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Adopting Drone Technology in Mathematical
Education
1st Siditë Duraj
Department of Mathematics
University of Shkodra “Luigj
Gurakuqi"
Shkodër, Albania
sidite.duraj@unishk.edu.al
2st Lekë Pepkolaj
Department of Engineering
Albanian University
Tirana, Albania
l.pepkolaj@albanianuniversity.edu.al
3st Gerald Hoxha
Department of Engineering
Albanian University
Tirana, Albania
gerald.hoxha@agsco.al
Abstract - Mathematics has always been one of the most
challenging topics in education. Many factors come into
play, one of which is the lack of use of what technology
offers. The technology of drone aids learning that involves
concrete experience such as a lab, observations and
feedback, the formation of abstract concepts and the
application or testing of this knowledge in a real-life
context. This paper describes a theoretical framework
outlining the use of drone technology in mathematics
education in order to improve engagement and learning
with the experimental learning method. In particular, it
looks at the pedagogical aspect where drones can be used
in combination with a variety of learning techniques and
methods. The role of drones in mathematics education is
seen in two aspects, the mathematics topics that make the
drone work and the drone as a mathematical object in itself
and as a medium that can construct mathematical objects.
Various problems in trigonometry, geometry, linear
algebra, differential equations and statistics can be
covered. The basic design of the drone can be explored
geometrically, the 2D and 3D model of a drone can be
understood, designed and constructed. In the end, the
study can be used for educators and researchers interested
in the field of drone-based mathematics learning.
Keywords: drone technology, experiential learning,
mathematical education
I. INTRODUCTION
The main purpose of drones in the early stages was for
military purposes and today we have large surveillance
drones and small personal drones, which are smarter,
smaller, more reliable and easier to use. Although its
early use of drones was only for military purposes,
today’s drones have many applications like
thermography, agriculture [1], disaster management [2],
military drone [3], thermal observation [4], disaster
monitoring and management [5], state agency
investigations, landslide and other steep terrain
investigations, bridge inspections, roadside and roadway
inspections [6].
The rapid development of technology over the years has
made drones more accessible to the public, thus gaining
more attention in academics to create smart solutions for
the real-world applications. Despite the fact that drones
are becoming increasingly common in a variety of
industries, their use in education is still relatively new.
However, drones have previously been shown to have
educational benefits in the fields of environmental
chemistry education [7], geosciences, geology,
journalism education [8], and model-based learning [9]
used model-based learning to educate about situational
analysis, which involves students analyzing
circumstances and scenarios, setting up and flying the
drones and mapping them to create models. Other
applications are in the humanities to engage students in
history and archeology, in philosophy and ethics, in
domains of transportation, in particular, road safety,
traffic monitoring and highway infrastructure
management [10], [11], [12]. Drones can also be used to
collect bacteria from whale lungs [13].
Despite these educational advantages and possibilities,
few tools and models exist to guide the use of drone-
based learning in higher education [7]. Regarding the
fact that these studies [14], [15], [16] suggested drone-
based systems still use drones as learning outcomes
rather than providing frameworks to assist educators in
designing drone-integrated learning environments.
Many universities have recently engaged with drone
technology allowing students to work on their own to
make learning easier. Basically, drones can be used in
universities to teach different topics and subjects.
Learning activities can be planned with the use of a
drone to reinforce classroom learning. For example,
students can improve their programming skills by using
a computer program to direct a drone to fly in a certain
direction [17]. In this way, students are more motivated
to learn and more interested in the learning process when
they have been asked to apply their knowledge in an
authentic environment. As mathematics can be a really
challenging subject, drones can give a real-world
application to solve problems from different domains of
it such as trigonometry, geometry, linear algebra,
statistics, engaging students to see the actual result of
their work, as well as realizing the great power of
mathematics.
II. A FRAMEWORK OF DRONOLOGY IN EDUCATION
With the proposed framework of dronagogy, which is an
active learning ecosystem, the drone-based learning for
mathematical education links all three aspects which are
encouraged and enabled by each other [18]. The figure
1 describes the relationship between pedagogy-space-
technology aspects.
Fig. 1: Pedagogy-Space-Technology Framework
A. Technological aspect
Drones are unmanned aerial devices and they are
controlled remotely. For example, the simple
quadcopter drone consists of two pairs of counter-
rotating rotors and propellers, electronic speed controls,
a radio transmitter and receiver, a microcontroller such
as Multi-Rotor control Board, a battery, and a charger.
Drones are often classified according to their range,
which includes a very close range, close range, short
range, mid-range, and endurance drones. Very close-
range drones can fly up to 40-45 minutes and can travel
5-8 kilometers.
Drones have three levels of autonomy [19]. These levels
are:
• Reactive autonomy, where drones can react
according to the environment as avoiding
obstacles and coordinate with other moving
objects.
• Sensory-Motor autonomy, where drones can
perform high-level human commands as
moving according to global positioning
system.
• Cognitive autonomy, where drones are capable
of simultaneous localization and mapping,
object and human recognition, panning and
learning as well. The simultaneous localization
and mapping are illustrated in the figure 2.
Fig. 2: Simultaneous localization and mapping of cognitive
autonomy.
The educational affordances of small autonomous
drones can be classified into three categories [16]:
• Video shooting and monitoring based on active
tracking.
• Video shooting and monitoring based on
gestures.
• Video shooting and monitoring based on
controllers.
Video shooting and monitoring based on active tracking
refers to a video taken by the drone while it is tracking
an object or location. It is done by using geolocation and
video imagery monitoring [20]. These drones video
shoot on a fixed target and follow the target’s movement
without the need for human interference.
Video shooting and monitoring based on gestures is
accomplished by using local on-board cameras on
drones and machine vision techniques. When the drone
captures a hand gesture, it uses the approximate hand
direction and face score system to measure the angle and
distance.
Video shooting and monitoring based on controllers is
the common form of video shooting. The controller is
connected via wireless signals, and to a mobile phone or
tablet to visualize the video shooting.
Drone technology opens a world of opportunities in
school, allowing students to actively improve their
understanding by integrating analytical knowledge with
logic and cognitive skills. Drones have a lot to promise
in this regard, as they can help students study science,
technology, engineering, and math (STEM) topics in an
engaging and enjoyable way. Below are listed some
types of drones for educational purposes:
Size
Model
Level of
application
Medium
Aeromao-
Aeromapper EV2
Universities /
Research and
surveying.
Mini
DJI Phantom 4
High School
Mini
Parrot AR. 2.0
High School /
Universities
Mini
Robolink
CoDrone
High School /
Universities
Mini
Ryze DJI Tello
EDU
Universities
Mini
Yuneec Typhoon
H
High School /
Universities
Nano
Delphy Explorer
Primary school
Table. 1: Drone types, models, and suitability for education
Drones have a great potential to be used as an innovative
medium for studying and experimenting. Thanks to its
excellent quality and ease of use, the DJI Phantom 4 can
be used in high school level as well as to solve real world
problems. Other example, the Ryze DJI Tello EDU is an
excellent drone to engage students in coding. The Parrot
AR 2.0 can be used to educate students in real-world
problems that are covered in high school and college
science.
B. Pedagogical aspect
In order to optimize the ability of drones, suitable
teaching and learning methods and hypotheses must be
chosen, since drones provide fascinating educational
affordances that could be used in a variety of learning
contexts. In terms of pedagogy, drones may be used in
conjunction with a variety of learning techniques and
methods. As we saw in technological aspect, the three
levels of autonomy can be used in order to learn different
topics as well as different fields of study, allowing
educators to tailor their pedagogy to the educational
capabilities of drones, or use drones to fit their
pedagogy.
Integrating drone technology in and out of the classroom
has a lot of advantages [21], [22], [23], [24], [25]:
Develop Deep Understanding: This advantage improves
the ability to understand individual concepts as well as
relate many concepts at once, leading to further thought
and the creation of new concepts. Using drones can
reflect the same problem in different ways, this way
students can see various daces of the problem, which
aids in the construction of interpretation, the
advancement of ideas and the encouragement of making
sense of their ideas, all of which contribute to the
creation of new insights.
Motivation and Engagement through Hands-on Practice:
By incorporating drones into student task has the
potential to increase inspiration and engagement.
Involving students in hands-on experiences with drones
increases their interest in the subject while also
including the element of fun, resulting in learning that
takes place in light and friendly atmosphere with long-
lasting effects.
Technical Knowledge and Skills: Knowledge, learning
and creativity, life and job skills, media and technology
skills, are among them. As drones will be a big consumer
technology in the future, it is a wise idea to train and
educate students with technical skills and experience
that will be in demand in the future. Using drones for
educational purposes will improve student’s
technological expertise and problem-solving skills,
preparing them to meet the potential technical and
professional demands.
Potential for Developing Spatial Visualization Skills:
The ability to visually control two-dimensional or three-
dimensional structures is known as spatial visualization
[26]. Using drones in education can include three-
dimensional learning opportunities, similarly of the
three-dimensional interactive software. On the other
hand, it allows students to directly feel three-
dimensional sensations of their surroundings, as
opposed to seeing abstract objects on a computer screen.
Students use spatial logic to construct flight patterns by
considering the three-dimensional geographical details
surrounding them when learning to fly a drone [10].
Potential for Developing Sequencing Skills: The ability
to arrange objects or schedule actions in a linear order is
referred to as sequencing [27]. Young children’s play
games also reveal sequencing abilities. Sequencing has
also been linked to statistical and problem-solving
abilities. The researchers [28] suggested that the
sequencing capabilities shown through the use of
building blocks demonstrated the basic mathematical
thought by which children view the environment.
Critical Thinking: Drones allow students to be
motivated to improve problem-solving skills,
interpretation, imagination and critical thinking skills
while also being inspired to learn more about their
subjects by designing assignments in a creative way.
Students are involved in and engaging in an experience
that allows them to contribute subject knowledge and
skills, putting them in a position to build claims, reason,
and logically evaluate the problem when drones are used
in school.
Soft skills development, such as teamwork: Teamwork
can also be used as a pedagogical method, and students
working on a drone together will learn useful career
progression skills they can help them advance in their
careers. Drones in the classroom will boost student’s
morale and interest while also allowing them to learn a
variety of skills. The researchers [24] have developed a
drone-use in-classroom module that focuses on the user
interface while establishing research-based foundations
for ethical and practical drone use. Learning activities
can be planned for the use of drone to reinforce
classroom learning. Students were more inspired to
learn more and more interested in the learning process
when they were asked to apply their expertise in a real-
world environment. Drone helped to improve content
understanding, pedagogy and self-learning abilities [29].
Simulation for the students to self-control their work:
Students will use a simulation to learn how to self-
control their work [30]. Students working on several of
the examples use simulations to validate their responses.
A student attempting to place a satellite into
geostationary orbit, for example, can double-check his
or her answer by gazing back at the Earth from the
satellite to verify if the satellite reminds in place. This
provides a number of benefits.
• Students are interested in seeing if their
responses are correct.
• When students know they are correct, they are
more confident in their replies.
• Students are aware when their replies are
incorrect. They can usually figure out what
went wrong by watching a simulation
• Students are considerably more inclined to
preserve until a problem is solved correctly.
Debate: Drones themselves can be a lesson in
discussion. Drones provide the perfect background for
such a methodology because they are dealing with real
life problems [31].
C. Space aspect
An important part of space aspect is physical learning
spaces or places. “A learning space should be able to
motivate learners and promote learning as an activity;
support collaborative, as well as formal, practice;
provide a personalized and inclusive environment; and
be flexible in the face of changing needs” [18].
Physical learning environments can be divided into two
categories [32]:
• Structured. Teacher-centered education.
• Unstructured. As example, collaborative
learning, social learning and self-learning.
In terms of drones and learning environments, drones
have educational opportunities in both interactive and
simulated learning environments. Starting with physical
learning spaces, drones can be designed to be used in
both structured and unstructured learning environments.
Drones may be used for outdoor lab research and
fieldwork in structured learning environments. For
unstructured learning environments, in both indoor and
outdoor learning environments, drone capabilities like
active tracking-based video shooting can be used to
record community conversations [16], [20],
III. MATHEMATICS FOR DRONES
The drone can be used in and out of the classroom and
its advantages in teaching mathematics are numerous.
We will not dwell on the education that is taken in the
construction of the drone, nor on its integration with
various computer programs, but we will discuss the help
that mathematics provides in the operation of the drone
and the drone as an object where mathematics can be
taught.
A. The mathematics topics that make the drone work.
Drones are a wonderful setting in which to study
mathematics and technology. For example, the Parrot
AR 2.0, it is designed for indoor and outdoor use. This
model can be used to educate students in real-world
problems. For example, this drone can fit for
mathematics classes like trigonometry and Pythagorean
theorem, linear algebra, differential equations, statistics,
as well as for simple mathematics like addition,
subtraction, multiplication and division. Some
mathematical topics where the Parrot drone can be used
are listed below [33]:
Flight planning and trigonometry: This simulation can
be performed in an open environment where dimensions
are measured, illustrated in figure 3. The drone starts
from a fixed point with known dimensions in relation to
those of the environment. You can measure the
coordinates of the base from where the drone starts, then
its destination coordinates. Another item is the
measurement of the angle from the base to the
destination. Knowing the speed of the drone, we can
measure the flight time.
Fig. 3: Flight planning and trigonometry.
Pythagoras' theorem, division and addition: Suppose
you want to fly from a base to a position x meters north
of the base and back. The drone, for example, travels at
x meters per second. Simple math’s can be used to
answer the questions: First leg bearing, time spent at
destination, second leg bearing, total time spent at base.
Planning a wind speed of x meters per second blowing
from east to west on this mission, using a simulation [30]
to double check your work, Pythagoras' Theorem can be
used to address this question. Another question, the
mission should be planned with a wind speed of x meters
per second blowing from northeast to southwest, a
simulation can be used to double check your work, this
question may require the use of trial and error on
vectors.
The Language of Differential Equations, Position and
Velocity in Two Dimensions (2D): For instance, suppose
a pilot is flying from point (-x,0) on the other goal line
with a y meters per second wind blowing from south to
north. It takes m seconds to complete this flight. The
identical flight would take less than m seconds for a
mathematically oriented pilot. Students can use a
software to imitate a non-mathematical pilot [30]. Two
simulations are used:
• A “flying simulator” in which the user directs
the drone’s flying path.
• The students must finish a skeletal simulation.
For the second simulation, students can explain the
flight in differential equations. We will use the letters p
and d to represent the drone’s location and the
destination’s location respectively. Students can provide
the following differential equation (1):
()
This is a fantastic exercise on Differential Equation
language for vector-valued functions.
Matrices and Vectors: Drones can be controlled by and
iOS or Android device. These devices can tell where
they are and how they are being handled thanks to an
electronic compass and a three-axis accelerometer. The
foreword direction of the drone is determined by the
angle at which the front-facing camera is pointed. In the
context of understanding aerial images and back-and
forth translations between image coordinates and
ground coordinates, with its photography capabilities,
the drone naturally raises these issues. If a downward
looking camera takes an aerial snapshot while the drone
is flying level, then a transformation of the form (2)
transforms the coordinates x1 and x2 on the image to
coordinates y1 and y2 on the ground (or a map of the
ground).
y1 = a11 * x1 + a12 * x2 + b1 (2)
y2 = a21 * x1 + a22 * x2 + b2
or in a matrix-and-vector notation (3)
(3)
These two questions can be asked of students: Find the
transformation from ground coordinates to photograph
coordinates given the transformation from photograph
coordinates to ground coordinates above. Find the
transformation that transforms photograph coordinates
to ground coordinates given ground and photograph
coordinates of three recognized features. Combining the
drone with simple imaging software to better
comprehend the surroundings or implementing simple
transformations in mathematics courses using software
like MATLAB (or GNU Octave) for computing basic
image transformations might be intriguing (e.g.,
translation, rotation, dilation, affine transformations and
more) [30], [33].
Traveling Salesman Problems on Steroids: The
traditional traveling salesman issue is determining the
shortest path a traveling salesperson may take to visit a
group of customers. We frequently have a list of goals
for the drone – locations where we’d like to capture
aerial photos – and want to visit as many of them as
possible on a single battery charge [30].
B. Drone as an object that can teach mathematics
As the students’ progress and they have already
mastered the concepts taught in geometry class, now
they have the ability to study, adapt and observe
geometric concepts as they refer to drone technology.
Arguments that can be addressed:
Explore from geometric point of view the basic design
of the drone:
• Identify simple geometry concepts shown and
interpreted in the body and function of a drone,
make references to math or science principles.
• Understand 3D models and the use of geometry
in drone architecture and technology.
• Plan and build 2D and 3D model of a drone.
Some different measurements when the drone is flying.
Since the drone is attached to a small platform [34], the
angle of the tether will be established using a protractor
attached to the platform.
Fig. 4: Calculating the length of C.
As seen in the figure 4, different lengths and concepts
can be measured: e.g., length B, (length of the tether),
distance from tether to a student. Students should now
have two sides and an angle after using the protractor to
find C. To find the missing side C, use the law of
Cosines. Students should now be able to identify three
sides as well as an angle. To build up proportion and
solve for missing angles, use the law of Sines. Figures
such as cones and cylinders can be created, their surfaces
and the volumes can be found [34].
Another argument can be calculations such as battery
life.
The use of photogrammetry concepts is a pleasant way
for users to calculate from aerial imagery. We will
utilize a drone to take an autonomous snapshot, which
we will then use to determine distance and area. Students
may learn how to compute the area from the process of
taking a drone snapshot.
Students learn how to construct and understand graphs
by watching the drone’s movements and illustrating its
journey with a graph of distance and time.
Calculate the likelihood of occurrences and compare
data from observed models. Examine and analyze
alternative explanations for model differences. Observe
the motion of an item and use the facts and information
to assess and forecast future motion.
IV. CONCLUSIONS
New drone technologies continue to open up new
avenues of teaching/learning. Today's increased
functionality of drones has made their use in a wide
variety of fields much more practical and interesting,
and they can also be used to make mathematics more
fun. The use of drones in mathematics teaching has
many advantages:
• Both for the teacher and the student and should
be considered more seriously for inclusion in
educational curricula.
• Because of its potential, the use of drones
should be considered as an alternative teaching
strategy for learning environments where it can
Law of Cosines
Law of Sines
α
produce more effective learning than
traditional ones.
• Drones can play an important role in engaging
and motivating learners with attributes of being
confident, creative, active and informed.
• Content knowledge can be reinforced and there
are opportunities to build interactive and
collaborative activities by actively
participating in them.
• A learner-centered inquiry-based learning
environment can be created.
• One can address real world problems and
improve their critical thinking and reasoning in
the learning process.
Finally, we suggest these recommendations
• Challenge-based learning could be more
suitable and could yield other educational
possibilities of drones in the field of
mathematics.
• Further explore techniques for integrating
drone technology into mathematics
teaching/learning.
Drone technology is still in development, it has a lot of
potential for improvement in the future.
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