Teaching Mathematics Online: Emergent Technologies and Methodologies

415 Pages·2011·14.36 MB·English

by  Maria A. Huertas, Sven Trenholm, Cristina Steegmann

Teaching Mathematics
Online:
Emergent Technologies and
Methodologies
Angel A. Juan
Open University of Catalonia, Spain
Maria A. Huertas
Open University of Catalonia, Spain
Sven Trenholm
Loughborough University, UK
Cristina Steegmann
Open University of Catalonia, Spain
Senior Editorial Director: Kristin Klinger
Director of Book Publications: Julia Mosemann
Editorial Director: Lindsay Johnston
Acquisitions Editor: Erika Carter
Development Editor: Michael Killian
Production Editor: Sean Woznicki
Typesetters: Christen Croley, Adrienne Freeland
Print Coordinator: Jamie Snavely
Cover Design: Nick Newcomer
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Editorial Advisory Board
Elena Barbera, eLearn Center, Open University of Catalonia, Spain
Antonio F. Costa, Universidad Nacional de Educación a Distancia, Spain
Thanasis Daradoumis, University of the Aegean, Greece
Kellie S. Grasman, Missouri University of Science & Technology, USA
Susan L. Kutryb, Hudson Valley Community College, USA
Fernando Pestana da Costa, Universidade Aberta, Portugal
Francesca Pozzi, Istituto Tecnologie Didattiche CNR, Italy
Teresa Sancho, Open University of Catalonia, Spain
List of Reviewers
Buma Abramovitz, Ort Braude College, Israel
Giovanna Albano, Universita degli Studi di Salerno, Italy
Juan G. Alcazar, Universidad de Alcala, Spain
Matthew Badger, University of Birmingham, UK
Elena Barbera, eLearn Center - Open University of Catalonia, Spain
Barry Cherkas, Hunter College - CUNY, USA
Antonio F. Costa, Universidad Nacional de Educación a Distancia, Spain
Hans Cuypers, Eindhoven University of Technology, The Netherlands
Thanasis Daradoumis, University of the Aegean, Greece
Blazenka Divjak, University of Zagreb, Croatia
Kellie S. Grasman, Missouri University of Science & Technology, USA
Boris Horvat, University of Ljubljana, Slovenia
Jiri Hrebicek, Masaryk University, Czech Republic
Maria Antonia Huertas, Open University of Catalonia, Spain
Daniel H. Jarvis, Nipissing University, Canada
Angel A. Juan, Open University of Catalonia, Spain
Manuel Juarez-Pacheco, CENIDET, México
Susan L. Kutryb, Hudson Valley Community College, USA
Agata Lapedriza, Open University of Catalonia, Spain
Birgit Loch, Swinburne University of Technology, Australia
Rafael Lopez, Universidad de Granada, Spain
Ciarán Mac an Bhaird, National University of Ireland Maynooth, Ireland
Maria Meletiou-Mavrotheris, European University of Cyprus, Cyprus
Travis Miller, Millersville University, USA
Morten Misfeldt, The Danish School of Education, Denmark
Diana S. Perdue, Pride Rock Consulting, USA
Fernando Pestana da Costa, Universidade Aberta, Portugal
Francesca Pozzi, Istituto Tecnologie Didattiche - CNR, Italy
Jordi Saludes, Universitat Politecnica de Catalunya, Spain
Teresa Sancho, Open University of Catalonia, Spain
Fernando San Segundo, Universidad de Alcala, Spain
Jason Silverman, Drexel University School of Education, USA
Cristina Steegmann, Open University of Catalonia, Spain
Dirk Tempelaar, Maastricht University School of Business and Economics, The Netherlands
Sven Trenholm, Loughborough University, UK
Alexander Vaninsky, Hostos Community College - CUNY, USA
Joe Ward, Loughborough University, UK
Table of Contents
Foreword .............................................................................................................................................viii
Preface .................................................................................................................................................... x
Acknowledgment ................................................................................................................................ xiv
Section 1
Blended Experiences in Mathematics E-Learning
Chapter 1
A Model for Asynchronous Discussions in a Mathematics Content Course .......................................... 1
Travis K. Miller, Millersville University, USA
Chapter 2
A Blended Learning Approach in Mathematics .................................................................................... 22
B. Abramovitz, ORT Braude College, Israel
M. Berezina, ORT Braude College, Israel
A. Berman, Technion Israel Institute of Technology, Israel
L. Shvartsman, ORT Braude College, Israel
Chapter 3
Screencasting for Mathematics Online Learning: A Case Study of a First Year Operations Research
Course at a Dual Delivery Mode Australian University ....................................................................... 43
Birgit Loch, Swinburne University of Technology, Australia
Chapter 4
Mathematics Education: Teaching and Learning Opportunities in Blended Learning ......................... 60
Giovannina Albano, Università di Salerno, Italy
Chapter 5
Best Practices for Hybrid Mathematics Courses .................................................................................. 90
Diana S. Perdue, Intare Educational Resources, USA
Chapter 6
Implementation of Learning Outcomes in Mathematics for Non-Mathematics Major by Using
E-Learning .......................................................................................................................................... 119
B. Divjak, University of Zagreb, Croatia
Section 2
Pure Online Experiences in Mathematics E-Learning
Chapter 7
Online Communities of Practice as Vehicles for Teacher Professional Development ........................ 142
Maria Meletiou-Mavrotheris, European University, Cyprus
Chapter 8
Mathematics Bridging Education Using an Online, Adaptive E-Tutorial:Preparing International
Students for Higher Education ............................................................................................................ 167
Dirk T. Tempelaar, Maastricht University School of Business & Economics, the Netherlands
Bart Rienties, University of Surrey, UK
Wolter Kaper, Universiteit van Amsterdam, the Netherlands
Bas Giesbers, Maastricht University School of Business & Economics, the Netherlands
Sybrand Schim van der Loeff, Maastricht University School of Business & Economics, the Netherlands
Leendert van Gastel, Universiteit van Amsterdam, the Netherlands
Evert van de Vrie, Open Universiteit Nederland, the Netherlands
Henk van der Kooij, Universiteit Utrecht, the Netherlands
Hans Cuypers, Technische Universiteit Eindhoven, the Netherlands
Chapter 9
Teaching Mathematics Teachers Online: Strategies for Navigating the Intersection of Andragogy,
Technology, and Reform-based Mathematics Education .................................................................... 187
D. H. Jarvis, Nipissing University, Canada
Chapter 10
Developing Teachers’ Mathematical Knowledge for Teaching through Online Collaboration .......... 200
Jason Silverman, Drexel University, USA
Ellen L. Clay, Drexel University, USA
Chapter 11
Self-Regulated Learning and Self Assessment in Online Mathematics Bridging Courses ................ 216
R. Biehler, University of Paderborn, Germany
P. R. Fischer, University of Kassel, Germany
R. Hochmuth, University of Kassel, Germany
Th. Wassong, University of Paderborn, Germany
Chapter 12
Long-Term Experiences in Mathematics E-Learning in Europe and the USA ................................... 238
Sven Trenholm, Loughborough University, UK
Angel A. Juan, Open University of Catalonia, Spain
Jorge Simosa, Massachusetts Institute of Technology, USA
Amilcar Oliveira, Universidade Aberta, Portugal
Teresa Oliveira, Universidade Aberta, Portugal
Section 3
Mathematics Software & Web Resources for Mathematics E-Learning
Chapter 13
My Equations are the Same as Yours! Computer Aided Assessment Using a Gröbner Basis
Approach ............................................................................................................................................. 259
M. Badger, University of Birmingham, UK
C. J. Sangwin, University of Birmingham, UK
Chapter 14
Interactive Web-Based Tools for Learning Mathematics: Best Practices ........................................... 274
Barry Cherkas, Hunter College of the City University of New York, USA
Rachael M. Welder, Hunter College of the City University of New York, USA
Chapter 15
NAUK.si: Using Learning Blocks to Prepare E-Content for Teaching Mathematics ........................ 307
M. Lokar, University of Ljubljana, Slovenia
P. Lukšič, University of Ljubljana, Slovenia
B. Horvat, University of Ljubljana, Slovenia
Chapter 16
Software Tools Used in Math Refresher Courses at the University of Alcala, Spain ......................... 327
J. G. Alcázar, University of Alcalá, Spain
M. Marvá, University of Alcalá, Spain
D. Orden, University of Alcalá, Spain
F. San Segundo, University of Alcalá, Spain
Chapter 17
Formula Editors and Handwriting in Mathematical E-Learning ........................................................ 350
Morten Misfeldt, Aarhus University, Denmark
Anders Sanne, Norwegian University of Science and Technology (NTNU), Norway
Chapter 18
The Role of Technology in Mathematics Support: A Pilot Study ....................................................... 367
Ciarán Mac an Bhaird, National University of Ireland Maynooth, Ireland
Ann O’Shea, National University of Ireland Maynooth, Ireland
About the Contributors .................................................................................................................... 384
Index ................................................................................................................................................... 394
viii
Foreword
There is a global perception that mathematics is surrounded by a special aura that places this discipline
in a rather unbalanced position. On the one hand, mathematics is all around us, permeating everything,
and has been created to simplify our world by building models to better explain and understand our
reality. On the other hand, mathematics appears rather complex due to a broad range of variables; for
example, its intricate notation. Nevertheless, and also quite inexplicably, mathematics enjoys a massive
consensus around the world: theorems, formulas, principles, methods, and so forth are identical from
one continent to the other. As if to further enhance its splendour, mathematics is known as the “queen
and servant of the sciences” because it supplies the needs of other sciences (physics, economics, geol-
ogy, engineering, etc.).
However, teaching mathematics is not mathematics itself, it is a completely different issue. Curricula,
teachers, and institutions must deal with the perceptions and emotions provoked by this divine discipline.
Hundreds of papers have been written on the subject of the anxiety caused by mathematics learning,
not to mention feelings of frustration and a lack of self-confidence experienced along the path towards
accomplishment. Linked to this, a common belief can be identified in the educational arena: students
have the sense that one is either good or bad at mathematics. Those who are gifted at mathematics are
believed to be blessed with divine inspiration thus making their locus of control more external. These
feelings are virtually exclusive to learning mathematics. Of course, successfully learning mathematics
is also related to enthusiasm and empowering, both of which are required for achieving the prized goal
of autonomy. It is not easy to find the middle ground when talking about perceptions and behaviour and
that makes the teaching work in a mathematics classroom all the more difficult. It is also a challenge for
teachers to gracefully cross the bridge separating mathematics (as a discipline) from teaching mathemat-
ics (making it meaningful), tackling Chevallard’s evocative “didactical transposition”.
In this context, teaching and learning mathematics has an ally in online education. Like all partner-
ships it can be for better or for worse; in the end, it mostly depends on one’s willpower and ability.
In equal shares, online education has the ability to either improve or worsen mathematical teaching.
It is taken for granted that when designing an online course the ultimate aim is to improve teaching and
learning and not simply to reach the greatest possible number of students with the least amount of effort
by breaking through barriers of space and time. However, in both the design and implementation phases
there are important decisions to be made that have no routine answers, much less any solutions that can
copied directly from face-to-face education. On the contrary, teaching delivered using a technological
medium is supposed be extremely considerate due to the fact that technology mediation is able to take
us from one extreme to the other without us really being aware of the journey. Teachers are able to bring
together a group of elements that enhance significant learning all in the same course. In this modality of
ix
learning it is easy to succumb to the temptation of teaching large numbers of students while expending
the minimum amount effort. Nonetheless, quality online education without a reasonable and continuous
investment is as yet unknown.
In consequence, different online decisions can standardise teaching and make it poorer, but in their
favour they are easy to implement in online classes. Some examples of these decisions include: choosing
increased automation; making teaching homogeneous; a preference for quantity as opposed to person-
alisation; opting for formalisation that reduces flexibility; or an inclination towards poor feedback and
study based on repetition and low skills levels.
Putting aside this negative aspect, online teaching has been called to do much more and to truly
provide an amplifier for teachers that goes beyond borders and offers an authentic study framework to
help students better understand and live in today’s world.
Among its more positive aspects, the online alternative has the potential to catapult the educational
community towards providing more transparent ideas and processes that present facts and events neatly
from the inside. Moreover, thanks to its ability to bring the real world into online classrooms by simu-
lating or capturing everyday situations, students are encouraged to develop high-level skills, such as
argumentation and reflection relating to the processes they have experienced at first hand.
This approach does not ignore the scalable values technology provides education in terms of measur-
ability, counting, and more, but we all need to go a step further. More than simply building a stereotype
of online mathematical education by making the most basic choices, we aspire to use online education
as a mindtool as a whole (extending on Jonassen’s terminology) thus giving teachers and students the
right to expand their competences when working with technology to carry out tasks that they would not
be able to do alone. Online mathematics education defined as a mindtool in this manner helps to better
capture, visualise, and manipulate hidden processes, reasoning, and facts that otherwise exist only in
teachers’ minds and are barely intuited by students. The opportunity we have in online mathematical
education nowadays is precious, and our decisions are open to innovation.
Elena Barbera
Universitat Oberta de Catalunya, Spain
December 16, 2010
Elena Barbera is a Doctor in Psychology from the University of Barcelona (1995). She is currently Director of Research
for the eLearn Center at the Universitat Oberta de Catalunya in Barcelona (Spain). She is also an Adjunct Professor for the
international doctorate in Nova Southeastern University in Florida (USA). Her research activity is specialised in the area of
educational psychology, a field in which she has more than a hundred publications, conferences, and educational courses,
relating in particular to knowledge-construction processes and educational interaction in e-learning environments, evaluating
educational quality and assessing learning, distance learning using ICT, and teaching and learning strategies. As head of the
EDUS (Distance School and University Education) research group, she participates in various national and international proj-
ects related to online teaching and learning and student assessment. She is an external and independent evaluator of research
projects promoted by local, national, and European Union (e-learning and lifelong programme) bodies, and she collaborates
with international organisations in developing knowledge by organising congresses and international awards as a member of
their scientific committees.