Creative and Dynamic ConCreative and Dynamic Contents Evolve tents Evolve When Technological Tools AdvanceWhen Technological Tools Advance

Wei-Chi YANGWei-Chi YANG

PresenterPresenter• Wei-Chi YANGWei-Chi YANG• URL: http://www.radford.edu/wyangURL: http://www.radford.edu/wyang

– E-mail: E-mail: wyang@radford.eduwyang@radford.edu• Professor of Math/Stat at Radford University, Professor of Math/Stat at Radford University,

Virginia. 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 Editor-in-chief, The Electronic Journal of

Mathematics and Technology (Mathematics and Technology (eJMTeJMT, , http://ejmt.MathAndTech.org) http://ejmt.MathAndTech.org)

• Founder of the Founder of the Asian Technology Conference in Asian Technology Conference in MathematicsMathematics ( (ATCMATCM,, http:atcm.MathAndTech.org)http:atcm.MathAndTech.org)

• Editor-in-Chief, Editor-in-Chief, majoritymajority of refereed Proceedings of refereed Proceedings of of ATCMATCM..

Outline of the lectureOutline of the lecture

• Graphical and Geometrical Animations can make learning Graphical and Geometrical Animations can make learning mathematics more mathematics more accessible accessible by leaving complicated by leaving complicated Algebraic computations behind.Algebraic computations behind.

• Generalization with Dynamic Geometry with CAS makes Generalization with Dynamic Geometry with CAS makes mathematics more mathematics more challengingchallenging and lead to and lead to Analytical Proofs. Analytical Proofs.

• Dynamic contents evolve when technological tools advance. Dynamic contents evolve when technological tools advance. – Traditional static problems versus New dynamic problems.Traditional static problems versus New dynamic problems.

• Dynamic contents can be created and used for inter-Dynamic contents can be created and used for inter-disciplinary fields. disciplinary fields.

• International collaboration (through eJMT and ATCM) is a key International collaboration (through eJMT and ATCM) is a key to narrow the gap for each individual country.to narrow the gap for each individual country.

• Conclusion.Conclusion.

Technology issues around the Technology issues around the worldworld• In 2000, Ministry of Education in Taiwan: In 2000, Ministry of Education in Taiwan:

1.1. Mathematics should be accessible to Mathematics should be accessible to 80%80% of students; of students;2.2. complicated algebraic manipulationscomplicated algebraic manipulations can be replaced by using calculators or/and can be replaced by using calculators or/and

computers.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:

1.1. The “fill the duck style” (rote learning style) should be replaced by The “fill the duck style” (rote learning style) should be replaced by solving more real life solving more real life problemsproblems. Disappeared: Compare 6^0.7, 0.7^6, log[0.7](6); Find log(20)+log[100](25).. Disappeared: Compare 6^0.7, 0.7^6, log[0.7](6); Find log(20)+log[100](25).

2.2. Diverse standards of measuringDiverse standards of measuring students’ success instead of basing on testing alone. students’ success instead of basing on testing alone. • Ministry of Education in Singapore: All A-level will be using graphics calculator in Ministry of Education in Singapore: All A-level will be using graphics calculator in

mathematics starting from 2006. O-level will follow soon. (**mathematics starting from 2006. O-level will follow soon. (**New syllabus rolled New syllabus rolled out in January, 2006out in January, 2006))

• In the Australian In the Australian state of Victoriastate of Victoria: From 2000 onwards, student access to an : From 2000 onwards, student access to an approved graphics calculator has been assumed for Victorian Certificate of approved graphics calculator has been assumed for Victorian Certificate of Education (VCE) mathematics examinations in Further Mathematics, Education (VCE) mathematics examinations in Further Mathematics, Mathematical Methods and Specialist 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 France: MOE has added technology competency as part of the oral examination for

future teachers in 2004.future teachers in 2004.

Technology issues in the Technology issues in the U.S.U.S.• The No Child Left Behind Act, passed in January 2002, requires each The No Child Left Behind Act, passed in January 2002, requires each

state to demonstrate that it has developed challenging standards for state to demonstrate that it has developed challenging standards for students in reading and math and science in future years. Each state students in reading and math and science in future years. Each state must annually test every child's progress in reading and math in must annually test every child's progress in reading and math in third third through eighth gradesthrough eighth grades and at least once during and at least once during 10th through 12th10th through 12th grades. grades.– More measurements on students’ achievementMore measurements on students’ achievement. . (More like systems in Asian (More like systems in Asian

Pacific regions??)Pacific regions??)– Hold teachers accountable. Hold teachers accountable. – (NCTM)” (can be found at (NCTM)” (can be found at http://http://nctm.org/standards/pressrelease.htmnctm.org/standards/pressrelease.htm) on April ) on April

13, 2000 “13, 2000 “Technology is essential in teaching and learning mathematicsTechnology is essential in teaching and learning mathematics; it ; it influences the mathematics that is taught and enhances student learning”. influences the mathematics that is taught and enhances student learning”.

• Students must be fluent in arithmetic computationStudents must be fluent in arithmetic computation -- they must have -- they must have efficient and accurate methods, and understand them. efficient and accurate methods, and understand them.

• ****In my views, we may not have enough qualified math teachers in the In my views, we may not have enough qualified math teachers in the U.SU.S..

•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.

Mathematics can be Mathematics can be Fascinating 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 Algebra • *Cylinders (file 1) , (*Cylinders (file 1) , (more cylindersmore cylinders))•Height of the Pole Height of the Pole (Java applet)(Java applet)•Mathworld.wolfram.com-offlineMathworld.wolfram.com-offline•A donut, a yo-yo, a sphere and etc.A donut, a yo-yo, a sphere and etc.

– ****((152f01->revol.mws152f01->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 angles1 – About angle2 About angle2 – Handwriting technologyHandwriting technology..

AAccessibleccessible

When Dynamic Geometry is integrated with CASWhen Dynamic Geometry is integrated with CAS• Derivative functions by animation (Derivative functions by animation (classpadclasspad-derivative-sine-derivative-sine) ) • **Paper FoldingPaper Folding• *A *A ladderladder problem problem• *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 A Rope ; Journal file; Journal file• A unit circle.A unit circle.• Polar equations and its derivativePolar equations and its derivative• Shortest distanceShortest distance• Maximum and Minimum problem Maximum and Minimum problem • Derivative at one pointDerivative at one point

CChallenginghallenging• Problem 1Problem 1: Look at the distance problem-Lagrange Multipliers-again. : Look at the distance problem-Lagrange Multipliers-again.

– Between two curves and two surfaces.Between two curves and two surfaces.– Among three curves (CP) (Maple) (link to 3d-(file1; file 2)Among three curves (CP) (Maple) (link to 3d-(file1; file 2)– Four surfaces (thanks to Four surfaces (thanks to Jean-Jacques DAHAN’s construction)Jean-Jacques DAHAN’s construction) – Lagrange Multipliers.Lagrange Multipliers.

• Problem 2: Problem 2: Shrinking circle-James Stewart’s text book, Shrinking circle-James Stewart’s text book, – 2d (classpad->shrinking.vcp).2d (classpad->shrinking.vcp).– A A textex file. file.– An ellipse (An ellipse (gspgsp) (Thanks Nicholas Jackiw and Daniel Scher)) (Thanks Nicholas Jackiw and Daniel Scher)– Back to 2d (Back to 2d (textex), *(), *(non-symmetric casenon-symmetric case) () (an animationan animation-open shifted) (-open shifted) (

tentative prooftentative proof))– A generalization, a paper co-authored with Douglas Mead, published at eJMT.A generalization, a paper co-authored with Douglas Mead, published at eJMT.

• Problem 3. 3D geometry will impact us Problem 3. 3D geometry will impact us – *Shrinking Sphere*Shrinking Sphere (Thanks to Jean-Marie and Colette Laborde (Thanks to Jean-Marie and Colette Laborde))– 2d and 3d interactive files.2d and 3d interactive files.– 3d by a student (Kevin Thompson)3d by a student (Kevin Thompson)

• Problem 4. Problem 4. *Mean-Value Theorem (video clip) *Mean-Value Theorem (video clip) and Cauchy Mean Value (a and Cauchy Mean Value (a video clip) video clip) – *Cauchy Mean Value Theorem and Lagrange Remainder (refer to ICTMT-8 *Cauchy Mean Value Theorem and Lagrange Remainder (refer to ICTMT-8

Proceedings).Proceedings).– Converse of Cauchy Mean Value Theorem and Largrange Remainder.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).

TTheoreticalheoretical

o EXPERIMENT => CONJECTURES =>VERIFY=>EXPERIMENT => CONJECTURES =>VERIFY=> DISCOVER NEW THEOREMS.DISCOVER NEW THEOREMS.

Traditionally Traditionally (Lecture (Lecture Theorem) ->Theorem) ->Memorize Memorize the the Proof->Proof->Do the Do the Homework.Homework.

o Existence of a solution is not good enough=> Can Existence of a solution is not good enough=> Can we attempt to find the solution?we attempt to find the solution?o Paper on minimum distance by Yang, Gao and Paper on minimum distance by Yang, Gao and

Yang.Yang.o Paper (shrinking sphere) by Mead and YangPaper (shrinking sphere) by Mead and Yang

o Theoretical and Computational Integration. Theoretical and Computational Integration. o Animations will lead to more discovery.Animations will lead to more discovery.o Geometry in 2D and 3D + CAS are crucial areas.Geometry in 2D and 3D + CAS are crucial areas.

Experiment Conjecture VerifyDiscover New Theorem

Some scenarios that will make Some scenarios that will make contents accessible and Challengingcontents accessible and Challenging

• Wireless connectionWireless connection with notebook/tablet pc for students with notebook/tablet pc for students and teachers.and teachers.– Both teacher and students have access to necessary Both teacher and students have access to necessary

mathematical software packages.mathematical software packages.– Teacher experiment mathematics and project contents to the Teacher experiment mathematics and project contents to the

computer screen computer screen wirelesslywirelessly..

• A mini-conference. A mini-conference. Breeze5 will allow us to view our Breeze5 will allow us to view our computer screens.computer screens.– Is this what they (US-India) are doing?Is this what they (US-India) are doing?

• **Distance education. **Distance education. Teachers can record their lectures Teachers can record their lectures and post them on the internet-Camtasia3.1.and post them on the internet-Camtasia3.1.– Those with innovative ideas integrating technology with Those with innovative ideas integrating technology with

mathematics teaching, learning and research materials will be mathematics teaching, learning and research materials will be useful. (an example) (one more useful. (an example) (one more example)example)

Role of Technology and Role of Technology and MathematicsMathematics• Connect Abstract concepts to Concrete Connect Abstract concepts to Concrete

applications with Technologyapplications with Technology• Graphical and Geometrical animations Graphical and Geometrical animations

makes mathematics accessiblemakes mathematics accessible• CAS provides the thrust to analytical CAS provides the thrust to analytical

proofs and make mathematics challengingproofs and make mathematics challenging• Traditional Contents maybe shorten with Traditional Contents maybe shorten with

more real-life applicationsmore real-life applications• Make Math more accessible but more Make Math more accessible but more

challenging at the same timechallenging at the same time

Where do we go from here?Where do we go from here?

A.A. A graduate program needs to allow A graduate program needs to allow technologytechnology to play a pivotal role. to play a pivotal role.

– Students should be competent in using Students should be competent in using various technological tools and know when to various technological tools and know when to use which technological tools.use which technological tools.

– Technology will unite Math, Math Ed and Technology will unite Math, Math Ed and people from other disciplines. people from other disciplines.

– There will be There will be many research problemsmany research problems that that will arise thanks to technology.will arise thanks to technology.

B. Electronic JournalB. Electronic Journal in English and other local in English and other local languages should be available simultaneously with languages should be available simultaneously with collaborations among institutions.collaborations among institutions.– Are traditional textbooks too expensive?Are traditional textbooks too expensive?– New technological tools will push more dynamic New technological tools will push more dynamic

contents.contents.– The eJMTThe eJMT..

C. Distance Education is crucial.C. Distance Education is crucial.– Are universities too expensive?Are universities too expensive?– Delivering Mathematics contents through the web Delivering Mathematics contents through the web

will include videos.will include videos.– Lectures on demands.Lectures on demands.

D. CollaborationsD. Collaborations among universities/individuals: among universities/individuals:– Internet will make our communications in research Internet will make our communications in research

and teaching more efficient->and teaching more efficient->more collaborations.more collaborations.

ObservationsObservationsMath can be approached as Math can be approached as FFascinating, ascinating, AAccessible, ccessible, CChallenging hallenging

and then and then TTheoretical-FACT.heoretical-FACT.

Technology makes mathematics to go high, low and wide.Technology makes mathematics to go high, low and wide.

Technology becomes a bridge to make us rethink the Technology becomes a bridge to make us rethink the mathematics curriculum all over again-how to make math a mathematics curriculum all over again-how to make math a cross-disciplinary subject.cross-disciplinary subject.

Exploration with technology is the key, examination is Exploration with technology is the key, examination is only one way to measure students’ understandings.only one way to measure students’ understandings.

Learners expand their knowledge horizons with technology in various stages.

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www.fractalus.com/paul/www.fractalus.com/paul/checkered.jpgcheckered.jpg

www.fractalus.com/paul/www.fractalus.com/paul/technophile.jpgtechnophile.jpg

Observations-continueObservations-continueTraining competent teachers is an ongoing process.Training competent teachers is an ongoing process.

Think Global and Act Local.Think Global and Act Local.

Learn more mathematics through innovative use of Learn more mathematics through innovative use of technology.technology.

International collaboration is a crucial component to International collaboration is a crucial component to shape up a program in the internet era.shape up a program in the internet era.

Making Exploratory activities (contents) available to all.Making Exploratory activities (contents) available to all.

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Thank you very muchThank you very much

• Danke schönDanke schön• Merci beaucoup Merci beaucoup • Muchas graciasMuchas gracias• Arigato gozaimasu Arigato gozaimasu • KamsahamnidaKamsahamnida• Shukran gazilan (Arabic)Shukran gazilan (Arabic)• Terima KasihTerima Kasih• (Xie xie)(Xie xie)