October 26, 2020

### Animated geometric shapes and/or graphs changing in form with changing numbers will expand the world of mathematics

Kazushi Ahara

Professor, School of Interdisciplinary Mathematical Sciences,

Meiji University

In Fiscal 2022, the revised National Curriculum Standards will come into effect for senior high school curriculum formulation. One of the features in the revision is giving weight to the development of the “abilities to think by oneself, make a judgement, and express oneself.” In response to this, teaching approaches to individual subject areas will be reformed. For mathematics, one approach is that learning and teaching using computers will become essential. Here in this paper, let us take a look at how that learning and teaching approach works.

##### Developing the ability to think by oneself by capturing social issues from a mathematical perspective

The “abilities to think by oneself, make a judgement, and express oneself” are considered essential for surviving and playing an active role in modern society. With the introduction of the new National Curriculum Standards giving weight to these abilities, reform will be required for the methods of teaching and assessment in classes.

I specialize in mathematics, so, here in this paper I would like to talk about mathematics. The subject area of mathematics has focused to date on learning to cultivate knowledge and skills, thus giving weight to learning about methods for calculations and problem solving.

In contrast, the new educational policy, which makes much of the abilities to think by oneself, make a judgement, and express oneself, will call for not only solving problems by calculation but also denoting individual events by formula, devising a plan for solving problems, and feeding the calculation results back to the events. In short, pupils are expected to become competent in grasping social issues and natural phenomena from a mathematical perspective, address them through mathematics, and feed the consequences back to the social issues and natural phenomena.

For example, in microeconomics, there is the concept of the demand curve, which is known to help determine the price of a product at which the product is most likely to be purchased. This involves applying mathematics conventionally, but the subject area of mathematics in senior high school has seldom treated mathematics premised for a useful purpose as referred to above.

In another familiar example, a great concern is growing about preventing the splashing of droplets in the spread of the novel coronavirus. In this case, regarding how far the droplets could splash and disperse by sneezing, we can consider a guideline for physical distancing through mathematics based on simulation using a super computer rather than trying ambiguously to make a calculation based on our daily experience.

This suggests that a need would arise for developing human resources that are competent to not only identify the events happening before their very eyes based on the rule of thumb or their own sensation but also discover a fixed law in the events, derive the formula expressing the law, make a problem-solving assumption based on the calculation results of the formula, and feed it back to the events.

As the keywords for realizing this, “developing the abilities to think by oneself, make a judgement, and express oneself” become very important.

I specialize in mathematics, so, here in this paper I would like to talk about mathematics. The subject area of mathematics has focused to date on learning to cultivate knowledge and skills, thus giving weight to learning about methods for calculations and problem solving.

In contrast, the new educational policy, which makes much of the abilities to think by oneself, make a judgement, and express oneself, will call for not only solving problems by calculation but also denoting individual events by formula, devising a plan for solving problems, and feeding the calculation results back to the events. In short, pupils are expected to become competent in grasping social issues and natural phenomena from a mathematical perspective, address them through mathematics, and feed the consequences back to the social issues and natural phenomena.

For example, in microeconomics, there is the concept of the demand curve, which is known to help determine the price of a product at which the product is most likely to be purchased. This involves applying mathematics conventionally, but the subject area of mathematics in senior high school has seldom treated mathematics premised for a useful purpose as referred to above.

In another familiar example, a great concern is growing about preventing the splashing of droplets in the spread of the novel coronavirus. In this case, regarding how far the droplets could splash and disperse by sneezing, we can consider a guideline for physical distancing through mathematics based on simulation using a super computer rather than trying ambiguously to make a calculation based on our daily experience.

This suggests that a need would arise for developing human resources that are competent to not only identify the events happening before their very eyes based on the rule of thumb or their own sensation but also discover a fixed law in the events, derive the formula expressing the law, make a problem-solving assumption based on the calculation results of the formula, and feed it back to the events.

As the keywords for realizing this, “developing the abilities to think by oneself, make a judgement, and express oneself” become very important.

##### In Europe, learning mathematics using computers is increasingly adopted

In math teaching in Europe, experiments and verification using computers are used effectively in classes.

The individual’s ability to make a judgement by manually performing calculations and then applying the calculation results practically will be limited by nature. However, simulation by a computer facilitates the performed experiments and/or verification under various conditions and the sharing of results among many people. Experiments and verification using computers provide an opportunity to expand the mathematical way of thinking.

Try drawing a graph of a function as a simple example, and you should be able to visually identify that the value of y changes with the value of x, following definite laws. It can also be visualized that differentiation refers to the slope of a tangent line. In contrast, where pieces of data for something are found lined up in a row in a scatter diagram, what should we do in order to consider the data as constituting a function? To address this problem, humans draw up a plan. In practice, however, if we could rely on computer software for the actual work of approximating with a function, we would be freed from the troublesome work of calculations and become concentrated on experiments and verification of the properties of the data.

In fact, the new National Curriculum Standards require more active use of computers than ever in the teaching of mathematics in senior high school classes. To date, utilization of ICT (information and communication technology) in senior high-school education have received a lower priority for the unfortunate reason that learning using ICT-based teaching materials would not work effectively for university entrance examinations. However, in the Common Test for University Admissions based on the revised National Curriculum Standards, questions are expected to be set about the relationship between social phenomena and mathematics and the discussion about mathematics using mathematics software. Thus, math teaching in senior high school classes will require addressing those questions.

In Europe, learning and teaching mathematics using computers is already quite advanced. In Germany, one of the countries where the math teaching using computers is most advanced, a math learning and teaching application called GeoGebra has been developed. The GeoGebra is now used widely throughout the world, and it has the largest share among educational software.

GeoGebra is free software that allows users to run it for any purpose. It has a lot of teaching materials available publicly in the cloud, and math teachers can download and use the necessary teaching materials from the Internet for their own teaching in classes.

This application software features the capabilities of drawing geometric shapes and graphs accurately with ease, which are troublesome when attempted manually, and the interactive (bidirectional) environment in which users are allowed to recalculate and change the once created geometric shapes and graphs in real time with mouse manipulation.

For example, let us construct a triangle by connecting any three points. Drag one of the three apexes of the triangle with the mouse to any free position, and the sides will move accordingly and the triangle changes in form. Using a tablet PC, geometric shapes could be changed intuitively with the fingertips. It is quite interesting that mathematics can be learned with finger movements owing to the interactive application. The capability of changing geometric shapes with hand movements will facilitate the users in understanding the importance of efforts to experiment and verify mathematics.

However, GeoGebra was rarely used in Japan. This software has long been available in a Japanese language version, but the teaching materials published on the cloud are all explained in English. The teaching materials given in English are difficult to use in math teaching in senior high school classes in Japan. (I think that even English notation may serve useful for the promotion of globalization.)

Then, in around 2014, I developed teaching materials in Japanese for GeoGebra. They were organized to correspond to individual units of math textbooks and posted on our website. A short time later, they were adopted gradually by domestic places of education in Japan. Before long, teachers appeared who reported the practice of the teaching materials using GeoGebra in the regular training of teachers at prefectural and national levels.

The classroom teachers can download from the cloud the small-sized teaching materials I developed and customize them for easy application to their own classes. The customized teaching materials may be uploaded to the cloud. The customizing has been repeated, and a community of ICT-based teaching materials has been formed through GeoGebra. My small attempt at setting up the Web pages has led to an expanding circle of a wide variety of ICT-based teaching materials available in Japanese.

The individual’s ability to make a judgement by manually performing calculations and then applying the calculation results practically will be limited by nature. However, simulation by a computer facilitates the performed experiments and/or verification under various conditions and the sharing of results among many people. Experiments and verification using computers provide an opportunity to expand the mathematical way of thinking.

Try drawing a graph of a function as a simple example, and you should be able to visually identify that the value of y changes with the value of x, following definite laws. It can also be visualized that differentiation refers to the slope of a tangent line. In contrast, where pieces of data for something are found lined up in a row in a scatter diagram, what should we do in order to consider the data as constituting a function? To address this problem, humans draw up a plan. In practice, however, if we could rely on computer software for the actual work of approximating with a function, we would be freed from the troublesome work of calculations and become concentrated on experiments and verification of the properties of the data.

In fact, the new National Curriculum Standards require more active use of computers than ever in the teaching of mathematics in senior high school classes. To date, utilization of ICT (information and communication technology) in senior high-school education have received a lower priority for the unfortunate reason that learning using ICT-based teaching materials would not work effectively for university entrance examinations. However, in the Common Test for University Admissions based on the revised National Curriculum Standards, questions are expected to be set about the relationship between social phenomena and mathematics and the discussion about mathematics using mathematics software. Thus, math teaching in senior high school classes will require addressing those questions.

In Europe, learning and teaching mathematics using computers is already quite advanced. In Germany, one of the countries where the math teaching using computers is most advanced, a math learning and teaching application called GeoGebra has been developed. The GeoGebra is now used widely throughout the world, and it has the largest share among educational software.

GeoGebra is free software that allows users to run it for any purpose. It has a lot of teaching materials available publicly in the cloud, and math teachers can download and use the necessary teaching materials from the Internet for their own teaching in classes.

This application software features the capabilities of drawing geometric shapes and graphs accurately with ease, which are troublesome when attempted manually, and the interactive (bidirectional) environment in which users are allowed to recalculate and change the once created geometric shapes and graphs in real time with mouse manipulation.

For example, let us construct a triangle by connecting any three points. Drag one of the three apexes of the triangle with the mouse to any free position, and the sides will move accordingly and the triangle changes in form. Using a tablet PC, geometric shapes could be changed intuitively with the fingertips. It is quite interesting that mathematics can be learned with finger movements owing to the interactive application. The capability of changing geometric shapes with hand movements will facilitate the users in understanding the importance of efforts to experiment and verify mathematics.

However, GeoGebra was rarely used in Japan. This software has long been available in a Japanese language version, but the teaching materials published on the cloud are all explained in English. The teaching materials given in English are difficult to use in math teaching in senior high school classes in Japan. (I think that even English notation may serve useful for the promotion of globalization.)

Then, in around 2014, I developed teaching materials in Japanese for GeoGebra. They were organized to correspond to individual units of math textbooks and posted on our website. A short time later, they were adopted gradually by domestic places of education in Japan. Before long, teachers appeared who reported the practice of the teaching materials using GeoGebra in the regular training of teachers at prefectural and national levels.

The classroom teachers can download from the cloud the small-sized teaching materials I developed and customize them for easy application to their own classes. The customized teaching materials may be uploaded to the cloud. The customizing has been repeated, and a community of ICT-based teaching materials has been formed through GeoGebra. My small attempt at setting up the Web pages has led to an expanding circle of a wide variety of ICT-based teaching materials available in Japanese.

##### Working adults also have a good opportunity to acquire a mathematical way of thinking

I have also been involved in the development of PointLine software intended specifically for geometric constructions in addition to GeoGebra. You may wonder why additional software is required despite of the availability of GeoGebra. To tell the truth, GeoGebra is unable to draw any circumcircle of a triangle without correct prerequisite knowledge even if performing the correct drawing of geometric shapes is attempted. This is due to the fact that regarding geometric constructions, the software is designed to give preference to teaching the methods and techniques for drawing geometric shapes. I was sure that this would create a barrier for senior high-school students in their attempts at experiments and verification based on geometric constructions.

To a given triangle, drawing the circumcircle based on the conventional method requires extra knowledge of perpendicular bisectors. This is also referred to in the math textbooks. In practice, without the extra knowledge, an attempt to experiment with circumcircle fails. With PointLine, a triangle and a circle may be drawn first in any positions and then the condition that the circle should pass through an apex can be additionally specified later. Apply the condition to the individual apexes of a triangle, and the simple circle should become the circumcircle of the triangle.

Can you understand the differences from the conventional learning about geometric constructions? This concept of the geometric constructions has never been seen in the conventional math teaching. Originally, mathematics was considered as a study system based on a building-up approach, thus not giving consideration to any experimental testing by trial and error.

Drawing a circumcircle of a triangle has had to start with the listing of knowledge like a top-down system such as using the perpendicular bisector and so forth. (This may have been giving rise to the dislike of math.) This does not contribute to developing the ability to think by oneself through free experiments and verification. In the learning about geometric constructions, even GeoGebra is still insufficient because of the ICT materials used to discover through experiments the fact that a circumcircle is present for every triangle and the law to find the center (circumcenter) of the circle.

When I had PointLine used in a demonstration lesson at a senior high school, the students were fascinated by the freedom of geometric constructions and started to develop their own tricks for drawing geometric shapes. The lesson was clearly different from a conventional class, in which teaching is provided only with reference to paper textbooks. Adopting learning and teaching using ICT, which allows the students to conduct experiments and verification as they wish, will provide students with opportunities to formulate original ideas and express them in classes that have been centering on providing the students with knowledge.

Work and daily life, events and happenings considered vaguely problematic, and connections between those matters that lead to social problems – there are a lot of things that you can consider logically through trial and error by using the tools of mathematics. Mathematics is a study which has been evolving to not only handle knowledge and skills for calculation but also solve problems in real life and allow experiments and verification without restrictions.

The National Center Test for University Admissions will replace the Common Test for University Admissions, and questions are expected to be set that call for solving social phenomena by means of mathematics. In addition, over the next several years, the generation of students who learn in accordance with the new National Curriculum Standards will enter society. Also, for everyone who is already a member of society, it is essential to become conscious of the practice of thinking about the events and happenings in society based on mathematical principles.

To this end, it is also important to establish a good relationship with computers. Understand that events and happenings in society could be tested and verified using computers and mathematics. It is recommended to utilize computers not only as tools for documentation but also for searching on the Internet for content that will encourage learning and then again studying the mathematics which appears in the content.

In recent years, artificial intelligence (AI) has come to serve the decision making in our everyday life, which is apt to rely on the rule of thumb and/or intuition. AI may be defined as being made up of massive content of mathematics. Thus, it is not preferable to use AI as a simple black box. It is preferable to be interested in mathematics at the level provided in senior high school and think about how AI has been developed, and then the way of perceiving things should change dramatically.

Get experience with GeoGebra and PointLine if possible. You can come into contact with the teaching materials that your children and the generations who will go out into the world in the future will use. As an example, let us look at a page in the teaching material, allowing you to really understand how to construct a circumcircle of a triangle. On the display screen, check that you can move the apexes with the mouse.

https://www.geogebra.org/m/em4pWN9C

PointLine is content currently under research and development, but it can be downloaded from the following site:

http://www.aharalab.sakura.ne.jp/PointLine/

I hope that readers will be interested in the software.

* The information contained herein is current as of October 2020.

* The contents of articles on Meiji.net are based on the personal ideas and opinions of the author and do not indicate the official opinion of Meiji University.

* I work to achieve SDGs related to the educational and research themes that I am currently engaged in.

To a given triangle, drawing the circumcircle based on the conventional method requires extra knowledge of perpendicular bisectors. This is also referred to in the math textbooks. In practice, without the extra knowledge, an attempt to experiment with circumcircle fails. With PointLine, a triangle and a circle may be drawn first in any positions and then the condition that the circle should pass through an apex can be additionally specified later. Apply the condition to the individual apexes of a triangle, and the simple circle should become the circumcircle of the triangle.

Can you understand the differences from the conventional learning about geometric constructions? This concept of the geometric constructions has never been seen in the conventional math teaching. Originally, mathematics was considered as a study system based on a building-up approach, thus not giving consideration to any experimental testing by trial and error.

Drawing a circumcircle of a triangle has had to start with the listing of knowledge like a top-down system such as using the perpendicular bisector and so forth. (This may have been giving rise to the dislike of math.) This does not contribute to developing the ability to think by oneself through free experiments and verification. In the learning about geometric constructions, even GeoGebra is still insufficient because of the ICT materials used to discover through experiments the fact that a circumcircle is present for every triangle and the law to find the center (circumcenter) of the circle.

When I had PointLine used in a demonstration lesson at a senior high school, the students were fascinated by the freedom of geometric constructions and started to develop their own tricks for drawing geometric shapes. The lesson was clearly different from a conventional class, in which teaching is provided only with reference to paper textbooks. Adopting learning and teaching using ICT, which allows the students to conduct experiments and verification as they wish, will provide students with opportunities to formulate original ideas and express them in classes that have been centering on providing the students with knowledge.

Work and daily life, events and happenings considered vaguely problematic, and connections between those matters that lead to social problems – there are a lot of things that you can consider logically through trial and error by using the tools of mathematics. Mathematics is a study which has been evolving to not only handle knowledge and skills for calculation but also solve problems in real life and allow experiments and verification without restrictions.

The National Center Test for University Admissions will replace the Common Test for University Admissions, and questions are expected to be set that call for solving social phenomena by means of mathematics. In addition, over the next several years, the generation of students who learn in accordance with the new National Curriculum Standards will enter society. Also, for everyone who is already a member of society, it is essential to become conscious of the practice of thinking about the events and happenings in society based on mathematical principles.

To this end, it is also important to establish a good relationship with computers. Understand that events and happenings in society could be tested and verified using computers and mathematics. It is recommended to utilize computers not only as tools for documentation but also for searching on the Internet for content that will encourage learning and then again studying the mathematics which appears in the content.

In recent years, artificial intelligence (AI) has come to serve the decision making in our everyday life, which is apt to rely on the rule of thumb and/or intuition. AI may be defined as being made up of massive content of mathematics. Thus, it is not preferable to use AI as a simple black box. It is preferable to be interested in mathematics at the level provided in senior high school and think about how AI has been developed, and then the way of perceiving things should change dramatically.

Get experience with GeoGebra and PointLine if possible. You can come into contact with the teaching materials that your children and the generations who will go out into the world in the future will use. As an example, let us look at a page in the teaching material, allowing you to really understand how to construct a circumcircle of a triangle. On the display screen, check that you can move the apexes with the mouse.

https://www.geogebra.org/m/em4pWN9C

PointLine is content currently under research and development, but it can be downloaded from the following site:

http://www.aharalab.sakura.ne.jp/PointLine/

I hope that readers will be interested in the software.

* The information contained herein is current as of October 2020.

* The contents of articles on Meiji.net are based on the personal ideas and opinions of the author and do not indicate the official opinion of Meiji University.

* I work to achieve SDGs related to the educational and research themes that I am currently engaged in.

**Kazushi Ahara**

Professor, School of Interdisciplinary Mathematical Sciences, Meiji University

**Research fields:**

Geometry software

**Research themes:**

To develop software related to a wide range of mathematics centering on geometry. To address a wide variety of software from that intended to support mathematicians for their research to that intended to support mathematics learners in their learning.

[Keywords] Topology, mathematics software, mathematics teaching using ICT

**Main books and papers:**

◆“Sakuzu de Minitsuku Sokyoku-kikagaku” (Mastering Hyperbolic Geometry through Geometric Constructions) Kyoritsu Shuppan Co., Ltd., 2016

◆“Konpyuta Kika” (Computational Geometry) Sugaku Shobo Co., Ltd., 2014

◆“Keisann de Minitsuku Topology” (Mastering Topology through Calculation Practice) Kyoritsu Shuppan Co., Ltd., 2013