Welcome to Summer!

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  The Summer Solstice has officially passed, and many of us have finished, or are finishing up the school year.  It is nice to put the books away for a while - or at least to pretend that they aren't still lurking in the corner of the dining room waiting for me to finish issuing grades and get started planning for next year...
  While we are taking a shorter summer break that usual this year, we have spent the last few weeks enjoying not having to visit the school table first thing after breakfast.  We have spent more time playing in the backyard and visiting the library.  We've been to the pool and eaten popsicles.  We have even done an art project or two.  We are enjoying the comparatively leisurely days of summer.

What about you?  Are you taking a summer break?  How are you enjoying it? 

  On behalf of all of us at the PHEA office, I'd like to say, "Well done Moms and Dads!"  Being responsible for your child's education is a big undertaking.  We hope you enjoy your summer break.
  And to all our graduates this spring: Congratulations!

Great Resources: The Library

Greenville County Main Library

  This being the age of the internet, with all the information we could ever need (and more!) at our fingertips, it seems a little odd that I should tout the library - home of books - as a great resource.  But the truth is, I have found the library to be an excellent place to supplement our homeschool. 
  We use a very literature heavy curriculum and while I do pick up a lot of the readers we need at book sales or online, there are usually several titles each year that I borrow from the library.  This helps save us some money.
  We rely heavily on the library for leisure reading books for the kids.  My eight year old is a voracious reader, so it is great to be able to borrow enough books to keep her busy for a week or two and then trade those in for new books.  The same is true of picture books for our five year old.  I think this is more for my sanity that for his sake - one can only read "Stop That Ball" so many times before a little madness sets in.  We have a great time discovering new favorites and checking out old friends.  And when we pick out a book that we don't enjoy it is easy enough to return it to the library.
  We also use the library for research projects.  It is often difficult to find enough information on the internet written at a level a third or fourth grade level.  But the library has plenty of books on most topics written specifically for children.  Often we do our initial research on the library website, looking up suitable books and reserving them online.  Then we go to the library to pick up the books we have reserved and to consult the encyclopedias or magazines that we cannot check out.   
  In addition to books, our library also has a nice selection of movies and music.  I like that we can pick up a few videos for the kids to supplement our own collection.  They have episodes from popular children's shows - we particularly like the Magic School Bus.  They also have a pretty good selection of older movies, BBC films and book adaptations - Pride and Prejudice, Swiss Family Robinson, the Railway Children.  We get science movies from time to time to go along with our studies as well.
  As for music, the library is a great way to expose your students to different types of music without purchasing a whole collection.  Last summer, for example, we checked out a CD of the Boston Pops 4th of July music and had a marvelous time marching around the house to patriotic music.  There are many other classical collections available by composer, by theme and by performer.
  In addition to books, music and movies, the library has many other great resources.  It is a great place to find all kind of activities.  This summer our library has a reading programs for toddles to teens.  Each child gets a medal for reading a certain number of books as well as other goodies.  Most libraries also have story hours and crafts available at various times as well as book discussions and other activities for teens.  Check with your librarian for a complete list, or check out the events page of the library's website.

  If it has been a while since you have been to the library I encourage you to stop by your nearest branch and see what they have to offer!

 

Guest Post - The Narrative in Number: Teaching the Story in Mathematics By Daniel Maycock

Everyone loves a good story. In fact, narrative is so essential to our humanity that God reveals himself to the world in a story. Art and the humanities are full of narrative elements. Symphonies and string quartets develop themes and motives to create sonic stories which include rising tension, a climax, and a resolution. Even 30-second commercials present stories of despair-turned-to-joy and promise us the same happily-ever-after if we buy their products.

Mathematics too is like a story. That may seem a strange idea, yet it seems strange only because we don’t usually teach mathematics as if it were one of the humanities––which it is. We are happy to teach literature and history and theology through discussion and essay assignments, but suddenly change tactics for mathematics. But what if we taught mathematics in a more human (and perhaps humane) way? What if we taught math in narrative contexts? First, however, I should explain what I mean by saying that math is like a story.

One of the clearest examples of the narrative quality of mathematics can be found in the greatest geometry text to come out of the ancient world: Euclid’s Elements. The Elements is divided into 13 books, the first 6 of which form the basis for what is still taught in high school geometry.

Euclid begins The Elements by presenting definitions, postulates, a few constructions, and the side-angle-side theorem for the congruence of triangles––and with that foundation laid, a world is opened for discovery. I can’t help but compare this beginning to the opening of Genesis. God creates the world out of nothing in six days, and Euclid creates a world out of concepts in 4 propositions.

After this grand opening Euclid unfolds several themes: first the triangle, then parallel lines, and finally parallelograms. Euclid’s handling of these themes is much like the development in a mystery novel where little observations soon uncover a complex web of intrigue. The climax of Book I occurs in the second to last proposition, which we know as the Pythagorean Theorem. Here, theorems on parallelograms and triangles suddenly unite to prove a beautiful and surprising truth: that squares built on the two sides of a right triangle are equal to the square built on the hypotenuse. Taken out of context and put in plain language as I have just done, the proposition seems hardly surprising or beautiful. But this too The Elements shares with the mystery novel. Unless we have followed the story it is not shocking to find out that a cook murdered a millionaire. Likewise, Euclid’s propositions are not surprising to anyone who is merely given them; the story must be read through from the beginning.

But reading the story through from the beginning is not an opportunity frequently given to students. As beautiful and precise as The Elements is, it is no longer used to teach geometry. The reasons for this are twofold. First, it is inefficient to wade through 46 propositions in order to teach the Pythagorean Theorem. It is far easier to simply tell students that A² + B² = C². Second, textbooks are usually organized to present material in the most expedient way, rather than the way in which concepts actually developed and were discovered.

In other words, most textbooks take a CliffNotes approach and tell you all you need to know about the narrative without allowing you to encounter the narrative itself. This may seem an efficient way to prepare for tests but it misses the story and gives the student the impression that mathematics, like Athena, sprang full-grown from the head of Zeus (or from the textbook writers, in this case). The hidden tragedy is that test answers are much sooner forgotten than stories.

As a result, we often teach mathematics as if it were merely a list of skills to learn and concepts to memorize, and consequently treat students as robots to be programed rather than as souls to be cultivated. Students usually have no idea why certain concepts and formulas were developed, and even less idea where they came from. Consequently, students grow bored and frustrated and decide that they hate math.

This is precisely what happened to me in high school. I grew annoyed and confused and decided that math was less important––and certainly less interesting––than literature. Literature had a narrative I could understand; mathematics was nothing but a list of abstractions. Thus, teaching mathematics without properly introducing the narrative and expecting students to remain interested is like telling the punchline without telling the joke and expecting to get a laugh.

For the textbook, it doesn’t matter that Viete’s algebra was too bound to geometry (he didn’t believe one could add a squared number to a cubed number because it doesn’t make sense to add an area to a volume). Nor does it matter that a few years later Descartes simultaneously merged algebra and geometry and, through greater abstraction, liberated algebra from the constraints of geometry. If narratives like these do not matter to the textbook writers, we should not find it surprising when mathematical concepts and rules matter little to our students. But to a student who understands and appreciates the story, concepts do matter, because Descartes’s revision of Viete becomes a triumph on the level with the French discovery of the Rosetta Stone.

Despite my dislike of textbooks, there are several things they do well. They relay information efficiently and home schooling would be more difficult without them. Teaching from primary texts is, after all, messy and inefficient (but then, so is raising children). If you introduce a primary text into your math curriculum, you’ll find that it won’t fit neatly anywhere. Book I of The Elements won’t fit into your geometry textbook in any one place, nor will Viete’s The Analytical Art fit neatly into Algebra 1 nor Descartes Geometry into Algebra 2. The concepts from these and other works fall into textbooks in distilled, abridged, and reorganized forms––like meat in Spam or bologna.

Yet there is good news! Although textbooks remain the only option for many, growing numbers of home schoolers and classical Christian schools are finding ways to put primary texts back into math curricula. A good place to begin is with the first book of Euclid’s Elements. Although, as I said, it won’t fit neatly into your curriculum, Book I makes a great supplement to any high school math curriculum. It presents the basics of high school geometry in a logically complete system with fewer than 50 propositions, which makes it relatively easy to use alongside a modern textbook.

You may find, however, that the narratives, even in Euclid, are subtle and difficult to discover. But the work is worth the reward. When perception dawns, you may feel, as I have felt, that for a silent moment you stand upon a mountain overlooking a sun-splashed valley. And when this happens, you are thinking no longer as a mere student, but as a mathematician.

Daniel Maycock is the founder of Polymath Classical Tutorials (www.polymathclassical.com) where he teaches Classical Mathematics and offers summer workshops in writing and mathematics. Daniel also works for Memoria Press Online Academy where he teaches Composition, Literature, and Material Logic.

A Lesson from Paul Revere

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  I took a few minutes the other night to read “And Then What Happened, Paul Revere?” by Jean Fritz.  When I started flipping through this forty-five page book written for elementary school students, I didn’t think I’d learn anything I didn’t know before, but I was wrong.  I knew that Paul Revere was by trade a silversmith.  What I did not know was that in his ever-present need to provide for his large family, Revere was also at various points in his life a bell ringer, engraver, dentist and maker of false teeth.  During the Revolution, not only did he take part in the Boston Tea Party and go on his famous midnight ride, but he was also a messenger for the Committee of Safety, lieutenant colonel in the Massachusetts militia, and commander of the fort at Castle Island.  He also printed paper money for Massachusetts, helped set up a powder mill, and learned how to make brass and iron cannon.  After the war Revere went back to his silver work, but he also opened a hardware store, set up a foundry where he made a variety of items from pumps and cogs to stoves and church bells.  He also learned how to roll sheet copper and made copper sheathing for ships and roofs.   During his 83 years, Paul Revere did a remarkable number of things.  It seemed that by sheer force of will he was able to make a new opportunities for himself at every turn.

  And he was by no means the only man of his time that pursued a variety of careers.  In the course of our study of Early American History this year, we have had the opportunity to read about men like Benjamin Franklin, Eli Whitney and Robert Fulton (though I should point out that Whitney and Fulton came after the Revolution, but they are good examples nonetheless).  All of these men applied their keen minds to a variety of pursuits.  They had a great ability to adapt to the great changes taking place in America and to make something out of the new opportunities presented.

  After I read through the book about Paul Revere, I started thinking about how different the expectations for a modern career path are.  For the most part we send our students off to college where they quickly select a major and then narrow their focus within that major to an even smaller field of expertise.  For the most part, our students are not encouraged to diversify their studies during high school and college.  We send them off to college and the work world trained to do one specific job.

  I’m not saying this is necessarily a bad thing.  If I ever have a brain tumor, I’d certainly like a surgeon who has made neurosurgery his specialty!

  But for most of us, I think more diversity would be a good thing.  During the recent recession we saw many young people either just emerging from college or only a few years out of college who were unable to find jobs in their field of study.  They were often forced to take jobs requiring skills well below their level of training.  Or in some cases they discovered that they had not specialized enough to be eligible for certain positions and that more schooling would be required to secure a job. 

  It has been a long time since the days of “company men” – men or, to a lesser degree, women who worked at the same company from college to retirement ; sometimes working for the same company for thirty or forty years.  The average tenure these days is 4.4 years according to Forbes.  This means that the average person could hold fifteen or more different jobs during his or her lifetime.  It seems unlikely that each job will require the exact same skill set.

  I wonder if we are going back to the days of Paul Revere, when the ability to diversify one’s business pursuits will be the key to steady employment?

  As our children approach high school, and especially as we are guiding them into college and beyond, to keep this concept of diversity in mind.  If we train our students to be adaptable and to look for ways to create their own opportunities – as Paul Revere did – we will help set them up for a more promising future in an increasingly uncertain business environment.