### Abstracts for Plenary Talks and Invited Papers

- 16936, Practice and think about teaching dynamic geometry course in Normal University, Jingzhong ZHANG, Xicheng PENG, Mao CHEN
- 17150, Applications of the Internet Archive to Dynamic Geometry, Jen-Chung Chuan
- 17196, From friezes and wallpapers to ornaments of rods and further to circular and spiral mosaics, Bogumila Klemp-Dyczek
- 17198, Analysis of a Tumour Growth Model with MATLAB, Keng Cheng Ang
- 17224, A Journey through Chinese Windows and Doors -- an Introduction to Chinese Mathematical Art, Miroslaw Majewski, Jiyan Wang
- 17296, Volumes of Solids of Revolution via Summation Methods, Tilak de Alwis
- 17301, Mobile Calculating Laboratory in Everybody's Hands, Changpei Wang
- 17310, ICT and Mathematics - a multi-lingual approach, Douglas Butler
- 17333, Content Knowledge, Creativity and Technology, Wei-Chi Yang
- 17348, What has changed and how, in Math and Math Education, since recent and unprecedented moves in Informatics?, Jean-Marie Laborde
- 17399, School Mathematics and Popularization of Mathematics, Sung Je Cho
- 17700, Elementary PDEs with CAS technology, Matthias Kawski

### Abstracts for Full Papers

- 17068, Visualizing Statistical Concepts with the Aid of Technology, Tower Chen
- 17077, Graphing Calculator Strategy in Teaching and Learning of Mathematics: Effects on conceptual and procedural knowledge performance and instructional efficiency, Nor'ain Mohd. Tajudin, Rohani Ahmad Tarmizi, Wan Zah Wan Ali, Mohd. Majid Konting
- 17080, Better learning of geometry with computer, Tyger Yegambaram, Ramu Naidoo
- 17088, Why College or University Students Hate Proofs in Mathematics? Zacharie M'Baitiga
- 17098, Role Playing Game Based Learning in Mathematics, Wan Fatimah Wan Ahmad, Afza Shafie, Mohd Hezri Amir Abdul Latif
- 17164, Problem Posing and its Environment with technology, Chieko Fukuda, Kyoko Kakihana
- 17181, The Mathematics Conceptual Game, Nunnapad Toadithep
- 17199, Learning mathematics through computer games, Chang Suo Hui
- 17208, Applying GC to Problem Solving: A Case Study of Transfering Handheld Game into Classroom, Chang Wenwu, Chang Wencui
- 17215, An Analysis of Students' Research on Model Lessons that Integrate GeoGebra into School Mathematics, Jack Carter, Beverly Ferrucci
- 17221, Blended Assessment: A Strategy for Classroom Management, Josefina Barnachea Janier, Afza Bt. Shafie
- 17251, The Use of R Language in the Teaching of Central Limit Theorem, Wai Kwong Cheang
- 17259, The effect of CAS calculator usage on the Algebra Achievement of Low Mathematics achievement students, IN KYUNG KIM, Hee-Chan Lew
- 17260, Using MCL to discover more mathematics from real life, Xici Wang
- 17261, The Dimensionality Analysis of the Specialized Mathematics Open-Ended Questions using The Multidimensionality Model for DIF Framework, Leong Chee Kin, Warabhorn Preechaporn, Teoh Boon Tat
- 17263, Folding and Unfolding a Cone with Cabri 3D : a Process of Modelling Generating New Problems to Be Explored, Jean-Jacques Dahan
- 17265, A System for Helping Concept-Building of 3D Linear Algebra by Connecting Graphic, Symbolic, and Verbal Representations, Hitoshi Nishizawa, Bernhard Zgraggen, Takayoshi Yoshioka
- 17266, Handwriting Tool For Note Takers In Mathematical Classes, Ryoji Fukuda, Koji Nii
- 17267, Undergraduate Mathematics Assessment in an Integrated Technology Environment, Greg Oates
- 17268, Comparison study for Primality testing using Mathematica, Hailiza Kamarulhaili, Ega Gradini
- 17270, The elementary school teachers' belief of integrating calculator into Mathematic instruction, Yi-Ching Lin, Yuan Yuan
- 17274, Teachers' dealing with technology-enhanced mathematics in teaching practice, Mette Andresen, Morten Misfeldt
- 17277, The Three-Joint Lamp and Vectors, Nunnapad Toadithep
- 17282, Using Multimedia Learning System for Teaching Media Math, Ruzlina P Adni, Hanafizan Hussain
- 17288, Teaching and Learning Trigonometry with Technology, Barry Kissane, Marian Kemp
- 17293, Analysis of Discrete Models Using an Electronic Spreadsheet--A Survey of Examples, Thomas McMillan
- 17307, A New Block Method and Their Application to Numerical Solution of Ordinary Differential Equations, Rei-Wei Song, Ming-Gong Lee
- 17308, Teaching and Learning Calculus with the TI-Nspire: A Design Experiment, Wee Leng Ng, Wee Chuen Tan, Meow Leng Nancy Ng
- 17320, Exploring and Visualizing Hilbert Geometry in a Triangle, Mauricio Herrera, Rubén Preiss, Gonzalo Riera, Hernán Carrazco
- 17324, Enhancing Students' Understanding Statistics with TinkerPlots: Problem-Based Learning Approach, Krongthong Khairiree, Piromya Kurusatian
- 17325, Construction of Quadratic Curves Using Analysis, Hee Chan Lew, Soo Youn Je

### Papers with Abstract Only

- 16869, How to Effectively Integrate a 1:1 programme into the Mathematics classroom, Jennifer Wathall
- 17070, Hooked on Webquests, Denise LeGrand
- 17075, Entertainment Product Packaging for Distribution to Maximize Revenue, Bruce Wang, Yihua Li, Kai Yin
- 17083, Educational Software in Compensation of Geometric Background Insufficiency for Computer Graphics Studies, Vladimir Nodelman
- 17084, Effectiveness of STestMAP as a tool for selection of Statistical Tests, Kien Kheng Foo, Howe Eng Tang, Ung Hua Lau, Chin Ying Liew, Yee Hui Robin, Chang
- 17085, Role of new technology on education of mathematics, Saeed Hashemi
- 17087, Why I Love Subtraction: Using Error Analysis in First Year Calculus, Mark Howell
- 17092, Periodic Orbits in Triangular Billiards as a Minimization Problem, Felino Pascual
- 17094, The effect of CAS calculator usage on the Algebra Achievement of Low Mathematics achievement students, IN KYUNG KIM, Hee-Chan Lew
- 17096, Journey to Discover More Mathematics -- Additional Mathematics Project Work, Hong Eng Ding
- 17102, Discovering Fresh Aspects of Geometry, Algebra and Calculus in PNG Stamps, String Bags and Traditional House Wall Patterns, Thadreina Abady & Lorna Denema, Lorna Denema
- 17105, Teachers' Conceptualizations of Data Collection, GT Springer
- 17108, A System for Helping Concept-Building of 3D Linear Algebra by Connecting Graphic, Symbolic, and Verbal Representations, Hitoshi Nishizawa, Bernhard Zgraggen, Takayoshi Yoshioka
- 17113, Rethinking Calculus, Michael Livshits
- 17115, Cryptography and the DES, Wolfgang Baltes
- 17116, New Approaches to Mathematical Exposition with Excel, D. E. Arganbright
- 17127, Using PPF to Design Mathematical problems for Students, Da Liu
- 17130, Dynamic Simulation for "The Nine Chapters": Using Dynamic Geometry System and Interactive WebPages, Feng-Lin Lu, Tai-Yih Tso
- 17134, The Teaching Experiment of Multiplication of Complex Numbers in Dynamic Geometry Environments, Chi-Chiang Hsu, Tai-Yih Tso
- 17135, Trigonometry through the Geometer's Sketchpad (GSP) Make a Difference, Warabhorn Preechaporn
- 17136, Enhancing Teaching and Learning of Quadratic Function using GeoGebra, Warabhorn Preechaporn, Leong Chee Kin, Teoh Boon Tat
- 17161, E-resources for the undergraduate engineering mathematics student, Aruna Palipana
- 17168, Helping Students Succeed in Learning Mathematics With Geogebra, Premjit Singh
- 17206, Application of Finite Element Method to Design an Electric Generator, Wooi Ping Hew, K Chandran
- 17207, Exploring some interesting theorems in number theory using a graphing calculator, Michael Grasse
- 17209, Seeing is Believing in Inferential Statistics:Visual Remedies for Common Misconceptions in Decision Making, Marian Kemp, Jen Bradley
- 17211, Lab Sessions for Linear Algebra with MATLAB, Victor Tan
- 17212, GeoGebra: Powerful Tool to Explore in Mathematics, Leong Chee Kin, Teoh Boon Tat, Warabhorn Preechaporn
- 17248, Computer simulation and rigorous proofs in some perturbation problems, Chunqing Lu
- 17272, The comparison between different teaching styles of using mathematical techniques, Siming Gui
- 17299, Using Casio Scientific Calculator to Solve Mathematical Problems in Vietnam Schools, Minh Tran
- 17330, Technology-based Activities which Promote Student-centered Learning, Ma.Louise Antonette De Las Penas
- 17300, Using Technology in Mathematics for the Technologically Disenfranchised II, Monos Naidoo
- 17334, 数学教学与CAI---"信息技术与课程整合"改变了什么(Mathematics Education and CAI, What have changed by "Integration of Information Technology, with Curricula"?), Liu Jijie
- 17385, Technology Enhanced 3D Visualization, Zhonghong Jiang
- 17338, Using CASIO fx-82ES, 991ES & ClassPad 330, Discovering more mathematics in China, Jiyan Wang, Chongyi Xin, Lei Lin
- 17357, Stability of Boundary Layer Solution to the Boltzmann Equation with Diffuse Boundary Condition, Qianzhu Tian
- 17371, The Algebraic Properties of Geometric Objects: The Case of SSP, Chuan-Bo ZUO
- 17375, Gevrey regularity of solutions to the spatially homogeneous Boltzmann equation for non Maxwellian molecules without angular cutoff, Shi you Lin
- 17380, Spectral Domain Immitance Matrix Using Dyadic Green Function, Kassem Ghulam, Kassem Ghulam, Asif Zakariyya
- 17383, 超级画板在研制中考动点问题中的应用与研究, Chuan-Bo ZUO, Gui-Guang Yan
- 17395, ICT-Based Mathematics Learning: Realistic Mathematics Education using Games Based on Papua's Environment and Culture, Benidiktus Tanujaya
- 17404, The Role of Geometry in Architecture: A case of study with 9th to 12th grade High school students in Japan, Maryam Behnoodi, Tadashi Takahashi
- 17407, Teaching Sketching Graphs of Trigonometric Functions Using Graphmatica: an Alternative Approach, SARIPAH AHMAD, PUAN SALMAH ABU BAKAR
- 17408, Enhancing Form 5 Students Mastery Of Skills In Identifying The Angle Between Lines And Planes, As Well As The Angle Between Two Planes Using The `WON TECHNIQUE', NORAINI KASSIM, AINI HAZIAH AMIRULLAH
- 17415, "O.I.C" METHOD IN COMPOSITE FUNCTION, SITI AZLINA HAIRUDIN, SALHALIDA ABD KUDUS
- 17419, Maplets for Calculus as an Electronic Study Guide, Douglas Meade, Philip Yasskin
- 17420, Analytic, Geometric, and Numeric Analysis of the Shrinking Circle and Sphere Problems, Douglas Meade, Wei-Chi Yang
- 17421, The new era of Mathematica technology, Wolfram Research

### Abstracts for Hands-on Workshops

- 16834, The Electronic Spreadsheet as a Problem Solving Tool, Thomas McMillan
- 17082, An experimental approach to analytic geometry with the TI NSpire, Jean-Jacques Dahan
- 17090, Exploring and Discovering Mathematics with The Dynamic Mathematics Software: SSP, Chuan-Bo ZUO
- 17091, 利用Z+Z超级画板进行数学探索与发现, Chuan-Bo ZUO
- 17107, An introduction of Magic Board: a Web-based virtual manipulatives for teaching elementary mathematics, Yuan Yuan
- 17117, Discovering and Teaching Mathematics with Excel, D. E. Arganbright, Lorna Denema, Thadreina Abady, Susan Arganbright
- 17125, 激發式動態教學 ( Trigger-based animated Instruction), Mingjang Chen
- 17131, Using Graphing Technology to engage students in Mathematical Investigations for understanding, Caroline Lee
- 17213, Introducing Calculus using an HP Graphing Calculator, Colin Croft
- 17214, Teaching statistics and probability using an HP Graphing Calculator, Craig Davis
- 17216, Using CAS in your classroom, Christopher Longhurst
- 17217, Using Graphic calculators in the classroom, Christopher Longhurst
- 17227, Connecting Geometry, Algebra and Calculus with the Geometer's Sketchpad, Krongthong Khairiree
- 17241, Exploring Advanced Calculus with ClassPad Manager, Wei Ching Quek
- 17246, Introduction to Probability and Statistical functions of ClassPad Manager (In Mandarin), Wei Ching Quek
- 17250, GeoGebra Workshop, Morten Misfeldt, Mette Andresen
- 17311, Autograph for the younger students, Douglas Butler
- 17312, Autograph for advanced students, Douglas Butler
- 17316, How to Turn a Cube into a Rhombic Dodecahedron?, Jen-Chung Chuan
- 17317, 3D Poncelet Porism and Steiner Porism, Jen-Chung Chuan
- 17318, Learning about equations with a graphics calculator, Barry Kissane
- 17319, Learning about functions with a graphics calculator, Barry Kissane, Marian Kemp
- 17321, Learning data analysis with a graphics calculator, Marian Kemp
- 17322, Learning calculus with a graphics calculator, Marian Kemp, Barry Kissane
- 17323, Enhancing Mathematics Teaching Through Handheld Graphing Technology, Wee Leng Ng
- 17362, Interactive visualization tools for div, curl, Stokes theorem and differential equations, Matthias Kawski
- 17363, "Autograph" 精 画 - 适用于中级数学教师的ATCM专题讨论会, Douglas Butler
- 17396, Getting Started with Maple and Maplets for Calculus, Douglas Meade, Philip Yasskin

### Abstracts for Poster Sessions

- 17109, Software Module Showing Dynamic Link of 3D Graphic Objects and Vector Equations, Norihide Kishi, Keisuke Ishiguro, Takayoshi Yoshioka, Hitoshi Nishizawa
- 17110, Constructing a Game for Introducing 3D Vector Operations, Yoshihiro Yamada, Norihide Kishi, Takayoshi Yoshioka, Hitoshi Nishizawa
- 17111, Cabri 3D Module to Visualize the Construction of 3D Vector Equation, Takahiro Asano, Takayoshi Yoshioka, Hitoshi Nishizawa
- 17178, Simplifying Calculus, Michael Livshits
- 17249, Mathematics in Quantum Physics with Mathematica, Woei-Yun Ho
- 17382, Computer technology in Korean Secondary Mathematics Textbooks Based on the 2007 revised curriculum, Lew Hee-chan, Tae Seon Cheon
- 17384, ON PLC-GEOMETRY, Hirotaka Ebisui

### Abstract for 16936

Practice and think about teaching dynamic geometry course in Normal University

#### Authors: Jingzhong ZHANG, Xicheng PENG, Mao CHEN

#### Affiliations: Central China Normal University

According to the demand for modernization of education, combined with math education requirements of middle school, we provide the dynamic geometry course for students majoring in mathematics education, i.e., the pre-service teachers of mathematics, in Huazhong Normal University. This paper records our experience of teaching dynamic geometry course in 2008. First, the classroom teaching details is presented so that the situation of course is better understood, and the problems exposed in the practice are also shared in this paper, which deserve our reflection. The strong incentive to learn dynamic geometry and the skilled using of dynamic geometry software -- Super Sketchpad after training gives us surprises and pleasure. All of those records present valuable experience for pre-service teacher's education. The Affiliations and feedback of the students to the course, as well as the expectation and prospect for the future are presented in the end of the paper.

### Abstract for 17150

Applications of the Internet Archive to Dynamic Geometry

#### Authors: Jen-Chung Chuan

#### Affiliations: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300

As of this date, The Open-Access Text Archive contains 1,450,319 items. The collection is a part of the Internet Archive which serves as a valuable source of intellectual capital for researchers and students alike. We shall make a brief survey of the collection, its academic value and to discuss works of particular interests in Dynamic Geometry.

### Abstract for 17196

From friezes and wallpapers to ornaments of rods and further to circular and spiral mosaics

#### Authors: Bogumila Klemp-Dyczek

#### Affiliations: Nicolas Copernicus University

The aim of the talk is to describe a relationship between frieze groups, wallpaper groups, rod groups and symmetry groups of circular and spiral mosaics.

All is known about frieze and wallpaper groups and they were approached in many ways. However, rod groups are far less popular and symmetry groups of circular and spiral mosaics of the Gaussian plane were not so extensively investigated and popularised.

The relationship between these groups is given by means of an interpretation dictionary translating isometries of the Euclidean space into isometries of the Euclidean plane restricted to a stripe and further, into the inversive transformations of the Gaussian plane. In this dictionary axes of transformations are important since f.e. distinct translations of the Euclidean plane may correspond to translations, rotations and screws of the Euclidean space.

We thoroughly investigate a few significant examples of creating rod groups and symmetry groups of spiral mosaics from wallpaper groups.

The lecture uses little formalism and illustrates every notion by numerous pictures and diagrams.

### Abstract for 17198

Analysis of a Tumour Growth Model with MATLAB

#### Authors: Keng Cheng Ang

#### Affiliations: Nanyang Technological University, 1 Nanyang Walk, Singapore 637616

Mathematical modelling can play a very important role in cancer research. In particular, modelling the growth of tumour has the potential of shedding light on the mechanisms of tumour cell growth and proliferation. In this paper, we examine and analyze one such model with the aid of MATLAB.

The model, first proposed by Sherratt and Chaplain in 2001, is based on a set of partial differential equations. The equations describe the growth, movement and death of tumour cells, accompanied by a supply of nutrients. This spatial-temporal model depends on a number of parameter values as well as rate functions. The model is solved numerically using finite difference method implemented on MATLAB. Effects and influence of the parameter values and rate functions are analyzed. Results are validated against a set of known experimental data, and good agreement is observed.

### Abstract for 17224

A Journey through Chinese Windows and Doors -- an Introduction to Chinese Mathematical Art

#### Authors: Miroslaw Majewski, Jiyan Wang

#### Affiliations: New York Institute of Technology, Abu Dhabi Campus, Department of Mathematics East China Normal University Shanghai, China

We know quite a lot about mathematical art in the West. Hundreds of papers and books were written about Islamic art. Some researchers explored mathematical patterns in tribal art from both Americas as well as from Africa. However, we know very little about mathematical art in Chinese culture.

There are numerous examples of artistic creations in China that have very strong mathematical background. We can find them in geometric patterns on silk fabrics, on Chinese ceramic ware, in furniture as well as in windows and doors of Chinese buildings.

In this paper we are going to look on some of these patterns from a transformation geometry point of view. Our main target will be geometric patterns in wooden lattices in windows and doors. We will be interested which of the plane symmetry groups are represented in these patterns and how they can be modeled with dynamic geometry software. All examples analyzed in this paper are modeled with Geometer's Sketchpad. The paper will demonstrate numerous examples taken from real buildings as well as their computer models.

### Abstract for 17296

Volumes of Solids of Revolution via Summation Methods

#### Authors: Tilak de Alwis

#### Affiliations: Southeastern Louisiana University

In this paper, we will show how to calculate volumes of certain solids of revolution without using direct integration. The traditional method of such volume computation uses definite integrals as given by Disk Method or Shell Method in a calculus course. However, instead of direct integration, we will calculate these volumes as a limit of a summation. Even though somewhat longer then the traditional method in general, this method emphasizes the fundamental idea behind a definite integral, i.e. the definite integral as the limit of a sum. We will also use the computer algebra system Mathematica to facilitate and verify our calculations.

### Abstract for 17301

Mobile Calculating Laboratory in Everybody's Hands

#### Authors: Changpei Wang

#### Affiliations: Beijing Institute of Education, MCL, Beijing Mathematics Association

Graphing calculators have become one of the most widely adopted handheld technologies in mathematics education, but it is not the case in China. A series of questions has to be answered: what role should GC played in today's new curriculum? What is the feasible way in which GC can be effectively integrated into the new curriculum? What are the specific suggestions for development of mathematics curriculum? ...etc. It is critical to answer those questions above in the context of new curriculum in China. This is why MCL emerged.

The presentation is going to discuss those questions based on the two years of practice of project MCL. Mobile Calculating Laboratory (MCL) has double meaning. On one hand it means a calculator based laboratory (CBL) that consists of graphing calculator and together with other mobile devices. Even one set of graphing calculator can also be seen as a laboratory for mathematics exploring.

On the another hand, MCL has another meaning ¨C it is the short title of the project of Integrating hand held technology into new curriculum. The project aims to answer the questions raised above by means of developing a model that show how to integrate the MCL (graphing calculator in particular) into the secondary mathematics curriculum and create new generation of school mathematics in China.

### Abstract for 17310

ICT and Mathematics - a multi-lingual approach

#### Authors: Douglas Butler

#### Affiliations: iCT Training Centre (Oundle School), UK

Mathematics is one of the world's common languages - just about every 15 year old that goes to school anywhere in the world is learning mathematics. There are particular challenges to the software community when it comes to representing the localized notation on screen.

A surprising number of countries use what we know as "Western Notation", though there are some notable special exceptions. For example the decimal point and list separator can be "." and "," or "," and ";". Microsoft tries to help here by setting this according to the language "locale" chosen. The important objective has to be to make all programs able to accept and interpret either convention, but this is not helped by Microsoft who give Argentina and Vietnam the same symbols for both!

The most challenging test is representing fully right-to-left Arabic notation for mathematics on screen. The first challenge is a complete lack of Arabic mathematics symbols in the Unicode font set, and the fact that Arabic Hindi numerals are written left-to-right!

It is not just the mathematics language that is localized: nearly every country has a slightly different physical keyboard, and this need to be addressed too.

This paper will describe how these challenges were met when Autograph was localized, and explore some of the interesting differences in the mathematics curriculum in secondary schools in the various countries. Languages explored have included Traditional Chinese, Simplified Chinese, Vietnamese, Thai, Japanese , Russian, and Arabic.

### Abstract for 17333

Content Knowledge, Creativity and Technology

#### Authors: Wei-Chi Yang

#### Affiliations: Radford University

It is always debatable how much math content knowledge we expect from a future math teacher before he or she starts teaching. Many educators do not agree that we solely rely on examinations to test one's mathematical knowledge, which gives reasons for stuffing lots of contents into a math curriculum. The real question is how a person can build life long interests in mathematics and be creative when it comes to solving their daily problems. Could innovative use of technological tools provide some possible solutions? In short, we can come to the following consensus for mathematics curriculum in general:

1. We shall see more innovative use of technological tools to be integrated in a mathematics curriculum to enhance one's mathematics knowledge horizon.

2. An important task is motivating a learner to appreciate mathematics as early as possible, so a learner can be motivated in discovering more mathematics.

### Abstract for 17348

What has changed and how, in Math and Math Education, since recent and unprecedented moves in Informatics?

#### Authors: Jean-Marie Laborde

#### Affiliations: Cabrilog, University of Grenoble and CNRS

During last decades Computer Science underwent two major revolutions. One match the way computers have changed in size, from room size to palm size, keeping the same, if not augmenting, and their computing power. The other is the Direct Manipulation revolution where "command line" practice has moved away for the possibility (through innovative metaphors), to "directly" act on mathematical objects and concepts, allowing us for a deeper comprehension and understanding of the matter. I will illustrate my presentation using (good, bad and ugly) examples from Cabri, including the new Cabri LM mixing in one unique environment 2D and 3D, numeracy, algebra and geometry, as well from many other math computer environments.

### Abstract for 17399

School Mathematics and Popularization of Mathematics

#### Authors: Sung Je Cho

#### Affiliations: Seoul National University

Mathematics is vital not only in ordinary person's everyday life but also in the cutting-edge technology of the information age. Mathematics is everywhere from counting objects to telecommunication and image processing. Unfortunately most of mathematics in everyday life is invisible to many people's eyes. This is partially because mathematics used in technology is usually deep results of ablest researchers of the field. The details of mathematics are even harder to mathematicians whose specialties are different from the materials used in the particular technology. The result is that mathematics is getting far and far away from general publics¡¯ attention and is confined to specialists. Eventually this will undermine the whole enterprise. We mathematicians need to act against the public ignorance of mathematics.

### Abstract for 17700

Elementary PDEs with CAS technology

#### Author: Matthias Kawski

#### Affiliations: School of Mathematical and Statistical Sciences, Arizona State University,

Partial differential equations (PDEs) are arguably one of the most powerful tools for mathematical modeling of diverse phenomena that include such diverse applications as traditional mechanical and electro-magnetic settings, bio-medical and financial applications, and even social dynamics. Ever since the days when technology meant pencil and paper, the study of PDEs and their solutions has been a major driving force for the development of new mathematical theories that underlie analytical and computational tools for solving and simulating solutions of PDEs. For elementary PDEs such as the wave and heat equation on very simple, regular domains have traditionally been solved using methods related to Fourier analysis. These start with elementary separation of variables techniques, identification of bases of eigenfunctions and representations of solutions by infinite series. For less regular domains, descendants of the Riemann mapping theorem and harmonic analysis are an example of analytic tools. Nonetheless, analytic techniques are not only arduous to use, but often also do not provide the desired hands-on information about solutions. Thus, the numeric simulation of solutions of diverse PDEs has in many areas become the predominant approach, and it has spawned the new discipline of modern scientific computing.

## Abstracts for Full Papers

### Abstract for 17068

Visualizing Statistical Concepts with the Aid of Technology

#### Authors: Tower Chen

#### Affiliations: University of Guam

Most professional statistics programs only display the data, assumptions, and final numerical output. These programs that omit the intermediate steps and calculations are, from an educational perspective, not the best tools for learning statistics. The use of a spreadsheet framework will aid students in visualizing relationships between data sets and statistical formulas, in exploring data, and in interpreting data. In our new statics course, students learn to develop programs in Excel to solve homework problems and projects. Using their programs, they can review the output from each step without performing repetitive, manual calculations. This allows students to focus more on the statistical concepts rather than the details. Initial assessment results of our new statistics course which utilizes a spreadsheet framework are very positive.

### Abstract for 17077

Graphing Calculator Strategy in Teaching and Learning of Mathematics: Effects on conceptual and procedural knowledge performance and instructional efficiency

#### Authors: Nor'ain Mohd. Tajudin, Rohani Ahmad Tarmizi, Wan Zah Wan Ali, Mohd. Majid Konting

#### Affiliations: Universiti Pendidikan Sultan Idris, Institute for Mathematical Research, Universiti Putra Malaysia

Two series of quasi-experimental study with non-equivalent control group post-test only design were conducted to investigate the effects of using graphic calculators in mathematics teaching and learning specifically in the Straight Line topic, on Form Four(16 year-old) Malaysian secondary school students' conceptual and procedural knowledge performance and 3-D instructional efficiency index. Experiment in Phase I was conducted to provide an initial indicator of the effectiveness of graphic calculator strategy on students' performance. Experiment for Phase II was further carried out incorporating measures of mathematical performance, namely the conceptual and procedural knowledge performance and measures of instructional efficiency. There were two instruments used in this study namely, the Straight Lines Achievement Test and the Mental Effort Rating Scales. The data were analyzed using independent t-test and planned comparison test. The results of this study showed that the graphic calculator strategy group had better conceptual knowledge performance as compared to conventional instruction strategy group and most important they did not lose procedural knowledge performance. In addition, the study also showed that the graphic calculator instruction increased the 3-dimensional instructional efficiency index in learning of Straight Lines topic. These findings indicated that the graphic calculator instruction is superior in comparison to the conventional instruction, hence implying that it was more efficient instructionally than the conventional instruction strategy.

### Abstract for 17080

Better learning of geometry with computer

#### Authors: Tyger Yegambaram, Ramu Naidoo

#### Affiliations: AMESA, Durban University of Technology

The purpose of the research is to determine whether the use of computer assisted instruction and classroom intervention will significantly improve the primary school learners " geometric understanding of concepts. The research focuses on the learners" ability to engage in computer generated instruction for learning outcome 3- SPACE AND SHAPE.

### Abstract for 17088

Why College or University Students Hate Proofs in Mathematics?

#### Authors: Zacharie M'Baitiga

#### Affiliations: Department of Media Information Engineering, Okinawa National College of Technology

Empirical studies have shown that students emerge from proof-oriented courses such as high-school geometry, introduction to proof and complex variable are unable to construct anything beyond very trivial proofs. Furthermore, most university students do not know what constitutes a proof and cannot determine whether a purported proof is valid. To motivate students hating proofs and to help mathematics teachers, how a proof can be taught, we investigated in this study the idea of mathematical proofs. To tackle this issue, the modified Moore method and the author method called Z. Mbaitiga method are introduced follow by two cases of studies on proof of triple integral. Next a survey is conducted on fourth year college students on which of the proposed two cases of studies they understand easily. The result of the survey showed that more than 95% of the students who responded pointed out the proof that is done using details explanations of every theorem used in the proof construction, the case study 2. From the result of this survey, we learned that mathematics teachers have to be very careful about the selection of proofs to include when introducing topics; and filtering out some details which can obscure important ideas and discourage students.

### Abstract for 17098

Role Playing Game Based Learning in Mathematics

#### Authors: Wan Fatimah Wan Ahmad, Afza Shafie, Mohd Hezri Amir Abdul Latif

#### Affiliations: Universiti Teknologi PETRONAS

Computer based role-playing games are able to provide a fun and motivating environment for teaching and learning of certain subjects. Role- playing games allow students to assume the role of a character in the game world and to determine the actions of their characters based on the characterization. This would provide an exciting and motivating strategy for students to practice skills that they have already learned. This paper describes the development of a role-playing game in learning mathematics. Due to its interactive and stimulating nature, the game is suitable for school children in learning this subject. The focus of the game is on fraction since most school children have difficulties in understanding this topic. This game is able to engage these children and at the same time assist them in learning the topic. Progressive spiral methodology together with Flash 8, Photoshop CS2 and vector image editor have been employed in the development of the game. Educational elements have also been added from time to time without compromising the enjoyment of playing the game. Results from Heuristic Affiliations will be discussed.

### Abstract for 17164

Problem Posing and its Environment with technology

#### Authors: Chieko Fukuda, Kyoko Kakihana

#### Affiliations: Teikyo University, JAPAN, Tsukuba Gakuin University, JAPAN

The OECD reported a `mathematising ability' to extract mathematics from a problem situation as a mathematical literacy, is important. In this research we gave students the task of problem posing and analyzed their perspectives in creating problems. Their posed problems were almost all very similar and the students' passive attitudes toward mathematics were shown in an environment with paper and pencil. In order for students to be able to pose various kinds of own problems, they must change from a passive to an active approach. We will show the new possibility in the learning environment with technology as an appropriate method for these activities. An ideal environment to search various problems from one situation is one in which various expressions can be considered at the same time. Such environments include graphs, tables, mathematical expressions, geometric figures, and etc., and students can manipulate them in a problem situation dynamically. A leaning environment with technology encouraged students to find two or more methods to propose a problem and to confirm the forecasted solution through experiment. We will show a task which includes various ways for posing new problems and exploring them through the use of technology.

### Abstract for 17181

The Mathematics Conceptual Game

#### Authors: Nunnapad Toadithep

#### Affiliations: Department of Computing, Faculty of Science, Silpakorn University, 73000, Thailand.

Abstract: Some suitable or appropriate games can make children improve their thinking in a numerical way or in a mathematical way. This may be the way to create the Great Mathematician. This paper explores conceptual thinking in playing a Pick1-2-3 game. This game needs two players to take turn in picking either 1, 2, or 3 balls in their turn. In this game, the last player who picks the last ball will lose the game. One thing to remember is not to leave 4 balls or less (except 1) to the opponent. To have a chance of winning the game, a player should leave the multiple of 4 plus one balls, for example 5, 9, 13, 17, 21,..etc.

### Abstract for 17199

Learning mathematics through computer games

#### Authors: Chang Suo Hui

#### Affiliations: National Institution of Education, Ministry of Education, Singapore

Since the insurgence of computer games in the 1980s, it created an unstoppable wave that invades the lives of many, in particular, pupils. It is undeniable that pupils spend bulk of their out-of-school hours to play computer games. Though computer games are often perceived as unhealthy and detrimental to the young minds, there are also many studies which indicate otherwise. They advocate and highlight the potential of using computer games as a medium of learning. This paper illustrates an attempt by a primary school in Singapore to explore the viability of learning mathematics through playing commercially produced computer games in an internet café constructed within the school premise. Three different types of computer games namely, simulation, role-play and quest or adventure, were selected for the study. Four pupils spent at least two hours a day over a week period to play all three computer games and separate interview sessions were conducted for each pupil at the end of the week. The focus of the study is to examine, through the three selected computer games, how they (1) serve as possible avenues for the acquisition and application of various mathematical skills and (2) motivate pupils to learn mathematics. Interview results revealed that all four pupils applied some form of mathematical skills or processes, such as reasoning, deduction and pattern looking, while playing the computer games. The benefits and problems faced with employing such a strategy are examined. Recommendations for further work include the need to examine collaboration between educators and game designers.

### Abstract for 17208

Applying GC to Problem Solving: A Case Study of Transfering Handheld Game into Classroom

#### Authors: Chang Wenwu, Chang Wencui

Affiliations: Shanghai Putuo Modern Educational Technology Center, Master teacher of Intel TTF & TEO Program, Computer Science Department Jinzhong College Shanxi China

This report present a case of high school mathematics project in which graphic calculators (GC) was used as a tool in two ways: one is to use providing small handheld game ¡°Lights Out¡±; the other way is to use it in solving this puzzle. two classical methods were offered during the solving phase: solving a linear system or applying an algorithm to search the answer. This makes the same game useful in teaching two different parts of mathematics curriculum. This design engages students in team activities and ICT using£¬which coincides with the spirit of the mathematics reforming movement at present in a full swing in China. In addition, it can be seen as an example of math edutainment or mobile learning. By encourging students to take part in these project activities, the author believes that math course can be made more interesting and fruitful. The pattern of ¡°Choose-Analysis-Design¡± method is most likely to be a common way in applying game to mathematics projects.

### Abstract for 17215

An Analysis of Students' Research on Model Lessons that Integrate GeoGebra into School Mathematics

#### Authors: Jack Carter, Beverly Ferrucci

#### Affiliations: California State University, East Bay, Keene State College

After participating in a GeoGebra orientation, prospective elementary and secondary school mathematics teachers completed a senior-level course project in which they researched and prepared model lessons that incorporated GeoGebra into the curriculum they were preparing to teach. The future teachers then concluded the project by presenting their model lessons to classmates in the course. Presentations were videotaped and analyzed for content and categories of tools and activities that were used in the lesson. All model lessons emphasized multiple geometric constructions and measurement activities. Project presentations showed that the prospective teachers exhibited considerable expertise in using GeoGebra's coordinate axes, sidebar display of measures, and a various measurement tools in presenting the results of the research. Results showed that over 80% of the future teachers used some form of dynamic geometry in their presentations, although there were often missed opportunities wherein dynamic geometry could have provided demonstrations that were more extensively and illustratively presented. Student presentations also used dynamic geometry as aids in constructing figures for the presentations, as a means for adjusting lengths to integer values, and as possible extension activities for the model lessons. Conclusions include suggestions for increasing the effectiveness of dynamic geometry and the use of sliders in future student research projects.

### Abstract for 17221

Blended Assessment: A Strategy for Classroom Management

#### Authors: Josefina Barnachea Janier, Afza Bt. Shafie

#### Affiliations: EE Department, Universiti Teknologi PETRONAS, Fundamental of Applied Sciences Department Universiti Teknologi PETRONAS

The classroom is an environment where teachers constantly need to make decisions on planning instructional activities, monitoring of students" progress and evaluating teaching effectiveness. One way to monitor students" progress is by giving assessment. Assessments which can be formal or informal, are used to gather information about students" achievement. Previously, lecturers used the traditional paper and pencil assessment which proved to be tedious and time consuming for large classes. To facilitate a more efficient and quicker marking and to provide feedback to students immediately, the researchers looked into the potential of blended assessment in Probability and Statistics which comprised of 500-600 second year engineering students at Universiti Teknologi Petronas. Implemented in this course for January 2009 semester, blended assessment is the use of technology specifically the Wiley Plus and MOODLE combined with the traditional mode of testing. Findings of this study revealed that this method has significantly reduced the number of hours spent on marking and enabled immediate feedback for students.

### Abstract for 17251

The Use of R Language in the Teaching of Central Limit Theorem

#### Authors: Wai Kwong Cheang

#### Affiliations: National Institute of Education, Nanyang Technological University

The Central Limit Theorem (CLT) is probably the most important theorem in statistics. In the teaching of CLT, issues encountered include: (i) How large should the sample size n be for CLT to hold? Is the rule of thumb n >= 30 adequate? (ii) How important is the condition of normal population for the convergence in distribution of the t-statistic to normal? (iii) Will CLT still hold if the sample observations are not independent? To provide students with better insight into these issues, we can perform empirical study by simulating samples from different distributions, and exploring the behaviours of the z-statistic and t-statistic as the sample size gets large. Statistical software packages such as SAS can be used to perform simulation. However, these packages are not free and may not be cost-effective to implement. The R language is powerful software for data analysis within which many statistical procedures have been implemented. It is an official part of the Free Software Foundation's GNU Project (and so R is free). The strength of R derives from its many capabilities besides being a data analysis tool. In this paper, we explore the simulation capability of R in teaching the Central Limit Theorem, as well as its graphing capability in presenting the simulation results. With its ease of adaptability according to user's need, R has the potential to be an effective teaching tool for other topics of statistics.

### Abstract for 17259

The effect of CAS calculator usage on the Algebra Achievement of Low Mathematics achievement students

#### Authors: IN KYUNG KIM, Hee-Chan Lew

#### Affiliations: Korea University of Education

This paper observed the effect of CAS calculator usage while studying algebra on the achievement of low-achievement students in mathematics, here in referred to as low-achievement students. Participants were composed of 70 low-achievement tenth graders from a high school located in a metropolitan city that had never used a mathematics educational calculator before. Target participants were divided into two groups: an experiment group that studied activity papers with the aid of a CAS calculator, and a control group that studied the same activity papers using only paper and pencil. The activity papers were used in 14 consecutive classes. The content of the activity papers for the two groups was the same, but the structure differed. Content consisted of numbers and operations, equations and inequalities (character and expressions), and functions. The activity papers of the control group were solved with the use of only paper and pencil and solutions were compared to those presented by the teacher. Students in the experiment group first solved problems with paper and pencil and then again using a CAS calculator. They were told to compare their two problem-solving processes; i.e., compare the paper and pencil procedure with the CAS calculator. Students in the experiment group exhibited matacognition learning using a CAS calculator usage method. The activities were carried out once a day for about one mouth. The two groups completed mathematics achievement tests both before and after the activity papers. Most students had similar overall Mathematics results, however, on the algebra section, the average scores of the experiment and control groups were very different. Therefore, ANCOVA analysis results showed that compared to the pretest, results of the experiment group improved considerably more than the control group.

### Abstract for 17260

Using MCL to discover more mathematics from real life

#### Authors: Xici Wang

#### Affiliations: Key Laboratory of Mathematics and Complex Systems, Ministry of Education, China, Beijing Normal University, Mathematical Education

The Authors of this paper has designed an example using music to support the connection between mathematics and real life practice, with the help of the technology of Mobile Calculator Lab (MCL). The example, which is highly recognized by experts, teachers and student, was carried out in a class by one of the Authors. The example showed that, with the help of MCL, students could collect and see the relative data more conveniently, handle it more flexibly and explore the rules relatively easier than conventional practices. It provided more opportunities for students of different levels to make good use of mathematics to organize the exploration of natural phenomenon, and accumulate more mathematics experiences. Most likely, it will deeply influence the future development of science curriculums in China, including integrated science courses.

### Abstract for 17261

The Dimensionality Analysis of the Specialized Mathematics Open-Ended Questions using The Multidimensionality Model for DIF Framework

#### Authors: Leong Chee Kin, Warabhorn Preechaporn, Teoh Boon Tat

#### Affiliations: SEAMEO RECSAM, Penang, Malaysia, SEAMEO Regional Centre for Education in Science and Mathematics, Penang, Malaysia , SEAMEO Regional Centre for Education in Science and Mathematics, Penang, Malaysia

In this paper, we report the dimensionality investigation of the specialized mathematics open-ended questions using conventional approaches and the multidimensional model for Differential Items Functioning (DIF) framework. The dimensionality of these questions was initially analysed exploratory utilizing the Principal Components Factor Analysis and model fitting in the Confirmatory Factor Analysis. The open-ended questions were verified to be multidimensional using the exploratory analysis method. We then advocated the use of the Poly-SIBTEST Differential Test Functioning (DTF) analysis, which was based on the multidimensional model for DIF framework. This was to confirm that the open-ended questions were multidimensional and to identify a possible secondary dimension. The possible secondary dimension was hypothesized to assess a distinct cognitive dimension to supplement the dominant ability being measured, namely the formal reasoning ability beside the dominant construct, the general mathematics ability. The DTF analyses revealed that the open-ended questions were multidimensional and the multidimensionality was due to the inclusion of the formal reasoning ability. The exploratory and model fitting analyses also revealed that there could be more than one secondary dimension involved. Further studies should investigate the other possible secondary dimensions. Knowledge of the secondary dimensions assessed by the open-ended questions will have an impact on the way the teachers generally teach. Teachers will need to give attention to the other secondary abilities. Another significant implication is that the single score being normally reported for multidimensional questions test influences the validity of inferences and decisions being made about the examines.

### Abstract for 17263

Folding and Unfolding a Cone with Cabri 3D : a Process of Modelling Generating New Problems to Be Explored

#### Authors: Jean-Jacques Dahan

#### Affiliations: IREM of Toulouse

As Cabri 3D does not contain the tool "net of a cone", a middle school teacher complains to me because she needed it to help her students to understand that the net of a cone is not always half a disk. I have explored this problem and present a process of modeling the (un)folding of a cone. This result allows opening a window on new problems that can be explored: some invariants about the net of a cone and some 3D curves obtained when unfolding the cone. By the way we enrich the cooperation between a researcher and a teacher. Some of the problems explored thanks to my model can be posed to the students of the middle school teacher and enhance their experimental practice of mathematics using ICT.

### Abstract for 17265

A System for Helping Concept-Building of 3D Linear Algebra by Connecting Graphic, Symbolic, and Verbal Representations

#### Authors: Hitoshi Nishizawa, Bernhard Zgraggen, Takayoshi Yoshioka

#### Affiliations: Toyota National College of Technology, University of Applied Sciences Rapperswil

Many students in the college two of the Authors work for have suffered lower performance in three-dimensional linear algebra for a long time. It is partially because they have learned linear algebra as a set of symbolic procedures isolated from graphical or verbal representations, and also from real world applications. Procedural knowledge without understanding the concept seems to evaporate just after the examinations.

To help students building conceptual knowledge of linear algebra, we have been constructing a learning system, which shows the dynamic relations of three representations: vector equations, 3D graphic objects represented by the equations, and verbal explanations of the features of equations and graphic objects. The system serves a series of activities, where students are able to touch or transform some parts of the graphic objects in virtual space, observe animated transformations of the graphic objects and simultaneous changes of symbolic expressions, and connect them to a list of verbal explanations. By experiencing the activities in the system, the students are expected to recognize the links between the three representations in 3D linear algebra.

The system is a combination of several software programs including web-browser/server, a database management system, a dynamic geometry software, and a computer algebra system. The structure and several activities of the system are presented in this talk.

### Abstract for 17266

Handwriting Tool For Note Takers In Mathematical Classes

#### Authors: Ryoji Fukuda, Koji Nii

#### Affiliations: Faculty of Engineering Oita University, Graduate School of Computer Science and Systems Engineering Kyushu Institute of Technology

We made a hand writing input system for dictation to support hearing impaired students in a class. In mathematical class there are some formulas and figures even in conversations, then hand writing input method must be most suitable. These tasks must be done very quickly, then we made some special edit functions to input contents using premade one. To adjust individual usability there is a function to suggest improving the system parameters.

### Abstract for 17267

Undergraduate Mathematics Assessment in an Integrated Technology Environment

#### Authors: Greg Oates

#### Affiliations: The University of Auckland

Effective integration of technology into the teaching and learning of mathematics presents a significant challenge to tertiary mathematics educators. Assessment issues in particular are widely considered in the literature as a critical factor in technology implementation, and this was confirmed in a PhD study investigating the overall use of technology in undergraduate mathematics. This paper briefly discusses the taxonomy developed as a part of this wider study to describe and compare technology use within individual courses and departments. Assessment is one of six overarching characteristics identified in this taxonomy for consideration in the construction of an Integrated Technology Mathematics Curriculum (ITMC). The paper then presents evidence with specific reference to assessment practices gathered from an observational study of technology implementation at The University of Auckland, and examines this evidence against the taxonomy. The findings of this study suggest that aspects of assessment such as curricular congruency, equity and the advantages and affordances provided by different technologies require continued attention, if integrated technology is to be successfully implemented and sustained.

### Abstract for 17268

Comparison study for Primality testing using Mathematica

#### Authors: Hailiza Kamarulhaili, Ega Gradini

#### Affiliations: Universiti Sains Malaysia, Universiti Sains Malaysia, Minden, Penang,

In this paper, we present four primality testings. Mathematica software is used to carry out the primality tests. The application of Fermat's Theorem on the tests are also discussed in a great detail and this leads to the concept of pseudoprime. We also showed some examples of pseudoprimes with certain range and do quantitative comparison and in addition to this, we discussed the distribution of carmichael numbers and Mersenne primes.

### Abstract for 17270

The elementary school teachers' belief of integrating calculator into Mathematic instruction.

#### Authors: Yi-Ching Lin, Yuan Yuan

#### Affiliations: graduate student, professor

This study uses a questionnaire to investigate elementary school teachers¡¦ beliefs of applying calculators in teaching mathematics. Participates were 131 elementary school teachers in Tao-yen County. The questionnaire used to collect data was revised from the instrument designed by Brown et al. (2007). Elementary school teachers' beliefs of applying calculators in teaching mathematics were divided into three factors. They are catalyst beliefs, teacher knowledge and crutch beliefs. The result showed that elementary school teachers in Tao-yen County do not seem to agree with using calculators in teaching mathematics. Elementary school teachers thought that calculators could not promote students' learning of mathematics and students will depend on using calculators and they impede students' learning. Meanwhile, the teachers thought that they had low teaching efficiency when using the calculator in their classes. The other result indicated that teachers who had technology efficiency of information tool can help the students construct not only knowledge but also conceptual practice and desire, who participated in related training activities, and who had the willing to popularize the calculator application into mathematical teaching in classes agreed to the thought that using calculators can promote mathematical learning.

### Abstract for 17274

Teachers' dealing with technology-enhanced mathematics in teaching practice

#### Authors: Mette Andresen, Morten Misfeldt

#### Affiliations: Aarhus University, National Knowledge centre for Math. Education, UCC

Abstract: The aim of the study is to throw light on how experienced, Danish upper secondary teachers deal with technology enhanced mathematics in their teaching practice. In particular, we find the relations between these teachers' personal view on mathematics and the use of ICT tools may be of interest. We have chosen the very first lessons with the new tool as object for this study because explicit arguments and reflections on the use of ICT were most likely to be given by the teacher at these occasions. Besides, we expected the teachers to be especially aware of the students impression of the role of ICT when they prepared these first lessons and, thereby, revealing their viewpoints.

The study aims to contribute to the rich variety of examples of good practice, to inspiration for colleagues with more or less experience themselves. In contrast, the study does not aim at setting up a strict correspondence between teachers' beliefs and their handling ICT in practice. Neither, we aim to rate different approaches or to assess them with regard to students' learning outcome.

### Abstract for 17277

The Three-Joint Lamp and Vectors

### Authors: Nunnapad Toadithep

### Affiliations: Department of Computing, Faculty of Science, Silpakorn University, 73000, Thailand.

This work is to express the relationship between the position of movement and its vector representation by an ordinary tool used in everyday life. So that ordinary people can understand the concept of vectors easily. Force is a good example of vector. A force that is applied to any object is composed of magnitude and direction. In physics, a work occurs if one pushes an object with some force for some distances. Forces that are applied to household tools such as doors, windows, lamps, etc, can be represented by vectors. This paper shows how to implement a vector object in VPython. An example of how to determine the best composition of the 3 limbs of a three-joint lamp by vector rotation is given.

### Abstract for 17282

Using Multimedia Learning System for Teaching Media Math

#### Authors: Ruzlina P Adni, Hanafizan Hussain

#### Affiliations: Multimedia University

In this paper, we will focus on Multimedia University's own system is known as Multimedia Learning System (MMLS). It is a web-based teaching and learning tool, supported by multimedia tools, which create a virtual interactive teaching and learning environment. Architecturally, MMLS comprises three sub systems, of the front end system for the user to interact, the main engine, which manage the content and interactivity and lastly the back end systems, which stores all data and history of the users. This study will look into the interface of MMLS for teaching Media Math in Faculty of Creative Multimedia (FCM) at Multimedia University. It also will identify the lecturer's preference in approaching the MMLS environment and it will comprise basic essential features of a successful web-based teaching and learning tool.

### Abstract for 17288

Teaching and Learning Trigonometry with Technology

#### Authors: Barry Kissane, Marian Kemp

#### Affiliations: Murdoch University

Modern school classrooms have access to a range of potential technologies, ranging from calculators to computers to the Internet. This paper explores some of the potential for such technologies to affect the curriculum and teaching of trigonometry in the secondary school. We identify some of the ways in which the teaching of trigonometry might be supported by the availability of various forms of technology. We consider circular measures, graphs of functions, trigonometric identities, equations and statistical modeling and focus on activities that are not possible without the use of technology. Modern technology provides an excellent means of exploring many of the concepts associated with trigonometry, both trigonometric and circular functions. Many of these opportunities for learning were not available before technology development and access within schools we enjoy today. This paper suggests some of the avenues for exploration.

### Abstract for 17293

Analysis of Discrete Models Using an Electronic Spreadsheet--A Survey of Examples

#### Authors: Thomas McMillan

#### Affiliations: University of Arkansas at Little Rock

This paper surveys examples in which questions are answered by developing recurrence relations that model certain situations and then exploring those recurrence relations using an electronic spreadsheet. The goals of this approach are (1) to develop an understanding of and a facility for using recurrence relations as a modeling tool to motivate the need to derive analytical methods for solving recurrence relations, (2) to increase awareness of the spreadsheet as a problem solving tool, (3) to validate intermediate steps in analytical solutions using the spreadsheet, and (4) to use the spreadsheet's computational and graphics capabilities to explore and analyze questions that do not lend themselves to an easy analytical solution. The examples described here have been used in discrete mathematics courses and professional development courses for licensed teachers.

### Abstract for 17307

A New Block Method and Their Application to Numerical Solution of Ordinary Differential Equations

#### Authors: Rei-Wei Song, Ming-Gong Lee

#### Affiliations: College of Engineering, Chung Hua University, Department of Applied Mathematics Chung Hua University Hsinchu, 30012 Taiwan

A class of multistage and multistep integration methods which can obtain r new values simultaneously at each integration step was developed. Their stability regions were derived and sketched by MATLAB, and their regions are either A-stable or -stable. Their applications to numerical solutions of non-stiff and stiff equations by predictor-corrector scheme were also studied.

### Abstract for 17308

Teaching and Learning Calculus with the TI-Nspire: A Design Experiment

#### Authors: Wee Leng Ng, Wee Chuen Tan, Meow Leng Nancy Ng

#### Affiliations: National Institute of Education, Nanyang Technological University, Dunman High School

A design experiment was conducted to examine the role of the TI-Nspire, the latest graphing calculator from Texas Instruments, in teaching and learning calculus. This paper reports details on, and preliminary results of, the design experiment involving the design and conduct of a TI-Nspire Intervention Programme for an intact class of thirty-six secondary four students (15-16 years) from a secondary school in Singapore. Use of the TI-Nspire was integrated into teaching and learning Calculus concepts with the aid of the TI Navigator, a wireless classroom network system that enables instant and active interaction between students and teachers. Mathematics attitudes surveys and structured interviews were administered to assess the effects of the use of the TI-Nspire on students' attitudes towards mathematics. It was found that appropriate use of graphical, numerical and algebraic representations of Calculus concepts using the TI-Nspire could enable the subjects to better visualize the concepts and make generalizations of relevant mathematical properties. Results of paired samples t-tests and interviews with students suggest that there the use of the TI-Nspire has a positive effect on students' confidence in and perceived usefulness of mathematics.

### Abstract for 17320

Exploring and Visualizing Hilbert Geometry in a Triangle

#### Authors: Mauricio Herrera, Rubén Preiss, Gonzalo Riera, Hernán Carrazco

#### Affiliations: Universidad Diego Portales, Universidad Diego Portales, Facultad de Ingeniería, Instituto de Ciencias Básicas, Santiago. Chile., Pontificia Universidad Católica de Chile, Facultad de Matemáticas, Santiago. Chile., Universidad Internacional SEK

In this paper we explore Hilbert geometry in a triangle, using Maple to illustrate some concepts such as Hilbert distant, projective and affine coordinates, unitary circle, etc. and to introduce new "trigonometric functions" for this geometry.

### Abstract for 17324

Enhancing Students' Understanding Statistics with TinkerPlots: Problem-Based Learning Approach

#### Authors: Krongthong Khairiree, Piromya Kurusatian

#### Affiliations: Thaweetapisek School Bangkok, Thailand

The purpose of the research study was to explore the effectiveness of using TinkerPlots dynamic software and problem-based learning approach in statistics classes. The research study was conducted in the Education Year of 2009. The subjects were students and teachers from secondary school in Bangkok Thailand. The pre-test and post-test control group design was used in the study. The experimental group learned statistics using TinkerPlots dynamic software and problem-based learning approach. The control group learned statistics through problem-based learning. The data collection included quantitative and qualitative methods. The research finding showed that if it is appropriately employed, TinkerPlots dynamic software can be used as an effective tool in enhancing active learning and students' understanding in statistics. In addition, the students had positive attitude toward statistics after they learned statistics using TinkerPlots dynamic software.

### Abstract for 17325

Construction of Quadratic Curves Using Analysis

#### Authors: Hee Chan Lew, Soo Youn Je

#### Affiliations: Korea National University of Education, Department of mathematics education Korea National University of Education

Abstract: This research aims to observe the process in which students participate in constructing three quadratic curves tended to be overlooked in the current climate of school instruction. In this process, our aim was to have the students learn how to construct the quadratic curves logically by using the Analysis, and recognize its definition and properties of several geometric figures that they had learned while proving justification of the constructing. We taught two 10th grade girls who hadn't learned quadratic curve, about the parabola, ellipse, and hyperbola. Then, we had those girls construct quadratic curves on a plane. In conclusion, the Analysis suggested effective processes for the students to find out how to construct quadratic curve. Finally, it was found that GPS facilitated the students to learn the construction more conveniently by offering the numerical and visual feedback

## Papers with Abstract Only

### Abstract for 16869

How to Effectively Integrate a 1:1 programme into the Mathematics classroom

#### Authors: Jennifer Wathall

#### Affiliations: Island School, English Schools Foundation

With the rapid technological advancement in the world, educators have a duty to enhance the teaching and learning in their classrooms by incorporating learning technologies that not only stimulate learning but also give the learner deeper cognitive connections. Many schools world wide have adopted a 1:1 programme but what impact has this had on learning? How do we effectively integrate a 1:1 programme in a mathematics classroom? How can we use the internet effectively to enhance the learning of mathematics? A number of problem based learning activities utilizing the 1:1 will be discussed including the use of Mathematical Webquests. Mathematical Webquests require higher order thinking skills and enables the learner to synthesize, evaluate and criticize.

### Abstract for 17070

Hooked on Webquests

#### Authors: Denise LeGrand

#### Affiliations: University of Arkansas at Little Rock

Traditional methods of teaching are taking on new facets attributed to technological advances. Incorporating new communication techniques such as Blogs, chat, and Facebook can significantly enhance an online mathematics course. Mathematics software has greatly improved with the addition of applets, tutorials and other resources. Webquests are useful in blending these communication tools, software and other resources together. Here I will discuss the use of webquests in an online calculus course. In the usual sense, webquests require the students to search the internet for solutions to real life problems, to make deductions and form a conclusion from that search. I will present how webquests can be used a little differently: students search for tools, software and resources they can utilize to augment learning material based on specific criteria, purpose and categories and then discuss pros and cons. Specific webquest examples will be presented with the resulting online dialogues, providing students with a door that opens into further explorations of calculus.

### Abstract for 17075

Entertainment Product Packaging for Distribution to Maximize Revenue

#### Authors: Bruce Wang, Yihua Li, Kai Yin

#### Affiliations: Texas A&M University, Walt Disney World

In this paper, we study a product packaging and distribution problem in which a chain store selects from a large number of available products (e.g. DVDs) a subset to combine into packages to distribute to its local stores. Each package only contains a small number of items. Only a limited number of different combinations are packaged due to the economy of scale in packaging and distribution. We decide what products should be in each package and which package to distribute to each store for revenue maximization. A column generation method is proposed to solve this large scale optimization problem. Two heuristic methods are tested for the nonlinear sub-problem. Bounds are developed for both the optimal number of different packages and the total revenue in order to expedite the solution process. The algorithm developed shows promise with partial data from a chain store which has several thousand retail locations.

### Abstract for 17083

Educational Software in Compensation of Geometric Background Insufficiency for Computer Graphics Studies

#### Authors: Vladimir Nodelman

#### Affiliations: Holon Institute of Technology

Geometry at all and geometric models of algebraic notions in particular are very often disappearing from the fundamental mathematics studies. Anyway, some essential for computer graphics course geometric gaps are traditionally closed by their explicit studies as a part of the course.

Space transformations provide an obvious example of such subject. Introduction of homogeneous coordinates allows reaching a general approach to 3D space transformations' interpretation by 4x4 matrices. Students fluently perceive the concrete matrices of basic transformations but confuse in case of

• Discovering appropriate sequences of transformations (figure's positioning and animations),

• Not invertible transformations (shadow matrices),

• Projective transformations matrices' analytic featuring (in general and in order to receive proper 2D image of the 3D scene).

Students spend a lot of time trying to find out fitting solution by indigested parameters' changes followed by compilations and executions (especially in the most frequent task of transformations' sequences definition for correct positioning of each one of figures in the 3D scene) instead of their analytic recognition.

Presented here an approach to ease and accelerate digestion on necessary knowledge and correspondent skills by application of educational software for mathematics studies. While using ready models and creating their own ones with following interaction with them students discover essential characteristics and their interrelations in the studied matter. These activities are based on an intensive utilization of some features provided by the software:

• Fluent parameters' value change by GUI,

• Synchronized separated multiple views for presenting results of space transformation by different matrices and their compositions with mapping investigation facilities, • Mathematical objects' definition by built-in C-like script.

Several examples of final course's projects that reflect fruitful results of students' deliberate control on dynamics of the scene elements' geometry are presented also.

### Abstract for 17084

Effectiveness of STestMAP as a tool for selection of Statistical Tests

#### Authors: Kien Kheng Foo, Howe Eng Tang, Ung Hua Lau, Chin Ying Liew, Yee Hui Robin, Chang

#### Affiliations: Universiti Teknologi MARA, Kuala Pilah Campus, Negeri Sembilan, Universiti Teknologi MARA Samarahan Campus Kota Samarahan Sarawak Malaysia

One major problem faces by many researchers and students when confronted with the hypothesis testing procedure is to select the relevant statistical test for the purpose of calculating the test statistics. To facilitate the acquisition of this selection skill, a mapping model called STestMAP was designed for this purpose. This study aims to determine the effectiveness of STestMAP as a tool to help in the selection of the test. It also aims at investigating the flexibility afforded by the STestMAP procedure in catering to the needs of both linear and non-linear users. The research design divides the study into Phase I where quantitative analysis dominates while qualitative analysis is used in the second phase. Four instruments were adopted and administered. Data collection was carried out through paper and pencil tests and interviews. Both quantitative (descriptive and inferential analysis) and qualitative data analysis techniques were utilized. A random sample of 50 respondents from two public universities in Sarawak, Malaysia was chosen out of which 6 respondents were purposively selected and interviewed. The Wilcoxon Signed Rank Test showed that there was a statistically significant difference in pre and post test for control group, Z=-3.978, p=0.001 as well as for the treatment group, Z=-4.275, p=0.001. In comparing the gains between the treatment and control group however, the test indicated insignificant difference in the gain scores, Z=-0.142, p=0.887. A more detailed analysis with program of studies and Cumulative Grade Point Average as independent variables revealed statistical significance differences for certain group comparisons. In phase 2 of the study, within-case and cross-case analysis were employed. The findings clearly showed that STestMAP provides a flexible structure in assisting users to successfully identify the statistical test. A short discussion of the limitations and some suggestions for further research concludes the paper.

### Abstract for 17085

Role of new technology on education of mathematics

#### Authors: Saeed Hashemi

#### Affiliations: School of International Relation Ministry of Foreign Affairs

History of science epically history of mathematics shows to us that all civilizations during early educations thought mathematics for that reason they had civilization .If we are interested to have good civilization we must teach nice mathematics to have good engineer, economist, physics. In this article I try to show how we can have nice mathematics classes with use of new technologies .I use mathematical software to teach better and better understanding and also I use internet in classes in each classes I show to student how to prepare small research work on application of mathematics on interest field of student in this article I will compare numbers of two classes which one I used soft ware and internet and other which I did not used these technologies.

### Abstract for 17087

Why I Love Subtraction: Using Error Analysis in First Year Calculus

#### Authors: Mark Howell

#### Affiliations: Gonzaga College High School, Washington, DC

Technology affords opportunities to explore the errors resulting from various approximations in first year Calculus. This session focuses on analyzing such errors to gain insight into fundamental Calculus concepts. Examples include comparing the tangent line to another line with almost the same slope; comparing higher-order polynomial approximations; investigating why machines use the symmetric difference quotient for approximating the derivative, as opposed to a one-sided difference quotient; and various methods for approximating definite integrals, including midpoint and trapezoidal sums. All of these are rich settings to investigate deeply some key Calculus concepts.

### Abstract for 17092

Periodic Orbits in Triangular Billiards as a Minimization Problem

#### Authors: Felino Pascual

#### Affiliations: Mathematics/Statistics, Winona State University

The example of polygonal billiards in mechanics and dynamical systems can be used as an introductory topic for investigating periodic orbits and the permutations that can be associated with the "bounces" that result when a point mass or billiard "ball" hits a side of a polygonal billiard table. The simple case when the billiard table is triangular provides an excellent starting point for the study of the principles of reflection and geometric ideas related to these "bounces". Using Geometer's Sketchpad, we will highlight these ideas and a few basic results such as those concerning the existence and uniqueness of Fagnano or 3-period orbits, 4-, 5-, and 6-period orbits. One such result, that periodic orbits in triangular billiards correspond to paths of minimal length, a fact that is related to isosceles triangles and their bases, can be easily shown using basic calculus and geometry.

### Abstract for 17094

The effect of CAS calculator usage on the Algebra Achievement of Low Mathematics achievement students

#### Authors: IN KYUNG KIM, Hee-Chan Lew

#### Affiliations: Korea University of Education, PME

This paper observed the effect of CAS calculator usage while studying algebra on the achievement of low-achievement students in mathematics, here in referred to as low-achievement students. Participants were composed of 70 low-achievement tenth graders from a high school located in a metropolitan city that had never used a mathematics educational calculator before. Target participants were divided into two groups: an experiment group that studied activity papers with the aid of a CAS calculator, and a control group that studied the same activity papers using only paper and pencil. The activity papers were used in 14 consecutive classes. The content of the activity papers for the two groups was the same, but the structure differed. Content consisted of numbers and operations, equations and inequalities (character and expressions), and functions. The activity papers of the control group were solved with the use of only paper and pencil and solutions were compared to those presented by the teacher. Students in the experiment group first solved problems with paper and pencil and then again using a CAS calculator. They were told to compare their two problem-solving processes; i.e., compare the paper and pencil procedure with the CAS calculator. Students in the experiment group exhibited matacognition learning using a CAS calculator usage method. The activities were carried out once a day for about one mouth. The two groups completed mathematics achievement tests both before and after the activity papers. Most students had similar overall Mathematics results, however, on the algebra section, the average scores of the experiment and control groups were very different. Therefore, ANCOVA analysis results showed that compared to the pretest, results of the experiment group improved considerably more than the control group.

### Abstract for 17096

Journey to Discover More Mathematics -- Additional Mathematics Project Work

#### Authors: Hong Eng Ding

#### Affiliations: Faculty of Education, University of Malaya, Kuala Lumpur, Malaysia

Additional Mathematics Project Work is introduced in Malaysian curriculum to enhance the understanding of concepts and mastering of skills. Project Work enables Form 4 and Form 5 students to apply what they have learned in the classroom to solve real-life problems, to make use of various technologies to further discover mathematics. It provides opportunities for students to explore the mathematical problems and gain a lot of benefits. For instance, it activates their minds, makes the learning of Mathematics more meaningful, develops their communication skills and higher order thinking skills. This presentation aims to explain how technologies like Graphing Calculators, Geometer's Sketchpad, Spreadsheet and Internet are used to solve challenging mathematics problems creatively, critically and innovatively. It also gathers teachers' and students' opinions and suggestions towards the implementation of project work.

### Abstract for 17102

Discovering Fresh Aspects of Geometry, Algebra and Calculus in PNG Stamps, String Bags and Traditional House Wall Patterns

#### Authors: Thadreina Abady & Lorna Denema, Lorna Denema

#### Affiliations: Divine Word University

In this paper we consider some fresh aspects of studies that integrate geometry, algebra, and calculus with PNG cultural features such as traditional house wall patterns, string bags which are also called `bilums', and PNG stamps through the use of Microsoft Excel. We represent these features visually in Excel by inputting XY values, and using Excel's mathematical functions such as cosine and sine to generate series for XY scatter charts to represent the values in a graphical form which illustrates features that we discuss in this paper. The traditional houses or huts which still exist today in all native villages of the country are built mostly by using bush materials. These house walls display patterns formed from such common basic geometrical shapes as diamonds, squares and triangles. The string bags are hand woven using knotting techniques and are passed on from generation to generation. They are used not only to carry locally grown food and other goods, as well as babies in the same manner as modern factory made bags. The traditional construction concept is adapted to integrate new styles, patterns, colors, and factory made wool to create bilums. Bilums adapt names according to the patterns that they display either the geometrical shapes or other real world objects. PNG stamps are part of modern art, and the characters that the stamps depict are rare species of PNG nature such as orchids, birds of paradise, and others that we can create by using mathematical functions from trigonometry and calculus.

### Abstract for 17105

Teachers' Conceptualizations of Data Collection

#### Authors: GT Springer

#### Affiliations: Hewlett-Packard

In this presentation, the speaker will share over a decade of experience in teacher professional with U.S. mathematics teachers. The presentation will focus on issues related to learning to collect data using scientific probes and sensors with graphing calculators. Specifically, the author will focus on the different experiences teachers have when learning to use data loggers vs. data streamers. The discussion will touch on ease of use issues, but the focus of the presentation is the teacher's own conceptualization of how these systems work. The speaker will share some of his own current conjectures about how these conceptualizations align (or fail to align) with key concepts from secondary mathematics education, and the implications for their use in the classroom.

### Abstract for 17108

A System for Helping Concept-Building of 3D Linear Algebra by Connecting Graphic, Symbolic, and Verbal Representations

#### Authors: Hitoshi Nishizawa, Bernhard Zgraggen, Takayoshi Yoshioka

#### Affiliations: Toyota National College of Technology, Universitiy of Applied Sciences Rapperswil, Switzerland

Many students in our college have suffered lower performance in three-dimensional linear algebra for a long time. It is partially because they have learned linear algebra as a set of symbolic procedures isolated from graphical or verbal representations, and also from real world applications. Procedural knowledge without understanding the concept seems to evaporate after the examinations.

To help the students building conceptual knowledge of linear algebra, we have been constructing a learning system, which shows the dynamic relations of three representations: vector equations, 3D graphic objects represented by equations and verbal explanations of their characteristics to the students. The system serves a series of activities, where the students are able to touch or transform some parts of graphic objects in virtual space, observe animated transformations of graphic objects and simultaneous changes of symbolic expressions, and connect them to a list of verbal explanations. By experiencing the activities in the system, the students are expected to recognize the links between the three representations in 3D linear algebra.

The system is a combination of several software programs including web-browser/server, a database management system, a dynamic geometry software, and a computer algebra system. The structure and several activities of the system are presented in this talk.

### Abstract for 17113

Rethinking Calculus

#### Authors: Michael Livshits

#### Affiliations: Cambridge, MA, USA

Calculus is considered a finished piece of mathematics, based on the late 19th century classical analysis. Many computer programs that help people learn it have been written. On the other hand, little has been done to simplify and clarify the fundamental concepts of calculus; in fact many claim that it is impossible, since calculus is already perfect. I will challenge this point of view and discuss some recent progress, done by Hermann Karcher, Karl Dovermann, Peter Lax, Mark Bridger, Qun Lin and myself. I will start with differentiating polynomials by factoring p(x)-p(a) through x-a, and show how calculus can be developed directly and naturally, with no use of limits and continuity, and hardly any use of real numbers. Some of the material for this talk is discussed in my preprint at http://arxiv.org/abs/0905.3611 and references there. My purpose is to convince the audience that calculus is rather elementary, that it can be understood without wading through unnecessary mathematical abstractions, that learning and teaching it can be made more satisfying for students and teachers alike.

### Abstract for 17115

Cryptography and the DES

#### Authors: Wolfgang Baltes

#### Affiliations: Qatar University

Cryptographic tools are increasing its relevance and importance in our daily life since any kind of networks are growing up quickly with a huge variety. In this process, protocols and ciphers play an important role because they establish secure and reliable connections between clients to exchange data.

In this context, the Data Encryption Standard (DES) serves as a classic and famous example of a cipher, that is reaching its end of lifetime but that is still widely used. To introduce basics about ciphers and especially the DES, a 15 minutes introduction is completed with a DES-visualization in form of a java-applet. The DES-design is explained and the applet visualizes explicitly all bits and bytes in each of the 16 DES' internal cipher rounds and also its related roundkeys.

Moreover we demonstrate in statistics and in a colored differential-matrix the effect in the ciphertext and in S-boxes if bitwise changes undergo the plaintext. The related field is well known under the name "Differential Cryptanalysis". The visualization serves as an introduction to what cryptography is about and uses the DES as example. The presented program simplifies many aspects quite explicitly that normally are either difficult or very time consuming to grasp. In this way it motivates beginners or advanced levels to understand some underlying ideas of ciphers and to get an explicit insight of this charismatic field.

### Abstract for 17116

New Approaches to Mathematical Exposition with Excel

#### Authors: D. E. Arganbright

#### Affiliations: Divine Word University

This paper examines innovative ways in which we can use a spreadsheet to invigorate the teaching and dissemination of mathematics. For many years, the spreadsheet Microsoft Excel has gained increasing popularity in the mathematics classroom, especially for implementing algorithms, designing mathematical models, and creating interactive graphics. Here we illustrate an additional use of Excel to enhance the visualization and explanation of mathematical concepts by using it in the manner of a PowerPoint presentation. Besides being able to generate static pictures and text, we enhance our displays with animated spreadsheet graphics. Although we do all work using Excel and its standard features, our visual output appears in a standard mathematical format rather than in the row/column format of Excel. In the process, we employ some of the relatively unexploited capabilities of the spreadsheet display itself. We create pictorial representations of mathematical structures, algorithms, and models. As we interrogate them using Excel's "What if …?" capability, we interactively observe our displays being updated instantly. We use Excel's array layout to produce such mathematical formats as matrices with brackets and Venn diagrams directly on the spreadsheet itself, with headings and gridlines removed. This paper provides classroom-tested illustrations of algorithms, models, and graphs coming from calculus, discrete mathematics, linear algebra, numerical analysis, and statistics. One valuable application of this approach is the application of Excel to create material for upgrading the knowledge and skills of novice university mathematics lecturers as they further their understanding of the theoretical aspects of calculus and other courses.

### Abstract for 17127

Using PPF to Design Mathematical problems for Students

#### Authors: Da Liu

#### Affiliations: Putuo Educational College, Shanghai, China

PPF is the acronym for production possibilities frontier, one of the economic models. PPF is a curve that shows the different rates of production of two goods that an individual or group can efficiently produce with limited productive resources. The simplified PPF model can be realized as a function of mathematics by the students of secondary schools. In this paper, we designed a mathematical problem about PPF based on the essential knowledge in high school math levels. With Graphic Calculator or only pen and paper operations are the two methods for the students to solve the problem. Furthermore, we also lead students to deeply think about how to use Graphic Calculators to find a common way to solve the problems of this kind.

### Abstract for 17130

Dynamic Simulation for "The Nine Chapters": Using Dynamic Geometry System and Interactive WebPages

#### Authors: Feng-Lin Lu, Tai-Yih Tso

#### Affiliations: Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan

The purpose of this study is to construct a dynamic network resource which includes simulating multiple representations of mathematics texts and interactive classroom activities. By integrating school mathematics and new technology, people can make sense of mathematical ideas of ancient mathematics texts.

Since "The Nine Chapters", as famous as "The Element", shows mathematical problem solving by using algorithms, which is similar to the developments of mathematical thinking? Thus, we choose "The Nine Chapters" as our main digital content in the dynamic network resource. Learners can realize the mathematical ideas of "The Nine Chapters" by experiencing dynamic simulation and interacting classroom activities. Therefore, dynamic geometry system is not only the platform about simulation and manipulation, but a virtual partner for learning.

According to cognitive load theory (CLT) and multiple representations, the dynamic network resource is composed of five blocks:

1. Mathematics Texts: this block displays the original texts of "The Nine Chapters" and learners can realize the Affiliations of mathematics.

2. Interpretation: the block contains translations of "The Nine Chapters" and some algebraic expressions.

3. System: this block shows outlines of "The Nine Chapters" and introductions to this resource.

4. Dynamic Geometry System: the block represents simulation for mathematics texts and the process of solutions.

5. Integration: this area contains the interactive classroom activities based on syllabus.

We expect this resource not only represents digital content, but guides people to understand the meaning of mathematics texts and enjoy mathematics.

### Abstract for 17134

The Teaching Experiment of Multiplication of Complex Numbers in Dynamic Geometry Environments

#### Authors: Chi-Chiang Hsu, Tai-Yih Tso

#### Affiliations: National Taiwan Normal University Math Department

The purpose of this study is to analyze the teaching and learning within multiplication of complex numbers in the classes of the senior high school, to establish the learning system in computers with dynamic multiple representations, and to evaluate the effect of students" learning and application of dynamic linking multiple representations in the dynamic Geometry environments. Samples consist of the students from two classes of the grade two and one class of the grade three in the senior high school. The dynamic Geometry environments are designed by the GeoGebra software, embedded into the web pages and operated by JavaScript statements. Users can move points to see the effects in the box of GeoGebra software by the mouse, and type some numbers or press some buttons in the webpage to change the graph in the GeoGebra box. Students proceed as easily as to explore the internet. Students in the experimental group are instructed by DGE in the computer classroom, and make dynamic linking with dynamic multiple representations. Students in the control group are instructed by the traditional teaching in the ordinary classroom, and make motionless separation with motionless representations. The result of the study reveals that the three classes of the students make progress apparently. The advancement score of the experimental group is more than the advancement score of the control group.

### Abstract for 17135

Trigonometry through the Geometer's Sketchpad (GSP) Make a Difference

#### Authors: Warabhorn Preechaporn

#### Affiliations: SEAMEO RECSAM, Penang, Malaysia

Educational technology has a significant positive impact on achievement in all subject areas. Across all levels of school, and in regular classrooms as well as those for special-needs students. The Geometer's Sketchpad (GSP) is an interactive and dynamic computer program that can be used to help students learn and understand geometrical concepts and principles. The GSP allows the user explore simple, as well as highly complex, theorems and relations in geometry.

This paper is a brief attempt to propose how to help the students explore and investigate trigonometry related to the right triangles that were used to define the trigonometric functions by placing one of the acute angles of a right triangle on a coordinate plane and connection to the geometry on the unit circle. The students explore its measurements using the GSP that helps students develop meaningful understanding of the concept of solving trigonometry.

### Abstract for 17136

Enhancing Teaching and Learning of Quadratic Function using GeoGebra

#### Authors: Warabhorn Preechaporn, Leong Chee Kin, Teoh Boon Tat

#### Affiliations: SEAMEO RECSAM, Penang, Malaysia

The powerful environment for the journey of exploration and extension of a problem is GeoGebra. GeoGebra is dynamic mathematics software. The user can create dynamic worksheets that can be used with any Internet browser supporting Java on any operating system. The software was developed to enable multiple representations and visualization of mathematical concept. Thus GeoGebra lends itself to create activities incorporating multiple representations of mathematical concepts that are linked dynamically. Without having to spend a significant amount of classroom time on drawing figures, objects or function, students can explore mathematical concepts and dynamically connect algebraic, graphic and numeric representations of these concepts.

In this paper we present by illustrative of ideas how to enhance teaching and learning of quadratic function using GeoGebra. Then we discussed the importance of creating dynamic worksheets by emphasizing that the use of various representations and visualization.

### Abstract for 17161

E-resources for the undergraduate engineering mathematics student

#### Authors: Aruna Palipana

Affiliations: UK Centre for Excellence in Mathematics and Statistics Support (SIGMA), Mathematics Education Centre, Loughborough University, UK, Maths, Stat and OR Network (MSOR) UK, IMechE (UK)

For the engineering undergraduate learning mathematics, various e-resources have been made available by the Maths Education Centre (MEC) at Loughborough University, UK. 48 Mathematics workbooks developed under the ‘Helping Engineers Learn Mathematics’ project are available electronically as PDF files. These workbooks cover first year engineering mathematical and statistical content and some second year concepts too. They also contain engineering examples and case studies. About 80 interactive e-lessons, supporting 23 workbooks, are provided to aid understanding. A web based computer aided assessment regime with over 5000 questions in question banks, is available to support the formative and summative assessment of maths concepts covered by the workbooks. Students are also encouraged to use video lessons, lesson leaflets in PDF format and quizzes developed under MathCentre and MathTutor projects. These videos are available in DVD format and can be viewed online as well. The same videos broken down into smaller ‘chunks’ are also available to download in formats suitable for video iPods and mobile phones. Some lecturers at the MEC also provide audio embedded, hand written PowerPoint shows. The handwriting and the voice of the lecturer are already familiar to the students and this makes these shows more personalised. Some lessons have also been prepared as video podcasts, which are computer generated slides containing text, graphs and equations, backed up by the lecturer’s voice. Video screen captures, later suitably edited to enhance the lesson flow are also being made available. Students have responded extremely positively about these resources.

### Abstract for 17168

Helping Students Succeed in Learning Mathematics With Geogebra

#### Authors: Premjit Singh

#### Affiliations: Dept. of Mathematics, Ohio University

This presentation will discuss how GeoGebra can help motivate students to do mathematics and strengthen their critical thinking skills by visualizing mathematics.

### Abstract for 17206

Application of Finite Element Method to Design an Electric Generator

#### Authors: Wooi Ping Hew, K Chandran

#### Affiliations: University of Malaya

Finite Element Method (FEM) transforms the continuous partial differential equations that describe the natural phenomena associated with an engineering problem into many small discrete ordinary differential equations. This method is often applied to investigate the structural strength in civil and mechanical engineering designs. In electrical engineering, FEM is used to design electrical machines and to investigate the electromagnetic and thermal properties of the final design. This paper describes how Finite Element Magnetic Method (FEMM) software can be used to design a 4 kW electric generator. The FEMM software allows the designer to optimize the physical dimensions of the generator in order to generate the desired power output without causing thermal stress in the machine.

### Abstract for 17207

Exploring some interesting theorems in number theory using a graphing calculator

#### Authors: Michael Grasse

#### Affiliations: Elk Grove High School, Hewlett-Packard Company

Theorems from number theory can be difficult to rigorously prove but their illustrations are often quite accessible. This talk will show how to use a graphing calculator to explore modular congruencies, divisibility rules, Fermat's theorem, Euler's phi function and Euler's theorem. The content will be suitable for teachers of students from junior high school to early tertiary.

### Abstract for 17209

Seeing is Believing in Inferential Statistics:Visual Remedies for Common Misconceptions in Decision Making

#### Authors: Marian Kemp, Jen Bradley

#### Affiliations: Murdoch University

Students in their first encounter with statistical inference often develop several misconceptions. These take many different forms and can be related to what a probability is, what a P-value is, what test statistic should be used, what level of significance should be used and not least -- what decisions should be made to draw appropriate conclusions. This paper focuses on the latter. Teachers need to enable students to not only make critical decisions in an informed way for their own reports but also to interpret statistical results in published papers in their discipline area. The main misconceptions with decision making tend to arise from confusion over precisely which two values should be compared and how. This paper proposes a visual way of developing accuracy in decision making; firstly comparing test statistics and critical values and secondly comparing P-values and significance levels. Whilst there are many forms of technology to help generate numerical test statistics and P-values, from simple scientific calculators through to the most sophisticated statistical packages, this paper proposes ways of utilising the "DRAW" facilities on the graphics calculators. The Authors suggest some practical ways of harnessing this capability for teachers to develop templates to use with their students in the classroom. In addition, the Authors suggest a variety of ways that students can engage with the calculators themselves. The use of the visual representations should lead to the development of fluency with decision making using only numerical values.

### Abstract for 17211

Lab Sessions for Linear Algebra with MATLAB

#### Authors: Victor Tan

#### Affiliations: National University of Singapore, Singapore Mathematical Society

It is common these days that instructors incorporate computer algebra system in the teaching and learning of a first Linear Algebra undergraduate course. At the National University of Singapore, we have been using MATALB to conduct Lab sessions as a component of such courses for many years. Over the years, the format of lab sessions has evolved. In this talk, I will share my experience of designing MATLAB worksheets and quizzes for my Linear Algebra course.

### Abstract for 17212

GeoGebra: Powerful Tool to Explore in Mathematics

#### Authors: Leong Chee Kin, Teoh Boon Tat, Warabhorn Preechaporn

#### Affiliations: SEAMEO RECSAM, Penang, Malaysia, SEAMEO RECSAM Penang, Malaysia

GeoGebra is free, open source, dynamic mathematics software that links Geometry and Algebra, and can be used at practically all levels of secondary mathematics. It is a very versatile, easy to use piece of software for the mathematics classroom bringing together dynamic geometry, coordinate geometry, algebra and calculus. The coordinate geometry or algebra component addresses a wider range of topics than the usual dynamic geometry software packages.

In this paper we propose the use of GeoGebra by illustrating how a teacher can use to explore mathematical conjectures through an activity exploring from the field of geometry.

### Abstract for 17248

Computer simulation and rigorous proofs in some perturbation problems

#### Authors: Chunqing Lu

#### Affiliations: Southern Illinois University Edwardsville

Computer simulation can help estimate the boundary layers for singular perturbation problems. For simple equation ay"" + (1+a) y" + y = 0 subject to y(0) = 0 and y(1) = 1, where a is a sufficiently positive small parameter and the prime denotes d/dx, the numerical simulation by MATLAB shows that the boundary layer of the equation is at the left end of the interval [0,1], x = 0, and the thickness of the boundary layer is O(a) as a approaches zero. Furthermore, a rigorous analysis proved that the computer simulation on this problem gave a good estimation about the boundary layer. However, there are some singular perturbation problems the computer simulation may only indicate the location of the boundary layer, because the boundary layer is too complex. To determine the thickness of the boundary layer more delicate analyses are needed. An equation modeling the laminar flow in a rectangular channel, ay"""" = yy"""- y"y"" with y(0) = y""(0) = 0 and y(1) = 1 and y"(1) = 1, is an example. In this case, the computer simulation showed that the boundary layer is at the right end of the interval [0,1], x = 1, and the thickness of the boundary layer is o(a), much smaller than a. Further rigorous analysis showed the boundary layer has the thickness O(Exp(-1/(4a))/a^(7/4)), which is exponentially small as a approaches zero.

This paper also explains how the computer simulation worked on finding chaos on the equation y"" + (1+sin(at)) sin(y) = 0 (a is a positive small parameter) with MATLAB and the shooting method showed the existence of chaotic solutions of this equation modeling the pendulum with oscillatory force on its support.

### Abstract for 17272

The comparison between different teaching styles of using mathematical techniques.

#### Authors: Siming Gui

Affiliations: CAOYANG No.2 HIGH SCHOOL SHANGHAI CHINA

This article will give out the effect of the technique of using computer and graphic calculator on the education of mathematics in Chinese high school. Several pieces of illustrations are given to show the way that teacher using the techniques and the different effect of different teaching style, ¡°display¡± and ¡°activity¡±. This reflects the currents state of using techniques in teaching. And, the result of the comparison between different styles of teaching provides teachers some ideas on how to use the new technology in mathematics education.

### Abstract for 17299

Using Casio Scientific Calculator to Solve Mathematical Problems in Vietnam Schools

#### Minh Tran

#### Department of Applied Mathematics, Ho Chi Minh City University of Technology, Vietnam

The author presents how CASIO scientific calculator is used in Vietnam schools. Major topics of his lecture include teaching arithmetic, equations, and programming on CASIO fx-570MS,fx -570ES , and TI calculators.

### Abstract for 17300

Using Technology in Mathematics for the Technologically Disenfranchised II

#### Authors: Monos Naidoo

#### Affiliations: South African Mathematical Society, South African Gravitational Society, International Society on GR and Gravitation, Tshwane University of Technology

Years of systematic exclusion of the vast majority of the South African population from the arena of Science and Technology has left huge demographic imbalances in these fields. With Universal Franchise only achieved in 1994, these inequities persist and present challenges in training at tertiary level. Students from previously economically and politically disadvantaged backgrounds remain underprepared for tertiary Mathematics, Science and Technology. Most of these students have had little or no exposure to technology in their school career. Large classes in Mathematics at undergraduate level necessitate the use of technology. It is almost unavoidable including technology in the delivery of lectures as transfer and exposition are greatly enhanced by the mathematical software currently available. In any event, any graduate student in Mathematics or related disciplines will be hindered in progressing with their studies or entering mathematics related professions if they have failed to encounter technology in Mathematics in their undergraduate training. We investigate the inclusion of technology in undergraduate Mathematics. In this paper we consider second year students who have already been exposed to the use of technology in the learning of Mathematics in their first year (for the first time).

### Abstract for 17330

Technology-based Activities which Promote Student-centered Learning

#### Authors: Ma.Louise Antonette De Las Penas

#### Affiliations: Ateneo de Manila University

This talk reports on the use of technology to support and enhance student-centered learning, an integral part of the way of teaching advanced by our university.

The first half of the talk will focus on classroom activities using calculators, designed to promote independent inquiry and allow for an active participation from the students. In the second part, we present an interactive website for the study of Algebra and Trigonometry where students gain additional learning experiences by performing mathematical investigations and studying real world problems outside the classroom. Each applet in the website come hand-in-hand with guide questions to facilitate student explorations and student-centered learning.

### Abstract for 17334

数学教学与CAI---"信息技术与课程整合"改变了什么(Mathematics Education and CAI, What have changed by "Integration of Information Technology with Curricula"?)

#### Authors: Liu Jijie

#### Affiliations: Yantai Vocational College

信息技术与数学学科的有效整合，需要有一定的软件环境，这软件环境通常叫做教学工作平台。(The effectiveness of Integration of Information Technology with Mathematics Curricula depends on the specific environment created by the software. Such environment is called Mathematics Platform.) 教学工作平台的选择对信息技术是否能有效的辅助教与学意义重大。(The choice of information technology employed bears important consequence to the effectiveness of the CAI.) 张景中院士的Z+Z系列软件---超级画板，让我们老师和学生去观察、探索、发现规律，合理有效地实现信息技术与学科教学的整合，从而去达成学科教育目标。(The Z+Z Software System "Superpad" developed by Professor Zhang Ching-Chung enables teachers and students to observe, explore and to discover scientific rules. The dream of integration of Integration of Information Technology with Mathematics Curricula is thus realized. The professional educational goal is thus reached.)

### Abstract for 17385

Technology Enhanced 3D Visualization

#### Authors: Zhonghong Jiang

#### Affiliations: Texas State Univesrity, NCTM, SSMA

This presentation will describe some of the work the presenter has done with colleagues and previous students on the use of technology in 3D geometry teaching and learning. The Geometer's Sketchpad and Cabri 3D software were used to engage students in developing their visualization abilities as they solved challenging geometry problems. Examples will show how the dynamic geometry learning environment provided opportunities to have a positive impact on students" learning of 3D geometry.

### Abstract for 17338

Using CASIO fx-82ES, 991ES & ClassPad 330, Discovering more mathematics in China

#### Authors: Jiyan Wang, Chongyi Xin, Lei Lin

#### Affiliations: East China Normal University

In China more and more teachers and students have considered it is very important to make the integration between the modern information technology and mathematics teaching and learning. Almost every teachers and students fell the permeability of modern information technology, for example CASIO fx-82ES, 991ES or ClassPad 330, will make the mathematics learning become more significant and more interesting.

The scientific calculators, CASIO fx-82ES and 991ES, have been used to learn mathematics by more and more students in many middle schools. Especially in Shanghai CASIO fx-82ES has been allowed to do mathematics in the entrance examination of university from 2000.

The graphic calculators, CASIO ClassPad 330, have been known as a powerful tool for their teaching and learning by the teachers and students of some middle schools and universities in some places.

The paper will shows some examples about teaching, learning, exploring and problem-solving to illustrate that CASIO fx-82ES, 991ES & ClassPad 330 will help teachers and students to discover more mathematics:

Mathematics is a lively and interesting subject which has rich culture glamour, it is not dry as dust, not a mixed object simply by many form of symbols, digits and figures. These calculators can be used to solve the complex problems rapidly and succinctly in the operation of numbers and expressions.

Mathematics stems from practice and daily life, it is real not false. These calculators can be used to establish and solve mathematics models with real data by the inducing and reasoning methods. They are also used to do some experiments about random happening. Especially the table function of these calculators can be used to deal with many mathematics problems on inequalities, functions, equations, etc.

Mathematics is full of beauties which include the beauties of symmetry, form, connotations, etc. These calculators can be used to draw many geometry figures accordingly to show the mathematics properties of the problems.

In a word, CASIO fx-82ES, 991ES and ClassPad 330 can help every teachers and students to understand mathematics, to discover more and more mathematics!

运用CASIO fx-82ES, 991ES和ClassPad 330 - 发现更多的数学于中国

王继延 忻重义 林磊

华东师范大学 数学系

中国 上海

在中国，越来越多的教师和学生认识到现代信息技术与数学教学的整合是十分重要的。几乎所有的教师和学生感到现代信息技术，如CASIO fx-82ES, 991ES或ClassPad 330的渗透会使数学学习变得更为有意义，更为有趣。

科学型计算器CASIO fx-82ES和991ES已经为越来越多的中学生运用于数学学习，特别是在上海，CASIO fx-82ES自2000年以来一直被允许带入高等学校的入学数学学科的考试考场。

图形计算器CASIO ClassPad 330已经被一些地方的中学与大学的教师和学生所接受，作为他们教学与学习的强有力的工具。

本文将通过具体实例，包括教师的数学课堂教学，学生的常规学习与研究性学习，还包括某些数学问题解决，说明CASIO fx-82ES, 991ES或ClassPad 330将有助于教师与学生发现更多的数学：

数学是一门生动有趣的，且富有文化背景的学科，它不是枯燥乏味的，不是简单地由符号、数字与图形组成的混合体。

这些计算器可以运用来迅速而又简洁地解决许多复杂的数与代数式的运算。

数学来源于生活与实际，它是真实的，而非虚假的。借助于这些计算器，可以运用演绎与推理的方法建立并解决带有实际数据的数学模型。

它们也可以用来进行关于随机事件的模拟试验。特别是可以借助于这些计算器所带有的由函数生成数字表格的功能，处理许多有关不等式、函数与方程的数学问题。

数学充满着美，包括对称美、形式美与内涵美。这些计算器可以运用来描绘数学问题的几何图形，从而展示它们的数学性质。

总之，CASIO fx-82ES, 991ES或ClassPad 330有助于教师和学生更好地理解数学，发现越来越多的数学！

### Abstract for 17357

Stability of Boundary Layer Solution to the Boltzmann Equation with Diffuse Boundary Condition

#### Authors: Qianzhu Tian

#### Affiliations: Joint Advanced Research Center of University of Science and Technology of China and City University of Hong Kong

In this talk, the stability of boundary layer solutions to the Boltzmann equation with diffusive effect at the boundary for hard sphere model is considered. When Mach number less then -1, the exponential decay in time is proved for linearized operator firstly. At last, based on this property, nonlinear stability of the boundary layers is obtained by bootstrapping argument.

### Abstract for 17371

The Algebraic Properties of Geometric Objects: The Case of SSP

#### Authors: Chuan-Bo ZUO

#### Affiliations: Guangzhou University

We can easily draw a point on a circle in Dynamic Geometry System (DSG), and then drag the point on the circle dynamically. But it may need some techniques and several complementary objects to make the point on the exact top of the centre of circle or any other specific and accurate positions of the circle. There are plenty of problems as above to be solved when we develop materials for mathematics classes with DGS.

In this talk I will introduce the algebraic properties of geometric objects of Super Smart Platform (SSP), dynamic mathematics software developed in China. Through several examples we can find that with this characteristic of SSP:

1. The process for these tasks is more simple and easier.

2. Students can deeply understand geometry and algebra by denoted each other.

3. We are doing mathematics but not clicking when we make our ideas to be realized.

### Abstract for 17375

Gevrey regularity of solutions to the spatially homogeneous Boltzmann equation for non Maxwellian molecules without angular cutoff

#### Authors: Shi you Lin

#### Affiliations: Room 221, Zhishan Building, USTC-CityU Joint Advanced Research Center, Renai Road No. 166, Dushu Lake Higher Education Town, Suzhou, 215123

In this paper, we study the Gevrey class regularity in velocity variable for solutions of the spatially homogeneous Boltzmann equation without Grad's cut-off assumption. By using pseudo-differential calculus and Cauchy integral theorem, we prove the Gevrey smoothing property of the solutions of the linear and the nonlinear Cauchy problems in the inverse power law potential case (the part of $0 < \nu < 1$ in the following section) with non-Maxwellian molecules.

### Abstract for 17380

Spectral Domain Immitance Matrix Using Dyadic Green Function

#### Authors: Kassem Ghulam, Kassem Ghulam, Asif Zakariyya

#### Affiliations: International Centre of Wireless Collaborative Research Shanghai (WirelessCoRe) Shanghai China P.R, Int'l centre of wireless Collaborative Research Shanghai China P.R, Institute of Computation and Information Technology (ICIT) Gomal University D.I.Khan PAKISTAN

Dyadic Green's function (DGF) find an important role in natural sciences and technology more specifically, in communication engineering to model microstrip lines and numerous kinds of antennas at radio frequencies. The DGF relates electromagnetic fields with the source current in free space. A simple configuration is used to define the dyadic Green's function (DGF) in free space and consequently its components in spectral domain can be derived by simply suppressing one dimension. In turn, used frequently in determining the parameters such as resonant frequency, input impedance, radiation characteristics and scattering matrix etc. for simple as well as the stratified structures.

### Abstract for 17383

超级画板在研制中考动点问题中的应用与研究

#### Authors: Chuan-Bo ZUO, Gui-Guang Yan

#### Affiliations: Guangzhou University, Nanping Institute of Teaching, Fujian

近年来，动点问题已经成为中考数学命题的热点，并且在大多数情况下以压轴题的形式出现。动点问题考查的是当一个点或几个点按照指定的路径和方式运动时，

探索几个变化的对象之间的数学关系。动点问题以中考试题的形式出现，既能够对结果性目标达成状况的考查，同时又能够对一些过程性目标达成状况的考查。

具有很好地发挥考试在学业评价和人才选拔方面的功能，同时对初中数学教学也具有良性循环的导向，有利于课程改革的持续发展和有效推进。

因为动点问题探索的是运动变化过程中对象之间的数学关系，这就给考试命题者提出了更高的要求：如何保证所研制的动点问题正确无误？

如何检验已知的条件是否限制得过于狭窄？在中国大部分省市都施行自主命题、每年出现150套新的中考数学试卷的情况下，

如何才能不断研制出具有创造性的试题？这些问题是数学中考命题工作者所面临的越来越严峻的考验。基于作者近几年研究和研制中考数学动点问题的实践与经验，

从几个方面介绍了动态数学软件《超级画板》在研制和研究动点问题中的重要作用,阐释了动态数学软件在中考压轴题命题工作中的应用价值。

### Abstract for 17395

ICT-Based Mathematics Learning: Realistic Mathematics Education using Games Based on Papua's Environment and Culture

#### Authors: Benidiktus Tanujaya

#### Affiliations: The State University of Papua

Mathematics is considered as the most unpopular and hatred subject for students. In their opinion, mathematics is very difficult subject, worthless, and even boring. Accordingly, students experience difficulties in learning mathematics, and thus they find it difficult to understand its concept and application. Characteristic of mathematics which is believed to be abstractive, axiomatic, symbolic, and deductive makes mathematics as a confusing subject and thus inapplicable. This misleading judgment is a crucial problem in education since mathematics is a basic for science and technology. Hence there is an essential need to create various efforts philosophically to make mathematics as an interesting and enjoyable subject.

One of the mathematics learning is method with particular orientation to mathematize of everyday experience and to apply mathematics in daily life is Realistic Mathematics Education (RME). The characteristics of RME are the use of real life context, models, student's production and construction, interactive and intertwinement learning. Accordingly, this paper is aiming at describing the implementation of RME using Information and Communication Technology (ICT). In this regard, RME is performed by creating ICT-based games.

Game as a learning tool is created based on local environment and culture. In this case, Papua with its unique life, environment, and culture become a vast, everlasting potential source for creating ICT-based games as a mathematics learning tool. For RME at elementary level, introduction to basic numbers is learned by constructing games on biodiversity; algebra operations is learned by constructing games on ethnic warrior dance; the concept of mixed operations on decimal fraction is learned using games on land allocation based on "ulayat" community rights, etc. The application of RME in higher education level, games could be created based on many favorite aspects of life in Papua, like sports such as athletics and football as a mathematic model and learning tool for explaining geometry and trigonometry.

### Abstract for 17404

The Role of Geometry in Architecture: A case of study with 9th to 12th grade High school students in Japan

#### Authors: Maryam Behnoodi, Tadashi Takahashi

#### Affiliations: KOBE University-Japan

The great thing about Geometry is that it is all around us! Architecture has in the past contributed to a better understanding and measures the lands; it was the people's need to have building that first allowed them to investigate the theory of form and shape. Therefore Geometry is one of those parts of mathematics that can be visualized for students. It decided to do an architectural geometry project related to environment sciences with one of the international Japanese high school students from 9th to 12th grades. In this case of study, students get practical activities for designing a favorite high school for the future to be strong against the natural disasters by using curve shapes in the structure. And they should design the school that makes them near to each other in friendship and social life as we know the technology causes people become far and far from each other. The students learn how to use mathematics tools, software and multimedia to present their project for the other students and teachers. They learn to recognize various geometric concepts in the field of environmental sciences and architecture. The result of the questionnaire shows that an architectural project in group activities can promote the imagination ability of students and the power of creation by using multimedia and software. Also it will show the role and necessity of human development of the environment in future building designs from the student's view point.

### Abstract for 17407

Teaching Sketching Graphs of Trigonometric Functions Using Graphmatica: an Alternative Approach

#### Authors: SARIPAH AHMAD, PUAN SALMAH ABU BAKAR

#### Affiliations: SEK MEN SAINS MUZAFFAR SYAH MELAKA, SMK AYER KEROH MELAKA, MALAYSIA

The Graphmatica is a powerful equation plotter with numerical and calculus features. Graph Cartesian functions, relations, and inequalities, plus polar, parametric, and ordinary differential equations. Numerically solve and graphically display tangent lines and integrals. Find critical points, solutions to equations, and intersections between Cartesian. An action research on the use of Graphmatica has been carried out on a mixed ability group of 32 Form 5 students from two different schools and it has been found to be more effective in understanding the graph of trigonometric functions. In the post test shows 90% of the students got grad A. The Graphmatica can helps the students understand the topic better. Results of the post-test indicated improvement in students' achievement. Students also showed interest to learn the subsequent topic.. This software made it possible to help students to answer questions on the topics of sketching graph of trigonometric functions more accurately. It also helps to minimize teacher preparation time and to optimize the teaching and learning session in the classroom

### Abstract for 17408

Enhancing Form 5 Students Mastery Of Skills In Identifying The Angle Between Lines And Planes, As Well As The Angle Between Two Planes Using The `WON TECHNIQUE'

#### Authors: NORAINI KASSIM, AINI HAZIAH AMIRULLAH

#### Affiliations: SEKOLAH MENENGAH SAINS SEREMBAN, SEREMBAN, NEGERI SEMBILAN, MALAYSIA, SEK MEN SAINS SEREMBAN, SEREMBAN NEGERI SEMBILAN, MALAYSIA

This study is carried out to identify and rectify the problems faced by Form 5 students in identifying the angle between lines and planes, as well as the angle between two planes. 120 out of 215 Form 5 students were involved in this study. Strategy for the research was focused on identifying the angle between lines and planes, as well as the angle between two planes. Early observation was carried out during the revision of topics and chapters. It was found that 44.19% of students scored 70% and above, while the remaining 55.81% scored below 70%. A teaching and learning session which lasted for 80 minutes was carried out, where the students studied the skills stated above using the `WON Technique' ( Which One is the Nearest Technique) with power point presentation for enhancement of the learned skills. Results of the post-test indicated improvement in students' achievement. Students also showed interest to learn the subsequent topic and chapter. This method made it possible to help students to answer questions on the topics of lines and planes as well as angle between two planes more accurately. It also helps to minimize teacher preparation time and to optimize the teaching and learning session in the classroom. This particular topic is one of the most critical parts for students to master and for teachers to teach.

### Abstract for 17415

"O.I.C" METHOD IN COMPOSITE FUNCTION

#### Authors: SITI AZLINA HAIRUDIN, SALHALIDA ABD KUDUS

#### Affiliations: SEKOLAH MENENGAH SAINS SEREMBAN, SEREMBAN, NEGERI SEMBILAN, MALAYSIA., SEKOLAH MENENGAH SAINS SEREMBAN,SEREMBAN,NEGERI SEMBILAN,MALAYSIA.

Sekolah Menengah Sains Seremban, SASER performed well in the SPM examinations, specifically in Additional Mathematics. The school managed to achieve Grade Point Average of 1.68, 1.48 ,1.47 and 1.52 respectively for 2004, 2005, 2006 and 2007. This excellent achievement has spurred the Additional Mathematics Panel to continue its performance-enhancing programmes which have been carried out for the past few years to ensure that this momentum is continued in the future. One such programme is the Additional Mathematics Learning Skills Workshop. The workshop is tailored especially to discuss and cover short topics in the Additional Mathematics syllabus, such as Composite Functions, Probability as well as Permutation & Combination. This study is carried out to identify and rectify the problems faced by Form 5 students in determining one of the functions given a composite function and the other related function. In this workshop the students studied the skills stated above using the `OIC Method' ( Outside function, Inverse method and Inside function, Composite method ) with power point presentation. The method really helps the students whereby they are able to determine one of the functions given a composite function and the other related function without confusing the students. The workshop uses the mastery learning approach as well as station formation which encourage cooperative learning.

### Abstract for 17419

Maplets for Calculus as an Electronic Study Guide

#### Authors: Douglas Meade, Philip Yasskin

#### Affiliations: Department of Mathematics, University of South Carolina, Industrial Mathematics Institute, USC, Department of Mathematics, Texas A&M University

Not enough time to grade hand-written solutions or to give all students one-on-one attention? Frustrated by online homework systems with multiple choice questions or that check only the final answer? Not satisfied with the finite and static content of a paper study guide or Schaum's Outline? Looking to avoid the high cost (and uneven quality) of a private tutor? Electronic study guides address each of these concerns. Maplets for Calculus (M4C, see http://m4c.math.tamu.edu) is a collection of over 115 Maple applets designed to ask a question and to provide step-by-step guidance through the solution, requiring correct answers to intermediate steps. Hints are available and when errors are detected, instructive feedback is provided. There is an endless supply of problems with significant variation. The Maplets use symbolic, graphic (2D or 3D, some animated), numeric and verbal devices to investigate problems. Problems are algorithmically generated or entered by the student or instructor. The M4C can be used for self-study, in a computer lab, or as lecture demonstrations. All of this enables students to use M4C as "a tutor without the tutor." The M4C received the 2008 ICTCM Award for "excellence and innovation in using technology to enhance the teaching and learning of mathematics" at the 20th International Conference on Technology in Collegiate Mathematics and are supported in part by NSF DUE CCLI Grants 0737209 (Meade) and 0737248 (Yasskin). This talk will feature an extended demonstration of Maplets for Calculus designed to illustrate both advantages of an electronic study guide and weaknesses that remain to be addressed.

### Abstract for 17420

Analytic, Geometric, and Numeric Analysis of the Shrinking Circle and Sphere Problems

#### Authors: Douglas Meade, Wei-Chi Yang

#### Affiliations: Department of Mathematics, University of South Carolina, Industrial Mathematics Institute, USC; Radford University

The Shrinking Circle Problem is an example of a simple-to-state geometry problem that is visually appealing yet quite challenging to solve. A combination of geometry and analysis is used to completely solve the general problem in the plane, and its extension to three dimensions: the Shrinking Sphere Problem. We show why traditional numerical attempts to answer even the simplest problem are futile. The original problem was generalized based on visual evidence produced by dynamic geometry and computer algebra software. The complete proofs utilize insights obtained with dynamic geometry software and further analyzed with a computer algebra system.

### Abstract for 17421

The new era of Mathematica technology

#### Authors: Wolfram Research

#### Affiliations: Wolfram Research Inc.

This seminar will provide an overview of Mathematica functionality that makes it easy for educators to integrate the software into precollege, community college, and higher education classrooms. Whether you have used Mathematica for years or have no technical computing experience, you'll see many examples of Mathematica use for education that can be implemented immediately. Resources and presentation materials are made available to participants.

Mathematica is also the power behind Wolfram Alpha (http://www.wolframalpha.com/). The unique, next-generation computational knowledge engine was developed and deployed entirely in Mathematica.

## Abstracts for Hands-on Workshops

### Abstract for 16834

The Electronic Spreadsheet as a Problem Solving Tool

#### Authors: Thomas McMillan

#### Affiliations: University of Arkansas at Little Rock

The electronic spreadsheet is ubiquitous. Some version of this powerful tool is installed on nearly every personal computer. Originally conceived as a computerized model for the accountant's spreadsheet, the electronic spreadsheet's computational tools make it a powerful problem-solving tool applicable to problems that range beyond the field of accounting. Spreadsheet formulas are naturally applied to problems that can be analyzed recursively, or inductively. The purpose of this workshop is to sensitize teachers and students to a problem-solving tool that can be used yield numerical solutions, with minimal programming, to problems that are difficult or impossible to solve analytically. The spreadsheet's graphical capabilities will be used to present and analyze numerical solutions. The workshop will also illustrate, by example, how the spreadsheet can be used to reinforce the derivation of analytic techniques in discrete mathematics. No prior experience with electronic spreadsheets will be assumed.

### Abstract for 17082

An experimental approach to analytic geometry with the TI NSpire

#### Authors: Jean-Jacques Dahan

#### Affiliations: IREM of Toulouse

We will discover the TI NSpire environment through the power of linking its different applications. We will see how to create variables that can be used in the CAS, graphs and geometry, and spreadsheet applications (including the slider tool). We will present a new way of approaching the equations of lines, circles and conics and also a new way of exploring transformations with the spreadsheet and geometry applications.

### Abstract for 17090

Exploring and Discovering Mathematics with The Dynamic Mathematics Software: SSP

#### Authors: Chuan-Bo ZUO

#### Affiliations: Guangzhou University

Technology plays a more and more significant role in mathematics teaching and mathematics learning, because it makes mathematics accessible and fun. Furthermore, technology promotes students¡¯ enthusiasm in mathematics. Super Smart Platform (SSP) is dynamic Mathematics Software joint with geometry, algebra, calculus, probability, statistics, and so on, designed for learning, teaching and research in mathematics from elementary through college level. Participants of this workshop will be introduced series of materials developed with SSP for mathematics learning and research that is suitable for students ages 9-18 years old. Then the participants will learn how to develop these materials with SSP in hands-on activities.

### Abstract for 17091

利用Z+Z超级画板进行数学探索与发现

#### Authors: Chuan-Bo ZUO

#### Affiliations: Guangzhou University

技术在数学教学和数学学习活动中正发挥着越来越大的作用。一方面，技术让数学变得更容易理解、更生动有趣；另一方面，技术能够促进学生更深层地思考数学问题，

激发学生研究数学的热情。Z+Z超级画板是为数学教学、数学学习和数学研究而设计、开发的动态数学软件，它能够同时处理几何、代数、微积分、随机过程、

数据统计等方面的知识内容，在一定程度上能够满足小学、中学和大学在数学教育方面的需求。在本作坊中，

首先向参与者介绍一系列利用超级画板所开发的用于数学学习和数学研究的教学资源，这些资源所涉及到的数学知识由浅入深、从容易到复杂，

可以说能够适合9至18岁这个群体的学生使用。另外，通过本作坊的活动，参与者还能学习到利用超级画板自己动手开发这些教学资源。

目标受众：数学教师和数学教育工作者

建议参加人数：30人

建议作坊的时限：1.5至2个小时

### Abstract for 17107

An introduction of Magic Board: a Web-based virtual manipulatives for teaching elementary mathematics

#### Authors: Yuan Yuan

#### Affiliations: Chung Yuan Christian University

Magic Board is a web-based environment that includes three important components: tools center, problem posing center, and instructional materials center. The Magic Board tools center contain a virtual collection of physical manipulatives that elementary teachers can use to teach mathematics. Elementary school teachers can use this on-line virtual environment to pose and organize instructional materials for their students. Website members also have access to a variety of instructional materials that have been posed or organized by other Magic Board members.

### Abstract for 17117

Discovering and Teaching Mathematics with Excel

#### Authors: D. E. Arganbright, Lorna Denema, Thadreina Abady, Susan Arganbright

#### Affiliations: Divine Word University, Divine Word University, Papua New Guinea

This hands-on workshop provides participants with a wide range of creative ideas and techniques for using Microsoft Excel in teaching mathematics and in discovering new mathematical insights. Through a wide range of useful examples, the focus is on how Excel provides us with a natural environment to discover, create, and investigate new mathematical ideas, rather than simply to implement already known mathematics. The workshop features a component on designing interactive graphics and producing animation effects using scroll bars and simple macros. Novel mathematical uses for Excel's data table, pivot table, and 3-dimensional graphics are also provided. The skills encountered in the workshop can equip our students to use the principal mathematical tool of the workplace effectively in their own study of mathematics. A highly-experienced 4-person team provides immediate assistance for participants in their workshop experience. Participants who are unfamiliar with Chinese language programs can use their own notebook computers. Each person receives a collection of Excel files to use in the workshop and in their later classroom teaching. Each file provides a previously designed template, enabling the workshop experience to remain focused on the most important aspects of the creation process. Each file also contains additional annotated sheets that discuss all aspects of the creation process, and a finished version that illustrates formatting and other enhancement techniques. Examples are drawn from many areas of mathematics and applied fields, including calculus, linear algebra, statistics, geometry, discrete mathematics, number theory, and mathematical modeling in the natural and social sciences.

### Abstract for 17125

激發式動態教學 ( Trigger-based animated Instruction)

#### Authors: Mingjang Chen

#### Affiliations: National Chiao Tung University

激發式動態呈現 (Trigger-based Animation)就是以一個物件當激發器來控制一連串的動態呈現，一個物件可以被一個以上的激發器所控制，這是一個基於認知心理學、

認知負荷理論及媒體教學理論而發展出的數位內容設計方法；其操作簡易、有彈性且功能強大，可以協助教師依據教學現場的狀況，有彈性地調整教學的順序與步調，

引導學習者的注意力，引導認知，並降低認知負荷，進而達到教學成效。

AMA (Activate Mind Attention)是一個外掛在PowerPoint上的一個增益集(addins)，激發式動態呈現是其核心功能之一，此一活動理論與實務兼顧，前三十分鐘，

我們將運用激發式動態呈現來介紹激發式動態教學的理論，示範如何運用激發動態呈現，讓您親身體會注意力引導的作用，接著將介紹激發式動態呈現的基本模式，

現場的聽眾可以一邊聽講一邊操作，只要您具備使用PowerPoint的能力，就可參加此一活動。我們將運用注意力導引的手法來進行整個活動，過程緊湊、內容豐富有趣。

激發式動態呈現適用於各種領域之教學及一般的展演，本次活動針對中學數學教學而設計，尤其歡迎對資訊科技融入教學有興趣的數學教師參加！

### Abstract for 17131

Using Graphing Technology to engage students in Mathematical Investigations for understanding

#### Authors: Caroline Lee

#### Affiliations: Texas Instruments Singapore (Pte) Ltd

Packaged in the right way, graphing technology has the potential to provide the opportunities for learning through Mathematical investigations.

Using practical examples of activities in graphing, calculus, and statistics, participants will experience how the TI-Nspire graphing handheld can be used to engage students and promote understanding in a Mathematics classroom.

### Abstract for 17213

Introducing Calculus using an HP Graphing Calculator

#### Authors: Colin Croft

#### Affiliations: St Hilda's Anglican School for Girls, Hewlett Packard

In this session we will look at how student understanding of the early concepts of calculus can be enhanced through the use of an HP graphical calculator. This will include tips and tricks for the effective use of the Function applet, as well as some of the pitfalls commonly encountered by teachers in this area. We will also cover some of the applets available on the internet which can aid in the process of introducing key concepts in differentiation, curve sketching and areas under curves.

### Abstract for 17214

Teaching statistics and probability using an HP Graphing Calculator

#### Authors: Craig Davis

#### Affiliations: St Hilda's Anglican School for Girls

In this session we will develop techniques of modeling bivariate data using different regression functions available in the Statistics applet. This will include tips and tricks for manipulation of columns, extraction of coefficients of lines of regression and production of columns and graphs of residuals. The session will also develop strategies for producing random numbers to simulate chance processes. We will also use the Solve applet to solve problems involving probability distributions.

### Abstract for 17216

Using CAS in your classroom

#### Authors: Christopher Longhurst

#### Affiliations: Hewlett Packard

CAS stands for Computer Algebra Systems. Some countries are currently using CAS in their classrooms, mainly for senior grades eg. Australia (Victoria and Western Australia), Germany, Scotland, Austria to name a few and in the junior high school eg. New Zealand.

This workshop will assume that the teacher has knowledge of the use of the graphic calculator. However, the workshop will also cater for teachers who are interested in pursing knowledge about what CAS is and how it can be used in the mathematics classroom. This hands-on workshop will guide the participants step by step the calculator steps in using CAS. Each participant will be given a document with steps on how to perform basic CAS functions in relation to basic algebra. The presenter will constantly refer to how this powerful and useful technology can be used in the classroom for all secondary levels. Some investigations and worksheets specifically designed for CAS will be given and used.

### Abstract for 17217

Using Graphic calculators in the classroom

#### Authors: Christopher Longhurst

#### Affiliations: Hewlett Packard

The workshop will begin with a brief explanation of the function applet (graph) and the solve applet. The participants will work through the process of downloading applets from the internet and use the power of transferring downloaded and saved applets to other members of the class (this is also a practical application of a geometric progression). Finally, a practical investigation using the power of the calculator will be worked though using the finance applet.

### Abstract for 17227

Connecting Geometry, Algebra and Calculus with the Geometer's Sketchpad

#### Authors: Krongthong Khairiree

The workshop is designed for participants to experience in using the Geometer's Sketchpad (GSP) with constructivist approach in mathematics lessons. The workshop will be a hands-on activity and the workshop will be conducted in such a way as to simulate a mathematics class.

Prior experience with the Geometer's Sketchpad is not necessary. The knowledge on how to use GSP and the important features of GSP will be covered in this workshop. The participants will learn both basic and advanced features of GSP.

Primary emphasis will be on learning how to use GSP effectively in integrating geometry algebra and calculus. The participants will learn how to build and animate parameters with functions. The workshop also provides participants learn how to use GSP for classroom demonstrations and explanations.

At the end of the workshop conducted the participants will:

• gain knowledge on using GSP to help their students better understand mathematical concepts;

• learn how to integrate GSP in mathematics class;

• know how to use and when to use GSP in mathematics classes; and

• have an opportunity to develop competence by using GSP for teaching mathematics.

### Abstract for 17241

Exploring Advanced Calculus with ClassPad Manager

#### Authors: Wei Ching Quek

#### Affiliations: Singapore Polytechnic

One way to motivate and inspire students to study mathematics is to provide them with opportunities to interpret the mathematics and how it is related to their discipline of studies. The ClassPad Manager is a popular learning tool. We will explore how the tool will assist the learning of advanced undergraduate mathematics.

This workshop is consists of three activities:

1. Getting Started

Getting started with the ClassPad Manager, a popular handheld CAS calculator. The workshop intends to share with teachers essential features of the ClassPad Manager and explore the potentials to solve advanced calculus problems. No previous experience with the ClassPad is assumed.

2. Problems Solving

Examine some interesting problems in engineering model and participants will explore the problem from different perspectives, numerically, graphically symbolically and provide further insights to the problem. 3. Lesson Plan

Discuss and demonstrate possible development of a lesson plan which integrate with the CAS technology and e-activities of the ClassPad.

Participants will be able to experience the interactive and dynamic interface of ClassPad Manager. The clever interface will help teachers and students work with ease on numeric, algebraic and geometric representations.

### Abstract for 17246

Introduction to Probability and Statistical functions of ClassPad Manager (In Mandarin)

#### Authors: Wei Ching Quek

#### Affiliations: Singapore Polytechnic

This workshop will introduce the basic features of ClassPad Manager and then focus on probability and statistics computation.

Participants will learn how to organize, present, summaries and interpret data. Teachers will find that ClassPad Manager will make the lesson more interesting and can have more interactions with students.

No technology or experience with ClassPad Manager is assumed.

### Abstract for 17250

GeoGebra Workshop

#### Authors: Morten Misfeldt, Mette Andresen

#### Affiliations: National Knowledge Centre for Mathematics Education, UCC, Danish School of Education, Aarhus University

GeoGebra is introduced by working hands-on with materials from a teaching experiment on symmetry. The workshop aims to demonstrate didactic potentials and special features of this open - source software. Besides, the workshop is linked to the Authors presentation of a model for design of professional development sessions for teachers in primary and upper secondary school and teacher education.

### Abstract for 17311

Autograph for the younger students

#### Authors: Douglas Butler

#### Affiliations: iCT Training Centre (Oundle School), UK

This hands-on workshop will be conducted in English and in traditional Chinese (with translation). Autograph offers teachers and students exciting opportunities to explore coordinate geometry in 2D and 3D, and also probability and statistics. There will be many opportunities to bring the subject to life through images and data off the web.

### Abstract for 17312

Autograph for advanced students

#### Authors: Douglas Butler

#### Affiliations: iCT Training Centre (Oundle School), UK

This hands-on workshop will be conducted in English and in traditional Chinese (with translation). Autograph offers teachers and students exciting opportunities to explore coordinate geometry in 2D and 3D, and also probability and statistics. There will be many opportunities to bring new life to topics in calculus, vectors, differential equations and hypothesis testing.

### Abstract for 17316

How to Turn a Cube into a Rhombic Dodecahedron?

#### Authors: Jen-Chung Chuan

#### Affiliations: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300

1) Start with a cube:

2) Construct an inverted pyramid by taking the center as the vertex and the square on top as the base:

3) Construct a diagonal of the cube:

4) Construct a 270-degree arc:

5) Rotate the pyramid along the axis of the arc:

6) Rotate the pyramid in 5) with respect to the diagonal by 120 degrees in both directions:

7) Construct the pyramids point symmetric to those in 6):

8) The cube is now turned into a Rhombic Dodecahedron by grabbing the large vertex:

Software environment: Cabri 3D installed in Windows XP. Network connection will enhance the demo but is not required.

### Abstract for 17317

3D Poncelet Porism and Steiner Porism

#### Authors: Jen-Chung Chuan

#### Affiliations: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300

The plane version of the various forms of the Poncelet porism and Steiner porism has been experimented with Cabri Java Applets: http://steiner.math.nthu.edu.tw/disk3/cabrijava/index.htm

Here we wish to extend the analogous phenomena in 3D. In this workshop we are to explore the mathematical properties with the Cabri 3D files prepared before-hand:

1) Inversion of Circles Inscribed in Faces of Pentagonal Cupola:

2) Poncelet Porism for Escribed-Inscribed Quadrilaterals:

3) Poncelet Porism for Quadrilaterals: 4) Sphere Inversion:

5) Steiner Porism on Sphere:

6) Inversion of Circles Inscribed in Faces of Cuboctahedron:

7) Inversion of Circles Inscribed in Faces of Cuboctahedron II:

8) Inversion of Circles Inscribed in Faces of Square Cupola:

9) Inversion of Circles Inscribed on Faces of Dodecahedron:

Software environment: Cabri 3D installed in Windows XP. Network connection will enhance the demo but is not required.

### Abstract for 17318

Learning about equations with a graphics calculator

#### Authors: Barry Kissane

#### Affiliations: Murdoch University

This hand-on workshop will illustrate how graphics calculators can be used to support learning about equations in mathematics classrooms. The use of graphics calculators enables students to develop key concepts in an interactive and exploratory way not possible without such technology. The workshop will use the Casio fx-9860GII to examine the opportunities provided by the calculator to represent and solve equations of various kinds, including polynomial equations and simultaneous linear equations We will build upon this experience to consider the place of exact and numerical solutions of equations in the school curriculum. No previous experience with these calculators will be assumed; beginners are very welcome.

### Abstract for 17319

Learning about functions with a graphics calculator

#### Authors: Barry Kissane, Marian Kemp

#### Affiliations: Murdoch University, Student Learning Centre, Murdoch University Australia

This hand-on workshop will illustrate how graphics calculators can be used to support learning about functions in mathematics classrooms. The use of graphics calculators enables students to develop key concepts in an interactive and exploratory way not possible without such technology. The workshop will use the Casio fx-9860GII to examine the opportunities provided by the calculator to represent functions graphically, symbolically and numerically. We will also consider families of functions, transformations and functions and links between functions and equations. No previous experience with these calculators will be assumed; beginners are very welcome

### Abstract for 17321

Learning data analysis with a graphics calculator

#### Authors: Marian Kemp

#### Affiliations: Murdoch University

The availability of graphics calculators in mathematics classrooms allows students to have different experiences than would otherwise be the case without this kind of technology. This hands-on workshop will focus on data analysis to illustrate and investigate some of these different experiences. Participants will be introduced to informal and intuitive approaches to data analysis. The workshops will use Casio fx-9860GII graphics calculators to illustrate ways to represent and interpret data including both numerical and graphical approaches. No previous experience with these calculators will be assumed; beginners are very welcome.

### Abstract for 17322

Learning calculus with a graphics calculator

#### Authors: Marian Kemp, Barry Kissane

#### Affiliations: Murdoch University

This hand-on workshop will illustrate how graphics calculators can be used for learning introductory calculus in mathematics classrooms. The use of graphics calculators enables students to develop key concepts in an interactive and exploratory way not possible without such technology. The workshop will use the Casio fx-9860GII to examine calculus concepts including derivatives, continuity and limits and discuss how teachers can use informal and intuitive methods to build students' understanding. No previous experience with these calculators will be assumed; beginners are very welcome.

### Abstract for 17323

Enhancing Mathematics Teaching Through Handheld Graphing Technology

#### Authors: Wee Leng Ng

#### Affiliations: National Institute of Education, Nanyang Technological University

Handheld graphing technology, if used appropriately in the mathematics classroom, has the potential to enhance teaching and learning of mathematics by empowering students to learn across different visual representations of a mathematical problem. With the aid of such technology, teachers have the means to help students develop a deeper understanding of mathematical concepts, broaden their critical thinking skills and discover meaningful real-world connections.

In this workshop, participants will explore ways to enhance teaching of mathematics through the TI-Nspire Technology.

### Abstract for 17362

Interactive visualization tools for div, curl, Stokes theorem and differential equations

#### Authors: Matthias Kawski

#### Affiliations: Arizona State University

The JAVA Vector Field Analyzer II is a powerful JAVA applet that runs on any JAVA enabled WWW-browser, and it is freely available on-line at http://math.asu.edu/~kawski/vfa2/vfa2sample.html. It was created to help students understand the dynamic meaning of curl and divergence, and to develop a deep understanding of the fundamental connections between differential equations, linear algebra, vector calculus and complex analysis. The start-up costs are minimal, experiments and results are immediate, typically leading to profound theoretical discussions: Students ask for, demand, proof!

This hands-on workshop introduces the participants to the basic functionalities of this applet. Participants will become students and typically will develop completely new understandings of vector calculus and its connections with other subdisciplines. They will also experience a dramatic inquiry-based approach to learning and teaching.

Selected topics that may be addressed include: See the curl of a vector field by zooming. See the dynamic meaning of curl and of divergence free. What is the derivative of a linear field? How do you get pi from a triangle? The variational equation and the solution of every linear DE is linear. Different views for covariant and contravariant fields. Quick images for Poincare-Bendixson theory. Lyapunov stability. Zoom to see the Cauchy Riemann equations. This workshop is targeted at faculty who teach vector calculus and differential equations, at advanced students in these classes, and at anyone interested in seeing technology-enabled inquiry-based learning in action.

This work was partially supported by the National Science Foundation through the grants DUE 97-52453 and DMS 09-08204.

### Abstract for 17363

"Autograph" 精 画 - 适用于中级数学教师的ATCM专题讨论会

#### Authors: Douglas Butler

#### Affiliations: iCT Training Centre (Oundle School), UK

"Autograph" 精 画 - 活用 数学 学习 软件

由Douglas Butler （英国）主持

精画（自动绘图仪）的主要作者

这次实地的专题讨论会会使用新的简体中文版本的精画（自动绘图仪）。精画给老师们和学生们提供了探索2D和3D版本的解析几何，概率论和统计学的机会

这是一个机会，在网络之外把这个课题通过图像和数据，还有微积分，向量，微分方程和假设检验的主题的效果图带进生活。

www.autograph-maths.cn

debutler@argonet.co.uk

### Abstract for 17396

Getting Started with Maple and Maplets for Calculus

#### Authors: Douglas Meade, Philip Yasskin

#### Affiliations: Department of Mathematics, University of South Carolina, Industrial Mathematics Institute, USC, Department of Mathematics Texas A&M University

The Maple computer algebra system is both powerful and flexible. In this hands-on workshop, designed for both novice and experienced users, you will learn some of the different ways in which Maple's graphic, numeric, and analytic features can be used to solve a variety of problems. Examples will focus on classroom use, and how Maple can be used to improve student's conceptual understanding of the underlying mathematics. This part of the workshop will include an introduction to the Maplets for Calculus, a collection of more than 100 Maple-based applets covering topics in pre-calculus and calculus.

## Abstracts for Poster Sessions

### Abstract for 17109

Software Module Showing Dynamic Link of 3D Graphic Objects and Vector Equations

#### Authors: Norihide Kishi, Keisuke Ishiguro, Takayoshi Yoshioka, Hitoshi Nishizawa

#### Affiliations: Toyota National College of Technology, Toyota National college of Technology

Students in our college do not understand liner algebra very well. It is especially difficult for them to visualize three-dimensional objects expressed by vector equations. One of the causes is that they only learn vector equations separately from the graphic objects. Discrete knowledge generally evaporates in a short term.

To change this situation, we have been developing a learning system that helps to recognize the relationship between vector equations and the graphic objects. In this poster presentation, we demonstrate a module of the system based on the new feature of computer algebra system Mathematica version 7, which connects the change of graphic objects, for example, lines and planes in 3D-space with the simultaneous change of vector equations. When a learner change a parameter in an equation with the slider on the screen, the vector equation change its form, and the related graphic object also changes the position or direction in the 3D-space on the screen. In addition, there are mode-selecting buttons on the screen which show/hide additional attributes of the target graphic objects for the learner to concentrate certain attribute of the objects.

The activity helped reluctant learners to recognize the link between graphic objects and vector equations, and to start examining the link by themselves.

### Abstract for 17110

Constructing a Game for Introducing 3D Vector Operations

#### Authors: Yoshihiro Yamada, Norihide Kishi, Takayoshi Yoshioka, Hitoshi Nishizawa

#### Affiliations: Toyota National College of Technology

A possible cause of our engineering students' lower performance in linear algebra was that they could not find the link between abstract mathematical operations and the applications in real world. In the past, linear algebra was taught only as symbolic procedures: useful but too abstracts ones. To improve the situation, we had installed some graphical explanation of vector operations in 2D and 3Dspace with the help of graphic software. But they did not motivate our students to learn linear algebra enough. So we needed more attractive introduction.

In this poster presentation, we would like to demonstrate a game as an introduction of vector operations. By attending the game, each student has his/her own player as avatar whose characteristics can be manipulated by 3D vectors. The players compete each other in various kinds of battlefields in the game. When two players fight each other in a battlefield, the strength of each player is calculated by dot product of characteristic vector of each player and one of the battlefield, and the winner of the game is decided by comparing the strength of two competitors.

We have installed the game as a network system with the help of CAS; Mathematica and DBMS; Postgress. Mathematica or Mathematica player are the front end program with customized interface and functions, and Postgress stores all the data necessary for the game.

### Abstract for 17111

Cabri 3D Module to Visualize the Construction of 3D Vector Equation

#### Authors: Takahiro Asano, Takayoshi Yoshioka, Hitoshi Nishizawa

#### Affiliations: Toyota National College of Technology

Understanding three-dimensional vector equations to express lines or plane is not easy for many college students, when they learn without connecting them with the graphic objects. That is why we have started to build a Web-system that shows the link between graphic objects and vector equations. An embedded module in a Web-page has 3D graphic objects that could be directly manipulated with mice. The modules are constructed with Cabri 3D and works with the help of Plug-in software on Web-pages.

Each module shows several vectors to characterize the position and direction of the graphic object in virtual 3D space, and the vectors can be manipulated by dragging one of the end-points with a mouse.

In this poster presentation, we would like to visualize the construction of vector equation of a plane in 3D space, which most of our students have difficulty to understand the concept. With the experience of manipulating the vector on the screen, the student becomes gradually ready to accept the idea of using vectors to describe the position and direction of target graphic object in 3D space.

### Abstract for 17178

Simplifying Calculus

#### Authors: Michael Livshits

#### Affiliations: Cambridge, MA, USA

I will explain how calculus can be developed directly and naturally, starting with polynomials and using uniform estimates instead of limits and continuity. This approach is mathematically simpler and more relevant to practical computations. It also may be easier to understand than the traditional approach based on classical analysis. Most of the material is described in my preprint at http://arxiv.org/abs/0905.3611 and references there.

### Abstract for 17249

Mathematics in Quantum Physics with Mathematic

#### Authors: Woei-Yun Ho

#### Affiliations: Department of Applied Physics National Pingtung University of Education

The abstract Hilbert space and skills required for solving the Schrodinger equation in quantum physics are the two major obstacles for many physics students. For example, in the standard solutions of the hydrogen atom and the simple harmonic oscillation, the student is often lost in the mathematics. Besides, traditional physics education tends to limit its choice of problems which can be solved in symbolic form, but modern research heavily depends upon computational tools because relatively few interesting problems can be solved analytically. With a traditional approach, few quantum physics problems can be reduced to carrying out both numerical and symbolical calculations.

Unsatisfied with traditional `paper and pencil' approach, I conducted a study to aim at developing my student's competences in the use of Mathematica. This study was targeted at undergraduate physics students and was focused on how student approach problems with Mathematica. I endeavored to provide a wide variety of problems within Mathematica to my students. These course materials were designed to help prepare them for both numerical and symbolical calculations. Although such materials are essential to the development of understanding and physical insight, I found that few students could produce realistic Mathematica codes for solving problems without assistance. The problem appeared to be that solving these problems often required conceptual understanding of quantum physics and procedural programming approach would bring in new obsession to students. Adaptations from classic Mathematica approach for students to get ideas on how to use Mathematica to solve the problems was explored in this study.

### Abstract for 17382

Computer technology in Korean Secondary Mathematics Textbooks Based on the 2007 revised curriculum

#### Authors: Lew Hee-chan, Tae Seon Cheon

#### Affiliations: Korea National University of Education

Recently many countries have encouraged mathematics teaching and learning to make use of computer technology in all grade levels. Korea has not been exceptional. The 7th mathematics curriculum revised in 2000 already had encouraged using computer technology in mathematics classrooms. Furthermore even in the 2007-revised curriculum, the proper use of various computer technologies is emphasized to make effect of mathematics teaching and learning and to evaluate mathematics achievement.

The purpose of this study is to analyze the technology use appeared in 8 kinds of secondary mathematics textbooks developed in 2008 for the 2007 revised curriculum, which were authorized by Ministry of Education, Science and Technology. In algebra, calculator is mainly used to calculate complex calculation. In analysis, Equation Grapher is mainly used to draw a graph y=ax, y= a/x. In statistics, spreadsheet is mainly used to organize data and to make frequency distribution table, histograms and to calculate the average. In geometry, GSP is used to draw figures and to measure interior angle and outer angle of polygon and Wingeom and poly are used to observe solid figures.

### Abstract for 17384

ON PLC-GEOMETRY

#### Authors: Hirotaka Ebisui

#### Affiliations: Oval Research Center

In mathematics field, there is geometry. It is developing day by day. Such a situation, we found a new field on Geometry. This field is named as PLC-Geometry that is defined by constructing with points, lines, and circles. Till now, we found Rose theorem, Sun flower theorem and Warp theorem, etc. Number of the theorems has exceeded 1600. Here, we show about 20s compositions (diagrams) whish is constructed by Rapid CAD soft. These compositions may be applied to consider in order finding new principals of Geometry. And, we can enjoy the order of construction which are the steps of a theorem from the condition to the conclusion. Let's study PLC-Geometry.