Hee-chan LewKorea National University of Education

2011.6.2 Singapore

MTC 2011(AMS)

Contents

Background of This LectureThe Characteristics of LOGO for Project-

based mathematics learningConcepts and CharacteristicsRole for mathematics education

Project-based Mathematics Learning through LOGO activitiesOverview & MethodologyTasks & ActivitiesResults

Conclusion

Background of This LectureThe Characteristics of LOGO for Project-

based mathematics learningConcepts and CharacteristicsRole for mathematics education

Project-based Mathematics Learning through LOGO activitiesOverview & MethodologyTasks & ActivitiesResults

Conclusion

Background

Background

LOGO was developed 30 years ago, in 1980 by S. Papert. It is very old software.

But, why today?It is a good environment suitable for today’s

conference theme: Communication, Reasoning & Connections communication among teachers and students inductive & deductive reasoning connection between mathematics and other

subjects like art, geometry and algebra, and old action(thinking) and new action(thinking)

LOGO was developed 30 years ago, in 1980 by S. Papert. It is very old software.

But, why today?It is a good environment suitable for today’s

conference theme: Communication, Reasoning & Connections communication among teachers and students inductive & deductive reasoning connection between mathematics and other

subjects like art, geometry and algebra, and old action(thinking) and new action(thinking)

Background

Why LOGO was chosen as my research topic?

It is a good environment for pursuing mathematics educational objectives Korean Government has emphasized currently.

For example, mathematics education policy issued by Korean MEST, May 17, 2011.Emphasizing connection between mathematics

and other subjects like Science, Technology, Engineering, and Art. (STEAM)

Inducing positive image of students and public toward mathematics by designing more interesting action-based mathematics education

Why LOGO was chosen as my research topic?

It is a good environment for pursuing mathematics educational objectives Korean Government has emphasized currently.

For example, mathematics education policy issued by Korean MEST, May 17, 2011.Emphasizing connection between mathematics

and other subjects like Science, Technology, Engineering, and Art. (STEAM)

Inducing positive image of students and public toward mathematics by designing more interesting action-based mathematics education

Background

Training creative manpower through integrated thinking/reasoning and problem solving ability

Constructing democratic society through reasonable communication in mathematics classrooms

Today, I will introduce LOGO as a good environment suitable for current common concerning of mathematics education of two countries: Korea and Singapore.

Particularly, I will introduce LOGO activities used for project-based mathematics learning as a good methodology for communication, reasoning & connections

Training creative manpower through integrated thinking/reasoning and problem solving ability

Constructing democratic society through reasonable communication in mathematics classrooms

Today, I will introduce LOGO as a good environment suitable for current common concerning of mathematics education of two countries: Korea and Singapore.

Particularly, I will introduce LOGO activities used for project-based mathematics learning as a good methodology for communication, reasoning & connections

Background

Conclusively I believed that LOGO combined with project-based mathematics learning environment can strengthen Communication, Reasoning, Connection and eventually can develop positive attitude of mathematics.

Conclusively I believed that LOGO combined with project-based mathematics learning environment can strengthen Communication, Reasoning, Connection and eventually can develop positive attitude of mathematics.

LOGO as an environment for project-based mathematics learning

LOGO is a “math-land” for the connection of abstractness and concreteness: LOGO is a good place to make students understand abstract concepts like angle, length, variables, functions through concrete activities and to make these concrete activities be a matrix of further higher abstract concepts.LOGO is a micro-world to provoke a “mind-storm”: In the micro-world, students can enjoy “thinking” freely. LOGO is a language for designing thinkingTo carry out these concepts, LOGO has four kinds of special characteristics.

LOGO is a “math-land” for the connection of abstractness and concreteness: LOGO is a good place to make students understand abstract concepts like angle, length, variables, functions through concrete activities and to make these concrete activities be a matrix of further higher abstract concepts.LOGO is a micro-world to provoke a “mind-storm”: In the micro-world, students can enjoy “thinking” freely. LOGO is a language for designing thinkingTo carry out these concepts, LOGO has four kinds of special characteristics.

Concepts of LOGO

Characteristics of LOGO 1

First, LOGO is closely related to students' actions: body-syntonic. The movement of a turtle implemented by a LOGO command is matched easily to students' actions in the thinking level.

First, LOGO is closely related to students' actions: body-syntonic. The movement of a turtle implemented by a LOGO command is matched easily to students' actions in the thinking level.

Student’s Drawing

Action

Student’s Drawing

Action

LOGO CommandLOGO Command

Characteristics of LOGO 1

FD 50 RT 90FD 50 RT 90FD 50 RT 90FD 50 RT 90 orRepeat 4 [FD 50 RT 90]For example, when the above command is given,

even students who are less experienced with computer can imagine easily the result is a square with the length of 50.

And, when the following square with the length of 50 is given, students can make an above command for the square easily.

FD 50 RT 90FD 50 RT 90FD 50 RT 90FD 50 RT 90 orRepeat 4 [FD 50 RT 90]For example, when the above command is given,

even students who are less experienced with computer can imagine easily the result is a square with the length of 50.

And, when the following square with the length of 50 is given, students can make an above command for the square easily.

Characteristics of LOGO 2

Second, LOGO is a "procedural" language. That is, once a program is written, LOGO can store it as a "procedure" or a “name” to use it in the future with basic commands for another programming task.

To Triangle

Repeat 3 [FD 100 RT 120]

End

To Square

Repeat 4 [FD 100 RT 90]

End

To House

Square FD 100 RT 30

Triangle LT 30 BK 100

End

Second, LOGO is a "procedural" language. That is, once a program is written, LOGO can store it as a "procedure" or a “name” to use it in the future with basic commands for another programming task.

To Triangle

Repeat 3 [FD 100 RT 120]

End

To Square

Repeat 4 [FD 100 RT 90]

End

To House

Square FD 100 RT 30

Triangle LT 30 BK 100

End

Characteristics of LOGO 2

Thus, a “structured” programming is possible in LOGO: A whole programming process can be divided into several functional units and each unit can be analyzed independently. This method makes the programming process much easier and makes debugging process simpler.

To flower

Repeat 8 [petal RT 45]

End

To petal

Repeat 2 [repeat 9 [FD 5 RT 10] RT 90]

End

Thus, a “structured” programming is possible in LOGO: A whole programming process can be divided into several functional units and each unit can be analyzed independently. This method makes the programming process much easier and makes debugging process simpler.

To flower

Repeat 8 [petal RT 45]

End

To petal

Repeat 2 [repeat 9 [FD 5 RT 10] RT 90]

End

Characteristics of LOGO 3

LOGO is a mathematical language. That is, in the programming process, turtle movement or the shapes turtle makes is determined by the values of the variables.

To Triangle :X

Repeat 3 [FD :X RT 120]

End

Triangle 100 Triangle 80 Triangle 50

LOGO is a mathematical language. That is, in the programming process, turtle movement or the shapes turtle makes is determined by the values of the variables.

To Triangle :X

Repeat 3 [FD :X RT 120]

End

Triangle 100 Triangle 80 Triangle 50

Characteristics of LOGO 3

TO POLY :SIDE :MULTI REPEAT :SIDE [FD 100 RT 360 * :MULTI / :SIDE]END POLY 7 2 POLY10 3 POLY 12 5

More “broad” mathematical concepts of variables and functions can be easily and naturally learned in LOGO environment. Many students think that the domain and co-domain of function should be a real number only.

TO POLY :SIDE :MULTI REPEAT :SIDE [FD 100 RT 360 * :MULTI / :SIDE]END POLY 7 2 POLY10 3 POLY 12 5

More “broad” mathematical concepts of variables and functions can be easily and naturally learned in LOGO environment. Many students think that the domain and co-domain of function should be a real number only.

Characteristics of LOGO 4

LOGO is a "recursive" language. That is, one procedure can be used as a command in itself. Recursion sets up a never-ending process in its character.

To POLY1 FD 100 RT 72 POLY1End

Continuous Repeat of [FD 100 RT 72]It is a pentagon!!

LOGO is a "recursive" language. That is, one procedure can be used as a command in itself. Recursion sets up a never-ending process in its character.

To POLY1 FD 100 RT 72 POLY1End

Continuous Repeat of [FD 100 RT 72]It is a pentagon!!

Characteristics of LOGO 4

A recursion is particularly able to evoke an excited response because the idea of "going on forever" touches on every child's fantasy and makes children feel like mathematicians.

TO STAR :X :R :N IF :N = 0 [STOP] LT 126 REPEAT 5 [FD :X LT 18 STAR :X * :R :R :N - 1 RT 18 RT 144 FD :X LT 72] RT 126END

STAR 30 0.4 1 STAR 30 0.4 2 STAR 30 0.4 3 STAR 30 0.4 4 STAR 30 0.4 5

A recursion is particularly able to evoke an excited response because the idea of "going on forever" touches on every child's fantasy and makes children feel like mathematicians.

TO STAR :X :R :N IF :N = 0 [STOP] LT 126 REPEAT 5 [FD :X LT 18 STAR :X * :R :R :N - 1 RT 18 RT 144 FD :X LT 72] RT 126END

STAR 30 0.4 1 STAR 30 0.4 2 STAR 30 0.4 3 STAR 30 0.4 4 STAR 30 0.4 5

Role of LOGO for mathematics education

What needs to be noted here is a role of LOGO to make children "reflect on" their own thinking.

Because it is body-syntonic, procedural, mathematical, and recursive, LOGO can give an easy and natural environment that encourages children

to be aware of their actions, to analyze or criticize them, to generalize them to control them to synthesize some actions performed previously,

In reflecting their own thinking, communication, reasoning and connection can be strengthened.

What needs to be noted here is a role of LOGO to make children "reflect on" their own thinking.

Because it is body-syntonic, procedural, mathematical, and recursive, LOGO can give an easy and natural environment that encourages children

to be aware of their actions, to analyze or criticize them, to generalize them to control them to synthesize some actions performed previously,

In reflecting their own thinking, communication, reasoning and connection can be strengthened.

Piaget theory

Reflecting on their own thinking is same as the reflective abstraction clarified by Piaget.

According to Piaget, intellectual development is a continuous process of this reflective abstraction. Thus, what is the most important in mathematics education is how to organize an educational environment to evoke the reflective abstraction.

Papert believes that, like an environment to learn the mother tongue, LOGO is a micro-world which is called as “Math-land” in which children can learn mathematics naturally and spontaneously based on reflecting their own thinking.

Reflecting on their own thinking is same as the reflective abstraction clarified by Piaget.

According to Piaget, intellectual development is a continuous process of this reflective abstraction. Thus, what is the most important in mathematics education is how to organize an educational environment to evoke the reflective abstraction.

Papert believes that, like an environment to learn the mother tongue, LOGO is a micro-world which is called as “Math-land” in which children can learn mathematics naturally and spontaneously based on reflecting their own thinking.

Polya-style problem solving

Programming in LOGO can be considered as a Polya-style problem solving process itself. It passes steps of understanding, planning, carrying out, and looking back.

LOGO can provide students with a natural environment for improving problem solving ability.

Problem solving strategies like "to subdivide" and "to relate to the already known facts" can be exercised in a natural setting.

Particularly, by reflecting on their own planning process and results, meta-cognition or managerial skills can be fostered.

Programming in LOGO can be considered as a Polya-style problem solving process itself. It passes steps of understanding, planning, carrying out, and looking back.

LOGO can provide students with a natural environment for improving problem solving ability.

Problem solving strategies like "to subdivide" and "to relate to the already known facts" can be exercised in a natural setting.

Particularly, by reflecting on their own planning process and results, meta-cognition or managerial skills can be fostered.

Poincare’s mathematical esthetic sense

In LOGO environment, students' intuitive and creative thinking can be improved by training them to grasp the situations in their own 'eyes'.

As a result, an improvement in attitude toward mathematics and mathematics education can be expected.

According to Poincare, the role of mathematics education is to train students’ mathematical attitude, their own mathematical intuition and their own mathematical esthetic sense.

In LOGO environment, students' intuitive and creative thinking can be improved by training them to grasp the situations in their own 'eyes'.

As a result, an improvement in attitude toward mathematics and mathematics education can be expected.

According to Poincare, the role of mathematics education is to train students’ mathematical attitude, their own mathematical intuition and their own mathematical esthetic sense.

Synthesis

Papert = Piaget + Polya + Poinacare

LOGO = reflecting on thinking + problem solving + mathematical esthetic sense

= communication, reasoning and connection

Papert = Piaget + Polya + Poinacare

LOGO = reflecting on thinking + problem solving + mathematical esthetic sense

= communication, reasoning and connection

Overview and Methodology

This study was undertaken with three sixth grade students chosen at the Education Center for the Scientifically Gifted of the Seoul National University of Education, and one a fifth grade student chosen from a group of 20 high ranked students in mathematics at the Education Center for the scientifically gifted of the Gang-Nam District Office of Education in Seoul. The four students made two groups of two students.The students learned the basic MSWLOGO through various project-based tasks during a total of 12 experimental classesIn the first 2 classes, the students learned the basic commands and defined the procedures to draw several diagrams.

This study was undertaken with three sixth grade students chosen at the Education Center for the Scientifically Gifted of the Seoul National University of Education, and one a fifth grade student chosen from a group of 20 high ranked students in mathematics at the Education Center for the scientifically gifted of the Gang-Nam District Office of Education in Seoul. The four students made two groups of two students.The students learned the basic MSWLOGO through various project-based tasks during a total of 12 experimental classesIn the first 2 classes, the students learned the basic commands and defined the procedures to draw several diagrams.

Overview and Methodology

In the 3rd class, each group was asked to determine the theme for project which will be done in the 10th to 12th classes and to plan how to design and how to approach to the final product. In the 3th to 9th classes they cooperated to accomplish various tasks given by teachers and selected their own diagrams required for the project for each group and built up programming of the diagrams. In the 10th to 12th classes, students integrated or modified several diagrams previously made by each group and made new diagrams if necessary.

In the 3rd class, each group was asked to determine the theme for project which will be done in the 10th to 12th classes and to plan how to design and how to approach to the final product. In the 3th to 9th classes they cooperated to accomplish various tasks given by teachers and selected their own diagrams required for the project for each group and built up programming of the diagrams. In the 10th to 12th classes, students integrated or modified several diagrams previously made by each group and made new diagrams if necessary.

Overview and Methodology

Final result of the group 1Final result of the group 1

Overview and Methodology

Final result of the group 2Final result of the group 2

Project Learning 1st - 9th Classes

Introduction to MSWLOGO and 3 Kinds of screen: MSWLogo Screen, Editor, Commander

Introduction to MSWLOGO and 3 Kinds of screen: MSWLogo Screen, Editor, Commander

1st class1st class

Basic commands FD, BK, RT, LT PU, PD, PE HT, ST, Home, SHOW Pos CS, CT FD 50 PU FD 50 PD FD 50

Repeat command Repeat 4 [FD 80 RT 90] How to make a command to draw a regular triangle,

regular Pentagon … a circle (Turtle Journey Theorem)

Repeat 3 [FD 80 RT 120] Repeat 5 [FD 80 RT 72] Repeat 360 [FD 1 RT 1] Double repeats: Repeat 18 [Repeat 360 [FD 1 RT 1]

RT 20]

Repeat command Repeat 4 [FD 80 RT 90] How to make a command to draw a regular triangle,

regular Pentagon … a circle (Turtle Journey Theorem)

Repeat 3 [FD 80 RT 120] Repeat 5 [FD 80 RT 72] Repeat 360 [FD 1 RT 1] Double repeats: Repeat 18 [Repeat 360 [FD 1 RT 1]

RT 20]

2nd class2nd class

Procedure Square Triangle House Repeat of House Repeat 36 [house RT 10]

Procedure with variables Regular n-polygon Polygon :n Polygon :x :n

Command for Coloring Blocks

Procedure Square Triangle House Repeat of House Repeat 36 [house RT 10]

Procedure with variables Regular n-polygon Polygon :n Polygon :x :n

Command for Coloring Blocks

2nd class2nd class

Application of Square Procedure(Basic)

Parallelogram, Rectangle

Application of Square Procedure(Basic)

Parallelogram, Rectangle

3rd class3rd class

Application of Square Procedure(Advanced)

Make Command: Make “Name X Label Command: Label “Word or sentence

Application of Square Procedure(Advanced)

Make Command: Make “Name X Label Command: Label “Word or sentence

3rd class3rd class

Application of Triangle Procedure(Basic) Regular triangle Tree Butterfly

Application of Triangle Procedure(Basic) Regular triangle Tree Butterfly

4th class4th class

Application of Triangle Procedure(Advanced)

Big House, Rocket, Daisy, Wheel

Moving Turtle: SETXY 90 30, SH 45, SH 0

Application of Triangle Procedure(Advanced)

Big House, Rocket, Daisy, Wheel

Moving Turtle: SETXY 90 30, SH 45, SH 0

4th class4th class

Application of Circle Procedure(Basic) Flower, Translation of circle, Circles with a same

center,

Application of Circle Procedure(Basic) Flower, Translation of circle, Circles with a same

center,

5th class5th class

Application of Circle Procedure(Advanced)

Face, Flower, Snowman, Petal

Application of Circle Procedure(Advanced)

Face, Flower, Snowman, Petal

5th class5th class

Recursive Procedure(Basic) Spiderweb Maze Buildings

Recursive Procedure(Basic) Spiderweb Maze Buildings

6th class6th class

Recursive Procedure(Advanced) Rotation of Square Maze, Rotation of Flowers Rotation of Polygons

Recursive Procedure(Advanced) Rotation of Square Maze, Rotation of Flowers Rotation of Polygons

6th class6th class

Recursive Procedure(Advanced 2) Line-circle 1, Line-circle 2, Line-circle 3, Line-circle 4 Dragon

Recursive Procedure(Advanced 2) Line-circle 1, Line-circle 2, Line-circle 3, Line-circle 4 Dragon

7th class7th class

FractalKoch curvesTO KOCH :N :SIF :N = 0 [FD :S STOP]KOCH :N - 1 :S / 3 LT 60KOCH :N - 1 :S / 3 RT 120KOCH :N - 1 :S / 3 LT 60KOCH:N - 1 :S / 3END

Snow flake To Snow :n :sRepeat 3 [Koch RT 120]end

FractalKoch curvesTO KOCH :N :SIF :N = 0 [FD :S STOP]KOCH :N - 1 :S / 3 LT 60KOCH :N - 1 :S / 3 RT 120KOCH :N - 1 :S / 3 LT 60KOCH:N - 1 :S / 3END

Snow flake To Snow :n :sRepeat 3 [Koch RT 120]end

8th class8th class

Fractal Sirpinski Triangle

TO CROSS4 :X :R :N IF :N = 0 [STOP] REPEAT 3 [FD :X RT 120 CROSS4 :X * :R :R :N - 1] End CROSS4 80 0.5 3 CROSS4 80 0.5 4 CROSS4 80 0.5 5

Tree

Fractal Sirpinski Triangle

TO CROSS4 :X :R :N IF :N = 0 [STOP] REPEAT 3 [FD :X RT 120 CROSS4 :X * :R :R :N - 1] End CROSS4 80 0.5 3 CROSS4 80 0.5 4 CROSS4 80 0.5 5

Tree

8th class8th class

Animation To carpdsetpencolor [0 0 0]fd 30 rt 90 fd 30 lt 60 fd 30 rt 60 fd 50 rt 60fd 30 lt 60 fd 30 rt 90 fd 30 rt 90 fd 140 rt 180 fd 30 lt 90 setfloodcolor [255 125 0] pu fd 10 pd fillpu bk 10 pdrepeat 360 [fd 1/5 rt 1] rt 90 fd 5 pu lt 90 fd 5 pdsetfloodcolor [0 0 0] fillpu bk 10 pd fillpu fd 5 rt 90bk 5 pd fd 60 lt 90repeat 360 [fd 1/5 rt 1] rt 90 fd 5 pu lt 90 fd 5 pdsetfloodcolor [0 0 0] fillpu bk 10 pd fillpu fd 5 rt 90 bk 5 bk 90 lt 90 pdrt 90pu fd 50 lt 90 fd 30 pdfd 20 rt 90 fd 40 rt 90 fd 20 rt 90 fd 40 rt 90pu rt 45 fd 10 setfloodcolor [255 255 255]pd fill pu bk 10 lt 45 bk 30 lt 90 fd 50 rt 90 pdrt 90end

Animation To carpdsetpencolor [0 0 0]fd 30 rt 90 fd 30 lt 60 fd 30 rt 60 fd 50 rt 60fd 30 lt 60 fd 30 rt 90 fd 30 rt 90 fd 140 rt 180 fd 30 lt 90 setfloodcolor [255 125 0] pu fd 10 pd fillpu bk 10 pdrepeat 360 [fd 1/5 rt 1] rt 90 fd 5 pu lt 90 fd 5 pdsetfloodcolor [0 0 0] fillpu bk 10 pd fillpu fd 5 rt 90bk 5 pd fd 60 lt 90repeat 360 [fd 1/5 rt 1] rt 90 fd 5 pu lt 90 fd 5 pdsetfloodcolor [0 0 0] fillpu bk 10 pd fillpu fd 5 rt 90 bk 5 bk 90 lt 90 pdrt 90pu fd 50 lt 90 fd 30 pdfd 20 rt 90 fd 40 rt 90 fd 20 rt 90 fd 40 rt 90pu rt 45 fd 10 setfloodcolor [255 255 255]pd fill pu bk 10 lt 45 bk 30 lt 90 fd 50 rt 90 pdrt 90end

9th class9th class

To delsetpencolor [255 255 255]setpensize [300 300]pdfd 3 bk 3setpensize [1 1]end To movie1repeat 100 [del pu fd 10 lt 90 car]end

Animationto deletesetpencolor [255 255 255]setpensize [125 125]fd 1 bk 1endto movie :speedhtrepeat (abs(int(360 / :speed))) [delete rt :speed pinwheel wait 1]stendto pinwheelsetpencolor[255 0 0]setpensize [2 2]setfloodcolor [255 125 0]wheelfillendto wheelpu fd 30 pd rt 120repeat 6 [fd 30 repeat 3 [lt 120 fd 30 lt 120] rt 60]lt 120 pu bk 30 pdEnd

Animationto deletesetpencolor [255 255 255]setpensize [125 125]fd 1 bk 1endto movie :speedhtrepeat (abs(int(360 / :speed))) [delete rt :speed pinwheel wait 1]stendto pinwheelsetpencolor[255 0 0]setpensize [2 2]setfloodcolor [255 125 0]wheelfillendto wheelpu fd 30 pd rt 120repeat 6 [fd 30 repeat 3 [lt 120 fd 30 lt 120] rt 60]lt 120 pu bk 30 pdEnd

9th class9th class

The Results of Projective Learning

In this study, LOGO is incorporated in the dynamic project-based learning that provides students with opportunities to apply and develop their mathematical knowledge and engage in diverse creative activities through the integration of mathematics and art as a positive way to foster higher levels of thinking for gifted students.

Today, I will focus on what kinds of thinking the mathematically gifted elementary students use to plan, implement and debug in the programming as a problem solving process.

In this study, LOGO is incorporated in the dynamic project-based learning that provides students with opportunities to apply and develop their mathematical knowledge and engage in diverse creative activities through the integration of mathematics and art as a positive way to foster higher levels of thinking for gifted students.

Today, I will focus on what kinds of thinking the mathematically gifted elementary students use to plan, implement and debug in the programming as a problem solving process.

Strategic thinkingStrategic thinking

Analogy(6th class)How can we draw these two diagrams ?

Analogy(6th class)How can we draw these two diagrams ?

AnalogyAnalogy

Students could analogical thinking based on the procedure to draw the following figure.

Students could analogical thinking based on the procedure to draw the following figure.

TO POLYGON :S :A :IIF :S>200[STOP]FD :S RT :APOLYGON :S +:I :A :IEND

POLYGON 5 90 2

POLYGON 5 91 2

POLYGON 5 89 2

AnalogyAnalogy

Generalization (5th class)Soohyun proposed to input a variable in the vehicle procedure control the size.They decided to change each number to represent distance to the numbers divided 10 multiplied :XThey did not change the number to represent anglesThey made a procedure as a general formula.

Generalization (5th class)Soohyun proposed to input a variable in the vehicle procedure control the size.They decided to change each number to represent distance to the numbers divided 10 multiplied :XThey did not change the number to represent anglesThey made a procedure as a general formula.

GeneralizationGeneralization

9th class9th class

Critical thinking (5th class) Students thought the procedure Repeat 360 [FD 1 RT 1 ] to make a circle is sometimes uncomfortable because it does not show its radius. They criticized the problem to make another circle procedure using a radius.

2∏r / 360

Critical thinking (5th class) Students thought the procedure Repeat 360 [FD 1 RT 1 ] to make a circle is sometimes uncomfortable because it does not show its radius. They criticized the problem to make another circle procedure using a radius.

2∏r / 360

9th class9th class

9th class9th class

To circle :rpu fd :r rt 90 pdRepeat 360 [fd 2*3.14* :r/360 rt 1]lt 90 pu bk :r pdEnd

Circle 50 Circle 100

Progressive thinking

1 Juseong: What is this? What happened? 2 Jihan: A twisted tree, oh! How about we call it the tree of Pisa?3 Juseong: If so, shall we try drawing the triangle to become smaller while slowly rotating? 4 Jihan: Then it could have the shape of a golden spiral.5 Juseong: I have a good idea. Let’s draw the golden spiral with the triangle located in the center. 6 Juseong: Then, how can we draw the golden spiral?

Progressive thinking

1 Juseong: What is this? What happened? 2 Jihan: A twisted tree, oh! How about we call it the tree of Pisa?3 Juseong: If so, shall we try drawing the triangle to become smaller while slowly rotating? 4 Jihan: Then it could have the shape of a golden spiral.5 Juseong: I have a good idea. Let’s draw the golden spiral with the triangle located in the center. 6 Juseong: Then, how can we draw the golden spiral?

9th class9th class

9th class9th class

Soohyun: It might be possible to use a square instead of a triangle. It is a complete a golden spiral!

Soohyun: It might be possible to use a square instead of a triangle. It is a complete a golden spiral!

9th class9th class

Debugging using Visualization

Jeho: Teacher! I tried to draw the windmill rotating in the opposite direction of the windmill to rotate toward the right direction, but it came out like this. But, I think it also looks nice.

Debugging using Visualization

Jeho: Teacher! I tried to draw the windmill rotating in the opposite direction of the windmill to rotate toward the right direction, but it came out like this. But, I think it also looks nice.

9th class9th class

Repeat 12 [fd 100 lt 90 fd 100 bk 100 lt 90 bk 100 rt 30]

Repeat 12 [fd 100 lt 90 fd 100 bk 100 rt 90 bk 100 rt 30]

Repeat 12 [fd 100 lt 90 fd 100 bk 100 lt 90 bk 100 rt 30]

Repeat 12 [fd 100 lt 90 fd 100 bk 100 rt 90 bk 100 rt 30]

9th class9th class

Debugging by Empirical Inference

Soohyun made the following procedure after suggesting that we shall have the procedure to change the flower size.

But, Soohyun: (After running “flower 50 30”) Why do we have the shape like this? It should be a flower…. The value of the variable does not seem right.

Debugging by Empirical Inference

Soohyun made the following procedure after suggesting that we shall have the procedure to change the flower size.

But, Soohyun: (After running “flower 50 30”) Why do we have the shape like this? It should be a flower…. The value of the variable does not seem right.

9th class9th class

14 Soohyun: I have a good idea. Let’s execute in sequence from “flower 1 1”. Oh, I know what is wrong. The input of radius and rotation angle were too large. Now I have it completed. Let me show you a great flower (After inputting flower 2 12 RT 15 flower 2.5 12 RT 15 flower 3 12) Take a look!

14 Soohyun: I have a good idea. Let’s execute in sequence from “flower 1 1”. Oh, I know what is wrong. The input of radius and rotation angle were too large. Now I have it completed. Let me show you a great flower (After inputting flower 2 12 RT 15 flower 2.5 12 RT 15 flower 3 12) Take a look!

9th class9th class

9th class9th class

Conclusion:

First, LOGO combined with project learning can be utilized in an integrated curriculum customized for some students in mathematics classroom. The work for project tasks consisted diverse types of graphics can provide a meaningful learning experience that integrates logical elements of mathematics and aesthetic elements of art in completing a final product in each class. Therefore it is an overall integrative, cohesive and systematical mathematics education program to encourage communication, reasoning and connection.

First, LOGO combined with project learning can be utilized in an integrated curriculum customized for some students in mathematics classroom. The work for project tasks consisted diverse types of graphics can provide a meaningful learning experience that integrates logical elements of mathematics and aesthetic elements of art in completing a final product in each class. Therefore it is an overall integrative, cohesive and systematical mathematics education program to encourage communication, reasoning and connection.

Conclusion Conclusion

Second, LOGO project learning improve the creative mathematics problem solving skills like analogical thinking, generalization, critical thinking, progressive thinking, debugging by visual inference and empirical inference. It was because learners’ experience of programming in various way to construct diagrams presented by their teacher or independently planned is so closed with such thinking. Here we need to indicate that such an experience is strongly related with communication, reasoning and connection witch is a theme of this conference.

Second, LOGO project learning improve the creative mathematics problem solving skills like analogical thinking, generalization, critical thinking, progressive thinking, debugging by visual inference and empirical inference. It was because learners’ experience of programming in various way to construct diagrams presented by their teacher or independently planned is so closed with such thinking. Here we need to indicate that such an experience is strongly related with communication, reasoning and connection witch is a theme of this conference.

Conclusion Conclusion

Third, analysis on higher thinking skills displayed in the LOGO project learning can be used as the basis for developing curriculum materials for teaching and learning in mathematics. Materials used in this LOGO project learning can used as the instructional model for considering what type of strategic thinking should be emphasized for LOGO learning project and how learning and teaching process should be accomplished in school mathematics.

 

Third, analysis on higher thinking skills displayed in the LOGO project learning can be used as the basis for developing curriculum materials for teaching and learning in mathematics. Materials used in this LOGO project learning can used as the instructional model for considering what type of strategic thinking should be emphasized for LOGO learning project and how learning and teaching process should be accomplished in school mathematics.

 

Conclusion Conclusion

Thank you very much for your attention!!