Research Article
BibTex RIS Cite

A Design Study To Develop The Proof Skills Of Mathematics Pre-Service Teachers

Year 2022, Volume: 16 Issue: 1, 189 - 226, 30.06.2022
https://doi.org/10.17522/balikesirnef.1081825

Abstract

Mathematical proof, addition to its duties such as verifying and explaining a claim, aids the understanding of mathematics and the formation of new information, thereby increasing its importance in mathematics education. Proving skills need to be developed to learn mathematics and find effective solutions in real-life problem situations. Given the importance of proof in mathematics education, it is critical to develop appropriate teaching practices to increase pre-service teachers' knowledge of proof, proving, and understanding levels, particularly in departments of mathematics teachers, to provide effective and long-term mathematics instruction to future students. The effect of designed proof tasks on the development of pre-service mathematics teachers' proving skills was investigated in this study. This research, the design-based research, was carried out with the participation of three volunteer mathematics teacher candidates studying at a state university in Ankara. The data obtained from the proof tasks and individual interviews were used in the study. Proof tasks designed according to the findings obtained from the data analyzed with the qualitative analysis method contributed to the development of the participants' proving skills.

References

  • Almeida, D. (2000). A survey of mathematics undergraduates’ interaction with proof: Some implications for mathematics education. International Journal of Mathematical Education in Science and Technology, 31(6), 869-890.
  • Arslantaş İlter, E. (2020). Matematik öğretmenlerinin sözsüz ispat becerilerinin pisagor teoremi bağlamında incelenmesi. Yayınlanmamış Yüksek Lisans Tezi, Sivas Cumhuriyet Üniversitesi, Sivas.
  • Atwood, P. R., (2001) Learning to construct proof in a first course on mathematical proof. Doctoral Dissertation, Western Michigan University, Kalamazoo, Michigan.
  • Baki A. ve Kutluca T. (2009). Dokuzuncu sınıf matematik öğretim programında zorluk çekilen konuların belirlenmesi. e-Journal of New World of Science Academy, 4(2), 604-619.
  • Balacheff, N. (1988). Aspects of Proof Pupils‘ Practice of School Mathematics. Mathematics, Teachers and Children, Pimm, D. (ed.) Hodder ve Stoughton. 216-235. London.
  • Benkhalti, A., Selden, A., & Selden, J. A. (2017). Slowly Develops Self-Efficacy with Writing Proof Frameworks, but Her Initial Progress and Sense of Self-Efficacy Evaporates When He/she Encounters Unfamiliar Concepts: However, It Eventually Returns.
  • Branch, R. M. (2009). Instructional design: The ADDIE approach (Vol. 722). Springer Science & Business Media.
  • Brodahl, C., Larson, N., Wathne, U., & Bjørkestøl, K. (2020). Developing Further Support for In-Service Teachers' Implementation of a Reasoning-and-Proving Task and Their Identification of Students' Level of Mathematical Argumentation. Journal of the International Society for Teacher Education, 24(2), 73-87.
  • Brown, A.L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning Sciences, 2(2), 141–178.
  • Cihan, F. (2019). Matematik öğretmen adaylarının ispatla ilgili alan ve pedagojik alan bilgilerini geliştirmeye yönelik bir ders tasarımı (Unpublished doctoral dissertation). Marmara University, İstanbul.
  • Collins, A. (1992). Towards a design science of education. E. Scanlon ve T. O’Shea (Eds.), New directions in educational technology içinde (15–22). Berlin: Springer.
  • Çontay, E. G. ve Duatepe-Paksu, A. (2019). Ortaokul matematik öğretmeni adaylarının ispatın doğasına ilişkin görüşleri. Sınırsız Eğitim ve Araştırma Dergisi, 4(1), 64-89.
  • Dane, A. (2008). İlköğretim matematik 3. sınıf öğrencilerinin tanım, aksiyom ve teorem kavramlarını anlama düzeyleri. Kastamonu Eğitim Dergisi, 16(2), 495-506.
  • Dean, E. E., (1996). Teaching the proof process, a model for discovery learning. College Teaching, 44 (2), 52-55.
  • de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.
  • Dreyfus, T. (1999). Why Johnny can’t prove? Educational Studies in Mathematics,38, 85-109.
  • Doorman, M., Bakker, A., Drijvers, P., & Wıjaya, A. (2016). Design-Based Research In Mathematıcs Educatıon. In Sriwijaya University Learning and Education International Conference 2(1), 21-46.
  • Greenberg, M.J. (1993). Euclidean and non-Euclidean geometries: Development and history. New York: W.H. Freeman.
  • Hammack, R. (2013). Book of Proof, Privately published, downloadable from http://www.people.vcu.edu/~rhammack/BookOfProof.
  • Hanna, G., ve de Villiers, M. (2008). ICMI Study 19: Proof and proving in mathematics education. ZDM: International Journal on Mathematics Education, 40(2), 329–336.
  • Harel, G., Selden, A. & Selden, J., (2006). Advanced Mathematical Thinking. Handbook of Research on the Pschology of Mathematics Education: Past, Present and Future, Gutierrez, A. and Boero, P. (eds.), pp.147-172.
  • Harel, G., ve Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234-283). Providence, RI: American Mathematical Society.
  • Harel, G. & Sowder, L., (2007). Toward comprehensive perspectives on learning and teaching proof. Handbook of Research on Teaching and Learning Mathematics, Lester, F. (Ed.), Greenwich, CT: Information Age Publishing. 805-842.
  • Healy, L., ve Hoyles, C. (2007). Curriculum change and geometrical reasoning. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice, 81-116. Rotterdam, The Netherlands: Sense.
  • Houston, K. (2009). How to Think Like a Mathematician. (Mehmet Terziler ve Tahsin Oner, ¸Cev.). Cambridge: Cambridge University Press.
  • Karaoğlu, Ö. (2010). Matematik Öğretmen Adaylarının Anahtar Nokta ve Fikirlerle Desteklenmiş İspatları Yapabilme Performansları. Yayınlanmamış Yüksek Lisans Tezi, Gazi Üniversitesi, Ankara.
  • Knapp, J., (2005). Learning to prove in order to prove to learn. http://mathpost.asu.edu/~sjgm/issues/2005_spring/SJGM_knapp.pdf
  • Ko, Y. & Knuth, E., 2009, Undergraduate mathematics majors‘ writing performance producing proofs and counterexamples about continuous functions. Journal of Mathematical Behavior, 28, 68-77.
  • Mariotti, M.A.(2006). Proof and proving in mathematics education. A. Gutierrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education. Past, present and future (173-204). Rotterdam: Sense Publishers.
  • Miyazaki, M. (2000). Levels of proof in lower secondary school mathematics. Educational Studies in Mathematics, 41(1), 47-68.
  • 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: Author.
  • Pala, O., Aksoy, E., & Narli, S. (2021). Can the Proof Image Exist in the Absence of the Formal Proof?: Analyses of an Unsuccessful Proving Attempt. Online Submission, 15(1), 1-31.
  • Polat, K. ve Akgün, L. (2016). Ortaöğretim matematik öğretmeni adaylarının ispat kavramına ve ispat yapmanın zorluklarına yönelik görüşleri. The Journal of Academic Social Science Studies, 43, 423-438.
  • Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52(3), 319-325.
  • Sarı, M. (2011). Üniversite öğrencilerinin matematiksel kanıt ile ilgili güçlükleri ve kanıt öğretimi (Doktora Tezi), Hacettepe Üniversitesi Fen Bilimleri Enstitüsü, Ankara.
  • Sarı Uzun, M. ve Bülbül, A. (2013). Matematik öğretmen adaylarının kanıtlama becerilerini geliştirmeye yönelik bir öğretme deneyi. Eğitim ve Bilim, 38(169), 372-390.
  • Sarı Uzun, M. (2020). Öğrenenlerin ispat yapma davranışları/ispat şemaları. In. I. Uğurel (Ed.), Matematiksel İspat ve Öğretimi. Okul Yıllarında İspat Öğretimini Destekleyen Çok Yönlü Bir Bakış, 189-226.Ankara: Anı Yayıncılık.
  • Schabel, C. J., 2001, An Instructional Model To Improve Proof Writing In College Number Theory. Doctoral Dissertation,. Portland State University, USA.
  • Selden, A. ve Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36.
  • Selden, J. ve Selden, A. (2009). Teaching proving by coordinating aspects of proofs with students’ abilities. In D. A. Stylianou, M. L. Blanton & E.J. Knuth (Eds.), Teaching and learning proof across grades: A K-16 perspective, 339-354. New York/Washington, DC: Routledge/National Council of Teachers of Mathematics.
  • Selden, A., Selden, J., & Benkhalti, A. (2017). Proof frameworks: A way to get started. PRIMUS, 28(1), 31-45.
  • Selden, J., Selden, A. ve McKee, K. (2008). Improving advanced students proving abilities. Paper presented at 11th International Congress of Mathematical Education (ICME-11), Monterrey, Mexico.
  • Skott, J., Larsen, D. M., & Østergaard, C. H. (2020). Learning to teach to reason: Reasoning and proving in mathematics teacher education. In Professional Development and Knowledge of Mathematics Teachers (pp. 44-61). Routledge.
  • Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.
  • Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the task of proving. Journal of mathematics teacher education, 11(4), 307-332.
  • Stylianides, A. J. ve Stylianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72, 237-253.
  • Stylianides, A. J. (2016). Proving in the elementary mathematics classroom. Oxford University Press.
  • Velleman, D. J. (2008). Kanıt nasıl yapılır. Çev., Terziler, M ve Öner, T. Ankara: Nobel Yayıncılık.
  • Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101–119.
  • Weber, K.. (2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course. Journal of Mathematical Behavior, 23, 115–133.
  • Weber, K. (2005). A procedural route toward understanding aspects of proof: Case studies from real analysis. Canadian Journal of Science, Mathematics, and Technology Education, 5(4), 469–483.
  • Weber, K. (2012). Mathematicians’ perspectives on their pedagogical practice with respect to proof. Int J Math Educ Sci Technol. 43(4), 463–482.
  • Yan, X., Mason, J., & Hanna, G. (2017). An exploratory teaching style in promoting the learning of proof. In CERME 10.
  • Yıldırım, A. ve Şimşek, H. (2008). Nitel Araştırma Yöntemleri. (7. Baskı). Ankara: Seçkin Yayıncılık.
  • Zeybek Şimşek, Z. (2020).İspatın matematik öğretim programları ve starndartlardaki yeri ve önemi. In. I. Uğurel (Ed.), Matematiksel İspat Ve Öğretimi. Okul Yıllarında İspat Öğretimini Destekleyen Çok Yönlü Bir Bakış, 69-88.Ankara: Anı Yayıncılık.
Year 2022, Volume: 16 Issue: 1, 189 - 226, 30.06.2022
https://doi.org/10.17522/balikesirnef.1081825

Abstract

References

  • Almeida, D. (2000). A survey of mathematics undergraduates’ interaction with proof: Some implications for mathematics education. International Journal of Mathematical Education in Science and Technology, 31(6), 869-890.
  • Arslantaş İlter, E. (2020). Matematik öğretmenlerinin sözsüz ispat becerilerinin pisagor teoremi bağlamında incelenmesi. Yayınlanmamış Yüksek Lisans Tezi, Sivas Cumhuriyet Üniversitesi, Sivas.
  • Atwood, P. R., (2001) Learning to construct proof in a first course on mathematical proof. Doctoral Dissertation, Western Michigan University, Kalamazoo, Michigan.
  • Baki A. ve Kutluca T. (2009). Dokuzuncu sınıf matematik öğretim programında zorluk çekilen konuların belirlenmesi. e-Journal of New World of Science Academy, 4(2), 604-619.
  • Balacheff, N. (1988). Aspects of Proof Pupils‘ Practice of School Mathematics. Mathematics, Teachers and Children, Pimm, D. (ed.) Hodder ve Stoughton. 216-235. London.
  • Benkhalti, A., Selden, A., & Selden, J. A. (2017). Slowly Develops Self-Efficacy with Writing Proof Frameworks, but Her Initial Progress and Sense of Self-Efficacy Evaporates When He/she Encounters Unfamiliar Concepts: However, It Eventually Returns.
  • Branch, R. M. (2009). Instructional design: The ADDIE approach (Vol. 722). Springer Science & Business Media.
  • Brodahl, C., Larson, N., Wathne, U., & Bjørkestøl, K. (2020). Developing Further Support for In-Service Teachers' Implementation of a Reasoning-and-Proving Task and Their Identification of Students' Level of Mathematical Argumentation. Journal of the International Society for Teacher Education, 24(2), 73-87.
  • Brown, A.L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning Sciences, 2(2), 141–178.
  • Cihan, F. (2019). Matematik öğretmen adaylarının ispatla ilgili alan ve pedagojik alan bilgilerini geliştirmeye yönelik bir ders tasarımı (Unpublished doctoral dissertation). Marmara University, İstanbul.
  • Collins, A. (1992). Towards a design science of education. E. Scanlon ve T. O’Shea (Eds.), New directions in educational technology içinde (15–22). Berlin: Springer.
  • Çontay, E. G. ve Duatepe-Paksu, A. (2019). Ortaokul matematik öğretmeni adaylarının ispatın doğasına ilişkin görüşleri. Sınırsız Eğitim ve Araştırma Dergisi, 4(1), 64-89.
  • Dane, A. (2008). İlköğretim matematik 3. sınıf öğrencilerinin tanım, aksiyom ve teorem kavramlarını anlama düzeyleri. Kastamonu Eğitim Dergisi, 16(2), 495-506.
  • Dean, E. E., (1996). Teaching the proof process, a model for discovery learning. College Teaching, 44 (2), 52-55.
  • de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.
  • Dreyfus, T. (1999). Why Johnny can’t prove? Educational Studies in Mathematics,38, 85-109.
  • Doorman, M., Bakker, A., Drijvers, P., & Wıjaya, A. (2016). Design-Based Research In Mathematıcs Educatıon. In Sriwijaya University Learning and Education International Conference 2(1), 21-46.
  • Greenberg, M.J. (1993). Euclidean and non-Euclidean geometries: Development and history. New York: W.H. Freeman.
  • Hammack, R. (2013). Book of Proof, Privately published, downloadable from http://www.people.vcu.edu/~rhammack/BookOfProof.
  • Hanna, G., ve de Villiers, M. (2008). ICMI Study 19: Proof and proving in mathematics education. ZDM: International Journal on Mathematics Education, 40(2), 329–336.
  • Harel, G., Selden, A. & Selden, J., (2006). Advanced Mathematical Thinking. Handbook of Research on the Pschology of Mathematics Education: Past, Present and Future, Gutierrez, A. and Boero, P. (eds.), pp.147-172.
  • Harel, G., ve Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234-283). Providence, RI: American Mathematical Society.
  • Harel, G. & Sowder, L., (2007). Toward comprehensive perspectives on learning and teaching proof. Handbook of Research on Teaching and Learning Mathematics, Lester, F. (Ed.), Greenwich, CT: Information Age Publishing. 805-842.
  • Healy, L., ve Hoyles, C. (2007). Curriculum change and geometrical reasoning. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice, 81-116. Rotterdam, The Netherlands: Sense.
  • Houston, K. (2009). How to Think Like a Mathematician. (Mehmet Terziler ve Tahsin Oner, ¸Cev.). Cambridge: Cambridge University Press.
  • Karaoğlu, Ö. (2010). Matematik Öğretmen Adaylarının Anahtar Nokta ve Fikirlerle Desteklenmiş İspatları Yapabilme Performansları. Yayınlanmamış Yüksek Lisans Tezi, Gazi Üniversitesi, Ankara.
  • Knapp, J., (2005). Learning to prove in order to prove to learn. http://mathpost.asu.edu/~sjgm/issues/2005_spring/SJGM_knapp.pdf
  • Ko, Y. & Knuth, E., 2009, Undergraduate mathematics majors‘ writing performance producing proofs and counterexamples about continuous functions. Journal of Mathematical Behavior, 28, 68-77.
  • Mariotti, M.A.(2006). Proof and proving in mathematics education. A. Gutierrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education. Past, present and future (173-204). Rotterdam: Sense Publishers.
  • Miyazaki, M. (2000). Levels of proof in lower secondary school mathematics. Educational Studies in Mathematics, 41(1), 47-68.
  • 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: Author.
  • Pala, O., Aksoy, E., & Narli, S. (2021). Can the Proof Image Exist in the Absence of the Formal Proof?: Analyses of an Unsuccessful Proving Attempt. Online Submission, 15(1), 1-31.
  • Polat, K. ve Akgün, L. (2016). Ortaöğretim matematik öğretmeni adaylarının ispat kavramına ve ispat yapmanın zorluklarına yönelik görüşleri. The Journal of Academic Social Science Studies, 43, 423-438.
  • Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52(3), 319-325.
  • Sarı, M. (2011). Üniversite öğrencilerinin matematiksel kanıt ile ilgili güçlükleri ve kanıt öğretimi (Doktora Tezi), Hacettepe Üniversitesi Fen Bilimleri Enstitüsü, Ankara.
  • Sarı Uzun, M. ve Bülbül, A. (2013). Matematik öğretmen adaylarının kanıtlama becerilerini geliştirmeye yönelik bir öğretme deneyi. Eğitim ve Bilim, 38(169), 372-390.
  • Sarı Uzun, M. (2020). Öğrenenlerin ispat yapma davranışları/ispat şemaları. In. I. Uğurel (Ed.), Matematiksel İspat ve Öğretimi. Okul Yıllarında İspat Öğretimini Destekleyen Çok Yönlü Bir Bakış, 189-226.Ankara: Anı Yayıncılık.
  • Schabel, C. J., 2001, An Instructional Model To Improve Proof Writing In College Number Theory. Doctoral Dissertation,. Portland State University, USA.
  • Selden, A. ve Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36.
  • Selden, J. ve Selden, A. (2009). Teaching proving by coordinating aspects of proofs with students’ abilities. In D. A. Stylianou, M. L. Blanton & E.J. Knuth (Eds.), Teaching and learning proof across grades: A K-16 perspective, 339-354. New York/Washington, DC: Routledge/National Council of Teachers of Mathematics.
  • Selden, A., Selden, J., & Benkhalti, A. (2017). Proof frameworks: A way to get started. PRIMUS, 28(1), 31-45.
  • Selden, J., Selden, A. ve McKee, K. (2008). Improving advanced students proving abilities. Paper presented at 11th International Congress of Mathematical Education (ICME-11), Monterrey, Mexico.
  • Skott, J., Larsen, D. M., & Østergaard, C. H. (2020). Learning to teach to reason: Reasoning and proving in mathematics teacher education. In Professional Development and Knowledge of Mathematics Teachers (pp. 44-61). Routledge.
  • Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.
  • Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the task of proving. Journal of mathematics teacher education, 11(4), 307-332.
  • Stylianides, A. J. ve Stylianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72, 237-253.
  • Stylianides, A. J. (2016). Proving in the elementary mathematics classroom. Oxford University Press.
  • Velleman, D. J. (2008). Kanıt nasıl yapılır. Çev., Terziler, M ve Öner, T. Ankara: Nobel Yayıncılık.
  • Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101–119.
  • Weber, K.. (2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course. Journal of Mathematical Behavior, 23, 115–133.
  • Weber, K. (2005). A procedural route toward understanding aspects of proof: Case studies from real analysis. Canadian Journal of Science, Mathematics, and Technology Education, 5(4), 469–483.
  • Weber, K. (2012). Mathematicians’ perspectives on their pedagogical practice with respect to proof. Int J Math Educ Sci Technol. 43(4), 463–482.
  • Yan, X., Mason, J., & Hanna, G. (2017). An exploratory teaching style in promoting the learning of proof. In CERME 10.
  • Yıldırım, A. ve Şimşek, H. (2008). Nitel Araştırma Yöntemleri. (7. Baskı). Ankara: Seçkin Yayıncılık.
  • Zeybek Şimşek, Z. (2020).İspatın matematik öğretim programları ve starndartlardaki yeri ve önemi. In. I. Uğurel (Ed.), Matematiksel İspat Ve Öğretimi. Okul Yıllarında İspat Öğretimini Destekleyen Çok Yönlü Bir Bakış, 69-88.Ankara: Anı Yayıncılık.
There are 56 citations in total.

Details

Primary Language English
Journal Section Makaleler
Authors

Sema Er 0000-0001-9775-8915

Şenol Dost 0000-0002-5762-8056

Publication Date June 30, 2022
Submission Date March 2, 2022
Published in Issue Year 2022 Volume: 16 Issue: 1

Cite

APA Er, S., & Dost, Ş. (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. https://doi.org/10.17522/balikesirnef.1081825