Monday, May 27, 2013

The Pitfalls of Reason

Two weeks ago, I shared a situation that had bogged me down. Focus on self had blinded me from alternatives. Amazing things began to unfold once I exercised "the way of will" and focused on God "instead" of the morass. Once God opened my eyes to see beyond my nose, new options arose. The Holy Spirit softened hearts and led us to a resting place from which we began afresh. In a clash of unbending wills, I see how the way of will and the way of the reason each played a part. In this situation, the persons involved aimed for a noble outcome (helping others). Selfish motives, rash thinking, and haste kept them from consulting more experienced guides. Once the will was set, the reason kicked in and justified deceit, betrayed trust, and led them into broken relationships. These consequences could have been avoided had the will chosen to seek wiser counsel than their own.

The reason operates on two fronts: (a) the logical playing fields of mathematical truth and (b) the real world of accepting or rejecting competing ideas. Some frivolous ideas are not even worth our time (dog versus cat, over or under orientation of toilet paper, back or middle squeezer of toothpaste, etc.)—or are they? Trying to apply mathematical reason to a sticky situation is as ridiculous as Sheldon's flow chart for a friendship algorithm on Big Bang Theory. The social arena fails to play by the rules of logic. Reason also overlooks sticky consequences to justify what the will desires. Charlotte Mason gives several poster children for reason run amuck: Macbeth, Cain, Richard III, etc. "Well-reasoned arguments are brought into play for a wrong course as for a right.... It is only when [the will] chooses to think about some course or plan, as Eve standing before the apples, that reason comes into play."

You might think that scientists are the most reasonable people in the world for their livelihood depends upon logic. One of the discoverers of the double helix structure of DNA thinks otherwise. He explains in his personal account of the journey to a Nobel peace prize, "Science seldom proceeds in the straightforward logical manner imagined by outsiders. Instead, its steps forward (and sometimes backward) are often very human events in which personalities and cultural traditions play major roles." Parents in the autism world will appreciate his description of skeptics, "Many were cantankerous fools who unfailingly backed the wrong horses.... In contrast to the popular conception supported by newspapers and mothers of scientists, a goodly number of scientists are not only narrow-minded and dull, but also just stupid."

Dr. Watson is quite candid about his own moral failings. Laziness and carelessness in the lab in the form of an explosion drove away any interest in biochemistry. He headed to Copenhagen on someone's dime to spend a year doing research with a renowned biochemist and return home with enough knowledge to map out the structure of viruses. Indifference to the biochemist's work and poor communication between the two drove Watson to work with a virologist. Clearly violating the terms of his fellowship made him a bit uneasy at first. Yet, three months into his stay, reason told Watson to fudge plans for the following year. His purpose shifted from working with the biochemist to working in the stimulating environment of Copenhagen. Reason helped him feel justified about his shiftiness because his boss trained another biochemist back in the states.

At first, Watson felt guilty about avoiding the biochemist who must have some concerns about his frequent absences. His conscience cleared up when he learned of the biochemist's divorce. Reason reassured him that the poor man would not be able to focus on science anyway. Watson was doing him a favor by not asking to be taught nucleic-acid biochemistry. Deceiving the fellowship electors was morally superior to forcing the divorced man to talk about biochemistry. In fact, reason congratulated him on doing great work: he had collected enough experimental data on bacterial-viruses to take a break for the rest of the year. Reason assured him that he could publish a paper and still look productive. He left the bitter cold of Copenhagen and spent April and May in sunny Naples. The poor biochemist was healing his broken heart doing work at a zoological station in Naples anyway. To ease a twinge of guilt, reason suggested that Watson could still look busy doing work on the embryos of marine animals and quietly study genetics on the side. His request for permission to join the biochemist in nature was rewarded with a cheery letter and a two-hundred-dollar check to cover travel expenses. Watson concluded, "It made me feel slightly dishonest as I set off for the sun."

Watson's story illustrates the relationship between will (the ruler) and reason (the servant). The scientist had no desire to learn biochemistry. His will chose to continue studying virology instead. Then, reason stepped in and justified why he should continue deceiving the fellowship electors. In retrospect, as Watson wrote his account, reason cooed that the trip to Naples led him to a researcher who did fire his enthusiasm for biochemistry and eventually lead his prize-winning research. His lies led to a noble outcome (pun intended).

Moreover, our sanitized version of scientific research leads us to believe that reason and objectivity manage to push out ambition, pride, deceit, etc. Reading a subjective and literary account of how research really happens reveals the truth of the matter. Even the most logical, reason-loving people on the planet make morally flawed decisions. This is why I chose a literary approach to science.

There is no single point upon which two persons may reason,––food, dress, games, education, politics, religion,––but the two may take opposite sides, and each will bring forward infallible proofs which must convince the other were it not that he too is already convinced by stronger proofs to strengthen his own argument. Every character in history or fiction supports this thesis; and probably we cannot give a better training in right reasoning than by letting children work out the arguments in favour of this or that conclusion. ~ Charlotte Mason

Monday, May 20, 2013

The Pointlessness of Precalculus

I know! Algebra is not used every day. People mock its pointlessness all the time. This showed up on Facebook the other day. You can spend your life escaping algebra! Woo hoo!

What we learned in history, science, and poetry is not useful either. When do you use a historical date? Recite Poe? Balance chemical equations? Calculate acceleration? But, I digress.

I was tutoring a senior (Jay) in precalculus while a junior in the same class (Kay) watched. The following conversation ensued.

Jay: "Mrs. Tammy, when am I ever going to use this?"

Me: "I used this sort of thing to learn high level statistics, and my husband did to learn engineering."

Jay: "I'm not going to major in any of that. This is useless!"

Kay: "My mom teaches math. She never uses this stuff."

So, I admitted what everyone knows. We'll never need to prove trigonometric equations in real life. I shared what I love about these problems. It's like the Gordian's knot. Tug at the right spot, and the whole thing unravels quickly. The same thing that drives Sherlock Holmes to solve mysteries drives me to prove equations. A puzzle. A mystery. A quest. How exciting when you find the right key (or two) and the knots disappear!

Jay had to prove the sides of the equation to be equal even though they look different. Many manipulations shows they are the same. This looks much scarier than it really is!

I ask students what to do first. If I hear a confident "cross multiply," we continue. If I hear a faltering "Cross multiply?" doubt arises. Some students speak jargon that confuses them. So, they launch a buzz word, hoping it's the right one. Then, I ask why cross multiplication. If the explanation makes sense, we move ahead. Otherwise, we linger.

I ask what they can do to make the equation easier. "I don't like fractions!" [That is why we cross multiply.} Then, I ask how to dump the fraction. Most know to multiply the fraction by its denominator (the bottom). I probe further to see if they know why you multiply both sides by the denominator.

The reason so many students falter in math is that they do not understand the why behind what they are doing!

Imagine the left side of the equation sits on the left side of a balance, and the right side on the other. Multiplying the left by a number greater than 1 increases its weight. Doing the same thing to each side keeps the balance. This applet shows this for simple equations! Some pre-algebra materials sow the idea of seeing equations on a physical scale and viewing them as puzzles.

The joy of math is found in unraveling complex mysteries.

Multiplying both sides of the equation by sec θ - 1 scares some students. Working with fractions, however, is scarier! We treat this ugly expression the same as any simple number. Including parenthesis avoids confusion.

We multiply by sec θ - 1 to create sec θ - 1 over sec θ - 1.

Why? Well, what is sec θ - 1 divided by sec θ - 1? Relax! Take deep breaths. What is 2 ÷ 2? One! What is 300 ÷ 300? One! What is a million ÷ a million? One! What is cheese ball ÷ cheese ball? One! So, what is (sec θ - 1) ÷ (sec θ - 1)?


See that's not so hard even if the fraction looked ugly. Now, we have rid ourselves of fractions and all is well.

Some eye the right side of the equation nervously. Why would anyone want to times one number by the difference between two other numbers? Suppose you find your favorite microwave lunch on sale for $2.94 and you want to buy one for each day of the week. You have $20.50 cash on hand. Do you have enough money? While Pamela might be able to multiply 2.94 by 7 in her head, lesser minds like mine can alter the problem slightly and solve mentally, too. The product of 2.94 and 7 is really seven rows of 2.94 as illustrated below.

If we alter 2.94 a tad, we have something easier to manage mentally. What? How can 3.00 - 0.06 be easier than 2.94?

Mulitiplying 3.00 - 0.06 times 7 looks like this.

We can multiply in our head! Yes, 7 times 3 dollars is 21 dollars and 7 times 6 cents is 42 cents. Taking 42 cents away from 21 dollars yields $20.58. Alas, you can only buy six meals.

On the left is another way to illustrate the distributive property. This little rabbit trail has a point for those who do not like the look of cot Θ (sec Θ - 1). You can try to multiply vertically if horizontally worries you.

Sadly, the equation is still ugly. Do you see the next step? Take one step, and the equation looks prettier. Try to focus on what you can remove completely. Think about the analogy of the balance and what you can take off both sides.

Each side of the equation has - cot Θ. Are you stuck? Pretend it's something less scary like - 2. What can you do to - 2 to turn it into zero? Add two! What is - 2 + 2? Zero! What is - 300 + 300? Zero! What is negative million + positive million? Zero! What is negative cheese ball + positive cheese ball? Zero! So, what is negative cot θ + positive cot θ?

ZERO, zip, nada!

The new form of the equation looks almost friendly compared to the original rubbish. At this point in the game (to me, it is a game), I convert everything on the right side to sin θ or cos θ and let the chips fall where they may. These little identities are things that you memorize with use and can always look up if you aren't suffering through a test: csc θ = 1/sin θ and cot θ = cos θ/sin θ and sec θ = 1/cos θ. I simply plug and chug on the right side from here on out!

Thursday, May 16, 2013

Patience Rewarded

Something funny happened on the way to blogging math. Pamela surprised me again, and several perfectly timed articles got me to thinking.

Right now, our readings address the end of the 19th century and the beginning of the 20th. I chose In My Father's House, the autobiography of Corrie ten Boom, as a bridge from the period to World War II. Before leaving for Kansas last week, we read about her encounter with a mentally-ill man named Thys. As a child, Corrie prayed for the alcoholics and homeless people living on the streets near her home. She and her sister Nollie came across a crowd of children picking on Thys. The passage we read began,
I was so full of pity for poor Thys and angry at the cruel children that I shouted, "You leave him alone, do you hear!"

The children stopped at my bold challenge. They looked for his defender and saw a little girl, less than half his size. Suddenly he walked toward me and stooped down. I could smell the unpleasant odor of his unwashed clothes and matted beard. He put his hand under my chin and kissed me on both cheeks.
Corrie went on to say that her sister Nollie, appalled at the kiss, whisked Corrie home. After hearing about what happened, their aunt scrubbed her face. At bedtime, their mama noted that Jesus was the source of her pity and kindness and that praying for street people might be a safer thing for little girls to do.

Yesterday—about a week later—Pamela read the next section. First, she narrated our last reading. Her retelling of the heated emotion of the scene came out in a burst of loud, angry words. "Corrie sees bullies. ANGRY! Leave him alone! Go away! Went to bed. Mom said obey. That's citizenship." Many things struck me about her account in that Pamela:
  • Clearly felt Corrie's passion.
  • Accurately labeled the children as bullies, a word not used in the passage.
  • Recognized what Corrie did as a good deed even though her mother was worried.
  • Picked a perfect word to describe Corrie's behavior: citizenship.
We do not work on vocabulary directly. Children glean word meaning from context. They do not need to copy the definition and use the word in three sentences. Is that how you taught babies new words? Is that how you beef up your vocabulary?

Pamela enjoys looking up words in the red discovery book (dictionary) and encyclopedia for fun. Curious about her knowledge of citizenship, I asked her where she read about it. She said, "In encyclopedia." I asked why she thought Corrie was doing citizenship. She explained, "Citizenship stop the war. Corrie stop the fighting." Then, I began to wonder. She reads my planning spreadsheets where I loosely categorize her school books as subjects. I had this book listed as World History, not Citizenship! Her summation of Corrie's action was truly an original thought!

Many students with autism prefer textbooks. Expectations are predictable. Having to answer the questions of others means more black-and-white thinking. Much of the work required is mindless busywork. Even though Pamela would find textbooks easier, I think going the living book route promotes more flexible thinking and more experience sharing.

Several articles that appeared in my Facebook feed yesterday affirm the elements of our approach to education:

  • Reading Aloud to Older Children Is Valuable - Pamela and I read books in the manner described in the article, "And they read the Bard’s plays together, divvying up the parts, because 'that's how they are meant to be experienced.'" (We are reading aloud Macbeth right now!) Reading aloud to older children offers academic and emotional benefits. It broadens the menu by adding more challenging books. One mom describes how more pleasurable reading becomes for her daughter with dyslexia, "Reading together–with her watching the words as I read, and then her reading to me–is a way to be together, to experience the world, to enjoy a common pleasure." Retired teachers who reconnect with former students online find the most memorable thing they did as a class was reading aloud.

  • Multi-Tasking Equals Failure to Filter - The jury is out on whether inattentive folks are born that way or are the product of chronic media multitasking. When several modes of data stream in, heavy multitaskers cannot ignore the irrelevant. They find organizing and storing information into memory difficult. They do not even switch attention faster. In a nutshell, information overload slows down their ability to process. That sounds a lot like autism!

  • The Mind Is Made for Story - Modern textbooks promote multitasking. My son switched to public school as a junior. I was appalled at his precalculus textbook. The math book looked like it had ADHD: graphics, sidebars, pop-ups, font variability, etc. When looking up information to help David with his questions about math, I found it hard to follow a train of thought. Living books embed the information contained in graphics and sidebars into a narrative account. We can follow the train of thought in living books because our minds respond to stories.

  • Lingering a Living Book in a Term or Year Supports Long-Term Memory - Modern students have to pack a textbook into one semester, and their classes can take up to ninety minutes. Packing in information over a short, intense period of time is called massed studying. Science backs up common thinking that cramming is ineffective. Yet, students today have to learn that way thanks to block scheduling. My high school spread out a math textbook over two semesters in five fifty-minute periods five times a week. Distributed study is far more effective. My husband's high school offered one math class. It weaved algebra, geometry, trigonometry, statistics, etc. over the course of four years. He never took long breaks from algebra in order to learn geometry. He aced his math classes in college!

  • The Real Issue Might Not Be Inattention! - Students today have so many entertainment choices they can avoid boredom. When something loses their attention, they tune into something else. Having no other option forces you to stick with what you are doing. Jennifer Roberts, professor of art and architecture history, assigned her students to write an in-depth research paper based upon a three hour study of any painting in a nearby museum. Continued looking at one thing revealed something new. The longer students the painting, the more they discovered. Patience was rewarded. Living books are another way to encourage sustained attention, and the joy of reading them is the reward.

And, folks, I am not the only person not giving up teaching an adult person in the autism spectrum to communicate! "the neuroplasticity necessary for new language learning that was not supposed to exist in this population, did exist. Many of the students desperately wanted to crack the code of conventional communication, and their brains were capable."


Sunday, May 12, 2013


In case you haven't noticed, I've had a three-week hiatus from blogging. A situation that has been percolating for a long time has boiled over and spent my mental energy. I have several blog posts on the back burner. I kept replaying the situation, imagining new scenarios, and rehearsing my response instead of living. Last Friday, I started reading the text for next week's Charlotte Mason Blog Carnival and suddenly an idea hit me! I needed to apply "the way of the will" to myself.

God has prepared me for this situation through two Sunday school books read in the past year. Counterfeit Gods by Tim Keller has helped me identify idols in my life while Facing Your Giants by Max Lucado helped me realize the importance of focusing on God instead of my giants. He is also sustaining me through it for He keeps pointing me back to Beth Moore's James, Mercy Triumphs, a Bible study we just finished.

Until Friday, I did not apply head knowledge to real life.

God is encouraging me. Two Sundays ago, our pastor, unaware of the issue, addressed the situation perfectly in his preaching. I found myself texting Steve during the sermon. A few days later, John Piper posted a conference message on the very same verses! As I listened to that message, God orchestrated the sun to peek out of the clouds and fill me with delight just as He did five years ago when I was sitting on my back porch reading the book Desiring God. Friends on Facebook have posted my favorite Bible verses. In one case, I had been meditating on a passage falling asleep the night before and the next morning it was in my message feed. God is there when we seek Him instead of trying to fix a problem.

Even on the way to Kansas, when traffic bogged down due to road construction, He let my eyes rest on these words and on this brilliant sunset.

My local friends have built me up instead of tearing me down: one stopped by bearing the gift of a fragrant basil plant and a card that quoted Plutarch! A friend who can find the perfect card is truly constant! I dislike contradicting the wisdom of Plutarch but constant friends really are not all that rare when you have friends in Christ. God is reminding me of His love through my spiritual family at church, homeschool friends, and long-term friends.

On Friday, I chose "I will" (keep my eyes on God) instead of "I want" (having my way in a situation beyond my control). Each time my thoughts drifted off, I studied a Bible verse I hope to memorize. I turned to the book of James. Since then, I have slept better than I have in weeks. I feel more peace.

God is encouraging me to choose instead. He put that word on Pamela's heart—imagine Him giving such a word to a person who has struggled with words most of her life. Yesterday, whenever she did not like a choice we were offering, she deliberately used it in a sentence. "I want to go to the park instead." "I'm doing instead: Wednesday, not Sunday." Whenever she says, "Instead," I am reminded to be more mindful about my choices. By choosing to dwell on God instead, I am dwelling with Him. When I let Him handle my burdens instead of me, He gives me rest.

Jesus, I am resting, resting, in the joy of what Thou art; I am finding out the greatness of Thy loving heart. Thou hast bid me gaze upon Thee, and Thy beauty fills my soul, for by Thy transforming power, Thou hast made me whole.

O, how great Thy loving kindness, vaster, broader than the sea! O, how marvelous Thy goodness, lavished all on me! Yes, I rest in Thee, Beloved, know what wealth of grace is Thine, know Thy certainty of promise, and have made it mine.

Simply trusting Thee, Lord Jesus, I behold Thee as Thou art, and Thy love, so pure, so changeless, satisfies my heart; satisfies its deepest longings, meets, supplies its every need, compasseth me round with blessings: Thine is love indeed!

Ever lift Thy face upon me as I work and wait for Thee; resting ’neath Thy smile, Lord Jesus, Earth’s dark shadows flee. Brightness of my Father’s glory, sunshine of my Father’s face, keep me ever trusting, resting, fill me with Thy grace. ~ Jean Pigott