MATHLOVE
ABSTRACT
As students, we see today’s youth shying away from math because of its association with hard
work and no play. Technology has attempted to solve the problem, but most online tutorials or apps are
dry, uninteresting, hard, or otherwise un-useful. MathLove combines both the artistry of designing new
fractals with the fundamentals of mathematics. MathLove takes the user through the world of fractals,
designing fractals and moving through levels that teach them about math concepts like recursion, selfsimilarity, functions, and imaginary numbers. MathLove will almost certainly be measured by two
metrics: degree of learning and fun among users. Our app will benefit mostly students looking to enrich
their understanding of math concepts.
INTRODUCTION
For students, math is a subject most are reluctant to learn. In light of this, we plan to implement
a gamified pseudo-story, explaining math and problem solving concepts relating to fractals. This could
easily be extended to other parts of science, but for the purpose of this project we are narrowing the
scope to fractals. They are relatively unknown to the general public while being readily visible in daily
life (based on our survey findings). Fractals appear everywhere in our day-to-day lives, ranging from
delicate snowflakes and lush plant life to the rhythm of one’s heartbeat.
As a gamified math-learning tool, the program may be similar in nature to games like
ShakyTower and Angry Birds. These games, at heart, teach the physics concepts of center of gravity and
projectile motion. However, our goal would be to focus more intently on the math concepts in our
game. While both the aforementioned games obviously draw on the mentioned concepts, neither is
particularly mentioned within the games. Our game would allow players to tie in the concepts
adamantly without becoming a reference quid like Wolfram Alpha.
In order to work on this project, we were required to understand more about fractals. In short, a
fractal is a never-ending pattern. Fractalsimple clouds and ferns are infinitely complex patterns that are
that are self-similar across different scales. Repeating a simple process over and over in an ongoing
feedback loop creates them. Driven by recursion, fractals are images of dynamic systems - the pictures
of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely
familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells,
hurricanes, etc.
This project will be a success if we are able to make learning about fractals and math fun. We
fully intend to introduce an app into the market that does what we define in this document.
1
BACKGROUND AND PAST WORK
EXISTING WORK WITH EDUCATION, FRACTAL AND GAME APPLICATIONS
There have been various attempts to include fractals in middle and high school education since
Benoit Mandelbrot presented them in the 1980s. In the book “Fractals, Graphics, and Mathematics
Education,” Michael Frame and Benoit Mandelbrot describe the impacts of including fractals in
mathematics education along with several case studies conducted in different levels of education. They
present one motivation through the observation: “that fractals—together with chaos, easy graphics, and
the computer—enchant many young people and make them excited about learning mathematics and
physics” [6]. The idea is to use fractal geometry to get students motivated to study mathematics and,
specifically for undergraduates, to reduce the number of students that have to suffer through low-level
mathematics classes. “The belief is that this excitement can help make these subjects easier to teach to
teenagers and to beginning college students. This is true even of those students who do not feel they
will need mathematics and physics in their professions” [6]. Fractal geometry is the most visual subject
in mathematics and science and it provides instant gratification as the beautiful and complex fractals are
based on simple mathematical concepts [7].
One existing fractal application related to education is WolframAlpha’s Fractal Reference
App[13], which presents various common fractal patterns and allows users to display them using a
specified number of iterations. As a reference guide this is a great introduction to what fractals are; by
letting users pick the number of iterations for drawing fractals, it becomes interactive but remains
encyclopedic in nature. If we could incorporate a quarter as much of mathematical information in our
game, it would likely be sufficient for a student’s basic understanding of fractals.
Another application, Recursive Drawing[9], takes a more hands-on artistic approach, with
minimal guidance. It allows users to drag and drop images into a canvas, drag the same canvas into itself,
then start a recursion from which fractal patterns evolve. Because of this, It becomes very simple to
create exponential and Fibonacci growth. As a case study of alternative programming, it implements
parts of the drawing language Context-Free. However, its actual relevance to fractals isn’t readily
apparent. There is an entire scientific publication about it and alternative programming, which would
not be ideal for high school students, who would rather fiddle around with the drawing tool until they
got bored. From an aesthetics point of view, the tool is great with the exception of its total lack of color.
In addition, the video tutorial is clear, but the text tutorial is long winded and cramped. If our program
had a free drawing mode, this would be a nice example of how it might be implemented with the
exceptions of a better tutorial and broadened program abilities like color gradients.
ShakeyTower[11] is a game for Android where the main goal is to move blocks to meet various
challenges. The most common challenge is to stack blocks to reach a specific height, dealing greatly with
center of balance. Reinforcing this, tilting the phone adjusts where the origin of the gravitational pull on
the blocks is. Occasionally, users are required to use this along with a basic understanding of projectile
motion to move the blocks through various courses. Similarly, challenges also include dealing with ice
and various surfaces with minimal friction. The app of course rarely, if ever, mentions the actual
concepts of center of gravity, projectile motion and friction.
2
Angry Birds[1] lets users catapult birds across fields to destroy objects. This requires them to
analyze the projectile motion of the launched birds. If the game mentioned measurements of how far
the bird traveled or its current speed, it might lend more to its potential as an educational app.
However, without this information, it is more a fun trial and error game rather than being educational.
Ideally, our project would prove to be as enjoyable as Angry Birds while meeting its goal of being
educational.
Julia’s kaleidoscope[8] encapsulates a large number of applications on the market that allow
users to fiddle with a common fractal such as the Mandelbrot set and Julia set. Users can cycle between
two Julia Sets and the Mandelbrot set. In some apps, the color scheme shifts when the user shakes their
iPod. Tilting the iPod also causes the tolerance of the gradient displayed to change and lastly there is the
coveted feature of zooming in and out of the fractal. The user can also readily take screenshots of the
fractal image they have created, letting them share it or make it their wallpaper. Unfortunately, within
the app there is no indication of what the Mandelbrot and Julia sets are other than cool looking patterns,
which is what our project wants to avoid. Our project should look intriguing while having an educational
element.
Soulwire[12] hosts a program simply called “Recursion Toy” that has a few basic patterns that
have a fractal nature like frosts and vines. By adjusting different factors about the pattern it’s about to
draw you can alter how the image will turn out with slight changes in attributes leading to large changes
in the display it serves as a superb example of the butterfly effect and aspects of chaos theory that often
tie in with fractals which by their self-similar nature are non-differentiable and chaotic. The tool
however can actually grow boring quickly because there is no way to recurse on its own you can only
adjust the presets and draw the images they create, this chaos and recursion is almost completely predetermined in actuality while for instance in recursive drawing you can create your own patterns and
build your own chaotic patterns. This program will be further discussed in the Formative Research
Results as it was one of the websites we chose to analyze further to assess its educational capacity.
Apophysis[2] is a tool for drawing and rendering fractal art. It focuses on using recursing fractal
frames that can be edited. Taking the seed image and creating a histogram of the image spawn the
flames. From there, it renders the histogram and repeats the process on the newly rendered image.
Additionally, the editor included in the software can edit the different segments of a fractal, leading to
intricate and variable designs like that in figure 1. This would be the ideal way to implement our freehand fractal drawing, due to the flexibility of the editor.
3
FIGURE 1 [2] A FLORAL IMAGE
CREATED IN APOPHYSIS
FractaAnimation4Lite[3] is probably the best example of what we want to avoid. It tried to have
the movement of Julia’s Kaleidoscope while also inserting a game mechanic by segmenting the whole
fractal into pieces of a sliding puzzle. It was an interesting concept, but the colors and movement are
dizzying and adds on to the inflexible controls. Most the comments are along the lines of “horrible app
you don’t do anything[sic]” and ”its not even full screen.” Both comments suggest the players never
made it past the home screen and didn’t realize it was a puzzle game, since the puzzle is full screen and
very much something you can do. The resolution was also poor which we believe is due to the memory
intensive nature of rendering animated fractals.
Fractal Designer[4] is an application on Android similar to Recursive Drawing. The main
difference is Recursive Drawing let you use different canvases and has shapes to draw with, while Fractal
Designer has only lines. The Designer also allows users to select and use gradients like the green
gradient of the fern in figure 2. If we could implement a free range area for doodling and sandboxing
fractal images, it would be useful to utilize color gradients. Furthermore, the app’s “How to Draw
Fractals” tutorial is very interactive and shows off the functionality well. It also allows the user to save
and open drawings, which RecursiveDrawing lacks.
Figure 2 [4] a fern created in fractal design
Fractal Foundation’s mission is to use “the beauty of fractals to inspire interest in Science, Math
and Art” [5]. Their aim is to teach both children and adults about fractals through many different
resources and events. Their website contains a vast amount of information ranging from fractal related
news, fractal shows dates and tickets, an Educator’s Guide, to fractal software and a fractal course. The
fractal course is aimed at high school students and explains fractals in nature, the math behind fractals,
4
and some different types of fractal sets. The concepts are explained in a simple manner and after each
section, there are questions that the users can answer, testing their understanding of the current
section. However, it is structured like a book in that it doesn’t allow the students to interact with the
fractals. Without interactivity, many students could have a hard time finding the fun aspect of learning
about fractals. The Fractal Foundation’s website is analyzed further in the Formative Research Results
section.
Oracle ThinkQuest[11] is an educational foundation for elementary, middle and high schools,
that offers a fractal tutorial. It is divided into lessons about different concepts needed to understand
fractals, such as self-similarity, Brownian self-similarity, fractal dimension, iteration, and others. Each
lesson contains text, images and related links at the end. To analyze the effectiveness and impact of this
tutorial, we included it in the interviews we conducted. This is discussed in the Formative Research
Results section.
TARGET USERS
MathLove is intended to be a platform for math learning and teaching, and as such there is a long
and varied list of users, including the following:









Middle/High School students (weak/strong)
Undergraduate students
Parents
Middle/High school teachers
Day Care Providers
Educational Decision Makers
Math Enthusiasts
Students with Disabilities
Students with English as a second language
The primary user group we are focusing on are weak and strong middle/high school students. We
reason that these students would be impacted the most by an app that is designed to get them
interested in math. After some brief research, we found that there isn’t one clear reason why students
“fail”, but there is heavy weight placed on factors outside of the classroom; friend groups, time
management, parental support, focus on grades rather than learning, and teacher enthusiasm were
recurring themes in a lot of the research we read.
MathLove will not address all of these issues, but certain aspects of the app will be very appealing.
Firstly, we aim for a fun educational experience focused on student learning rather than teaching. This is
a subtle difference, but can make all the difference in the world in how effective the lessons are. Next,
the inherent flexibility in the platform means that the users are free to learn at their own pace, on their
own time. Creating social change is a very lofty goal, although one can not discount the popularity and
social change brought on by applications such as Twitter and Facebook. If our app is able to enable
student success in math education, it may become a tool that can address other student issues like time
management or social pressure to not fail.
5
FORMATIVE USER RESEARCH
SURVEY
INTRODUCTION
Team Neon Bats decided to do a brief survey to understand the impact of our project, while the
interviews provided in-depth understanding of what our user base wanted. The survey, however, was
designed to answer the question of “is this a need?” To this end, we designed a survey to see how many
people we could find who knew what a fractal was, and also if they had ever used an educational app
like our project.
PARTICIPANTS
The participants in the survey were 50+ undergraduate students at the University of Maryland.
This may seem counter-intuitive- if we wanted to know about high schoolers, why target undergraduate
students?
Our thought was that if we show that if a concept like fractals is foreign to college students,
then we know that this sort of education is lacking in our age range. The survey was also a chance for us
to do quick market research on what sorts of apps people use for self-education, and we figured college
kids would likely know more/better apps that do this. Of course, there is also a convenience to surveying
people who you know well, and who are very accessible.
PROCEDURE
Our interviews were conducted online, via Google Form. We reached out to individuals we knew
would help us out, and had them complete our survey. The survey (in addition to demographics)
included the following questions:



Do you know what a fractal is? If so, describe what one is.
Have you used an educational app that teaches you a concept through a story or structured
learning plan? If so, which ones?
How interested would you be in an educational app to teach you fractals, or other math
concepts?
DATA AND ANALYTICAL METHOD
After an appropriate amount of time (1 week), we closed the survey, and downloaded the
results into Microsoft Excel to do some analysis. We calculated the descriptive statistics for relevant
questions, and qualitatively discussed the responses for the entered text. This discussion was largely
driven on how this research relates to the interviews and background research.
INTERVIEWS
6
INTRODUCTION
The other method we employed was conducting interviews with middle and high school
students about current fractal-related offerings in math education. Interviews are most appropriate for
researching our target users’ understanding of fractals and perspectives on current available tutorials
and fractal drawing applications. Making open ended questions allowed us to gather more information
about the students’ grasp of fractal concepts, which aided us in focusing on specific content.
PARTICIPANTS
The participants consisted of one middle school student and four high school students, each in a
different grade. They were recruited through one of our group members who tutors middle and high
school students in math. Some students are tutored consistently and the others are students she helped
once or twice with specific math problems they were having. These participants were selected because
they represent the varying levels within our target user group: grade level (age), and math level (weak
versus strong).
Student
Grade
Math Level
Gender
1
11th
Algebra 2
Female
2
10th
Geometry
Male
3
7th
Middle School Math (High)
Male
4
12th
AP Calculus AB
Female
5
9th
Algebra 1
Female
PROCEDURE
The interviews were conducted in each student’s house. They lasted about ten to fifteen minutes
and the answers were recorded in a notebook. Three of the interviews were conducted after the
students’ tutoring sessions. The students were asked whether they had heard about fractals before.
They were then instructed to look at and play around with three-fractal education/drawing related
websites for three minutes while the interviewer observed and took down notes in a notebook. Among
the three websites were Soulwire’s Recursion Toy [12], Fractal Foundation [5], and the Oracle
ThinkQuest Tutorial [11]. After the three minutes had passed, the students were asked a series of
questions about the websites:
7




Rank the websites in order of which one you liked the most, and discuss why you ranked them
the way you did.
Which site taught you the most about fractals? How would you describe your user experience in
this site? What do you know about fractals now?
If you could pick one element you liked the MOST of any of the websites, what would it be?
What did you like the LEAST about each of the websites?
DATA AND ANALYTICAL METHOD
During the interviews, the observational notes and responses to the questions were taken in a
notebook and later typed up. After compiling the individual responses by question, we discussed the
findings as a group and analyzed the positive and negative aspects that the students pointed out about
each website. We were mainly interested in the aspects of each method (drawing- Recursion Toy,
tutorial- Oracle ThinkQuest, or multiple resources- Fractal Foundation) that were most effective in
teaching the fractal concept and the math behind it.
FORMATIVE USER RESEARCH RESULTS
SURVEY
The survey results show us the significance of our problem:




The percentage of people who didn’t know what a fractal was: 71.4%
The percentage of people who haven’t used an educational app: 85.7%
Average interest level in app like we are proposing (scale 1-5): 3.5
Median interest level in app like we are proposing (scale 1-5): 4
These statistics provided an example of the lack of fractal education in our school system. The
two most telling statistics were the percentage of people who haven’t used an education app, and the
median interest level.
Since this was a survey of college-aged students, it is a little astounding that 85% of them
haven’t used an education app. With the technology available today, one would guess that most college
students use some sort of education app.
With our research, it was pretty clear to see that there are educational apps out there, just none
that are in popular use. This alone motivated us to add educational functionality to our fractal program
in order to facilitate academic progress.
Specifically, the median of 4 lets us know that half of our participants would have marked an
interest level of 4 or above. The fact that the average score is lower than the median leads us to believe
that the interest in our project was bipolar, with a slight preference towards more interest.
INTERVIEWS
8
The interview responses were helpful in identifying the pros and cons of each website and the
best way to teach the mathematical concepts involved in the fractal education process.
When asked to rank the three websites in order of preference, three students ranked the
Recursion Toy as first because they said it was “cool” and “fun.” Compared to the other two websites,
the Recursion Toy allows more interaction and exploration on the users’ side. One student ranked the
tutorial as first because it is the most straightforward one, the lessons are bulleted and there seems to
be an order, unlike in Fractal Foundation and the Recursion Toy. Lastly, the other student ranked Fractal
Foundation as number one as it shows a bigger picture of the lesson due to all of the resources it
provides.
Although most students liked the Recursive Toy the most, they found it was difficult to figure
out what to do. There are no clear instructions provided and no topics listed, so they didn’t know how to
get started. A couple of students also pointed out that there should be more colors and different shapes.
From observing the participants play around with this site, the interviewer saw that they were confused.
Only a couple of students found and played around with the “Render Style” and “Preset Behaviors”
options and none of them realized it was possible to moderate the levels on the right by clicking on the
blue bars.
When asked about the website that taught them the most about fractals, four students
responded that it was the Oracle Tutorial because of the reasons described above, mainly because it is
organized into sections. However, when asked what they liked the least about this site, most students
said that it was boring, too advanced (didn’t really understand it), and lacked color.
Lastly, some participants pointed out Fractal Foundation has too much information distributed
everywhere and there is too much decoration/color that is distracting. When they opened up this
website, they didn’t know where to start either. Most of them went into the pictures section. This
website is designed for people searching for a variety of resources to learn about fractals as opposed to
just someone who doesn’t know what a fractal is and is looking for a definition. Nevertheless, a couple
of students responded that the element they liked the most about any of the websites were the large
amount of pictures of fractals in Fractal Foundation. The 12th grader liked that the Foundation
presented fractal shows. She was impressed by the fact that people attend these shows and have fun
with Mathematics.
Asking them what they learned about fractals allowed us to understand the relative amount of
knowledge gained about fractals in the short period of time (three minutes) they were provided with
these resources. None of the students had previously heard about fractals and each of them drew their
conclusions about fractals from a different site. The Fractal Foundation only gives a general idea that
fractals are related to science and art and the tutorial was too advanced, only one student actually read
through half of it. In general, the Recursion Toy was used as a basis for their explanation, but this tool
did not prove effective in teaching what a fractal really is to students. As opposed to the Oracle Tutorial
that is too advanced, the online fractal course that the Fractal Foundation provides (mentioned in the
Background and Review of Past Work section) has a more appropriate level for middle and high schools
students.
9
From these interviews, we were able to conclude that MathLove would need to teach the
necessary concepts in a categorized, step by step manner and should include an interactive tool that
allows users to draw their own fractals and play around with different colors, shapes and other options.
CONCLUSION
Upon reviewing several current fractal drawing applications and educational resources, we were
able to identify the pros and cons of each. The survey we conducted confirmed the need for a fun fractal
education application and the interviews helped us identify which teaching methods and features would
be most effective in the MathLove application. We will incorporate varying difficulty levels in a map like
structure that the user will be able to navigate through while learning about fractals in a fun way.
REFERENCES
1. "Angry Birds." Angry Birds. Rovio Mobile Ltd., Dec. 2009. Web. 30 Sept. 2013.
2.
3.
4.
5.
<https://play.google.com/store/apps/details?id=com.rovio.angrybirds>.
Draves, Scott, Ronald Hordijk, Mark Townsend, Peter Sdobnov, and Piotr Borys.
"Apophysis.org." Apophysis.org. N.p., 10 Sept. 2009. Web. 30 Sept. 2013.
<http://apophysis.org/>.
"Fractal Animation 4 Lite -." Fractal Animation 4 Lite. SingularPoint, 5 Feb. 2011. Web.
30 Sept. 2013.
<https://play.google.com/store/apps/details?id=sglpnt.FractalAnimation4Lite>.
"Fractal Designer - Android Apps on Google Play." Fractal Designer - Android Apps on
Google Play. Resonos, 25 May 2012. Web. 30 Sept. 2013.
<https://play.google.com/store/apps/details?id=com.resonos.apps.fractal.ifs>.
"FractalFoundation.org." FractalFoundationorg RSS. Fractal Foundation, 2013. Web. 30
Sept. 2013. <http://fractalfoundation.org/>.
6. Frame, M. L., & Mandelbrot, B. B. (2002). Chapter 3. Fractals, Graphics, and
Mathematics Education (21-28). Washington D.C.: Mathematical Association of America.
7. Frame, M. L., & Mandelbrot, B. B. (2002). Chapter 1: Some Reasons for the
Effectiveness of Fractals in Mathematics Education. Fractals, Graphics, and
Mathematics Education (3-9). Washington D.C.: Mathematical Association of America.
8. "Julia's Kaleido." App Store. : Beijing Yihelin Science & Technology Co., Ltd., 23 Jan.
2013. Web. 30 Sept. 2013. <https://itunes.apple.com/us/app/juliaskaleido/id425996462?mt=8>.
9. Schachman, Toby. (2012). “Alternative Programming Interfaces for Alternative
Programmers”. Proceedings of the ACM international symposium on New ideas, new
paradigms, and reflections on programming and software (Onward! '12). ACM, New York,
NY, USA. Web. 30 Sept.2013<http://doi.acm.org/10.1145/2384592.2384594>
10
10. "ShakyTower (physics Game)." ShakyTower (physics Game). Mobillness, June 2011.
Web. 30 Sept. 2013.
<https://play.google.com/store/apps/details?id=com.mobillness.shakytower.full>.
11. "ThinkQuest : Think.com, Oracle Education Foundation." ThinkQuest : Think.com,
Oracle Education Foundation. Oracle, n.d. Web. 30 Sept. 2013.
<http://www.thinkquest.org/en/>.
12. Windle, Justin. "Recursion Toy." Recursion Toy. Soulwire, 25 Aug. 2011. Web. 30 Sept.
2013. <http://soulwire.co.uk/data/experiments/recursion-toy/>.
13. "Wolfram Fractals Reference App." App Store. Wolfram Alpha LLC, 04 June 2013. Web.
30 Sept. 2013. <https://itunes.apple.com/us/app/wolfram-fractalsreference/id439739180?mt=8>.
APPENDIX
ORIGINAL ELEVATOR PITCH
FractalDrawing on Android Phones and Tablets
Fractals are really awesome to view and create.Unfortunately most programs for drawing fractal
patterns on Android are limited in scope[2,3]. For instance the app Fractoid only lets the user
zoom in and out on finite number of algorithms to draw fractals like the Mandelbrot and Julia set
with a limited number of predefined color schemes and possible iterations[2]. Another
application Fractal Designer is also used for making fractals. Its far more fine‐tuned than
Fractoid, but still limited in scope since there is only one layer of fractals you can create and edit,
which makes it like Fractoid not as fine‐tuned as it could be[3]. In this project we propose
creating a program to better allow fractal drawing on mobile phones following recursive drawing
similar to that Toby Schachman[4] where users can select base shapes and drag them into a
canvas, dragging the same shape onto itself would start a recursion that iterates a predefined
amount, users could specify a specific color for the shapes at different iterations mod the value
of different colors to create their own gradients in a side menu and if we can do so efficiently
even allow users to select images from their gallery for gradients (eg. As a slightly creepy
example they could take oval cut outs of faces from photos on their phones and repeat these in
the recursion in place of colors for the gradients see figure 1). This would most benefit artists
and kids who want a funny, slightly silly app for creating fractal art. Ideally if someone wanted to
make a more serious piece they might be able to with the application. To evaluate this
application we plan to have users try this application alongside others like those mentioned
above and any others we might encounter as the project moves forward and have users rate the
applications on ease of use and flexibility in designing images.
Figure 1[1]
References
11
1. Buschmann, G. (2011).Juliasetsdkfieldlines4.
http://en.wikipedia.org/wiki/File:Juliasetsdkfieldlines4.jpg
2. Byrne,D.(2011). Fractoid.GooglePlay.
https://play.google.com/store/apps/details?id=byrne.fractal&hl=en
3. Resonos.(2012). FractalDesginer.Google Play.
https://play.google.com/store/apps/details?id=com.resonos.apps.fractal.ifs&hl=en
4. Schachman,T.(2012). Alternative Programming Interfaces for Alternative Programmers.
Onward!’12. http://dl.acm.org/citation.cfm?id=2384594
BRAINSTORMING NOTES
12
13
14
15
Kids/Education
● Geared towards kids or math enthusiast
● Interactive for kids to learn
● Story of Mr. Fractal (Story to explain fractals)
● Tiered Education Levels
16
●
●
Small goal oriented tutorial
Mesmerizing effect for little kids
System
● Live wallpaper
● 3d web in fractal
● Rainmeter fractal skin
● Fractal home screen
● Fractal file browser
Gamify
● Fractal Battle
● Palette Picker
● Collaborative fractal design
● Kinect movement through fractal
● Unlocking Shapes
Applications
● Physical fractal
● Todo List visualization fractal
● Responsive fractal design
● Website blocking fractal
● 3d shapes
● Specific application examples
● Model objects by fractals
Input Methods to design fractal
● Images
● Sounds
● Light
● DJ's
● Pictures
RAW SURVEY RESULTS
Timestamp
Do you
If yes, explain
know what a to us what a
fractal is?
fractal is.
Have you used
an educational
app that
teaches you a
concept
through a
story or
structured
learning plan?
If yes, which
ones?
How interested
would you be
in an
educational
app to teach
you fractals, or
other math
concepts?
17
9/27/2013
11:29:14
No
No
5
9/27/2013
11:46:26
No
No
2
9/27/2013
11:57:32
No
No
3
9/27/2013
12:02:27
No
No
4
9/27/2013
12:08:07
No
No
3
9/27/2013
12:12:21
No
No
3
9/27/2013
12:14:23
No
No
4
9/27/2013
12:19:43
No
No
4
9/27/2013
12:44:47
No
No
1
9/27/2013
12:55:05
It's a recursive
picture when you
zoom into the
picture it's the
same picture so
you can look at
the picture while
you look at the
picture while you
look at the
Yes
picture.
Yes
Luminosity
4
9/27/2013
13:13:06
No
No
3
9/27/2013
13:25:13
No
No
3
No
4
9/27/2013
15:00:00
Yes
geometric
figure/pattern
that repeats
itself once you
zoom in...
18
9/27/2013
16:52:21
It's a recursive
Yes view of a picture
Yes
Duolingo
5
9/27/2013
18:18:56
No
No
4
9/28/2013
14:11:23
No
No
4
9/28/2013
14:10:36
No
No
2
9/28/2013
14:11:12
Yes
No
3
9/28/2013
14:11:38
No
No
3
9/28/2013
14:12:19
A pattern of
repeating units.
I think it's a
mathematical
concept that
creates a
Yes patterned image
Learnsmart with
Yes
McGraw-Hill
2
Fractal is a math
thingie where it
splits up into the
same shape (the
one what is it on
a macro level)
infinitely.
9/28/2013
14:12:45
9/28/2013
14:13:41
9/28/2013
14:15:41
9/28/2013
I.e. a piece of
broccoli (i think)
or a coast line's
Yes
shape
No
A self-similar
repeating
mathematical
structure like the
Mandelbrot Set
and Koch's
Yes
snowflake
No
No
4
No
3
No
4
Yes
MyPearson.
3
19
14:16:13
9/28/2013
14:16:20
No
Yes
9/28/2013
14:20:06
No
No
4
9/28/2013
14:25:03
No
No
5
9/28/2013
14:27:31
No
No
5
9/28/2013
14:37:43
No
No
5
9/28/2013
14:50:51
Not 100% sure
of what is it, but I
briefly read
about it for my
CMSC131 class
to do a project
involving it. All I
know is that it is
a pattern or
function of some
sort that the
more you zoom
in, the less
straight the lines
become, which
is opposite of
conventional
functions or
Yes
curves.
Yes
Duolingo
Not sure how
related this is,
but I use Khan
Academy for a
lot of
mathematical
concepts in
calculus and
linear algebra.
3
4
9/28/2013
15:00:52
No
No
3
9/28/2013
15:03:28
No
No
3
9/28/2013
15:09:19
No
No
4
9/28/2013
15:10:26
Yes
Colors, shapes,
balsamic
vinegarette.
Yes
Khan academy
3
20
Word of mouth.
Jk. It's the
infinite shapes
that can be
generated by
computers.
9/28/2013
15:17:14
No Literally no idea
No
5
9/28/2013
15:22:54
No
No
4
9/28/2013
15:29:58
No
Yes
4
9/28/2013
15:43:04
Yes
Self-similar
patterns.
No
5
9/28/2013
15:47:03
Yes
A shape that is
defined
recurisively
Yes
9/28/2013
15:47:03
No
No
2
9/28/2013
15:49:29
No
No
4
9/28/2013
15:54:45
Yes
No
1
9/28/2013
16:04:05
No
No
4
9/28/2013
16:24:09
No
No
5
9/28/2013
16:32:27
Yes
No
3
9/28/2013
17:07:40
No
No
3
No
3
9/28/2013
17:35:24
self-similar
pattern
A math thing
where a function
is within itself,
almost
recursively.
Yes usually a really
TED Talks
4
21
cool looking
graph
9/28/2013
17:51:10
No
No
4
9/28/2013
17:51:18
No
No
5
9/28/2013
19:32:54
No
No
3
9/28/2013
20:02:03
A repeating
geometric
pattern that is
similar under
various levels of
Yes
magnification.
Yes
McGraw-Hill
online learning
software, Wiley
online learning
software.
3
9/28/2013
20:17:58
No
No
3
9/28/2013
20:20:28
No
No
4
9/28/2013
21:02:35
No
No
1
9/28/2013
21:18:59
No
No
2
9/28/2013
21:59:43
No
No
3
Not that I
remember. This
question would
be more clear
with examples of
some of these
No
apps.
4
No
4
9/28/2013
22:03:24
I know a bit
about them.
They are
geometrical
constructs that
have repeating
patterns as you
zoom into them.
I no longer
remember their
Yes
applications.
9/28/2013
22:10:21
A repeatable
pattern that at
different stages
Yes
22
can appear like
different shapes.
There is an
interesting
theory that
anything and
everything can
be represented
as a fractal -this could have
huge data
compression
uses.
9/28/2013
22:55:10
9/28/2013
23:27:10
No
A fractal is an
image that has
similarities with
itself. Fractal
patterns are
pretty common
in nature such
as snowflakes
can have fractal
Yes
patterns.
No
2
No
5
9/28/2013
23:51:01
No
No
5
9/29/2013
0:14:30
No
No
3
No
4
No
2
9/29/2013
9:33:39
9/29/2013
2:17:26
A looped
equation that
results in a cool
graph with a
repeated
pattern. Useful in
antennae, video
game
landscapes, and
a lot of
environmental
Yes
science.
No
23
9/29/2013
11:43:55
No
No
2
9/29/2013
12:51:55
No
No
4
9/29/2013
15:23:10
Yes
9/29/2013
18:05:37
No
math pattern for
experts
Fractal...is that
that thing that's
4! so it's
4*3*2*1?
Yes
kahn academy
5
No
1
No
5
9/29/2013
18:41:26
No
9/29/2013
20:30:09
No
No
3
9/30/2013
0:28:50
No
No
5
RAW INTERVIEW RESPONSES
INDIVIDUAL RESPONSES
Student 1 – High School, Algebra 2 (11th grade) - Desktop Computer
Have you heard of what a fractal is?
No
Behavior notes while looking at websites:
First opened Recursive Toy, “whoa! What is this?”
Tutorial, read through some of it (until section where logs appear approximately)
Foundation, first clicked on Pictures section, looked through pictures, “we did this in elementary school
right?”
1. Rank them in order of which one you liked most, and discuss why you ranked them the way you did.
1 – Fractal Foundation because it explains more the point of it all
2 – Recursion toy because it was fun
3 – Tutorial
2. Which site taught you the most about fractals? Was it enjoyable? Do you understand what a fractal is
and how it works? --------------NOTE: this question was changed in the following interviews to avoid
yes/no answers!!
24
Tutorial, yes, yes
3. If you could pick one element you liked the MOST on any of the websites, what would it be?
The pictures and examples in Fractal Foundation. There were a lot of them.
4. What did you like the LEAST about any of these websites?
·
Tutorial – it got very complicated at the end
·
Foundation – too much information everywhere and it is not well distributed
·
Recursion toy – it doesn’t explain anything
Student 2 – High School, Geometry (10th grade) - Laptop
Have you heard of what a fractal is?
No
Behavior notes while looking at websites:
1. Rank them in order of which one you liked most, and discuss why you ranked them the way you did.
1- Recursive Toy, it’s cool
2- Tutorial, explains math with drawings
3- Fractal Foundation
2. Which site taught you the most about fractals? How would you describe the experience? (enjoyable?)
What do you know about fractals now? (do you understand what a fractal is and how it works?)
Tutorial, boring, learned that you can make drawings with equations
3. If you could pick one element you liked the MOST on any of the websites, what would it be?
Toy, draw fractals
4. What did you like the LEAST about the websites?
·
Recursive toy – no colors
·
Tutorial – boring
·
Foundation – too much information
Student 3 – Middle School (7th grade, 2 math levels: high) - Laptop
Have you heard of what a fractal is?
No
Behavior notes while looking at websites:
Played around with Recursive Toy, clicked on different things (played around with “Preset Behaviors” and
“Render Style”)
Briefly looked at Tutorial
Didn’t know what to click on in Foundation, clicked on everything
1. Rank them in order of which one you liked most, and discuss why you ranked them the way you did.
1- Recursive Toy – you had to do an activity (more interactive)
2- Foundation – shows more things, has pictures
3- Tutorial – too advanced
2. Which site taught you the most about fractals? How would you describe the experience? (enjoyable?)
What do you know about fractals now? (do you understand what a fractal is and how it works?)
- Fractal Foundation and Recursive Toy (could see what a fractal is). Tutorial is too advanced.
25
- Experience too short, could see the news in fractal foundation, overall interesting.
- It's a picture that when you click you get another picture that expands
3. If you could pick one element you liked the MOST on any of the websites, what would it be?
That you could do an activity in the recursive toy
4. What did you like the LEAST about the websites?
·
Tutorial – too advanced, didn’t understand
·
Recursive Toy – needs more color, shapes, more help on how everything works for smaller kids
·
Foundation – only gives an idea of fractals, doesn’t show you what to do (don’t know where to go)
Student 4 – High School, Calculus AB (12th grade) – Laptop in kitchen
Have you heard of what a fractal is?
No
Behavior notes while looking at websites:
1. Rank them in order of which one you liked most, and discuss why you ranked them the way you did.
1- Recursion Toy – looks interesting subject
2- Foundation – is organized in categories, has a lot of info: shows, videos, examples, guide
3- Tutorial
2. Which site taught you the most about fractals? How would you describe the experience? (enjoyable?)
What do you know about fractals now? (do you understand what a fractal is and how it works?)
- Tutorial, it is divided into sections, shows examples, step by step
- boring
- “when looking at a whole thing you can’t differentiate” ??
3. If you could pick one element you liked the MOST on any of the websites, what would it be?
Fractal Foundation has fractal shows, seems like people go and have fun with math
4. What did you like the LEAST about the websites?
·
Recursive Toy – no colors, don’t know where to go, what to do
·
Tutorial – boring, white background, small letters
·
Foundation – disorganized
Student 5 – High School, Algebra 1 (9th grade) – Laptop in kitchen
Have you heard of what a fractal is?
No
Behavior notes while looking at websites:
1. Rank them in order of which one you liked most, and discuss why you ranked them the way you did.
1- Tutorial – straight forward, first bullet: “what is a fractal”
2- Foundation – hard to look for something
3- Recursive Toy – didn’t understand it
2. Which site taught you the most about fractals? How would you describe the experience? (enjoyable?)
What do you know about fractals now? (do you understand what a fractal is and how it works?)
Tutorial, boring, science and art
3. If you could pick one element you liked the MOST on any of the websites, what would it be?
26
Pictures Found (foundation)
4. What did you like the LEAST about the websites?
·
Recursive Toy – has no pictures/topics, don't know where things are
·
Foundation – too much decoration, no organization
·
Tutorial – doesn’t have any decoration
COMPILED RESPONSES
Subjects:
·
middle school – 7th grade (2 levels: high)
·
high school – 11th grade (Algebra 2)
·
high school – 12th grade (AP Calculus AB)
·
high school – 10th grade (Geometry)
·
high school – 9th grade (Algebra)
*None of the interviewed students had heard of fractals before.
Behavior notes:
- When they were looking at Fractal Foundation they first looked at “Pictures”
- In Recursive Toy, nobody realized it was possible to moderate levels on the right
- Tutorial, only one student read through it, rest skimmed or clicked on the different parts but didn’t read
anything
1. Rank them in order of which one you liked most, and discuss why you ranked them the way you did.
#1 – Foundation 1, Tutorial 1, Toy 3
#3 - Foundation 1, Tutorial 3, Toy 1
2. Which site taught you the most about fractals? How would you describe the experience? (enjoyable?)
What do you know about fractals now? (do you understand what a fractal is and how it works?)
Foundation 1/2, Tutorial 4, Toy 1/2
3. If you could pick one element you liked the MOST on any of the websites, what would it be?
- The pictures and examples in Fractal Foundation (quantity)
- Recursive Toy – there was an activity you could do
- Fractal Foundation has fractal shows, seems like people go and have fun with math
4. What did you like the LEAST about any of these websites?
·
Tutorial: very complicated at the end, too advanced, no colors, boring, white background, small
letters
·
Fractal Foundation: too much info everywhere, not well distributed, not well organized, too much
decoration
·
Recursion Toy: no instructions, needs more colors/shapes, don’t know where to go/what to do, no
decoration, no topics
27