 Miroslaw Majewski, JenChung Chuan, Nishizawa Hitoshi
The New Temple Geometry Problems in Hirotaka's Ebisui Files
 Keng Cheng Ang
Teaching and Learning Mathematical Modelling with Technology
 Noraini Idris Assessment for a KnowledgeBased Era  Issues and Challenges
 Markus Hohenwarter and Zsolt Lavicza
 GeoGebra, its community and future
 Celia Hoyles 
Constructing and Collaborating through ICT
 Bernd Zimmermann, Torsten Fritzlar, Lenni Haapasalo  et al. 
Possible gain of IT in problemoriented learning environments from the viewpoint of history of mathematics and modern learning theories
 Bogumila KlempDyczek
Polyhedral tensegrity structures
 GT Springer
Questions Answers; and Mathematical Discourse
 WeiChi Yang  Xiaofeng Ding
Fundamental Theorem of Calculus and Computations on Some Special HenstockKurzweil Integrals
 Scott Steketee
Variables and Functions: Using Geometry to Explore Important Concepts in Algebra
 Tilak deAlwis
Bilinear Transformations via a Computer Algebra System
 Antonio R. Quesada
Does Technology Empower Students to do Research at the Secondary and Undergraduate Levels?
The New Temple Geometry Problems in Hirotaka's Ebisui Files
Miroslaw Majewski, JenChung Chuan, Nishizawa Hitoshi
mirek.majewski@yahoo.com
New York Institute of Technology; Abu Dhabi Campus
For quite a while we are witnessing development of Hirotaka files. These are PDF documents with hundreds of geometry problems developed by the Japanese mathematician Hiroraka Ebisui. At the same time there are very few mathematicians, or even geometers, aware about existence of these files, and even fewer number of people who recognize their value. The objective of this paper is to make a brief analysis of these files, examine roots of the problems described there, and point out their value.
Teaching and Learning Mathematical Modelling with Technology
Keng Cheng Ang
kengcheng.ang@nie.edu.sg
Nanyang Technological University; 1 Nanyang Walk; Singapore 637616
In the last few decades, there have been abundant discussions among mathematicians and mathematics educators on promoting mathematical modelling (a process of using mathematics to tackle real world problems) as a classroom practice. Mathematics educators and curriculum planners have been advocating the teaching of mathematical modelling in schools for some time now. Despite the consensus on its importance and relevance, mathematical modelling remains a difficult activity for both teachers and learners to fully engage in. In this paper, we examine some of these difficulties and discuss how technology can play a pivotal role in providing the essential support to make mathematical modelling a more accessible mathematical activity amongst students. Through a series of examples drawn from different fields and topics, we illustrate how a range of technological tools may be successfully and efficiently utilized in modelling tasks. In addition, we discuss the need for an optimal use of technology to balance between achieving the objectives of the tasks and attaining the goals of learning mathematics.
EASSESSMENT FOR A KNOWLEDGEBASED ERA
Noraini Idris
noridris@um.edu.my
University of Malaya
Assessment continues to be a crucial and controversial issue in education. Assessment of student achievement is changing, largely because today’s students face a world that demands new knowledge and abilities. In the global economy of the 21st century, students will need to understand the basics, but also to think critically and analyze, to judge logically, to communicate, to reason, and to make inference. Helping students develop these skills will require changes in assessment at the school and classroom level, as well as new approaches to largescale high stakes assessment. Assessment is changing for several reasons. Many debate the merits or demerits of assessment especially with respect to the effect of national examinations. This presentation discusses the issues involving assessment and how changes in the skills and knowledge needed for success, in our understanding of how students learn, and in the relationship between assessment and instruction are changing our learning goals for students and schools. It is suggested that changing our assessment strategies will tie assessment design and content to new outcomes and purposes for assessment. Presenter will also share the award winning system on how to assess and change students’ performance.
Intuitions for Exploring Curves in 17th century: Technology in Classroom with dbook
Masami Isoda
isoda@criced.tsukuba.ac.jp
CRICED; University of Tsukuba
Seventeen century is known as the century of Scientific Revolution. In this century, two type of mathematics has been originated from the translated mathematics from Arabic and Ancients Greek Mathematics until Renaissance. First one is the Universal Mathematics based on new analysis with algebraic notation beyond Arabic approach. Second one is the Extensions of Geometry keeping visual images of Ancients Greek Mathematics. Through the century, curves has been explored both approaches. Both of them related with the locus problems including the problem of tangent. Before defining curves by algebraic notation, curves has been constructed by their geometric definitions and properties. Not only limited to the ruler and compass, physicalmechanical tools had used for drawing curves. Astronomy, Music, Perspectives, Projection, Mechanism, Motion and Kinematics provided special models for mathematical intuitions at this era. School mathematics used to have these intuitions until WWII and lost most of them after New Math. In this lecture, dbook, the freeware to develop etextbook using historical textbook with Dynamic Geometry Software, is used for demonstration to recognize lost intuitions through the exploration of curves using historical textbooks.
GeoGebra, its community and future
Markus Hohenwarter and Zsolt Lavicza
zl221@cam.ac.uk
University of Cambridge; University of Pecs
Information technology has undergone fundamental changes during the past decades. In particular, the emergence of mathematical software, Computer Algebra Systems (CAS) and Dynamic Geometry Systems (DGS), has opened unanticipated opportunities for both the mathematical sciences and disciplines applying mathematics. The use of CASDGS in sciences and their potentials in mathematics teaching and learning have accelerated their incorporation into mathematics curricula at all levels. As a result of the improvement of the software, CASDGS became applicable in almost all phases of teaching and learning, for instance, in classroom demonstrations, laboratory experimentations, practicing problems, assessment, and students'' selfassessment. These wideranging applications originate from the fact that mathematical software can, simultaneously, both media and motivator of mathematics learning. This dual role poses substantial challenges to development and requires continuous work from researchers on both software and didactical aspects of CASDGS. Thus, mathematicians and educational researchers must aim to surmount these challenges in order to enhance the efficient incorporation of CASDGS into mathematics classrooms. The endeavour of our first three Computer Algebra and Dynamic Geometry Systems in Mathematics Education conferences was to contribute to solving the challenges mentioned above by becoming familiar with the work of fellow researchers, and with new software applications, as well as to develop new research directions. In our paper, we will offer an overview of the trends of CASDGS research in Europe based on the talks, papers, and workshops at CADGME and other related conferences in Europe.
Constructing and Collaborating through ICT
Celia Hoyles
C.Hoyles@ioe.ac.uk
University of London
Professor Celia Hoyles, London Knowledge Lab, Institute of Education, University of London, U.K We are seeing rapid developments in the ways that it is possible for students and teachers to share resources and ideas, and to collaborate through technological devices, both within the same physical space and at a distance. These developments are becoming more and more available as the Web becomes almost universally accessible. I will explore the potential and challenges for teaching and learning mathematics of these new levels of connectivity, both within and between classrooms and from first the student perspective and then the teacher perspective. First, I will present the MiGen project, which starts from what is well known: that is that students have difficulties in appreciating and expressing mathematical generalisations. I will discuss the development of a microworld, the eXpresser, designed as part of this project. This is intended to support students'' awareness of structures underpinning figural patterns by providing scaffolds to help them express these structures and generate rules for them. I will then focus on the functionalities introduced to facilitate the sharing of constructions and rules and how they have been used in classroom trials with students aged from 12 14years. In particular the students were used these tools to assess together the equivalence of symbolic rules, as a precursor to algebraic manipulation. Second, I will turn to the work of the National Centre for Excellence in the Teaching of Mathematics, which I have directed since 2007. The Centre provides a national infrastructure to support the professional development of teachers of mathematics across all phases and in all schools and colleges in England. In particular, the Centre encourages schools and colleges to learn from their own best practice and has developed tools to assist collaboration and sharing of good practice locally, regionally and nationally through interactions on the NCETM portal, www.ncetm.org.uk (augmented by facetoface interactions). Again I will illustrate some of the tools that have been put in place, how they have been used and to what effect.
One Computer in the Classroom – some experiences of embedding ICT into teaching and learning mathematics direct from the ‘chalk face’
Alan Catley
alan@catley.org
Tyne Metropolitan College; NCETM
10 years after making a start at adapting my teaching style in an effort to make constructive use of the interactive whiteboard that was installed into my classroom is a good time to reflect on what has worked well. This talk will include examples from all areas of the (UK) secondary school curriculum that have proved extremely successful in motivating learning and improving both understanding and examination performance of my students of all levels of ability. The key lesson for me to learn at the outset was to become far less of a ‘teacher’ and adopt more of a ‘facilitator of learning’ role. I will draw on experiences of using carefully selected resources, including Autograph, Excel etc. and illustrate the benefits of the ‘what if . . . ?’ approach to learning new concepts as opposed to ‘show and tell’.
Possible gain of IT in problemoriented learning environments from the viewpoint of history of mathematics and modern learning theories
Bernd Zimmermann  Torsten Fritzlar  Lenni Haapasalo  et al.
bernd.zimmermann@unijena.de
University of Jena; University of HalleWittenberg; University of Eastern Finland; Technical University of Braunschweig
In the first part we will outline that in history of mathematics eight activities proved to be fundamental for generating new mathematical knowledge. They can be taken as a framework for scaffolding mathematical learning environments in classrooms of today. By this, modern learning theories about constructivism as well as procedural and conceptual learning could be augmented and enriched. In the second part we will demonstrate by some mathematical examples for the middle and upper grades of high school the use of technology which might help to foster productive problem solving and thought processes. Furthermore ideas for a new computer based tool for measuring mathematical problem solving abilities in a PISAlike test are described which simulates some aspects of oral examinations. Finally we try to highlight in which way a computersimulation of a mathematical lesson might help preservice teachers to improve their abilities to teach mathematical problem solving.
Polyhedral tensegrity structures
Bogumila KlempDyczek
klemp@mat.uni.torun.pl
PTM
The concept of tensegrity structures is relatively new although with no name it appeared for ages in weaving, in mechanics, in architecture and in constructions. Living creatures, the most perfect systems and mechanisms in the nature are also examples of tensegrity structures. When external forces act on such structures they always tend to and regain the equilibrium state unless the forces destroy them. If wind acts on stems of cane, they bend at first but straighten up later. If an earthquake acts on some constructions they temporarily change their shape, though later came back to their equilibrium state and remain undistroyed . Some constructions, even they might appear as very strong, perish in a few seconds. What is the scientific model that allows to distinguish between them? We understand them as systems built from bars, cables and springs, in which tensional and compressive forces are distributed and balanced within the structure. Here we deal with polyhedral tensegrity structures. They allow to discover interesting relationships between distinct Platonic and Archimedean polyhedra. In particular, the concepts of rihgt and left handedness of such polyhedra are investigated. The other aspect refers to the concept of flexible polyhedra, which in fact gave impact for mathematical investigations of tensegrity structures.
Questions Answers; and Mathematical Discourse
GT Springer
gt.springer@hp.com
HewlettPackard
Technology has often been accused of just giving students answers without the students knowing why. In this talk, we examine problems in which this is a very good thing. Specifically, we discuss problems in which technology provides the answer very quickly. At that point, the initial conditions are changed but the answer remains the same! As a result, the answer becomes less important than figuring out why the answer is not changing as conditions change. What will emerge from this pattern is that these problems require students to focus on proof and reasoning, and engage in mathematical discourse.
Fundamental Theorem of Calculus and Computations on Some Special HenstockKurzweil Integrals
WeiChi Yang  Xiaofeng DING
wyang@radford.edu
Radford University; College of Computer and Information Engineering; Hohai University
The constructive definition usually begins with a function f, then by the process of using Riemann sums and limits, we arrive the definition of the integral of f over an interval [a,b]. On the other hand, a descriptive definition starts with a primitive function F satisfying certain condition(s) such as FŒ=f and F beining absolutely continuous if f is Lebesgue integrable and F is generalized absolutely continuous if f is Henstock Kurzewil integrable. For descriptive integrals, the deficiency is that we need to know primitive F such that FŒ=f (and satisfying some properties). For constructive integration, we need a good computation scheme to handle the Riemann sum using an uneven partition to get a broader family functions which will include some Henstock integrable functions. In this paper, we describe how we can start with the constructive definition and reach a description definition for some improper integrable functions and highly oscillating nonabsolute functions.
Does the software matter?
JeanMarie Laborde
jeanmarie.Laborde@cabri.com
Cabrilog; Univ. Joseph Fourier  France; CNRS France
Since the turn of the 80s, Computer Aided Instruction has been around and according to conservative estimates CAI industry is worth at least 40 billion euros worldwide. At the same time the actual use of Technology Enhanced Learning (another way of speaking of CAI) is still far from being dominant in most of the classrooms, especially math classrooms. Nevertheless several international studies tend to prove the high efficiency of computers in the classroom. This presentation will address various aspects, especially the one expressed in the title, of such amazing apparent contradiction. Among other examples and casestudies the paradigmatic example of Cabri technology, including the new one for Elementary Mathematics, will be considered.
Students find the concepts of variables and functions in algebra very challenging. Part of the challenge arises from the use of static media, and part arises from insufficient emphasis on how variables change and functions behave. A geometric approach provides students with experiences in which they manipulate variables directly and continuously, and in which they observe, directly and immediately, the behavior of functions as they vary the independent variables. Sketchpad® dynamic mathematics software makes it easy for students to create geometric variables and functions and to use them to study domain, range, composition, and inverses. Through these experiences, students can more easily progress from an atomic view of function (taking a single input variable to a single output variable) to a collective view (mapping an entire domain to an entire range).
Bilinear Transformations via a Computer Algebra System
Tilak dealwis
talwis@selu.edu
Southeastern Louisiana University
In this paper, we will investigate several properties of bilinear transformations using a computer algebra system. Bilinear transformations, also known as linear fractional transformations or Möbius transformations belong to a wider class of functions known as conformal mappings in complex analysis. It is wellknown that bilinear transformations carry a circle or a straight line in one complex plane to a circle or a straight line in another complex plane. Due to the graphical nature of these transformations, a computer algebra system such as Mathematica is an ideal tool to further study their properties. In the process we will observe several new nonstandard results, which can be proved using traditional methods without resorting to any computer algebra system. The paper uses Mathematica version 7.0 on a Windows XP platform, but identical results can be obtained by using any other computer algebra system of reader’s choice.
Does Technology Empower Students to do Research at the Secondary and Undergraduate Levels?
Antonio R. Quesada
aquesada@uakron.edu
The University of Akron; Department of Mathematics
Traditionally, most students at the undergraduate, not to mention at the secondary level, are rarely exposed to any research in mathematics. This is mostly due to their lack of mathematical background and the time involved in such activity, in an already crowded curriculum. However, we live in a time when information is becoming readily available, and when basic technology such as handheld graphing technology (HHGT) allows bridging over cumbersome calculations, and facilitates the access to a variety of new and relevant topics at basic levels. It would seem reasonable to expect that the integration of these innovations is followed by an increased focus on conceptual understanding, applications, and exploration in the mathematics classroom. But, is it reasonable to expect that regular students at these levels will get involved in research and discovery? In this article, we will show some new mathematical results obtained by mostly secondary students empowered by technology. We will also look at some basic conditions that we can adopt in our teaching that seem to foster students’ exploration and discovery in mathematics. In addition we will show two new results obtained using multiple representations illustrating the capability that the latest handheld graphing technology provides for linking in every platform, a variable defined in one of them.
