way to place four animals in four cages?


Whenever he read a sentence of the form, “A was not put in B”, Robert wrote in the B rectangle, “no A”.  To sound less abstract: when Robert read. ” Elephant was not the last”, Robert wrote in the last rectangle, “NO elephant.” Moreover he made sure not to put elephant there.  That was the progress as during our previous sessions, a year or two ago, he (and thus I) often made mistakes there.  Another sign of  progress was the fact that Robert slowed down and injected a few seconds of thinking between reading direction and following it.  That was mostly visible when he had to choose simultaneously two out of the three spaces.

To find where to put zebras, a parrot, an elephant and a hippo, Robert read all three sentences before placing pictures in order:

1.The elephant is fed last.

2.The zebras are fed after the hippo.

3. The hippo is not fed first.

Those few seconds of hesitation squeezed  between reading directions and acting upon them was the most important developmental improvement.

Momentum Math F, Unit 1 Multiplying and Dividing Natural Numbers.

It is an easy unit for Robert.  He knows how to multiply and divide numbers. Moreover, he knew that three years ago and five years ago. well, the matter of fact, he knew that seven years ago too. But when four and five years ago Robert was working with this textbook/workbook, I decided to skip some topics. For instance, I decided not to teach him multiplying by partial products.  I was afraid that learning another method might impact negatively his ability to use standard multiplication algorithm.  This time, I knew that his skills were strong enough to resist possible confusion. Moreover, over the years Robert practiced writing a number in expanded form (247=200+40+7) thus he had prerequisite skills allowing mu to grasp the method of partial products.

During our previous work with Momentum Math, I introduced for the first time the prime factorization.  I did it because it was a chapter in the book and I tried to follow the curriculum.  I wasn’t sure if Robert would benefit from knowing it.  But mainly, I wasn’t sure if Robert could grasp it. Well, it was rather easy for him to get down to the prime factors. However, he had difficulties presenting a number as a product of its prime factors – those which were on the ends of the tree. He wanted to write all factors, not only the prime ones.

This time, he didn’t make that mistake, but there is still something amiss.

Piece of Rock

March 7, 2015

I thought it would be a piece of cake.  The task  so easy that Robert could do it without my help or even without my presence (Those are not exactly the same things for Robert as my silent presence gives him the courage to undertake more difficult problems.) Using five line segments of equal length, each divided into different parts (halves, thirds, fourths, sixths, and eights) Robert was supposed to compare fractions.  I though the problem was self-explanatory. Just decide which line is longer or if they have the same length.  It was so much simpler endeavor than comparing fractions by replacing them with equivalent fractions with the same denominators.  That Robert could do.  He might  need one example as a reminder, but then he can follow with the same algorithm and find the correct answers.

Using the number lines, however, seemed so much simpler and quicker for ME, so I was led  to the conclusion that it would be also easier and faster for Robert. It wasn’t.

Five parallel number lines completely confused him.  I was not able to detect what exactly was a reason for his errors and I was not able to help him get on track.  I was telling him, that the longer segments represent larger fractions, that the same length segments represent equivalent fractions. that didn’t help.

Moreover, I knew that without those five number lines, Robert would easily point to the pairs of equivalent fractions.  Still, he couldn’t do that WITH the visual HELP of number lines.  I am still not sure why, but I might consider these reasons:

1. Robert was looking not at the length of segments assigned to the whole fraction (for instance 3/4) but at the length of one part – that is 1/4.  And thus decided than 2/3 was larger than 3/4 because 1/3 segment was longer than 1/4/

If that was the case, I should have asked him to measure the length of the segment starting from 0.

2. Placing the equivalent fractions on top of each other might have suggested to him that those which were on top were larger than those on the bottom. Again, I should have asked Robert to measure the segments starting from 0.

I also could ask him to draw vertical lines through some of the fractions.

I am not sure what was the reason for Robert’s confusion.  I am sure, however, that I  should refrain from deciding for Robert what is easy and what is not as he processes signs differently than I do.


Short Confusion and Quick Recovery

February 26, 2015

This evening, when Robert was changing improper fractions into mixed fractions, he was suddenly startled by the fraction 49/9.  So instead of dividing in his head, he rewrote the problem in a long division format. He still didn’t come with the answer. He hesitated and then he wrote on the side of the paper: “9, 18, 27, 36, 45”  when he got to 45, he already knew the answer: 5 and 4/9

I don’t know why Robert was confused by this particular problem. He was changing many other improper fractions.  Many operations he performed in his head (  23/7, 5/4) and many he completed with the help of  long divisions (82/3, 47/2).  I don’t know either why he forgot how much 49 divided by 9 was. It just happened and there is no point in delving on the causes of his confusion.

The important thing is that Robert used an appropriate tool to help himself recover from a short confusion. He did that without any prompting coming from me.  Writing multiples of a divisor was a tool I introduced to Robert years ago when I was teaching him to divide with reminder. Over the years, I had an opportunity to suggest to Robert to do just that whenever he had difficulties with division.  Today, however, Robert used this tool of the mind (To use Vygotsky’s very appropriate terminology) all by himself.

He clearly appropriated this tool.


Still Counting Coins. Why?

January 12, 2015

Yes, I was against teaching counting coins as so-called “functional” skill.  Besides using a vending machine, it is hard to find another place where ATM or credit card wouldn’t be easier and more practical.  And yet, Robert spent a few hours over the period of two weeks doing exercises from Kumon workbook, “Dollars and Cents”. Why?

The exercises had value for Robert and for me.  For Robert because he was “learning” to pay attention while counting.  I was learning what were the obstacles to Robert’s calculations and how to remove them.

Robert never made an error when all the coins were of the same value. All quarters or all nickles.  He made errors mainly when the quarters were followed by nickles often assigning to  the first nickel the value of the quarter.

I couldn’t figure out what caused Robert to write 57 instead of 75.  Was that an error in counting or just in writing.

Another error Robert kept making was to write the sum of $1 and 10 dimes as $1.100 despite the fact that he could write 100c as $1.

I also noticed that my effort to slow Robert down by writing the values of coins under them before counting was leading to more confusion.  Robert began by writing the value of the singular coins, but soon he switched to writing the added value of coins, as if he was counting them together. For instance he wrote 10, 10 under two dimes, but then under pennies that followed the dimes he wrote 21, 22.

The irony is, that he was making between 0 and 2 errors on the page of 10 problems, but after I “helped” him with my suggestion to write the values of each coin, he kept making 8 errors on the page.

What is shows, that it is hard to help when the nature of student’s thinking is not understood by the teacher.  We install doubts instead of providing tools leading to independence.