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Cultivating Creative and Innovative Mathematical Thinking with Technologies


ATCM 2019, Leshan, China

  1. Abstracts for Invited and Plenary Papers
  2. Abstracts for Full Papers
  3. Abstracts for Presentations with Abstract Only
  4. Abstracts for Hands-On Workshops
  5. Abstracts for the Poster Session

Abstracts for Invited and Plenary Papers


Abstract for 21684 


Building the Russian Nesting Dolls by Dissecting the Five Platonic Solids 


Author:                Jen-Chung Chuan 

Affiliation:           Department of Mathematics, National Tsing Hua University, Hsinchu, TAIWAN 300 


Motivated by the Nesting Russian Dolls, we are to make this follow dissections 

1.     Staring with a regular dodecahedron D 

2.     D is dissected in to a cube C and 6 mutually congruent "roofs 

3.     C is dissected into a regular tetrahedron T and 6 mutually congruent non-regular tetrahedra 

4.     T is dissected into a regular Octahedron O and 4 mutually congruent tetrahedra 

5.     O is dissected into a regular Icosahedron I and 6 mutually congruent non-convex hexahedra 

We thus have the nested polyhedra D, C, T, O, I. 


Abstract for 21708 


Steiner Ellipse and Marden’s Theorem 


Author:                Jean-Jacques Dahan 

Affiliation:           IRES of Toulouse, FRANCE 


In this paper we will see the richness of both algebraic and dynamic approaches of a problem related to the three zeroes of a third-degree complex polynomial and the two zeroes of its derivative. Visualization with DGS in the Argand plane of these five points with the help of CAS will facilitate the exploration leading to the discovery of a very special ellipse (the Steiner ellipse).We will first prove (via a geometric proof) the existence and the uniqueness of such an ellipse (tangent to the triangle defined by the zeroes of the third degree polynomial at the midpoints of these zeroes) and then prove (complex proof) that the foci of this ellipse are the zeroes of the derivative polynomial (Marden’s theorem). Connecting easily DGS and CAS within the TI-Nspire environment is a crucial tool of this work as well as the use of sliders to summarize the different stages of the investigations, the conjectures and their corroboration (experimental) or their validation (proof). This paper aims to show ideas of challenging mathematics to everybody at all levels. It aims also to show rich techniques of investigation within DGS and CAS environments. A final problem in relation to Marden’s theorem will be investigated within DGS, leading to a very nice conjecture, which is proved, with a blend of DGS and CAS. An original way to prove that the Steiner ellipse is the ellipse of maximum area inscribed in a triangle will be shown in an important part of this paper: the way this very last work was conducted could be an important resource for techniques of investigation or proofs to know in order to use DGS and CAS relevantly. 


Abstract for 21709 


An Introduce of Dynamic Mathematical  
Digital Resources Open Platform 


Author:                Guan Hao 

Affiliation:           Chengdu Institution of Computer Application, CHINA 


 In response to the principle of achieving shared improvement through collaboration, relevant actions taken on the aspect of the digital education resources have become the important contents and research highlights of the present information and communications technology (ICT) in education. After years of researches and constructions, the digital education resources still have not been fully valued, and what is worse is that comments such as the so-called "resources island" and "digital ruins" are presented. This phenomenon is mainly because the construction of the existing resource platform lacks sufficient openness and scientific correlation. This paper proposes a model of the educational resources open platform based on the PaaS structure. By combining the techniques such as the dynamic geometry, the computer algebra, and the automated geometric theorem proving, a dynamic mathematical educational resources open platform is designed and implemented by means of a micro-service architecture. Moreover, the OAuth2.0 technology is applied to provide authentication and authorization to a third-party application and a user, as well as hierarchically customized source materials and capabilities for editing dynamic mathematical contents. This paper proposes an overall design of the open platform, gives a hierarchical strategy and an authentication process for applications and user permissions thereof, and introduces the process of customizing the open resources and the tools. Currently, the open platform serves a plurality of third-party applications and is widely applied. 

A collection of video clips associated with this presentation is available online. They can be found at 


Abstract for 21710 


Limit and Continuity of a Function. Software aspect 


Author:                Vladimir Nodelman 

Affiliation:           Holon Institute of Technology, ISRAEL 


The studies of fundamental concepts of limit and continuity of a function cause difficulties: students for the first time meet here definition with quantifiers. 

This paper illustrates the methods of teaching and learning these concepts with intensive use of software. In general, the types of student’s activities do not depend on the specifics of functions (whether they are real or complex), whereas, the supporting models are significantly different. 

By means of author’s non-profit software, "VisuMatica" students not only use some ready models but construct proper models by themselves and explore with their help the studied contents. 


Abstract for 21712 


Realizing Computational Thinking in the Mathematics Classroom: Bridging the Theory-Practice Gap 


Authors:               Weng Kin Ho, Chee Kit Looi, Wendy Huang, Peter Seow, Longkai Wu 

Affiliations:          Nanyang Technological University, SINGAPORE 


With the impact of computer technology on human civilization in the 21st century, Computational Thinking has become an important topic of research and discussion in the area of education over various non-computer science disciplines; for example, mathematics, sciences, languages, social sciences, etc. By now, Computational Thinking has often been referred to as a paradigm of knowledge and problem solving by which the problems and their solutions can be implemented using an effective means, e.g., by a computer, though it never really possess a universal definition. In this paper, we invite the reader to revisit the original meaning of the word “Computational Thinking” as intended in 1980 by its inventor, Seymour Papert. By so doing, we bring into focus the concrete issues that a mathematics teacher need to consider so as to realize Computational Thinking in his or her lessons. We analyze the design principles put forth earlier by the authors, and understand these principles in the context of mathematics teachers’ professional development and classroom implementation. 


Abstract for 21713 


Poncelet's Porism and Its Connections with Algebra, Analysis, Mechanics, and Dynamical Systems: A Technology-Based Approach 


Author:                Guillermo Davila-Rascon 

Affiliation:           University of Sonora, Department of Mathematics, MEXICO 


In 1822, Jean-Victor Poncelet published his fundamental work in which he demonstrated a remarkable result that became known as Poncelet Closure Theorem or Poncelet''s Porism, and can be enunciated as follows: Let C and D be two smooth conics in the projective plane. If there is a polygon of $n$ sides inscribed in C and circumscribed about D, then there are infinitely many such polygons having the same number of sides n. Moreover, each point of C is a vertex of such a closed polygonal line. 

In this talk, we will explore some of the remarkable connections of this result with some others in different fields, showing us the richness of ideas and methods that originated from this theorem. For example, we will connect the Poncelet's porism with the theory of elliptical billiards, especially; we are interested in deriving conditions for the periodicity of billiard’s trajectories. In addition, we will present the relation of the theorem with the theory of elliptic integrals, as well as algebraic geometry, etcetera. 

We will be using GeoGebra and Maxima for exploring some of the scenarios presented in the talk. For example, modelling the reflections of light beams inside ellipses and the integrability of two dimensional elliptical billiards. 


Abstract for 21719 


Robot for Mathematics College Entrance Examination 


Authors:               Hongguang Fu, Jingzhong Zhang, Xiuqin Zhong, Mingkai Zha, Li Liu 

Affiliations:          University of Electronic Science and Technology of China,  
School of Computer Science and Engineering,  
University of Electronic Science and Technology of China, 
Southwest Minzu University, Chengdu, CHINA 


Automated mathematics has attracted more and more interests to mathematicians and computer scientists. In this article, we propose a new solution, Robot for Mathematics College Entrance Examination (RMCEE), to automatically solve elementary mathematical problems. RMCEE took the college entrance exam of China in 2017 and scored 105 points out of 150. RMCEE takes as input Chinese natural language and outputs human-readable solution processes. It leverages artificial intelligence technologies in multiple domains such as the knowledge graph, natural language understanding, cognitive reasoning, and deep learning. Our project outperforms other similar projects such as Todai Robot of Japan in that RMCEE proposes a natural language understanding model of entity combinations based on knowledge graph, and implemented human-readable solution processes based on a novel cognitive reasoning model. 


Abstract for 21723 

(The Role of Dynamic Mathematics in Mathematics Education) 


Authors:               Chuan-Bo Zuo, Jian-Lan Tang 

Affiliations:          Hawgent Technology Centre in Mathematics, Gangxi Normal University, CHINA 


 动态数学系统(Dynamic Mathematics System)在数学教育中能够发挥什么作用?这取决于数学问题的类型以及解决问题的策略. 在这个报告当中,我们根据不同的数学教学问题进行了分析、研究与思考,从而提出了动态数学系统在数学教育教学过程中所具有三个层次或方面的价值与作用:一、动态数学系统是一种工具;二、动态数学系统仅仅是一种工具;三、动态数学系统从不仅仅是一种工具. 


(What role can the Dynamic Mathematics System play in mathematics education? It depends on the type of mathematics problem and the strategy to solve the problem. In this report, we analyze, study and think according to different mathematics teaching problems, and then propose three levels of dynamic mathematics system in the process of mathematics education and teaching. First, the dynamic mathematics system is a tool. Second, the dynamic mathematics system is only a tool. Third, the dynamic mathematics system is more than just a tool.) 


Abstract for 21726 


Some Opportunities for Computational Thinking in the Mathematics Classroom 


Author:                Jonaki Ghosh 

Affiliation:           Lady Shri Ram College, Delhi University, INDIA 


Computational thinking has been identified as an important skill for students who wish to pursue mathematics and mathematics related disciplines as a career. This has called for a special focus on CT activities in the K – 12 curriculum. While many definitions of CT may be found in the literature, all of them seem to focus on specific skills, such as, the ability to deal with challenging problems, representing ideas in computationally meaningful ways, creating abstractions for the problem at hand, breaking down problems into simpler ones, assessing the strengths and weaknesses of a representation system and engaging in multiple paths of inquiry. These skills are also critical for mathematics learning and there is a common consensus on the understanding that CT skills have to be developed in mathematics classrooms right from the school years. However finding appropriate tasks which help to develop and elicit such thinking remains to be a key pedagogical challenge. Further teachers need to be empowered to create as well as integrate CT tasks in their lessons. This article hopes to suggest a guide map for preparing and integrating CT based activities in the mathematics classroom. It also attempts to highlight how technology can play an pivotal role in engaging students and in steering them towards computational thinking. Examples of tasks ranging from iteration and recursion in fractal constructions to exploration of chaos and simulation of queues and games which were conducted with secondary school students in the Indian context will be discussed. 


Abstract for 21737 


Fixing Education for the AI Age 


Author:                Conrad Wolfram 

Affiliation:           Wolfram Research, UK 


The importance of quantitative understanding for jobs, society and management has exploded over the last few decades. Meanwhile, the mainstay of preparation for this computational thinking--mathematics education--increasingly struggles to match up with these requirements. Why has this chasm opened up and how can it be bridged? Understanding the changing role of computers and automation of knowledge is crucial. What will be the future role of humans in solving problems--in the workplace or in society--given increasing artificial intelligence? Which human skills must we focus on developing for everyone for day-to-day survival? What are the top value-added skills for the highest value contribution?  

Conrad Wolfram will address these questions, explaining the role of computers and a renewed, broad, core computational subject in transforming education for the AI age. He will give live demos throughout the talk and answers questions afterwards. 




Abstract for 21738 


The Mathematics behind Cryptocurrencies and Blockchain 


Authors:               Juan Medina Molina, Elena Soledad Jiménez-Ayala 

Affiliations:          Universidad Politécnica de Cartagena, SPAIN, 
Universidad Politécnica de Cartagena, SPAIN 


In day-to-day education, when we introduce a new mathematical concept in class, students often ask: what is its purpose? It is then useful and enriching to be able to present some interesting current technology to which the mathematics can be applied. We show some of the mathematics that underlies cryptocurrencies, specifically bitcoin and the technology behind it, known as the blockchain. Many people talk about the blockchain as being one of the most important innovations of recent times, since it can be applied in a wide variety of fields, from economy to education. 

A vital aspect of the mathematics present in cryptocurrencies and the blockchain is cryptography, mainly in connection with elliptic curves. Having some knowledge of cryptography, therefore, may be appropriate for economists, engineers, scientists in general, and mathematicians. However, the way to present it to each of these groups should be different. 

We examine the approach and the contents by which this topic should be introduced in class, and distinguish to whom the training is aimed. 


Abstract for 21748 


Proving Nonnegativity of Polynomial with Computer 


Authors:               Bican Xia, Haokun Li, Lu Yang 

Affiliations:          Peking University, School of Mathematical Sciences, CHINA, 
Peking University, CICA, CHINA 
Chinese Academy of Sciences, CHINA 


Proving a polynomial is nonnegative (either globally or under some constraints) is a basic and important problem in the field of real algebraic geometry and has many applications in other fields. This article reviews some of our works on this topic with emphasis on how to use our tools to prove nonnegativity of polynomial with computer. Furthermore, a concept of block SOS decomposable polynomials is introduced which characterizes a class of polynomials whose SDP matrices (corresponding to their SOS decompositions) can be block-diagonalized. It is proved that the set of block SOS decomposable polynomials is measure zero in the set of SOS polynomials. 


Abstract for 21752 


Creative Technology-Assisted Approaches to Foster Innovative Mathematics Learning Environments in South Africa 


Author:                Werner Olivier 

Affiliation:           Nelson Mandela Metropolitan University, First Rand Chair in Maths Education, SOUTH AFRICA 


In this paper the author describes and demonstrates a number of creative components of an offline Techno-Blended teaching and learning model (TBM) for mathematics. This model was developed by the Govan Mbeki Mathematics Development Centre at the Nelson Mandela University with the aim of addressing some of the key education challenges in under-resourced schools from rural regions in South Africa. It incorporates modern interactive digital materials with innovative technological features alongside Android applications to create innovative 21st century offline learning environments. Successes with Tablet-assisted learner incubation programmes will be discussed with reference to qualitative and quantitative impacts of recent large scale pilot projects. The use of the TBM model to implement an in-service professional learning network programme for teachers that focuses on Technology Pedagogy and Content Knowledge (TPACK) will also be discussed. Finally, aspects of the TBM that focuses on STEAM education awareness and creative problem-solving skills for 4IR will be discussed and demonstrated.  


Abstract for 21756 


Augmented Reality and Blended Learning: Engaging Students Learn Word Problems with Bar Model and the Geometer’s Sketchpad 


Author:                Krongthong Khairiree 

Affiliation:           International College, Suan Sunandha Rajabhat University, Bangkok, THAILAND 


The purpose of this study aims to explore the students’ perceptions of teaching approaches using Augmented reality, blended learning in mathematics via smartphone. In the 2019, action research was conducted in mathematics class of a lower secondary school in Bangkok, Thailand. The total of 35 Secondary Year 1 students participated in this study and the duration of the action research was about three months. In mathematics classroom, the researcher created mathematics lessons, activities and learning instructions of her lectures. The students studied mathematics word problems and used the Geometer’s Sketchpad outside classroom. They used smart phone to scan QR Code Reader to explore mathematics animations and mathematics activities prior attending class. The research findings indicated that the blended learning was new teaching strategy that combined the learning on mathematics in the classroom and outside classroom via technology. The students brought assignments/exercises of mathematics concepts inside the classroom via learning activities. The research findings shown that the students’ engagement in blended learning were higher than the using traditional classroom. Based on the students’ interviews they revealed that using blended learning incorporated mathematics word problems, animations using Geometer’s Sketchpad via smart phone, Augmented reality and QR Code Reader methods they were able to manage their learning pace and learned remotely. This learning methods made learning mathematics fun and challenging. 


Abstract for 21757 


Exploring Reflections That Are Inspired by A Chinese Exam Problem 


Author:                Wei-Chi Yang 

Affiliation:           Radford University, USA 


We use technological tools to explore the reflections on circles. The discussions in this paper were inspired by a college entrance practice exam from China. We see if we choose the incoming and outing light beams are at a specific angle within a circle, we shall create many nice geometric patterns. Secondly, we replace the straight lines for light beams by two symmetric curves with respect to the corresponding normal line at the point on the circle, we will create nice patterns involving curves. Finally, we explore some known facts regarding the reflections along ellipses with technologies. 


Abstract for 21758 


Innovative Strategies: Visualizations for Teaching and Learning Mathematics 


Author:                Lila Roberts 

Affiliation:           Clayton State University, USA 


The last 20 years have seen a dramatic evolution of software and hardware tools that can be used for creating demonstrations and models for teaching and learning mathematics. From flat 2D representations of three-dimensional objects, to robust 3D representations using sophisticated graphics software, to tactile objects manufactured using 3D printing, the important role of visualizations in teaching and learning mathematics cannot be overstated. Visualizations increase student understanding and learner motivation. 3D models of mathematically significant objects provide access to students who have low vision and have benefits for all students. In this paper, we investigate various free and proprietary software tools for generating 3D visualizations, both graphical representations and models created using 3D printers. Some practical aspects of 3D printing are also discussed. 


Abstract for 30005 


Future Knowledge in Engineering Mathematics and  PCET Calculator - Latest Research Results 


Author:                Peter Chew 

Affiliation:           PCET Multimedia Education, Bukit Mertajam, MALAYSIA 


In the talk, Peter Chew will discuss some of his vision regarding his research in Engineering Mathematics and his development in the PCET calculator latest research results. One area is to create new rules, methods or theorems to complement necessary information that can enhance certain area of engineering mathematics. The new discovery can make solving certain engineering mathematics problems easier, which in turn can assist our mathematics teaching and learning.  He also will share his PCET calculator special functionalities, where he will cover the following two main areas, Peter Chew Rule in solutions of triangle, as well as some special features of the PCET calculator, uses future knowledge in PCET calculator design such as Peter Chew Rule. He will present current issues, techniques and methods in his field of research and introduce his latest research findings. 

Recently, mathematics has become a challenging course for high school students around the world. Some mathematics researchers and educators have adopted different approaches to solve this challenge. Technical tools have had a major impact on advanced mathematics teaching and mathematics learning. Today's online calculators and math applications, photo math applications and calculators contain only the knowledge already stated in the book such as the sine and cosine rule. However, some areas of mathematics today are not complete, which leads to a certain field where some of the same questions are difficult to solve or cannot be solved. Therefore, a new product with some complete mathematical fields is needed today to help teach and learn in higher education, which is the main goal of the PCET calculator. In author’s opinion, the PCET calculator covers some specific areas of mathematics, and its design contains some knowledge that has not yet been covered in any textbook as of today. Another goal of the PCET calculator is to help students to gain interests and promote effective learning in mathematics. 

(摘要-在演讲中,周贻德将讨论他对工程数学的研究以及在PCET计算器最新研究成果。未来知识研究的目标是创建新的定律,方法或定理,以补充必要的信息,从而增强工程数学的某些领域。新发现可以使解决某些工程数学问题变得更加容易,从而可以帮助我们进行数学教学。他还将分享PCET计算器的特殊功能,其中将涵盖以下两个主要领域 周贻德定律在任意三角形,以及PCET计算器的一些特殊功能,使用未来知识在 PCET 计算器设计例如周贻德定律.他将在他的研究领域介绍当前的问题, 技术和方法,并介绍他最新研究成果。 

最近, 数学已成为全世界中学学生的挑战性课程。一些数学研究和教育工作者采用了不同的方法来解决这一挑战。技术工具对高等数学教学和数学学习产生了重大影响。现今的在线计算器和数学应用程序,照片数学应用程序和计算器只包含已经在书中陈述的知识例如正弦定律和余弦定律. 可是现今数学的某些领域还没有完整,这导至该领域某些同题难以解决或无法解决。因此,现今需要一个具有某些完整数学领域的新产品,以帮助高等教育中的教导与学习,这是PCET计算器的主要目标。 PCET计算器具某些完整数学领域, PCET计算器设计包含一些未来的知识,这些未来的知识还没有说明在现今任何高等教育书籍中。一些未来知识可以解决一些目前无法解决的问题。 PCET计算器的別一个目标是帮助学生维持兴趣,促进数学的有效学习。 

Video clips can be downloaded from the following URLs: ( (English) or (Chinese). 


Abstract for 30010 


Towards Dynamic Mathematics with Cabri: Where do we stand? Where do we go? 


Author:                Jean-Marie Laborde 

Affiliation:           CNRS Univ. Grenoble Alpes  Cabrilog, FRANCE 


It is common to say that technology and its impact on society is moving fast. Is that so sure? In the early 1980s, the rise and success of the Graphical User Interface or GUI concept, in direct manipulation, made possible the invention of dynamic geometry. 

It took some time for Dynamic Geometry Systems (DGS), namely Cabri and just after Geometer's Sketchpad, to spread throughout the various educational systems. For a long time, only the most innovative teachers have understood the real potential of the revolution brought by such environments. 

Nowadays, we must admit that dynamic geometry, and more and more widely and recently, dynamic mathematics have not yet completely revolutionized the learning and teaching of mathematics. 

In my presentation, I will explore the limitations of making profound, technology-based changes in mathematics education. I will present evidence of these constraints, but will also explain the evolutions and the solutions that the Cabri team is working on, especially with regard to the availability and maturation of the new Cabri-Express. Cabri Express being free of charge and running on the Internet on any device. 



Abstract for 30100 


The Evolution of Computer Algebra in the Teaching and Learning of Mathematics: Differential Equations 


Author:                Douglas B. Meade 

Affiliation:           University of South Carolina – Columbia, USA 


In this talk I will provide a quick tour through some of the different ways in which I have used Maple to improve student understanding of traditional topics in an introductory differential equations course, such as direction fields, the phenomenon of beats, and developing an understanding of solutions to first-order systems in phase space. In addition to highlighting improvements that became possible as technology, particularly Maple, has improved I will also discuss additional technological advances that can bring additional educational benefits for differential equations. 

The following two screenshots show two of the interactive examples that will be discussed in this talk. 

A collection of video clips and other resources ssociated with this presentation is available online. They can be found at 


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Abstracts for Full Papers


Abstract for 21689 


Scientific Calculators to Improve Students’ Critical Thinking Skills: An Evidence from Mathematical Exploration in Mathematics Classroom 


Authors:               Ariyadi Wijaya, Heri Retnawati,  
Wahid Yunianto, Pasttita Ayu Laksmi,  
Mutia Meilina, Pientha Glenys Amanti 

Affiliations:          Universitas Negeri Yogyakarta, SEAMEO QITEP in Mathematics, INDONESIA 


 In Indonesia, most teachers hold negative perspectives toward the use of scientific calculators for learning mathematics. Many teachers are worried if calculators will weaken students’ fluency in performing calculation and hinder students’ understanding of mathematics concepts. Considering this situation, the present study is aimed at exploring scientific evidences that calculators are beneficial for developing students’ conceptual understanding and critical thinking skills. The present study used a quasi-experimental research with a pretest-posttest control-group design. A total of 940 tenth graders from 21 schools in nine provinces in Indonesia participated in the study. The experimental groups learnt the topic of linear and quadratic functions by using scientific calculator Casio Classwiz, whereas the control groups learnt the same topic in regular activities utilizing paper-and-pencil approach. The use of Casio Classwiz in the experimental groups was not as a calculation tool, but as a tool to explore and investigate mathematics concept. The data analysis shows a significant difference for students’ critical thinking in which the experimental group (M =17.40, SD=21.13) outperformed the control group (M = 12.97, SD = 20.00); p = .000. This finding serves as a scientific evidence that using scientific calculators in exploratory mathematics activities could improve students’ critical thinking. 


Abstract for 21692 


Visualizations of Vector Fields and Ordinary  
Differential Equations with KeTCindy 


Authors:Takeo Noda, Setsuo Takato 

Affiliations:Toho University, JAPAN. 
Toho University, JAPAN 


 In this paper, we present some examples of visual teaching materials which are made by using KeTCindy and used in courses of ordinary differential equations and vector calculus. In undergraduate mathematics education, simple and accurate figures sometimes help learners to understand mathematical concepts. KeTCindy is a powerful tool for generating such mathematical figures. It uses Cinderella as a graphical user interface and creates TeX codes for the graphics. The outputs can be implemented not only in printed materials, but also in slides for screen presentation. Furthermore, KeTCindyJS, which is an extended version of CindyJS, can make those figures into interactive content, which can be viewed and manipulated on web browsers.


Abstract for 21718 


Connecting with Data: Exploring Statistical Concepts Using R 


Author:Leslie Chandrakantha 

Affiliation:John Jay College of Criminal Justice of CUNY, USA 


R is a language and environment for statistical computing and graphics. R has proven to be an excellent tool for understanding and visualizing statistical concepts. In this paper, we present how to use R for data analysis, elementary simulation, and understanding concepts in an introductory level statistics course. We consider examples from descriptive statistics, normal distributions, sampling distributions, confidence intervals, and hypothesis testing to demonstrate the use of R. 


Abstract for 21722 


The Delivery Role and Assessment Role of Computer-Based Technology in a Flipped University Mathematics Course 


Authors:               Wee Leng Ng, Kok Ming Teo, Khoon  
Yoong Wong, Kang Ling Michelle Kwan 

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


 In recent years, computer-based technology (CBT) has enabled university lecturers to teach their courses using non-traditional pedagogies. One such pedagogy is the flipped learning model. Under this model, students learn the basic content on their own using pre-class tasks and then come to class to engage in more challenging work such as solving difficult problems. CBT can play two important roles in flipped learning, namely to deliver learning materials efficiently and to assess student achievement effectively. This paper describes how these two roles were applied to a flipped Linear Algebra II course in the National Institute of Education (Singapore), taken by a group of student teachers (n = 15) over a 12-week period from January to April 2018. Their perceptions of flipped activities were gathered using weekly surveys, mid-semester survey, end-of-course survey, and end-of-course interviews. They generally agreed that flipped learning using CBT was helpful and enjoyable. As flipped learning becomes more common among university lecturers in Asian countries, it is beneficial to share experiences of utilising CBT to promote active learning of mathematics among university students. 


Abstract for 21727 


What does Research Say on the Use of Calculator to Improve Indonesian Students’ Mathematics Achievement? 


Authors:               Pasttita Ayu Laksmiwati, Ariyadi Wijaya,  
Heri Retnawati, Wahid Yunianto,  
Pientha Glenys Amanti 

Affiliations:          SEAMEO QITEP in Mathematics, INDONESIA, 
Yogyakarta State University, INDONESIA,  
Casio Singapore Pte Ltd, SINGAPORE 
Jakarta Rep Office INDONESIA 


 Technology has essential roles in mathematics teaching and learning. So, it is crucial to facilitate students with access to technology, such as the use of calculators in their learning process. This study was aimed to investigate the effectiveness of the Classwiz scientific calculator on the improvement of students’ mathematics achievement. This study was conducted in nine provinces of Indonesia, involving eleven senior high schools. The researchers employed a quasi-experimental research study with pretest-posttest control-group design with both qualitative and quantitative data collected and analyzed. Five lessons with context-based activities as interventions in the experimental group were designed to offer students OTL. The qualitative data were collected by using a mathematics test involving pre-test and post-test. Moreover, the study used classroom observation and field notes to collect qualitative data. The main focus of the research was students’ mathematics achievement. The students’ mathematics achievement was also analyzed based on the perspective of gender and school location (western, central, and eastern Indonesia). The investigation showed that the use of the calculator gave a significant impact on the students’ mathematics achievement. In conclusion, the finding of the study suggests that the use of calculators in mathematics learning could improve students’ mathematics achievement. 


Abstract for 21729 


How to Enhance Students’ Participations in Mathematics Learning Using Calculator? 


Authors:               Heri Retnawati, Ariyadi Wijaya,  
Wahid Yunianto, Pastita Ayu Laksmiwati,  
Mutia Meilina, Pientha Glenys Amanti 

Affiliations:          Universitas Negeri Yogyakarta, Mathematics Education Department Mathematics and Science Faculty, INDONESIA, 
Universitas Negeri Yogyakarta, INDONESIA,  
SEMEO QITEP in Mathematics, INDONESIA,  
Casio Education, SINGAPORE 


 One of strategy to improve the quality of mathematics learning is increasing students'' participations in learning. During this process, students are guided to find mathematical concepts through exploration, so that concepts can last longer in their minds to be utilized in problem solving. One strategy that can be used to enhance students in mathematics learning is integrating calculator utilization. This study was to describe the strategy to enhance students’ participations in mathematics learning in high school and vocational school using a calculator. This research is a narrative qualitative study. Data was collected by observation of 21 teachers from 8 provinces in Indonesia who integrate teaching and learning using calculator and documenting students’ responses to worksheets developed by researchers. Data analysis was carried out using Cresswell’s steps. The results showed that the teacher’s keywords to enhance students in mathematics learning on the topic function was integration the use of a calculator starting from the lessons design, prepared appropriate work sheets and carry out learning, guided students to discover concepts through learning activities, and facilitated students to collaborate and communicate during the process learning. The steps of learning that integrate calculator to explore mathematical concepts in detail are then discussed. 


Abstract for 21735 


Making the Mathematics Enjoyable in Science 


Author:                Minoru Ito 

Affiliation:           Tokyo University of Science, JAPAN 


 An introduction to current Japanese mathematics and science education for enhancing children’s motivation to concern about the nature of phenomena around us; for example, the size of the universe etc. We discuss how teachers can apply mathematics to science using materials that I have developed. We shall see some of those contents in details, with which science teachers can apply and make their mathematics and science education classes more enjoyable. 


Abstract for 21769 


Teaching Binary Number Concepts using Mathematic Magic Card Trick 


Author:                Janchai Yingprayoon 

Affiliation:           International College, Suan Sunandha Rajabhat University, THAILAND 


Mathematic Magic Card Trick is a creative mind-reading trick based on binary system that uses 5 tables. The tables are constructed from a principle of conversion between decimal and binary system. This magic cards can be used as an induction set before teacher starts introducing the binary system. This can make classroom fun and meaningful. 


Abstracts for Presentations with Abstract Only


Abstract for 21743 


Comparison of Some Properties of Triangles and Tetrahedrons 


Author:                Younbae Jun 

Affiliation:           Kumoh National Institute of Technology, SOUTH KOREA 


 The triangle is the simplest polygon, and the tetrahedron is the simplest polyhedron. They have very similar characteristics, and in many cases, similarity can be explained by analogy from 2D to 3D. 

Many properties of a triangle can be constructed with a ruler and a compass, but their construction in a tetrahedron is relatively difficult. Thanks to advanced technological tools such as interactive and dynamic geometry software, 2D and 3D figures can be easily constructed. 

In this paper, we compare some properties of triangles and tetrahedrons in terms of analogy, construct them using software, and improve students'' creativity by learning analogies. 


Abstract for 21744 


Mining Cryptocurrency for Mathematics Applications 


Author:                Russel Carson 

Affiliation:           BYU-Hawaii, USA 


 Cryptocurrencies, such as Bitcoin, provide practical examples of a host of different kinds of mathematical tools. However, many math educators are not familiar enough with them to use them as examples when teaching math. Some of the mathematical tools involved in cryptocurrencies are non-invertible and non-linear functions, modular arithmetic, convergent geometric series, and Boolean algebra. The basic mathematical structure of a cryptocurrency will be discussed, followed by some examples of applications that could be used in mathematics classes. 


Abstract for 21745 


Development of Animation of Sine Curves to Teach the Concepts of Trigonometric Functions 


Authors:               Koji Nishiura, Setsuo Takato, Kunihito Usui 

Affiliations:          National Institute of Technology, Fukushima College, Faculty of Sciences, Toho University, JAPAN,
Control Engineering, National Institute of Technology, Kisarazu College, JAPAN 


In this study, we develop an effective method for teaching mathematics and enhanced materials for teaching problematic concepts to students of upper secondary and higher education. 

Students often find understanding the concepts of trigonometric functions difficult; therefore, we selected trigonometric functions as a suitable topic to create enhanced teaching materials. We developed the animation of sine curves using KeTCindy mathematical software. 

The second author developed KeTpic to simplify the process of producing high-quality graphics in TeX documents. Further, he developed KeTCindy, which serves as an interface between KeTpic and Cinderella dynamic geometry software, and KeTCindyJS, which integrates the functions of KeTCindy to CindyJS, enabling easy creation of various interactive materials. 

Using KeTCindy and KeTCindyJS, we created a teaching material that includes the animation of sine curves along with the explanation of some mathematical concepts and problems. Furthermore, we conducted experiments involving eye movement measurement to analyze the effectiveness of the created teaching material. In this talk, we will present the animation of sine curves and the results of the experiments. 


Abstract for 21747 


Visualization of Complex Function Integral with GeoGebra 


Author:                Shigeki Ogose 

Affiliation:           Kawaijuku, JAPAN 


 You can visualize not only the result of a complex function integral but also how the partial sum (∑f(z)dz ) while z is a complex number) converges to the integral along its pass with GeoGebra. And it' is easy. Once you write the main program with GeoGebra, all you have to do is to input a function and compose a path with a mouse for almost all integrals found in undergraduate textbooks. 


Abstract for 21767 


Animations and Dissections of Rhombic Polyhedra 


Author:                Jia Yao 

Affiliation:           National Tsing Hua University, TAIWAN 


 In this paper, we will show 18 sets of animation of dissection of rhombic polyhedra: (1) Animation of Rhombic Polyhedra of type (2k+1,2k), k>1; (2) Animation of Rhombic Polyhedra of type (n,n+1), n>3; (3) Dissection of Rhombic Polyhedra of type (2k+1,2k); (4) Dissection of Rhombic Polyhedra of type (n,n+1); (5) Animations of Dissection of Rhombic Polyhedra of type (2k+1,2k) into Rhombohedra; (6) Animation of Dissections of Rhombic Polyhedra of type (n,n+1) into Rhombohedra; (7) Animation Connecting a cube of 48-face Rhombic polyhedra; (8) 120-face Rhombic Polyhedra; (9) 48-face Rhombic Polyhedra; (10) Six Cylinders each touching four others at one point; (11) Linkage; (12) Diagonal Burr; (13) Space Filling Polyhedra; (14) Pennyhedra in the form of Dodecahedra; (15) Pennyhedra in the form of Tricontahedron; (16) 12 Sticks; (17) Rhombic Polyhedra Complex; (18) Two-Piece Interlocking Puzzle. 

This work is valuable for 3D demonstration in a science exhibition. The geometric construction and animation serve as a good training for hand-eye coordination. Visual Geometry has attracted people’s attention for a long time. This work shows how the interactive geometry software Cabir3D can concretely develop new ideas in this field. 


Abstract for 30002 


The Different Meaning of the Pythagorean Theorem and the Area 


Author:                Michel Carral 

Affiliation:           Université Toulouse, FRANCE 


Through the different demonstrations of the theorem of Pythagoras, we show the different meanings of the Pythagorean Theorem even though they have the same statement. We look vertically the notion of “area” as a Euclidian notion and algebraic notion (is an affine invariant) from the primary to the university. We show the links between the trilinear (Cartesian, Barycentric, normal …), and some actions of the affinity. 


Abstract for 30200 


Integrating Hawgent Dynamic Mathematics into Teaching “The Formula for the Area of a Circle” from a TPACK Perspective 


Authors:               Jian-Lan Tang, Jun-Yu Wang, Jia-Zhen Song 

Affiliations:          Guangxi Normal University, 15 Yucai Road, Qixing District, Guilin, P.R.CHINA 541004 


The formula for the area of a circle is an important content of the sixth-grade mathematics, and it is a good material for students to cultivate their mathematical reasoning. Reasoning of the formula for the area of a circle has been a difficult point and focus of teaching. Aiming at the issue "why is the formula for the area of a circle like S= πr²? What is the relationship between the formula for the area of a circle and that of a rectangle or a triangle?" more than 90% of teachers and students only know the formula S= πr², but don’t know why the formula is like this, in a "knowing it but do not know why, let alone knowing how to know it" . This paper attempts to explore this issue through integrating Hawgent Dynamic Mathematics into teaching “the Area of a Circle” from a TPACK perspective. 


Abstracts for the Hands-On Workshops


Abstract for 21675 


Generating Educational Videos 


Author:                Juan Medina 

Affiliation:           Universidad Politécnica de Cartagena, SPAIN 


Educational videos are very popular nowadays, from MOOC courses to videos that can be found on online platforms such as Youku or YouTube. In this workshop session we will tell of our own experience in the creation of mathematics videos, which began in December 2005 with our portal – one of the first educational video portals in the world. We will discuss a variety of procedures for the creation of educational videos and also talk about how to translate videos into other languages efficiently. 


Abstract for 21679 


Exploring and Visualizing Statistical Concepts Using R Programming Environment 


Author:                Leslie Chandrakantha 

Affiliation:           John Jay College of Criminal Justice of CUNY, USA 


This workshop session will highlight the use of the R programming environment as a teaching tool in understanding basic statistical concepts. Introductory statistics concepts are relatively abstract for many students. Research has shown that the use of technology enhances the understanding of basic concepts. The following topics will be explored: descriptive statistics and charts, probability distributions, sampling distributions and the central limit theorem, hypothesis testing and confidence intervals. Participants will have hands-on experience in learning how R can be used to reinforce their teaching of statistical concepts in the classroom.


Abstract for 21681 


Revolutionary Maths with a Revolutionary Freeware: Cabri Express 


Authors:               Jean-Jacques Dahan, Jean-Marie Laborde 

Affiliations:          IRES of Toulouse, FRANCE 
Cabrilog, FRANCE 


In the same platform, Cabri Express provides an environment for 2D and 3D DGS, for coding and for algebra : you will approach math in a so different and motivating way. You will see how to use the online as well its offline version. 


Abstract for 21686 


Design and Implementation of Web-Based Dynamic Mathematics Intelligence Education Platform 


Author:                Guan Hao 

Affiliation:           Chengdu Institution of Computer Application, CHINA 


The information and communications technology (ICT) in education can effectively promote the development of ICT in subject by going to depth of a subject, and can practically assist the teaching process and improve the teaching quality. A dynamic mathematics intelligence education platform (DMIEP) is a tool of ICT in subject that goes to the depth of the mathematics subject. We will briefly reviews the development of the web-based dynamic mathematics software; proposes that the design principles of the web-based, especially a mobile internet-based, DMIEP software should be “opening and sharing”; and provides goals to be achieved, such as seamless integration with a system or a third party application in a cross-terminal manner, good expansibility, and intelligence. Under the guidance of the principles and goals, an opening and sharing DMIEP has been designed and implemented, having functions such as relationship-based drawing, geometric graphic transformation, dynamic measurement, dynamics and parameters iteration, geometrical theorem automatic reasoning, and opening interfaces. At present, the platform has been widely used in the mathematics teaching process of primary and secondary schools. 


Abstract for 21691 


Cabri 3D Workshop on Circumscriptible Hexahedron of type (4,4,3,3,3,3) 


Author:                Jen-Chung Chuan 

Affiliation:           Department of Mathematics, National Tsing  Hua University, Hsinchu, TAIWAN 300 


In this workshop, we are to construct, with Cabri 3D, a polyhedron that has two quadrilateral and 4 triangular faces, such that all edges are tangent to a sphere. 

Step 1: Planar construction 

Construct this configuration: 


Here PB/BQ = Golden Ratio 

Step 2: Inversion 


Invert points A, B, C to A’, B’, C’ respectively w.r.t. a sphere (not shown) tangent to the xy-plane at O. 


Step 3: Quadrilateral face 

Construct a the plane U parallel to the xy-plane passing through the center of inversion. 

Let PA’ meet the plane U at S. 

Let T be the reflection of S across the z-axis. 

Let TC’ meet PB’ at U. 

Construct the quadrilateral PSTU. 

Step 4: Convex hull 

The convex hull of PSTU and its reflection across the z-axis is the required hexahedron 


Step 5: Verification 

Verify that each edge of the hexahedron meets the sphere (with center of inversion as its south pole) at exactly one point. 



Abstract for 21618 


Discovering the Online Freeware Cabri Express 


Authors:               Jean-Jacques Dahan, Jean-Marie Laborde 

Affiliations:          IRES of Toulouse, FRANCE 
Cabrilog, FRANCE 


We will discover the possibilities of this free online calculator which environment contains both a 2D and a 3D DGS. We will see how to record the created files online and to open online files created by anybody. We will also see that this freeware allows the users to open activities (that can be multipage activities) created with the New Cabri. 


Abstract for 21703 


A Potpourri of Activities for the Classroom by using Free Software 


Author:                Guillermo Davila-Rascon 

Affiliation:           University of Sonora, Department of Mathematics, MEXICO 


This hands-on workshop is intended for undergraduate students and we will present some computer activities with GeoGebra and Maxima, for 

1) applying modular arithmetic for residue design 

2) exploring the reflections of light rays in curves, especially, in conics 

3) modeling and synchronizing chaos 

The workshop material is intended for two sessions of one hour each, and the materials (programs) will be provided by the instructor. 


Abstract for 21730 


How to Use KeTCindyJS to Produce Interactive Materials in HTML Format 


Authors:               Setsuo Takato, Koji Nishiura, Takeo Noda 

Affiliations:          Toho University, JAPAN, 
National Institute of Technology, Fukushima College, JAPAN, 
Toho University, JAPAN 


LaTeX is a great tool to produce high-quality educational materials containing finely-tuned mathematical expressions. However, it is not that easy for mathematics teachers to handle scientific artwork on their classroom materials. KetCindy enables them to overcome this limitation and prepare high-quality mathematical figures in PDF format through intuitive, user-friendly operations. In short, KeTCindy is a computer collaborative system based on two major components: the dynamic geometry software Cinderella, and LaTeX drawing tool KeTpic, the latter being a tool developed by ourselves. We organized a workshop on KeTCindy in ATCM2016. 

Recently, we have developed a powerful follow-up of KetCindy, called KeTCindyJS. It is a collaborative system of KeTCindy itself and CindyJS ( to produce interactive materials in HTML format from KeTCindy. KeTCindyJS does not create LaTeX files, but HTML files, which are closely related to LaTeX files from KeTCindy. Lots of samples of KeTCindy and KeTCindyJS can be found at the URL: 

Both systems, KeTCindy and KeTCindyJS, are provided for free, so they can be freely used anytime and anywhere. The package KeTCindy, including KeTCindyJS, is fully downloadable from CTAN (Comprehensive TeX Archive Network). Go to CTAN ( and search for ``ketcindy", or proceed directly to: CTAN/ketcindy (URL: 

This workshop will be organized as a brief introduction to KeTCindy and KeTCindyJS. KeTCindy needs LaTeX, R and Maxima to produce PDF files via LaTeX, so some samples will be only briefly demonstrated. The rest of the time will be assigned to experiences of KeTCindyJS. Teachers who are not necessarily frequent LaTeX users, but are interested in using better figures in their interactive materials, are highly welcomed. 


Abstract for 21755 


Designing Mathematics + Computational Thinking Lessons 


Author:                Weng Kin Ho 

Affiliation:           Nanyang Technological University, SINGAPORE 


This workshop provides the participants with a hands-on experience in designing lesson using the four design principles (Complexity, Data, Mathematics, Computability) that teach mathematics to high school students, harnessing on computational thinking paradigm. 


Abstract for 21765 



(Creating Personalized Dynamic Mathematics Software with Hawgent) 


Authors:               Chuan-Bo Zuo, Chu-Biao Lin 

Affiliations:          Hawgent Technolgoy Centre in Mathematics, CHINA, 
Foshan Guicheng Middle School, CHINA 


动态数学软件发展至今已有30多年,功能也越来强大、越丰富. 但是,我们发现,无论选择哪一款动态数学软件,都会感到有些非常实用的功能它没有实现. 事实上,现在或将来任何一个团队开发的任何一款软件都无法做到这一点,即:满足所有使用者的千变万化甚至是不断变化的需求. 那么允许每一位使用者创建个性化的、满足个性需求的工具与菜单就显得非常重要. 



(Dynamic mathematics software has been in development for more than 30 years, and its functions are becoming more powerful and richer. However, we found that no matter which dynamic math software you choose, you will find some very useful functions that are not implemented. In fact, now or in the future, any software developed by any team cannot do this, that is, to meet the ever-changing needs of all users. It is very important that the software allow each user to create personalized tools and menus that meet individual needs. 

In Hawgent, you can directly call all of her internal functions, or you can directly define your own functions, and even run all the functions in C# directly, thus serving us to create personalized dynamic math software. 

In this workshop, we will guide you how to use Hawgent to create personalized dynamic math tools and menus through several specific cases.) 


Abstract for 21768 


Creative Mathematics Hands-On Activities in the Classroom 


Author:                Janchai Yingprayoon 

Affiliation:           International College, Suan Sunandha Rajabhat University, THAILAND 


Many children find that Mathematics is difficult and boring. Nevertheless, they are curious and they love to have fun with exciting things around them. Appropriate activities can be found to stimulate them to have fun and love to learn Mathematics. The workshop will show ways to develop creativity in Mathematics and Technology Education to increase intellectual curiosity, to develop problem solving and thinking skills, to promote discovery as well as to unleash creativity. In the workshop, the participants will share with each other how to make Mathematics lessons more meaningful, effective and interesting, how to cultivate intrinsic motivation for learning Mathematics, and how to develop thinking abilities, problem-solving skills and creativity. 

Every participant will receive a fun and creative activity pack. Samples of creative hands-on activities will be demonstrated as follow: Curves in Nature, Reaction Time Test, Simple Balance, Mathematics of Robot arms. 


Abstract for 30000 


Mathematical Explorations using Spreadsheets 


Author:                Jonaki B Ghosh 

Affiliation:           Department of Elementary Education, Lady Shri Ram College for Women University of Delhi, INDIA 


Spreadsheets such as Microsoft Excel or LibreOffice Calc can be very useful tools for performing scientific calculations and statistical data analysis. They enable us to simulate problems and also create, visualise and analyse mathematical models. The power of spreadsheets can be leveraged in a course on mathematical modeling where problems may be explored through simulations and graphical analysis. This hands-on workshop will focus on the use of a spreadsheet for exploring some mathematical concepts and problems, which lend themselves to investigations. Problems discussed in this session will include interesting paradoxes related to probability such as the Birthday Paradox and the Monty Hall problem. It will also include explorations in chaos and dynamical systems, simple regression analysis and applications of mathematics to cryptography. Participants will get a hands-on experience 


Abstracts for the Poster Session


Abstract for 21707 


About Decimal Expansion (Approximate Value) of Irrational Numbers 


Author:                Kimio Seishi 

Affiliation:           Tokyo University of Science, JAPAN 


It is easy for the junior high school students to understand the size of displaying fractions and irrational numbers as decimal approximations. However, understanding is difficult because the definition is complicated, especially in the case of irrational numbers. 

In junior high school in Japan, the image of the square root is described using a figure such that the length of one side of the square is the square root of the area of the square. However, in order to calculate an approximate value, the length of one side is estimated and it is calculated by calculating how close the square is to the area of the square to be obtained. In other words, it uses the method of calculating the approximate value by calculating many times each time while increasing the decimal places one by one. 

Here, I will introduce two methods systematically to calculate the approximate values. We set the length of one side of a square to the square root of its area. 

First, we consider a rectangle of the same area instead of a square, and from the average of the vertical and horizontal lengths of this rectangle. We find the next new vertical and horizontal length. Repeat this operation. However, the rectangle is transformed into a square to obtain the length of one side. (I will refer this as Horizontal and Vertical Averaging.) 

Second, we consider a rectangle in which one square is known and the other square to be determined is mixed and halved. The length of the square, which is known to be half the area and the other side. To create a new square, we repeat this operation to find the length of one side. (I will call this as Square Dichotomy.) 

From the above, the approximate value of P, for example, can be easily calculated systematically as follows by using a graph calculator. 

At first, after entering the length of the first rectangle, enter (ANS + P / ANS) / 2 and repeat until the value no longer changes. 

Second, enter the length of one side of the square you know, then enter (ANS^2 + P) / 2 / ANS and repeat until the value no longer changes. 

          © Douglas B. Meade, University of South Carolina, USA