Research Article
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Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’

Year 2019, Volume: 9 Issue: 1, 21 - 39, 06.05.2019

Abstract

The importance of both proof and
visualization has been frequently emphasized in mathematics education. Visual
proof or nonverbal proofs are defined as diagrams or illustrations that help us
to see why a mathematical expression is correct and how to begin to prove the
accuracy of this statement. The aim of this research is to examine non-verbal
proof skills of preservice mathematics teachers’. The study was carried out
with case studies from qualitative research designs. The participants of the
study consisted of 53 preservice mathematics teachers in a state university in
Central Anatolia. The data were collected with a sample of 3 non-verbal proof
samples directed to preservice teachers. The analysis of the data was made by
classifying the replies of the pre-service teachers according to their
similarities and differences. The findings showed that preservice teachers
generally associate images with geometric figures. In addition, it was also
seen that those who saw the visual relationship between the given visual and
mathematical expression used to show that that the expression is correct
instead of proofing the visual.

References

  • Altıparmak, K. & Öziş, T. (2005). Matematiksel ispat ve matematiksel muhakemenin gelişimi üzerine bir inceleme. Ege Eğitim Dergisi, (6) 1, 25–37.
  • Alsina, C. & Nelsen R. (2010). An invitation to proofs without words. European Journal of Pure and Applied Mathematics, 3(1), 118-127.
  • 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 907–920). Beijing: Higher Education.
  • Bardelle, C. (2009). Visual proofs: an experiment. In V. Durand-Guerrier et al (Eds), Paper presented at the annual meeting of CERME6. (pp. 251-260) Lyon, France. INRP.
  • Bell, C. (2011). Proofs without words: A visual application of reasoning and proof. Mathematics Teacher, 104(1), 690-695.
  • Britz, T., Mammoliti, A. & Sørensen, H. K. (2014). Proof by picture: A selection of nice picture proofs. Parabola, 50(3), 1-8.
  • Casselman, B. (2000). Pictures and proofs. Notices of the American Mathematical Monthly, 47(10), 1257-1266.
  • Demircioğlu, H. & Polat, K. (2016). Ortaöğretim matematik öğretmeni adaylarının “sözsüz ispatlar” ile yaşadıkları zorluklar hakkındaki görüşleri. Uluslararası Türk Eğitim Bilimleri Dergisi, Ekim, 7, 81-99.
  • Demircioğlu, H. & Polat, K. (2015) Ortaöğretim matematik öğretmeni adaylarının “sözsüz ispatlar” yöntemine yönelik görüşleri. The Journal of Academic Social Science Studies, 41, Winter II, 233-254, DOI: http://dx.doi.org/10.9761/JASSS3171
  • Doruk, M. & Güler, G. (2014). İlköğretim matematik öğretmeni adaylarının matematiksel ispata yönelik görüşleri. Uluslararası Türk Eğitim Bilimleri Dergisi, Ekim,71-93.
  • Doruk, B., Kıymaz, Y. & Horzum, T. (2012). İspat yapma ve ispatta somut modellerden yararlanma üzerine sınıf öğretmeni adaylarının görüşleri. X. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi, Niğde
  • Foo, N., Pagnucco, M. & Nayak, A.(1999) Diagrammatic Proofs, Proceedings of the16th International Joint Conference on Artificial Intelligence, (IJCAI- 99). (pp. 378 – 383).Morgan Kaufmanpp.
  • Gierdien, F. (2007). From “Proofs without words” to “Proofs that explain” in secondary mathematics. Pythagoras, 65, 53 – 62.
  • Hammack, R. H. & Lyons, D. W. (2006). Alternating series convergence: a visual proof. Teaching Mathematics and its Applications, 25(2), 58.
  • Heinze, A. & Reiss, K. (2004). The teaching of proof at lower secondary level – a video study. ZDM Mathematics Education, 36(3), 98–104
  • Iannone, P. & Inglis, M. (2011). Undergraduate students' use of deductive arguments to solve 'prove that . . . ' tasks. In E. Swoboda (Hrsg.), Proceedings of the 7thCongress of the European Society for Research in Mathematics Education. (pp.2012-2022).
  • 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. (2002a). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.
  • Knuth, E. J. (2002b). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal for Mathematics Teacher Education, 5, 61–88.
  • Kulpa, Z. (2009). Main problems of diagrammatic reasoning. Part I: The generalization problem”, Foundations of Science 14: 75–96.
  • Miller R. L. (2012). On Proofs without Words, Retrieved from: http://www.whitman.edu/mathematics/SeniorProjectArchive/2012/Miller.pdf
  • Miyazaki M. (2000) Levels of proof in lower secondary school mathematics. Educational Studies in Mathematics, 41(1), 47-68.
  • Nelsen. R. (1993). Proofs without Words: Exercises in Visual Thinking. Washington: Mathematical Association of America.
  • Nelsen. R (2000). Proofs without Words II: More Exercises in Visual Thinking. Washington: Mathematical Association of America.
  • Özdemir, M. (2010). Nitel Veri analizi: sosyal bilimlerde yöntembilim sorunsalı üzerine bir çalışma.Eskişehir Osmangazi Üniversitesi Sosyal Bilimler Dergisi, 11(1), 323- 343.
  • Polat,K, & Demircioğlu, H.(2016).Matematik eğitiminde sözsüz ispatlar: kuramsal bir çalışma. Ziya Gökalp Eğitim Fakültesi Dergisi, Eylül, 28, 129-140 DOI: http://dx.doi.org/10.14582/DUZGEF.686
  • Pease, A. Colton, S. Ramezani, R. Smaill, A. & Guhe, M. (2010) Using analogical representations for mathematical concept formation. Model-Based Reasoning in Science and Technology. pp 301-314|
  • Rösken, B. & Rolka, K. (2006). A picture is worth a 1000 words - the role of visualization in mathematics learning. In J. Novotná, H. Moraová, M. Krátká &N.Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 441-448). Prague, Czech Republic: PME.
  • Štrausová, I., & Hašek, R. (2013) Dynamic visual proofs using DGS. The Electronic Journal of Mathematics and Technology, 7 (1),130-142
  • Stucky, B.(2015). Another Visual Proof of Nicomachus' Theorem. Retrieved from: http://bstucky.com/bstucky_files/nico.pdf
  • Thornton, S. (2001). A picture is worth a thousand words. New ideas in mathematics education. Proceedings of the International Conference of the Mathematics Education into the 21st Century Project
  • Uğurel, I., Moralı, H. S. & Karahan, Ö. (2011). Matematikte Yetenekli Olan Ortaöğretim Öğrencilerin Sözsüz İspat Oluşturma Yaklaşımları, I. Uluslararası Eğitim Programları ve Öğretimi Kongresi, 5-8 Ekim, Eskişehir.
  • Yıldırım, A. & Şimşek, H. (2005). Sosyal bilimlerde nitel araştırma yöntemleri (5. bs.). Ankara: Seçkin Yayıncılık.
  • Yin, R. K. (2003). Case study research design and methods (3rd ed.). Thousand Oaks: Sage Publications.

Investigation of Nonverbal Proof Skills of Preservice Mathematics Teachers’: A Case Study / Matematik Öğretmen Adaylarının Sözsüz İspat Becerilerinin İncelenmesi: Bir Durum Çalışması

Year 2019, Volume: 9 Issue: 1, 21 - 39, 06.05.2019

Abstract

The importance of both proof and visualization
has been frequently emphasized in mathematics education. Visual proof or
nonverbal proofs are defined as diagrams or illustrations that help us to see
why a mathematical expression is correct, and how to begin to prove the
accuracy of this statement. The aim of this research is to examine nonverbal
proof skills of preservice mathematics teachers. The study was carried out with
case studies, one of the qualitative research designs. The participants of the
study consisted of 53 preservice mathematics teachers at a state university in
Central Anatolia, Turkey. The data were collected with a sample of three
nonverbal proof samples directed to preservice teachers. The analysis of the
data classified the preservice teachers’ responses according to their
similarities and differences. The findings showed that preservice teachers
generally associate images with geometric figures. In addition, it was also
seen that those who saw the visual relationship between the given visual and
mathematical expression used it to show the expression as correct instead of
proofing the visual.

References

  • Altıparmak, K. & Öziş, T. (2005). Matematiksel ispat ve matematiksel muhakemenin gelişimi üzerine bir inceleme. Ege Eğitim Dergisi, (6) 1, 25–37.
  • Alsina, C. & Nelsen R. (2010). An invitation to proofs without words. European Journal of Pure and Applied Mathematics, 3(1), 118-127.
  • 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 907–920). Beijing: Higher Education.
  • Bardelle, C. (2009). Visual proofs: an experiment. In V. Durand-Guerrier et al (Eds), Paper presented at the annual meeting of CERME6. (pp. 251-260) Lyon, France. INRP.
  • Bell, C. (2011). Proofs without words: A visual application of reasoning and proof. Mathematics Teacher, 104(1), 690-695.
  • Britz, T., Mammoliti, A. & Sørensen, H. K. (2014). Proof by picture: A selection of nice picture proofs. Parabola, 50(3), 1-8.
  • Casselman, B. (2000). Pictures and proofs. Notices of the American Mathematical Monthly, 47(10), 1257-1266.
  • Demircioğlu, H. & Polat, K. (2016). Ortaöğretim matematik öğretmeni adaylarının “sözsüz ispatlar” ile yaşadıkları zorluklar hakkındaki görüşleri. Uluslararası Türk Eğitim Bilimleri Dergisi, Ekim, 7, 81-99.
  • Demircioğlu, H. & Polat, K. (2015) Ortaöğretim matematik öğretmeni adaylarının “sözsüz ispatlar” yöntemine yönelik görüşleri. The Journal of Academic Social Science Studies, 41, Winter II, 233-254, DOI: http://dx.doi.org/10.9761/JASSS3171
  • Doruk, M. & Güler, G. (2014). İlköğretim matematik öğretmeni adaylarının matematiksel ispata yönelik görüşleri. Uluslararası Türk Eğitim Bilimleri Dergisi, Ekim,71-93.
  • Doruk, B., Kıymaz, Y. & Horzum, T. (2012). İspat yapma ve ispatta somut modellerden yararlanma üzerine sınıf öğretmeni adaylarının görüşleri. X. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi, Niğde
  • Foo, N., Pagnucco, M. & Nayak, A.(1999) Diagrammatic Proofs, Proceedings of the16th International Joint Conference on Artificial Intelligence, (IJCAI- 99). (pp. 378 – 383).Morgan Kaufmanpp.
  • Gierdien, F. (2007). From “Proofs without words” to “Proofs that explain” in secondary mathematics. Pythagoras, 65, 53 – 62.
  • Hammack, R. H. & Lyons, D. W. (2006). Alternating series convergence: a visual proof. Teaching Mathematics and its Applications, 25(2), 58.
  • Heinze, A. & Reiss, K. (2004). The teaching of proof at lower secondary level – a video study. ZDM Mathematics Education, 36(3), 98–104
  • Iannone, P. & Inglis, M. (2011). Undergraduate students' use of deductive arguments to solve 'prove that . . . ' tasks. In E. Swoboda (Hrsg.), Proceedings of the 7thCongress of the European Society for Research in Mathematics Education. (pp.2012-2022).
  • 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. (2002a). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.
  • Knuth, E. J. (2002b). Teachers’ conceptions of proof in the context of secondary school mathematics. Journal for Mathematics Teacher Education, 5, 61–88.
  • Kulpa, Z. (2009). Main problems of diagrammatic reasoning. Part I: The generalization problem”, Foundations of Science 14: 75–96.
  • Miller R. L. (2012). On Proofs without Words, Retrieved from: http://www.whitman.edu/mathematics/SeniorProjectArchive/2012/Miller.pdf
  • Miyazaki M. (2000) Levels of proof in lower secondary school mathematics. Educational Studies in Mathematics, 41(1), 47-68.
  • Nelsen. R. (1993). Proofs without Words: Exercises in Visual Thinking. Washington: Mathematical Association of America.
  • Nelsen. R (2000). Proofs without Words II: More Exercises in Visual Thinking. Washington: Mathematical Association of America.
  • Özdemir, M. (2010). Nitel Veri analizi: sosyal bilimlerde yöntembilim sorunsalı üzerine bir çalışma.Eskişehir Osmangazi Üniversitesi Sosyal Bilimler Dergisi, 11(1), 323- 343.
  • Polat,K, & Demircioğlu, H.(2016).Matematik eğitiminde sözsüz ispatlar: kuramsal bir çalışma. Ziya Gökalp Eğitim Fakültesi Dergisi, Eylül, 28, 129-140 DOI: http://dx.doi.org/10.14582/DUZGEF.686
  • Pease, A. Colton, S. Ramezani, R. Smaill, A. & Guhe, M. (2010) Using analogical representations for mathematical concept formation. Model-Based Reasoning in Science and Technology. pp 301-314|
  • Rösken, B. & Rolka, K. (2006). A picture is worth a 1000 words - the role of visualization in mathematics learning. In J. Novotná, H. Moraová, M. Krátká &N.Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 441-448). Prague, Czech Republic: PME.
  • Štrausová, I., & Hašek, R. (2013) Dynamic visual proofs using DGS. The Electronic Journal of Mathematics and Technology, 7 (1),130-142
  • Stucky, B.(2015). Another Visual Proof of Nicomachus' Theorem. Retrieved from: http://bstucky.com/bstucky_files/nico.pdf
  • Thornton, S. (2001). A picture is worth a thousand words. New ideas in mathematics education. Proceedings of the International Conference of the Mathematics Education into the 21st Century Project
  • Uğurel, I., Moralı, H. S. & Karahan, Ö. (2011). Matematikte Yetenekli Olan Ortaöğretim Öğrencilerin Sözsüz İspat Oluşturma Yaklaşımları, I. Uluslararası Eğitim Programları ve Öğretimi Kongresi, 5-8 Ekim, Eskişehir.
  • Yıldırım, A. & Şimşek, H. (2005). Sosyal bilimlerde nitel araştırma yöntemleri (5. bs.). Ankara: Seçkin Yayıncılık.
  • Yin, R. K. (2003). Case study research design and methods (3rd ed.). Thousand Oaks: Sage Publications.
There are 34 citations in total.

Details

Primary Language English
Subjects Studies on Education
Journal Section Research Article
Authors

Handan Demircioğlu

Publication Date May 6, 2019
Published in Issue Year 2019 Volume: 9 Issue: 1

Cite

APA Demircioğlu, H. (2019). Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’. Eğitim Bilimleri Araştırmaları Dergisi, 9(1), 21-39.
AMA Demircioğlu H. Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’. EBAD - JESR. May 2019;9(1):21-39.
Chicago Demircioğlu, Handan. “Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’”. Eğitim Bilimleri Araştırmaları Dergisi 9, no. 1 (May 2019): 21-39.
EndNote Demircioğlu H (May 1, 2019) Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’. Eğitim Bilimleri Araştırmaları Dergisi 9 1 21–39.
IEEE H. Demircioğlu, “Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’”, EBAD - JESR, vol. 9, no. 1, pp. 21–39, 2019.
ISNAD Demircioğlu, Handan. “Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’”. Eğitim Bilimleri Araştırmaları Dergisi 9/1 (May 2019), 21-39.
JAMA Demircioğlu H. Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’. EBAD - JESR. 2019;9:21–39.
MLA Demircioğlu, Handan. “Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’”. Eğitim Bilimleri Araştırmaları Dergisi, vol. 9, no. 1, 2019, pp. 21-39.
Vancouver Demircioğlu H. Investigation of Non-Verbal Proof Skills of Preservice Mathematics Teachers’. EBAD - JESR. 2019;9(1):21-39.