The Finnstitute

We’ve been studying, and have created a main lesson book about the Finger Lakes. This winter at a holiday event, a parent of a former student at Stone Circle School said, “ She just sang us, word for word, the song you wrote for her class  about the Finger Lakes in third grade! “ (This student has just graduated college.) I had been thinking about reviving that study, and this gave me the impetus.

So the first thing we did was learn the song together. Some of you might have heard your children singing it; it’s a melodic way to remember the names of all the lakes.  “Cay,Cay uga, Cay,Cay,uga…”

Then we did a two-page  map of the whole Finger Lakes region including all 11 lakes and notable cities and towns in the area. Next we learned about the East Lakes and the West lakes, and we did a chart of the Finger Lakes showing for each lake: altitude of surface above sea level (ft.), greatest depth(ft.), altitude of bottom above or below sea level(ft.). The marked difference in depths and altitude of surfaces would be better-explained later on when we studied the geology of how the lakes were formed.

Geology: The Pleistocene Epoch in N. America was 2 million years ago. 10,000 years ago the last ice sheets retreated, after covering all of NY State but a small area near what is now Allegheny State Park. The ice was completely melted in Canada 7,000 years ago. These retreating glaciers filled with mud, sand, gravel and boulders ground and scoured the rock. South-flowing V-shaped water valleys were gouged into U-shaped valleys and troughs. Water trapped at North and South ends by glacial debris became the Finger Lakes, flowing North! Keuka is the only lake that flows into another lake(Seneca). The glacier moved at a speed of 1 meter a day. We did an experiential exercise and moved as a group at the speed(?) of a glacier…Drumlins, eskers, kames, kettle lakes (Dryden Lake) and moraines were formed by the depositions of the glaciers

To localize it a little more we drew beautiful, powerful pictures of Taughannock Falls before 1888, as shown in an old photo, and after, as shown in a later photo,1892,  where the crest broke in the Sherburne flags to the re-entrant angle now seen, drastically changing the water flow and appearance of the falls. Also we drew in our books the beautiful, 3D maps of the geological features and glacial footprints of our area as developed by the Cornell Geology Dept. These show the southern, eastern and western drainage areas flowing into Cayuga Lake at Ithaca, the southern highlands and the lower topography and drumlins north of the lake.

History: Pre-Iroquois habitations were found at Bluff Point (Keuka Lake). Seneca and Cayuga tribes were joined by other southeren tribes during colonial times, for protection. The Iroquois held the strongest military force in North America at that time, and were able to hold off settlement by whites for almost two centuries. The Siouan, Tutelo-Suponi tribes came up from what is now Maryland, to build a stockade village at Coreogonel, under Cayuga protection. ( where Lick Brook inlet meets Rt. 13). The Iroquois adeptly played the French off against the British. But later, during the American Revolutionary War, some Iroquois tribes sided with the British, and some with the Americans, resulting in the eventual civil war and break-down of the once-mighty Iroquois Nation.

Here we interjected into our studies the Seneca creation myth, the Evil-minded and the Good-minded, with a startling drawing of the first sprouting of the tobacco, corn, beans, squash and potatoes from the mother. We also heard the tale of Drop Star, a child raised by the old chief Skenandoh, along the shores of what is now Kayuta Lake, south of Seneca Lake. Irish Settlement road was built on the North\South trail that separated the Cayugas from the Onondagas. The Irish came over to flee the famine that killed a million in Ireland. Painted Post was so named because of a bloody post in that spot covered with scalps taken by the tall, fierce Seneca Warriors. (Now it’s a road sign on Rt. 17).

Architecture: After the Seneca Longhouse declined with the onset of white settlers, styles from abroad became adapted and popular, such as the Federal Style. But it was the fervent madness about all things from the Greek Classics that overtook the region with  the style known as Greek Revival that showed itself in towns sprouting up all around the Finger Lakes, towns with names like Aurora, Ulysses, Ithaca. One prominent feature on all these structures was  the use of columns, in the Classical style. We drew very careful beautiful examples of Doric, Ionic and Corinthian columns that are prevalent around the area. We studied the many museums and gardens that display these various architectural achievements, and the many prominent educational institutions in the Finger Lakes that are famous for their architecture. We talked about and listed many of the Historical museums such as the Women’s Rights National Historic Park in Seneca Falls, home of the Suffrage movement in America, and the Harriet Tubman House, home of the heroine of the Underground Railroad, Harriet Tubman.

Erie Canal: Certainly one of the chief historical events in the history of America was the building of the Erie Canal, Clinton’s Ditch, a project so preposterously massive at the time that President Jefferson called it sheer madness. We learned about the struggles to convince backers to fund it and governments to support it, the many new immigrants that built it, the amateur engineers that visualized and planned it, and the enormous effect it had on NY State, the US  and the world. It passed along the northern edge of the Finger Lakes. Our Lake, Cayuga, feeds into the Montezuma Wildlife Refuge; but at the time of the Canal Constructuion, the only stoppage and really protracted delay came from an outbreak of malaria in what was then Montezuma Swamp, killing thousands of workers and shutting down the project till the cool weather abated the disease. The Hudson was an Estuary, America’s only fjord. (I remember eating  sturgeon caught in the Hudson as a kid) eminently navigable all the way from NYCity to Albany. “From Albany to Buffalo !” We learned that famous folk song and sang it as we worked on drawings, drawings of how a lock worked on the Canal, drawings of canal boats being pulled along the towpath, typically north side, by a horse or mule; along the 40 ft. wide, 4 ft. deep trench dug hundreds of miles through wilderness, ascending some 565 ft from Albany at sea level to Lake Erie. With access from NYC to the Great Lakes and the Midwest farms  via the Erie Canal, Buffalo became the grain capital of the world! Rochester became flour capital, and New York City became the most important harbor on earth. We drew an interesting pie chart of goods shipped along the Erie Canal in 1849, grain and flour being the most, but other goods like bacon, sugar and butter in the millions of pounds! We also drew a bar graph of the years and amounts of commercial activity along the Erie Canal, which told the mathematical story of its’ rise and fall.

Obviously we did many other things this month, drama, geometric drawing with compass and rulers, craypas drawing\painting, creative writing, using some of my original poetry as a starting, structure point,  but that will be for another re-telling.

Volcanoes: Shield, Fissure, Composite, Cone-shaped, Crater Volcanoes
Strombolian, Vulcanian, Peleean, Plinian Eruptions
Volcano zones around the earth.
Famous eruptions: Mt. St. Helens, 1980, Thera destroys Akroteri, 1645BC (possible Atlantis myth), Mt.Vesuvius buries Pompeii, 79AD, Krakatoa near Java,1883, heard ¼ around the world, Tambora, Indonesia chilled the whole earth  with a sky full of ash. Turner’s famous, dazzling skyscapes were attributed to this ashen atmosphere! 1816.
Earthquakes:  Epicenter, Hypocenter,  Techtonic plates, Isoseismic lines
Deep quakes , shallow quakes, primary, secondary waves
Mercalli Scale, Richter Scale, measures strength of shock waves, Chile 1960, 9.5 highest registered. We studied these phenomenon two weeks before a 7.8 shallow quake (most dangerous; 10 H-bombs of force) moved Katmandu 10 ft. laterally and 3 ft. upward. Mt. Everest 100 miles away fell by 1 inch, and aftershocks  caused deadly avalanches. Thousands perished in this natural disaster. Also concurrently a huge volcano erupted three times in South America, causing mass evacuations and shutting down all air travel in the continent. We decided after studying these phenomena  and then history following fast with major disasters, we better study something like babies or rainbows. And later on you’ll see we did study the perception of rainbows.
Earth’s Core, Mantle and Crust:  We made clay paintings of the earth’s layers, with a focus on how shallow and fragile the life-supporting surface is.
Gravity: We did a number of fun and informative experiments about Gravity.
Magnetism, Loadstone.
Walk to Columbia Street Footbridge: We saw the north-flowing creek flowing down from the high ground north  towards the lake, and under the magnificent bridge.
Maps Tompkins County, Finger Lakes Parks Maps – We’re studying the Finger Lakes, learned the Finger Lakes song,  and by studying the various state parks around the lakes, we’re beginning to make plans for a class trip.
Glaciers and glaciated landscapes:  Finger Lakes East/West- We were surprised to learn the very different glacial effects on theEast vs.West Finger Lakes; the glaciers hit the high ground south of the east lakes and then retreated and dragged the effluvial drainage of the east lakes north towards lake Ontario; the glaciers break through the soil in the west and create a drainage basin that works south towards the Susquehanna, with a clear demarcation between Seneca (east) and Keuka (west).
Rainbow Perceptions: Humans, dogs, crabs and butterflies all have very different perceptions of the colors of the rainbow.
No Blue in Odyssey – Prime Minister Gladstone of 1800’s England was obsessed with Homer’s Odyssey. He noticed some very strange, unnatural color descriptions; so he did a thorough cataloging of all the colors mentioned in the work: black-100, white – 95, red-45, yellow – 20 green – 17, and blue —– zero! As blue as the Aegean Sea and the Grecian sky is to us modern people, blue is never mentioned! Scientists and ethnologists found, after studying Gladstone’s work, that in the Scandinavian Sagas, in the Hindu Vedas, in ancient Chinese stories, in early Biblical texts, no blue is ever mentioned!
Himba Tribe  – This Namibian Tribe has many subtle names for green, and in a scientific experiment could perceive the slightest differences; but they had no word for blue and so could not perceive blue as different from green! Without language we seem to have an absence of consciousness. Which makes you wonder what we’re not noticing that’s obviously present in our surrounding.
We’ve begun moving forward in the History of Mathematics, going from the
Golden Mean, and it’s 1:1.618 proportions to Equiangular spiral that exists in the
Chambered Nautilus. Then we followed the startling and world-changing biographer of
Descartes, and his transformation of human thought, through Mathematics.

Pulley, Pendulum, Pressure, Potential Energy

Pulleys – We’ve drawn diagrams of various styles of rigging, and then built them in the classroom and recorded their properties.
Here are some of the setups we looked at:
Fixed Pulley
Movable Pulley W/2 advantage
Gun Tackle Show W/2 advantage
Gun Tackle, “ Rove to Advantage “, W/3 advantage
Luff Tackle, adds fixed pulley to rove advantage, very smooth, W/3 advantage
Double Tackle Show, 2 fixed pulleys, 2 movable pulleys, W/4 advantage
The children enjoyed the smooth, easy lifting power of the more advanced pulleys, and would try them out during breaks.

Pendulums – We did an extensive study of pendulums, starting with tying washers, nuts and bolts of various sizes and weights to one yard of string and thread. Then we also tied string to a large ball of clay, creating a pendulum obviously much larger than the other various-sized pendulums we used. We set the pendulums swinging from fixed points, working in groups of three students. We developed and filled out a chart. From a 45degree angle we dropped the pendulum and counted the movements in one minute (using a stopwatch). We repeated this from 90 degrees and 25 degrees. We used heavier pendulums and lighter pendulums, and finally the enormous clay pendulums. Then we calculated the average of all the findings of each group for the many different trials. No matter what the angle was, or the weight, all the pendulums swung about 70 times per minute. The children were surprised. Then we talked about Galileo discovering pendulum motion sitting in church, and the astounding regularity they seem to swing with, and the physics of it. The first pendulum clocks replaced mechanical spring clocks and greatly increased their time-telling accuracy and regularity, from 15 minutes lost in a day in 1500 to 5 seconds lost in a day in 1600; a good quality in a clock. I also mentioned Poe’s  “Pit in the Pendulum“, much to their excitement, and Mrs. Finn’s chagrin.

Is Air Heavy? – We balanced two balloons (that the students inflated and tied) on a suspended dowel. The kids worked in pairs. One child popped their balloon with a pin. The dowel dipped decidedly. Then the other student popped their balloon with a pin, and the dowel was level again. As you can imagine, they hated popping balloons with pins.

The Power of Air Pressure – We placed a large sheet of paper on a table, then placed a wooden ruler or piece of wooden lath under the paper with about two inches of wood exposed and hanging over the table edge. I asked the children to hit the ruler sharply with their fists to try and lift the paper. They couldn’t ; there seemed to be an enormous weight on the thin paper sheet making it immovable. The next day I brought a hammer and more expendable wooden straight edges. The students were asked to strike the rulers sharply with the hammer. Instead of lifting the flimsy paper, the ruler snapped cleanly off at the table edge! They realized that there was a column of air pressing down on that sheet. Everyone had a turn to hit with the hammer and snap a ruler. Believe it or not it wasn’t that hard to get them to try it, some more than once.

Down – Using our pendulums tied to the center of  dowels, the class, in pairs, holds both ends of the dowel  in such a way that the dowel is level and the pendulum hangs straight down. Then one student lifts and the other dips their end of the dowel. Then they reverse that motion and the other lifts and dips. In each case the pendulum remains straight up and down, plumb to the center of the earth.

Equal – Using scales we measured out exactly 50 grams of clay and then, in pairs, dropped the clay balls into the hand of  partners, from 1 ft., 2 ft., 3 ft., and 4 ft. We observed the difference in force at each height. We learned about  Energy (E), Work (W), Potential Energy (PE), Gravitational Potential Energy (GPE) , Kinetic Energy (KE), and Joules, a measurement of work.

Earth Pressure – We fastened balloons to a bicycle air pump and then, using charts, counted the pumps and monitored the appearance of the balloons as each child pumped them up. We continued until the balloons got so big and transparent, and the tension in the room got so excited, fearful, and insanely expectant — until – SUDDENLY – POP!!!. Though we did it twelve times, and the  number of pumps varied from 85-115, the students went crazy with tension each time we got to around 70 pumps. It was a palpable, measurable tension and insanity. Nothing prepared anyone for the sudden, loud, upsetting explosion, even after the first eleven times. It was remarkable, scientifically but also psychologically. This segued easily into our study,   which we will be continuing with, of Volcanoes, and Earth Forces,  and  Earth Science.
Needless to say, this was another arduous experiment, wherein I had to convince children to explode things.

The Human Face through History
We’ve been very deliberately studying how the Human Face has been presented and perceived by the many great cultures through history. Each cultures’ artistic representation of the portrait gives a telling doorway into and through the eyes of its’ society. After writing in our Main Lesson books about the aesthetics, history and unique meanings and perspectives of each culture, the students did very developed drawings  in their books of  striking examples from each culture and period. I helped them refine and focus on their drawings, helping them to be more accurate in seeing and drawing of brow lines, mouths, nose shapes, eye-line levels and skull shapes, so they could develop the skills to achieve a likeness. These are not facile pieces or cartoon sketches, but refined, classic and revealing works of art, that require extra work.

The Babylonian mask: The delicate, striking mask, thousands of years old, has the empty eyed stare of the mask, but the most extraordinarily; modeled mouth, subtle smile and soft cheeks. It is a face so supple and so present; thousands of years falling away from someone we recognize.

The Burial Sculpture of the Pharaoh: Because the Pharaoh was believed to be a god, he could not be taken away by death and forgotten, as mere mortals are; his image monumental and with a careful, but ennobling likeness needed to keep the god pharaoh alive after his body was mummified and entombed. These are heroic, culturally desperate, aesthetically imposing efforts to transcend death.

The Chinese Sacrificial Prisoner and Monk: The hollow eye cavities and grim, fearful mouth in  the kneeling figure of the prisoner, hands tied behind his back, are so compellingly anonymous, unimportant, of the slave class. The noble Buddhist priest’s portrait is so delicately drawn in fine graceful lines with great attention to detail and likeness; he is clearly of a class worth the artistic effort.

The Greek Head from Benevento: The beautiful Greek head is idealized, serene androgynous and  perfectly proportioned. This is the idea of a face that uplifts and has no ugliness. This is a portrait of the democracy, the republic, not an individual but a symbol of the Greek citizen.

The Roman Senator: The strong-jawed, willful,  lifelike, lined, accurate portrait head, so unidealized, unflattering , truthful, and powerful.

The Painted Coptic Death Portraits: The dark, huge mesmerizing eyes of the painted on wood portraits are so compelling; you can’t look away. Yet what a transformation, evolution in consciousness, that in the third century AD, after the sculptural realism of all the earlier cultures, that the soul and true inner nature of these beloved dead could be best captured in a painting, in two dimensions, in the first threshold of the Abstract!

The Mayan Whistle Priest: The elaborate head-dress tells the rank of this high priest, and his costume. Compare this with the beautiful, serene mask with the pointed hat and the lifelike portrait of a woman in jade. The hierarchical class structure of the Mayan culture is carefully catalogued in how each  is clothed, bejeweled, and dignified.

This is the beginning of a study that will go through African masks, the medieval and Giotto, the Renaissance, and Raphael, through the Dutch and English portraits, Goya’s condemning royal portraits, master of the self-portrait Rembrandt, the Impressionists and Van Gogh, the Expressionists and Munch, the Cubists and Picasso, etc. We will visit the Johnson Art Museum, where we can see examples of ancient and modern busts and portraits, including a show of Dutch Portraiture, now on exhibit.

Math Puzzles and Number Conundrums
As a balance to the intensive drawing, history, and emotional exploration of  the human face, we’ve been working on many wonderful, some easy, some more difficult, and some impossible math puzzles. Also simple and complex math mysteries have been fun, challenging, frustrating; and revealing as to the students’ grasp of number patterns and processes.

American History, Social Studies and Government Questions
By quizzing the students on varied American history and government questions, and then re-questioning, the children were able to slowly build up a rapport and repertoire of American History knowledge. This also gave me clear pictures of the areas where the students need study and deepening.

Symbols
We began a careful study of the many powerful symbols that create our culture and consciousness, from the ancient to the modern. What is so startling to the children is how similar and derivative the old and new iconographies are. We started with the braiding of the Celtic Knot,  US Biohazard, Cave Ideogram, Octogram of Creation, Seal of Solomon, Trinity Braid, Chinese Shou Long Life, Gnostic Star, Star of Ishtar Goddess, Star of Venus, Cross of St. John, Cross of Endlessness, Atomic Energy-Nuclear Reactor(modern), the eerily similar Dangerous Power, Highest Power (ancient), etc. We’re working carefully on these symbols which involve geometry, number, specificity and beauty, and we’re creating little books with this work.

Measurement – our measurement studies in December included:

Dry Measure – Using fruit, vegetables and grain we studied market measure, pints and quarts of berries, ½ pecks and pecks, bushels of apples, barrels of grain, bales of cotton and tobacco, bales of straw vs. hay and wagonloads. Also dry measure in teaspoon, tablespoon, cups, for flour and spices.
Liquid Measure – Using many  cooking and baking measures of varied sizes, we filled gallons of water using 1/3 teaspoons, tablespoons, ½ cups, cups, pints, quarts, etc. getting our hands wet and experiencing the differences in measure. We also studied firkins and caldrons and barrels and the old wine measures.
Distance Land – Starting with the old style measures using the human body, i.e., the digit, the palm, the hand, the cubit, the Biblical cubit, the common pace, the military pace, we discovered that everyone in the class came up with different measurements for objects in the room and the distance across it. We learned about the standard “ ruler “ Charlemagne demanded, and the story that 12 inches made one foot, the size of the King of England’s foot, the “ Ruler “.  The need for standardization became quickly apparent as arguments over goods and lands needed to be settled.
So we came to the inch, foot, yard, rod, chain, furlong, mile, league. Using rulers, yardsticks, tape measures and 100ft. Tape wheels the children measured all sorts of distances.
Nautical- We studied the fathom,  cable, nautical mile, nautical league, knots, 1/6 degree, etc. and discussed how these measures were used during the Age of Exploration.
Area – We discussed acreage, people’s yards or land,  square numbers, survey maps, the Holland Land Company. ( I’d like to take them to the Assessors map office where my friend Jay works to look at maps, and if allowed, to the County Clerks office to look at old deeds in large books, written in chains and rods .)
As luck would have it, the tiles in the atrium in front of the studio are one square foot in size; so we measured the square footage in the atrium using the 100 foot tape, and the concept of square feet was visible and clear, right before our eyes.
Cloud Formations – We studied the various cloud formations found at different altitudes, and their configurations and properties.
Wind Speed- Nautical Beaufort Scale -We wrote up as a table the colorful and descriptive wind speed table developed by Admiral Beaufort for sailors.
Water Temperature for Fish – we wrote up a table listing many common species of fish and the water temperature ranges they are most commonly found in.
Value – Bartering, Trade, Monetary Systems – We looked at the various ways humans traded and measured the value of goods and commodities, and the development of monetary systems.
Currency Comparison Tables – Ruble Falling – The students looked at and saw how different currencies were holding a comparative value, currently, as in that one day. ( We didn’t discuss nano, high-speed trading ). They all wanted to go to where they could get 12,500 rupiahs for a dollar, and buy stuff. We discussed the collapsing ruble, and the falling oil prices, and they began to understand how volatile some of these systems could be; and how the value of things can change.

Excursions
Cemetery, Celtic Cross – The cemetery between university and Stewart Ave. is a complex, fascinating place, rich in histories, headstones, mausoleums, and magnificent trees.  One large Celtic Cross was full of intricate braiding patterns that we sat and tried to draw.
Cascadilla Gorge-  We hiked once to the first bend in the gorge and drew what we saw. The next excursion we hiked the entire way up the refurbished gorge trail, stopping many times to observe, all the way up to Collegetown, and then back down through the neighborhood and E.Seneca Street.
Stewart Point  Atmospheric Drawing Trip –  We hiked in behind the Boat Club to the point and sat on logs looking out  north at the foreground, middle ground and distant treed background across the expanse of lake. We observed the distinct shapes, light and shadow contrasts in the foreground steadily muted as the vision went back through middle ground of lake waves, geese and trees, then back to the uniform, grey, muted hill in the distance.
Cass Park Measurement trip – We all walked the  hiking, fitness trail from the Cass Park playground, along the inlet to the boat launch and back. We all (those who didn’t lose count!) had a different number of paces, further reinforcing the need for standard measurement.
Osprey Point Drawing Class – After a wonderful walk past the Osprey nest on the knoll we went to the shore to draw driftwood and then walked back to draw the marina and Cornell towers in the distance.
Treman Park Playground trip
Buttermilk Falls Playground trip – After some strenuous academic mornings studying the Renaissance, and the history of Mathematics, we would take the kids to the new playground at lower Treman and at Buttermilk Falls park. The playground at Buttermilk was designed by my former co-worker, designer, organizer at Leathers and Assoc. Architects, Steve Lauzan.  Steve branched out on his own and slowly grew his PG design company. His dynamic, creative, inclusive designs are now in parks and communities all over the country. It was great to see the children playing on structures sprung from a mind I knew.
Six Mile Creek Water Drawing – We are lucky to have a wonderful, dynamic body of moving water just outside our back door and across the street. Often we go there to skip rocks, make rock sculptures, or to draw, through careful observation, the movement of water.
Henry St. John Playground Recesses – We often go there for a short break to climb around, have snack or lunch and play games. It’s just a few minute walk from our class.

Fine Art projects –  We’ve worked on geometric drawing, using compass, ruler and colored pencils, exploring various configurations of circles spirals, logarithmic spirals, triangles, squares and complex polygons. We  also did colored rubber band drawings on twenty four division of a circle boards made by previous Finnstituters.
The children loved doing oil pastel paintings, where they cover the entire page with color so it’s painting not drawing; they asked to continue and didn’t stop for snack.
We’ve done some work with pen and ink, a form that requires attention to detail, constant dipping and cleaning; and a quiet peacefulness that comes over the group. We will continue this skill and expand into printmaking.
Using the floor to ceiling blackboards we’ve done drawings of Michelangelo’s Moses statue; Form Drawing where we work on repeating patterns; the children love the mirroring exercises where one child tries to follow a pattern created on one side of a vertical line using curved and straight lines. We’ve also created windows on the board which the children fill in with their own views their own imaginations, recollections, explorations.
We did quick sketch studies of the children in an active pose for one minute or two minutes.
One Creative Writing class built out of grammar studies evolved into a silent story-writing classroom that lasted an hour, in peaceful, earnest concentration. Was I dreaming?
Another class had the children sitting at the big window writing down notes and sketches of the occasional person or groups of people walking by just below. The children were fascinated by this, and got very good at making quick notes on clothing, mannerisms, and speculation about their destinations or character.

September Main Lesson   The Renaissance

The Humanists, Petrarch, “a Classical Education”, Machiavelli’s  “The Prince”, the often cited and emulated book on Governing, Castiglione “The Book of the Courtier” an immensely popular book on Courtesy.

Scientific Revolution:  Principle of Doubt, the need for Proof, Scientific Method, Descartes “Discourse on Method”, Sir Francis Bacon, theory based on repeatable experiments.  The Telescope and the Microscope; two world-altering inventions.

Copernicus “On the Revolution of the Celestial Spheres.” Heliocentric theory countering Ptolemy’s geocentric theory.  Galileo discovers the moons of Jupiter revolving around the planet. His “Dialogue on the Two Great Systems of the World” causes him to be called before the Inquisition. Kepler, mathematician, discovers the elliptical orbits can mathematically prove the heliocentric theory.

Vesalius “ On the Fabric of the Human Body” becomes a landmark work in medicine. Harvey describes the heart and circulation of the blood through arteries and veins, the blood vessels. Using the microscope Hooke discovers cell structure in cork, Leeuwenhoek discovers bacteria.
Gutenberg’s Printing Press makes information available to many; growth of knowledge leads to the founding of many universities.

Art:  Florence, Italy,  Giotto, Masaccio,  Botticelli “La Primavera”, “The Birth of Venus”,
Leonardo da Vinci, “Mona Lisa”,  “The Last Supper”, prototypic flying machine, anatomical drawings, Michelangelo, Buonarroti, the sculptures “Moses”, “David”, “The Pieta”, The Sistine Chapel’s “Creation of Adam”.
Brunelleschi’s theories of Perspective, Raphael’s “School of Athens”,  Savonarola’s art burnings.
Dutch Master: Durer, Hals, Van Dyke, Brueghel,  Vermeer, Bosch.
Religious Reformation : Martin Luther, 95 Theses, nailed to Cathedral door,  spread by printing press; prompted by Pope’s extravagances, and burdensome Indulgences, requiring common people to pay for their salvation, religious  sects,  Protestantism spreads, the Catholic Church and the Pope fight back.
Calvinists and Puritans, Henry VIII, King of England divorces daughter of Ferdinand and Isabella of Spain, against the Catholic Pope’s strict orders, and founds the Church of England, so he can remarry.

The Age of Discovery
Technological advances: mapmaking,  navigational instruments, astrolabe, compass, galleys, caravel ships, cannons, latitude and longitude.
The Commercial Revolution, standardized money, joint-stock companies, mercantilism, balance of trade, favorable balance of trade, tariffs, subsidies to promote manufacturing,  colonies to supply resources.
Portugal and Prince Henry the Navigator; Dias to the Cape of Good Hope, Vasco da Gama went around the Cape, across the Indian Ocean to India, bringing home spices and jewels.
Columbus, Ferdinand and Isabella, thought to reach India sailing West.  They landed on San Salvador, and what is now the West Indies, (not the East Indies or India as Columbus believed till his death).
Vespucci sails to what is now Brazil, writes about this new world corroborated by Balboa’s discovery of the South Sea (the Pacific Ocean) after crossing the isthmus of Panama, and a mapmaker calls this new world “America” after the explorer. Magellan makes the arduous journey around South America, and through the Straights of Magellan out into this very vast peaceful sea he called “Pacific”(Latin for peaceful) Ocean, and all the way to the Philippines and then on to Borneo where he was killed in a fight with the islanders. His crew member Elcano and 18  sailors  made it back to Spain, the first circumnavigation of the world.

The Columbian Exchange: the new world gave the old world potatoes, corn and tomatoes, turkey, grey squirrels and muskrat, the old world gave the new world horses, cattle, goats, chickens, cats, measles, typhus, plague and smallpox.
Portugal, Spain, England, France and Netherlands all colonized the Americas, Asia and Africa and traded places as most powerful nation over the next two centuries. Some overextended defending colonies, lost too many men at sea.  The mighty Spanish Armada seeking to wrest revenge and restore Catholicism to England, was defeated in rough seas in the English Channel. The triangular trade between Europe, Africa and the Caribbean,  including the slave trade, was carried on by all those nations, for profit and national power. The Dutch East India and Dutch West India Companies thrived. King Louis XIV, the “Sun King”, Absolute Monarch, as ordained by God, ruled France absolutely, built Versailles, to keep his nobles watched, and sought to expand France to all its natural borders including the North Sea and the Rhine.
The next time we visit world history we’ll explore what happened in North and South America, and in Asia, as a result of this profligate, empire-building by these European nations.
We’ve worked on one-point perspective drawings, and put our rulers and pencils to copies of School of Athens by Raphael, done pencil drawings of Michelangelo’s Moses, and lifesize chalk drawings of the statue. We’ve drawn colored pencil drawings of the Caravel Explorer, the most popular ship in world exploration of the times we studied, and had lively discussions of the remarkable personalities, and ramifications of the events I’ve mentioned above.

 Here is a review of our work this month in the History of Mathematics:

COUNTING
Early tribes used a base 2 to count: 1,2,2-1,2-2,2-3, or a base 3: 1,2,3,3-1,3-2,3-3; then they advance to a number system using their fingers and toes, a base 20 system; remnants of these ancient systems still are echoed in the quatre-vingt and quatre-vingt-dix , four twenties and four twenties plus ten, 80 and 90 in French  counting, and the 20 shilling pound in Britain. The abstruse British monetary system, with ha‘pennies, sixpence, et-al, went ½,1,3,6,12,30,240,252, until quite recently.
Babylonian astronomers, astrologers and mathematicians used a base 60, divisible by  l,2,3,4,5,6,10,12,15,20,30and 60. It fit well into their division of the year into 360 days, and the circle into 360 degrees, concepts still very much used in timekeeping, astronomy and geometry.
Because of the physiological uniformity, the ten base system appeared throughout many cultures, from Tehran to Hong Kong, and in the abacus. The  Hindus developed our 1-9, further augmented by the Ghobar numerals, and decimal system. Florence banned the use of decimal numbers in the 13th century. We studied and drew the number symbols of the Arabic, Chinese, Mayan, Egyptian, Greek, Roman, Bank check numerals.
PROOF
We studied the ideas of Proof and Abstraction as developed by the Greeks. We explored general premises, axioms, and specific premises, postulates; then induction and deduction, theorems and corollaries. They also readily employed reductio ad absurdum to make a Proof (reduction to the absurd ).
THALES
The first early Greek geometer to make a decisive contribution to geometry was Thales, a retired, wealthy trader. His five Propositions are the base for classic mathematics.  The class did a careful geometric rendering of Thales’ fourth principle, and were surprised and delighted while discovering its validity. Thales was no less thrilled and was said to have sacrificed a bull to celebrate his discovery.
Angles in a semicircle – Any angle inscribed within a semicircle is a right angle ( 90 degrees ).
PYTHAGORUS
Under Thales’ recommendation his student Pythagorus travelled widely among the Mesopotamians, Persians and priests of Zoroaster.  Then in 540 B.C. in Crotona, Pythagorus founded his cult. He taught his disciples, along with Math, to worship numbers, to believe in reincarnation, to never eat beans, and to remain anonymous. They believed the universe was made up of whole numbers, with ther even numbers female and the odd numbers  male; thus marriage was 5, the sum of 3 and 2.
Although the Chinese and Babylonians had similar concepts around the same time, Pythagorus was the first to prove his most famous theorem of the right triangle, a square + b square  = c square. This is still used widely in science, architecture and carpentry. We drew the 3-45 triangle and
Pythagorean Theorem using squares.
Pythagorus also found that musical intervals are governed by ratios of whole numbers. The Pythagorean religion of whole numbers and the music of the spheres was dealt a rude awakening with the discovery pf the Irrational, such as square root of 2.
One of our most comprehensive and time-consuming but absorbing lessons was drawing  Pythagorus’ proof , using compass and ruler,  of Inscribing a Pentagon in a Circle. Each succeeding step had to be carefully achieved in order to insure that the last step was geometrically correct. It took some redoing but the children persevered and did good work.
ZENO
The Eleatics, rivals of Pythagorus, were interested in science. Their leader, Zeno presented his famous Paradox, of Achilles never catching the tortoise.
EUCLID
Around 300B.C. in Alexandria, Egypt, the greatest of all Geometers collected the theorems of his
Predecessors, including Democritus, Hippocrates, Archytas, into terse, clear terms, and a single whole. His masterpiece “The Elements”, is considered one of the primary books in human history, like the Bible. “ It’s commanding lucidity and style “ make it one of the most clearly reasoned works ever assembled.
The “Elements” contains 13 books which prove all the human race knew or still knows about lines, points, circles and solids, brilliantly deduced by Euclid , from 5 axioms and 5 postulates. Algebra, infinity, number theory and calculus, all find their foundation in this remarkable work.
APOLLONIUS
Apollonius wrote “ Conics“, which details his work with cones sliced to create circles and ellipses, parabolas and hyperbolas.
These proofs led to the science of ballistics, missiles and rockets,  the paths which satellites and moons follow influenced by  the gravity of planets and stars.
PLATO
The famed and influential philosopher was devoted to geometry and insisted that all proofs be done with only straight-edge and compass. Over the gates of his famed Academy he is purported to have inscribed “ Let no man ignorant of geometry enter “. Because of his prestige and high standing in Greek society all mathematicians had to work under the narrow bounds of Platonic discipline, even the modern-thinking Archimedes
ARCHIMEDES
Considered by most, along with Newton and Gauss, one of the three greatest mathematicians of all time,
he tackled the task of writing really large numbers, in his treatise “ The Sand Reckoner “. Using the Greek Myriad, or 10,000, he called a myriad of myriads, or 100,000,000, the First Order of Numbers.
100,000,000 to the 100,000,000 th power, or a myriad of myriads multiplied a myriad of myriad times,  he called Numbers of the First Period. He then multiplied this number by itself a myriad of myriad times, and came up with a number with 80 million billion zeros!
Archimedes discovered how to calculate the volume of a sphere as 2/3 the volume of the smallest possible cylinder which will enclose it. The sphere and cylinder diagram was engraved, at his request , on his tombstone. He was also a physicist and engineer of the highest order. He is most known as the inspired, absent minded discoverer, who ran naked through the streets of Syracuse crying “ Eureka! Eureka! “ when in the bath he had discovered the basic laws of hydraulic engineering. He used these laws, when hired by King Hieron of Syracuse to find out if the court jeweler was substituting silver for gold in his crowns.
Archimedes wrote the proofs of the mathematical laws of the lever still used, and the laws of pulleys and finding the center of gravity that enable to this day every skyscraper to stand and every bridge to span.
The historian Plutarch writes of how Archimedes kept an invading Roman fleet at bay for three years with his catapults and other inventions.
DIOPHANTES
Known as the Father of Algebra, Diophantine equations, indeterminate equations, and what has evolved into Theory of Numbers, Diophantes gave us his famous autobiographic riddle to figure out his age; x=x/6+x/12+x/7+5+x/2+4.
84 years.
HYPATIA
After  the Roman onslaught the mathematics of creative inquiry went dark. One brave Greek intellectual, Hypatia, lectured at the University of Alexandria around 400A.D. in mathematics and had a huge following. A sectarian Christian mob dragged her to their church, skinned her with oyster shells and burned her, for the terrible heresy of geometry. Math went dark for centuries.
AL-KHOWARIZMI
In 825A.D. al-Khowarizmi wrote the first clear textbook on algebra “al-jabr w’al-muqabala’’ or the art of bringing together unknowns to match a known quantity. The word in the title “al-jabr” bringing together, gave us  our word “algebra”.
FIBONACCI
Leonardo da Pisa, known as Fibonacci, was on of the first mathematicians to openly consider the existence of and importance of negative numbers . Despite Fibonacci’s recognition that an equation might have a negative solution, most mathematicians doubted the validity of negative numbers until the Renaissance.
CUBIC EQUATION
Friar Luca Pacioli, in “Summa de Arithmetica”, 1494, challenged that no one had solved the cubic equation. This challenge was taken up by Ferro, Univ. of Bologna, who secretly developed x to the third + ax = b. He gave this to his student, Fior, who used it in a famous algebra match with Tartaglia, the Stammerer, who had his own secret weapon for solving cubics,x to third + ax to the second = b. With a crowd gathered in the inn, and a big purse of gold on the line, Tartaglia won.  But the most noteworthy antagonist to confront Tartaglia, was Cordano, astrologer for kings, physician, compulsive gambler, teetering on bankruptcy and prison. Cordano cajoled and flattered Tartaglia’s secret solution from him; the Pope gave Cordano a pension.  But what incensed Tartaglia was Cordano publishing the Stammerer’s solution in his famous, monumental treatise “ Ars Magna”, 1545. In this work Cardano, ( who credited Tartaglia ) gave the world the solution to cubics and quartics ( solved by his student Ferrari ).
In “Ars Magna” Cardano formally accepted negative numbers and gave the laws which govern them.
DURER
Known as perhaps the greatest engraver, etcher and graphic artist of all time, Durer was a wonderful amateur mathematician.
The class drew, using compass  and ruler, Durer’s proof for inscribing a regular pentagon in a circle. Using external circles, this solution proved to be much easier than Pythagorus’. We wrote down and explored Durer’s famous Magic Square, wherein numbers in every direction and in many geometric configurations add up to 34. The children tried to invent their own Magic Squares.
SQUARES
We made a times table square where the square numbers make the diagonal. We explored the relationships of the number families 1,2,3,4,5,6,7,8,9,0 for the ones, 9,8,7,6,5,4,3,2,1,0 for the nines; 2,4,6,8,0 for the twos, 4,8,2,6,0 for the fours,6,2,8,4,0, for the sixes, 8,6,4,2,0, for the eights; 3,6,9,2,5,8,1,4,7,0,for the threes, 7,4,1,8,5,2,9,6,3,0 for the sevens; 5,0,5,0,5,0, for the fives, 0,0,0,0,0,0 for the tens.
PRIMES
We discussed Prime numbers, and the use of “complex primes”, a prime times a prime, as a way of protecting computer programs from being hacked and entered. We discussed the enormous paper two feet thick thousands of pages long, which is the largest known single prime number.

We’ve seen how exciting and animated the history of Mathematics is. We’ll continue with Newton, a play about Gauss, Einstein, quantum physics, topology, applied mathematics, etc. when we revisit this exploration.

In our study of geodesic domes, we studied the tetrahedron and constructed one out of card stock. The incredible strength and simplicity impressed them.  We then moved on to the octahedron, and the eventual icosahedron. In further studies we used our compasses to create the Almond of overlapping circles; using these skills we explored and carefully constructed with compass and ruler the harmonic structure of the musical string. We found the full tone (C) the octave, 1/2, (C), the fifth, 2/3, (G), and the fourth, 3/4, (F); all using compass almonds and ruler. The children worked assiduously and enthusiastically.

In our reading and acting out of the stories from the Sun Dance Movement, (which swept through the Plains Tribes at the end of the 1800’s) we found many characters, animal and human that children frantically waved their hands to be chosen to be. These vision quest stories resonate with the children. We’ve also drawn, with great care and colorfully, the dramatic geometric shields of these teachings.

Interestingly, concurrently I began reading some anthologies of Russian Tales and they are so fantastic, down to earth and magically understandable; quite more like the Native American stories we’ve been living with than the European Folklore; and the children love them.