Int. J. Cont. Engineering Education and Life-Long Learning, Vol. 18, Nos. 5/6, 2008 657
Copyright © 2008 Inderscience Enterprises Ltd.
Technological challenges of teaching mathematics
in a blended learning environment
Petr Sojka*
Faculty of Informatics,
Masaryk University,
Brno 60200, Czech Republic
E-mail: sojka@fi.muni.cz
*Corresponding author
Roman Plch
Faculty of Science,
Masaryk University,
Brno 61137, Czech Republic
E-mail: plch@math.muni.cz
Abstract: This paper describes the following technological aspects of blended
learning of mathematics: effective preparation of electronic teaching materials
suited for different students’ needs (PDF, HTML and XML + MathML parallel
generation), videotaping of mathematics lectures, automated (self)testing of
subjects taught using the computer algebra system Maple and preparation of
interactive teaching materials with the MapleNet technology. Authors describe
results achieved and the experience gained during preparation and
implementation of these challenges in a Calculus course taught at the Faculty of
Science, Masaryk University in Brno, in autumn 2006.
Keywords: blended learning; calculus; format conversion; Maple; MapleNet;
maplets; MathML; PDF; publishing technologies; teaching mathematics;
technological challenges; TEX; tex4ht; video.
Reference to this paper should be made as follows: Sojka, P. and Plch, R.
(2008) ‘Technological challenges of teaching mathematics in a blended
learning environment’, Int. J. Continuing Engineering Education and Life-Long
Learning, Vol. 18, Nos. 5/6, pp.657–665.
Biographical notes: Petr Sojka received his PhD in Computational Linguistics
at the Masaryk University, Brno, Czech Republic. Currently, he is an Assistant
Professor at the Faculty of Informatics, Masaryk University in Brno. His main
research interests include electronic publishing, textual information systems
and blended learning support. He has published over 70 papers.
Roman Plch received his PhD in Mathematics at the Masaryk University, Brno,
Czech Republic. Currently, he is an Assistant Professor in the Faculty of
Science, Masaryk University in Brno. His main research interests include
computer algebra systems, technology in teaching mathematics and blended
learning support.
658 P. Sojka and R. Plch
1 Motivation
Teaching of mathematics is specific as are the technological challenges of our digital age
for its support. Teaching of mathematics has its traditions, settled over centuries and
glorified by traditional professors, in stark contrast to the possibilities of Information
Communication Technologies (ICT) change every year.
We have prepared several electronic materials and tried several new approaches to
support students of mathematics at Masaryk University in Brno, who are enrolled in a
Calculus course.
2 Objectives and their realisation
The Information System of Masaryk University (http:/is.muni.cz, IS MU) is currently
being enhanced to support blended learning methods which includes an extensive usage
of ICT. It allows authenticated access to structured study materials, there is support for
students’ (self-)testing and examination, and new functions are continuously being added
with the aim of it becoming fully fledged Learning Management System (LMS). As
several specific needs for supporting the teaching of mathematics were missing, we have
investigated the possibilities of their realisation in the pilot project of support of the
course M3501 Calculus III (Metric spaces and Calculus of several variables) in the
autumn semester 2006.
2.1 Study materials in different formats
Every student is different. ‘Put yourself in the reader’s place,’ is the oft repeated
incentive. Students study by reading the course materials on their notebook or computer
screens, doing exercises with pen and paper or using a Computer Algebra System (CAS)
on the computer, testing themselves by solving special tests in IS MU or even hearing
and watching the lectures repeatedly.
Mathematics textbook authors prefer writing their text and exercises once only
(usually in TEX), rather than changing their setup every semester. On the other hand,
there are the new possibilities: of high resolution computer screens, internet and PC
availability almost everywhere by almost every student. New formats brings new
possibilities: PDF allows high fidelity of textbook delivery, MathML (or OMDoc) soon
to be accepted as the structured text format for exchanging and delivery of mathematics
electronic documents. On the other hand, TEX notation remains preferred by authors for
editing and authoring.
For Calculus III course there were two textbook materials prepared in LaTeX,
i.e. Došlá and Došlý (2000) and Došlá, Plch and Sojka (1999). We have evaluated that at
least four formats are of interest to the students:
1 PDF suitable for printing: electronic copy of the printed textbooks (no colours).
2 PDF suitable and designed for PC screen reading: PC screen has different aspect
ratio, resolution, allows usage of colours and easy searching.
3 HTML for reading in the ‘old-fashioned’ web browsers, with math formulas
as pictures.
4 XHTML/MathML for reading in ‘new generation’ web browsers, allowing
‘cut&paste’ functionality with CAS systems.
Technological aspects of blended learning of mathematics 659
We have managed to automate the generation of all four formats of the two textbooks
(more than 300 pages) from single LaTeX sources. We have used pdfTEX
(http://en.wikipedia.org/wiki/PdfTeX) and tex4ht (http://www.cse.ohio-state.edu/~gurari/
TeX4ht/) programs.
We have added conditionals in the single LaTeX source file for every book to switch
settings for these four versions, and prepared Makefile to automate the processes as much
as possible.
PDF suitable for printing. Printing without colours is sufficient for both textbooks,
however pictures need to be adjusted as high-resolution bitmaps or vectors only. Several
versions were generated for different paper formats (A4 with wide margin for comments
and B5). Navigation and references in the text were implemented by standard LaTeX
citation macros, see Figure 1.
Figure 1 Design suitable for printing (PDF)
660 P. Sojka and R. Plch
PDF designed for screen reading. A special version for reading on standard computer
screen (width:height ratio 4:3, cca 100 ppi, colour) has been designed – see Figure 2.
Colour is used extensively for the reader’s navigation in the hypertext. Animations have
been added too, based on the possibilities of embedded Java-Script in the PDF
(Sojka, 2003).
Figure 2 Design for computer screen (PDF) (see online version for colours)
2.1.1 Conservative HTML
Having documents published as web pages in HTML 4 which is one of the safest and
most foolproof options. We have tested several possible options for automatically
generating HTML versions of the books from LaTeX sources: LaTeX2html, TEX4ht
(http://www.cse.ohio-state.edu/~gurari/TeX4ht/), home-grown converter in Python,
tex2page (http://www.ccs.neu.edu/home/dorai/tex2page/), hyperlatex, Hevea and others.
We ultimately chose TEX4ht for its stability and power, although it is a rather complex
system and has to be enriched for new authors’ markup. For mathematics not expressible
in HTML, small transparent images in PNG format are generated, see Figure 3.
2.1.2 Progressive MathML
MathML is a W3C proposal for math in XML on the internet, already supported by
several browsers and almost all computer algebra systems. There are several tools
converting TEX to present MathML (Tralics (http://www-sop.inria.fr/apics/tralics/),
LaTeXML (http://dlmf.nist.gov/LaTeXML/), Omega and others). We have stayed with
TEX4ht, as it can be configured for MathML generation as well. This version is
especially useful for blind people, as math covered by MathML does not use any picture
and is the text rendering scales well and quickly in web browsers. It fulfills Web
Accessibility Initiative, http://www.w3.org/WAI (WAI) suggestions, see Figure 4.
Technological aspects of blended learning of mathematics 661
Figure 3 Design for web (HTML) (see online version for colours)
Figure 4 Design for web (MathML) (see online version for colours)
662 P. Sojka and R. Plch
All conversions were done by free software and tools, available as part of the
TeXlive project (http://www.tug.org/texlive). The support of MathML has not yet been
fully implemented in all web browsers, but we expect that the situation will improve
as the benefits are easily seen and widely accepted. We envisage that TEX notation
will remain preferred in math authors community for authorship. This trend is
supported by a plethora of tools supporting TEX math notation on the web (ASCIIMath
http://www1.chapman. edu/~jipsen/asciimath.xml, JSMath http://www.math.union.edu/
~dpvc/jsMath/, IBM texplorer) or publishing systems that allow TEX notation for input
(MathType plugin for Word, 3B2, etc.). For details see Sojka and R ži ka (2007).
2.2 Videotaping lectures
Mathematics professors are used to deliver their lectures ‘the good old way’ using the
blackboard table and chalk. This has several advantages: introducing new definitions and
proof deductions cannot be increased to an incomprehensible speed of reading
Powerpoint bulleted items Tufte (2003).
There are students who benefit from hearing the teacher, and those who have to see
the teacher talking or writing on the board. We have videotaped all the lectures using the
two 3CCD miniDV cameras (Canon DM-XM2). One view was from static camera and
the second was by the cameraman taking the details of the teacher and the blackboard.
The teacher uses a wireless micro-port microphone to grab high quality voice, but in
general, very few constraints are imposed on the teacher. Every videotaped lecture is
available in the AVI format to students in the IS within a few days. A short example of
video has been put on the web so that reader may have closer look at current videotaping
possibilities http://www.fi.muni.cz/usr/sojka/videos/M3501examples.avi.
Last but not least, we have experimented with Flash technology, where PDF and
downsampled videos were loaded in separate windows. We made both materials mutually
crosslinked – seeing a new page in PDF searches for an appropriate sequence in the video
window and vice versa.
2.3 Automated testing and evaluation by Maple
The greatest obstacle to automated exercising in LMS IS (and to the best of our
knowledge in other systems, too) is the automated evaluation of results of mathematical
exercise and subsequent checking of the results. We have designed and implemented a
new type of testing exercises using the computer algebra system Maple. It evaluates the
math the student types into the browser form.
There is a standard function testeq in Maple for testing of equivalence, but it
works only for simple algebraic expressions. For testing other mathematics objects
we wrote a new Maple code, which is able to test algebraic expressions, lists, sets,
matrices, vectors, etc. for equivalence. The corresponding Maple worksheet is available
at http://www.math.muni.cz/~xsrot/mws/newtesteq.mws (Šrot, 2007). The benefit is, that
it should be able to test also nested objects.
So that students were not tempted to cheat by using the powerful Maple functions
(ad diff for derivation tasks), a system of allowed and forbidden function names
(white-list and black-list) was introduced. The functions listed in the black-list are not
allowed for an exercise solution. In the case where a teacher does not explicitly lists the
function names allowed, a white-list is generated from the canonical solution.
Technological aspects of blended learning of mathematics 663
Figure 5 shows an Example test with results. The first question is solved by means of
Maple function diff, which is not allowed in this case. The second question has a wrong
answer. The third question is solved succesfully although the equation is written in a
form different from the given result.
Figure 5 Example test with results (see online version for colours)
2.4 Interactive study with MapleNet
MapleNet (http://www.maplesoft.com/products/maplenet/) is a technology allowing
on-the-fly Maple computations in web applications. With MapleNet, one can create
interactive teaching materials with Java applets, Maplets or Maple worksheets. To read
these materials, the user needs a web browser with Java support enabled and a plug-in
client for connection to the MapleNet server. Students do not have to have Maple
installed on their computer, but may use the power of symbolic computation on their
desk: an important new portable tool for blended learning. Special interactive graphic
applications called Maplets, written in the Maple programming language bring exciting
new possibilities. Maplets are platform independent, thus we can use them on a computer
with Windows operating system as well as on a Linux/Unix machine.
These interactive graphical applications were prepared for support of Calculus III
course (functions of several variables, limits, partial and directional derivatives,
differential, Taylor polynomilas, maximum and minimum values, Lagrange
multipliers and implicit functions). In most cases, the user can display not only the final
answer or the graphical output, but also all solution steps. Maplets are posted in
LMS IS MU using the interactive structured curriculum tool. Their overview can be
viewed on http://melian.ics.muni.cz:8080/maplenet/glozar/, Figure 6 shows partial
derivatives maplet. The purpose of this Maplet is to help the user to compute and
664 P. Sojka and R. Plch
visualise partial and directional derivatives to a surface in three dimensions. The Maplet
is directly accessible from the adress http://melian.ics.muni.cz:8080/maplenet/glozar/
ParcDerE:14444/. For details see Glozar (2007).
Figure 6 Partial Derivatives Maplet (see online version for colours)
3 Conclusion and further work
Evaluation of the new generation of support for the Calculus course shows that new
aspects of the of course are used by the students differently. New possibilities allow
students to choose studying materials according to their preferences, skills and studying
methods. Maplets are powerful visualisation tools that help with the understanding of
covered topics.
Technological aspects of blended learning of mathematics 665
Stepping in the new technology e-learning possibilities is an adventurous exercise
with many ‘no way’ experiences. However, once set up and the students start using them,
we are certain that learning productivity increases considerably. Rather marginal changes
in the teachers’ uses of available technologies may have a drastic impact on the
usefulness and accessibility of math teaching materials.
Acknowledgements
This work has been partially supported by grant 1ET208050401 of Academy of
Sciences R.
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