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Innovative
Technology in Mathematics: New Ways for Learning, Teaching, and Researching
Mathematics
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ATCM 2018,
Yogyakarta, Indonesia
- Abstracts for Invited and Plenary Papers
- Abstracts for Full Papers
- Abstracts for Presentations with Abstract Only
- Abstracts for Hands-On Workshops
Abstracts for Invited and
Plenary Papers
ABSTRACT FOR 21599
Computational
Thinking in the Mathematics Classroom
AUTHORS: Jonaki
B Ghosh
AFFILIATIONS: Lady Shri Ram College for
Women, University of Delhi, New Delhi, India
In
recent years computational thinking has been identified as one of the key
analytical abilities required for mathematics and science learning. The
rapidly changing nature of scientific and mathematical disciplines and the
need to prepare students for careers in these disciplines have been the
primary motivation for bringing computational thinking into classroom
practices. Papert (1980) was the first to stress on
the importance of computational thinking by referring to the affordances of
computational representations for highlighting powerful ideas. Over the
decades many researchers have attempted to define computational thinking.
According to Wing (2006), computational thinking involves solving problems,
designing systems and understanding human behaviour
by drawing on the concepts fundamental to computer science. However operationalising computational thinking for the k –
12 classroom remains a key pedagogical challenge.
In
this paper we shall argue that computational thinking is critical to learning
mathematics. The ability to deal with challenging problems, represent ideas
in computationally meaningful ways, create abstractions for the problem at
hand, break down problems into simpler ones, and assess the strengths and
weaknesses of a representation system are some important aspects of
computational thinking for mathematics learning. In order to encourage this
kind of thinking, specific investigatory tasks need to be integrated into the
curriculum. This article describes two research studies where participants engaged
in various aspects of computational thinking by working on investigatory
problems. In the first study, 30 pre-service teachers explored the Tower of
Hanoi puzzle through multiple representations. As they engaged with the
puzzle through pictorial, numerical, symbolic and graphical representations,
they displayed multiple paths of inquiry, used co-operative problem solving,
dealt with recursive and explicit relations and developed greatly in their
mathematical thinking. In the second study, a grade 12 student researched one
– dimensional Cellular Automata using Mathematica, a computer algebra
system and NICO, an open access software tool. The exploration included
generating multiple representations of the 256 Elementary Cellular Automata
(ECA) and categorising them based on their
evolutionary patterns. The studies highlight the nature and characteristics
of mathematical tasks which require computational thinking and illustrate
that such tasks need to find their place in school mathematics curricula.
ABSTRACT FOR 21632
Synchronization
of Chaos
AUTHORS: Guillermo Davila-Rascon
AFFILIATIONS: University of Sonora
Synchronization
phenomena pervades our daily lives and even the whole cosmos. For example,
many of our bodies physiological functions are synchronized to the day-night
cycle (circadian rhythm); thousands of pacemaker cells n
a cluster called sinoatrial node, fire at unison in order to maintain the
regular beats of our hearts; around the world, thousands of fireflies come
together along riverbanks and synchronize their flashes in an amazing
spectacle that has been noticed and reported for over three centuries; laser
beams are also examples of perfect synchronization of trillions of atoms;
electrons flowing in a superconductor is another.
We
will talk about these phenomena and present some mathematical models of
synchronization for pedagogical purposes and with the aid of technology, in
order to motivate our main topic, namely, the synchronization of chaotic
systems.
At
first glance, and since chaotic dynamics is characterized by its sensibility
to initial conditions, it is striking that two chaotic dynamical systems can
get synchronized and have a common behavior and the same evolution, through a
coupling mechanism or by means of a forcing. We will explore some instances
of synchronization of coupled chaotic systems and how this topic can be used
for a deeper understanding of chaos. We will also mention some relevant
applications in physiology and fluid dynamics.
ABSTRACT FOR 21634
Friend
or Foe: The Dangers of Dependence on Online Platforms for Conceptual
Understanding
AUTHORS: Haitham SOLH
AFFILIATIONS: Holon Institute of Technology
A counterexample is an example that refutes the fidelity of some
statement. For a mathematician, constructing counterexample is a common way
to disproof mathematical conjectures. Counterexamples also help her to
establish the constraints imposed on theorems.
This report shows that in mathematics education counterexamples can and
should be applied at the earliest stages - in the study of concepts, long
before the first acquaintance with the theorems and proofs. Herewith, the use
of software becomes an organic element of the learning process.
ABSTRACT FOR 21636
Mathematical
Walks with Mobile Technology
AUTHORS: Ron Lancaster
AFFILIATIONS: Ontario Association for
Mathematics Education, National Council of Teachers of Mathematics, Origami
USA, International Brotherhood of Magicians
Imagine
if students went out for a walk on a regular basis to view the world through
a mathematical lens and to get some exercise. Imagine if they went outside to
make measurements, to collect data, to observe how things change, and to
notice the little things that we miss all the time. Imagine if students used
their mobile device to take photos and videos of their encounters with
mathematics. Imagine if you can, a world where students do not wonder why
they are studying mathematics - they know because they have seen where
mathematics lives.
(Edited on January 10, 2019)
Mathematical walks
with mobile technology
http://tinyurl.com/23ATCMRL27
Posing Powerful Questions
by Exploring World Flags
http://tinyurl.com/23ATCMRL82
Experimenting with equations
and their graphs
http://tinyurl.com/23ATCMRL41
ABSTRACT FOR 21641
Correcting
Errors: Putting Elementary Topology to Work
AUTHORS: Juan Medina, Jose A Vallejo
AFFILIATIONS: Universidad Autonoma
de San Luis Potosi, Departmento de Matem ticas y Estadıś tica
Universidad Polit cnica de Cartagena
Nowadays,
virtually any field of Mathematics has some interesting application to
technology, and topology is no exception. We show an example of a math lab
session illustrating the use of basic notions of metric spaces (and a little
bit of probability) in the context error correction codes, through the use of
Hamming’s distance. The CAS Maxima is very well suited to this task,
and we also show how to write a simple set of commands for solving some basic
exercises. Finally, we also present some online digital resources
complementing the contents presented here, so the combined materials can be
used in a blended learning program.
ABSTRACT FOR 21642
Creating
a Symbiotic Relationship between Epistemology of Combinatorics and STEM
Teaching Process
AUTHORS: Haslinda Ibrahim
AFFILIATIONS: School of Quantitative Sciences,
college of Arts and Science, Universiti Utara
Malaysia, 06010 Sintok Kedah, Malaysia
Combinatorics
is a domain in mathematics that concerns with procedures of arranging of
objects into patterns of specified rules. Meanwhile, STEM (science, technology,
engineering, and mathematics) is a term used to group these related subjects.
Recently, educators are looking for practical ways to teach STEM. In
combinatorics domain there are steps that need to be followed to design any
specific problems. We are optimistic that these steps in combinatorics will
be the basis to establish a framework for teaching STEM. Thus, in this paper,
we would like to present the historical roots of combinatorics towards
proposing a framework for teaching STEM disciplines. Based on combinatorics
approach, we attempt to establish a practical method when resolving problems
namely explore, discover and develop (ExDiD). This
method was demonstrated to a group of students from several schools in Kedah,
Malaysia who attended the “I C D’ BEAUTY IN STEM” workshop.
We illustrated the three steps involved to establish the procedure/method
needed in resolving any complex scenario. The method was well received by the
participants of the workshop.
ABSTRACT FOR 21651
Can
Secondary School Mathematics Students be Taught Computational Thinking?
AUTHORS: Weng Kin Ho, Wendy Huang, Chee
Kit Looi, Suranteran Longanathan
AFFILIATIONS: Nanyang Technological
University, Ministry of Education, Singapore
The
thrust of this paper is to demonstrate that secondary school mathematics
students can indeed be taught to think computationally. We do so by applying
APOS theory to design mathematics lessons that intentionally integrate
Computational Thinking (CT) and Mathematical Thinking (CT), and implementing
these tailor-made lessons in authentic classroom environment of a Singapore
secondary school. Based on lesson observation notes and post-lesson
discussions with teachers, we present evidence to confirm the hypothesis that
integrating CT and MT deepens students'' understanding of mathematical
concepts.
ABSTRACT FOR 21654
An
Expert Approach of the Different Ways to Use the New Cabri
and the Richness of its Connections with the Online Freeware Cabri Express
AUTHORS: Jean-Jacques Dahan
AFFILIATIONS : IRES
The
acceleration of the provision of more and more sophisticated technological tools
and more sharp on- line applications, instead of encouraging the teachers of
mathematics to integrate more technology into their teaching might lead them
rather to take refuge in their fundamental approaches. What I mean is that
the profusion of tools like tablets and smartphones, the accessibility to a
multitude of online courses (for the teachers as well as for their students),
the poverty of their in-service training and a final examination which does
not practically take into account the use of the technologies recommended by
the curriculum together might bring the teachers to return to a more classic
teaching leading to the assessment of technical skills. So we must have a
pertinent approach of the role of the use of technology during the math lessons
compatible with the actual needs of the teachers. We (experts) must rethink
the role of the calculators, the role of software (DGS or CAS) and especially
simplify them in order to provide to teachers and students a microworld
enhancing a normal but rich practice of mathematics. The new Cabri is an environment especially dedicated to authors
who want to create multipage activities for students. These activities can be
modified very easily by teachers (there are three available modes: the
author, the teacher and the student modes). Recently the developers of this Cabri, put online a software free of charge, Cabri Express, containing a part of the tools of the New Cabri : this freeware is a calculator connected to a 2D
environment that can be extended to a 3D one. This revolutionary environment
can be a response to those who want simple, accessible and powerful tools for
primary and middle schools. What is exciting is that the files created online
can be downloaded on your computer and also files available on your computer
can be opened online. More than that, activities created with the new Cabri can be opened with Cabri
Express. We aim in this paper to show with examples all the didactical
possibilities of both Cabris. We will also present
how the restricted 3D environment provided by these Cabris
can be extended to a richer one closer to the Cabri
3D one. We will see that the proposals of the expert allow teachers to show
the known techniques of DGS to enhance a link with the real world,
creativity, discovery and pleasure of proof when it is possible.
ABSTRACT FOR 21657
Strong
Algebraic Manipulation Skills are not Adequate for Cultivating Creativity and
Innovation
AUTHORS: Wei-Chi Yang
AFFILIATIONS: Radford University
In
this paper, we use some college entrance exam practice problems from China to
highlight some essential algebraic manipulation skills that are required by
high school students from China. Next we explore various scenarios by
assuming if technological tools are available to learners, how we may see
many unexpected surprising outcomes. For many countries, requiring college
entrance exams is inevitable and sometimes is the only fair channel of
selecting qualified students to enter a college. However, we hope examples
provided in this paper can serve a purpose of advising the decision makers in
education systems by allowing students to explore mathematics with available
technological tools. After all, creativity and innovation do not come by
giving one correct answer alone.
Exploiting
Excel’s Data Table Creatively in the Study of Mathematics
AUTHORS: Graham Supiri and Deane Arganbright
AFFILIATIONS : Divine
Word Univ., PNG (Emeritus)
A counterexample is an example that refutes the fidelity of some
statement. For a mathematician, constructing counterexample is a common way
to disproof mathematical conjectures. Counterexamples also help her to
establish the constraints imposed on theorems.
This report shows that in mathematics education counterexamples can and
should be applied at the earliest stages - in the study of concepts, long
before the first acquaintance with the theorems and proofs. Herewith, the use
of software becomes an organic element of the learning process.
ABSTRACT FOR 21664
MYMA
Calculus: A Comprehensive e-Textbook and Online Homework System
AUTHORS: Douglas MEADE, Phillip YASSKIN
AFFILIATIONS: Department of Mathematics,
University of South Carolina; Department of Mathematics, Texas A&M
University
While
many calculus textbooks are now available in an electronic form, most are
glorified PDF files. The MYMA Calculus project is developing a complete
three-semester calculus book whose native format is purely electronic. The
only pages are webpages; one webpage for each theme. The content of each
theme’s webpage is dynamically generated. While some content is static,
viz. definitions and statements of general rules and theorems, many of the
accompanying figures involve a combination of user control (via sliders) and,
when appropriate, animation.
For
example, the MYMA Calculus project replaces a sequence of figures showing
secant lines through two points on the graph of a function approaching the
tangent line through a single point on the graph of a single function with an
interactive animation of this convergence process. Users can control the
animation, changing the points and specific functions involved. In a similar
manner, the examples used to demonstrate an idea are not static either. To
encourage students to work through the examples, and to develop a sound
approach to applying the concepts, the steps in the solution process appear
one at a time. And, if a student wants additional examples, MYMA Calculus can
produce additional examples algorithmically. In the same manner, not only are
homework problems generated algorithmically but student work can be assessed
step-by-step, providing context appropriate feedback at each step in the
solution process. This presentation showcases the current state of MYMA
Calculus, discusses the various technologies used, and outlines the remaining
tasks to complete the project.
ABSTRACT FOR 31000
Doing
and Learning Mathematics with ClassPad.net
AUTHORS: Barry Kissane
AFFILIATIONS: School of Education, Murdoch
University, Australia
Previous
Asian Technology Conferences in Mathematics have highlighted at least three
important aspects of technology. Firstly, the place of computer software of
various kinds for both students and teachers has been regularly demonstrated.
Secondly, all conferences have also made clear the importance of personal
technology in the form of hand-held calculators, offering a range of
mathematical capabilities to suit the wide needs of students from the middle
years through to the undergraduate years. Thirdly, especially in recent
years, conferences have illustrated that the Internet has become increasingly
evident and important in Asian communities, improving access to information
as computers and tablets become more widely accessible. In this session, we
will explore an amalgam of these three important aspects of technology, in
the form of a new online web-based facility for mathematics developed by
CASIO and located online at www.classpad.net .
The ClassPad.net
online
app allows users to regard their computer or tablet screen as a form of
virtual scratch paper onto which intelligent sticky notes can be placed, used
and manipulated, as the following examples illustrate. The first example
shows some graphing and a calculation:
Sticky
notes use various mathematical capabilities, especially graphing, geometry
and statistics, as well as general calculation, and provide access to various
mathematical tasks relevant to school and early undergraduate mathematics.
 
The
preceding example shows how a junior secondary school student might undertake
geometric drawings, and then use the software to explore the inherent
relationships involved. Geometric drawings are dynamic, so that the
measurements involved will change as the object is changed, and thus provide
a stimulus for understanding why the relationships are evident and how to
formally establish them.
The following
example suggests that key statistical work, such as that encountered by
students in introductory courses, is included within the software.
Guest
access to the ClassPad.net
app
is freely available through a standard browser, while more sophisticated
capabilities (such as those below which are reliant on CAS capabilities and
others such as the evaluation of Poisson probabilities shown above) require a
subscription.
The
focus of the Special Session will be a demonstration of some of the
capabilities provided by this new environment, as well as a consideration of
their significance for mathematics education together with some discussion of
the question of what kinds of mathematical affordances are required to
address the needs of typical learners across a suite of secondary and
undergraduate courses.
Many
ATCM delegates learned and taught mathematics in the second half of the
twentieth century, when the available equipment was often limited to a pencil
and some paper. We are now in the first half of the twenty-first century, where
the available equipment often includes a computer or a tablet and the
Internet. Teaching, learning and assessment might reasonably be expected to
change in this new environment and the major purpose of this session is to
explore one kind of possible direction for change.
ABSTRACT FOR 31001
Using
Dynamic Geometry Software in High School in Vietnam: The Case of Geometrical
Construction Problems in Space
AUTHOR: Nguyen Chi Thanh
AFFILIA TION: Vietnam National University-University
of Education
Mathematical
software in general and Dynamic geometry software in particular (DGS) becomes
a little bit popular in Vietnam secondary education, especially in urban
areas where school teachers can afford to by materials or to have access in
Internet. With the comprehensive and radical educational reform which is
carried out in Vietnam, several directions are given by the different level
of educational managements, from Ministry of education and training (MOET) to
Department of education and training (DOET). This paper aims to investigate
the use of DGS in teaching and learning geometry in school. After analyzing
curriculum and mathematic textbook in grade 11, more precisely in the topic
“geometry in space”, this study addresses the following
questions: what are the characteristics of geometrical knowledge introduced
in grade 11; what are the difficulties and obstacles of using DGS in learning
and teaching geometry in space; what are the conditions of using effectively
a DGS in teaching and learning regarding geometrical construction problem at
high school in Vietnam.
Abstracts for Full Papers
ABSTRACT FOR 21601
E-learning
System with Computer Algebra based on JavaScript programming language
AUTHORS: Takuya Kitamoto,
Masataka Kaneko, Setsuo Takato
AFFILIATIONS: Yamaguchi University, Toho
University
Usually,
e-learning system provides multiple choice questions and students choose the
correct answer. However, some e-leaning systems, such as moodle
with STACK plugin, are equipped with computer algebra systems and can deal
with mathematical expressions. Such e-learning system let users input an
answer directly in the form of mathematical expressions, hence is friendlier
to the users. One of the problems in those systems is that they often
requires a special web server that has computer algebra systems installed, which
is not an easy condition to elementary, junior, and senior high schools. This
prevents classroom teachers from creating their own e-learning materials.
In
this paper, we present a web-based e-learning system which can deal with
mathematical expressions. Our system is constructed with JavaScript
programming language, and only requires a web server with PHP support
(database and standalone computer algebra system are unnecessary). More
concretely, our e-learning system has the following features;
1.
(1) It contains computer algebra JavaScript
library, and can deal with mathematical expressions.
2.
(2) It only requires a web server with PHP
support (database and standalone computer algebra system are unnecessary).
3.
(3) It runs on standard browsers such as Google
Chrome, Firefox and Opera, and can be used on PCs, smartphones, tablets.
It has editing functions itself, and a new e-learning material can be created
on browsers. The above features make it easy for classroom teachers to create
their own e-learning materials.
We
illustrate how to create e-learning materials and give some examples created
by our system.
ABSTRACT FOR 21604
Overcoming
Identified Learning Difficulties in Trigonometry Using Casio Classwiz fx-991EX
AUTHORS: Ika Wulandari,
Triana Harmini, Siti Suprihatiningsih
AFFILIATIONS: SMK N 2 Wonosari,
Darussalam University of Gontor, STKIP PamaneTalino
The objective
of research was to find out the trigonometry learning difficulty and the
solution to the learning difficulty with classwiz
fx-991EX help. The subject of research consisted of 96 10th graders of Public
Vocational High School 2 Wonosari in the school
year of 2017/2018. The research method employed was descriptive qualitative
one. The result of research showed that learning difficulty was related to:
(a) reasoning & understanding of prerequisite material (geometry and real
number operation) including: symbol, distance, Pythagorean theorem, circle,
triangle, angle in radian and degree, pi concept, and root form
rationalization, (b) reasoning and understanding of function including:
representation (numeric, graphic, symbolic/algebraic, verbal), and inter-representation
relation, (c) reasoning and understanding of trigonometry function including:
definition of function based on unit, definition of function based on right
triangle sides ratio, multiple representation and property in trigonometry
main function. Data of diagnostic test result was supported with result of
interview with informants selected representing any types of error in
diagnostic test. Learning difficulty revealed was improved using remedial
teaching with casio classwiz
fx-991EX help.
ABSTRACT FOR 21605
Improvement
of the Usability of Figures in STACK by Use of CindyScript
and CindyJS
AUTHORS: Kenji Fukazawa, Yasuyuki Nakamura
AFFILIATIONS: National Institute of
Technology, Kure College, Nagoya University
Last
year at ATCM2017, we reported the enhancement of figures by appending the
capability of interactive manipulations in STACK, which is the e-learning
system for mathematics. In this enhancement, we fix the parameters and
functions to create figures that can be manipulated interactively, but it is
favorable that parameters and functions are selected randomly when a question
is presented to students. In this paper, we explain how randomly selected
parameters and functions can be used to create figures with interactiveness by use of CindyScript
and CindyJS. Finally, several examples are
presented.
ABSTRACT FOR 21612
Effects
of the Traditional Flipped and the In-Class Flipped Classroom Models on the
Students'' Performance in Geometry: A Comparative Analysis
AUTHORS: MAXIMA ACELAJADO, MIGUEL
PUNONGBAYAN
AFFILIATIONS: DE LA SALLE UNIVERSITY -
MANILA, DE LA SALLE UNIVERSITY - DASMARINAS, Philippine Council of
Mathematics Teacher
Educators, Mathematical Society of the Philippines, Statefields
School, Inc Bacoor, City Cavite
This
study focused on analyzing and comparing the effects of the Traditional
Flipped Classroom Model and the In-Class Flipped Classroom Models on the
students’ performance in Geometry. A quasi- experimental design was
utilized with 60 Grade 9 students from two sections of Statefields
School, Inc., during the third term of school year 2017 – 2018 as
respondents. Two sections from Grade 9 were randomly chosen to form the
control group and the experimental group of the study. The control and
experimental group were exposed to the Traditional Flipped Classroom Model
and the In-Class Flipped Classroom Model respectively. Pretest and posttest
on selected topics in Geometry were administered to determine the
mathematical performance of the respondents who were matched according to
their pretest scores. Findings revealed that learning took place among
students in both Flipped Classroom Models as evidenced by the increase in
their mean score. Furthermore, through the use of the paired samples t-test,
significant gains were observed in the mathematical performance of the
students under both Flipped Classroom Models. Thus, the Flipped Classroom
Models are both effective in improving the students’ performance in
Mathematics. The use of the t-test for independent samples pointed out that
there is no significant difference between the mathematical performance of
the respondents when they were exposed to the Traditional Flipped Classroom
Model and the In-Class Flipped Classroom Models. No significant difference
was observed between the gain scores of the respondents as well. Therefore,
both classroom models are equally effective in improving the students’
performance in Mathematics.
ABSTRACT FOR 21613
Blended
Learning Station-Rotation Model: Effects on Grade 10 Students’ Performance
in and Attitude Toward Mathematics
AUTHORS: MAXIMA ACELAJADO, MARIANNE
MONDRAGON
AFFILIATIONS: DE LA SALLE UNIVERSITY -
MANILA, DE LA SALLE UNIVERSITY - DASMARINAS, Philippine Council of
Mathematics Teacher Educators, Mathematical Society of the Philippines, City
of Bacoor National High School – Green Valley Cavite Philippines
This
experimental study was conducted to determine the effects of Blended Learning
Station-Rotation Model on Grade 10 students’ performance in and
attitude toward Mathematics. A quasi-experimental design was utilized to two
intact classes consisting of 60 students of City of Bacoor National High
School – Green Valley. Two groups of students were exposed to different
strategies: the control group was taught using the traditional approach while
the experimental group was taught using the integration of Station- Rotation
Model. A 40-item multiple choice type DepEd validated test and a modified Fennema-Sherman Mathematics Attitude Scale were used in
gathering data.
The
respondents of both groups were given pretest and posttest to measure their
performance in and attitude toward mathematics and then the t-test for dependent
and independent samples were used to determine whether there exists
significant difference between the respective mean gain scores of the two
groups. Findings showed that there is a significant difference in the
respective pretest and posttest mean performance of each group although only
the experimental group showed significant difference in their attitude toward
mathematics. Moreover, there exists a significant difference in the mean gain
scores of the two groups as well as in their attitude toward mathematics in
favor of the experimental group. Also, a weak positive correlation between
the performance and attitude toward mathematics was noted from the experiment
group.
ABSTRACT FOR 21615
Learning
and Teaching of Group Theory through Visualization using Graphs
AUTHORS: Faqir Bhatti, Khawaj
fahd
AFFILIATIONS: Riphah Institute of Computing and
Applied Sciences, Riphah Institute of Computing and
Applied Sciences, Riphah International University, Lahore.Pakistan
Group
Theory is one of the difficult courses in undergraduate level. This is not
only difficult for the students from learning point of view, but it is also
difficult for the instructors from teaching perspective. The main difficulty
in both teaching and learning is due to the abstract nature of this course.
The topics are mainly based on proving theorems about abstract structures
without having any visual representation. This lack of visualization limits
the capability of most students to understand basic concepts of group theory.
In
this paper we discuss a model that provides visualization in the form of a
directed labeled graph. Using this graph it becomes easier for the
instructors to teach and for the students to learn fundamental concepts of
group theory. Moreover, in this paper we explain how different group axioms
like closure property, identity element and the inverse can be visualized
using the proposed graph representation.
ABSTRACT FOR 21617
On
Applications of Technology to Understanding Hierarchies of Elementary
Geometry
AUTHORS: Yosuke Sato, Ryoya Fukasaku, Katsusuke Nabeshima
AFFILIATIONS: Tokyo University of Science,
Tokushima University
Problems
and theorems of elementary geometry are categorized roughly into four
hierarchies, affine, metric, Hilbert and Tarski geometry. Difference between
the latter three is especially hard to make out.
In
this paper, we give algorithmic descriptions for these hierarchies. Our
descriptions together with sophisticated programs of computer algebra systems
such as Grobner basis computation, primary
decomposition of a polynomial ideal and real quantifier elimination enable
teachers to understand these hierarchies. They also could help teachers to
make high quality problems of elementary geometry.
ABSTRACT FOR 21620
Simulating
Confidence Intervals for Mean and Variance using Real Data in R Programming
Environment
AUTHORS: Leslie Chandrakantha
AFFILIATIONS: John Jay College of Criminal
Justice of CUNY
Research
has shown that the use of computer simulation methods as an alternative to
traditional methods enhances the understanding of the statistical concepts.
The increasing availability of technology allows instructors and students to
use computationally intensive methods such as simulation. This paper presents
the use of real data and a simulation approach to help students understand
the confidence intervals for population mean and variance. Use of real data
makes the concepts more real for students and enhances their ability to
ground the new concepts in their existing knowledge. We use the R programming
environment for simulating repeated sampling from a fairly large dataset and
compute the approximate sampling distributions of sample mean and variance.
We notice that the confidence intervals for population variance work poorly
if the normality assumption is violated.
ABSTRACT FOR 21625
Improving
Student Learning Outcomes by Using an Applet in Class
AUTHORS: Russel Carlson
AFFILIATIONS: BYU-Hawaii
With
the availability of computers in the classroom, a variety of active learning
methods can be used to teach mathematics. This paper presents an example of
using an applet to teach a mathematical principle during a unit on
cryptology. As part of this unit, two sections of a general education math
course were taught how to break the Vigenère
cipher using an applet, one by lecture and demonstration of the applet, and
the other by allowing the students to experiment with the applet for
themselves. The students who had learned by experimenting with the applet
retained more knowledge of the process used to break the code than those who
had learned by lecture and demonstration. This paper discusses the Vigenère cipher and the method of teaching it using
the applet, and summarizes the difference in outcome between the two sections
of the course.
ABSTRACT FOR 21627
Empirical
Research on Mathematical Inquiry Based Dynamic Software in the Lesson Study
Training
AUTHORS: Li HongYun,
ChunLan Wu
AFFILIATIONS: Beijing Institute of
Education
This
research designed activities using theories of teachers¡ ̄ learning, two
teachers from different schools participated in the teaching of the law of
triangle altitude using dynamic software. The research showed that students
were interested in mathematics inquiry using GeoGebra or geometer¡ ̄s
sketchpad, found the exploratory direction using the dynamic of the software,
built experiences in mathematical activities, and posed good mathematic
problems under the proper guidance. The performance of students had
encouraged the teachers who participated in the research, changed the belief
in the integration of technology into the teaching, deepened the
understanding of the content of mathematics, and influenced the understanding
of the students¡ ̄ ability to learn mathematics. The research showed
that there are some points in an effective mathematical inquiry based on
technology: negotiation with the trainees who participate in the teaching to
a consensus(motivation); discussion of teaching strategies according the
content of mathematics, analysis of the students and the characteristics of
technology(understanding); teaching practice using technology ( practice);
reflection process based evidence (reflection).
ABSTRACT FOR 21630
Integration
of Classwiz Calculator to Deliver Hots Concepts of Mathematics
Learning in Indonesian Senior High School
AUTHORS: Khairuddin Budiman
AFFILIATIONS: SMA Negeri
Nurussalam Aceh Timur
Indonesia
The
National Examination of senior high school in Indonesia began to apply some
higher order thinking skills (HOTS) type problems. Among the four subjects
that were tested in the National Examination, Mathematics is the lowest.
Higher order thinking learning process directed at critical thinking,
logical, metacognition, and creative is still a new thing. Not only students
but teachers also face difficulties. Higher-order thinking involves
demonstrating an understanding of information and reasoning rather than
merely recalling information. Higher order thinking skill is one of the
priorities in students'' thinking skills in the Indonesian curriculum.
Mathematics textbooks present HOTS type questions to be discussed together in
the classroom. The use of affordable technology is a must in helping students
understand the concept of Mathematics. Scientific calculator Casio Claswizz can be used as a learning tool to deliver HOTS
concept.
ABSTRACT FOR 21631
How
To Increase Lower Order Thinking Skills To Higher Order Thinking Skills
Through Modifiying Problems Using The fx-991 ID
Plus Scientific Calculator
AUTHORS: Chandra Sri Ubayanti
AFFILIATIONS: SMA Negeri
1 Fakfak Papua Barat, Ikatan
Guru Indonesia (IGI), Master Casio Indonesia
The
demands of high-level thinking are not only the goal of learning in school
but so for students to solve the problem in their life in the 21st century
which is full of challenges. In fact the possession of high level thinking
skills based on Bloom’s taxonomy revision requires the fulfillment of
three lower order thinking skills i.e knowing (C1),
understanding (C2) and applies (C3). Without the mastery of the three, higher
level thinking skills comprising analyzing (C4), evaluating (C5) and being
creative (C6) cannot be reached. This paper will exemplify how to modify
low-level math questions into high-level questions assisted by the fx 991 ID Plus scientific calculator. This paper can be
considered by teachers who want their students to think critically in
mathematics.
Papers with Abstract Only
ABSTRACT FOR 21570
Spaced
Learning Strategy in Teaching Mathematics
AUTHORS: Ace Ceremonia,
Remalyn Casem
AFFILIATIONS: Don Mariano Marcos Memorial
State University, Mathematics Teachers Association in the Philippines, Don Mariano
Marcos Memorial State University, Mid La Union Campus
Students’
low mastery of the lesson in Mathematics is one of the alarming problems
confronted by Mathematics teachers (Department of Education, 2016). It is in
this light that this study was conceptualized to determine the effectiveness
of spaced learning strategy on the performance and mastery of DMMMSU
Laboratory High School students (Grade 7) in Mathematics. This study used the
descriptive-experimental research design, specifically the pretest-posttest
control group design. The main instrument used to gather data was the
pretest-posttest which was subjected to validity and reliability tests. It
was found out that the experimental and control groups were comparable in the
pretest and posttest. Comparison on their gain scores revealed significant
difference with the performance of the experimental higher than the control
group. It was also found out that the effect size of using the spaced
learning strategy was large. This indicates that the intervention is
effective in increasing the performance and mastery of high school students
in Mathematics. It is recommended the use of the Spaced Learning Strategy to
improve the performance of the high school students in Mathematics.
ABSTRACT FOR 21579
Order
of Addition: Theory and Algorithms
AUTHORS: Rahul Mukherjee
AFFILIATIONS: Indian Institute of Management
Calcutta
Consider
a process where the output is influenced by the order of addition of m
different components. Examples abound in chemistry where the performance of a
reaction depends on the order of adding the reagents. Similarly, in genetics,
the properties a phylogenetic tree depend on the order of addition of taxa.
Interest lies in understanding precisely how the order of addition exerts its
influence on the output. For instance, does the addition of component 1
before component 2 or vice versa have a significant impact?
Answering
questions like the above calls for testing various possible orderings. It is,
however, impractical to include all such m! orderings in an experiment,
because m! grows very fast with m. Therefore, one needs to select carefully a
subset of the possible orderings so that by testing these alone one can draw
valid inference on the issues of interest in the most efficient manner. This
leads to the problem of optimal selection of a subset of orderings from a
statistical perspective.
The
present paper develops a theoretical foundation for solving the aforesaid
optimization problem which is highly nonlinear. Analytical determination of
an optimal subset of orderings is, however, found to be intractable beyond a
certain point. Hence, we devise step-down and exchange algorithms which are
easy to understand and employ, and which are also seen to work very fast.
Thus, these can form the basis of a dedicated software for handling this
important problem, with immediate applicability to chemical and biological
sciences.
ABSTRACT FOR 21582
A
Mathematical and Experimental Studies of a Free Surface Wave
AUTHORS: Jeongwhan Choi, Sungha
Yoon
AFFILIATIONS: Korea University
This
research concerns forced surface waves on an incompressible, inviscid fluid
in a two-dimensional channel with a small bump on a horizontal rigid flat
bottom. We present the mathematical and experimental results of the surface
waves when a non-dimensional wave speed, called Froude number, is near 1. A KdV equation with forcing is derived and studied
mathematically and numerically. Various numerical results are given and some
corresponding experimental results, generated by using a thin elliptical
moving obstruction on the bottom of a long water tank, are also given.
ABSTRACT FOR 21585
Clustering
Induced Binary Classification with Asymmetric Misclassification Cost
AUTHORS: Uttam Kumar Sarkar
AFFILIATIONS: Indian Institute of Management
Calcutta
In
binary decision problems where cause-effect relationship is not understood
well, the use of classification tools and techniques of artificial
intelligence and machine learning in decision making is increasingly gaining
importance among domain experts. The fields of application are as diverse as
education, psychology, medicine and industry, to name only a few.
Commonly
used classification methods often express predictive accuracy as percentages
as a simplification effort which ignores the asymmetry that may be inherent
in the underlying misclassification matrix. In reality, such asymmetry can
have serious implications. For example, in an educational or psychological
test, the impact of wrongly assessing an inferior student as superior can be
grossly different from that of doing the other way round. Similarly, in
prediction of cancer, the cost implications of predictive error is asymmetric
as suspecting a patient with benign cancer as one having malignant cancer,
vis-à-vis certifying a patient with malignant cancer as having a benign
variety, will have asymmetry in terms of financial and mental trauma
befalling the patient and her family. Improving predictive accuracy becomes
very challenging when this asymmetry, expressed as a ratio, is specified a-
priori.
This
paper proposes a novel classification method where the specified asymmetry
ratio is incorporated in a mathematical function which is used in evaluating
and improving the quality of a binary clustering problem whose outcome then
influences the original classification algorithm for fine-tuning its
predictive accuracy. A classifier for solving such problems of relevance to
society is built and its performance is compared with commonly used
classifiers. The proposed method is computationally feasible for commonly
occurring problems involving large data sets.
ABSTRACT FOR 21586
On
estimation of Gini Index for grouped data using asymmetric loss
AUTHORS: SAIBAL CHATTOPADHYAY
AFFILIATIONS: INDIAN INSTITUTE OF MANAGEMENT
CALCUTTA
Gini
inequality index is one of the most famous and widely used measures of income
inequality derived from the Lorenz curve. It is also used as an inequality measure
in several other social and economic development parameters such as health,
educational attainment, business concentration, etc.
A
large literature exists dealing with estimation of Lorenz curve and Gini
index for grouped data. It is well known that in the absence of information
about within class variability, grouped data leads to an overestimation of
the curve and an underestimation of the Gini index. Two approaches have been
used by researchers to address this problem – either by reducing the bias
through a model based approach, or by defining non-parametric bounds on the
Gini index.
The
present article attempts to address this problem by introducing an asymmetric
loss function to estimate the Gini index. The loss function is suitably
chosen so that underestimation becomes more serious than overestimation. An
alternative estimator of the Gini index is proposed with respect to such
asymmetric loss function, and its dominance over the usual estimator in terms
of estimation risk is studied. Simulation studies are carried out to
demonstrate the procedure.
ABSTRACT FOR 21602
Data
Simulation with Markov Chain Monte Carlo, Gibbs Sampling, and Bayes
(Beta-Binomial) Methods as the Parameter Estimations of Spatial Bivariate Probit Regression Model
AUTHORS: Dewi Retno
Sari Saputro, Yuanita
Kusuma Wardani, Shaifuddin
Zuhdi, Purnami Widyaningsih
AFFILIATIONS: Department of Mathematics, Universitas Sebelas Maret, Department of Mathematics, Faculty of Mathematics
and Natural Sciences, Universitas Sebelas Maret
A probit regression model is a regression with categorical
dependent variable in dichotomy or binary form. The dependent variable value
of the probit regression states the probability of
certain issue. In some cases, the application of this model considers the
influence of area (spatial effect). The dependency tendency to the close
regions is known as autocorrelation in spatial data. Due to this matter, the parameter
estimations of ordinary least square (OLS) method cannot be used thus it is
substituted by the simulation technique, which is a method in randomly
raising data. The method includes direct and indirect simulations. The latter
has Markov Chain Monte Carlo (MCMC) method with Gibbs Sampling algorithm,
which is the order in conducting certain distributed random data sampling by
understanding the required distribution. In this case, the beta binomial
distribution is applied. A data simulation with Gibbs Sampling algorithm can
be conducted by knowing the required distributions of each variable used for
R software beforehand. This research purposes to define the parameter
estimation value of spatial probit regression by
applying MCMC and Gibbs sampling methods with R software. The results show
that the parameter estimations of spatial bivariate probit
regression model by simulating through Gibbs sampling algorithm (R software),
in which β ̂
is the
independent variable parameter and ρ ̂ is the spatial lag autoregressive
coefficient. The simulation with the first-value determination rise result
data β=〖(0,1,-1)〗^ ,ρ=0.7
deciding n=400, m=10, and k=6 show the estimations for β ̂=(0.01205,0.98709,-0.9675)
and ρ ̂=0.68523.
ABSTRACT FOR 21626
Misconceptions
of Differential and Integration Technique among Engineering Students: A Case
Study l
AUTHORS: Prof.Dr. Sarwar Jahan Abbasi, Ambreen
Zehra
A F F I L I A T I O N S : University of Karachi, Karachi-Pakistan
The aim of
this study is to identify the problems of engineering students in Pakistani
universities, face to learn derivatives and integrations techniques in their
first semester calculus course. The study based on data collected of 500 engineering
students of Hamdard University, Karachi-Pakistan.
In the process of data analyzing students’ misconception lie as
follows:
1.
1)
Product and quotient rule of derivatives was applied on integration
problems.
2.
2)
ln function was integrated by using derivative technique.
3.
3)
Integration of trigonometric function was evaluated by using derivative
technique.
4.
4)
In substitution technique of integration students observed confused in
both derivatives and integration.
The results
of this study will identify the students’ problems and will help them
improving their technology aided learning skills.
ABSTRACT FOR 21629
Unlocking
the Power of Data: Teaching Statistics to First- Year Life Science Students
AUTHORS: Maree Skillen
AFFILIATIONS: Western Sydney University
We
live in a data-driven world, and yet many students are likely to have trouble
with statistics due to non-cognitive factors such as negative attitudes or
beliefs towards this area of mathematics. Students can be puzzled when faced
with questions which may include: How does a statistical model differ from a
mathematical model? What are the differences among the sample distribution,
the sampling distribution, and the population distribution? In an experiment,
what effect does the sampling method have on the results? What are the
implications of the use of processes of random selection and random
assignment? Can a small sample yield accurate estimates of population
parameters?
Changes
to the structure, content and teaching approaches including less theory and
more data, making data analysis central, building intuition, incorporating
technology, fostering active learning, and using context to develop
statistical inference have supported student engagement and improved
performance levels in an undergraduate statistics subject.
Specific
areas of focus to be addressed in this presentation will centre
on summarising curriculum developments implemented;
explore how the application of real-data aided to support student
understanding; consider possible challenges, limitations and implications for
teaching statistics to undergraduate students; and, identify possibilities
for further research. Examples used will relate to a first-year University
subject which focused on introducing statistical concepts and techniques to
support decision making in the life sciences.
ABSTRACT FOR 21655
Professional
Development for Mathematics Teachers: Using a New Hypermedia Video-Case Tool
AUTHORS: Hang Chen, Jian-sheng Bao
AFFILIATIONS: School of Mathematical Sciences,
East China Normal University, Shanghai, China
The
purpose of this paper is to introduce a new model for teacher professional
development, which focus on teaching practice based knowledge and competency.
The model is structured into three main parts: the knowledge base for
instruction; the process of classroom teaching and learning; and, the
teachers¡ ̄ believe system. This programme
involves the creation of several series of hypermedia video-cases on teaching
and learning designed to facilitate mathematics teachers¡ ̄ professional
development. Each of these video-cases consists of lesson clips, case
questions, interviews with experts, comments by peers, responses by students
and other related resources. The study has implications pertaining to the use
of technology in teacher development, the production of hypermedia
video-cases, as well as research on case-based pedagogy and pedagogy in
general.
ABSTRACT FOR 21660
Conjugacy
Maps and Dynamics of a Family of Optimal Fourth-Order Multiple-Zero Finders
AUTHORS: Young Hee
Geum
AFFILIATIONS: Dankook University
We
have developed a family of optimal fourth-order multiple-zero finders with
rational weight functions and discussed the stability of the strange fixed
points for conjugate map. In addition, the stability surfaces for the fixed
points are displayed and the relevant dynamical analysis has been handled
from the viewpoint of stability analysis as well as in terms of parameter
spaces and dynamical planes associated with basins of attraction.
ABSTRACT FOR 21663
The
Mathematics Class Using the Tablet PCs
AUTHORS: Tsutomu Ishii
AFFILIATIONS: Bunkyo University, Japan.
In this
paper, we will consider about the reason of using Tablet PCs. Japanese
teachers don’t use Tablet PCs so much. The reason is the research
question. Japanese Government is supporting financially for teachers to use
the projectors and computers in the classroom. Recently, teachers using
Tablet PCs are increasing. But there are teachers who don’t use Tablet
PCs, too. In the 1st step, several studies were considered. And we checked
its educational possibility and expectation of Tablet PCs. For example, in early
childhood showed that teaching with Tablet PCs contributed significantly to
the development of children’s mathematical ability. In the 2nd step, we
analyzed the mathematics class using Tablet PCs about the percentage of the
elementary school 5th grade in Japan. We analyzed us carefully about
understandings of children and instructions of teacher. In the 3rd step, as
the case study, we considered whether instruction by Tablet PCs was proper in
detail. And we argued the result based on several studies. In this paper, at
the end, we pointed out 2 points. First, children’s understanding
deepens by Tablet PCs. Next there is little contribution to bring the
flexibility and creativity up.
Abstracts for the Track of
Hands-On Workshops
ABSTRACT FOR 21573
Using
a Graphics Calculator for Learning Mathematics
AUTHORS: Barry Kissane
AFFILIATIONS: Murdoch University
Graphics
calculators were developed more than thirty years ago to meet the needs of
students learning mathematics in the secondary school and the early years of
college; significant research has established that they can be effective
tools for both students and teachers. Recent models, such as the CASIO fx- CG50, can be used to support student learning and
teachers teaching about many mathematical ideas related to real numbers,
functions, equations, statistics, probability, matrices, sequences and
calculus. The graphics environment allows students to generate and use graphs
of functions and data, explore the use of spreadsheets, investigate random
phenomena, and explore significant mathematical concepts in powerful ways.
This introductory hands-on workshop will allow participants to experience
some of the learning opportunities available, and become aware of free
resources to support their work. Previous experience with graphics
calculators will not be assumed.
ABSTRACT FOR 21574
Learning
School Mathematics with ClassWiz, an Advanced Scientific
Calculator
AUTHORS: Barry Kissane
AFFILIATIONS: Murdoch University
While
scientific calculators have been available since the 1970s, early use in
schools often concentrated on numerical computation alone. Recent advanced
versions have been developed explicitly to suit the needs of mathematics
education throughout the secondary school years and offer more powerful
capabilities, potentially valuable for learning; modern advanced calculators
provide powerful learning opportunities for many aspects of mathematics in
the secondary school and early college years. In addition, capabilities such
as tabulation, spreadsheets, equation solving and other advanced features
give students access to efficient calculation. This workshop will focus on a
variety of ways in which the CASIO fx-991EX ClassWiz
calculator can be used to enhance both teaching and learning, drawing on
published materials and videos developed by the author for free use by
teachers. A variety of topics will be addressed, including calculus,
functions, equations and statistics, as well as aspects of mathematical
modelling. Previous experience with these calculators will not be assumed.
ABSTRACT FOR 21575
ClassPad.net:
A New Online Environment for Doing and Learning Mathematics
AUTHORS: Barry Kissane
AFFILIATIONS: MurdochUniversity
ClassPad.net
is a new online software environment from CASIO that provides access to
useful tools for calculation, graphing, geometry and statistics, designed to
support secondary school mathematics. In this hands-on workshop, participants
will use the software in a browser to explore its potential for students
learning mathematics
and for teachers teaching mathematics. ClassPad.net provides users with
virtual Paper and allows them to attach Sticky Notes to the paper for
mathematical purposes. Sticky Notes are used to add mathematical expressions,
graphs, tables, spreadsheets, and geometric shapes. They can be attached
wherever users want on Paper to create original materials for learning and
teaching. ClassPad.net can be accessed from computers or tablets; participants
are welcome to bring their own devices to the session if they wish, equipped
with a browser and Internet access, although it is not necessary to do so.
Previous experience with ClassPad.net will not be assumed.
ABSTRACT FOR 21584
Teaching
Statistics and Visualization with QR Codes
AUTHORS: Wei Ching Quek
AFFILIATIONS: Singapore Polytechnic
The
New Classroom Standard : Casio Classwiz is loaded
with useful spreadsheets, statistics functions and is able to generate
relevant QR codes to retrieve additional graphical information. This workshop
is consists of two activities:
1.
(1) Getting Started
Begin with introduction to new features in ClassWiz,
followed by exploring essential features of the statistics and spreadsheet
function, and to visualisations of charts by QR
codes generated.
2.
(2) Problems Solving/Discussion
Explore some applications of in statistics, distribution and spreadsheet.
Participants will be able to gain further insights to the problem.
ABSTRACT FOR 21598
STACK:
an Online Assessment System for Mathematics
AUTHORS: Yasuyuki Nakamura, Yuko Ichikawa,
Takahiro Nakahara, Yoshinori Miyazaki, Saburo
Higuchi, Kentaro Yoshitomi
AFFILIATIONS: Nagoya University, National
Institute of Technology, Tokyo College, Sangensha
LLC., Shizuoka University, Ryukoku University,
Osaka Prefecture University
STACK
is an online assessment system for mathematics, science and related disciplines.
STACK, developed by Sangwin (2013) of the
University of Edinburgh, uses Maxima as its CAS to evaluate students’
answers. STACK not only assesses the mathematical equivalence of
students’ answers but also generates outcomes, such as providing feedback
according to the mathematical properties of students'' answers. The feedback
function is implemented using the potential response tree (PRT) mechanism.
PRT is an algorithm that establishes the mathematical properties of
students'' answers and provides feedback specifically to each student. STACK
is an open source system and users can develop required functions. For
example, a new input type, FlickMath, was developed
for using STACK on mobile devices (Nakamura and Nakahara 2016). FlickMath allows the input of mathematical expressions by
the flick operation. The flick operation is carried out by placing a finger
on the prepared keyboard, shifting the finger vertically or laterally, and
subsequently releasing it.
We
would like to demonstrate STACK and have a workshop in which participants can
solve questions and make some typical questions.
The
STACK demonstration website should be accessed via: www.stack.ed.ac.uk
ABSTRACT FOR 21618
Discovering
the Online Freeware Cabri Express
AUTHORS: Jean-Jacques Dahan, Jean-Marie Laborde
AFFILIATIONS: RES of Toulouse, Cabrilog
We
will discover the possibilities of this free online calculator which
environment contains both a 2D and a 3D DGS. We will see how to record the
created files online and to open online files created by anybody. We will
also see that this freeware allows the users to open activities (that can be
multipage activities) created with the New Cabri
ABSTRACT FOR 21619
Modelling
the Moebius Strip with Cabri 3D
AUTHORS: Jean-Jacques Dahan, Jean-Marie Laborde
AFFILIATIONS: RES of Toulouse, Cabrilog
Let
us enter in the 3D space allowed by Cabri 3D. We
will learn how to use Cabri 3D in modelling with
mathematical but tricky constructions a Moebius strip
ABSTRACT FOR 21639
Exploring
Concepts and Applications of Mathematics Using Casio Teaching Tools
AUTHORS: Jonaki Ghosh
AFFILIATIONS: Lady Shri Ram College, Delhi
University
This
workshop session will highlight the use of the CASIO CG - 20 graphics
calculator as a teaching tool in the high school mathematics classroom. The
emphasis will be on developing conceptual understanding in various
mathematical topics and also in enabling students to perform mathematical
investigations. Participants will be given a hands-on experience on the
calculator. Problems will be taken up from the topics of functions, calculus,
matrices, simulation of experiments in probability and statistical inference.
Applications of mathematics to topics such as cryptography, genetics and
chaos theory which can motivate the high school student towards
investigations will be highlighted.
ABSTRACT FOR 21640
Image
Processing with the CAS Maxima
AUTHORS: Guillermo Davila-Rascon, Jose Antonio Vallejo
AFFILIATIONS: University of Sonora,
Autonomous University of San Luis Potosi
This
hands-on workshop is intended to provide some pedagogical tools for math
teachers in order to motivate students on the utility of elementary matrix
calculus and some of its applications, by using the computer algebra system
Maxima
The
workshop is divided in two parts: An elementary one where basic Maxima
commands for producing and manipulating binary matrices, as well as their
image rendering, are introduced. We also show how the Maxima palette can be
used for handling colored images (2 hours).
In a
second and more advanced part we show how to handle and manipulate more
complex pictures with Maxima, focusing on some of its characteristics like
color and contrast. We introduce, for this purpose, some Maxima commands that
allow us to make a deeper analysis of a given picture, like getting the
frequency distribution values of its coloring and the respective histogram.
We also perform an equalization process that renders a softer image in which
extreme colors are replaced with softer tones, closer to the average (2
hours).
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