The Voice in Mathematics

Today is the birthday of Oronce Fine born 1494 in Briancon, France. Fine published major works in mathematics and astronomy.

Today's quote is by Raoul Bott who said "There are two ways to do great mathematics. The first is to be smarter than everyone else. The second way is to be stupider than than everybody else - but persistent."

I am of the belief that mathematics and written language are more similar than different. Each has its structure and its form of creativity. I have touted this belief to my English teaching colleagues using author, John Greene, and his views on the education continuum as a supportive example. I often hear from some of these colleagues that they do not possess the "math gene". I am not a believer in a math gene. I am not any more a genetic math mutant than I am a lobster fisherman. Keith Devlin states in the prologue of his book, The Math Gene, How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, "One of my aims in this book is to convince you of just how remarkable and powerful - and uniquely human - language and mathematics are. Let me once again quote Neil Armstrong. When the lunar module broke free from the command ship that would remain in orbit above the moon during the course of the moon walk, Armstrong declared that "The Eagle has wings." The acquisition of language and mathematics gave humanity the wings to soar above our fellow creatures. My other aim is to argue that these two faculties are not separate: both are made possible by the same feature of the human brain."

My first educational love was books. Each day as a youth I visited the library and each day my imagination soared with the descriptive voice of the author. I was transported from my sleepy hometown to bustling cities and exotic locales snuggled in a world that were filled with intrigue and adventure that I was sure did not exist in my community and most definitely, in my life. During my high school years, I took classes that were titled Short Stories and Novels which were taught by Bill Nems and with his facilitation, my perception of what I read deepened.

As my formal education continued, I gained a mistress. Her name was Mathematics, the queen of sciences. She offered me structure, organization, predictability, and the allure of a right answer. She eventually introduced me to my drug of choice, solving complex mathematics problems. I had my first initial rush from this drug in my 8th grade algebra class. I do not recall the problem but I do remember solving it in my sleep. This was my first dream in which I awoke with the solution in hand. I was ecstatic. The thrill was overwhelming and I have been addicted ever since. I have continued to solve math problems in varied states of consciousness.

Language is not complete without the written word. I started to appreciate writing as I was completing my master's degree. Within the composition of papers that detailed the educational practices of teaching mathematics and examinations of mathematical curriculum, I discovered my voice. I could be insightful and humorous, sarcastic and tactful, gregarious and prudent. I became a logophile, a lover of words. Writing has offered me creativity in structure and the tantalizing mystism of solving the problem of organizing my thoughts and voicing my passion.

As I stated earlier, I constantly discuss with the English teachers in my school, the parallels that exist in mathematics and English. Recently, an English teacher stopped me in the hallway, approached me resolutely, and stated "Where is the voice in mathematics?" I was delivered a knock out punch. I stood speechless and wandered away muttering to myself, "Where is the voice of in mathematics?"

My first question to address is what is meant by voice in writing. I found two sites that offered descriptions of an author's voice. Understanding Voice and Tone in Writing by Julie Wildhaber and Voice in Writing: Developing a Unique Writing Voice by Cris Freese. Freese states "A writer's voice is something uniquely their own. It makes their work pop, plus readers recognize the familiarity." Wildhaber defines voice as "the distinct personality, style, or point of view of a piece of writing or any other creative work." She elaborates, "Many musicians have played the 'Star-Spangle Banner,' for instance, but there's a world of difference between the Boston Pop's performance and Jimi Hendrix's, even though the basic melody is the same."

I asked my English colleagues their opinions of voice.

Beth Gadola: "In writing, voice can be expressed when a writer puts him or herself 'into' the words, providing a sense that a real person is speaking and cares about the message. When a writer is personally engaged with the topic, he/she imparts a personality to the piece that unmistakably his/hers alone. And it is that individual personality - different from the personalities of all others - that we call voice. Voice is the distinct personality of a piece of writing."

Stan Berg: "For me, it's a sense of authenticity, the idea that a real person is letting his/her real self come through in the writing so that it doesn't sound like a canned response that could have been written by anyone."

Lyndsy Schwantes: "I would say that a writer's voice is what makes the writing stand out to me. It's not only their word choice, but the way their words work together to tell a story and create characters the reader can connect with. When I read a story by an author I consider to have a good "voice" I don't think about the words or why their choosing their words, but I become lost in the story."

Maria Burnham: "Voice, simply put, is the thing that breathes life into writing. It's the formula of writing that tells my brain's inner voice how to read something. Is the writing factual and simply stated? There is voice in that style of writing. Is the writing full of vivid description and flowery language? There's voice in that style as well. Is the writing full of questions followed by answers? You've got it, there's voice in that, too.

For what it's worth, the reason I majored in English in college was that I fell in love with voice in writing. I loved that the choice in words and the way in which they were arranged changed how the writing sounded. In writing there's cadence, emotion, meaning, all because of an author's choice.

I find myself gravitating toward specific kinds of voices, particularly those of poets and essayists like Whitman and Thoreau. But I also love humor, sappy love, and even the simplicity of technical writing.

Letters and words in isolation have meaning but little depth unless put together by an artist, a wordsmith, a writer."

A common thread winds through these definitions. Voice, like beauty, is in the eye of the reader. Even a boring voice in one's mind may conjure up an image of Ben Stein lecturing in Ferris Bueller's Day Off . A voice that paints images, resonates emotion, and is layered with complexity may only be appreciated by a few experienced readers. I watched a rich and vivid dance performance recently. A strong female voice emanated from that performance but I knew that my perception only heard whispers of her messages.

Mathematics has beauty. "Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." Bertrand Russell,  The Study of Mathematics

A February 2014 article by James Gallagher in the BBC News entitled Mathematics: Why the brain sees maths as beauty, discusses the brain activity of emotion that is triggered equally by a beautiful equation, a great painting, or a classical piece of music. The equation that is believed to be the most beautiful is e^(iπ) + 1 = 0. This equation is referred to as Euler's Identity. The proof of the identity resonates in me wonderment and awe.

Does beauty in a medium indicate a voice? I don't necessarily think so. A rose can be appreciated for its beauty but it remains silent.

I was at a loss in describing a voice in mathematics until a recent Math League practice. I sat and listened to my students work on various problems. I believe these three problems spoke to my students.

1.

I was asked my students where to start on this problem and told them to rationalize the denominator. As one student proceeded, she paused, "Oh, cool! This is a cute problem!"

2. 

As we worked on this problem, several students remarked how this was a "sneaky" problem. When I asked them to define "sneaky", they described problems that first appear to relatively direct to approach but as those problems are solved, they entail twists, turns, and surprises that require persistence and flexibility by the solver.

3.

In this problem, the graph is defined as the second derivative and students were asked to draw the first derivative and the original function from which the second and first derivative were formed. Students described this problem as a simply stated but multilayered in its complexity. I compare their description of this problem to a single stanza poem. A poem that offers new insight to a reality that surrounds us each time it is read.

As I contemplate what specific mathematical ideas speak to me, I have a few the quickly come to mind.

The proof that the square root of 2 is an irrational number has always evoked a voice of sarcasm. I can hear my son, Sam saying,"So you don't believe it to be true? Let's assume its not and see where that leads us!"

The Pythagorean Theorem speaks to me in a voice of security, loyalty, and strength. The Pythagorean Theorem is in algebra, geometry, trigonometry, and construction. When I am stuck on a problem, I can hear John Wayne lean in and say "Hey, little pilgrim have you tried the Pythagorean Theorem?"

I can hear the imaginary i screaming at mathematicians, "You thought that negative numbers were imaginary and you found application. You define me as the square root of a negative one and call me imaginary, yet you found application for me! What is it with you guys?!"

Finally, the controversial number 0, the number that apparently does nothing in addition but is the mighty destroyer in multiplication. The number that can be divided but is not allowed to the dividing. The number that was considered the null and void and repulsed by religions but without it we would not have calculus. The number that says, "Go ahead, make my day."

The voice in mathematics exists but it is different than the voice in writing, the voice in music, the voice in dance, and the voice in art. Mathematics is creative. Its voice may be more restrained and its voice may take more work to hear but the voice exists, nonetheless.

Deja Vu All Over Again

Today is the birthday of Franz Aepinus born in Rostock, Germany, 1724. Aepinus made contributions in the area of electricity and magnetism.

Today's quote is by Niccolo Tartaglia. This quote is a poem written to Jerome Cardan. In this poem, Tartaglia reveals to Cardan the secret to solving a cubic equation.

"When the cube and the things together

Are equal to some discrete number,

Find two other numbers differing by this one,

Then you will keep this as a habit,

That their product will always be equal,

Exactly to the cube of a third of the things.

The remainder then as a general rule

Of their cube roots subtracted

Will be equal to your principal thing."

In my calculus classes, I talk about my dog, Rosie, and she has become quite a celebrity. Last year and again this year, I showed them a graph that I entitled My Walk With Rosie. I assigned them the task of detailing the calculus concepts that are embedded in the graph. I have listed those responses. The responses are both mathematical and creative. One student included cartoons of Rosie and myself.

Rosie patiently waiting for a walk.

Rosie spending some quality time with my son.

The graph of My Walk With Rosie.

1) The limit of f(t) as t approaches 45 from the right is 150.

2) At 45 minutes there is a relative maxima (vertex), where the derivative is zero. This represents the point at which you turned around to return home because for a moment you were not moving away from or toward you house.

3) There are x-intercepts at (0, 0) and (60, 0). These points represent times when you were at your house. Because there are only two points, we can infer that you left at (0, 0) and returned at (60, 0) and did not go to your house at anytime between those two points.

4) There are two points on this graph (30, 50) and (45, 150). We can find a secant line with those two points. The slope of this secant line is 20/3 or 6 2/3 feet per minute. Because his rate (slope) is positive we can tell that you’re moving away from your house.

5) After coming home from a grueling day of teaching, Kruger came home and decided to take his dog, Rosie, on a walk. Starting at the end of his driveway, he started walking at an average rate of 3 1/3 feet per minute^* for a duration of 15 minutes. This took him a total of 50 feet. Being too tired to continue, he had to stop and sit down on the curb. They sat on the curb for 15 minutes when Rosie got up and started chasing a passing car. Alarmed, Kruger got up and pursued his runaway dog at the slow average rate of 6 2/3 feet per minute^*. Even though this was double his previous pace you would think that if he was concerned about Rosie, he would pick up the pace. He grew discouraged after 100 feet and 15 minutes and turned around to go home to call the police and report Rosie missing. On the return trip of 150 feet he averaged a speed of 10 feet per minute^* and arrived home after 15 minutes. When he got to his front door he found Rosie there waiting for him.

^* Secant lines can be used to find his average speeds at different intervals of his walk. The part of the graph with a slope of zero means he is not walking.

6) I can tell you took a break from 15 minutes to 30 minutes because you stayed at 50 feet the whole time. The slope is zero because the tangent line is a horizontal.

7) I can tell your started off slow and once you got closer to 15 minutes you moved faster. The slope of the tangent line is steeper towards the end.

8) 150 feet is the furthest you walked. This shown at the top of the parabola.

9) It took you 45 minutes to walk Rosie the first half of your walk and 15 minutes to walk her back. The graph is increasing the it starts decreasing that's how I can tell.

10) It was a find day outside an on October 12, 2014. The sun was out and the chill of autumn was in the air. Being that math nerd that he was, Mr. Kruger decided to make a very well drawn graph of his little adventure. With his handy dandy calculator in hand and his sidekick Rosie by his side, Kruger began his adventure.

Mr. Kruger started off his walk at a good pace. After 15 minutes, he had already gone 50 feet! This fast pace was due to the fact that Rosie saw a squirrel ahead and she would stop at nothing to chase it. Well, Rosie did stop for the squirrel ran up an oak tree to seek refuge.

Rosie's sudden outburst almost caused Mr. Kruger to drop his calculator right down the storm drain near by. Thankfully, Rosie was jumping after the squirrel at the same time the calculator air born and their trajectories collided and the calculator deflected off of Rosie's side. The calculator then soared back into Mr. Kruger's waiting hands.

Mr. Kruger wept at the sight of the mathematical miracle that had happened here. He tried to go on with his walk but the heat from his tears had caused the poor man's glasses to fog to the point where he couldn't see. While waiting for his glasses to defog, he decided to figure out the equation for the first 15 minutes of his walk. After fiddling with this calculator for 12 minutes, he did some math like this: Start point (0, 0), End point (15, 50), m (slope) = (50 - 0)/(15 - 0) = 50/15 = 10/3, 50 = 10/3(15) + b, 50 = 50 which means the secant line of line for the first 15 minutes of his journey was: y = 10/3x. 

Kruger was about to continue on his walk when he realized he had just wasted 15 minutes standing still, finding no challenge in determining an equation to the next section of his walk. The equation was y = 50. For those 15 minutes he was nothing but a sad constant.

Mr. K quickly got over the fact that he was a constant and continued on his merry way. After another 15 minutes, Kruger stopped to pick up a coin on the ground - after all it was heads up - and stuck the coin in his pocket. Little did he know that this move would alter the equation of his walk for the day.

Mr. Kruger, being the math genius he is, quickly figured out the equation to this piece of his walk was y = -2/9(x - 45)² + 150.

Kruger then started to increase his speed. After all, he had been walking Rosie for 45 minutes now. Mr. Kruger pushed with all his might to go another 15 minutes. He was out of breath and huffing and puffing and wheezing and squeezing and Rosie may have had to drag hime a bit but he made it a whole hour.

Mr. Kruger sat in his kitchen wondering what else he could conclude from his walk. He noticed his graph was continuous. Since time never stops, there was no holes or asymptotes. Mr. Kruger then decided to dig a little deeper and find the derivative portions of his graph.

Farming Fibonacci

Today is the birthday Guido Grandi born 1671 in Cremona, Italy . Grandi was an Jesuit that made contributions in the area of geometry and hydraulics.

My math quote of the day is attributed to an anonymous contributor. "Mathematics may not teach us how to breathe oxygen or exhale carbon dioxide; or to love a friend and forgive an enemy. It may not even help us to find our way to our one true love but it gives us every reason to hope there is a solution to every problem."

I grew up on the edge of a small rural town in southeastern Minnesota. My dad was a self-employed welder who employed my siblings and I at a young age. On the first day of my employment I was given the task on cleaning a manure spreader so that my dad could repair it. I initially was a reluctant employee but I eventually became accustomed to the apperception, timbre, and aroma that defined for me the life of a farmer.

My first teaching job was in a rural community and I commuted to that district for 18 years. I enjoyed the 45 minute commute. I was able to view the rhythmic patterns of farming; preparation, planting, and harvesting. There was a strong FFA presence in that school community much like there is in the one I am teaching at now. FFA no longer means Future Farmers of America but is an organization focused on Agriculture Education. I have purchased fruit from the members of the FFA for 34 years and will again this year. The selling of fruit around Christmas has been a fundraiser for the FFA ever since I can remember. I have noticed a shift in the membership since I have started teaching. More female students are involved and they usually have taken on leadership positions within the organization. The "ag" teachers that recently have been recently hired have been female. Our current FFA advisor (nicknamed Taylo) is an energetic, dynamo graduate of the University of Minnesota. She suggested to me the idea of blogging about the connection between mathematics and agriculture. I suggested to her the topic of Fibonacci numbers.

I first learned of the Fibonacci sequence when I was a freshman in college. Fibonacci was an Italian mathematician who did not discover the sequence but used the sequence to spread the use of the Hindu-Arabic numeral system. The sequence is as follows: 0, 1, 1, 2, 3, 5, 8, … . After 0 and 1 each new term is formed by adding the two previous terms. The explicit form is also called Binet's Formula and is an = [(1 + √5)^n - (1 - √5)^n]÷(2^n√5). If a decimal representation is formed by dividing the new term by the previous term the following sequence is formed: 1/1 = 1; 2/1 = 2; 3/2 = 1.5; 5/3 = 1.667; 8/5 = 1.6; 13/8 = 1.625; 21/13 = 1.615; 34/21 = 1.619; 55/34 = 1.617; 89/55 = 1.618; 144/89 = 1.618. (1 + √5)/2 is called the golden ratio. 

Fibonacci numbers occur in nature such as in the growth pattern of rabbits, cows, and bees. My particular interest is how the sequence appears in cones, particularly, the hop cone. One of my hobbies is brewing beer. I have been doing this for 21 years. My oldest son has convinced me of growing hops. 

The hop flower or seed cone is picked, dried, and then dry hopped into a container holding the fermenting beer. The hops remains in that wort for 1 to 3 weeks. After which the beer is bottled or kegged. Using hops in this manner enhances the aroma of a beer.

In the above photo of a common pitch pine cone, the spirals have been highlighted for easy counting. There are 8 red spirals opening to the left and 13 blue spirals opening to the right. 8 and 13 are consecutive Fibonacci numbers. Other species of pine cones also have spirals that are consecutive Fibonacci numbers such as 5 and 8. I have not examined the hop cone in particular nor have I found any research that details the existence of consecutive Fibonacci numbers but I will continue to examine each hop that I grow in my quest of producing the elusive perfect beer.

The Fibonacci sequence is truly fertile ground for mathematical exploration and growth. Taylo, herself, was bursting in pride when seeing my Fibonacci t-shirt, baffled her boyfriend when she detailed her knowledge of Fibonacci.

The Big Cheese Dog

He called himself "The Big Cheese Dog". The purpose of this reference was to insure to his students, his audience, that what he was about to state was to be written down, that this nugget of information was somehow to be seared into the recesses of their minds. His classes were, generally speaking, attentive, whether they were seventh graders or seniors or somewhere in between. The words, "The Big Cheese Dog", were simply the icing on the cake. Prior to the conclusion that included those words were a series of gestures, gyrations, stories, analogies, graphs, demonstrations, and other illustrations that kept attention and attempted to make the complex real. As a former math student of Paul Edward's stated, "I'll never forget the chalk all over Mr. Edwards' pants as he would rise up and down on his tippy toes, flailing his arms and scribbling on the board as he lectured". I remember when he took his math class outside for "a wake up call". The students were drowsy and didn't meet Paul's expectation of attentiveness so his class took a brief field trip into the Minnesota winter, a crisp -10°F awakening that attributed to the legend of his intensity and his commitment to the education of his students.

Paul passed away August 6th, 2014. With his passing, a hole opened in my soul. I taught with Paul in two school districts. We commuted for 7 years to one of those districts. We had an hour and a half each day to share ourselves. I loved and admired Paul. He showed me that math could be cool. He made me aware that I was not alone, there were others that were as passionate about mathematics I was. He being one. During our commutes we would be irrelevant, challenging, blasphemous, contemplative, and flippant. Our mantra was that "only fools took themselves seriously." We really believed that the two of us could change the world, as Paul often quoted, "Chuck, its us against the world of ignorance." We once convinced a colleague that Ozzy Osbourne was the son of Bob Keeshan, Captain Kangaroo.

Practical jokes were not limited to "outsiders". I convinced students that Paul's real name was Polycarp Whitecloud Edwards and registered him as such for a national mathematics convention. As he picked his name card and realized what was printed on it, a series of expletives were machine gunned at me. The resulting expense of calming beers was worth his reaction. In turn, he frequently enjoyed placing my car keys in strategic locations in my room prior to the end of our work day. The most memorable being just 18 inches above my head resting on my document camera. The calming beers on that Friday were well received.

Paul and I worked on our masters degrees together. Our degrees are in mathematics. Paul started ahead of me. He saw the potential of a masters degree in mathematics. He eventually left teaching and applied what he learned in the area of Statistics. Our relationship became more focused on the success of learning a higher level of mathematics. Our individual strengths complemented each other. Paul was excellent at taking notes. I, in contrast, listened. We would share our perspectives, mine, a view of the "big picture" and his, a detailed account on how to get from point A to point B. His wife quipped that she overheard our discussions and believed we were making up the words, "real people didn't really talk that way."

Paul's life and mine coincided for 13 years, merging at one district and diverging at another. He changed the way I teach and eventually how I view my own life. He could be intense and childlike within the same moment. I will miss him. He told me that he loved me as a brother. I will always cherish his love.

2014 Calculus Presentations

Today is the birthday of Karl Gruenberg born 1928 in Vienna, Austria. Gruenberg worked in finite group theory.

Today's quote is from Kareem Abdul-Jabbar who said, "Music rhythms are mathematical patterns. When you hear a song and your body starts moving with it, your body is doing math. The kids in their parents' garage practicing to be a band, may not realize it but they are also practicing math."

For my AP Calculus Final, I assign the students to find a topic they love to do or have a strong interest in exploring. I am trying to convey the idea that mathematics exists in everything. All a person has to do is look for the mathematics. The students form groups and give their presentations on their agreed upon topic. These presentations are linked to their Youtube videos. Not all the topics have links due to video production scheduling. At the end of this blog, I also have links to the powerpoint presentations. The following are the opics various groups of students presented: Airplanes, Bowling, Cats, Counter-Strike, Disney, DQ Ice Cream, Golf, Hurdling, Income Gaps, Juggling, Math on the Beach, Pickleball, Ping Pong, Quadcopter, Searching for Sasquatch, Stellar High 5, and Volleyball.

The power points for the following topics are also linked at: Calculus Presentations.

Taking a Risk

Today is the birthday of Edward Titchmarsh born 1899 in Newbury, Berkshire, England. Titchmarch work was in analysis. He studied Fourier Series and Fourier Integrals. Titchmarsh is the author of today's quote. He said, "It can be of no practical use to know that π is irrational, but if we can know, it surely would be intolerable not to know."

I constantly lecture to my students the importance of taking a risk with the understanding that failure is a consequence. I must confess I am not a risk taker. My wife and my sons are. If I had not fell in love with my wife, I could very well see my self living in northern Minnesota, in some remote cabin, teaching mathematics in a small community.

I did take a risk this past week. I spoke at our high school's graduation ceremony. Speaking in front of a large group of people has never been on my bucket list and now, I can say that it never will. I am sharing my speech both in written form and as a Youtube video. I do so because I feel that this blog is a diary of sorts. I will let you be the judge of the speech. I am not a public speaker. I am aware of my mistakes and flaws.

Speech 5/27/2014

"Dr. Bittman, Mr. Martens, School Board Members, Staff and Faculty, Friends and Family of the graduates, and the Class of 2014.  (Cell phone rings) Hello?  Hi Mom. Thanks but the speech is not tomorrow night.  When? Ah… tonight.  Yes, oh right now… No, that’s ok. I think he’ll understand.  Yes, Mr. Martens has a mother too. Thanks again. Say hi to Dad. Yes, I love you, too.

I suppose you guessed that was my mom. She loves me. She always has. Both my dad and mom worked at creating an environment that prepared me for adulthood. Growing up, I looked forward to adulthood and its freedoms; but adulthood has its responsibilities.

When I was asked to speak at this graduation ceremony I pondered a great deal what would be my theme. I know math; really not much else. I thought of graduation speeches I had heard. One speaker sang, while another danced. When I suggested these ideas to my family. They reminded me that as a youth I was told to mouth the words in choir and in college, I almost failed ballroom dancing. I was dismayed. So I went for a walk with Rosie. Rosie is my dog and is a great listener. When I told her of my dilemma, her response was to wag and pant in her nonverbal dog way. I had to agree with Rosie. I would talk about what I knew: math and poetry. You see, Rosie knows that prior to our morning walk I work on a math problem and read a bit of poetry. Rosie is a pretty smart dog.

Mathematics is a way to solve problems. In a recent conservation with my dad, he conveyed to me his concern of the future generations’ ability to solve problems. He believes as do I that upon graduation we transition from celebration to responsibility. I assured him that our graduates, you who are in front of me now have the tools to solve problems, the problems within our relationships, the problems within our communities, our society, our world. I would not be in front of you; the faculty, the school board and the administration would not be in front of you; if we did not believe that to be true. We have placed our stamp of approval that you are ready for the next step; to help us solve problems, to be the stewards of our communities.

I believe problems are solved with the continued acquisition of knowledge, the ebb and flow of communication, hard work, and creativity.

You need to be life long learners; learning through experience and through education spawned by curiosity. Albert Einstein said, “Intellectual growth should commence at birth and cease only at death” and Henry Ford quipped “Anyone who stops learning is old, whether at twenty or eighty.  Anyone who keeps learning stays young.”

Knowledge is a powerful tool.

Being able to communicate is a two way street. You need to be understood, to be able create a conversation that is reasonable and sound, a conversation that is clear and easy to understand, and a conversation that is open to a response. Communication requires the ability to listen. That ability not only includes listening to those with whom you agree with but also listening to those who offer an alternative opinion.

As with real math problems, the problems that surround us will take time, hard work, perseverance, and focus but you have the tools and have the ability to gather more tools to address the problems before us. I suggest to my students that they work in groups. There is power within groups. When groups work at solving problems, they develop cohesiveness.  They become accustomed to each other’s rhythm… each other’s poetic meter. You have experienced these communities in our school whether you participated in any of our co-curricular or extra-curricular activities. As these communities solved problems , you became connected to your peers.  You became part of the larger dynamical community we call high school. Last week was the birthday of Edward Lorenz, a-would be mathematician who decided instead to study meteorology. Lorenz initiated the idea of chaos theory.  Lorenz termed chaos theory as “the butterfly effect” in his presentation, “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” You are part of the “butterfly effect”, your actions affect the dynamical system we call community.

A community has gathered around you tonight. They solved the problem of your education. Did it take time? Yes, about 13 years. Did it take hard work? Yes. Look around you, look at this community that is proud you, you are worth their time and their efforts.

Be creative in your problem solving. Allow yourself to take a risk, to view the problem from different vantage points. Being innovative and entrepreneurial will mean you will come to dead ends and at times, fail but creative people are resilient and strong.

I am aware that not everyone becomes problem solvers in the same manner. You will create your own path or take paths less traveled but we are all of the same fabric, the cast of the same play. I was struck this year by an advertisement from Apple for the IPAD. The ad had Robin Williams reciting a scene from the movie in “The Dead Poets Society” in which he plays John Keating.  In this scene he quotes one of my favorite poets, Walt Whitman.

Robin Williams states, "We don't read and write poetry because it's cute. We read and write poetry because we are members of the human race. And the human race is filled with passion. And medicine, law, business, engineering, these are noble pursuits and necessary to sustain life. But poetry, beauty, romance, love these are what we stay alive for."

By your birth you became members of the human race, with this graduation you have become part of its solution.  You will determine how you will be part of it.

I will finish with this poem by Walt Whitman

'O ME! O life!... of the questions of these recurring;
Of the endless trains of the faithless--of cities fill'd with the
foolish;
Of myself forever reproaching myself, (for who more foolish than I,
and who more faithless?)
Of eyes that vainly crave the light--of the objects mean--of the
struggle ever renew'd;
Of the poor results of all--of the plodding and sordid crowds I see
around me;
Of the empty and useless years of the rest--with the rest me
intertwined;
The question, O me! so sad, recurring--What good amid these, O me, O
life?

Answer.

That you are here--that life exists, and identity;
That the powerful play goes on, and you will contribute a verse.

You will contribute a verse.

What will your verse be?'"

http://youtu.be/OopKEYQus9Y

My Struggle With Velocity and Mom

Today is the birthday of Pafnuty Chebyshev born 1821 in Okatovo, Kaluga Region, Russia. Chebyshev investigated number theory, mechanics, and invented orthogonal polynomials.

Today's mathematician's quote is from G. Simmons who said "To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls."

The lives of my younger brother's are a study of contrast.  My younger brothers are identical twins. Their births were two minutes apart.  The oldest of the pair just recently became a grandfather while my "youngest" brother is teaching his 15 year old daughter how to drive. I was fortunate to have the majority of the driver's education instruction fall on the slender but resilient shoulders of my wife.

My real behind-the-wheel experience of learning how to drive fell on my mother. I have a number of experiences that I can recall during that time, many of them are not pleasant.  During the 1970's seat belts were in cars but were not used by anyone in my family except for the time I slammed on the brakes in the middle of a busy intersection catapulting my mother's head to the windshield of our banana colored 1972 Ford LTD.

This is the car that my brothers and I referred to as the "banana boat".

I am sixteen and have hair.

A constant discussion between my mother and myself was my obsession with speed and lack of the proper use of the "gas pedal".  The following sketch represents my mother's lament about the use of the accelerator.  She would often describe me as a "heavy foot" driver.

The graph that is labeled "Chuck" illustrates a quick gain of distance in a short period of time while the other graph labeled "Chuck's mom" indicates a gradual gain of distance in a greater time span.  Both of these graphs are defined as position functions.  They depict the distance traveled over an interval of time.  The graphs are both increasing; one quickly, the other gradually both they differ in terms of mathematicians call concavity. The "Chuck" graph is concave up and my velocity is increasing.  In calculus, velocity can be viewed as the slope of a line tangent to a position curve. I have examples below.

Notice for the same time value, x = 3, the slope of the green line on the upper graph is 0.2886 and the lower graph it is 6. When time increases as in the next two graphs, the slope decreases to 0.25 in the upper graph but increases to 16 in the lower graph.

In the upper graph, my velocity has decreased for another unit of time.  I have negative acceleration. My mother is relaxed, viewing the dafodils blooming in the neighbor's garden, and sets aside the belt buckle she had placed in her lap.  I turn down the volume on the radio as Perry Como  sings "Mama Loves Mambo".

In the lower graph, my velocity has increased for another unit of time.  I have positive acceleration. My mother has been forced further back into her seat, her eyes are closed, her teeth are clenched, and she is gripping tightly to the door handle.  I, on the other hand, have the window open, and am looking for tanned coeds that are enthralled by the power in front of me.

When velocity and acceleration have the same signs (they are both positive or both negative), speed is increasing.  When velocity and acceleration have opposite signs, speed is decreasing.  Here is another example.

This is a graph of two velocity functions.  Each graph is moving towards the x-axis.  On the x-axis, each will have a value of zero so each velocity function is decreasing.

The slopes of the lines tangent to each of these graphs represents the acceleration of an object at a particular time, x.

These diagrams indicated that the acceleration is decreasing (-2 to -3),velocity is positive, and speed is decreasing.

The above diagrams show that acceleration is increasing, velocity is negative, and again speed is decreasing.

I have mellowed out in my quest for speed. I have become accustomed to letting the other drivers pass me by while I reflect on the mathematics I see around me.  :-)

Tractor, Slide Rule, Calculator, and The Digital World

Today is the birthday of Heinrich Jung born 1876 in Essen, Germany.  Jung made contributions in the field of algebraic functions.

Today's quote is from Herbert Turnbull.  He quipped, "The usefulness of mathematics in furthering the sciences is commonly acknowledged: but outside the ranks of the experts there is little enquiry into its nature and purpose as a deliberate human activity."

My dad and I were having a discussion about innovation.  He grew up in South Dakota in the 1920's and 30's during the Dust Bowl.  He remarked to me that he remembered the turmoil that a purchase of a tractor caused among neighboring farmers.  I found an article that supported his perspective of that time. There are very few farmers that would subsist without the tractor.

When I was in high school our choice of a calculating device was the slide rule.  I still have my slide rule but am no longer proficient in using it.  The logarithmic concepts on which it is based have stayed with me.  The terms, mantissa and characteristic, have remained a common part of my vocabulary. :-)  I did not use a calculator my high school mathematics classes even though the scientific calculator existed.  I have colleagues that are approximately my age and they did use calculators in their high school math classes. I attributed my lack of technology to my mathematics teacher.  He was an excellent teacher and was a strong influence in my career path, I don't believe he was of the nature to jump to the latest "trend" in teaching.  He explained to me that he was confident that I would do well in college mathematics because the curriculum and instruction I received my K-12 experience built a solid foundation.

My parents purchased my first digital calculator as my high school graduation gift.  The cost was $150 and it was a Hewlett Packard 35s.   This calculator was in lieu of the traditional gift of suitcases.

I started collecting mechanic calculators a few years ago.  The image below is a particular model I purchased at a flea market.  A hardware/lumber yard store in my community has the same model on display.  Their calculator was the original adding machine used when the company was first established.

My current calculator of choice is the Casio Prizm FX-CG10.  This calculator has everything I want in a calculator.  The graphics are in color and has a minimal number of keystrokes needed to accomplish specific tasks.

Our school to transitioning to 1 to 1.  In mathematics, functions that are defined as 1 to 1 are those whose inverses are also functions.  In the classroom, 1 to 1 means that each student is equipped with a digital device.  As with any innovation, whether it is a tractor or a calculator there is a period of struggle to find its proper place.  As with tractors, there were losses and gains as the calculator entered the mathematics classroom.  Some mathematics concepts have fallen to the side while others have been advanced.  I used to teach linear interpolation as a method to approximate nonlinear results.  That topic is no longer applicable.  I am able to give more meaning to the richness of functions in graphic and tabular ways.  There are appropriate times to use the calculator in class but the use of the calculator exists and cannot be denied.  I believe the same will be true in 1 to 1 classrooms.  As educators, we will have our period of struggle, we will find appropriate times to use the devices in our classrooms but there is no denying that we are in a digital world and we must use all means available to us to engage our students.