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Creative and Dynamic Contents Evolve When Technological Tools Advance Wei-Chi YANG

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Presenter Wei-Chi YANG Wei-Chi YANG URL: URL: – Professor of Math/Stat at Radford University, Virginia. Professor of Math/Stat at Radford University, Virginia. Ph. D. in Math (University of California at Davis) Ph. D. in Math (University of California at Davis) Editor-in-chief, The Electronic Journal of Mathematics and Technology (eJMT, Editor-in-chief, The Electronic Journal of Mathematics and Technology (eJMT, Founder of the Asian Technology Conference in Mathematics (ATCM, Founder of the Asian Technology Conference in Mathematics (ATCM, Editor-in-Chief, majority of refereed Proceedings of ATCM. Editor-in-Chief, majority of refereed Proceedings of ATCM.

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Outline of the lecture Graphical and Geometrical Animations can make learning mathematics more accessible by leaving complicated Algebraic computations behind. Graphical and Geometrical Animations can make learning mathematics more accessible by leaving complicated Algebraic computations behind. Generalization with Dynamic Geometry with CAS makes mathematics more challenging and lead to Analytical Proofs. Generalization with Dynamic Geometry with CAS makes mathematics more challenging and lead to Analytical Proofs. Dynamic contents evolve when technological tools advance. Dynamic contents evolve when technological tools advance. – Traditional static problems versus New dynamic problems. Dynamic contents can be created and used for inter- disciplinary fields. Dynamic contents can be created and used for inter- disciplinary fields. International collaboration (through eJMT and ATCM) is a key to narrow the gap for each individual country. International collaboration (through eJMT and ATCM) is a key to narrow the gap for each individual country. Conclusion. Conclusion.

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Technology issues around the world In 2000, Ministry of Education in Taiwan: In 2000, Ministry of Education in Taiwan: 1.Mathematics should be accessible to 80% of students; 2.complicated algebraic manipulations can be replaced by using calculators or/and computers. In 2001, the standard of the National Curriculum is set preliminarily in China: In 2001, the standard of the National Curriculum is set preliminarily in China: 1.The fill the duck style (rote learning style) should be replaced by solving more real life problems. Disappeared: Compare 6^0.7, 0.7^6, log[0.7](6); Find log(20)+log[100](25). 2.Diverse standards of measuring students success instead of basing on testing alone. Ministry of Education in Singapore: All A-level will be using graphics calculator in mathematics starting from O-level will follow soon. (**New syllabus rolled out in January, 2006) Ministry of Education in Singapore: All A-level will be using graphics calculator in mathematics starting from O-level will follow soon. (**New syllabus rolled out in January, 2006) In the Australian state of Victoria: From 2000 onwards, student access to an approved graphics calculator has been assumed for Victorian Certificate of Education (VCE) mathematics examinations in Further Mathematics, Mathematical Methods and Specialist Mathematics. In the Australian state of Victoria: From 2000 onwards, student access to an approved graphics calculator has been assumed for Victorian Certificate of Education (VCE) mathematics examinations in Further Mathematics, Mathematical Methods and Specialist Mathematics. In India, 7000 high schools are experimenting MathLab projects. In India, 7000 high schools are experimenting MathLab projects. Malaysian MOE also actively providing graphics calculators to school teachers. Malaysian MOE also actively providing graphics calculators to school teachers. France: MOE has added technology competency as part of the oral examination for future teachers in France: MOE has added technology competency as part of the oral examination for future teachers in 2004.

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Technology issues in the U.S. The No Child Left Behind Act, passed in January 2002, requires each state to demonstrate that it has developed challenging standards for students in reading and math and science in future years. Each state must annually test every child's progress in reading and math in third through eighth grades and at least once during 10th through 12th grades. The No Child Left Behind Act, passed in January 2002, requires each state to demonstrate that it has developed challenging standards for students in reading and math and science in future years. Each state must annually test every child's progress in reading and math in third through eighth grades and at least once during 10th through 12th grades. – More measurements on students achievement. (More like systems in Asian Pacific regions??) – Hold teachers accountable. – (NCTM) (can be found at on April 13, 2000 Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances student learning. Students must be fluent in arithmetic computation -- they must have efficient and accurate methods, and understand them. Students must be fluent in arithmetic computation -- they must have efficient and accurate methods, and understand them. **In my views, we may not have enough qualified math teachers in the U.S. **In my views, we may not have enough qualified math teachers in the U.S.

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Theoretical Mathematics can be made accessible by many undergraduate students to inspire them to further research in the future (a need for eJMT). The eJMT gives people opportunities to read and write anywhere and anytime. The adoption of technology varies from country to country, we need a forum for discussion Technology evolves fast enough that we should meet annually Technological tools will enhance our teaching, learning and research in Mathematics.

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Mathematics can be Fascinating *Fish-maple file (Thanks to Ebisui from Japan). *Fish-maple file (Thanks to Ebisui from Japan). Hat-maple file (Thanks to Ebisui from Japan). Hat-maple file (Thanks to Ebisui from Japan). *Linear Algebra *Linear AlgebraLinear AlgebraLinear Algebra *Cylinders (file 1), (more cylinders) *Cylinders (file 1), (more cylinders)more cylindersmore cylinders Height of the Pole (Java applet) Height of the Pole (Java applet) Height of the Pole Height of the Pole Mathworld.wolfram.com-offline Mathworld.wolfram.com-offline A donut, a yo-yo, a sphere and etc. A donut, a yo-yo, a sphere and etc. –**(152f01->revol.mws) ()() 3d wave; Open box; 2d animations; 3d wave; Open box; 2d animations; Pointwise and Uniform Convergence (tex file) Pointwise and Uniform Convergence (tex file) (Trigonometry): (Trigonometry): –About angles1 About angles1About angles1 –About angle2 About angle2About angle2 –Handwriting technology. Handwriting technologyHandwriting technology

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Accessible When Dynamic Geometry is integrated with CAS Derivative functions by animation (classpad-derivative-sine) Derivative functions by animation (classpad-derivative-sine)classpad-derivative-sine *Paper Folding *Paper FoldingPaper FoldingPaper Folding *A ladder problem *A ladder problemladder *Riemann Sum-1 ; Riemann Sum-2 *Riemann Sum-1 ; Riemann Sum-2 *Finding the limit geometrically and polar graphs *Finding the limit geometrically and polar graphs A Rope ; Journal file A Rope ; Journal file A Rope A Rope A unit circle. A unit circle. A unit circle. A unit circle. Polar equations and its derivative Polar equations and its derivative Shortest distance Shortest distance Shortest distance Shortest distance Maximum and Minimum problem Maximum and Minimum problem Derivative at one point Derivative at one point Derivative at one point Derivative at one point

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Challenging Problem 1: Look at the distance problem-Lagrange Multipliers-again. Problem 1: Look at the distance problem-Lagrange Multipliers-again. – Between two curves and two surfaces. Between two curves and two surfaces. Between two curves and two surfaces. – Among three curves (CP) (Maple) (link to 3d-(file1; file 2) – Four surfaces (thanks to Jean-Jacques DAHANs construction) – Lagrange Multipliers. Problem 2: Shrinking circle-James Stewarts text book, Problem 2: Shrinking circle-James Stewarts text book, – 2d (classpad->shrinking.vcp). – A tex file. tex – An ellipse (gsp) (Thanks Nicholas Jackiw and Daniel Scher) gsp – Back to 2d (tex), *(non-symmetric case) (an animation-open shifted) (tentative proof) texnon-symmetric casean animationtentative prooftexnon-symmetric casean animationtentative proof – A generalization, a paper co-authored with Douglas Mead, published at eJMT. A generalization A generalization Problem 3. 3D geometry will impact us Problem 3. 3D geometry will impact us – *Shrinking Sphere (Thanks to Jean-Marie and Colette Laborde) – 2d and 3d interactive files. 2d and 3d interactive files. 2d and 3d interactive files. – 3d by a student (Kevin Thompson) Problem 4. *Mean-Value Theorem (video clip) and Cauchy Mean Value (a video clip) Problem 4. *Mean-Value Theorem (video clip) and Cauchy Mean Value (a video clip)video clipvideo clipvideo clipvideo clip – *Cauchy Mean Value Theorem and Lagrange Remainder (refer to ICTMT-8 Proceedings). – Converse of Cauchy Mean Value Theorem and Largrange Remainder. For all x in (a,b), there is a y in (c,d) such that F(x)=G(y). For all x in (a,b), there is a y in (c,d) such that F(x)=G(y).

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Theoretical o EXPERIMENT => CONJECTURES =>VERIFY=> DISCOVER NEW THEOREMS. DISCOVER NEW THEOREMS. Traditionally (Lecture Theorem) ->Memorize the Proof->Do the Homework. Traditionally (Lecture Theorem) ->Memorize the Proof->Do the Homework. o Existence of a solution is not good enough=> Can we attempt to find the solution? oPaper on minimum distance by Yang, Gao and Yang. oPaper (shrinking sphere) by Mead and Yang Mead and YangMead and Yang o Theoretical and Computational Integration. o Animations will lead to more discovery. o Geometry in 2D and 3D + CAS are crucial areas. ExperimentConjecture Verify Discover New Theorem

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Some scenarios that will make contents accessible and Challenging Wireless connection with notebook/tablet pc for students and teachers. Wireless connection with notebook/tablet pc for students and teachers. –Both teacher and students have access to necessary mathematical software packages. –Teacher experiment mathematics and project contents to the computer screen wirelessly. A mini-conference. Breeze5 will allow us to view our computer screens. A mini-conference. Breeze5 will allow us to view our computer screens.Breeze5 –Is this what they (US-India) are doing? **Distance education. Teachers can record their lectures and post them on the internet-Camtasia3.1. **Distance education. Teachers can record their lectures and post them on the internet-Camtasia3.1. –Those with innovative ideas integrating technology with mathematics teaching, learning and research materials will be useful. (an example) (one more example)

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Role of Technology and Mathematics Connect Abstract concepts to Concrete applications with Technology Connect Abstract concepts to Concrete applications with Technology Graphical and Geometrical animations makes mathematics accessible Graphical and Geometrical animations makes mathematics accessible CAS provides the thrust to analytical proofs and make mathematics challenging CAS provides the thrust to analytical proofs and make mathematics challenging Traditional Contents maybe shorten with more real-life applications Traditional Contents maybe shorten with more real-life applications Make Math more accessible but more challenging at the same time Make Math more accessible but more challenging at the same time

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Where do we go from here? A. A graduate program needs to allow technology to play a pivotal role. –Students should be competent in using various technological tools and know when to use which technological tools. –Technology will unite Math, Math Ed and people from other disciplines. –There will be many research problems that will arise thanks to technology.

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B. Electronic Journal in English and other local languages should be available simultaneously with collaborations among institutions. –Are traditional textbooks too expensive? –New technological tools will push more dynamic contents. –The eJMT.. C. Distance Education is crucial. –Are universities too expensive? –Delivering Mathematics contents through the web will include videos. –Lectures on demands. D. Collaborations among universities/individuals: –Internet will make our communications in research and teaching more efficient->more collaborations.

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Observations Math can be approached as F ascinating, A ccessible, C hallenging and then T heoretical-FACT. Technology makes mathematics to go high, low and wide. Technology becomes a bridge to make us rethink the mathematics curriculum all over again-how to make math a cross-disciplinary subject. Exploration with technology is the key, examination is only one way to measure students understandings. Learners expand their knowledge horizons with technology in various stages reference:

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Observations-continue Training competent teachers is an ongoing process. Think Global and Act Local. Learn more mathematics through innovative use of technology. International collaboration is a crucial component to shape up a program in the internet era. Making Exploratory activities (contents) available to all reference:

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Thank you very much Thank you very much Danke schön Danke schön Merci beaucoup Merci beaucoup Muchas gracias Muchas gracias Arigato gozaimasu Arigato gozaimasu Kamsahamnida Kamsahamnida Shukran gazilan (Arabic) Shukran gazilan (Arabic) Terima Kasih Terima Kasih (Xie xie) (Xie xie)

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