Friday, September 21, 2007

PRESENTING RESEARCH AT CMU





Today, I am Chiang Mai University. I visit here for present my research to faculty of education. My research is " A study of geometric reasoning in Open-ended problem Solving :Focusing on small-group discussion"

Inspite of the faculty, my student who were engaged with me as advisee involve in the room of presentation also. I am happy to do so much. At least, I propose something although have no one can here, it may be concrete in the future.

So I would like to present the abstract following.


Research Title : A study of Geometric Reasoning in Open-Ended Problem

Solving : Focusing on Small-Group Problem Solving

Author : Jensamut Saengpun

Abstract

The purpose of the present research was to study the characteristics of geometric reasoning in open-ended problem solving focused on small-group problem solving. The study employed the qualitative, case-study research method. The collected data were analyzed by means of protocol analysis and analytic description. The study group was consisted of three third year students of faculty of Education, Chiang Mai University majored in mathematics of the 2006 academic year. They were organized into small-group. They were asked to solve one geometric problem about congruence and similarity. The activity was organized outside the classroom and was without the researcher’s interference. While the students were working on the problem the researcher and his co-researcher made videotape and tape recording and took field notes of the activity.
The empirical data which has been for analysis consisted of 1) one protocol analysis transcribed from videotape and tape recording during solving the problem , 2) six protocols analysis transcribed from videotape and tape recording during individual interviewing, 3) their written works, 4) fieldnotes, and 5) the students’ background information about their character and working in group attitudes. The obtained data were analyzed by means of Alice F.Artzt and Eleanor Armour-Thomas’s (1992) episode analysis and Raymond Duval’s (1998) theoretical frameworks for learning of geometric reasoning.
The research results showed that geometric reasoning in open-ended problem solving focused on small-group problem solving has cognitive processes in each episode of problem solving as following:
1) Geometric reasoning in episode of understanding involved visualization processes.
2) Geometric reasoning in episode of exploring involved visualization and construction processes.
3) Geometric reasoning in episode of analyzing involved reasoning and visualization processes.
4) Geometric reasoning in episode of planning involved construction processes.
5) Geometric reasoning in episode of implementing involved reasoning and construction processes.
Moreover, the research founded that in episode of reading, verifying and watching and listening have not any cognitive process also.

Tuesday, September 4, 2007

Terence Tao


ผมเพิ่งรู้จักศาสตราจารย์ Terence Tao ได้สามวัน จาก MY MATH Magazine รู้สึกทึ่งและประทับใจมากว่า คนที่แก่กว่าเราไปแค่ปีเดียว จะเป็นศาสตราจารย์แล้ว ชีวิตของ Tao น่าสนใจในความมุมานะพยายาม ความสำเร็จไม่ได้มาจาก พรสวรรค์เพียงอย่างเดียว มีมุมแง่งามของ Tao ที่ใครๆ ก็ตามก็เรียนรู้ชีวิตของเขาได้


Terence Tao, ‘Mozart of Math,’ Is UCLA’s First Mathematician Awarded the Fields Medal, Often Called the ‘Nobel Prize in Mathematics’
Terence Tao became the first mathematics professor in UCLA history to be awarded the prestigious Fields Medal, often described as the "Nobel Prize in mathematics," during the opening ceremony of the International Congress of Mathematicians in Madrid on Aug. 22. In the 70 years the prize has been awarded by the International Mathematical Union, only 48 researchers ever have won it.

"Terry is like Mozart; mathematics just flows out of him," said John Garnett, professor and former chair of mathematics at UCLA, "except without Mozart's personality problems; everyone likes him. Mathematicians with Terry's talent appear only once in a generation. He's an incredible talent, and probably the best mathematician in the world right now. Terry can unravel an enormously complicated mathematical problem and reduce it to something very simple."
"I'm not surprised," said Tony Chan, dean of the Division of Physical Sciences and professor of mathematics. "Someone like Terry comes along once every few decades. People all over the world say, 'UCLA's so lucky to have Terry Tao.' He has solved important problems in several areas of mathematics that have stumped others for a long time. The way he crosses areas would be like the best heart surgeon also being exceptional in brain surgery. What is also amazing is that Terry is still so young.

"The best students in the world in number theory all want to study with Terry," Chan added. "He's a magnet attracting the best students the same way John Wooden attracted outstanding basketball players." Chan said he is known as "the dean of the university where Terry Tao works." He described the International Congress of Mathematicians as "the World Cup or Olympics of mathematics."

Christoph Thiele, UCLA professor and chair of the mathematics department, said outstanding graduate students from as far as Romania and China, as well as throughout the United States, have come to UCLA for the chance to study with Tao.

Born and raised in Adelaide, Australia, Tao was awarded the Fields Medal "for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory." In honoring Tao, the organization said, "Terence Tao is a supreme problem-solver whose spectacular work has had an impact across several mathematical areas. He combines sheer technical power, an other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view that leaves other mathematicians wondering, 'Why didn't anyone see that before?' "

Like the summer Olympics and the World Cup, the Fields Medal is awarded every fourth year. Along with Tao, the Fields Medal also was presented to Andrei Okounkov, professor of mathematics at Princeton University; Grigori Perelman, formerly a Miller Fellow at University of California, Berkeley; and Wendelin Werner, professor of mathematics at the University of Paris-Sud in Orsay.

Tao's genius at mathematics began early in life. He started to learn calculus when he was 7, at which age he began high school; by 9 he was already very good at university-level calculus. By 11, he was thriving in international mathematics competitions. Tao, now 31, was 20 when he earned his Ph.D. from Princeton University, and he joined UCLA's faculty that year. UCLA promoted him to full professor at age 24.

One of the branches of mathematics on which Tao focuses is theoretical field called harmonic analysis, an advanced form of calculus that uses equations from physics. Some of this work involves, in Garnett's words, "geometrical constructions that almost no one understands." Tao also works in a related field, nonlinear partial differential equations, and in the entirely distinct fields of algebraic geometry, number theory and combinatorics — which involves counting. His research has been supported by the David and Lucille Packard Foundation and the Clay Mathematics Institute.

"Terry wrote 56 papers in two years, and they're all high-quality," Garnett said. "In a good year, I write three papers."

Discover magazine praised Tao's research on prime numbers, conducted with Ben Green, a professor of mathematics at the University of Bristol in England, as one of the 100 most important discoveries in all of science for 2004. A number is prime if it is larger than one and divisible by only itself and one. The primes begin with 2, 3, 5, 7, 11, 13 and 17.

Euclid proved that the number of primes is infinite. Tao and Green proved that the set of prime numbers contains infinitely many progressions of all finite lengths. An example of an equally spaced progression of primes, of length three and space four, is 3, 7, 11; the largest known progression of prime numbers is length 23, with each of the numbers containing 16 digits. Green and Tao's discovery reveals that somewhere in the prime numbers, there is a progression of length 100, one of length 1,000, and one of every other finite length, and that there are an infinite number of such progressions in the primes.

To prove this, Tao and Green spent two years analyzing all four proofs of a theorem named for Hungarian mathematician Endre Szemerédi. Very few mathematicians understand all four proofs, and Szemerédi's theorem does not apply to prime numbers.

"We took Szemerédi's theorem and goosed it so that it handles primes," Tao said. "To do that, we borrowed from each of the four proofs to build an extended version of Szemerédi's theorem. Every time Ben and I got stuck, there was always an idea from one of the four proofs that we could somehow shoehorn into our argument."

Tao is also well-known for his work on the "Kakeya conjecture," a perplexing set of five problems in harmonic analysis. One of Tao's proofs extends more than 50 pages, in which he and two colleagues obtained the most precise known estimate of the size of a particular geometric dimension in Euclidean space. The issue involves the most space-efficient way to fully rotate an object in three dimensions, a question of interest to theoretical mathematicians.

"Terry is the world's expert on this set of five problems, and has been since he finished graduate school," Garnett said. "When Terry made a new estimate of how big the dimension must be, he also produced the solutions, or partial solutions, to many other problems."

Tao and colleagues Allen Knutson at UC Berkeley and Chris Woodward at Rutgers solved an old problem (proving a conjecture proposed by former UCLA professor Alfred Horn) for which they developed a method that also solved longstanding problems in algebraic geometry and representation theory.

Speaking of this work, Tao said, "Other mathematicians gave the impression that the puzzle required so much effort that it was not worth making the attempt, that first you have to understand this 100-page paper and that 100-page paper before even starting. We used a different approach to solve a key missing gap."

Tao found a surprising result to an applied mathematics problem involving image processing with California Institute of Technology mathematician Emmanuel Candès; their collaboration was forged while they were taking their children to UCLA's Fernald Child Care Center. Chan said that Tao and Candès work is providing important insights into how to compress images, which has applications for medical imaging.

"A lot of our work came in the preschool while we were dropping off our kids," Tao said.
"Outstanding mathematicians love working with Terry," Garnett said. "You could build the best mathematics department in the world by hiring his co-authors."

What are Tao's secrets for success?
Tao offered some insight. "I don't have any magical ability," he said. "I look at a problem, and it looks something like one I've done before; I think maybe the idea that worked before will work here. Nothing's working out; then you think of a small trick that makes it a little better but still is not quite right. I play with the problem, and after a while, I figure out what's going on.

"Most people, faced with a math problem, will try to solve the problem directly," he said. "Even if they get it, they might not understand exactly what they did. Before I work out any details, I work on the strategy. Once you have a strategy, a very complicated problem can split up into a lot of mini-problems. I've never really been satisfied with just solving the problem. I want to see what happens if I make some changes; will it still work? If you experiment enough, you get a deeper understanding. After a while, when something similar comes along, you get an idea of what works and what doesn't work.

"It's not about being smart or even fast," Tao added. "It's like climbing a cliff: If you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools. You still need a plan — that's the hard part — and you have to see the bigger picture."

His views about mathematics have changed over the years.

"When I was a kid, I had a romanticized notion of mathematics, that hard problems were solved in 'Eureka' moments of inspiration," he said. "With me, it's always, 'Let's try this. That gets me part of the way, or that doesn't work. Now let's try this. Oh, there's a little shortcut here.' You work on it long enough and you happen to make progress towards a hard problem by a back door at some point. At the end, it's usually, 'Oh, I've solved the problem.'"

Tao concentrates on one math problem at a time, but keeps a couple dozen others in the back of his mind, "hoping one day I'll figure out a way to solve them."
"If there's a problem that looks like I should be able to solve it but I can't," he said, "that gnaws at me."
Most of Tao's work is pure theoretical mathematics. Of what use is that to society?

"Mathematicians often work on pure problems that do not have any applications for 20 years, and then a physicist or computer scientist or engineer has a real-life problem that requires the solution of a mathematical problem and finds that someone already solved it 20 years ago," Tao said. "When Einstein developed his theory of relativity, he needed a theory of curved space. Einstein found that a mathematician devised exactly the theory he needed more than 30 years earlier."

Will Tao become an even better mathematician in another decade or so?
"Experience helps a lot," he said. "I may get a little slower, but I'll have access to a larger database of tricks. I'll know better what will work and what won't. I'll get déjà vu more often, seeing a problem that reminds me of something."

What does Tao think of his success?

"I'm very happy," he said. "Maybe when I'm in my 60s, I'll look back at what I've done, but now I would rather work on the problems