柳本 朋子

結び目の数学は1980年代に入って様々な分野の最先端研究と関連して急速に発展してきた学問である。また一方で,結び目は私たちの日常生活において身近なものである。ここでは,この「結び目の数学」の教育的意義を考察することを目的とし,小学生の空間図形に対する認識を調査した。その結果,三葉結び目は中学年の段階からほぼ正しくかけるようになり,立方体に比べるとより早い時期からほぼ正しくかけることがわかった。また,立方体の図表現とその図形の認識についての関係を調べたところ,正しく表現している子どもの方が,見えていない面の数や辺の数などを正しく認識している場合が多いことがわかった。このことから,交点数の少ない結び目を扱えば,子どもが自分で図をかきながら,空間図形の動的な扱い(回転移動・対称移動など)を考察したりすることが中学年の段階から可能であり有効ではないかという結論を見いだしたThis report outlines the importance of teaching knot theory to students from elementary school. The investigation of the pupils' representation and cognition of the spatial figures was carried out. It is found that the pupils in the middle grade of elementary school are easier to represent the trefoil knot than the cube. It is also found that pupils who represent the cube well can recognize the cube well. We think that when pupils learn spatial figure, it is effective to use the one which can be represented easier will be more effective. Therefore it is found the possibility of learning spatial figure with using knot as a teaching material in the middle grade of elementary school.

われわれは,ここ数年来,小学校におけるパソコン利用として,教具として,またプログラミングについての実験研究を2,3人の子どもを対象に行ってきた。ここでは,これらの成果をふまえて,教室にパソコンを20台もちこんで,1クラスの児童全員にプログラミングの授業をしたことについて報告する。We tried the following two instruction of LOGO and BASIC in elementary school.(1)LOGO-Third grade1Making fundamental procedures(lespecially in Turtle Graphics).2Studying length,degree in making maps.3Making Patterns by parallel transformation,rotation,symmetric transformation of a figure.4Making original works in LOGO.(2)BASIC-Sixth grade1Studying fundamental contents of BASIC programing(especially in Graphics)2Calculation of surface area of Takihata Dam,and quantity of using water of Kawachinagano City in a day,and by these things to caluculate how descend the surface of the Dam in a day.3Making original graduate memorial warks in BASIC.

われわれは,1980年来,数学教育とコンピュータのかかわりを追求している。本稿では前号の「数学教育におけるパソコンの利用とそのプログラミング(I)」の中で述べたプログラミング教育について,さらに授業実践を継続的に行ったことについて報告する。In our preceding paper「Using Personal Computer in Mathematics Education and It's Programming(I)」,we tried the following instructions.

われわれは、1980年来、数学教育とコンピュータのかかわりを追求している。本稿では、それらの一連の研究と実践として、小学生にパソコンを教具として適切に利用させることの方法と、教室における中学生のパソコンを使ってのプログラミング指導の実践について報告する。One of the nowaday problems is how can we change the chilren's cognition by using personal computer in the classroom.So we tried the following instructions.First we used the personal computer as the educational tool after work in elementary school.(1)First grade:positional notation(2)Second grade:time(3)Fourth grade:curvature(4)Fifth grage:ratio,circle graph,band graph.(5)Fourth-Sixth grade:quadrangle.After these trials we made the following clear.In elementary school,personal computer is very effective by the proper use with relation to other educational tools and work.And it is necessary to study about the following point hereafter.How can we change the children's cognition through the screen of computer.Second we gave lessons to a class of computer programming for the second grade students in junior high school.In this lesson,we taught by the way of subject solving form by which we give a subject every lesson and have the students program by using needed commands.Through this trials we got a following as a result.(1)In learning computer programming it is necessary for the students to learn by systematic teaching.(2)By teaching computer programming,it is necessary for us to examine of present and future institutional contents of mathematics.And after now we have to examine about following adding to ii).In the class the problem of the number of teachers and computers,the method of group-learning and pupils'interest.

第2報では,数学教育におけるコンピュータ・プログラミング,とくにLOGO言語による指導について,小学校・中学校・高等学校での実践を通して,その可能性と方向について論じた。この小論では,小学校における図形教育の中でのLOGO言語の活用,とくに子どもにとって価値のある作業-天守閣の測量・カブトの作成・万華鏡模様の作成-の中での実践を通して,子どもの学習へのかかわり方について考察する。We tried the following instruction of LOGO in elementary school particularly in valuable works for th children.(1)Fifth grade:figure's similarity - measuring the height of Kishiwada Castle and Mizuma Temple -(2)Fifth grade:figure's parallel,symmetric,or rotational movement - making a Japanese helmet(Kabuto)-(3)Fifth grade:figure's movement - making a wall accessory with a kaleidoscopic desigh - Preceding trial made the following clear.(i)In fifth grade,pupils can use English letter of the commands in LOGO easily.Because they use them as only marks.(ii)Even in computer programing,it is sometimes necessary for the pupils to use mathematics.For example,when pupil draw a symmetric figure in LOGO they must not only replace「FD」,「RT」and「LT」with「BK」,「LT」and「RT」by considering the property of symmetry.(iii)Pupil's intution is sometimes developed the stable recongnition by doing works or by using LOGO.Based on these thing,we think that it is necessary to study about the following points hereafter.(1)How should we place the learning LOGO in the school education?(2)How does LOGO,particulary the turtle graphics,change the pupil's recongnition of figure's property that have been learned in school?

われわれは先に「数学教育におけるコンピュータ・プログラミングの指導について(I)1)」においてBAISIC言語を中心としたコンピュータのプログラミングを数学教育の中で指導することを通して,その方法や意義について考察を試みた。本稿においては,LOGO言語を中心としたコンピュータ・プログラミングを算数・数学教育の中で位置づけることを試みた。1.In elementary school,we attempted to teaching of movements of figures and elementary concept of curvature by using the function of turtle graphics,and teaching of programing by using the function of list.2.We projected that junior high school's students learned the base of LOGO language and applied to the elementary geometry.3.Making use of LOGO's three features,that is,1turtle graphics,2recursive procedure,and3taking procedure to pieces and the reverse,we taught the limit of geometrical series in senior high school.The students study earnestly and understand the foundation of LOGO easily.