European Journal of Education Studies
ISSN: 2501 - 1111
ISSN-L: 2501 - 1111
Available on-line at: www.oapub.org/edu
Volume 3 │Issue 4│2017
doi: 10.5281/zenodo.345617
ASSOCIATING MATHEMATICAL STORIES THAT ARE WRITTEN
BY THE 8TH GRADE STUDENTS WHO ARE STUDYING AT
ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS'
SCHOOLS WITH THEIR MATHEMATICAL PERCEPTIONS:
ISTANBUL CASE
Elif Bahadıri
Department of Elementary Mathematics Education, Yıldız Technical University, Turkey
Abstract:
In this study, mathematical stories written by 50 middle school students were analyzed.
The study group consisted of two different student groups who were living in
advantageous and disadvantageous regions in Istanbul. At the first stage, the students
were presented a mathematical story called My Fractal Tree , then told about what the
mathematical story was and asked to write a mathematical story about any subjects. 43
of the stories have story characteristics. The stories were separately analyzed under the
headings of "involving a mathematical subject", "having mathematical characteristics or
not" and "the math topics students used in their stories". The findings about the
contents of the stories were analyzed on
main themes "Mathematics relations with
other subjects", "Perceptions towards mathematics", "Mathematical level" and
"Creativity". Students can be seen to have mentioned about the math s relations with
other subjects. Students abilities to use mathematical elements correctly are quite low,
their learning about concepts is weak and their misconceptions are too high. Generally,
a very plain language is seen in the stories in terms of language expression, but
transitions between topics seem weak. Considering the creativity element in the
students writings, advantageous group was seen to have created highly creative
stories.
Keywords: mathematical stories; 8th grade; mathematical advantageous and
disadvantageous regions' student groups; mathematical level
Copyright © The Author(s). All Rights Reserved.
© 2015 – 2017 Open Access Publishing Group
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ASSOCIATING MATHEMATICAL STORIES THAT ARE WRITTEN BY THE 8TH GRADE STUDENTS WHO ARE
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
1. Introduction
Many educators and researchers have recently put emphasis on the willingness of
students to learn instead of the structure of information. Therefore, learning
environments should be organized by factoring in the willingness of students to learn.
In order to provide for an effective learning process, it is necessary to provide a context
that will help students develop mathematical skills and competence and provide
sufficient physical space and suitable material to make it easier for students to learn
mathematics as suggested in the learning environment standards mentioned in NCTM.
The easiest way to keep students willing is to relate relevant topics to daily life. The
purpose of narratives, which become increasingly popular in education, is to relate to
daily life, help students develop a better approach to and admiration for sciences
through the presentation of scientific concepts in daily situations and raise scientifically
literate individuals. Such approach provides students with an opportunity to make
their learnings meaningful and to participate actively in helping them improve their
information of scientific concepts. Such teaching materials also contribute in students to
assume more responsibilities in their own learning process.
According to the Constructivist Learning Theory, learning takes place when an
individual constructs information in his own mind as a result of direct interaction with
his environment (Baker & Piburn, 1997; Brooks & Brooks, 1999). It is emphasized that
existing information of each student is very important to attribute a meaning to new
information or simulations since each student constructs his information and concepts
on his own through his skills and experience (Duffy & Jonassen, 1991; Wittrock, 1974).
Furthermore, many educators and researchers have recently laid a lot emphasis on
motivating students to help them become responsible for their own learnings and
making them more willing to learn. Therefore, learning environments should be
organized by factoring in not only the previous information of students but also their
willingness to learn. All children have an innate willingness to learn and need to be
supported specifically by high-quality educational environments and experiences.
Mathematics learning is based on the curiosity and enthusiasm of children and
developed through natural experiences. If children are given the chance to learn
mathematics in an ideal way during this period, they will be more ready for school or
provided with the opportunity to make a smoother transition to primary education
(NCTM, 2000).
According to Kitt & Leitze (1992), students not only perceive mathematics as a
difficult but also as a boring course. Other studies also suggest that students consider
mathematics as a difficult discipline to understand since it involves very abstract ideas
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ASSOCIATING MATHEMATICAL STORIES THAT ARE WRITTEN BY THE 8TH GRADE STUDENTS WHO ARE
STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
(Kee & McGovan, 1998; Reid, 2000). However, mathematics is an entertaining area that
helps students understand the world they live in aside from being a course necessary to
be taught in schools to prepare students for their careers. Nevertheless, when concepts
learned do not go beyond theory, they exist as an abstract expression that must be
memorized for exams. Therefore, there is a need for different teaching materials that
will make learning meaningful for students and address the concepts studied along
with what they stand for in daily life. Even though many different methods are used in
mathematics teaching, the most common method is still the conventional plain lecturing
method. Most students try to learn mathematics by heart even though it is necessary for
life, and most teachers direct students towards memorization through the teaching
methods they use. Teachers should teach mathematics by allowing students to
maximize their levels of understanding and not forgetting mathematical concepts. The
contemporary understanding of education presents teachers with the obligation and
responsibility to choose and implement the teaching methods that will ensure a
maximum level of learning (Yilmaz, 2001).
According to Bahadir & Ozdemir (2013), it is necessary to save children from the
limits of thinking within strict patterns and take them to a multi-optional world where
processes like imagination, intuition and emotion exist in a tangled pattern. According
to Yenilmez & Bozkurt (2006), it is necessary to ensure that students develop a positive
approach towards mathematics first in order to help them be more successful at this
course. Educational processes must be organized to change the negative approach of
students towards mathematics and facilitate understanding the subject. Principles and
Standards for School Mathematics (NCTM, 2000) suggest that mathematics, which is
identified with abstract concepts, must be rendered more understandable through the
use of physical materials and concentration models, and then, abstract concepts may be
taken to the level of abstract ideas.
Zemelman, Daniels & Hyde (1998) recommend that students discuss, read, listen
and write about difficult mathematics topics in order to understand them. Some
educators claim that Mathematics teaching is a social process. Communication plays an
important role in this process. Pirie (1998) categorizes ways of communication in
mathematics teaching under six groups. These are ordinary language, mathematical
verbal language, symbolic language, visual representations, unspoken but shared
assumptions, and quasi-mathematical language. Mathematical language constitutes a
dimension of such communication. Syllabuses and in-class activities started to be
reviewed after it was realized that communication played an important role in
mathematics teaching. In this context, the commission founded in the United States by
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ASSOCIATING MATHEMATICAL STORIES THAT ARE WRITTEN BY THE 8TH GRADE STUDENTS WHO ARE
STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
the National Council of Teachers of Mathematics (NCTM) to lay down the standards of
school mathematics defined communication and language in mathematics teaching as
a standard in 1986 and presented its final version in the report published in 1989. This
report states that one of the purposes of mathematics teaching is to ensure that the
student learns how to speak mathematically. To that end, it is expected from a student
to use the mathematical language, namely his mathematical vocabulary.
The activities carried out to enrich the classroom environment have an important
place in using language in mathematics teaching. These activities include readingwriting activity and problem creation (Bali & Alvarez, 2003). It is not common for
mathematics teachers to conduct writing activities and give writing assignments in
mathematics teaching. Thus, students may not quite relate writing to mathematics. In a
research conducted in this area by Liedtke & Sales (2001) on 11 females and
males
7th-grade students in American schools, the students were asked about their opinions
on writing in mathematics courses, and it was observed that students at first could not
relate writing to mathematics at all. Students were later observed to have changed their
opinions after certain writing techniques and assignments.
What will ensure improvement in the language of mathematic teaching is
reading mathematics books and giving mathematical reading assignments. Orton &
Frobisher (1996) put forth in their research on frequency and readability of mathematics
books, that the frequency and readability of mathematics books was low. Use of
mathematical language plays an important role in mathematics teaching whereas
creation of word problems has an important role in improving the mathematical
vocabulary and organizing and presenting the mathematical idea. Orton & Frobisher
(1996) indicate that the purpose of using word problems in a mathematics course is to
define a daily life situation as an area of implementation and turn it into a mathematical
problem. Thus, it is purported that mathematics is not only a course lectured at school
but also a study that can be related to real life. It is important to encourage students to
use mathematical stories in class and write mathematical stories in order to improve
mathematical communication in students. It is one of the most effective methods to
make topics more entertaining and more relating to life. Stories are extremely important
tools that try to give meaning to information and comprise of related and consistent
information (Millar & Osborne, 1998). Mathematics stories have a setting that is
integrated with mathematical terms, characters, a scenario and a plot (Borasi, Sheedy &
Siegel, 1990).
Storytelling method improves verbal language use of children, provides fun
learning experiences for students, enhances the use of words and realizes their social
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STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
and emotional development through social experiences (Belet, 2011). Storytelling helps
children imagine, feel and learn more meaningfully (Goral & Gnadinger, 2006).
Children create mental images while listening to a story and make some personal
connections because young children are in the age of imagination, and because of this,
our teaching must be delivered to them through images (Steiner, 1997). Kurtz &
Ketcham
believe that stories are the vehicle that moves metaphor and image into
experience . Stories are extremely important tools that try to give meaning to
information and comprise of related and consistent information (Millar & Osborne,
1998).
2. Using a Story Context to Teach Mathematics
According to many cognitive scientists, stories contain a great deal of natural pieces of
information that have their well-organized versions in the cognitive systems. Thus,
information can be learned more meaningfully and permanently through a story
context (Bruner, 1987).
Therefore, it can be said that stories as teaching materials have a rather strong
effect on remembering information and improving concentration (Bower & Clark, 1969;
Graesser al., 1980). In narrative writing, the aim is to allow a student to tell his/her
experience based on true events or make him/her create a story based on imaginary
events. According to Bereiter-Scardamalia (1982), inexperienced writers only describe
information in their writings or tell a simple story. There is no need to plan or set a goal
in such writings. Starting from these powerful characteristics of the stories,
mathematical stories in which we can review so many components together related to
students are important tools in terms of our comprehending the students mathematical
understandings.
A mathematics story should not be confused with a mathematics problem. The
goal is not to ask a question to students but to present the solution to the problem in an
interesting manner based on the plotline. Narrative texts are the texts that are read
during reading-understanding studies and for many purposes such as satisfying the
curiosity of students and enabling learning while having fun. The literature indicates
such texts are easier to read and understand when compared to informative texts and
readers are more successful at reading and understanding such texts (Akyol & Temur,
2006; Bastug 2012; Hiebert, 2003).
NCTM (1989, 2000) has recommended the use of children's trade books (story
books) as a way of introducing mathematical ideas through literature. Such trend has
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
become quite popular in the last 2 decades (Lewis, Long & Mackay, 1993; Thatcher,
2001). This phenomenon occurs for children as well: a narrative context helps them
learn and remember information better (Lucariello & Nelson, 1985). Cordova & Lepper
(1996) found that for fourth and fifth graders, a fantasy story context substantially
improved performance on a math test. Recent researches on stories have been
conducted empirically in order to understand the role of stories in mathematics
teaching. For example, (Jennings, Jennings, Richey, & Dixon-Krauss, 1992; YoungLoveridge, 2004) detected a very high mathematical progress in their research on the
mathematical literacy of low-income groups through teachings conducted with the aid
of mathematical stories.
In his research on students with high socioeconomic statuses, Hong (1996)
detected positive effects on their mathematical literacy motivations and did not detect
any nonstandard increase in their mathematical scores. There are studies suggesting
that ethnic culture, socioeconomic status, and racial structure are effective in
mathematical achievements of students. Albert (2000) detected that students from
African American and Hispanic cultures were disadvantageous at mathematical
learning through verbal lecturing. Furthermore, Haynes & Gebreyesus (1992)
concluded that music-based activities and cooperative learning are far more effective in
the learning of African American children.
3. Fractals and Fractal Teaching
Fractals in teaching program, which also appear in pattern decorations, are important
structures to understand the mathematics structure and examine the patterns and their
relations (Hargreaves, 1999). Fractals may be expressed as repeated patterns that are
always self-similar or sometimes randomly different in some parts. A fractal has a
structure that is different from the patterns consisting of usual Euclidian figures. It can
be said that fractals generally have four basic features, which are complexity, iteration,
self-similarity and fractal dimension. Complexity is mostly a feature of natural fractals.
Natural fractal shapes are irregular, indented and complex in structure (Lornell &
Westerberg, 1999).
Fractal objects are formed as a result of iteration rules instead of algebraic
formulae, unlike Euclidian objects. Iteration is defined as a continuous repetition
process where the result in one step is the beginning in another (Kelley & Allison, 1999;
Korvin, 1992). Each result derived from the iteration process presents a beginning for
the next iteration process. Even though it is clear how the steps of fractals are iterated,
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STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
the structure becomes complex; therefore, the mathematical expression of fractals
should be noted (Lee, 1999). In its report in 1989, NCTM recommends students work
with non-Euclidian geometries in order to get to know and identify the universe better.
Furthermore, the supplemental reports published by NCTM between the years 1991
and 1993 emphasize a series of new mathematics topics, which will increase the
interests and needs of students from all levels in mathematics, help them feel positive
about mathematics, relate mathematics to the nature and allow use of technology in
such relationship, aside from traditional mathematics topics. One of the topics
recommended is fractal geometry.
Considering the studies in literature conducted regarding the teaching of fractals,
it is seen that these studies are mostly intended to improve the activities to be utilized
during fractal teachings. For example, in his study, Thomas (1989) uses Logo program
to form fractal shapes and discusses activities regarding two basic properties of fractals,
which are self-similarity and fractal dimension. Similarly, Adams & Aslan-Tutak (2006)
form the Sierpinski s triangle in the worksheets they have prepared for th to th-grade
students and define the patterns within the triangle.
Naylor (1999) gives activity examples to create fractal structures, examine their
self-similarity and iteration properties and calculate the circumference and area of
fractals. Fraboni & Moller (2008) shortly define fractals in their study and explain their
properties that differ from Euclidian geometry. Lornell & Westerberg (1999) introduce
the activities that they use in 9th and 12th-grade courses and have developed regarding
the teaching of complexity, repetition, and self-similarity, which are basic properties of
fractals. In addition, they include an activity to calculate the circumference and area of
the Sierpinski s triangle.
In some studies, conducted on fractals, it was found out that students did not see
mathematics as a whole (Simmt & Brent, 1998; Lornell & Westerberg, 1999; Karakus,
2007; Fraboni & Moller, 2008). As understood from the aforementioned studies, fractals
can be said to have helped the students to grab a holistic approach.
4. Overview and Goals of the Study
This study is intended to create the classroom environment at the Learning
environment standards defined by NCTM (2000) and ensure the 5 standards
(information of mathematical problem solving, information of reasoning and proof,
information of mathematical communication, information of mathematical connections
and information of Mathematical Representation) sought for the principles and
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ASSOCIATING MATHEMATICAL STORIES THAT ARE WRITTEN BY THE 8TH GRADE STUDENTS WHO ARE
STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
standards for school mathematics as defined by NCTM (2012). In order to reach these
standards in the study, an activity that includes a mathematical story is prepared and
then secondary school students are asked to create a mathematical story. With this way,
mathematical story creating skills of students are analyzed.
This process is intended to build a relation between mathematics and daily life
by benefiting from the power of stories by means of making students feel that they can
create a mathematical story as well through the creation of an effective learning
environment. In addition, the study aims to help students make an interpretation of the
mathematics concept of the mathematical stories they write in their minds, learn the
skills to clearly and consistently convey their mathematical thoughts to their friends,
teachers and others and get an idea about the ability to interpret physical, social and
mathematical phenomena together.
In the study, the mathematical story narrated to the students is about fractals.
Fractal geometry has been commonly used in many different areas such as art,
astronomy, biology, chemistry, physics, computer, economics, engineering, geology and
genetics; and the increasing use of fractals in many areas makes it necessary to learn
and teach within the frame of mathematics education in school. Besides, it is believed
that this study will contribute to the literature in terms of teaching fractals through
storytelling technique, which is not so common in mathematics teaching, and analyzing
story creation skills of students.
In this study, which will assess the practicality of My Fractal Tree story when
teaching the fractals topic and examine the mathematical stories created by students,
answers will be sought for the following sub-problems:
1. Is the mathematical story called My Fractal Tree , prepared for fractals topic, an
effective activity that can be used to explain the topic to middle school students?
2. Could middle school students write a mathematical story?
3. What are the characteristics of the mathematical stories created by middle school
students?
4. Do the stories exhibit different characteristics when the stories created by the
students in advantageous and disadvantageous regions are compared? Were
there any differences when these characteristics were categorized?
In the student stories, their levels of "ability of writing a story included a math
subject", "whether the written text has a characteristics of a mathematical story" and "the
math topics which students choose to use in their stories", "Mathematics relationship
with other courses", "perceptions towards Mathematics", "Mathematical level" and
"Creativity" have been analyzed.
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ASSOCIATING MATHEMATICAL STORIES THAT ARE WRITTEN BY THE 8TH GRADE STUDENTS WHO ARE
STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
5. Method
The research was conducted during the fall semester of the Academic Year 2015-2016. A
qualitative method was used in the research. This study has been modeled as an action
research since it is an interventional qualitative study. A case may be a program, an
event, an activity or a group of individuals limited to a certain period of time and space
(McMillian & Schumacher, 2001). Observational data collection techniques and written
sources were used during the research. In the research, data collection tools were used
and constant comparison method was adopted to analyze and interpret the data
collected.
5.1 Participants
The research was conducted in two different schools comprising students from
Advantageous and Disadvantageous groups in Istanbul city in the Academic year 20152016. For academic success and creative thinking skills of students, the structure of
society they grew in has been known to have a great importance. The recent studies
have been revealed that there is a significant relationship between the student s socioeconomic infrastructure and his/her success in school (Hanushek, 2010; Hanushek &
Woessmann, 2010; Lacour & Tissington, 2011; Maughan, 1988; OECD, 2011; UNESCO,
2006). Considering the fact that mathematical story writing activity which is the focus of
this study is based on creative thinking, mathematical skills, and verbal expression
skills, this study was conducted with two study groups as advantageous and
disadvantageous, by thinking parent support and socio-economical structure were also
important. On one hand, students living in advantageous regions are supported by
their parents effectively, encouraged to think creatively and promoted socially, on the
other hand, students living in disadvantageous regions are known to reach these
opportunities very limitedly. Conducting the study in both advantageous and
disadvantageous regions has been seen to be important in this study in terms of
students' verbal expression and creativity skills.
The purposeful sampling method was used to determine the participants of the
research. The students in the advantageous group were selected among the students
residing in a region that was populated by high-income parents whose social statuses
were above average in Turkey. A private school located in Etiler region was selected as
the advantageous region. The students in the disadvantageous group were selected
among the students residing in a region that was populated by low-income parents
whose social statuses were at or below average in Turkey. A public school located in
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STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
Esenler region was preferred as the disadvantageous region. Since fractals are included
in the curriculum of 8th grade, the activity was conducted in the 7th grades of both
schools. The research was carried out in an 8th-grade class comprising of 30 students in
the advantageous region school and an 8th-grade class comprising of 29 students in the
disadvantageous region school. The study is conducted with 59 students in total.
5.2 Analysis of the Data
Categorical context analysis was used in the analysis of the research data. The key
objective of this analysis is to attain concepts and connections that can explain the data
collected. In general, categorical analysis as a sub-category of context analysis involves
division of a certain message in units first and then grouping of these units into
categories according to predefined measures.
The data gathered was evaluated by two independent specialists in order to
ensure the reliability of the research. The effect of the agreements and disagreements
between the researchers and specialists on the reliability of the research was analyzed
using
the
formula
of
Miles
&
Huberman
(1994),
[Agreement/(Agreement+Disagreement) x100]. As a result of the analysis, the
agreement (reliability) between the decisions of the researcher and the two specialists
was found out to be %91 and %87, respectively.
5.2.1 Analysis and Interpretation of the Data
The stories created by the students were collected and used as part of a set of data.
Descriptive analysis was used to analyze the research data. The order of the steps
carried out in the analysis of the data is as follows;
A. Inventory of data: The data gathered from each story created by the students were
inventoried in the data inventory forms, each line in the form was enumerated and the
interpretation of the researcher was written down in relevant sections. At the end of this
process, all inventories were submitted to a specialist to ensure the validity of the
inventoried data. Creating coding keys and coding data: This step of the analysis
involved reading and organizing all the data collected and gathering them logically.
Thus, it was defined under which themes the research data would be gathered. The
data derived from the stories written by the students in the research was separately
coded by the researcher and a field specialist and an agreement was reached on the
themes created. Comparison of coding and reliability: Following the coding process
independently performed by the researcher and the specialist of the field, the researcher
and the field specialist met and compared their analyses and defined the points of
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STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
agreement and disagreement. The analysis of the data was conducted by the researcher
and two field specialists, after which the data was submitted to other researchers for
control.
B. Definition and interpretation of the findings
This was the step where all the data were analyzed and determined based on the
research questions and the findings were transferred, explained and made meaningful.
The findings of the research were directly offered through references and interpreted by
relating them to different research findings. The study was conducted with 59 students,
and 52 of them were able to create a text. 43 out of 52 texts have been identified to have
had story characteristics by the researchers.
The general findings of the data analyzed from the story texts were determined
under these headings separately: "to be able to write a story including mathematical
subject , "whether the written text has a mathematical story characteristics" and
"mathematical subjects that students used in the story texts". In addition, the findings
based on the text contents were analyzed under
relationship
with
the
other
subjects",
main themes "Mathematics
"Perceptions
towards
Mathematics",
"Mathematical level" and "Creativity".
5.2 Activity
The activity was implemented in three stages. At the first stage of the activity, students
were handed out various fractal examples and asked to examine them. Students were
told about the fractal concept. ‚t the second stage of the activity, the story my fractal
tree was narrated with the aid of a presentation supported with the visual materials
prepared. At the end of the story, students were expected to arrive at a conclusion
about exponential numbers on their own. At the last stage of the activity, the students
were asked to write a mathematical story. The stories of the volunteering students were
shared in the class.
During the first stage of the activity, students were also asked what they
understood from the fractal concept and the ideas of the students were written on the
blackboard. These ideas were kept on the blackboard until the end of the activity. Since
the students were instructed on the fractals topic during the semester, a majority of
them answered this question. Some of the answers given by the students included they
are decoration art, geometric shapes, interwoven shapes, symmetry, mosque ornaments, and
laceworks are fractals… Some examples prepared from fractals were shown to students
and they were given examples of fractals found in nature. Students were given a
worksheet containing simple fractals in order to help them remember fractals. By going
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STUDYING AT ADVANTAGEOUS AND DISADVANTAGEOUS REGIONS' SCHOOLS WITH THEIR
MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
over the fractal samples handed out, it was emphasized that fractals actually had a
regular structure.
My fractal tree story was enriched with a visual presentation and narrated to
students (Appendix-1). The story used in the study is as follows: Murat, the character of
the story, draws a fractal tree and wonders how many branches he has drawn on the
last order he has drawn. Murat divides each figure into 2 repeatedly in order to draw a
tree; he comes to the last order having repeated this operation regularly for 9 times. At
this stage of the story, the students are asked how many branches there are. Some
students mention how difficult it would be to count the branches while others easily
solve the problem by multiplying 9 by 2. In the rest of the story, our character solves the
problem using exponential numbers. After the story was completed, the ideas of the
students about the fractals written on the blackboard at the beginning of the course
were reexamined.
In the study conducted, students solved the problem given in the story using
basic mathematics skills even though they were not very experienced in it. Even though
what the students experienced was a fictional story, the character of the story had a real
problem to solve and use of exponential numbers helped the solution of the problem.
At the last stage of the activity, the students were asked if they liked the mathematical
story. Most of the students indicated that they liked the story. The students were asked
to write a story about any mathematics topic
6. Findings
The data obtained from the written stories by the students were analyzed, and the
results obtained from the study were analyzed in two separate titles. Firstly, the general
findings on the data analyzed from the stories were analyzed separately under these
headings: "to be able to write a story including mathematical subject , "whether the
written text has a mathematical story characteristics" and "mathematical subjects that
students used in the story texts". Secondly, in addition, the findings based on the text
contents were analyzed under
main themes "Mathematics relationship with the other
subjects", "Perceptions towards Mathematics", "Mathematical level" and "Creativity".
6.1 General Findings Related to the Data Analyzed from the Story Texts.
First of all, students were asked if they wrote any stories about any subjects.
Distribution of the given answers for these questions is seen in Table 1.
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
Table 1: Distribution of the answers to the question of
Have you ever written a story about any subjects?
Have written a story
Have never written
a story
Have you ever written
a
story
about
(f)
(%)
(f)
(%)
Advantageous
3
10
27
90
Disadvantageous
0
0
29
100
3
5,08
56
94,92
any
subjects?
Total
It has been reached the conclusion that only 3 out of 59 students have written a story
before.
Then, the students were asked the question of Can you create a story including a
math subject? . Distribution of the answers for this question is seen in Table 2.
Table 2: Distribution of the answers to the question of
Can you create a story covering a math subject?
Can write a story
Cannot write a story
(f)
(%)
(f)
(%)
Advantageous
23
76,66
7
23,33
Disadvantageous
29
100
0
0
52
88,13
7
11,86
Can you create a story
covering a math subject?
Total
As seen in Table 2, 7 students have indicated that they could not create a story, while 57
of them have indicated that they could create a story.
The students who said that they could not create a story have been in the
advantageous group. According to the given answers and observations; many of the
students who could not create a story could understand the subject, enjoy listening to
the stories, but did not want to write a story. However, many of them stated that they
wanted to write a story but would not be successful writing it. In the disadvantageous
group, there was no student could not create a story. In the texts which were written by
the students who indicated that they could write a story, the texts had the story
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characteristics and the texts did not have the story characteristics were classified
according to the advantageous and disadvantageous regions and presented in Table 2.
Table 3: Percentage and number of students according
to the written text had a story characteristics or not
Advantageous
Disadvantageous
Total
(f)
(%)
(f)
(%)
(f)
(%)
the
17
73,91
26
89,65
43
82,69
The written text did not have the
6
26,08
3
10,34
9
17,30
The
written
text
had
characteristics of a story
characteristics of a story
As seen in Table 3, written texts were evaluated in terms of whether they are
mathematical story or not. 43 written texts by the students showed mathematical story
characteristics, 9 of them showed no mathematical story characteristics. The texts had
no mathematical story characteristics were found to have different states. Some of the
students included Word problems in their stories (31-9-34). They have described those
word problems as a story. Besides, it has not been seen a specific story line in the texts
including simple operations and arithmetic calculations (24-36). The texts that were
referred to a particular topic, but were not a mathematical story and only had simple
narration and samplings were taken into the mathematical story category (5-42-44-50).
The number of advantageous group students whose stories did not have the
characteristics of a story (%26,08) is seen more than the ones (%10,34) in the
disadvantageous group. In the written story texts, it was observed that there was a
piling up in certain subjects. Data related to the topics covered in the text of the stories
written by the students were classified according to the advantageous and
disadvantageous regions and presented in Table 3.
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
Table 4: Mathematical Subjects that the Students used in their stories
Disadvantageous (f)
Advantageous (f)
f
%
Square root expression
7
3
10
20,83
Square root expression and integer
0
1
1
2,08
Fraction
4
3
6
12,5
Histogram
2
0
2
4,16
Fractal
3
0
3
6,25
Ratio
2
0
1
2,08
Number sets
0
1
1
2,08
Natural Number
3
1
4
8,33
Integer
0
2
2
4,16
Algebraic expression
1
1
2
4,16
Equations
1
1
1
2,08
Exponential expression
3
4
7
14,58
Non-story content texts
3
6
8
16,66
25
23
48
100
TOTAL
As seen in Table 4, mathematical topics that were focused on the written texts have
shown a piling up in certain subjects. 10 students have seen to be interested in the
square root expressions, 7 of them have been interested in exponential expressions and
6 of them have preferred fractions.
6.2 The Findings Related to the Contents of the Stories
The findings analyzed from the story texts of the students were examined under 4 main
themes. A total of 9 texts have not been categorized since they did not have a story
characteristics. The themes have been created from 43 texts.
Students stories have been shown by the initials of their groups and the
numbers,
such as;
D-19
(disadvantageous
group-19
numbered
story),
A-34
(Advantageous group- 34 numbered story). In addition, the first 29 rows represent
disadvantageous group numbers and 30 through 52 represent the advantageous group
numbers. Themes and codes related to mathematics relations with other courses were
revealed on the basis of the analysis of the students stories and presented in Table 5.
Table 5: The Findings related to Mathematics relations with other courses
Theme
Math s relations with other courses
Codes
Story Numbers
f
With Arts course
D-16
1
With Music course
D-21
1
With Science course
A-48
1
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
When Table 5 is analyzed, it is seen that students could establish a relationship between
mathematics and other disciplines, and could express their thoughts along with the
examples from everyday life. Figure 1 and Figure 2 include these story examples.
Figure 1: A story from one of the students
It was Monday, again. Ceren didn t want to go to school that day, but she set out the
school way by puffing. Their first course was Music. They were going to learn about
Notes and Note ‛eats . Their teacher entered the classroom and started directly talking
about the topic. She started with the note of Do and said that it had
beats. She also
said that the other notes also increase and decrease depending on it. She wrote 4x2=8
8x2=16 16x2=32 on the white board. Ceren liked her to tell this subject by making a
story. She understood the subject. She has begun to like Mondays since then.
In figure 1, the student expressed the connection between the note beats and
mathematics in his story.
Figure 2: A story from one of the students.
One day, in the art lesson, we painted some strings with watercolor, then folded
drawing paper in two and put those strings between papers and pulled them out. There
was the same figure on the both sides of the paper. In mathematics, while doing addition
operation, changing the places of numbers does not matter and the result is the same.
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
In figure 2, the student expressed the connection between changing feature of
string print and addition in his story. With reference to the texts written by students,
the revealed themes and codes related to students perceptions towards mathematics
course are presented in Table 6.
Table 6: The Findings related to Perceptions towards Mathematics
Theme
Perceptions
Codes
Mathematics is a difficult subject
towards
Story Numbers
f
43-23-39-35-30-4-20-18-41-17-
Advantageous: 5
22-11-19-
Disadvantageous: 8
Math
Total: l3
It is a difficult, unlikable, but
10-27-29-46-2-25-48
necessary subject
Advantageous: 2
Disadvantageous: 5
Total: 7
It is a lovable subject when
7-12-14-48-51-38-49-1-8-15-
Advantageous: 8
understood and solved
26-40-49-37-13-21-52-32-6
Disadvantageous: 10
Total: 18
It is a subject in which calculation
and arithmetic operations are
Advantageous: 2
3-16-28-45-33
studied
Disadvantageous: 3
Total: 5
In table , when themes and codes that were revealed about students perceptions
towards mathematics course were analyzed, it has been seen that mathematics was
described as a difficult course by them, and the story characters in many of their stories
had difficulties in mathematics course. In general, in the stories, story characters were
seen to have used such sentences as; "he could not calculate it", "he made this operation
wrongly". ‚lso, the ideas of if studied, it is achievable and it is a must course have
been emphasized. In the stories, some characters were seen to have been bored of
studying mathematics, but continued studying it. There were not significant statistical
differences between disadvantageous and advantageous groups.
With reference to the texts written by students, the findings related to the
mathematical level in the stories are presented in Table 7.
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Table 7: The Findings related to Mathematical Level in the Stories
Theme
Classification
Mathematical
Low
level
Story Numbers
f
10-27-35-30-4-20-18-41-17-22-11-29-46-2-25-
Advantageous: 7
19-43-23-39-48
Disadvantageous: 12
Total: 19
Middle
38-49-1-8-15-26-40-49-37-13-21-52-32-6
Advantageous: 6
Disadvantageous: 7
Total: 13
High
3-7-12-14-48-51-16-28-45-33
Advantageous: 4
Disadvantageous: 7
Total: 11
Students stories are assessed by experts, scrutinized in a holistic way, and divided into
three mathematical level categories as low, medium and high. While experts
categorized the stories, it has been considered if mathematical elements used
appropriately, consistent transitions were made between the topics, mathematical
operations were calculated correctly -if applicable, and the preferred concepts were
used properly. As shown in Table 7, 19 students are in the lower level. It can also be
seen that the students abilities to use mathematical elements correctly is quite low,
their learning about concepts is weak and their misconceptions are too high. As well as
misconceptions, there are also students who calculated operations inaccurately. The
texts in which misconceptions have been detected are 10-27-35-30 and the texts in which
incorrect operations have been detected are 4 and 35.
With reference to the texts written by students, the findings related to the
creativity levels in the stories are presented in Table 8.
Table 8: The Findings related to Creativity Levels in the Stories
Theme
Creativity
Classification
Low
Story Numbers
11-29-46-2-25-10-27-35-30-4
f
Advantageous: 3
Disadvantageous: 7
Total:10
Middle
1-8-15-26-13-3-7-21-14-6-28-20-43-23-18-41
Advantageous: 2
Disadvantageous: 12
Total: 15
High
38-49-12-48-51-16-45-40-37-49-33-17-22-19-39-48-52-
Advantageous: 13
32
Disadvantageous: 5
Total: 18
In the stories, originality of the topics and the narrations is taken into account as a
creativity. What mathematically notable in the student stories is students relating
mathematics with the examples of daily life nicely in their stories.
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
In addition to the findings that are identified mathematically, there are some
findings available common to both groups in terms of language and expression. In
general, a very plain language is seen in terms of language expression, but transitions
between topics seem weak. Students often used their own names in their stories. The
stories are original and the plots are independent of each other. Story characters have
always succeeded. The given problem has been solved despite the obstacles. 13 out of
the 18 stories classified as having high creativity were written by students in the
advantageous region. The findings showing differences between the two groups are as
follows.
In the advantageous region, students wrote stories in fantastic topics. In the
stories they wrote, there were events only might have happened in fairy tales and
stories often begin with the phrase of 'once upon a time ...'. Numbers were personified
and spoken. Some events could be portrayed and a concrete example showing the
student s imagination was created by drawings. In the advantageous region, stories had
titles and were used to arouse curiosity about the content of the story. (For example;
The Secret of Equivalence Mirror, The Numbers that cannot be Grouped, The Wizard of
x and y, etc.) In addition, numerical and algebraic expressions were used for entitling
the stories in the advantageous regions. The studies using real people were less. There
are differences between time and events in the stories written in advantageous and
disadvantageous regions. While real events that may relate to real life were chosen in
disadvantageous regions, advantageous region students focused on solving problems in
imaginary places (War, Interplanetary Journey, etc.). Figure 3 and Figure 4 show the
examples of these stories.
Figure 3: A story from one of the students
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
Atmosphere in the Planet - One day,
/
got on the spaceship and went to Mars. He
understood that there was no atmosphere and suddenly simplified and became a
/
and
converted into an alien. Then he sees his own copy in front of him. He was exactly the
same as him. He was an alien, too. When he understood it was from the simplification of
the atmosphere in Mars, he was not afraid. They started to talk and wondered about what
would happen if they went to another planet. Then they went to Jupiter. However,
because there was an enlargement in its atmosphere, they suddenly became bigger and
became a
/
. They liked this situation and tried it in other planets. Even though
they changed in each planet, they realized that they were the same when they went to the
Earth.
In figure , the student who wrote this story mentioned about the numbers
traveling to space, and the fractions simplifying and enlarging in his story.
Figure 4: A story from one of the students
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The Secret of Equivalence Mirror - Once upon a time, there was a wizard living on an
island called Obscurities . This wizard was responsible for keeping all communities
away from obscurities. He was doing this with an equivalence mirror. For instance; X
person who forgot about himself and his equation (his family) to the wizard, and he takes
him to the equivalence mirror and X person sees his equinumerous equation in the
mirror, then finds himself. One day, forgetting about themselves, X and Y found
themselves in front of the wizard s door. ‚s always, Wizard took them to the mirror, but
it was not there. There was a small note instead of it. That note was a map, indeed.
Wizard, X and Y hit the road. When they arrived at the point which was shown on the
map, they found another note saying: Mirror is inside of the cave. If you want to find it,
you should take a x²+y number of steps.3x+y=21 and 2x+3y=42 They started to solve the
equation together and finally the wizard found the answer.
They found it by calculating the number of steps. They took the mirror and went
back to their island. But, Wizard was wondering about who stole the mirror. This is
another obscurity to solve
In figure 4, the student who wrote the story mentioned about some fantastical
elements such as equivalence mirror and wizard in his story he designed about
equations.
7. Discussion
The research has been conducted with the students from two different schools where
were located in both advantageous and disadvantageous regions in three stages. At the
first stage of the activity, students were handed out various fractal examples and asked
to examine them. Students were told about the fractal concept. At the second stage of
the activity, the story my fractal tree was narrated with the aid of a presentation
supported with the visual materials prepared. At the end of the story, students were
expected to arrive at a conclusion about exponential numbers on their own. At the last
stage of the activity, the students were asked to write a mathematical story. The stories
of the volunteering students were shared in the class. The students were asked whether
they understood the concept of fractals at the first stage. There were students who
defined all ornaments given in textbooks as ornament examples since the answers given
by students mentioned fractals on parallel displacement and ornamentation topic (such
as lacework, mosque ornaments, etc.).
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
These findings have shown that the fractals were mixed with ornamentations.
The reason of this finding obtained from the study is that the self-similarity
characteristics of fractals were implicitly emphasized, and the concept of repeating
which is an important factor of fractal was not addressed, and the eternal patterns of
fractals were not given in the fractal definition given in the 8.grade mathematics
textbooks according to Karakus & ‛aki,
s research. For these reasons, the
students had difficulties to make decisions whether an object was fractal or not.
In the second stage, the story of My fractal tree was presented to the students.
The question for the exponential expression in the story was the final stage of the story
and was easily solved by the students. Students making an operation that is considered
a simple operation for an 8th grader eagerly and curiously draw attention. There are
studies in which students showed high achievements and attitudes in the situations
when mathematical problems were introduced with everyday life. The subjects
analyzed in the field studies about this topic are followings mathematics teaching s
being incompatible in schools with the real life, students inabilities of using knowledge
and skills they acquired in the school in real life and solving problems, their acting to go
quickly to the results instead of thinking about the problems and creating solution
strategies (De Hoyos al., 2002; Fitzpatrick, 1994; Schonfeld, 1985; Selden al, 2000; cited:
Nancarrow, 2004; Verschaffel al, 1999;). The section went through the final stage of the
study was conducted with the total of 59 students who were in the groups identified as
advantageous and disadvantageous groups. However, at the last stage which is also the
stage of creating a story, 52 of them were able to create a text. Then, these 52 texts were
analyzed by the researchers and only 43 of them were determined to have had a story
characteristic. All of 7 students who could not create a story were in the advantageous
group.
The reason why those students did not want to participate in the given
assignment is that they were set free in non-obligatory tasks throughout the year. It is
also noteworthy that 6 out of 9 written texts that did not have a story characteristics
were in the advantageous group. The majority of students were able to write at least
one story integrated with a mathematical subject, which also included event-situation,
place, people and time elements. Thus, the students can be said that they understood
the description of a mathematical story. In the texts excluded a mathematical story
characteristics, students were seen to have written regular stories that did not contain
any mathematical subjects, or to have written the texts describing their attitudes
towards mathematics, or to have expressed a word problem as a mathematical story. In
the created stories, a plain language can be seen in terms of language and expression in
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general, but transitions between topics were weak. The reason of these findings has
been thought the fact that the majority of students had never written any stories in
mathematics and other subjects. Since this study was conducted with 8th graders, the
students knew the most subjects of mathematics. It was observed that some specific
topics were mostly chosen in the written story texts. Topic distributions have serious
differences in terms of advantageous and disadvantageous groups. In particular, square
root numbers and exponential numbers were new subjects for students and introducing
them to the students enabled students to create various simulations (For example; a
square root expression was thought as a prison and a square number was escaped from
there). One of the other mostly chosen issues was the fractions. Fractions are one of the
topics that are widely covered in the mathematics curriculum. Because students can
often encounter the fractions topic in daily life, its use in stories might have been more
than other topics. Although diversity is seen in the topics, geometry subjects seem not
to be preferred by the students to write a story. Since geometrical subjects are related to
shapes, the students may have had difficulties to make these subjects stories.
Many components could be studied in the student stories. It is seen that students
have referred to mathematics relations with other subjects. Mathematics can be said to
have an effective role in the establishment of relations with other disciplines and daily
life. There are many studies that mention the importance of this case in a realistic
teaching of mathematics and constructive approach (Hein, 2002; Lewis al., 2002; Smith,
Winn,
. The students perceptions of mathematics course were also reached
in their writings. The ideas of "Mathematics is often difficult and unlovable, but if
studied, it is achievable" and "it is a must course" have seen to have emphasized in the
texts. In the stories, some characters were seen to have been bored of studying math but
still continued studying it. The perception that mathematics is usually a difficult course
is a common perception reached between disadvantageous and advantageous groups.
Students perceptions of mathematics being a difficult course can also be found in some
other sources of research. They suggest that mathematics being taught in different
game activities instead of boring classroom environments can resolve this problem
(Kubinova, Novotna & Littler, 1998; Ticha & Kubinov, 1998).
The results obtained on mathematical levels of the students show that the overall
level is low. Although the numerical data findings that showed the disadvantageous
group was relatively lower than the advantageous group was reached, it has been
thought that the data was similar to each other in two groups when the stories were
analyzed. This situation suggests that since the number of advantageous group
students who did not participate in the activity of writing stories and whose stories
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
were not accepted as a story was high, the students whose math level was low were
thought to have been eliminated at first.
That their levels of being able to use mathematical elements correctly are quite
low, their learning of concepts is weak, and their misconceptions are excessive can be
also seen. Besides these misconceptions, there are also some students who made
operation mistakes. This case shows that students are weak in learning the concepts.
This kind of incomplete and inaccurate learning of concepts enables students to
misunderstand the topic and to establish a link between other topics. There are also
many studies available which emphasized that incomplete learning of mathematical
concepts brought along the failure (Baker, 1996; Tall, 1993; Yusof & Rahman, 2001;
Zachariades al., 2002; Zaslavsky & Peled, 1996).
In general, students stories were seen to have a very plain language in terms of
language expression, but they were seen to be weak in transitions between the topics.
Based on these findings, the importance of mathematical literacy has emerged. Akyuz &
Pala (2010), in their study they conducted by using PISA 2003 data belonged to Turkey,
Finland, and Greece, found a significant relationship between mathematical literacy and
problem-solving, which shows a positive relationship between mathematical literacy
and learning mathematics. Furthermore, in order to determine the level of mathematical
literacy, Uysal & Yenilmez (2011) conducted a study based on PISA 2003 mathematics
questions and evaluations of 8th-grade students and suggested that the majority of
students participating in the test were found to have been below the third level. This
case reveals that obtaining a sufficient level of information and skills, each individual
needs to be literate and strong in mathematics. With the new regulations in 2004,
raising students as literates in mathematics in the elementary mathematics curriculum
was taken into consideration and necessary arrangements were made in the program
(MEB, 2005). However, according to the PISA assessment of EARGED (2007); 76,4% of
our students is on the second level or lower in the PISA 2006 mathematical literacy
scale. With regard to the results of PISA 2006 report, the average performance of our
students was on the second qualification level, while the average performance of OECD
countries was on the third level. However, according to the results of PISA 2009 report,
in mathematical literacy levels in our country, the percentage of students who are
below 2. level, which is considered basic proficiency level by experts, is 42,2%.
Moreover, Turkey's average mathematical literacy score is (445) and it is below the
OECD average (EARGED, 2010).
Considering the creativity element in the students writings, the advantageous
group was seen to have created highly creative stories. This case can be considered that
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
those students were more socioeconomically advantageous, because, their creativities
were supported and they were offered these kinds of activities from the pre-school
period. They were also in a supportive social environment. It has also been thought that
students choosing fantastical topics in their stories came from the effects of computer
games they played and movies they watched. In fact, some studies suggest that some of
computer and video games affect students in a positive way (Cordova & Lepper, 1996;
Ocel, 2002; Schie & Wiegman, 1997).
Such findings of the research are in agreement with the findings of Gaber-Katz in
literature that the fact that students read, tell and write stories offer an alternative way
to improve the critical and creative reading skill (1991).
It was found out in the research that, storytelling method could be functionally
used at some stages of Math teaching process. Also, the prepared activities increased
the active participation of students in the course, and students could cooperate as a
result of activities which helped them to learn and actively used their prior information.
In addition, it also supported students speaking and writing among language skills.
Another conclusion related to the use of storytelling method in Math courses was that
students had fun when learning, included many descriptions in their stories and
actively used mathematical subjects. The findings of the research supported the views
and findings given in literature (Kabadayi, 2005). Storytelling and story writing
methods improve verbal language use of children provide entertaining learning
experiences for students, increase their use of words and enhance their social and
emotional developments through social experiences.
In addition, it could be said that students gained the courage to express
themselves, easily adapted to activities and exhibited great willingness to participate in
them. Such findings of the research also support the findings that suggest that
storytelling method intended for improving reading-writing skills in the early years of
primary education ensures willingness of students when participating in such activities
(Palmer al., 2001).
8. Recommendations
In order to develop mathematical literacy and to determine students mathematical
perceptions and their preliminary information, mathematical stories can be put in the
programs further.
Since fractal geometry is in close correlation with so many traditional
mathematical subjects such as; sequence of numbers, symmetry, proportion,
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measurement and fractions in primary level; logarithm, composite functions, Pascal's
triangle, geometric series and complex numbers in secondary level. It has been known
that fractal geometry is in 33 close correlations. Fractal is an important issue in terms of
students being able to see mathematics whose subject exists in nature. There are many
studies that were conducted for fractals to have been integrated into existing
mathematics curriculum, by the reasons of its helping students to establish a correlation
between mathematics and nature, to establish relations between other disciplines, to
explore school mathematics with non-analytical ways and to see the latest
developments in mathematics (Fraboni & Moller, 2008; Goldenberg, 1991; Lornell &
Westerberg, 1999; Vacc, 1999;). A proposal of addressing more to the teaching of fractals
can be suggested. Mathematical stories can be involved in the programs for the
development of mathematical literacy and the findings of students' mathematical
perceptions and preliminary information.
Many studies suggest that fractal geometry is in close relationship with the
traditional mathematical subject such as; sequence of numbers, symmetry, proportion,
measurement and fractions at primary level, and logarithm, composite functions,
Pascal's triangle, arithmetic sequence, geometric sequence and complex numbers in
secondary level. This fact helps students to establish relations with math and nature and
between mathematics and other disciplines, as well as them to see the latest
developments in the field of mathematics and to discover the school mathematics with
non-analytical way. Because of these reasons, fractal geometry should be integrated into
existing mathematics curriculum (Fraboni and Moller, 2008; Goldenberg, 1991; Lornell
& Westerberg, 1999; Vacc, 1999). The proposal of mentioning about teaching of fractals
more can be offered.
In the direction of the research results, following suggestions can be made
towards the development of mathematical literacy skills of students in mathematics
course: some activities and applications can be arranged for students to engage in social
activities which are at their final stage of mathematical literacy process.
The researches in which storytelling method is used for students turning critical
reading into a habit and observing their usage of strategy in a more detailed way can be
conducted in longer terms in order to reveal the effect of teaching experiment
application. A study can be fulfilled by using qualitative and quantitative research
approaches, as well as using newspaper articles, and television programs along with
qualified texts. Integrating mathematics curriculum with mathematical reading and
writing skills, some applications can be made towards students developing these skills.
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MATHEMATICAL PERCEPTIONS: ISTANBUL CASE
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Appendix 1
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