1. Combining Various Technologies in Effective Online
Math Instruction – A Nationwide Secondary Level
Math Project
Przemysław Kajetanowicz
przemyslaw.kajetanowicz@pwr.wroc.pl
Jędrzej Wierzejewski
jedrzej.wierzejewski@pwr.wroc.pl
Institute of Mathematics and Computer Science
Wrocław University of Technology
Wrocław, Poland
Abstract—Since January 2010, Wrocław University of Technol-
ogy has been involved in an educational project under the name
of “Mathematics – Reactivation”. The second-named author is
the project coordinator and both authors play the primary role
as content creators. The project goal is to provide Polish secon-
dary schools with a complete e-course in mathematics, covering
the whole math curriculum in secondary schools. Automatic as-
sessment of the learner’s progress is one of crucial elements of
the course. The demand for such additional teaching support
emerged in view of the continuing decline in the quality of math
instruction together with recent restoration of obligatory gradua-
tion math exam in Polish education system. The target group of
project beneficiaries amounts to 14000 students. The project goes
far beyond mere dissemination of electronic content. The partici-
pants are involved in a sophisticated system of regular electronic
testing, with the results stored in a database and later processed
statistically. In that way, the participating schools receive a com-
prehensive, interactive supplement to the traditional teaching
process. The e-learning solutions that the authors had previously
implemented at the university level have now been enhanced. A
variety of technologies have been combined. The usefulness of the
whole system is continuously being confirmed by the so-far gath-
ered feedback on the part of participating teachers and students.
Keywords-online mathematics;secondary education;
I. INTRODUCTION
E-learning solutions developed by the authors in Wroclaw
University of Technology over past six years are now an im-
portant element of math courses delivered to first-year students.
Various aspects of those solutions (regarding both technology
and pedagogical issues) can be found in [1] and [2].
In 2010, the compulsory graduation exams in mathematics
were restored in Polish secondary-level education system. Prior
to that, for over 20 years the exit exam in mathematics had
played a diminished role, as it was being (or being not) taken at
the discretion of a graduating secondary-school student, with-
out any obligation. Typically, the exam was being chosen by
those students who planned their professional careers as engi-
neers, scientists or economists. For the overwhelming remain-
ing part of Polish secondary school students, the mathematics
appeared for years as a troublesome subject of no importance.
The long-lasting lack of obligatory math graduation exams led
to continuously decreasing motivation among the students to
learn and among the teachers to effectively teach.
The restored math graduation exam requirement was thus
suddenly confronted with years of deterioration of math in-
struction level in Poland. It was obvious that any means to im-
prove the teaching process and to lever the students’ motivation
would be of great use.
In 2010, an educational project whose aim was to support
secondary level math instruction was launched at Wrocław
University of Technology. The project is financed by the EU
Human Capital Operational Programme and by the Polish Min-
istry of Education.
The second-named author is the project coordinator. Both
authors are primary content developers. The first-named author
is additionally responsible for supervision of the content aggre-
gation and dissemination. The number of persons involved in
the project either full-time or part-time totals to 13.
In the paper, we are not going to discuss formal aspects of
the project, like reporting requirements, strict procedures of
enrollment process and other paperwork-related elements. We
will address functional, technological and pedagogical issues
instead. In general, we wish to give a pretty complete picture of
online-assisted teaching process where traditional in-class in-
struction can be (and actually is) enhanced through the pres-
ence of well-designed and attractive electronic content.
We first describe the way that electronic content functions.
After that, we discuss a few of behind-the-scenes technological
solutions and standards that have been employed in the process

2. of content creation and aggregation. Finally, we give a brief
overview of the resulting instructional system in which a
teacher-learner interaction is blended with online support.
II. CONTENT ORGANIZATION AND
FUNCTIONALITY
The main idea behind the process of content design was
that a learner should work in learning environment that
strongly encourages active study.
Accordingly, the content is highly interactive and tries to
keep a learner busy while studying. That goal is attained by
offering a great deal of opportunity to experiment and to prac-
tice.
A. Content organization
The mathematical subject matter of the e-course is divided
into 14 chapters covering the secondary school math curricu-
lum. Speaking traditionally, the content can be viewed as an
electronic textbook with strictly linear organization on the one
hand, but with a very high interactivity level on the other. We
wish to emphasize at this point that the very nature of mathe-
matics forces the linearity of the teaching process. Individual
portions of material have to be built one upon the other and
there is no escape from that. There exist, of course, some ex-
ceptions from that rule. For example, two distinct chapters can
be independent of each other in the sense of the lack of mathe-
matical reliance of one on the other. However, this does not
change the fact that predominant part of mathematics must be
taught on the “A first, B next” basis.
From functional viewpoint, each chapter consists of:
• lecture notes,
• e-exercise pages,
• practice e-exam,
• chapter-specific user guide,
• description of grading algorithms,
• index of chapter contents.
The lecture notes provide a learner with a detailed exposi-
tion of math concepts and methods. The same “keeping a
learner busy” approach has been assumed that proved so effec-
tive in academic e-courses created by the authors in the past.
When a learner comes across a new math concept or method,
he or she is immediately offered the opportunity to self-check
whether the particular portion of material „clicks in place” in
the learner’s mind. This goal is being accomplished twofold.
First, suitable e-exercises follow the presentation of a specific
portion of math (we discuss e-exercises in greater detail later in
the paper). Secondly, a great deal of mathematics presented in
the lecture notes is additionally accompanied by interactive
simulators that give a learner the self-experimentation opportu-
nity. E-exercises and simulators have strong positive impact on
developing active learning attitudes during the study.
The exercise pages contain e-exercises that address an indi-
vidual learning unit. They mimic what is called “review prob-
lems” at the end of a chapter section of a traditional textbook.
A practice e-exam provides a learner with a collection of e-
exercises typical for the chapter contents. We discuss the de-
tails of e-exams functionality later in the paper.
The sophistication of user interface in an e-exercise often
goes far beyond simple edit fields. Therefore each chapter pro-
vides a detailed user guide as to how to use respective user
interface elements.
Through e-exercises, a learner works math problems of
many different types. The learner’s solution is immediately
graded, according to a grading algorithm dedicated to a specific
problem type. To give a learner a better insight into how
his/her solutions are evaluated, each chapter includes a detailed
description of the grading typical to chapter-specific math
problem types.
B. E-exercises
The e-exercises play a crucial role as self-assessment tools.
As the design and the creation of consecutive chapters pro-
ceeds, the corresponding math material is being inspected from
the point of view of the optimal choice of related math prob-
lems. It is assumed that by the end of the project, over a 1000
math problem types will have been put in action.
From a learner’s perspective, an e-exercise is an electronic
object capable of the delivery of practically unlimited number
of math problems of common type. A learner reads the prob-
lem formulation in an e-exercise window, then works the prob-
lem by hand and finally enters his or her solution through user
interface. For some problem types, the requirements of the user
input are limited to entering numerical values. In many prob-
lems, however, the solution is supposed to be provided in a
more complicated way (e.g., as a graph object, in which case
specially designed graphing tools are available in the exercise
window).
We will discuss some technology behind the scenes later –
at this point we only wish to mention that an individual e-
exercise acts as a Java applet that is embedded either in a lec-
ture notes XHTML page or in an HTML exercise page. An e-
exercise behavior mimics the process of giving a learner one
math problem after another with immediate grading upon the
user solution input. Below we list the most important features
of sophisticated functionality of an e-exercise:
• ability to of generate a (theoretically) unlimited se-
quence of problems of a given type,
• immediate grading of a learner’s solution,
• availability of a short hint to help a learner,
• availability of short help on the user interface,
• presence of virtual calculator,
• availability of the step-by-step presentation of the cor-
rect solution process,

3. • completeness check of the user input (this means that
a learner is warned that a part or the whole of the re-
quired input has not been provided),
• initial correctness check (this means that a learner is
given one or more chances to correct his/her answer
before the grading is carried out).
E-exercises can be easily fine-tuned through a special ex-
ternal configuration file (thus avoiding the necessity to tamper
with the source code). For example, the content designer can
decide on the specific problem’s difficulty level, or on the
maximum score for the correct solution. The designer can even
set probabilities with which certain math problems are gener-
ated within a given problem type. For example, the designer
can make sure that in the “solve the given quadratic equation”
e-exercise, the generated quadratic equations follow the rule of
uniform distribution of “no roots”, “one root” and “two roots”
cases.
C. E-exams
Individual e-exercises can be combined into e-exams. An e-
exam can be viewed as a collection of single e-exercises, each
addressing a specified math problem type.
The application of e-exams is twofold.
Firstly, they serve as practice utilities, as described earlier
(each chapter offers a practice exam at the end).
Secondly, e-exams constitute an important formal part of
the project: each participating school is obligated to give 14 e-
exams to its students in a school computer lab by the end of the
project. We are addressing this in the last section of the paper.
Each exam incorporates such features as time constraint,
variability in selecting the problem set (so that no two students
get identical problem sets), security and storing the student’s
results in a secure database. A virtual scientific calculator is
available in the exam window. Flexible grading systems can be
set up without the intervention in the source code. In that way,
an e-exam mimics the traditional in-class testing.
Which is most important, an e-exam is always configured
in such a way that a different collection of problems is pre-
sented to a learner each time that the exam is launched. This
means that a learner is motivated to take a practice exam more
than once after he or she completes the study of a chapter. In
the case when an e-exam plays the administrative role in the
assessment process of participating students, no two students
receive the same collection of problems to solve. Which is
more, even if two students happen to get a problem of the same
type on their exams (e.g., “solve the given quadratic equa-
tion”), the actual problem is different for each student (i.e., the
two quadratic equations have different coefficients).
D. Simulators
Interactive simulators are scattered throughout lecture notes
pages. They provide a learner with the opportunity to “play
with mathematics”, so to speak. The authors always take time
to consider whether an interactive illustration is applicable at a
given point of material exposition. As useful as “live” experi-
menting can be, it should not draw the learner’s attention from
the fact that mathematics cannot be learned just by playing
with gadgets. In general, the presence of simulators is limited
to the cases when presented math notions are likely to pose a
conceptual difficulty.
III. COMBINING VARIOUS TECHNOLOGIES
It is not a coincidence that the title of the present section is
a part of the paper’s title. The author’s experience gained dur-
ing years of math e-courses development shows that before
starting the process of content creation one has to identify
which kinds of technological tools to reach for and which stan-
dards to follow in order to get the most effective results at rela-
tively small cost of time and effort.
A. MathML
One of the main differences between presenting typically
descriptive knowledge and presenting mathematical content
lies in the fact that mathematics uses special notation. The mo-
tivating role of structurally consistent math display with good
quality cannot be overestimated; the more so as we deal with
an inexperienced young learner for whom to plough through
rigid mathematical notation and terminology is a problem in
itself.
The TeX standard is widely used for typesetting mathemat-
ics, which is perfect for print purposes. When mathematical
expressions are supposed to be part of a web page, however,
one needs to reach either for graphical (predominantly GIF)
representation of mathematics or to decide on MathML. The
problem is that even nowadays some browsers do not support
MathML, and other need plug-ins to display MathML-encoded
expressions in an accurate way.
While creating academic e-courses during 2004-2007, the
authors were publishing lecture notes pages with mathematical
expressions imbedded as GIF images. The outcome was satis-
factory, but not even close to anything that can be seen in
printed textbooks. Besides, the resulting HTML files had to be
accompanied in an incredible amount of graphics. In Java-
driven exercises, on the other hand, mathematics was being
displayed through special templates that had to be meticulously
developed from the floor.
In the present project, the authors decided to completely
rely on MathML both in lecture notes and in Java-driven math
problems.
McKichan’s Scientific Notebook was purchased for author-
ing purposes and proved to be an excellent choice. Not only
does it export mathematics in standard-compliant MathML
format, but it is also extremely productive when it comes to
entering mathematical expressions during the writing process.
Additionally, it also enables the user to seamlessly embed por-
tions of HTML code (e.g. Java applets invocations) in the
document under work.
The final content is displayed with true “printed mathemat-
ics” quality in Mozilla Firefox and Internet Explorer browsers
at the price of some additional browser configuration (the
MathPlayer plug-in for IE or a collection of dedicated fonts in
the case of Mozilla). Additionally, the MathPlayer provides a

4. nice feature that enables a learner to magnify the view of each
individual math expression on the web page.
A part of MathFlow SDK Java library was purchased from
Design Science, Inc. and combined with specially developed
Java classes to support MathML in e-exercises. The library’s
primary role is to dynamically render MathML encoded by a
programmer. As a result, a learner sees mathematics displayed
in very high quality in math problem formulations and solution
presentations.
B. Java
The so-far experience and the presence of new MathML-
related libraries made Java an unquestionable choice as the
programming language in which to code e-exercises, e-exams
and some of the simulators.
The results of e-exams that the participating students take at
their school computer labs are stored in a database that addi-
tionally hosts the students themselves. The database connec-
tivity is supported by Java, too.
C. Geogebra
Geogebra is open-source math software that has been earn-
ing fast-growing recognition among math teachers in many
countries over past years.
Geogebra is used in two ways for the project purposes. In
the first place, it serves as a tool for placing graphics (geomet-
rical drawings, graphs of functions etc.) in lecture notes. Sec-
ondly, most of the interactive simulators are created with the
use of one of Geogebra’s nicest features: the ability to export a
vivid, interactive mathematical illustration as a Java applet that
can be further embedded in a web page. It is interesting to note
that the presence of those simulators in the lecture notes in-
spired some teachers to reach for Geogebra in their class teach-
ing.
D. Learning Environment System
Individual chapters of the e-course are aggregated into IMS
packages and made available to participating teachers and stu-
dents through a Moodle platform. The administrative e-xams
(i.e. those that the participating schools are bound to give their
students at regular time intervals) are packed into SCORM
modules and placed in Moodle, too.
The platform itself has been tailored to specific project
needs. It is connected to an additional external database that
supports administrative e-exams. The platform look has been
designed in such a way that a learner’s attention is not unneces-
sarily distracted from the essential goal – learning.
With the exception of Scientific Notebook and MathFlow
SDK library, all the software used for the content creation is
non-commercial.
IV. PEDAGOGY: E-COURSE AT WORK
The above-described e-course is the main part, but still a
part of the “Mathematics Reactivation” project. The wording
“silent, patient and user-friendly teacher’s assistant” is proba-
bly the best short description of the role that the e-course plays.
The participating schools are obligated to carry out 14 e-
exams during the project. In that way, in addition to regular in-
class teaching, each student will have taken 14 online exams in
mathematics by the end of his or her 3-year secondary school
track. The fact that those exams are compulsory is an additional
motivating factor for participating students. The results of the
exams are available to teachers through a database and can be
used as an additional source of information as to their students’
progress.
The obligatory e-exams are designed by project developers
in accordance with secondary school math programme. Since
the pace and the order in which mathematics is taught can vary
from school to school, the respective chapter-specific e-xams
are published as soon as the corresponding content has been
published and are available to all schools throughout the pro-
ject duration. The decision as to specific dates on which respec-
tive exams are given is left to the discretion of each school.
The content itself can be used by a participating teacher in
many various ways. At the very minimum, the teacher may
treat the online material as a supplementary aid only, by just
encouraging the students for additional self-study. On the other
hand, the teacher may want to make an extensive use of the
online materials in class, e.g. by presenting selected definitions
or theorems on screen rather than writing them on the board by
hand. Some math teachers are already planning to give up tra-
ditional textbooks in their classes. Such kind of approach con-
siderably speeds up the pace of a lesson. Rather than hurriedly
putting down the instructor’s notes from the board, a student
can concentrate on listening. The teacher, in turn, has more
time for discussing various subtleties of math concepts and to
provide students with more help on coping with typical diffi-
culties. The time pressure being considerably decreased, class
lessons gain in value and can be handled in a more appealing
way.
The e-learning ideas and solutions implemented over past
years to enhance university level mathematics seem to be
working equally well in secondary education. As of this writ-
ing, the first group of nearly 7000 students has finished the first
year of the participation in the project. A similar number is in
the process of enrollment. The participants’ results during the
national graduation math exam in 2013 will provide further
verification whether ant to what extent the project had positive
impact of the effectiveness of secondary level teaching.
REFERENCES
[1] P.Kajetanowicz, J.Wierzejewski, “E-learning in Mathematics at
Wrocław University of Technology – the Present and the Future”, 9th
International Conference Virtual University, Bratislava, Slovakia,
December 2008
[2] P.Kajetanowicz, J.Wierzejewski, “Highlights and Downsides of E-
learning in Mathematcis – Wrocław University of Tehchnology
Experience”, 10th
International Conference Virtual University,
Bratislava, Slovakia, 2009