ICMS 2014 Session: Mathematical Web/Mobile Interfaces, Editing and Visualization

ICMS 2014: Home, Sessions


Aim and Scope

Since the last ICMS there has been rapid technological innovation in human-computer interaction. Touch-enabled interfaces, mobile devices, cloud computing, and Web 2.0 are changing the way users engage with technology. Aligned with this spectacular technological development, computer tools for mathematical editing and scientific artwork generation and visualization (such as scientific and graphical editors, word processors, mathematical utilities for the web, and others) are becoming ubiquitous in our current digital world. Today's authors are usually required to type scientific papers with embedded mathematical content and sophisticated pictures and other artwork, leading to an increasing demand of computer solutions for those purposes.

The aim of this session is to focus on the mathematical aspects of all these exciting advances, bringing together researchers, developers and users of mathematical web/mobile interfaces, mathematical editing and scientific visualization tools, to explore how evolving technologies may influence the direction of current developments in those fields. The session accepts papers addressing research and development issues and presenting technologies oriented to mathematical Web/Mobile interfaces, editing and scientific visualization. Papers exploring educational experiences by using these technologies in an original way are also welcome.

Topics (including, but not limited to)


Submission Guidelines


  1. Math Web Interfaces and the Generation Gap of Mathematicians

    Andrea Kohlhase (Jacobs University, Bremen, Germany)

    Abstract: New technologies and interfaces are changing the way users engage with technology, mathematicians are no exception. In a previous study we found some interesting attitudes/practices of professional mathematicians with respect to search interfaces, that sets them apart from other web searchers.

    For instance, they strategically use the search engine "Google" for finding specific objects of interest, i.e., specifically looking for identifiable information. This comprises that mathematicians not only know previous to the search what exactly they are looking for and thus, anticipating the exact search result, but also that they know how to formalize the search query. In contrast, Google is best-known for its browsing quality, i.e., getting an impression of what data are available (e.g., for an overview or for inspiration) and possibly refining the search as a consequence. Very simply put, mathematicians would look for a definition of "Cauchy sequence" instead of looking for information about "Cauchy sequence".

    But there were noticeable differences: some expected the Google experience in all text-based math search interfaces, others restricted it purely to Google itself. It was conjectured that the generation gap in the math community could account for this. This would have the particular consequence that math search interfaces for the new generation of mathematicians have to be different as the usability criteria inbetween generations are not shared.

    To explore mathematicians' reasoning for their adoption of Google and selected other math search interfaces, we present an elaboration of the above study by differentiating between younger and older mathematicians. In particular we analyzed a total of 16 repertory grid interviews with mathematicians in that respect. The method is of semi-empirical nature: On the one hand, it allows to get deep insights into the perception of math search interfaces thru deconstruction and intense discussion of each subject's idiosyncratic constructs. On the other hand, the interviews can be analyzed with a General Procrustes Analysis to obtain statistically significant correlations between either the elicited constructs or the interfaces.

    The analysis data indeed imply a generation gap that is, for example, explicitly not triggered by different technology competencies (as some general studies suggest), but by a time/experience-induced change of convenient helpers into tools into media as "extensions of man" (McLuhan). Such a statement has direct consequences for further developments of math interfaces. E.g., it is common design practice to favor a step-by-step adaptation strategy from one interface to a new interface. But such a strategy would be rather destructing for mathematical practices, as mathematicians would know exactly of the difference and had to reevaluate the helper as a medium.

    In a nutshell, this study explores whether and if so, how math search interfaces are distinctly perceived by younger and older mathematicians and we offer first design implications.

  2. An Implementation Method of a CAS with a Handwriting Interface on Tablet Devices

    Mitsushi Fujimoto (Department of Mathematics, Fukuoka University of Education, Japan)

    Abstract: In 2003, I developed AsirPad, a computer algebra system (CAS) on Linux PDA Zaurus [1]. It had a handwriting interface for mathematical expressions and could communicate with a CAS engine Risa/Asir through the OpenXM protocol. It was used to present a lecture on RSA cryptography at a junior high school. Ordinary calculator is not available because encryption and decryption in RSA use division of large numbers. The students learned how to encrypt/decrypt their messages through calculations by AsirPad. They could input mathematical expressions and calculate without any special training. This experience encouraged us to explore the possibility of mobile devices in math classroom.

    In 2010, we started a project to develop a Math e-Learning system Mathellan for pen-based mobile devices [2]. We are planning to rebuild AsirPad for a client of Mathellan. However, the mainstream of the current mobile devices is shifting from PDA to smartphones or tablet devices. Therefore, we needed a new development environment. AsirPad consists of two main components: a CAS engine and a handwriting interface. We used a cross-build environment by QEMU and chroot to make an executable binary of Risa/Asir for the Android platform. We can build C/C++ source code in this environment as if we are in a self -build environment. It means that we do not need to modify source code for cross-build. Furthermore, we adopted a cross-platform application framework Qt to build the handwriting interface of AsirPad. Qt can be used to build applications for various operating systems: Windows, MacOS X, Linux, Android and iOS. We can develop GUI for various mobile devices with the same source code by Qt.

    In this talk, I would like to explain the details of this implementation method.

    [1] M. Fujimoto and M. Suzuki, AsirPad -- A Computer Algebra System with a Pen-based Interface on PDA, Proceedings of the Seventh Asian Symposium on Computer Mathematics, Korea Institute for Advanced Study, (2005) 259--262. Demo movie, http://www.inftyproject.org/demo/AsirPad_Demo.zip
    [2] M. Fujimoto and S. M. Watt, An Interface for Math e-Learning on Pen-Based Mobile Devices, Proceedings of the Workshop on MathUI 2010, (2010) 10 pages.

  3. A Touch-based Mathematical Expression Editor

    Su Wei (Lanzhou University, China), Paul S. Wang (Kent State University, USA), Li Lian (Lanzhou University, China)

    Abstract: MathEdit (http://www.mathedit.org) is an interactive tool for creating and editing mathematical expressions on the Web. MathEdit is an open-source program implemented in standard XHTML and JavaScript to run in regular browsers. The tool supports both WYSIWYG editing and command-line editing operations. MathML is the primary internal representation for MathEdit. But other formats such as infix, OpenMath, and LaTeX are also supported. Recently, a touch version of MathEdit is under development. In MathEdit touch version, we design a virtual mathematical keyboard for users to enter mathematical symbols and expression templates conveniently. The layout of each key on the virtual mathematical keyboard is based on the utilization rate of mathematical symbols and templates. Such a layout of keys could make users enter the expressions more easily and quickly. Another important user interface aspect of a WYSIWYG editor is navigating to the precise point within the expression where an editing operation is to take place. The size of human fingers and the lack of sensing precision make precise touch screen interactions difficult. In MathEdit, editing operations, such as insert, edit, delete, covert, and replace, are relative to the current node, which is highlighted visually and visual navigation refers to moving the current expression to different positions in the expression being edited. The design, implementation, and application of MathEdit touch version will be introduced in the presentation.
  4. A Tablet-Compatible Web-Interface for Mathematical Collaboration

    Marco Pollanen (Trent University, Canada), Jeff Hooper (Acadia University, Canada), Bruce Cater (Trent University, Canada), Sohee Kang (University of Toronto, Canada)

    Abstract: Mobile devices and Web 2.0 are changing the way we interact and communicate with one another. Immediacy of communication has become the norm, and less technical users are increasingly using such communication technologies, in particular on mobile and touch-enabled devices. Most of the interfaces for these devices are text-based, and not directly suitable for mathematical communication and collaboration, but there is an increasing need for such tools.

    Online mathematical collaboration has traditionally been the domain of mathematical experts. Increasingly, mathematical novices, such as students in low-level mathematics service courses, need to communicate mathematics online and face great difficulties in communicating their ideas easily and accurately using currently-available interfaces. In particular, the shift towards mobile devices necessitates a re-examination of digital input methods and interfaces for mathematics.

    Input interfaces for mathematical expressions range from text-based TeX input to WYSIWYG palette-based graphical interfaces, but have largely been developed for document creation and not for real-time collaboration. Communication of mathematical expressions differs from text-based communication in that input of mathematics expressions requires the use of many symbols that are not represented in the computer keyboard and have a nested hierarchical structure (i.e., a 2-dimensional layout). This makes developing intuitive interfaces for novice mathematical users challenging, particularly on touch-enabled devices.

    Thus far, little attention has been paid to developing mathematical input methods for touch-enabled devices. Lack of a physical keyboard and small screen sizes are problematic, in particular, if we wish to create cross-platform interfaces consistent with PC-counterparts.

    In this talk we introduce a cross-platform synchronous communication interface for mathematical collaboration. It is designed to be intuitive to multiple user groups ranging from novice to expert. We demonstrate that it is possible to create a Web-based communication interface that simultaneously incorporates TeX-based, palette-based, and pen-based input methods, yet is compatible with both touch-enabled tablet and traditional keyboard-mouse user interface principles.

    While we concentrate on real-time communication, the design principles we introduce may be valuable for the design of other mathematical user interfaces on touch-enabled devices, such as with CAS interaction.

  5. Development and Evaluation of A Web-Based Drill System to Master A Basic Math Formula Using The New Interactive Math Input Method

    Shizuka Shirai (Mukogawa Women's University, Japan), Tetsuo Fukui (Mukogawa Women's University, Japan)

    We have proposed a new mathematical expression input method in terms of conversion from linear character strings using a colloquial style that is analogous to a language translation system. With this system, the list of candidates for the mathematical expression from the user input string is shown in WYSIWYG and if all the elements which user wishes for are chosen, then the formatting expression process is complete. As a result, with only a keyboard user will be able to input almost any mathematical expression without learning a new language or syntax.

    In this paper, we present a web-based drill system named DIGITAL-WORK which allows students to master some basic mathematics formulas using the formula editor which is implemented by this new interactive math input method. DIGITAL-WORK is an e-Learning system with the direct math input method above mentioned and with a function for evaluating an inputed answer in mathematical expression automatically.The purpose of this study is to investigate whether students can smoothly learn some basic mathematics formulas with our drill system. As a result of teaching the basic mathematics formula to 20 junior high-school students with our drill system, 85% of students have answered that learning with DIGITAL-WORK is more fun than learning on paper.

  6. GNU TeXmacs: Towards a Scientific Office Suite

    Massimiliano Gubinelli, Joris van der Hoeven, Francois Poulain, Denis Raux

    GNU TeXmacs is a free mathematical text editor, which can also be used as an interface for several computer algebra systems and other mathematical software, such as Scilab, GNU R, etc. Its primary aim is to offer an alternative to LaTeX, which achieves a similar typesetting quality, but also provides a user friendly WYSIWYG interface. This user friendliness makes TeXmacs suitable for a broader audience, such as high school education.

    In addition to the possibility to write structured text with sophisticated mathematical formulas, TeXmacs also integrates several other tools which are useful in the everyday workflow of scientists and beyond. For instance, we provide an integrated graphical drawing editor, a versioning tool for visualizing structured differences, as well as a tool for laptop presentations. Interfaces exist for several external computer algebra systems and other mathematical software. Such interfaces can be used via shell sessions, as computational engines for spreadsheets, and directly from within ordinary text or mathematical formulas.

    During the year 2014, we plan to release the entirely relooked version 2.1 of TeXmacs, with several major improvements. First of all, TeXmacs is now based on the Qt graphical toolkit, which allows us to provide native versions under Windows, MacOS and Linux. The graphical user interface has also been completely redesigned, to make it even more user friendly and compatible with standard interface conventions under several operating systems. The quality of our converters from and to LaTeX has also been greatly improved and we now provide a native converter to PDF.

  7. IFSGen4LaTeX: Interactive Graphical User Interface to Generate and Visualize Iterated Function Systems for LaTeX

    Akemi Galvez Tomida (University of Cantabria, Spain), Kiyoshi Kitahara (Kogakuin University, Japan), Masataka Kaneko (Toho University, Japan)

    Fractals are among the most exciting and intriguing mathematical objects ever discovered. In addition to their impressive visual beauty, they have many interesting and useful mathematical properties. Because of these reasons, they have also become the subject of great interest for the mathematical community. In fact, several computer tools for generating fractal images have been created during the last few decades. However, their manipulation and visualization for mathematical editing purposes is still challenging, since the typical approach of embedding them as figures is not very flexible and do not always perform properly in terms of computer storage and visual quality.

    In this talk we present a new interactive, user-friendly graphical user interface (GUI) for generation and visualization of a very popular class of fractals, called Iterated Functions Systems. The program, called IFSGen4LaTeX, is particularly designed for proper LaTeX mathematical editing in WYSIWYG mode. In a typical working session, IFS fractals are created interactively from scratch and visualized by using the IFSGen4LaTeX GUI; simultaneously, its engine generates source code that, once inserted in LaTeX and compiled, generates the same fractal images in LaTeX. This process leads to substantial savings in time, memory storage and visual quality.

  8. Computer Software for Representation and Visualization of Free-Form Curves with Bio-inspired Optimization Techniques

    Andres Iglesias (University of Cantabria, Spain & Toho University, Japan), Akemi Galvez Tomida (University of Cantabria, Spain)

    The accurate mathematical representation and graphical visualization of free-form shapes is a recurrent problem in many theoretical and applied domains, such as numerical analysis, data fitting, approximation theory, computer-aided design and manufacturing (CAD/CAM), geometric modeling and processing, computer graphics and animation, and so on. In fact, a number of different techniques have been developed during the last decades to tackle this issue. However, the problem has proved to be more elusive than it appeared at first sight, and the scientific community is still looking for new mathematical and computational methods to solve it.

    Among the myriad of methods proposed in the field, those based on bio-inspired optimization techniques are receiving increasing attention during the last few years, due it their ability to perform well under very unfavourable conditions, such as multimodal, multivariate, nonlinear optimization problems, noisy data points, little knowledge about the problem to be solved, and many others. Unfortunately, such methods are hard to use and require a certain level of expertise in order to apply them efficiently.

    In this talk we propose a new interactive, user-friendly computer software to deal with this problem. Our software consist of two major components:

    - a computational kernel, comprised of a set of modular and extensible libraries for several bio-inspired optimization techniques, to compute an accurate mathematical representation of the shape in terms of free-form curves.
    - a powerful graphical user interface (GUI). Given an input data consisting on a cloud of data points (acquired through a 3D laser scanner or other digitizing device), our GUI allows the user to choose the bio-inspired technique of his/her preference, set up the control parameters interactively, and obtain the mathematical representation and graphical visualization of the underlying shape.

  9. Creating Interactive Graphics for Mathematics Education Utilizing KETpic

    Shunji Ouchi (Shimonoseki City University, Japan), Yoshifumi Maeda (Nagano National College of Technology, Japan), Kiyoshi Kitahara (Kogakuin University, Japan), Naoki Hamaguchi (Nagano National College of Technology, Japan)

    Our project team has been developing effective teaching materials for mathematics (including statistics) education, predominately at the early grades of tertiary education.

    In teaching mathematics, there are instances when we need to graphically present mathematical concepts and solid figures to clarify students' understanding of them. For the last few years, we have been creating graphics that illustrate these various concepts dynamically through careful utilization of KETpic style files, ketlayer and ketslide. ketlayer enables us to precisely lay out graphics and symbols on LaTeX documents in the exact spot we wish them to be embedded. Using ketslide, we can easily produce LaTeX presentation slides incorporating high-quality graphics.

    Examples we will look at include an interactive graphic produced in order to dynamically present the correspondence relation between the z-plane and w-plane in a complex function w=f(z). Here, when clicking on different regions of a rectangular grid representing the z-plane we can see how the corresponding regions on the w-plane change accordingly. A second example is of an interactive graphic developed to clearly illustrate the line of intersection of two solid surfaces. In this case, we can easily show the cross-section of the intersection following the cut. Other graphics we have produced will also be introduced.

  10. Generating Data of Mathematical Figures for 3D Printers with KETpic and Educational Impact of the Printed Models

    Setsuo Takato (Toho University, Japan), Naoki Hamaguchi (Nagano National College of Technology, Japan) , Haiduke Sarafian (Pennsylvania State University, USA)

    Embedding 3D images created by popular Computer Algebra Systems (CAS) such as Mathematica, Maple, Mathlab, Scilab, R and etc. in TeX and LaTeX documents are a general practice. However, the documents are massive, reducing their Internet transportation efficiency. The main objective of creating a graphic code, KETpic, native to TeX and LaTeX language, is to overcoming this issue. KETpic is a macro package that rids on CAS. KETpic also has the capability of creating graphic objects such as frequently needed tables. Utilizing these features alone makes the KETpic an ideal portable code language for printing mathematical material for in-class use or publishing textbooks, for no cost. The latest version of KETpic supports easy to use 3D graphs. Recently new commands for generating data with obj format have been introduced in KETpic. The data can easily be converted to stl format making prints of 3D models possible. As a result, a TeX or LaTeX document with figure and 3D model are obtained simultaneously. The 3D printed models are solid and add visual dimensions to learning skills. It is the view of the authors that combining printed materials and 3D models are the preferred mode of approach in math education. In practice it is to be seen what the impact of this approach in math education is.

  11. Establishment of KETpic Programming Styles for Drawing

    Satoshi Yamashita (Kisarazu National College of Technology), Yoshifumi Maeda (Nagano National College of Technology, Japan), Hisashi Usui (Gunma National College of Technology, Japan), Kiyoshi Kitahara (Kogakuin University, Japan), Hideyo Makishita (Shibaura Institute of Technology, Japan), Kazushi Ahara (Meiji University, Japan)

    When collegiate mathematics teachers make their students understand a new mathematical concept, the teachers often hand them their original teaching materials with figures and tables. The materials must contain accurate and impressive figures, which urge students to understand the concept. In order to create accurate figures, teachers often use Computer Algebra System (in short, CAS), such as Mathematica, Maple, Maxima, Scilab and R, and make TeX documents used as the materials. The figures created by CAS are changed into the graphics files formatted into EPS or PDF, and are inserted in the TeX documents. It is difficult for teachers to insert satisfactory accessories, such as characters, expressions, ticks and scales, in graphics files.

    The authors have developed KETpic as a plug-in based on CAS since 2006 to enable teachers to create figures as they like. Because the figures created by KETpic are accurate and impressive line drawings, we think that these figures are suitable for mathematics class materials. We clarified the necessary functions for creating figures in mathematics class materials by questionnaires and interviews to collegiate mathematics teachers. To produce a figure using KETpic, teachers create a KETpic program in CAS along with mathematical drawing procedures. During this creation they must recognize the global image of the figure clearly and concentrate their energy on qualitative improvement of the figure. This thinking activity is called "symbolic thinking". In order to perform symbolic thinking, they have to master the programming manner of writing for drawing. By doing so, other teachers can use the KETpic program to create their original figures. In order to realize the above, both the TeX document of the class materials and the KETpic program for drawing must be written in code that every teacher can read. TeX programming styles by Knuth and the programming styles for programming languages by Kernighan and Plauger are famous as good programming styles. However, a good programming style for drawing is not yet known. It is necessary to establish KETpic programming style as a good programming style for drawing. The authors investigated many KETpic programs for drawing, which were written by collegiate mathematics teachers. As a result, they were able to find out requirements for a good programming style for drawings. Moreover, based on KETpic programming styles for drawing, they are going to start a portal site for supporting creation of mathematics class materials with figures.

  12. New Way of Explanation of the Stochastic Interpretation of Wave Functions and its Teaching Materials Using KETpic

    Kenji Fukazawa (National Institute of Technology, Kure, Japan)

    In Quantum Mechanics, it is difficult to introduce the stochastic interpretation of wave functions logically, because of no reasonable grounds. In this talk, we present the new way of explanation of the stochastic interpretation of wave functions and show some teaching materials for thie explanation using KETpic.

    We have found the finite period plane wave solutions for Schrödinger equations in some cases. In classical physics, the squared amplitude of a wave is proportional to its energy, which gives the position of the center of mass of the finite period plane wave. The assumption that this position is equal to the position of a particle gives the method to calculate the position of the particle, corresponding to the plane wave, at arbitrary time. Other physical quantities of the particle can be calculated in the same manner. One finds that the obtained formulas are equivalent to the formulas based on the ordinary stochastic interpretation. In this way, we think that students are able to accept the stochastic interpretation of wave functions smoothly.

    The square-well potential problem with infinite depth is the typical problem and steady state solutions are well known. We have found the dynamic solutions other than steady state solutions, which correspond to the classical solutions. We have produced the teaching materials for these solutions using KETpic and show some examples in this talk.

  13. Practice with Computer Algebra Systems in Mathematics Education and Teacher Training Courses

    Hideyo Makishita (Shibaura Institute of Technology, Japan)

    The PISA survey research evidenced that Japanese high school students have a problem in making use of mathematics. In response to the survey's results and with the aim of improving this situation, the Japanese Ministry of Education create a new subject named "Application of Mathematics" in which Mathematics is developed while closely involved in culture. The design of the subject "Application of Mathematics" is based on two fundamental pillars: "human activity and mathematics", and "mathematical considerations for social life".

    As it is well known, WASAN (Japanese Mathematics) evolved in its own unique way, especially during Edo period. It has a lot of problems of geometry and beautiful figures. In this talk the author wants to add a new point of view, which consists of "making use of mathematics in mathematics". Accordingly, the talk will show some examples from WASAN problems of this concept by using Computer Algebra Systems (in short, CAS), such as Mathematica, Maple, Maxima, Scilab and R, and TeX documents used as the classroom materials through KETpic. The talk embodies the author's practice in the teacher's training course and high school class with CAS. It is the opinion of the author that this approach is effective for the students in the field of geometry.

  14. On Some Attempts to Verify the Effect of Using High-quality Graphics in Mathematics Education

    Kiyoshi Kitahara (Kogakuin University, Japan), Tadashi Takahashi (Konan University, Japan), Masataka Kaneko (Toho University, Japan)

    According to the result of our questionnaire survey, one major opinion of collegiate mathematics teachers in Japan is that there is no need to use high-quality graphics in education. However, from our experience, using graphics seems to play a crucial role in some classroom situations. The aim of this research is to verify the effect of using high-quality graphics in collegiate mathematics education through the following two types of experiments:

    (1) Using materials containing graphics for the theme which is usually taught without using graphics and estimating its effect through statistical approach
    (2) Detecting the change of students' brain activity following the use of effective figures in materials

    In experiment (1), we pick up the case of materials concerning the law of logarithmic functions: log xy = log x + log y. The context of classroom is designed so that some hints are given step by step. To prepare these hints, we utilize some extension of TeX capabilities for flexible page layout. After the classroom, we asked students whether this explanation is easier to understand compared to the usual one or not. From the students' response, this material can be considered to be effective to improve the students' understanding this law.

    In experiment (2), we pick up the case of materials concerning the comparison of growth degree between exponential function y=2^x and power function y=x^4. We prepared some graphs of these functions by gradually changing the scale in y direction, so that students can recognize that the growth of y=2^x is greater than that of y=x^4 when x becomes sufficiently large. We showed these graphics step by step to three students and detected their brain activities through EEG (ElectroEncephaloGram) and NIRS (Near Infrared Spectroscopy) measurement. As a result, the judgment of these students changed when they saw a triggering figure, and some change in the trend of EEG or NIRS signal was observed at that time.

    These results indicate that using effective figures should give great influence to students' reasoning process.

  15. Development of Visual Aid Materials in Teaching the Bivariate Normal Distributions

    Toshifumi Nomachi (Yuge National College of Maritime Technology, Japan), Toshihiko Koshiba (Anan National College of Technology, Japan), Shunji Ouchi (Shimonoseki City University, Japan)

    In this talk, we calculate the orthant probabilities of bivariate normal distributions by using Scilab software. Moreover, we represent the tables of upper probabilities of bivariate standard normal distributions with the various correlation coefficients by applying Scilab software. Using these tables, we can solve many statistical problems which have bivariate normal distributions, even if those are not independent.

    It is well known that the random errors often have normal distributions. But it is difficult to calculate exactly these probabilities by definite integration. So we use the tables of standard normal probabilities by numerical integration using some computer software. Although there are very interesting many stochastic phenomena such as metabolic syndromes, it is more complicated and more difficult to calculate the bivariate probability distributions than univariate ones. So, during the fundamental studies courses, students have to study the univariate probability distributions. However, it is desirable to treat two random variables which have bivariate probability distribution in the case of two factors.

    Besides that in this work, we have made several 3D diagrams by using Scilab software. By applying them, we have evaluated visually the relationship between orthant probabilities and the correlation coefficients of bivariate normal distributions. Without using preceding tables, we can also evaluate the probabilities of some groups (e.g. overweight group, etc.), in a couple of ways. For instance, we can readily estimate the size of the groups.

  16. Some Problems of Making Educational Materials of Mathematics with CAS

    Kazushi Ahara (Meiji University, Japan)

    In Japan, introduction of electronic textbooks (in short, e-textbooks) is promoted rapidly in elementary schools and junior-high and high schools. Many textbook companies are developing standard e-textbooks of mathematics. On the other hand, it is pointed out that constructing educational materials for mathematics with Computer Algebra Systems (in short, CAS) is very important. In this article, we discuss some problems on developing materials for mathematics with CAS for high school lectures. Some of them are caused by specifications of CAS. The author considers that similar cases of troubles can occur not only in Japan but also in any other country under each domestic circumstances.

    The author has given lectures of calculus and linear algebra in universities, and he wrote some textbooks on mathematics for high school or university students. He uses Mathematica, Geogebra and Wolfram Alpha in order to show materials and to make experimentation in classes. Furthermore he is managing a website entitled 'Let's make Geogebra materials in Japanese' [1].

    All countries have their own custom of writing mathematical formulas. Textbooks are traditionally written accordingly. For example, in Japanese high-school, sqrt[5]-sqrt[2] is preferable to -sqrt[2]+sqrt[5] (because the front minus symbol is considered undesirable). Even apart from this, sqrt[2]/2 is preferable to 1/sqrt[2] and (sqrt[5]-sqrt[2])/3 is preferable to 1/(sqrt[5]+sqrt[2]) in Japan (because an irrational expression in the denominator is avoided). On the other hand, it is suspicious that there are not enough options of formatting in each CAS. Of course it might be rather important for novice to understand variety of formatting of expression, but in the viewpoint of making e-textbook with CAS, it is desirable for CAS to have such various options.

    In the class of geometry, we present Archimedes' polyhedrons and truncated polyhedrons. CAS allows us to watch animations (or interactive action of move) of the exact perspectives of such polyhedrons on screen. But the author thinks it is not enough to grasp three dimensional objects only on screen. For example, see the video in [2]. Can we count the number of faces of the polyhedron in it quickly? We hope a linkage of 3D materials and outputs of 3D printers (or ORIGAMI crafts).

    In lectures of mathematics, it is required that CAS show us not only correctness in mathematics but also comprehensibility in cognitive science. We want CAS to have many options of expressions to answer these requests.

    [1] http://www56.atwiki.jp/Geogebra_kyozai/
    [2] https://www.youtube.com/watch?v=6JT_udnpiOk