Hatalı Örnek Aracılığıyla Dikdörtgen ile Paralelkenar Arasındaki İlişkinin İncelenmesi

Abstract

Bu çalışmanın amacı matematik öğretmen adaylarının hatalı örnek aracılığıyla dikdörtgen ve paralelkenar arasındaki ilişkiyi nasıl anlamlandırdıklarını araştırmaktır. Araştırma Türkiye’de bir devlet üniversitesinde İlköğretim Matematik Öğretmenliği bölümünde ikinci ve üçüncü sınıfta öğrenim gören 90 öğretmen adayı ile yürütülmüştür. Örnek durum çalışması niteliğindeki çalışmanın veri toplama araçlarını hatalı bir örneğin yer aldığı bir etkinlik ile öğretmen adaylarının etkinlik önerileri oluşturmaktadır. Elde edilen veriler içerik analizi ile analiz edilmiştir. Bulgular öğretmen adaylarının sadece %13’ünün hatalı örneği analiz etme ve kabul edilebilir bir etkinlik önerisinde bulunarak yeterli kavramsal bilgiye sahip olduklarını göstermektedir. Katılımcıların büyük çoğunluğunun (%64) hatalı örneği analiz etmede başarısız oldukları; bu öğretmen adaylarının da %59’unun hatalı örnek içeren etkinlik önerisinde bulundukları tespit edilmiştir. Araştırmanın sonuçları hatalı örneklerin öğretmen eğitiminde pedagojik bir araç olarak kullanılabileceği gibi öğretmen adaylarının kavramları anlamlandırma düzeylerinin belirlenmesinde etkili bir araç olduğunu desteklemektedir. Özel olarak bu çalışmanın sonuçları hatalı örnek aracılığıyla öğretmen adaylarının sahip oldukları “Dikdörtgen paralelkenarın özel bir halidir.” bilgisinin birçoğu için kavramsal temellere değil görsel algıya dayalı olduğunu ortaya koymuştur.

Keywords

• hatalı örnekler
• dörtgenler
• dikdörtgen
• paralelkenar

Examining the Relationship Between Rectangles and Parallelograms Through an Erroneous Example

Abstract

The aim of this study is to investigate how pre-service mathematics teachers make sense of the relationship between rectangle and parallelogram through an erroneous example. The study was conducted with 90 pre-service mathematics teachers studying in the second and third grades in the Department of Elementary Mathematics Teaching at a state university in Turkey. The data collection tools of the case study consisted of a task involving an erroneous example and the task suggestions of the pre-service teachers. The data obtained were analyzed by content analysis. The findings show that only 13% of the pre-service teachers had sufficient conceptual knowledge to analyze the erroneous example and propose an acceptable task. It was found that the majority of the participants (64%) were unsuccessful in analyzing the erroneous example and 59% of these pre-service teachers proposed tasks that included erroneous examples. The results of the study support that erroneous examples can be used as a pedagogical tool in teacher education as well as an effective tool for determining pre-service teachers' level of understanding of concepts. In particular, the results of this study revealed that the knowledge of "Rectangle is a special case of parallelogram" held by pre-service teachers through erroneous examples was based on visual perception rather than conceptual foundations for most of them.

Keywords

• Erroneous Example
• quadrilaterals
• rectangle
• parallelogram

Ethical Statement

Ethical approval to report this case was obtained from Kilis 7 Aralık University with approval number 2022/07 at 21/04/2022.

References

1. Akkuşci, Y. E. (2019). Examining the effectiveness of in-class use of error-based task applications in mathematics teaching (Master's thesis, Graduate School of Educational Sciences).
2. Avcu, R. Pre-service middle school mathematics teachers’ personal concept definitions of special quadrilaterals. Math Ed Res J 35, 743–788 (2023). https://doi.org/10.1007/s13394-022-00412-2
3. Biza, I., Nardi, E., & Zachariades, T. (2007). Using tasks to explore teacher knowledge in situation-specific contexts. Journal of Mathematics Teacher Education, 10(4), 301-309.
4. Borasi, R. (1989). Students' Constructive Uses of Mathematical Errors: A Taxonomy.
5. Borasi, R. (1994). Capitalizing on errors as "springboards for inquiry": A teaching experiment. Journal for research in mathematics education, 25(2), 166-208.
6. Brodie, K., & Shalem, Y. (2011). Accountability conversations: Mathematics teachers' learning through challenge and solidarity. Journal of Mathematics Teacher Education, 14, 419-439
7. Bütün, M. (2005). “İlköğretim matematik öğretmenlerinin alan eğitimi bilgilerinin nitelikleri üzerine bir çalışma.” Karadeniz Teknik Üniversitesi, Yayımlanmamış yüksek lisans tezi, Trabzon.
8. Bütün, M. (2011). Matematik öğretmenlerinin alan eğitimi bilgi yapılarının incelenmesinde senaryo tipi mülakat sorularının kullanımı. Dicle Üniversitesi Ziya Gökalp Eğitim Fakültesi Dergisi, (16), 105-115.

9. Creswell, J. W. (2015). Educational research: Planning, conducting, and evaluating quantitative and qualitative research. pearson.
10. De Villiers, M. (1998, July). To teach definitions in geometry or teach to define? In PME conference (Vol. 2, No. 8).
11. Ding, M. (2007). Knowing mathematics for teaching: A case study of teacher responses to students' errors and difficulties in teaching equivalent fractions. Texas A&M University.
12. Durkin, K. (2012). The effectiveness of incorrect examples and comparison when learning about decimal magnitude (Doctoral dissertation).
13. Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206-214.
14. Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In Exploiting mental imagery with computers in mathematics education (pp. 142-157). Springer Berlin Heidelberg.
15. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational studies in mathematics, 61(1-2), 103-131.
16. Fischbein, E. (1993). The theory of figural concepts. Educational studies in mathematics, 24(2), 139-162.
17. Fujita, T., & Jones, K. (2006). Primary trainee teachers' understanding of basic geometrical figures in Scotland. Psychology of Mathematics Education.
18. Fujita, T., & Jones, K. (2007). Learners' understanding of the definitions and hierarchical classifıcation of quadrilaterals: towards a theoretical framing. Research in Mathematics Education, 9(1), 3-20.
19. Gal, H., & Lew, H. C. (2008). Is a rectangle a parallelogram-Towards A bypass of van Hiele level 3 decision making. In Proceedings of the 11th International Congress on Mathematical Education. Monterrey, Mexico, A paper presented at Topic Study Group (Vol. 18, pp. 34-48).
20. Gardee, A., & Brodie, K. (2015). A teacher's engagement with learner errors in her Grade 9 mathematics classroom. Pythagoras, 36(2), 1-9.
21. Gardee, A., & Brodie, K. (2022). Relationships between teachers' interactions with learner errors and learners' mathematical identities. International Journal of Science and Mathematics Education, 20(1), 193-214.
22. Große, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: Can this foster learning outcomes?. Learning and instruction, 17(6), 612-634.
23. Gürbüz, R., Yıldırım, İ., & Doğan, M. F. (2021). The Effect of Error-Based Tasks on the Success of 7th Grade Students in Some Statistics Concepts. Journal of Computer and Education Research, 9(18), 997-1021.
24. Hansen, A. (2011). Children's errors in mathematics: Understanding common misconceptions in primary school. (2nd edn.). Exeter: Learning Matters Ltd.
25. Heinze, A. (2005). Mistake-Handling Tasks in the Mathematics Classroom. International Group for the Psychology of Mathematics Education, 3, 105-112.
26. Horzum, T. (2018). Examining pre-service mathematics teachers' understanding of quadrilaterals through a concept map. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 9(1), 1-30.
27. Karakuş, F., & Erşen, Z. B. (2016). Investigation of pre-service primary school teachers' definitions and classifications of some quadrilaterals. Karaelmas Journal of Educational Sciences, 4(1), 38-49.
28. Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. In V. Richardson (Ed.). Handbook of research on teaching (4th edn., pp. 333–357). Washington, DC: American Educational Research Association.
29. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
30. Mason J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243–267.
31. Nesher, P. (1987). Towards an instructional theory: The role of student's misconceptions. For the learning of mathematics, 7(3), 33-40.
32. Ngu, B. H., Chen, O., Phan, H. P., Usop, H., & Anding, P. N. (2025). Can correct and incorrect worked examples supersede worked examples and problem-solving on learning linear equations? an examination from cognitive load and motivation perspectives. Education Sciences, 15(4), 504.
33. Özdemir, A., & Çekirdekci, S. (2022). Geometric habits of mind: The meaning of quadrilaterals for elementary school student teachers. International Journal of Educational Studies in Mathematics, 9(1), 49–66. https://doi.org/10.17278/ijesim.1033078
34. Sahin, S. (2023, October 28–30). Examination of pre-service primary school mathematics teachers’ levels of definition and figural representation regarding rectangle and parallelogram. In Proceedings of the 6th International Symposium of Turkish Computer and Mathematics Education (TURCOMAT-6) (pp. 33–40). Kızılcahamam, Ankara, Turkey: TURCOMAT-6 Organizing Committee.
35. Sahin, S. (2011). The influence of a professional development program related to students’ mathematical difficulties on teachers’ ways of handling difficulties (Unpublished Master Thesis), Gaziantep University, Gaziantep, Türkiye.
36. Santagata, R. (2005). Practices and beliefs in mistake-handling tasks: A video study of Italian and US mathematics lessons. Teaching and Teacher Education, 21(5), 491-508.
37. Shimizu, Y., Kang, H. (2025). Research on classroom practice and students’ errors in mathematics education: a scoping review of recent developments for 2018-2023. ZDM Mathematics Education. https://doi.org/10.1007/s11858-025-01704-0
38. Skinner, B. F. (1958). Teaching machines. Science, 128, 969-977.
39. Soncini, A., Visintin, E. P., Matteucci, M. C., Tomasetto, C., & Butera, F. (2022). Positive error climate promotes learning outcomes through students’ adaptive reactions towards errors. Learning and Instruction, 80, 101627. https://doi.org/10.1016/j.learninstruc.2022.101627
40. Swan, M. (2002). Dealing with misconceptions in mathematics. In Issues in mathematics teaching (pp. 147-165). Routledge.
41. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational studies in mathematics, 12(2), 151-169.
42. Toluk, Z., Olkun, S., & Durmuş, S. (2002). The effect of geometry teaching supported by problem-centered and visual models on the development of geometric thinking levels of classroom teaching students.
43. Tsovaltzi, D., Melis, E., McLaren, B. M., Meyer, A. K., Dietrich, M., & Goguadze, G. (2010). Learning from erroneous examples: when and how do students benefit from them?. In Sustaining TEL: From Innovation to Learning and Practice: 5th European Conference on Technology Enhanced Learning, EC-TEL 2010, Barcelona, Spain, September 28-October 1, 2010. Proceedings 5 (pp. 357-373). Springer Berlin Heidelberg.
44. Ubuz, B., & Gökbulut, Y. (2015). Classroom Teacher Candidates' Pyramid Knowledge: Creating Definitions and Examples. Journal of Kirsehir Education Faculty, 16(2).
45. Ulusoy, F., & Çakıroğlu, E. (2017). Middle School Students' Distinguishing Parallelograms: Over-specialization and over-generalization. Abant Izzet Baysal University Journal of Faculty of Education, 17(1), 457-475.
46. Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293-305.
47. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In Advanced mathematical thinking (pp. 65-81). Dordrecht: Springer Netherlands.
48. Vízek, L., & Samková, L. (2023). Investigating how lower secondary school students reason about quadrilaterals emerging in dynamic constructions. International Journal of Mathematical Education in Science and Technology, 56(1), 1–20. https://doi.org/10.1080/0020739X.2023.2255184
49. Von Glasersfeld, E. (1989). Cognition, construction of knowledge, and teaching. Synthese, 80, 121-140.
50. Watson, A., Mason, J. (2007). Taken-as-shared: a review of common assumptions about mathematical tasks in teacher education. J Math Teacher Educ 10, 205–215 (2007). https://doi.org/10.1007/s10857-007-9059-3.
51. Yap, J. B. K., & Wong, S. S. H. (2024). Deliberately making and correcting errors in mathematical problem-solving practice improves procedural transfer to more complex problems. Journal of Educational Psychology, 116(7), 1112–1128. https://doi.org/10.1037/edu0000850
52. Zaslavsky, O., & Sullivan, P. (2011). Setting the stage: A conceptual framework for examining and developing tasks for mathematics teacher education. Constructing knowledge for teaching secondary mathematics: Tasks to enhance prospective and practicing teacher learning, 1-19.