Technological Creativity and Innovation in Mathematics Applications

# Abstracts for Invited and Plenary Papers

### ABSTRACT FOR 21435

#### Is Technology Taking Mathematics Education in the Right Direction?

AUTHOR: Douglas BUTLER
AFFILIATION: iCT Training Centre (Oundle, UK), Autograph-Maths

There are numerous examples, in many fields, of spectacular advances in technology getting in the way of progress. In mathematics education, this can present itself through a plethora of new titles that can all too easily offer students clever short-cuts before they have a sound enough understanding. Most digital resources have their origins in the web, but the internet is a public and totally unregulated network, allowing open source software, schemes of work and entire courses to appear without any moderation. This presentation will attempt to tread a cautious path through all this, suggesting standards by which students and teachers should evaluate digital resources before they use them. It is particularly important to choose dynamic software that has a sympathetic user-interface, allowing the mathematics to shine through.

### ABSTRACT FOR 21454

#### Using the Riemann Sums to Evaluate Areas and Volumes within DGS and CAS environments (TI-Nspire and Cabri): Enriching Dialectic Between Math Knowledge and Technical Skills

AUTHOR: Jean-Jacques DAHAN
AFFILIATION: IRES of Toulouse

Creating math resources using technology is still an important issue for both teachers and experts in order to provide new tools for teaching that can enrich the approach of important concepts and enhance a motivating practice of math. But there is a huge difference between the work of a teacher and the one of an expert in such a process of creation. As we know that the purpose of the use of technology "is not only to do faster what was possible to do by hand but to do it differently" (Colette Laborde), we will show the work of an expert to have good command of the tool (here TI-Nspire and Cabri 2 Plus) before creating microworlds allowing students to discover a concept (we have chosen the concept of integral). The different stages of this creative work will illustrate a special dialectic between technological skills and math knowledge. At last, we will see that this work completed with collaboration with a math teacher can lead to resources well designed for a powerful use in the classroom.

### ABSTRACT FOR 21474

#### Counterexamples in Mathematics Education: Why, Where, and How? - Software aspect

AFFILIATION: Holon Institute of Technology

A counterexample is an example that refutes the fidelity of some statement. For a mathematician, constructing counterexample is a common way to disproof mathematical conjectures. Counterexamples also help her to establish the constraints imposed on theorems.

This report shows that in mathematics education counterexamples can and should be applied at the earliest stages - in the study of concepts, long before the first acquaintance with the theorems and proofs. Herewith, the use of software becomes an organic element of the learning process.

### ABSTRACT FOR 21500

#### Hawgent 皓駿數學實驗室，從設計、研發、建設到應用

AUTHOR: Chuan-Bo ZUO
AFFILIATION: Hawgent Technology Centre in Mathematics

### ABSTRACT FOR 21516

#### New Model for Calculating Sphere Volume

AUTHOR: Chang WENWU
AFFILIATION: PuTuo Modern Educational Technology Center of Shanghai

Though mankind's knowledge of sphere volume begun from great Archimedes 2000 years ago, the proof of corresponding formula has been indirect. Even if Chinese mathematician Zu Geng (lived in the fifth century) and Italian mathematician Cavalieri (lived in 14-15th century) arrived at a useful principle (so-called Cavalieri principle in western world and Zu Geng principle in China) independently, their models were not direct either. This paper introduces a new model of tetrahedron, whose volume equals to a sphere directly. This may benefit high school students to understand sphere volume formula more easily and without the preparation of calculus.

### ABSTRACT FOR 21517

#### Mathematical Analysis of Information Systems through Technology

AUTHOR: En-Bing LIN, Yu-Ru SYAU
AFFILIATION: Central Michigan University, IEEE, AMS, National Formosa University

We live in a world submerged with more information than ever before. We express information or data mathematically and it is growing faster than ever. If the data is imperfect, out of context or otherwise contaminated, it can lead to decisions that could undermine the competitiveness of an enterprise or damage the personal lives of individuals. Therefore, knowledge representation plays an important role in dealing with many aspects of problem solving. In order to obtain complete information systems, we use a mathematical tool, namely, Rough Set Theory (RST), which was introduced by Pawlak in 1982 as a way to deal with data analysis based on approximation methods in information systems. RST is a novel approach to cope with imperfect data analysis as well. In essence, the theory extends the classical crisp set to rough set by defining lower and upper approximations for any subset of a nonempty finite universe. The theory presupposes that with every object of the universe some information is associate with certain relationship. It has many applications in a number of different areas, such as engineering, environment, banking, medicine, bioinformatics, pattern recognition, data mining, machine learning and others. RST is intrinsically a study of equivalence relations on the universe. We intend to use some advanced computing technologies to implement the computations and find several properties of the characteristics of objects. We will show some advanced computing method that can solve our problems effectively. We also extend the results from one universe to two universes.

### ABSTRACT FOR 21523

#### Appreciating functional programming: A beginner's tutorial to HASKELL illustrated with applications in numerical methods

AUTHOR: Weng Kin HO
AFFILIATION: National Institute of Education, NTU, Singapore

This paper introduces functional programming to the ATCM community with the aim of popularizing this programming paradigm through a deeper appreciation for function as a mathematical concept and, at the same time, for its practical benefits. The functional language Haskell is chosen amongst several choices because of its lazy evaluation strategy and high-performance compiler WinGHCi. We demonstrate the elegance and versatility of Haskell by coding Haskell programs to implement well-known numerical methods.

### ABSTRACT FOR 21525

#### Teaching Mathematical Modelling in a Technology-Enabled Environment

AUTHOR: Keng Cheng ANG
AFFILIATION: Nanyang Technological University, 1 Nanyang Walk, Singapore 637616

This paper examines the use of computer simulation as a modelling approach in the teaching of mathematical modelling. In a technology-enabled environment, teachers can design modelling activities based on simulation models that are both suitable and accessible to their learners. We describe how an electronic spreadsheet equipped with coding and programming capabilities is used to investigate problems through suitable simulation models. Three examples of modelling activities will be discussed. Details of the problem situation, problem solution and construction of simulation models will be described in each case. Through these examples of tried and tested modelling activities, we demonstrate the underlying principles in designing simulation models appropriate for use in teaching mathematical modelling. In concluding the paper, we will briefly discuss the support that teachers need in designing, developing and delivering mathematical modelling lessons involving computer simulations.

### ABSTRACT FOR 21528

#### The Turing Bombe and its Role in Breaking Enigma

AUTHOR: Neil SIGMON, Rick KLIMA
AFFILIATION: Radford University, Appalachian State University

The work of the codebreakers at Bletchley Park in breaking the German Enigma cipher during World War II was one of the most extraordinary events in human history. Led by Alan Turing, the codebreakers employed an electromechanical device known as the bombe to regularly cryptanalyze and read German encrypted communications throughout much of the war. This work likely helped the Allies to win the war much sooner than expected and saved countless lives. Due to the extraordinary number of combinations that the Enigma could be set to, the Germans believed that the Enigma was impenetrable. However, Turing and the codebreakers were able to use the bombe to exploit the part of the Enigma that the Germans thought gave the device its most security. This paper will describe the logic behind how the bombe exploited the Enigma cipher and the process involved.

### ABSTRACT FOR 21531

#### Exploring Statistics with TinkerPlots in Flipped Classroom

AUTHOR: Krongthong KHAIRIREE
AFFILIATION: International College Suan Sunandha Rajabhat University

The purpose of this study was to explore the students' perceptions of teaching approaches using a flipped classroom incorporated with TinkerPlots dynamic software program. In the 2017, action research was conducted in business statistics class of International College, Suan Sunandha Rajabhat University Thailand. The total of 26 students participated in this study. In the flipped classroom, the researcher created online lessons of her lectures and the students viewed them prior to attending class. Students worked on problem-solving activities in their classrooms. The research findings indicated that the flipped model of instruction was new teaching strategy that moved the lecture on business statistics outside classroom via technology and brought assignments/exercises of business statistics concepts inside the classroom via learning activities. The research findings shown that if TinkerPlots dynamic software program is appropriately employed, the program can be used as an effective tool in enhancing active learning. In addition, the students' engagement in the flipped classroom were higher than the traditional classroom. Based on the students' interviews they revealed that using flipped classroom method incorporated with TinkerPlots program they were able to make learning statistics fun and challenging.

### ABSTRACT FOR 21533

#### Design Innovative Teaching Aids and Creative Activities to Help Children Understand Mathematics Concepts and Excited about Learning Mathematics

AUTHOR: Poh Yew TEOH
AFFILIATION: National Association for Gifted Children, Malaysia (NAGCM), Malaysian Association of Professional Speakers (MAPS), Malaysian Invention and Design Society (MINDS)

Helping children understand Mathematics concepts and getting them excited about learning mathematics is the teachers' constant challenge. The answer lies in incorporating creative techniques into our classroom practice. Spicing up our mathematics lessons with some magic tricks, puzzles and innovative teaching aids could easily help increase attention, understanding and retention significantly. Appropriate recreational activities/problems stimulate intellectual curiosity, develop problem-solving skills, promote discovery and thinking skills, as well as unleash creativity. In solving these problems, students actively participate in discovering or deciphering the hidden principles underlying each problem and in the process construct their own knowledge of mathematics. This presentation also stresses on the need for excitement to encourage mathematical thinking. Mathematical magic can serve as an effective means of motivation at almost all levels of instruction. The major purpose of its use in the classroom is to enable students to discover that mathematics is easy and interesting, and thus stimulate interest in their further study of mathematics.

Presentation outline:

• The importance of excitement need (Kano Model)
• How to be a more creative teacher
• Using Magic and Puzzle as Set Induction
• How to develop new teaching aids
• Doing revision the magical way
Target audience: Preschool and primary school mathematics educators

### ABSTRACT FOR 21538

#### The Third Generation of Calculus (Calculus without Limit Theory)

AUTHOR: Jingzhong ZHANG and Zengxiang TONG
AFFILIATION:Otterbein University, School of Computer Science and Educational Software, Guangzhou University

This paper presents calculus with an innovative approach. Without the help of limit and infinitesimal, it directly defines derivative and definite integral on an ordered field, proves the fundamental theorem of calculus with no auxiliary conditions. It easily reveals the common properties of derivatives and obtains derivative formulas for elementary functions. Further discussion shows that for continuously differentiable functions, our new definitions are in accord with the traditional concepts.

### ABSTRACT FOR 21543

#### General triangular arrays of numbers

AUTHOR: Hung-ping TSAO
AFFILIATION: San Francisco State University

We present here all foreseeable approaches to derive polynomial expressions for the power-sum of the natural sequence. Throughout, binomial coefficients play as the key role of linking together the product-sums and power-sums. We take the opportunity to sort out the intricate liaisons among Stirling numbers of both kinds and Eulerian numbers of two orders. WE further generalize the related numbers based on the natural sequence to those that are arithmetically progressive sequence-based. As a result, various structures of triangular arrays can be built on top of different underlying bases.

### ABSTRACT FOR 21550

#### The Importance of Adopting Evolving Technological Tools to Expand Content Knowledge to 3D

AUTHOR: Wei-Chi YANG

It is clear that many uninspired and uninteresting math problems are created mainly to test students' algebraic manipulation skills. In this paper, we shed some lights on how technological tools can be adopted in a classroom to stimulate students' interests in discovering mathematics. It has been demonstrated in [9] and [10] that college entrance exam problems from China sometimes can be daunting to students. However, if technological tools are adopted for explorations, those problems can become more accessible to more students. Furthermore, those explorations can inspire some deep and serious research activities with the help of technological tools. In this paper, we use examples to reaffirm that the first step to attract students' interests in a math problem is to interpret the problem in a more understandable way such as in a real-life setting. Next, we investigate if further 3D or higher dimension extensions are possible while exploring activities with technological tools. Consequently, many boring exercises can be made lively and appealing to broader students again. Moreover, real-life applications in 3D may become possible from these exploratory activities.

### ABSTRACT FOR 21551

#### Discovering New Tessellations Using Dynamic Geometry Software

AUTHOR: Ma. Louise Antonette DE LAS PEÑAS, Eduard TAGANAP
AFFILIATION: Ateneo de Manila University, Ateneo de Manila University and Central Luzon State University

In this paper we use dynamic geometry software to investigate a class of tilings called k-uniform tilings or tessellations. A tiling consisting of regular polygons whose vertices belong to k-transitivity classes under the action of its symmetry group (vertex-k-transitive) is said to be k-uniform. We also present constructions of tilings consisting of irregular polygons that are vertex-k-transitive.

### ABSTRACT FOR 21552

#### A litany of ladders: easy problems with hard solutions

AUTHOR: Alasdair McANDREW
AFFILIATION: Victoria University, Melbourne Australia

Mathematics is full of problems from the easy to the intractable; in fact, it may be quite fairly said that mathematics is the study of problems. Some problems are straightforward enough to be used as student exercises; others are of a difficulty and complexity to occupy the attention of scholars all their lives. The purpose of this paper is to look at a few problems which are neither trivially easy nor impossibly difficult. These problems all have the similarity that in their classical statements they are about the placement of ladders. However, they are difficult enough that solving them is more than an elementary routine exercise; also, they are perfect vehicles for the use of a computer algebra system. We thus show how standard, simple problems can be greatly expanded in scope by the use of technology, and these new problems, which may be seen as "difficult" in a classroom sense, are amenable to experimentation.

### ABSTRACT FOR 32000

#### Designing Circumscriptible Heptahera Inspired by Sangaku

AUTHOR: Jen-chung CHUAN
AFFILIATION: National Tsing Hua University, Taiwan

The Sangaku were painted in color on wooden tablets and hung in the precincts of Buddhist temples and Shinto shrines as a way to be thankful for the inspiration to solve the geometric construction problems. Heptahedra are 7-faced polyhedra. There are 34 topologically distinct heptahedra. A polyhedron is said to be circumscriptible if all edges are tangent to a fixed sphere. Since each edge of a circumscriptible polyhedron meets the sphere at exactly one point, the circles formed by taking the intersection of the sphere with the faces have at most one point in common with each other. In other words, the pattern formed by the circles resembles that in a Sangaku drawing! In this talk we shall show, with Cabri 3D, the construction of a circumscriptible heptahedron with following the steps:

1. Construct an appropriate pattern of 7 circles on the plane.
2. Take the inverse of the circles with respect to a sphere tangent to the plane, resulting in a pattern of 7 circles on another sphere 1/2 in diameter.
3. Take the convex hull of the tangent points of the circles.
4. Find the pole (with respect to the small sphere) of each face of the convex hull. These are the vertices of the circumscriptive heptahedron and its dual.

# Abstracts for Full Papers

### ABSTRACT FOR 21456

#### Graphing a quadrilateral using a single Cartesian equation

AUTHOR: Lai WEI, Weng Kin HO
AFFILIATION: No.22 middle school student, National Institute of Education, Nanyang Technological University

In this paper, we show that it is possible to graph an arbitrarily given quadrilateral (with known vertices) using only a single Cartesian equation. Crucially, we rely on matrix algebra; in particular, projective mappings which are commonly exploited in computer graphics but seldom mentioned in high school lessons or undergraduate matrix algebra courses. Our exploration is also helped by the use of a graphing calculator. Assuming no prior knowledge of matrices on the part of the reader, this paper introduces the necessary matrix-related machinery that discussion requires.

### ABSTRACT FOR 21467

#### An Alternative Model of Online Mathematics Instruction to Promote Student Support

AFFILIATION: Governors State University, Western Governors University

Limitations of classroom space, time, economics, and a growing global village turn many adults to electronic-based resources for learning. In this paper, we analyze models of online learning, particularly for adults learning mathematics as a part of a degree or certificate program. Several decades of research point to benefits of online learning, but also a number of concerns and limitations based on test performance, student retention, and the limitations derived from indirect human contact. As a preliminary study, we describe the structure of online mathematics courses at a four-year degree granting institution that has taken an approach unlike a traditional classroom. Instead of one teacher interacting with a group of students with homework, quizzes, lecture, and exams, the alternative model includes many faculty that support a student as they progress through self-paced instruction, competency-based learning, and criterion-referenced assessment. We detail the software and hardware used by the faculty, and a demonstration of how the technology used allows for fully online instruction, mentoring, advising, and assessing. Surprisingly, this model is more cost-effective for both the institution and its students. We conclude with recommendations for best practices in designing support for adults taking online mathematics courses.

### ABSTRACT FOR 21470

#### On Possible Use of Quantifier Elimination Software in Upper Secondary Mathematics Education

AUTHOR: Yosuke SATO, Ryoya FUKASAKU
AFFILIATION: Tokyo University of Science

Quantifier elimination (QE) is a powerful tool of computer algebra systems. It enables us to solve many mathematical problems in the areas of science, engineering, economics and education, etc. In this paper, we introduce our attempt to apply QE software of computer algebra systems in upper secondary mathematics education. We focus especially on the education of logical reasoning.

### ABSTRACT FOR 21471

#### Understanding Sampling Distributions Using Simulation in R

AUTHOR: Leslie CHANDRAKANTHA
AFFILIATION: John Jay College of Criminal Justice

This paper presents a simulation method using a programming language called R to help students understand the concepts of sampling distributions and the central limit theorem. We simulate an approximation of a sampling distribution by taking 1000 random samples from different populations, calculating the mean of each sample, and creating histograms to display the distributions of the sample mean. Normal probability plots of the sample mean are also created as a second tool for understanding the distribution of the sample mean. Students will observe the effects of sample size on the shape and spread of the approximate sampling distribution by varying the sample size.

### ABSTRACT FOR 21481

#### Pade Approximant Using ISCZ Method

AUTHOR: Haruka MISHIMA, Hiroshi KAI
AFFILIATION: Ehime University

Pade approximant is a rational approximation constructed from the coefficients of a power series of a given input function. Pade approximant may be obtained by the extended Euclidean algorithm, but the algorithm can be unstable if the computations are performed with floating-point arithmetic. In this paper, we utilize the ISCZ method to stabilize numerical computations of Pade approximants using extended Euclidean algorithm. The proposed method guarantees the solutions are correct. Furthermore, through experimental results, we will show computations by the proposed method are faster than those of a symbolic implementation of Pad\''e approximant using extended Euclidean algorithm in the case of the power series with coefficients over Q(e).

### ABSTRACT FOR 21483

AUTHOR: Wing Kin CHENG
AFFILIATION: Hong Kong Baptist University

In this paper, the researcher investigated techno-pedagogical task design in learning travel graph for Hong Kong primary 6 students and how teachers' knowledge about digital artifacts affects the task design and students' learning. This paper elicits an iterative cycle of the nested epistemic sequences for task design using digital tool. This paper also discusses the knowledge gap in learning with digital tool. It is suggested that didactic intervention plays a crucial role in relating the mathematics knowledge, the situated discourse with digital tool and learners'' physical world and thus narrows the knowledge gap in a digital learning environment

### ABSTRACT FOR 21484

#### Lindenmayer systems, fractals, and their mathematics

AUTHOR: Alasdair McANDREW
AFFILIATION: Victoria University, Melbourne Australia

Students are always asking for applications of mathematics. But more often than not, textbooks are filled with applications' which are unimaginative, contrived, unrealistic, and uninteresting. However, there are of course numerous areas in which mathematics can be, and is, deployed to great effect. The purpose of this article is to introduce one such area: Lindenmayer systems, which beautifully joins mathematics and graphics, and investigate some of the mathematics---in particular their fractal dimension---and also the beauty of some of the graphics. We claim that the simplicity of the systems, and the complexity of their outputs, make for a simple way to introduce complexity---and modelling of the natural world---into a mathematics course, especially a course on finite mathematics, geometry, or computational mathematics.

### ABSTRACT FOR 21489

#### Collaborative Summarizing Feature: Supporting Group Knowledge Construction in an Online Discussion Forum

AUTHOR: Kasiyah JUNUS, Harry Budi SANTOSO, Heru SUHARTANTO, Sri Hartati SURADIJONO, and Lia SADITA
AFFILIATION: The Faculty of Computer Science, Universitas Indonesia, The Faculty of Psychology, Universitas Indonesia

Experts agree that online discussion forum has the potential to support individual and group knowledge construction collaboratively. Previous studies show that challenges faced by students entering online collaborative learning are managing motivation, the feeling of uncertainty, difficulty in putting their thoughts in texts, summarizing large numbers of postings especially during exploration and integration phases of the Practical Inquiry Model. In the lens of instructors, monitoring discussion progress to provide prompt feedback when needed is not a trivial task. This study proposes a collaborative summarizing feature integrated to an online discussion forum to support group knowledge construction in the learning management system. The Community of the Inquiry Model is used to guide the study. Students evaluate, integrate, and extract the content of others' postings to compose a summary collaboratively. They can add, edit, delete, and restructure the summary. The contributors' names are automatically displayed to support social presence. The feature facilitates the presence of the CoI model in a different format compared to that of the online discussion forum.

### ABSTRACT FOR 21490

#### A Retrospective Study on the Effects of Flipping a Calculus Course

AUTHOR: Wee Leng NG, Kok Ming TEO
AFFILIATION: National Institute of Education, Nanyang Technological University

The purpose of this study was to examine the effects of a calculus course using the flipped classroom model on undergraduate students' achievement in mathematics which was measured by their scores on three quizzes, a test, and a final written examination, as well as their overall scores. The scores of a total of 58 second year students, comprising 17 students in the experimental group and 41 students in the control group, enrolled in a university degree programme in Singapore were analysed retrospectively using analysis of covariance (ANCOVA) so as to control for initial differences. The experimental group comprised students who took the flipped calculus course in the August 2016 semester while the control group comprised students who took the same calculus course taught using a lecture-tutorial approach in the August 2013 semester. Results of ANCOVA show that after controlling for initial differences the experimental group scored statistically significantly higher in the test but lower in the final examination than the control group.

### ABSTRACT FOR 21492

#### Vector Data Viewer for Distribution Glance

AUTHOR: Ryoji FUKUDA, Yuki TOYOSAKA
AFFILIATION: Faculty of Information Science, Kyushu Sangyo University, Japan, Faculty of Science and Technology, Oita University, Japan

We developed a software to plot vector data for a rough understanding of their distribution. For higher dimensional data, we created selection and change functions for view axes. Moreover, we analyzed approximation methods for determining the density of a vector distribution by using local information of vector data. In addition, we created a density viewer for vector distribution.

### ABSTRACT FOR 21495

#### Top-Down Expression of Mathematical Document for Nonvisual Communication

AUTHOR: Ryoji FUKUDA, Naoki TAMURA, Yuki TOYOSAKA
AFFILIATION: Faculty of Information Science, Kyushu Sangyo University, Faculty of Science and Technology, Oita University, Japan, Graduate School of Engineering, Oita University, Japan

Mathematical expressions are originally graphical contents, and there are several methods to express them through text contents or voice outputs. However, these are sometimes difficult to understand because of the information structure. In this study, we considered a top-down expression structure for mathematical documents. We prepared four expression patterns, by using which the whole document can be expressed as a nested structure. We also defined an expression method similar to natural language, and its translation from manual explanation is not difficult. This expression can be automatically translated to our top-down expression. By using these expression methods, we could realize easier scientific communication without visual information.

### ABSTRACT FOR 21497

#### Geometric Modeling as Spatial Thinking Approach among Prospective Teachers

AUTHOR: Diego LIEBAN, Zsolt LAVICZA
AFFILIATION: Instituto Federal do Rio Grande do Sul, Johannes Kepler Universität, Johannes Kepler Universität

The use of Dynamic Geometry Systems (DGS) in mathematics classes have increased steadily over the past years, connecting several topics in mathematics and applied sciences. This study developed a pilotto promote geometric modelling approaches among prospective teachers in STEM (Science, Technology, Engineering and Mathematics) subjects. Inspired by objects from students' lives and knowledge, participants built physical prototypes and modeled them using GeoGebra3D software. Modelling these machines digitally contributed to students' understanding of concepts in geometry and highlight previous held misconceptions. In addition, these activities helped to improve students' spatial thinking and assisted their transition from plane to space while using the software tools. All student models are available freely on the GeoGebra Materials online platform and particular samples and approaches will be outlined in this work.

### ABSTRACT FOR 21502

#### Constitution of the Proof using the Isabelle Theorem Prover

AFFILIATION: Konan University, Graduate School of Natural Science, Konan University, JAPAN

Congruence relation have been widely used in science. We use an Isabelle theorem prover to verify and specify some lemmata in proving a certain theorem of congruence relation. This paper is about using the Isabelle theorem prover to establish the proofs of some elementary number theoretic lemmata for modulo arithmetic.

### ABSTRACT FOR 21503

#### A framework for evaluating computer algebra systems for mathematics teaching and learning

AUTHOR: Alasdair McANDREW
AFFILIATION: Victoria University, Melbourne Australia

Computer algebra systems: software which can perform algebraic as well as numeric and graphical computations, have become central to mathematics teaching and learning over the past few decades. However, there is little consensus as to what constitutes the best system for the purpose, or by what standards such a system should be judged. Part of the difficulty is that there are many competing systems, from the commercial (and very expensive), to open source systems, as well as specialized systems and handheld systems. Deciding which system is `best'''' is not simply a matter of lining the systems up and deciding which has the greatest mathematical power, but rather which system best fits the needs of the students and the teachers. This paper investigates this problem, and describes a framework---in a sense a decision making process---to help mathematics educators make such a decision.

### ABSTRACT FOR 21505

#### Mathematics, Virtual Reality, and Programming

AUTHOR: Andy YEH
AFFILIATION: Queensland University of Technology

In this paper, an online virtual reality learning environment named VRMath2 is introduced. The epistemological underpinnings for the VRMath2 learning environment are presented to support its design of linking mathematics with programming in virtual reality microworlds. The fractal geometry, specifically the Koch snowflake and Siepinski triangle are demonstrated as applications in mathematics in this paper. By using the Logo programming language in VRMath2, many 2D or 3D fractals can be described, created and experimented in virtual reality space. And in doing so, mathematical understandings are developed and/or enhanced. There is great potential in using programming to create mathematical entities, particularly in 3D virtual reality space. VRMath2 is freely available online, and could be further explored as a teaching and learning resource for mathematics. However, more empirical studies about the use of VRMath2 learning environment should be evaluated to validate its theoretical underpinnings and to improve its practical applications in mathematics education.

### ABSTRACT FOR 32001

#### Study of Tones Characteristics in Thai Language using Fast Fourier Transform (FFT)

AUTHOR: Janchai YINGPRAYOON, Dr.rer.nat.
AFFILIATION: Suan Sunandha Rajabhat University, Bangkok, Thailand

The Thai language is a tonal language. Tones are the core of the language. Tones distinguish the meaning of one word from another. Thai language is one of the languages that uses tones for communication of information. Thai language uses consonant sound symbols to indicate how the words should be pronounced correctly. Otherwise we cannot understand the meaning of the words if they have the same tones but different symbols. In order to speak Thai correctly in terms of meaning the tones of the words have to be pronounced correctly. This work is to study the tone characteristics of Thai language using Fast Fourier Transform (FFT) to analyze the frequency patterns of the 5 tones in Thai language.

### ABSTRACT FOR 32002

#### Thai License Plate Recognition Using Proportional and Filtering Method

AUTHOR: Kann YINGPRAYOON
AFFILIATION: Thai - German Graduate School of Engineering King Mongkut's University of Technology North Bangkok

This paper is a preliminary study and development. We will focus on the detection of province name on the Thai license plate image and result the outcome in text. The procedure is to obtain a License Plate image and generate texts out of the lower resolution image. The image has to be cropped into the size of the plate which reveals only the image of texts. The image will be converted to grayscale first and then converted to binary after all. The process only needs binarized image. There are three main parts on Thai license plate which consist of alphabets, Arabic numbers and a province name. In order to detect a province name, we will consider just numbers and province name by collecting the size of the matrix and its values. Apart from detecting each letter, we will detect a whole word and save the values of its height, width, histogram, number of spaces between each letter, number of letters and number of vowels to create a proportion between each value and compare, evaluate and result the outcome into a certain province text. All the values will be stored in the program. Each province will be filtered by each proportional value to distinguish into different provinces. All the processes will be simulated using MATLAB software.

# Papers with Abstract Only

### ABSTRACT FOR 21419

#### Optimal bias control in causal inference via semidefinite programming and eigen-analysis

AUTHOR: Rahul MUKHERJEE
AFFILIATION: Indian Institute of Management Calcutta

This paper shows how semidefinite programming software as well as eigen-analysis can solve an important problem of optimal bias control in causal inference. Such inference, under a potential outcomes framework, has been of significant current interest due to its wide-ranging applicability in diverse fields like behavioral sciences, sociology, biomedical sciences, etc.

The issue of unbiased variance estimation is a key challenge in causal inference. While much of the existing literature revolves around the assumption of Neymannian strict additivity to overcome this difficulty, very recently, Mukerjee et al (2017, Journal of the American Statistical Association, in press) have shown that unbiased variance estimation is, indeed, possible under conditions milder than strict additivity. This opens up the problem of choosing from amongst a plethora of rival variance estimators, from the perspective of controlling the bias caused by these estimators when their underlying unbiasedness conditions, strict or mild, fail to hold.

It is shown that a mathematical formulation of the above problem calls for finding a positive semidefinite matrix Q, having zero row sums and certain additional constraints on its elements, so as to minimize a quadratic form given by Q. The variables in the quadratic form turn out to be unknown parametric functions, adding to the complexity of the problem. To overcome this difficulty, two approaches arise, based on minimizing (a) the average of the quadratic form under suitable prior specification, or (b) the maximum of the quadratic form over spherical contours. It is shown in the paper that (a) can be re-formulated as a problem in semi-definite programming. This allows the adaptation of software such as SDPT3 in MATLAB for obtaining its solution and thus paves the way for the development of a dedicated software for solving this crucial problem in statistics, with immediate practical applicability. Moreover, a careful eigen-analysis is found to yield an analytical solution to the minimaxity problem in (a).

### ABSTRACT FOR 21423

#### Exploring Computer Science with MicroworldsEX to Learn Geometry and Logo Programming Code

AUTHOR: Thomas WALSH
AFFILIATION: Iowa State University / Ames (IA) Schools

Future employment of computer-programming jobs will be best for applicants with experience in different languages and coding tools (Bureau of Labor Statistics, 2015). Empirical and meta-analysis research studies support of teaching Logo programming in developing student cognitive problem-solving skills has been documented. Using guided instruction with teacher-mediated scaffolding Exploring Computer Science with MicroworldsEX is found as an effective method in preparing students using the Logo code programming language to create geometric graphic, animation, and gaming projects. The presentation of this e-book constructivist curriculum integrated with student projects will include an informational handout and URL link to the author's (teacher's) web page providing access to educational materials.

Employment of software developers (i.e., work with or programming code) is projected to grow 17 percent from now through 2024, faster than the average for all occupations (Bureau of Labor Statistics, 2015). According to the report, computer-programming jobs will be best for those applicants with experience in different languages and programming tool skills. While the demand for coding specialists is increasing knowledge of basic IT skills will be a literacy requirement given the growth of technology. With this technology growth more countries are introducing programming as part of their syllabus (i.e., European countries and Canada) or introducing a digital curriculum as part of the STEM initiative (i.e., Australia and Singapore).

Given these trends students will need to be prepared to learn programming code language, preferably starting at an earlier age. Empirical and meta-analysis research studies of the problem-solving cognitive benefits of teaching Logo programming has been documented with student learning of near transfer of skills to other domains (e.g., geometry) and far transfer to other programming languages.

To prepare students for the future workplace and with research support for learning Logo code a curriculum program is needed. Exploring Computer Science with MicroworldsEX was developed as a structured learning methodology of learning activity lessons, with opportunities for discovery and exploration, to support student learning in a "Microworlds" project-based environment to create geometric graphics, animation, and gaming using the Logo programming language. The curriculum was developed from the author's over 25-year Logo teaching experience with elementary and middle school regular and gifted education students, along with dissertation research and journal publications supporting use of guided instruction for student learning programming code. Guided instruction was found for the potential cognitive benefits for teaching Logo to be achieved by implementing more carefully planned teacher-directed lessons balanced with student problem solving and discovery learning using teacher-mediated scaffolding.

The Microworlds research-based field-tested curriculum provides a lesson plan methodology beginning with introductory Logo commands and procedures progressing to graphics and animation programming activities. The e-book Exploring Computer Science with MicroworldsEX includes student project examples, teacher resource support activities, and is published with Logo Computer Systems, Incorporated (LCSI)* who developed the MicroworldsEX program. Walsh currently teaches adjunct Logo programming code sessions to high ability elementary and middle school students at Iowa State University (OPPTAG program) using the e-book lesson activities. Some students in OPPTAG have been able use Logo code in developing basic gaming program procedures.

The presentation will highlight the methodology for this constructivist curriculum integrated with examples of student "Microworlds" and includes a demonstration of the MicroworldsEX program showing student animation projects. A handout will provide information about the curriculum with URL link to the author's web page providing access to program materials including supplemental teaching resources from the author's over 35 years of classroom teaching experience.

### ABSTRACT FOR 21424

#### The Survey Toolkit Curriculum Methodology for Researching Information, Survey Questioning, and Analyzing Data with TinkerPlots®

AUTHOR: Thomas WALSH
AFFILIATION: Iowa State University / Ames Schools

In an era where social media traffics fake news websites that publishes misinformation it is imperative to provide students' experiences in The Survey Toolkit and TinkerPlots® curriculum teaching sound research principles and information gathering techniques. The field-tested program was found effective in guiding students choosing research questions, writing a research report using a paragraph cluster information strategy, developing unbiased survey questions using reliable sampling, analyzing survey data with TinkerPlots®, and sharing results. The presentation of this constructivist curriculum, integrated with student projects, includes an informational handout and URL link to the author's (teacher's) web page providing access to educational materials.

Providing students with a methodology for collecting information and facts from multiple sources for conducting research projects and developing survey questionnaires is needed more than ever. This is particularly true in an era where social media traffics fake news websites that publish hoaxes, propaganda, and misinformation (e.g., research findings) readers believe to be true. These fake news sites, over 50 listed on Wikipedia, along with use of "clickbait" web content publishing sensationalist headlines or click-thoughts over online social networks influences political views and orientation of personal values. Availability of these sites is quickly accessible to students on their phone using apps or other social media devices.

Given this trafficking of misinformation it is imperative to provide students The Survey Toolkit and TinkerPlots curriculum learning sound research principles and techniques for gathering information. This instructional program is a project-oriented learning pedagogy, developed primarily for elementary to middle school age students, and based on study of research instructional design methodology at Iowa State University. The program has been field-tested with the author's regular grade three and six grade gifted education students for over 10 years. The research method was found effective in guiding students in choosing a research question, writing a research report using a cluster paragraph information collection strategy, developing unbiased questions for giving a survey to a reliable sample, analyzing survey data using TinkerPlots, and sharing results on presentation boards. The Survey Toolkit curriculum was found to be applicable for researching topics across the curriculum. For example, student topics have included study of math in the context of Arabic culture, Chinese history, alternative fuels, ophthalmology, working physics, and nanotechnology to name a few. The Survey Toolkit is available through McGraw-Hill. * Walsh (2011) provides information about the development, implementation, and use of the curriculum with students.

The presentation will highlight the methodology for this constructivist curriculum integrated with examples of student projects and will demonstrate the TinkerPlots program using collected data from conference attendees. A handout will provide information about the research project with a URL link to the author's web page providing access to program materials (e.g., pdf download Resource Guide) including supplemental teaching resources from the author's over 35 years of classroom teaching experience.

### ABSTRACT FOR 21428

#### Making mathematics engaging with technology

AUTHOR: Maree SKILLEN
AFFILIATION: UTS:Insearch, Australia

Education is increasingly moving towards technology enhanced delivery, as a way to engage students in the teaching and learning process. Teachers are employing a pedagogical design capacity which involves the integration of technology with mathematics to transform the learning and application of concepts by students. Using this approach performance levels in Mathematics subjects has improved, and teachers have noted increased participation and commitment to learning by students. This presentation will discuss approaches utilised for the integration of technology into mathematics classrooms to ensure that student-centred technology-enabled learning is occurring. Several examples will also be demonstrated.

This presentation will consider how the use of technology provides a paradigmatic shift in the instructional focus of specific computer applications, to more sophisticated uses of general purpose software. Educational uses of technology will be examined as exemplars for a discussion of alternative modes of teaching to engage students.

### ABSTRACT FOR 21437

#### On the surveys of Flipped classes in Korea University

AUTHOR: Jeongwhan CHOI, Jiyee CHOI, Junkyung KIM
AFFILIATION: Korea University

As higher education has been changed dramatically in 21st century, many Korean universities are making efforts to changing teaching methods from lecture-based to student-centered. One of such efforts is the Flipped Class, which is a type of blended learning that reverses the traditional learning environment by delivering instructional content, often on-line, outside of the classroom. In a flipped classroom, students watch online lectures, collaborate in on-line discussions, or carry out research at home and engage in concepts in the classroom with the guidance of a mentor. In Korea University, the flipped classes have been offered since 2016 and 19 flipped classes have been opened for 1000 students in the spring semester of 2017. The purpose of this study is to suggest strategies for improving the Flipped Class based on student perceptions and other information gathered from satisfaction surveys.

### ABSTRACT FOR 21440

#### How Middle-School Mathematics Textbooks Suggest Technology Use in Mathematics Classrooms

AUTHOR: Mihyun JEON, Gooyeon KIM
AFFILIATION: Graduate School of Education, Sogang University

This study aims to explore how Korean middle-school mathematics textbooks suggest using technology in mathematics classrooms. For this purpose, we investigated Korean mathematics textbooks for middle grades in terms of which types of technology is included and how the technology suggested foster students¡¯ mathematical understanding. Technology can support students¡¯ learning mathematics and effective teaching of mathematics (National Council of Teachers of Mathematics, 2000) and Korea National Curriculum (2009) encourages that teachers incorporate technology in mathematics teaching and learning. Teachers tend to get advice from textbook examples or suggestions including online resources and their in-school colleagues in technology use in mathematics (Kissane, McConney & Ho, 2015). Although many teachers have some barriers such as lack of experience, confidence, time, skills, and technical or administrative support, they have enthusiastic and positive attitude toward using technology in classrooms (Forgasz, 2006; Pierce & Ball, 2009; Yu, 2013). However, Korean teachers seem to be not willing to use technology in their mathematics classroom (Kang & Kim, in review). Preliminary results from the textbook analysis reveal as follows: a) types of technology are limited to graphing calculator, excel, and applet; b) mathematical tasks integrated with technology require simple steps in ways that student only enter a value or equation in calculators, programs, applet according to the directions provided by the textbooks; and thus, c) it seems that students do not have opportunities to experience finding patterns, making generalizations, explaining connections the representations to the mathematical concepts or meaning and justifying their reasoning about various context of the tasks.

### ABSTRACT FOR 21444

#### Enhancement of Figures in STACK by Appending the Capability of Interactive Manipulations

AUTHOR: Kenji FUKAZAWA, Yasuyuki NAKAMURA
AFFILIATION: National Institute of Technology, Kure College, Graduate School of Information Science Nagoya University

STACK is the e-learning system for mathematics and can show static figures which are generated dynamically according to randomly selected parameters. In this talk, we explain the enhancement of figures by appending the capability of Interactive manipulations. In this enhancement, by making Maxima functions. We utilize CindyScript, which is the functional programming language for manipulating the interactive geometry software, Cinderella, and CindyJS, which is the framework to create interactive mathematical contents in Web pages. With the Interactive manipulations of figures, we believe that students get better understanding. Finally, several examples are presented.

Analysis of data including worksheets, DGS activities, and interviews showed effects of geometric understanding as a result of utilizing DGS. During the students had personally composed problem situations on their own through DGS, they had intuitively recognized beneath geometric meaning.

### ABSTRACT FOR 21445

#### Enhanced Teaching and Learning of Structural Dynamics with Computer Algebra Systems

AUTHOR: Thomas HU
AFFILIATION: Dept. of Civil and Environmental Engineering, The Hong Kong University of Science and Technology

A revolutionary approach has been adopted when teaching structural dynamics at HKUST in recent years, utilizing the powerful features offered by graphing calculators equipped with CAS. Traditionally demanding and tedious mathematical tasks such as solving differential equations, matrix inversion and eigenvalue analysis are greatly simplified by utilizing CAS. Moreover, the whole course is taught in a much more "dynamic" manner by allowing students to visualize structural behavior using their CAS calculators--often with the slider feature--to see how a series solution converges as the number of terms increase, how multiple-degree-of-freedom systems vibrate in time, and how a tuned-mass-damper system changes its behavior as the degree of damping increases, etc. An end-of-course student survey was also conducted, while student projects were completed requiring the use of CAS. The results show students welcomed and benefitted from the use of technology in teaching and learning.

### ABSTRACT FOR 21446

#### Origami and paper-folding activities as tools for teaching mathematical content with elements of programming

AUTHOR: Natalija BUDINSKI, Zsolt LAVICZA, Kristof FENYVESI
AFFILIATION: Johannes Kepler University, University of Cambridge, International GeoGebra Institute, Budapest Metropolitan University, University of Sarajevo, University of Jyvaskyla

This talk proposes how can paper-folding activities be used for teaching mathematical content and programming. The outlined activities have been already implemented in the high school classes in Serbia in 2017 through various curricular and extracurricular activities. The educational approach is based on introducing students with the dragon curve fractal. By following instructions, the concept of origami and folding the dragon curve fractal, students on the one hand discover properties of fractals, which are currently not a usual topic in the mathematical classroom, and on the other hand, they revise previously learned mathematical concepts such as the Pythagorean theorem or logarithms. Hands-on activities jointly with mathematical tasks can raise students' motivation and prepare ground for basics of programming with GeoGebra and Scratch. Students knowledge gained through paper folding can support understanding of mathematics, designing patterns in GeoGebra and coding the dragon curve in Scratch. In our presentation, we will discuss the benefits and obstacles we have been faced during this unusual combination of activities in mathematical classrooms and offer recommendation connecting physical, digital and virtual activities to promote mathematical understanding as well as computer programming.

### ABSTRACT FOR 21447

#### Technology-related Trends and Examples in STEM Education Research

AUTHOR: Zsolt LAVICZA, Kristof FENYVESI
AFFILIATION: Johannes Kepler University, University of Cambridge, International GeoGebra Institute, Budapest Metropolitan University, University of Jyvaskyla, Finland

Technology is increasingly becoming an important part of STEM teaching and learning in the 21st Century. There have been numerous attempts to integrate technology into education systems, but without serious development and research the success of these attempts had been limited. In our talk, we will highlight the importance of research in technology-supported education and describe some research projects attempting to make larger impact on STEM teaching and learning. We will particularly outline findings four projects making sizable contribution to STEM education: 1) GeoGebra, an open-source project offering accessible software for STEM education and currently used by more than 40 million teachers and students around the globe; 2) Geomatech, a large-scale EU funded project developing high-quality materials and teacher training resources with the aim to embed technology use into entire education systems; 3) KIKS (Kids Inspire Kids) project, endeavours to involve students in STEM projects and encourage them to develop programmes that inspires their peers to study STEM subjects; 4) Experience Workshop Movement, aims to integrate Arts into STEM teaching (STEAM) through physical and virtual spaces, which are developed together by researchers, artists, teachers and students. In addition, we will also describe the recently established international Doctoral Programme in STEM Education at Johannes Kepler University in Linz, Austria; highlight work of some current PhD students; and hope to encourage students and researchers to develop joint projects with us as well as to foster student and staff exchanges with the JKU STEM Education Centre.

### ABSTRACT FOR 21457

#### Application of Sharing Economy to the Teaching and Learning of Junior High School Geography

AUTHOR: Ming-Gong LEE, I-Hui CHEN
AFFILIATION: Department of Leisure Recreation and Management, Chung Hua University, Hsinchu Municipal Hsiang Shan Senior High School, Hsinchu, 300 Taiwan

This study is to apply the concept of "The sharing economy " (hereinafter referred to as " The shared concept ") to the teaching and learning of geography in junior high school, to build a model and to explore its application of teaching and learning attitude by the new Sharing concept. In the experiment, two classes in the seventh grade of the middle school were taken as the research object. The shared concept was implemented in geography teaching in the experimental group, and the ordinary teaching model was carried out in the control group. The study covers "Coast and Islands", and "Weather and Climate," two lessons in Unit 1 of the first semester of the Seventh grade. At the end of each lesson, each smaller group in the experimental group can pose problems and upload the questions to the KAHOOT!, which is acting as the platform for the shared concept and it can be shared them with the researchers. In contrast, students in the control group only work out the ordinary homework assignments on the subject. After 7 weeks, the second test was conducted to compare the learning achievement of two groups of students under different teaching methodologies, and to explore whether there were significant differences in learning effectiveness and attitude between the two groups of students by applying the shared concept in geography teaching.

Results obtained from this research:

1. The application of the shared concept has a significant effect on the geography learning performance of middle school students.
2. The application of the shared concept has a significant effect on the geography learning performance of students who are rated as medium learning level.
3. The shared concept applied to teaching has a significant impact on geography learning attitude of middle school students.

### ABSTRACT FOR 21462

#### Modified Penalty Method for Solving Transportation Model

AUTHOR: Rebecca TOLENTINO
AFFILIATION: University of the City of Manila

In this paper, a new method for solving transportation model is explored. The new method, called the Modified Penalty Method considered two measures associated with each shipment/distribution schedule: unit transportation cost per shipment and the row and column penalty associated with each shipment. The new method was used to solve transportation problems and the result was compared with the Vogel's Approximation Method which also considered penalty associated with each path. Comparisons were based on the total cost in the initial feasible solution (IFS), number of iterations to get to the optimal solution and computation time. Results showed that the new method has reduced the number of computations required to get to the optimal solution and has therefore reduced computation time. The new method works best with unbalanced transportation problems.

### ABSTRACT FOR 21477

#### Pre-service mathematics teachers' perceptions relating to the use of coding for teaching mathematics

AUTHOR: Hazel TAN
AFFILIATION: Monash University, Australia

The use of coding and algorithm is becoming popular in the mathematics curriculum of many countries and regions, including Victoria, Australia. In this presentation, the findings of a survey about pre-service secondary mathematics teachers'' perceptions relating to the use of coding for teaching mathematics are reported. The pre-service teachers were surveyed after going through a workshop on use of Scratch for teaching mathematics.

### ABSTRACT FOR 21478

#### Based on the New Curriculum Reform, Prime is Full of Charm in Teaching and Learning

AUTHOR: Yapin TIAN
AFFILIATION: Weishanlu Middle School, China

This article describes how the upgraded version of the graphic calculator, Prime, can narrow the gap between the reality and the reform from the following three aspects: (1) Prime makes a transform of the students'' teaching activities and reduces the burden of teachers and students, with students'' health improved. (2) Prime makes the abstract mathematics curriculum embodied and visualized, with students'' learning, development and innovation improved. (3) Prime strengthens the mutual learning between students. In the meantime, it also promotes the cooperation among the students.

### ABSTRACT FOR 21480

#### The practice and reflection on mathematical inquiry learning based on Geogebra--the example of Quadrilateral congruent conditions

AUTHOR: Wu CHUNLAN, Li HONGYUN
AFFILIATION: Beijing Institute of Education

Geogebra is an open-source software about mathematics teaching and learning suitable for all levels of education. This paper explains how students experience the process of posing a problem, giving a reasonable conjecture, inquiring and verifying it, getting a conclusion and reflection under Geogebra circumstance in the exploration of quadrilateral congruent condition as an example. It gives some suggestions in how to strengthen the deep integration between information technology and mathematics teaching and learning: the learning of information technology, awareness that integration of information technology and mathematics teaching including resources and information technology is a powerful tool for students to learn mathematics and solve problems.

### ABSTRACT FOR 21491

#### Visually Analyze of Stability of Dynamical System using Parallel Coordinate

AUTHOR: Hartono HARTONO
AFFILIATION: Yogyakarta State University, Indonesia

Communication is vital tool towards progress. Nowadays, varied mode of communications have surfaced even in the educational arena. A lot of educators have already been utilizing Facebook and text-messaging in facilitating teaching-learning process. In this paper, techniques and strategies were developed and tested in Mathematics teaching-learning process. Results of the investigations will be presented in this paper.

### ABSTRACT FOR 21493

#### A Framework for Primary School Mathematics Teachers to Decide when to Use Calculators with their Pupils

AUTHOR: YEO Kai Kow Joseph
AFFILIATION: National Institute of Education, Nanyang Technological University, Singapore

Calculator has commonly been viewed only as a tool to perform calculations and thus often considered with disapproval by mathematics teachers. Calculators do not "understand" mathematics but they do significantly facilitate the understanding of mathematics. Calculators empower children to focus more on the 'whys" of mathematics rather than on the "hows". Thus, it is inevitable that primary school mathematics teachers to make decision on when and how to use calculators in their lessons. The aim of the presentation is to share a framework to assist mathematics teachers decide when to use calculators with their primary school pupils. This framework helps the teacher focus less on the calculator and more directly on his or her own educational objectives and the pupils' needs and abilities. Examples are then generated within the framework to include product and process related to mathematics.

### ABSTRACT FOR 21498

#### 從簡單運算到函數概念，例談 Hawgent 皓駿支持下的中小學數學教學設計策略

AUTHOR: Chuan-Bo ZUO, Yan-Qin LIN
AFFILIATION: Hawgent Technology Centre in Mathematics, Shuiyinlu Primary School, Yuexiu, Guangzhou, Guangdong, China

### ABSTRACT FOR 21510

#### The Effect of Geogebra on Students' Performance in Algebraic Concepts: The Case of Applications of Kuwait Project

AUTHOR: Mamdouh SOLIMAN, Zsolt LAVICZA, Maryam Al-KANDARY
AFFILIATION: Kuwait University, Johannes Kepler University

Integrating the properties of computer algebra systems and dynamic geometry settings, Geogebra became an effective and powerful device for teaching and learning mathematics. One of the reasons that teachers use Geogebra in mathematics classrooms is to make students learn mathematics meaningfully and conceptually. Students in this project need to grasp the concepts of mathematics instead of memorizing formulae, in this research we concentrated in the learning of some concepts of algebra. This is important for them to further their knowledge in algebra. The purpose of the research was to determine whether GeoGebra software influences year nine students' performance in algebraic concepts. The research used quantitative approach with pre-test and post-test control and experimental groups. The experimental group (n= 63) students received instruction with Geogebra while the control group (n= 65) received traditional instruction in learning algebraic concepts. A paired sample t-test was conducted. The findings show a significant difference in the scores between the two groups in favor of experimental group. Similarly, the results also found statistically significant difference in scores among students in both groups in pre and post measures. In conclusion, the study implies using GeoGebra enhances students' performance in algebraic concepts. Implementing teaching and learning algebra using GeoGebra would help students to understand, interpret, and explore the concept more in details and help them to build and develop their own algebraic knowledge.

### ABSTRACT FOR 21549

#### Dynamical behavior of an iterative method

AUTHOR: Young Hee GEUM
AFFILIATION: Dankook University, South Korea

In order to find the root for the nonlinear equations, the numerical methods are sought. In this study, the dynamical analysis of the parametric fourth-order iteartive method is presented on quadratic polynomial to draw the fractal images. The relevant complex dynamics including the parameter spaces and the dynamical planes are displayed.

### ABSTRACT FOR 32003

#### Applications in Mathematics through Technology for Modeling Spread of Waterborne Disease in Networks

AFFILIATION: Foxcroft School, Middleburg, Virginia, USA, American International School, Chennai, India

Motivated by living conditions in rural India where waterborne diseases are endemic, we consider a network of villages that share a common water source and develop mathematical models for understanding spread of waterborne diseases through multiple transmission pathways. The latter includes direct transmission within each network and indirect transmission via a shared water source. These interactions are captured as a system of coupled ordinary differential equations that are solved using numerical algorithms. This application of mathematics is implemented using technology that employs automated spreadsheets running the algorithms coupled with real-time input feeds via social media tools. We will demonstrate how one can use crowdsourcing ideas combined with mathematics and technology to predict final outbreak sizes for the network model developed which can help make important policy decisions. Theoretical results to determine fraction of population affected by a potential outbreak as well as computational results for benchmark examples will also be presented.

### ABSTRACT FOR 32004

#### Using Mathematics with Technology to Control Gang Activity in Puerto Rico

AFFILIATION: Foxcroft School, Middleburg, Virginia, USA 2Inter American University of Puerto Rico-Bayamon Campus, USA

Crime fighting and gang activity are controversial social issues. To prevent and minimize gang spread in the youth of Puerto Rico, a new mathematical model was developed. In this work, gang membership is being treated as an infectious disease via coupled differential equations. This new model accounts for the possibility of determining how members of the youth community interact with the infected adult (gang) community using different mixing patterns. The numerical results of the implementation of the different mixing patterns using MATLAB software are analyzed. We observed that regardless of the mixing pattern, the greatest reduction of gang infection occurred when parameters such as gang activity, recruitment, conviction and rehab/recidivism rates, were varied in combination. A technology enhanced Graphical User Interface (GUI), integrated with the new gang mathematical model, was also developed that allows users to enter real-time data and perform predictive analysis. We hope to use the infrastructure developed from this project to make informed policy decisions that can help control the spread of gangs as well as bring the much needed awareness using mathematics and technology.

# Abstracts for the Track of Hands-On Workshops

### ABSTRACT FOR 21416

#### Having Fun with Augmented Reality

AUTHOR: Wei Ching QUEK
AFFILIATION: Singapore Polytechnic

Augmented reality (AR) changes the way we interact with the world by unlocking digital layers that are embedded into one-dimensional images. It will be exciting to explore how AR can be used in teaching and learning in classroom to engage the students and achieve higher learning outcomes. Using AR, we can create an overlay that demonstrate real experiments and applications linking to the mathematical concepts

This workshop consists of two activities:

1. Getting Started: Participants will explore and modify existing 3D models to convert the model to augmented reality. Tools such as Tinkercad, Augment will be used. No programming knowledge are needed.
2. Problems Solving/Discussion: Discussion of other existing apps for Augmented Reality for teaching/learning. Bringing the applications to "live" to the classroom. Introduce more advanced tools: Unity 3D and Vufuria.

### ABSTRACT FOR 21425

#### Investigating mathematics in the middle years with ClassWiz

AUTHOR: Barry KISSANE
AFFILIATION: Murdoch University, Australia

Calculators are frequently misunderstood solely as devices to allow students to undertake numerical calculations. The CASIO fx-82EX ClassWiz is a modern scientific calculator with significant functionality designed especially for school mathematics. In this workshop, we will focus on ways in which it can be used to provide a more engaging form of learning mathematics than merely undertaking calculations, especially for students in the middle years of schooling. We will consider some examples of student investigations that are 'low floor, high ceiling' in the sense that it is easy for students to make a start, while there is still significant opportunity for learning afforded by directed use of the calculator. This is a hands-on workshop in which participants will use calculators to experience this way of learning. Previous experience with calculators will not be assumed.

### ABSTRACT FOR 21426

#### Learning school mathematics with ClassWiz, an advanced scientific calculator

AUTHOR: Barry KISSANE
AFFILIATION: Murdoch University, Australia

While scientific calculators have been available since the 1970s, recent advanced versions have been developed explicitly to suit the needs of mathematics education throughout the secondary school years. So, these calculators provide powerful learning opportunities for many aspects of mathematics treated these days in secondary school and early university curricula. In addition, capabilities such as tabulation, spreadsheets, equation solving and other advanced features give students access to efficient calculation. This workshop will focus on a variety of ways in which the CASIO fx-991EX ClassWiz calculator can be used to enhance both teaching and learning, drawing on published materials developed by the author. A variety of topics will be addressed, including calculus, functions, equations and statistics. Previous experience with calculators will not be assumed.

### ABSTRACT FOR 21427

#### The graphics calculator is a tool for learning mathematics

AUTHOR: Barry KISSANE
AFFILIATION: Murdoch University, Australia

Now more than 30 years old, graphics calculators were developed to meet the needs of students learning mathematics in the secondary school and the first year of college. Recent models, such as the CASIO fx-CG50, can be used to support student learning and teachers teaching about many mathematical ideas related to real numbers, functions, equations, statistics, geometry, probability, matrices, sequences and calculus. The graphics environment allows students to generate and use graphs of functions and data, explore the use of spreadsheets, study geometric objects, investigate random phenomena, engage in scientific enquiries and write their own programs. This introductory hands-on workshop will allow participants to experience some of the learning opportunities available, and become aware of resources to support their work. Previous experience with graphics calculators will not be assumed.

### ABSTRACT FOR 21431

#### Heptahedra

AUTHOR: Jen-chung CHUAN
AFFILIATION: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300

There are 34 topologically distinct convex heptahedra. https://en.wikipedia.org/wiki/Heptahedron. Among these 34, eight have 7 vertices and 12 edges. Since V=7=F, these deserve the adjective "combinatorial self-dual". Among the 8, two are self-dual and 3 dual-pairs. The purpose of this workshop is to visualize such abstract concepts.

### ABSTRACT FOR 21432

#### Linkage formed by Skeletal Rhombohedra Complex

AUTHOR: Jen-chung CHUAN
AFFILIATION: Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300

We are to construct linkages based on the dual structure of the Platonic Solids and the Catalan-Archimedean solids. The animation of distortion of these linkage systems is also provided.

### ABSTRACT FOR 21438

#### Collaborative learning in the mathematics classroom

AUTHOR: Katrina NG
AFFILIATION: School of Arts Singapore

Often in the mathematics classroom, teachers are unable to evaluate students'' learning instantaneously. In this workshop, participants will learn to use handheld technology in a networked environment and experience how formative assessment can be done easily using modern technology.

### ABSTRACT FOR 21451

#### Using Digital Resources for Handling Large Data Sets and Understanding Basic Statistical Principles

AUTHOR: Douglas BUTLER
AFFILIATION: iCT Training Centre (Oundle, UK), Autograph-Maths

This hands-on workshop will start by exploring large data sets listed on Douglas' website, TSM Resources, ranging from earthquake data to the late Hans Rosling's "Gapminder". We will practice extracting this data to Excel, and then to Autograph for statistical analysis. The basic principles of linear regression will be explored, starting with a small cluster of points. We will then consider the statistical principles involved in studying the Binomial, Poisson and Normal distributions. Finally, the Central Limit Theorem will be put to the test starting from a simple uniform data set, and leading on to a complex application involving a large set of passenger data in aircraft. Participants in this workshop will receive a complimentary copy of Autograph 4.

### ABSTRACT FOR 21452

#### Using Dynamic Software to Help Students Visualise Key Principles Using a Sympathetic User Interface

AUTHOR: Douglas BUTLER
AFFILIATION: iCT Training Centre (Oundle, UK), Autograph-Maths

With the advent of widely available open source software it is important to discuss what represents a sympathetic user interface for dynamic software. This workshop will explore the user interface of the new Autograph 4 and consider its use in a number of key topics in secondary and college mathematics. Firstly, complex numbers: when explored dynamically on the Argand Diagram they will be 'imaginary' no more! The key principles of 3D vectors can be so much better understood by studying them in 2D first, but with 3D in mind. Another topic that benefits from Autograph's interface is first and second order differential equations - so visual, so relevant and motivational. Further topics will draw on the need to introduce students to problem solving. Participants in this workshop will receive a complimentary copy of Autograph 4.

### ABSTRACT FOR 21458

#### STEM Education - Hands-on Approach

AUTHOR: Thomas YEO
AFFILIATION: Texas Instruments

STEM (Science, Technology, Engineering, Mathematics) education is increasingly popular in many parts of the world. Many nation leaders have identified STEM related jobs and education as an important aspect of the future world. How do we do STEM in the classroom now? In this workshop, participants will have hands-on experience using engineer grade products and handheld technology to run STEM projects in the classroom. Participants will also learn about interesting ideas for teaching Math in the STEM environment.

### ABSTRACT FOR 21459

#### Euler's Method - A Programming Approach

AUTHOR: Thomas YEO
AFFILIATION: Texas Instruments

In this workshop, participants will have hands-on experience in writing a program on a graphing calculator to execute the Euler's Method to solve an initial value problem for differential equations. This activity discusses the real life application of this method to generate the steps required to improve accuracy.

### ABSTRACT FOR 21460

#### Solving Problems with CAS and HP Prime

AUTHOR: GT SPRINGER
AFFILIATION: Hewlett-Packard

In this hands-on workshop, participants will explore a number of problems from Algebra through Calculus using the Computer Algebra System (CAS) of the HP Prime graphing calculator. First, we will look at a problem involving equivalence using the CAS and the Advanced Graphing App. Then we will look at visualizations of Pascal's triangle using the CAS and the Spreadsheet App. As time permits, we will look at other problems involving the CAS and how it works with the rest of the HP Prime graphing calculator.

### ABSTRACT FOR 21463

#### Using Technology in Mathematics Teaching Calculus

AUTHOR: Yong Hor LEE
AFFILIATION: Singapore Sports School

The presentation will cover the use of technology in classroom mathematics teaching using handheld technology. Teachers usually encountered difficulties in explaining abstract concepts in mathematics with the traditional 'chalk-and-board' talk and students usually have difficulties in understanding these abstract concepts with the traditional way of teaching. The use of technology not only display abstract concepts in calculus, but also enhance a deeper understanding of calculus and its applications. The presenter will demonstrate the calculation of derivatives, higher derivatives, the relationships between functions f, and as well as calculating area enclosed by a curve and axes, area enclosed between curves, volume of revolution for area enclosed by a curve and the axes and those areas enclosed between curves. Participants will have a hands-on experience in learning how technology can be used to reinforce their teaching of calculus concepts in classroom.

### ABSTRACT FOR 21465

#### Discovering Cabri express

AUTHOR: Jean-Jacques DAHAN, Jean-Marie LABORDE
AFFILIATION: IRES of Toulouse, Cabrilog Grenoble France

During this workshop, you will discover this brand new environment of Cabri, free of charge. It allows to work in 2D and 3D always connected to a calculator. We will see what sorts of activities can be conducted with it. Cabri express is a very tiny part of the new Cabri which is a software dedicated to publishers in order to provide multipage and interactive activities.

### ABSTRACT FOR 21466

#### Modelling a submarine in action with Cabri 3D

AUTHOR: Jean-Jacques DAHAN, Jean-Marie LABORDE
AFFILIATION: IRES of Toulouse, Cabrilog Grenoble France

During this workshop, we will model a submarine in action. We will use transformations to perform this modelling. Even beginners can follow it without any problem. You will discover the power and the simplicity of Cabri 3D.

### ABSTRACT FOR 21468

#### Using Graphing Technology To Teach Trigonometry

AUTHOR: Yew Fook CHAN

Participants in the workshop will learn how graphing technology is used by teachers to facilitate students'' learning of trigonometric graphs through exploring and investigating the features of graphs of standard trigonometric functions. The teaching and learning ideas, curriculum content and pedagogical approaches will be presented from the perspective of an International Baccalaureate teacher.

### ABSTRACT FOR 21499

#### 利用Hawgent 皓駿訂制個性化的動態數學軟體

AUTHOR: Chuan-Bo ZUO, Chu-Biao LIN
AFFILIATION: Hawgent Technology Centre in Mathematics, Guicheng Middle School, Foshan, Guangdong, China

### ABSTRACT FOR 21520

#### Programming Fractals in 3D Virtual Reality (Part 1 of 3)

AUTHOR: Andy YEH
AFFILIATION: Queensland University of Technology

In this hands-on workshop, participants will use a Logo programming language to create fractal geometry in an online 3D virtual reality learning environment named VRMath2. VRMath2 is freely available at https://vrmath2.net.

Using the Logo programming language in VRMath2, fractals such as Fern leaves, Trees, Koch curve (snowflake), Sierpinski triangle and carpet, Dragon and Peano curves etc. can be described, experimented, and created using the Logo turtle geometry, and recursive and random capabilities of the programming language. Further, these fractals can be extended to 3D (not to confuse with fractal dimension) in VRMath2's virtual reality interactive space, and presented online in web browsers and/or viewed with Cardboard VR.

Participants will be introduced with a basic structure of recursive function and Logo programming, then they will be able to experiment and invent variations of fractals in 3D. All these will be done online in a web browser. Participants can also publish their fractals online in the VRMath2 website.

In this Workshop, there will be a brief introduction to Logo programing and turtle geometry, then we will focus on creating fractals in the nature, specifically on creating:

1. Fern leaves and variants in 2D and 3D.
2. Trees variants in 2D and 3D.
Examples of Logo programs and 3D fractals can be found at https://vrmath2.net/content/fractals-virtual-reality.

### ABSTRACT FOR 21521

#### Programming Fractals in 3D Virtual Reality (Part 2 of 3)

AUTHOR: Andy YEH
AFFILIATION: Queensland University of Technology

In this hands-on workshop, participants will use a Logo programming language to create fractal geometry in an online 3D virtual reality learning environment named VRMath2. VRMath2 is freely available at https://vrmath2.net.

Using the Logo programming language in VRMath2, fractals such as Fern leaves, Trees, Koch curve (snowflake), Sierpinski triangle and carpet, Dragon and Peano curves etc. can be described, experimented, and created using the Logo turtle geometry, and recursive and random capabilities of the programming language. Further, these fractals can be extended to 3D (not to confuse with fractal dimension) in VRMath2's virtual reality interactive space, and presented online in web browsers and/or viewed with Cardboard VR.

Participants will be introduced with a basic structure of recursive function and Logo programming, then they will be able to experiment and invent variations of fractals in 3D. All these will be done online in a web browser. Participants can also publish their fractals online in the VRMath2 website.

In this Workshop, there will be a very brief introduction to Logo programing and turtle geometry, then we will focus on creating recursive and random fractals of:

1. Koch snowflake, Sierpinski triangle, and variants in 2D and 3D.
2. Sierpinski carpet (2D) and Menger's sponge (3D).
Participated in Workshop 1 or some Logo programming experience. Examples of Logo programs and 3D fractals can be found at https://vrmath2.net/content/fractals-virtual-reality.

### ABSTRACT FOR 21522

#### Programming Fractals in 3D Virtual Reality (Part 3 of 3)

AUTHOR: Andy YEH
AFFILIATION: Queensland University of Technology

In this hands-on workshop, participants will use a Logo programming language to create fractal geometry in an online 3D virtual reality learning environment named VRMath2. VRMath2 is freely available at https://vrmath2.net.

Using the Logo programming language in VRMath2, fractals such as Fern leaves, Trees, Koch curve (snowflake), Sierpinski triangle and carpet, Dragon and Peano curves etc. can be described, experimented, and created using the Logo turtle geometry, and recursive and random capabilities of the programming language. Further, these fractals can be extended to 3D (not to confuse with fractal dimension) in VRMath2's virtual reality interactive space, and presented online in web browsers and/or viewed with Cardboard VR.

Participants will be introduced with a basic structure of recursive function and Logo programming, then they will be able to experiment and invent variations of fractals in 3D. All these will be done online in a web browser. Participants can also publish their fractals online in the VRMath2 website.

In this Workshop, there will be a very brief introduction to Logo programing and turtle geometry, then we will focus on creating recursive and space-filling fractals of:

1. Dragon curves in 2D and 3D.
2. Peano (Hilbert) curves in 2D and 3D.

### ABSTRACT FOR 21535

#### A beginner's tutorial to functional programming via HASKELL

AUTHOR: Weng Kin HOW
AFFILIATION: National Institute of Education, NTU, Singapore

This workshop provides the participants with a hands-on experience of writing simple computer codes in a user-friendly functional language, Haskell. The functional programming paradigm appeals to most people who have some acquaintance with the mathematical concept of functions: every program is a function! It is preferably, but not necessary, that participants have basic knowledge of what a function is. We do not require the participant to possess any background knowledge in writing computer programs; in fact, the less you know about computer programming the better it would be! In this workshop, we demonstrate (1) how computational thinking can be developed and nurtured in the vehicular language of HASKELL, and (2) how mathematics manifests directly in computer science.

Target audience: High school teachers and university instructors of mathematics

### ABSTRACT FOR 21539

#### Learning Fractions the Fun and Magical Way

AUTHOR: Poh Yew TEOH
AFFILIATION: National Association for Gifted Children, Malaysia (NAGCM), Malaysian Association of Professional Speakers (MAPS), Malaysian Invention and Design Society (MINDS)

Fraction and its various operations are very important mathematics concepts for later mathematics achievement but the amount of rules there are to learn about fractions make it difficult for many children. However, there is an effective way to overcome this. Children simply love to have fun, therefore one of the most effective ways to help them engage in learning fraction is through recreational mathematics. In this hands-on and fun-filled workshop, participants will be engaged in exciting activities such as magic tricks, puzzles, art and craft, games on mobile app, etc. These activities may appear to be recreational or magical in nature but they are actually based on sound mathematical concepts related to fraction. The presenter will also share how she develops such huge collection of activities. The participants will them be encourage to come up with their own variations of activities to help children understand fraction more effectively.

Target audience: Primary school mathematics educators.

### ABSTRACT FOR 21548

#### Iterate in GeoGebra and realize by origami

AUTHOR: Chang WENWU, ?? ?
AFFILIATION: PuTuo Modern Educational Technology Center of Shanghai, ????????

During this workshop, we will first show that how a iteration figure can be drawn through the GeoGebra software. Then an origami process will be brought to the audience that arrives to that iterate paper structure. This process will imitate a famous dress designed by Issey Miyake called 132 5.

### ABSTRACT FOR 33001

#### Exploring Technology in the International Baccalaureate

AUTHOR: Christopher LONGHURST
AFFILIATION: Hewlett Packard

The International Baccalaureate has become a very popular program worldwide because of its focus on modelling, investigation and technology in mathematics.In countries like Australia, the IB program is challenging the traditional matriculation programs in mathematics available to schools and students.

In this workshop, I will provide a hands-on view of how to use Technology (Graphing calculator) to enhance teaching and learning in the International Baccalaureate program and other mathematical programs. The focus of the workshop will be using technology in examinations (exam mode); applications of algebra; projectile motion and calculus. Investigative worksheets will be used to consolidate the workshop presentation. At the end of this workshop each participant will be shown how to download a HP Prime graphing calculator emulator for the use in the classroom

# Abstracts for the Track of Poster Sessions

### ABSTRACT FOR 21496

#### Application of mathematics to mathematics for Geometric Construction using by CUI and GUI

AUTHOR: Hideyo MAKISHITA
AFFILIATION: Shibaura Institute of Technology, Japan

When mathematical material is added to geometric construction using rulers and compasses, the use of dynamic geometry (DG) software is one option, whereas ketcindy is used for this study because ketcindy is equipped with DGS Cinderella as GUI (Graphical User Interface) and can be used for drawing figures by Script as CUI (Command Line User Interface). Therefore, mathematically precise figures can be drawn with ease, producing beautiful results. This paper explains figure drawing while the quadratic curve concept is added to geometric construction. The author considers that figure drawing by Script is extremely useful for mathematics education from the viewpoints of application of mathematics to mathematics. This point will be discussed hereinafter.

### ABSTRACT FOR 21501

#### Hawgent 皓駿數學實驗室建設方案及專案案例

AUTHOR: Chuan-Bo ZUO, Hui-Jiao LI, Yan-Dong LIU
AFFILIATION: Chuan-Bo ZUO, Hui-Jiao LI, Yan-Dong LIU

### ABSTRACT FOR 21508

#### 240 Distinct Soma Cubes

AUTHOR: Hui Hsiang CHEN
AFFILIATION: National Tsing Hua University

This poster is to display all 240 distinct Soma Cubes.

### ABSTRACT FOR 21524

#### The effect of instruction by mathematics competence-based grouping for grade eight students

AUTHOR: Pei-Jung HSIEH, Hsin-Ying HUANG
AFFILIATION: National Academy for Educational Research

Previous studies revealed that competency-based instruction might be an effective strategy for improving mathematics achievement. It has been a national education policy in Taiwan since 2009. However, few official documents mentioned about whether ability grouping is a common practice in middle schools. The purpose of this study is to investigate the proportion of ability grouping in eighth grade mathematics instruction and its effect for academic performance. Participants were 8805 grade students from 269 schools by two-stage stratified cluster sampling. The instruction design of the class was answered by teachers from sampling schools. Mathematics performance were measured at the end of the semester by the items from Taiwan Assessment of Student Achievement, which is a nationally representative assessment. 78 multiple choice items and 13 constructed response items were selected in this study. Ten plausible values of mathematics achievement were estimated by R package version 3.4.0. 95% confidence intervals of the group mean with jackknife resampling method were used to test the hypothesis of mean difference. Teachers reported the frequency of ability grouping's use in eighth grade mathematics instruction was 15.80% (1393 students). In contrast to NAEP data, the percent is quite low. Among them, 37.62% students were assigned in between-class and 62.38% students were arranged in within-class ability grouping for mathematics class. In the within-class grouping classrooms, 91.14% was heterogeneous ability grouping and 8.86% was homogeneous ability grouping. At the end of the semester, no statistically significant difference was found between ability and non-ability group. Among the competence-based grouping students, there was no statistically significant difference between within-class and between-class ability grouping. Furthermore, students in homogeneous within-class ability grouping performed better mathematics achievement than heterogeneous within-class ability grouping. The resulting survey found that appropriate ability grouping might help students'' learning. The Ministry of Education could do more effort to arouse teachers from the traditional instruction model.

### ABSTRACT FOR 21532

#### The Path Analysis of Structure, Intrinsic Motivation, and Performance of Polynomial Multiplication in Junior High School

AUTHOR: Hsiu Ju CHANG
AFFILIATION: Department of Education, National Chengchi University, Taipei County Shu-Lin High School

This paper presents the path analysis model among the structures, intrinsic motivation, and performance to support and enhance the perceptible information for learning polynomial multiplication in digital material. In this path model, we wanted to insight the sensory information within learning polynomial multiplication in junior high school. Participants were 51 junior high school learners (grade 8~9, 13~16 years old, 26 males, 25 females). The model analysis is based on partial least squares (PLS) which is an exploration or construction technology to predict the causal model form the latent variables for reasoning and comparing. And the causal model maintains the relationships among the latent variables and constructs. The values of measurement model are Cronbach's alpha (Intrinsic motivation, 0.642615; Structure satisfaction, 0.675581; Performance satisfaction, 0.834461), AVE (Average Variance Extracted) (Intrinsic motivation, 0.59893; Structure satisfaction, 0.754647; Performance satisfaction, 0.754514), and composite reliability (Intrinsic motivation, 0.805702; Structure satisfaction, 0.860139; Performance satisfaction, 0.901407) We find the loading factors are higher than 0.5, the reliabilities are higher than 0.7, and the average variance extracted values are higher than 0.5. And the discriminant validities are verified by the square root of average variance extracted values for each construct. The path analysis (Figure 1) showed that the positive relationships among structures, intrinsic motivation, and performance significantly.

### ABSTRACT FOR 30100

#### Hamiltonian Cycles Associated with Polyhedron

AUTHOR: Sutee PAKAVALEETORN
AFFILIATION: Department of Mathematics, National Tsing Hua University

The improvement in technological tools provides various applications on mathematical research and education. Geometry is one of the areas that gains considerable benefits from the advanced technology. In general, human would perform better when visual aids are presented, since it makes the abstract ideas become more tangible. Historically, William Rowan Hamilton solves Hamilton's puzzle or icosian game by icosian calculus, an associative but not commutative algebraic structure with roots of unity. This paper demonstrates a synthetic Hamiltonian cycle, the solution of the icosian game. This cycle travels on the edge of a polyhedron, passes each vertex once, and returns to the starting point. The same procedure can be used for any type of convex regular polyhedron. By the help of interactive geometry software, Icosian game can be solved without algebra.

(c) Douglas B. Meade, University of South Carolina, USA