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Название публикации: «EXPERIENCES OF APPLYING
PROBABILITY THEORY IN PRIMARY SCHOOL»
ФИО авторов: Abdig’afforov Davron Erkin o‘g‘li
The academic lyceum of Termez state engineering and agrotechnology
university
Abstract. In recent years, many countries have tried to incorporate the
probability concepts into the curriculum of primary school. The researchers
disagree as to what the age of children dealing with probability contents should
be. The aim of this study was to investigate the grade of understanding of the
probability concepts in primary school students depending on their age and their
gender. It has been concluded that the majority of students was able to recognize
different events and categorize them depending on their likelihood. The major
difference in their abilities was noticed between the children of the second grade
and those of the third grade whereas it has been experienced that girls performed
better in all the tasks and in all the grades except for the fourth grade in which
boys present a slightly better score.
Keywords: teaching probabilities, primary school, probability concepts,
probability tasks
Among the new development and trends in mathematics education
internationally at primary level, is the introduction of probability. Nowadays,
there is a need to work on the Probability Theory because we are obliged to make
predictions and to take decisions under uncertain situations having to evaluate
at the same time a big amount of information. Researchers from many countries
(like Farnworth, 1991; Fischbein & Schnarch, 1997; Freudenthal, 1973; Gardner,
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1989; Jones, 1995; Koshy, Ernest & Casey, 1999; Shaughnessy, 1992; Sobel &
Maletsky, 1988), notice that probability is very important branch in mathematics
education because:
• it provides opportunities for students to engage in interesting and
purposeful learning activities,
• it is easily understood by students that are in different ages and have
different capabilities as well as
• wherever it has been taught, the results were positive and enhanced the
mathematical thinking (Skoumpourdi, 2004).
According to Fischbein (1984) the reason to introduce probability are
“dealing” with uncertain situations, predicting, deciding among different
probabilities (critical implementation), problem solving (deliberate actiontaking)
and developing the thinking ability different from the deterministic one. Gal
(2005), Franklin et al (2005) and Jones (2005) state that the reasons for including
probability in schools are related to the usefulness of probability for daily life,
its instrumental role in other disciplines, the need for a basic stochastic
knowledge in many professions, and the important role of probability reasoning
in decision making. Students will meet randomness not only in the mathematics
classroom, but also in biological, economic, meteorological, political and social
activities (games and sports) settings. As evident, teaching contents in probability
have numerous advantages, which other mathematical disciplines lack.
Through dealing with the mentioned contents children learn to accept the fact
that also negative situations can be encountered, which are not possible to be
precisely predicted. The only thing to be done is to critically interpret all the
possibilities and choose the one which is most likely to happen. In this way
children gather experiences for real life situations, in which it is necessary to
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decide on the best option out of many on a daily basis. At the same time children
have to accept the fact that some events are impossible to happen. So, it is
necessary to act deliberately and solve the problem, whereby one should make
use of his mode of thinking, different from the one applied at learning other
mathematical disciplines. (HodnikCadez & Skrbe, 2011). There was very
intensive research conducted into the establishment of understanding the
probability contents of the early school children in the 70-ies and 80-ies of the
last century. These research works, which shall be presented in detail in
continuation, were the key starting points of our research. It should be
emphasized that the subject matter of understanding the probability contents with
the youngest population is still topical, as it also being delved into nowadays.
Let us shortly mention some such research works: Gelman and Glickman (2000)
researched the importance of the demonstration and concrete experience with
teaching probability contents and established that children better understood
more difficult concepts if they actively participated in the corresponding
demonstrations. Mills (2007) delved into the attitudes of teachers towards
probability contents establishing their positive attitude to the statistics and
probability contents, and their wish to be offered the possibility of suitable
additional training. Ashline and Frantz (2009) dealt with the connection between
proportionality and probability contents, while Click (2010) was engaged in
probability games played at lessons. Van Dooren et al. (2003) was interested in
pupils’ misconceptions pertaining to the probability contents. As the most
absorbing discussion on the understanding of the probability contents was
conducted a long time ago among the scientists, such as, e.g. Piaget, Inhelder,
Fischbein, Davies and others, we shall focus more precisely on their findings, and
on some conclusions of the most recent research in this field, which are topical for
our research. The opinions of various researchers about the abilities of children
with regard to solving probability tasks differ a lot. Piaget as well as Inhelder
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(1951) state that a child in a concrete-operational period is neither able to
differentiate between certain and random predictions nor formulate predictions,
taking into account his experiences from previous similar situations. In their
opinions a child first encounters the concept of probability at the level of his
concrete operations, at which time he starts to differentiate between a certain and
a possible event (Piaget, Inhelder, 1951; Goldberg, 1966) They also note that
the systematic understanding of probability starts not earlier than between the
ages of 9 and 12 years and even during that period children solve problems
intuitively, and not on the basis of formal reasoning. Many researchers
contradicted their findings on abilities of children regarding their perception of
the probability contents and argued the converse, among them Fischbein et al.
(1984) and also Davies (1965) and Yost et al (1962) criticized.
Piaget’s research mainly because it was based on a child’s verbal abilities;
they developed “the decision making method”, which was not based on verbal
abilities; using it children decide between two boxes (children draw out of the
very bow, from
which they believe to extract the chip of the desired color), but they do not
need to use the expression “most probable”, whereas in Piaget’s research the
box contained chips of two colors and children had to choose the color, which
was more probable to be extracted (Yost et al., 1962). Fischbein and his
numerous colleagues also elaborated on teaching and learning the probability
concepts, thereby concluding that it was possible to teach probability without
any major efforts, which had a positive influence on the child’s prejudices and
misconceptions about the sequence of events and uncertain situations
(Fischbein, Gazit, 1984; Fischbein, Pampu, Manzat, 1970).
Among other things he found out that under certain conditions learning of
probability concepts may have a negative impact (children taught probability
topics performed worse at some tasks compared to those children who were not
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presented with these topics); nevertheless, Fischbein believes it would be
possible to avoid this by presenting children with tasks including relationships
calculations and probability estimations (Fischbein, Gazit, 1984). Lately also
Gurbuz et al. (2010) dealt with probability teaching and learning. They were
trying to establish the effectiveness of the teaching approach based on pupils’
active participation, whereby pupils were making numerous experiments
pertaining to probability, and were discussing their findings among themselves
in the followup. In the control group the pupils were deprived of this possibility
at lessons (Gurbuz et al. 2010). In the research, in which 50 children participated,
it was established that children who were provided with the teaching approach
based on the discussion between pupils and teachers performed better than those
children who were provided with lecturing lessons only (Gurbuz et al, 2010).
Also Andrew (2009) stresses the importance of concrete experience, as he
believes that pupils better understand probability contents if they perform
experiments related to probability in advance. Thus, it is important that pupils
gain experience also by drawing out, thus trying to determine the more likely
event.
Concepts in probability can be more readily understood if pupils are first
exposed to probability via experiment. Performing probability experiments
encourages pupils to develop understandings of probability grounded in real
events, as opposed to merely computing answers based on formulae (Andrew,
2009). Andrew (2009) further states that pupils who have gained concrete
experience in probability develop their understanding on this basis and wish to
define the starting points to calculate probability of certain events (HodnikCadez
& Skrbe, 2011). Apart from researchers, many constitutions, like Unesco (1972)
and Ceeb (1958), have recognized the important role that probability plays in
our society and they suggest to include them in the primary school’s
curriculum. Probability was introduced to the academic lesson series in the
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western world as a part of the “new mathematics” changes in the 1960s. They
are one of the few things that have been preserved in the last forty years. But
only in the late 1970s did the educational importance of probability in the
primary and junior high school become apparent. This was due to the fact that
until that time the grounds of their introduction was based more on practical and
academic reasons rather than on some educational belief having to do with
their importance as a part of the overall education. At exactly the same period
(1975) when the book written by Piaget & Inhelder was translated into English,
Fischbein (1975) published his significant work on intuition and probability that
is based on research made over many years. Thus, at about 1975 probability
teaching and learning began to interest the researchers (Truran, 2001). The
study of probability, in the 1970s, became initially an object of teaching in
junior high schools, while later on it was also introduced in the fifth and sixth
grade of the primary school. However, since the 1990s, probability seem to
hold an integrated part in the mathematics curriculum from even the smaller
grades in primary schools internationally. Recent curriculum recommendations
of the USA and Canada (NCTM, 2000), England (DfEE - Department for
Education and Employment, 2001), Cyprus (CIE, 1997) and Greece (1987 and
1998) have recognized the importance of having all students develop an
awareness of probability constructs and applications. Specifically, they mention
that the students should understand and apply basic concepts of probability using
the property vocabulary for them depending of course, their age. Because of
this emphasis on probability in the school curriculum, there has been
considerable research into student’s probabilistic thinking. The current tendency
even for primary school level is towards a data-orientated teaching of
probability, where students are expected to perform experiments or simulations,
formulate questions or predictions, collect and analyze data from these
experiments, propose and justify conclusions and predictions that are based on
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data (Carmen & Carmen, 2012). Based on the views that distinguished scientists
maintain and the suggestions that a number of organisms have made, it becomes
apparent that the introduction of probability in primary education is necessary as
well as feasible. Of course, an important condition for the introduction of the
probability concepts in primary school is the use of the appropriate
probabilistic expressions and the suitable vocabulary for the different age groups
of children (Skoumpourdi & Kalavassis, 2003). The purpose of the present study
was to investigate the level of understanding the probability concepts in primary
school students and to establish any potential statistically significant differences
among different age groups of children and between genders at solving
individual tasks.
The results of children of different age groups at solving the tasks of the
Test are presented below. All the tasks relating to the children’s differentiating
among certain, possible and impossible events were correctly solved by more
than half of the respondents, which is well evident from Figure 1. In addition,
many of the tasks were correctly solved by more than the 75% of the students
of all the grades. In the first task, we observe that the lowest score belongs to
younger participants (75.1%) and as the age grows, the score is getting higher as
well. Students of the third grade obtained 85.2% and children of the fourth grade
have a better score (95.2%) than the kids of the fifth grade (92.1%) and the kids
of the sixth grade (94.6%). A statistically significant difference is observed in
this task between the grades (F4, 399 = 24.907, p
< 0.001) according to the technique of One Way ANOVA which was used in
order to examine possible differences among the score of the five grades of the
elementary school. In the second task, children of the second grade obtain the
lowest score (78.7%) and they are followed by the students of the third grade
(90.6%). The highest score belongs to the sixth grade students (97.1%). Students
of the fourth grade (94%) performed better than children of the fifth grade
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(93.6%). A statistically significant difference is experienced in this task (F4,
399 = 19.295, p < 0.001) as well. In the
third task, children of the fourth grade made it better (82.5%) whereas some
small differences are observed among the third, the fourth, the fifth and the
sixth grade as the statistically significant difference is observed to a lesser
extent (F4, 399 = 3.025, p = 0.018 < 0.05).
In the research, it was established that the majority of children of second
to sixth grade of the elementary school was able to compare the probability of
various events. Also, some statistically significant differences have been
experienced between the different ages of the students and the gender at solving
probability tasks. It can be concluded that our results are similar to the ones of
the researchers, such as Fischbein et al. (1984) and Davies (1965), who believe
that children in elementary school are able to solve certain probability tasks. In
addition, these results agree with the outcomes of the research of HodnikCadez
& Skrbe (2011) who support that children of the younger grades of the primary
school . In all the mentioned activities children predicted and assessed the
likelihood of an event. The situations differ among themselves, they are related
to everyday life, to common language, they are presented in different ways and
offer children many possibilities for discussion, assessment and arguing the
likelihood of an event. Alongside the vocabulary development and familiarity
with recording conventions, all of the different types of activity offered to
primary aged children in mathematics lessons were supposed to bring with
them some aspect of a mathematical perspective on the relationship between
possibilities and probabilities. That is, after all, the main point of introducing
probability into mathematics classes .
References
[1]
Andrew, L., 2009. Experimental probability in elementary school. Teaching
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[2]
Ashline, G., Frantz, M., 2009 Proportional reasoning and probability.
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[3]
Athanassiadis, E., Skoumpourdi, C., & Kalavassis, F., 2002. A Didactical
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[4]
Carmen, B., & Carmen, D., 2012. Training school teachers to teach
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