Atcm 2014 proceedings

Innovation and Technology for
Mathematics Education
Proceedings of the Nineteenth
Asian Technology Conference in Mathematics
November 26-30, 2014,
State University of Yogyakarta,
Yogyakarta, Indonesia

Editors
Wei-Chi Yang, Radford University, USA
Miroslaw Majewski, New York Institute of Technology, UAE
Tilak de Alwis, Southeastern Louisiana University, USA
Wahyudi Wahyudi, SEAMEO QITEP in Mathematics Indonesia
Published by Mathematics and Technology, LLC (http://mathandtech.org)
ISBN: 978-0-9821164-8-7 (hard copy)
ISSN 1940-2279 (CD)
ISSN 1940-4204 (online version)

III
Introduction
On the behalf of the State University of Yogyakarta, Yogyakarta, Indonesia, members of the International
Program Committee of the ATCM conference, we are honored to introduce the papers of the ATCM 2014 –
“Innovation and Technology for Mathematics Education”.
We are fortunate to have presenters from a wide spectrum of scientists and educators, whose presentations
and workshops will demonstrate the most current trends in technology for mathematics and mathematics with
technology. Papers and presentations address a very wide spectrum of topics and ideas. We can find papers
concentrating on using computer software in teaching mathematics, papers on using Internet, multimedia, and
other tools for interactive and online mathematics courses delivery, as well as research papers from pure
mathematics where technology was used to produce some new results.
Thanks to evolving technological tools, we are able to explore more interdisciplinary areas such as science,
technology, and engineering with Mathematics which we could not before. Therefore, integrating technology
into mathematics teaching, learning and research will definitely allow us to expand our knowledge horizon in
mathematics. We encourage all authors and readers to contribute your new findings to our next ATCM or the
Electronic Journal of Mathematics and Technology (eJMT: https://php.radford.edu/~ejmt/).
We would like to express our appreciation to the local organizers, in particular to the State University of
Yogyakarta for the enormous task of planning and preparation of ATCM 2014 – one of the most enjoyable and
instructive conferences in the World. We thank also the members of the International Program Committee and
external reviewers for their great contribution in reviewing papers.
Editors of ATCM 2014
Personal note
I have been involved in organization of ATCM conference and ATCM matters for quite a while. Since September
2013 I started my retirement hoping that I will have more time for freelance writing and doing research. Very
soon I found myself even more busy than while having a regular employment. I do no longer teach regular
mathematics courses. However, I started teaching courses related to Islamic mathematics and Islamic
geometric patterns, conducting workshops and attending professional meetings. My cooperation with Istanbul
Design Center under umbrella of ENSAR Foundation and Turkish Ministry of Culture turned into a long term
research project on Islamic geometric ornaments in Turkey. This involves frequent trips to various parts of
Turkey and spending a lot of time in libraries, mosques and old madrasa buildings.
After the last workshop at IDC in September 2014 I realized that I am not able to copy with all my duties and
interests. Thus I have to drop something from my back in order to do the remaining tasks appropriately and
enjoy doing them. As I do no longer teach mathematics, I also do not use much of technology –Geometer’s
Sketchpad, Inkscape and sometimes SketchUp are all what I need, therefore I decided to withdraw from my
ATCM duties and involvement. I hope that somebody else will take my duties and have a joy doing them.
I wish you enjoyable ATCM conferences, I wish you luck in transforming mathematics education in all Asian
countries and having fun with technology.
Mirek Majewski

IV
International Program Committee (IPC)
International Program Co-Chairs
Tilak de ALWIS , Southeastern Louisiana University, U.S.A.
Miroslaw MAJEWSKI , New York Institute of Technology, United Arab Emirates
Wei-Chi YANG , Radford University, U.S.A.
Members of the International Program Committee
Paul ABBOTT , University of Western Australia, Australia
Keng Cheng ANG , NIE/ Nanyang Technological University, Singapore
Yiming CAO Beijing Normal University, China
Jen-Chung CHUAN , National Tsing Hua University, Taiwan
Jean-Jacques DAHAN , Paul Sabatier University Toulouse France
Ma. Louise Antonette De Las Penas , Ateneo De Manila University, Philippines
Hongguang FU , University of Electronic Science and Technology-Chengdu (UESTC), China
Lenni HAAPASALO , University of Joensuu, Finland
Masami ISODA , University of Tsukuba, Japan
Erol KARAKIRIK, Abant İzzet Baysal University, Turkey
Matthias KAWSKI , Arizona State University, U.S.A.
Krongthong KHAIRIREE , Suan Sunandha Rajabhat University, Thailand
Barry KISSANE , Murdoch University, Australia
Ichiro KOBAYASHI , International Educational Software, Japan
Hee-chan LEW , Korea National University of Education, South Korea
Alasdair McANDREW , Victoria University, Australia
Douglas MEADE , The University of South Carolina, U.S.A.
Eva Milková , University of Hradec Králové, Czech Republic
Vladimir Nodelman , Holon Institute of Technology, Israel
Antonio R. QUESADA , The University of Akron, U.S.A.
Inder K RANA , Indian Institute of Technology, Powai, India
Tadashi TAKAHASHI , Konan University, Japan
Yuan YUAN , Chung Yuan University, Taiwan
Bernd ZIMMERMANN , Friedrich Schiller University, Germany
Local Organizing Committees
Local Steering Committee
1. Prof. Dr. Rochmat Wahab, M.Pd (Rector of Yogyakarta State University)
2. Prof. Dr. Syawal Gultom, M.Pd (Head of Board of Human Resources Development in Education and Culture
and Education Quality Assurance, Ministry of Education and Culture Indonesia)
3. Prof. Dr. Soewarsih Madya (Vice Rector for International Cooperation, Yogyakarta State University)
4. Prof. Subanar, Ph.D (Director of SEAMEO QITEP in Mathematics Indonesia)
5. Prof. Dr.rer.nat. Widodo, M.S. (Head of PPPPTK Matematika Indonesia)
6. Prof. Dr. Marsigit, M.A. (Mathematics Education Department Yogyakarta State University)
7. Dr. Hartono (Dean of Faculty of Mathematics and Natural Sciences Yogyakarta State University Indonesia)
Local Organizing Committee
1. Dr. Wahyudi (Chair: wahyudiw@yahoo.com) , SEAMEO QITEP in Mathematics Indonesia
2. Punang Amari Puja, SEAMEO QITEP in Mathematics Indonesia
3. Dr. Sugiman, Department of Mathematics Education, Yogyakarta State University
4. Sumaryanto, M.Pd, PPPPTK Matematika Indonesia
5. Sahid, M.Sc, SEAMEO QITEP in Mathematics Indonesia
6. Dr. Ali Mahmudi, Department of Mathematics Education, Yogyakarta State University

V
Advisory Committee
Bruno BUCHBERGER, LINZ, Austria
Peng Yee LEE, NIE/ Nanyang Technological University, Singapore
Lu YANG, Guangzhou University, China
External Reviewers
Alasdair McAndrew
Barry Kissane
Chong Chee Keong
Douglas Meade
Chew Cheng Meng
Gary Fitz-Gerald
Gerard De Las Penas
Greg Oates
Hailiza Kamarulhaili
Jean-Jacques Dahan
Jen-Chung Chuan
Jozef Hvorecky
Krongthong Khairiree
Lila Roberts
Luisito Lacatan
Ma. Louise Antonette De Las Penas
Maria Ailynn Diansuy
Matthias Kawski
Miroslaw Majewski
Noor Hasnah Moin
Tilak dealwis
Wei-Chi Yang
Yashpau Chugh
Zacharie M. Baitiga

VI
Table of Contents
Invited and Plenary Papers
1. Bill Blyth, Teaching Experimental Mathematics: Digital Discovery using Maple _________________________1
2. Wei-Chi YANG, Technological tools have enhanced our teaching, learning and doing mathematics, what is
next? ___________________________________________________________________________________16
3. Keng-Cheng Ang, Liang-Soon Tan, Professional Development for Teachers in Mathematical Modelling _____33
4. Antonio R. Quesada, A Capstone Course to Improve the Preparation of Mathematics Teachers on the
Integration of Technology___________________________________________________________________43
5. Colette Laborde, Interactivity and flexibility exemplified with Cabri _________________________________51
6. Paulina Pannen, Integrating Technology in Teaching and Learning Mathematics ______________________62
7. Marsigit, Re-conceptualizing Good Practice of Mathematics Teaching Through Lesson Study in Indonesia __77
8. Leong Chee Kin, Educating the Educators: Technology-Enhanced Mathematics Teaching and Learning ____85
9. Hongguang Fu, Xiuqin Zhong, Zhong Liu, Mathematics Intelligent Learning Environment ________________94
10. Widodo, Muh.Tamimuddin H., Three Training Strategies for Improving Mathematics Teacher Competences
in Indonesia 2015-2019 based on Teacher Competency Test (TCT) 2012-2014 ________________________104
Regular Papers
1. Yoichi Maeda, Visualization of special orthogonal group SO(3) with dynamic geometry software ________119
2. Glenn R. Laigo, Abdul Hadi Bhatti, Lakshmi Kameswari P., Haftamu Menker GebreYohannes, Revisiting
Geometric Construction using Geogebra _____________________________________________________124
3. Jean-Jacques Dahan, Examples and Techniques of Morphing within CAS and DGS Environments (Cabri
and TI-Nspire) A Way of Enriching our Teaching at all Levels _____________________________________132
4. Chin Kok Fui, Sharifah Norul Akmar bt Syed Zamri, Effects of spreadsheet towards learners’ usage of
mathematical language ___________________________________________________________________142
5. Al Jupri, Paul Drijvers, Marja van den Heuvel-Panhuizen, The impact of a technology-rich intervention
on grade 7 students’ skills in initial algebra ___________________________________________________152
6. Maria Isabel T. Lucas, Erlinda A. Cayao, Effects of using Casio FX991 ES PLUS on achievement and
anxiety level in Mathematics ______________________________________________________________162
7. Rebecca C. Tolentino1 and Janette C. Lagos, Integration of Products using Differentials _______________170
8. Chee-Keong Chong, Marzita Puteh, Swee-Choo Goh, Use of Lecture Capture in the Teaching and
Learning of Statistics _____________________________________________________________________179
9. Tsutomu ISHII, Consideration on the Effect of the Lesson in Problem Solving by Few Children ___________187
10. Ryoji Fukuda, Masato Kojo, Non-visual Expression Method for Mathematical Documents in Elementary
Geometry _____________________________________________________________________________195
11. Ryoji Fukuda, Junki Iwagami, Takeshi Saitoh, Applicability of Gaze Points for Analyzing Priorities of
Explanatory Elements in Mathematical Documents ____________________________________________203
12. Barry Kissane & Marian Kemp, A model for the educational role of calculators _______________________211
13. Yang Jianyi, Liu Chengyang, A Practical Case for e-Mathematical Experiment with “Geometry Apps”
from HP Prime __________________________________________________________________________221
14. Ren-shou Huang, Yuan-jing Xia, Zhen-xin Yang, Ling Yan, Applications of Information Technology to the
"Five Points" Conjecture __________________________________________________________________229

Teaching Experimental Mathematics: Digital Discovery
using Maple
Bill Blyth 1, 2
bill.blyth@rmit.edu.au
1
Australian Scientific & Engineering Solutions (ASES)
2
School of Mathematical and Geospatial Sciences
RMIT University
Australia
Abstract
One of the “Recommendations (by Klaus Peters) on Jon Borwein’s LinkedIn page
starts with: “Jonathan Borwein is a world-renowned research mathematician and one
of the leaders in the emerging field of experimental mathematics to which he has made
many original contributions. He has written several books that have helped to define the
field. Examples include finding closed forms of integrals by numerically computing high
precision approximations (accurate to hundreds of decimal digits) which is used to identify
the answer. Knowing the correct answer facilitates the proof of its correctness.
Here this experimental mathematics approach is taken with a couple of traditional
“find the maximum . . . ” problems suitable for senior school or first year undergraduate
students (who could even be pre-calculus students). We use the Computer Algebra Sys-
tem, Maple, for experimentation (and visualization) to compute high accuracy numerical
approximations to the answer. Maples identify( ) command is used to identify the exact
answer. Knowing the answer, we prove that it is correct - without using calculus. We
use small group collaborative learning in the computer laboratory, so we parameterize the
problem and recommend the use of Computer Aided Assessment (such as provided by the
package MapleTA).
Students engage with the visualization, are active learners with deep learning of the
concept of maximum and gain a gentle introduction to the important new field of exper-
imental mathematics.
1 Introduction
Since the mid 1990s, we have often stated our view that all mathematics graduates from uni-
versities should be expert users of one of the two professional level Computer Algebra Systems,
CAS: Maple being one of these. A recent study [9] of UK university students reported “that
59% of students find their lectures boring half the time and 30% find most or all of their lec-
tures boring.” A range of different teaching methods were investigated: the most boring were
lab sessions; computer sessions ranked as the second most boring. “Computer sessions too have
PROCEEDINGS OF THE 19th ASIAN TECHNOLOGY CONFERENCE IN MATHEMATICS
Page 1 of 238

the potential to be stimulating or tedious; the findings of this study suggest that too many fall
into the later category. This could be due to the manner in which sessions are conducted (e.g.
are the computer tasks relevant and interesting?), . . . etc.”
These findings are not surprising given that most courses are teacher centric and use a tra-
ditional teaching approach (lectures) with “tutorials” and assignments which are mostly done
without use of technology. When technology such as CAS is used (at both school and under-
graduate levels), it is often a minor supplement where the usual “by hand” exercises completed
by using CAS following the “by hand” approach. We suggest that innovative teaching materi-
als which exploit the strengths of the CAS for tightly integrated e-Learning and e-Assessment,
such in the Maple component of our first year course (and in higher year courses taught entirely
using Maple), and a pedagogy of collaborative learning in small groups as active learners in a
computer lab need to replace (at least in part) boring lectures and boring computer lab sessions.
In this paper we consider a standard calculus problem: find which cylinder inscribed in
a unit sphere has the maximum volume. We provide visualization and show how to obtain
an accurate decimal approximation to the radius that gives the maximum. This leads to an
identification of the possible exact answer. By foregoing calculus, we give a gentle introduction
to experimental mathematics: we provide several algebraic proofs that introduce important
ideas in mathematics, such as approximation in the small. We also support our intuitive notion
that the maximum of a graph of a function “looks like a parabola” near the maximum by
algebraically calculating the quadratic approximation of the function near its maximum.
This paper is organized as follows. Section 2 introduces experimental mathematics. Sec-
tion 3 has a brief overview of first semester, first year Maple sessions which provide a suitable
preparation to undertake the Cylinder in a Sphere problem described here. Section 4 pro-
vides a description of the Cylinder in a Sphere problem and visualization (including multiple
representation and animations). Section 5 zooms in on the plot of volume versus radius of
the cylinder to approximate the r that maximizes the volume; and a possible exact answer is
identified. Several proofs without using calculus are discussed: in Section 6, volume squared is
maximized; in Section 7, volume is maximized. Several more proofs are indicated in Section 8.
Section 9 introduces a parameterized problem (of a Cylinder in a Spheroid) and discusses
e-Marking and Computer Aided Assessment. This is followed by a brief Conclusion.
2 Experimental Mathematics
Jon Borwein is a pre-eminent research mathematican who is one of the leaders in developing
and popularizing experimental mathematics. We suggest that every aspiring or practising
mathematician should read some of his work: we particularly recommend the book [6] and the
short article [7]. In the later, Borwein shares Polya’s view on intuition:
Intuition comes to us much earlier and with much less outside influence than formal
arguments . . . Therefore, I think that in teaching high school age youngsters we
should emphasize intuitive insight more than, and long before, deductive reasoning.
George Polya (1887–1985) [10], p.128
Borwein continued with
[Polya did go] on to reaffirm, nonetheless, that proof should certainly be taught in
school. [ . . . ] My musings focus on the changing nature of mathematical knowledge
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and in consequence ask questions such as How do we come to believe and trust
pieces of mathematics?, Why do we wish to prove things? and How do we teach
what and why to students?. [7]
In our experience with our own mathematics undergraduate and PhD students is that they
have little appreciation of intuition in mathematics: my discussions about the importance of
intuition (and the development of it) come as a surprise to them.
Borwein [7] gives an informal summary of what is meant by experimental mathematics:
• Gaining insight and intuition;
• Discovering new relationships;
• Visualising math principles;
• Testing and especially falsifying conjectures;
• Exploring a possible result to see if it merits formal proof;
• Suggesting approaches for formal proof;
• Computing replacing lengthy hand derivations;
• Confirming analytically derived results.
Another eminent mathematician, Doran Zeilberger [11] states that
The purpose of mathematical research should be the increase of mathematical
knowledge, broadly defined. We should not be tied up with the antiquated notions
of alleged “rigor”. A new philosophy of and attitude toward mathematics is devel-
oping, called “experimental math” (though it is derided by most of my colleagues;
I often hear the phrase, “Its only experimental math”). Experimental math should
trickle down to all levels of education, from professional math meetings, via grad
school, all the way to kindergarten. Should that happen, Wigners “unreasonable
effectiveness of math in science” would be all the more effective!
Lets start right now! A modest beginning would be to have every math major
undergrad take a course in experimental mathematics.
We agree, but note that many senior school mathematics courses use CAS so we suggest
“the beginning” should sensibly occur in the secondary school.
3 Preliminaries
For about a decade, we designed and ran Maple topics for our first year university “Calculus”
course comprised of four traditional lectures and one Maple lab session each week. The weekly
lab sessions had no lectures: students work collaboratively in small groups. For the first
semester, the topics were secondary school level mathematics. Students start by taking two
sessions to work through an Introduction to Maple which includes a simple assignment that is
printed and submitted as a hardcopy, but all other work is completely electronic. An Animation
Page 3 of 238

worksheet is followed by a major Animation Assignment: students pick a project (from a list)
that is on a school level topic, see [1].
After these activities, students are expert at preparing plots and animations, so these are
recurring themes. The next activity is Spot the Curve. The Maple file Spot the Curve is
accompanied by another Maple binary file which contains all the procedures for randomly
generating the curves, their translated curves and automatic marking procedures: this binary
file is read into the student Spot the Curve file so that all of the required procedures are invisible
but active in the student file. Each student group is presented with several plots of a curve
and a translated copy of that curve: the task is to identify what the translation is (by using
plots and animations). Students use the provided automatic marking procedure to immediately
mark their result. Feedback shows that students really enjoy this activity: for details, see [2].
Another style of auto marked problem is the FishPond problem, a straightforward applica-
tion of using the trapezoidal rule to approximate area. It has multiple steps with numerical
answers: the marking procedure is used by students to immediately mark their work: for de-
tails, see [2]. The FishPond problem is followed by working through a Maple file of a structured
Polya approach to “finding the maximum area”, followed by an assignment on finding the max-
imum area of the Norman Window problem. For a discussion of the Norman Window problem,
multiple representations, e-Marking and Computer Assisted Assessment, see [3]. With this (or
similar) background, students are prepared to investigate various solution methods (proofs) of
a particular related problem, see [5].
These activities provide students with an appropriate background to undertake the maxi-
mum problem(s) described next.
4 The Cylinder of Maximum Volume inscribed in a unit
Sphere
This problem is a classic one that appears in many textbooks. We introduce the experimental
mathematics methodology by providing a Maple worksheet for student groups to work through.
[After completing this, they are presented with a parameterized version as an assignment.]
The Problem: For a cylinder inscribed in a sphere of radius 1, find the exact radius of the
cylinder which has the maximum possible volume.
For this problem, visualization of the unit sphere with an inscribed cylinder is wanted, but
students start with plots in two dimensions and should do a lot of work with 2D plots before
using plots in three dimensions (other than simplistic plots such as straightforward plots of
z = f(x, y)). Since the details on how to produce the 3D plot are not relevant to the problem
being addressed here, we recommend that the 3D plot is provided in the worksheet. It is even
better to provide an animation as the radius of the cylinder, r, varies from 0 to 1: see Fig. 1
for one frame of such an animation.
Note that a cut-out of the sphere is used so that the inscribed cylinder is easily visualized.
Some understanding of spherical polar coordinates is needed for this: we introduce parameti-
zation of surfaces properly in a university third year course on Geometry of Surfaces (taught
entirely in the computer lab using Maple). Students work through a worksheet on Quadric
Surfaces. We introduce cut-outs of 3D objects in our course on Vector Calculus (in second or
third year, also entirely in the computer lab using Maple). The use of a cut-out is very useful
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Figure 1: A cylinder inscribed in the unit sphere (with cut-out).
for visualization of “slicing” the 3D object so that triple integrals can be correctly set up. We
do not advocate that students at high school or first year university be required to do this.
However we always provide the Maple code since some students do want to experiment with
the code. This is easy in Maple: the plot or animation should be done in a Table in which the
code is hidden. Students who are interested can un-hide the code. Since graphics and anima-
tions use a lot of memory, all of the output should be removed for storage and transmission
of the Maple file (greatly reducing the file size). The user can then execute the whole file or
sections when working on the file.
Another “common” way of visualizing the cylinder inscribed in the sphere is to use trans-
parency (for the sphere). This is easier than using a cut-out and so this is recommended for less
experienced students. We have not included a plot in this paper since it does not transfer well
to paper, but it is adequate on a computer monitor. It is relatively easy since Maple provides
a package called plottools which plots objects such as spheres and cylinders (without the user
needing to know about parameterizations of these surfaces). Using provided Maple commands,
the sphere can be obtained (using the default with centre at the origin and unit radius)
[> with(plottools): with(plots):
[> sph := sphere( style=patchnogrid, transparency=0.8):
This specifies the options that the grid lines are removed and a (high) transparency is used.
To define an inscribed cylinder, we need to calculate the height of the cylinder that has a
given radius. It is easy to use right angle triangles to show (see below) that for r = 11/16
the half height is hh = 3
√
15/16. The coordinates of the centre of the base of the cylinder is
[0, 0, −hh], the radius is 11/16 and the height is 2 hh, so the cylinder is defined by
[> cyl:=cylinder([0, 0, −hh], 11/16, 2*hh, strips=100, style=patchnogrid):
To make the cylinder’s curved surface looks smooth, the number of vertical strips has been in-
creased from the default of 24 to 100 by using the option strips=100. The plot seems to be more
attractive if the gridlines are also removed from the cylinder, so the option style=patchnogrid
have also be used for the cylinder. The partially transparent sphere and the cylinder are dis-
played together by
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Figure 2: One frame of the animation of the multiple representation of the 3D plot of the sphere
and the cylinder and the plot of the volume of the cylinder. The radius of the cylinder is the
animation parameter.
[> display(sph, cyl, scaling=constrained, axes=none, orientation=[50,75]);
To give a slightly better view, the orientation of the displayed plot has been changed from
the default. Note that the orientation can be changed interactively by using the mouse on
the displayed plot. Students do not need much encouragement to do this and so engage as
active learners which is much to be preferred to the lecturer just showing the visualization as
a demonstration in a lecture.
Once students have experimented with the animation of the 3D plot, they have established
their visualization of the problem. We want to help them to further develop some intuition
about the problem so we provide a multiple representation where the 3D plot has placed
alongside the plot of the corresponding Volume (of the inscribed cylinder) for the given radius
- as an animation where the cylinder’s radius is the animation parameter. See Fig. 2 for one
frame of such an animation.
First year students who have followed the type of sequence of Maple lab sessions on anima-
tion; the Polya approach to problems of finding maximum area; an “Advanced” Polya approach
to finding maximium area (including multiple representations), see [3], could produce multiple
representation animation (using transparency for the 3D plots). However for less experienced
Maple users, we recommend providing this animation.
Of course, the formula for the volume of the inscribed cylinder as a function of its radius
has not yet been derived: this is done in the next section.
5 Volume of the Inscribed Cylinder and Zooming In
A cylinder of radius r is inscribed inside a sphere of radius R=1. To derive a formula for the
volume of the cylinder, we plot a cross section obtained by cutting the sphere and the inscribed
(vertical) cylinder by a plane which includes the z axis, see Fig. 3. This plot is straightforward.
An even easier alternative is to use Maple’s Drawing tools: insert a Canvas and use the tools
Page 6 of 238

Figure 3: A cross section obtained by cutting the sphere and the inscribed (vertical) cylinder
by a plane which includes the z axis.
for lines and curves. Use freehand drawing to put in arrows and variables - this is particularly
easy using a stylus (or pen) on a touch screen (but harder just using a mouse!).
From the plot it is easy to see that r and the half height h of the cylinder form sides of a
right angle triangle with hypotenuse 1 (for a unit sphere). Thus Volume of the cylinder is area
of the circular base times the height, so
V = πr2
× 2 h2
= 2π r2
√
1 − r2 .
This is the volume as a function of r which was used to generate the plot in the right hand
side of Fig. 2. The curve is skewed to the right, clearly the maximum is for r somewhere
between 0.7 and 0.9 so we can obtain a better approximation by zooming in, see the plot on
the left in Fig. 4.
Students at secondary schools in the State of Victoria in Australia have been using graphics
calculators (and then CAS calculators) for about two decades, so students are very familiar with
the idea of zooming in. This is a very useful skill, but is hardly ever used at university level
(except for education studies) since calculators are hardly ever used in science and engineering
courses.
From (the plot on the left in) Fig. 4, near the maximum, the curve “looks like a parabola”
with a maximum (read off the graph - place cursor on plot and right click - choose Probe Info,
Nearest point on line - position the cross hairs and read off the point’s coordinates) of about
0.816 or 0.817, say?
With three more zoom ins, we estimate that r is about 0.8164967. This is probably accurate
enough, but (if needed) we could get higher acuracy by reading the graph better to see where
the maximum is. Thus we zoom in again as well as choose a view window with restricted “y”
values (we experiment with the y values to use):
[> rL, rR:=0.8164965, 0.8164968:
[> plot(V, r=rL .. rR, view=[rL .. rR, 2.4183991523122 .. 2.4183991523123] );
The plot is displayed on the right in Fig. 4. Hence the approximate answer is r = 0.81649658
accurate to about the last digit!
To try to identify the (possible) exact answer from the numeric answer by estimation from
the graph:
Page 7 of 238

Figure 4: A zoomed in plot of the volume of the cylinder on the left; and a four times zoomed
in plot on the right.
[> identify(0.8165);
This gives the result 1
3
√
6. Since the exact ans is
q
2
3
, this has given the correct identification
of the exact answer. (We prove that this is the correct result below.)
5.1 Comparison of Maple and Mathematica
Maple and Mathematica are the only professional level CAS. For identification of numbers from
floating form numbers (possibly of very high precision), the identify( ) command in Maple is
very superior to the limited functionality provided in Mathematica. The Mathematica (version
8) commands are Rationalize[ ] and RootApproximate[ ] with option RootApproximant[x,n]
which “finds an algebraic number of degree at most n that approximates x.” Note that “For
degrees above 2, RootApproximant generates Root objects.”
Using Mathematica 8,
[> RootApproximate[0.8165] gives 1633/2000,
[> RootApproximate[0.816496] gives 51031/62500 and
[> RootApproximate[0.816497] gives the correct root.
Using Maple 16,
[> identify(0.8165); gives the correct result (see above).
Also, Maple correctly identifies nearby approximations: 0.81649, 0.81651 and 0.816496 are all
correctly identified.
For this simple problem, Mathematica could be used, but we note that higher precision
is required for the approximation. Since the approximation is obtained by graphical means,
this is a disadvantage. However Mathematica is not capable of identification of transcendental
numbers. For example, 1 + e ≈ 3.718281828.
In Maple: [> identify(3.718); gives the correct result of 1 + e.
With Mathematica,
[> RootApproximate[3.718] gives 1859/500 and
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[> RootApproximate[3.718282] returns its answer as the third root of a degree 9 polynomial!
Mathematica cannot identify non algebraic numbers.
Research using an experimental mathematics methodology may need to compute an ap-
proximation accurate to many hundreds of decimal digits to identify a possible exact answer
(which might be transcendental), which we then seek to prove is correct, see [6, 7].
6 Proof without Calculus: maximize Volume Squared
Now that we have the (possible) exact answer, we prove that it is the correct answer. By not
using calculus, we engage with the experimental mathematics methodology and connect several
topics in mathematics as well as enhance our understanding and visualization of how functions
behave and are approximated. The objective function is Volume, V , as a function of the radius,
r, where V = 2πr
p
(1−r2
). The presence of the square root complicates the mathematics, but
we notice that when V is a maximum, then V 2
is also a maximum. Thus we work with V 2
:
[> V Sq:= 4π2
r2
(1 − r2
);
Now let r =
p
2/3 + δ and we postulate that δ = 0 gives the maximum V 2
. Note that the
graph of V 2
(δ) corresponds to a horizontal translation of the graph of V 2
(r). We have V 2
(δ):
[> V Sq delta:=subs(r=sqrt(2/3)+delta, V Sq); which expands to
V Sq delta = 4π2

4/27 − (8/3)δ2
− (28/9)
p
(6)δ3
− 9δ4
− 2
p
(6)δ5
− δ6

.
This is a sixth degree polynomial. How does it behave near the origin (that is, for δ small)?
Algebraically we know that if δ is small (in magnitude) then δ2
is smaller (than |δ|). Similarly
|δ3
| is even smaller. Thus, for δ small, then δ2
is small (compared to δ, and δ3
is small (compared
to δ2
), etc. Thus we know that, for δ small, V 2
can be approximated by neglecting all terms
with powers of δ higher than δ2
: we call this approximation V Sq delta 2.
Maple has a command, coeff( ), to give the coefficients of a polynomial, so it is easy to
obtain V Sq delta 2:
V Sq delta 2 =
16
27
π2
−
32
3
π2
δ2
.
This is a quadratic polynomial with a maximum at δ = 0 which proves our result. The maximum
volume is also easy to obtain: take the square root of the constant term in V Sq delta 2 to
give 4/9
√
3 π.
Note that we can rewrite the quadratic approximation to V 2
in terms of r by translating with
the substitution of δ = r −
p
2/3. The plot of the sixth degree polynomial V 2
(r) is displayed
with the quadratic approximation (as derived above) on the left of Fig. 5. This visualization
reinforces our intuition that the graph of a function “looks like a parabola” near the function’s
maximum.
7 Proof without Calculus: maximize Volume
We now directly show that our value for r does give the maximum of the function V =
2πr
p
(1 − r2
). As mentioned in the previous section, the presence of the square root com-
plicates the mathematics, but we indicate how this can be done. Again, we postulate that
Page 9 of 238

Figure 5: On the left, a plot of volume squared of the cylinder and its quadratic approximation
(about the maximum) and, on the right, a plot of volume and its quadratic approximation.
rMax =
p
2/3 maximizes the volume. We let r = rMax + δ (initially leaving rMax as unde-
fined to avoid the algebra looking too messy) and we postulate that δ = 0 gives the maximum
V . Note that the graph of V (δ) corresponds to a horizontal translation of the graph of V (r).
We have:
V (δ) = 2π(rMax + δ)2
p
1 − (rMax + δ)2
For a maximum, the curve is approximately a parabola for δ small. So we need to approximate
the square root for δ small (up to powers of 2). Now the generalized Binomial Theorem for a
power of 1/2 where |x/a| 1 is
p
1 − x/a = 1 −
1
2a
x −
1
8a2
x2
+ O(x3
)
where O(x3
) means terms with x3
and higher powers. This is easily proven by algebra: let
√
1 − x = a + bx + cx2
+ · · · , square, and hence result (by equating coefficients of each power
of x).
Exercise: Prove this result.
Now, rewrite: using the Binomial Theorem above and assuming δ is small (but only keeping
terms up to δ2
since higher powers of δ are negligible), it requires a little algebra to obtain the
approximation as a quadratic in δ. The next step is to multiply the two quadratic expres-
sions in the formula for V (δ), discarding all higher powers (than δ2
) to obtain the quadratic
approximation. This algebra is straightforward, but a little messy, giving the approximation
2π rMax2
√
1 − rMax2 −
2π rMax(3 rMax2
− 2)
p
(1 − rMax2)
δ +
(6 rMax4
− 9 rMax2
+ 2)π
(1 − rMax2)3/2
δ2
.
This could be obtained by using the high level command series( ), then convert the result to a
polynomial, followed by using the simplify( ) command:
Page 10 of 238

[ series(V delta, delta, 3); (where V delta is the V (δ) above)
[ convert( %, polynom); (where % is the previously calculated result)
[ simplify( % )
Teaching Note. We support the White-Box/Black-Box Principle first articulated by Bruno
Buchberger in 1989 [8]. Students using CAS should initially include every step of their working
until they are secure in their understanding (the White Box), after which they should use the
higher level commands provided by the CAS (the Black Box). Maple provides a command that
computes the series. Students should initially master the notion and the skills to calculate the
series “by hand” (either by pen on paper or step by step calculation using the CAS) before
using the high level commands such as series( ).
Now substitute our postulated value
p
2/3 for rMax to obtain:
V delta 2 =
4
√
3
9
π − 4
√
3π δ2
.
This quadratic polynomial clearly has a maximum at δ = 0 as postulated and hence our result
that the volume has a maximum when the radius is
p
2/3.
The plot of V (r) is displayed with the quadratic approximation (as derived above) on the
right of Fig. 5. This visualization reinforces our intuition that the graph of a function “looks
like a parabola” near the function’s maximum.
7.1 The quadratic approximation for V implies that for V 2
and vice
versa
Knowing the quadratic approximation for V , it is simple to derive the quadratic approximation
for V 2
.
Exercise. Do it.
It is also possible to derive the quadratic approximation for V from knowing the quadratic
approximation for V 2
.
Exercise. Do it.
(Hint: Let the quadratic approximation for V be c+dδ +eδ2
; square this, neglecting all powers
of δ greater than 2; equate to the known result and equate coefficients of powers of δ.)
8 Further Solution Methods (Proofs)
It is common for there to be several methods of solving a mathematical problem. For the “find
the maximum . . . ” problems common in secondary school and first year mathematics, calculus
is invariably used. However it is good for students to develop their mathematical maturity
by following several methods, and this is the focus in [5]. Calculus provides the most efficient
method here, but does not provide the connection to (and development of) other important
topics and methods. The problem here can be solved by many different methods. We indicate
another two objective functions that can be obtained: they both result in cubic polynomials.
First. In Sec. 6, V 2
= V Sq, we notice that only powers of r2
appear. Thus if we consider
V 2
(r2
), the objective function is a cubic polynomial in r2
. It is easy to identify that the
maximum is given by r2
= 2/3. This is also easy to find directly since it is almost as easy to
Page 11 of 238

find where the maximum (or minimum) lies for cubic polynomials as for quadratics: see Section
4 of [4].
Second. In Sec. 5, the initial volume function depends on both r and h. We choose to
eliminate the h variable, but if r is eliminated instead, the volume function becomes a cubic
polynomial in h, giving a problem which is much easier to solve than using V (r).
We also indicate a further variation on the solution using V (r), by using the approach
introduced in [4]. In Sec. 7, the quadratic approximation for V (δ) was obtained with rMax
unspecified. We derive the value for rMax by noticing that the maximum for the quadratic
approximation for V (δ) must occur when the coefficient of δ is zero. Thus, we must have
−
2π rMax(3 rMax2
− 2)
p
(1 − rMax2)
= 0 ⇒ 3 rMax2
− 2 = 0
and hence the result.
9 A Parameterized Problem and Automatic Marking:
Computer Aided Assessment, CAA
After students groups have worked through the Cylinder in Sphere problem (in one or two
sessions), they are asked to complete an assignment (for assessment), where students could
be asked to follow the method above to maximize the volume or the volume squared. We
parameterize the assignment problem since presenting the same problem to the whole class
invites copying. Our experience over many years with a variety of courses and year levels is
that students consider parameterized problems to be different problems. We always try to
include a lot of visualization: this requires plots and animations for which parameters cannot
be arbitrary: they must have assigned values. Thus, there is not a problem with copying (even
with high level courses where the assignments contribute up to 80% of the assessment).
For the Cylinder in Sphere problem, an obvious parameterization is to use the radius of the
sphere R. A less obvious (but closely related) problem is to inscribe a (vertical) cylinder in a
Spheroid (a special case of an ellipsoid usually used to model the Earth’s surface):
x2
a2
+
y2
a2
+
z2
c2
= 1 .
The r to maximize the volume of the inscribed cylinder is
p
2/3 a. Choosing the values of the
parameter(s) to use is always important to ensure that some students do not encounter some
unintended difficulty. It is difficult to predict what precision (that is, number of correct decimal
digits) will be required to identify a given number. By experiment, we have established that
higher precision is required in order to identify a rational multiple of
p
2/3 than a multiple of
√
2 or
√
3 (or their reciprocals). For example, if a = 37/10, then 23 decimal place accuracy
is required to identify r that gives the maximum. Thus we recommend setting a as a rational
multiple of
√
2 or
√
3. We have checked several values, including
for a = 11
p
(2), 7 digits are required to identify r that gives the maximum,
for a = 13/2
p
for a = 23
p
Page 12 of 238

for a = 11
p
for a = 16/3
p
(3), 5 digits are required to identify r that gives the maximum.
So we recommend setting a to be multiple (integer, half integer, or third integer) times
√
2
or
√
3. The other parameter c only appears as a multiple in the formula for volume, so we
recommend choosing c as a convenient rational which is not very different in value to a so that
the spheroid looks like a sphere that is not deformed very much from the sphere.
With parameterized assignments, the marking load increases. Initially we used e-Marking,
see [3] for a discussion and references. The basic idea is to have a Maple solution/marking file
in which the student group’s parameter value is entered and the Maple worksheet is executed.
In this file, a Marking Section is progressively developed so that the Marking Report and the
feedback comments (that are saved in the student’s file that is returned to them) are saved
for later use. With significant content being visualization with animations, using e-Marking
(without CAA) is an attractive option, particularly for small classes.
By CAA we mean CAS-enabled CAA in which some CAS is used by the CAA system for
marking: thus answers that are symbolic (as well as numeric and many other types, including
multiple choice) can be marked. The only commercial CAS-enabled CAA using a professional
level CAS is MapleTA which uses Maple. We could mark the Cylinder in Spheroid assign-
ment final answer and intermediate steps (such as the quadratic approximation for volume)
symbolically (writing a provided procedure within Maple, or by using MapleTA). The number
of decimal places needed to correctly identify the answer could also be asked for. Using CAA
would reduce the marking load of staff, which becomes a major problem for large classes. Many
universities have between 1000 to 2000 students in the normal first year calculus class and are
faced with having to choose between assessing less student work, or implement CAA. Similar
issues arise in schools which can have large enrolments in the senior school. However many
teachers might be concerned that key parts of the assignment are not marked (and so ignored
by students): for example, here, a zoomed in plot is neither seen nor marked!
We emphasize visualization and believe that visualization needs some assessment: otherwise
the message for students is that visualization is not important. We have recently designed a
CAS-immersed CAA in which plots can be marked efficiently within an automatic marking
context: this is implemented for the Norman window problem, see Section 6 of [3] and references
therein.
The recent development of MapleTA’s “Question Designer” has made it much easier to
author questions (as we have demonstrated for the Norman Window problem in several unpub-
lished presentations). An advantage with using MapleTA is that all of the student adminis-
tration of results is dealt with automatically. Although plots are not marked, using MapleTA
requires less Maple expertise than writing CAS-immersed CAA. Most academics would prefer
to mark more than the final answer and provide more feedback than the answer is right or
wrong. For e-Marking and CAS-immersed CAA, the extra feedback is desirable (and reason-
ably straightforward to achieve). However in both cases students understand that they have to
use the required method (since their Maple file has to be submitted). For CAS-enabled CAA,
the issue is more complicated because the student work is not available. This issue can be ad-
dressed by careful design of multiple part (or Multiple Response) questions: the new Question
Designer in MapleTA (from version 8) has made the authoring of Multiple Response questions
easy.
Page 13 of 238

10 Conclusion
A CAS such as Maple provides a rich environment for e-Teaching and e-Marking or CAA of
mathematics: innovative approaches to curriculum, pedagogy and assessment at school and
university are supported. Standard problems solved in the standard way using CAS risk being
another activity (such as lectures and computer sessions) that students find boring. Well de-
signed teaching and assessment is effective, efficient and even fun. The development of these
CAS based materials requires a lot of staff time, but they give high returns with staff satis-
faction, student performance and attitude. With large classes, the reduced marking load using
automatic marking more than repays the development cost. We recommend more collaborative
learning by students and more collaborative development of CAS materials by academics.
Experimental mathematics is now an established area of mathematics: it has had its own
research journal for over 20 years, and there have been some graduate level courses. Although
the use of CAS is relatively common, not all use of CAS constitutes experimantal mathematics:
for example, routine use of CAS to execute simple evaluations (such as evaluating integrals
that are typically asked for in secondary or first year undergraduate courses) are no more
experimental mathematics than the evaluation of
p
(2) with a calculator. In this paper we
show that elementary optimization problems (such as “find the maximum volume of a cylinder
inscribed in a sphere”) can introduce the experimental mathematics methodology and be a
rich source of mathematics if the usual (and efficient) method of calculus is not used. We
show advanced visualization to build intuition; a graphical method (zooming in) of obtaining a
good numerical answer; the use of the Maple command identify( ) to identify a possible exact
solution; and several different methods using algebra to obtain the quadratic approximation of
the objective function about its maximum: hence proving the postulated maximum is correct.
This approach introduces an important topic in mathematics: approximation in the small. It
also provides strong support for (and visualization of) the intuitive notion that the function
“looks like a parabola” near its maximum.
The Maple activities described here are all web mediated, so they could be part of an online
course or a blended learning course: students have different learning styles and some do take
advantage of the flexible learning that the approach here supports (since attendance in the
computer lab is recommended but not required). Students enjoy the collaborative learning
using Maple and learn (surprisingly quickly) to do the calculations; use and adapt advanced
animations; and use automatic marking. Students are engaged, active and collaborative learners
with these Maple sessions.
Acknowledgements
The author thanks Dr Asim Ghous, the Director of ASES, for support with Maple over many
years (and MapleTA more recently) and ASES for financial support to prepare this paper and
to be a participant at the ATCM 2014 Conference.
References
[1] Blyth, B., “Animations using Maple in First Year”, Quaestiones Mathematicae, Suppl., 1,
Supplement, 2001, 201–208
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[2] Blyth, B. and Labovic, A., “Active Learning and Fun in First Year using Maple”, in Buffel-
spoort TIME2008 Peer-reviewed Conference Proceedings, P.H. Kloppers, C.S. Joubert and
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noti1061
Page 15 of 238

Technological tools have enhanced our
teaching, learning and doing mathematics,
what is next?
Wei-Chi YANG
wyang@radford.edu
Department of Mathematics
and Statistics
Radford University
Radford, VA 24142
USA
Abstract
In this paper, we give an overview why technological tools have advanced so fast
since ATCM 1995, and yet the adoption of exploration activities have not been widely
implemented in many mathematics curricula. We also give examples to demonstrate
the types of exploration that require the integration of CAS and DGS. We outline the
components needed when developing an interactive online system, which is crucial for
communications and collaborations on mathematical …elds now and in the near future.
1 Introduction
Several speakers at various ATCM occasions have shown that technological tools not only can
lead learners to deeper understandings on abstract and complex concepts of mathematics, but
also encourage learners to discover more mathematical applications in real life. Indeed, with
current technological tools, a mathematical problem or a project can be explored with the
help of a computer algebra system (CAS), a dynamic geometry system (DGS) or combinations
of both. We have discovered that new concepts or knowledge can be acquired by exploring
mathematics with rapidly evolving technological tools. On one hand, many researches have
recognized the positive impacts of proper adoption of technological tools in enhancing teaching,
learning and doing mathematics and agree that curriculum heavily based on teaching for the
test will hinder students’creative thinking skills and opportunities for innovation. However,
because mathematics curriculum in many countries still examinations driven, the adoption of
technological tools in teaching, learning and research is still far from desired.
In section 2, we describe typical di¢ culties for teachers when adopting technological tools in
a classroom and emphasize the importance of creating video clips. In section 3, we use examples
to demonstrate a mathematical problem can be made interesting and yet challenging at the same
Page 16 of 238

time. We also give examples to demonstrate why integrating the knowledge of both CAS and
DGS is crucial for training students in current environment. Finally in section 4, we outline the
needs of an online learning tutorial system which allow students to explore mathematics online,
allow students to do online practice problems and receive instant feedbacks through a server. We
urge educators, researchers, software and hardware developers to work cooperatively to make
doing mathematics online a reality someday. Only when an online interactive mathematics
learning environment becomes available, can we deliver dynamic interactive contents to users,
and encourage instant discussions on mathematics at anytime and anywhere.
2 Supporting materials for teachers
The bene…ts of implementing technological tools in classrooms have been long discussed since
the …rst ATCM in 1995. We have seen many governmental initiatives to boost the use of
technological tools in math and sciences areas around the world ever since. However, we still
see many teachers simply use computers or technological devices for PowerPoint presentations
in a regular classroom. The problem exists because many teachers feel uncomfortable using
technological tools for discussing or exploring mathematics in a classroom. Simply, it is still
a non-trivial task for beginners learning how to use a technological device. So what are the
possible reasons?
1. For hand-held devices, there are di¤erent brands and di¤erent keystrokes, author had al-
ways dreamed about all brands of calculators have the same basic keyboard display. This
way teachers and students can get familiar with a calculator very quickly without worrying
about how to manipulate a device. The place where di¤erent manufacturing companies
may compete is how their computation engines reach an answer or how additional func-
tionality of a calculator are being o¤ered. Just like driving a rental car, consumers are
allowed to choose di¤erent brands of cars, styles and options but the renter can easily
start driving a car without worrying about what the brand of such car is.
2. For computer software packages with di¤erent capabilities, we have a lot to choose from,
for complete introduction regarding this, we refer readers to [10]. In short, we can group
them into software packages that can do CAS, DGS, statistics and etc. Any software
packages have their own learning curves, it takes time to train teachers to be comfortable
using them; not to mention nowadays, we need teachers to be able to integrate di¤erent
capacities of software together, for example it is important to know how to integrate CAS
with DGS. We will see examples in section 3.
The professional trainings for teachers to be competent of utilizing a hand-held device and
computer software packages should be an ongoing process but it takes time and resources to
attain predetermined objectives. It is known that video clips provide e¤ective way of delivering
lectures anywhere anytime such as those from [7] Video clips are useful for learners to recap
content being covered in a lecture or to review steps when learning how to use a technological
tool. It is worth mentioning a project from South Africa directed by Professor Werner Oliver,
which contains all learning and teaching contents, to improve the quality of students learning
and teachers teaching in school environments where internet bandwidth is not su¢ cient and
also not a¤ordable. His projects are aimed at providing o¤-line 21st century sca¤olding support
Page 17 of 238

platforms for learners and teachers that are curriculum aligned and ‡
exibly accessible. One of
the aims of the projects is to improve the national pass rate of students who obtain more than
of 50% for their …nal Mathematics school exams as this is normally the minimum requirement
for access to science, engineering and technology related study programmes at higher education
in this country(currently this rate is only 6%).
The project utilizes syllabus aligned video lessons, student workbooks, cell phone technology
for formative assessment and peer tutoring with tablets for also improving teaching and learning
mathematics and physical sciences subjects, see Figures 1(a) and 1(b) below:
Figure 1(a) Students bring tablets
to classrooms
Figure 1(b) Teachers and students
have access to contents
Since internet is not widely available currently in South Africa, the project is using an indepen-
dent o- ine package which runs in a local Windows or Android browser environment. A local
web browser menu system of hyperlinks allow user-friendly and ‡
exible access to (see Figure 2)
all math and sciences content components (video lessons, PowerPoint slides, student workbooks
and solutions) by merely a click of a mouse or touch of a screen. The content video lessons
consist of narrated concept explanations, sets of related examples, tutorial exercise problems
and corresponding solutions. It also contains some interactive applets. Although the resource
materials of this project is Windows or Android based, we believe this is still very valuable
for those countries where internet is not widely available. Moreover, all the content video clips
are narrated and animated using dynamic graphical software to enhance visualization of math-
ematics or science concepts and problems. This makes the model even more attractive as it
provides a wonderful teaching and learning tool for teachers and students that is complete and
Page 18 of 238

freely accessible on a local device - anywhere, anytime.
Figure 2 Intranet webpages which contains topics in math
and sciences
3 Skills needed for students and teachers
Many educators and researchers believe that creativity does not come from rote or repetitive
learning but from exploration and play. Those open-ended or out of textbook real-life problems
are excellent resources for students to develop their creative thinking skills. The existence of
a solution often is not enough. Instead, we may ask how we can approximate a solution. Can
we use a technological tool such as DGS to simulate or conjecture what a possible solution
may be? Do we have real-life applications for the open-ended project? However, we often see
mathematics curriculum (in most part of the world) is geared toward exam-based. Too much
emphasis on testing alone when measuring a student’
s understanding in mathematics will hin-
der student’
s interests in mathematics and their abilities for innovation. Here we note that
Finnish students do not take a national, standardized high-stakes test until they matriculate
secondary school and then only if they intend to enter higher education. Instead, the purpose
of assessment in Finland is to improve learning; it is “encouraging and supportive by nature”
(Finnish National Board of Education, 2010, “Encouraging Assessment and Evaluation, para.
1). Therefore, we may say that allowing time for exploring is essential for students’learning
and success in mathematics for Finnish system. We have seen many presenters showing that
evolving technological tools in recent years have allowed learners to increase and enhance ones’
mathematics content knowledge. In addition, math can be approached as Fun, Accessible,
Challenging and then Theoretical. Learners can expand their knowledge horizons with tech-
nology in various stages. Exploration with technology is the key, examination is only one way
to measure students’understandings.
We use the following example (see Figure 3) to demonstrate how mathematics can be made
interesting and yet challenging at the same time. In the mean time, it also demonstrated
some basic skills needed for students and teachers when it comes to solving an open-ended ex-
ploratory type of activities in current mathematics current. The following graph is contributed
by V.SHELOMOVSKII using [5] and its animation can be found at [6].
Page 19 of 238

Figure 3 Floating …sh with their
respective space curves
We can ask the following simple questions to those students who have learned (multivariable)
calculus:
1. Please describe how computer program draw a 3D …sh in this case. Do we need a two or
three variables function or parametric equation?
2. How can we make a 3D …sh swim in a space? How many variables do we need in this
case?
3. Please describe, in mathematics term, how two …sh can swim without running into each
other.
The following is a simple example showing basic skills of manipulating a CAS is needed
both teachers and students when solving a problem.
Example 1 Consider r = sin 2 versus r = sin 160 ; where 2 [0; 2 ]: (a) What do you see the
di¤erences between these two polar graphs? (b) Would you like to make your conjecture base on
what you see? [Do you see the graph of r = sin 160 ; where 2 [0; 2 ] almost …ll up the whole
circle of r = 1?] (c) Can you prove if your conjecture is true or false? Some may conjecture
the area of r = sin n when n ! ; where 2 [0; 2 ]; is equal to that of r = 1; can you prove
or disapprove this? It is a fun problem for those who just learned the plot of a polar graph,
and using a graphics calculator can even trace the direction of the curve such as r = sin 2 :
The Figure 4(a) shows the plot of r = sin 2 for 2 [0; 2 ] and Figure 4(b) shows the plot of
r = sin 160 ; for 2 [0; 2 ]: Some may ask if we let n ! 1; the graph of r = sin n may …ll
out the whole circle and thus they may conjecture that the area of r = sin n ; for 2 [0; 2 ] and
when n ! 1; will approach that of the circle of r = 1; which is : However, we know that the
area of r = sin n ; for 2 [0; 2 ] is 1
2
R 2
0
(sin n )2
d ; and Maple returns the answer of
1
2
2 (cos n )3
sin(n + sin (n ) cos (n ) + n
n
: (1)
Page 20 of 238

After taking the limit of equation (1) for n ! 1 using [8] we obtain the answer of 2
; which
means that our earlier conjecture was false.
Figure 4(a) Polar graph of
y = sin 2
Figure 4(b) Polar graph of
y = sin 160
The following problem was posted by Ph.D. students during my research project at Guangzhou
University after discussing the problem [15] with them.
Example 2 Consider the following three curves given by C1 : y = sin(x); C2 : y = x2
+ 2; and
C3 : (x 3)2
+(y 3)2
1 = 0, see Figure 5(a) below. We need to …nd approximate points A; B
and C on C1; C2 and C3 respectively, so that the total distance of AB + BC + CA achieves its
minimum.
Figure 5(a) Total shortest distance
obtained from a CAS
Figure 5(b) Total shortest
distance obtained from a
DGS
We remark that Ph.D. students have access to Chinese developed DGS so they are excellent in
manipulating a DGS; however, some do not have access to a CAS or not comfortable using a
CAS, so it is di¢ cult for them to verify if their answers are correct. They …rst reduced their
problem to a simpler case by considering three circles as follows (see Figure 6): C4 : (x 2)2
+
Page 21 of 238

(y 2)2
1 = 0; C5 : (x+3)2
+(y+3)2
4 = 0; and C6 : (x 3)2
+(y+2)2
1=4 = 0: We need to …nd
appropriate points A; B and C on C4; C5 and C6 respectively, so that the total distance of AB +
BC+CA achieves its minimum. Students were able to use geometric constructions (will be pub-
lished in a separate paper) to show that AB+BC+CA achieves its minimum when we move A to
A0
= 1:76761632020814; 1:0273758046571); B to B0
= ( 1:20628618061282; 2:115358414871);
and C to C0
= (2:5843451027448; 1:72209532859668) accordingly in Figure 6. The normal
vectors at A0
; B0
and C0
respectively, pass through the incenter of triangle A0
B0
C0
:
Figure 6 Solution obtained
using a DGS
However, because of lacking the access to a CAS, they were not able to extend their result to the
original problem (Example 2). In [16], we show that for the general case described in Example
2, the normal vectors at A; B; C respectively should be the angle bisectors of BAC; ABC and
BCA respectively and they should pass through the incenter of triangle ABC: We also obtain an
analytical solution by using Lagrange multipliers and with the help of Maple (see Figure (a)).
We con…rm that the analytical solution is consistent with the geometry construction using a
DGS [2] (see Figure 5(b)).
It is natural to generalize the 2D problem to the following 3D case: We are given four
surfaces shown in Figure 7, where S1 : x2
+ y2
+ z2
1 = 0; S2 : x2
+ (y 3)2
+ (z 1)2
1 = 0;
S3 : z (x2
+y2
) 2 = 0 and S4 : (4(x 3)+(y 3)+(z 1))(x 3)+((x 3)+4(y 3)+(z
1))(y 3) + ((x 3) + (y 3) + 4(z 1))(z 1) 3 = 0: We want to …nd points A; B; C and D
on the surfaces S1; S2; S3 and S4 respectively, so that the total distance AB + BC + CD + DA
Page 22 of 238

achieves its minimum.
Figure 7 A total shortest distance
problem in 3D
Remark: We may think of this problem as a light re‡
ection from one surface to the other. If
we think BA as an incoming light toward the point A on surface S1 and DA as an outgoing
light at A, then NA; the normal vector at A on S1 should be their angle bisector. Similarly, we
should have the followings:
1. The line segments BC; AB and the normal vector at B; denoted by NB; should lie on the
same plane and NB is the angle bisector for AB and BC.
2. The line segments BC, CD and the normal vector at C; denoted by NC; should lie on
the same plane and NC is the angle bisector for BC and CD.
3. The line segments CD; DA and the normal vector at D; denoted byND; should lie on the
same plane and ND is the angle bisector for CD and DA.
We apply Lagrange Method with the help of Maple to …nd the shortest total distance
AB + BC + CD + DA algebraically. The desired points A; B; C and D on S1; S2; S3; and S4
respectively obtained from Maple are A = [s11; s12; s13]; B = [s21; s22; s23]; C = [s31; s32; s33] and
D = [s41; s42; s43]; where
s11 = :33389958127263692867; s12 = :82675923225542380177; s13 = 0:45274743677505224502;
s21 = 0:13641928282913798603; s22 = 2:0175068585849490438; s23 = 1:1268739782018690446;
s31 = 0:28488630072773836633; s32 = :62315636129160757197; s33 = 2:4694840549605319295;
s41 = 2:3607938428457871966; s42 = 2:4901551912577622423; s43 = 1:3195995907854243548:
We show geometrically (see Figure 8) that the blue dotted lines are respective normal
vectors at respective points A; B; C and D; and the dark red line segments are respective angle
bisectors in Figure 8, they coincide with respective normal vectors when the minimum total
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distance AB + BC + CD + DA is achieved. In other words, our geometry constructions
coincide with our algebraic analysis using CAS.
Figure 8 Total shortest distance 3D
problem veri…ed using a DGS
The preceding problem demonstrates why integrating the knowledge of both CAS and DGS
is crucial for training students in current environment. The following example comes from a
university entrance exam practice problem from China. We use the dynamic software [5] to
demonstrate how the problem can be made accessible to most students and how the problem can
be generalized to more challenging ones in 2D and corresponding ones in 3D. We remark that
these open ended projects are excellent choices for adopting technological tools for exploring
mathematics.
Example 3 Consider the following Figure 9:
Figure 9 A college
entrance practice problem
from China
Page 24 of 238

The circle is given with radius r, assuming the center is at (0; 0), and A = (x; 0) and x 2 [0; r].
The points D and E are any two points on the circle such that the angle of EAD forms a
…xed angle : For the college entrance exam practice problem from China, students are asked
to conjecture the maximum and minimum lengths for DE. The following values were those
given at the practice college entrance problem from China: r = 2; = 2
and A = (1; 0). It is
not di¢ cult for one to conjecture after exploring with a DGS that the maximum length of DE
occurs when DE is perpendicular to AO and A is on the opposite of DE, see Figure 10(a);
and the minimum of DE occurs when DE is perpendicular to AO but A is on the same side of
DE; see Figure 10(b).
Figure 10(a) Maximum
length DE
Figure 10(b) Minimum
length DE
It is not di¢ cult to generalize this problem from a circle to a sphere in this case. However,
the problem should be rephrased as follows: Given a sphere of a …xed radius r of the form
x2
+ y2
+ z2
= r2
and pick the point A = (d; 0; 0), where d 2 [0; r]: Let B be a point on the
sphere and rotate AB with a …xed angle to form a cone, see Figure 11. We want to …nd the
maximum and minimum intersecting surface areas between the cone and the sphere.
Figure 11 Generalize the
circle probem to a sphere
After exploring with [5], it is not di¢ cult to see that the maximum intersecting surface area
occurs when the normal vector at B is parallel to the vector OA and A is on the opposite of B
(see Figure 12(a)). The minimum intersecting surface area occurs when the normal vector at
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B is parallel to the vector OA but A is on the same side of B (see Figure 12(b)).
Figure 12(a) Maximum
intersecting surface area
Figure 12(b) Minimum
intersecting surface area
We may now ask students to extend Example 3 to an ellipse of this form x2
a2 + y2
b2 = 1 and
use a DGS to make conjectures where the maximum and minimum lengths of DE could be, see
Figure 13. We note that the answers will vary depending on the values of a; b; the …xed angle
and the position of the point A: Complete solutions on the ellipse case have been discovered,
see [17]
Figure 13 Generalize the 2D
problem to an ellipse
The problem becomes even more challenging and is still an open problem if we extend
the ellipse to an ellipsoid. For example, given an ellipsoid of the form x2
a2 + y2
b2 + z2
c2 = 1; let
A = (x; 0; 0) with x 2 [0; a]. Pick B be on the ellipsoid and if the cone determined by rotating
Page 26 of 238

a …xed angle around AB; see Figure 14 below:
Figure 14 An ellipsoid and a
cone
We may ask the question: For a …xed point A and …xed angle for the cone, …nd the point
B that will result in the maximum or minimum intersecting surface area between the ellipsoid
and the cone. Just like the case for ellipse, the answers vary depending on the values of a; b; c;
the …xed angle and the position of the point A:
We summarize a general strategy of how we may approach solving a problem and how we
can generalize it to higher dimensions with the helpful of a DGS and CAS.
1. We use DGS or similar technological tool to simulate animations in 2D.
2. We make conjectures through your observations from step 1.
3. We verify our results using a DGS or CAS for 2D case.
4. We extend our observations to a 3D scenario with technologies if possible.
5. We prove our results for 3D case using a DGS or CAS.
6. We extend our results to …nite dimensions or beyond if possible.
It is interesting to note the following scenarios while author conducted lectures to students
of di¤erent backgrounds:
1. For students who have access to a CAS but not DGS: It is impossible for them to repro-
duce di¤erent animations with geometric motivations. They can only modify the CAS
worksheet author provides them when validating their algebraic answers.
2. For students who have access to a DGS but no CAS: Students will use their favorite DGS
to reproduce author’
s problems and make conjecture about the validity of the solutions;
however, their conjectures cannot be veri…ed since they have no CAS to verify solutions
analytically.
3. For those students who have no access to either a CAS or DGS, they can only appre-
ciate the graphical representations of authors’lectures, they have no available tools to
experiment on their own.
Page 27 of 238

4 Delivering interactive online contents
It is interesting to see that the internet technology has advanced rapidly since the …rst ATCM
in 1995. However, delivering interactive online content still has room to improve. To develop a
truly online dynamic learning system for mathematics, we need computation engines that are
capable of doing both CAS and DGS. With such a system in place, we will be able to create
online interactive textbooks, we can also allow open-ended exploratory type of problems such as
those described in Section 3. We describe some of the desired components for online interactive
learning in mathematics and related subjects so people can easily discuss mathematics.
1. Allowing instant exploration using CAS such as WolframjAlpha [13], the system provides
an excellent system where users can explore various areas using Mathematica engine
online. For example, we can explore the torus knots (see Figure 15 [14]). Author has seen
teachers from China developing teaching contents using [13]. More interactive approaches
using [13] can be found in the invited speech by [1]. After all, this is a CAS system where
users have no way of experimenting problems when a DGS system is desirable.
Figure 15 An
interactive torus knots
using Wolfram Alpha
2. For exploring DGS online, we direct readers to the page of [12], where [4] is used as the
DGS engine to describe a parabola Caustic Rays emanate from D, and their re‡
ections
in the parabola form a caustic (see Figure 16). The family of rays by pressing Show, then
the caustic curve by pressing Show again. You can drag C and D. However, this is a DGS
Page 28 of 238

based not CAS based so we cannot experiment problems when CAS are needed.
Figure 16 An online
interactive caustic using
Geometry Expression
3. Using applets: If we look at the webpage [9], it describes an online content in Calculus
subject using Java applets. For example the following Figure 17 gives a dynamic way
of learning the spherical coordinate system, users can drag ; and , and can see the
e¤ects of these parameters on the position by dragging the red dot. However, we know
that applets using Java script is platform dependent, which also presents a problem for
developers when they consider designing interactive online contents.
x = sin cos ;
y = sin sin ;
z = cos (2)
Figure 17 An
interactive online
content using Java
applet
4. Many book publishers are interested in an online assessment system because these are
what schools and universities desire for assessment purpose. Ideally a good system should
Page 29 of 238

allow teachers to create randomized questions containing graphs, and students are able to
see their results with feedbacks instantaneously. Therefore, a system requires an authoring
tool for teachers to design problems and an users’testing system. The following shows
a question generated by the Exam Builder of Scienti…c Notebook (see [11]), which is a
Window-based and has been available in mid 90s: However, there is no compatible online
testing system so far. We observe in Figure 18, the quantity a in sin ax can be randomized
using the authoring system; accordingly the graph of sin ax will vary accordingly because
there is a built-in CAS. After choosing the answers and clicking ‘
click to grade’
, we
receive a feedback from the system as seen in Figure 19. This is a Window based program
which author adopted during 1997-1998 while visiting Singapore, and yet it is surprising
that a comparable ‘
engine-based’online system is still not available as of now. The invited
talk by [3] introduces a mathematics interactive learning environment, it remains to be
seen if the system can ful…ll the possibilities of online discussions on those open ended
problems that are discussed in Section 3.
Figure 18 An authoring tool from Exam Builder
Figure 19 A feedback form using Exam Builder
Page 30 of 238

5 Conclusions
It may be true that many math textbooks (including those from US) are not based on explo-
ration, and schools buy the books that are aligned to the test questions instead of the books
that promote deep mathematical understanding and creativity. This is rather a true reality but
common sense tells us that teaching to the test can never promote creative thinking skills in
a classroom. We know the continual e¤orts for all ATCM communities to address the impor-
tance and timely adoption of technological tools in teaching, learning and research can never be
wrong. Therefore, we encourage ATCM communities continue creating innovative examples by
adopting technological tools for teaching and research and in‡
uence your colleagues, communi-
ties and decision makers in your respective countries. We also think by developing technology
based textbooks which describe contents in continuos, connected and creative fashion is impor-
tant. Selecting examples that can be explored from middle to high schools, university levels
or beyond, and they should be STEM related by linking mathematics to real-world applica-
tions. Allowing users to explore mathematics with technological tools online is crucial step
for developing a social media of discussing mathematics through internet. Many agree that
examination should not be the only way to measure students’understandings in mathematics.
Technology has become a bridge to make us rethink how to make mathematics an interesting
and a cross disciplinary subject. Through the advancement of technological tools, learners
will be able to discover more mathematics and its applications through an interactive online
learning environment.
6 Acknowledgements
Author would like to thank Vladimir Nodelman for using his own DGS to create Figures 7
and 8 and Author would also like to express his sincere thanks and appreciation to following
hosts for providing excellent research environments during his professional leave in 2014: Chi-
nese Academy of Sciences in Beijing, National Institute of Education-Nanyang Technological
University of Singapore, Guangzhou University of China, Beijing Normal University of China,
and Center for Research on International Coorporation in Education Development-University
of Tsukuba of Japan.
References
[1] Abbott, P., an invited speech at ATCM 2014, http://atcm.mathandtech.org/EP2014/Abbott/Abbott_
[2] ClassPad Manager for ClassPad II Series, 90 days trial download, see http://classpad-
manager-for-classpad-ii-series.software.informer.com/
[3] Fu, H., an invited speech at ATCM 2014, http://atcm.mathandtech.org/EP2014.
[4] Geometry Expression, see http://www.geometryexpressions.com/.
[5] Geometry in Mathematical Arts (GInMA): A Dynamic Geometry System, see
http://deoma-cmd.ru/en/Products/Geometry/GInMA.aspx.
Page 31 of 238

[6] GInMA animation: An animation of two …sh. see
http://www.radford.edu/wyang/Tsukuba/…sh.MOV.
[7] Khan Academy: https://www.khanacademy.org/.
[8] Maple: A product of Maplesoft, see http://maplesoft.com/.
[9] Math Insight: Interactive webpages underlying few topics in mathematics, see
http://mathinsight.org/about/mathinsight.
[10] M. Majewski, A plenary speech at ATCM 2014, see http://atcm.mathandtech.org/EP2014.
[11] Scienti…c Notebook by MacKichan Software, Inc. see
http://www.mackichan.com/index.html?products/snb.html~mainFrame.
[12] Todd, P., An interactive webpage, see http://euclidsmuse.com/members/polymnia/apps/app/768/
[13] Wolfram Alpha: http://www.wolfram.com/mathematica/how-mathematica-made-
wolframalpha-possible.html
[14] Wolfram Alpha: http://library.wolfram.com/webMathematica/Mathematics/Knots.jsp.
[15] Yang, W.-C. Some Geometric Interpretations of The Total Distances Among Curves and
Surfaces, the Electronic Journal of Mathematics and Technology (eJMT), Volume 3, Num-
ber 1, ISSN 1933-2823, Mathematics and Technology LLC.
[16] Yang,W.-C., Fu, Y., Shortest Total Distance Among Curves, Surfaces and Lagrange Mul-
tipliers, accepted to appear at the Electronic Journal of Mathematics and Technology
(eJMT), ISSN 1933-2823, Mathematics and Technology LLC.
[17] Yang,W.-C., Shelomovskii, V. Thompson, S., Extreme Lengths for Chords of an Ellipse,
accepted to appear at the Electronic Journal of Mathematics and Technology (eJMT),
ISSN 1933-2823, Mathematics and Technology LLC.
Page 32 of 238

Professional Development for Teachers in
Mathematical Modelling
Keng-Cheng Ang
kengcheng.ang@nie.edu.sg
Liang-Soon Tan
liangsoon.tan@stdmail.nie.edu.sg
National Institute of Education
Nanyang Technological University
1, Nanyang Walk, Singapore 637616
Abstract:
The importance of mathematical modelling and its value in mathematics education has been
discussed and emphasized by various researchers in the field. However, it is widely acknowledged
that mathematical modelling can be demanding for students and teachers. Many teachers, including
experienced practitioners in Singapore, do not have any formal training in mathematical modelling
as a student either at school or in the university, and teaching it has been very challenging. Very
often, a teacher’s decisions, from planning of a mathematical modelling task to its execution in the
classroom, depend on his orientation, resources and goals. Unless a teacher has been properly
trained, adequately prepared and well resourced, his decisions are unlikely to result in a successful
mathematical modelling lesson.
In this paper, we discuss a school-based professional development (SBPD) programme aimed at
preparing teachers to teach mathematical modelling in the classroom. This programme has been
successfully carried out in several schools in Singapore over a period of about two years. A case
study will be presented to illustrate the key principles of the programme. Examples of student work
and modelling tasks designed by participating teachers will be presented. In addition to the
observation that there is a definite paradigm shift in the teachers’ orientation and goals in teaching
mathematical modelling, it is also evident from these examples that technology had played a crucial
role in the success of the modelling lessons designed by the participants of the SBPD programme.
Introduction
In recent years, mathematical modelling has become an important area of growing interest in
mathematics education. Many researchers have discussed the relevance of teaching mathematical
modelling and the value of learning mathematical modelling in the classroom [1], [2], [3], [4], [5].
In Singapore, the Ministry of Education (MOE) has also supported the teaching of mathematical
modelling, seeing it as a “worthwhile learning experience that will benefit students and prepare
them well for the future” [6].
However, despite the emphasis and constant attention, mathematical modelling has yet to become a
common or popular approach of mathematical study in the Singapore classroom. One reason could
be that mathematics teachers here have limited knowledge and hence little confidence in
mathematical modelling. At the same time, there is a lack of relevant resources and tested
exemplars of mathematical modelling for local teachers to tap on [7].
Page 33 of 238

Teachers’
DECISION
MAKING
COMPETENCE in
teaching mathematical modelling
Knowledge of
modelling,
Framework for
modelling
instruction
Transformative
learning cycle
School context
and culture
Content Process Context
It is widely acknowledged that mathematical modelling can be demanding for students and teachers
[5]. Therefore, for mathematical modelling to succeed in the classroom, teachers will need to be
properly prepared and suitably trained. Other researchers have emphasized the importance and
nature of pedagogical content knowledge (PCK) in effective teachers [8], [9], [10]. Hence, to be an
effective teacher of mathematical modelling, one not only needs to know mathematical modelling,
but also possess the pedagogical acumen to deliver it at the classroom level.
Unfortunately, most of our teachers do not have any formal training or experience in mathematical
modelling as a student, whether in school or at the university. It is understandable then that
teaching something one has no experience in or knowledge of can be challenging and daunting.
Moreover, the unpredictable nature of a mathematical modelling lesson or class has further deterred
and discouraged teachers from trying to implement mathematical modelling lessons in the
classroom [11].
This problem of limited knowledge and experience in the subject area is compounded by the fact
that there is a general lack of relevant and usable resources, as well as support for the teacher who
wishes to plan, design and implement modelling lessons [7], [12]. Meaningful mathematical
modelling tasks involve problem solving in an unfamiliar setting, often requiring cross-disciplinary
knowledge. Thus, the teacher needs to be really well prepared, well informed, and well resourced.
With this in mind, a teacher-centric, school-based professional development (SBPD) programme in
the teaching of mathematical modelling was designed and implemented in a few pilot schools. The
objective is to help teachers develop the appropriate orientation, adopt a positive mindset and
acquire the requisite competencies to teach mathematical modelling in the classroom.
The SBPD
The SBPD programme consists of three dimensions, namely, Content, Process and Context, that
contribute towards developing teachers’ confidence and enhancing teachers’ decision making in
planning, designing and teaching mathematical modelling. The essential components are illustrated
in Figure 1 below.
Figure 1. Essentials of the SBPD programme
Page 34 of 238

The content of the programme will include the subject matter (that is, mathematical modelling,
approaches to modelling, the modelling process and so on). More importantly, pedagogical content
knowledge of modelling will also be discussed and this would include knowledge of modelling task
space, knowledge of the broad spectrum of tasks, diagnosing students’ learning difficulties during
modelling process, strategies of intervention modes, appropriate beliefs and so on. In addition, a
new framework for the teaching of mathematical modelling will be introduced and used as a tool in
translating teachers’ ideas into modelling tasks or lessons [13]. This framework essentially consists
of five questions that will guide teachers as they develop and design their modelling tasks (see
[14]).
As noted by Thomson and Zeuli, the process of transformative learning in professional
development promises to bring about “changes in deeply held beliefs, knowledge, and habits of
practice” [15], which is more desirable and impactful. To achieve this effect of transformative
learning, the project team will work closely with teachers, providing advice and support while
teachers plan and design mathematical modelling lessons for their students. This active
engagement in planning, designing and classroom enactment was video-recorded for both review
and analysis. Review was necessary to provide the basis and platform for teachers to discover and
resolve cognitive dissonance, if any, in the process of applying their new knowledge. To further
strengthen and deepen the learning, the whole process of applying was repeated.
It is not uncommon to hear teachers say that while they may have learnt something from some
professional development courses, they find it hard to apply what they have learnt in their school.
This is especially so in mathematical modelling. Customized support, as well as adequate
allocation of professional development time, can foster the coherence and meaningfulness of
teachers’ professional learning experiences in a particular school context and culture [16]. The
SBPD comprises a planning sequence that incorporated goal setting, planning, applying and
reflecting. These sub-processes were influenced by various aspects of a teacher’s work in school.
In order to align these processes with the school’s multiple areas of focus, placing the programme
within the school context, both physically and philosophically, had brought about support from the
school leaders and participating teachers. It had made the entire exercise more relevant and useful
for teachers, their students and their school.
The SBPD programme consists of three phases (Training, Applying and Reflecting), with two
learning cycles spread over a period of between six and eight months. The activities in these
phases are summarized in Table 1 below.
Table 1: SBPD Programme for teachers in mathematical modelling
Programme Milestones Activities
Baseline Checking Pre-programme discussion to ascertain baseline knowledge of
participating teachers
Phase 1: Training Learning to be a modeller
Framework for teaching of mathematical modelling
Phase 2: Applying Planning and designing modelling lessons
Implementing modelling lessons
Phase 3: Reflecting Analysing modelling lessons
Post-lesson discussions
Page 35 of 238

A Case Study
The school chosen for the case study reported here is an autonomous secondary school. School
leaders of an autonomous school have some level of freedom in providing programmes outside of
the national curriculum. In this case, it was decided that mathematical modelling would be planned
to be integrated into the Secondary 1 and 2 (Grades 7 and 8) mathematics curriculum.
Ten teachers (four male and six female), each of whom had between one and 23 years of teaching
experience, participated in the SBPD programme. At the preliminary interviews, although the
participants had expressed concerns about their lack of understanding in teaching mathematical
modelling, they welcomed the opportunity to develop their competencies in the SBPD programme.
In Phase 1 of the SBPD programme, three 4-hour training sessions were carried out. Participating
teachers engaged in independent modelling experiences involving three tasks in the first two
sessions. These tasks were based on common modelling approaches suggested by Ang [17]. The
framework for teaching mathematical modelling (as described in [13]) was introduced in the third
and last of these training sessions.
In Phase 2, participating teachers worked their groups (or pairs) to plan and design modelling
lessons for their students. One focus group discussion (FGD) was held to discuss their design for a
Level 3 task, as part of the transformative learning cycle.
Phase 3 consists of implementation of the task, and meetings and discussions with participating
teachers. A FGD was held to discuss lesson issues that arose from their enactment of modelling
lessons. This was followed by a post-programme interview with all participating teachers.
Sample Modelling Tasks
A number of ideas for good modelling tasks arose during Phase 2 of the programme. Participating
teachers were able to find real and relevant problems that may be suitably constructed into
modelling tasks for their students. In this section, two such tasks are highlighted.
The Darts Game
The task was posed as follows.
This is a real problem and students were given a $100 budget to run the stall in a real carnival. As a
trigger, students were given the proposed design shown in Figure 2, but this was not attractive, and
rather arbitrary in terms of choice of rewards.
Students worked on the problems in groups, and were asked to upload their work on Google Docs
for sharing and discussions. Sharing their work on Google Docs had facilitated discussions and the
level of participation was heightened. In addition, students quickly learned from one another, and
were able to modify or justify their own designs.
This task could also involve running a simulation using the proposed designs, to determine the
experimental probabilities of each landing space, and hence the relative value of the rewards.
Our school is organizing a carnival to raise funds. Our class has been assigned to be in
charge of a “Darts” game stall. Instead of the usual bulls-eye type of dartboard, design
a new dartboard and provide rewards that will be attractive enough and yet strike a
balance between players winning and the class earning a profit.
Page 36 of 238

Figure 2: A proposed design for the dartboard
Figure 3: A sample of a shared work uploaded on Google Docs
Page 37 of 238

S
P
Relay Race Problem
This problem was posed as follows.
The succeeding runner should begin running from her position, S, and pick up speed when she sees
the preceding runner approaching and reaching position P (see Figure 4). This is so that when she
actually receives the baton, she will be well on her way in reaching her maximum running speed.
The question is, where should Point P be, relative to Point S?
Figure 4: Preceding and Succeeding runners in a relay race
This problem was discussed by Osawa, who noted that in order to solve this problem, “students
were required to have large amounts of mathematical knowledge, and the skills to apply it” [18].
However, when this problem was posed as a modelling task, participating teachers in the SBPD
programme had planned for sufficient scaffolding, and made good use of technology to bridge the
cognitive gap.
This task had involved students thinking about the factors that influence or determine the position
of Point P. Eventually, students realized that they would need to know the runners’ “running
profile” – that is, how one picks up speed starting from rest – and to assume that runners are able to
maintain their top speed for a period of time (at least a few seconds). A runner’s “running profile”
is simply the “distance-time” graph over a short distance, and to obtain this, some practical data
collection exercise had to be carried out. Cones were placed at regular (10 m) intervals, and each
runner is video-recorded as he sprints a distance of about 50m (see Figure 5).
Figure 5: Data collection for relay race problem
In a 4 x 100m relay, a preceding runner has to pass the baton to the succeeding runner
within a certain range of distance. At what point should the preceding runner be when
the succeeding runner starts his sprint so as to reduce the overall timing.
Page 38 of 238

Technology and Modelling
In both the tasks described and carried out by teachers in the SBPD programme, some form of
technological tools was used.
In the first task, students created and edited online documents to facilitate better sharing and
discussion. The technological tool had enabled a much richer and more focused discussion
amongst students as they worked on the modelling task. Teachers were also able to monitor
students’ thought process and assess their progress throughout the lesson and activity.
The task also provided opportunities for students to engage in simulating and outcome. Some of
the dartboards had designs that contained irregular shapes and it was not easy to calculate the
theoretical probabilities of a randomly thrown dart landing in them. An estimate needed to be
found and this involved having to run simulations on a computer, which required skills and
experience in coding.
In the second task, after collecting data, students needed to construct a function to represent the
runner’s running profile. A graphing calculator with regression tools may be used to construct the
required function. Alternatively, as in this case, MS Excel’s Solver tool may also be used to fit a
function to the collected data. In the present study, students had earlier been taught the Solver tool
technique through examples similar to that described in [19]. A sample of students’ work on this
problem is shown in Figure 6.
(a) MS Excel worksheet utilizing Solver tool
(b) Graphical output showing that baton
should be exchanged when preceding
and succeeding runners are running at
the same speed.
In this case, Point P should then be 8 m
from Point S
Figure 6: Sample of students’ work for the Relay Race problem
Page 39 of 238

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