I wanted to add this commentary to my previous post about teaching Robert to write and draw but after thinking it over, I decided to add this short post instead. It took us, his teachers and me, a lot of time to deal with the problem I described below, so it might warrant a separate post.
When Robert was very young (4-6 years old), he was unable to draw any picture with angles. Squares and triangles were “rounded” in such a way that they resembled deformed circles and not the polygons they were. Robert couldn’t stop at the polygons vertices even for a nano second. In smooth, continuous motion he slid to the next side leaving the curve where the vertex should be.
I thought about a few remedies to address that.
One was to ask Robert to raise his hand after completing each single segment as this movement assured that he stopped. Robert used this approach when he was asked, for instance, to draw a house.
Another one was to suggest to Robert to begin with placing all the vertices on paper and then connecting them. When Robert saw those black end-points, he considered them his cues to stop and start anew with a next side of a polygon.
It took long time and many trails for Robert to master that skill.
Moreover, although he doesn’t need to use it to draw triangles or square, he still uses it to “plan” other drawings.
Lately, for instance, he learned to place five dots in a way that helps him to draw a five sided star. Quite an achievement for him. He also places appropriate number of dots on a circle to draw a hexagon, pentagon, or octagon.
When he was learning cursive writing, he encountered most difficulties while writing lowercase “s” as it required drawing slant segment (drawing aslant line is problem in itself) and then turning it into a curve at the top of the letter. Again, Robert tended to “round” that corner. He still does this, if he is not reminded not to do so.
All posts in category Math
Math related
Rounding Angles
Posted by krymarh on February 24, 2014
https://krymarh.com/2014/02/24/rounding-angles/
Still Learning Together
I have neglected writing on these pages. But I have not neglected learning and teaching Robert.
1. Everyday, Robert is copying pictures of people from a little book I Can Draw People. The pictures are very simple. They are mainly basic shapes connected together with some extensions and additions. For last three days, Robert copied twice the picture of a soccer player, added a gallery of spectators, and started coloring. Maybe, he will finish today. In the past, we spent a lot of time on copying different pictures. Usually, after Robert was done, he took he drawing to a recycle bin. That of course is not a good outcome as it demonstrates how Robert treats his artwork. But then, since he just concentrated on drawing only one object, there was not much to the picture itself. It is different this time around. I hope that today Robert will hang his picture on the refrigerator.
2. Three days ago, we started reading My first Book of Nature, How Living Things Grow by Dweight Kuhn. Every day we spend 5 minutes looking at pictures and reading short paragraphs related to them. There are not many new facts for Robert to learn from this book. But the great pictures might fill the gap in his understanding of some of the words. Moreover, bringing together pages about different living things might result in Robert better appreciating of the richness of the nature that surrounds us all.
3. Every few days, Robert reads two very short texts (one paragraph each) from Power Practice Science grades 3-4. After reading, he answer simple questions either related to the text or requiring additional knowledge. The workbook is rather dry. I am using it instead of a curriculum. I simply don’t know what to teach and this workbook shows me the topics and general direction. I use, however, many books I bought over the years, as a main tool in teaching. For instance, before Robert read a short paragraph Structure of the Earth, he and I looked at two colorful pictures (one from a pop-out book The Earth Pack by Ron van der Meer , and one from a flap book Amazing Earth by Heather Maisner)
4. We continue with Reasoning and Writing level C . On some days, I ask Robert to just talk and on some days to write, what he said. At this time we are concentrating on Robert noticing small differences in what people do, where they are, or what they wear. With the help of those pictures, Robert builds sentences that would first address the difference, and then they would state the main things the characters do. For instance: Robert has to notice that the character X in the first picture (A) has a parrot on his shoulder, and in picture B does not. That should lead to a sentence, “X has a parrot on his shoulder.” Then, Robert notices that in both pictures the character A is opening a treasure chest, the fact that Robert should describe in the next sentence. And so on. It has been a struggle to build sentences that address the differences.
5. And of course, we still work on Saxon Math, level 4, repeating it yet again. All computations come easy, everything else needs prompts. For different problems, different prompts. For finding an average of a few numbers in a math problem, it suffices that I emphasize the word, “average”. For balancing a checkbook knowing interests and service fee, I would have to write on a separate page that interests we add, the service fee we subtract. Of course, Robert doesn’t know that, as I still failed to practice that skill in his real checkbook. For Robert to find the estimate of 5 times 78, I would have to start with drawing a horizontal line. Robert, then, places 70 and 80 on both ends of that line and 75 in its center. He decides that 80 is a better approximation of 78 and without difficulties chooses the right answer, 400. But without me drawing this horizontal line, Robert wouldn’t know what to do. When he has to find the value of a mixed number A presented on the number line with each unit divided into small parts, I would began counting those parts from 0 to 1. Robert continues to find into how many parts the unit was divided and what is the denominator. After that,he doesn’t have problems finding that for instance A = 3 and 2/5. I still don’t know how to make Robert rely on his memory and his own deductions and not on my prompting.
Hopefully, I will learn, and so might he.
Posted by krymarh on January 31, 2014
https://krymarh.com/2014/01/31/still-learning-together/
Unexpected Reward
December 16, 2013
Usually, when Robert returns from school, the worksheets and textbooks are waiting for him. Robert won’t take an evening bath before first completing all assignments I prepared for him. Today, I forgot to assemble papers and books. Robert took it as an indication that there won’t be any evening learning. When I tried to make up for my negligence, he protested repeating softly and quickly, “No, no, no, no.” He watched me opening books and taking out worksheets from a folder full of copies, then convinced by inescapable fact, – the worksheets spread on the table-, he surrendered, picked up his favorite pencil and got ready to study.
1.We read two pages about organisms competing for scarce resources. I am not sure what Robert understood. He encountered the word population in a new context. In the past the word related only to people, now to all organisms of the same kind. He heard and read the word “resources” before, mainly in the context of the conservation through recycling, reusing, and reducing. Today, for the first time, he read about possible scarcity of resources and their grave consequences. As I said, I am not sure what he understood. Below the text there were questions, but I did not use them as a way of checking Robert’s retention of information but as a way to review and reteach. I mostly fed the answers to Robert.
2. Robert practiced talking in sentences. The hardest thing for him. I asked the questions related to pictures and Robert should answer in full sentences. Although he had to apply the same sentence structure, he too often put the words in a wrong order. I asked, “Which animals have stripes (whiskers, bushy tails, pouches…) He responded with one word and then reluctantly followed with a full sentence. I noticed, to my surprise, that as we went on, Robert demonstrated more difficulties, not less in speaking in sentences. I stopped this exercise as soon as he strung words correctly in his answer so he could finish on a high note.
3.A page from Reasoning and Writing level C, provided him with some relief. Robert was supposed to write a paragraph about the picture and he did it without verbal prompting. Still, I kept pointing to different parts of the picture to help Robert build sentences. After he did, he read the whole paragraph and was pretty pleased with himself.
4. Robert floated through lesson 85 from Saxon Math. Among other thing, he properly aligned numbers for addition: $3.26 + $45 +36c changing units, using decimal points, placing decimal points under each other and adding properly.
Then, he smiled. He smiled at himself. He smiled because he felt he mastered the skill, because he wasn’t tricked by the lack of decimal points in cents and dollars and he knew exactly where he should put them, because he was proud, because he could do it all by himself.
His joy of his accomplishment!
And my joy of Robert RECOGNIZING his accomplishment.
Posted by krymarh on December 17, 2013
https://krymarh.com/2013/12/17/unexpected-reward/
The Vertex in the Middle
A year ago, in the post https://krymarh.wordpress.com/2012/04/18/importance-of-little-words/ , I noticed that the word that impeded Robert’s learning of division of fractions was not “multiplication” , not “reciprocal” but “instead”. I used seemingly simple direction, “Instead of dividing, multiply by reciprocal.” I also concluded that this rule was much more important not as an advice on how to divide, but as an example of the meaning of the word “instead”.
A few days ago, I worked with Robert on naming angles using three letters. We, encountered the same problem we had done a few months ago, on our first try. Robert didn’t understand my direction, “Vertex has to be in the middle.” Robert knew “vertex. He could point to it without a problem. He didn’t understand “in the middle” in the context of the three letters (two of them naming points on angle’s arms and one naming the vertex).
It has to be said that part of Robert’s problem was the way I introduced this task to him. Without thinking, I just followed the problem from Saxon Math 4. The angle with a vertex A which Robert was supposed to name using three letters was one of the four angles in the quadrilateral. All the letters A, B, C, and D named vertices. The way I tried to help Robert, confused him even more.
I put the textbook aside, and drew many angles not attached to any other shapes. Now, there were three points but only one vertex. Still, it took two days of practice, before the direction, “vertex in the middle”, resulted in the correct answers. This time the culprit was, “In the middle.”.
The most important gain of those lessons was not that Robert learned how to name angles with three letters, but that he understood the concept of being in the middle in one more context.
Posted by krymarh on October 3, 2013
https://krymarh.com/2013/10/03/the-vertex-in-the-middle/
Identifying Errors, Diagnosing Problems, Designing Intervention
I noticed, while teaching Robert to divide decimals, that he made quite a few mistakes. It was clear that he lost some of the previously acquired skills.
He made most, if not all, errors while dividing large numbers by 7, 8, or 9. I went back to dividing two digit numbers with reminder. I soon found out that Robert had difficulties with these and similar problems: 61:8, 60:8, 62:8 but he was fine with those problems 64:8, 65:8, 67:8.
I noticed a pattern. When the dividend was a digit, two or three below 64, Robert had problems. When dividend was slightly larger than 64, Robert didn’t make mistakes. This pattern repeated itself with other dividends and divisors.
At first, I just wanted to do reteaching using the old approach. Upon hearing direction, “Help yourself”, Robert wrote the multiples of eight (the divisor): 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, placed 61 (60 or 62) between 56 and 64 , found a quotient of 7, and finished dividing without problems. I believe that repeating this strategy many times would at some point lead to an improvement.
Still, I wanted Robert to grasp the idea behind choosing 7 or 8 as a quotient . I designed other worksheets. In the center of the page I wrote in big numbers, 64:8. I drew a line through the center of the page (but not through the division). Above the line I wrote all the divisions with dividends larger than 64 (65, 66, 67, 68, 69), below the line I wrote the ones with dividends smaller than 64 (60, 61, 62, 63).
I made similar pages for different problems.
Why I did that? What was the difference?
Since Robert could find most of the quotients and reminders, I did not want to lose too much time by reteaching using the old method. I believed (this is all domain of beliefs not knowledge yet) that the new approach would help Robert relearn quickly.
I knew, however, that this method could be helpful only if Robert develops better understanding of numbers. On the other hand, Robert’s understanding of numbers could be greatly improved by mastering this approach.
Posted by krymarh on October 2, 2013
https://krymarh.com/2013/10/02/identifying-errors-diagnosing-problems-designing-intervention/
As of September 14, 2013
Robert and I continued doing the exercises that prepare for finding differences between 100 (or 60) and another number in memory. We still begin with writing the subtraction problems, for instance 100-38. Then, Robert rewrites the problem replacing one subtraction with two partial subtractions: 100-30-8. With this visual support, he doesn’t have problems finding the solutions. As we continue, I ask him not to write, but to say aloud what he is supposed to do, “100 minus 30 is 70 minus 8 is 62.” But he rushes through without saying the whole sentence, just the final answer. That would be great if it did not lead to more errors down the line. As long as he is required to say the whole sentence, the errors are rare.
After two pages of such exercises Robert returns to problems involving time and money. I thought that the introductory exercises would help Robert now. This is not the case. The first problem on the page, expressing 3;48 as 12 minutes before 4:00, confuses him. But with every problem, Robert becomes better. Unfortunately, tomorrow, the whole process will be replayed. It will take a couple of weeks, before Robert solve correctly the first task on the page.
Today, we have finished the last chapter of Real Science 2. A relatively easy book for Robert with not many new vocabulary words, but with many topics that apply to real life.
One of the advantages of teaching such topics to your own child is that it allows you to support reading of the text with those points of reference your child already has but only you are aware of their existence. During reading, I can remind Robert those experiences that relate to the text. I can also bring the concepts from the book to add to any non-learning activity and enrich the experience. For instance the meaning of a new word “friction” was literally felt by Robert during our driving through a road under construction. The phrases, “less friction” and ” more friction” were associated with car either gliding on the new surface or bumping during the ride over an uneven pavement.
For the last few days, Robert was practicing pronouns with the help of pages from No Glomour Grammar 1 and 2. It was rather relaxing activity for him. The emphasis was of course on proper usage and understandable pronunciation. The last one, was as always more difficult than the first one.
For Robert, the hardest were exercises in listening comprehension also from No Glomour series. It was easier for Robert to answer four wh questions (who, what, where and when) as they related to two sentence long texts supported by illustrations, then to answer one “WHEN” question as it applied to one sentence with a picture.
That one sentence was longer and more complicated than the two short sentences in the previous texts. From the picture, it is easy to deduce who did what and where, but it is much harder to “see” WHEN something took place. Moreover, the vocabulary describing the time of events is larger and more diverse than vocabulary related to subjects, actions, and places. Not just “Monday, 8:15 PM, in the evening, last year”, but also “before or after something, while doing something else” and so on. The answer to the “when” question can be found at the beginning, at the end, and in the middle of the sentence. Robert was able to seize such words as ” on Monday”, “in the morning” but not “in the middle of the ride” and a few similar. Whenever the phrase related to time (to WHEN) was harder to find, I asked Robert to read the text and find it himself as that was much easier for him than attend to my speech and finding the relevant part of a sentence.
As I watched Robert’s struggles with the questions, it occurred to me that although Robert’s listening comprehension was always delayed (He did not have one receptive word until he was four and a half years old) the gap is getting wider. It has a lot to do with the fact that the people who talk to Robert utter fewer words than they would use with young children on a similar developmental level. In doctors offices and restaurants, the nurses and waitresses ask Robert only one question and when he doesn’t immediately answer, they turn to me waiting for my answer. i don’t answer. I translate which means I pose the same question to Robert. he answers me. Of course, if the nurse or a doctor repeats the question to Robert, chances are he answers them. But they very rarely do that.
Robert is pretty good at following directions. When another person tells Robert what to do, he will comply. When the other person demands a reply, he won’t answer. Robert’s ability to follow directions is a result of a very strong emphasis that his private school and I put on this aspect of communication. I can only speculate, that if similar emphasis was placed on Robert’s intraverbal skills, he would listen more attentively and answer the questions much better.
Posted by krymarh on September 14, 2013
https://krymarh.com/2013/09/14/as-of-september-14-2013/
Counting Coins
Chances are that when you ask the teacher (or administrator) leading the transition classroom what exactly the students are learning in “functional mathematics”,the response will be, “Counting coins.”
I don’t have anything against counting coins. I think that counting by dimes, nickles, quarters, and pennies can teach many important math concepts. Through counting by tens, fives,and twenty fives the students can get a better grasps on decimal system and get an important introduction to fractions (1/4. 1/2, 3/4) and decimals.
Counting coins, however, is not a functional skill. At least not AS functional as its dominance in many transition programs indicates. In today’s world this skill can be utilized only while buying snacks or drinks in vending machines. Since, however, most of those items are unhealthy, the skill of counting coins should not be practiced there. When Robert returns cans and bottles, the machines do all the counting for him.
Even if he were using cash and not his debit card to pay his bills, Robert should not concentrate on counting coins in the change received, but on counting dollars. Counting the change in coins would only distract him from paying attention to bigger bills, and might lead (I don’t believe that this might really happen) to being cheated of much more then a few pennies.
It is possible, that at some point, in this world which day after day becomes less compassionate, Robert will be homeless. He might beg for money. He might look for coins on the pavements of sidewalks all over the city, maybe then the skill would come handy. But as I learned, when Robert considers something very important, he learns quickly, often without ANY instruction from others.
Despite everything I have written above, the skill of counting coins will come handy today, as I plan to open Robert’s piggy bank and place all the coins in the appropriate rolls before taking them to the bank and exchange them for paper bills. That means that Robert will be practicing counting by ones, fives, tens, and 25s. upon returning from the bank, he will do something much more valuable. He will count his dollar bills.
The fact that in so many school programs and programs for adults with developmental disabilities, counting coins is the pillar of “functional academics” is really depressing. It demonstrates the great disconnect between the real needs of the individuals with disabilities and the people who, in one way or another, are responsible for them. It sadly indicated that those people are not learning.
Not learning, because, there are teachers and the professionals invested in special education, who already came to understanding, that there are other math skills which are much more needed. Much more functional.
The first and the most basic is to round the amount of money to the higher dollar amount. This way, when Robert is buying something that costs $3.28 or $3.99, he knows that he should give at least $4.00 (or $5 or $10…)
The collaborative program, my son attended in 2005/2006 school year was teaching just that. That was not,sadly, something any teacher in his public school TRANSITION program has ever done. For the last 3+ years they have been practicing counting coins. Over and over and over.
As I said, I don’t have anything against practicing counting coins from the mathematical point of view. But this is not a functional skill even in those situation when Robert has to use money instead of his debit card.
I realized that clearly when Robert was trying to buy his watermelon flavored frozen lemonade in Roger William Park and Zoo. It cost $3.50. Robert had dollars and a few coins. He did not know what to give to the vendor.
Luckily, I could give him some clues. Robert learned quickly and during the next visit to the zoo he handed seller $4.00. Oh, well, he forgot to wait for the change, but that is another lesson and another story.
Posted by krymarh on September 13, 2013
https://krymarh.com/2013/09/13/counting-coins/
Time and Time Again Help Yourself
During teaching Robert to substitute digital times with equivalent verbal phrases, I noticed that he had difficulties finding the number of minutes missing to the full hour. He did not have difficulties with easy times: 10;55, 10:50, 10:45. But to replace 10:47 with an expression, “13 minutes before 11 o’clock”, he had to first find the difference 60-47. Finding the difference is not the problem for Robert. Problem was that he did not know he had to find it in the first place. Every time he encountered similar time, he was startled as he kept forgetting what to do.
So I kept reminding him, “Find the difference 60-47”. Then I switched to telling Robert, “Find the missing minutes.” Finely, my direction was, “Help yourself”.
I often use the phrase, “Help Yourself” as the last, the least (?) invasive but the most general prompt, hoping that by the time I use it, Robert would establish a strong connection between this phrase and the step he needs to take.
I realized, however, that if Robert was able to calculate the difference 60-47 in his mind then the whole problem would become straightforward. As long as Robert doesn’t immediately see that 47 minutes are 13 minutes away from the full hour, he is distracted and not often sure what to do next.
Before zeroing on mental computation I checked what Robert could and couldn’t do, I noticed some strange results. For instance, Robert didn’t have any problems subtracting one digit number from two digits one: 56-8, 22-5, and so on except finding those differences which seemed the easiest for me: 30-7, 100-5.
I also observed that when Robert wrote the subtraction 60-47 vertically and I didn’t let him write anything else: no regrouping, no crossing, and no “borrowing” but asked him to LOOK at the numbers, IMAGINE what he should do, and, TELL me the answer, he could do that.
But when the subtraction was written horizontally, the same directions did not bring any results.
Yet, the problem with vertical subtraction was that without seeing the numbers Robert was unable to calculate their difference and there was no next step that would lead to solving problems mentally. So I decided to apply the same method I used a year ago with subtracting from 100. (As a preparation for counting the change from a dollar.)
To transfer Robert’s ability from subtracting on paper to mental calculation I followed those steps:
I presented a model, 100-47=100-40-7.
Robert first mentally subtracted 100-40 and wrote the answer, 60, above the minus sign. Then he mentally subtracted 7 from 60 and wrote =53 at the end of the expression.In the following problems he wrote the model himself.
During the next step, Robert still wrote, “100-40-7 but he was not allowed to write 60 above the first difference but he had to keep it in his mind and use it for the second operation.
During the third step, Robert was not allowed to write 100-40-7 but he had to say, “Hundred take away forty is sixty. Sixty take away 7 is fifty-three.”and write =53 at the end of the problem.
Now, I replaced 100 with 60, and Robert practiced finding the differences: 60-47, 60-32, 60-59 with the help of the expressions: 60-40-7, 60-30-2, 60-50-9 either written or said aloud.
Robert easily mastered the first and the second step but we are still working on the third. It might be that Robert’s difficulties with saying long sequences of words affect his thinking performance. I will try to reduce the number of words. Maybe that will help.
Despite the fact that Robert still has some difficulties with mental computation, after a page of subtractions from 60 , we return to the page with digital times. When Robert stumbles, I just tell him, “Help Yourself.” and he does.
Posted by krymarh on September 11, 2013
https://krymarh.com/2013/09/11/time-and-time-again/
Round and Round the Clock
Not once, while teaching Robert, I stumbled upon difficult to explain obstacles in passing information/skill to Robert. Unfortunately, it often takes me (and others) long time to understand the nature of the problem before I could design a method to address it. In one of my previous posts I reported on Robert’s difficulties in memorizing addition facts.
I discovered that he was not able to remember (or pay attention to) three different numbers and two signs in the expression that makes an addition sentence. For instance 3+8=11. He was, however, able to remember (or pay attention to ) an addition sentence where the addenda were the same. For instance 7+7=14. I wrote about this in the post: https://krymarh.wordpress.com/2012/03/31/looking-for-variables/
Alas, I encounter such difficulties almost daily.
I am not always aware why Robert cannot understand what i am explaining him so simply and clearly. Because that is what I believe I do. I am often convinced that I explain or demonstrate a new fact/skill in the simplest possible way. And because I don’t believe I can do it in any simper way, I repeat the same approach, the same drawing, the same words again and again with the same negative results.
Many times I have tried to explain to Robert that if the minute hand on a clock makes a full circle that means that 60 minutes have passed. Somehow there is a disconnection between my words and Robert’s understanding.
When I ask Robert to make a full circle with minute hand starting, for instance, at 3: 45, Robert stops the minute hand on any full hour. It might be 4:00 or 5:00.
To make it more confusing, he knows that one hour after 3:45 is 4:45. He just doesn’t connect that with a full movement around circumference.
My directions are not understood. I explain(?) “You have to end at the same point you started” “Just go around and stop at the same place.” Robert pushes minute hand on a Judy clock well past mark for 45 minutes.
I give up explaining and return to counting elapsed time by subtracting times. As long, the two times are not on both sides of 12:00 Robert is fine.
It happened so many times, that I begun to consider it a problem in itself. I feel that if Robert understands how to move a minute hand one hour from any time on the Judy Clock, he would also understand something else as Robert would gain a new thinking tool.
Today, I continued to work with Robert on time skills. He counted elapsed time by subtracting the time the activity ( flight) begun from the time the activity ended. He knew how to regroup minutes. To subtract 3h 30min-1h 40min, he changed the expression to 2h 90min-1h 40min. He did a few similar operations. That went surprisingly well.
But again,Robert had problems moving minute hand for exactly one hour. My words did not seem to carry any meaning. “From here to here.” “To the same place” “You start here, and you end here.” “If you leave at 15 minutes mark, you have to return to 15 minute mark.” I kept saying it one way, another way, many times, and Robert kept turning the minute hand up to full hour mark.
I am not sure yet, why my words are so confusing for Robert, but they are.
So i try a different approach.
I drew a few clocks on a piece of paper. I asked Robert to draw a circle that began at 45 minutes mark. He ended at the same mark. Now I asked him to make the same movement with a minute hand. Robert passed 45 minute mark just for 5 minutes and stopped. He realized that he went to far. We repeated the sequence. He drew a circle on a paper that started and ended at the same mark and then copied that movement with a minute hand on a Judy clock stopping at 45 minutes mark.
I did not push for more. One success is enough for today. Tomorrow, we will repeat those sequences :drawing full circles on paper clocks, and moving minute hands on Judy Clock. I will ask for that, not because I want Robert to have one more tool to count elapsed time. I will ask Robert to do that, as a way to explain to him what it means when I say, ” Start and end at the same point /place” . It sounds so simple, but, as I learned already, the simplest words are the hardest to explain.
Posted by krymarh on September 1, 2013
https://krymarh.com/2013/09/01/round-and-round-the-clock/
krymarh 