European Journal of Education Studies ISSN: 2501 - 1111 ISSN-L: 2501 - 1111 Available on-line at: www.oapub.org/edu 10.5281/zenodo.225656 Volume 2│Issue 12│2016 PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS Cigdem Arslani, Hatice Nur Erbay, Pinar Guner Department of Mathematics, Teacher Education, Hasan Ali Yucel Faculty of Education, Istanbul University, Turkey Abstract: In the present study we try to highlight prospective mathematics teachers ability to identify mistakes of sixth grade students related to angle concept. And also we examined prospective mathematics teachers knowledge of angle concept. Study was carried out with 30 sixth-grade students and 38 prospective mathematics teachers. Sixth grade students required to define the concept of angle with their own statements and describe what it brought to their minds by writing their responses on a paper. Students written responses were examined by the researchers and students mistakes in their definitions were determined. A data collection form that included students definitions of angle (correct, partly correct and incorrect) was obtained. Prospective teachers were required to define the concept of angle with their own statements and evaluate whether sixth grade students responses were correct by explaining the reasons in written form. The data obtained from written forms of prospective teachers were analysed through the descriptive analysis technique. Prospective mathematics teachers comments they had made in response to each student are coded. The results of the research show that prospective mathematics teachers didn t have problems in determining students' definitions of angle as correct, incorrect or partially correct. But they have problems about expressing the students failures and to clearly identifying the mistakes in the incorrect definitions of students. Additionally, almost all of the candidates have made only the static definition of the angle concept. Key words: prospective mathematics teachers, angle concept, students mistakes, teacher knowledge about students mistakes i Correspondence: email arslanc@istanbul.edu.tr Copyright © The Author(s). All Rights Reserved Published by Open Access Publishing Group ©2015. 190 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS 1. Introduction Geometry is one of the most important branches of mathematics education and its place in education cannot be discussed. It plays a bridge role in establishing connections between daily life and mathematical concepts. It is thought that the mathematics is cumulative and previous knowledge and concepts constituted a step for the latter in mathematics, so to determine the misconceptions of concepts and lack of knowledge, and to find solutions to eliminate these mistakes and deficiencies is crucial. Angle concept is one of the basic concepts in geometry. But because of manysided nature, it is clear from the research literature that school students have great difficulty learning the angle concept and grasping the multifaceted nature of the concept of angle (Biber, Tuna & Korkmaz, 2013; Butuner & Filiz, 2016; Clements & Battista, 1992; Clements & Burns, 2000; Dane & Baskurt, 2012; Devichi & Munier, 2013; Erbay, 2016; Mitchelmore, 1998; Mitchelmore & White, 2000; White & Mitchelmore, 2003, Keiser, 2004). Mayberry (1983) indicated that students mostly learn geometric concepts based on a rote learning approach. The properties, scopes, associations, and meanings contained in geometric expressions cannot be taught satisfactorily. One of the important reasons for this is the experience of teachers in this issue who are going to teach the concept of angle. There are also several studies indicating the difficulties prospective mathematics teachers (PMTs) had experienced with angle concept (Ipek, Atasoy & Okumus 2010; Silfverberg & Joutsenlahti, 2014; Tuluk 2015; Yazgan, Argun & Emre, 2009; Yigit 2014). Gokkurt, Sahin, Soylu and Dogan (2015) stated that in order to perform meaningful learning in mathematics teaching, teachers should be aware of learning difficulties and mistakes of their students related to geometric concepts. When we look at the studies about angle concept with student teachers; Silfverberg and Joutsenlahti (2014) investigated the comprehension of plane angle concept of 191 Finish prospective teachers. The results of their examination showed that even the adults who have completed their years with mathematics studies still cherish various notions (beliefs) on such basic concepts of elementary mathematics as an angle, and these different notions and beliefs can remain very much alive. Yet, these concepts were used in mutual discussions regularly. Ipek, Atasoy and Okumus (2010) described the perceptions of the pre-service mathematics teachers on the angle concept with a case study. And according to their findings pre-service teachers generally emphasized the static dimension of the angle concept rather than its dynamic dimension. Additionally, their angle perception has also negative effects on their angle types and angle measurement perceptions. European Journal of Education Studies - Volume 2│Issue 12│2016 191 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS The reason of angles being difficult to understand may stem from the fact of multiple definitions used for the concept. An angle can be defined using a static angle representation; as a part of the plane included between two rays meeting at their endpoints or a dynamic representation; as the amount of rotation necessary to bring one of its rays to the other ray without moving out of the plane (Kieran, 1986). The definitions of angle concept in the modern sense falls into one of three categories according to Keiser s interpretation are given below. 1. The angle is the difference of direction between two straight lines. 2. The angle is the quantity or amount (or the measure) of the rotation necessary to bring one of its sides from its own position to that of the other side without its moving out of the plane containing both. 3. The angle is the portion of a plane included between two straight lines in the plane that meet in a point. The grade 6 mathematics textbooks in Turkey provide only one definition of the angle concept and it is a static definition (Aydin & Gundogdu, 2016). The definition of the angle concept given as textbook format can be seen in Figure 1. *Angle is an intersection of two rays at the same end point. Figure 1: Usual definition of angle concept in 6th grade textbook of Turkey Because of many different definitions in the field related to the concept of angle, the boundaries of the concept cannot be determined precisely. Developing understanding in mathematics is an important but difficult goal and in order to achieve this goal, teachers must be aware of student difficulties, the sources of the difficulties and design instruction to diminish them (Yetkin, 2003). Therefore, there is a need for new research to be done in this regard about angle concept. In this context the purpose of this study was to investigate the PMTs ability to identify mistakes held by elementary students in angle concept definitions and their proposed corrections to overcome those mistakes. Another purpose was to determine PMTs knowledge about angle concept. European Journal of Education Studies - Volume 2│Issue 12│2016 192 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS 2. Methodology 2.1 Research Design In current research, case study which is one of the qualitative research designs was used because it answered how and why questions. Case study is a detailed description and analysis of case or cases (Merriam, 2009). It provides to examine a case, a relation or a process in all its aspects (Cepni, 2012). Because of that the aim of this study was to investigate conceptual knowledge of PMTs regarding the concept of angle and their abilities to determine students mistakes in their definitions of angle, this approach was preferred for detailed examination. 2.2 Participants The participants in this study were 38 prospective middle school mathematics teachers who were enrolled in 4th grade of elementary mathematics education program at a public university in Turkey. Senior PMTs were selected since participants were expected to have necessary knowledge about the concept of angle and main features of it and enough ability to link what they learned to students answers. It can be said that these participants took most of both mathematics and education courses so that their levels were appropriate to define a mathematical concept and determine students mistakes related to the same concept by using their current knowledge. 2.3 Data Collection Tools The process of data collection consisted of two phases. In first phase, one open ended question related to the concept of angle was asked to 30 6th grade students in a middle school. They were required to define the concept of angle with their own statements and describe what it brought to their minds by writing their responses on a paper. In second phase, in order to prepare data collection form for PMTs, students written responses were examined by three researchers and students mistakes or deficiencies in their definitions were determined. 6 of these responses, 2 correct, 2 partly correct and 2 incorrect, were chosen with consensus. It was paid attention not to choose similar mistakes or deficiencies and unclear definitions. Therefore, a form that included students definitions of angle was obtained. This form was given each PMT by the researchers. PMTs were required to define the concept of angle with their own statements and evaluate whether students responses were correct by explaining the reasons in written form. They were also expected to determine the mistakes in students responses if there is any. Besides, PMTs were given enough time to examine each student s response and to give their own answers. European Journal of Education Studies - Volume 2│Issue 12│2016 193 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS 2.4 Data Analysis The data obtained from written forms of PMTs were analysed through the descriptive analysis technique. Descriptive analysis is a technique which provides to analyse, organize and interpret verbal and written data with the help of predetermined categories Yıldırım & Simsek, . In this direction, PMTs definitions of angle were analysed with classification of Mitchelmore and White and PMTs explanations about students responses were analysed with through using the coding categories of Gokkurt, Sahin, Soylu and Dogan (2015). These codes and categories were presented below. 2.5 The Codes and Categories for Determining Mistakes  Not to determine the mistakes: It includes the cases which PMTs are not able to determine the mistakes in students answers. They make wrong explanation or  do not give answers. Determine the mistakes partly true: It involves the cases which PMTs are not able to determine students  mistakes in their definitions completely. Their explanations include some ideas which are not true completely or insufficient. Determine the mistakes completely true: It reflects the cases which PMTs are able to determine students mistakes. They offer true explanations and gave expected answers. 2.6 The Codes and Categories for Definitions of Angle   an amount of turning about a point between two lines  the region formed by the intersection of two half-planes  a pair of rays with a common end-point inappropriate definitions Classification of PMTs angle definitions was made by considering the mathematical concepts such as point, line, ray, end-point, region, intersection, halfplanes. The values of percentage and frequency for each category were calculated. PMTs answers were coded based on the properties of categories. The values of percentage and frequency for each category were calculated. The data were labelled separately by two researchers and the percentage of conformity was found to be %91. Researchers reached a consensus after a meeting and matched the responses and categories. European Journal of Education Studies - Volume 2│Issue 12│2016 194 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS 3. Findings In this section, the findings of the study and relevant explanations were presented. The data were analysed in two steps. At first, a correction table was designed for incorrect or partly correct expressions in the definition of the angle concept of the sixth grade students on the data collection form. Table shows that the definitions of “yşe and Figen were incorrect because they did not include the essential concepts which were the same origin and two rays instead they talked about incorrect or irrelevant concepts such as three rays and two line segments . According to table, the definitions of Beyza and Deniz were partly correct because although they mentioned two rays which was one of the essential features for identifying angle, they did not refer to the concept of the same origin in their definitions. If these deficiencies and mistakes are corrected, these students definitions regarding angle concept will be true. At second step of this study, the PMTs were expected to determine the mistakes in students responses and offer similar correction expressions that were indicated on Table 1. Table 1: Incomplete or wrong expressions in the definition of the angle concept of sixth grade students 6th Grade Students’ Nicknames The correctness of “yşe Incorrect Beyza Partly correct The same origin Cenk Correct --- Deniz Partly correct The same origin Efe Correct --- Figen Incorrect students answers Corrections The same origin Two rays instead of two or three rays The same origin Two rays instead of two line segments PMTs answers related to the correctness of the students definitions of angle were analysed under three categories which were Not to determine the mistake , Determine the mistake partly true and Determine the mistake completely true . The frequencies of their answers were presented in Table 2. European Journal of Education Studies - Volume 2│Issue 12│2016 195 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS Table 2: The frequencies of PMTs answers according to categories 6th Grade Students’ Not to determine the mistake Nicknames Determine the mistake Determine the mistake partly true completely true Wrong Empty “yşe 2 1 20 15 Beyza 3 2 21 12 Cenk 2 1 3 32 Deniz 1 1 28 8 Efe 2 1 1 34 Figen 6 0 24 8 It has been seen that PMTs generally have no problems while determining the students who defined the angle concept correctly. Of the 38 prospective teachers, 32 PMTs understood that Cenk's definition was correct whereas 34 of them accepted Efe's definition as correct. PMTs have difficulty in expressing where the mistake is although they can identify it as false. Prospective teachers were aware of that “yşe and Figen's angle descriptions were wrong. prospective teachers for the definition of “yşe and prospective teachers for the definition of Figen were able to address the mistakes completely. But a large part of them could not make sufficient explanations about what these mistakes were. Similar findings were also seen for the angle definitions that students did partially correct. Prospective teachers might have felt that there were shortcomings in their responses, but they could not express them fully. 12 prospective teachers determined that Beyza's response was partly correct, while 8 identified the deficiencies in the definition of Deniz. The examples from PMTs responses to students definitions of angle concept were presented below. In the figures, * represents the students definitions of angle concept whereas ** refers to prospective teachers responses regarding the correctness of students answers and their correction expressions for these definitions. European Journal of Education Studies - Volume 2│Issue 12│2016 196 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS *An angle is scratch consists of two rays that meet each other, called degree. **The answer is deficient. The angle is composed by a pair of rays with a common end-point. ”eyza didn t tell the beginnings here. *An angle consists of two line segments meet each other. The measure between two line segments is called angle. **It is false. An angle consists of two rays with a common end-point meet each other, not two line segments. Figure 2: The examples for the PMTs who determine the mistake completely true The examples of PMTs answers which he could determine the students mistakes completely true and expressed the reasons clearly can be seen in Figure 2. *An angle is scratch consists of two rays that meet each other, called degree. **”eyza s answer is partly true. The part of scratch called degree is false. *An angle is an area between two rays that going out from the same point. **It has to be the measure of the area, not the area. Figure 3: The examples for the PMTs who determine the mistake partly true European Journal of Education Studies - Volume 2│Issue 12│2016 197 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS Figure 3 presents the examples from the responses of PMTs who could partially identify the students mistakes, explain some of the corrections or simply indicate that there was a mistake without providing sufficient explanation. *An angle is an area between two rays that going out from the same point. **The interior region of the angle is given. *An angle consists of two line segments meet each other. The measure between two line segments is called angle. **How two line segments meet each other is ambiguous. Figure 4: The examples for the PMTs who determine the mistake wrong Figure 4 shows the examples from the answers of PMTs who were not able to identify student mistakes and made irrelevant explanations. The frequencies of PMTs definitions of angle concept according to the categories of Mitchelmore and White (2000) which were indicated in method can be seen in Table 3. Table 3: Descriptive analysis for the definitions of angle concept made by PMTs Definition f An amount of turning about a point between two lines 3 A pair of rays with a common end-point 25 The region formed by the intersection of two half-planes 9 Inappropriate definition 1 TOTAL 38 When the angle definitions of the prospective teachers were examined, all PMTs except one of them correctly defined this concept. 34 of these definitions were static whereas only 3 of them were in the definition of dynamic angle. As seen in Table 3, most of the European Journal of Education Studies - Volume 2│Issue 12│2016 198 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS PMTs defined the angle as a pair of rays with a common end-point. Three PMTs defined the angle concept as an amount of turning about a point between two lines. The examples of PMTs definitions of angle concepts are as the following. *The arc drawn in counter clockwise between two rays with a common end-point is called angle. Figure 5: The example for the PMTs who define angle as an amount of turning about a point between two lines For Figure , it can be said that PMTs these kinds of angle definitions were correct. As mentioned before, only 3 prospective teachers could make the definition of angle concept by referring to an amount of turning about a point between two lines and in terms of being dynamic. *An angle is a geometric figure composed by intersecting of two rays at the same end point. Figure 6: The example for the PMT who define angle as a pair of rays with a common end-point Figure 6 included an example of angle definition of PMTs which was in the category of a pair of rays with a common end-point . It is seen that this definition addressed the essential concepts which were two rays and same point and represented an example of static angle definition. Totally 25 PMTs defined the angle concept as a pair of rays with a common end-point. 15 of prospective teachers defined the concept of angle correctly; however, 10 of them indicated the concept of angle in a way that had the same meaning as the measure of angle. European Journal of Education Studies - Volume 2│Issue 12│2016 199 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS *An angle is the measure between two rays with a common end-point, measured by protractor, expressed with degree. Figure 7: The example for the PMT who define angle as a pair of rays with a common end-point confused the angle and angle s measure concepts with each other Figure 7 shows the example of PMTs who confused the definition of angle with the measure of angle concept. It is seen that prospective teachers identified angle base on the concepts of two rays and same point but they also used the concept of measure . Therefore, these definitions were accepted under the category which was named as a pair of rays with a common end-point . *An angle is the region formed by two rays that going out from the same point. It is expressed with degree. Figure 8: The example for the PMT who define angle as the region formed by the intersection of two half-planes In Figure , the example of PMTs definitions of angle which were considered in the category of the region formed by the intersection of two half-planes was presented. It is seen that prospective teachers used the concepts such as degree and region two rays , same point , in order to identify angle concept. These definitions further exhibited the features of static angle definition because these concepts did not refer to any movement. 4. Discussion and Conclusion The purpose of this study was to investigate the PMTs ability to identify mistakes held by elementary students in angle concept definitions and their proposed corrections to overcome those mistakes. The findings show that PMTs didn t have problems in European Journal of Education Studies - Volume 2│Issue 12│2016 200 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS determining students' definitions of angle as correct, incorrect or partially correct. Although they notice the correct angle definitions, they have problems about expressing the students failures and to clearly identifying the mistakes in the incorrect definitions of students. The findings of Kilic (2010) also revealed that preservice teachers had difficulty in both identifying the source of students misconceptions, and errors and generating effective ways to eliminate such misconceptions. To correct the students mistakes PMTs should expand their understanding of angle concept, to help them to think of the concept in a flexible way. A student can use her mistakes/errors to develop a deeper understanding of a concept as long as the error can be recognized and appropriate, informative feedback can be obtained. Pedagogical methods that systematically address common student errors produce significant gains in student learning (Fisher & Lipson, 1986). Therefore, students need teachers competent to identify their mistakes and can translate into their own benefits. When we looked to the findings related to PMTs angle concept definitions almost all of them define the angle as a pair of rays with a common end-point, which is a static definition. They could not be able to inform their students about the different definitions of angle concept because they do not have them. This may be due from the inclusion of static definitions in the textbooks in Turkey. Angles need to be represented as both static images of pointedness and as dynamic examples of turns in order to develop understanding of the topic (Barmby, Bilsborough, Harries & Higgins, 2009). If students cannot adequately comprehend basic geometric concepts they will not understand and succeed in the subsequent subjects of geometry and this may reduce the achievement of an individual in both school life and daily life (Alkan & Altun, 1998). Results of this research suggest that; textbooks can be rearranged to include multidimensional definition of the concept of angle in Turkey. PMTs could receive more detailed training on how to teach the concept of angle during teacher education. PMTs must gain experience in assessing student responses, it may be included more intensively in the course of teaching practice . The conclusions of the present study are limited by the small sample size and the small number of contexts investigated. A more comprehensive study is currently underway and the results will be shared with another study. European Journal of Education Studies - Volume 2│Issue 12│2016 201 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS References 1. Aydin, E. & Gundogdu, L. (2016). 6th Grade Elementary Mathematics School Book. Sevgi Yayınları, “nkara. 2. Barmby, P., Bilsborough, L., Harries, T. & Higgins, S. (2009). Primary Mathematics: Teaching for Understanding. McGraw-Hill Education (UK). 3. Biber, C., Tuna, A., & Korkmaz, S. (2013). The Mistakes and the Misconceptions of the Eighth Grade Students on the Subject of Angles. European Journal of Science and Mathematics Education, 1(2), 50-59. 4. Butuner, S. O. & Filiz, M. (2016). Exploring High-achieving Sixth Grade Students Erroneous Answers and Misconceptions on The Angle Concept. 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Teacher Sceneries Related to “ngle Concept : Turkey case. Procedia-Social and Behavioural Sciences, 1(1), 285-290. 24. Yetkin, E. (2003). Student Difficulties in Learning Elementary Mathematics. ERIC Digest. ERIC Clearing house for Science Mathematics and Environmental Education. 25. Yildirim, A. & Simsek, H. (2013). Qualitative Research Methods in Social Sciences. Ankara: Seckin Yayincilik. 26. Yigit, M. (2014). An Examination of Pre-service Secondary Mathematics Teachers' Conceptions of Angles. The Mathematics Enthusiast, 11(3), 707-736. European Journal of Education Studies - Volume 2│Issue 12│2016 203 Cigdem Arslan , Hatice Nur Erbay, Pinar Guner PROSPECTIVE MATHEMATICS TEACHERS’ ABILITY TO IDENTIFY MISTAKES RELATED TO ANGLE CONCEPT OF SIXTH GRADE STUDENTS Creative Commons licensing terms Author(s) will retain the copyright of their published articles agreeing that a Creative Commons Attribution 4.0 International License (CC BY 4.0) terms will be applied to their work. 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