to lecture slides - Mathematics and Mathematics Education

MTC 2011(AMS)
Hee-chan Lew
Korea National University of Education
2011.6.2 Singapore
 Contents
Background of This Lecture
The Characteristics of LOGO for Projectbased mathematics learning
Concepts and Characteristics
Role for mathematics education
Project-based Mathematics Learning through
LOGO activities
Overview & Methodology
Tasks & Activities
 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)
 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
 Background
Training creative manpower through integrated
thinking/reasoning and problem solving ability
Constructing democratic society through
reasonable communication in mathematics
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.
LOGO as an environment for projectbased mathematics learning
 Concepts of LOGO
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 “mindstorm”: In the micro-world, students can enjoy
“thinking” freely. LOGO is a language for designing
To carry out these concepts, LOGO has four
kinds of special characteristics.
 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.
Student’s Drawing
LOGO Command
 Characteristics of LOGO 1
FD 50 RT 90
FD 50 RT 90
FD 50 RT 90
FD 50 RT 90
Repeat 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]
To Square
Repeat 4 [FD 100 RT 90]
To House
Square FD 100 RT 30
Triangle LT 30 BK 100
 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]
To petal
Repeat 2 [repeat 9 [FD 5 RT 10] RT 90]
 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
 To Triangle :X
Repeat 3 [FD :X RT 120]
Triangle 100
Triangle 80
Triangle 50
 Characteristics of LOGO 3
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
FD 100 RT 72
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.
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 126
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.
 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.
 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.
 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
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
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 classes
In the first 2 classes, the students learned the basic
commands and defined the procedures to draw several
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.
Overview and Methodology
Final result of the group 1
Overview and Methodology
Final result of the group 2
Project Learning
1st - 9th Classes
1st class
 Introduction to MSWLOGO and 3 Kinds of screen:
MSWLogo Screen, Editor, Commander
 Basic commands
HT, ST, Home, SHOW Pos
FD 50 PU FD 50 PD FD 50
2nd class
 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 class
 Procedure
Repeat of House
Repeat 36 [house RT 10]
 Procedure with variables
 Regular n-polygon
 Polygon :n
 Polygon :x :n
 Command for Coloring Blocks
3rd class
 Application of Square Procedure(Basic)
 Parallelogram, Rectangle
3rd class
 Application of Square Procedure(Advanced)
 Make Command: Make “Name X
 Label Command: Label “Word or sentence
4th class
 Application of Triangle Procedure(Basic)
 Regular triangle
 Tree
 Butterfly
4th class
 Application of Triangle Procedure(Advanced)
 Big House, Rocket, Daisy, Wheel
 Moving Turtle: SETXY 90 30, SH 45, SH 0
5th class
 Application of Circle Procedure(Basic)
 Flower, Translation of circle, Circles with a same center,
5th class
 Application of Circle Procedure(Advanced)
 Face, Flower, Snowman, Petal
6th class
 Recursive Procedure(Basic)
 Spiderweb
 Maze
 Buildings
6th class
 Recursive Procedure(Advanced)
 Rotation of Square Maze,
 Rotation of Flowers
 Rotation of Polygons
7th class
 Recursive Procedure(Advanced 2)
 Line-circle 1, Line-circle 2, Line-circle 3, Line-circle 4
 Dragon
8th class
 Fractal
Koch curves
IF :N = 0 [FD :S STOP]
KOCH :N - 1 :S / 3 LT 60
KOCH :N - 1 :S / 3 RT 120
KOCH :N - 1 :S / 3 LT 60
KOCH:N - 1 :S / 3
 Snow flake
To Snow :n :s
Repeat 3 [Koch RT 120]
8th class
 Fractal
 Sirpinski Triangle
IF :N = 0 [STOP]
REPEAT 3 [FD :X RT 120 CROSS4 :X * :R :R :N - 1]
CROSS4 80 0.5 3 CROSS4 80 0.5 4 CROSS4 80 0.5 5
 Tree
9th class
To car
setpencolor [0 0 0]
fd 30 rt 90 fd 30 lt 60 fd 30 rt 60 fd 50 rt 60
fd 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 fill
pu bk 10 pd
repeat 360 [fd 1/5 rt 1] rt 90 fd 5 pu lt 90 fd 5 pd
setfloodcolor [0 0 0] fill
pu bk 10 pd fill
pu fd 5 rt 90
bk 5 pd fd 60 lt 90
repeat 360 [fd 1/5 rt 1] rt 90 fd 5 pu lt 90 fd 5 pd
setfloodcolor [0 0 0] fill
pu bk 10 pd fill
pu fd 5 rt 90 bk 5 bk 90 lt 90 pd
rt 90
pu fd 50 lt 90 fd 30 pd
fd 20 rt 90 fd 40 rt 90 fd 20 rt 90 fd 40 rt 90
pu rt 45 fd 10
setfloodcolor [255 255 255]
pd fill pu bk 10 lt 45 bk 30 lt 90 fd 50 rt 90 pd
rt 90
To del
setpencolor [255 255 255]
setpensize [300 300]
fd 3 bk 3
setpensize [1 1]
To movie1
repeat 100 [del pu fd 10 lt 90 car]
9th class
 Animation
to delete
setpencolor [255 255 255]
setpensize [125 125]
fd 1 bk 1
to movie :speed
repeat (abs(int(360 / :speed))) [delete rt :speed pinwheel wait 1]
to pinwheel
setpencolor[255 0 0]
setpensize [2 2]
setfloodcolor [255 125 0]
to wheel
pu fd 30 pd rt 120
repeat 6 [fd 30 repeat 3 [lt 120 fd 30 lt 120] rt 60]
lt 120 pu bk 30 pd
The Results of Projective Learning
Strategic thinking
 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.
(6th class)
How can we draw these two diagrams ?
Students could analogical thinking based on
the procedure to draw the following figure.
IF :S>200[STOP]
POLYGON 5 90 2
POLYGON 5 91 2
POLYGON 5 89 2
(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 :X
They did not change the number to represent
They made a procedure as a general formula.
9th class
9th 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
2∏r / 360
9th class
To circle :r
pu fd :r rt 90 pd
Repeat 360 [fd 2*3.14* :r/360 rt 1]
lt 90 pu bk :r pd
Circle 50
Circle 100
9th class
Progressive thinking
1 Juseong: What is this?
What happened?
2 Jihan: A twisted tree,
oh! How about we call it the tree of
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
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 class
9th class
Soohyun: It might be possible to use a square
instead of a triangle. It is a complete a golden
9th 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.
9th 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]
9th class
Debugging by Empirical Inference
Soohyun made the following procedure after suggesting that we
shall have the procedure to change the flower size.
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 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!
9th 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
 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.
 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.
Thank you very much
for your attention!!
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