Öz
This study was carried out in order to determine how the 3rd grade students of the Department of Elementary Mathematics Education structured their "if and only if propositions”. The data were obtained by examining the students' answers given to the midterm exam questions and discussing the solutions with the students in the classroom. The study is a case study. As a result of the application, it was found out that the students had difficulty in determining the parts of the hypothesis that are included in “if and only if” proposition and therefore dividing the proposition into two “if” proposition. Some students think that the part or parts given as hypothesis should also be proved. When defining propositions, in addition to their “if and only if propositions”, it is suggested to define new types of proposition in the form of “hypothesis-containing if and only if propositions”.
Anahtar Kelimeler
Theorem if and only if proposition cognitive structuring definition proof
Kaynakça
- Anapa, P. & Şamkar, H. (2010). Investigation of undergraduate students’ perceptions of mathematical proof. Procedia Social and Behavioral Sciences, 2, 2700–2706.
- Arsac, G. (2007). Origin of mathematical proof: History and epistemology. In P. Boero (Ed.), (pp. 27-42). Theorems in schools: From history, epistemology and cognition to classroom practice. Rotterdam, The Netherlands: Sense Publishers.
- Barendregt, H. & Wiedijk, F. (2005). The challenge of computer mathematics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 363, 2351-2375.
- De Villiers, M. (1999). Rethinking proof with the Geometer's Sketchpad. Emeryville, CA: Key Curriculum Press.
- Doruk, M., Özdemir, F. & Kaplan, A., (2015). The relationship between prospective mathematics teachers’ conceptions on constructing mathematical proof and their self-efficacy beliefs towards mathematics, K. Ü. Kastamonu Eğitim Dergisi, 23 (2), 861-874.
- Doruk, M. (2019). Preservice mathematics teachers' determination skills of the proof techniques: The case of integers. International Journal of Education in Mathematics, Science and Technology, 7(4), 335-348.
- Doruk, M., & Kaplan, A. (2015). Preservice mathematics teachers’ difficulties in doing proofs and causes of their struggle with proofs. Bayburt Üniversitesi Eğitim Fakültesi Dergisi, 10(2), 315-328.
- Gibson, D. (1998). Students’ use of diagrams to develop proofs in an introductory analysis course. Students’ proof schemes. In E. Dubinsky, A. Schoenfeld & J.Kaput (Eds.), Research in Collegiate Mathematics Education, III, 284–307. AMS.
- Gökkurt, B., Deniz, D., Akgün, L., & Soylu, Y. (2014). A corpus-based study on the process of proof within the field of mathematics. Baskent University Journal of Education, 1(1), 55-63.
- Güner, P. (2012). Matematik öğretmen adaylarının ispat yapma süreçlerinde DNR tabanlı öğretime göre anlama ve düşünme yollarının incelenmesi [The investigation of preservice mathematics teachers' ways of thinking and understanding in the proof process according to DNR based education. (Yayımlanmamış Yüksek Lisans Tezi), Marmara Üniversitesi, İstanbul.
- Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5- 23.
- Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from an exploratory study. In A. H. Schoenfeld, J. Kaput & E. Dubinsky (Eds.), Research in College Mathematics Education III (pp. 234-283). Providence, RI: AMS
- Hemmi, K. (2010). Three styles characterizing mathematicians' pedagogical perspectives on proof. Educational Studies in Mathematics, 75(3), 271-291.
- Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49, 283-312.
- Jones, K. (2000). The student experience of mathematical proof at university level, International Journal of Mathematical Education in Science and Technology, 31(1), 53 -60.
- Knuth, E. J. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics, Journal of Mathematics Teacher Education, 5(1), 61-88.
- Moore, R.C. (1994). Making the transition to formal proof. Educational Studies in Mathematics. 27, 249-266.
- National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston VA.
- Oflaz, G., Bulut, N., & Akcakin, V. (2016). Pre-service classroom teachers’ proof schemes in geometry: a case study of three pre-service teachers. Eurasian Journal of Educational Research, 16(63).
- Polster, B. (2004). Q.E.D. Beauty in mathematical proof. New York: Walker Publishing Company,
- Reis, K. & Renkl, A. (2002). Learning to prove: The idea of heuristic examples, Zentralblattfür Didaktik der Mathematik (ZDM), 34 (1), 29- 35.
- Sema, E. R. & Şenol, D.(2022). A design study to develop the proof skills of mathematics pre-service teachers. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 16(1), 189-226.
- Sevgi, S. & Kartalcı, S. (2021). Investigation of university students’views on mathematical proof and conceptual-operational approaches, Baskent University Journal of Education, 8(1), 275-291
- Tall, D. (1999). The cognitive development of proof: Is mathematical proof for all or for some? In Z. Usiskin (Ed.), Developments in School Mathematics Education around the World, vol, 4, pp.117-136. Reston, Virginia: NCTM.
- Tall, D. (2014). Making sense of mathematical reasoning and proof. Mathematics & mathematics education: Searching for common ground, 223-235.
- Usiskin, Z. (1982). Van hiele levels and achievement in secondary school geometry. Final Report, Cognitive development and achievement in secondary school geometry project. Chicago: University of Chicago.
- Varghese, T. (2011). Balacheff’s 1988 taxonomy of mathematical proofs. Eurasia Journal of Mathematics, Science & Technology Education, 7(3), 181-192.
- Yopp, D. (2011). How some research mathematicians and statisticians use proof in undergraduate mathematics, Journal of Mathematical Behavior, 30(2), 115-130.
Öz
This study was carried out in order to determine how the 3rd grade students of the Department of Elementary Mathematics Education structured their "if and only if propositions”. The data were obtained by examining the students' answers given to the midterm exam questions and discussing the solutions with the students in the classroom. The study is a case study. As a result of the application, it was found out that the students had difficulty in determining the parts of the hypothesis that are included in “if and only if” proposition and therefore dividing the proposition into two “if” proposition. Some students think that the part or parts given as hypothesis should also be proved. When defining propositions, in addition to their “if and only if propositions”, it is suggested to define new types of proposition in the form of “hypothesis-containing if and only if propositions”.
Anahtar Kelimeler
Theorem if and only if proposition cognitive structuring definition proof
Kaynakça
- Anapa, P. & Şamkar, H. (2010). Investigation of undergraduate students’ perceptions of mathematical proof. Procedia Social and Behavioral Sciences, 2, 2700–2706.
- Arsac, G. (2007). Origin of mathematical proof: History and epistemology. In P. Boero (Ed.), (pp. 27-42). Theorems in schools: From history, epistemology and cognition to classroom practice. Rotterdam, The Netherlands: Sense Publishers.
- Barendregt, H. & Wiedijk, F. (2005). The challenge of computer mathematics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 363, 2351-2375.
- De Villiers, M. (1999). Rethinking proof with the Geometer's Sketchpad. Emeryville, CA: Key Curriculum Press.
- Doruk, M., Özdemir, F. & Kaplan, A., (2015). The relationship between prospective mathematics teachers’ conceptions on constructing mathematical proof and their self-efficacy beliefs towards mathematics, K. Ü. Kastamonu Eğitim Dergisi, 23 (2), 861-874.
- Doruk, M. (2019). Preservice mathematics teachers' determination skills of the proof techniques: The case of integers. International Journal of Education in Mathematics, Science and Technology, 7(4), 335-348.
- Doruk, M., & Kaplan, A. (2015). Preservice mathematics teachers’ difficulties in doing proofs and causes of their struggle with proofs. Bayburt Üniversitesi Eğitim Fakültesi Dergisi, 10(2), 315-328.
- Gibson, D. (1998). Students’ use of diagrams to develop proofs in an introductory analysis course. Students’ proof schemes. In E. Dubinsky, A. Schoenfeld & J.Kaput (Eds.), Research in Collegiate Mathematics Education, III, 284–307. AMS.
- Gökkurt, B., Deniz, D., Akgün, L., & Soylu, Y. (2014). A corpus-based study on the process of proof within the field of mathematics. Baskent University Journal of Education, 1(1), 55-63.
- Güner, P. (2012). Matematik öğretmen adaylarının ispat yapma süreçlerinde DNR tabanlı öğretime göre anlama ve düşünme yollarının incelenmesi [The investigation of preservice mathematics teachers' ways of thinking and understanding in the proof process according to DNR based education. (Yayımlanmamış Yüksek Lisans Tezi), Marmara Üniversitesi, İstanbul.
- Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5- 23.
- Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from an exploratory study. In A. H. Schoenfeld, J. Kaput & E. Dubinsky (Eds.), Research in College Mathematics Education III (pp. 234-283). Providence, RI: AMS
- Hemmi, K. (2010). Three styles characterizing mathematicians' pedagogical perspectives on proof. Educational Studies in Mathematics, 75(3), 271-291.
- Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49, 283-312.
- Jones, K. (2000). The student experience of mathematical proof at university level, International Journal of Mathematical Education in Science and Technology, 31(1), 53 -60.
- Knuth, E. J. (2002). Teachers’ conceptions of proof in the context of secondary school mathematics, Journal of Mathematics Teacher Education, 5(1), 61-88.
- Moore, R.C. (1994). Making the transition to formal proof. Educational Studies in Mathematics. 27, 249-266.
- National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston VA.
- Oflaz, G., Bulut, N., & Akcakin, V. (2016). Pre-service classroom teachers’ proof schemes in geometry: a case study of three pre-service teachers. Eurasian Journal of Educational Research, 16(63).
- Polster, B. (2004). Q.E.D. Beauty in mathematical proof. New York: Walker Publishing Company,
- Reis, K. & Renkl, A. (2002). Learning to prove: The idea of heuristic examples, Zentralblattfür Didaktik der Mathematik (ZDM), 34 (1), 29- 35.
- Sema, E. R. & Şenol, D.(2022). A design study to develop the proof skills of mathematics pre-service teachers. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 16(1), 189-226.
- Sevgi, S. & Kartalcı, S. (2021). Investigation of university students’views on mathematical proof and conceptual-operational approaches, Baskent University Journal of Education, 8(1), 275-291
- Tall, D. (1999). The cognitive development of proof: Is mathematical proof for all or for some? In Z. Usiskin (Ed.), Developments in School Mathematics Education around the World, vol, 4, pp.117-136. Reston, Virginia: NCTM.
- Tall, D. (2014). Making sense of mathematical reasoning and proof. Mathematics & mathematics education: Searching for common ground, 223-235.
- Usiskin, Z. (1982). Van hiele levels and achievement in secondary school geometry. Final Report, Cognitive development and achievement in secondary school geometry project. Chicago: University of Chicago.
- Varghese, T. (2011). Balacheff’s 1988 taxonomy of mathematical proofs. Eurasia Journal of Mathematics, Science & Technology Education, 7(3), 181-192.
- Yopp, D. (2011). How some research mathematicians and statisticians use proof in undergraduate mathematics, Journal of Mathematical Behavior, 30(2), 115-130.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik Eğitimi |
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Erken Görünüm Tarihi | 7 Ekim 2023 |
| Yayımlanma Tarihi | 21 Ekim 2023 |
| Gönderilme Tarihi | 7 Aralık 2022 |
| Yayımlandığı Sayı | Yıl 2023 Cilt: 7 Sayı: 15 |
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