No school due to great weather: Seattle principal gives students rare 'sun day' off - Washington Times

No school due to great weather: Seattle principal gives students rare 'sun day' off - Washington Times

This has been an idea of mine for about 30 years.  A built-in snow day that is not used by the end of April.  In May, the superintendent announces that on Friday (an 80 degree sunny day) that there will be no school due to awesome weather ... precious if only it could happen in Minnesota ! 

AP Calculus Presentations

Today is the birthday of Albrecht Dürer born 1471 Imperial Free City of Nurnberg (Germany).  Dürer was an artist who is known for his work as an engraver.  The foundations of descriptive geometry are based on his work on human proportions.
 
Dürer called the curve "muschellini," which means conchoid. 
 
 Dürer designed several such machines.

My AP Calculus students are assigned a research project for their final.  The task they are given is to find the mathematics in something they enjoy doing.  The students usually form groups of two or threes.  In their presentation, they must find at least one higher level mathematics concept that exists in their area of enjoyment, give a historical background of that concept, and a future application of that concept.

This year's topics are:  Mathematics in Marching Band, Easy as Cake, Spiderman, Cards, Color Guard, Basketball, Horses, and Martial Arts/Pakour.

What does this have to do with Dürer?   Dürer found the mathematics in his art and made excellent use of its application.  I do feel that mathematics exists in everything.  Sometimes effort and insight is needed to discover its complexities.  Future blogs will include the mathematics my students have found in their lives.
 

Logic

Bertrand Russell born May18, 1872 Trellech, Monmouthshire, UK.   His work has had a considerable influence on logic, mathematics, set theory, linguistics, computer science, and philosophy.  He said,"A stupid man's report of what a clever man says can never be accurate, because he unconsciously translates what he hears into something he can understand."

Logic is little tweeting bird chirping in meadow. Logic is wreath of pretty flowers that smell bad. --Spock in 'I, Mudd'

Logic: The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding.  Ambrose Pierce

A logic puzzle:
1. The first question with B as the correct answer is:

A. 1
B. 4
C. 3
D. 2

2. The answer to Question 4 is:

A. D
B. A
C. B
D. C

3. The answer to Question 1 is:

A. D
B. C
C. B
D. A

4. The number of questions which have D as the correct answer is:

A. 3
B. 2
C. 1
D. 0

5. The number of questions which have B as the correct answer is:

A. 0
B. 2
C. 3
D. 1

Hint:  Look closely at question 1.  It can't have A or B as its answer.

Buckets of Rain

"Buckets Of Rain"

Buckets of rain
Buckets of tears
Got all them buckets coming out of my ears
Buckets of moonbeams in my hand
You got all the love honey baby
I can stand.

I been meek
And hard like an oak
I seen pretty people disappear like smoke
Friends will arrive friends will disappear
If you want me honey baby
I'll be here.

I like your smile
And your fingertips
I like the way that you move your hips
I like the cool way you look at me
Everything about you is bringing me
Misery.

Little red wagon
Little red bike
I ain't no monkey but I know what I like
I like the way you love me strong and slow
I'm taking you with me honey baby
When I go.

Life is sad
Life is a bust
All ya can do is do what you must
You do what you must do and ya do it well
I'll do it for you honey baby
Can't you tell? 

Bob Dylan

 

Rain Gutter Calculus Optimization Problems

1.  A strip of copper wire is 8 inches wide and needs to be made into a rain gutter by turning up the sides to form a trough with a rectangular cross section. Find the dimensions of the cross section if the carrying capacity must be maxed out. 

 

2. A rain gutter is to be constructed from sheet metal "30 cm" wide by bending it "10 cm" in from both ends. Find the angle of those bends that will result in the maximum water-carrying capacity.

 

 

My optimization lesson :-)

 

 

Mathematical Community

Today is the birthday of Pierre Brocard born 1845 Vignot, France.  He was best known for his discovery of the so-called Brocard points of a triangle.

In a triangle ABC with sides a, b, and c, where the vertices are labeled A, B and C in counterclockwise order, there is exactly one point P such that the line segments AP, BP, and CP form the same angle, ω, with the respective sides c, a, and b, namely that angle PAB = angle PBC = angle PCA.

Point P is called the first Brocard point of the triangle ABC, and the angle ω is called the Brocard angle of the triangle.  The Brocard point is constructed at the intersection of  3 circles.

I had mentioned previously that I was prodded by a colleague to start working on my master's degree in mathematics.  Another factor in my motivation was the regular mathematical dialogue between us while we commuted.  He was taking a class on problem solving and I had not yet began working on my degree.  As we commuted, he would pose a problem and we would generate solutions without paper, pencil, and most definitely without a calculator.  If our answers differed, we would justify our solutions with the mathematical knowledge we had at hand.  I found that our mathematical discourse to be gratifying.  I had to be part of a community where I could have those types of conversations so I began that educational journey.

I suppose the reason why people choose to form book clubs or musicians gather to jam is to share and learn.  The exchange of intellectual thought appears to be part of our genetic code.  A Greek Gymnasium was a place for socializing and engaging in intellectual pursuits.  After we satisfy our basic needs, we seek an intellectual community.

I often feel that mathematicians are viewed as isolationists but in reality mathematicians search for a community in which the enjoyment of mathematics can be shared.  I attended a presentation on the derivation of the cubic formula, a formula similar to the quadratic formula but used in cubic functions. In the middle of the presentation, I paused my attention and observed the audience.  The audience hung on every word.  The questions and discourse that followed was lively and excited.

At the state Math League Competition, hundreds of students were also filled with excitement and anticipation watching what is dubbed the Math Bowl, eight students on stage, answering mathematics questions within a 90 second time period sometimes not using a calculator.  They were part of a mathematical community where others like themselves were gathered doing math.

The Binary Tree of Life

Today is the birthday of Edna Kramer born 1902 Manhattan, New York, USA.  Kramer was a prolific writer of mathematics.  Her works include but are not limited to Mathematics Takes Wings (1942) which related aeronautics to many different topics in the high school mathematics curriculum and The Integration of Trigonometry with Physical Science (1948) which showed how trigonometry could be taught with applications to electricity, sound and light. As co-author of Experiences in Mathematical Discovery (1966), she developed special materials for the student of general mathematics. 

Kramer's parents believed strongly in the importance of education and her father served on the New York City Board of Education.

My father also served on the Plainview Board of Education without having earned a high school diploma.  His active involvement imprinted the importance of education into my belief system.

My wife's grandmother sold eggs to support private piano lessons for her daughter, Rita.  As a young girl, Rita walked once a week from their farm house to the town of Plainview...a walk of at least 3 miles.  Commitment and eggs infused the importance of music in this young girl's life.  The 10 children raised by Rita were not only taught the value of hard work but also importance of music.  My sons received a rich music education because of their mother.  My oldest married an orchestra teacher and I do not doubt that the music lessons will continue.

A student of mine showed me a diagram earlier this year that he had drawn illustrating a binary effect of taking or not taking a particular action.  The student's diagram was similar to the example at the right.

My student feels that a binary tree is formed every time we act or don't act on a decision.  That action or inaction ripples through the future.  He plays with the thought that perhaps a binary code exists for life and future events could be predicted with a given probability.  I think that this freshman has quite the future ahead of him.

Moments of Mathematical Inspiration, Part II

Today is the birthday of Augustin Fresnel born 1788 in Broglie, France.  Fresenel did important work on optics where he was one of the founders of the wave theory of light. At age twelve Fresnel began his studies at the École Centrale in Caen. He was introduced to science and he began to show a liking for mathematics.  He contributed his interest to the inspiration provided by his teachers. He decided on a career in engineering.  He entered the École Polytechnique in Paris in 1804 where he began his analysis on optics.  He often quoted that "Nature is not embarrassed by difficulties of analysis." 

Inspiration, passion, perseverance, and enjoyment are ingredients that are blended in the biographies of many mathematicians.

Another story of a moment that I knew that mathematics was my chosen field.  I had just started my masters degree in mathematics.  I had put off this journey for some time.  I had doubts about my ability in mathematics but I had recently found a paper I had written as an undergrad to gain acceptance into a student teaching position.  In this paper, I had listed a number of life goals, one of which was obtaining my masters degree by the age of 40.  I was 35 at the time.  Also, I was commuting to work with a friend who had started his graduate work and he prodded me, relentlessly so I began the journey.

One of the my early classes was on teaching geometry and in this class the students were required to solve several proofs.  A proof that was assigned was:

 Given a convex quadrilateral and midpoints on each side, prove that the midpoints are the vertices of a parallelogram.

The following is my proof:


 1.  Segment FB and segment EC are parallel because of the triangular midline theorem.
 2.  Segment HG and segment EC are parallel because of the triangular midline theorem.
 3.  Segments FB and EC are parallel because of the transitive property.
 4.  The length of FB is 1/2 the length of EC as is HG by the triangular midline theorem.
 5.  Thus FB and HG are of equal length and parallel.
 6.  Quadrilateral FHGB is a parallelogram since opposite sides congruent and parallel is a sufficient condition.

I had worked on this proof the next day and finally solved it as I was driving on the way home.  I was literally driving and deriving :-)  When I completed the proof, I looked up and saw that a milk truck was barreling in my direction in the lane I was occupying - the left lane.  I immediately corrected course and avoided the collision, however, I felt at peace.  I was ecstatic by my solution and was resigned to any resulting consequence, even one that may have been considered by others as a tragic end.  I was the only one the next week that had determined the proof.  Obviously, my fellow students disapproved of solving math problems as they commuted.

Inspiration, passion, perseverance, and enjoyment coincided at that moment and resulted in a powerful moment I will remember for a long time.

Moments of Mathematical Inspiration, Part I

Today is the birthday of Karen Smith born 1965 Red Bank, New Jersey, USA.  Smith's research is in the areas of algebraic and commutative algebra.  Algebraic geometry is a branch of mathematics, classically studying zeros of polynomial equations.  Algebraic geometry has applications in coding theory which is used in compact discs and spline theory which is used in computer graphics.

I have a favorite quote from Karen Smith which I would like to share.  She states, "One cool thing I remember was learning how to "cast out nines"* to check my multi-digit multiplication in elementary school. ... This trick intrigued me already in elementary school, but later in middle school, when we learned about arithmetic in bases other than the standard base 10, I tried to apply this trick to check my work in base 8, and realized it no longer worked. I was very excited to realize that in base 8, one needs to cast out sevens instead! Finally, in high school, when I learned a little modular arithmetic ..., I was thrilled when I finally figured out the complete generalization to any base, including a formal proof. I finally fully understood something that has intrigued me for years! A great feeling."

I have had cool moments like Ms. Smith's in my education that intrigued me to study mathematics and become a mathematics teacher.  A simple example is when my eighth grade teacher, Mr. Schiller did an algebra proof of how 1 and 0.999... are equivalent.  The proof is as follows:
Let x = 0.999...
then 10x = 9.999...
thereby 10x - x = 9.999... - .999...
resulting in 9x = 9
so x = 1 and x = 0.999...

I later learned of infinite series and limits was able to derive that 1 and 0.999... are the same.  The best, and possibly the simplest proof that was given to me was by a seventh grader named Ben.  Ben, looking very studious, stated, "Mr. Kruger is seems quite obvious 2/3 is 0.666... and 1/3 is 0.333... so their sum, 1, is equal to 0.999..."  A professor was impressed when I told him of this story.

*Cast Out Nines

Today is AP Calculus Day

Today is the birthday of Allen Shields born 1927 New York, New York, USA.  Shields worked on a wide range of topics such as measure theory, complex functions, functional analysis, and operator theory.



Today my AP Calculus students and students across the nation and world will be taking the exam.  This test is a culmination of an entire year of tears and laughter.  These students have worked extremely hard to gain college credit.  Students are scored from a range of 1 to 5.  A score of 3, 4, or 5 usually earns college credit for the student.  There will be students that have chosen to endure the class who will not score high enough to receive college credit.  I believe that all students that have taken this class are well-prepared for a post-secondary experience.  My son, Sam, did well in AP Calculus and earned college credit.  He was at one time considering studying Neurology.  He was confident that he had the education background to explore that major.  He decided to close that door and became an actor.  When he did so, he remarked to me, "Gee, Dad I won't have to take another math class!"  Oh, well . . . I have even heard that remark from former students that are becoming engineers :-)

My goal is to energize students about the beauty of mathematics.  However, I feel that I overwhelm some students and actually cause math anxiety.  Is that anxiety a result of their lack of strong fundamentals in mathematics, the pressure of gpa and class rank, or is it the intensity that I bring to the classroom?  Is my relentless questioning and probing intimidating?

To all my former calculus students, thanks for your efforts, and to this year's class GOOD LUCK :-)

Bent Christiansen and The Classroom

Today is the birthday of Bent Christiansen born 1921 in Aalborg, Denmark.  Christiansen was a teacher of mathematics and a teacher of teachers in Denmark.  In December 1969 he addressed the First International Congress on Mathematical Education in Lyons, France. Here is part of the introduction to his address:

"It is my belief that the very necessary changes in the millions of class-rooms with regard to the approach to mathematical education will not take place unless we explain ourselves at many levels of language. At one level we will have to convince the students at universities and training colleges of the necessity of using new means. At other levels of communication, we will have to motivate for debate the participants in the in-service training, the students in the schools and certainly also the parents and the authorities. While the research increases with regard to mathematical teaching, thereby providing sharper and stronger answers to important educational problems, it is thus in my opinion - for implementation purposes - still necessary to discuss in a general way the philosophy of mathematical education."

The instruction that happens in the classroom is the focal point of the school.  The connection between the teacher and the student can transcend time and space.  There are moments that connection is so strong that I am oblivious to the rest of the world.  I am literally high.  Those moments are my drug of choice.

I do believe that there are necessary changes that need to occur in the classroom.  Students need to be engaged and as instructors we have the tools to engage them in and out of the classroom.  The engagement needs to be personal.  The engagement needs to be challenging.  A positive component about MCA and AP testing is that there is a line drawn into the sand for both student and teacher to cross.  That crossing requires focus and effort by student and teacher.

I am frustrated that a call for change was announced in 1969 and now the year is 2013, a span of 44 years.  I am completing year 33.  Time is becoming short.  In every grade level, there is a large group of students that needs a new type of teacher.  Those students are bored.  They need to realize the opportunities that available to them.  I teach students in a college that have come to that realization, however, math is a struggle for them because their weak background.

Today is the birthday of Jean Hachette born 1769 in Mézières, Ardennes, France.  He worked on descriptive geometry.  Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions, by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art.  I have included examples of four different 2D representations of the same 3D object.

I draw a picture almost every time I solve a problem.  I actually sketch  a diagram on a coordinate plane, label it, and then apply either algebra or trigonometry to solve the problem.  I also use tables to solve problems.  I look for patterns in the tables.  My last step is to graph it on my Casio Prism.  The calculator allows me to examine detail.

I learned using diagrams to solve problems from my father.  I remember when I was young working in his welding shop how he reacted to customers asking him to do a particular job.  He had a pad and pencil in his pocket and as the customer started to talk, my dad would thrust the pad and pencil toward the customer.  Many of the diagrams were rather crude but my father was able to discern the scribble and create the item to the satisfaction of his client.

I recently asked my dad what he enjoyed about his job.  He told me that the challenge of solving a new problem . . . something he had never seen.  I know that the thrill of solving a hard problem continually brings me back to mathematics.  That success does not happen often.  I try to keeping pushing the limits of what I can solve.

A Foot in the Mechanical and A Foot in the Digital

Today is the birthday of Johann Faulhaber born in Ulm, Germany, 1580.  He was an algebraist and is known for his work in explaining logarithms.

Logarithms brings back memories of using slide rules.  I am no longer proficient in their use but my high school mathematics experience was one void of calculators.  I did buy a calculator for college.  It was an HP that I plugged into an outlet nightly.  I struggle at times with the digital world.  I do not game for my enjoyment.  I read and watch television.  My son, Sam, believes I struggle between the transition from analog to digital.  "Your feet", he quips to me in a recent excursion to Minneapolis, "are firmly planted in both worlds."

I recently attended the MCTM conference in Duluth and went to a presentation by a middle school teacher who constructed a class management system based on gamification . . . earning points, moving up to levels, and gaining rewards.  "Are these rewards extrinsic?", he concluded, "Yes but what are grades?"  He is a teacher that has grownup in the digital.  He games.  I do not, have not, and probably will not.   I play pinball, real pinball, not virtual pinball.

A discussion in at a recent 6-12 district math teachers focused on the use of graphics calculators as aps on cell phones.  Alarming to me - I'm still trying to understand my "dumb" phone and my Casio Prism calculator.

What does this mean for my students?  I am creating digital media clips of instruction on various standard based units sans Khan Academy . . . not a truly flipped classroom - a hybrid of sorts.  A mathematics teacher at Tech high school does nothing digital and has a great deal of success.  Sam, I do have a foot firmly planted in the past but only a toe testing the waters of the future :-)