Opportunities to Learn Reasoning and Proof in Eighth-Grade Mathematics Textbook

Year 2019, Volume: 20 Issue: 2, 601 - 618, 31.08.2019

Muhammed Fatih Dogan

https://doi.org/10.17679/inuefd.527243

Cited By: 4

Abstract

In this study,
reasoning and proof activities in the eighth-grade mathematics textbook were investigated.
The reasoning and proof activities of the entire textbook were evaluated
separately based on the different learning areas, sections of the book, and
purpose of the activities. According to the results, these activities
constituted 7.7% of all activities in the book. The proportion of the
reasoning and proof activities differed by learning area, with 11.8% of the
total activities in Numbers and Operations, 7.8% in Probability, 7.4% in
Geometry and Measurement, and 5.3% in Algebra. In the learning area of
Statistics, no such content was found. When the sections of the book were
evaluated, most of the proof-related activities were observed in the Warning
section (55%); followed by 38% in the Information and Activity sections and
29% in the Warming-up section. This ratio was only 2.9% in the Examples
section, and there was no proof related content found in the Exercises
section. In examining the purpose of proof activities, it was mostly used for
investigating claims/conjectures (49 tasks), and making claims/conjectures (20
tasks). There were only 8 activities for evaluating an argument, but none for
producing arguments. The results conclude that reasoning and proof were not
sufficiently evident in the textbook; therefore, the engagement of students with
such activities may be limited.

Keywords

Reasoning and Proof, Textbook Analysis, Curriculum

Sekizinci Sınıf Matematik Ders Kitabındaki Matematiksel Akıl Yürütme ve İspatı Öğrenme Olanakları

Abstract

Bu çalışmada 8. Sınıf matematik ders kitabında
matematiksel akıl yürütme ve ispat etkinliklerinin ne oranda ve nasıl yer
aldığı araştırılmıştır. Bunun için okullarda yaygın olarak okutulan bir ders
kitabının tüm içeriği incelenmiştir. Bu incelemede akıl yürütme ve ispat
etkinlikleri öğrenme alanlarına, kitabın bölümlerine ve etkinliklerin amacına
göre ayrı ayrı değerlendirilmiştir. Elde edilen sonuçlara göre bu etkinlikler
kitapta yer alan tüm etkinliklerin % 7,7’sini oluşturmaktadır. İspat ile
ilgili etkinliklerin öğrenme alanlarına göre oranlarının Sayılar ve İşlemlerde
% 11,8; Olasılıkta %7,8; Geometri ve Ölçmede % 7,4 ve Cebirde % 5,3 olduğu
tespit edilmiştir. Veri İşleme öğrenme alanında ise ispat ile ilgili bir
içerik bulunamamıştır. Kitabın bölümlerine göre değerlendirildiğinde ispat
etkinliklerinin en çok Uyarı kısmında (% 55) yer aldığı; Bilgi ve Etkinlik kısımlarında
% 38, Hazırlık Çalışması kısmında ise % 29 oranında ispata değinildiği
görülmüştür. Kitabın Örnekler kısmında ispat etkinliklerine % 2,9 oranında yer
verilirken Alıştırmalar kısmında ispatla ilgili herhangi bir kavrama
ulaşılamamıştır. İspat etkinlikleri amacına göre incelendiğinde ise çoğunlukla
varsayımları araştırma amacıyla (49 etkinlik) sunulabileceği görülmüş; kitapta
varsayımda bulunma amacıyla sunulabilecek 20 etkinlik; bir argümanı
değerlendirmeye yönelik de 8 etkinlik mevcut olduğu belirlenmiştir. Öte
yandan, kitapta argüman oluşturma amacına uygun herhangi bir etkinliğe
rastlanmamıştır. Bu araştırma ders kitabında akıl yürütme ve ispata yeterli
düzeyde yer verilmediğini; dolayısıyla öğrencilerin bu tür etkinliklerle
etkileşimlerinin sınırlı kalabileceğini göstermektedir.

Keywords

Akıl Yürütme Ve İspat, Ders Kitabı Analizi, Öğretim Programı

References

• Alibert, D. (1988). Toward new customs in the classrooms. For the Learning of Mathematics, 8(2), 31-43.
• Alibert, D., & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.) Advanced Mathematical Thinking (pp. 215-230). Kluwer: The Netherlands.
• Altun, M., Arslan, Ç., & Yazgan, Y. (2004). Lise matematik ders kitaplarının kullanım şekli ve sıklığı üzerine bir çalışma. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 17(2), 131-147.
• Balacheff, N. (1991). The benefits and limits of social interaction: The case of mathematical proof. In Mathematical knowledge: Its growth through teaching (pp. 173-192). Springer, Dordrecht.
• Begle, E. (1973). Some lessons learned by SMSG. Mathematics Teacher, 66, 207–214. Bell, A. (1976). A study of pupils’ proof – explanations in mathematical situations. Educational Studies in Mathematics, 7, 23-40.
• Bieda (2010). Enacting proof-related tasks in middle school mathematics: challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351-382.
• Bieda, K. N., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. International Journal of Educational Research, 64, 71–80.
• Cai, J., & Cirillo, M. (2014). What do we know about reasoning and proving? Opportunities and missing opportunities from curriculum analyses. International Journal of Educational Research, 64, 132–140.
• Cai, J., Ni, Y., & Lester, F. K. (2011). Curricular effect on the teaching and learning of mathematics: Findings from two longitudinal studies in China and the United States. International Journal of Educational Research, 50, 63–64.
• Chazan, D., & Lueke, H. M. (2009). Exploring tensions between disciplinary knowledge and school mathematics: Implications for reasoning and proof in school mathematics. Teaching and learning mathematics proof across the grades, 21-39.
• Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Council of Chief State School Officers.
• Dede, Y.& Karakuş, F. (2014). Matematiksel ispat kavramına pedagojik bir bakış: Kuramsal bir çalışma. Adıyaman Üniversitesi Eğitim Bilimleri Dergisi, 4(2), 47-71.
• Department for Education. (2013). Mathematics: Programmes of study: Key Stages 1-2 (National Curriculum in England). Retrieved, 2018, from https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/239129/PRIMARY_nati onal_curriculum_-_Mathematics.pdf
• Dole, S., & Shield, M. J. (2008). The capacity of two Australian eighth-grade textbooks for promoting proportional reasoning. Research in Mathematics Education, 10(1), 19–35.
• Epp, S. S. (1998). A unified framework for proof and disproof. Mathematics Teacher, 91, 708–713. http:// www.nctm.org/publications/mt.aspx
• Eisenmann, T., & Even, R. (2011). Enacted types of algebraic activity in different classes taught by the same teacher. International Journal of Science and Mathematics Education, 9(4), 867–891.
• Fan, L., & Kaeley, G. S. (2000). The influence of textbooks on teaching strategies: An empirical study. Mid-Western Educational Researcher, 13(4), 2–9.
• Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: development status and directions. ZDM, 45(5), 633-646.
• Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2011). How to design and evaluate research in education. New York: McGraw-Hill Humanities/Social Sciences/Languages.
• Fujita, T., & Jones, K. (2014). Reasoning-and-proving in geometry in school mathematics textbooks in Japan. International Journal of Educational Research, 64, 81–91.
• Grouws, D. A., Smith, M. S., & Sztajn, P. (2004). The preparation and teaching practices of United States mathematics teachers: Grades 4 and 8. In P. Kloosterman & F. K. Lester Jr. (Eds.), Results and interpretations of the 1990–2000 mathematics assessments of the National Assessment of Educational Progress (pp. 221–267). Reston, VA: National Council of Teachers of Mathematics. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13.
• Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23.
• Hanna, G. (2018). Reflections on proof as explanation. In Advances in Mathematics Education Research on Proof and Proving (pp. 3-18). Springer, Cham.
• Hanna, G., & de Bruyn, Y. (1999). Opportunity to learn proof in Ontario grade twelve mathematics texts. Ontario Mathematics Gazette, 37(4), 23–29.
• Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education, III (pp. 234-283). Washington DC: Mathematical Association of America.
• Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. Second handbook of research on mathematics teaching and learning, 2, 805-842.
• Healy L. & Hoyles C., (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428.
• Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students?. Cognition and Instruction, 24(1), 73-122.
• Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399.
• Isler, I. (2015). An Investigation of Elementary Teachers’ Proving Eyes and Ears (Unpublished Doctoral dissertation, The University of Wisconsin-Madison).
• Jones, D. L., & Tarr, J. E. (2007). An examination of the levels of cognitive demand required by probability tasks in middle grades mathematics textbooks. Statistics Education Research Journal, 6(2), 4–27. doi:10.1.1.154.6160
• Knuth, E. (2002). Teachers conceptions of proof in the context of secondary school mathematics. Journal for Research in Mathematics Education, 5(1), 61-88.
• Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153-170). New York, NY: Routledge.
• Levin, S. W. (1998). Fractions and division: Research conceptualizations, textbook presentations, and student performances (Doctoral dissertation, University of Chicago, 1998). Dissertation Abstracts International, 59, 1089A.
• Li, Y. (2000). A comparison of problems that follow selected content presentations in American and Chinese mathematics textbooks. Journal for Research in Mathematics Education, 31, 234–241. doi:10.2307/749754
• Mariotti, M. A. (2006). Proof and proving in mathematics education. In A. Gutiérrez & P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics Education: Past, present and future (pp. 173-204). Rotterdam, The Netherlands: Sense Publishers.
• Milli Eğitim Bakanlığı (MEB) (2018). Matematik Dersi Öğretim Programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar). Ankara.
• Miyakawa, T. (2012). Proof in geometry: A comparative analysis of French and Japanese textbooks. Proceedings of PME 36, 3, 225-232.
• Moore, R.C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27, 249-266.
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
• National Research Council. (2004). On evaluating curricular effectiveness: Judging the quality of K–12 mathematics evaluations. Washington, DC: National Academy Press.
• Newton, D. P., & Newton, L. D. (2007). Could elementary mathematics textbooks help give attention to reasons in the classroom? Educational Studies in Mathematics, 64, 69–84. doi:10.1007/ s10649-005-9015-z
• Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16(1), 51–79.
• Ozgur, Z. (2017) Relationships Between Students' Conceptions of Proof and Classroom Factors (Unpublished Doctoral dissertation, The University of Wisconsin-Madison).
• Pickle, M. C. C. (2012), Statistical content in middle grades mathematics textbooks. Unpublished doctoral dissertation, University of South Florida, USA.
• Reid, D. (2005). The meaning of proof in mathematics education. In M. Bosch (Ed.), Proceedings of the 4th Conference of the European Society for Research in Mathematics Education (pp. 458-468). Sant Feliu de Guixols, Spain.
• Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.
• Reys, B. J., Reys, R. E., & Chávez, O. (2004). Why mathematics textbooks matter. Educational Leadership, 61(5), 61–66. http://www.ascd.org/publications/educational-leadership.aspx
• Seah, W. T., & Bishop, A. J. (2000). Values in mathematics textbooks: A view through two Australasian regions. Paper presented at the 81st annual meeting of the American Educational Research Association, New Orleans, LA.
• Schoenfeld, A.H. (1994). What do we know about mathematics curricula? Journal of Mathematical Behavior, 13(1), 55-80.
• Stacey, K., & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. Educational Studies in Mathematics, 72(3), 271–288.
• Stein, M. K., Remillard, J. T., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester, Jr., (Ed.). Second handbook of research on mathematics teaching and learning (pp. 319–369). Charlotte, NC: Information Age Publishing.
• Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.
• Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education. In A. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 315–351). Rotterdam: Sense Publishers.
• Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258–288.
• Stylianides, G. J. (2014). Textbook analyses on reasoning-and-proving: Significance and methodological challenges. International Journal of Educational Research, 64, 63–70.
• Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237–266). Reston, VA: National Council of Teachers of Mathematics.
• Stylianou, D. A., Blanton, M. L., Knuth, E. J. (Eds.). (2010). Teaching and learning proof across the grades: A K-16 perspective. Routledge & National Council of Teachers of Mathematics.
• Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43, 253–295.
• Uğurel, I.; Moralı, S. (2010). Bir Ortaöğretim Matematik Dersindeki Ispat Yapma Etkinliğine Yönelik Sınıf içi Tartışma Sürecine Öğrenci Söylemleri Çerçevesinde Yakından Bakış, Buca Eğitim Fakültesi Dergisi, 28, 135 – 154.
• Vincent, J., & Stacey, K. (2008). Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS video study criteria to Australian eighth-grade mathematics textbooks. Mathematics Education Research Journal, 20(1), 82–107.
• Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht, The Netherlands: Kluwer Academic Publishers.