GEOMETRIC ORNAMENTS IN ISTANBUL
Authors: Miroslaw Majewski
Affiliations: New York Institute of Technology, Abu Dhabi Campus
Islamic ornaments are one of the greatest achievements of
ancient geometers in the Middle East, Turkey, India, Spain, and North Africa. These ornaments are frequently used to decorate secular and civil buildings,
books, and furniture. Istanbul, in particular, is a place where such ornaments
were, and still are, frequently used.
In this lecture we will explore selected geometric
ornaments from Istanbul. Many of these ornaments were created using very
precise geometric constructions. We will analyze structure of these ornaments
and show how grids used to draw these ornaments were constructed. We will use
Geometer’s Sketchpad to construct these grids and complete ornaments. Many of
presented in this lecture examples have a serious didactical value. While creating
them students can learn a number of traditional topics in geometry –
constructions of regular polygons, constructions of figures circumscribed or
inscribed in a circle, division of angles and segments into a given number of
parts, transformations of figures, symmetry groups, topics related to coloring
maps, and many other.
Tilings, Patterns and Technology
Authors: Ma.Louise Antonette De Las Penas, Angela Fatima Guzon
Affiliations: Ateneo de Manila University
Tilings and patterns theory has been an interesting field
of study by mathematicians for more than 100 years. The mathematical theory of
tilings includes various branches of mathematics such as geometry, algebra and
number theory.
In this talk, we discuss the different roles of technology
in the mathematical study of tilings and patterns. We show how colored tilings
and patterns, generated through the aid of technology, can be used as tools in
teaching and learning algebraic and geometric concepts, and in the development of
critical thinking. In the past years we also have seen new development in
tiling theory such as the emergence of nonperiodic tilings (e.g. the Penrose
tiling), nonEuclidean tilings, and tessellations in higher dimensions. We also
present how it is possible, through technology, to understand the interesting
properties of these tilings, to explore the deeper mathematical ideas presented
by these tilings and to discover a wide variety of their connections to nature,
architecture, art and the study of crystal structures. In addition, we
illustrate how technology facilitates the solution and proof of challenging
mathematical problems suggested by tilings. A focus of this talk will also be
the mathematics of Islamic patterns and tilings. Part of the discussion will be
technology as a means to exploring the cultural connections of these tilings;
and as a bridge to linking historical tilings and modern emergence of these
tilings.
How to Double the Rate of Discovery with Cabri 3D?
Authors: Jenchung Chuan
Affiliations: National Tsing Hua University
Joseph Gergonne and JeanVictor Poncelet independently
noted that given any theorem or definition in projective geometry, substituting
point for line, lie on for pass through, collinear for concurrent, intersection
for join, or vice versa, results in another theorem or valid axiom, the
"dual" of the first. The principle of duality has thus doubled the
rate of discovery. In this talk we are to examine what other principles are
doing the same using Cabri 3D as a tool of discovery.
Cases in TechnologyEnriched School Mathematics
Authors: Beverly Ferrucci, Ngan Hoe Lee
Affiliations: Keene State College, National Institute of Education Singapore Nanyang Technological University
This presentation is a report of three case studies
intended to advance the integration of technology into school mathematics. The
first study used interactive computer software and visual manipulatives to
enhance primary school students’ understanding of equivalent fractions and
comparing fractions in Singapore. The second study reviews the performance of
preservice secondary school teachers on proportional reasoning lessons that
incorporated various technologies. The third study reports on the mathematics
lessons presented by preservice secondary school teachers using Smart Board
technology.
Analysis of these case studies included performance data
and video assessments of students’ presentations. Results showed that (1)
technologies provided an excellent platform for primary school students to
build their understanding of mathematical concepts, principles, and
applications; (2) preservice secondary teachers used a variety of appropriate
technologies in proportional reasoning lessons; and (3) future mathematics
teachers were quick to develop expertise in using the Smart Board to design and
implement lessons in secondary school mathematics.
Presentation of these cases includes samples of student
computer interactions and video clips of students’ lessons. We conclude the
presentation with suggestions for further research involving the integration of
technology with school mathematics.
New Content, Methods, and Approaches Promoted by the Integration of
Technology in Secondary Mathematics
Authors: Antonio R. Quesada
Affiliations: The University of Akron, Department of Mathematics
In this presentation we review relevant changes that the
integration of technology in general and of handheld graphing technology
(HHGT) in particular, facilitates at the secondary and basic college
mathematics levels. We will show how numerous topics and concepts,
traditionally reserved for upper level mathematics courses, are now accessible
at lower levels. Likewise, the different representations and data types that
HHGT provides, and lately the possibility of recognizing in one platform a
variable defined in any other, makes possible the introduction of different
approaches to solve a variety of problems, as well as some new approaches
inherent to the new technology. Finally, more than ever, technology makes it
possible to implement the use of the “explorediscovertestconjectureprove”
model at every level, providing the possibility of increasing the inquirybased
approach in the way we teach and learn mathematics. This approach may help
students become independent learners, in a time when it is becoming
increasingly important to do so, since new discoveries and the access to data
is increasing exponentially. The implications of these potential changes in
content, approaches to problem solving, and methods affect most aspects of the
teaching and learning process. The curriculum in secondary mathematics is already
full; however, given the importance and scope of the applications now
accessible, it is important to ask: does our curriculum responds to these new
topics at this level? What criteria can be used to decide what topics should be
added and which ones should be removed? Does the preparation of our future
mathematics teachers respond to these new realities? Do our current textbooks
take into consideration these possible changes? If time allows, we will provide
some initial research results to how these questions can be answered in the
USA.
Interactivity in dynamic mathematics environments: what does that
mean?
Authors: Colette Laborde, JeanMarie Laborde
Affiliations: University Joseph Fourier, Cabrilog, France
It is commonly accepted that an important feature of
technology and technology based tasks lies in their interactivity and in the
possibility of providing feedback to students’ actions. The talk will address
the notion of interactivity and feedback from a twofold perspective, on the one
hand the design of the features of a dynamic mathematics environment and on the
other hand the design of tasks based on this environment and of different types
of feedback provided to students. It will be shown that the degree of
interactivity may greatly vary and that interactivity affects many aspects of
the use of such environments. The discussion will be illustrated by the various
Cabri technologies.
Imagery and The Geometric Imagination
Authors: Nicholas Jackiw
Affiliations: KCP Technologies, Inc., Simon Fraser University
Ubiquitous digital cameras enrich discussion in the math
classroom by providing both handy images of relevant applications, and the
opportunity for motivating connections between students'' personal lives and
their mathematical activity. At a deeper level, geometry is our best language
for thinking about shape and space, and in turn helps us construct, transform,
dissect, and analyze images. Technology is the vehicle that brings these ideas
together accessibly and affordably. In this talk, I'll use The Geometer's
Sketchpad Version 5 to describe some of the geometric and algebraic
possibilities that arise, up and down the curricular ladder, as one begins to
explore mathematical ideas using digital pictures. In the accompanying paper, I
investigate in more detail some of the software functionality and design
decisions that permit these explorations in Sketchpad.
Redefining school as a pit stop: It is the free time that counts
Authors: Lenni Haapasalo
Affiliations: University of Eastern Finland
Instrumental genesis in a modern society together with a
redefined conception of teaching and learning have caused that mathematics
appears more learner (community) centered, and more distributive (i.e. free
from time, location and formal modes). On the other hand, it has been
recognized that the general lack of enjoyment of institutional mathematics
teaching is one of the basic reasons behind the bad reputation of mathematics
in society. Increasing students’ motivation to make mathematics through
enjoyment and playing, especially in their free time, might therefore be a
relevant research focus. This contribution discusses with concrete examples how
instrumental genesis pushes educators on all levels to adapt a thorough shift
in their views as regards where the optimal learning of mathematics is
allocating and which kind of assessment would fit this new paradigm. It also
discusses reasons why the school institution might encounter stalemate in assessment,
and severe problems to orchestrate especially technologybased investigation
spaces which allow students to explore the facility of real and virtual
environments which are both meaningful to them and their society and which
naturally motivate a greater use of mathematical language in its different
forms. The value of these environments is assessed from the perspective of the
challenges of instrumental orchestration, represented by the author in his
plenary in ATCM 2008.
Perspectives of Interactive Geometry Software
Authors: Ulrich Kortenkamp
Affiliations: Pädagogische Hochschule Karlsruhe, CERMAT, Cinderella
Interactive (or dynamic) geometry software seems to be
wellestablished by now. Being invented more than 20 years ago, with roots in
the sixties of the last century already, we might be tempted to say that
"the problem has been solved" and move on. There might be some
innovation, bringing IGS to more devices, improving the presentation of
sketches, integration with other mathematics software  but all these are
rather evolutionary and not revolutionary. The great impact of the first real
geometry software systems  Geometer''s Sketchpad and Cabri Geometry  does not
seem to happen again.
On the other hand, even that impact did not really change
the situation in schools. In my talk I want to point out some of the real
challenges we face. How can we substantially change the way teachers and
students use the computer for teaching and learning? What methodology will help
us to find best practices? What are the chances for research in mathematics
education? I may raise more questions than I will be able to answer, but as the
ubiquity of computers cannot be stopped, it is time to ask them.
Technology Has Shaped Up Mathematics Communities
Authors: WeiChi Yang
Affiliations: Radford University
There is no doubt that technological tools have greatly
impacted our mathematics teaching, learning and research in recent years. The
exciting innovative ways of presenting learning, teaching and research
materials on the internet have prompted educators and researchers to rethink
the importance of taking global views to solve local problems. In this paper,
we use several examples to demonstrate some abstract mathematical concepts can
be conveyed to students graphically, and the needs to conduct collaborative
research because the existence of a solution is simply not adequate when
computational tools are available nowadays. Furthermore, we emphasize that
technology can be implemented effectively to enhance preservice teachers’
content knowledge. Finally, we urge all software and hardware developers to
work together to make the learning tools more uniformly accessible; after all,
beginners should be concentrating in how mathematics content is learned instead
of worrying about what software syntax or hardware keystrokes they should be
using. When interests in math and science are genuinely cultivated, one can
truly appreciate mathematics and discover exciting mathematical theories.
Constructivist in Mathematics in Virtual Class with Moodle and The
Geometer's Sketchpad
Authors: Krongthong Khairiree
Affiliations: International College, Suan Sunandha Rajabhat University
Bangkok Thailand
This paper is a report of research explored how Moodle and
the Geometer’s Sketchpad can enhance students construct their mathematics
knowledge. The samples of this research were students of International College,
Suan Sunadha Rajabhat University. Moodle website of mathematics course was
designed. The components of the Moodle website were course description,
mathematics contents and topics, worksheets, assignment, and chat room.
Mathematics activities using the Geometer’s Sketchpad were designed and embedded
in this Moodle website. Research finding showed that students were able to
construct their mathematics knowledge through Moodle in virtual classes or
online learning environment by communicating and receiving helping from peers.
The students explored their mathematics activities with the Geometer’s
Sketchpad; they interacted by dragging and animating as many as they wanted.
Through Moodle, the shy students who never asked questions in normal class were
able to ask and received feedback from lecturer and their friends. The students
reflect their thinking by chats, write answers, informal talk and discuss with
their friends. The students had positive attitude toward mathematics.
My JournEy From Logo To Geogebra
Authors: Adnan Baki
Affiliations: Karadeniz Technical University, Turkey
This presentation deals with my journey within the world of
educational computing. This journey started with Logo in 80''s and continues
with Cabri and Geogebra. At the beginning of the journey I came up with the
book titled "New horizons in educational computing". In this book
Saymor Papert enthusiastically says that computers as powerful learning tools
will change tomorrow's classrooms. In 80''s and 90''s, I actually had
difficulty to see this potential of computers in changing teacher's role and
practice. I tried to compromise my teaching approach based on telling and
showing with the approach based on Papert’s constructivist ideas. It was not
easy for me to shift from traditional notions of teacher to constructivist
teacher using Logo, Cabri and Geogebra as primary tools for doing and exploring
mathematics in classrooms. In this presentation, I illustrate how I linked Logo
with Piaget’s assimilationaccommodation process. Then I give examples showing
how we as mathematics teachers can initiate problem solving and exploration
experiences though Cabri and Geogebra in mathematics classrooms together with
our students. So, our journey does not finish, we continue to run towards new
educational horizons.
Graphing Calculators in Secondary Mathematics Teaching and Learning
Authors: Chang Pei Wang, GT Springer
Affiliations: HewlettPackard, China
The teaching of mathematics is by necessity mediated by
representations of mathematical objects. In some sense, this makes mathematics
unique among the academic subjects. Indeed, the U.S. National Council of
Teachers of Mathematics (NCTM) clearly recognized the central role of
representations when they included a representation strand in their Principles
and Standards for School Mathematics. U.S. secondary mathematics teachers have
long acknowledged the graphing calculator for its ability to quickly create,
manipulate, and switch between various representations of mathematical objects.
Chinese mathematics teachers share similar ideas about representation; indeed,
the expression of "shu xing jie he" (combine number with graph) has
already been widely accepted as a golden principle for mathematics teaching
since the 1960’s. A recent project in China, Integration of Handheld Technology
with the New Mathematics Curriculum, gives us an opportunity to compare and
contrast the ways Chinese teachers and students use the graphing calculator
with the ways of their American counterparts. The similarities are as interesting
and informative as the differences. Come join us to hear about our preliminary
findings and hypotheses.
Deriving Geometry Theorems by Automated Tools
Authors: Pavel Pech
Affiliations: University of South Bohemia
Derivation of geometry theorems belongs to mighty tools of
automated geometry theorem proving. By elimination of suitable variables in the
system of algebraic equations describing a geometric situation we get required
formulas. The power of derivation is presented on computation of the area of
planar polygons given by their lengths of sides and diagonals. This part we
conclude with derivation of a formula of Robbins for the area of a cyclic
pentagon given by its side lengths. Searching for loci of points of given
properties is a special case of derivation. This topic belongs to the most
difficult parts of school mathematics all over the world. New technologies DGS
and CAS enable to overcome this problem. We demonstrate it in a few examples
from elementary geometry.
StudentFriendly TechnologyAided Calculus Applications with
Minimal Overhead
Authors: Douglas Meade
Affiliations: Department of Mathematics, University of South Carolina,
Industrial Mathematics Institute, USC
Many students do not appreciate or relate to the theoretical
basis of the mathematics that they are learning. Instead, they are motivated by
applications that can be analyzed with the mathematics that they are learning.
Because of the diversity of our students, and limited time, the applications
should not require too much additional background from another discipline.
Unfortunately, many of the classical calculus applications do not meet these
requirements.
The speaker has assembled a collection of applications that
have proven to be accessible and appealing to students. This talk will
highlight a few of these that benefit from the use of various modes of
technology, including computer algebra, dynamic geometry, and graphical
visualization. Examples include designing a roller coaster, an exploration of
fractal curves that are continuous everywhere but differentiable nowhere (Koch
snowflakes), a model for a parachute jump that is physically realistic and
meets safety guidelines, and a model for the trajectory of the rear wheel of a
scooter
Each of these applications involves no mathematics beyond
what is found in traditional singlevariable calculus. The use of technology
allows these applications both to be introduced earlier in the curriculum and
to reach mathematicallyinteresting conclusions than otherwise possible. As a
result, more students are attracted to study more mathematics  both
theoretical and applied.
Abstracts for Full Papers
Dynamic construction of the common perpendiculars in the
threesphere
Authors: Yoichi Maeda
Affiliations: Tokai University
Construction problems are good exercises to understand
geometry deeply. Using dynamic geometry software, we can easily check whether
some conjecture is true or false. In this paper, we introduce a construction of
the common perpendiculars to two great circles in the threedimensional sphere.
We will see that the foci of some hyperbola play an important role in the
construction.
The Development of Multimedia Courseware of Lines and Planes in
3Dimensions: an Application of van Hiele’s Levels
Authors: Wan Fatimah Wan Ahmad, Syazwan Noordin
Affiliations: Universiti Teknologi PETRONAS, Computer & Information Sciences
Department Universiti Teknologi PETRONAS, 31750 Tronoh, Perak, Malaysia
The role of visualization in Mathematics has been a subject
of current research. This paper is inspired by and utilizes the van Hiele’s
levels for teaching and learning Lines and Planes in 3 Dimensions. Based on an
early study, students have identified of having problems in visualizing the
figures. Therefore, a multimedia courseware was developed and applied the van
Hiele’s levels on the visualizing the 3D models. This paper also points out
features of van Hiele’s Levels and shows that they are also characteristics of
the proposed levels of Lines and Planes in 3D. The tools used in the
development were 3DS Max 7 and Macromedia Director MX. The results of the
courseware development will be discussed. This paper concludes that, the
development of multimedia courseware with 3D models is mainly based on van
Hiele theory which is believed to solve the students’ problems in
visualization.
Development of Creativity Using 3D Dynamic Geometry System InMA
Authors: Shelomovskiy Vladimir
Affiliations: Deoma, Murmansk State University
Experience of teaching mathematics with Interactive
Mathematical Art (InMA) software is described in this article. InMA project has
been used for teaching mathematics in Russian schools with indepth study of
mathematics since 2005. Courses in algebra and geometry have been developed for
secondary school based on the traditional textbooks. Now about 1500 students
and 60 teachers use InMA software, 20 teachers act as experts, i.e.,
periodically make suggestions to improve certain topics. Teachers have a
special interest in interdisciplinary methodical sets, examples of which are
demonstrated in this article. InMA project includes computer algebra system
(CAS), singlestepping system (SSS), graphing tools for all kinds of functions
and two kinds of dynamic geometry systems (DGS). That allows creating
interactive 2D and 3D geometric images, interactive graphics and text with
changeable parameters. InMA is a tool for creating electronic textbooks and
methodical sets. All the constructions in the geometric part of InMA project
are carried out on the screen. However, teachers prefer to use readymade
teaching packages on the topics, only adding some more lines if it is need to
answer questions of curious students. Many complex geometric problems can be
represented as logical chains and a process of constructing of such a logical
chain helps to develop student`s creativity. Dynamic geometry systems help to
build logic chain, allowing stepbystep to provide a common part of
overlapping solids. In this paper we consider the samples of methodical sets
created using dynamic geometry system GInMA, the part of InMA project.
The Gambler's RuinAnalysis by Spreadsheet
Authors: Thomas McMillan, James Fulmer
Affiliations: University of Arkansas at Little Rock
The “Gambler's Ruin” problem is stated as follows. Two
players have a total of N coins, they win or lose a coin based on the outcome
of a coin toss, and they continue to flip a coin until one or the other player
has won all N coins. We use this problem to illustrate how the spreadsheet can
be used to give a detailed analysis of this game, giving numerical answers to
the following questions: (1) What is p(n,t), the probability that a game ends
in t coin tosses given that one of the players has n coins? (2) What is q(n,t),
the probability that a player with n coins loses the game in t coin tosses? (3)
What is p(t), the probability that a game will last t coin tosses? We
illustrate using the spreadsheet to check our models for internal consistency
and for consistency with known results. Our examples illustrate how using the
spreadsheet motivates the study of recurrence relations and the advantages of
this approach in an introductory discrete mathematics class.
AN INNOVATIVE APPROACH TO LEARNING PROCESS: EFFECTS OF DYNAMIC
MODELING ON TEACHING OF MATHEMATICS
Authors: Halil ARDAHAN
Affiliations: Selcuk University
The purpose of this study was to examine prospective
teachers’ views on dynamic modeling activities for the renewed curricula. Data
collection was done with the Dynamic Modeling Activities Scale (DMAS) developed
by the researcher in the year 2008. The survey is a 5point likert type scale.
Oneway analysis of variance was used to test differences between first and
last tests. Paired Samples and Wilcoxon test were used to examine further
significant differences. Results were discussed in accordance with the research
questions being addressed. Discussion and implications of the results were
articulated. This study focuses on the prospective teachers’ qualified learning
and teaching mathematics and effects of dynamic modeling on learning process of
them. The sample included 77 secondary mathematics prospective teachers, 48 of
them are female and 29 are male attending the Pedagogical Formation Programs in
Secondary Mathematics Education in Selcuk University in 2011. To collect data
Dynamic Modeling Activity Scale (DMAS), including five point Likert type ten
items, developed by the author was used. Reliability coefficients of the scale
is 0,84 and 0,78 respectfully by the pretest and post test. Descriptive
statistics and Wilcoxon signed rank test for paired samples revealed that
dynamic modeling has more positive effects on qualified learning and teaching
(QLT) than our expectations. The degree of positive effects of quality factors
on QLT outcomes was studied and ordered as follows; thinking mathematically and
expressing with the mathematics language have the first degree positive effect
on qualified learning via dynamic modeling. Meaningful and anchored learning
have second degree, constructing reliable knowledge has third and constructing
and discovering relations among the concepts and content with the real life
situations have forth degree positive effect on qualified learning which the
desired result all over the world is. Also, you will see an innovative approach
on designing the learning process and interactive instructional materials which
enable our metacognitive skills. We are glad to announce that mathematics
modeling designed dynamically have very positive effects on inquiry learning
process.
Self Learning Laboratory Sessions for Engineering Mathematics
Authors: Pee Choon Toh
Affiliations: National Institute of Education, Nanyang Technological
University
In this paper, we will discuss the implementation of self
learning computer laboratory sessions for a first year mathematics course
designed for students from the Faculty of Engineering. Computer laboratory
sessions have been part of this course for several years. It typically consists
of allocating students into small groups to be taught by instructors. The self
learning model significantly reduced the amount of resources required and
introduced flexibility and was well received by the students. This was made
possible by the availability of free open source software.
Developing Mathematical Teaching Materials of Fundamental Analytic
Geometry and Conic Sections Using The Geometer’s Sketchpad (GSP) on Students in
Grade 10
Authors: Ubol Klongkratoke
Affiliations: Department Informatics Mathematics. Faculty Science &
Technology.University Rajabhat Suan Sunundha, Bangkok ,Thailand.
This research aims to develop Mathematical teaching
materials on the topic of the introduction to Analytic Geometry and Conic
Section using the Geometer's Sketchpad (GSP) on the students in grade 10 in
order to have the performance efficiency of 75/75 by the comparison on the
student achievement before and after these useful teaching materials are used
in classes and to study the attitude of students on Mathematics after taking
classes with the application of these materials. The sample group in the research
is 33 students studying in grade 10 at the Demonstration School of Suan
Sunandha Rajabhat University by simple random sampling. Tools used in this
study include lesson learning plans on the introduction to Analytic Geometry
and Conic Section using the Geometer's Sketchpad (GSP), a 30question
achievement test on Analytic Geometry and Conic Section and a 20question
attitude test on Mathematics. The collected data are statistically analyzed in
terms of mean, standard deviation and coefficient of variation. The results
reveal that (1) the effectiveness in using the Geometer's Sketchpad (GSP) on
the introduction to Geometry and Conic Section on 10th grade students is at the
level of 76.40/76.50 (2) students’ academic achievement on the post test is higher
than that on the pre test at the significant level of 0.05 (3) students’
attitude toward Mathematics after the use of these Mathematical teaching
materials is at a good level.
Revisiting the teaching of perimeter, area and volume at a middle
school level with Cabri environments
Authors: JeanJacques Dahan
Affiliations: IREM of Toulouse
Two years ago a joint research project between my Institute
of research and a french middle school in Casablanca had been started. The aim
of this project was to combine my skills as an expert in dynamic geometry and
experimental math with the skills of a colleague of this middle school as a
teacher and a technology user. We aimed to respond to her needs for resources
involving using Cabri environments to teach some contents of the French middle
school syllabus: perimeter, area and volume. This paper describes the files
created by the expert, how they were designed with the teacher and how they
were used to enhance the more experimental practice of mathematics in the
classroom. The theoretical background of this work generated the results I have
obtained concerning the experimental process of discovery using Cabri.
On Some Advantages of Using Mathematics Software
Authors: Tuncay Kör, Ali Hikmet Deger, Murat Beþenk, Bahadýr Ö. Güler
Affiliations: Karadeniz Technical University Faculty of Science
Department of Mathematics, Karadeniz Technical University
Science Faculty Department of Mathematics, Rize University Arts & Science
Faculty Department of Mathematics Keywords: In this paper,we want to share our
observations in some courses. We give some example about that how increase
students'' interest in the case of giving an opportunity to use some
mathematics software exploring some concepts in the content. For studying group
action which one of the standard mathematical subject in algebra, we give some
data to students and want to produce their conjectures. Results show us that
our students are more willing to prove their own conjecture and try to explain
eagerly.
Introducing algebra with interactive geometry software
Authors: Kate Mackrell
Affiliations: Institute of Education, University of London, UK
In order to consider the ways in which interactive geometry
software might facilitate the expression of generality, an exploratory
inspection using a series of tasks related to finding a connection between the
radius and area of a circle was performed using Cabri II Plus, Casyopée,
Cinderella, GeoGebra, Geometer’s Sketchpad and TINspire (CAS). It was found
that the programs specifically designed to include algebraic functionality had
less potential for enabling students to develop the concept of variable and to
generate symbolic representations between variables than did the programs with
less overt algebraic functionality. An implication is that more attention needs
to be paid to the needs of learners in the early stages of algebraic thinking
when incorporating algebra within IGS.
Mobile Calculating Lab (MCL) based Mathematics Application
Authors: Ling Yiguo
Affiliations: Beijing No. 15 high school
Since 2007, Beijing Education carried out curriculum
reform. Mathematics education was performed according to newly revised
curriculum standards and adopted new text books. In our fresh curriculum standards,
we emphasize “Develop the students” mathematics application consciousness and
propose. The connection between mathematics application and practice need in
our senior school mathematics education needs to be strengthened, and try to
turn mathematics application teaching into practical operating process. Mobile
Calculating Lab is one of many various mathematics application approaches.
While I was teaching mathematics application, I adopted this teaching model and
got some experiences. I summarized the general formula of this teaching model
should contain the following four parts: question raising; experiment designing
and data collecting; data analysis; data obtaining. The value behind this
teaching model is that it helps the students foster their abilities on data
collecting, data sorting and data processing; enhance their consciousness of
mathematics application; deepen their understanding of the intrinsic side of
the mathematics knowledge.
MCL is?
Authors: Yapin Tian
Affiliations: Weishanlu Middle School
After two years use of graphing calculators Hp39gs, MCL
gradually has stimulated the students’ interest in learning, enriched students
learning methods and improved their capability of learning. This paper aims to
argue that MCL does promote the overall development of students at all levels
through typical and specific cases from the three aspects under the new curriculum,
and mainly reflects in how: (1) MCL makes those advanced math learners find a
helping hand. (2) MCL helps those who are weak in math to find selfconfidence;
(3) MCL helps those who have no interest in mathematics students to find a
starting point.
A Preliminary Study on the Use of Online Resources in Quantitative
Techniques for First Year Business Students in a Malaysian Private University
Authors: CheeKeong Chong, YouHow Go, YingYin Koay, CheeHeong Lee
Affiliations: UTAR, Universiti Tunku Abdul Rahman, Malaysia
Keywords: In this paper we intend to find out the awareness
of students and the use of online resources provided by text book publisher.
The online resources identified are PowerPoint slides (plain and narrated),
audio lectures, Excel templates and Java applets on statistical concepts. The
usage level of these was found to be low except plain PowerPoint slides which
are common among students. On the other hand after the demonstration by the lecturer,
most of students considered these to be useful in their Quantitative Techniques
studies. Recommendations were made to integrate Excel and Java applets for the
study of Quantitative Techniques.
Creating computer graphics and animations based on parametric
equations of lines and curves  proposals for mathematics education at upper
secondary level
Authors: Andreas Filler
Affiliations: HumboldtUniversitaet zu Berlin, Gesellschaft fuer Didaktik
der Mathematik
Creating computer visualizations, especially animations, can
help students to understand geometric objects (especially straight lines and
curves), which are described by parametric equations, as point sets and to
discover functional relationships and dynamic aspects. Because creating
computer animations is very attractive for students it can help to motivate
them to figure out features of parametric descriptions. This paper makes
proposals for creating graphics and animations on lines, circles, spirals,
parabolas and other curves by describing these curves with parametric equations
and shows some examples created by students at upper secondary level. As a
prerequisite the students should have basic knowledge in trigonometry and
elementary analytic geometry. Computer animations based on parametric equations
of lines and curves can be created using computer algebra systems or
photorealistic 3d graphics software (like POVRay). Examples using both kinds
of software will be shown and described.
To Make APLET according to Actual Condition
Authors: Chengyang Liu, Jianyi Yang, Zhenping Rao
Affiliations: Quanzhou No.7 High School
When we use APPLET program carried by HP39GS itself, we feel
it compact, convenient and considerate. These classical programs are thought over
and refined again and again then developed basing on plentiful teaching
experience and they are consummated after being launched into the market. But
teaching content and test of math all over the country are different, for
example, there is no division for Art and Science in Jiangsu Province, what
more, dihedral angel is a point in the College Entrance Examination in Sichuan
Province but is not required in Fujian. On the other hand, HP39GS has not
programs like Vector Applet and solid geometry which are needed but not
available. We therefore have to develop the Applets that suit local education
according to the actual situation so as to ensure the students use it
frequently and correctly. In Fujian Quanzhou No. 7 Middle School, we use Applet
as often as possible even create the possibility to use while there is no.
Complying with ¡°Fujian College Entrance Exam Illustration¡± writer integrate
study situation in the class and develop independently several Applet program
along with accumulating in normal teaching. With many revision and improvement
in teaching practices, it leads to a good effect.
Developing Concepts in Linear Algebra and Analytic Geometry by the
Integration of DGS and CAS
Authors: Ana Donevska Todorova
Affiliations: Humboldt Universität zu Berlin, Deutschland, MIT
University Skopje, Macedonia Keywords: Facilitation of the
Computer Algebra Systems (CAS) in pedagogic purposes is an incentive for any
mathematics researcher, yet integration of the Dynamic Geometry Software (DGS)
with CAS in teaching mathematics is even a greater challenge. This paper,
throughout created applets in the DGS GeoGebra and the CAS wxMaxima, sustained
by additional materials, refers accurately to implementation of the integrated
DGS and CAS in obtaining new teaching approaches in the course Linear Algebra
and Analytic Geometry. Created teaching/ learning resources aim to facilitate
the transition period from the upper secondary to lower tertiary level of
this course. For this reason two and three dimensional, dynamic worksheets
have been designed and furthermore, they have been implemented in the
mathematics classroom. The goal of creating these interactive dynamic
worksheets is to develop important concepts in Linear Algebra and Analytic
Geometry, about which students have no previous knowledge. Thus, the paper
presents an indepth research in discovering new concepts for teaching and
learning this essential mathematics discipline.
The Effectiveness of ICTassisted Approach in Learning 3D Linear
Algebra
Authors: Hitoshi Nishizawa, Yoshihiro Yamada, Takayoshi Yoshioka
Affiliations: Toyota National College of Technology
Keywords: Linear algebra was one of underachieved fields of
mathematics for the students of colleges of technology in Japan. The low scores
at achievement tests were partially because, in the former lessons, the
procedures of symbolic manipulation were taught separately from the features of
associated graphical objects or how they were used in real world applications.
The isolated procedural knowledge was easily forgot just after the
examinations. To compensate this situation and deepen the students’ conceptual understanding,
we redesigned our lessonplan to be directed from actual applications towards
Abstract mathematical ideas; from handling graphic objects and observing their
characteristics towards building vector equations and manipulating them
symbolically. In the new lessons, several software programs are used to
demonstrate the close relation of graphic objects and vector equations
interactively. This paper reports how the new lessons are changing our students’
learning styles from blackbox approach and deepening their conceptual
understanding of vector equations. Their written answers have richer
explanations than their former students who learnt in traditional lessons.
Their scores in annual INCT achievement test, which stayed low for previous
four years, also increased significantly in 2011.
The Effectiveness of Using Scientific Calculator in Solving
NonLinear Equations by NewtonRaphson Method
Authors: Kian Boon Lim, Grace Joy Yong, Tau Han Cheong, Kim Gaik Tay
Affiliations: Universiti Pendidikan Sultan Idris, Universiti
Teknologi Mara, Universiti Tun Hussein Onn Malaysia In this
paper, we will report the result of the study on the effectiveness of using
scientific calculator in solving nonlinear equation by Newton Raphson method
from the aspect of students'' marks and duration taken in solving question. A
total of 38 mathematics students who enrolled for numerical analysis course in
the second semester session 2008/2009 in the Sultan Idris Education University
are involved in this research. The data taken from the pre test and post test
were analyzed using the Statistical Package for Social Science (SPSS).The findings
show that using scientific calculator gives positive effect on students''
ability to obtain the correct solution as well as reduces time to solve the
problems.
An Anchored Instruction Case Study: Developing Fifteens Puzzle as a
Graphical Calculator Class Task
Authors: Chang Wenwu, Xinsheng Lu
Affiliations: Shanghai Putuo Modern Educational Technology Center,
Master teacher of Intel TTF & TEO Program, Mathematics
Science Department Shanghai Normal University P.R.China Keywords: In this paper
we lead a class of 20 to rebuild a classical mathematics game called Fifteens
in HP39gs graphical calculator. This task covers 4 classperiods when high
school students of grade 11 work in team solving every kind of real problems.
This practice is guided under the theory of anchored instruction first raised
by John Bransford. We find this design has rich mathematics connotation and
suitable for high school students.
Solving NonLinear Equation by Newton Raphson Method using Built in
Derivative Function in Casio fx570ES Calculator
Authors: TAU HAN CHEONG, KIAN BOON LIM, Tay Kim Gaik
Affiliations: UNIVERSITI TEKNOLOGI MARA, Universiti Pendidikan
Sultan Idris, Universiti Tun Hussein Onn Malaysia Keywords:
This paper studied the difference between the used of built in derivative
function of scientific calculator and self derivative function in solving
nonlinear equation by means of NewtonRaphson method. Since this method
requires the derivative of the function, some basic differentiation skills are
needed. However, functions especially highly nonlinear functions are hardly to
be differentiated. Students may find it is difficult and thus quit from it.
Now, with the help of the built in derivative function in scientific
calculator, the problem can be easily solved regardless of the analytical
expression. Results obtained using builtin derivative function and exact
derivative are compared and discussed. A total of 350 engineering students
(2009) from School of Technology in Kolej Tunku Abdul Rahman who enrolled in
subject ATGE3083 Mathematics V have been taught using this method to solve
nonlinear equations. Analyses were done using Statistical Package for Social
Science (SPSS) and the finding shows there is a great improvement where
students more confident in solving nonlinear problem using lesser time. Casio
fx570ES was used in this study as this model of calculator is widely used in
Malaysia.
An experiment in Mathematics
Authors: Li Qiuxia
Affiliations: Gaolu
This paper is mainly about: During the teaching process in
middle school mathematics, a case of inquiry learning, on the student's
learning interests and creativity cultivation. Use HP39gs graphic calculators
and MCL sensor, for a sound intensity certain sound source, measure the sound
intensity in different distance, analysis and fitting the data collected,
explore the relationship between attenuation of sound intensity with and
distance. Use the fitting function got to analysis problems. It's important
that we saw the students'' learning enthusiasm and discovery spirit after this
small experiment, they explore the relevant knowledge of sound spontaneously,
such as Language spectra and it''s usage. Application is the ultimate goal of
learning mathematics. In the mathematics learning, using MCL, to carries on the
observation, analysis and exploration of life phenomenon, enthusiastic the
motivation of students to learn mathematics, and cultivate students'
creativity.
Influence of using KETpic graphics on the development of collegiate
students'' proof schemes
Authors: Masataka Kaneko, Setsuo Takato
Affiliations: Kisarazu National College of technology, Toho University
The influence of using graphics on mathematics education is
a complicated and delicate theme, because the effect of using graphics depends
on not only the quality of figures used but also many other factors such as
teachers'' classroom design or students'' mathematical ability. In case of
proof, this theme is especially difficult to analyze, since proof is entirely
subjective and can vary from person to person according to their attitude to
mathematics. In this paper, the authors will illustrate how geometric models
could be effective in students'' shaping the conception of proof through some
examples. Their approach is based on the classification of students'' proof
schemes proposed by G. Harel, and their considerations indicate that Harel's
classification should be improved. The graphical device to produce figures in
this paper is KETpic, which is a macro package designed for computer algebra
system to generate highquality graphical images in highquality mathematical
documents edited by LaTeX .
WE WELCOMED M.C. ESCHER IN TURKEY’S NEW GEOMETRY PROGRAMME
Authors: BURCU AKAYDIN, MUJDE UYSAL
Affiliations: ISTEK OZEL ACIBADEM ANADOLU LISESI/ ISTANBUL, ISTEK OZEL
ACIBADEM ANADOLU LISESI / ISTANBUL
Studying mathematics through patterns is an opportunity for
students of all levels to develop mathematical knowledge connecting different
subjects like geometry/geometric transformations and algebra. Considered
mathematics as the science of patterns (Biggs e Shaw, 1985; Devlin, 2002;
Goldenberg, 1998; Mottershead, 1985; Orton, 1999) it was the start point for
our work. Patterns gave many opportunities for the study of mathematical
concepts and the development of mathematical process as problem solving,
communication, reasoning and proof. We propose a new method of teaching the
principles of geometry to tenth grade students. The students focus on a field
of design in which geometry is the design: tessellation. The students define
their tessellations using the GEOGEBRA software. This procedure enables them to
understand the mathematical principles on which graphical tools are built upon.
It moves the Abstract concepts of math into the real world, so that the
students can experience them directly, which provides a tremendous reward to
them.
The Role of the Graphing Calculator in the Qualitative Analysis of
Function
Authors: Li Hong
Affiliations: NO.22 High School
This paper focuses on several cases of student’s
qualitative analysis of function under the support of graphing calculator and
other forms of technology. This paper proposed the following points. With
graphing calculator, students can get abundant material for qualitative
analysis, a process of revealing the essence of the research object. Since study
of function has very important effects on students’ understanding of the
characteristics of mathematics and the mathematical research methods and the
qualitative analysis is to emphasize the holistic understanding of the pattern
of variation expressed by the dependency between the two variables of function,
graphing calculator as a technology indeed helps students understand the
essence of mathematics, apply mathematics in a better way and even has a great
impact on students’ development.
On the Changes of Middle School Mathematics Teaching under the
conditions of MCL
Authors: Wu Shaobin
Affiliations: Suqian College
The experiments show that the integration of MCL
and middle school mathematics new curriculum brings some profound changes in
many aspects, namely, the change of teaching content  modern mathematics
comes into student’s domain of study, the change of teaching target aims of
emotion are realized, the change of teaching methodologies the active self
study under the teachers’ guidance becomes common pattern, the change of the
users of information technology students become the subject users of
information technology.
On Construction of Teaching Mode of Mathematics Experiment Based on
MCL
Authors: Lu Mingming
Affiliations: Suqian Highschool
MCL is a new instrument to teach and study mathematics. Its
primary characteristics are its portability, low price, easy operation and
various functions. A student can get or use it at any time or in any place, so
that he/she may be pleased to spend more time in exploratory mathematical
activities. Mathematics is not just involved logic reasoning, but is also
related to experimenting. Therefore, in a mathematics teaching class, the
teacher should fully represent these two sides. Studying mathematics is not
just about to learn the deduction or to complete a formal verification, but
also involved to learn a mathematical process, or to conduct a series of
experimental and conjectural thinking and exploratory activities prior to the
formal verification. Besides, the best way to conduct a mathematical
exploratory activity is to do Mathematics Experiments. With MCL, the students
may construct the teaching mode of a Mathematics Experiment through
experimental validation, experimental exploration or experimental construction.
While using MCL, the following principles need to be obeyed: grasping
appropriate opportunities, adopting appropriate modes, and achieving
appropriate goals.
Interactive Estimator for Stochastic Differential Equation
Authors: Tatsumune Abe, Ryoji Fukuda
Affiliations: Kyushu Institute of Technology, Faculty of Engineering Oita
University
Stochastic differential equations (SDE) are being applied
widely; however, theory behind the concept is difficult to understand.
Therefore, we designed an educational system for simple SDEs. The SDEs used in
this system are determined by two linear functions with constant coefficients.
Then, four real numbers are used to define this equation. In our system, a
graph of a sample path with respect to an SDE is given, and the purpose is to
estimate the four real numbers. The system suggests these values and some
provides a few graphs of sample paths for an SDE corresponding to the given parameters.
Using our system, a user should be able to understand the role of these SDE
parameters.
Ubiquitous Mathematical Graphic Viewer for Visually Impaired
Students
Authors: Ryoji Fukuda, Akihiro Miura
Affiliations: Faculty of Engineering Oita University, Faculty of
Engineering, Oita University
In a science class, a teacher may provide some temporal
graphical information, which cannot be understood by visually impaired
students. We have assumed the availability of support staffs for visually
impaired students, such as notetakers for aurally impaired students, and we
have designed a graphic input system for them. In this system, drawn curves are
recognized and corresponding curve types and parameters as well as compensation
curves are displayed. This system also has learning functions to improve the
evaluating functions for recognition and reference points for compensation.
Transversal Mathematical Teaching Focus Across other Sciences
Authors: Horacio E. Bosch, Noemi Geromini, Mercedes Bergero, Leonor
Carvajal, Mario Di Blasi Regner
Affiliations: Universidad Tecnológica Nacional, Fundación FUNPRECIT,
Academia de ciencias aeronáuticas y espaciales, Council of Industrial Research
Associations of the Americas
Engineering is very important to solve Society problems.
Students, in general, do not feel attraction for engineering careers, as they
find them very difficult and bored. In order students feel more interested, the
current teaching habits and procedures must be changed, particularly for
mathematics teaching. New focus related to experiments and models of real 
life problems must be introduced. In order to turn over the present situation,
the authors propose a new methodology to teach mathematics as transversal
subject with other sciences. In this work the authors show how to experiment
with new technical resources and solve a reallife problem through a model
presentation and its predictions. The example is the study of a light falling
body (coffee filter) within the atmospheric air. Ultrasonic radar coupled to an
interface records both the body displacement and velocity. The limit speed is
registered. Two physical models are presented, one, the friction force acting
on the body is proportional to velocity, and the other, proportional to the
velocity squared. The corresponding algorithm’s solutions are obtained using a
Computer Algebra System. The corresponding” models” predictions are displayed
in the same graphics and compared with the experimental velocity. The
conclusion is that the first assumption is the correct one.
A Mathematics Research of a Project Case under the Conditions of
MCL
Authors: Zhao Tao, Yuanxun Sun, Xiaoling Yang
Affiliations: Hainan Overseas Chinese Middle School
This paper expounds the process of setting foot on a
chemistry problem to reflect the application value of mathematics research
project by deliberation under the conditions of MCL. It also shows the process
of students’ applying mathematics thoughts to solve problems under the
conditions of MCL. This paper indicates the handheld technology great effect in
cultivating students’ application awareness and creative consciousness in
teaching. The change from mathematics learning to practicing has come out in
the new curriculum.
On the Effective Use of GeoGebraCAS in Mathematics Education
Authors: Hirono Naotoshi, Takahashi Tadashi
Affiliations: Kobe University, Nagata Senior High School, Konan
University, JAPAN
The computerbased mathematics education was studied
enthusiastically and the effective use of Computer Algebra System was a part of
this attempt in Japan. The incorporation of Computer Algebra System in
mathematics education in Japan, however, has not become able to become an
indispensable tool in teaching mathematics. On the other hand, we have seen
that the use of Dynamic Geometry Software in mathematics education has got many
results in recent years. GeoGebraCAS is interactive geometry software that includes
Computer Algebra System functions. In this paper, we show the effective use of
GeoGebraCAS and consider how the incorporation of Dynamic Geometry Software in
addition to Computer Algebra System can make positive impacts in mathematics
education.
Integration of Interactive Resources into the Teaching of
Mathematics in Primary Education in Mexico
Authors: Ivonne Sandoval, Edda Jimenez
Affiliations: Universidad Pedagógica Nacional, México, International Group
for the Psychology of Mathematics Education, National Pedagogical University
This article reviews an interactive resource used in a
national project in Mexico. The intention of its design is analyzed, as is the
use to which teachers put it in math class. International research has shown
that learning opportunities offered by Digital Technologies (DT) depend on the
teacher’s mediation supported by knowledge of the content to be taught and its
technique which, together with the characteristics of the students, permit
generation of new knowledge and mathematically useful learning activities,
rather than mere adaptations of paper and pencil situations. We understand
technological tools as an active part of the construction of mathematical
knowledge. Instruments are not mere auxiliary components or neutral elements to
the teaching of mathematics; they shape student actions. Every tool generates a
space for action, while at the same time posing on users certain restrictions.
Such limitations make it possible for of new kinds of action to emerge. The
results of this study demonstrate that integration of technology into the
classroom demands of the teacher not only knowledge of the tool, but also
mathematical knowledge for teaching: pedagogical content knowledge, subject
matter knowledge and the use of didactic teaching methods.
Innovative Uses of Excel in Linear Algebra
Authors: Deane Arganbright
Affiliations: currently retired
Over the years many new mathematical applications have been
developed for spreadsheets such as Microsoft Excel. This paper examines some
new illustrations of the use of Excel in linear algebra. Here Excel is used to
reinforce definitions and concepts, as well as to carry out computations and to
produce eyecatching interactive graphics. Three categories of interest are
highlighted – solving systems of linear equations, creating illustrations of
lines and planes, and investigating eigenvalues and eigenvectors.
Develop Students' Visualization and Understanding of Functions
Through Geometry and Pictures with Sketchpad 5
Authors: Scott Steketee, Steven Rasmussen
Affiliations: KCP Technologies, University of Pennsylvania Graduate School
of Education
Students understand the concept of function more deeply by
using dynamic mathematics software to manipulate an independent point and
observe the behavior of the dependent point. This approach gives students an
important visual window on the behavior of functions, on domain and range, and
especially on relative rate of change and on composition of functions. An
important Abstraction in students'' understanding occurs when they realize that
they can use a function to map an entire set of input values to a corresponding
set of output values. By working with geometric rather than numeric functions,
students can see this process as one of mapping a shape to a corresponding
transformed shape, or mapping a picture to a corresponding transformed picture.
By considering shapes or pictures not only as collections of points, but also
as recognizable visual objects, students can more easily understand the
important duality that functions can operate both atomically (transforming a
single input value or point) and collectively (transforming an entire set of
input variables). Students can use Sketchpad 5 to define such transformations
using isometries, similarity transformations, affine transformations, or
arbitrary geometric constructions. Two particular classes of functions that
interest them are transformations that remind them of those they see in popular
media, and the transformations that artists use to paint a realistic 3D scene
on a flat surface (anamorphic street art).
The strategic Thinking of Mathematically Gifted Elementary Students
in LOGO Project Learning
Authors: InOk Jang, HeeChan Lew
Affiliations: Korea National University of Education
In this study, LOGO is incorporated in the dynamic
projectbased learning that provides students with opportunities to apply and
advance their knowledge and engage in diverse creative activities beyond
understanding geometric or arithmetic concepts through the integration of
several disciplines not only mathematics but also art and others as a positive
way to foster higher levels of thinking for gifted students. This study will
investigate what strategic thinking the mathematically gifted elementary
students use to plan, implement and debug in the programming process as a
problem solving process.
A Technology Friendly Mathematics Teaching Methodology
Authors: Mohamed Watfa, Diana Audi
Affiliations: University of Wollongong in Dubai, American University of
Sharjah
Mathematics fear is an unaddressed reason that has a great
effect on the unexploited capacity of young students. All over the world, the
problem is so vast that huge amounts of money are being pumped into research
grants to find out why competent students are terrified of Mathematics even
when they get good grades in other subjects. In this research, we demonstrate
and rigorously analyze a number of innovative and new teaching methodologies
that incorporate the use of modern technology to encourage freshman students to
participate and take an active role in Mathematics courses More precisely, we
introduce for the first time two innovative teaching methodologies: 1) Dynamic
Lecture Notes: A lecturing technique that automatically changes the next
lecture slide based on the live student response to in class questions using
wireless voting systems and 2) 24/7 StudentTeacher Portal: A Mobile Social
Networking (MSN) application that attempts to bridge the gap between the
students and the teachers outside the walls of a classroom. Both methodologies
were researched thoroughly in a number of local university class rooms and the
results were collected to investigate whether they would lead to a dramatic
increase in the overall performance and therefore successfully enhance the
learning experience of the students.
Focusing Learning on Concepts of Introductory Linear Algebra using
MAPLE Interactive Tutors
Author: Leonard Louis Raj, leonard.raj@zu.ac.ae
Affiliation: Department of Mathematics and Statistics, Zayed University,
United Arab Emirates.
This paper describes the author’s experiences on the
application of MAPLE’s builtin interactive tutors to explore and reinforce
fundamental concepts in an introductory Linear Algebra course with students at
Zayed University in the United Arab Emirates. Students are allowed to work
interactively stepbystep through standard problems, and become engaged in
their learning without getting caught up in the arithmetic. The basics of the
functionality of interactive tutors for Gaussian Elimination, GaussJordan
Elimination, Matrix Inverse, Eigenvalues and Eigenvectors are discussed.
TRAINEE TEACHERS’ ATTITUDES ABOUTMATERIALS AND TECHNOLOGY USE
INMATHEMATICS EDUCATION
Mustafa Doğan, mudogan@selcuk.edu.tr
University of Selcuk, Faculty of Education, Konya, TURKEY
This study is planned to determine mathematics trainee
teachers’ attitudes about technology and material use in mathematics education.
The study is conducted with a selfdeveloped questionnaire as a survey. The
second part of the survey is a Likert Type Attitude Scale which contains 31
items. Sample is a total of 125 students from a primary teacher training
department. This paper includes findings from the scale. Descriptive
statistical techniques (f, %,) were used to analyze collected data for the
sample. The results show that the trainee teachers’ attitudes are quite
positive about materials and technology use in mathematics education. They
stated that they are going to use the technology and materials in their
professional mathematics teaching as well.
Abstracts for Papers with Abstract Only
Spreadsheet as an innovative tool for Traditional Counting
Authors: Thadreina Abady
Affiliations: Divine Word University
This paper seeks to highlight Excel spreadsheet as a
mathematical tool in the creative teaching and learning of traditional counting
systems in Papua New Guinea (PNG). PNG is known for its diversity in culture
and tradition and so this paper wants to take advantage of the available
technology such as spreadsheet to express the sort of diversity that exists in
our traditional counting systems. In this paper 3 different types of
traditional counting systems in certain parts of PNG are being discussed. The
first traditional counting system from Mikarep village in the Madang Province, uses base 2 approach and is taught in the native language call “Aruamu” or
“Big Man”. The second one uses a base 5 counting system and comes from Manam in
the Madang Province. And finally from East New Brititan Province, the
traditional counting system is taught in the native language call ‘Kuanua’ and
uses a base 10 approach. Each of this traditional counting system has been
taught in the past and at present but without the use of computerbased
learning technologies. It is our intention however, that with the current
expansion of the One Laptop Per Child project (OLPC) in Primary schools around
the country, spreadsheet can be better utilized as a dynamic tool for
mathematics education especially in this area of traditional counting. The
paper concludes that the way forward for mathematics education in Papua New Guinea, is to embrace technological tools like spreadsheet and use it creatively.
Using Spreadsheet to create different rug design
Authors: Maryanne Bagore
Affiliations: Divine Word University
A spreadsheet is a powerful mathematical tool that is widely
used by math educators and learners to comprehend and solve many mathematical
problems. This paper will illustrate an inventive way on how Excel can used to
create different artistic rug designs or patterns using the concepts of
Geometry, Algebra and Calculus in a Spreadsheet Application. Mathematical
models are created to show how a particular rug patterns or designs can be
created in Excel. My examples of rug patterns or designs would be taken from
the traditional Papua New Guinean mats or rugs which are made mostly from the
pandanas plant and also other examples will include rug patterns or designs
from different countries and cultures such as the kilim from Turkey. Through my examples, the main point is on how Excel can be seen as an application that can
used to create interesting, creative or odd things and not just for the fun of
doing mathematics. The concept of this paper can be used in both the classroom
teaching and teacher development.
The Effectiveness of ICTassisted Approach in Learning 3D Linear
Algebra
Authors: Hitoshi Nishizawa, Takayoshi Yoshioka, Yoshihiro Yamada
Affiliations: Toyota National College of Technology
Linear algebra was one of underachieved fields of
mathematics for the students of colleges of technology in Japan. The low scores
at achievement tests were partially because, in the former lessons, the
procedures of symbolic manipulation were taught separately from the features of
associated graphical objects or how they were used in real world applications.
The isolated procedural knowledge was easily forgotten just after the
examinations.
To compensate this situation and deepen the students’
conceptual understanding, we redesigned our lessonplan to be directed from
actual applications towards abstract mathematical ideas; from handling graphic
objects and observing their characteristics towards building vector equations
and manipulating them symbolically. In the new lessons, several software
programs are used to demonstrate the close relation of graphic objects and
vector equations interactively.
This paper reports how the new lessons are changing our
students’ learning styles from blackbox approach and deepening their
conceptual understanding of vector equations. Their written answers have richer
explanations than their former students who learnt in traditional lessons.
Their scores in annual INCT achievement test, which stayed low for previous
four years, also increased significantly in 2010.
The generalization of the area of an internal Polygon for the use of
Mathematically Gifted Students
Authors: Kwangho Lee, Heon Soo Lee
Affiliations: Korea National University of Education, Graduate School of
Chonnam National University
This study investigates how the GSP helps gifted and
talented students understand geometric principles and concepts during the
inquiry process in the generalization of the internal polygon, and how the
students logically proceeded to visualize the content during the process of
generalization. Four mathematically gifted students were chosen for the study.
They investigated the pattern between the area of the original triangle and
quadrilateral and the area of the internal triangle and quadrilateral, with the
ratio of each sides on m:n respectively. Digital audio, video and written data
were collected and analyzed. From the analysis the researcher found four
results. First, visualizing using GSP helps the students to understand the
geometric principles and concepts intuitively. It also helps in verifying the
various triangle and quadrilateral cases and analyzes the geometric structure,
as well as reveals the synthesizing insight. Second, GSP helps the students to
develop their inductive reasoning skills by proving the various cases. Third,
lessons using GSP increases interest in apathetic students and improves their
mathematical communication and selfefficiency. However, the measurement
function of the GSP, which cannot express fraction, gives the students had
difficulties on transforming a decimal to a fraction for the generalization of
the area.
Internal polygon is defined as any polygon which is formed
by connecting any points on another polygon''s perimeter.
Integrating Maplets and Other Technology into an Online Calculus
Course
Authors: Denise LeGrand, Thomas McMillan
Affiliations: University of Arkansas at Little Rock, University of Arkansas
at Little Rpck
Teaching mathematics online has offered many challenges. We
meet these challenges headon with the appropriate use of technology to help
our students develop a deeper understanding of Calculus, through active
learning and communication, visualization, reflection and motivation. A major
factor in the success of our online Calculus courses is the use of Maplets
developed by Dr. Douglas Meade and Dr. Philip Yasskin. A Maplet is an exercise
or guided tutorial with a graphical user interface. A Maplet may include graphs
as well as formulas and equations. Although in the beginning we were hesitant
to introduce more materials into an already overfilled course, we quickly
recognized that Maplets contribute to the understanding and appreciation of
Calculus. Using them as required tutorials thoroughly engaged our students.
Students voluntarily included Maplet examples to illustrate their explanations
in the required discussion postings. We will present how we and our students
use the Maplets for our online Calculus courses, and how they are graded. We
will also talk about other technological resources for mathematics readily
available on the internet (at little or no cost). We will concentrate on the
Maplets but will also discuss other internet tools in relation to different
student learning styles. It is exciting to see our students’ participation and
enthusiasm increase mainly as a result of the Maplets, but also through the
integration of other online resources such as Webassign, Wimba chat and
Webquests.
Amicable triangles and perfect circles
Authors: Michael Sejfried
Affiliations: METAL UNION
Perfect circles in the triangle ABC have unusual properties.
Their family spreads from the FermatPoint across the incircle of the triangle
ABC, until the circumcircle of the same triangle. Into perfect circles one can
inscribe amicable triangles and then we will receive the construction with
dozens of additional properties, which can be described with equations. Most
important however in it is that using perfect circles we can generalize the
problems related to SoddyCircles, with the SoddyLine and with the
GergonneLine. We can also generalize GergonnePoint and EppsteinPoints,
RigbyPoints, GriffithPoints and NobbsPoints.
VertexCircles of Soddy are tangent to themselves on the
sides of the reference triangle ABC in the points A1, B1 and C1. The cevians
based on these points intersect in the GergonnePoint. The straight line
connecting GergonnePoint with the incenter of the triangle ABC is called
SoddyLine. On this line lie several important centres in the triangle such, as
two SoddyPoints, two EppsteinPoints, two GriffithPoints and two
RigbyPoints. These centres generalized by perfect circles as before lie on the
generalized SoddyLine. After the generalization we receive however four EppsteinPoints
instead of two. Perfect circles are strictly connected with NobbsPoints, which
for these circles become the kind of focal points. The special attention
deserves some curve which in certain sense is connected with the Pascal
Theorem, but also with perfect circles and amicable triangles. This curve
exists exclusively for the circles, but appends to the Pascal Theorem the
equality of sums of some angles. I mentioned earlier about the amicable
triangles and also I would like to devote them the moment of the time. There
are triangles formed by 2 triplets of cevians so, that each of these triplets
creates the single triangle inscribed into the same circle situated inside the
triangle ABC. I'd like to present also the new interesting theorem concerning
of the amicable triangles.
The lecture will be demonstrated using Geometer's Sketchpad.
The names "amicable triangles" and "perfect circles" are an
original nomenclature used by the author.
Using Clickers to Review Prerequisite Material and Reading
Assignments
Authors: Thomas McMillan, James Fulmer
Affiliations: University of Arkansas at Little Rock
It is almost a universal complaint among mathematics faculty
that students in an advanced class have an inadequate understanding of material
that was covered in a prerequisite course. In this discussion, we will describe
how individual response systems (clickers) were used in a differential
equations class (1) to test students’ recall of required material from
calculus, and (2) to encourage discussion among students about concepts in
calculus. The activity took from five to ten minutes at the beginning of each
75minute class. We will discuss student reactions to this activity and its
affect on the conduct of the class as a whole. We will also discuss a similar
activity in a discrete mathematics class in which clickers were used to test
students’ understanding of assigned reading and to encourage discussions of
questions raised.
Visualization of Rational Numbers Commensurability by Using Dynamic
Geometry Software
Authors: InSun Shin, JiEon Kim
Affiliations: Dept. of Mathematics Education Korea National University of
Education
This study investigated middle school students’
understanding of rational numbers and irrational numbers. We developed teaching
and learning materials to visualize rational numbers commensurability by using
dynamic geometry software (the Geometers Sketchpad). It is expected that the
developed materials enhance students to understand irrational numbers incommensurability.
Models based on the area, the ratio of the length, overlapping and the
similarity were developed and applied to students. We observed and analyzed the
change of students perception about rational numbers and irrational numbers.
Development of Online Learning Sources on Applied Statistics
Authors: Somruay Apichatibutarapong, Chookait Pudprommart
Affiliations: Applied Statistics Department, Faculty of Sciences & Technologies, Suan Sunandha Rajabhat University, Bangkok, Thailand, Applied Statistics Department Faculty of Sciences & Technology Suan Sunandha Rajabhat University Bangkok Thailand
In recent years there have been rapid social changes and globalization in the fields of Economics, mass transportation, Information Technology (IT) and the environment. To prepare graduates who can take advantage of these developments and contribute further to them, we want to train them to have skills in computers, media technology and critical thinking, emphasizing lifelong, cooperative and flexible learning. Web technologies and Internet provide opportunities for such a global education. The purpose of this study was to develop online learning sources on Applied Statistics. The researchers used Visual Basic.Net with ASP.Net for applying on Web and used Microsoft Access for managing database system with 207 of the statistical vocabularies consisted of meaning, formulae, examples and usage in researches and the 40 statisticians. The research was conducted to study online learning to normal class. The quantitative and qualitative approach was used to investigate the quality of online learning sources and the opinions in using them. This study was intended to be the beginning step for new strategies in learning Statistics by using the Web technology. It would be beneficial to those looking for alternative strategies in learning Statistics.
Understanding and applying parametric equations of lines and curves
in computer graphics and animations
Authors: Andreas Filler
Affiliations: HumboldtUniversitaet zu Berlin, Gesellschaft fuer Didaktik
der Mathematik
Parametric equations of lines and planes are parts of higher
mathematics education in a lot of countries. According to the most curricula
the students only get to know parametric equations of these linear objects and
use it mainly for calculating intersections. As a result they see lines and
planes basically from a static point of view although parametric equations
include functional relationships between values of parameter(s) and points in
plane or space. To understand these relationships is a relevant objective
because functions and functional relationships are main ideas in mathematics
education. They can especially show students connections between analytic
geometry, trigonometry, precalculus and calculus.
Creating computer visualizations, especially animations, can
help students to understand geometric objects, which are described by
parametric equations, as point sets and to discover functional relationships
and dynamic aspects of lines and also of some interesting curves. Creating
computer animations is very attractive for students; therefore it can help to
motivate them to figure out features of parametric descriptions and to discover
functional aspects, which are essential for interesting animations.
Interpreting the parameter in equations of lines or curves
as time makes it possible to create computer animations using graphics software
(e.g. POVRay or Maxon Cinema 4D) or computer algebra systems (e.g.
Mathematica, Maple or MuPAD). Changing positions of objects can be explored
depending on the variation of the timeparameter in animations created this
way. The conference paper will include proposals for how graphics and
animations on lines, circles, spirals, trajectory parabolas and other curves
can be created by describing these curves with parametric equations. The paper
will also show some examples created by students at upper secondary level.
SECONDARY SCHOOL PRESERVICE MATHEMATICS TEACHERS’ EXPERIENCES OF
GEOGEBRA FOR GEOMETRY TEACHING
Authors: FATMA ASLANTUTAK, GUNEY HACIOMEROGLU
Affiliations: BOGAZICI UNIVERSITY, Canakkale Onsekiz Mart University,
Turkey
The technological development in society is developing
rapidly. New technologies also play an important role in teaching and learning
(e.g. Cabri, GeoGebra, and Geometry Sketchpad). Among the technological tools
the ones for geometry stands out because of the difficulties in learning
geometry. Duval (1998) identified three categories of cognitive processes of
geometric thinking: visualization, construction and reasoning. All three of
these components should be addressed during the geometry instruction in order
to enhance spatial reasoning. In that sense, dynamic geometry software (DGS)
provides a rich environment for development of all three components of the
geometric thinking (Goos, Stillman, & Vale, 2007). On the other hand,
teachers tend to teach the way they have been taught. When inservice or
preservice teachers were students at K12 schools, they were not introduced to
DGS for their geometry learning. Also, teachers have limited experience in
geometry as K12 students or even as preservice teachers (AslanTutak, 2009,
Jones, 2000). Therefore, the role of teacher education in implementation of
technological tools for instruction is important. The implementation of DGS in
teacher education may be viewed from two aspects; training of teachers
(preservice or inservice) on use of the tools and on implementation in the
instruction (KokolVojc, 2007). Both of these aspects would provide information
about not only how teachers perceive DGS but also how they enact their content
knowledge and pedagogical content knowledge of mathematics (Niess, 2005).
The purpose of this study is to investigate preservice
teachers’ perception of GeoGebra for geometry teaching and their use of it.
This study took place in a geometry course designed for secondary school
mathematics teachers at a northeastern public university in Turkey. The data
collection consists of artifact collection of preservice teachers enrolled in
the course. First students were asked to explore GeoGebra before any training
on using it. The purpose of this practice was to emphasize the GeoGebra as
being opensource and to improve preservice teachers’ skills to initiate using
GeoGebra which is vital for longterm learning of a DGS. Participants were
asked to write their first impression on the program. They were also asked
about their previous experiences with using DGS. Later, the preservice teachers
participated in training on using GeoGebra. Throughout the semester, learning
activities for GeoGebra was used during the instruction to provide examples. At
the end of the semester participants were asked to prepare a learning activity
for a geometry topic from two perspectives; with dynamic geometry software and
without any dynamic tool. Preservice teachers presented their work to other
participants and received peerreview on their presentation. Furthermore, the
preservice teachers were asked to write a reflection paper on using GeoGebra
for instruction by comparing using a dynamic software vs nondynamic environment.
The finding of this study reveals that preservice teachers
have limited experience with DGS when compared to their experiences with other
technological tools. Using an opensource DGS was helpful for them for two
reasons; availability out of university and transfer of their GeoGebra
knowledge for their teaching practice where they may have limited resources.
Furthermore, participants were able to identify the difference between using
dynamic environment for using geometry and nondynamic environment. They
reported that they would prefer to use GeoGebra for their instruction.
PRESERVICE TEACHERS’ REFLECTIONS ON CREATION OF GEOGEBRA MATERIALS
AND LESSON PLANS
Authors: Guney Haciomeroglu, Fatma Aslan–Tutak
Affiliations: Canakkale Onsekiz Mart University, Bogazici University
Challenge for teacher education programs is to prepare
preservice teachers to integrate technology into their teaching as they are
educated to become effective teachers of mathematics. Since technology has
become an integral part of learning, use of educational software has changed
the nature of classroom. Integrating technology into lessons also alter how the
teachers utilize their knowledge for teaching mathematics (Mishra &
Koehler, 2006) as well as how they create instructional materials to design a
lesson. Correspondingly, teacher education programs put great emphasis on
integrating technology into teaching of mathematics with using educational
softwares (e.g., GeoGebra, Cabri, and Geometer Sketchpad). However, several
factors impact on one’s implementation of technology into their teaching such
as skills and previous experiences, time and opportunities to learn, and
knowledge of how to integrate technology into mathematics teaching. As
preservice teachers plan to teach lessons and develop instructional materials,
they utilize their content knowledge for teaching and begin to understand what
it means to teach with technology (Niess, 2005). This new compression leads to
development of Technological Pedagogical Content Knowledge (TPCK) defined as
connection of content, pedagogy and technology (Mishra & Koehler, 2006;
Niess, 2005; Niess, 2008).
This study examined preservice secondary teachers’ views of
learning and teaching mathematics as they create instructional materials to
design a lesson plan with use of GeoGebra.
Forty preservice teachers who were enrolled in secondary
teacher education program at a university in the northeastern Turkey
participated in the study. Data were collected through the preservice teachers’
written reflections, lesson plans, class discussions, and observations in a
semester–long study. Due to the lack of experience in designing a lesson plan,
the preservice teachers were asked to use a lesson plan from Ministry of
Education as a guideline. Preservice teachers created GeoGebra materials for
their lesson plans. They wrote preand postreflections about their lesson
plans. These reflections provided perspectives about what factors influenced
preservice teachers’ planning as well as whether or not their views about
teaching and learning of mathematics with use of GeoGebra changed as a result
of their participation in this study.
Results of the study revealed that most of the preservice
teachers claimed that use of technology should be incorporated into lessons for
teaching and learning of mathematics. However, they claimed that students first
master basic skills and procedures with traditional teaching before using
Geogebra. Some of them considered that technology should be incorporated into lessons
to make learning more enjoyable for students rather than making learning of
mathematics meaningful. Findings from this study suggested that methods courses
should emphasize appropriate use of technology to support use of mathematical
knowledge for teaching and learning (Goos, 2005; KokolVoljc, 2007) in teacher
education since it may lead to development of technological pedagogical content
knowledge (TPCK).
Integrating Weblog in Teaching of Higherorder Thinking
Authors: Rossukhon Makaramani
Affiliations: Suan Sunandha Rajabhat University
Weblog has been widely used as one of viable tools in social
networks and learning society. It can enhance student learning experiences when
applying for instructional purposes. This paper describes a welldesigned
instruction that integrates a weblog system aiming to teach higherorder
thinking to 198 undergraduate students in a pedagogy course entitled Innovation
of Learning. The designed instructional model consists of six themes: I am,
IntellectVirtueHappiness, Problems are Wonderful, Knowing Death, Save the
world, and Human Rights. Data includes analysis of student works and reflection
essays. The evaluation uses an interpretive methodology to investigate 4
research areas: Higherorder thinking, metacognitive awareness/knowledge, team
work/collaboration, and ownership of learning. Examples of student and teacher
blogs are provided to illustrate how such technology can effectively promote
higherorder thinking.
Developing Mathematical Teaching Materials of Fundamental Analytic
Geometry and Conic Sections Using The Geometer’s Sketchpad (GSP) on Students in
Grade 10
Author: Ubol Klongkratoke
Affiliations: Department Informatics Mathematics. Faculty Science &
Technology, University Rajabhat Suan Sunundha, Bangkok ,Thailand.
This research aims to develop Mathematical teaching
materials on the topic of the introduction to Analytic Geometry and Conic
Section using the Geometer's Sketchpad (GSP) on the students in grade 10 in
order to have the performance efficiency of 75/75 by the comparison on the
student achievement before and after these useful teaching materials are used
in classes and to study the attitude of students on Mathematics after taking
classes with the application of these materials. The sample group in the
research is 33 students studying in grade 10 at the Demonstration School of
Suan Sunandha Rajabhat University by simple random sampling. Tools used in this
study include lesson learning plans on the introduction to Analytic Geometry
and Conic Section using the Geometer's Sketchpad (GSP), a 30question
achievement test on Analytic Geometry and Conic Section and a 20question
attitude test on Mathematics. The collected data are statistically analyzed in
terms of mean, standard deviation and coefficient of variation. The results
reveal that (1) the effectiveness in using the Geometer's Sketchpad (GSP) on
the introduction to Geometry and Conic Section on 10th grade students is at the
level of 76.40/76.50 (2) students’ academic achievement on the post test is
higher than that on the pre test at the significant level of 0.05 (3) students’
attitude toward Mathematics after the use of these Mathematical teaching
materials is at a good level.
Curiosity killed the cat. Creativity have mathematics classrooms
killed it?
Authors: Christopher Longhurst
Affiliations: Hewlett Packard
Is creativity lost in education, particularly in mathematics
education? I'll start with some IDs from the famous educationalist Sir Ken Robinson.
Then, we'll look at creating problemsolving, visualization using technology.
The MCL (mobile classroom laboratory) and the graphing calculator tools
available to us to use to allow students to enhance their creativity and
visualize problems in mathematics. Too often we concentrate on the final
product that is passing the exam without understanding or thinking on the half
of the student. Using examples I will stretch your creativity and give it some
activities that may be able to be used in your classrooms.
Teaching design based on Inquiry learning——Find the way to make
acceptable "statistic"
Authors: Yiming Cao
Affiliations: School of Mathematical Sciences Beijing Normal University;
Pengju Chen School of Education Beijing Normal University
Email: pengjuchen2010@gmail.com
As for the purpose of statistical teaching is to develop the
statistical literacy of the students but not to impart existing knowledge
experience, the teacher may do more afford to let the students seek the knowledge
source through the experience of data processing and statistical regularities''
discovering based on the problem situation. Information Technology（IT）gives
a great support in data processing and statistical characteristics description,
which make the experience process available. This paper tries to use modern
information technology on designing new statistical teaching. Through the
process of making, contrasting and correcting "statistic", students
will learn the practical significance of statistical law though the discovering
experience. As for students in different study levels, we may deepening the
statistical concepts and applications. Information Technology not only makes
the classroom research activities available, but also increases the students''
learning interests.
Case analyses of “use MCL to promote the students’ practicable
ability of mathematics”
Authors: Guangming Wang Guanghui Li
Affiliations: Tianjin Normal University;
email:bd690310@163.com
MCL（Mobile
Calculator Lab）is a kind of
portable and digital lab, which can be used to promote the students,
practicable ability of mathematics. In this article, we introduced the use of
MCL in promoting the students’ practicable ability of mathematics by showing some specific examples, like
collecting data and dealing with data, plotting graphics and fitting,
establishing and solving mathematical model, using and generalizing the
conclusion of the model etc.
Applications using Graphics Calculators
Authors: Tom Button
Affiliations: The Further Mathematics Support Programme (UK)
Title: Tasks for using multiple representations to improve
students’ mathematical understanding.
This workshop will focus on how dynamic multiple
representations can improve students’ understanding of mathematics. A series of
tasks will be presented that utilize the facility of a HP40gs graphical
calculator to represent objects algebraically, graphically and numerically. The
tasks used will be drawn mainly from the UK curriculum for year 12 and 13
(preuniversity). There will be an opportunity to trial some tasks and a
general discussion of the opportunities for using technology featuring dynamic
multiple representations.
The first steps of an online forum project to help students in
mathematics
Authors: Abdulkadir Erdogan
Affiliations: Anadolu University
Internet infrastructures have largely developed throughout
the world. Its speed of data transfer has increased and its cost has decreased.
These evolutions have made possible the emergence of various online teaching
tools and resources. Online forums, which make part of these tools, have some
specific potentialities for the teaching and learning. They are especially
characterized by the written interaction that can be used in order to satisfy a
particular request related to the school matters. For such potentialities,
nowadays one can find numerous online forums from mathematics to philosophy
courses. However, as the majority of these forums do not adopt specific approaches
and relevant teaching methods, their positive effects on students’ learning
have become questionable.
On the basis of this report, we have carried out an online
forum project  supported by Anadolu University  to help students in
mathematics, especially in properly doing homework and other school work such
as revising lessons and preparing for exams. The project addresses to the
students of 6th, 7th and 8th grades (1215 years old) and 15 preservice
teachers are involved in the project with the responsibility of answering to
the students’ questions. In this communication, the first steps of the project
will be presented and the theoretical and methodological questions will be
discussed.
IMPLEMENTATION OF WEBQUESTS IN MIDDLE SCHOOL MATHEMATICS COURSES
Authors: Aytaç Kurtuluþ, Tuba Ada, H. Bahadýr Yanýk
Affiliations: Eskiþehir Osmangazi Üniversitesi, Anadolu University
The purpose of this study was to explore how two mathematics
teachers who participated in a Professional development project regarding the
use of webbased projects (WebQuests) in middle schools implemented WebQuests
in their own classrooms. Teachers designed two separate Webquests. One of the
teachers used the WebQuest for performance task and the other teacher implemented
it as project task. While performance task was completed in the computer
laboratory under the guidance of the teacher, Project task was completed by the
groups of students out of the classroom. Both teachers thought that WebQuests
drew students’ attention and helped them focus on the learning goals of the
mathematics course. Moreover, the teachers thought that students used Internet
effectively for problem solving purposes. The data further suggest that
Webquests provided a learning environment in which students worked
cooperatively, and shared and discussed mathematical ideas. At the end of the
study, both teachers thought that WebQuests should be initially introduced to
the students as a performance tasks with the limited number of learning goals
and later it could be used for Project tasks. This presentation will focus on
the issues emerged from the implementation of WebQuests.
Interactive Mathematics on multitouch tablet PCs
Authors: Alfred Wassermann, Matthias Ehmann, Michael Gerhaeuser
Affiliations: University of Bayreuth, University of Bayreuth, Bayreuth,
Germany
The availability of tablet PCs are a great opportunity for a
better integration of technology into mathematics education. But most existing
Dynamic Geometry Systems will not run on these computer architectures. In 2008
the authors started the project JSXGraph. This opensource software library
makes Dynamic Geometry available on practically every computing device that
supports a stateoftheart web browser. Up to now, JSXGraph lacks a user
interface. Here, we show first results on realizing a minimalistic user
interface for tablet PCs. It is heavily based on gesture recognition. This
allows simplifying the traditional structure of user interfaces based on nested
menus and icons enormously.
Effects of Technology Use on Geometrical Construction
Authors: Nilüfer Y. Köse, Emel Ozdemir Erdogan, Tuba Yuzugullu Ada, Dilek
Tanýslý
Affiliations: Anadolu University
Geometry is particularly well placed for helping people
develop several ways of thinking and also ideally placed for helping to expand
a student’s conception of mathematics (Goldenberg, Cuoco & Mark, 1998).
Geometrical thinking consists of three processes: Visualization, construction
and reasoning (Duval, 1998). The primary and high school geometry curriculum in
Turkey highlight these processes. However, for many reasons such as the
influence of the national selective exams and the lack of the time, the construction
process does not have important place in geometry classes. The aim of this
paper is to study the effects of technology use on students’ ability of
geometrical construction in paper and pencil environment.
The students’ ability of geometrical construction was
identified before and after geometry lessons supported by the use of TINspire
CAS handheld. In this paper, findings about the effects of technology use on
students’ geometrical construction ability will be discussed. The study will be
illustrated through examples from students’ work and implications for teaching
geometry will be drawn.
Mathematics curriculum consequences of calculator choice
Authors: Barry Kissane, Marian Kemp
Affiliations: Murdoch University
Developers of mathematics curricula make choices regarding
the kinds of technology that are to be used by students, which in turn
influences the work of both students and teachers to learn and teach
mathematics. This paper analyses the choices made in mathematics curricula
regarding calculators, and examines their implications for what can be learned.
Three different levels of calculators are considered in the paper: basic
scientific calculators, advanced scientific calculators and graphics
calculators. An additional level involves a choice to allow no calculators at
all. Significant consequences of these choices are described and exemplified
through a consideration of a number of mathematical topics that are commonly
taught in many curricula in Asian countries.
THE ROLE OF INSPIRATION SOFTWARE IN TEACHER CANDIDATES’ PREPARATION
OF CONCEPT MAPPING
Authors: Mevhibe Kobak, Sevinç MERT UYANGÖR
Affiliations: Balikesir University Necatibey Educational Faculty Secondary
Mathematics Education
: The importance of information increases rapidly, thus the
concept “information” and the understanding of “science” change, and the
technology advances globally. Novel information and technologies affect
education process. Accordingly, this study aims to define the effect of
“Inspiration” software on computers and concept maps preparation studies. The
study carried out with 71 preservice teachers who study at third grade at the
faculty. Experimental group were 35 students and the control group were 36
students. First, the software was introduced to the students in the study
experimental group and examples of concept maps prepared by the programme on
various subjects were shown. Then, the students were asked to prepare a concept
map on “Prisms”, an 8th –grade Geometry Learning Domain.
After assessment of the student's concept maps, according to
the findings can be said that “Inspiration” software does not make a difference
to create concept maps. In addition, the students reported positive opinions
about the program.
A Comparative Study towards Bypassing of Quadrilateral Pairs in the
Dynamic Geometry Environments
Authors: Samet OKUMUÞ, Bülent GÜVEN
Affiliations: Rize University, Karadeniz Technical University
Elementary second level students (68th grade) generally
show a series of behaviors like creating basic definitions for geometric
shapes, knowing the mathematical properties of the shapes and distinguishing
them with their properties and thinking them independently with the logic of
exclusive definitions, etc. But, to express any geometrical shapes with its
minimum properties and also to create inclusive definitions for geometrical
shapes rather than exclusive definitions by taking hierarchical relationships
between geometrical shapes into consideration, will be more advantageous for
students before starting their high school educations. In particular, the topic
of quadrilaterals is very popular to test students’ logical deduction abilities
and classification preferences, to examine their definition type, etc. But
first off all, to determine how students could best bypass quadrilateral pairs
has a key role. In this context, we designed two different structured geometry
courses at 7th grade level. In the first learning environment the students used
the dynamic geometry software of Cabri Geometry while the second used the
tangible materials (geostripes, geoboards and dot paper). In the two
different learning environments, the students explored the quadrilaterals’
length, angle and diagonal properties in pairs with the worksheets designed for
each quadrilateral type. The treatments took 8 class hours. Before and after
the treatment, the clinical interviews were conducted to 1 high level, 1 middle
level and 1 low level student, who were identified in each group, to compare
the learning environments in terms of the students’ bypassing of quadrilateral
pairs. At the end of the study the clinical interviews analysis indicated that
the students used the dynamic geometry software identified better hierarchical
relationships between the quadrilateral pairs and outperformed the students
used the tangible materials in terms of bypassing of the quadrilateral pairs.
On the other hand, while the first group students used 3 different approaches
to the quadrilateral pairs, the second group showed four different approaches.
DEVELOPING PRESERVICE TEACHERS’CREATIVITY IN THE GEOMETER’S
SKETCHPAD ENVIRONMENT
Authors: Jale ÝPEK BÝNTAÞ
Affiliations: Ege University
A research study was conducted in the fall semester
20102011 within the course “Special Teaching Method II “ for 3th grade
students in computer and instructional technology department at Ege University.
The sample of the study is 55 preservice teachers. The researcher is also
lecturer of this course. The aim of the study is to teach to preservice
teachers how to use a new computer program, to see the course affect to
geometry course, to discover the different uses of this program and to develop
students’ own creativity that they can realize the impact of this program and
work environment. This study was conducted in 7 weeks including 3 hours in each
week and totally was completed in 21 hours in the computer laboratory. The
students use the books prepared by the researcher during the study. Students
also use their own learning speed and discuss with each other. They listened to
classical music every lesson. Both students’ homework and theirs thoughts about
GSP was gathered regularly every week. The researcher observed the students
during the study. At the end of the study, students were asked to prepare the
final project. This study shows that all preservice teachers can make a better
sense of the geometric descriptions, specifications and proofs through GSP than
through classical methods that they had learned earlier. They also stated that
they are encouraged against mathematic courses. Besides, almost all preservice
teachers stated that they could make very beautiful shapes, motives, wall
ornaments, frames, prints etc. through “Transform” tool menu features in GSP.
In addition, they explained that they could obtain new several shapes through
program dynamics that help them to develop their creativity and imagination and
students state that they cannot notice how time passed during the study.
INTEGRATION OF TECHNOLOGY INTO TEACHING MATHEMATICS: LEVELS OF
TECHNOLOGY USE AND ROLE OF PARTICIPANTS
Authors: Mehmet Fatih Ozmantar, Servet Demir, Erhan Bingolbali, Ali Bozkurt
Affiliations: University of Gaziantep
In this study we present excerpts obtained from mathematics
teachers’ classroom practices of teaching mathematics in technology rich
environments. The excerpts are analyzed with regard to Hughes’ (2005) three
levels of technology use: replacement, amplification and transformation.
Replacement refers to the use of technology in terms of changing the teaching
medium without changing the learners’ routine. Use of technology at an
amplification level indicates the utilization of technological tools to perform
actions faster and easier such as doing an algebraic calculation quickly and
correctly with a calculator. At transformation level, technology is used in
such a manner that brings changes in students’ learning routines via
establishing links among different mathematical structures. In this study we
focus on two cases for comparative purposes in terms of levels of technology
use. In the first case, the teacher is able to reach the transformation level
while using technology. The second teacher was not able to go beyond
replacement level. The role that the teachers attributed to the students was
seemingly the same: getting them to answer questions directed by the teachers.
However, management of students’ attention stands out as a critical feature to
achieve a high level of technology use. During the presentation, the classroom
practices of two teachers will be compared in terms of selection of particular
technologies, the manner in which technology is used with regard to Hughes’
levels and teachers’ management of student attention. Following this, several
questions will be raised and discussed with the participants, including: what
are the roles given to technology in achieving a transformative level of
technology use? What roles should be given to the students at this level of
technology use? What is the role of teachers in increasing the level of
technology use? To what extent does the level of technology use depend on the
roles given to the students?
WHAT KNOWLEDGE TEACHERS NEED FOR EFFICIENT TECHNOLOGY INTEGRATION
INTO MATHEMATICS TEACHING – AN EXAMINATION THROUGH TPCK FRAMEWORK
Authors: Erhan Bingolbali, Servet Demir, Fatih Ozmantar, Ali Bozkurt
Affiliations: Gaziantep University, Gaziantep University, Turkey
Technology integration into teaching mathematics is a
complex issue and requires a great deal of competencies and different types of
knowledge on the parts of the teachers. Technological Pedagogical Content
Knowledge (TPCK) framework is utilized in this study to first identify what
types of knowledge teachers need to know for technology integration and then
utilized to reveal teachers’ views regarding the types of knowledge that they
think that they need to know for efficient technology integration into teaching
mathematics. According to TPCK framework, for efficient technology integration,
teachers need to have the knowledge of pedagogy, content and technology as well
as the intersections of them namely; Pedagogical Content Knowledge (PCK),
Technological Pedagogical Knowledge (TPK) and Technological Content Knowledge
(TCK) (Koehler and Mishra, 2008). The intersection of PCK, TPK and TCK
constitutes TPCK and it is, to us, an operational knowledge which can be
usefully employed to evaluate the effectiveness of technology integration. For
the purpose of the study, approximately 120 elementary classroom and
mathematics teachers are applied a questionnaire including an item asking them
to respond to the question of ‘what knowledge a teacher needs to have for
efficient technology integration into teaching mathematics?’ The data analysis
reveals that the majority of the teachers refer to the pure technological
knowledge such as ‘being able to use computers’, ‘projectors’ and ‘internet’.
The other knowledge components receive little citation from the teachers. Of
course, basic technological knowledge is a prerequisite for any technology
integration. Nevertheless, as the TPCK framework reveals technology integration
is rather complex and requires such knowledge as TPK, TCK, PCK on the parts of
the teachers for an effective integration as well. The data hence demonstrate
that teachers’ views of technology integration are rather simplistic and
superficial. During the presentation, some other themes that the findings raise
as well as the implications of these findings will be discussed with the
participants. Especially the issue of teacher training and professional
development with regard to technology integration will be one of the foci.
ANALYSIS OF TURKISH WEB PLATFORMS OF VIRTUAL MATHEMATICS
MANIPULATIVES
Authors: Emin Aydin
Affiliations: Marmara University, Mathematics Education Dept.
The developments in the computer technologies provide
opportunities for enhancing students’ conceptual understanding. It is
imperative to concretize abstract concepts and relationships especially in the
teaching of mathematics. For that reason, the use of concrete objects is
encouraged by the 2005 Ministry of Education program document. Encouraged by
this, there have been efforts by some educational technologists to develop
platforms (with the support either from the national or from university
research funds) that aims to develop virtual mathematics manipulatives and
share them with the practitioners. Although these platforms have not been
reached to a wide audience, they carry potential in the near future for the
circulation of such materials. The aim of this paper is to analyze these
virtual manipulatives that exist in different platforms in terms of their
mathematical content and the type of reasoning they aim to develop. In doing
so, we intend to compare the Turkish platforms with each other and with
universally known platforms such as NVLM and WisWeb.
A Technological Approach to Provide Professional Developments to Teachers
on Using Technology
Authors: Serkan Ozel, Zeynep Ebrar Ozel, Tufan Adiguzel
Affiliations: Bogazici University, Fatih University, Bahcesehir University
Over the last few decades, collaborations between schools,
universities, and community partners on the Science, Technology, Engineering,
and Mathematics (STEM) pipeline are driven by funding initiatives from federal,
state and private agencies. These programs have increased the communication
between university faculty, K12 teachers and students; resulted in innovative
science and math curricula; and provided K12 teachers workshops to enhance
their pedagogy and content knowledge. However, there is need to improve the
effectiveness of these universityschool collaborations.
Clift, Veal, Johnson, and Holland (1991) defined
collaboration as the explicit agreement among two or more persons to meet and
accomplish a particular goal or goals (p. 54). The common goal of STEM
collaborations is to improve science, technology, engineering, and math
education of all students. However, models of collaborations toward this goal
are not well established yet.
The key factor found in successful universityschool
collaborations is recognizing and utilizing the expertise of both parties since
onesided partnerships where the flow of instruction is from university faculty
to K12 teachers were found unsuccessful (Moreno, 2005; Sternheim, 2003;
Tomanek, 2005). University faculty are the content experts, and K12 teachers
know the school culture, pedagogy, and students (Sternheim, 2003). Ultimately,
teachers are the ones who will decide if and how the products will be used.
Therefore, teachers need to be involved in both the planning and the
implementation of the collaboration. Beneficial partnerships arise out of
honest and supportive environments for the benefit of students.
Teachers should continually be improving their pedagogical
practices to insure students learn. However, teachers often only experience
significant pedagogical growth through sound professional development. There is
an emerging consensus among researchers on features of quality professional
development (Elmore, 2002). Explicitly, one of those features involves
developing, sustaining, and reinforcing group work. Yet, it is not simply
forming a group, but creating a professional learning community (e.g.,
KoellnerClark & Borko, 2004) to address other features of quality
professional development (PD) (Elmore, 2002).
This study illustrates and discusses the new approach in
dissemination of PDs and the schools’ needs. First, PDs on learning to operate
technology (e.g., interactive white boards, graphing calculators, and clickers)
and creating effective learning tasks for students will be provided to the
teachers. Second, an online portal will be developed to offer PDs, which will
eliminate travel difficulties and limitations. Third, teachers will find
opportunities to enhance their technology integrations in collaboration with
university faculty. The focus will be the STEM activities to create an
interdisciplinary learning community.
Programming Using SSP
Authors: RAO Yongsheng, WANG Ying
Affiliations: School of Computer Science and Educational Software,
Guangzhou Universiy, China, School of Information of Science and Technology,
Sun Yatsen University, Guangzhou, China, South China Institute of Software
Engineering, Guangzhou, China( 510990)
SSP (Super Smart Platform) is an excellent dynamic geometry
system. In China, it is the most popular mathematic educational software, many
teachers and students use it in class. Programming is one of its most unique
features compared to other dynamic geometry system. The special function is
very powerful, which can be used to draw dynamic geometric graph, realize
algorithm, do numeric and symbolic computation, load courseware, and define
custom functions. With programming, teachers can draw repetitive, mechanical
graphics easily and quickly, e.g. draw hundreds of curves in one second; we can
save a courseware as text code, and then reload it anytime, anywhere, just like
drinking instant coffee. It can be also used as a platform for teaching
students programming. In this talk, we will show how to write and run programs
on SSP.
Dynamic Geometry System for Mobile Devices
Authors: WANG Ying, RAO Yongsheng
Affiliations: School of Computer Science and Educational Software,
Guangzhou Universiy, China, School of Information of Science and Technology,
Sun Yatsen University, Guangzhou, China, South China Institute of Software
Engineering, Guangzhou, China, 510990
Now, smart mobile devices, e.g. smart phone, tablet PC, are
becoming more and more popular, and become an integral part of life as
information processing platform. Although there are more than 40 kinds of
dynamic geometry system in the world, there is still no one for mobile devices.
In this session, we will introduce our system MDGS for mobile device, which is
developed based on Flash AS 3.0. MDGS is crossOS, not only supports the
desktop operating system as Windows, Linux, also supports the mobile operating
systems such as Windows Phone, android, iOS with special plugin. We can use
the system by Web Browse without any plugin except Apple OS, and store data in
cloud. So, we can enjoy dynamic geometry anytime, anywhere.
Creativity in The Mathematics Education
Authors: NAHID MOAYERI, Mohsen Mohammadizadeh
Affiliations: Teacher of Mathematical at Fatemiyeh High School Educational Management of Sirjan –Educational Organisation of Kerman IRAN, Faculty Member of Islamic Azad University SirjanBranch,IRAN, Member of Faculty Islamic Azad University SirjanBranch
Interesting and controversial topic of creativity is that its effect on the success and advancement of people has been proven. But whether creativity can be paid to education and training people gave it up by its own educational programs on inducing most experts have confirmed the possibility of such training. Most of what educational psychologists in schools and in schools of cognitive behavioral believe that creativity can be taught and creativeness are. Below are the creative learning process. Many math teachers are faced with the question of why, despite having some of the students educational and welfare facilities in math lessons are failed? math classroom experience in 14 years me that according to several factors affects their academic achievement. The example with the words easier to express acceptance by the audience is so successful in this paper to the field of education theories about creativity in math achievement are discussed. Method which has been offered ways that are willingly students to study more math and makes them more curious. Society today requires that people think the prevailing wisdom of his actions and thoughts are being so innovative and problems of their own way they are.
PERIODIC FUNCTIONS and THEIR APPLICATIONS
Authors: Ferhat Öztürk, Ahmet Iþýk, Tuba Kaplan
Affiliations: Atatürk University, Atatürk University
Periodic functions have started to be frequently used in the
field of mathematics, in application areas of mathematics and in other applied
sciences such as physics, physics engineering, electrical and electronic
engineering and statistics. Periodic functions are a topic on which many
scientific studies and researches are conducted. If we define technology, which
is different from science, as ‘form of offering science to the public service’,
we can say that periodic functions have a great part in formation and
application of technology. In this context, it is a fact that mathematics
closely affects the human life, particularly periodic functions are an
important factor in formation of technology, and application areas of periodic
functions extend in accordance with theoretical bases of mathematics. This
study aims to examine periodic functions, which has a large application area in
applied mathematics, especially in physics, engineering and computer systems,
from a different perspective.
Within the scope of the study, national and international
literature was reviewed, and survey model was used. As a conclusion, in Fourier
analysis, it was found that there are many application areas which are formed
by extensive use of periodic functions and which make daily life easy (medical
imaging systems, image processing, material processing technology, machine
failure detection, design of modern musical instruments and talking computers
etc.).
SYSTEMS of LINEAR EQUATIONS and THEIR APLICATIONS
Authors: Tuba Kaplan, Ahmet Işık, Ferhat Öztürk
Affiliations: Atatürk University
Mathematics is closely associated with other disciplines
although it is a discipline which has its own facts. It is seen that
mathematics is used as a means in other disciplines. Linear algebra course,
which is covered in mathematics curricula, becomes more and more important for
engineering, physics, social and behavioral sciences. In addition, linear
algebra has undeniable contributions to the fields such as aviation and
aerospace industry, electrical circuits, communication networks, archeology,
weather forecasts, population change and trade. When one deals with a problem
about applied sciences, linear equation systems generally come into mind. In
the course of historical development of linear equation systems, studies were
conducted on various methods for solution of these systems besides general
theory of them. It is known that these methods are important for formation of
various current models in parallel with development of computer and technology.
Nature of current models is generally based on many factors. However, solutions
of problems emerging out of the field of mathematics require the interaction of
other branches of mathematics and linear algebra.
This study focused on linear algebra systems and their
theoretical bases, pointed out solution methods, and offered examples from
different applications available in engineering, physics, chemistry, economics
and many other disciplines. Within the scope of the study, national and
international literature was reviewed, and survey model was used. As a
conclusion, it was found that, in almost all disciplines, especially in
engineering, there are many application areas which are formed by extensive use
of linear equation systems and which make daily life easy.
Represent of Mathematic Model for Forming of Optimizing the Concrete Domes
Authors: MOHSEN MOHAMMADIZADEH, Mohsen Mohammadizadeh, Nahid Moayeri
Affiliations: Faculty Member of Islamic Azad University SirjanBranch,IRAN, Teacher of Mathematical at Fatemiyeh High School Educational Management of Sirjan –Educational Organisation of Kerman IRAN
In this article , a forming model for performing of concrete domes presented and by required of accuracy can do this dome. For this means, as the radius of dome be determined and Y=F(X) be related to top of dome to be huge the floor of circle .So rotation of this function around Z axis , the shape of dome will be handed ,which for forming of dome in " n " times , If more number''s n , nearer to dome .In here , the mathematic model for accounting the height of each form will be give. however , if the height of all forms be equal , the shape of dome will be in conical form and for performing the main shape , it's essential the height of each form will be count in a particular method
A geometric eye for dynamic geometry
Authors: Zlatan Magajna
Affiliations: Faculty of Education, University of Ljubljana
We present a technology supported approach that helps
students to understand the concept of proof and to prove geometric theorems.
The approach is implemented in a software program called OK Geometry (Observing
and Knowing in Geometry, available at htpp://www.zmaga.si) and is based on
Toulmin’s model of argumentation. According to this model, the process of proof
construction consists of conjecture production and of finding warrants (or
rebuttals) for the conjectures. The aim of the software is thus to help
students in conjecture production, in organizing the conjectures into a proof,
and also in producing possible warrants (and, indirectly, also rebuttals) to
conjectures.
Using dynamic geometry software (e.g. Cabri or GeoGebra) the
students are, in general, able to visualise geometric propositions and check
their correctness. But proving them is a different story. In such situations OK
Geometry may be used as a geometric eye. OK Geometry reads a dynamic construction
and produces a list of geometric properties of the construction. As an
extremely sensitive geometric eye (that observes constructions and not
drawings) OK Geometry detects and visualizes several geometric properties one
may not be aware of (e.g. congruent angles, collinear points, harmonic
quadruples). In the proof generation process students, using OK Geometry as a
geometric eye, look among the detected properties for conjectures and for
warrants for the chosen conjectures and gradually construct a proof. There are
obstacles that are immanent to this approach, but they can be relieved and
overcome by technological means.
Promoting investigative math classrooms through SAMAP manipulatives
Authors: Erol KARAKIRIK
Affiliations: ATCM 2011
SAMAP is Turkish acronym for a nationwide Turkish Science
Foundation, TUBÝTAK, project that aimed to develop a comprehensive set of
computer manipulatives for primary and secondary school mathematics. SAMAP
manipulatives consist of around 100 small java applets specifically developed
to highlight certain mathematics concepts and relations and are combined within
a single graphical user interface. They have been used for a few years in
Turkish math classrooms by many teachers throughout Turkey. They aim to promote
higher order thinking skills by providing an environment for investigating
mathematical concepts and relations rather than focusing on simple calculations
and mathematical operations. However, many teachers and students are uncertain
how to make use of SAMAP manipulatives since they are used to make repetitive
calculations instead of investigating mathematical concepts. Furthermore, many
teachers only make use of them as extracurricular activities. In this paper,
we aim to demonstrate how SAMAP could be used to promote discussions, to
increase students’ participation and enhance their conceptual understanding in
math classrooms. Author will exemplify, by the help of a few selected SAMAP
manipulatives, how they could be used to ask students conceptspecific
interpretive and estimation questions not requiring any calculations and verify
their answers immediately by manipulating certain variables and how one could
solve the same problem in different ways by the help of builtin expert systems.
Touch2Learn  Using MultiTouch technology on content focused
learning in Mathematics
Authors: Christian Dohrmann
Affiliations: University of Education Karlsruhe, CERMAT (http://cermat.org)
There is the longterm trend in mathematics education to pay
more attention to the understanding of concepts instead of building up a large
repertoire of knowledge and skills. According to this trend, the stated
requirements have to be considered for ICTbased teaching.
Traditional computer input/output devices are technically
restricted to single point interactions. Therefore, the user interface of a
computer tool is restricted to a singlepoint interaction. There is only one
mouse for every computer. Even if students work in pairs only one student is in
charge.
In general, discussion and arguments are indispensable
elements of teaching and learning. Hence, it is important to support group
work, not only to develop social skills, but also to be able to discuss on
mathematical problems. In ICT supported environments these important
interactions are often neglected.
With MultiTouch technology, several users can work
simultaneously on the same screen. It is even possible to develop cooperative
exercises that can only be solved by two or more students helping each other.
User studies have shown: MultiTouch enabled learning environments provoke
taskfocused discussions with emphasis on content rather than turntaking
agreements. Thus, MultiTouch has the potential to enhance communication
skills.
The presentation will include first results of research
activities with students using MTenabled environments as well as the
introduction of a theoretical framework for the development of MTenabled
computer tools in mathematics (DGS).
Abstracts for Handson Workshops
Dynamic Geometry, Dynamic Art
Authors: Kate Mackrell
Affiliations: Institute of Education, University of London, UK
The connections between art and static geometry are rich,
diverse, and well known. Dynamic geometry software enables us to explore some
of the additional possibilities that arise when representations of geometric
objects are set in motion; beautiful objects emerge, evolve, and transform,
sometimes in quite unexpected situations. In this session we will create a
variety of simple, beautiful objects using Cabri II Plus, Cabri 3D, Cinderella,
and Geometer’s Sketchpad 5, and discuss some of the mathematics behind these
objects.
Exploring Mathematics with the Geometer’s Sketchpad Version 5
Authors: Krongthong Khairiree
Affiliations: International College, Suan Sunandha Rajabhat University
Bangkok Thailand
The workshop is designed for participants to experience in
using the Geometer’s Sketchpad Version 5 (GSP V.5) with constructivist approach
in mathematics lessons. The workshop will be a handson activity and the
workshop will be conducted in such a way as to simulate a mathematics class.
Prior experience with GSP V.5 is not necessary. The
knowledge on how to use GSP V.5 and the news tools of GSP V.5 will be
introduced in this workshop. Primary emphasis will be on learning how to use
GSP V.5 effectively in integrating geometry and algebra. The workshop also
provides participants learn how to use GSP V.5 for classroom demonstrations and
explanations.
Graphing surfaces z = f(x,y) with Cabri 3D
Authors: JeanJacques Dahan
Affiliations: IREM of Toulouse
Even Cabri 3D does not contain a 3D grapher, we will during
this workshop achieve very realistic representations of surfaces z = f(x,y) in
using cleverly the tools “trajectory” and “animation”. We will also show how to
create macros (even this tool is not available in Cabri 3D) to improve the
previous representations.
The 3D grapher of TI N’Spire
Authors: JeanJacques Dahan
Affiliations: IREM of Toulouse
The last improvement of TI N’Spire contains a 3D grapher
using coloured representations. We will discover this new tool during the
workshop after a short presentation of the modelling of folding and unforlding
cylinders and cones in military perspective with the geometry application of TI
N’Spire.
Surfaces with Autograph
Authors: JeanJacques Dahan
Affiliations: IREM of Toulouse
During this workshop we will discover the 3D application of
Autograph to represent surfaces z = f(x,y) and z = f(x,y,m) and others. The
very special “Content controller” tool will allow us to link analytic and
geometric understanding of surfaces.
Creating Interactive Mathematics with Cinderella and CindyScript
Authors: Ulrich Kortenkamp
Affiliations: Pädagogische Hochschule Karlsruhe, CERMAT, Cinderella
In this workshop we will explore the new facilities of the
Interactive Geometry Software Cinderella 2 and work with physics simulations
and custom scripts. For a sampler of what is possible see
http://www.mathevital.de
A basic knowledge of any geometry software is required.
Participants will receive a temporary license for the software and can use
their own laptops or the lab facilities at the conference.
How to Turn a Polyhedron Inside Out?
Authors: Jenchung Chuan
Affiliations: National Tsing Hua University
In this workshop we will guide the participants to construct
an animation displaying the inside out process for the cube, the regular
tetrahedron and the regular dodecahedron. To prepare for the workshop, the
participant is invited to contemplate on these problems: what happens to a
solid cube when turned inside out completely? Is it possible to have the
components remain connected during the process?
How to Dissect a Polyhedron into Congruent Pieces in Infinitely Many
Ways?
Authors: Jenchung Chuan
Affiliations: National Tsing Hua University
1) Can a solid cube be cut into six congruent pieces, other
than the "obvious" symmetric ones?
2) Can a solid regular tetrahedron be cut into four
congruent pieces, other than the "obvious" symmetric ones?
In this tutorial we are to construct animations with Cabri
3D showing there are infinitely many such possibilities to construct the
dissections for each of 1) and 2).
The Educational Use of Scientific Calculators
Authors: Kian Boon Lim, Tau Han Cheong
Affiliations: Universiti Pendidikan Sultan Idris, Universiti Teknologi Mara
Casio fx570ES scientific calculator is widely used in
Malaysia’s secondary school. This scientific calculator is equipped with 403
functions. Many functions of the scientific calculator are very useful to the
students in learning mathematics but few of them know how to apply it. This
workshop will show how teacher can use fx570ES scientific calculator for the
classroom effectively in teaching mathematics, some features of Casio fx570ES
scientific calculator that can be used in teaching and learning of mathematics
will be shared, which will include the following: (1) Trigonometric function
and solution of triangles (2) Rectangular – Polar Coordinate conversion (3)
Quadratic Equations and Simultaneous Equations (4) Statistics and Normal
Distribution (5)Matrix Calculation (6) Calculation involving specific number
systems (binary, octal, decimal, hexadecimal) (7) Integration and
differentiation (8) Complex number (9) Numerical Integration (10) Roots of
nonlinear equation.
Experiencing the interactive digital resources “1 2 3… Cabri” for
middle school
Authors: Colette Laborde, JeanMarie Laborde
Affiliations: University Joseph Fourier, Cabrilog, Grenoble, France
The collection of multimedia interactive activity books “1 2
3 … Cabri” meant for middle school students will be presented. An activity book
comprises several pages proposing a sequence of tasks to students with various
feedback.
The collection covers the key notions of mathematics
curriculum of middle school from numbers and arithmetic to geometry through
measurement. Participants will explore different activity books in order to
analyze the tools and the various types of feedback offered to students
Advanced Sketchpad Version 5 Workshop: Focus on Transformations
Authors: Nicholas Jackiw, Steven Rasmussen
Affiliations: KCP Technologies, Key Curriculum Press
This workshop will continue Sketchpad learning (see
Workshop: "Introduction to Sketchpad") by focusing on the topic of
transformations across the curriculum, while emphasizing new functionality of
Sketchpad Version 5. Topics will include transforming shapes, functions, and
digital pictures through basic isometries and more advanced or unusual
geometric transformations such as inversion and morphing. Prior Sketchpad
exposure (such as completing the "Introduction to Sketchpad Version
5" Workshop) desirable. Participants will receive a 60day preview edition
of Sketchpad Version 5 in English or Turkish.
(ORGANIZERS: If no computer lab will be certainly available
for this workshop, please add "Bring your own laptop!" to the
Workshop.)
Introduction to TinkerPlots Version 2
Authors: Steven Rasmussen
Affiliations: KCP Technologies, Key Curriculum Press
This workshop will focus on teaching concepts in data and
chance using TinkerPlots, a dynamic data software for middle school students.
Topics will include analyzing data, simulating data to observe variability in
samples, and concepts in probability. No prior experience with TinkerPlots is
needed. Participants will receive a 60day preview edition of TinkerPlots
Version 2 in English.
(ORGANIZERS: If no computer lab will be certainly available
for this workshop, please add "Bring your own laptop!" to the
Workshop description.)
Introduction to Sketchpad Version 5
Authors: Nicholas Jackiw, Steven Rasmussen
Affiliations: KCP Technologies, Key Curriculum Press
This workshop will introduce teachers to the rich
mathematical feature set of The Geometer''s Sketchpad 5, including Dynamic
Geometry construction tools and analytic and algebraic modeling tools. Workshop
content will focus on basic software functionality and support resources. No
prior Sketchpad experience is necessary. Participants will receive a 60day
preview edition of Sketchpad Version 5 in English or Turkish.
(ORGANIZERS: If no computer lab will be certainly available
for this workshop, please add "Bring your own laptop!" to the
Workshop description.)
TechnologyRich Learning Experiences from the Web for the
Mathematics Classroom
Authors: Ngan Hoe Lee, Beverly Ferrucci
Affiliations: National Institute of Education, Nanyang Technological
University, Keene State College
This workshop will share some ways in which technology can
be incorporated into mathematics classes. Resource ideas and examples will
range from whole class activities and interactive projects to digital
libraries. The first part of the workshop will focus on examples for the
primary levels while the second part will draw on examples for the secondary
levels.
Getting Beyond the Visual Hurdles of Calculus in Three Dimensions
Authors: Drew Ishii
Affiliations: Sage Hill School, California Mathematics Council
Visualizing threedimensional objects and surfaces can be
difficult for students especially if they are not visual learners or
artisticallyinclined. Many of those students are accustomed to doing
twodimensional graphs by hand, but that process breaks down when it comes to
doing calculus in threedimensions. Students should not be constrained or
stifled in their learning of advanced calculus topics because they either lack
visualization skills that have never been required of them in previous courses
or do not have programming knowledge. I present a session that explores the
threedimensional coordinate system, quadratic surfaces, vector functions, the
TNB frame, and multiple integrals with the userfriendly software Grapher by
Apple that does not require any knowledge of programming. Participants will see
how intuitively students can investigate these multivariable calculus topics
by experimenting with equations and their graphs.
Learning mathematics with an advanced scientific calculator
Authors: Marian Kemp, Barry Kissane
Affiliations: Murdoch University
While scientific calculators have been available since the
1970s, advanced versions have been developed recently to extend the
mathematical capabilities to equations, vectors, matrices, series, complex
numbers, probability and statistics, as well as elementary calculus operations
of integration and differentiation. Consequently, these calculators provide
powerful learning opportunities for many aspects of mathematics treated these
days in senior secondary school and university curricula, as well as giving
students access to efficient calculation. In this workshop a variety of
examples will be used to consider ways in which sophisticated mathematical and
statistical concepts can be developed, through student use of these modern
calculators. We will use Casio fx991ES PLUS calculators, but do not expect
that participants will have prior experience with this calculator.
First steps in learning mathematics with a graphics calculator
Authors: Marian Kemp, Barry Kissane
Affiliations: Murdoch University
Graphics calculators provide opportunities for both students
and their teachers to engage with mathematics in new ways. This workshop is
intended to offer an introduction, for those new to this technology, of some of
the possibilities open to classrooms in which such technology is present. A key
aspect is that graphics calculators can be used to support student learning of
mathematics, even in situations for which external examination rules do not
permit their use in formal assessment. We will illustrate the possibilities
through drawing upon a range of mathematics areas, including the study of
functions, equations, elementary statistics, probability, trigonometry and
differential calculus. The focus will be on teachers of senior secondary or
early undergraduate mathematics. We will use the Casio fx9860GII calculator,
but do not expect that participants will have prior experience with this
calculator.
Learning mathematics with a scientific calculator
Authors: Barry Kissane, Marian Kemp
Affiliations: Murdoch University
For about forty years teachers and students have used
scientific calculators for simple and more complex arithmetic calculations and
for tasks involving logarithmic, exponential and trigonometric functions. More
recently, modern scientific calculators have been developed to become more
userfriendly and to extend their mathematical capabilities to suit modern
curricula. While calculators are sometimes regarded as merely devices to
produce numerical answers, in this workshop we will also consider instead some
ways in which the development of mathematical ideas can be supported with such
a calculator, focusing on the secondary school. We will use the Casio fx82 ES
PLUS calculators, but do not expect that participants will have prior
experience with this calculator.
Applications of Trigonometry and Polar Functions in Grade 1011 Math
Authors: Tonguc Ozdas, Seda Eren
Affiliations: ENKA Schools, Enka Schools
Technology integrated classes help students to be aware of
real life applications of mathematics and make sense of the world that we live
in by means of mathematics. Most of the time mathematics is considered to be an
abstract sequence of operations by a large student body rather than a science
that helps us understand the world that we live in. With the activities
designed, we aim at changing the perspective of our students and appreciate
mathematics and think out of the box. The importance of technology as an interactive
teaching tool will be emphasized.
Turkish and International Curriculum will be compared in the
light of these activities.
Investigating Investigations
Authors: Christopher Longhurst
Affiliations: Hewlett Packard
 What is a
mathematical investigation?
 How do you
put an investigation together?
 What
should we investigate?
 Are
investigations a good learning tool?
In this workshop I will attempt to answer the above
questions by starting at the beginning and developing an investigation using
tools such as the internet, and graphing calculators. The steps to producing a
good, meaningful mathematics investigation will be taken and the group will
perform the investigation. I will also add a couple of mathematics magic tricks
into the presentation that can be used in the classroom which provide
motivation and fun learning experiences for the students.
Developing open teaching supported by the implementation of graphing
calculators in the UAE
Authors: Pat Tunnicliffe, Chris Olley
Affiliations: King's College, London, ADEC (Abu Dhabi Education Council)
CfBT (Centre for British Teachers)
This workshop will present the early findings of a project
to develop open and exploratory approaches to the teaching of mathematics in
secondary schools in Abu Dhabi, UAE. Two schools (one boys only, the other
girls only) have been working with advisory teachers to expand the range of
teaching styles. Graphing calculators with a computer algebra system (HP40GS)
were introduced through a three day workshop to the teachers, with the
principal orientation on investigative modes of learning. The teachers had
different personal orientations to this pedagogy and to the technology. The
workshop will discuss the extent to which teachers have been able to take up
the technology as a pedagogic tool and use it to support a more investigative
approach to their teaching. Delegates will be able to try out activities
designed by the teachers using this technology to help frame discussion.
Using Graphical Software in Mathematical Modeling Tasks.
Authors: PINAR ÖZKUL SEZGÝN
Affiliations: MARMARA PRIVATE HIGH SCHOOL
This is a hands on workshop, which will show participants how to draw graphs using graphical software like Logger Pro, Graph 4.3. It will focus on one mathematical modeling task. Data contains National CO2 Emissions from FossilFuel Burning in Turkey which is taken from The Carbon Dioxide Information Analysis Center will be given. Finding a model function for this given data will be shown step by step. Scopes and limitations will be discussed. Multiple graphs will be drawn that contains the given data, model function and the function found by regression tools. Comparison of the data with data produced by the developed model will be shown. After finding the best model function, future predictions will be made in the context of the task. Hence attendees would be able to learn how to find properties of graphs such as extremum points, zeros, yintercepts, horizontal asymptotes, vertical asymptotes etc. Teachers will be encouraged to design mathematical modeling tasks in which technology is used to enhance the development of the task.
Introduction to Maple 15
Authors: Douglas Meade
Affiliations: Department of Mathematics, University of South Carolina,
Industrial Mathematics Institute, USC
Receive a handson introduction to Maple 15 that will
demonstrate its abilities to work with symbolic, numeric, and graphic
information. The focus will be on interactive use of Maple 15 to explore a
variety of mathematical topics chose from precalculus, calculus, linear
algebra, and differential equations. There will be specific instruction in the
creation of customized studentfriendly user interfaces that minimize the
amount of syntax students need to know. Participants are encouraged to think
about specific types of problems they would like to learn how to use Maple to
solve. No prior experience with Maple or another CAS will be assumed.
Handheld technology and calculus
Authors: Christopher Longhurst
Affiliations: Hewlett Packard
Calculus is often one of the most interesting strands of
mathematics for students. The reasons are varied but they include the fact that
it is a combination of algebra, number, dynamics as well as often it appears to
be a practical side of mathematics which brings real meaning to real world
problems. In this workshop I will take many of the aspects of calculus
including introduction, theory and practical aspects and show how I us the
graphing and CAS calculator to consolidate the students learning and
understanding of calculus.
Mathematical Modeling: A pedagogic context or a boundary?
Authors: Chris Olley
Affiliations: King's College, London
This session will present existing work using handheld data
streaming technology, investigating the cooling of pizzas. The basic activity
comes from materials developed by the presenter and others at King's College,
London for the Bowland Trust, whose purpose was to develop open extended tasks
in mathematical problem solving. The activity has developed as having a focus
on engaging learners with the practice of mathematical modelling in itself. The
emphasis being on critique and reformulation. Much of what appears as modelling
in English school's curricula, present mythogised versions of ''realworld''
problems validated only within the school mathematics context. The session will
explore the principles of evaluation deployed in the school mathematic setting
and the ''realworld'' setting and consider whether the activity of
mathematical modelling can be constructed as a separate pedagogic context, or
act as a boundary between the two. The session will contain live data streaming
on HP40GS graphing calculators and modelling by the participants to engage with
the issues.
Abstracts for Poster Sessions
Design Decisions in Interactive Geometry Software
Authors: Kate Mackrell
Affiliations: Institute of Education, University of London, UK
NOTE 1: This poster is based on a paper to appear in ZDM in
summer 2011. The poster will present these results schematically and also
incorporate two further softwares (Casyopee and TI’Nspire) and calculation operations,
not mentioned in the paper.
NOTE 2: I will be submitting a paper as well, looking at the
algebraic affordances of the IGS programs mentioned here: the poster
complements this paper.
Problem Definition and Objectives:
There is an increasing awareness that the details of the
design of pedagogical tools are significant and should be researched (Butler,
Jackiw, Laborde, Lagrange, & Yerushalmy (2009)). An analysis of seven
interactive geometry programmes (Cabri II Plus, Cabri 3D, Casyopee, Cinderella,
GeoGebra, Geometer’s Sketchpad and TI’Nspire was hence performed in order to
identify some of the design decisions necessary to enable basic geometrical
constructions, measurement, and calculation.
Strategy, methods, theory: The possible interactions in a visual
mathematics representation identified by Sedig and Sumner (2006) were used to
develop a categorization of the operations that can take place in an IGS
program. An exploration of the design decisions involved in several fundamental
operations was then undertaken with each software by performing a task
involving creating a circle and exploring its area.
Results: The main part of the poster will show schematically
the types of design decisions necessary (such as how a tool is used), the
different decisions made (such as the order in which objects need to be chosen
in using a tool), and the potential impact of decisions on the affordances of
the programs. A number of tensions (such as between user choice and default
behaviour) will also be shown.
Conclusions: There is an extraordinary diversity in both the
type of decisions that need to be made and in the decisions possible. For the
most part, the effect of these decisions upon student learning is currently
unknown: the poster hopefully provides a framework within which future research
may be situated.
The Technology in the Teaching of Calculus. An Experience with
Social Science Students
Authors: Jose Ortiz
Affiliations: University of Carabobo
The mathematical training of future graduates in economics
and social sciences should help strengthen the relationship between mathematics
and reality, with the expectation of providing students and conceptual tools
for understanding and addressing functional phenomena relevant to their future
professional field. This position is contrary to the current specification of
the curriculum at the university, which focuses on mathematics as an isolated
activity of the physical and social world of the student. In that sense, it
goes to an innovation in which changes in the approach to teaching and learning
of mathematics. In this paper, we analyze the activities of a group of
students, who participated in an environment of teaching and learning,
supported by the use of graphing calculator and a CAS in the context of
calculus. Subjects participating in the study were students attending
mathematics basic cycle of the Faculty of Economics and Social Sciences at the
University of Carabobo, Campus La Morita, Maracay, Venezuela. It is considered
a qualitative approach in the analysis of the productions of the participants,
the product of his performance in the classroom and the answers to problems set
conducted outside the classroom. The reports show changes in the mathematical
knowledge of students, expressed in the understanding and application of
concepts in the interpretation of the calculation of the physical world of
phenomena, natural and social. The participants used CAS in the achievement of
complex calculations and understanding of concepts and ideas involved in
solving problems relating to the discipline and the social environment. The
results revealed the use of several systems of representation of mathematical
concepts and properties, as well as conceptual understanding and practical use
of calculus in the real world.
TEACHER’S OPINION TOWARDS EMPLOYING GEOGEBRA SOFTWARE IN THE
TEACHING OF MATHEMATICS
Authors: Hutkemri Zulnaidi, Effandi Zakaria
Affiliations: Universiti Kebangsaan Malaysia
A study has been conducted to observe teacher’s evaluation
towards GeoGebra software in teaching Mathematics. The study comprises of four
respondents who are currently enrolled in Mathematics Education course at
Universiti Kebangsaan Malaysia. The study generates similar opinion among the
respondents based on their experience, with regard to GeoGebra software
content. All respondents agree that the software employs straightforward and
comprehensible instructions in addition to accurate and simple information
offered. As for technical aspects, respondents are clearly of the same opinion.
They find that the software is user friendly. The respondents on the whole have
positive opinions towards the idea of employing GeoGebra software in the
teaching of mathematics.
ACTIVITIES FOR TEACHING CONCEPT OF SYMMETRY WITH THE SOFTWARE
“SIMETRIA”
Authors: Pelin Turan
Affiliations: Anadolu University
The concept of symmetry is one of the most important
application areas in geometry. People need the idea of symmetry as a way of
understanding nature and environment which they live in. We see the most
beautiful examples of symmetry in architecture, in arts, in the world of plants
and animals, etc. Mainly, symmetry is a movement of rotation and translation.
So it is a tool to analyze mathematical situations. Also the study of symmetry
offers one approach to analyze patterns. These important two concepts (symmetry
and patterns) are included all the classes in geometry curriculum of primary
school in Turkey.
The aim of this study, with using geometry software
“Simetria”, is to give examples of activities that include the concept of
symmetry in patterns into technologybased environment for teachers. In this
study, firstly the potential of geometry software “Simetria” is introduced,
secondly it is explained how steps of activities can be constructed and finally
applications of activities in class are discussed.
MultiplexR: Basic concepts of numbers and operations through
multitouch & linked multiple representations
Authors: Silke Ladel
Affiliations: University of Education Karlsruhe, Germany
Many difficulties that children have in learning arithmetic
are based on the fact, that their concept of numbers and operations is
insufficiently developed. E.g. addition and subtraction is only a demand to
count forward or backward. Those children fail to build up relations between
number triples. They lack the partwhole concept of numbers, where a numeral
identifies a quantity which can be decomposed in several parts. As a
consequence of the insufficiently developed concept of operations certain
children are not able to link different forms of representations. They are
acting with mathematical symbols without any visualization of the meaning of
these operations.
With multitouch technology we are now able to link
mathematical representations so that children can experience the relations.
They act with the virtual material and see the consequences of their operations
on the symbolic side.
Our research question is how a multitouch learning
environment has to be designed to support the development of the partwhole
concept and to give the opportunity to experiment with interconnected
mathematical representations in due consideration of the fundamental theories
of mathematical learning. To implement prototypical environments and for
recording experimental data of children‘s interactions we use the interactive
geometry software (IGS) Cinderella, which acts as a standard tool for rapid
prototyping of learning environments. The integrated scripting language
CindyScript can be triggered by user actions and so it is possible to change
the standard behaviour of this IGS into the required interaction for an
experiment.
Introduction of Complex Geometry by Learning Visually with
Technology
Authors: Chieko Fukuda, Kyoko KAKIHANA
Affiliations: Teikyo University, JAPAN, Tsukuba Gakuin University
Euclidean geometry has traditionally been studied by figure
description. On the other hand, it has also been studied analytically using
numerical formulas. This analytical method connects geometry with other
mathematical fields, and many applications of geometry are created, for example
computer graphics, object recognition, and so on. In high school mathematics,
analytical geometry using real numbers is an important academic unit but
students have little chance to study it using complex numbers. It can be
appreciated, for example, that some kinds of calculation become simple by using
conjugate complex numbers and there are materials containing interesting topics.
Also, learning complex geometry is an important way to familiarize students
with complex numbers before they study complex analysis.
The purpose of this paper is to show educational materials
to help students’ understanding of geometrical meaning of complex numbers by
using software. We developed materials by using software to teach a visual
approach to complex geometry. First, students sort the basic characters of
complex numbers from a geometrical view. Then, they solve some geometrical
problems which are easy to manipulate by transforming geometrical characters
into complex formulas. In those problems, we take up inversion from the
viewpoint of transformation by complex function and, more generally, linear
fractional transformation.
Maplets for Calculus: Effective Resource for Teaching and Studying
Calculus
Authors: Douglas Meade, Philip Yasskin
Affiliations: Department of Mathematics, University of South Carolina,
Industrial Mathematics Institute, USC, Department of Mathematics, Texas A&M
University
Learning calculus is not a passive activity. As university
resources continue to be stretched, section sizes have increased and grading
support has declined. With limited resources, more courses are making use of
computerbased homework systems. Unfortunately, most of these systems still
have pedagogical limitations.
Maplets for Calculus (M4C) is an electronic study guide that
consists of 129 customized applets for specific topics in precalculus,
univariate calculus and multivariate calculus. Each applet presents an
algorithmicallygenerated problem, requires correct intermediate responses
before moving on to the next step, employs computer algebra to analyze student
responses and provides customized hints and feedback. Graphics (2D, 3D, animation
and stereo) are used whenever possible to reinforce the symbolic mathematics.
In short, M4C is a "tutor without the tutor".
Students appreciate the stepbystep guidance through
problems and the way algebraic, graphic, numeric and verbal approaches support
diverse learning styles. Instructors like the interactions that arise when
students in a lab have different versions of similar problems and frequently
use the applet graphics as lecture demonstrations. Initial assessment of M4C's
effectiveness is underway.
2*Nprimes defined by unceasing Ntwin primes
Authors: Hirotaka Ebisui
Affiliations: Oval Research Center
We found 12 unceasing primes which are 6 twin primes. Now
One set is found as Followings.
[[325267931,325267933],[...37,...39],[...49,...51],[...61,...63],[...79,...81],
[325267991,325267993]]
We found this primes sequence using Maple software and
consuming 30 hours. Before we found these prime, we found unceasing 5twin
primes. So, we can expect the existence of 2*N (N is arbitrary natural number)
unseasing primes consisted by NTwins. This fact shows any long dense primes in
natural numbers and NTwins behave like compound number. In poster session, we
show various number tables.
Thank you for your reading of this abstract.
Choosing Technology Applications: Considering Students’ Mathematical
Developmental Backgrounds
Authors: Agida Manizade, Marguerite Mason
Affiliations: Radford University, The College of William and Mary
In this paper we are using van Hiele theory of students’
geometric development as an underlying structure of our framework for teachers’
decision making when assessing appropriateness of technology applications for
teaching mathematics at the high school level. When planning their mathematics
lessons teachers have to consider many factors, including but not limited to
how appropriate is a given online technology application for students’
mathematical background and their levels of mathematical development. We
propose a framework for teachers’ decision making when using free, open source
GeoGebra applications when teaching high school geometry topics.
