P by D at WGA1 (ICME 9)

Day 3 "Supporting Teachers in Understanding, Assessing,

Developing Children's Mathematical Abilities"

ABSTRACT

In present society, the content of education tends to be far removed from the natural environment. Under this situation, we hope that children will know the real world more widely and more deeply through the learning.

We have been studying to create materials of geometry in elementary school through our experiments. As an example we would like to introduce an experiment of geometry education below. This experiment was named the "Geometry of the Globe". The characteristics of this geometry were related to ideas of straight-lines, angles, tangent planes and etc on a spherical surface.

The aim of this experiment was to make clear the following points.

•  
• Can children understand the concepts of geometry on a spherical surface?

   

• Can children take an interest in the learning contents or not?

   

• Can the experiments give them a deeper understanding of geometry on a plane?

   

• Can children develop their cognition?

The method of this experiment was as follows; we taught the contents to our children (the 5^th graders) in 1999. We set about 20 hours for our class.

Teaching of the "Geometry of the Globe" was developed as follows; first, children regarded an apple as a globe. They drew some lines (latitude and longitude) on the apple. Then they cut off the apple along latitude and longitude. In this way, they could understand the coordinate system on a spherical surface. Next, children studied to measure the distance from the point of A to the point of B (i.e. points A and B represent cities.). Moreover, they learned the concept of a sundial (equator type). And, they created several works in cooperation with their friends. Their works were "A time table of the world", "World flight timetables around Osaka" and so on using the idea of a time difference. They thus studied the concepts of straight lines on a spherical surface. Last, they made sundials (equator type).

As a result of the experiment, we found the following facts.

•  
• They took an interest in the study of the "Geometry of the Globe", because the contents were very realistic and very compact.

   

• They understood the concepts of geometry on a spherical surface well, and above all they understood the concepts of straight lines on a spherical surface well.

   

• They could learn the content meaningfully and vividly.

   

• They changed their viewpoints of geometry on a plane. That is, they could develop their cognition.

Introduction

Since many children are not good at geometry in math education, we should reconstruct the contents of geometry in elementary school. We hope that children will learn the real world more widely and more deeply through their geometrical learning.

We have been studying to create materials of geometry through our experiments in elementary school for several years.

As an example we would like to introduce an experiment of geometry education below. The experiment was named the "Geometry of the Globe". The characteristics of this geometry were related to ideas on straight-lines, angles, tangent planes and etc on a spherical surface.

In this paper, we report on the experiment that we taught the "Geometry of the Globe" to children of 5^th school year in 1999.

The experiment of the "Geometry of the Globe"

• The aim of the experiment

The aim of the experiment was to make clear the following points.

• Whether children can understand the concepts of geometry on a spherical surface or not?
• Whether children can take an interest in the learning contents or not?
• Whether the experiment can give them a deeper understanding of geometry on a plane or not?
• Whether children can develop their cognition of space or not?

2. The method of the experiment

The subjects of this experiment were 5^th graders (27 pupils) in Shijyonawatehigashi Elementary School. The experiment amounted to 20 hours.

We made a textbook for the experiment. The contents of this experiment involved 4 stages.

(1^st stage): The coordinate system on a spherical surface

Children studied latitude and longitude using a concrete thing such as an apple. They studied how to locate a position on a spherical surface through activities such as cutting an apple and making a globe.

(2^nd stage): The concepts of a straight line on a spherical surface

Children measured the distance between A and B (A and B represent cities of the world) on a spherical surface. Furthermore they studied the azimuth on a spherical surface by using an "Azimuth Board". They studied the "Time difference" using a "Tangent Plane" and a "Sun Cone".

(3^rd stage): The concepts of a sundial (equator type)

Children learned the principle of a sundial (equator type) by using an apple, a tangent plane and so on.

(4^th stage): The application learning

Scenes of the experiment

Children made some products in cooperation with their friends. They put their mathematical knowledge to practical use, such as "A timetable of the world", "A distance table around Osaka" and "World flight timetables around Osaka". Furthermore, they made sundials (equator type).

3. Children's Reaction

In order to confirm reaction of children, we asked them to write down some descriptions of their impression. Descriptions of their impressions were as follows;

1. It was very glad that I made the sundial by myself.

(B) I have understood the structure of the globe though cutting an apple.

(C) The most surprised thing was "Sunrise and Sunset" as if sun was turning around the base of a cone. I understood that there were 4 seasons in Japan because the sun changed its movement.

(D) More people should understand the earth we live on. I was surprised that I could understand the structure of the globe through the activity of cutting an apple.

(E) I understood the meaning of latitude and longitude well through this learning. I made out the concepts of "Sunset and Sunrise" well. But it was difficult for me to relate to time differences.

(F) The more I studied, the more I came to be interested!

We concluded the following from these impressions.

• Children understood well the meaning of the coordinate systems on a sphere through this learning.
• Children understood the concepts of straight lines on a spherical surface.
• Children took an interest in the learning.
• Children came to be able to have good thinking as follows. "Many people must know our earth, and we must think about our earth."
• Children understood the mathematical conception in the contents. Furthermore some children applied the knowledge to make some products.

Conclusion

We found out the following facts through the experiment.

• Children took an interest in the study of the "Geometry of the Globe", because the contents were very realistic and very compact.
• Children could understand the concepts of geometry on a spherical surface well, above all they could understand the concepts of straight lines on a spherical surface well.
• Children could learn the content meaningfully and vividly.
• Children changed their viewpoints of geometry on a plane. That is to say, they could develop their cognition.

• We have to investigate their attitudes toward learning mathematics continuously in detail.
• We have to continue to create another teaching materials for geometrical education.

Address of thanks

I would like to express my gratitude to Prof. Yokochi Kiyoshi (Visiting Professor of Beijing Normal Univ.), and Prof. Suzuki Masahiko (Professor of Osaka Kyoiku Univ.) whom I consulted when I wrote this paper.

References

• Yokochi Kiyoshi: "Cultural history of mathematics", Yokochi Kiyoshi ed. Perspectives on school mathematics toward the 21 century (Seibundou Sinkousya 1994).
• Yokochi Kiyoshi: "Geometry of the globe", Yokochi Kiyoshi ed. An introduction of Mathematics education (the last vol.) (Gyousei 1981) Click to return to main WGA1 page.