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Ortaokul Öğrencileri Kanıt Yapabilir mi?

Year 2021, Volume: 11 Issue: 2, 852 - 880, 01.07.2021
https://doi.org/10.18039/ajesi.923938

Abstract

Öğrencilerin muhakeme becerilerinin gelişiminin desteklemesi için küçük yaşlardan itibaren matematiksel kanıtla tanışmaları ve bu süreçteki eylemleri gerçekleştirmeleri oldukça gerekli ve önemli olmakla birlikte bu durum “küçük yaştaki öğrenciler kanıt yapabilir mi?” sorusunu akla getirmektedir. Bu soru doğrultusunda bu çalışmanın amacı ortaokul öğrencilerinin verilen problem ve önermeleri kanıtlama sürecindeki muhakemelerini incelemektir. Bu amaçla 2019-2020 eğitim öğretim yılında, bir devlet ortaokulunda öğrenim gören yedinci sınıf öğrencileri katılımcı olarak seçilmiştir. Nitel araştırma yaklaşımının benimsendiği bu çalışmada veriler açık uçlu görevler aracılığıyla toplanmıştır. Çalışma bulguları görevleri anlamayan öğrencilerin görevi yanıtsız bıraktıklarını, tekrar yazdıklarını, görevdeki verileri eksik, hatalı kullandıklarını ya da verilerin dışına çıktıklarını, bazen de görevdeki önermenin öncülünü anlamadıklarını ortaya koymuştur. Görevleri anlayan öğrencilerin muhakemeleri incelendiğinde, doğrudan kanıt yapmayı gerektiren sayı görevlerinde genel olarak deneysel doğrulama yaptıkları belirlenmiştir. Doğrudan kanıt yapmayı gerektiren geometri görevlerinde ise çoğunlukla bir dörtgenin kapsadığı dörtgeni aksine örnek vererek hatalı gerekçelendirme, prototip çizim üzerinde sembolle gerekçelendirme ya da hatalı çizime dayalı gerekçelendirme yaptıkları, bazı öğrencilerin ise tümdengelimsel muhakeme yaptığı belirlenmiştir. Aksine örnek vererek kanıt yapmaları gereken görevlerde bazı öğrencilerin ön bilgi eksikliğinden kaynaklı mantıksal olmayan gerekçelendirmeler yaptıkları, bununla birlikte pek çok öğrencinin tümdengelimsel muhakeme yaptığı görülmüştür. Bu çalışmada genel olarak öğrencilerin kanıtlama sürecinde argüman üretmede zorlandıkları, çoğunlukla özel durumlar kullanarak genelleme yapma eğiliminde oldukları ve önceki öğrenmelerini dönüşüm yapmadan kullandıkları söylenebilir.

References

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  • Ball, D.L., Hoyles, C., Jahnke, H.N. & Movshovitz-Hadar, N. (2002). The teaching of proof. In L.I. Tatsien (Ed.), Proceedings of the International Congress of Mathematicians (pp. 907-922). Beijing: Higher Education Press. Retrieved June 15, 2019, from http://www.lettredelapreuve.org/OldPreuve/Newsletter/03Printemps/teaching_proof.pdf
  • Bayazit, N. (2009). Prospective mathematics teacher’s use of mathematical definitions in doing proof. Unpublished doctoral dissertation. Florida State University. Retrieved November 7, 2018, from https://diginole.lib.fsu.edu/islandora/object/fsu%3A253932
  • Blanton, M., Stylianou, D. & David, M. (2009). Understanding instructional scaffolding in classroom discourse on proof. In D. Stylianou, M. Blanton & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (1nd ed., pp. 290-306). Routledge.
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  • Flores, A. (2002). How do children know that what they learn in mathematics is true? Teaching Children Mathematics, 8(5), 269-274. Retrieved April 20, 2018, from https://dx.doi.org/10.5951/TCM.8.5.0269
  • Glesne, C. (2013). Nitel araştırmaya giriş. (A. Ersoy ve P. YALÇINOĞLU, Çev.; 2. Baskı). Anı Yayıncılık
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  • Hersh, R (2009). What ı would like my students to already know about proof. In D. Stylianou, M. Blanton & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (1nd ed.,pp. 17-21). Routledge.
  • Jeannotte, D. & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1-16. Retrieved April 8, 2020, from https://link.springer.com/article/10.1007/s10649-017-9761-8
  • Knuth, E.J. & Elliott, R.L. (1998). Characterizing students’ understanding of mathematical proof. The Mathematics Teacher, 91(8), 714-731. Retrieved July 1, 2019, from https://dx.doi.org/10.5951/MT.91.8.0714
  • Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1), 151-161. Retrieved January 5, 2019, from https://dx.doi.org/10.1023/A:1012793121648
  • Liu, P. (2003). Do teachers need to incorporate the history of mathematics in their teaching?. Mathematics Teacher, 96(6), 416–421. Retrieved June 19, 2018, from https://www.researchgate.net/publication/281223989
  • Mason, J., Burton, L. & Stacey, K. (2010). Thinking mathematically. (2nd ed.). Pearson Education Limited.
  • Milli Eğitim Bakanlığı. (2019). PISA 2018 Türkiye ön raporu. http://www.meb.gov.tr/meb_iys_dosyalar/2019_12/03105347_PISA_2018_Turkiye_On_Raporu.pdf. adresinden 01.09.2020 tarihinde erişilmiştir.
  • Mubark, M.M. (2011). Mathematical thinking: Teachers perceptions and students performance. Canadian Social Science, 7(5), 176-181. Retrieved March 18, 2018,from https://dx.doi.org/10.3968/J.css.1923669720110705.502
  • National Council of Teachers of Mathematics. (2000). Principles and Standards for school mathematics. NCTM
  • Perry, P., Camargo, L., Samper, C. & Echeverry, O.M. (2009). Assigning mathematics tasks versus providing pre-fabricated mathematics in order to support learning to prove. In F.L., Lin, F.J., Hsieh, G., Hanna, M. & De Villiers (Eds.) ICMI Study 19-Proof and Proving in Mathematics Education. Retrieved August 15, 2018,from https://www.researchgate.net/publication/282571052_Proceedings_of_the_ICMI_Study_19_Conference_Proof_and_Proving_in_Mathematics_Education_Volume_2.
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  • Sowder, L. & Harel, G., (1998). Types of students' justifications. The mathematics teacher, 91(8), 670-675. Retrieved January 20, 2019, from https://doi.org/10.5951/MT.91.8.0670.
  • Staples, M.E., Bartlo, J.& Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior, 31(4), 447–462. Retrieved January 15, 2018, from http://dx.doi.org/10.1016/j.jmathb.2012.07.001
  • Stefanowicz, A. (2014). Proofs and mathematical reasoning. University of Birmingham Mathematics Support Center. Retrieved March 5, 2018,from https://www.birmingham.ac.uk/Documents/college-eps/college/stem/Student-Summer-Education-Internships/Proof-and-Reasoning.pdf
  • Stylianides, A.J. (2007a). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321. Retrieved February 15, 2018,from https://psycnet.apa.org/record/2007-06531-004
  • Stylianides, A.J. (2007b). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1-20. Retrieved February 15, 2018,from https://dx.doi.org/10.1007/s10649-006-9038-0
  • Stylianides, G.J. & Stylianides, A.J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314-352. Retrieved April 18, 2018,from https://doi.org/10.5951/jresematheduc.40.3.0314
  • Şen, C. & Güler, G. (2015). Examination of middle school seventh graders’ proof skills and proof schemes. Universal Journal of Educational Research, 3(9), 617-631. Retrieved March 25, 2019,from https://dx.doi.org/10.13189/ujer.2015.030906
  • Tanisli, D. (2016). How do students prove their learning and teachers their teaching? Do teachers make a difference?. Eurasian Journal of Educational Research, 66, 47-70. Retrieved February 12 2019, from http://dx.doi.org/10.14689/ejer.2016.66.3.
  • Tanışlı, D. ve Yavuzsoy Köse, N. (2020). Etkinlikler yoluyla matematiksel muhakemenin desteklenmesi, Y. Dede, M. F. Doğan, F. Aslan Tutak (Eds.), Matematik Eğitiminde Etkinlikler ve Uygulamaları içinde (363-393). Pegem Akademi.
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Can Middle School Students Make Proof?

Year 2021, Volume: 11 Issue: 2, 852 - 880, 01.07.2021
https://doi.org/10.18039/ajesi.923938

Abstract

In order to support the development of reasoning skills, it is very necessary and important for students to meet mathematical proof from an early age and to carry out actions in this process, and this is why "can young students make proof?" brings to mind the question. In line with this question, the purpose of this study is to examine the reasoning of middle school students in the process of proving the given problems and propositions. For this purpose, in the 2019-2020 academic year, seventh-grade students studying at a state middle school were selected as participants. In this study, in which qualitative research approach was adopted, the data were collected through open-ended tasks. The study findings revealed that students who did not understand the tasks left the task unanswered, rewrote it, used the data in task incompletely, incorrectly or went outside the data and sometimes did not understand the premise of the proposition in the task. When the reasoning of the students who understood the tasks was examined, it was determined that they generally made empirical verification in number tasks that require direct proof. In geometry tasks that require direct proof, it was determined that they mostly made incorrect justification by giving an example as opposed to another quadrilateral covered by a quadrilateral, justification with symbols on the prototype drawing, or justification based on incorrect drawing, and some students make deductive reasoning. In tasks in which they need to give counterexamples, was observed that some students made non-logical justifications caused by a lack of prior knowledge in the tasks, while many students made deductive reasoning. In this study, it can be said that students in general have difficulty making arguments in the proving process, often tend to generalize using special situations and use their previous learnings without transformation.

References

  • Arslan, Ç. (2007). İlköğretim öğrencilerinde muhakeme etme ve ispatlama düşüncesinin gelişimi. [Doktora tezi, Uludağ Üniversitesi.] YÖK Ulusal Tez Merkezi. https://tez.yok.gov.tr/UlusalTezMerkezi/tezSorguSonucYeni.jsp adresinden 03.09.2019 tarihinde erişilmiştir.
  • Aylar, E. ve Şahiner, Y. (2016). Yedinci sınıf öğrencilerinin ispat becerileri ve tercihlerinin incelenmesi. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi, 17(3), 559-579. https://www.researchgate.net/publication/311953562 adresinden 01.05.2019 tarihinde erişilmiştir.
  • Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In. D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). Hodder and Stoughton.
  • Balacheff, N. (1991). The benefits and limits of social interaction: The case of mathematical proof. In A. Bishop, S. Mellin-Olsen & J. Van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (1nd ed., pp. 175–192). Kluwer.
  • Ball, D.L., Hoyles, C., Jahnke, H.N. & Movshovitz-Hadar, N. (2002). The teaching of proof. In L.I. Tatsien (Ed.), Proceedings of the International Congress of Mathematicians (pp. 907-922). Beijing: Higher Education Press. Retrieved June 15, 2019, from http://www.lettredelapreuve.org/OldPreuve/Newsletter/03Printemps/teaching_proof.pdf
  • Bayazit, N. (2009). Prospective mathematics teacher’s use of mathematical definitions in doing proof. Unpublished doctoral dissertation. Florida State University. Retrieved November 7, 2018, from https://diginole.lib.fsu.edu/islandora/object/fsu%3A253932
  • Blanton, M., Stylianou, D. & David, M. (2009). Understanding instructional scaffolding in classroom discourse on proof. In D. Stylianou, M. Blanton & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (1nd ed., pp. 290-306). Routledge.
  • Bogdan, R.C. & Biklen, S.K. (1992). Qualitative research for education: Introduction and methods. (2nd ed.). Allyn and Bacon.
  • Brodie, K. (2010). Teaching mathematical reasoning in secondary school classrooms. Springer Science and Business Media.
  • Csíkos, C.A. (1999). Measuring students’ proving ability by means of Harel and Sowder’s proof- categorization. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 233–240). PME Retrieved June 10, 2020, from, http://www.igpme.org/wp-content/uploads/2019/05/PME23-1999-Haifa.pdf
  • Dawson, J. (2006). Why do mathematicians re-prove theorems? Philosophia Mathematica, 14(3), 269-286. Retrieved November 10, 2019, from https://doi.org/10.1093/philmat/nkl009
  • Derek, M. (2011). Teaching and learning of proof in the college curriculum. [Master’s thesis, San Jose State University]. Retrieved November 5, 2018, from https://doi.org/10.31979/etd.ytyx-fs6x
  • De Villiers, M. (2003). Rethinking proof with the geometer’s sketchpad. (1nd ed.). Key Curriculum Press.
  • De Villiers, M. (1994). The role and function of a hierarchical classification of quadrilaterals. For the learning of mathematics, 14(1), 11-18. Retrieved September 05, 2020, from https://flm-journal.org/Articles/58360C6934555B2AC78983AE5FE21.pdf
  • De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24(24), 17–24. Retrieved June 6, 2018, from https://www.researchgate.net/publication/264784642_The_Role_and_Function_of_Proof_in_Mathematics
  • Epp, S. (1998). A unified framework for proof and disproof. Mathematics Teacher, 91(8), 708-713. Retrieved April 28, 2018, from https://condor.depaul.edu/~sepp/MathTeacher.Nov98.pdf
  • Flores, A. (2006). How do students know what they learn in middle school mathematics is true?, School Science and Mathematics, 106(3), 124-132. Retrieved April 12, 2019, from https://dx.doi.org/10.1111/j.1949-8594.2006.tb18169.x
  • Flores, A. (2002). How do children know that what they learn in mathematics is true? Teaching Children Mathematics, 8(5), 269-274. Retrieved April 20, 2018, from https://dx.doi.org/10.5951/TCM.8.5.0269
  • Glesne, C. (2013). Nitel araştırmaya giriş. (A. Ersoy ve P. YALÇINOĞLU, Çev.; 2. Baskı). Anı Yayıncılık
  • Hanna, G. (1990). Some Pedagogical Aspects of Proof. Interchange, 21(1), 6-13. Retrieved May 8, 2018, from https://dx.doi.org/10.1007/BF01809605
  • Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44 (1-2), 5–23. Retrieved May 8, 2018, from https://dx.doi.org/10.1023/A:1012737223465
  • Hersh, R (2009). What ı would like my students to already know about proof. In D. Stylianou, M. Blanton & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (1nd ed.,pp. 17-21). Routledge.
  • Jeannotte, D. & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1-16. Retrieved April 8, 2020, from https://link.springer.com/article/10.1007/s10649-017-9761-8
  • Knuth, E.J. & Elliott, R.L. (1998). Characterizing students’ understanding of mathematical proof. The Mathematics Teacher, 91(8), 714-731. Retrieved July 1, 2019, from https://dx.doi.org/10.5951/MT.91.8.0714
  • Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1), 151-161. Retrieved January 5, 2019, from https://dx.doi.org/10.1023/A:1012793121648
  • Liu, P. (2003). Do teachers need to incorporate the history of mathematics in their teaching?. Mathematics Teacher, 96(6), 416–421. Retrieved June 19, 2018, from https://www.researchgate.net/publication/281223989
  • Mason, J., Burton, L. & Stacey, K. (2010). Thinking mathematically. (2nd ed.). Pearson Education Limited.
  • Milli Eğitim Bakanlığı. (2019). PISA 2018 Türkiye ön raporu. http://www.meb.gov.tr/meb_iys_dosyalar/2019_12/03105347_PISA_2018_Turkiye_On_Raporu.pdf. adresinden 01.09.2020 tarihinde erişilmiştir.
  • Mubark, M.M. (2011). Mathematical thinking: Teachers perceptions and students performance. Canadian Social Science, 7(5), 176-181. Retrieved March 18, 2018,from https://dx.doi.org/10.3968/J.css.1923669720110705.502
  • National Council of Teachers of Mathematics. (2000). Principles and Standards for school mathematics. NCTM
  • Perry, P., Camargo, L., Samper, C. & Echeverry, O.M. (2009). Assigning mathematics tasks versus providing pre-fabricated mathematics in order to support learning to prove. In F.L., Lin, F.J., Hsieh, G., Hanna, M. & De Villiers (Eds.) ICMI Study 19-Proof and Proving in Mathematics Education. Retrieved August 15, 2018,from https://www.researchgate.net/publication/282571052_Proceedings_of_the_ICMI_Study_19_Conference_Proof_and_Proving_in_Mathematics_Education_Volume_2.
  • Rossi, R. J. (2006). Theorems, corollaries, lemmas and methods of proof. (1nd ed.). Wiley Interscience.
  • Sowder, L. & Harel, G., (1998). Types of students' justifications. The mathematics teacher, 91(8), 670-675. Retrieved January 20, 2019, from https://doi.org/10.5951/MT.91.8.0670.
  • Staples, M.E., Bartlo, J.& Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The Journal of Mathematical Behavior, 31(4), 447–462. Retrieved January 15, 2018, from http://dx.doi.org/10.1016/j.jmathb.2012.07.001
  • Stefanowicz, A. (2014). Proofs and mathematical reasoning. University of Birmingham Mathematics Support Center. Retrieved March 5, 2018,from https://www.birmingham.ac.uk/Documents/college-eps/college/stem/Student-Summer-Education-Internships/Proof-and-Reasoning.pdf
  • Stylianides, A.J. (2007a). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321. Retrieved February 15, 2018,from https://psycnet.apa.org/record/2007-06531-004
  • Stylianides, A.J. (2007b). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1-20. Retrieved February 15, 2018,from https://dx.doi.org/10.1007/s10649-006-9038-0
  • Stylianides, G.J. & Stylianides, A.J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314-352. Retrieved April 18, 2018,from https://doi.org/10.5951/jresematheduc.40.3.0314
  • Şen, C. & Güler, G. (2015). Examination of middle school seventh graders’ proof skills and proof schemes. Universal Journal of Educational Research, 3(9), 617-631. Retrieved March 25, 2019,from https://dx.doi.org/10.13189/ujer.2015.030906
  • Tanisli, D. (2016). How do students prove their learning and teachers their teaching? Do teachers make a difference?. Eurasian Journal of Educational Research, 66, 47-70. Retrieved February 12 2019, from http://dx.doi.org/10.14689/ejer.2016.66.3.
  • Tanışlı, D. ve Yavuzsoy Köse, N. (2020). Etkinlikler yoluyla matematiksel muhakemenin desteklenmesi, Y. Dede, M. F. Doğan, F. Aslan Tutak (Eds.), Matematik Eğitiminde Etkinlikler ve Uygulamaları içinde (363-393). Pegem Akademi.
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There are 50 citations in total.

Details

Primary Language Turkish
Journal Section Research Article
Authors

Tuğba Yulet Yılmaz 0000-0003-2872-4062

Nilüfer Köse 0000-0001-7407-7498

Publication Date July 1, 2021
Submission Date April 21, 2021
Published in Issue Year 2021 Volume: 11 Issue: 2

Cite

APA Yılmaz, T. Y., & Köse, N. (2021). Ortaokul Öğrencileri Kanıt Yapabilir mi?. Anadolu Journal of Educational Sciences International, 11(2), 852-880. https://doi.org/10.18039/ajesi.923938
AMA Yılmaz TY, Köse N. Ortaokul Öğrencileri Kanıt Yapabilir mi?. AJESI. July 2021;11(2):852-880. doi:10.18039/ajesi.923938
Chicago Yılmaz, Tuğba Yulet, and Nilüfer Köse. “Ortaokul Öğrencileri Kanıt Yapabilir Mi?”. Anadolu Journal of Educational Sciences International 11, no. 2 (July 2021): 852-80. https://doi.org/10.18039/ajesi.923938.
EndNote Yılmaz TY, Köse N (July 1, 2021) Ortaokul Öğrencileri Kanıt Yapabilir mi?. Anadolu Journal of Educational Sciences International 11 2 852–880.
IEEE T. Y. Yılmaz and N. Köse, “Ortaokul Öğrencileri Kanıt Yapabilir mi?”, AJESI, vol. 11, no. 2, pp. 852–880, 2021, doi: 10.18039/ajesi.923938.
ISNAD Yılmaz, Tuğba Yulet - Köse, Nilüfer. “Ortaokul Öğrencileri Kanıt Yapabilir Mi?”. Anadolu Journal of Educational Sciences International 11/2 (July 2021), 852-880. https://doi.org/10.18039/ajesi.923938.
JAMA Yılmaz TY, Köse N. Ortaokul Öğrencileri Kanıt Yapabilir mi?. AJESI. 2021;11:852–880.
MLA Yılmaz, Tuğba Yulet and Nilüfer Köse. “Ortaokul Öğrencileri Kanıt Yapabilir Mi?”. Anadolu Journal of Educational Sciences International, vol. 11, no. 2, 2021, pp. 852-80, doi:10.18039/ajesi.923938.
Vancouver Yılmaz TY, Köse N. Ortaokul Öğrencileri Kanıt Yapabilir mi?. AJESI. 2021;11(2):852-80.