Teaching and Learning Mathematics, Science and Engineering through Technology

# Abstracts for Invited and Plenary Papers

### ABSTRACT FOR 21193

#### Can Technology make a difference to school Mathematics Teaching?

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

Douglas is in the fortunate position of having the benefit of 30 years teaching, 20 years developing mathematics software, and 20 years promoting TSM training workshops to help teachers make the most effective use of technology. Despite its obvious advantages, and considerable investment from Education Ministries, the majority of mathematics teachers do not allow technology to enliven their teaching. This is mainly through lack of access, but also lack of training.

At the same time the software and hardware spectrum is changing rapidly, and many teachers simply fail to keep abreast of all the change. The gradual shift from teacher-centric fixed hardware to student-centric mobile technology is a massive challenge to teachers and to students, and also to those developing appropriate software. This presentation will include some lesson plans that maximise the potential of technology using web resources, spreadsheets and dynamic software, and delivering exciting visualisations and engagement.

### ABSTRACT FOR 21209

#### Rhombohedra Everywhere

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

A rhombohedron is a 6-face polyhedron in which all faces are rhombi. The cube is the best-known example of the rhombohedron. We intend to show that other less-known rhombohedra are also abundant. We are to show how the rhombohedra appear in the algorithm of rhombic polyhedral dissections, in designing 3D linkages and in supplying concrete examples in mathematics amusement.

### ABSTRACT FOR 21256

#### Mathematics, secrets and smartphones

AUTHORS: Jose A Vallejo, Lina Rubí Ipiña
AFFILIATION: Universidad Autonoma de San Luis Potosi

It is commonplace that Mathematics are everywhere, and we recurrently mention to our students some examples of this occurrence. One of the most used is the mathematics of cryptography. Of course that is true: modern cryptographic methods are strongly mathematics-based, and there are many excellent texts and web resources explaining at any imaginable level of sophistication their foundations. However, still it is difficult to find resources with a hands-on approach, in which the student can do more than merely academic computations and examples. Taking advantge of the interplay between mathematics, free software and the computational capacity of current smartphones, we will illustrate how these ingredients can be cooked to build an elementary course on number theory and cryptography with a twist: everything will be done in your (Android) smartphone.

### ABSTRACT FOR 21271

#### Exploiting Digital Technologies for Learning Mathematics

AUTHOR: Celia Hoyles, Richard Noss
AFFILIATION: UCL Institute of Education, London, UK

A key challenge for task design in mathematics education and an organising design principle is to enhance engagement with mathematics. One way to achieve this is to exploit digital technology to reveal more of what mathematics actually is; first, by offering a glimpse of the mathematical models underlying a given (and carefully chosen) phenomenon; and second, by fostering an approach to mathematical tasks that transcends the purely procedural. We describe in this paper how we have attempted to address these challenges.

### ABSTRACT FOR 21272

#### Outbox Centroid Theorem: An Episode of Dynamic Geometry Exploration

AUTHOR: Weng Kin Ho
AFFILIATION: National Institute of Education, Singapore

An outbox of a given convex quadrilateral is a rectangle such that each vertex of the quadrilateral lies on one side of the rectangle and all the vertices lie on different sides, with all the sides of the rectangle external to the quadrilateral. This paper reports on a new geometrical result concerning outboxes of convex quadrilateral -- the Outbox Centroid Theorem, and gives a new proof of an existing result of M. F. Mammana. Interestingly, the investigation that leads to this new result comes from dynamic-geometry explorations.

### ABSTRACT FOR 21274

#### From One to Infinity: What DGS Has or Could Have Changed in our Teaching and Learning

AUTHOR: Jean-Jacques Dahan
AFFILIATION: IRES of Toulouse, France

The collaborative work enhanced by the communication tools developed in the last decade is perceived by everybody as a progress for teaching and learning. We will see in this paper that this way of working avoids to use all the knowledge developed by experts especially in the use of the technological tools. We will focus our analysis on « dynamic geometry software » and especially on the Cabri environments. We will remind the special role of DGS in a more experimental way of practicing maths (researching, teaching and learning) sustained by the theoretical framework I have developped in my PhD thesis (different stages of an experimental process of discovery, different levels of techniques of investigations known as praxeologies G1, G1 informatique, G2 and G2 Informatique, the heuristic power of dynamic approach of figures.). We will give a lot of examples showing the new tools of exploration or investigation provided by DGS (such as traces, loci, animation, redefinition, macro, sliders.) to understand the difference between a paper and pencil approach or a DGS approach of a problem (that can be not necessarily a geometric problem). In these examples, we will present different techniques that must be taught in order to be used appropriately by the users of DGS. At last we will present my YouTube channel where lots of situations are provided to help teachers and students to use DGS without ignoring all the work of the experts during the last 30 years.

### ABSTRACT FOR 21279

#### Quadratic and Cubic Polynomials in Applied Problems: Finding Maximum - no Calculus, using CAS (Maple)

AUTHOR: Bill Blyth
AFFILIATION: Australian Scientific & Engineering Solutions, RMIT University, Australia

We use CAS to provide a new algebraic approach in some optimization applications where the objective function (to be minimized or maximized) is a quadratic polynomial. These problems can be solved just by knowing properties of quadratics and so give context to why we want to complete the square. Without calculus, to find the max/min of cubics, we introduce a new straightforward algebraic method (no calculus). The use of a Computer Algebra System, CAS, such as Maple easily deals with any messy algebra! The key property of a function is that, near a maximum or minimum, the function "looks like a quadratic". This visual idea is combined with some straightforward algebra to find this local quadratic approximation of a cubic near the maximum. Traditional "find the maximum ..." problems are introduced to senior school or first year undergraduate students in their calculus course. With our approach, these applied problems and the Polya method of problem solving can be introduced to pre-calculus students. We use the CAS, Maple, for algebra and visualization. We use small group collaborative learning in the computer laboratory, so we parameterize the problem and recommend the use of Computer Aided Assessment (such as provided by the package MapleTA). Students engage with the visualization and algebra, are active learners with deep learning of the concept of maximum and have fun doing so.

### ABSTRACT FOR 21294

#### Challenges in Integrating Technology in Teaching and Learning Mathematics in Basic Education

AUTHOR: Yuriko Yamamoto Baldin
AFFILIATION: Universidade Federal de São Carlos, Brazil

This paper presents considerations about the role of technology in teaching and learning mathematics from the perspective of teacher education programs that is required to prepare new generation of teachers for the potentialities of diverse uses of technology in the teaching mathematics content on one side, and in the learning new classroom dynamics on other side. The reflections retrieve some insightful initiatives and foresights of pioneering researchers about the educational possibilities of technology tools from late 90´s and 2000´s, bring frameworks from researches in mathematics education, and are illustrated with examples from the author´s work about development of teaching materials for prospective and in-service teachers, regarding the contemporary needs of mathematics education. The paper discusses the concept of experimental mathematics as a core concept that connects the learning of mathematics content, at student level, with the learning by teachers of new methods of teaching, at professional level. The examples, not exhaustive, will be given with CAS, DGS, and Calculators. The challenges to integrate effectively the technology in educational context are commented considering the presence and advantages of information and communication technologies that imply necessarily the rethinking of teacher education. The reflections of this paper take into account the power of technology towards the educational needs of developing countries that strive for a quality education.

### ABSTRACT FOR 21297

#### Technology: Inquiry based learning, inverse questions, and control

AUTHOR: Matthias Kawski
AFFILIATION: Arizona State University

New computational tools become available at every faster rates. A fundamental question asks how such tools can help achieve clearly defined learning objectives. This article argues, in a sequence of examples, that computing technologies can much support the implementation of modern pedagogy. The focus is on enhancing a learner centered environment. Of critical importance is that the learner takes the key role of asking further questions, to take ownership of the discovery experience. We highlight the special role of asking inverse questions with examples ranging from math circles to vector calculus.

### ABSTRACT FOR 21298

#### Technology in Mathematics Education: A Stocktake & Crystal-Ball Gazing

AUTHOR: Greg Oates
AFFILIATION: The University of Tasmania

This paper seeks to conduct a stocktake of the current use of technology in mathematics education, and engage in some crystal-ball gazing as to how it might be used in the future. First, it briefly discusses the history of the use of digital technologies in mathematics education, focusing primarily on the period of growth from the mid-1990's. It will consider some of the theoretical perspectives that have emerged over that period, and using the framework developed by the author [x; x], will attempt to describe the 'current state of play' for the effective integration of technology into the teaching and learning of mathematics. Then using this position and framework as a starting point, the paper postulates what might be some significant challenges ahead for teachers and institutions in the continuing search for effective meaning-making in mathematics with technology.

### ABSTRACT FOR 21300

#### Locus, Parametric Equations and Innovative Use of Technological Tools

AUTHOR: Wei-Chi Yang
AFFILIATION: In this paper, we discuss two problems found from Chinese college entrance exam practice problems [8]. We see how original problems in 2D, stated in an exam static and somewhat uninspired setting, can be extended to other interesting cases in 2D and more challenging corresponding problems in 3D for students to explore with the help of a Dynamic Geometry Software (DGS) and a Computer Algebra System (CAS). We use a DGS to construct the locus or locus surface geometrically, and use a CAS to verify our locus or locus surface analytically. We shall see with the innovative use of technological tools, mathematics can be made more fun, accessible, challenging and applicable to broader group of students. Finally, we attempt to make these problems relevant to real-life applications, we invite readers to investigate how these problems can be interpreted differently.

### ABSTRACT FOR 30001

#### Teaching and Learning Mathematics with Mobile Technology

AUTHOR: Ma.Louise Antonette N.De Las Peñas
AFFILIATION: Ateneo de Manila University, Philippines

This talk will discuss a current government funded project in the Philippines on the creation of software applications and applets for mobile technology that will enhance the teaching and learning of mathematics. The Department of Education in the Philippines has recently implemented the K to 12 Basic Education program. The K to 12 curriculum recognizes that calculators, computers, including mobile devices such as smartphones and tablets can be appropriate tools in teaching mathematics. These can help learners demonstrate understanding and appreciation of key concepts and principles of mathematics as applied in problem solving, communicating, reasoning, making connections, representations and discussions in real life (Dep Ed, 2012).

To fully support the implementation of Mathematics Grade 7-10, a comprehensive resource has been created to aid teachers develop their competencies in teaching various strands in mathematics and promote critical thinking and problem solving in the classroom. In this talk we present segments of this resource, which includes interactive software for mobile technology together with a framework for the design of teaching guides, arranged according to competencies prescribed by the curriculum. The extent and manner for which technology can be used in the teaching of mathematics depends not only on the availability of resources (software, internet) but on the teacher's ability and disposition to use the devices. The teaching guides are aimed to optimize the use and benefits of technology. Suggested activities for the students are given which promote student centered learning.

The mobile apps are designed to support student investigations, promote student centered learning and at the same time enhance teaching practices and student engagement. These cover topics in algebra, geometry and statistics. These include apps for proving in Geometry. Related literature studies show the novelty of these proving applications.

# Abstracts for Full Papers

### ABSTRACT FOR 21147

#### Use Technology in Secondary Mathematics Teaching: Preparing Teachers for the Future

AUTHOR: Munindra Mazumdar
AFFILIATION: MPS College, Guwahati, India

Technology is an essential tool for learning mathematics in the 21st century and all Modern school classrooms have access to a range of potential technologies, ranging from calculators and computers to the Internet. This paper examines the use of Information and Communication Technology (ICT) for supporting the teaching and learning of transformations of functions in terms of linear, quadratic, cubic and trigonometry in the secondary school; and identifies some of the ways in which the teaching of functions might be supported by the availability of various forms of technology. An exam was applied and performance comparison between computer and paper-and-pencil groups. Modern technology provides an excellent mean of exploring many of the concepts associated with functions. This paper suggests some of the avenues for exploration.

### ABSTRACT FOR 21160

#### Homothetic centers of three circles and their three-dimensional applications

AUTHORS: Yoichi Maeda
AFFILIATION: Tokai University, Japan

In this paper, we recall the famous Mongef's theorem of three circles. There are several proofs for the theorem. One of the proofs is that of using three similar right cones. Inspired by the proof, we propose a three-dimensional problem of the same angles of elevation: For three similar right cones on the ground, find the places from where three angles of elevation are equal to each other. There are at most two places. With dynamic geometry software, we can simply construct the solutions. In addition, the relation between two solutions is cleared.

### ABSTRACT FOR 21179

#### Two examples on discrepancy potentials of dynamic geometry software

AUTHORS: Yip Cheung Chan
AFFILIATION: Department of Curriculum and Instruction, The Chinese University of Hong Kong

It is unavoidable that a computer program has its constraints. In a mathematics pedagogical program such as dynamic geometry software (DGS), the constraints may lead to deviations of mathematical representations from its intended mathematical concepts. If it is used properly, the deviations can open up pedagogical opportunities and hence may be called as discrepancy potentials of the tool. In this paper, examples of developing mathematical discussions by using the discrepancy potentials in two DGS tasks are discussed. It aims at elaborating the idea of discrepancy potentials in the design principles for effective DGS-based tasks. This paper relates to the topic ¡§Mathematics Education using Information Communication Technology.

### ABSTRACT FOR 21183

#### Connection Between Multiplication and Division of Fraction Using GeoGebra

AUTHORS: Boo Jia Yi, Leong Kwan Eu
AFFILIATION: University of Malaya

Several studies had shown that fraction multiplication and division were the most challenging concepts to teach at the primary level. Previous research also found that elementary pupils lacked conceptual understanding when learning fractions. The purpose of this study was to identify Year Six pupils' understanding of the connection between multiplication and division of fraction using GeoGebra. The theoretical framework of this study was based on the radical constructivism theory of integrating mental image, representation, interpretation and problem solving about fraction multiplication and division using GeoGebra. The case study research design was utilized and intended to support pupils building their own understanding among the connection between multiplication and division of fraction. Data for this study includes both verbal and non-verbal behaviors were gathered from seven pupils based on five clinical interview sessions and used to collect data to answer the research questions. The findings revealed that (1) pupils performed better after using mental image with GeoGebra, (2) pupils were able to construct visual representation with GeoGebra to explain its connection between two operations, (3) pupils were able to give meaning about the connection between multiplication and division of fraction after using GeoGebra and (4) pupils faced difficulty in solving problems of multiplication and division with the use of GeoGebra. Interestingly, this study would benefit primary school teachers' teaching and learning methods on the conceptual understanding of multiplication and division of fraction. This study would also further develop pupils' conceptual understanding of fractions using GeoGebra.

### ABSTRACT FOR 21191

#### The Courselab VLE: Its Effect on the Achievement and Students' Perception Toward Learning Algebra

AUTHOR:<\strong> Carmelita Ragasa
AFFILIATION: University of the East Manila, National Research Council of the Philippines, MTAP-TL, Philippine E-Learning Society

Courselab, a Virtual Learning Environment was used to design classes in algebra. Its effect on the achievement of the students was tested. Pretest and posttest were given to the experimental and control groups. The findings showed that there was no significant difference in the achievement of students taught with lecture and the use of Courselab. A survey questionnaire showed that majority of the students disagree that they learned better using computer based materials. Seventy two percent agree that they learned better by reading a good textbook and with face to face discussion. Seventy seven percent agree that they learned better by listening and by reading and rereading materials. The perception of the students on the effectiveness of Courselab in learning college algebra was not favorable. Fifty seven percent disagree that it was clear how Courselab modules fitted into learning algebra and that they also disagree that the scoring system in Courselab was useful for students in preparing their subject's test. On the other hand, half of the students think that they can easily understand and deal with the interface and more than half think that the platform tools were easy for students to use. This means that Courselab was not the problem but it was learning algebra in Courselab where the difficulty was.

### ABSTRACT FOR 21192

#### Introducing MATLAB into High School Mathematics

AUTHORS: Ian Allan Thomson
AFFILIATION: Ormiston College, Australia

In order to solve cubic equations by Euclidean means, the standard ruler and compass construction tools are insufficient, as was demonstrated by Pierre Wantzel iin the 19th century. However, the ancient Greek mathematicians also used another construction method, the "neusis", which was a straightedge with two marked points. We show in this article how a neusis construction can be implemented using dynamic geometry software, and give some examples of its use.

### ABSTRACT FOR 21196

#### Using Dynamic Geometry Software to Enhance Student Understanding of the Concept of Speed

AUTHOR: Wee Leng Ng, Foo Him Ho
AFFILIATION: National Institute of Education, Nanyang Technological University, Ministry of Education, Singapore

The authors of this paper have conceptualized and constructed a tool, using dynamic geometry software, to simulate scenarios of word problems involving speed, and used it to help students acquire the concept of speed and solve speed problems involving two objects moving toward each other. This paper demonstrates how such a tool can be created using GeoGebra, and describes how the use of this tool had improved understanding of speed concept, and ability to solve speed problems for a class of sixth grade students in Singapore.

### ABSTRACT FOR 21198

#### On Simple Representation of Locally Closed Sets

AUTHORS: Yosuke Sato, Ryoya Fukasaku, Katsusuke Nabeshima
AFFILIATION: Tokyo University of Science, Tokushima University

A locally closed set of an affine space is defined as a difference of two varieties. Simple representation of locally closed sets is of great importance in many areas of computational mathematics. We presents a practical simplification algorithm of locally closed sets. Our algorithm consists only of computations of Groebner bases, it does not use any heavy computation of polynomial ideals such as a primary decomposition.

### ABSTRACT FOR 21201

#### Finding the signature matrix of minimizing the Cayley transform by using Computer Algebra

AUTHORS: Dexuan Zhang, Yongbin Li, Haocheng Zhou
AFFILIATION: School of Mathematical Sciences, University of Electronic Science and Technology of China, School of Mathematical Sciences, University of Electronic Science and Technology of China

Given an orthogonal matrix Q, we can choose a diagonal matrix D with diagonal entries such that I+QD is nonsingular and then that the Cayley transform (QD)=(I+QD)(I-QD)^(-1) is well defined. Evan O''Dorney has proven the existence of the diagonal matrix $D$ with diagonal entries plus or minus 1 (called a signature matrix) to make sure every entry of QD is less than or equal to 1 in absolute value. The remaining question is how to compute D directly. In this paper, we present a method for computing the signature matrix D based upon Grobner basis and Real-Root-Classification in the case of $n=2$. Our approach is helpful to develop the interest of learning computer algebra and using computer algebra systems in researching.

### ABSTRACT FOR 21202

#### Impact of multimedia in Teaching Mathematics

AUTHORS: Haftamu Menker Gebreyohannes, Abdul Hadi Bhatti
AFFILIATION: Middle East College, Oman

In this paper, we are going to examine issues within the new frontier of integrating technology into mathematics education. We present an approach on how to teach mathematics courses by integrating meaningful multimedia technology to foster the learning process. Specifically, this paper focuses on how the integration of multimedia based teaching approach into a Calculus and Numerical Methods module impact on student's performance and their attitudes toward educational technology. Empirical data will be collected from controlled and experimental group students enrolled into this mathematics module which include students' engagement using traditional and multimedia technology teaching and learning process.

### ABSTRACT FOR 21203

#### Mathematical Regression Modeling For Smart Environmental Weather forecasting

AUTHOR: Haftamu Menker Gebreyohannes
AFFILIATION: Middle East College, Oman

Environmental weather forecasts have been increase the city's ability to be self-sustaining and lowering environmental impact as well as metrological support for critical decisions as climate change brings more extreme weather events. To predict weather meteorologists form models based on the land's geography and starting weather conditions and they can calculate future forecast by entering it into a computer to be processed. In this paper we will describe a survey of Mathematical techniques and Practices used to model weather forecasting today..

### ABSTRACT FOR 21204

#### Hexagrammums

AFFILIATION: University of Technology-Papua New Guinea, Deoma-Russia

In this paper we consider hexagrammums, plane geometric configurations based on six points. Pairs of given points form straight lines. Points of intersection of these lines form the new daughterly points. We get the net configuration of points and lines. The work is based on firstly, delightful drawings of Hirotaka Ebisui [1], and secondly, the GInMA software [2], which makes it easy to explore the geometric configurations and exercise their conversions. Typically the only one unusual point found by Ebisui has served the source for the study. As a result, the net has been found. In each case, the method of barycentric coordinates has been used for the formal proof. The evident solutions have been found in many cases with the use of collineations. This may introduce solutions of problems into standard courses of geometry. All the pictures in the paper are interactive. So they come to life, install on your computer GInMA software from the website [2]. Free basic version will allow you to get acquainted with all the materials, to conduct a study, to create your file, but not to keep it. See the video How to convert pictures from the text in interactive drawings.

### ABSTRACT FOR 21205

#### Enhancement of Plotting Environment of STACK with Gnuplot

AUTHORS: Kenji Fukazawa, Yasuyuki Nakamura
AFFILIATION: National Institute of Technology, Kure College, Graduate School of Information Science, Nagoya University, Japan

In this paper, we explain our approach to building an enhanced plotting environment in which, for example, graphs of implicit functions and surfaces described by functions containing two variables could be visualized. This enhancement is achieved by creating Maxima functions, which produce the appropriate scripts of gnuplot, which is then called with these scripts. This approach enables us to make full use of the power of gnuplot, and can be applied to any plotting software other than gnuplot, such as matplotlib, and Pyxplot. Several examples of questions with enhanced plots are presented.

### ABSTRACT FOR 21206

#### A Study of gamification on GeoGebra for remedial pupils in primary mathematics

AUTHORS: KAI CEAT LIM, KWAN EU LEONG
AFFILIATION: FACULTY EDUCATION, UNIVERSITY MALAYA

Remedial pupils are often more engaged in their learning when the content delivered uses games-based-activities. This study integrated gamification using technology in learning mathematics which is one of the 21st century learning approaches. In this paper we report the results of a study on lessons using 2 learning games based on GeoGebra with reference to remedial pupils' motivation in learning primary mathematics. The respondents consisted of 4 remedial pupils from a primary school in Selangor, Malaysia. Data were obtained through observation on lessons and questionnaire on students' perception in gamification on GeoGebra. Observation by using video recording indicated that the sample showed high motivation in learning mathematics. Questionnaire result also showed students' positive interest in using gamification on GeoGebra. In conclusion, this study found that using gamification on GeoGebra is highly effective in helping remedial pupils to learn primary mathematics besides increasing their motivation in learning. This study strongly recommends the implementation of gamification on GeoGebra to facilitate mathematics learning at the primary level especially in geometry.

### ABSTRACT FOR 21210

#### IMPACT OF USING TINKERPLOTS IN STATISTICAL REASONING

AUTHORS: Leong Kwan Eu, Pavethira Seloraji
AFFILIATION: Tenby Schools Setia Eco Park, Universiti Malaya

In this 21st century of learning, students are not only required to know how to do the calculation procedures but more importantly they must be able to justify their solutions by reasoning. This is particularly important in the teaching and learning of statistics in schools. This study aims at investigating the impact of using TinkerPlots in statistical reasoning among Year Five pupils. The research was conducted in an international school. The research utilized the quasi experimental research design. Two intact classrooms were selected with one classroom as the control group while the other was the experimental group. The experimental group went through intervention using TinkerPlots whereas the control group went through the traditional method. The research conducted pre- test and post-test for both the groups. The findings show that a significant difference existed between pre-test and post-test for the experimental group. In conclusion, the students' statistical reasoning results improved using TinkerPlots. The study implies that using TinkerPlots enhances students' reasoning skills. Implementing teaching and learning statistics using TinkerPlots would help students explore data, concepts and test their conjectures. This would support the student-centered learning where teachers facilitate the students to develop their ideas and knowledge.

### ABSTRACT FOR 21211

#### Exploring generalizations of a result about cubic polynomials

AUTHORS: Alasdair McAndrew
AFFILIATION: Victoria University, Melbourne Australia

For a real cubic function with three distinct roots, the tangent at the mean of any two of them passes through the third. This elegant and elementary result seems to have first surfaced as an investigation for students studying the International Baccalaureate. There are number of ways for generalizing this result; in this article we look at two of them: higher degree polynomials, and polynomials over the complex numbers, and the quaternions. Thus we show how the result can be defined in considerable generality over general domains. This may be considered as a case study in the use of a CAS to generalize a simple result.

### ABSTRACT FOR 21216

#### Are Mathematics Students' Learning Styles Related to Their Preferred Method of Learning How to Use Advanced Calculators?

AUTHORS: Hazel Tan
AFFILIATION: Monash University, Australia

One of the learning styles models used for investigating students' preferences is based on the Visual, Aural, Read-Write or Kinesthetic (VARK) modalities for receiving information. This paper presents the research findings of a survey study on Singaporean and Australian students, contrasting students' VARK preferences with their instructional learning preferences when using graphing calculators and calculators with computer algebra systems (advanced calculators). Students filled in an adapted 7-item instrument about their VARK preferences, and were also asked to indicate their most preferred instructional method of learning how to use the calculators to solve mathematics problems. It was found that students generally preferred visual and kinesthetic instructional methods when learning how to use calculators, regardless of their VARK preferences. Results also show that there were regional differences in students' VARK preferences. Overall, the results suggest that students adapt their learning preferences to different contexts, and that advanced calculators lend themselves to visual and kinesthetic modalities of learning.

### ABSTRACT FOR 21217

#### Teaching methodology for smart students with less-developed abstract thinking skills

AFFILIATION: University of Technology, Deoma

The author has been teaching statistics to students in Papua New Guinea for three semesters. These students are smart but their mathematical knowledge and abstract thinking skills require considerable development. The author is forced to build teaching on the basis of binding mathematical concepts with objects of the real world. The subject of the paper are examples of realizing this idea.

### ABSTRACT FOR 21219

#### Parallel and Distributed Boolean Grobner Bases Computation in SageMath

AUTHORS: Akira NAGAI, Yosuke SATO
AFFILIATION: Tokyo University of Science, NTT Secure Platform Laboratories, Japan

We introduce a parallel and distributed computation method of Boolean Grobner bases, which are implemented in the computer algebra system SageMath using the PolyBoRi library. We present an easy way to parallelize Boolean Grobner bases computation using a decorator for parallel computation supported in SageMath, we also present a way of distributed computation using a multiprocessing module in Python. Our software achieves satisfactory speed-up comparing to our sequential implementation in SageMath.

### ABSTRACT FOR 21222

#### Exploring Derivative Functions using HP Prime

AUTHORS: Betty, Wan Niu Voon, Ling Shing Wong
AFFILIATION: Universiti Tenaga Nasional, Faculty of Health and Life Sciences, INTI International University, Persiaran Perdana BBN, Putra Nilai, 71800 Nilai, Negeri Sembilan, Malaysia

Derivative is one of the essential topics in calculus, which is an important basic skill should be mastered before integration comes in. Learning derivative in higher education can be achieved by using interesting graphing-calculator based learning. In this paper, activities using HP Prime to explore derivative function are outlined. The usage of Function and Statistics 2Var applets are emphasized and used in these activities. Through these activities, students are able to learn how to sketch the graph of a derivative from its original function, through observation on the changes of slopes and tangent lines in HP Prime.

### ABSTRACT FOR 21227

#### Teaching and Learning of Geometry in Primary School Using GeoGebra

AUTHORS: Boo Jia Yi, Leong Kwan Eu
AFFILIATION: University of Malaya, University Malaya

The purpose of this article is to discuss how GeoGebra can be used to teach the concept of angle in Geometry in elementary level. This result is obtained after 2 weeks of lesson exploration. Teachers used GeoGebra as a teaching tool to make the lesson more creative and innovative in order to show how geometrical shapes relate with different angles in different polygons. For students, they can use GeoGebra to construct, drag or apply the actual shape instead of drawing on a piece of paper. Besides that, all the works created by students can be saved as documents for future reference. At the end of two weeks' exploration, pupils were asked to answer survey question regarding their experience with using GeoGebra. Analysis of the survey showed that pupils are able to express their geometric imagination and understanding of mathematical concepts before and after the exploration. Hence, using GeoGebra can make the classroom lesson more enjoyable and interesting.

### ABSTRACT FOR 21261

#### Solving Sudoku Puzzle by Evolutionary Algorithm

AUTHORS: Kedar Nath Das, Sumit Bhatia, Shubhin Puri, Kusum Deep
AFFILIATION: Assistant Professor, Department of Mathematics, NIT Silchar, Assam, India, Department of Computer Science and Engineering, The Pennsylvania State University, USA, Department of Chemical Engineering, IIT Roorkee, India

"Sudoku" means "Single number". "Sudoku Puzzle" is an interesting and popular Japanese game, where the non-givens need to be filled by a single number (from 1 to 9) provided no repetition occur in the corresponding rows, columns and sub-squares. Solving a Sudoku puzzle is challenging due to its easy rules and difficult inherent phenomenon. Although a number of approaches exist for solving a given Sudoku puzzle, it becomes a challenge among the researchers to solve it by using evolutionary algorithms. In this paper, a Retrievable Genetic Algorithm is proposed to solve a given Sudoku puzzle. A new fitness function is designed with puzzle-character-dependent constraints. The Genetic Algorithm is made "Retrievable", since the population is reinitialized after a certain number of generations in order to escape from the premature convergence or from being trapped in the local minima. A set of 9 sample puzzles of different difficulty label have been considered for comparison. The superiority of Ret-GA is ensured from the comparative results and discussions.

# Papers with Abstract Only

### ABSTRACT FOR 21153

#### Modeling with Netlogo

AUTHOR: Crista Arangala
AFFILIATION: Elon University, USA

Netlogo is a free software in which can be used to learn agent based modeling. In addition, there is a package for importing data from Netlogo into Mathematica so students can then explore the similarities and differences between agent based models and traditional differential equation models. This presentation will give an overview of how to introduce Netlogo programming in the classroom, examples of predator prey and SIR agent based models developed by students, and how this technology inspires further study in applied mathematics.

### ABSTRACT FOR 21154

#### THE COMBINED EFFECT OF OUTCOME-BASED EDUCATION (OBE) AND BLENDED LEARNING ON THE STUDENTS PERFORMANCE IN GENERAL MATHEMATICS

AUTHOR: Restituto Rodelas
AFFILIATION: Andres Bonifacio Integrated School, Philippines

The need to improve student learning in response to the increasing demands of globalization prompted studies on and pilot testing of alternatives to the traditional approach to delivering education. This study evaluated the effectiveness of the combined use of blended learning and outcomes-based education in Grade 11 General Mathematics classes at Andres Bonifacio Integrated School in the first semester of AY 2016 - 2017. Each consisting of two classes, the control group was handled using blended learning while the experimental group was handled using blended learning and outcomes-based education.

Blended learning consisted of 4 face-to-face classes and one online module per week. With outcomes-based education, various instructional strategies, class requirements, and assessments were used to ensure that the students were able to do the learning outcomes. The same test was given as pretest and posttest and the posttest-pretest score difference was used to measure the student learning. All results were statistically significant at a significance level of 0.05. The groups were equal in prior knowledge on the topics. There was positive learning in each group, but both classes in the experimental group had greater score difference than those in the control group. Based on this, the combined blended learning and outcomes-based education had positive impact on student learning.

### ABSTRACT 21166

#### Exploring Mathematics with Technology

AUTHOR: Maree Skillen
AFFILIATION: UTS:Insearch, Australia

Technology has been referred to as having an integral role in the reform of mathematics education, and is rapidly changing how we teach and how we learn. Educationalists are required to support changes in mathematical content and instructional methods, as well as incorporating and supporting changes in technology. A new approach to instructional design is needed to combine and integrate these changes so that technology can be utilised to allow students to explore and discover mathematical concepts through appropriate computer applications.

Recent trends emphasise the importance of learning with technology, rather than learning about technology. Computers should be viewed as a cognitive tool to enhance student learning of content material, rather than a tool which allows an individual to acquire isolated skills in basic applications or specific programming languages.

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 21168

#### Enriching the Teaching and Learning of Linear and Non-linear Systems of Equations through Developing a Module using HP Prime Graphing Calculator

AUTHOR: Loreto Delizo, Joanne Olos
AFFILIATION: Saint Pedro Poveda College, HP Calculators, Saint Pedro Poveda College HP Calculators

In response to the needs of the 21st century learners, efforts have been made to integrate technology in the High School Mathematics classroom. Various ways of teaching and learning linear and non-linear systems of equations have been done, which can be further deepened with the aid of graphing calculators. This paper explores the development of a module on linear and non-linear systems of equations, which aims to enhance the teaching-learning experience using the HP Prime. The teacher-made module intends to develop students' better understanding of the concepts involved in the given systems in order to solve real-life applications.

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 21169

#### The Occurrence of Passive Intermodulation and Troubleshooting in Thailand Mobile Industry

AFFILIATION: Faculty of Industrial Technology, Rambhai Barni Rajabhat University, TEST & MEASUREMENT CO. , LTD, Thailand

This paper describes the problem of the passive intermodulation (PIM) in mobile communications. We focus the problem occur between the transmitter and antennas. The paper start from the theory of intermodulation, mathematics concept and effect in mobile channels. The problem occurs when two or more signals frequencies are transmitted at the same time in the same passive device. The non-linear behavior produces spurious signals where frequencies are linear combinations of the frequencies of the original signals. Then we measure PIM level and find the problem solving methods. Finally, we got the PIM level that can be acceptable due to IEC 62037 standard.

### ABSTRACT FOR 21176

#### Improving Primary Mathematics Teaching and Learning with ORIGO Stepping Stones Online Program

AUTHOR: Korakot Chonvichit

This study investigated the impact of ORIGO Stepping Stones online program on the teaching and learning of mathematics at Dara Academy Chiang Mai, Thailand. The targeted population consisted of 400 students and 12 teachers in grade 2. To assess the effects of the intervention, a mix of classroom observation; semi-structured interviews with the mathematics teacher; and students' artifacts were used. The ten-month intervention of ORIGO Stepping Stones consisted of Lesson Study activities, and Mathematics Professional Leaning workshops. Even though this study was limited in duration and scope, the outcomes clearly support earlier research on positive effects of ORIGO Stepping Stones on teachers' mathematical content knowledge and student motivation. The study revealed that the students had positive attitudes toward Stepping Stones online program with increases in motivational intensity as well as their desire to learn the mathematics. The outcomes also showed some obstacles in implementing electronic and online program in Thai school with reference to both the teacher and the students. Based on these findings, some recommendations for better outcomes are discussed and a suggestion is included for future research.

### ABSTRACT FOR 21213

#### On the Visualizing Activities of Analytic Expressions of Lines and Circles via DGS

AUTHOR: Minshik Cho, Minji Kim
AFFILIATION: Department of Mathematics, Korea National University of Education

The equations of straight line and circle are one of the most important contents of Korean high school mathematics courses. They are expressed analytically as the solutions of equations in the coordinate plane. Hence students mostly learn the topic only practicing and calculating equations. Some topics include parameters meaning pencil of lines like kx+k+y=0. Since they do not mention the geometric meaning of parameters of straight line and circle equations in Korean textbooks, many students don¡¯t understand involved geometric meanings in analytic problems.

The purpose of this study is to investigate, through case study, how to apply DGS(Dynamic Geometry Software) and to find out effective methods for learning equations of straight line and circle with technological assistance. We developed some problems in order to investigate students'' level about connecting algebra with geometry. Only a few students could solve those problems analytically. Many students could not even understand written solutions. One of the reason is that the lack of understanding on the relation between analytic expression and its geometric meaning.

Learning worksheets including DGS activities were designed so that student's understanding of the straight lines and circles can be improved. During the class with DGS activities, students could correct their misconceptions and understand the geometric meanings of parameters of straight line and circle equations.

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 21218

#### Fostering a Teaching Culture with Technology in Mathematics

AUTHOR: Maree Skillen
AFFILIATION: UTS:Insearch, Australia

With education increasingly moving towards technology enhanced delivery we need to identify and use alternative approaches to teaching and learning to support our students. To sustain the engagement of students in mathematics classrooms, teachers are employing a pedagogical design capacity. This involves the mobilisation of mathematics and embedding of real-time assessments to transform the learning and application of concepts by students. Performance levels in Mathematics subjects have improved, and teachers have noted increased participation and commitment to learning by students.

This presentation will provide examples of teaching and learning approaches where technology has been incorporated to ensure that student-centred technology-enabled learning is occurring at UTS: INSEARCH, a pathways provider preparing students from diverse language and learning backgrounds for study in Australia.

### ABSTRACT FOR 21220

#### Development Students'' Understanding the Formula of the Surface Area of Sphere through the SAK hands-on Activity

AUTHOR: Cherdsak Pakdeeviroch
AFFILIATION: King's College, Samphran, Nakhon Pathom, Thailand, Institute for Innovative Learning, Mahidol University, Thailand

Sphere is a three-dimensional geometric shape of which its formula of a surface area is difficult for students to understand. In this study, students in grade 9 were allowed to participate with the Surface Area Kit (SAK) as a hands-on activity integrating the POE strategy. The dynamic mathematics software GeoGebra was applied as learning tool and representative of the concrete entities. The results showed that student' developed understanding about the meaning of the surface area of the sphere formula and there are significantly different understanding between the students who did and did not study with SAK hands-on activity. Moreover, the three processes about the students' understanding gains were also identified and discussed.

### ABSTRACT FOR 21221

#### Project Based Learning in Science with Graphing Calculator Coupled Portable Sensors

AUTHOR: Wong Ling Shing, Voon Betty Wan Niu
AFFILIATION: INTI International University, Malaysia, Universiti Tenaga National, Malaysia

Project based learning allows students to participate in the learning process, by allowing them to construct questions, draw hypotheses, find explanations, and discuss their idea collaboratively, which all together polish the students' thinking competency. Through simple graphing calculator coupled portable sensors, project based learning can be carried out in teaching biology related topics. In this paper, an example of project based learning in biology is reported. The students were given a few portable sensors, including light, temperature, pressure, and pH sensors which could be coupled with graphing calculator to collect data on a life culture of living algae. The possible designs from of the experiment, and the benefit of conducting the project for students are highlighted.

### ABSTRACT FOR 21223

#### Pre-service Teachers'' Designing of Mathematics Activities Integrated the Use of the Geometer's Sketchpad Software

AUTHOR: Sasiwan Maluangnont, Pilaluck Thongtip, Teerawit Tinprapa, Poonyapon Chanfoy
AFFILIATION: The Institute for the Promotion of Teaching Science and Technology, Suan Sunandha Rajabhat University

The purpose of this study is to investigate how pre-service mathematics teachers could design activities that used the Geometer's Sketchpad software to teach mathematics. This study was conducted with a group of 20 pre-service mathematics teachers who enrolled in a mathematics teacher preparation program in Thailand. This mathematics preparation program provided a course for pre-service teachers to experience the use of technology, including the Geometer's Sketchpad, in mathematics classroom teaching. As pre-service mathematics teachers learned to use the Geometer's Sketchpad software, they were provided opportunities to experience a set of mathematics activities that used the Geometer's Sketchpad software to enhance mathematical learning. At the end of the intervention, pre-service mathematics teachers had to design mathematics activities that integrated the use of the Geometer's Sketchpad. Results of this study showed that pre-service mathematics teachers, who learned to use the Geometer's Sketchpad software and experienced the mathematics activities that integrated the use of the Geometer's Sketchpad software, were able to design mathematics activities that used the Geometer's Sketchpad software to enhance students' mathematical learning. Besides the activities in geometry, the pre-service mathematics teachers were able to design mathematics activities in other mathematical areas such as numbers and algebra.

### ABSTRACT FOR 21229

#### STEM in Mathematics

AUTHOR: Thomas Yeo
AFFILIATION: Texas Instruments

With the increasing popularity of STEM education around the world, educators are now more interested in how a STEM activity or lesson looks like. In this session, participants will hear the perspective of a Math educator implementing a STEM activity and how can STEM education help teach Math more effectively.

### ABSTRACT FOR 21253

#### The Art of Using Modern Communication Technologies in Mathematics Teaching-Learning Process: Facebook and Text-Messaging

AUTHOR: Rodulfo T. Aunzo, Jr.
AFFILIATION: Faculty, USC Cebu Philippines

Communication is vital tool towards progress. Nowadays, varied mode of communications have surfaced even in the educational arena. A lot ot 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 21270

#### Multiple Language Converter AUTHOR: Joe Marlou OpellaAFFILIATION: Cavite State University - Carmona Campus, Philippines The developed Programming Language Converter serves as an important tool to students, researchers, professors and to anyone who wants to learn different programming languages because it allows various programming constructs to be differentiated. It is a standalone programming platform that allows user to create, edit, and compile three different programming languages. It can convert Java to C++ programming source code and C++ to C programming source code and vice versa. For easy and fast construction of program source code an auto help function that lists down correct suggested syntax is designed which also contains program samples in data structures. A menu bar is included for the user to easily create, open and save file and a help tool that guides the users on how to use the programming platform. It can be run on Windows based operating system not older than Windows 7. The software was developed using Python 64 bit 3.4.1.1 as the main programming language and Adobe Photoshop CS6 for design and stitching. A hardware specification of at least dual core processor and 2GB RAM for faster and efficient performance are required. The developed software was assessed according to Technological University of the Philippines (TUP) instrument on software evaluation and obtained a mean rating of 4.31 with an adjectival description of very good which signifies that the platform performs according to the desired functionality and specifications for a helpful programming environment. ### ABSTRACT FOR 21276 Knowledge for early childhood educators to facilitate children's mathematics learning under the influence of technology AUTHOR: Kam Ling Lao AFFILIATION: Open University of Hong Kong Teacher knowledge is always one of the research focuses in education. Shulman drew our attention to the importance of pedagogical content knowledge (PCK) (1986). Ball, Thames and Phelps (2008) attempted to refine Shulman's proposed categories and introduced the concept of Mathematical Knowledge for Teaching (MKT). In response to the rapid growth of technology, teachers' knowledge in technology integration is also a concern. Based on Shulman's category, Koehler and Mishra (2009) put forward a unique construct of teacher knowledge, technological pedagogical content knowledge (TPACK) and research related to the TPACK framework has been proliferating. Since learning and teaching of mathematics in primary and secondary education are different from those kindergartens, the two teacher knowledge frameworks may not applicable to early childhood education. A teacher knowledge framework with consideration of the unique early childhood context is necessary for and beneficial to the developmental needs of teacher education. In this presentation, a research project aims at investigating the knowledge early childhood educators require to develop children's mathematics concepts under the influence of technology will be presented. This project modifies Ball's (2008) bottom-up approach in framework development. In this session, the preliminary teacher knowledge framework (MtEceK framework) and its development process will also be introduced. Acknowledgements: The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project Reference No. UGC/FDS16/H06/15). ### ABSTRACT FOR 21286 Online Interactive Visualization Tools for Learning Linear Algebra AUTHOR: Victor Tan, Bao Rong Cham AFFILIATION: National University of Singapore, Singapore Mathematical Society Linear algebra is a subject that lends itself well to visualisation, through its relationships with objects in the 2-dimensional and 3-dimensional world, such as lines, planes, points and vectors. In many tertiary textbooks covering the subjects, there is also an emphasis on helping students to visualise the geometrical properties of Linear Algebra, especially the more abstract concepts such as vector spaces, linear independence, basis, span and dimension. Traditional ways of visualisation are limited to the 2-dimensional space. This may pose difficulties in students' visualisation of 3-dimensional representations on the 2-dimensional space. There are many visualisation tools for linear algebra concepts being developed and available on the internet. However, many of them are not interactive. This limits the effectiveness on the learning of the subject. This paper aims to introduce a series of online interactive visualisation tools that we have developed. We also describe a preliminary study on the use of these tools in a first undergraduate linear algebra course. ### ABSTRACT FOR 21292 PERFORMANCE-BASED LEARNING EXPERIENCES ALGEBRA 1: INPUTS FOR THE ENHANCEMENT OF THE K-12 GRADE 7 TEACHING GUIDE AUTHOR: Maris Lasco AFFILIATION: Couples for Christ Member, Mathematics Society of the Philippines-Caraga Chapter Member This study sought to determine the performance-based learning experiences and achievement in Elementary Algebra 1 of the Grade VII freshmen. The said study was conducted during the second quarter or grading period in. The descriptive method of research was used as it looked into the profile of the respondents in terms of age, Math VI grade and English VI grade. The achievement of the students in the learning experiences such as board work, seat work, group activity, oral recitation, journal writing and reporting were gathered to determine possible effects on the performance and achievement of the respondents in the different domains of learning which are the knowledge, process, understanding and performance in Elementary Algebra I. The statistical tools used in this research were frequency, percentage, arithmetic mean, correlation analysis and partial correlation. The personal profile of the students in terms of age shows that majority are aged 12 years old the typical age. The majority of the respondents Math VI and English VI obtained the very satisfactory level performance. The data on the students' performance in board work, seat work, group activity, oral recitation, journal writing and math reporting activities showed that majority achieved a very satisfactory performance rating. The data on the level of performance of the respondents in Elementary Algebra indicated in the data that majority of the student-respondents obtained a very satisfactory performance in the knowledge domain of learning but poor in the over-all performance. Oral recitation, group activity, board work and seatwork were found to be highly correlated with the students' performance in Elementary Algebra 1. When Math grade was held constant the degree of relationship was reduced such that only group activity and board work activities remained to have a high correlation with the students' performance in Elementary Algebra 1. The result conveyed the scenario that makes it imperative to formulate an enhancement of the current K-12 Teaching Guide for teachers handling Grade 7 mathematics. ### ABSTRACT FOR 21293 Applying math software to the real world AUTHOR: Ogose Shigeki AFFILIATION: High School Attached to Northwest Normal University Unlike main stream subjects- like algebra, analysis, or geometry, probability & statistics have a greater potential to appeal to general public. But, even then students often in the class room think those subjects have nothing to do with their real life. This time I will use Mathematica & Geogebra to help students visualize this connection and have them consider probability & statistics as useful subjects to learn. ### ABSTRACT FOR 21295 PUPILS' GEOMETRICAL ACHIEVEMENT USING VAN HIELE''S PHASE BASED INSTRUCTION AUTHOR: Kwan Eu Leong, Geik Tieng Poh AFFILIATION: University of Malaya It is pertinent to develop pupils geometrical understanding at the primary level. Using dynamical software like Geometers'' Sketchpad would not only help pupils'' visualise the shapes but also improve their understanding in geometry. This study investigated the impact of Van Hiele''s phase-based instruction using Geometer's Sketchpad on pupils' geometrical achievement in angles. This study utilised a quasi experimental research design. The participants were 73 Year Four pupils from a primary school in Malaysia. Eight different pre-sketched phase-based activities were designed to guide the students in the experimental group. The experimental group were taught using the phase based instruction while the control group using the chalk and talk approach. The analysis was done using the mean difference test. Result of the pretests showed that there was no significant differences in both groups. After the intervention, the pupils in the experimental group performed statistically significant better than the pupils in the control group. In short, phase-based instruction using Geometer's Sketchpad was useful in assisting students improve their geometrical achievement in angles. ### ABSTRACT FOR 21325 Discover Pythagoras triangles in WASAN-The role of the technology in the problem discovery AUTHOR: Shin Watanabe AFFILIATION: The Mathematics Certification Institute of Japan Do you know the Japanese mathematics? WASAN? About 500 years ago, Japanese mathematics was made by mathematics lovers. The mathematics lovers were not professors at university. They only enjoyed learning mathematics. This Japanese mathematical knowledge was close to western mathematics but the WASAN had no axiom, no theorem and no proof. The scholarly system of mathematics was different from the western mathematics. But they had no the society of mathematics. They learned mathematics diligently and often discuss mathematics among them. After they solved a mathematical problem, they displayed the problem and its solution at the precincts of the shrine. This kind of displaying mathematics was named SANGAKU. In Japan, many mathematical loners studied WASAN, but modern Japanese changed from WASAN to western mathematics. ### ABSTRACT FOR 21341 Exploration of Linear Programming Problems with a Feasible Region as a Polygon Region AUTHOR: Xi Wu AFFILIATION: High School Affiliated to Shaanxi Normal University Explore a series of linear programming problems containing absolute value, from thoughts and methods to solve problems based on dynamic demonstration and rational analysis and develop exploration, innovation and rationalism spirits of students. ### ABSTRACT FOR 21354 Incorporate Information Technology and Geometric Iteration into Constructing Pythagorean Tree and other iterated graphs AUTHOR: Jiaqi Tao, Puyuan Li AFFILIATION: Beijing No.22 Middle School As high school students we have showed great interests in the verifications of Pythagorean theorem; In the process of discovering and proving it, we created Pythagorean Tree which is made of squares and triangles, however, the similarity between each right triangle could not be guaranteed if they were drawn by hand. As a result, we considered whether it is possible to use modern technology to improve our realizations. We used the iteration function in Geometer's Sketchpad to make the Pythagorean Tree. At the end, we improved our understandings about iteration greatly and were able to draw more colorful and beautiful patterns to find beauty in mathematics. ### ABSTRACT FOR 21355 The Use of Math Experiment in Students' Mathematics Learning: Case Study AUTHOR: Hong Li AFFILIATION: Beijing No.22 Middle School In this paper I will discuss the use of math experiment in the process of students' mathematics learning. As Freudenthal pointed out, the use of math experiment enables students to have free space in realizing their imaginative math ideas and doing math experiments according to their interests. This paper includes two cases in my real mathematics teaching and deeply discusses the relationship exists between math experiment, math teaching and math learning. ### ABSTRACT FOR 21356 Express Quadrilaterals by One Equation AUTHOR: Lai Wei AFFILIATION: Beijing No.22 Middle School As high school students, we were surprised that we could draw a square by using one equation with a graphics calculator in a math class. We were also wonder whether we can use only one equation to express a diamond, rectangle, parallelogram, or even a quadrilateral. In this paper, we conduct further research on answering these questions. We first prove in theory that we can create equation for a diamond, rectangle, parallelogram or quadrilateral. Second, we manipulate the graph of a square by using the scale and rotation to transform the square to the graphs we desire to achieve. Finally, we write those transformations as algebraic forms and rearrange the equation for the square. We make use of the advanced graphing functionality in HP graphing calculator to verify the equations we had created are correct or not. ### ABSTRACT FOR 21357 Explore the Definition of the Distance from a Point to a Parabola by using Mathematical Experiment AUTHOR: Siqi Jia AFFILIATION: Beijing No.22 Middle School Starting with the definitions of the distances from a point to a line and to a plane, in this article we consider defining the distance from a point to a parabola. We conduct experiments with the help of software packages. The experiments include measure, observation, comparison, analysis, guess, induction, verification, inference and conclusion. After that we try to give a reasonable definition of the distance from a point to a parabola. The experiments reveal the importance of technology in supporting mathematical research. ### ABSTRACT FOR 21359 About the Chinese Traditional Toy: Spirograph AUTHOR: Xinyue Zhang and Xuan Yao AFFILIATION: Beijing No.22 Middle School The Spirograph is one of the traditional toys in China. It has inspired us in doing a great deal of research as high school student. In this paper, we discovered the relationship between the Spirograph and astronomy. We applied the idea of "Combination of number and form", the Pythagorean Theorem and the golden section in solving some problems. We found interesting results when the ratio between the radius of the parent's ruler (big circle) and that of the children ruler (small circle) is different. Basing on the original foundations, we tried to trace more points and drew more beautiful patterns. In this research, we understood the relationship between Spirograph and astronomy. We deeply appreciate technological tools have inspired us doing some interesting mathematical research. ### ABSTRACT FOR 21360 The Applications of Geometer's Sketchpad in the Process of Learning Geometry AUTHOR: Yubing Leng and Zixuan Wang AFFILIATION: Beijing No.22 Middle School In this paper, the center of gravity, or the center of mass, of different types of polygonal regions in XY-plane is considered. Geometric idea is a convenient method to calculate the center of gravity of the polygonal regions. Although the regions are with different nature, the locus of their center of gravity is still in the XY-plane. The locus of the center of gravity of a certain region can be considered as a hyperbola with an axes parallel to the coordinate axes. From this observation, a new definition of a hyperbola, without involving eccentricity, can be deduced. A novel construction of a hyperbola, therefore, can be presented by using the dynamic geometry software Geometer's Sketchpad. By using a circle as example, its own properties, the extension and relationship among other geometrical elements, are investigated in this paper by employing the Geometer's Sketchpad. Based on the definition of the circle, a new operation is defined, the properties and characteristics of the derived geometry is then analyzed. After the analysis by using the Geometer's Sketchpad, following conclusion can be drawn as: The locus of the sum, the difference, the product, and the ratio of the distance between a fixed point and a moving point are ellipse, hyperbola, Cassini's Oval and circle, respectively. In addition to the preliminary study, the investigation is extended from the "circle" to the "conic". During the investigation, the technology plays an important role, which virtually shows us a clear understanding about the "Connection between the Geometry and the operation" and "Generation of a graphic". ### ABSTRACT FOR 31000 STEM-WHAT IS THE BIG DEAL AND THE IMPLICATIONS ON OUR MATHEMATICS CLASSROOMS AUTHOR: Dr. Chris Longhurst AFFILIATION: Hewlett Packard Australia What is STEM? Why has it suddenly appeared in our school vocabulary? It is all about connectedness and making mathematics real in the world. The traditional approach to teaching mathematics is a rote learning approach. Research shows that students learn better when they have a growth mindset. I will look at how the mathematics classroom can be used to teach using a STEM focus through stories, technology and a growth mindset model. ### ABSTRACT FOR 31001 Proving Apps via Mobile Technology AUTHOR: Debbie Marie Verzosa*, Alva Aberin** and Ma. Louise D Las Peñas** AFFILIATION: Ateneo de Davao University* and Ateneo de Manila University** One of the main challenges in the mathematics curriculum is teaching and engaging students in proving geometric concepts. Students often have difficulty understanding the basics on how to prove, and find it challenging to connect various concepts to formulate a sound geometric proof. This talk will focus on software applications for mobile technology that have been designed to provide visual feedback as students complete geometric proofs. Proofs are limited to those reflected in the Philippine secondary school curriculum; such as, triangle congruence and similarity, and theorems on quadrilaterals. ### ABSTRACT FOR 31002 Making Mathematical Models with HP Prime AUTHOR: GT Springer AFFILIATION: Ateneo de Davao University* and Ateneo de Manila University** Using mathematical representations as models is a core feature of many standards for the teaching of mathematics. But mathematical modeling can be interpreted to mean a number of different things. For example, mathematical modeling could mean using a function as a model for real-world bivariate data. It could also mean using dynamic geometry software to model a mathematical process. In this talk, we examine a number of mathematical models that can be easily created using the HP Prime graphing calculator. In each case, the pedagogical purpose behind the model will be discussed as well. # Abstracts for the Track of Hands-On Workshops ### ABSTRACT FOR 21148 Spreadsheets and QR Codes with Casio CLASSWIZ AUTHOR: Wei Ching Quek AFFILIATION: Singapore Polytechnic The Casio Classwiz is the new classroom standard. It has a new iconic display. In addition to usual functions in Advanced Scientific Calculators, it is also loaded with useful spreadsheets, tables, statistics functions. It also can generate QR codes for students to retrieve relevant graphical information. This workshop is consists of two activities: 1. Getting Started Begin with introduction to new features in ClassWiz, this includes tables, statistics, followed by exploring essential features of the spreadsheet function, and generation of QR codes. Participants will use their mobile devices to retrieve additional resources from the QR codes generated. Problems Solving/Discussion Examine some applications of spreadsheet to solve Riemann Sums and numerical modelling of population growth. Participants will explore the solution to gain further insights to the problem. ### ABSTRACT FOR 21155 Introducing MATLAB into High School Mathematics AUTHOR: Ian Allan Thomson AFFILIATION: Ormiston College, Australia MATLAB (Matrix Laboratory) is the language of technical computing that is used by millions of engineers and scientists. It is widely used in STEM-related courses across the world. In this workshop, participants will find out how MATLAB can be introduced into high school mathematics. Participants will receive a collection of ten modelling and problem solving tasks suitable for use with MATLAB or other forms of technology. ### ABSTRACT FOR 21156 Mathematical Problem Solving with a Graphing Calculator AUTHOR: Wee Leng Ng AFFILIATION: National Institute of Education, Nanyang Technological University Enabling students to acquire and apply the necessary mathematical concepts and skills to solve problems in a wide range of situations is one of the primary aims of mathematics education. On the other hand, empowering students to make effective use of a variety of mathematical tools, including technological tools, in the learning and application of mathematics is identified as an important objective of many school mathematics curricula. Problem solving activities involving the use of technological tools have the potential to contribute to meeting the two aforementioned goals of mathematics education. In this workshop, participants will engage in activities which explore effective uses of the TI-84 Plus graphing calculator, at different stages of the problem-solving process, in solving a collection of mathematical problems suitable for upper secondary students. ### ABSTRACT FOR 21157 Enhancing the Teaching and Learning of Mathematics Through Handheld Technology AUTHOR: Wee Leng Ng AFFILIATION: National Institute of Education, Nanyang Technological University Handheld graphing technology, if used appropriately in the mathematics classroom, has the potential to enhance the teaching and learning of mathematics by empowering students to learn across different visual representations of mathematical concepts. With the aid of such technology, teachers have the means to help students develop a deeper understanding of abstract mathematical concepts and sharpen their critical thinking skills. In this workshop, participants will explore the use of the TI-Nspire CX handheld in helping students develop relational understanding of concepts in calculus and statistics at upper secondary and pre-university levels. ### ABSTRACT FOR 21161 Coding in Mathematics AUTHOR: LC Mooi AFFILIATION: Ministry of Education, Singapore This workshop introduces participant to a possible pedagogical approach of increasing mathematical content knowledge through coding. A mathematics topic for primary school pupils will be used to demonstrate how technology (viz computer software application) can play a role in enhancing mathematics content knowledge. ### ABSTRACT FOR 21162 Learning about equations with ClassWiz AUTHOR: Barry Kissane AFFILIATION: Murdoch University, Australia While scientific calculators have been available since the 1970s, advanced versions have been developed recently to suit the needs of mathematics education. So, these calculators provide powerful learning opportunities for many aspects of mathematics treated these days in secondary school and university curricula, as well as giving students access to efficient calculation. This workshop will focus on a variety of ways in which the CASIO ClassWiz calculator can be used to enhance both teaching and learning related to equations in particular. Previous experience with ClassWiz is not assumed. ### ABSTRACT FOR 21163 Modelling with spreadsheets on ClassWiz AUTHOR: Barry Kissane AFFILIATION: Murdoch University, Australia Sophisticated spreadsheets are widely used in mathematics and in learning mathematics, but they require students to have individual access to a computer. The CASIO ClassWiz is a hand-held advanced scientific calculator that includes a spreadsheet facility, thus offering learning opportunities only previously available to those with a computer. In this hands-on workshop, we will use some of the activity materials that have been recently developed for the ClassWiz to consider some of the ways in which a spreadsheet can be used for various kinds of modelling. Previous experience with ClassWiz or with other spreadsheets is not assumed. ### ABSTRACT FOR 21164 Investigating mathematics 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 ClassWiz is an advanced scientific calculator with significant functionality designed for secondary 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. We will consider some examples of student investigations that are 'low floor, high ceiling' in the sense that it is easy for secondary 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 the ClassWiz to experience this way of learning. Previous experience with ClassWiz will not be assumed. ### ABSTRACT FOR 21165 Applying Instructional Technology Creatively In Mathematics Education (to Deliver Content Knowledge and Develop Higher Order Thinking Skills Effectively) AUTHOR: Poh Yew Teoh AFFILIATION: Creative Wizard Pte Ltd, Malaysia, International Group for Mathematical Creativity and Giftedness MCG Getting students excited about learning is the teachers' constant challenge. The answer lies in designing creative teaching aids and incorporating creative teaching techniques into our classroom practice. Spicing up our mathematics lessons with some fun and magical, hands-on or animated activities could easily help increase attention, understanding and retention significantly. Appropriate used of instructional technologies will also stimulate intellectual curiosity, develop problem-solving skills, promote discovery and cultivate higher order thinking skills. In this workshop, participants will learn how to: use 5 simple definitions of creativity as a tool to come up with creative ideas to develop effective teaching aids using inexpensive/discarded material. begin their lessons with meaningful set induction. engage students using ICT-based instructional technologies to learn concepts, solve problems or develop higher order thinking skills decipher some interesting mathematical magic and puzzles create new magic tricks to help students learn mathematical concepts Target participants: Upper primary and lower secondary mathematics educators Sub Theme: Increase Mathematics Content Knowledge through Technology. ### ABSTRACT FOR 21167 Harnessing the Power of HP Prime graphing calculator in Teaching Senior High School Mathematics AUTHOR: Helma Mesa AFFILIATION: Saint Pedro Poveda College, HP Calculators Saint Pedro Poveda College's Personalized Education Program employs teaching strategies that are learner-centered and most-suited to the students. The graphing calculator technology proved to be useful in attaining a more student-centered learning environment in the classroom. In this workshop, the participants will explore how the power of HP Prime graphing calculator can be harnessed to reinforce or enhance teaching practices in Senior high school Mathematics. They will explore how Math concepts may be learned by students through investigative/ exploratory approach with the aid of the features/functions of the HP Prime. This will include Math topics in Precalculus, AP Calculus, and Statistics. Examples of modules/worksheets and exploratory activities integrating the use of HP Prime will be presented to show how the teaching-learning process is enhanced and to give the participants the opportunity to integrate HP's technology in teaching Mathematics in the 21st century classroom. ### ABSTRACT FOR 21177 Deeper Understanding of Mathematics Through Effective Use of Parameters AUTHOR: Yew Fook Chan AFFILIATION: School of the Arts In this workshop, we will discuss how Texas Instruments TI-Nspire CX hand-held technology can be used to facilitate the learning of functions and graphs through the use of parameters. Lesson examples will be shared. ### ABSTRACT FOR 21178 Improving Student Engagement Through the Use of Handheld Technology AUTHOR: Yew Fook Chan AFFILIATION: School of the Arts Singapore In this workshop, we will learn how to design, construct and use learning activities for students using Texas Instruments TI-Nspire CX hand-held graphing technology that fosters student engagement and understanding in learning mathematical concepts. ### ABSTRACT FOR 21180 Exploring Graphs using HP Prime AUTHOR: Betty, Wan Niu Voon AFFILIATION: Universiti Tenaga Nasional Graphing calculators, are meant to portrait graphs. In this workshop, a few aplets will be explored to plot different types of graphs. Graphs of functions, inequalities, conic sections, parametric equations, polar equations and even sequence and series will be explored. In HP Prime, certain aplets enable integration of commands. Furthermore, commands from aplets can be executed from homepage and the outcome can be imported from homepage into applets as well. These features can be time saving while avoiding mistakes. Hence, participants will be guided in this hands-on workshop to grasp the skills to play around with these features of this graphing calculator. Handouts will be provided. ### ABSTRACT FOR 21185 Assembling 10 Rhombic Hexahedrons into a Rhombic Icosahedron AUTHOR: Jen-chung Chuan AFFILIATION: National Tsing Hua University, Taiwan In this workshop with Cabri 3D we are to assemble 10 rhombic hexahedrons to form a rhombic icosahedron. To see more details, you may visit http://atcm.mathandtech.org/EP2016/jc_21185.pdf. ### ABSTRACT FOR 21186 An Animation of Six Identical Cylinders Each Touching Exactly Four Others AUTHOR: Jen-chung Chuan AFFILIATION: National Tsing Hua University, Taiwan In this workshop with Cabri 3D, we are to construct an animation displaying six identical cylinders each touching exactly four others. For more details, you may visit http://atcm.mathandtech.org/EP2016/jc_21186.pdf. ### ABSTRACT FOR 21187 How Jitterbug inspired and INteresting Dancing Routine for Skeletal Thombic Hexahedrons? AUTHOR: Jen-chung Chuan AFFILIATIONS: National Tsing Hua University, Taiwan In this Cabri 3D workshop we are to construct: An animation of a conformal polyhedral linkage known as the Buckminster Fuller's Jitterbug. A linkage movement for eight skeletal rhombic hexahedrons each sharing a common edge with three others. <\ol> For more details, you may visit http://atcm.mathandtech.org/EP2016/jc_21187.pdf. ### ABSTRACT FOR 21190 Technology-Advantaged Mathematics Teaching-Learning Process: CASIO fx-991EX Classwiz AUTHOR: Rodulfo T. Aunzo, Jr. AFFILIATION: Faculty, USC Cebu Philippines Research studies show that students with a non-positive attitude towards Mathematics don't find learning Mathematics fun. Learning Mathematics to them doesn't sound interesting. In the context of the K-12 General Mathematics and Statistics and Probability, learning will be made easier and fun upon the integration of calculator in the teaching-learning process. Students display an awe expression upon arriving exactly the same answer when using few manual computations as compared to their calculator-aided computations. This paper talks about how CASIO fx-991EX CLASSWIZ calculator, together with its emulator, is useful in handling computational learning activities in Business Mathematics, Statistics, and Advanced Algebra. Thus, making mathematics learning to be fun and much easier. ### ABSTRACT FOR 21195 Autograph v.4 for probability, data handling and statistics AUTHOR: Douglas Butler AFFILIATION: iCT Training Centre (Oundle, UK), Autograph-Math This session will start by exploring the TSM Resources web site for exciting real data sets that can be downloaded to Excel and analysed there or in the new Autograph v.4. Difficult topics such as histograms and frequency density will be covered, moving on to probability distributions, hypothesis testing and scatter diagrams. Delegates will receive a free copy of the software and the chance to sample the pedagogical strength of the Autograph user-interface. ### ABSTRACT FOR 21199 Creating and designing furniture with Cabri 3D AUTHORS: Jean-Jacques Dahan, Jean-Marie Laborde AFFILIATION: IRES of Toulouse, Cabrilog Grenoble France We will use the creations and the transformations tools of Cabri 3D to create in 3D tables, chairs, beds, chests of drawers, bedside lamps . This workshop will stimulate your creativity and give you some tools and techniques in order to create and design the fruits of your own imagination. ### ABSTRACT FOR 21200 Riemann sums with Cabri 2 Plus and the new Cabri AUTHORS: Jean-Jacques Dahan, Jean-Marie Laborde AFFILIATION: IRES of Toulouse, Cabrilog Grenoble France We will discover a simple trick to represent easily the Riemann sums with Cabri 2 Plus and the new Cabri. We will also show how to represent the dynamic generation of solids of revolution and the approximations of their volumes with cylinders got by the rotation of the Riemann sums around the axis of the axes of these solids. ### ABSTRACT FOR 21231 Mathematical Modelling Using Handheld Technology AUTHOR: Thomas Yeo AFFILIATION: Texas Instruments Very often, it is more effective for students to learn Math in an authentic context to make meaning of what they are learning in the lesson. In this workshop, participants will learn how to design and implement a mathematical modelling activity/task, which will allow students to apply concepts learnt in real life. Handheld technology will be used in this workshop in a 1-to-1 classroom concept/setting. ### ABSTRACT FOR 21232 Mathematics Modelling in Real Life AUTHORS: Katrina Ng AFFILIATION: School of Arts Singapore Often in a mathematics classroom, students are unable to relate Mathematical concepts to real life. In this workshop, participants will learn how to conduct an activity using temperature sensor to collect data and model the function. This activity consists of using the concept of Newton Law of Cooling and Exponential function. ### ABSTRACT FOR 32001 The Pedagogical Usability of GSP in Designing and Creating Mathematics Instructional Materials AUTHORS: Ronnachai Panapoi,Ph.D.; Danita Chunarom; Pilaluck Thongtip; Siriwan Jantrkool; Woranart Yoosook; Jannapa Uttama AFFILIATION: Thailand ### ABSTRACT FOR 32002 Fun Math with GeoGebra AUTHORS: Pinyada Klubkaew; Amarisa Chantanasiri; Nuanchan Ritkham; Pathamaporn Awachai; Puttoei Talawat, Ph.D.; AUTHORS: Maluangnont, Ph.D.; Sutharot Nilrod, Ph.D.; Pattanachai Rawiwan AFFILIATION: Thailand ### ABSTRACT FOR 32003 Instructional Materials for Primary Mathematics AUTHORS: Bhimmawajna Thammachai; Benjamas Laokwansathit; Ausanee Wongarmart AFFILIATION: Thailand ### ABSTRACT FOR 32004 Math Circle demo: Catalan counting AUTHORS: Matthias Kawski AFFILIATIONS: Arizona State University, USA Math Circles have a long tradition in Bulgaria and Russia, and since the late 1990s have been taking over the US and other countries. Math Circles typically are complementary to formal school education, and provide after school settings in which students in grades K-12 explore advanced mathematical scenarios. Math Circles successfully compete with piano lessons, chess clubs, and organized sports - get kids excited about math, and Math Circles provide them with a place where they can flourish while being mentored by professional mathematicians. The emphasis is on fun and students' self-directed inquiry into ever more advanced mathematics. Having observed the changes in attitude by students participating in such inquiry based learning environment, which is not driven by standardized tests and curricular benchmarks, many teachers have formed their own Math Teachers' Circles to rejuvenate their own scholarly explorations of mathematics beyond the routine classroom drudgery. In this workshop participants will experience an active learning experience that is typical for the exploratory environment in Math Circles. This particular workshop will focus on counting problems, and Catalan numbers, and is an example of a Math Circle for advanced high school students, just as it is an excellent example for a Math Teachers' Circle for teachers from middle school grades to college levels. # Abstracts for the Track of Poster Sessions ### ABSTRACT FOR 21214 On the Visualization of Three Perpendicular Theorem Using Projection Tool of GeoGebra AUTHORS: Minji Kim, Minshik Cho AFFILIATION: Korea National University of Education The three perpendicular theorem is a fundamental content of Korean high school geometry courses and powerful tool for the solving problems on space geometry. But most students have difficulties to understand the theorem and its applications. It seems that students have not sufficiently experienced space geometry. The three perpendicular theorem is mostly presented with 2-dimensional images by standard textbooks and teachers explain it with brief axiomatic proof. However, most students have various misconceptions due to the lack of spatial visualization and don¡¯t understand involved geometric meaning in written problems. Because it is not easy to grasp the 3-dimensional figures from the 2-dimensional drawings in the text, it is desirable for students to have experiences the 3-dimensional activities. GeoGebra is one of the optimal dynamic geometry tool for this study. Students can explore space figures in any directions, which could not be possible in 2-dimensional environment. Particularly, the 3D projection tool of GeoGebra is a nice gadget for students to understand space figures intuitively because the tool construct most likely to 3-dimensional space figures. So we will developed teaching materials using the projection tool of GeoGebra to help students¡¯ understanding of three perpendicular theorem and adapt them to problems as well. ### ABSTRACT FOR 21225 On the misconceptions and enforcements of visualizing images of tangent line AUTHORS: Seong Hwan Kim, Minshik Cho AFFILIATION: Korea National University of Education The tangent line is important to understand the geometric meanings of various curves. Since most curve can be represented as a graph of a differential function, the meaning of a differential coefficient are explained as the slope of the tangent line in textbooks. Many students just memorize it without understanding its concepts. It seems that students feel difficulties in solving problems due to lack of visualizing images of the tangent line. This may not have been provided sufficient opportunities for students to form a visualizing image for the tangent line. Students have learned the tangent line several times from middle school to high school. Tangent line is accessible from the view of Synthetic Geometry in middle school, the view of Analytic Geometry in high school. And it is noteworthy that the definition of the tangent line is not taught, but the properties of the tangent line in modern sense. It is not easy to connect its meaning to one in association with each other, so it sometimes turn into misconceptions. Hence it is necessary to enhance the visualizing image of the tangent line, so as to reduce the misconceptions about the tangent line. GeoGebra is one of the optimal dynamic geometry tool for this study. In the text book-based environment, it is difficult to experience dynamic activities, such as moving a point or a straight line, but can be done via GeoGebra. So we will developed teaching materials to help students understand the tangent line. ### ABSTRACT FOR 21237 Steady Critical Surface Waves over a Bump with Surface Tension AUTHORS: Jeongwhan Choi, Sangwon Lee, Junkyung Kim, Sungim Whang AFFILIATION: Korea University, Ajou University We consider steady forced surface waves propagating on a two-dimensional incompressible and inviscid fluid with a small bump placed on a rigid flat bottom. If the surface tension coefficient $$T$$ on the free surface is not zero and the wave is moving with a constant speed $$C$$, the wave motion is determined by two non-dimensional constants, $$F=\sqrt{gh}$$ and $$\tau=T/(\rho g h)^2$$ where $$g$$ is the gravity constant and $$h$$ is the height of the fluid at infinity. It has been known that $$F = 1$$ and $$\tau = 1/3$$ are the critical values of $$F$$ and $$\tau$$, respectively. In the critical case $$F = 1 + \lambda_1\epsilon^2$$ and $$\tau = 1/3 + \lambda_1\epsilon$$ with $$\epsilon > 0$$ a small parameter, a time-dependent forced Kawahara (F-Kawahara) equation is derived to model the wave propagation on the free surface and the steady F-Kawahara equation is studied both theoretically and numerically. It is shown that the steady F-Kawahara equation has many different kinds of one and multi-hump solutions when $$\lambda_1$$ and $$\tau_1$$ vary. In particular, for a fixed $$\tau_1$$, there is a $$\lambda_0 < 0$$ such that if $$\lambda_1<\lambda_0$$, two one-hump steady solutions can be obtained, one with small amplitude and the other with large amplitude. By using the unsteady F-Kawahara equation, it appears that the small one-hump solution is stable while the large one is unstable. In addition, two-hump solutions are unstable. ### ABSTRACT FOR 21265 Intention analysis in explainer indications for elementary geometry documents using laser pointer images AUTHOR: Yuki Toyosaka, Naoki Tamura, Ryoji Fukuda AFFILIATION: Faculty of Engineering Oita University, Faculty of Information Science, Kyushu Sangyo University, Japan , Graduate School of Engineering, Oita University, Japan We analyzed human intentions during explanations of elementary geometry using graphical contents. The lecturer indicates graphical contents by laser pointer during the explanation, and we create BMP images by using digital video camera. We prepared position data of the graphical elements in the graphical contents, and the feature values for the analysis are defined by using relative position of indicated pointers with respect to the graphical elements and their moves. We divided an explanation into several parts considering connections of meaning, and we succeeded rough discrimination for the parts using these feature values. ### ABSTRACT FOR 21266 Visualization of accuracy and reliability of approximation of density function in a discriminant analysis AUTHOR: Yuki Ogino, Yuki Toyosaka, Ryoji Fukuda AFFILIATION: Faculty of Engineering Oita University, Graduate School of Engineering, Oita University , Faculty of Information Science, Kyushu Sangyo University Japan For a discriminant analysis using one feature value, conditional probability with respect to a given value must be a good evaluation value, in the case where we correctly know the density functions for two groups. In an actual data analysis, approximations of density function values sometimes are not correct enough, especially in cases where data size is small or the appearance ratio is small. We developed a software to visualize these situations, for example, estimation of density functions or reliability of discrimination. We evaluate the performance by a simulation using random variables which laws are contaminated normal distributions. ### ABSTRACT FOR 21324 Geometric Constructions with Cinderella and ketcindy -- Application of mathematics to mathematics in senior high school AUTHOR: Hideyo Makishita AFFILIATION: Shibaura Institute of Technology Geometric constructions normally mean generation of a figure suited for given conditions using rulers and a pair of compasses only for a finite number of times. Hereinafter, this is referred to simply as figure drawing. This poster presents a discussion of, in addition to geometric constructions, the practice of geometric constructions by adding mathematical contents. When mathematical material is added to geometric constructions 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 DG''s Cinderella as GUI and can be used for drawing figures by Script as CUI. Therefore, mathematically precise figures can be drawn with ease, producing beautiful results. The author explains geometric constructions while the quadratic curve concept is added to figure drawing. The author considers that geometric constructions by Script is extremely useful for mathematics education from the viewpoints of application of mathematics to mathematics. The author would like to discuss with you on this point. ### ABSTRACT FOR 21352 Experimental attempt to practice flip teaching with ShowMe AUTHORS: Mahiko Takamura AFFILIATION: The second high school at Takashima, Thailand The poster session will clarify the advantages and disadvantages of using materials created by a learning application software ShowMe.

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