Research Article
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Analysis of mathematical reasoning processes: The case of middle school students

Year 2024, Volume: 13 Issue: 3, 663 - 677, 29.09.2024

Hilal Güler Baran , Gönül Yazgan Sağ

https://doi.org/10.30703/cije.1409847

Abstract

The aim of this qualitative research is to examine the mathematical reasoning processes of secondary school 7th grade students in the context of mathematical problems. This research was designed as a phenomenology design. The participants of the research were determined by the maximum diversity sampling method, one of the purposeful sampling methods, in the 2021-2022 academic year, consisting of six 7th grade students studying in a big city in the Central Anatolia Region. In order to reveal the mathematical reasoning processes of the students, 8 mathematical problems determined in the light of the literature and finalized by pilot interviews were used. Data were collected through individual interviews with each participant and these interviews were recorded with a video camera. These collected data were analysed using the descriptive analysis method, with using the relevant literature. According to the findings, it is observed that the participants displayed all of the analysis, generalization and justification reasoning processes in mathematical problems at various levels. In addition, it has been found that the reasoning indicators of some problems differentiated students who are characterized as high academic levels to students whose academic achievement is characterized as medium and low at different levels.

Keywords

Reasoning mathematical reasoning qualitative research middle school students problem solving

References

  • Artzt, A. F., & Armour-Thomas, E. (1992). Development of a cognitive metacognitive framework for protocol analysis of mathematical problem solving in small groups. Cognition and Instruction, 9(2), 137–175. https://doi.org/10.1207/s1532690xci0902_3
  • Bragg, L. A., & Herbert, S. (2018). What can be learned from teachers assessing mathematical reasoning: A case study. In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces: Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia (pp. 178–185). Auckland: MERGA.
  • Bragg, L. A., Vale, C., Herbert, S., Loong, E., Widjaja, W., Williams, G. & Mousley, J. (2013). Promoting awareness of reasoning in the primary mathematics classroom. In A. McDonough, A. Downton, & L. A. Bragg (Eds.), Mathematics of the planet earth: Proceedings of the MAV 50th annual conference (pp. 23–30). Melbourne: Mathematical Association of Victoria.
  • Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.
  • Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41–53. https://doi.org/10.1080/0141192940200105
  • Desoete, A., Roeyers, H., & Buysse, A. (2001). Metacognition and mathematical problem solving in grade 3. Journal of Learning Disability, 34(5), 435–449. https://doi.org/10.1177/002221940103400
  • Ellis, A. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–22. https://doi.org/10.2307/30034866
  • Ericsson, K. A. (2006). Protocol analysis and expert thought: Concurrent verbalizations of thinking during experts' performance on representative task. In K. A. Ericsson, N. Charness, P. Feltovich, & R. R. Hoffman, R. R. (Eds.). Cambridge handbook of expertise and expert performance (pp. 223–242). Cambridge, UK: Cambridge University Press.
  • Francisco, J., & Maher, C.A. (2011). Teachers attending to students’ mathematical reasoning: Lessons from an after-school research program. Journal of Mathematics Teacher Education, 14, 49–66. https://doi.org/10.1007/s10857-010-9144-x
  • Girit, D. & Akyüz, D. (2016). Farklı sınıf seviyelerindeki ortaokul öğrencilerinde cebirsel düşünme: Örüntülerde genelleme hakkındaki algıları. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 10(2), 243–272. https://doi.org/10.17522/balikesirnef.277815
  • Herbert, S. & Williams, G. (2023). Eliciting mathematical reasoning during early primary problem solving. Mathematics Education Research Journal, 35, 77–103. https://doi.org/10.1007/s13394-021-00376-9
  • Herbert, S. & Bragg, L. A. (2021). Elementary teachers’ planning for mathematical reasoning through peer learning teams. International Journal for Mathematics Teaching and Learning, 22(1), 24–43. https://doi.org/10.4256/ijmtl.v22i1.291
  • Herbert, S., Vale, C., White, P., & Bragg, L. A. (2022). Engagement with a formative assessment rubric: A case of mathematical reasoning. International Journal of Educational Research, 111, 101899. https://doi.org/10.1016/j.ijer.2021.101899
  • Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1–16. https://doi.org/10.1007/s10649-017-9761-8
  • Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven, CT: Yale University.
  • Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258. https://doi.org/10.1207/s15327833mtl0703_3
  • Lannin, J., Ellis, A. B., & Elliot, R. (2011). Developing essential understanding of mathematics reasoning for teaching mathematics in prekindergarten-grade 8. Reston: NCTM.
  • Loong, E., Vale, C., Widjaja, W., Herbert, E.S., Bragg, L. & Davidson, A. (2018). Developing a rubric for assessing mathematical reasoning: A design-based research study in primary classrooms. In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces: Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia (pp. 503–510). Auckland: MERGA.
  • Mason, J. (1982). Thinking mathematically. London: Addison-Wesley.
  • Mata-Pereira, J., & Ponte, J. P. (2017). Enhancing students’ mathematical reasoning in the classroom: Teacher actions facilitating generalization and justification. Educational Studies in Mathematics, 96(2), 169–186. https://doi.org/10.1007/s10649-017-9773-4
  • Meyer, M. (2010). Abduction–A logical view for investigating and initiating processes of discovering mathematical coherences. Educational Studies in Mathematics, 74(2), 185–205. https://doi.org/10.1007/s10649-010-9233-x
  • Miles, M, B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded Sourcebook. Thousand Oaks, CA: Sage.
  • Milli Eğitim Bakanlığı (MEB) (2013). Ortaokul Matematik Dersi (5, 6, 7 ve 8. Sınıflar) Öğretim Programı. Ankara: Milli Eğitim Bakanlığı.
  • Milli Eğitim Bakanlığı (MEB) (2018). Ortaokul Matematik Dersi (5, 6, 7 ve 8. Sınıflar) Öğretim Programı. Ankara: Milli Eğitim Bakanlığı.
  • National Council of Teachers of Mathematics (NCTM) (2014). Principles to actions: Ensuring mathematics success for all. Reston, VA: National Council of Teachers of Mathematics.
  • Patton, M. Q. (2002). Qualitative research and evaluation methods. (3rd Edition). USA: Sage.
  • Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41. https://doi.org/10.1007/s10649-006-9057-x
  • Peker, S. (2020). Ortaöğretim öğrencilerinin genelleme becerilerinin incelenmesi. [Yayımlanmamış yüksek lisans tezi]. Sivas Cumhuriyet Üniversitesi.
  • Reid, D. A. (2003). Forms and uses of abduction. In M. A. Mariotti (Ed.). Proceedings of 3rd Conference of European Societyfor Research in Mathematics Education (pp.1–10). Bellaria: CERME.
  • Rivera, F. D. (2008). On the pitfalls of abduction: Complicities and complexities in patterning activity. For the Learning of Mathematics, 28(1), 17–25. https://www.jstor.org/stable/40248593
  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.). Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan. Schraw, G. (1998). Promoting general metacognitive awareness. Instructional Science, 26(1–2), 113–125. https://doi.org/10.1023/A:1003044231033
  • Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1–20. https://doi.org/10.1007/s10649-006-9038-0
  • Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.
  • Swan, M. (2011). Improving reasoning: Analysing alternative approaches. http://nrich.maths.org/7812/index sayfasından erişilmiştir. Toulmin, S. E. (2007). The uses of argument. New York: Cambridge University.
  • Umay, A. (2003). Matematiksel muhakeme yeteneği. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24, 234–243.
  • Umay, A. & Kaf, Y. (2005). Matematikte kusurlu akıl yürütme üzerine bir çalışma. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 28, 188–195.
  • Vale, C., Widjaja, W., Herbert, S., Bragg, L. A., & Loong, E. Y. K. (2017). Mapping variation in children’s mathematical reasoning: The case of ‘what else belongs?’. International Journal of Science and Mathematics Education, 15(5), 873–894. https://doi.org/10.1007/s10763-016-9725-y
  • Yakut Çayir, M. & Akyüz, G. (2015). 9. sınıf öğrencilerinin örüntü genelleme problemlerini çözme stratejilerinin belirlenmesi. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 9(2), 205–229. https://doi.org/10.17522/nefefmed.66921
  • Yeşildere, S. & Türnüklü, E. B. (2007). Examination of students’ mathematical thinking and reasoning processes. Ankara University Journal of Faculty of Educational Sciences (JFES),40 (1), 181–213. https://doi.org/10.1501/Egifak_0000000156
  • Yıldırım, A. & Şimşek, H. (2018). Sosyal bilimlerde nitel araştırma yöntemleri. (11. Baskı). Ankara: Seçkin.
  • Widjaja, W., & Vale, C. (2021). Counterexamples: Challenges faced by elementary students when testing a conjecture about the relationship between perimeter and area. Journal on Mathematics Education, 12(3), 487–506. https://doi.org/10.22342/jme.12.3.14526.487-506
  • Widjaja, W., Vale, C., Herbert, S., Loong, E. Y., & Bragg, L. A. (2021). Linking comparing and contrasting, generalising and justifying: A case study of primary students’ levels of justifying. Mathematics Education Research Journal, 33(2), 321–343. https://doi.org/10.1007/s13394-019-00306-w

MATEMATİKSEL AKIL YÜRÜTME SÜREÇLERİNİN ANALİZİ: ORTAOKUL ÖĞRENCİLERİ ÖRNEĞİ

Year 2024, Volume: 13 Issue: 3, 663 - 677, 29.09.2024

Hilal Güler Baran , Gönül Yazgan Sağ

https://doi.org/10.30703/cije.1409847

Abstract

Bu araştırmanın amacı, ortaokul 7. sınıf öğrencilerinin matematiksel akıl yürütme süreçlerini matematik problemleri bağlamında incelemektir. Bu araştırma nitel araştırma desenlerinden biri olan fenomenoloji deseni olarak tasarlanmıştır. Araştırmanın katılımcıları da amaçlı örnekleme yöntemlerinden maksimum çeşitlilik örneklemesi yöntemi ile belirlenen 2021-2022 eğitim-öğretim yılında İç Anadolu Bölgesinde bulunan büyük bir ilde öğrenim gören altı 7. sınıf öğrencisidir. Öğrencilerin matematiksel akıl yürütme süreçlerini açığa çıkarmak için literatür ışığında belirlenen ve pilot görüşmeler yapılarak son halini alan 8 matematik problemi kullanılmıştır. Veriler, her bir katılımcı ile yapılan bireysel görüşmeler yoluyla toplanmış ve bu görüşmeler video kamera ile kayıt altına alınmıştır. Toplanan bu veriler ilgili literatür göz önüne alınarak betimsel analiz yöntemi kullanılarak analiz edilmiştir. Verilerden elde edilen bulgulara göre, katılımcıların matematik problemlerinde analiz etme, genelleme ve gerekçelendirme akıl yürütme süreçlerinin tamamını çok çeşitli düzeylerde sergiledikleri görülmüştür. Ayrıca bazı problemlerin ortaya çıkardığı akıl yürütme göstergeleri; akademik başarısı yüksek olarak nitelendirilen öğrencileri, akademik başarısı orta ve düşük olarak nitelendirilen öğrencilerden ayrıştırdığı tespit edilmiştir.

Keywords

Akıl yürütme matematiksel akıl yürütme nitel araştırma ortaokul öğrencileri problem çözme

References

  • Artzt, A. F., & Armour-Thomas, E. (1992). Development of a cognitive metacognitive framework for protocol analysis of mathematical problem solving in small groups. Cognition and Instruction, 9(2), 137–175. https://doi.org/10.1207/s1532690xci0902_3
  • Bragg, L. A., & Herbert, S. (2018). What can be learned from teachers assessing mathematical reasoning: A case study. In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces: Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia (pp. 178–185). Auckland: MERGA.
  • Bragg, L. A., Vale, C., Herbert, S., Loong, E., Widjaja, W., Williams, G. & Mousley, J. (2013). Promoting awareness of reasoning in the primary mathematics classroom. In A. McDonough, A. Downton, & L. A. Bragg (Eds.), Mathematics of the planet earth: Proceedings of the MAV 50th annual conference (pp. 23–30). Melbourne: Mathematical Association of Victoria.
  • Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.
  • Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41–53. https://doi.org/10.1080/0141192940200105
  • Desoete, A., Roeyers, H., & Buysse, A. (2001). Metacognition and mathematical problem solving in grade 3. Journal of Learning Disability, 34(5), 435–449. https://doi.org/10.1177/002221940103400
  • Ellis, A. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–22. https://doi.org/10.2307/30034866
  • Ericsson, K. A. (2006). Protocol analysis and expert thought: Concurrent verbalizations of thinking during experts' performance on representative task. In K. A. Ericsson, N. Charness, P. Feltovich, & R. R. Hoffman, R. R. (Eds.). Cambridge handbook of expertise and expert performance (pp. 223–242). Cambridge, UK: Cambridge University Press.
  • Francisco, J., & Maher, C.A. (2011). Teachers attending to students’ mathematical reasoning: Lessons from an after-school research program. Journal of Mathematics Teacher Education, 14, 49–66. https://doi.org/10.1007/s10857-010-9144-x
  • Girit, D. & Akyüz, D. (2016). Farklı sınıf seviyelerindeki ortaokul öğrencilerinde cebirsel düşünme: Örüntülerde genelleme hakkındaki algıları. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 10(2), 243–272. https://doi.org/10.17522/balikesirnef.277815
  • Herbert, S. & Williams, G. (2023). Eliciting mathematical reasoning during early primary problem solving. Mathematics Education Research Journal, 35, 77–103. https://doi.org/10.1007/s13394-021-00376-9
  • Herbert, S. & Bragg, L. A. (2021). Elementary teachers’ planning for mathematical reasoning through peer learning teams. International Journal for Mathematics Teaching and Learning, 22(1), 24–43. https://doi.org/10.4256/ijmtl.v22i1.291
  • Herbert, S., Vale, C., White, P., & Bragg, L. A. (2022). Engagement with a formative assessment rubric: A case of mathematical reasoning. International Journal of Educational Research, 111, 101899. https://doi.org/10.1016/j.ijer.2021.101899
  • Jeannotte, D., & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1–16. https://doi.org/10.1007/s10649-017-9761-8
  • Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven, CT: Yale University.
  • Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258. https://doi.org/10.1207/s15327833mtl0703_3
  • Lannin, J., Ellis, A. B., & Elliot, R. (2011). Developing essential understanding of mathematics reasoning for teaching mathematics in prekindergarten-grade 8. Reston: NCTM.
  • Loong, E., Vale, C., Widjaja, W., Herbert, E.S., Bragg, L. & Davidson, A. (2018). Developing a rubric for assessing mathematical reasoning: A design-based research study in primary classrooms. In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces: Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia (pp. 503–510). Auckland: MERGA.
  • Mason, J. (1982). Thinking mathematically. London: Addison-Wesley.
  • Mata-Pereira, J., & Ponte, J. P. (2017). Enhancing students’ mathematical reasoning in the classroom: Teacher actions facilitating generalization and justification. Educational Studies in Mathematics, 96(2), 169–186. https://doi.org/10.1007/s10649-017-9773-4
  • Meyer, M. (2010). Abduction–A logical view for investigating and initiating processes of discovering mathematical coherences. Educational Studies in Mathematics, 74(2), 185–205. https://doi.org/10.1007/s10649-010-9233-x
  • Miles, M, B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded Sourcebook. Thousand Oaks, CA: Sage.
  • Milli Eğitim Bakanlığı (MEB) (2013). Ortaokul Matematik Dersi (5, 6, 7 ve 8. Sınıflar) Öğretim Programı. Ankara: Milli Eğitim Bakanlığı.
  • Milli Eğitim Bakanlığı (MEB) (2018). Ortaokul Matematik Dersi (5, 6, 7 ve 8. Sınıflar) Öğretim Programı. Ankara: Milli Eğitim Bakanlığı.
  • National Council of Teachers of Mathematics (NCTM) (2014). Principles to actions: Ensuring mathematics success for all. Reston, VA: National Council of Teachers of Mathematics.
  • Patton, M. Q. (2002). Qualitative research and evaluation methods. (3rd Edition). USA: Sage.
  • Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41. https://doi.org/10.1007/s10649-006-9057-x
  • Peker, S. (2020). Ortaöğretim öğrencilerinin genelleme becerilerinin incelenmesi. [Yayımlanmamış yüksek lisans tezi]. Sivas Cumhuriyet Üniversitesi.
  • Reid, D. A. (2003). Forms and uses of abduction. In M. A. Mariotti (Ed.). Proceedings of 3rd Conference of European Societyfor Research in Mathematics Education (pp.1–10). Bellaria: CERME.
  • Rivera, F. D. (2008). On the pitfalls of abduction: Complicities and complexities in patterning activity. For the Learning of Mathematics, 28(1), 17–25. https://www.jstor.org/stable/40248593
  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.). Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan. Schraw, G. (1998). Promoting general metacognitive awareness. Instructional Science, 26(1–2), 113–125. https://doi.org/10.1023/A:1003044231033
  • Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1–20. https://doi.org/10.1007/s10649-006-9038-0
  • Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.
  • Swan, M. (2011). Improving reasoning: Analysing alternative approaches. http://nrich.maths.org/7812/index sayfasından erişilmiştir. Toulmin, S. E. (2007). The uses of argument. New York: Cambridge University.
  • Umay, A. (2003). Matematiksel muhakeme yeteneği. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24, 234–243.
  • Umay, A. & Kaf, Y. (2005). Matematikte kusurlu akıl yürütme üzerine bir çalışma. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 28, 188–195.
  • Vale, C., Widjaja, W., Herbert, S., Bragg, L. A., & Loong, E. Y. K. (2017). Mapping variation in children’s mathematical reasoning: The case of ‘what else belongs?’. International Journal of Science and Mathematics Education, 15(5), 873–894. https://doi.org/10.1007/s10763-016-9725-y
  • Yakut Çayir, M. & Akyüz, G. (2015). 9. sınıf öğrencilerinin örüntü genelleme problemlerini çözme stratejilerinin belirlenmesi. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 9(2), 205–229. https://doi.org/10.17522/nefefmed.66921
  • Yeşildere, S. & Türnüklü, E. B. (2007). Examination of students’ mathematical thinking and reasoning processes. Ankara University Journal of Faculty of Educational Sciences (JFES),40 (1), 181–213. https://doi.org/10.1501/Egifak_0000000156
  • Yıldırım, A. & Şimşek, H. (2018). Sosyal bilimlerde nitel araştırma yöntemleri. (11. Baskı). Ankara: Seçkin.
  • Widjaja, W., & Vale, C. (2021). Counterexamples: Challenges faced by elementary students when testing a conjecture about the relationship between perimeter and area. Journal on Mathematics Education, 12(3), 487–506. https://doi.org/10.22342/jme.12.3.14526.487-506
  • Widjaja, W., Vale, C., Herbert, S., Loong, E. Y., & Bragg, L. A. (2021). Linking comparing and contrasting, generalising and justifying: A case study of primary students’ levels of justifying. Mathematics Education Research Journal, 33(2), 321–343. https://doi.org/10.1007/s13394-019-00306-w
There are 42 citations in total.

Details

Primary Language Turkish
Subjects Mathematics Education
Journal Section Articles
Authors

Hilal Güler Baran 0000-0003-4575-6517

Gönül Yazgan Sağ 0000-0002-7237-5683

Publication Date September 29, 2024
Submission Date December 25, 2023
Acceptance Date May 21, 2024
Published in Issue Year 2024Volume: 13 Issue: 3

Cite

APA Güler Baran, H., & Yazgan Sağ, G. (2024). MATEMATİKSEL AKIL YÜRÜTME SÜREÇLERİNİN ANALİZİ: ORTAOKUL ÖĞRENCİLERİ ÖRNEĞİ. Cumhuriyet Uluslararası Eğitim Dergisi, 13(3), 663-677. https://doi.org/10.30703/cije.1409847
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