Introduces several mathematicians who contributed significantly to the history of geometry. Pythagoras and his cousins want to win a music contest, but first they must figure out how to play their instruments in tune, something that's never been.
How do children learn? How are they taught? These are two fundamental questions in education. Caleb Gattegno provides a direct and lucid analysis, and concludes that much current teaching, far from feeding and developing the learning process, actually stifles it. Memory, for instance, the weakest of the mental powers available. A Baby Boomer's Decision Making drives the nation's 75 million baby boomers to examine life through the decisions they make, whether they are considering finances, changing national priorities or the needs of aging parents.
Boomers will explore decisions of love, school, career, family and fun. From Chevrolets to Martin Luther King. This survey of topics in Non-Euclidean Geometry is chock-full of colorful diagrams sure to delight mathematically inclined babies. Non-Euclidean Geometry for Babies is intended to introduce babies to the basics of Euclid's Geometry, and supposes that the so-called "Parallel Postulate" might not be true. Mathematician Fred Carlson believes that it's. This classic text, written by a distinguished mathematician and teacher, focuses on a fundamental theory of geometry.
Display it nicely on a Smartboard, speech teletherapy platform, or tablet as the story is being read to your students. It can be adapted for whole group or small group language and literacy instruction. It is full color and intended for digital use.
However, print it for a fun interactive activity. Finally, subscribe to my newsletter to be the first to know about new resources.
Newsletter subscribers are also the first to know about promos, blog posts, tutorial videos, and more! Known as Akousmatikoi , these students were permitted to eat meat and own personal belongings. According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organizing factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory.
Hermippus is quoted as saying about Pythagoras: "In practicing and repeating these precepts he was imitating and appropriating the doctrines of Jews and Thracians. In fact, it is actually said that that great man introduced many points of Jewish law into his philosophy. The history of the Pythagorean theorem that bears his name is complex.
Whether Pythagoras himself proved this theorem is not known, as it was common in the ancient world to credit a famous teacher with the discoveries of his students. The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch.
It is also believed that the Indian mathematician Baudhayana discovered the Pythagorean Theorem around B. E, about years before Pythagoras and is written in Sulba Sutras. Pythagoras also made an important achievement in astronomy; he was one of the first people to realize that Venus as the morning star and Venus as the evening star were the same planet.
He believed in the mistaken geocentric world-view of his age, but on the other hand he recognized that the moon's orbit around the earth was inclined towards the equator of the earth. Diogenes Laertius records more than one variant of a humorous tale that bases the manner of Pythagoras' death on his respect for beans. According to one account, Pythagoras had joined the fight of the Agrigentines against the Syracusans. When the Agrigentines were put to flight, Pythagoras, not wishing to cross a bean field, went around it and was killed by the Syracusans.
Pythagoras's followers were commonly called "Pythagoreans. The Pythagoreans observed a rule of silence called echemythia, the breaking of which was punishable by death. This was because the Pythagoreans believed that a man's words were usually careless and misrepresented him and that when someone was "in doubt as to what he should say, he should always remain silent". Another rule that they had was to help a man "in raising a burden, but do not assist him in laying it down, for it is a great sin to encourage indolence", and they said "departing from your house, turn not back, for the furies will be your attendants"; this axiom reminded them that it was better to learn none of the truth about mathematics, God, and the universe at all than to learn a little without learning all.
In his biography of Pythagoras written seven centuries after Pythagoras's time , Porphyry stated that this silence was "of no ordinary kind. Porphyry wrote "the mathematikoi learned the more detailed and exactly elaborate version of this knowledge, the akousmatikoi were those which had heard only the summary headings of his Pythagoras's writings, without the more exact exposition. The akousmatikoi were not allowed to see Pythagoras and they were not taught the inner secrets of the cult.
Instead they were taught laws of behavior and morality in the form of cryptic, brief sayings that had hidden meanings. The akousmatikoi recognized the mathematikoi as real Pythagoreans, but not vice versa. After the murder of Pythagoras and a number of the mathematikoi by the cohorts of Cylon, a resentful disciple, the two groups split from each other entirely, with Pythagoras's wife Theano and their two daughters leading the mathematikoi.
Theano, daughter of the Orphic initiate Brontinus, was a mathematician in her own right. She is credited with having written treatises on mathematics, physics, medicine, and child psychology, although nothing of her writing survives. Her most important work is said to have been a treatise on the principle of the golden mean.
In a time when women were usually considered property and relegated to the role of housekeeper or spouse, Pythagoras allowed women to function on equal terms in his society. The Pythagorean society is associated with prohibitions such as not to step over a crossbar, and not to eat beans. These rules seem like primitive superstition, similar to "walking under a ladder brings bad luck". The abusive epithet mystikos logos "mystical speech" was hurled at Pythagoras even in ancient times to discredit him.
The key here is that akousmata means "rules", so that the superstitious taboos primarily applied to the akousmatikoi , and many of the rules were probably invented after Pythagoras's death and independent from the mathematikoi arguably the real preservers of the Pythagorean tradition. The mathematikoi placed greater emphasis on inner understanding than did the akousmatikoi , even to the extent of dispensing with certain rules and ritual practices.
For the mathematikoi , being a Pythagorean was a question of innate quality and inner understanding.
0コメント