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Math Workbench A tangible user interface to explore Fractal Geometry

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Table of Contents

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INTRODUCTION

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THESIS STATEMENT & RESEARCHABLE QUESTION

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TARGET AUDIENCE

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IDEATION

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PROTOTYPE

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BIBLIOGRAPHY

How I arrived at my concept

Exploration in the work

Why children?

Why math?

Why physical interaction?

Reference materials and documents

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Introduction

How I arrived at my concept

I am very interested in the challenges children are facing with math, and as a designer I have made it a personal mission to communicate better ways to help children change their way of learning.

Far too many students in America hate math and for many it is a source of anxiety and fear. One group of parents and educators believe that mathematics should be taught traditionally with the teacher explaining methods and the students watching and then practicing them, in silence and the second group believe that students should be more involved discussing ideas and solving applied problems with their hands and objects. Many students in America hate math and for many it is a source of anxiety and fear. I think part of the problem with current elementary school math is that children are not getting any cool topics. That statement has been lingering in my mind for a while now, which has prompted me to think how might I develop a game or tangible device to tackle the toughest learning hurdles? How might I introduce a new topic in the elementary school curriculum? In my constant pursue to figure out why I don’t enjoy math could be the reason that I need to connect what I learn in school to real life. Everything around us is changing rapidly before our eyes. I strongly believe educators in the elementary school level need to prepare their students for an ever-changing future. We must begin now to demonstrate an appropriate beginning for children to be immersed in the study about fractals, chaos, and dynamical systems.

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Thesis Statement

The purpose of my exploration in the work

How can I teach kids between the ages of 7 to 10 to explore the concept of fractal geometry using a tangible media approach to solve problems?

Kinesthetic learning occurs when people widen their thinking to interact with the information and experiences with their hands. People learn more profoundly and retain knowledge longer when they have opportunities to connect actively with the information and experiences. I am interested in producing math games/ exercises that could be used at home where students are not pressured by the routines of regular class time. By getting children and parents to realize that the thought process used for problem-solving math exercises in the classroom is the same thought process used in solving problems in other areas of their studies, this will encourage the children to become productive thinkers.

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Target Audience

Why children? “If you want to build a ship, don’t drum up people together and work, but rather teach them to long for the endless immensity of the sea.” — Saint-Exupéry

Children of the 21st century are born into an age where technology is part of their daily experience – from simple mobile phones to playing computer games. Creating, sharing and viewing are naturally within their vocabulary – buttons and gadgets are an endless source of fascination. Exposure to and, later, training in math will benefit children in various ways. It can e.g. positively effect concentration, patience, self-esteem, school performance and expression. With this in mind, an exploration of children and music was the starting point. It became apparent that in most cases there is a gap between the average age children become interested in music and when their motor skills, maturity and local offerings allow them to start taking music lessons. In many cases, the lessons on offer are also less focused on the children’s creativity and engagement than skill-acquisition, i.e. learning how to play a certain instrument. Children’s education tools/toys like Leap frog and education software, has fascinated me ever since I started playing with them. We want to prepare the next generation of children for the better. To inspire them to like math so in the future they will appreciate it more. I had a constant fear that my project might not work from the start or fall apart in the middle. I began by interviewing a number of parents that I know who are interested in helping their children appreciate math more. Specifically I was seeking a better understanding of the frustration they either have for the current education school system or why the society thinks children shouldn’t be allow to learn advanced math topics at an early age.

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Ideation Why math?

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These books were my primary sources of information and influences in developing my thesis proposal.

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Why math? Now that a mathematical language exists that can be applied to highly irregular and fragmented special patterns, the daily environment of children, which consists of both fractal patterns and classical geometric shapes, can no longer be overlooked by educators. Discussions of objects with non-classical geometric shapes do not need to be avoided by teachers. Rather, attention needs to be given to ways in which this new mathematics can be addressed in the K-12 curriculum. I believe this is the appropriate time of studying basic concepts of fractal geometry in the elementary grades. In the elementary mathematics cirriculum, discussions about shape continue to focus with classical geometric shapes only. This behavior is not unexpected, given the “newness” of fractal geometry. However, it seems that, as with classical geometry, some basic concepts of fractal geometry could be introduced to young children, including shape, iterations, and measurement.

The best way to teach someone something is to instill in them a love and fascination for the thing so that they are motivated and continue to pursue it on their own. How this relates to math: school math does not instill a love of math in kids. In fact, it does just the opposite by giving kids boring repetitive tasks that don’t seem to relate to anything in real life. How do we get kids to love math? Show them something cool! Fractals is a great candidate for cool math because they relate to real life.

A. Low entry floor: They are easily understood without confusing equations. in fact, Mandelbrot himself is a proponent of first gaining an intuitive understanding of fractals before delving into proofs and formulas. This means that kids can understand fractals without difficulty especially because...

B. They relate to the real world: Snowflakes, trees, broccoli, coastlines. Can give kids an exhilarating new lens through which to view the world

C. High ceiling: You can delve deeply into the field. It’s “real math” as opposed to toy examples given to schoolchildren today. Having an intuitive initial understanding is just the foundation to delve into a whole world of fascinating concepts.

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EXPLORE congruence and similarity

EXPLORE congruence and similarity

INVESTIGATE, describe, and reason about the results of subdividing, combining, and EXPLORE congruence and similarity transforming shapes

EXPLORE congruence and similarity

MAKE AND TEST conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.

MAKE AND TEST conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.

Analyze characteristics and properties of 2D & 3D geometric shapes and develop mathematical arguments MAKE ANDabout TEST conjectures geometric relationships about geometric properties and relationships and develop logical arguments to justify Analyze characteristics and conclusions. properties of 2D & 3D geometric shapes and develop ANDabout TEST conjectures mathematical MAKE arguments about geometric properties geometric relationships and relationships and develop logical arguments to justify IDENTIFY, COMPARE, AND ANALYZE conclusions. attributes of 2D & 3D shapes and develop vocabulary to describe the attributes

INVESTIGATE, describe, and reason about the results of subdividing, combining, and DESCRIBE location and movement using transforming shapes common language and geometric vocabulary

CLASSIFY 2D & 3D shapes Specify locations and describe according to their properties and spatial relationships using develop definitions of classes coordinate geometry and other of shapes such as triangles and representational systems pyramids

geometric relationships

MAKE AND USE coordinate systems to CLASSIFY 2D & 3D shapes specify locations and to describe paths according to their properties and develop definitions of classes of shapes such as triangles and pyramids

CLASSIFY 2D & 3D shapes according to their properties and IDENTIFY, COMPARE, AND ANALYZE develop definitions of classes attributes of 2D & 3D shapes and develof shapes such as triangles and op vocabulary to describe the attributes pyramids

IDENTIFY, COMPARE, AND ANALYZE IDENTIFY, COMPARE, AND ANALYZE attributes of 2D & 3D shapes and develattributes of 2D & 3D shapes and develop vocabulary to describe the attributes op A vocabulary PREDICT AND DESCRIBE the results of DESCRIBE MOTION to or describe a series the attributes PREDICT AND DESCRIBE the results ofthat will show that two sliding, flipping, and turning 2D shapes of motions sliding, flipping, and turning 2D shapes shapes are congruent Geometry Standard

for Grades 3-5 DESCRIBE PREDICT AND DESCRIBE the results of A MOTION or a series of shapes motions that will show that two sliding, flipping, and turning 2D shapes are congruent Apply transformations and use symmetry to analyze mathematical situations

Apply transformations and use symmetry to analyze mathematical situations IDENTIFY AND DESCRIBE line and rotational symmetry in 2D & 3D shapes and designs

IDENTIFY AND DESCRIBE line and rotational symmetry in 2D & 3D shapes and designs

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Apply transformations and use symmetry to analyze mathematical situations

Apply transformations and use symmetry to analyze mathematical situations

FIND THE DISTANCE between points DESCRIBE location and movementalong usinghorizontal and vertical lines of a common language and geometric vocabulary coordinate system

Specify locations and describe INVESTIGATE, describe, andspatial relationships using reason about the results of coordinate geometry and other subdividing, combining, and representational systems transforming shapes

INVESTIGATE, describe, andand Analyze characteristics reason about the of geoproperties ofresults 2D & 3D subdividing, combining, metric shapes and and develop transforming shapes mathematical arguments about geometric relationships CLASSIFY 2D & 3D shapes according to their properties and Analyze characteristics and develop definitions of classes properties of 2D & 3D geoof shapes such as triangles and metric shapes and develop pyramids mathematical arguments about

Geometry Standard for Grades 3-5

PREDICT AND DESCRIBE the results of sliding, flipping, and turning 2D shapes

FIND THE DISTANCE between points along horizontal and vertical lines of a DESCRIBE location and movement using common language and geometric coordinate vocabulary system

DESCRIBE location and movement using common language and geometric vocabulary

Geometry Standard MAKE AND USE coordinate systems to RECOGNIZE GEOMETRIC IDEAS specify locations and to describe paths for Grades 3-5

and relationships and apply them to other disciplines and to problems that arise in the classroom or in DESCRIBE A MOTION or a series everyday life of motions that will show that two Geometry Standard shapes are congruent

for GradesGEOMETRIC 3-5 RECOGNIZE IDEAS

and relationships and apply them to other disciplines and to problems BUILD AND DRAW classroom or DRAW in DESCRIBE A MOTION or a series that arise in the BUILD AND geometric objects of motions that will show that two everyday life geometric objects shapes are congruent Use visualization, spatial reasoning, and geometric modeling to solve problems BUILD AND DRAW geometric objects USE GEOMETRIC MODELS to solve problems in other areas of mathematics, such as number and measurement

USE GEOMETRIC MODELS to IDENTIFY AND DESCRIBE line and rotational solve problems in other areas symmetry in 2D & 3D shapes and designs of mathematics, such as number and measurement

IDENTIFY AND DESCRIBE line and rotational symmetry in 2D & 3D shapes and designs

FIND THE DISTANCE between points along horizontal and vertical lines of a coordinate system

FIND THE DISTANCE between points along horizontal and vertical lines of a coordinate system

Specify locations and describe spatial relationships using coordinate geometry and other representational systems

Specify locations and describe spatial relationships using coordinate geometry and other representational systems MAKE AND USE coordinate systems to specify locations and to describe paths

MAKE AND USE coordinate systems to specify locations and to describe paths RECOGNIZE GEOMETRIC IDEAS and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life CREATE AND DESCRIBe mental images of objects, patRECOGNIZE GEOMETRIC IDEAS terns, and paths and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life CREATE AND DESCRIBe Use visualization, spatial mental images of objects, patreasoning, and geometric terns, and paths modeling to solve problems

CREATE AND DESCRIBe mental images of objects, patterns, and paths

CREATE AND DESCRIBe mental images of objects, patterns, and paths

BUILD AND DRAW geometric objects Use visualization, spatial reasoning,USE and GEOMETRIC geometric MODELS to problems in other areas modeling solve to solve problems of mathematics, such as number and measurement IDENTIFY AND DRAW a two-dimensional representation a three-dimensional USEof GEOMETRIC MODELS to object solve problems in other areas of mathematics, such as number and measurement IDENTIFY AND DRAW a two-dimensional representation of a three-dimensional object

Use visualization, spatial IDENTIFY AND BUILD a threereasoning, and geometric dimensional object from twomodeling to solve problems dimensional representations of that object

IDENTIFY AND DRAW a two-dimensionIDENTIFY AND BUILD a of threeal representation a three-dimensional dimensional object from twoobject dimensional representations of that object

IDENTIFY AND BUILD a threedimensional object from twodimensional representations of that object

IDENTIFY AND BUILD a threedimensional object from twodimensional representations of that object

IDENTIFY AND DRAW a two-dimensional representation of a three-dimensional object

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Brainstorming

Boring Memorization & Drills

Fractal Geometry

School Math Doesn't relate to real things in life

Pencil and Paper can become tedious

Interesting Patterns

Fractals can relate to real objects in life

Fun for children

Infinite ways to explore one problem

Thick textbook

I think a lot of people consider math to be boring probably because it can be very tedious and since there is always one right answer, there isn’t really any opportunity to be creative with it, unlike writing or the arts where there isn’t always a right and wrong answer.

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Visually beauitful

Math

Geometry

Fractal Geometry

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What I first started off...

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I noticed what I was doing was starting to look like a static interface...

and not a tangible interface.

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Learning from my mistakes

I feel like the strength’s of my table were fairly scattered. While I clearly displayed a strong passion for the project’s purpose, I had failed at designing the table with a clear and accurate portrayal of the audience the table was catered to. Instead, I had jumped quite quickly into the aesthetics of the interface, the technology/hardware I’m going to use, and “woooo look how pretty and cool fractals are!!” of the project without giving a serious thought to the structural backbone that is asked of projects appealing to kids. What did I do?? As such, the past 5 days after our March 20th meeting, I meticulously worked and sketched out several different edits to the interactive fractal geometry table and pages after pages of more interaction ideas and “new” lesson plan. In general they are: 1) revision of the story; 2) a sense of “leveling up” in order to give rewards for continued interaction with the table (so they don’t get bored with my table after 10 minutes), and; 3) the creation of gainful multi-player interaction. I went back to the questions I wrote for myself and added new ones to help 22

me continue this project: 1) What was the goal of the project? 2) What was the advantage of putting the project into an interactive table? 3) How does this better serve the education of fractal geometry? 4) How do you make this table continuously appealing, even to returning users? 5) Where is the sense of progression? Where is the storytelling? 6) How have you differentiated this from a glorified verbal lesson? There is still a majority of the creative design left to do, I am coding in Processing of some of the rudimentary display models that would enable me to have a visual aid when pitching the fractal geometry table to my peers. Another reason why pencil and paper alone aren’t enough: Fractals are repetitive and require generating the same thing at smaller and smaller scales with a pretty high degree of accuracy. Computers are good at doing repetitive tasks accurately. Children are less so. Pencil, papers, and rulers could get very tedious for kids and reduce their enjoyment of the concepts. However, I’m questioning my thinking on why tangibles are absolutely necessary. I can just do a touch screen interface by dragging around virtual shapes, which might actually decrease

the technical difficulties so I don’t have to deal with vision (ReacTIVision and tracking id markers on tangibles). Ok. I will prototype a small portion of my lesson using “touch screen” interface. 2 more weeks left. It’s doable! Once I sort out my sketches on loose leave papers, I will reply under my own post, my new demo on my interactive fractal geometry table. By the way, I am taking out the video part (where I’m showing children where are real fractals found). I feel like showing a video is just giving them a “tour” of what are fractals. I came up with a different promising way to have children explore and use classical geometric polygons to real life fractals. Thinking a step ahead: some people out there might be thinking that teaching kids “hard math” would further frustrate them, causing them to hate math even more. Arguments against that: this is precisely why I’m building my experiment. Fractals aren’t inherently hard to understand. They just aren’t usually presented to kids, so it’s really the existing media, not the message itself that are unfriendly for kids. I am building a way for kids to see how cool fractals are in an interface that is not intimidating.

than clicking operational buttons on a computer screen with a mouse was an important goal for me. I went down a rabbit hole trying a number of Processing and Java frameworks that promised a live, multitouch interface with the added bonus of realistic Microsoft Surface-like gestures. Without much knowledge of Processing, I dove in to the tutorials and spend a few weeks educating myself about the code that needed to go smoothly in my design. It was about three weeks into my building process when I realized that I was moving at a snail’s pace with the code as I was having a great deal of difficulty customizing the pre-designed frameworks to match with the visual design and simulations I had envisioned. The complex code was over my head, and I wasn’t picking it up as quickly as I had anticipated. It was then that I decided I had to give up on the dreams of having a real working multi-touch table, and proceeded down a new path of using my visual design in a prototyping tool called Apple Keynote and Adobe After Effects for minor animation effects. Apple keynote is believed to be one of the best tools available for designers to prototype any interface.

The biggest challenge in getting my tabletop interface design into a working prototype was the technological hurdles I encountered on the way. I knew I wanted a realistic, tap-through multi-touch demo playing right on the [Math Workbench] tabletop surface [device]. Using your fingers and tangibles in your hand rather 23


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Prototype

Why physical interaction?

Sketches

Physical education plays a critical role in educating the whole student. Research supports the importance of movement in educating both mind and body. Physical education contributes directly to development of physical competence and fitness. It also helps students to make informed choices and understand the value of leading a physically active lifestyle. The benefits of physical education can affect both academic learning and physical activity patterns of students. The healthy, physically active student is more likely to be academically motivated, alert, and successful. In the preschool and primary years, active play may be positively related to motor abilities and cognitive development. As children grow older and enter adolescence, physical activity may enhance the development of a positive self-concept as well as the ability to pursue intellectual, social and emotional challenges. Throughout the school years, quality physical education can promote social, cooperative and problem solving competencies. Quality physical education programs in our nation’s schools are essential in developing motor skills, physical fitness and understanding of concepts that foster lifelong learning lifestyles.

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Building the tangible object and table + Cut out geometric wooden pieces made of pine, then painted over + Table top made of plexiglas and then a layer of acetate paper on top

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Started building my interface in Processing but realized it was getting to complicated and I was running out of time.

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How it works When you place the tangible in the middle of the circle, it will recognize it’s a triangle (and any shape you put on there). Every time you turn the triangle 360 degrees, you will see a new iteration.

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Bibliography

Reference materials and documents

Albarn, Keith, Smith, Jenn Mial, Steele, Stanford, and Walker, Dinah. The Language of Pattern. New York: Harper & Row, 1974. Baglivo, Jenny A. and Graver, ack E. Incidence and Symmetry in Design and Architecture. New York: Cambridge University Press, 1983 Henderson, Linda. The Fourth Dimention and Non-Euclidean Geometry in Modern Art. Princeton,NJ: Princeton University Press, 1983. Guthrie, Kenneth. The Pythagorean Sourcebook and Library. Grand Rapids: Phanes Press, 1987. The Geometer’s Sketchpad: Dynamic Geometry for the 21st Century, Key Curriculum Press Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman and Company, 1983. Edgar, Gerald A. Measure, Topology, and Fractal Geometry. New York: Springer Verlag New York Inc., 1990. Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman and Company, 1983.

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Written & Designed by Connie Wang Advisor: Brian Lucid Massachusetts College of Art and Design Senior Degree Project 2012

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Math Workbench process book