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Today's quote is from Jack Nitzsche and Sony Bono.
My first love was books. I was at an early age, a library groupie. Our city librarian was Mrs. Herman and in today's vernacular, was a rock star. She was the wife of Vern Herman, a legendary educator, a coach of a multitude of sports, a physical education teacher, and a driver's education instructor during the twilight of his career. Mrs. Herman became my literacy advisor many years before Vern guided me through the travails of parallel parking. I visited the library at least twice a day during the summer months of my preteen years. Each time I was greeted with a warm smile and was shepherd through a list of intriguing possibilities. The back of each possible book was secured with a code. That code was a dewey decimal which gave the appearance of the "Good Housekeeping Seal of Approval", assuring that this budding reader would be gratified.
Reading is one of my sanctuaries. When I read, I am transported to exotic locales, struggle with vile scoundrels, peek into the minds of historic figures, and am stunned by plot twists as in Gone Girl by Gillian Flynn, Fight Club by Chuck Palahniuk or The Red Wedding in A Game of Swords by George R. R. Martin. I cherish the unexpected. The surprise twist or ending in a book is like the shock that vibrates through your brain as an unsuspecting snowball strikes the side of your head as you turn the corner of garage and contains that shiver. The shiver that trickles down your spine like the slow melting snowball whose cold stream meanders the rise and fall of the quivering vertebra. This shiver, this prickly sensation, feels like needles and pins.
This blog is the second in my series of Point A to B. My calculus students would be remiss if I didn't identify this particular moment in my progression as C1, somewhere in the interval, [A, B], but closer to A than to B.
In 1972, I was in 8th grade and a student at Immanuel Lutheran School in Plainview, Minnesota. My teacher was Evan Schiller. His enthusiasm as he taught math captured my attention. I was an average mathematics student. My scores in previous grades in the subject oscillated between B's and C's. The concentration of my math experience to this point was with arithmetic and I had been applying those skills for 4 years in my dad's welding shop. My teacher was introducing algebra and generalizing arithmetic concepts with letters. He was moving numbers to a different and more abstract level.
The lesson of the day was fractions and decimals. All rational numbers (fractions) can be written as decimals and vice versa. 0.5 is an example of a terminating decimal because it ends with a repeating 0. All decimals that end this way are called terminating decimals. The fraction equivalent for 0.5 is 1/2. 0.666... is a repeating decimal because it ends with a repeating digit other that 0. Its fractional representation is 2/3. I knew that, but Schiller continued, "What about 0.999...?" He extended the algebra that he just taught us to demonstrate.
This blog is the second in my series of Point A to B. My calculus students would be remiss if I didn't identify this particular moment in my progression as C1, somewhere in the interval, [A, B], but closer to A than to B.
In 1972, I was in 8th grade and a student at Immanuel Lutheran School in Plainview, Minnesota. My teacher was Evan Schiller. His enthusiasm as he taught math captured my attention. I was an average mathematics student. My scores in previous grades in the subject oscillated between B's and C's. The concentration of my math experience to this point was with arithmetic and I had been applying those skills for 4 years in my dad's welding shop. My teacher was introducing algebra and generalizing arithmetic concepts with letters. He was moving numbers to a different and more abstract level.
The lesson of the day was fractions and decimals. All rational numbers (fractions) can be written as decimals and vice versa. 0.5 is an example of a terminating decimal because it ends with a repeating 0. All decimals that end this way are called terminating decimals. The fraction equivalent for 0.5 is 1/2. 0.666... is a repeating decimal because it ends with a repeating digit other that 0. Its fractional representation is 2/3. I knew that, but Schiller continued, "What about 0.999...?" He extended the algebra that he just taught us to demonstrate.
x = 0.999...
10x = 9.999...
10x = 9.999...
- x = -.999...
9x = 9
x = 1
1 = 0.999...
I was flabbergasted. Really, how can this be true? 1 is the same as 0.999... I experienced for the first time the exact same feeling I treasured in reading, the staggering "needles and pins". This meant that 2.999... was equivalent to 3 but more importantly, mathematics contained a mystery, a twist and turn, and I could find the joy of surprise. I could feel "needles and pins".
In 1978, I found out that 89.999996 was not 89.999... and definitely not 90. I was in college and taking a computer programming class entitled FORTRAN. Prior to the final, my percentage was 91. 91% was important. To earn an A, I had to have a 90% or greater. 89% grade was the same as 80%, a B. I did not do well on the final and when I received my grade report, the course notated a grade of a B. I did a little arithmetic and determined that I had a percentage of 89.999996. I mustered some courage and approached my instructor, Dr. Jim. Dr. Jim was amenable when I nervously suggested that I would like to examine the computation of my grade. Dr. Jim pulled out his red grade book and his Hewlett Packard 35 pocket calculator. "Mmm," he pondered. His eyes squinted and an eye brow raised to form a question mark. "Mmm... 89.999996%," he summarized, "What is your question?" I looked into his clear, blue, stoic eyes. "Nothing," I murmured. I knew that 89.999996% is not 90%.
This nightmarish decimal would resurface again in college when a calculus discussion about divergent and convergent series emerged. The instructor was demonstrating how a repeating decimal could be written in its fractional form using a series and the example being used was 0.999.... He wrote it as a sum of a sequence of 9's, each multiplied by 0.1. 0.9 + 0.09 + 0.009 + 0.0009 + ... = 0.999... This series is called a geometric series and using limits its sum could be found by the following computation: 0.9/(1 - 0.1) and yes, that sum is 1!
Finally, in 1998, I was giving same lesson to my class of seventh graders as Schiller gave me 26 years prior when one of my students, Ben Stommes, raised his hand and offered his proof. "Mr. Kruger, it appears quite obvious," Ben confidently pointed out.
In 1978, I found out that 89.999996 was not 89.999... and definitely not 90. I was in college and taking a computer programming class entitled FORTRAN. Prior to the final, my percentage was 91. 91% was important. To earn an A, I had to have a 90% or greater. 89% grade was the same as 80%, a B. I did not do well on the final and when I received my grade report, the course notated a grade of a B. I did a little arithmetic and determined that I had a percentage of 89.999996. I mustered some courage and approached my instructor, Dr. Jim. Dr. Jim was amenable when I nervously suggested that I would like to examine the computation of my grade. Dr. Jim pulled out his red grade book and his Hewlett Packard 35 pocket calculator. "Mmm," he pondered. His eyes squinted and an eye brow raised to form a question mark. "Mmm... 89.999996%," he summarized, "What is your question?" I looked into his clear, blue, stoic eyes. "Nothing," I murmured. I knew that 89.999996% is not 90%.
This nightmarish decimal would resurface again in college when a calculus discussion about divergent and convergent series emerged. The instructor was demonstrating how a repeating decimal could be written in its fractional form using a series and the example being used was 0.999.... He wrote it as a sum of a sequence of 9's, each multiplied by 0.1. 0.9 + 0.09 + 0.009 + 0.0009 + ... = 0.999... This series is called a geometric series and using limits its sum could be found by the following computation: 0.9/(1 - 0.1) and yes, that sum is 1!
Finally, in 1998, I was giving same lesson to my class of seventh graders as Schiller gave me 26 years prior when one of my students, Ben Stommes, raised his hand and offered his proof. "Mr. Kruger, it appears quite obvious," Ben confidently pointed out.
1/3 = 0.333...
+ 2/3 = 0.666...
1 = 0.999...
There have been other mysteries that have opened their cocoon unsuspectedly. The derivative of e^x is e^x and e^(iπ) + 1 = 0. These moments I have cherished and have attempted to replicate in my teaching. This year in my BC Calculus class when I used Taylor Series to show e^(iπ) + 1 = 0, my students hushed and then in unison murmured "wow". This was my third greatest moment this year. I had two presentations for finals. My greatest were yet to come. One was entitled Calculus in Nature and the other, The Gaussian Integral. Both of these gave me "needles and pins". Nature connected for me Ecology and mathematics using the Lotka-Volterra Equations. These differential equations model prey versus predator. The Gaussian Integral bridged the assumed crevice that separated Calculus and Statistics. The ending of this presentation redirected my isolationist view of statistics.
The unsuspected realities that arise in literature and mathematics should not be confused with the "aha" moments that can also come to fruition in both subjects. Those epiphanies are topics to be examined at a later date. The mysteries that result in "needles and pins" abound in literature and mathematics. Those unexpected nuggets invite me to explore the continuous depth and complexities in those disciplines and have moved me along my path. The path that I have followed to this point in time.
The unsuspected realities that arise in literature and mathematics should not be confused with the "aha" moments that can also come to fruition in both subjects. Those epiphanies are topics to be examined at a later date. The mysteries that result in "needles and pins" abound in literature and mathematics. Those unexpected nuggets invite me to explore the continuous depth and complexities in those disciplines and have moved me along my path. The path that I have followed to this point in time.
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