EXCITING NORMAL DISTRIBUTION - uni-graz.at

EXCITING NORMAL DISTRIBUTION

Karl Josef Fuchs1) ([email protected])
Reinhard Simonovits2) ([email protected])

Bernd Thaller3) ([email protected])

1) University of Salzburg, Department of Mathematics and Informatics Education
2) Handelsakademie (HAK), Grazbachgasse, Graz
3) University of Graz, Institute for Mathematics and Scientific Computing

TABLE OF CONTENTS

TABLE OF CONTENTS..............................................................................................2
ABSTRACT ................................................................................................................3
1 INTRODUCTION ...................................................................................................4
2 THE SOFTWARE M@TH1 DESKTOP ..................................................................5
3 STUDENT’S EVALUATION OF THE SOFTWARE ..............................................8
4 THE PEDAGOGICAL PATTERN........................................................................10
5 A TYPICAL LESSON ..........................................................................................12
6 THE STUDENT’S PROJECT ..............................................................................14
7 SUMMARY ..........................................................................................................16
ACKNOWLEDGEMENT ...........................................................................................17
APPENDIX: PROJECT’S TIME TABLE...................................................................18
REFERENCES.......................................................................................................... 19

1 The math teaching and learning software M@th Desktop 2.0 (MD) contains the modules
Basics including tools for teachers and students and fitting routines, Differenziation,
Integration and MDTools. The homepage of the project is www.deltasoft.at. MD’s engine is
Mathematica. An evaluation copy of MD is downloadable.

2

ABSTRACT

This paper describes a high school project where the math teaching and learning
software M@th Desktop (MD) based on the Computer Algebra System Mathematica
was used for symbolical and numerical calculations and for visualization. After a
short introduction delineating the initial situation, the aims of the project, and the
methodological frame, we describe the teaching process and final outcomes. First
results indicate that computer-based teaching intensifies the students’ activities, their
level of attention, and their contentment with mathematics.

3

1 INTRODUCTION

In this paper we want to report on a project with the title "Testing e-Learning
Resources on Normal Distribution”. Its main goal was to investigate how the teaching
of mathematical statistics could benefit from the use of computer software. The
project was carried through in the school year 2004/5 with math students from a
business high school in Graz, Austria (fifth grade 5bk of the Handelsakademie "HAK"
Grazbachgasse). The math teacher was R. Simonovits, the project was
accompanied, supervised, and evaluated by K. Fuchs and B. Thaller.

The fifth grade is the last grade of the business high schools in Austria. The average
age of the students is eighteen. In recent years, the math curriculum of business high
schools has been reduced to a total of 20 semester hours. Therefore, students have
to attend only two units of math instruction per week (50 minutes per unit). Statistics
is still a mandatory part of the curriculum in the fifth grade, and the discussion of the
normal distribution is its most important chapter.

The 5bk of HAK Graz consisted of eleven girls and four boys. It was a special
information technology class where each student owned a laptop and was already
experienced in computer-assisted mathematics lessons. The students volunteered
unanimously to participate in the project. Their teacher described this class as
extremely talented.

The e-Learning resources used for this project belong to the educational software
M@th Desktop (MD), which was developed under the supervision of R.Simonovits at
HAK Graz in collaboration with the University of Graz. MD is a Mathematica-based
application, with associated content, for teaching mathematics to high-school
students. The user interface of MD consists of notebooks and palettes providing
easily accessible and user-friendly templates for exploring various mathematical
topics (see Section 2). The design of MD lends itself in particular to a blended
learning scenario (see Section 3). The participating students were already familiar
with MD in general.

One outcome of the project should be the answer to the question whether the design
of MD will live up to the requirements of teaching statistics. We evaluated not only
content and technical aspects of the statistics module in MD, but also the impact of
the suggested teaching strategy. The following questions should be answered in
detail:

Concerning technical aspects:

• Are notebooks and palettes instrumental in teaching statistics?
• Is MD appropriate for students to carry out a small project in statistics?
Concerning pedagogical aspects:

• To what extend are the students satisfied with blended learning as a teaching
method?

4

2 THE SOFTWARE M@TH DESKTOP

The software M@th Desktop (MD) is based on the Computer Algebra System
Mathematica. One key advantage of using Mathematica in an educational context is
the following: As it comes with an overwhelming number of built-in functions and
support for various types of visualizations, it is relatively easy (and therefore cheaper)
to develop high-quality educational material, e.g., interactive animations. Moreover,
all Mathematica-based teaching material retains a high degree of adaptability and
can be adjusted to personal needs by teachers and students alike.
MD provides teachers and students with a simplified user interface to Mathematica,
so that only very basic skills are necessary to use the software. Using the tutorial
provided via Mathematica's built-in help browser, a teacher should be able to
familiarize himself with MD within a few hours. If the students are new to
Mathematica, one would need about four one-hour units for the necessary
introduction.
Each module of MD consists of a collection of Mathematica notebooks (interactive
worksheets containing the descriptions of problems and definitions) and associated
palettes (floating windows with embedded buttons for specific calculations and visual
presentations). MD offers various tools for teachers: It renders possible tests,
animations and tables, and provides a large pool of exercises.
The mathematical content of the current version comprises modules for
differentiation, integration, and statistics and thus covers most fields of senior high
school and freshman math, with more modules in the pipeline.
Especially for this project we created the palettes "Normal Distribution - Basics" and
"Normality of Data". The methodologically most interesting idea of coding specific
palettes for the teaching process by teachers and students is discussed in a
dissertation by Alfred Dominik (2003).
Figure 1 shows a screenshot of the notebook about the normal distribution. It
contains definitions, animations and different examples. The associated palette floats
on top of the notebook.
The buttons on the palette give access to the Mathematica commands which are
most frequently used for a particular type of problems. Moreover they provide
intelligent masks for user input. This feature of MD virtually eliminates typing errors,
which is essential for a successful use in a classroom.

5

Figure 1 --- Notebook and palette for the MD unit "Basics of Normal Distribution".

Let us illustrate the functionality of the palette "Normal Distribution - Basics" with the
following typical textbook example:
The length of 3 year old trouts in a basin have a normal distribution with mean-value
µ = 45.2 cm and standard deviation σ = 3 cm. What is the probability that a
randomly caught trout is smaller than 46 cm?
The palette contains the button P(X≤x) that provides an input mask (see Figure 2):

Figure 2 --- The button "P(X≤x)" provides an input mask for the parameters of the
normal distribution.

With the tab-key, the placeholders can be selected and replaced by the user's input.
After that, the user just has to evaluate the input. This gives a numerical answer and
a visualization of the result (see Figure 3).

6

Figure 3 --- The answer to the trout problem - graphical and numerical
representation.

In order to complete the solution, a student would have to type a sentence describing
the result.
Actually, the Austrian educational scenery is dominated by the discussion of
educational standards and standardisation (bmbwk, 2005). We think that MD
contributes essentially to a standardised use of Computer Algebra Systems in
teaching mathematics when integrating such palettes for problem solving.
MD was chosen as the adequate software in several EU projects. Eight schools from
Austria, Germany and Spain took active part in a Comenius 1 project from 2001 to
2004, a new Comenius 1 project ‘Economics, Physics and Mathematics’ which will
last from 2005 to 2008.
Universities and high schools from 7 countries will collaborate in a Comenius 2.1
project “LTM - Learning Tools for Mathematics” from 2005 to 2008 in order to create
additional multilingual modules for MD (LTM, 2005).
During the past few years several diploma theses about the educational profit of MD
were written at the University of Graz (Diploma theses, 2001 – 2003).

7

3 STUDENT’S EVALUATION OF THE SOFTWARE

After two weeks of getting acquainted with the Normal Distribution notebook and
palette, the students were handed a questionnaire. They should evaluate this
notebook and further technical aspects including design issues of MD, but not the
teaching method. The students were already familiar with the M@th Desktop system
in general. The students could answer each question by choosing from five options
on a scale between ‘I do not agree’ to ‘I agree’. Here are the questions.

1) Questions concerning the Normal Distribution Notebook

a. Do the prepared notebooks offer sufficient opportunities

(a1) to experiment (a2) to make conjectures (a3) to solidify your observations and
insights?

b. Do you think that animations showing the effect of parameter changes on function
graphs or significant values like µ and σ are a suitable way to understand facts and
their relations?

2) Questions concerning structure and representation of the Normal
Distribution Notebook

a. Are the boxes with the formulas helpful?

b. Do the explanations of exercises remind you of texts in schoolbooks?

c. Do the different problems often assume a knowledge of concepts and techniques
that you do not possess?

d. Do you think that repeated graphical representations (Density function – Normal
Distribution) are helpful?

e. Do pictures rather distract you from the underlying topic?

3) Questions concerning the use of pocket calculators

a. Do you think that “computation by hand” and “computer usage” occur in the right
proportion?

b. Is the additional use of pocket calculators for intermediate calculations confusing?

4) Questions concerning the Normal Distribution Palette

a. Are the prepared palettes helpful for solving problems?

b. Do the palettes rather hide the underlying mathematics?

c. Will you be able to solve the various problems without palettes, that is, with pure
Mathematica functions?

Let us now sketch the results. It must be seen as a qualitative analysis as only 15
students took part in the experiment.

Part 1: We noticed a high degree of acceptance and a positive attitude with respect
to all parameters (experimenting, argumenting, conceptual consolidation, and
practising). This outcome underlines the statement of Karl Fuchs (1998) that the use
of Computer Algebra in mathematics teaching awakes the students’ delights and
interests in doing mathematics.

8

Part 2: These questions deal with different forms of representations (Bruner, 1970).
The majority of the students think that the symbolic representation with a Computer
Algebra System is very helpful. The students could not give a clear answer to the
question of comparing explanations in traditional textbooks and MD notebooks.
Some individual answers indicate that these students have rarely been working with
traditional textbooks. Graphical representations are in high demand by the majority of
students.
Part 3: The discussion of similarities or mutual interference of notebook/palettes and
pocket calculator exposes clearly that these tools do not exclude but complement
one another in a wonderful way.
Part 4: The students highly appreciate the convenience offered by the palettes in
general. Only few students think that the palettes hide the underlying mathematics or
let mathematics appear as a Black Box (Buchberger, 1989). They accept it in the
same way as they accept the result of a pocket calculator computing the square root
of two without revealing the algorithm.
The use of the buttons of the Normal Distribution Palette is strongly influenced by the
mathematical knowledge of the students. This means that students’ problems with
the palette have its sources in a misunderstanding of the mathematical concepts.
Many students claim that their abilities are not sufficient to solve problems with pure
Mathematica functions. We think that this is an important statement as it shows that
the usage of palettes emphasizes the understanding of mathematical concepts, while
Mathematica with all its complexity takes a back seat.

9

4 THE PEDAGOGICAL PATTERN

The MD software has been designed in view of a blended learning strategy, in which
computer-based activities are used to complement other approaches, such as
teacher-led work or paper-based exercises.

There is no universally accepted definition of the blended-learning method. Here we
quote the definition of Peter Mayr and Sabine Seufert (2002), which is close to the
method in this project.

Blended Learning characterizes concepts of teaching and learning which aim for a
meaningful educational combination of traditional teaching and learning in the
classroom and Virtual- or Online-Learning based on new Information- and
communication- media.

The two authors Martin Ebner and Andreas Holzinger (2002) see the added value of
these new forms of teaching – where Blended Learning is explicitly included – in an
increase of learning efficiency. Thereby growth of efficiency is determined by
permanence of learning outputs when reducing learning efforts on one hand or by
improvements of learning outputs when keeping constant efforts on the other hand.

The blended-learning concept was chosen because it should be our goal to make
students think about mathematical concepts rather than to create an automated
pattern of problem solving without thinking. Both the purely conventional teaching
method and a purely computer-based teaching method run the risk of creating such
behaviour. For example, conventional math instruction is likely to produce a
behaviour of inserting expressions into pre-fabricated formulas without thinking about
their meaning. Purely PC-based instruction is likely to produce a behaviour of
automatically pressing buttons in a certain order.

In order to address these problem we applied the following teaching strategy (or
pedagogical pattern), which we call the 50:50 chalk-PC pattern.

This pedagogical pattern consists of two phases:

1. The "chalk phase": For about 50% of the time the teacher stands in front of the
students, teaching in the conventional way, i.e., by talking to the students and writing
with chalk on the blackboard. Theory and abstract concepts are introduced as
answers to questions arising in the applications. Finally, the students have to solve
one or two very simple problems of each type by hand. The correct solution is
discussed by the whole group.

2. The "PC phase": The PC enters the teaching process only after it has become
clear that the students have mastered the basic skills and are able to solve simple
problems by hand. The teacher now shows how the computer may be used to
achieve the same goals. Finally, the students try more complex examples from "real
life", making benefit of the PC's computational power, but applying the same
concepts as before. Moreover, they may use computer graphics for the visualization
of the results.

We believe that this pattern has distinctive advantages both over a purely
conventional approach or a purely computer-based approach. It is clear that
particularly during the chalk phase the students have occasion to make typical errors
like computational mistakes or errors in reasoning. Hence, this phase is essential for
the students to acquire the right mathematical conceptions.

10

Problems and examples designed for the chalk phase have to be numerically simple,
for obvious reasons. During the PC phase the students learn to attack more realistic
and hence more interesting problems. The computer allows us to choose examples
from everyday life with numbers taken, e.g., from newspapers.
The 50:50 chalk-PC pattern has the advantage that students have to think about the
concepts twice, because one and the same problem is treated once by hand and
once by PC. The actions are quite different, but the mathematical ideas remain the
same. This teaching method therefore puts an emphasis on concepts and
understanding. The PC will not become the only tool for solving a problem. We have
made the observation that even weak students keep doing simple problems by hand,
checking their solution with the help of the PC and visualizing the solution as needed.
Indeed, the students were highly satisfied with this approach (see Section 4 below).
We believe that M@th Desktop is not suitable for the reverse procedure where
computer-assisted problem solving precedes a conventional phase of exploring
mathematical concepts through calculations by hand. This creates dissatisfaction
among the pupils, as they do not know exactly what the computer is doing. Students
should first acquire a reasonable understanding of the mathematics behind, before
they use the computer to get rid of repetitive tasks.

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5 A TYPICAL LESSON

This chapter is based on notes taken by B. Thaller during two inspections on March 4
and March 18, 2005. Two double units from 7:45 am onwards were at our disposal.
Here (http://www.deltasoft.at/english\project2005/Fotogallerie/) are some pictures of
the lessons with M@th Desktop and the evaluation.

The class begins with a conventional phase in which the teacher explains the theory
on the blackboard. As usual, students write in their school exercise books. One
example with simple numbers is calculated by hand. Only after this, the computer
supported part of the lesson begins. The laptop of one student is connected to a data
projector. First the teacher dictates the most important results of the theoretic part,
which the students write in a M@th Desktop worksheet on their computers. Each
student works for him-/herself. Sometimes they take a look at the projection on the
wall. The students are now getting more and more attentive and the talking is only
relevant for the subject, as the computer does not forgive any mistakes.

The M@th Desktop-worksheet, now functioning as a school exercise book, has input
formats for texts, examples, formulas, and notes. Students thus create an attractive
learning aid by themselves. The teacher continually observes the data projection and
corrects the notes, if necessary, thereby creating an approbated version of the
tutorial. Although all students write in their own worksheets, they can always get the
“official“ version over the network, in case the software crashed or a student is late.

The importance of the graphic representation of the numeric data and the
visualisation of mathematical objects is increasing in computer-supported teaching.
The software enables graphical representation of the statistical data, the distribution
function, or the regression line with only a few clicks and inputs. With this graphic,
typical characteristics and initial data can be easily visualized.

Even during the computer-supported phase, the teacher is the centre of the class
and determines speed and content. Activities for which there is no time during class
are postponed to the project phase (see below). The students appear to realize, how
much easier the calculation of the example becomes with the help of the provided
software. They are pleased with the result.

The attention has turned from technical problems of the calculation to the content of
the example itself. Questions and contributions of the students show interest in the
interpretation of the result. For example, they want to know the relation of the points
on the regression line with the actual data and they want to discuss various features
of the visualization.

Symbolic, numeric, textual, and graphic representations of mathematical content are
contained in the students’ worksheets in balanced proportion. Mathematical symbols
are used for the formulation of the example and for user input. Results are
represented graphically as well as numerically. Moreover, the teacher insists that the
problem as well as the result have to be described by the students in clear
sentences.

What do the students think about this type of lessons? Has the computer-aided
instruction created a positive mood in the class? On March 4, 2005, the following
short questionnaire was handed to the students:

Please, assess on a scale from 1 (true) to 5 (false) the correctness of the following
statements (mean value of the answers is given in parenthesis)

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1) I'm interested in mathematics (2.67)
2) I think, mathematics will play a major role in my later professional life (3.73)
3) As a matter of principle, computer-aided instruction is a good thing (1.33)
4) The computer helps me to understand mathematics better (2.67)
5) I think it is confusing to work with notebooks and palettes (4.47)
6) A good teacher would do better without a computer (3.87)
7) Today's lesson was typical for computer-assisted instruction by our teacher
(1.73)
8) Occasionally, I have searched the internet for learning aids in mathematics
(4.27)
Finally, give a grade to your teacher (2.13)
The evaluation was supported and rendered more precisely in the course of a
discussion between the evaluator and the students in absence of their teacher. It
showed the following results: Students are quite indifferent to mathematics. Most of
them neither dislike the subject, nor do they think that it is particularly interesting. The
opinion that maths will probably not be of any importance in the students’ later life in
business is even more dominant. The students welcome computer-supported
lessons and highly appreciate them, but they do not search the internet for additional
tools or information.
However, as far as the question is concerned, whether the computer enables the
students to understand mathematics in a better way, the students’ opinions are not
very distinctive. They rather do not think that a very good teacher could explain
maths better without the help of the computer. The students liked their teacher (he
received a grade “B”) and confirmed unanimously that the lesson visited by the
evaluator was a rather typical one.
For the students themselves, the use of the computer and M@th Desktop is not
difficult in any way. Actually they have been used to working with the computer for
several years. Before, they had to feed Mathematica with instructions directly, which
was considered rather annoying, but now they are very pleased with M@th Desktop
and its palettes and the way of working that results from it. They think that they have
got a useful tool for the solving of mathematical problems and that they are quite
good in applying this tool.

13

6 THE STUDENT’S PROJECT

In order to encourage an explorative approach, the students finally had to realize a
larger project in small groups of two. Each group should present a typical application
of the normal distribution and the regression line. The problems had to be chosen by
the students themselves. Problem, theory, and analysis of the data were to be
presented in a M@th Desktop worksheet. The students had one month, from 18.2. to
18.3.2005 to complete their projects.

These are the instructions handed over to the students:

The content of the project: The project should apply statistical methods to
interesting current problems. The group is choosing their own topic. The chosen
topics should contribute to the awareness of realistic problems. A project may include
sampling and analysis of statistical data or the search for correlations within already
acquired data. Publicly available data (e.g., from the central bureau of statistics or
from the internet) may be used. Projects that contribute to improving procedures in
your school (HAK) are especially important.

Examples of topics of the projects: Volume of HAK-student's side jobs; budget of
HAK students; customer satisfaction at the HAK cafeteria; academic success and its
dependence on personal qualities and preliminary work; salary of managers and its
dependence on education; etc. In all these cases it has to be considered how to
measure the interesting variables in the best way.

The project report of about 7 pages has to be printed in color and be presented in
form of a M@th Desktop Notebook by 18.3.2005. The report must describe the
problem, the acquisition of the data, the statistical analysis and the interpretation of
the results. The paper should also discuss possible weak points and futher related
problems.

The topics chosen by the students dealt with the financial situation of HAK-students,
the distribution of the height of 120 girls and 80 boys, or with the connection between
health expenditures and life expectancy in the countries of the world. Here is a
particularly interesting example (by Wolfgang Kiegerl and Harald Urwalek):
"Comparison of the Ping-data of the network game ‘Natural Selection’. From the idea
to the diagram".

The students presented their projects with commitment and enthusiasm. One could
clearly feel that they were pleased with the stylish way of presentation made possible
by M@th Desktop. The students chose interesting and original problems, for
example, the distribution of ping-values in computer network-games, the connection
between health expenditures and life expectancy in the countries of the world, or the
financial situation of the students, etc. The analysis of the data was done by
mathematical and numerical means, whereas the explanation of the results was
mainly done with the help of graphical representations made possible through M@th
Desktop.

12 students volunteered to participate in the group projects (3 students asked for
special treatment, because they needed more time to prepare for their final school
leaving examination). Students in the same group received the same grade. The
grade was determined by assessing

- the quality of the problem (originality and connection to reality) - 20%,
- the quality of data and statements (plausibility of the reasoning) - 30%,

14

- the presentation (quality, use of media, discussion) - 20%,
- and the report (quality, size) - 30 %.
Among the 6 teams, two finally received grade A (“sehr gut”), three grade B (“gut”),
and one team got a “D” (“genügend”).
All the students declared that they had fun working on the project. In particular they
liked the practical aspects of their work, like designing questionnaires to collect
statistical data. Even an otherwise very weak student pronounced after a successful
presentation that she had learned much more when working on the project than
during preparation for the written exam. Her team got a “B” for their project.

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7 SUMMARY

(1) The M@th Desktop e-learning resources on normal distribution provide a useful
and attractive way to support teaching.
(2) The computer cannot be used to save time in class. The combination of
conventional and computer-assisted instruction rather requires additional time.
(3) Alternation between conventional and computer-aided methods of instruction was
an essential element of the project's success. This method enables the teacher to
present the same ideas repeatedly in two different ways and helps students to
acquire knowledge with completely different activities. Moreover, this method
improves the student’s level of attention as well as their satisfaction with the results.
(4) The software encourages students to use symbolical, numerical, graphical and
textual forms of representation in problem solving in a way that is balanced and
appropriate for the particular purpose at hand.
From our experience with the student’s projects, we would like to add the following
general recommendations:
The size of a student’s project should be rather small and concentrate on one task or
example. Less is more. The project should enhance the student’s understanding but
not present them with additional teaching material. Work on the project one should
not take more than two or three weeks. Students should be told exactly what they are
supposed to do (how detailed the solution should be, how long the summary has to
be). Give them an exact schedule and describe in advance how the grades will be
determined. The teacher should be aware that they might be expecting too much
from the students. There is always the risk that students fail to accomplish their
tasks. Do not put pressure on the students when they run out of time. Then it is better
to leave out one task of the project. Students should present their results in class and
there should be enough time for discussion.

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ACKNOWLEDGEMENT

The project was well received and accomplished by the class 5bk. For this, we
express our gratitude to the students. A few months later, all students of this class
have successfully passed the final exams (Matura).
The further development of M@th Desktop is funded by the European Commission,
Contract No: 226391-CP-1-2005-1-AT-COMENIUS-C21. This article reflects the
views only of the authors, and the Commission cannot be held responsible for any
use which may be made of the information contained therein.

17

APPENDIX: PROJECT’S TIME TABLE

The project took place from November 2004 to March 2005.
During the last two months, the students had to team up in pairs. Each team had to
work on a self-chosen real-life problem and present it in class.
Here is a short record of the project lessons’ contents

1.) Friday, 19.11.2004: Why Normal Distribution? First example
2.) Friday, 26.11.2004: The cumulative PHI – function based on the probility

densitiy function phi of the normal distribution, two further examples
3.) Friday, 3.12. 2004: The standardisation of the normal distribution
4.) Friday, 17.12.2004: Discussion with assessment of students’ contributions,

further example
5.) Friday, 14.1.2005: Replication of the previous lessons
6.) Friday, 21.1.2005: Written exam in mathematics about the standard

distribution
7.) Friday, 28.1.2005: Assessment of a sampling of normality, simulation of data
8.) Friday, 4.2.2005: Assessment of a sampling of normality, simulation of data
9.) Friday, 11.2.2005: Discussion of the project and homework
10.)Friday, 18.2.2005: Making up of the project model, beginning of the students

project
11.)Friday, 18.3.2005: Students present their projects. The project is graded. End

of students project.
Evaluation

1.) Friday, 3.12.2004, Evaluation of the software M@th Desktop (questionnaire
1)

2.) Friday, 4.3.2005, Evaluation of the project regarding the student-teacher
activity (questionnaire 2)

3.) Friday, 18.3.2005, Evaluation of the presentation of the student projects,

18

REFERENCES

Altrichter, H., Posch, P. (1998): Lehrer erforschen ihren Unterricht. Eine Einführung
in die Aktionsforschung. Verlag Klinkhardt Bad Heilbrunn.

Baumann, T. (2003): Definition of Blended Learning in the Journal ‘Lehren und
Lernen mit neuen Informations- und Kommunikationstechnologien I im Text E-
Learning in der Lehrerinnen- und Lehrerbildung’, p. 52
http://beat.doebe.li/bibliothek/b01876.html.

Bruner, J. (1970): Der Prozeß der Erziehung. Verlag Schwann Berlin, Düsseldorf.

Buchberger, B. (1989): Should Students Learn Integration Rules? Technical Report.
RISC-Linz.

Cobb, P., Confrey, J., Disessa, A., Lehrer, R., & Schauble, L. (2003): Design
experiments in education research. Educational Researcher, 32 (1), 9-13.

Cobb, P., Mcclain, K., & Gravemeijer, K. (2003): Learning about statistical
covariation. Cognition and Instruction, 21, 1-78.

Dominik, A. (2003): Mathematica Paletten als Lern- und Experimentiertools im
Mathematisch – Naturwissenschaftlichen Unterricht. (Mathematica Palettes as a
Learning- and Experimentation- Tool in Teaching Mathematics and Natural Sciences)
Dissertation at the University of Salzburg.

Ebner, M., Holzinger, A. (2002): E-learning – Multimediales Lernen des 21.
Jahrhunderts, in: proceedings of the conference “Tag der Neuen Medien”, Leoben, 7
pages.

Fink, R. (2001): Auf Mathematica basierende Entwicklung von Lerneinheiten mit
M@th Desktop auf dem Gebiet der Differenzialrechnung (The Design of Learning
Sequences with M@th Desktop in the Field of Differential Calculus), Diploma thesis,
University of Graz.

Fuchs, KJ. (1998): Computeralgebra - Neue Perspektiven im Mathematikunterricht.
(Computer Algebra – New Points of View in teaching Mathematics) Habilitation
theses at the Natural Science Faculty of the University of Salzburg for the
requirement of a Venia Docendi in mathematics education.

Führer, L. (1997): Pädagogik des Mathematikunterrichts. Verlag vieweg,
Braunschweig, Wiesbaden.

Kainhofer, R., Simonovits, R. (2003): M@th Desktop and MD Tools: Mathematics
and Mathematica Made Easy for Students. Reinhold Kainhofer, Reinhard Simonovits,
conference proceeding of PrimMath[2003], Zagreb: 123 – 138.

19

Mayr, P.; Seufert, S. (2002): Definition of Blended Learning in the e-learning
encyclopedia, http://beat.doebe.li/bibliothek/b01275.html.

Roth, G. (2003): Auf Mathematica und M@th Desktop basierende
Unterrichtsequenzen zur Volumsbestimmung von Rotationskörpern sowie
Anwendungen des Student-t-Tests (Teaching Sequences concerning the solids of
revolution based on Mathematica and M@th Desktop), diploma thesis, University of
Graz.

Siller, H-St. (2002): Auf Mathematica basierende Lerneinheiten zur fundamentalen
Idee der Modellbildung, illustriert an Extremwertbeispielen und Beispielen der
Integralrechnung mit M@th Desktop (Learning Sequences featuring the idea of
modeling based on Mathematica, illustrated by examples of Differential- and Integral
Calculus), diploma thesis, University of Graz.

Simonovits, R. (2000): Projekt M@th Desktop. Didaktikhefte Österreichische
Mathematische Gesellschaft] 32, 172-179.

Simonovits, R. (2001): Differentialrechnung mit M@th Desktop. Didaktikhefte der
Österreichischen Mathematischen Gesellschaft 33, 130-139.

Simonovits, R. (2002): EU-Projekt mit M@th Desktop, basierend auf Mathematica.
Didaktikhefte der Österreichischen Mathematischen Gesellschaft 34, 101-110.

Simonovits, R. (2003): Daten fitten und approximieren mit Mathematica und M@th
Desktop. Didaktikhefte [Österreichische Mathematische Gesellschaft] 35, 98-103.

Simonovits, R. (2004): Lineare Regression mit Mathematica und M@th Desktop.
Didaktikhefte der Österreichischen Mathematischen Gesellschaft 37, 137-154.

Smole, M., Simonovits, R., Thaller, B. (2004): Messen von Radwegen mit
Mathematica und M@th Desktop. MNU Jg 57,5: 271-276.

Smole, M. (2003): Auf Mathematica und M@th Desktop basierende
Unterrichtsequenzen zur Approximation von Radwegen sowie zur linearen
Regression und Korrelation (The Approximation of cycle tracks and linear regression
and correlation based on Mathematica and M@th Desktop), diploma thesis,
University of Graz.

Wittmann, EChr. (2002): Grundfragen des Mathematikunterrichts. Veralg vieweg,
Braunschweig, Wiesbaden.

Welik, W. (2002) Auf Mathematica basierende Entwicklung von Lerneinheiten zu
Anwendungen der Integralrechnung unter M@th desktop verbunden mit dem

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methodisch zielgerichteten Einsatz eines Help Browser – Systems (Learning
sequences inforcing the application of Integral Calculus with M@th Desktop joined
with a methodological – purposeful use of the Help Browser – System), diploma
thesis, University of Graz.
Additonal references
All Diploma theses can be downloaded at
http://math.uni-graz.at/diplomarbeiten/index.html (June 2006)
Keyword LTM (2005)
Learning Tools for Mathematics, Contract No: 226391-CP-1-2005-1-AT-COMENIUS-
C21. See http://math.uni-graz.at/ltm/
Keyword: standardization bmbwk (2002)
http://www.bmbwk.gv.at/schulen/unterricht/ba/bildungsstandards.xml

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