EVALUATION OF EXPLORATORY APPROACHES IN
LEARNING PROBABILITY BASED ON COMPUTATIONAL
MODELLING AND SIMULATION
Markus Kuhn
University Duisburg-Essen
Lotharstraße 63, D-47057 Duisburg
kuhn@collide.info
Ulrich Hoppe
University Duisburg-Essen
Lotharstraße 63, D-47057 Duisburg
hoppe@collide.info
Andreas Lingnau
University Duisburg-Essen
Lotharstraße 63, D-47057 Duisburg
lingnau@collide.info
Martin Fendrich
University Duisburg-Essen
Lotharstraße 63, D-47057 Duisburg
fendrich@collide.info
ABSTRACT
Learning about probability is a challenge for students in math lessons because it requires both new concepts and new
approaches, namely experimentation and modelling. To support this kind of mathematical experience in the classroom we
have developed learning activities that engage students in computational modelling using simulation tools. Our
collaborative modelling environment offers a toolbox to set up and simulate diverse experiments as e.g. throwing dice or
drawing balls, playing lottery games, and to facilitate validation. We see its potential to expand traditional math instruction assisting students to develop a sustainable conceptual basis of probability principles offering them a higher degree of
responsibility of their learning. This is confirmed by our evaluation of experiences made in a German secondary school.
KEYWORDS
modelling, collaborative learning, mind tools, action research, probability learning, integration of new media.
1. INTRODUCTION
The European project SEED (“Seeding cultural change ... ”, 2001-2004) aimed at developing and supporting
richer and more integrated forms of using interactive digital media in the classroom in cooperation with
groups of associated teachers in different countries. Based on the national curricula, the project partners want
to provide school education with collaborative software tools, which the teachers appropriate and use to
engage students in learning activities in different domains. In our approach, we extend Jonassen’s notion of
“mind tools” in which “computational objects to think with” are created and used to experiment and express
own ideas (Jonassen, 2000) towards collaborative group activities. In tradition of the NIMIS project which
was dedicated to early learning in primary school in computer integrated classrooms, we enrich learning
scenarios by “seamless integration of interactive media” (Lingnau, 2003). Based on the initiative of teachers
and their interest in innovating teaching and learning, essential part of the German research work in SEED
was action research in the learning domain of probability in secondary schools.
2. BODY OF PAPER
2.1 Approach
The attraction of learning probability is widely discussed. Nilsson (2003) claims that “insight into aspects of
probability can be a relevant factor as probability is both a part of mathematics (…) and a relevant part of
everyday situations”. Already in 1975 Fischbein (pp.93) points out that "practical experience with
probabilities provides an ideal way of familiarizing children with the fundamental concepts of science, such
as prediction, experiment and verification”. Its attraction comes not only from its chances but also from the
challenges in solving problems.
Teachers often begin their instructions initiating simple hands on experiments. This helps the students to
understand basic principles but is naturally limited. Computers are well suited to carry out and help analysing
experiments. Combining the advantages of students’ hands on experiments with computational capabilities
we have constructed a toolbox for our modelling environment. Experiments are explicitly modelled and
simulated, further means are provided to expand the limits of hands on experiments and to open new ways of
teaching and learning. Learning probability and modelling, both relatively modern aspects of math education,
go hand in hand and play an essential role to initiate learning processes. We assume that learning takes place
actively and that learning self regulated intensifies building own knowledge and competences which the
students can apply in real world situations.
Motivated by this challenge our group of teachers and researchers designs learning materials and didactic
sequences in a participatory process supported by the Collide research group at University Duisburg-Essen.
Based on the practical work in school we develop a software tool embedded in a collaborative modelling
environment. Over a period of two years we explored new possibilities of and effects on learning probability
in three didactic sequences. They were arranged as natural learning settings - compliant with the curriculum.
Besides design and exploration another purpose of this study was to specify research questions to be
evaluated in the nearer future.
The sequences were structured in different phases. At the beginning students get familiar with the
learning environment simulating elementary experiments e.g. throwing dice, activating their knowledge
about Laplace’s principle of equiprobability and about ways to calculate the probability of compound events.
These activities also help the students to validate and accept the software as a random generator. The design
of following phases enables students to act independently to solve concrete problems modelling them with
the probability toolbox. e.g. to experience Bernoulli’s ‘law of large numbers’ the students simulate a large
number of experiments and see that the frequencies converge towards the probability, as the theory predicts.
The concrete experimentation and processing is arranged in groups working collaboratively together
(Hoppe, 2002b); dealing with more complex problems like the ‘birthday problem’ is initiated in plenary form
guided and moderated by the teacher. In both arrangements, we make use of digital input devices as graphic
tablets and an interactive board to combine the ‘technical’ modelling with scratch pad features.
2.2 Modelling - a crucial part of learning processes
We see modelling as a crucial part of learning processes, not only in mathematics education. Following
Stachowiak’s general modelling theory (1973), modelling means the process of constructing a model in
which the ‘subject’ composes a model in regard to the ‘original’ to serve a specific purpose. So, the process
of modelling means setting ‘original’, ‘subject’ and ‘model’ in a specific relation. Modelling is therefore
more than reproduction: the whole process is a reflected transformation in which students organise actively
their own learning. The ‘subject’ decides which attributes and connections out of the context are accepted,
emphasised or neglected and how the results are applied to the real world.
Fig. 1: Adapted mathematical modelling cycle
In fig. 1 this process is specialised for the mathematical modelling cycle (Berry & Houston, 1995). In our
context it means solving concrete probability problems by modelling experiments, simulating and analysing
their outcomes to answer ‘real’ questions. In case of a positive validation the student presents his solution e.g.
in form of a prognosis to the audience otherwise he can repeat the cycle of scientific experimentation.
A group sets up and uses the model assisted by one or several computers, makes decisions and draws
conclusions together. Understanding learning as a social process, this sharing of ideas and discussing
opinions can be conducive for individual learning processes. As a consequence the environment supports
communication (shared workspaces, annotation, mind mapping), enables co-constructive work and facilitates
presentation through its handwriting and visualisation features.
2.3 The modelling environment ‘Cool Modes’
2.3.1 Visual languages in Cool Modes
Cool Modes (“COllaborative Open Learning, MOdelling and DEsigning System”) is an environment to
support co-constructive activities in the whole modelling cycle. Elements of so-called visual languages
(Pinkwart et al, 2001) are offered in form of toolboxes for specific domains e.g. for probabilistic
experimenting. In shared workspaces, the co-learners can synchronously and jointly elaborate models
represented as graphs in a specific visual language. The main characteristic of Cool Modes is the approach of
adding semantic structures to define co-operative visual languages without assuming a specific domain
semantics for the overall system. The semantics of a visual language is carried by its domain specific objects
and relationships. This enables the system to provide e.g. special tools, means of analysis or domain-related
support mechanisms.
Fig. 2: Models for a) a die experiment b) the birthday experiment
To have private and shared areas simultaneously or to separate independent cooperative tasks, Cool
Modes facilitates handling several workspaces represented in different windows. Each workspace can contain
elements of varying visual languages and offers a number of transparent layers for handwriting or images.
2.3.2 The toolbox ‘stochastic experiments’
The toolbox offers manipulable elements (fig. 3) to set up and simulate experiments and to display, analyse,
sort and filter outcomes. E. g. the simulation of throwing a die 100 times is shown in fig. 2a. The outcomes
are stored in a collector and visualised as a list or as bar diagrams showing the absolute or the relative
frequencies. In fig. 2b the birthday experiment is modelled by drawing a fixed number of dates out of the
calendar urn whereby each drawn element is put back immediately. Automatic repetition of this experiment
is enabled by the multiple drawing node.
Fig. 3: Visual language elements for stochastics
Searching for positive events can be done by hand (for example to identify ‘multiple birthdays’) or
automatically using filters - a must for analysing lottery games. In such models the exact calculation of
probabilities and a formal description of the model itself are available and represented by special display
elements.
Compared to Pratt’s Chance Maker Microworld (Pratt, 2000) experiments can be constructed and carried
out with a higher degree of freedom. Most of the elements are manipulable in several parameters and thus
enable creative and constructive input. As the toolbox is developed for natural settings in school several
generators are adjustable and new ones can be quite easily defined during the modelling process or in
advance. Different types of experiments can be specified by changing ‘drawing edges’ connecting the
generators and the drawing node.
2.4 Classroom experiences
With respect to conceptualising statistics our approach shows parallels to the French statistics curriculum
which is “grounded on the idea of observing the fluctuation of samples by simulating the repetition of a
random experiment and observing the stabilization of the frequency distribution of the possible outcomes, the
notion of probability being introduced later as a ‘theoretical frequency’” (Parzysz, 2003).
During the last two years we explored three learning sequences in which learning probability started in a
comparable way. They took place at Elsa-Brändström-Gymnasium in Oberhausen, a secondary school in
Germany. During the preparations and the sequences themselves we discussed arising problems in our group
of teachers and researchers often leading to design changes of the toolbox or to further development.
The researchers attended several lessons placed in different phases to get an overall impression of each
sequence. Documentation was done by recording of class activities or focused on selected groups. Snaps
were taken during those lessons. We used videotaping only in a few situations not to disturb the students
involvement in the task.
For evaluation purposes we utilised the writings on the interactive board which were archived
consecutively. Also the students’ files containing the annotated models were collected after the lessons. After
each sequence we had a reflecting discussion with the teachers on the basis of the collected material,
recording their impressions, experiences, ideas, etc. During our work several questions arose:
- Do the students trust the computational random generators?
- Is the modelling of experiments explicit enough and easy to learn?
- Does the representation of the toolbox elements support the modelling process?
- Can the tool be used to solve real probability problems?
- Can we find indicators for a deeper understanding of probability principles?
- Were lies the added value for students - and teachers? …
2.4.1 Learning setting
The sequences took place in a room containing six in a round table integrated computers and a plenary area.
Here an interactive board was used intuitively like a chalk board with additional archiving functions. It
allows to control the software through a special pencil which is used for writing and provides the
functionality of the mouse. Some lessons took place in the normal classrooms where a data projector and a
laptop with a pen based input device were used as an ‘interactive overhead projector’.
Fig. 4: Work room with interactive board and computer area
2.4.2 Sequence A: Modelling to investigate the “birthday experiment”
In the birthday problem ‘How high is the probability that the event at least one common birthday occurs in a
group of n people?’ it is not evident how to compute the probability directly but using the complementary
event ‘no common birthday’ simplifies. The topic is ideal to leave the normal ways of math instruction and to
initiate learning processes. Following the crucial steps of Konold (1991) “test own beliefs against the beliefs
of others,(…) against own beliefs about other related things,(…) against empirical evidence” (Nilsson 2003)
its easy to start with a betting contest. Modelling and simulating the problem provides the missing steps.
Setting: June 2002, a class of 24 9th graders, 7 lessons
Observation: After the students made their bets and added arguments for computation they were invited
to use the modelling environment to explore the problem. For modelling they transformed it to drawing dates
and soon decided to use the calendar urn from which n dates are to be drawn (fig. 2b). One girl immediately
pointed out to use the edge with ‘put back’ to make sure that same dates are possible. To test their beliefs
against their own produced empirical evidence they split into groups, using the computers to carry out many
experiments and count the frequency of positive events. The results were shared, cumulated on the interactive
board and discussed. They formed the basis for a related challenge in which they should find out the group
size for the percentage of 50% probability for ‘at least one common birthday’ in a collaborative setting (for
details: Lingnau et al., 2003). Here the students had not only to analyse and count positive outcomes but
modify the “micro-world” e.g. decreasing or increasing the group size. This time the pupils should report
their outcomes permanently via a shared table. So each group could make use of all reported outcomes and
alter its proceedings. The explored empirical results differed from the prior beliefs and motivated the
theoretical computation.
Conclusions: The results indicate that students were able to use the toolbox effectively right from the
start. The representation of its elements makes their usage evident as the episode concerning the choice of the
right drawing edge highlights. Starting with a bet, modelling and simulating the problem enable the students
to validate their beliefs and lead to a cognitive conflict (Tall, 1988) that motivates them for the theoretical
work. The accordance of the calculation and the simulation strengthens the trust in the fairness of the
software. In the collaborative setting feed back effects were not obvious – only slight signs of influence were
observed and could be affirmed by the collected data.
2.4.3 Sequence B: Working with ‘the law of large numbers’
Setting: July 2003, a class of 27 8th graders, 7 lessons
Observation: Based on the refreshed previous knowledge a female student initiates the experimental work
with an own formulation of the law of large numbers ‘While executing a large number of experiments the
relative frequency approximates the probability of the event’. Comparing the results of two groups (A + B)
throwing real dice and seven groups (C … I) simulating this experiment immediately showed an increased
motivation through the work with the software. Whereas the groups A and B have thrown the real die 250
times and 105 times resp. during 20 minutes, the others simulated far more experiments and observed the
convergence of the frequencies.
An episode concerning the convergence of the frequencies with students S1 (group A), S2 (group D) and
S3 may prepare further conclusions:
S1: We have counted how often we found the “six”. In 55 throws we had 16.3% but in 105 throws 18%.
S2: We have noticed something similar. But later on it approaches again.
Teacher: Well, isn’t this allowed?
S3: Sure, it’s no problem, we have to throw more often.
In the following discussion other students report their results. They found a frequency of 17.3% in 1111
and 16.8% in 100000 experiments. One student concludes that carrying out more experiments would lead to
the expected value of 16.6%.
Conclusions: These students use results of the simulations to explain problems of the real experiments.
This underlines their trust in the software. The students’ observation that even many simulated experiments
can show a high fluctuation saves more hands on experiments and underlines the importance of (a lot of)
simulated experiments. In our study situation the ‘law of large numbers’ emerges and is accepted for solving
the problem. This acceptance of the abstract rule can be interpreted as its validation as a part of the ‘real
world’ and closes the modelling cycle.
2.4.4 Sequence C: Exploring German lotto - the challenge of ‘real questions’
Problems out of the context of daily life are normally motivating for the students leaving the abstractness of
standard math instruction. But their complexity often causes great difficulties to mathematise, to find the
adequate mathematical model. Lottery games are such a fascinating and complex topic. The game itself is
easy enough, but e.g. two understand the distribution of the price money is challenging.
Setting: July 2003, a class of 21 10th graders, 4 lessons
Observations: The toolbox was used to help the pupils construct methods to calculate the probabilities for
3 up to 6 hits in a simplified lotto “6 out of 10” and control themselves empirically. Afterwards they
transferred their calculation methods increasing the number of “balls” to original German lotto and again had
the opportunity to validate their results by constructing a model simulating millions of lotto drawings,
filtering the positive ones automatically.
Conclusions: To convey the methods to calculate the probability in German lotto “6 out of 49” affords
endurance on both sides but can be facilitated by our environment. Affirmed by their experiences from
sequence A, modelling and analysing in groups now enable the students to get confident with the new mathematical methods and beware them of frustration. Having understood the theoretical background and accepted
the environment as a suitable or even accurate lotto microworld (fig. 5) the students have the chance to find
out the exact probabilities for 3,4,5 or 6 hits using a calculation node and set them in relation to the accordant
price money.
Fig. 5: Model for German lotto – frequency for 5 hits in a million experiments
3. CONCLUSION
In contrast to Parzysz’s (2003) assumption we don’t find any hints for mistrust of the computerised random
generator. The students observed the meaning of ‘the law of large numbers’ in different contexts and
conclude on their own that carrying out experiments by hand will not lead to ‘exact’ probabilities whereas the
simulation of many experiments leads to more reliable results.
The students were able to model independently and know how to handle the build models to explore more
or less complex problems. Results show that students were able to construct meaningful representations and
had little or no difficulties using the tool and the devices. Therefore we suggest the implementation of
described learning activities also with younger students.
Our teachers had used the toolbox to try out new ways of teaching by giving students more responsibility
of their learning process. They worked in groups in a new experimental way. Complex problems of different
topics were explored and solved successfully, facilitated and supported through annotation and visualisation.
This can indicate that the process of modelling, simulating, analysing and presenting leads to a greater
confidence in and a deeper understanding of methods to calculate probabilities.
In this article, we concentrated on the potential of new approaches for teaching probability based on the
stochastics toolbox integrated in the Cool Modes modelling environment. Yet, we also pursue more general
goals concerning the integration of interactive media in a networked classroom environment. Cool Modes, in
this sense, is more than a single-user learning environment, it is a distributed system which offers
communication and cooperation functions (Hoppe, 2002a). This is concretely exemplified in the stochastics
scenarios by the seamless transition of information between small group work and a teacher-moderated
plenary discussion shown in fig. 4. Specifically, the cumulation of results mentioned in scenarios A and B is
facilitated by the provision of shareable toolbox elements. Here, technology supports a smoother flow of
information and activities.
The development of the stochastics toolbox will be completed in 2004. Other learning sequences are in
preparation. Until now the toolbox was mainly used to simulate and get empirical results. It also can be used
to calculate probabilities after modelling a concrete problem. Parallel one can get a formal description of the
model itself and may have a look on the positive events. We are curious what consequences on learning this
mix of information will have.
Refined evaluation methods, lab experiments and further didactic sequences may help to find out how the
chosen representation of the toolbox and the explicit modelling are involved. To explore how collaborative
modelling influences the way of conceptualisation is another challenge for further research in the nearer
future.
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