November 20, 2017

I choose to teach!

apple web size copyOur culture is the exponentially growing collection of knowledge and invention that we pass from one generation to the next.  Science and mathematics are especially rich components of our culture with millenia of contributions from many cultures and all regions of the globe.  Unlike other forms of inheritance which must be subdivided and are often distributed with inequity, this inheritance is the birthright of every person, and every person may consume as large a share as he or she would wish. Our world is one of science, mathematics and technology; literacy in these areas is essential for any individual who is to be engaged with and empowered in our world.  The curriculum is a subset of this inheritance which is distilled down and shared with students in the environment we call school.

The America that we know today was built by those who left their homelands because they were not first-born sons  and therefore had very limited birthrights.  Others came from situations where even first-born sons had few if any prospects.  With an education in America a person has prospects to not only survive but to thrive and to provide for children.  But if we measure the value of an education solely in terms of economic benefit, then we ignore the greater part of its value.Owl Box

In times past you might have been a  carpenter because  your father was a carpenter whether or not you were well suited for that occupation.  You might have been a mother and a housekeeper because it was expected,  but even learning to read was not a possibility.  Perhaps you would have been a field hand because that was the only option available.  Carpentry, parenthood  and farm work  are all important roles, and they bring joy to those who choose them.  However, with education you may choose to be a nurse, or a doctor or a software engineer because  the work suits you well.  And along the way you may sculpt your own identity;  you need not accept the identity that others force upon you.

Students do not always appreciate the awesome value of this cultural inheritance, so  a primary function of the teacher is to help the student understand the meaning of the curriculum with respect to the life of the student.  Once the student is engaged the teacher serves as  a guide.  Success is  achieved when the student is  able to function as the most important person in his or her own education.  A successful education system produces graduates who are life-long learners that are able to learn and grow with or without formal schooling.   I teach as a matter of social justice. I teach to empower students. I teach so that students may understand their world, be part of it, contribute to it and reap its rewards.  I teach because the rewards of working with students greatly outweigh the economic remuneration that comes with teaching.

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Book Review: Learning to Love Math by Judy Willis, MD

Love Math

I just bought the book Leaning to Love Math by Judy Willis MD.  The reason I bought it is because I so thoroughly appreciate some of her videos such as What Makes the Adolescent and Teen Brain So Different and What Should Educators Do About These Differences? available on the ASCD Professional Development website.  In that video she describes adolescents as dopamine junkies always in need of a dopamine fix.  As an educator you can understand the physiology of the adolescent brain and work constructively with nature, or you can adopt an alternative ideology which promotes and perpetuates counter-productive power struggles.  The former option promotes learning while the latter obstructs it.

I have barely cracked the cover of Leaning to Love Math, but I am very confident that it will be excellent.  I promise to update this post when I have more to say!

Book Review: Descartes’ Secret Notebook by Amir D. Aczel

Descartes Notebook

Descartes’ Secret Notebook by Amir D. Aczel was thoroughly enjoyable.  Even if mathematics is not your primary interest this book is still a fascinating piece of history.  Descartes lived in fear of the Inquisition from the Catholics, but having been born Catholic he lived equally in fear of persecution from the Protestants.  When he was young he also had serious health problems, so there was no shortage of adversity.  Yet an amazing intellect was more powerful.  Through the many journeys and associations of Descartes one may learn much about the Renaissance world.

From the perspective of a scientist and mathematician it is delightful to see the relationships between Descartes and many of his contemporaries including Kepler, Newton, Leibnitz and others.  It appears that Descartes’ may have been on the verge of discovering calculus but feared the heresy of discussing infinity.  Additionally he had insight into the future work of mathematicians such as Euler.

Descartes appears elsewhere on this site, namely on the page for Cartesian Battleship.

Cartesian Battleship

Cartesian Battleship

Descartes sank my battleship?  Well, not exactly.  But, the Cartesian coordinate system is very useful for so many things.  Children get a sense of the game from the Battleship game.  Milton Bradley used a simplified approach with letters along one axis and numbers along the other.

For teaching mathematics to students a full Cartesian coordinate system including an origin and positive and negative numbers is useful.  The great thing is, students are intrinsically motivated to learn the coordinate system in order to play the game.

Read the full article about Cartesian Battleship, and download the free templates to transform your own game.

Compute Integrals by Taking Derivatives!

When I first learned this technique in undergraduate Quantum Mechanics at the University of Minnesota — Minneapolis I was truly blown away.  Richard Feynman references this technique in his memoirs.  I am pretty sure it was in Surely You’re Joking Mr. Feynman!  When I have taught this technique to my physics students at SUNY Geneseo nd The College at Brockport, well they have shown impressive enthusiasm for the technique as well.  It is a very cool technique, and it applies not only to Gamma and related functions that appear in Quantum Mechanics.  It is also very useful for computing otherwise nasty integrals in four-dimensional space-time that appear in loop-level Feynman integrals in Quantum Field Theory.  Whether or not there is an immediate application, it is an extremely cool technique to explore!  Read more in the full article.

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Group Theory and Indigenous People

The mathematics of Group Theory arises in so many contexts!  Not all of them are expected.  The kinship relations of the Warlpiri people of Australia are seemingly very complex, and one must wonder how the scheme was ever devised.  This is rather nicely covered in the book Ethnomathematics, A Multicultural View of Mathematical Ideas, by Marcia Ascher.  In my article at Warlpiri Groups I build upon the presentation from that book.  I work out a concrete representation of the group in terms of states and operators in an eight-dimensional real space.  This is a fairly advanced topic, but it is a great opportunity to deal with Group Theory in a pragmatic example.

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Why Lissajous Figures?


Lissajous figures may be drawn by moving some type of marking tool back and forth in one direction with one frequency while simultaneously moving it in a perpendicular direction at a second frequency.  The resulting curve will be closed when the frequencies are in an integer ratio.  The results can be somewhat amusing, like a child’s Spirograph toy.  At the same time, these curves are rich with mathematical content interesting to mathematicians, physicists and engineers.  For a more in depth discussion visit my Lissajous Figures article.  The curve you see plotted here was generated by a Matlab program with a Graphical User Interface which is available for free download to users of Matlab.  The program is useful for investigating the properties of Lissajous curves and their derivatives.  The discussion is rich with the periodic functions sine and cosine with various parametrizations.  The curves are parametric curves, and the derivative of a Lissajous curve is another Lissajous curve.  Jules Antoine Lissajous was inspired by these curves to create a mechanical device for projecting them on a screen, and others improved upon the concept in the form of the harmonograph.

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