ABSTRACT

The purpose of the study is to investigate Turkish elementary school students’ misconceptions and errors in circle and circle theorems . It is necessary to design specific study to determine and examine Turkish elementary school students’ misconceptions and errors in circle and circle theorems. In this research, elementary students’ misconceptions and errors in circle and circle theorems while teaching geometry will be examined and some suggestions may offer to the teachers according to finding results. Thus, qualitative descriptive research will be designed for this study. The target population of this study will be all seventh grade students in Sultanbeyli which is town of ”stanbul. The sample of the study will consist of randomly selected aprroximately 50 seventh grade students in Mevlana Ortaokulu located in Sultanbeyli which is town of ”stanbul. The research application will be carried out in 2016-2017 academic year second semester in this middle school. A descriptive methodology and student interview will be used in the study to analyze and interpret the results. The results from this study may show that seventh grade elementary school students have some misconceptions arising lack of background knowledge or mistakes resulting from reasoning and basic operation at circle and circle theorems. Data will be collected through an instrument that will developed by the researcher. The instrument will be a test consisting open-ended questions related to circle and circle theorems.The test shows that what is elementary students’ misconceptions and errors of circle and circle theorems in geometry. The test will be prepared in line with the objectives of seventh grade Turkish National Middle School Mathematics Education Curriculum related to circle and circle theorems.And content analysis will be employed in data analysis to determine misconceptions and errors in circle and circle theorems.

Keywords: elementary school students, misconceptions, errors, circle and circle theorems

1. INTRODUCTION

This study is a research assignment about seventh grade students’ misconceptions and errors in circle and circle theorems while teaching geometry. The reason of doing this study is that although researchers conducted the study of seventh grade students’ misconceptions in geometry, they didn’t detailed address seventh grade students’ misconceptions and errors in circle and circle theorems. So, it was determined to conduct this study which focuses on the circle and circle theorem in seventh grade students and analyze the misconceptions and errors related to this topics in seventh grade students.

Mathematics is one of important fields in the human life and also it has an important place in education of people. According to NCTM, it can be seen as a tool which provides opportunities and options that affected to people’s futures (NCTM,2000).Because of this importance of mathematics in human life, teaching mathematics has been gained an importance from pre-school to high school education in all levels of education. And also it can say that mathematics is divided into different areas such as geometry. (Y”lmaz, 2015).

Geometry is a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids and so it is one of the important fields of mathematics. When the history of geometry examined, geometry has accepted a important discipline since 2000 BC. Throughout the history, it has great importance in people’s lives with its origin in the need for human beings to specify quantities and to measure figures and lands. Nowadays, geometry still maintains its importance in mathematics curriculum. In order to make sense and solve problems in other topics of mathematics and in daily life situations, geometry knowledge is necessary. In addition to this, students begins to see, know and understand the physical world around them with learning geometry from early age and they continue their education with high level geometric thinking developing in deductive and inductive system towards older ages. Also, most of the goods and structures in our physical environment are geometric shapes and objects. According to National Council of Teachers of Mathematics (NCTM, 2000), geometry helps to describe, analyze and understand the world around us. In addition to this, Van de Walle (2001) stated that geometry helps us understand the world since it can be placed everywhere. Moreover, he stated that it is inevitable parts of world where humans construct have elements of geometric forms such as architecture, machines, cars, art and engineering (p.308). Similarly, geometry provides opportunities not only solving mathematical problems but also understanding important issues in different areas such as science, art and daily life (Akta” & Cans”z-Akta”, 2012). For these reasons, geometry occupy an important place in school education as a part of mathematics education from preschool education to higher school education (Ubuz, ”st”n, & Erba”, 2009). The aim of geomery education is not just learning the definitions or the properties of geometrical shapes but also to have the ability of applying these properties in real-world problems. And also this education should provide that students develop spatial reasoning and geometrical thinking by participating discussions related to geometry.

Although geometry has an important place in human life; Ubuz, ”st”n and Erba” (2009) stated that Turkish students’ geometry achievement level is lower than students’ geometry achievement level in other countries when examined the repeat of Trends in International Mathematics and Science Study (TIMSS-R) and Programme for International Student Assessment (PISA). The reason of this stuation is that students have some problems in geometry learning.These problems result from that students have some problems in perceiving geometric concepts.

In order to think more detailed and correctly, students have to apperceive what they have learned. Concepts are the abstract ideas representing the fundamental characteristics of objects, events, ideas and behaviors which have common specifications. Concepts reduce the complexity by simplifying the environment that students live. They help them in describing and expressing the situations and objects in their environment. Learning concepts cannot be determined only by defining objects or by classifying objects but also it provides interpreting, translating and transition of objects and situations . So, while teaching new knowledge to students , it should be made sure that the concepts are formed correctly and appropriately. In recent years, students’ understanding of the mathematical concepts is one of the subjects that researchers and teachers give importance to. The reason of this situation is that students have some problems in understanding mathematical concepts. In mathematics education, some terms use to explaning students’ these problems in mathematics learning. These problems are called as difficulty which is a outcome which results from lack of attention or knowledge of students’ in any mathematical topic. However, this term reflects to be general expression to understand students’ learning problems. Therefore, the terms ‘error’ and ‘misconception’ uses to express students’ problems in more detailed.

According to Luneta and Makonye (2010), although errors are closely associated with misconceptions, in fact they are completely different. For example, an error is a mistake or inaccuracy and a deviation from accuracy. Riccomini ( 2005) stated that errors are divided into two groups such as unsystematic and systematics errors. Unsystematic errors occur just once and unconscious such as calculation errors, whereas systematic errors are incorrect answers occurring repetitiously. Smith, diSessa & Roschelle (1993) pointed out that misconceptions are systematic errors in students’ learning. Misconception means ‘systematic error in students’ conception’ (Ke”an & Kaya, 2007, p. 27). Students’ conceptions consist of their beliefs , theories, explanations and meanings related to any topic.When in students’ mind there is a complication related to these conceptions, then a misconception has occurred (Osborne & Wittrock, 1983).According to explanations above , error and misconception are related, but they are different. Misconceptions are systematic errors occuring regulary incorrect answers. Therefore error could be accepted as the result of the misconception (Ke”an & Kaya, 2007). According to Ery”lmaz and S”rmeli (2002) all misconceptions could be accepted as errors; however not all errors could be accepted as misconceptions.Above mentioned, there are many factors that affect the learning process of students and their performance in educational settings. One of the major factors is that are the misconceptions of students. For effective and meaningful learning, describing the origins of misconceptions and finding ways to prevent them are important.Also, although geometry a important subject in human life, students have some misconceptions in learning geometry.So, conducting a study related to misconception in geometry can shape future of education world.Although there are some studies related to misconceptions in geometry, there are not enough studies related to misconceptions of circle and circle theorems in geometry.Whereas, circle is a difficult concept for both students and teachers.Because of these reasons, it is necessary to focus on misconceptions of circle and circle theorems in geometry.

”zerem ( 2012 ) conducted a study called as ‘Misconceptions in Geometry and Suggested Solutions for Seventh Grade Students’ which aims to determine the performances of 7th grade college students at geometry and to show the conceptual difficulties they face while learning. By doing this , the study tried to identify the misconceptions which arouse during the learning process of geometry. She studied 28 seventh grade students consisted of 12 males and 16 females at Turk Maarif Koleji in Cyprus. In this study, she used the descriptive methodology and student inteview to analyze and interpret the results.According to this study, seventh grade students have some misconceptions due to lack of knowledge in geometry subject. Although students don’t have any difficulties in examining, comparing, and analyzing properties of geometric figures, they have some difficulties in describe some concepts and remember formula related to geometry.In conclusion, due to lack in knowledge related to geometry subject, there are many misconceptions in seventh grade students.

Kaygusuz and ”zerba” (2012) conducted a study called as ‘Determining Misconceptions on ‘Circle Sub-learning Area ‘ In Primary School Mathematics Course Syllabus’ which aims to determine the misconceptions on ‘circle sub-learning area’ in primary school mathematics course syllabus. They studied 582 fifth grade students consisted of 298 female and 284male students studying primary schools located in Ankara. In this study, achievement test developed by reseracher was applied to 582 fifth grade students and it was analyzed the misconceptions in circle sub-learning area with definitional research. According to findings in this study, it can say that students have most misconceptions on radius and least misconceptions on center concepts on circle sub-learning area. Also, it stated that female students have fallen into error less than male students in circle concept. In addition to this, number of read books have meaningful effects on concept signification and increasing success on mathematics increases the significations of concepts.

Kemanka”l” and ”zsoy ( 2004 ) conducted a study as ‘Misconceptions and Main Errors of Secondary Education in Circle Subject’ which aims to examine and analyze secondary school students’ misconceptions, their main errors and their learning levels. Kemanka”l” and ”zsoy studied 70 eleventh grade students selected from high school. According to this study, it stated that students are not able to make contact with the concepts of interior, exterior, center and inscribed angles in a circle because they have confused definitons and properties of center and inscribed angles in circle. Furthermore, students have difficulty in applying properties of angle concepts on different regions such as trangular region into circle.

Purpose of the Study

The main purpose of this study is to investigate seventh grade students’ errors and misconceptions in circle and circle theorems while learning geometry. In other words, interest of this study is to examine how students understand the circle and circle theorems in geometry and also what misconceptions students have understanding of circle and circle theorems.

1.1. Research Questions

The following research question is in this study :

1. What are Turkish seventh grade students’ misconceptions and errors in circle and circle theorems ?

1.2. Significance of the Study

The significance of this study is that it provides to determine the errors and the misconceptions experienced by students during the learning of the subject of circle and circle theorems as well as the reasons for such mistakes and misconceptions, and to reveal the importance of taking into consideration these mistakes and misconceptions while giving a mathematics lesson.

Also, although there are the studies focusing the misconceptions and errors about geometry in Turkey, there are not enough studies focusing the understandings of circle and circle theorems and so nobody have enough knowledge related to misconceptions and errors of circle and circle theorems. Although there are some interpretations related to misconceptions and errors of understanding circle and circle theorems, there are not adequate knowledge related to misconceptions and errors of circle in Turkey. Also, when examined the literature in world, there are not enough studies related to misconceptions of understanding circle and circle theorems.So, this study aims to explore the patterns of the research articles about circle and circle misconceptions in Turkey and to reveal general tendencies in this area.It is hoped that this review will inform educators, teachers, pre-service teachers and curriculum developers about themes and patterns in misconception research. This collection of articles and reviewed literature provides information about the following:

‘ What kind of misconceptions students have

‘ How teachers can handle the misconceptions of students

‘ Which methods are preferred for correction of misconceptions

Definitions of the Important Terms

Turkish Elementary School Students; are the students who were in 5th, 6th, 7th and 8th grades in a public school.

Misconceptions; is defined ‘the perception of concepts by students in a different way than their scientifically accepted definitions’ (Ke”an & Kaya, 2007, p. 27).

Errors; is defined as ‘a mistake or inaccuracy and a deviation from accuracy’ (Luneta & Makonye, 2010).

Circle;is defined as a round shape consisting of a curved line that completely encloses a space and is the same distance from the centre at every point.

2. LITERATURE REVIEW

The Importance of Geometry

Geometry have a important role in mathematics curriculum and it has been common heritage for all countries from past to present. Geometry teaching in schools allows students to give insight to understand other significant concepts and ideas in different disciplines of mathematics (Mammana & Villiani, 1998; Muschla & Muschla; 2000; NCTM, 2000). In addition to this, it is helpful to the students realize unforseen properties of mathematics (Serra, 1993). Furthermore, geometry knowledge is very useful to solve everyday life problems like measurement of lengths, drawing, reading maps, etc. (Bussi & Boero, 1998; Kenney, Bezuszka, & Martin, 1992). For example, when studying how strong a building’s support will be, architects must use geometric reasoning. Similarly, geometry allows airplane designers to measure how aerodynamic a particular design is. Therefore geometry knowledge is very useful not only inside the school but also outside the school. As NCTM (2000) stated that while students engage with the topics of geometry, they have an understanding about other topics in mathematics and in art, science, and social studies.

In mathematics education, concept is important issue because mathematical concept underlies students’ mathematical understanding. Kaptan (1999) defined concept as common name that was given to ideas, events grouped according to some properties.When examined the meaning of concept, it is said that is an abstract idea indicating the fundamental properties of what it typifies. Concepts arise as abstractions or generalisations from experience or the result of a transformation of existing ideas. If concepts are different from scientific thoughts they are referred to as misconception. In other words, while students have some difficulties in understanding the mathematical concepts, misconception as barries to understand mathematics may occur. Therefore, in this paper, it is necessary to define the meaning of misconception.One of the definitions of misconception is ‘a student conception that produces a systematic pattern of error’ (Smith, diSessa & Roschelle, 1993, p.118) .According to Swan (2001), a ‘misconception’ is not a wrong thinking but is a concept in embryo or a local generalization that the pupil has made.In fact, it may be a natural stage of development.Also, the reasons of misconceptions are that students’ wrong and deficient perception of the background knowledge, language problems, no suitable teaching environment, no relationship between concepts and real life, and no determination of students’ misunderstandings.

Studies on Misconceptions of Geometry

There are many studies on misconceptions in geometry.These studies are examined in this part of the paper.

Ubuz (1999) conducted a study called as ’10th and 11th Grade Students’ Basic Mistakes and Misconceptions in Geometry’ which aims to investigate students’ understanding of angle concept in geometry according to their errors, misconceptions and gender. She studied 67 high school students consists of 10th and 11th grade students in a public school located in Ankara. She collected data through a test consisting 11 open-ended questions. The findings showed that male students were more extreme than female students in terms of approaches in questions. Also, although female students more succesful than male students, male students gave less incorrect answer than female students. Since instead of giving incorrect answers, male students didn’t answer the questions in this test.Also,female students gave more incorrect answers than male students since they are enterprising persons. In addition to this, there was an increase in achievement level of the students due to educational level. Also, students do not know the meaning of a triangle and the properties of its exterior and interior angles. In addition to this, students focus on geometric figures rather than their properties.

Da”l” (2010) conducted a study called as ‘Misconception of Elementary School Students in 5th Grades About The Perimeter, Area, and Volume Concepts’ which aims to investigate the 5th grades’ content knowledge about the perimeter, area and volume of the geometric shapes and objects, and also was to determine their misconceptions about these concepts. She studied 262 students included 5th grades at the seven elementary schools in U”ak city center. In this study, it used survey method which have one instrument with 40 items, the perimeter-area-volume test collected data. The findings showed that students can easily solve the problems asked the perimeter, however they can not do problems which require more geometric thinking. Also, although students do not have difficulty in problems included applying properties of the geometric figures, they do not exactly know the properties of some geometric figures. Some students are also confused by perimeter and area calculations. In addition to this, students have difficulty with questions about standard and non-standard measuring units, and have difficulty converting the measuring units to each other.

Ba”kurt (2011) conducted a study called as ‘The 6 th, 7th and 8th Grade Students’ Perceptions and Misconceptions on Point,Line and Plane Concepts ‘ which aims to determine the perceptions of 6th, 7th and 8th grade students on point, line, and plane concepts.He studied with 461 students included 6th, 7th, and 8th grade students in primary schools located in the Eastern Anatolia Region.Data were collected with a measurement tool of 12 open-ended questions. According to results of this study, it was discovered that some of the students cannot perceive point as a region on a plane, that they confuse line with line segment, ray with line and draw finite geometric shapes such as a square or rectangle to define a plane.

Ayy”ld”z (2010) conducted a study called as ‘ An Investigation of the Effect of Learning Logs on Remedying Students’ Misconceptions Concerning Mathematics Lesson’ which aims to investigate the effect of learning with diaries on overcoming students’ misconceptions in mathematics classes. For this study, it studied with 78 sixth grade students in 2009-2010 academic year. Ayy”ld”z compared misconceptions of students who keep a diary of their learning and those who don’t in this study. Data were collected through a Two-phased Open Ended Scale developed by the researcher for identifying misconceptions. The findings showed that the application of learning with diary have positive effect on overcoming students’ misconceptions In other words,students who keep a diary of their learning had less misconception than other students. It has also been found out that learning with diaries have been more effective to reduce female students’ misconceptions when compared to those of male students’.

”zerem ( 2012 ) conducted a study called as ‘Misconceptions in Geometry and Suggested Solutions for Seventh Grade Students’ which aims to determine the performances of 7th grade college students at geometry and to show the conceptual difficulties they face while learning. By doing this , the study tried to identify the misconceptions which arouse during the learning process of geometry. She studied 28 seventh grade students consisted of 12 males and 16 females at Turk Maarif Koleji in Cyprus. In this study, she used the descriptive methodology and student inteview to analyze and interpret the results.According to this study, seventh grade students have some misconceptions due to lack of knowledge in geometry subject. Although students don’t have any difficulties in examining, comparing, and analyzing properties of geometric figures, they have some difficulties in describe some concepts and remember formula related to geometry. Inother words,she stated that students get confused at recognizing the shapes and the reason for this is human perception. In conclusion, due to lack in knowledge related to geometry subject, there are many misconceptions in seventh grade students.

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