Thursday, September 20, 2012

A Turtle's Eye View of the World: A Metaphor for Autism

Last fall, Pamela and I made a new friend who loves turtles. The other day, Pat found an eastern box turtle (Terrapene carolina carolina) stranded in the road, closed up tight in her shell, too frightened to head out of traffic. She took the turtle home to make sure all was well. We paid our first visit on the day our friend found it (September 7). We only observed the shell for the turtle was not hungry enough to risk peeking out of her refuge. We took pictures, answered some of the questions from the Comstock book (our stock in trade for nature study), and later made an entry in our nature notebooks at home.

Seeing the turtle shut up tight in its shell reminded me of what happens to children with autism. Something in the environment (a sound, a small change, too much information, a fast pace) makes them feeling like a turtle trying to cross the road with cars racing by. If they cannot retreat into a shell (quiet place, repeated words, rocking), they meltdown and tantrum or they completely zone out. If we try to make the world too predictable and safe, as my friend Di described in her recent blogpost, then the whole family ends up boxed up into a very small world.

Many people wonder how to help children in the spectrum find the world less frightening. Our second visit (September 14) to the turtle illustrates how to make this happen. After a week of living in an alien world, she—her brown eyes, flat bottom shell, and high upper shell are clues to her gender—learned to come out of her shell. Because she could see us through the clear, blue bin, Pamela and I created a little bit of anxiety. She slowly crawled into one corner of the bin and turned her tail to us. She tolerated us as long as we sat very still. If we moved too much, the turtle would move away and, at one point, she retreated into her shell.

Di posted a graphic with guidelines for encouraging meaningful interaction. We applied these ideas in our observation of the turtle: stay calm, be aware of sensory issues, stop the action, and speak less. When Pamela feels anxious, I force myself to remain calm and quiet, even if a storm is blowing and the power has been out for awhile. When she seems inexplicably upset, I pay attention to the environment to search for a cause, so we can deal with it. When she blusters because I am not giving into an unrealistic demand of hers, I stop the action. Any action I take causes her to increase her intensity. Doing nothing helps her realize the ineffectiveness of storming. When she is ready to process so we can figure out a realistic option, I match my verbal communication to hers so that she feels like an equal partner. I communicate with my face, hands, and body because she can understand and express herself in like manner (Relationship Development Intervention helped us achieve that miracle). I speak with declarative language (rather than issuing commands) because she needs to think for herself.

Slowly, the turtle emerged. She was hungry for she hadn't eaten the day before (typical, for turtles). Pat dug up four juicy earthworms and plopped dinner into the bin. The first worm wiggled fiercely but the turtle ignored it. We sat quietly and waited and waited and waited and waited. Ants crawled across the pavement. And we waited. Hummingbirds buzzed overhead. And we waited. Pat dropped a juicy worm right in front of the turtle, who sat on the wiggly thing. Mosquitos sucked our blood. And we waited. During this downtime, we reminded Pamela to sit still and avoid swatting the bugs. We reminded ourselves to sit still. The turtle took forever to find her courage.

Finally, we noticed her eyes tracking a worm. She slowly shifted her body in the right direction. A spider walked near Pamela's hand. And we waited. The turtle's head stretched out slowly. Then, very suddenly, very snakelike, she struck! She snapped open her jaws, picked up the worm, and gummed it to death for turtles lack teeth. She was a messy eater, leaving pieces behind. One bit of worm hung out of her mouth like a cigar. She was full after two worms.

People with autism process at slower speeds. While many can react to prompts quickly, you have to give them plenty of time to observe, process, decide, and respond. My friend Di had to wait 45 seconds for her son to respond when she first changed her communication style to foster interactions. His processing speed has increased dramatically now that they have been working on this for awhile! Pamela needs about 10 seconds to think through simple tasks. If language is involved, it takes much longer. When you give our kids the time they need to process and think, they develop the habit of processing and thinking and learn to do it more efficiently.

Pat decided to release her turtle into the woods near her home. Pamela watches and wishes her well.

Saturday, September 15, 2012

Helping Others Find Joy in Math

Last spring, a friend of mine from church came to me a few weeks before finals for help in high school mathematics. Jay is a bright student who has earned passing grades in math by memorizing formulas without understanding and cramming for tests. I usually prefer to peel back the layers of knowledge to what a student understands and build up from there. With so little time, I had to focus on major habits and enough understanding to pass. I prioritized teaching what would give her the most bang for the buck in the little time she had before finals. She passed but we both knew she could have done better.

Even though she is not taking precalculus until next semester, Jay came to me for help this week. She has set a goal of earning a 90 or above in all her classes in her last semester of high school. We both know she has gaps in understanding, relies too much on the calculator, and mixes up formulas. Rather than revisiting difficult topics while under the pressure of mastering new material, she is going to build a solid base now when she can relax and learn.

I asked Jay what concept she would like to study first. "Negative numbers! I know what to do when you multiply and divide. I get confused about what happens when you add a negative and a positive. I never know what the sign should be."

I drew a number line, and we did some simple problems. I said, "Explain to me what happens when you add two positive numbers like +5 and +3." I illustrated the problem on the number line.

She told me, "When you add two positives, you head in a positive direction. They stay positive."

We did the same thing for adding two negative numbers like -5 and -3. Jay gave a similar explanation of heading in a negative direction and staying positive.

Then, we worked on the piece of the puzzle that had long mystified Jay. I plotted adding +5 and -3. Jay looked at it for a moment and said, "Oh, I see. The negative number is not large enough to cross zero. So, the answer is positive."

We then studied what would yield a negative answer: adding +3 and -5. Jay smiled, "I get it! Since 5 is larger than 3, it is going to cross zero and the answer is negative! Wow!!"

We continued pursuing this line thought by adding -5 and +3 and then -3 and +5. I gave her a couple of problems to make sure she could apply what she understood.

I like to give students other ways to understand a problem. Jacob's Algebra offered an alternative view of adding positive and negative numbers. The book depicts a number as a set of circles. A positive number has circles with no filling (white), and a negative number has filled-in circles. The picture below shows two numbers: +6 is the first row of circles and -6 is the second row. I drew a picture like this for Jay, and she saw immediately how adding them together results in an answer of 0.

I drew another picture with -2 in the first row of circles and +2 in the second row. Again, she saw immediately how adding them together results in an answer of 0.

The picture below illustrates -5 plus +16. Jay had no trouble explaining that the answer had to be positive since the number of positive circles is greater than the number of negative circles. Since there are 11 circles left, the answer must be 11.

The picture below illustrates +3 plus -8. Jay easily explained that the answer had to be negative since the number of negative circles is greater than the number of positive circles. Since there are 5 circles left, the answer must be -5.

I assigned several addition problems for her to solve. Jay got them all correct. Even better, she smiled the whole time because she didn't have to guess the sign of the resulting sum. Then I picked a more difficult problem from the book:

1 + -3 + 5 + -7 + 9 + -11

She drew a picture like the one below and said the answer had to be -6.

Since Jay needs practice with mental math, I processed the problem aloud in a different way for her, "Do you see that 1 plus 5 plus 9 is 6 plus 9, and that is 15?" Then, I wrote down the number 15. I said, "And, -3 plus -7 plus -11 is -10 plus -11 and that is 21." I wrote "+ -21 = -6" to finish the solution. To practice both ways of adding positive and negative numbers, I assigned the next problem in the book.

-1 + 3 + -5 + 7 + -9 + 11

She drew the following picture and told me the answer was 6. Then, she did some mental math and wrote down the equation, "21 + -15 = 6."

Then, she paused and said, "That's funny. You get the same answer only the signs are reversed. Oh, wait! All the signs are opposite. Had I thought about it, I wouldn't have had to do all that work. That's cool." She warmed my heart when she said, "I want to keep this paper! It finally makes sense!"

Before closing, I will summarize some of the important ideas that help people who may not care for math to experience joy.
  • Figure out what the student knows. I usually do this by asking them to explain things to me. When they get, I ask a question about a more foundational concept until I find the student on solid ground.
  • Take small steps of logic by demonstrating the process with objects or in pictures. Too many students think in memorized procedures and formulas but cannot explain the rationale behind them. Once they understand why, procedures and formulas become effortless.
  • Ask the student to explain the rationale behind what they are doing. If they cannot explain it, then they lack a solid foundation.
  • Show how to solve a problem in different ways. It helps students see how math can be a creative process.
  • If a student struggles with mental math, assign problems with simple numbers and avoid using the calculator. I find that students who rely on calculators for simple problems usually lack number sense.

Friday, September 14, 2012

Breathing the Rarified Air of a Mountainous Land

David is a student at Charleston Southern University, taking 16 credits in your average freshman schedule. The main reason why he dipped back into math last spring was that his ACT math scores were less than ideal. I will refrain from sharing the actual number, but the folks at CSU felt he needed remedial math. I agreed. I suggested he retake the ACT to boost his math score and avoid a class in which he would study for long hours but not receive any credit. He pointed out that his college of choice had already accepted him. I did some digging and let him know that CSU offered a math placement exam. If he passed the test, he could take a regular math class. He saw wisdom in that idea and spent the spring and summer diligently studying algebra.

Why would the son of an engineer (my husband has three engineering degrees) and a mathematician (I have two math degrees) fare so poorly in math? For a long time, I have suspected that, while David has the intelligence to do well in STEM subjects, he prefers more creative outlets. He tackled math by memorizing and cramming, even though we had always encouraged him to think. Since he always seemed to struggle with math, I could never prove my hypothesis.

Long story short, David studied Jacob's Algebra from cover to cover. He was surprised to find the book, both interesting and readable. David told me that Jacobs teaches you how to look at problems logically and to understand algebra. For years, he focused on cramming formulas, which he mixed-up on comprehensive tests. Rather than feeling anxious when he couldn't think of a formula, he learned to relax, write down ideas, and reason his way through problems. He also realized that asking for help is not a sign of weakness and reached out rather than stare at a page until his brain froze.

I think Jacob's Algebra is a fabulous book for many reasons. The author has not updated the book since the late 1970s. Unlike textbooks of today, it does not have ADHD, distracting students with irrelevant graphics, sidebars, etc. When I peeked at David's college math text, the glossy pictures of movie scenes in which mathematics played a role nauseated me. The money wasted on pop culture would be better spent showing students how to reason. I personally think all the eye candy in the world will not attract students to this much maligned subject. Force-feeding formulas to children without understanding and encouraging them to cram to get ready for standardized tests make them loathe math. Jacobs carefully unfolds algebraic thinking through brain teasers and puzzles before getting abstract. Learning to reason and apply logic changed David's attitude toward math.

He also struggled with fluency and accuracy. The month before taking the placement test, I wrote algebra quizzes based upon ACT problems. I had him run through problems at Khan Academy to practice fluency and accuracy. Because he understood more, he had to memorize less. He could focus his attention on practicing and improving these habits.

The end result is that he earned an 81 on the placement test and is taking freshman math. But, even more important, he enjoys mathematics now. When asked what about his favorite classes, he put English Composition right at the top followed by math: "Actually for some reason I'm preferring the math problems over reading... Pretty odd, huh?" He dislikes reading because he is on a steady diet of typical textbooks (yawn). He even added, "I might take some extra math classes as soon as I get all my required math courses out of the way."

This fall, I have another opportunity to win another student over to mathematics. I will share the beginning of her story in my next post.
Another realm open to Intellect has an uninviting name, and travelling therein is difficult, what with steep faces of rock to climb and deep ravines to cross. The Principality of Mathematics is a mountainous land, but the air is very fine and health-giving, though some people find it too rare for their breathing. It differs from most mountainous countries in this, that you cannot lose your way, and that every step taken is on firm ground. People who seek their work or play in this principality find themselves braced by effort and satisfied with truth. ~ Charlotte Mason

Wednesday, September 05, 2012

Plugging the Gettysburg Address into my GPS

By the bivouac's fitful flame,
A procession winding around me,
Solemn and sweet and slow;—but first I note,
The tents of the sleeping army,
The fields' and woods'dim outline,
The darkness, lit spots of kindled fire—the silence;
Like a phantom far or near an occasional figure moving;
The shrubs and trees,
(As I left my eyes they seem to be stealthily watching me;)
While wind in procession thoughts,
O tender and wond'rous thoughts,
Of life and death—of home and the past and loved,
and of those that are far away;
A solemn and slow procession there as I sit on the ground,
By the bivouac's fitful flame.
~ Walt Whitman
I just counted the number of miles I drove between July 30 and August 30 of 2012 (not including two deliveries of meals on wheels and the times I took a wrong turn): 4,700 miles. That is the equivalent of driving over 150 miles a day! Whenever we travel, we always try to find interesting places to visit. Our recent trip to Pennsylvania, we drove through Gettysburg, so we had to stop there on the way home. As is typical in a Charlotte Mason paradigm, we had just finished reading about Gettysburg and Lincoln's address the week before we headed into "Yankee land" as our parakeet sitter called it. Our stop in Gettysburg ties into a visit to Vicksburg, Mississippi five years ago and Fort Sumter (a place we will revisit when the weather is more humane) four years ago.

Who can resist a headshot with Honest Abe?

But, I digress.

As we walked up to the museum, we were greeted by a chipmunk! Can you see it sitting on the rocks in the picture below?

We are in the middle of several books and things to heighten Pamela's understanding of the Civil War. Our poet Walt Whitman helps set the mood of many battles. Last month, we listened to a slower-tempo version (thank you, Audacity) of the Pa's Fiddle version of When Johnny Comes Marching Home Again, which was published in 1863 (we used lyrics from The Laura Ingalls Wilder Songbook). We have nearly finished a book with six stories about slavery and the Reconstruction era, and you can see Pamela trying on chains in the picture on the left. This visit is perfectly timed with what we have been studying for two years.

We are reading the closing chapters of for South Carolina history, Yankee Girl at Fort Sumter, and we bought its companion at the museum in Gettysburg (Yankee Girl at Gettysburg). Here are two pictures I took to tie into the Civil War and South Carolina. The flag is especially meaningful because the heroine of the book got kicked out of school for refusing to salute the Palmetto flag.

A couple of things struck me about the visit. The cyclorama astounded us. What is a cyclorama, you ask? In the late 19th century, this popular entertainment venue featured ginormous oil paintings, created in the round. Imagine standing inside an empty circular swimming pool with the walls depicting a famous historical scene. Most cycloramas portrayed dramatic religious, historical, and literary events. The advent of motion pictures caused interest to wane, and most cycloramas were lost or destroyed.

The Gettysburg cyclorama survived. French artist Paul Philippoteaux depicted the final Confederte assault on July 3, 1863. He came to the city in 1882 and spent the next year exploring the area, sketching, hiring a panoramic photographer, and interviewing veterans of the battle. The painting debuted in Chicago in 1883, made even more realistic with a three-dimensional foreground of dirt littered with battlefield debris, stone walls, shattered trees, and broken fences. Bringing it into the 21st century, the National Park Service has added sound effects, smoke, flashing lights, and a recorded narrative to the experience. The skyline above the painting allows the lighting to change as the day begins and progresses.

Pamela and I followed fellow visitors from all over the world into a theater that played a twenty-minute film narrated by Morgan Freeman. The guides herded us through winding passages and up ramps and stairs to a platform inside the cyclorama. The lights dimmed, and we watched a gorgeous sunset in the east. Then, confusion erupted. Unlike soldiers and citizens trapped in the battle, we heard a narrative, explaining how the skirmishes unfolded. This experience was so impressive, that I plan to stop in Atlanta's cyclorama on our next trip to Kansas.

The biggest idea I learned during our visit is how the battle was set in the middle of the town of Gettysburg. Battles were fought in the middle of farmland, where there were fences, barns, and homes. Since many roads converge and meet in the city of Gettysburg, the location of this battle makes sense. The town provides cover in its natural ridges and hills. The names of various skirmishes reveal how interwoven the battle was into the town: the peach orchard, wheatfield, cemetery hill (a graveyard), seminary ridge (a Lutheran seminary was located there), etc. The fact struck me when we first drove into the park and noticed homes within the boundaries of the park, which this map illustrates well.

After the two armies moved on, two thousand citizens crawled out of cellars and hurried back from nearby towns. Their community was wrecked: damaged property, looted homes, destroyed crops, and stolen food. Even worse, they found tens of thousands suffering in homes, barns, and public building. Dead bodies littered the ground. Animals feasted on some dug up from shallow graves. The stench sickened everyone.

The people of Gettysburg quickly realized they were standing on holy ground, consecrated with soldier's blood. A month after the battle, attorney David McConaughy's purchased the heights of Cemetery Hill for reinterring the battlefield casualties. They dedicated Soldiers' National Cemetery in November 1863, which is when Lincoln delivered his famous lines, "Four score and seven years ago...."

At the museum bookstore, Pamela picked up two books and I paid for a truly living book on the Civil War recommended to me by one of my study group friends: The Civil War: A Narrative by Shelby Foote. I have already found myself delaying housework, laundry, and blogging reading the opening chapter narrating the life of Jefferson Davis and that of Abraham Lincoln. The author's take on literature and history are spot on to what Mason educators believe,
The point I would make is that the novelist and the historian are seeking the same thing: the truth — not a different truth: the same truth — only they reach it, or try to reach it, by different routes. Whether the event took place in a world now gone to dust, preserved by documents and evaluated by scholarship, or in the imagination, preserved by memory and distilled by the creative process, they both want to tell us how it was: to re-create it, by their separate methods, and make it live again in the world around them.
Pamela made her Book of Centuries entry and I love how she shows a ball flying out of the cannon.

Pamela made her Book of Centuries entry, and I love how she shows a ball flying out of the cannon.

Even though some of my Facebook friends believe Pamela was consoling Lincoln, I think she was exploring how bronze hair feels.