Fractal Geometry: Enhancing Computationalthinking with MIT Scratch

Vinayakumar R∗, Soman KP∗and Pradeep Menon†∗center for Computational Engineering and Networking (CEN), Amrita School of Engineering, Coimbatore,

Amrita Vishwa Vidyapeetham, India†Lakhshya Cyber Security Labs Pvt Ltd, Coimbatore

Email: vinayakumarr77@gmail.com

Abstract—Recent developments in pedagogy have focused uponcomputational thinking. Computational thinking provides a wayto solve the problem and it is a key practice of science education.The importance of computational thinking is rarely found in K-12 education. Computational thinking is a an important skilleveryone needs and it is correlated with many other concepts.These factors have made the development of new tools andsyllabus. In this paper, we aim to show the experiments of fractalgeometry using MIT Scratch. These computational exercisesfacilitate to learn many of computational thinking skills that arevery important for the people in the near future. Based on ourexperience with students, we claim that the concept of fractaland its implementation in MIT Scratch is the best practices toimprove computational thinking in K-12 school level students.

Index Terms—Computational thinking, fractal geometry,Scratch, K-12 education, block based programming, 21st centurylearning skills

I. INTRODUCTION

Computational thinking is very much required skill for

everyone in 21st century[1]. It can bit each and all people’s in-

tellect implicitly or explicitly. This is applicable for every user

to easily solve current world problems. The recent problems

have identified and reinforced the importance of computational

thinking to solve the problems. Seeing that the e-newsletter

of Wing’s essay, there was colossal curiosity in, and study

dedicated to, incorporating computational Thinking into K-12

education curriculum[2].

There are various approaches and attempts made by K-12

researchers to impart the term computational thinking into K-

12 education[3]. Many of them investigated the importance of

the term visualization in the process of computational thinking.

John Jungck discusses the importance of visualization in bio-

logical process to enhance computational thinking[4]. Robert

tinker[5] gave importance for visualization in temperature,

light and force to teach computational thinking.

People often have a little confusion between computational

thinking and ICT education. The problem is handled by

introducing block based programming languages to enhance

computational thinking. The LOGO[6] programming took a

part of this, but it was not that much successful. After that

there are a lot of developments with just remixing the existing

technology to impart computational thinking to K-12 educa-

tion by K-12 educational researchers[7],[8],[9],[10],[11],[12].

MIT Scratch is considered as a powerful tool to impart

computational thinking in K-12 education[13]. Scratch is a

visual and media rich programming environment which allows

users to create games, stories, greeting card design and more

by snapping different blocks together. Scratch puts user into

the computational world without having much programming

experience. This also can be used to introduce programming

to young children in the ages of 6 to 13. They feel more

enjoyable in the process of learning programming.

Integrating the term computational thinking to K-12 ed-

ucation is not only meets the required goals but also the

assessments. Course materials and software plays an important

role. Preparation of computational experiments is often consid-

ered as difficult task. In this paper, we propose computational

exercises related to fractal geometry using MIT Scratch. These

exercises and its documentation in Kannada language have

been used in practice to impart computational thinking to K-

12 education.

II. MIT SCRATCH

Scratch is a graphical and media rich programming environ-

ment developed by MIT for children in the ages of 6 to 13. It

allows the user to create stories, games, simulations, arts etc.

Scratch provides a visual appealing programming environment

for users where they can simply drag block and stack their

blocks on script area in Scratch programming environment.

Each blocks execution can be seen by the user. This stimulates

the user to understand the complex problems and find methods

to solve it in an easy manner. The design based environment

of Scratch attracts the user and enables the user to transform

their ideas for solving real time problems easily. When they

are getting into this, they will be into abstract thinking process.

Scratch has a media rich programming environment for

creating design based learning activities. This makes students

to express themselves. It is an easy to use environment for the

user who doesn’t have prior programming experience because,

user no need to worry about the syntax of the language. Instead

user is going to think logic behind the problem. This property

increases the user attention. It is a graphical and control flow

language which enables the user to see what each block is

exactly doing. This makes programming more understandable

to the novices. In Scratch, commands are run in a sequence

and a program counter keeps track of these commands to point

to the next command to be run.

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Fig. 1: Scratch GUI(Graphical user interface)

You can see from Fig 1, the MIT Scratch GUI is divided

into 5 basic parts

1) Palette: MIT Scratch contains 8 different block category

and each category has its own set of blocks. For exam-

ple, variables block has blocks related to data structures.

2) Blocks: Scratch contains more than 100 preprogrammed

blocks, which are grouped into 8 different blocks cat-

egory. This you can see from the top right side of the

Scratch GUI and by clicking on these blocks category

you can also find blocks related those categories. These

blocks are used to control the sprite. For example, using

’Move 10 steps’, a sprite can be moved 10 steps forward.

Scratch provides an immediate effect whenever a user

clicks on a block and outputs visible result to the user.

This makes learning process more tinker, where users try

out different things by easily clicking on different blocks

to know what exactly doing. Scratch provides different

shape and color of blocks to easily identify blocks and to

make learning process easier. In Scratch, blocks include

connectors at top and right to snap together. This makes

construction of program is easier. It follows the ’fail

soft’ principle, which means the parameter of block has

default values. This value will be taken in a program

even a value is not given by the user at run time. These

blocks also avoid the most common error ’Syntax error’

i.e. the set of blocks in a stack provides output even if

it is not logically correct.

3) Script area: Blocks are drag and drop into script area by

the user.

4) Stage: Output screen of the Scratch.

5) Sprite information area: Displays information about

sprite which user uses in project activity. By right

clicking on it, a user can export it to other projects,

delete, and also make a copy of it.

With Scratch, novices can clicks on a block and see its

reaction. Next, they combine blocks and execute it. This

creates a concept of procedure. This sequence of blocks

creation enables the user to test their actions and find their

pattern. If they don’t get they can test it again and it is going

to create playful activity. For example, drawing a square in

MIT Scratch. In Fig 2, the instructions are serialized by 30

steps 4 times and by turning the degrees.

A. Scratch online community

’Excellent programmers know what to jot down. Quality

ones understand what to rewrite (and reuse)’ Raymond (1999).

This introduces the term remixing. Remixing is the process of

creating new resource based on the existing one. In order to

support this, Scratch has a website[14] where young people

can share their creations and connect with other member’s

online. Currently, Scatch website contains millions of projects.

We found many remix of the projects that are curated by

Scratchers. The process of remixing forces the user indirectly

to read the existing property and tries to add other features.

MIT Scratch website helps to solve individual problems by

communicating with others on the internet.

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Fig. 2: Scratch GUI(Graphical user interface)

MIT Scratch online community allows all ages of people to

create and play together. Scratch community member can share

their projects, they can download others projects, remix it and

share it to the outside world and they also write comments

on other projects. This also provides discussion forums where

community members can ask question related to Scratch. This

has launched in 2007 and now avilable as a vibrant online

community. Moreover, Scratch acts like a social computing

network via the Scratch online community.

III. ACADEMIC SUPPORTS TO EDUCATORS

ScratchEd[15] is a website where it provides support to

Scratch. The educators and students can share Scratch tu-

torials. It provides an online support and conversation, and

provides help for everyone.

IV. EXPERIMENTS: TREE DRAWING

The tree is drawn by repeatedly changing its size, shown

in Fig 4. Most commonly drawn fractals in Scratch online

community is shown in Fig 3. The steps of tree drawing

process stimulate the learner to think computationally. Each

steps of tree drawing is visually seen by the user. Once the tree

is drawn, the message can be transferred to flower blooming,

this initiates how to avoid collision detection. At a time

the flower may bloom and fall, programming this sequence

introduces user into the concept of ’parallelism’. This gives

the concept of conditionals, operators and data.

A. Fractal geometry iterative process

Chaos game is a well-known example where it intro-

duces the concept of iterative process[10]. Iterative process

is everywhere from simple shapes to complex fractals. The

below examples are all based on the concept of chaos game

experiment.

To create a fractal, the computation is mapped into Scratch.

Most commonly used blocks for fractal creation are shown

Fig. 3: Frequently used Scratch programming blocks for

Fractal Drawing

Fig. 4: Scratch GUI(Graphical user interface)

in Fig 4. In chaos game throwing a dice can be mapped

using ’pick random block’ in Scratch. The next point is to

find the required x and y coordinates. These are based on

the ’pick random’ block for example if it is one, it chooses

first coordinate and vice versa. Using ’pen block’ the point is

placed for the newly calculated x and y coordinates. These all

blocks will be placed inside the ’forever’ block, because once

the iteration increases you will be able to get the complex

fractal. The iterative process of each step calculation gets the

user into computational process.

Fractals are self-similar in nature, a mixture of art and

geometry. They are recursively outlined and small sections of

them are similar to colossal ones. They can be created using

complex mathematical concepts, Iterated function systems

(IFS) and Lindenmayer system (L-systems). In this context,

we use IFS concept to create fractals.

B. Iterated function system (IFS)

IFS Fractals have been first explored by Michael Barnsley

and his co-employees at Georgia Institute of technology in

the eighties. They are used to create complex shapes such as

mountains, clouds, trees and more. The IFS fractal is made up

of a couple of almost certainly overlapping smaller copies of

itself (self-similarity), each of which can be made of copies

of itself.

{fi : X → X|i = 1, 2, · · ·N}, N ∈ N (1)

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TABLE I: Comparison of Educational Tools

Feature LOGO KTurtle Kojo Alice MIT Scratch

Application Development technologies LISP and PDP-1 (First version) Based on LOGO programming Scala and Java/Swing Java

Squeak,

Action Script

(Scratch 2.0)

Open source code Yes Yes Yes Yes Yes

Open to public Yes Yes Yes Yes Yes

Block based programming NO NO NO Yes Yes

Web application No No No No Yes

Supports local language (Indian languages) No Yes Yes No Yes

Project activities documentation in local

languages (Indian languages)No No No No Yes

Supports fractal creation No Yes Yes No Yes

Supports fractals creation based on recursion No No No Yes Yes

Each fi is a contraction on the complete metric space X .

Each IFS has a set of affine transformations such as rotations,

scaling and translations. All these are represented by following

the matrix form.

[xn+1

yn+1

]=

[a bc d

] [xn

yn

]+

[ef

](2)

The above matrix can also be represented as,

xn+1 = axn + byn + e (3)

yn+1 = cxn + dyn + f (4)

The concept ’Recursion’ is not used for creating fractals

but it often makes learning process difficult for novices.

Instead, we have followed the following steps to create fractals

using MIT Scratch. Creation of fractals application using

MIT Scratch and its documentation in ’Kannada’ language

is available at[16].

Step 1: Use ’When green flag is clicked’ block. This triggers

other scripts whenever a user click green flag on MIT Scratch.

Step 2: Before starting to draw a point on stage, clear the

stage using ’clear’ block and make it pen up, so do not draw a

line on stage whenever we move from old point to the newly

estimated point.

Step 3: Declare variables x, y, xn+1, yn+1 and a list of

functions and initialize to zero.

Step 4: Add 100 values (1, 2 and 3) and the repetition

of these values is taken from probability in IFS table (used

’repeat’ block in MIT Scratch).

Step 5: Within a ’forever’ loop

• Pick random values from 1 to 100, used as an index for

list function and get the corresponding value of the index.

• The previous result is checked and find out new points

using eqn (3) and (4) ,where a, b, c, d values are taken

from IFS Table. xn and yn are previously calculated xand y values, e and f are previous result values.

• Assign xn+1, yn+1 values to x and y respectively. Using

this values plot a single point.

The scripts and it’s output is displayed in Fig 5, and Fig 6

respectively.

V. ACTIVITIES WITH STUDENTS

The above examples are shown to different levels of stu-

dents. When it is shown to primary level students they are

try to remix the projects by adding sound, changing the color

of a pen etc. but when it is shown to middle level students,

they tried to get the main concept behind this. They started

to understand the concept of chaos game and they find out

the whole concept of fractal tree drawing and they started to

create other fractals which are more complex. Finally, they

started to use the concept of ’recursion’ in order to eliminate

the duplication of codes.

VI. RELATED FRACTALS CREATION TOOLS

LOGO: LOGO is a computer programming language de-

veloped by Seymour Papert and a colleague in the 1960’s

which was invented to bring the students into the world

of programming[6]. This has influenced others to develop

new educational platforms for K-12 education. Using LOGO,

complex mathematical fractals can be created but this suffers

to provide block based commands and visual environment to

the user. This factors have made the environment not familiar

among kids. This also not been designed to learn computa-

tional thinking skills by creating programmable media. Now,

people are often remembered because of ’Turtle graphics’.

KTurtle: It is an educational programming language de-

signed for creating ’Turtle graphics’[17]. The environment

is developed based on LOGO programming language. This

provides localized programming environment. Kturtle provides

a programming environment similar kind of LOGO which can

be used for creating complex fractals. This environment is only

for the users who have prior programming experience

Alice: Alice is an object oriented, visual and innovative

programming environment that allows the user to create

animation, games, fractals and more. It was designed and

developed by late Randy Pausch team at University of Virginia,

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Fig. 5: Script for sierpinski triangle

then Carnegie Mellon (from 1997)[18]. Alice is primarily de-

veloped for teaching object oriented concepts to the user with

the help of programmable media (i.e. using story telling). The

platform also provides block based programming environment

to the user to make programming more easy and accessible

without using complex semantics of languages, C, C++ and

more. The key differences in comparison with Scratch are (1)

primarily Alice is not designed and developed with keeping

in mind ’computational thinking’ (2) It doesn’t support local-

ization (3) It doesn’t contains complex data structure and it is

often difficult to create complex fractals.

Kojo: kojo is an educational programming environment

Fig. 6: Sierpinski triangle

developed by Lalit Pant[19]. It provides block based and

visual programming environment for users to create animation,

games, fractals, simulations and more. It draws suggestions

from LOGO and Processing and developed using Scala in-

terface using Java Swing. This also does not meet Mitchell

Resnick design concepts such as low-floor, high ceiling and

wide walls.

MIT Scratch: Scratch is a block based and visual program-

ming language developed by Lifelong kindergarten (LLK)

group at MIT media lab that team led out by Mitchell

Resnick[13]. Scratch is primarily developed for the children in

the ages of 6 to 13. It provides computational thinking skills

to the students by creating design based learning activities

particularly programming interactive media. We are interested

towards computational thinking using fractals. The visual and

block based property of MIT Scratch makes user to get

easily started programming fractals by understanding complex

mathematics. Scratch is a control-flow language which shows

each step of program execution pictorially. Another unique

property of Scratch is to add media to the project activities.

This makes learning process more enjoyable to young students.

The comparision of the above discssed educational tools are

reported in Table I.

VII. CONCLUSION

Computational thinking is a crucial conception to be studied

in current education. The introduction of Scratch facilitates to

practice computational thinking for childrens in the ages of

6 to 13 at an early age. The tool and activities discussed in

this paper are employed in conjunction with computational

thinking and geometry learning course. The development of

fractals provides the way to grasp the mathematical ideas. Con-

sequently incorporating computational thinking into schooling

in an early stage tries to get computational abilities which

might be very essential for the long run. This transformational

skill appears poised to contribute to a revolution in education.

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ACKNOWLEDGMENT

This research was supported in part by Paramount Computer

Systems and Lakhshya Cyber Security Labs Pvt Ltd. We are

grateful to Computational Engineering and Networking (CEN)

department for encouraging the research.

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