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Innovative Technology in Mathematics: New Ways for Learning, Teaching, and Researching Mathematics



ATCM 2018, Yogyakarta, Indonesia

  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

Abstracts for Invited and Plenary Papers


Computational Thinking in the Mathematics Classroom

AUTHORS: Jonaki B Ghosh

AFFILIATIONS: Lady Shri Ram College for Women, University of Delhi, New Delhi, India

In recent years computational thinking has been identified as one of the key analytical abilities required for mathematics and science learning. The rapidly changing nature of scientific and mathematical disciplines and the need to prepare students for careers in these disciplines have been the primary motivation for bringing computational thinking into classroom practices. Papert (1980) was the first to stress on the importance of computational thinking by referring to the affordances of computational representations for highlighting powerful ideas. Over the decades many researchers have attempted to define computational thinking. According to Wing (2006), computational thinking involves solving problems, designing systems and understanding human behaviour by drawing on the concepts fundamental to computer science. However operationalising computational thinking for the k – 12 classroom remains a key pedagogical challenge.

In this paper we shall argue that computational thinking is critical to learning mathematics. The ability to deal with challenging problems, represent ideas in computationally meaningful ways, create abstractions for the problem at hand, break down problems into simpler ones, and assess the strengths and weaknesses of a representation system are some important aspects of computational thinking for mathematics learning. In order to encourage this kind of thinking, specific investigatory tasks need to be integrated into the curriculum. This article describes two research studies where participants engaged in various aspects of computational thinking by working on investigatory problems. In the first study, 30 pre-service teachers explored the Tower of Hanoi puzzle through multiple representations. As they engaged with the puzzle through pictorial, numerical, symbolic and graphical representations, they displayed multiple paths of inquiry, used co-operative problem solving, dealt with recursive and explicit relations and developed greatly in their mathematical thinking. In the second study, a grade 12 student researched one – dimensional Cellular Automata using Mathematica, a computer algebra system and NICO, an open access software tool. The exploration included generating multiple representations of the 256 Elementary Cellular Automata (ECA) and categorising them based on their evolutionary patterns. The studies highlight the nature and characteristics of mathematical tasks which require computational thinking and illustrate that such tasks need to find their place in school mathematics curricula.  


Synchronization of Chaos

AUTHORS: Guillermo Davila-Rascon

AFFILIATIONS: University of Sonora

Synchronization phenomena pervades our daily lives and even the whole cosmos. For example, many of our bodies physiological functions are synchronized to the day-night cycle (circadian rhythm); thousands of pacemaker cells n a cluster called sinoatrial node, fire at unison in order to maintain the regular beats of our hearts; around the world, thousands of fireflies come together along riverbanks and synchronize their flashes in an amazing spectacle that has been noticed and reported for over three centuries; laser beams are also examples of perfect synchronization of trillions of atoms; electrons flowing in a superconductor is another.

We will talk about these phenomena and present some mathematical models of synchronization for pedagogical purposes and with the aid of technology, in order to motivate our main topic, namely, the synchronization of chaotic systems.

At first glance, and since chaotic dynamics is characterized by its sensibility to initial conditions, it is striking that two chaotic dynamical systems can get synchronized and have a common behavior and the same evolution, through a coupling mechanism or by means of a forcing. We will explore some instances of synchronization of coupled chaotic systems and how this topic can be used for a deeper understanding of chaos. We will also mention some relevant applications in physiology and fluid dynamics.


Friend or Foe: The Dangers of Dependence on Online Platforms for Conceptual Understanding


AFFILIATIONS: Holon Institute of Technology

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

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


Mathematical Walks with Mobile Technology

AUTHORS: Ron Lancaster

AFFILIATIONS: Ontario Association for Mathematics Education, National Council of Teachers of Mathematics, Origami USA, International Brotherhood of Magicians

Imagine if students went out for a walk on a regular basis to view the world through a mathematical lens and to get some exercise. Imagine if they went outside to make measurements, to collect data, to observe how things change, and to notice the little things that we miss all the time. Imagine if students used their mobile device to take photos and videos of their encounters with mathematics. Imagine if you can, a world where students do not wonder why they are studying mathematics - they know because they have seen where mathematics lives.

(Edited on January 10, 2019)

  • Mathematical walks with mobile technology
  • Posing Powerful Questions by Exploring World Flags
  • Experimenting with equations and their graphs

  • ABSTRACT FOR 21641

    Correcting Errors: Putting Elementary Topology to Work

    AUTHORS: Juan Medina, Jose A Vallejo

    AFFILIATIONS: Universidad Autonoma de San Luis Potosi, Departmento de Matem ticas y Estadıś tica Universidad Polit cnica de Cartagena

    Nowadays, virtually any field of Mathematics has some interesting application to technology, and topology is no exception. We show an example of a math lab session illustrating the use of basic notions of metric spaces (and a little bit of probability) in the context error correction codes, through the use of Hamming’s distance. The CAS Maxima is very well suited to this task, and we also show how to write a simple set of commands for solving some basic exercises. Finally, we also present some online digital resources complementing the contents presented here, so the combined materials can be used in a blended learning program.

    ABSTRACT FOR 21642

    Creating a Symbiotic Relationship between Epistemology of Combinatorics and STEM Teaching Process

    AUTHORS: Haslinda Ibrahim

    AFFILIATIONS: School of Quantitative Sciences, college of Arts and Science, Universiti Utara Malaysia, 06010 Sintok Kedah, Malaysia

    Combinatorics is a domain in mathematics that concerns with procedures of arranging of objects into patterns of specified rules. Meanwhile, STEM (science, technology, engineering, and mathematics) is a term used to group these related subjects. Recently, educators are looking for practical ways to teach STEM. In combinatorics domain there are steps that need to be followed to design any specific problems. We are optimistic that these steps in combinatorics will be the basis to establish a framework for teaching STEM. Thus, in this paper, we would like to present the historical roots of combinatorics towards proposing a framework for teaching STEM disciplines. Based on combinatorics approach, we attempt to establish a practical method when resolving problems namely explore, discover and develop (ExDiD). This method was demonstrated to a group of students from several schools in Kedah, Malaysia who attended the “I C D’ BEAUTY IN STEM” workshop. We illustrated the three steps involved to establish the procedure/method needed in resolving any complex scenario. The method was well received by the participants of the workshop.

    ABSTRACT FOR 21651

    Can Secondary School Mathematics Students be Taught Computational Thinking?

    AUTHORS: Weng Kin Ho, Wendy Huang, Chee Kit Looi, Suranteran Longanathan

    AFFILIATIONS: Nanyang Technological University, Ministry of Education, Singapore

    The thrust of this paper is to demonstrate that secondary school mathematics students can indeed be taught to think computationally. We do so by applying APOS theory to design mathematics lessons that intentionally integrate Computational Thinking (CT) and Mathematical Thinking (CT), and implementing these tailor-made lessons in authentic classroom environment of a Singapore secondary school. Based on lesson observation notes and post-lesson discussions with teachers, we present evidence to confirm the hypothesis that integrating CT and MT deepens students'' understanding of mathematical concepts.

    ABSTRACT FOR 21654

    An Expert Approach of the Different Ways to Use the New Cabri and the Richness of its Connections with the Online Freeware Cabri Express

    AUTHORS: Jean-Jacques Dahan


    The acceleration of the provision of more and more sophisticated technological tools and more sharp on- line applications, instead of encouraging the teachers of mathematics to integrate more technology into their teaching might lead them rather to take refuge in their fundamental approaches. What I mean is that the profusion of tools like tablets and smartphones, the accessibility to a multitude of online courses (for the teachers as well as for their students), the poverty of their in-service training and a final examination which does not practically take into account the use of the technologies recommended by the curriculum together might bring the teachers to return to a more classic teaching leading to the assessment of technical skills. So we must have a pertinent approach of the role of the use of technology during the math lessons compatible with the actual needs of the teachers. We (experts) must rethink the role of the calculators, the role of software (DGS or CAS) and especially simplify them in order to provide to teachers and students a microworld enhancing a normal but rich practice of mathematics. The new Cabri is an environment especially dedicated to authors who want to create multipage activities for students. These activities can be modified very easily by teachers (there are three available modes: the author, the teacher and the student modes). Recently the developers of this Cabri, put online a software free of charge, Cabri Express, containing a part of the tools of the New Cabri : this freeware is a calculator connected to a 2D environment that can be extended to a 3D one. This revolutionary environment can be a response to those who want simple, accessible and powerful tools for primary and middle schools. What is exciting is that the files created online can be downloaded on your computer and also files available on your computer can be opened online. More than that, activities created with the new Cabri can be opened with Cabri Express. We aim in this paper to show with examples all the didactical possibilities of both Cabris. We will also present how the restricted 3D environment provided by these Cabris can be extended to a richer one closer to the Cabri 3D one. We will see that the proposals of the expert allow teachers to show the known techniques of DGS to enhance a link with the real world, creativity, discovery and pleasure of proof when it is possible.

    ABSTRACT FOR 21657

    Strong Algebraic Manipulation Skills are not Adequate for Cultivating Creativity and Innovation

    AUTHORS: Wei-Chi Yang

    AFFILIATIONS: Radford University

    In this paper, we use some college entrance exam practice problems from China to highlight some essential algebraic manipulation skills that are required by high school students from China. Next we explore various scenarios by assuming if technological tools are available to learners, how we may see many unexpected surprising outcomes. For many countries, requiring college entrance exams is inevitable and sometimes is the only fair channel of selecting qualified students to enter a college. However, we hope examples provided in this paper can serve a purpose of advising the decision makers in education systems by allowing students to explore mathematics with available technological tools. After all, creativity and innovation do not come by giving one correct answer alone.

    Exploiting Excel’s Data Table Creatively in the Study of Mathematics

    AUTHORS: Graham Supiri and Deane Arganbright

    AFFILIATIONS : Divine Word Univ., PNG (Emeritus)

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

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

    ABSTRACT FOR 21664

    MYMA Calculus: A Comprehensive e-Textbook and Online Homework System

    AUTHORS: Douglas MEADE, Phillip YASSKIN

    AFFILIATIONS: Department of Mathematics, University of South Carolina; Department of Mathematics, Texas A&M University

    While many calculus textbooks are now available in an electronic form, most are glorified PDF files. The MYMA Calculus project is developing a complete three-semester calculus book whose native format is purely electronic. The only pages are webpages; one webpage for each theme. The content of each theme’s webpage is dynamically generated. While some content is static, viz. definitions and statements of general rules and theorems, many of the accompanying figures involve a combination of user control (via sliders) and, when appropriate, animation.

    For example, the MYMA Calculus project replaces a sequence of figures showing secant lines through two points on the graph of a function approaching the tangent line through a single point on the graph of a single function with an interactive animation of this convergence process. Users can control the animation, changing the points and specific functions involved. In a similar manner, the examples used to demonstrate an idea are not static either. To encourage students to work through the examples, and to develop a sound approach to applying the concepts, the steps in the solution process appear one at a time. And, if a student wants additional examples, MYMA Calculus can produce additional examples algorithmically. In the same manner, not only are homework problems generated algorithmically but student work can be assessed step-by-step, providing context appropriate feedback at each step in the solution process. This presentation showcases the current state of MYMA Calculus, discusses the various technologies used, and outlines the remaining tasks to complete the project.

    ABSTRACT FOR 31000

    Doing and Learning Mathematics with

    AUTHORS: Barry Kissane

    AFFILIATIONS: School of Education, Murdoch University, Australia

    Previous Asian Technology Conferences in Mathematics have highlighted at least three important aspects of technology. Firstly, the place of computer software of various kinds for both students and teachers has been regularly demonstrated. Secondly, all conferences have also made clear the importance of personal technology in the form of hand-held calculators, offering a range of mathematical capabilities to suit the wide needs of students from the middle years through to the undergraduate years. Thirdly, especially in recent years, conferences have illustrated that the Internet has become increasingly evident and important in Asian communities, improving access to information as computers and tablets become more widely accessible. In this session, we will explore an amalgam of these three important aspects of technology, in the form of a new online web-based facility for mathematics developed by CASIO and located online at . The online app allows users to regard their computer or tablet screen as a form of virtual scratch paper onto which intelligent sticky notes can be placed, used and manipulated, as the following examples illustrate. The first example shows some graphing and a calculation:

    Sticky notes use various mathematical capabilities, especially graphing, geometry and statistics, as well as general calculation, and provide access to various mathematical tasks relevant to school and early undergraduate mathematics.

    page14image13080  page15image408

    The preceding example shows how a junior secondary school student might undertake geometric drawings, and then use the software to explore the inherent relationships involved. Geometric drawings are dynamic, so that the measurements involved will change as the object is changed, and thus provide a stimulus for understanding why the relationships are evident and how to formally establish them.

    The following example suggests that key statistical work, such as that encountered by students in introductory courses, is included within the software.


    Guest access to the app is freely available through a standard browser, while more sophisticated capabilities (such as those below which are reliant on CAS capabilities and others such as the evaluation of Poisson probabilities shown above) require a subscription.

    The focus of the Special Session will be a demonstration of some of the capabilities provided by this new environment, as well as a consideration of their significance for mathematics education together with some discussion of the question of what kinds of mathematical affordances are required to address the needs of typical learners across a suite of secondary and undergraduate courses.

    Many ATCM delegates learned and taught mathematics in the second half of the twentieth century, when the available equipment was often limited to a pencil and some paper. We are now in the first half of the twenty-first century, where the available equipment often includes a computer or a tablet and the Internet. Teaching, learning and assessment might reasonably be expected to change in this new environment and the major purpose of this session is to explore one kind of possible direction for change.


    ABSTRACT FOR 31001

    Using Dynamic Geometry Software in High School in Vietnam: The Case of Geometrical Construction Problems in Space

    AUTHOR: Nguyen Chi Thanh

    AFFILIA TION: Vietnam National University-University of Education

    Mathematical software in general and Dynamic geometry software in particular (DGS) becomes a little bit popular in Vietnam secondary education, especially in urban areas where school teachers can afford to by materials or to have access in Internet. With the comprehensive and radical educational reform which is carried out in Vietnam, several directions are given by the different level of educational managements, from Ministry of education and training (MOET) to Department of education and training (DOET). This paper aims to investigate the use of DGS in teaching and learning geometry in school. After analyzing curriculum and mathematic textbook in grade 11, more precisely in the topic “geometry in space”, this study addresses the following questions: what are the characteristics of geometrical knowledge introduced in grade 11; what are the difficulties and obstacles of using DGS in learning and teaching geometry in space; what are the conditions of using effectively a DGS in teaching and learning regarding geometrical construction problem at high school in Vietnam.

    Abstracts for Full Papers

    ABSTRACT FOR 21601

    E-learning System with Computer Algebra based on JavaScript programming language

    AUTHORS: Takuya Kitamoto, Masataka Kaneko, Setsuo Takato

    AFFILIATIONS: Yamaguchi University, Toho University

    Usually, e-learning system provides multiple choice questions and students choose the correct answer. However, some e-leaning systems, such as moodle with STACK plugin, are equipped with computer algebra systems and can deal with mathematical expressions. Such e-learning system let users input an answer directly in the form of mathematical expressions, hence is friendlier to the users. One of the problems in those systems is that they often requires a special web server that has computer algebra systems installed, which is not an easy condition to elementary, junior, and senior high schools. This prevents classroom teachers from creating their own e-learning materials.

    In this paper, we present a web-based e-learning system which can deal with mathematical expressions. Our system is constructed with JavaScript programming language, and only requires a web server with PHP support (database and standalone computer algebra system are unnecessary). More concretely, our e-learning system has the following features;

    1.     (1)  It contains computer algebra JavaScript library, and can deal with mathematical expressions.

    2.     (2)  It only requires a web server with PHP support (database and standalone computer algebra system are unnecessary).

    3.     (3)  It runs on standard browsers such as Google Chrome, Firefox and Opera, and can be used on PCs, smartphones, tablets.
    It has editing functions itself, and a new e-learning material can be created on browsers. The above features make it easy for classroom teachers to create their own e-learning materials.

    We illustrate how to create e-learning materials and give some examples created by our system.

    ABSTRACT FOR 21604

    Overcoming Identified Learning Difficulties in Trigonometry Using Casio Classwiz fx-991EX

    AUTHORS: Ika Wulandari, Triana Harmini, Siti Suprihatiningsih

    AFFILIATIONS: SMK N 2 Wonosari, Darussalam University of Gontor, STKIP PamaneTalino

    The objective of research was to find out the trigonometry learning difficulty and the solution to the learning difficulty with classwiz fx-991EX help. The subject of research consisted of 96 10th graders of Public Vocational High School 2 Wonosari in the school year of 2017/2018. The research method employed was descriptive qualitative one. The result of research showed that learning difficulty was related to: (a) reasoning & understanding of prerequisite material (geometry and real number operation) including: symbol, distance, Pythagorean theorem, circle, triangle, angle in radian and degree, pi concept, and root form rationalization, (b) reasoning and understanding of function including: representation (numeric, graphic, symbolic/algebraic, verbal), and inter-representation relation, (c) reasoning and understanding of trigonometry function including: definition of function based on unit, definition of function based on right triangle sides ratio, multiple representation and property in trigonometry main function. Data of diagnostic test result was supported with result of interview with informants selected representing any types of error in diagnostic test. Learning difficulty revealed was improved using remedial teaching with casio classwiz fx-991EX help.

    ABSTRACT FOR 21605

    Improvement of the Usability of Figures in STACK by Use of CindyScript and CindyJS

    AUTHORS: Kenji Fukazawa, Yasuyuki Nakamura

    AFFILIATIONS: National Institute of Technology, Kure College, Nagoya University

    Last year at ATCM2017, we reported the enhancement of figures by appending the capability of interactive manipulations in STACK, which is the e-learning system for mathematics. In this enhancement, we fix the parameters and functions to create figures that can be manipulated interactively, but it is favorable that parameters and functions are selected randomly when a question is presented to students. In this paper, we explain how randomly selected parameters and functions can be used to create figures with interactiveness by use of CindyScript and CindyJS. Finally, several examples are presented.

    ABSTRACT FOR 21612

    Effects of the Traditional Flipped and the In-Class Flipped Classroom Models on the Students'' Performance in Geometry: A Comparative Analysis



    Mathematics Teacher Educators, Mathematical Society of the Philippines, Statefields School, Inc Bacoor, City Cavite

    This study focused on analyzing and comparing the effects of the Traditional Flipped Classroom Model and the In-Class Flipped Classroom Models on the students’ performance in Geometry. A quasi- experimental design was utilized with 60 Grade 9 students from two sections of Statefields School, Inc., during the third term of school year 2017 – 2018 as respondents. Two sections from Grade 9 were randomly chosen to form the control group and the experimental group of the study. The control and experimental group were exposed to the Traditional Flipped Classroom Model and the In-Class Flipped Classroom Model respectively. Pretest and posttest on selected topics in Geometry were administered to determine the mathematical performance of the respondents who were matched according to their pretest scores. Findings revealed that learning took place among students in both Flipped Classroom Models as evidenced by the increase in their mean score. Furthermore, through the use of the paired samples t-test, significant gains were observed in the mathematical performance of the students under both Flipped Classroom Models. Thus, the Flipped Classroom Models are both effective in improving the students’ performance in Mathematics. The use of the t-test for independent samples pointed out that there is no significant difference between the mathematical performance of the respondents when they were exposed to the Traditional Flipped Classroom Model and the In-Class Flipped Classroom Models. No significant difference was observed between the gain scores of the respondents as well. Therefore, both classroom models are equally effective in improving the students’ performance in Mathematics.

    ABSTRACT FOR 21613

    Blended Learning Station-Rotation Model: Effects on Grade 10 Students’ Performance in and Attitude Toward Mathematics


    AFFILIATIONS: DE LA SALLE UNIVERSITY - MANILA, DE LA SALLE UNIVERSITY - DASMARINAS, Philippine Council of Mathematics Teacher Educators, Mathematical Society of the Philippines, City of Bacoor National High School – Green Valley Cavite Philippines

    This experimental study was conducted to determine the effects of Blended Learning Station-Rotation Model on Grade 10 students’ performance in and attitude toward Mathematics. A quasi-experimental design was utilized to two intact classes consisting of 60 students of City of Bacoor National High School – Green Valley. Two groups of students were exposed to different strategies: the control group was taught using the traditional approach while the experimental group was taught using the integration of Station- Rotation Model. A 40-item multiple choice type DepEd validated test and a modified Fennema-Sherman Mathematics Attitude Scale were used in gathering data.

    The respondents of both groups were given pretest and posttest to measure their performance in and attitude toward mathematics and then the t-test for dependent and independent samples were used to determine whether there exists significant difference between the respective mean gain scores of the two groups. Findings showed that there is a significant difference in the respective pretest and posttest mean performance of each group although only the experimental group showed significant difference in their attitude toward mathematics. Moreover, there exists a significant difference in the mean gain scores of the two groups as well as in their attitude toward mathematics in favor of the experimental group. Also, a weak positive correlation between the performance and attitude toward mathematics was noted from the experiment group.

    ABSTRACT FOR 21615

    Learning and Teaching of Group Theory through Visualization using Graphs

    AUTHORS: Faqir Bhatti, Khawaj fahd

    AFFILIATIONS: Riphah Institute of Computing and Applied Sciences, Riphah Institute of Computing and Applied Sciences, Riphah International University, Lahore.Pakistan

    Group Theory is one of the difficult courses in undergraduate level. This is not only difficult for the students from learning point of view, but it is also difficult for the instructors from teaching perspective. The main difficulty in both teaching and learning is due to the abstract nature of this course. The topics are mainly based on proving theorems about abstract structures without having any visual representation. This lack of visualization limits the capability of most students to understand basic concepts of group theory.

    In this paper we discuss a model that provides visualization in the form of a directed labeled graph. Using this graph it becomes easier for the instructors to teach and for the students to learn fundamental concepts of group theory. Moreover, in this paper we explain how different group axioms like closure property, identity element and the inverse can be visualized using the proposed graph representation.

    ABSTRACT FOR 21617

    On Applications of Technology to Understanding Hierarchies of Elementary Geometry

    AUTHORS: Yosuke Sato, Ryoya Fukasaku, Katsusuke Nabeshima

    AFFILIATIONS: Tokyo University of Science, Tokushima University

    Problems and theorems of elementary geometry are categorized roughly into four hierarchies, affine, metric, Hilbert and Tarski geometry. Difference between the latter three is especially hard to make out.

    In this paper, we give algorithmic descriptions for these hierarchies. Our descriptions together with sophisticated programs of computer algebra systems such as Grobner basis computation, primary decomposition of a polynomial ideal and real quantifier elimination enable teachers to understand these hierarchies. They also could help teachers to make high quality problems of elementary geometry.

    ABSTRACT FOR 21620

    Simulating Confidence Intervals for Mean and Variance using Real Data in R Programming Environment

    AUTHORS: Leslie Chandrakantha

    AFFILIATIONS: John Jay College of Criminal Justice of CUNY

    Research has shown that the use of computer simulation methods as an alternative to traditional methods enhances the understanding of the statistical concepts. The increasing availability of technology allows instructors and students to use computationally intensive methods such as simulation. This paper presents the use of real data and a simulation approach to help students understand the confidence intervals for population mean and variance. Use of real data makes the concepts more real for students and enhances their ability to ground the new concepts in their existing knowledge. We use the R programming environment for simulating repeated sampling from a fairly large dataset and compute the approximate sampling distributions of sample mean and variance. We notice that the confidence intervals for population variance work poorly if the normality assumption is violated.

    ABSTRACT FOR 21625

    Improving Student Learning Outcomes by Using an Applet in Class

    AUTHORS: Russel Carlson


    With the availability of computers in the classroom, a variety of active learning methods can be used to teach mathematics. This paper presents an example of using an applet to teach a mathematical principle during a unit on cryptology. As part of this unit, two sections of a general education math course were taught how to break the Vigenère cipher using an applet, one by lecture and demonstration of the applet, and the other by allowing the students to experiment with the applet for themselves. The students who had learned by experimenting with the applet retained more knowledge of the process used to break the code than those who had learned by lecture and demonstration. This paper discusses the Vigenère cipher and the method of teaching it using the applet, and summarizes the difference in outcome between the two sections of the course.

    ABSTRACT FOR 21627

    Empirical Research on Mathematical Inquiry Based Dynamic Software in the Lesson Study Training

    AUTHORS: Li HongYun, ChunLan Wu

    AFFILIATIONS: Beijing Institute of Education

    This research designed activities using theories of teachersˇ ̄ learning, two teachers from different schools participated in the teaching of the law of triangle altitude using dynamic software. The research showed that students were interested in mathematics inquiry using GeoGebra or geometerˇ ̄s sketchpad, found the exploratory direction using the dynamic of the software, built experiences in mathematical activities, and posed good mathematic problems under the proper guidance. The performance of students had encouraged the teachers who participated in the research, changed the belief in the integration of technology into the teaching, deepened the understanding of the content of mathematics, and influenced the understanding of the studentsˇ ̄ ability to learn mathematics. The research showed that there are some points in an effective mathematical inquiry based on technology: negotiation with the trainees who participate in the teaching to a consensus(motivation); discussion of teaching strategies according the content of mathematics, analysis of the students and the characteristics of technology(understanding); teaching practice using technology ( practice); reflection process based evidence (reflection).

    ABSTRACT FOR 21630

    Integration of Classwiz Calculator to Deliver Hots Concepts of Mathematics Learning in Indonesian Senior High School

    AUTHORS: Khairuddin Budiman

    AFFILIATIONS: SMA Negeri Nurussalam Aceh Timur Indonesia

    The National Examination of senior high school in Indonesia began to apply some higher order thinking skills (HOTS) type problems. Among the four subjects that were tested in the National Examination, Mathematics is the lowest. Higher order thinking learning process directed at critical thinking, logical, metacognition, and creative is still a new thing. Not only students but teachers also face difficulties. Higher-order thinking involves demonstrating an understanding of information and reasoning rather than merely recalling information. Higher order thinking skill is one of the priorities in students'' thinking skills in the Indonesian curriculum. Mathematics textbooks present HOTS type questions to be discussed together in the classroom. The use of affordable technology is a must in helping students understand the concept of Mathematics. Scientific calculator Casio Claswizz can be used as a learning tool to deliver HOTS concept.

    ABSTRACT FOR 21631

    How To Increase Lower Order Thinking Skills To Higher Order Thinking Skills Through Modifiying Problems Using The fx-991 ID Plus Scientific Calculator

    AUTHORS: Chandra Sri Ubayanti

    AFFILIATIONS: SMA Negeri 1 Fakfak Papua Barat, Ikatan Guru Indonesia (IGI), Master Casio Indonesia

    The demands of high-level thinking are not only the goal of learning in school but so for students to solve the problem in their life in the 21st century which is full of challenges. In fact the possession of high level thinking skills based on Bloom’s taxonomy revision requires the fulfillment of three lower order thinking skills i.e knowing (C1), understanding (C2) and applies (C3). Without the mastery of the three, higher level thinking skills comprising analyzing (C4), evaluating (C5) and being creative (C6) cannot be reached. This paper will exemplify how to modify low-level math questions into high-level questions assisted by the fx 991 ID Plus scientific calculator. This paper can be considered by teachers who want their students to think critically in mathematics.

    Papers with Abstract Only

    ABSTRACT FOR 21570

    Spaced Learning Strategy in Teaching Mathematics

    AUTHORS: Ace Ceremonia, Remalyn Casem

    AFFILIATIONS: Don Mariano Marcos Memorial State University, Mathematics Teachers Association in the Philippines, Don Mariano Marcos Memorial State University, Mid La Union Campus

    Students’ low mastery of the lesson in Mathematics is one of the alarming problems confronted by Mathematics teachers (Department of Education, 2016). It is in this light that this study was conceptualized to determine the effectiveness of spaced learning strategy on the performance and mastery of DMMMSU Laboratory High School students (Grade 7) in Mathematics. This study used the descriptive-experimental research design, specifically the pretest-posttest control group design. The main instrument used to gather data was the pretest-posttest which was subjected to validity and reliability tests. It was found out that the experimental and control groups were comparable in the pretest and posttest. Comparison on their gain scores revealed significant difference with the performance of the experimental higher than the control group. It was also found out that the effect size of using the spaced learning strategy was large. This indicates that the intervention is effective in increasing the performance and mastery of high school students in Mathematics. It is recommended the use of the Spaced Learning Strategy to improve the performance of the high school students in Mathematics.

    ABSTRACT FOR 21579

    Order of Addition: Theory and Algorithms

    AUTHORS: Rahul Mukherjee

    AFFILIATIONS: Indian Institute of Management Calcutta

    Consider a process where the output is influenced by the order of addition of m different components. Examples abound in chemistry where the performance of a reaction depends on the order of adding the reagents. Similarly, in genetics, the properties a phylogenetic tree depend on the order of addition of taxa. Interest lies in understanding precisely how the order of addition exerts its influence on the output. For instance, does the addition of component 1 before component 2 or vice versa have a significant impact?

    Answering questions like the above calls for testing various possible orderings. It is, however, impractical to include all such m! orderings in an experiment, because m! grows very fast with m. Therefore, one needs to select carefully a subset of the possible orderings so that by testing these alone one can draw valid inference on the issues of interest in the most efficient manner. This leads to the problem of optimal selection of a subset of orderings from a statistical perspective.

    The present paper develops a theoretical foundation for solving the aforesaid optimization problem which is highly nonlinear. Analytical determination of an optimal subset of orderings is, however, found to be intractable beyond a certain point. Hence, we devise step-down and exchange algorithms which are easy to understand and employ, and which are also seen to work very fast. Thus, these can form the basis of a dedicated software for handling this important problem, with immediate applicability to chemical and biological sciences.

    ABSTRACT FOR 21582

    A Mathematical and Experimental Studies of a Free Surface Wave

    AUTHORS: Jeongwhan Choi, Sungha Yoon

    AFFILIATIONS: Korea University

    This research concerns forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small bump on a horizontal rigid flat bottom. We present the mathematical and experimental results of the surface waves when a non-dimensional wave speed, called Froude number, is near 1. A KdV equation with forcing is derived and studied mathematically and numerically. Various numerical results are given and some corresponding experimental results, generated by using a thin elliptical moving obstruction on the bottom of a long water tank, are also given.

    ABSTRACT FOR 21585

    Clustering Induced Binary Classification with Asymmetric Misclassification Cost

    AUTHORS: Uttam Kumar Sarkar

    AFFILIATIONS: Indian Institute of Management Calcutta

    In binary decision problems where cause-effect relationship is not understood well, the use of classification tools and techniques of artificial intelligence and machine learning in decision making is increasingly gaining importance among domain experts. The fields of application are as diverse as education, psychology, medicine and industry, to name only a few.

    Commonly used classification methods often express predictive accuracy as percentages as a simplification effort which ignores the asymmetry that may be inherent in the underlying misclassification matrix. In reality, such asymmetry can have serious implications. For example, in an educational or psychological test, the impact of wrongly assessing an inferior student as superior can be grossly different from that of doing the other way round. Similarly, in prediction of cancer, the cost implications of predictive error is asymmetric as suspecting a patient with benign cancer as one having malignant cancer, vis-à-vis certifying a patient with malignant cancer as having a benign variety, will have asymmetry in terms of financial and mental trauma befalling the patient and her family. Improving predictive accuracy becomes very challenging when this asymmetry, expressed as a ratio, is specified a- priori.

    This paper proposes a novel classification method where the specified asymmetry ratio is incorporated in a mathematical function which is used in evaluating and improving the quality of a binary clustering problem whose outcome then influences the original classification algorithm for fine-tuning its predictive accuracy. A classifier for solving such problems of relevance to society is built and its performance is compared with commonly used classifiers. The proposed method is computationally feasible for commonly occurring problems involving large data sets.

    ABSTRACT FOR 21586

    On estimation of Gini Index for grouped data using asymmetric loss



    Gini inequality index is one of the most famous and widely used measures of income inequality derived from the Lorenz curve. It is also used as an inequality measure in several other social and economic development parameters such as health, educational attainment, business concentration, etc.

    A large literature exists dealing with estimation of Lorenz curve and Gini index for grouped data. It is well known that in the absence of information about within class variability, grouped data leads to an overestimation of the curve and an underestimation of the Gini index. Two approaches have been used by researchers to address this problem – either by reducing the bias through a model based approach, or by defining non-parametric bounds on the Gini index.

    The present article attempts to address this problem by introducing an asymmetric loss function to estimate the Gini index. The loss function is suitably chosen so that underestimation becomes more serious than overestimation. An alternative estimator of the Gini index is proposed with respect to such asymmetric loss function, and its dominance over the usual estimator in terms of estimation risk is studied. Simulation studies are carried out to demonstrate the procedure.

    ABSTRACT FOR 21602

    Data Simulation with Markov Chain Monte Carlo, Gibbs Sampling, and Bayes (Beta-Binomial) Methods as the Parameter Estimations of Spatial Bivariate Probit Regression Model

    AUTHORS: Dewi Retno Sari Saputro, Yuanita Kusuma Wardani, Shaifuddin Zuhdi, Purnami Widyaningsih

    AFFILIATIONS: Department of Mathematics, Universitas Sebelas Maret, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret

    A probit regression model is a regression with categorical dependent variable in dichotomy or binary form. The dependent variable value of the probit regression states the probability of certain issue. In some cases, the application of this model considers the influence of area (spatial effect). The dependency tendency to the close regions is known as autocorrelation in spatial data. Due to this matter, the parameter estimations of ordinary least square (OLS) method cannot be used thus it is substituted by the simulation technique, which is a method in randomly raising data. The method includes direct and indirect simulations. The latter has Markov Chain Monte Carlo (MCMC) method with Gibbs Sampling algorithm, which is the order in conducting certain distributed random data sampling by understanding the required distribution. In this case, the beta binomial distribution is applied. A data simulation with Gibbs Sampling algorithm can be conducted by knowing the required distributions of each variable used for R software beforehand. This research purposes to define the parameter estimation value of spatial probit regression by applying MCMC and Gibbs sampling methods with R software. The results show that the parameter estimations of spatial bivariate probit regression model by simulating through Gibbs sampling algorithm (R software), in which β ̂ is the independent variable parameter and ρ ̂ is the spatial lag autoregressive coefficient. The simulation with the first-value determination rise result data β=(0,1,-1)^ ,ρ=0.7 deciding n=400, m=10, and k=6 show the estimations for β ̂=(0.01205,0.98709,-0.9675) and ρ ̂=0.68523.

    ABSTRACT FOR 21626

    Misconceptions of Differential and Integration Technique among Engineering Students: A Case Study l

    AUTHORS: Prof.Dr. Sarwar Jahan Abbasi, Ambreen Zehra

    A F F I L I A T I O N S : University of Karachi, Karachi-Pakistan

    The aim of this study is to identify the problems of engineering students in Pakistani universities, face to learn derivatives and integrations techniques in their first semester calculus course. The study based on data collected of 500 engineering students of Hamdard University, Karachi-Pakistan. In the process of data analyzing students’ misconception lie as follows:

    1.     1)  Product and quotient rule of derivatives was applied on integration problems.

    2.     2)  ln function was integrated by using derivative technique.

    3.     3)  Integration of trigonometric function was evaluated by using derivative technique.

    4.     4)  In substitution technique of integration students observed confused in both derivatives and integration.

    The results of this study will identify the students’ problems and will help them improving their technology aided learning skills.

    ABSTRACT FOR 21629

    Unlocking the Power of Data: Teaching Statistics to First- Year Life Science Students

    AUTHORS: Maree Skillen

    AFFILIATIONS: Western Sydney University

    We live in a data-driven world, and yet many students are likely to have trouble with statistics due to non-cognitive factors such as negative attitudes or beliefs towards this area of mathematics. Students can be puzzled when faced with questions which may include: How does a statistical model differ from a mathematical model? What are the differences among the sample distribution, the sampling distribution, and the population distribution? In an experiment, what effect does the sampling method have on the results? What are the implications of the use of processes of random selection and random assignment? Can a small sample yield accurate estimates of population parameters?

    Changes to the structure, content and teaching approaches including less theory and more data, making data analysis central, building intuition, incorporating technology, fostering active learning, and using context to develop statistical inference have supported student engagement and improved performance levels in an undergraduate statistics subject.

    Specific areas of focus to be addressed in this presentation will centre on summarising curriculum developments implemented; explore how the application of real-data aided to support student understanding; consider possible challenges, limitations and implications for teaching statistics to undergraduate students; and, identify possibilities for further research. Examples used will relate to a first-year University subject which focused on introducing statistical concepts and techniques to support decision making in the life sciences.

    ABSTRACT FOR 21655

    Professional Development for Mathematics Teachers: Using a New Hypermedia Video-Case Tool

    AUTHORS: Hang Chen, Jian-sheng Bao

    AFFILIATIONS: School of Mathematical Sciences, East China Normal University, Shanghai, China

    The purpose of this paper is to introduce a new model for teacher professional development, which focus on teaching practice based knowledge and competency. The model is structured into three main parts: the knowledge base for instruction; the process of classroom teaching and learning; and, the teachersˇ ̄ believe system. This programme involves the creation of several series of hypermedia video-cases on teaching and learning designed to facilitate mathematics teachersˇ ̄ professional development. Each of these video-cases consists of lesson clips, case questions, interviews with experts, comments by peers, responses by students and other related resources. The study has implications pertaining to the use of technology in teacher development, the production of hypermedia video-cases, as well as research on case-based pedagogy and pedagogy in general.

    ABSTRACT FOR 21660

    Conjugacy Maps and Dynamics of a Family of Optimal Fourth-Order Multiple-Zero Finders

    AUTHORS: Young Hee Geum

    AFFILIATIONS: Dankook University

    We have developed a family of optimal fourth-order multiple-zero finders with rational weight functions and discussed the stability of the strange fixed points for conjugate map. In addition, the stability surfaces for the fixed points are displayed and the relevant dynamical analysis has been handled from the viewpoint of stability analysis as well as in terms of parameter spaces and dynamical planes associated with basins of attraction.

    ABSTRACT FOR 21663

    The Mathematics Class Using the Tablet PCs

    AUTHORS: Tsutomu Ishii

    AFFILIATIONS: Bunkyo University, Japan.

    In this paper, we will consider about the reason of using Tablet PCs. Japanese teachers don’t use Tablet PCs so much. The reason is the research question. Japanese Government is supporting financially for teachers to use the projectors and computers in the classroom. Recently, teachers using Tablet PCs are increasing. But there are teachers who don’t use Tablet PCs, too. In the 1st step, several studies were considered. And we checked its educational possibility and expectation of Tablet PCs. For example, in early childhood showed that teaching with Tablet PCs contributed significantly to the development of children’s mathematical ability. In the 2nd step, we analyzed the mathematics class using Tablet PCs about the percentage of the elementary school 5th grade in Japan. We analyzed us carefully about understandings of children and instructions of teacher. In the 3rd step, as the case study, we considered whether instruction by Tablet PCs was proper in detail. And we argued the result based on several studies. In this paper, at the end, we pointed out 2 points. First, children’s understanding deepens by Tablet PCs. Next there is little contribution to bring the flexibility and creativity up.

    Abstracts for the Track of Hands-On Workshops

    ABSTRACT FOR 21573

    Using a Graphics Calculator for Learning Mathematics

    AUTHORS: Barry Kissane

    AFFILIATIONS: Murdoch University

    Graphics calculators were developed more than thirty years ago to meet the needs of students learning mathematics in the secondary school and the early years of college; significant research has established that they can be effective tools for both students and teachers. Recent models, such as the CASIO fx- CG50, can be used to support student learning and teachers teaching about many mathematical ideas related to real numbers, functions, equations, statistics, probability, matrices, sequences and calculus. The graphics environment allows students to generate and use graphs of functions and data, explore the use of spreadsheets, investigate random phenomena, and explore significant mathematical concepts in powerful ways. This introductory hands-on workshop will allow participants to experience some of the learning opportunities available, and become aware of free resources to support their work. Previous experience with graphics calculators will not be assumed.

    ABSTRACT FOR 21574

    Learning School Mathematics with ClassWiz, an Advanced Scientific Calculator

    AUTHORS: Barry Kissane

    AFFILIATIONS: Murdoch University

    While scientific calculators have been available since the 1970s, early use in schools often concentrated on numerical computation alone. Recent advanced versions have been developed explicitly to suit the needs of mathematics education throughout the secondary school years and offer more powerful capabilities, potentially valuable for learning; modern advanced calculators provide powerful learning opportunities for many aspects of mathematics in the secondary school and early college years. In addition, capabilities such as tabulation, spreadsheets, equation solving and other advanced features give students access to efficient calculation. This workshop will focus on a variety of ways in which the CASIO fx-991EX ClassWiz calculator can be used to enhance both teaching and learning, drawing on published materials and videos developed by the author for free use by teachers. A variety of topics will be addressed, including calculus, functions, equations and statistics, as well as aspects of mathematical modelling. Previous experience with these calculators will not be assumed.

    ABSTRACT FOR 21575 A New Online Environment for Doing and Learning Mathematics

    AUTHORS: Barry Kissane

    AFFILIATIONS: MurdochUniversity is a new online software environment from CASIO that provides access to useful tools for calculation, graphing, geometry and statistics, designed to support secondary school mathematics. In this hands-on workshop, participants will use the software in a browser to explore its potential for students

    learning mathematics and for teachers teaching mathematics. provides users with virtual Paper and allows them to attach Sticky Notes to the paper for mathematical purposes. Sticky Notes are used to add mathematical expressions, graphs, tables, spreadsheets, and geometric shapes. They can be attached wherever users want on Paper to create original materials for learning and teaching. can be accessed from computers or tablets; participants are welcome to bring their own devices to the session if they wish, equipped with a browser and Internet access, although it is not necessary to do so. Previous experience with will not be assumed.

    ABSTRACT FOR 21584

    Teaching Statistics and Visualization with QR Codes

    AUTHORS: Wei Ching Quek

    AFFILIATIONS: Singapore Polytechnic

    The New Classroom Standard : Casio Classwiz is loaded with useful spreadsheets, statistics functions and is able to generate relevant QR codes to retrieve additional graphical information. This workshop is consists of two activities:

    1.     (1)  Getting Started
    Begin with introduction to new features in ClassWiz, followed by exploring essential features of the statistics and spreadsheet function, and to visualisations of charts by QR codes generated.

    2.     (2)  Problems Solving/Discussion
    Explore some applications of in statistics, distribution and spreadsheet. Participants will be able to gain further insights to the problem.

    ABSTRACT FOR 21598

    STACK: an Online Assessment System for Mathematics

    AUTHORS: Yasuyuki Nakamura, Yuko Ichikawa, Takahiro Nakahara, Yoshinori Miyazaki, Saburo Higuchi, Kentaro Yoshitomi

    AFFILIATIONS: Nagoya University, National Institute of Technology, Tokyo College, Sangensha LLC., Shizuoka University, Ryukoku University, Osaka Prefecture University

    STACK is an online assessment system for mathematics, science and related disciplines. STACK, developed by Sangwin (2013) of the University of Edinburgh, uses Maxima as its CAS to evaluate students’ answers. STACK not only assesses the mathematical equivalence of students’ answers but also generates outcomes, such as providing feedback according to the mathematical properties of students'' answers. The feedback function is implemented using the potential response tree (PRT) mechanism. PRT is an algorithm that establishes the mathematical properties of students'' answers and provides feedback specifically to each student. STACK is an open source system and users can develop required functions. For example, a new input type, FlickMath, was developed for using STACK on mobile devices (Nakamura and Nakahara 2016). FlickMath allows the input of mathematical expressions by the flick operation. The flick operation is carried out by placing a finger on the prepared keyboard, shifting the finger vertically or laterally, and subsequently releasing it.

    We would like to demonstrate STACK and have a workshop in which participants can solve questions and make some typical questions.

    The STACK demonstration website should be accessed via:

    ABSTRACT FOR 21618

    Discovering the Online Freeware Cabri Express

    AUTHORS: Jean-Jacques Dahan, Jean-Marie Laborde

    AFFILIATIONS: RES of Toulouse, Cabrilog

    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 21619

    Modelling the Moebius Strip with Cabri 3D

    AUTHORS: Jean-Jacques Dahan, Jean-Marie Laborde

    AFFILIATIONS: RES of Toulouse, Cabrilog

    Let us enter in the 3D space allowed by Cabri 3D. We will learn how to use Cabri 3D in modelling with mathematical but tricky constructions a Moebius strip

    ABSTRACT FOR 21639

    Exploring Concepts and Applications of Mathematics Using Casio Teaching Tools

    AUTHORS: Jonaki Ghosh

    AFFILIATIONS: Lady Shri Ram College, Delhi University

    This workshop session will highlight the use of the CASIO CG - 20 graphics calculator as a teaching tool in the high school mathematics classroom. The emphasis will be on developing conceptual understanding in various mathematical topics and also in enabling students to perform mathematical investigations. Participants will be given a hands-on experience on the calculator. Problems will be taken up from the topics of functions, calculus, matrices, simulation of experiments in probability and statistical inference. Applications of mathematics to topics such as cryptography, genetics and chaos theory which can motivate the high school student towards investigations will be highlighted.

    ABSTRACT FOR 21640

    Image Processing with the CAS Maxima

    AUTHORS: Guillermo Davila-Rascon, Jose Antonio Vallejo

    AFFILIATIONS: University of Sonora, Autonomous University of San Luis Potosi

    This hands-on workshop is intended to provide some pedagogical tools for math teachers in order to motivate students on the utility of elementary matrix calculus and some of its applications, by using the computer algebra system Maxima

    The workshop is divided in two parts: An elementary one where basic Maxima commands for producing and manipulating binary matrices, as well as their image rendering, are introduced. We also show how the Maxima palette can be used for handling colored images (2 hours).

    In a second and more advanced part we show how to handle and manipulate more complex pictures with Maxima, focusing on some of its characteristics like color and contrast. We introduce, for this purpose, some Maxima commands that allow us to make a deeper analysis of a given picture, like getting the frequency distribution values of its coloring and the respective histogram. We also perform an equalization process that renders a softer image in which extreme colors are replaced with softer tones, closer to the average (2 hours).

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