STUDENTS' MEANING-MAKING PROCESSES OF RANDOM
PHENOMENA IN AN ICT-ENVIRONMENT
Kjærand Iversen, Agder University College
Per Nilsson, Växjö University
Abstract. This paper puts focus on the different ways lower secondary students
handle compound stochastic phenomena. The data analysis is based on clinical
interviews in which the participants are up to explore different several-step problems
in an ICT-environment. How the students understand the content within the situation
is regarded from the perspective of how their understanding varies with their
interpretation of the situation. From the data a leakage strategy and a division
strategy are identified as being of particular importance for several students in their
meaning-making processes.
BACKGROUND
People’s different ways of handling chance encounters has been the object for
enormous amount of studies. For a survey Shaughnessy (1992) and Gilovich et al.
(2002), among others, can be used. In the latter mainly the psychological approach,
regarding heuristics and biases, are discussed, whereas Shaughnessy also discusses
some results linked to educational issues. Despite this mapping, we claim that there
are some issues of peoples’ probabilistic reasoning that need further investigations. In
particular we will discuss issues of how students’ deal with probabilistic phenomena
in an explorative and interactive setting.
The current paper aims to put focus on students’ reasoning processes with respect to
compound stochastic encounters, actualized in a computer-based environment. More
precisely, our concern is on how students, in an ICT-environment, are up to explore
and make use of different strategies in handling random processes, which can be
divided into several steps.
In the following part we will explore and define, in a more precise way, the actual
stochastic phenomenon, which is of interest for us.
SIMPLE AND COMPOUND STOCHASTIC PHENOMENA
A stochastic phenomenon involves one or more stochastic objects (random
generators). In contrast to a deterministic phenomenon, where there is only one
outcome, several different outcomes are possible in a stochastic phenomenon.
The stochastic objects are often put into action by an external force or mechanism,
but it is the objects' own characteristics, such as form, symmetry, centre of gravity
and so on that decide the probabilities for the possible outcomes. A simple stochastic
phenomenon involves only one object and it is put into action only once. The
throwing of one die exemplifies this. If the phenomenon is not simple it is called a
compound stochastic phenomenon. This paper focuses on a particular kind of
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compound stochastic phenomena that we name one-object several-step stochastic
phenomena (OOSP). A problem used by Green (1983), the Robot problem, illustrates
this.
Figure 1. The maze in the Robot-problem in Greens survey.
In this situation a robot is walking into a maze. Sometimes the robot encounters a
crossroad and the continuation is decided by a random mechanism. Finally the robot
ends up in one of eight rooms. In Green’s survey the following question was asked:
“In which room is the robot most likely to finish up, or are all rooms equally alike?”
From the expert’s point of view this problem might be resolved by using the Product
Law: If two events A and B are given then:
Using the product law of probability, a compound problem may be divided up into
sub-steps; where after individual probabilities may be multiplied. Stochastic
situations where this procedure may be used are commonly called several-step
problems in probability.
In an attempt to establish the issue of the current study, regarding students’ ways of
encountering OOSP, we first present two prominent results from earlier research.
PREVIOUS RESEARCH
In the survey by Green (1983) the Robot problem was included. As reported by
Green, the result was surprisingly poor for all participating students (age 11-16).
Only 13% of the grade 10 students (age 15-16) gave a correct response, even though
these students had been working with tree diagrams and the product law of
probability.
Considering chance encounters Fischbein (1975) emphasizes the role of intuitive
reasoning. He distinguishes between primary intuitions as cognitive acquisitions,
derived from individual experiences, without systematical instruction, and secondary
intuitions as formed by education and linked to formal knowledge. According to
Fischbein, developing secondary intuitions can be seen as a process in which the
learner learns to make use of generative mental models. This line of reasoning he
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exemplifies with the structural potentialities in using the well-known tree diagram. In
connection to this, Fischbein et al. (1975) were using six different devices to explore
the students’ probabilistic intuitions. Their results were quite contrary to the results of
Green’s study (as pointed out by Green). In an experimental environment (with
manipulative materials) students were given problems similar to the Robot problem.
A majority of the students gave a correct response to most problems. In the last
OOSP, the pagoda, it was only among the oldest students that a majority (more than
50% of the students) gave a correct response to the question.
2a
Figure 2.
2b
a) A picture of the five "two-dimensional" devices used by Fischbein.
b) A schematic picture of a three-dimensional device used - the pagoda.
The study by Fischbein et al. (1975) included pre-school children (age 5-7), and
surprisingly two of the tasks (II and III) were solved better by them than by the oldest
students (age 13-14). Such a result the authors explain as an issue of schoolwork, in
that schoolwork often seems to orient the child towards deterministic interpretations
of phenomena.
In the current study the interest is not primarily on correct or incorrect responses. Our
particular interest is directed towards a more qualitative analysis of students’ ways of
dealing with OOSP, in order to better understand different kinds of responses given
and strategies used. In particular we are going to focus on how students’ interpret the
tasks, which problems they involve themselves in, and relate this to their articulated
explanations and strategies.
THEORETICAL CONSIDERATIONS
A long time ago Jerome Bruner stated, “…when children give wrong answers it is not
so often they are wrong as they are answering another question…” (Bruner, 1968, p.
4). The meaning of this statement is that, if we are going to understand students’
ways of encountering random phenomena we have to take into account the questions
the students are engaged in. If we are interested in learning objects we have to treat
the learners individually and, from time to time, try to ascertain what each individual
is trying to accomplish (Halldén, 1988). For this Halldén (ibid.) saw it fruitful to
introduce a distinction between task and problem. He defines task as: “what is
presented to the pupils by the teacher with the intention that they are to do something
and/or that they are to learning something” (p. 125), and problem as the learner’s
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personal interpretation of the task given. Viewing the activity from a normative
perspective may too much restrict the analysis, so that obstacles rather than
possibilities are considered. Instead, we consider it more fruitful to base the analysis
on students’ interpretation of the situation at hand, in order to illuminate strategies
and learning potentialities.
Adopting such a student oriented perspective, implies a necessity to reflect on
contextual influences on learning. In accordance with a constructivist view, we refer
context and contextual elements to students’ personal constructions. That is, we
consider context as a mental device, shaped by individual interpretations. If we let the
conceptual context denote personal constructions of concepts, embedded in a study
situation, as well as the situational context denotes interpretations of the setting in
which learning occurs, and the cultural context refers to constructions of discursive
rules and patterns of behavior, we can talk of students’ ways of handle a learning
situation as a problem of contextualization (Halldén, 1999; Nilsson, 2004).
Halldén (1999) stresses that these different kinds of contexts are in play
simultaneously as we are trying to solve a task but, depending on how we interpret
the situation, by focusing certain aspects, they get different priorities in the
contextualization process.
Considering situational contextualizations specifically, one could argue that an ICT
environment offers an actor interesting possibilities. It is an arena where the concrete
and the abstract, or the informal and formal, can be related. This gives good
possibilities for important integration. Pratt (1999) writes:
Computers provide a medium for designing activities that build and integrate pieces of
knowledge. A microworld may be able to integrate these fragments of knowledge by
offering opportunities for their use, enabling the construction of meaning. (p. 61)
Difficulties in handling tasks presented in a study situation, and in acquiring new
concepts and strategies, are thus seen as a problem of finding relevant contexts, by
which new information can be interpreted and new strategies be worked out (Caravita
and Halldén, 1994). The learners’ ability to discern which parameters in a learning
situation that are of relevance will be of crucial importance and also, their ability to
evaluate between different ways of looking at the world, that is, their ability to
differentiate between different contextualizations.
Learning is then a process of decentering, in the Piagetian sense, rather than the acquisition
of more embracing logical or conceptual systems replacing earlier less potent ones.
(ibid., p. 106).
By reasons of limitations, we are not going to dig into details regarding processes of
decentering. Instead we are going to focus on which strategies that appears in an
computer-based environment, and how these strategies are related to the students’
different ways of posing the problem, i.e. their interpretation of the study situation
and its involved learning material.
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OBJECT OF STUDY
The aim of the study is to describe students’ different ways of acting within a
computer-based environment. In particular we are interested in how students’
strategies appear when they are up to explore OOSP-tasks, in which static as well as
dynamic aspects of probability come to the fore in an ICT-environment.
THE SOFTWARE
In the software the user can choose between several different flexitrees (systems
where marbles can move within). At the start the marbles are at the top of the system.
When pushing RELEASE the marbles move downwards. The marbles may be
temporarily stopped (by pushing STOP) and then released again by pushing
CONTINUE. The marbles will finally end up in one of several boxes (at the bottom).
A table and a diagram keep track of the number of marbles in each box.
Figure 3. A picture of Flexitree with some "action" from flexitree 7
METHOD
Semi-structured interview were used to collect the data. In our view standard tests or
naturalistic observation would not give the in-depth data needed to consider our
research questions. If naturalistic observation were used the waiting time for a useful
spontaneous verbalization might be quite long and we agree with Ginsburg (1981) in
that "in most instances naturalistic observation is simply not practical" (p. 5).
Because of the complexity of cognitive processes involved neither a written test was
seen as suitable. Ginsburg writes:
When the underlying cognitive processes are numerous and complex, standard tests may be
ineffective or at least inefficient.... A small set of standard questions....will not suffice to
capture the richness and complexity of the relevant cognitive structure. (p. 7)
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The participants were students from lower secondary schools (age 14-16). They had
some experience with several-step phenomenon (but not much). The problems used,
were expected to be quite challenging for the students.
The students were sitting in front of a PC, working with Flexitree. The session started
by the interviewer explaining the features of the software.
The screen-image was videotaped along with the sound by using a Televiewer. These
data were then transferred to a PC. Both the transcript and the digitalized videos were
used in the analyses.
The data were collected in several iterations. Based on both methodological and
theoretical reasons the setting and protocol for the interview were modified from
iteration to iteration. This paper concerns the interview from the first two iterations in
the study; 4 students in the first iteration and 12 students in the second iteration. In
iteration one the students were individually interviewed. This setting seemed to make
the student less talkative and they were also strongly influenced by any suggestions
by the interviewer. We therefore decided to interview the students in pairs in iteration
two.
Questions
In each interview the main questions given to the students were:
Q1) Do you think the probability for coming to each box is equal or is it not?
Q2) What do you think the probability is for coming to box B?
The students were asked to explain their answers. If the students seemed stuck, or
were not doing any progress, the students were given new questions or suggestions.
When working with one particular problem the students were first asked to answer
the above questions without using Flexitree, that is, responding to a static
representation. Later on they were asked to use Flexitree, and thereby encountering
the dynamical part of the situation. The purpose of this was to promote variations in
contextualization.
RESULTS
From the analysis of data, two types of strategies became prominent in the students
activity. In both cases the crucial importance of the crossroads is reflected, yet with
different meanings given. The strategies identified we call the leakage strategy and
the division strategy. In the following we will explain those strategies further, in
which two particular cases will serve as an illustration.
1) The leakage strategy – the case of Oskar
Before entering setup seven, Oskar has tried setups one, two and four (see figure 3).
However, in his previous activities most of his work has been based on guesses,
which he has had difficulties to motivate.
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In line with the interview method, also the activity in flexitree seven begins with
Oskar asked to come up with a first hypothesis. His guess is that it will be about 50%
to B and 25% to A and B respectively. This response could possibly be explained by
previous contextualizations, in particular regarding the work with setup four and
specifically the frequencies encountered.
After that Oskar turns over to work with Flexitree. Observing some relative
frequencies, he almost immediately starts to doubt his first hypothesis. Continuing
working with Flexitree for a while, he suggests that the probabilities for ending up in
boxes A, B and C are 1/6, 2/6 and 3/6 respectively. When the observer then asks
Oskar for an explanation, Oskar has difficulties to answer. The interviewer then tries
to make Oskar focus a sub-task, namely to explain the low amount of marbles, ending
up in box A. In doing so, what we call the leakage strategy appears:
O:
R:
O:
R:
O:
R:
O:
R:
O:
R:
O:
R:
O:
R:
O:
Because when they go ... because the probability for coming to A is quite
small ... because ... on it its way ... there is only one path to A. And then
... on this path there are two paths that go like this [pointing to the two
paths going away from the path to A].
Mm.
And then it becomes less likely.
Yes.
And then ... the marbles are disappearing so to speak.
Yes.
So I can ...
Is it (the probability) lower or higher than one sixth?
Maybe it's one sixth. Now, at least, I think I understand why the chance
is so small for A.
Mm. It becomes fewer and fewer along that path?
Yes. It is holes in the path in some sense.
Yes.
And then the marbles ... they fall off the steep slope and then go to B and
C instead.
Yes. So, on the path downward it becomes fewer because some are
dropping off?
Yes. Only a few continue downward.
After the intervention Oskar is not trying to give an explanation to the probability
estimate 1/6. Instead our interpretation is that he problematizes the situation to be a
question of explaining why so few marbles end up in box A. He tries to do so by
focusing the meaning of the crossroads on the way to A. However, what could be
argued is that he seems to appreciate these passages as leaking holes rather than
ordinary crossroads. If we, based on this, interpret Oskar’s view of the rolling
marbles as similar to what is described in figure 4a, we could understand the
difficulties Oskar has had to give meanings to his probabilistic estimates. Even if he
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has noticed the importance of the crossroads, his contextualization at hand does not
give him the opportunity of putting a numerical meaning to them.
a
b
Figure 4. Two different interpretation models
2) The division strategy
To illustrate the division strategy we look at a pair of students (from iteration 2)
working with setup 2. After playing a while with setup 2, Ole and Björn are asked to
make a first suggestion about the probability for ending up in B. Their response also
included an explanation of their hypothesis.
Ole:
Bjørn:
Ole:
Bjørn:
Ole:
R:
Ole:
It is ... actually ... as much as ... I believe that it is ... it is a fifty
percent chance for coming there [pointing at C].
Yes, it is fifty -fifty [pointing at the first crossing]
Yes, but it is also ... it is ... then it becomes almost twenty-five
percent chance for coming to B and fifty percent chance for
coming to C.
It must be twenty-five.
I was about to say that it is because fifty is divided by two again.
The chance is smaller for coming to A and B, than it is for coming
to C.
Ok. What is the chance for coming to B?
I believe it is twenty-five.
The students are aware of that the probability is reduced in a crossroad and since two
paths lead out from a crossroad they choose to divide by two. The division strategy
seems to be dynamic in the sense that the students follow the marbles from the top
and towards the end, beginning with 100 percent and then using repeated division by
two an appropriate number of times. (In the case above: 100% Æ50% Æ 25%). Their
way of reasoning also seems to involve idealized aspects. Based on symmetrical
features, a crossroad is modelled as a fifty-fifty chance, as mediated in figure 4b.
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DISCUSSION
The aim of this paper was to investigate strategies that became actualized in students’
different ways of handling OOSP-tasks in an explorative setting. From the variation
in outcomes it can be argued that the participants interpret the situation in different
ways. The participants activate different contexts for interpretation in their meaningmaking processes. Regarding this, two main strategies appeared as prominent. The
first has been described in terms of “holes in the way” and we called this approach
the leakage strategy. Considering the other strategy the situation was interpreted from
a more computational point of view, in terms of the mathematical operation division,
which we refer to as the division strategy.
Using the leakage strategy, the students’ reasoning reflects that aspects of random
phenomena are taken into account. They seem to be aware of that the probability in
some sense is distributed, in that the marbles are distributed in a non-deterministic
manner. However, it seems unclear for the students to know precisely how such
phenomena work. As argued above, this seems to be due to that the students might be
involved in a physical interpretation of the situation, similar to what we have
modelled in figure 4a. Comparing with the robot problem, illustrated in figure 1, it
would be interesting to further explore this on the basis of the leakage strategy. Based
on our result it could be argued that a greater amount of more proper answers would
appear, if the ways out to room 1-4 had been directed downwards instead of upwards.
In such a representation of the problem, the crossroads may be interpreted as leaking
holes and by that lowering the chance of ending up in room 5-8.
In the division strategy the students saw a crossroad as a 50-50 percent chance. It
could be argued that they here in a direct way take into consideration the number of
paths leaving a crossroad, that is, divide by two. To gain even more information of
such reasoning it would be interesting to create a Flexitree situation, containing
features similar to those of Fischbein’s pagoda, in which marbles encounter
crossroads having three or more way outs.
Regarding teaching implications, the results above indicate some important aspects.
Firstly the teacher has to be aware of that a learning situation not in itself is static.
That is, not all students do recognize the intended learning object. Secondly, an
analysis of student’s interpretations in general and strategies in particular, has a
pedagogical potential to make teachers aware of possible directions in the students’
activity, which may improve teachers’ capability to intervene in the situation.
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