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(Math) Geometry: Euclid Returns to Maths Lessons

 
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adedios
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PostPosted: Thu Dec 22, 2005 11:41 am    Post subject: (Math) Geometry: Euclid Returns to Maths Lessons Reply with quote






Euclid returns to maths lessons
Netherlands Organization for Scientific Research
15 December 2005
Knowing how a mathematical theory developed improves a pupil's understanding of it. This is the conclusion of Dutch researcher Iris van Gulik, who investigated how the history of mathematics can help pupils to learn this subject.

Van Gulik developed two teaching methods in which a mathematical theory was taught based on the history of its development. Firstly for 13 to 15-year-old high school pupils, geometry was introduced by studying 17th-century Dutch surveying in small groups. Secondly 16 to 18-year-old high school pupils learnt about proofs in plane geometry by working in groups on the history of non-Euclidean geometry.

Deeper understanding

After the lessons had been completed, Van Gulik investigated the motivation of the pupils and their results, and the experiences of the teachers. The history of non-Euclidean geometry was particularly successful. The pupils acquired a deeper understanding and the teachers indicated that they found the subject challenging and inspiring. In addition to this the new teaching method led to a livelier learning process and higher motivation among the pupils.

The study of 17th-century surveying did not directly lead to a deeper understanding or a higher motivation among pupils. However the 14 to 15-year-old pupils responded more positively to the integration of history in mathematics lessons than the 13 to 14-year-old pupils. The practical assignment in the curriculum was experienced as positive. A particular disadvantage of this method was the use of many texts written in old Dutch. Moreover the cooperation between the teachers of mathematics and Dutch was better at some schools than at others.

The inclusion of historical sources in the teaching material for mathematics is definitely effective. However the extent to which such historical source materials need to be processed should be established. A detailed teacher's handbook for the teaching methods is also vitally important.

Historical development

At the turn of the 20th century it was common practice to use the history of how mathematics developed as a starting point for teaching this subject. Systematically following the most important steps in the development of mathematics was considered to be the most natural and efficient way of learning the subject. A century later these opinions have become more nuanced and new teaching methods have made their debut. However there are clear parallels between the mistakes pupils make in learning a mathematical theory and the problems encountered during the theory's development.

Iris Van Gulik's research was funded by NWO.

*************************************************************

Questions to explore further this topic:

What is geometry?

http://www.learner.org/teacherslab/math/geometry/

What is shape?

http://www.sanford-artedventur.....shape.html
http://www.sanford-artedventur.....hapes.html
http://www.bbc.co.uk/schools/r....._act.shtml
http://www.amblesideprimary.co.....ase/db.htm

What are lines?

http://www.mathleague.com/help.....cterms.htm

What are angles?

http://www.mathleague.com/help/geometry/angles.htm
http://www.bbc.co.uk/schools/r....._act.shtml

What are protractors?

http://www.madras.fife.sch.uk/.....ractor.swf

What are polygons?

http://www.mathleague.com/help.....lygons.htm
http://www.nes-lab.com/english.....th1-2.html

What are space figures and basic solids?

http://www.mathleague.com/help/geometry/3space.htm

Here are some pictures relating geometry to the real world:

http://library.thinkquest.org/.....tures.html

What is the history of geometry?

http://library.thinkquest.org/C006354/history.html

Who is Euclid?

http://www.historyforkids.org/.....euclid.htm
http://www-history.mcs.st-and......uclid.html

GAMES

http://www.alfy.com/teachers/t.....s/PS_1.asp
http://www.learner.org/teacher.....index.html
http://library.thinkquest.org/C006354/match.html
http://www.apples4theteacher.c.....metrygames
http://www.funbrain.com/cgi-bi.....NSTRUCTS=1
http://www.aplusmath.com/cgi-b.....s/geoflash
http://www.learner.org/teacher.....index.html
http://www.kidscom.com/games/tangram/tangram.html
http://www.learningwave.com/en.....metry.html
http://www.aaamath.com/B/geo318x1.htm#section3


Last edited by adedios on Sat Jan 27, 2007 3:11 pm; edited 1 time in total
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PostPosted: Fri Jan 20, 2006 1:43 pm    Post subject: Amazon Tribes Know Geometry Reply with quote

Amazon Tribes Know Geometry
By Bjorn Carey
LiveScience Staff Writer
posted: 19 January 2006
02:04 pm ET

While high school freshmen sometimes struggle with parallelograms and the Pythagorean Theorem, people deep in the Amazon quickly grasp some basic concepts of geometry.

Although these indigenous tribes had never seen a protractor, compass, or even a ruler, a new study found they understood parallelism and right angles and can use distance, angles, and other relationships in maps to locate hidden objects.

The finding suggests all humans, regardless of language or schooling, possess a core set of geometrical intuitions.

"While geometrical concepts can be enriched by culture-specific devices like maps, or the terms of a natural language, underneath this variability lies a shared set of geometrical concepts," said study co-author Elizabeth Spelke of Harvard University. "Those concepts allow adults and children with no formal education, and minimal spatial language, to categorize geometrical forms and to use geometrical relationship to represent the surrounding spatial layout."

The study is detailed in the Jan. 20 issue of the journal Science.

Rainforest math class

Spelke and her colleagues developed and administered two sets of tests during two visits to the Munduruku people, who live in remote areas along the Cururu River in Brazil.

They assessed comprehension of basic concepts such as points, lines, parallelism, figure congruence, and symmetry by presenting arrays of six images, one of which was subtly different from the rest.

For example, five similar trapezoids would be shown with a sixth non-trapezoidal quadrilateral of similar size and subjects were asked to point out which of the images was "weird" or "ugly."

Subjects as young as six years old pointed to the dissimilar image an average of 66.8 percent of the time, showing competence with basic concepts of topology, Euclidean geometry, and basic geometrical figures, researchers say.

"If the Munduruku share with us the conceptual primitives of geometry," the researchers write, "they should infer the intended geometrical concept behind each array and therefore select the discrepant image."

Against Americans …

In the second test researchers gave subjects a simple diagram and asked them to identify which of three containers arrayed in a triangle on the ground hid an object. Both Munduruku adults and children were able to relate the geometrical information on the map to the geometrical relationships on the ground, scoring a success rate of 71 percent.

The Mundurukus' scores matched the performance of American children, but were somewhat lower than educated American adults taking these tests.

This suggests that formal education enhances or refines geometrical concepts. However, the authors conclude, "the spontaneous understanding of geometrical concepts and maps by this remote human community provides evidence that core geometrical knowledge…is a universal constituent of the human mind."

In another recent study, Spelke recently found that young children can do certain math operations with no training.
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PostPosted: Sat Feb 18, 2006 12:58 pm    Post subject: Calculating Dogs Reply with quote

http://sciencenews.org/article.....thtrek.asp
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PostPosted: Sat Jul 08, 2006 5:08 am    Post subject: Composer reveals musical chords' hidden geometry Reply with quote

Princeton University
6 July 2006

Composer reveals musical chords' hidden geometry
Princeton, N.J. – Composers often speak of fitting chords and melodies together, as though sounds were physical objects with geometric shape -- and now a Princeton University musician has shown that advanced geometry actually does offer a tool for understanding musical structure.

In an attempt to answer age-old questions about how basic musical elements work together, Dmitri Tymoczko has journeyed far into the land of topology and non-Euclidean geometry, and has returned with a new -- and comparatively simple -- way of understanding how music is constructed. His findings have resulted in the first paper on music theory that the journal Science has printed in its 127-year history, and may provide an additional theoretical tool for composers searching for that elusive next chord.

"I'm not trying to tell people what style of music sounds good, or which composers to prefer," said Tymoczko (pronounced tim-OSS-ko), a composer and music theorist who is an assistant professor of music at Princeton. "What I hope to do is provide a new way to represent the space of musical possibilities. If you like a particular chord, or group of notes, then I can show you how to find other, similar chords and link them together to form attractive melodies. These two principles -- using attractive chords, and connecting their notes to form melodies -- have been central to Western musical thought for almost a thousand years."

Tymoczko's findings appear as a report in Science's July 7 issue.

Making graphical representations of musical ideas is not itself a new idea. Even most nonmusicians are familiar with the five-line musical staff, on which the notes that appear physically higher represent sounds that have higher pitch. Other common representations include the circle of fifths, which illustrates the relationships between the 12 notes in the chromatic scale as though they were the 12 hours on a clock's face.

"Tools like these have helped people understand music with both their ears and their eyes for generations," Tymoczko said. "But music has expanded a great deal in the past hundred years. We are interested in a much broader range of harmonies and melodies than previous composers were. With all these new musical developments, I thought it would be useful to search for a framework that could help us understand music regardless of style."

Traditional music theory required that harmonically acceptable chords be constructed from notes separated by a couple of scale steps -- such as the major chord, whose three notes comprise the first, third and fifth elements in the major scale, forming a familiar harmony that most audiences find easy to enjoy. Many 20th-century composers abandoned this requirement, however. Modern chords are often constructed of notes that sit right next to one another on the keyboard, forming "clusters" -- dissonant by traditional standards -- that to this day often challenge listeners' ears.

"Western music theory has developed impressive tools for thinking about traditional harmonies, but it doesn't have the same sophisticated tools for thinking about these newer chords," Tymoczko said. "This led me to want to develop a general geometrical model in which every conceivable chord is represented by a point in space. That way, if you hear any sequence of chords, no matter how unfamiliar, you can still represent it as a series of points in the space. To understand the melodic relationship between these chords, you connect the points with lines that represent how you have to change their notes to get from one chord to the next."

One of Tymoczko's musical spaces resembles a triangular prism, in which points representing traditionally familiar harmonies such as major chords gather near the center of the triangle, forming neat geometric shapes with other common chords that relate to them closely. Dissonant, cluster-type harmonies can be found out near the edges, close to their own harmonic kin. Tymoczko said that composers have traditionally valued a kind of harmonic consistency that does not require that the listener jump far from one region of the space to another too quickly.

"This idea that you should stay in one part of space," he said, "is an important ingredient of our notion of musical coherence."

To bring these ideas to life, Tymoczko has created a short movie that illustrates the chord movement in a piece of music by 19th-century composer Frederick Chopin. His E minor piano prelude (Opus 28, No. 4) has charmed listeners since the 1830s, but its harmonies have not been well explained.

"This prelude is mysterious," Tymoczko said. "While it uses traditional harmonies, they are connected with nonstandard chord progressions that people have had trouble describing. However, when you plot the chord movement in geometric space, you can see Chopin is moving along very short lines, staying primarily within one region."

The movie is available at http://www.princeton.edu/pr/me.....3_350k.mov

Tymoczko said that the geometric approach could assist with our still-murky understanding of music ranging from the mid-1800s through the contemporary period, including the cluster-based compositions of Georgi Ligeti, whose work formed a dramatic part of the soundtrack to the film "2001: A Space Odyssey."

"What all this implies is that you can begin with any sort of harmony your ear enjoys, whether it's a familiar chord from a 300-year-old hymn or the most avant-garde cluster you can imagine," he said. "But once you have decided where to start from and what region of space your harmony inhabits, very general principles of musical coherence suggest that you stay close to that region of space."

Tymoczko, whose compositional influences include classical music, rock and jazz, said he does not expect people will start writing music by "connecting the dots" as a result of his research. But he hopes it will at least provide a new tool for understanding the relationships behind music.

"Put simply, I'm a composer and I like to write and play music that sounds good," he said. "But what does it mean to 'sound good'? That's a question that the musical community has grappled with for centuries. Our understanding of the Chopin piece, for example, had previously been very local -- as if we were walking in a heavy fog and could only see a few steps in front of our feet at any one time. We now have a map of the whole terrain on which we can walk, and can replace our earlier, local perspective with a much more general one."

Commenting on the significance of the work, Yale's Richard Cohn said that Tymoczko has made a useful contribution to a fundamental problem in music theory.

"Dmitri's solution is exhaustive, original, and expressed clearly enough to be meaningful even to those musicians and scholars who do not have Dmitri's mathematical abilities," said Cohn, who is the Batell Professor of the Theory of Music at Yale. "His work leads to a deeper understanding of why composers in the European tradition favor certain types of scales and chords, and it suggests that melody and harmony are more fundamentally intertwined than has been previously thought. His achievement will become central to future work in the modelling of musical systems."
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PostPosted: Sat Nov 25, 2006 7:47 am    Post subject: Geometry in Court Reply with quote

Geometry in Court
25 November 2006
Ivars Peterson

The TV series Numb3rs has highlighted how mathematics can play a role in solving crimes. Even though the episodes are sometimes rather fanciful, they still illustrate ways in which various types of math can help illuminate mysteries, confirm conjectures, and point to villains.

In real life, math can also be relevant in the courtroom or come up in legal disputes.

Last year, the Pythagorean theorem was a deciding factor in a case before the New York State Court of Appeals. A man named James Robbins was convicted of selling drugs within 1,000 feet of a school. In the appeal, his lawyers argued that Robbins wasn't actually within the required distance when caught and so should not get the stiffer penalty that school proximity calls for.

The arrest occurred on the corner of Eighth Avenue and 40th Street in Manhattan. The nearest school, Holy Cross, is on 43rd Street between Eighth and Ninth Avenues.

Law enforcement officials applied the Pythagorean theorem to calculate the straight-line distance between the two points. They measured the distance up Eighth Avenue (764 feet) and the distance to the church along 43rd Street (490 feet), using the data to find the length of the hypotenuse, 907.63 feet.

For the full article:

http://sciencenews.org/article.....thtrek.asp
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PostPosted: Sat Dec 23, 2006 8:45 am    Post subject: Crafty Geometry Reply with quote

Crafty Geometry
Mathematicians are knitting and crocheting to visualize complex surfaces

Erica Klarreich
23 December 2006

During the 2002 winter holidays, mathematician Hinke Osinga was relaxing with some lace crochet work when her partner and mathematical collaborator Bernd Krauskopf asked, "Why don't you crochet something useful?" Some crocheters might bridle at the suggestion that lace is useless, but for Osinga, Krauskopf's question sparked an exciting idea. "I looked at him, and we thought the same thing at the same moment," Osinga recalls. "We realized that you could crochet the Lorenz manifold."

For years, Osinga and Krauskopf, both of the University of Bristol in England, had been studying the Lorenz manifold, a complicated surface that emerges from a model of chaotic weather systems. The pair had created an algorithm to generate 2-dimensional computer visualizations of the surface, but Osinga found the flat images unsatisfying. When Krauskopf asked his question, she suddenly realized that the computer algorithm could be interpreted as crochet instructions. "I had to try it," she says. Eighty-five hours and 25,511 crochet stitches later, Osinga had a Lorenz manifold almost a meter tall and about 25 centimeters in diameter, which now hangs in the pair's house as a decoration.

For the full article:

http://sciencenews.org/articles/20061223/bob10.asp
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PostPosted: Sat Jun 02, 2007 7:45 am    Post subject: Trisecting an Angle with Origami Reply with quote

Week of June 2, 2007; Vol. 171, No. 22

Trisecting an Angle with Origami
Julie J. Rehmeyer

Many a mathematician has received a long letter from an unknown sender who claims to have found a way to trisect an angle using only a straightedge and compass. The mathematician may read passages of the letter aloud to the guffaws of colleagues at afternoon tea and then toss it out. Without even reading the details, any mathematician would know the writer is incorrect. In the early nineteenth century, the young French mathematician Évariste Galois proved the problem to be impossible.

More recently, mathematicians have found that it is possible to divide an angle into three equal parts by folding paper rather than using a straightedge and compass.

For the full article:

http://sciencenews.org/article.....thtrek.asp
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PostPosted: Mon Aug 13, 2007 10:59 am    Post subject: New Study Examines How Rearing Environment Can Alter Navigat Reply with quote

New Study Examines How Rearing Environment Can Alter Navigation
Blackwell Publishing
13 August 2007
--------------------------------------------------------------------------------

Many animals, including humans, frequently face the task of getting from one place to another. Although many navigational strategies exist, all vertebrate species readily use geometric cues; things such as walls and corners to determine direction within an enclosed space. Moreover, some species such as rats and human children are so influenced by these geometric cues that they often ignore more reliable features such as a distinctive object or colored wall.


This surprising reliance on geometry has led researchers to suggest the existence of a geometric module in the brain. However, since both humans and laboratory animals typically grow up in environments not entirely made up of right angles and straight lines, the prevalent use of geometry could reflect nurture rather than nature.


A new study published in the July issue of Psychological Science, a journal of the Association for Psychological Science, is the first attempt to examine whether early exposure to strong geometric cues influences navigational strategy.


Alisha Brown, a psychology graduate student at the University of Alberta, raised fish in either a rectangular tank, or a circular tank free of angular information. Brown and her colleagues later trained the fish to swim to one particular corner of a rectangular-shaped test arena with either all white walls (geometric information only), or one colored wall (featural and geometric information).


Their results demonstrated that the ability to use geometry to aid navigation did not depend on exposure to angular geometry during rearing: in the featureless test arena, fish from both rectangular and circular rearing tanks used geometry to navigate. However, when features were present to help navigation, the circle-reared fish were more likely to depend on the feature even if it meant choosing a geometrically incorrect corner.


The researchers concluded that the ability to learn about geometry for navigation seems to be innate, but the use of geometric cues to navigate is determined by both nature and nurture. When reared in the absence of rectangular geometric structures, fish show a greater dependence on features for navigational guidance.
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PostPosted: Sun Mar 23, 2008 1:25 pm    Post subject: Sacred Geometry Reply with quote

Week of March 22, 2008; Vol. 173, No. 12

Sacred Geometry
Julie J. Rehmeyer

Hundreds of years ago in Japan, people offered thanks to the gods by sacrificing a horse or a pig. Horses and pigs, however, were valuable and expensive, so poor folks had a hard time expressing their gratitude. So they came up with a solution: Rather than sacrificing a horse, they would simply draw a painting of a horse on a wooden tablet and hang it in the temple.

For the full article:

http://sciencenews.org/article.....thtrek.asp
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