### Abstract

104 grade 6 students drawn from four well-established elementary schools of Yozgat, Turkey were participated in this research. All participants were selected via a purposive sampling method and their mathematics grades were 4 or 5 out of 5, and. Data were collected through four questions prepared by considering the learning competencies set out in the grade 6 curriculum in Turkey and the findings of previous studies whose purposes were to identify students’ misconceptions of the angle concept. The findings were analyzed by two researchers, and their inter-rater agreement was calculated as 0.91, or almost perfect. Thereafter, coding discrepancies were resolved, and consensus was established.

The angle concept is a multifaceted concept having static and dynamic definitions.The static definition of the angle refers to “the space between two rays” or“the intersection of two rays at the same end point” (Mitchelmore & White, 1998), whereas the dynamicdefinition of the angle concept highlights that the size of angle is the amountof rotation in direction (Fyhn, 2006). Since both definitionsrepresent two diverse situations and have unique limitations (Henderson & Taimina, 2005), students may holdmisconceptions about the angle concept. In this regard, the aim of thisresearch was to explore high achievers’ knowledge regarding the definition ofthe angle concept as well as to investigate their erroneous answers on theangle concept.

104grade 6 students drawn from four well-established elementary schools of Yozgat,Turkey were participated in this research. All participants were selected via a purposive sampling method and their mathematics grades were 4 or 5 out of 5,and. Data were collected through four questions prepared by considering the learning competencies set out in the grade 6 curriculum in Turkey and the findings of previous studies whose purposes were to identify students’ misconceptions of the angle concept. The findings were analyzed by two researchers, and their inter-rater agreement was calculated as 0.91, or almost perfect. Thereafter, coding discrepancies were resolved, and consensus was established.

In the first question, students were asked to answer a multiple choice questions consisting of two statics definitions and one dynamic definition of the angle concept. Only 38 of 104 students were able to recognize these three definitions. Likewise, Mitchelmore and White (1998) investigated that less than10% of grade 4 students knew the dynamic definition of the angle concept. Additionally,the purpose of the second question was to figure out how well students could recognize 0-degree angle. We found that 49 of 104 students were unable to recognize MXW as an angle. While 6 students indicated that the size of MXW is0, other 6 students revealed that the size of MXW is 360. Therefore, 12 of 104students correctly answered this questions. On the other hand, 28 of 104students recognized the MXW angle as 180-degree angle. This finding demonstrated that these students have difficulties in naming the angles.Moreover, the third question consisted of three concentric circles with center O and two radiuses of the outer circle, and the intersection of the radiuses with these circles were named. Then, students were asked to compare the size of AOB, GOD and EOF angles. Only 36 of 104 students answered correctly by indicating that all three angles are equal, whereas 68 of 104 students incorrectly responded this question by revealing AOB<GOD< EOF. These students erroneously thought the size of the angle is related to either the size of the arc marking the angle or the area between the arms of the angle and the arc marking angle. These two erroneous strategies for determining the size of angles have been found by a few studies (Clausen-May,2008; Devichi & Munier, 2013; Kim & Lee, 2014; Mithcelmore, 1998;Wilson & Adams, 1992). The last question, whose aim was to determine how well students can adapt theangle concept to real life, consisted of an observer and a barrier, and students were asked to color the hidden area behind the barrier. Only 2 of 104students correctly responded this question, whereas 19 of 104 students drew rays from the observer to both sides of the barrier, and colored the area covered by the rays, the observer and barrier. While 35 of 104 students just colored behind the barrier without using any strategies, 33 of 104 students constructed two perpendicular lines at the both end of the barrier, and colored behind the barrier. Similarly, Munier, Devinci and Merle (2008) found that this incorrect strategy was used by 27% of students.

Consequently, we found that although the participants in this study were high achievers, they still held several misconceptions on the angle concept and had difficulties in adapting the angle concept to real life.

**Keywords:** the angle concept;misconceptions; erroneous answers; high achievers

**References**

Clausen-May, T. (2008). AnotherAngle on Angles. *Australian Primary Mathematics Classroom*, *13*(1),4–8.

Devichi, C., & Munier, V.(2013). About the concept of angle in elementary school: Misconceptions andteaching sequences. *The Journal of Mathematical Behavior*, *32*(1),1–19. http://doi.org/10.1016/j.jmathb.2012.10.001

Fyhn, A. B. (2006). A climbinggirl’s reflections about angles. *The Journal of Mathematical Behavior*, *25*(2),91–102. http://doi.org/10.1016/j.jmathb.2006.02.004

Henderson, D. W., & Taimina,D. (2005). *Experiencing geometry: Euclidean and non-Euclidean with history*(3rd ed.). New York, USA: Prentice Hall.

Kim, O.-K., & Lee, J. H.(2014). Representations of Angle and Lesson Organization in Korean and AmericanElementary Mathematics Curriculum Programs. *KAERA Research Forum*, *1*(3),28–37.

Mitchelmore, M. C., & White,P. (1998). Development of angle concepts: A framework for research. *MathematicsEducation Research Journal*, *10*(3), 4–27.

Mithcelmore, M. C. (1998). Youngstudents’ concepts of turning and angle. *Cognition and Instruction*, *16*(3),265–284.

Munier, V., Devichi, C., &Merle, H. (2008). A Physical Situation as a Way to Teach Angle. *TeachingChildren Mathematics*, *14*(7), 402–407.

Wilson, P. S., & Adams, V.M. (1992). A Dynamic Way to Teach Angle and Angle Measure. *ArithmeticTeacher*, *39*(5), 6–13.

Original language | English |
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Publication status | Published - Sep 2016 |

Event | III. International Dynamic, Explorative and Active Learning (IDEAL) Conference - Ondokuz Mayıs University, Samsun, Turkey Duration: 01 Sep 2016 → 03 Sep 2016 http://www.ideal-konferans.net/ |

### Conference

Conference | III. International Dynamic, Explorative and Active Learning (IDEAL) Conference |
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Country | Turkey |

City | Samsun |

Period | 01/09/2016 → 03/09/2016 |

Internet address |

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*Investigating high-achieving sixth grade students' erroneous answers on the angle concept in Yozgat*. Paper presented at III. International Dynamic, Explorative and Active Learning (IDEAL) Conference, Samsun, Turkey.