France: Day 2 - Germain

Today is the birthday of Paul Guldin born 1577 in St. Gall, Switzerland. Guiding made contributions in the areas of volumes and the center of gravity.

Today's quote is by Carl Friedrich Gauss. In a letter to Sophie Germain, Gauss wrote, "The enchanting charms of this sublime science reveal to only those who have the courage to go deeply into it. But when a woman, who because of her sex and our prejudices encounter infinitely more obstacles than a man in familiarizing herself with complicated problems, succeeds nonetheless in surmounting these obstacles and penetrating the obscure parts of them, without doubt she has the noblest courage, quite extraordinary talents and superior genius."

Sophie Germain

Sophie Germain was born April 1, 1776 in Paris, France. When she was 13, the Bastille fell and as a result she was required to stay inside. During this isolation, she taught herself Greek, Latin, and mathematics. Her parents disapproved of her passion for mathematics which at this time was considered an inappropriate field of study for women. Her parents attempted to restrict her studies by eliminating the fire in her room and by removing her clothes. However, Sophie's parents relented when they found her asleep with a frozen ink horn in hand and a slate of equations on her desk. 

Sophie was not allowed to attend to attend an university but was able to obtain lecture notes. She started send comments on the lecture notes under the pseudonym Monsieur Antoine-August Le Blanco. Using this pseudonym, she established a relationship with Carl Gaus and Joseph-Louis Lagrange, two prominent mathematicians. 

Germain made contributions in the areas of elasticity theory, differential geometry, and number theory. A Germain prime number and her additional work on Fermat's Last Theorem enhanced the exploration of the subject for hundreds of years.

I became aware of Sophie Germain through the movie, Proof. I wonder if the gender bias that has been prevalent in the STEM occupations has improved. Our mathematics department consists of 9 teachers, 5 of which are female. I wonder about nonteaching occupations and the pay inequity that exists. I know of at least 10 female, former calculus students that are deeply involved in STEM occupations. I wonder how they are coping?

France: Day 1 - Descartes

Today is the birthday of Charles Reyneau born 16546 in Brissac, Maine-et-Loire, France. Reyneau published one of the first influential textbooks on the newly invented calculus.

Today's quote is by Rene Descartes entitled, Cogito ergo sum. "... Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry, I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts (presentations) which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all objects (presentations) that had ever entered into my mind when awake, had in them no more truth than illusions of my dreams. But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be somewhat; and as I observed that this truth; I think, therefore I am, was so certain and of such evidence that no ground of doubt, however extravagant, could be alleged by the skeptics capable of shaking it, I counted that I might, without scruple, accept it as the first principle of the philosophy of which I was in search."

As witnessed when I watch a recent news segment, I am on "assignment" which means I am on vacation, and will be writing about my five favorite French mathematicians; Rene Descartes, Sophie Germain,  Pierre de Fermat, Emilie du Chatelet, and Cedric Villani.

Rene Descartes

Descartes was born on March 31, 1596 in La Haye, France. As a student, he studied rhetoric, logic, and the "mathematical arts". "Mathematical arts" included mathematics, music, astronomy, metaphysics, natural philosophy, and ethics. He attended the University of Poitiers and obtained a law degree.

After graduation, he studied medicine and theology. He traveled and joined the army. During his stint in the army, he met scientist and philosopher, Isaac Beeckman. Beeckman convinced Descartes that he should apply mathematics and logic to understand the natural world.

Descartes is considered the father of modern philosophy. He believed that all truths are ultimately linked and that the rational structure of mathematics and science could be used to uncover the meaning of the natural world.

Descartes introduced Cartesian geometry which is a fusion of geometry and algebra. I call Cartesian geometry, coordinate geometry. I love coordinate geometry and I believe its application allowed me to have an easy transition in to trigonometry and polar coordinates. Descartes also developed laws of refraction and formed a practical understanding of rainbows. Pope Alexander VII placed Descartes' writings on the Index of Prohibited Works.

Deriving While Driving

Today is the birthday of Max Zorn born in 1906 in Krefeld, Germany. Zorn is responsible for Zorn's lemma which states that if a partial ordered set exists and any subset of that set has an upper bound then the original set has a maximum element.

The quote of the day is by Srinivasa Ramanujan. Replying to G.H Hardy's suggestion that the number of a taxi (1729) was "dull", "No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being (1 + 1728) and (729 and 1000)."

A week ago, our school held its commencement for the class of 2015. During the week prior to graduation, I spent a great deal of time with the seniors that were in my class and in the organizations that I advised. Our conversations were reflective on their growth and their excited and nervous anticipation of their upcoming educational journeys. Each class attempts to leave a legacy their senior year and some classes do have a more resilient legacy to the corrosiveness of time.

Each numerical year seems to have its own identity. My high school graduation year, 1976, was the bicentennial year. The year 1984, was wrapped in the Orwellian concept of Big Brother. 1999 was made famous by the singer, Prince. The year 2000 was the millennial year. 2008 had the 8 rotated so that it became the infinity symbol. Members of the class of 2012 asked the question, "Don't you want to be 1  2?" 2013 ends in 3 and is also divisible by 3! 2015 was the year of π because on March 14 at 9:26 am, the first eight digits of that irrational number occurred. 

I was contemplating these numerical wonderments at a stoplight on my way to school the day after the commencement ceremony and was drawn to the class of 2016. In my meandering thoughts, I jumped to the factorization of the 2016 and was shocked in my determination that 2016 has 36 factors. Factors are natural numbers that 2016 can be divided by without any remainder. The factors of 2016 are: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016. I started to wonder about the number of factors in each of the years that I have taught. The chart below depicts those graduation years and the number of their factors.

To determine the number of factors, I found the prime factorization of each year. The prime factorization of 2016 would be 2 x 2 x 2 x  2x 2 x 3 x 3 x 7 or in exponential form, 2^5 x 3^2 x 7^1. I used the exponents in the exponential form to find the total factors that exist in 2016. The total number factors can be found by using the following arithmetic: (5 + 1) x (2 + 1) x (1 + 1). Using my high school graduation year, 1976, I found that its prime factorization is 2^3 x 13^1 x 19^1. The exponents are 3, 1, and 1. The number of factors in 1976 is 16; [(3 +1) x (1 + 1) x (1 + 1)].

When I examined the list, I noticed there are five prime years, 1987, 1993, 1997, 2003, and 2011. I believe the next graduating class that will have exactly 36 factors will be the class of 3168. The prime factorization for 3168 is 2^5 x 3^2 x 11^1. The last graduating class to have 36 factors was the class of 1800, 2^3 x 3^2 x 5^2. I was not teaching in 1800 and am not planning on teaching in 3168!

2015 Calculus Presentations

Today is the birthday of James Hutton, born 1726, in Edinburgh, Scotland. Hutton is famous for his theory of the age of the Earth.

The math quote of the day is by Rene Descartes who said, "Perfect numbers like perfect men are very rare."

Every year the final for my calculus classes consists of finding a partner, agreeing on a topic of interest, finding the mathematics that exists in it, and creating a presentation on that topic. This year's classes formed twenty presentations.

I have linked the title of each presentation to its Youtube site or the reader may visit my Youtube site: Charles Kruger.

The topics are: Aquariums, Aviation, Car Audio, Computer Animation, Dance 1, Dance 2, Discus, Farm, Golf, Guitars, Hydroelectric Power, Luminance in Art, Natural Disasters, Photosynthesis, Rollerblading, Soccer, Swimming, Tangled, Theory of Relativity, and Track & Field.

I enjoy these presentations for many reasons. Students take ownership of their presentations. The students and I usually learn something new. The students enjoy finding the mathematics in their topics. The presentations allow the students to be creative. The presentations are a showcase or a capstone of what the students have learned. The students use skills they have learned in other classes.

Enjoy!

Math In A Pea Cart

Today is the birthday of Abraham de Moivre born 1667 in Vitry-le-Francois, Champagne, France. de Moivre was instrumental in the development of Analytical Geometry and Theoretical Probability.

The math quote of the day is by Theodore Von Karman who said, "The scientist describes what is, the engineer creates what never was."

When I was sixteen I started my employment with Lakeside Packing Company. This company was a cannery in my home town that processed peas and corn. I worked every summer from the age of sixteen to twenty-one. My hours of work were from 6 am to 6 pm, seven days a week for the entire summer. There were a few days during the "pea pack" that the combines could not enter into the fields due to a large amount of rainfall, but generally speaking, I worked every day. The greatest and perhaps only incentive for working at Lakeside was its pay structure. I was paid overtime, time and half, for each hour over 48 hours in a work week. On the rare occasion that the peas were canned with carrots (a commodity provided by the federal government), work after 40 hours was considered overtime. My base pay was the current minimum wage of $2.10/hour and with overtime pay the hourly wage was boosted to $3.15/hour. My gross weekly wage was: 12 hours x 7 = 84 hours; 48 x $2.10 = $100.80; 36 x $3.15 = $113.40; $100.80 + $113.40 = $214.20.

At Lakeside, I was allowed to go to the restroom when I needed to, take morning, lunch and afternoon breaks, and most importantly, socialize. What I wasn't allowed to do was think. My entry-level job was to push pea carts into lines that were sorted by pea's size and density (determined by the pea's ability to float in a brine). 

A pea cart was a trapezoidal prism on rusted casters whose momentum was often impeded by B6's, peas the size of #Triball 12 buckshot with the hardness of diamonds. As a side note, when I asked who ate these type of peas, I was told unsuspecting institutions such as schools and nursing homes. These carts were filled with peas from tubes extending from the ceiling. The tubes were the recipients of peas garnered from trucks that had dumped their cargo into washers. From the washers, the peas had made a journey to the attic of the factory where they were washed, sorted, and flung down the appropriate tubes to the carts below. On the image above, there would be a small door on the side opposite of the slanted side. Once the carts were filled, I would push the cart to an assigned line and another employee would roll the cart to a predetermined position, lift up the door, and feed the peas through the rectangular opening into a small, swirling, bubbling vortex of steaming water. This vortex would take the peas to the roof of the factory where the peas coursed their way through a series of pipes. This corkscrew of rides was called the blanching process that all vegetables went through prior to being canned.

Every day, 12 hours each day for about 3 months, I pushed a cart. When I went back to school, my mind was like undercooked oatmeal. Constructing a thought was like wading through wet cement, and as a result of this mental decay, my mathematical processes were slowed. Also, my vocabulary was, well... not creative. I had heard a particular word so often that its usage had to be considered a fundamental rite of passage in this work environment. I had heard this word used as a noun, a verb, an adjective, and an adverb in one sentence. With this one word, employees described their frustrations and their joys. This monolexemic sentence was not conducive in a family or academic setting. This poisonous mixture of mental atrophy and constricted diction reduced me to a zombie with a hunger for abstract thought.

My second year at Lakeside Packing Company, I was promoted to the job of emptying the peas from the carts into the boiling whirlpool of churning water. As much as I had hated last year's job, I hated this worse. The temperature was hot and humid and no one wanted to spend any time discussing teenage topics within my restricted work area.

I was concerned about my deteriorating mental state and I devised a plan to solve math problems. I wanted to know how many A3 peas were in my cart. I first measured the dimensions of my cart and the radii of randomly selected of peas. I averaged those radii and formed an estimate of the volume of an A3 pea.  My initial answer was computed by dividing the volume of the cart by the volume of the pea. I knew my quotient was higher than the actual number due to the empty space between the peas.

I devised a new plan based on average rates. My peas eventually made it to the "fillers". The "fillers" were machines that actually placed the peas into a can, sealed the cans, and sent the canned peas to the "cookers". "Cookers" were a holding location where the contents were "cooked" for an appropriate time and temperature. At the beginning of my shift, my "filler" had no peas. I emptied my cart and stopped. When my assigned "filler" received the peas, it started up but stopped again when it ran out of peas. I timed the start and stop interval of the "filler". I knew the number of cans per minute that the machine sealed. Quality control randomly selected five cans from the "filler" to examine their contents. I had them count the number of peas in each can and determined an average number of peas per can. My formula: Minutes x Cans/Minute x Peas/Can = Total Peas in a Cart. I was pleased with the results. Both methods resulted in close estimates with the latter being slightly lower than the former.

I can't remember how long this problem took me to solve. I don't think it took more than a couple of days. I was hoping for a problem that I could work on for the entirety of canning season. I do know I was sad when I completed it. I believe I worked on mental math problems the rest of the summer but those activities did not give me the same satisfaction. One of the reasons I enjoyed this problem is that I involved other workers. I had the people working on the fillers giving me information about their machine and taking the risk of stopping and starting their machine which was not a popular idea among the mechanics and our superiors. I also had to work in concert with the quality control and convince them of doing the added work of counting peas. Perhaps, the problem temporarily broke everyone from the constant boredom that encased their daily routine. Many people were curious on how many peas were in a cart during that brief moment in the summer of 1975. Yes, for a brief time, doing math made me cool :-)

And so it begins, again

Today is the birthday of Henry Dudeney born 1857 in Sussex, England. Dudeney was interested in mathematical inquiries and past times. His publications about these topics lead to major mathematical research.

Today's quote is from Jospeh-Louis Lagrange who said, "As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when the two sciences have been united, they have lent each mutual forces, and have march together towards perfection."

This is my tenth year of teaching AP Calculus. I retired last year and was hired back part time. This year was also the year I had my official teacher observation. During the follow up conversation with my principal, I remarked that I would like to teach again next year if the opportunity arose. He was very open to the idea but did inquire why I didn't look to teaching opportunities in a different educational setting such as in a collegiate environment. I paused and then confessed that I was touched by the relationships that I formed with my students. He smiled and said, "Your not such a tough guy."

My classes are currently reviewing all the topics we have covered and preparing for the AP Calculus Exam that will be in May. I also had Parent-Teacher Conferences this past week and some serious discussions with parents on the probability that their child will pass the exam and receive college credit. I analyze the statistical data I have gathered on each individual student and have a fairly accurate predication model on that probability. I have a conservative model. By that I mean, most students due better that I predict but for me personally, this time of year is heart wrenching.

My students have worked hard, possibly harder than any math class they have taken previously. Many have given up the luxury of an A or B with little work and had to adjust to struggling to get a C. What has happened as I have become older, I agonize on how I can get them to squeak out 1 or 2 extra points on a question or how to see that problems they struggle with are related to the problems with which they solve with ease. The parents really want their child to succeed and not feel the pain of perceived failure.

I try to assure the parents that even if their child scores a 1 or a 2, they will be fluent in calculus when and if their child takes it again but the parents want more, they want assurance. I understand that, I'm a parent.

I have been with these students the entire year and every year it happens, I fall in love with them. They are delightful. They make me laugh, they make me think, they challenge me, and they teach me. I become the student. If I could, I would give them all 5's but that is not going to happen.

Bob Dylan wrote in the song "If You See Her, Say Hello" these lyrics: "Either I'm too sensitive or I'm getting soft. Sundown, yellow moon, I replay the past. I know every scene by heart, they all went by so fast." The years and classes do. I wish I could slow them down.

The Sands of Time

Today is the birthday of Clifford Dowker born 1912 in Parkhill, Ontario. Dowker made contributions in the area of Knot Theory.

Today's quote is from Joseph Serret who said "Algebra is, properly speaking, is the Analysis of equations."

"Like the sands through the hourglass so are the days of our lives." Quoted by MacDonald Carey in the opening credits of 'Days of Our Lives.'

"Sand is pouring into a conical pile of sand, at the constant rate of 6 m^3/second. Suppose the side of the cone makes a 60° angle to the cone's base. How fast is the height of the sand pile changing when the volume of the cone is 180 m^3?"

"Oil is filling a cylindrical tank at a rate of 2 gallons/minute. If the radius of the tank is 2.5 ft, at what rate is the height changing when the tank half full?"

These are the problems that I was thinking of at a recent wedding that I attended. I sometimes lose my focus and start thinking of mathematics at inappropriate times. The couple was pouring sand into a container. The colors of the sand were blue and white. The pouring of sand replaced the tradition of the unity candle. Although the pouring of sand was a new wedding experience for me, I have found out it is not unique in wedding ceremonies.

During my son and my daughter-in-law's wedding, the parents of the couple lit the candles the groom and bride used to light the unity candle. I found that act a moving experience.

As I watched this couple pour each of their respective sands into a truncated cone (often called a tapered glass/i.e pint glass), I noticed that there were times in which one paused to let the other pour. The pause that allowed the other their moment of time, their moment to pour.  Within the wait and pour, a unique formation of sand was formed.

I believe in marriage, each partner is required to pause. Whether the pause is to allow the other a moment in the sun, a moment of respite, a moment of encouragement, or a moment of solitude. Such is marriage, a series of give and takes that is formed by each individual grain of sand, each waiting their turn, each supporting the other, and forming its unique beautiful structure.