Investigation of prospective mathematics teachers’ noticing of student thinking related to probability

Year 2025, Volume: 14 Issue: 1, 93 - 118, 31.01.2025

Dilek Girit Yıldız , Esila Müftüoğlu

https://doi.org/10.19128/turje.1404971

Abstract

The aim of this study is to reveal and evaluate the attending and interpreting skills of student thinking of prospective teachers, as well as their instructional suggestions as responding skills. The current study was conducted with 29 prospective mathematics teachers (PMTs) within a qualitative design in the context of probability. First, three probability problems were asked to sixty-two 8th graders (13-14 years old) in a middle school, and their solutions were used to create tasks for PMTs. PMTs answered the tasks in a written report. Then, a class discussion was held, and PMTs were given the opportunity to revise the initial reports. Content analysis was used for data analysis. PMTs demonstrated partial or robust evidence for attending to and interpreting students' thinking. However, they struggled to respond to students' reasoning. In the revised reports, the PMTs' evidence for noticing skills was better with the support of the class discussion. This study provides an example of an approach that can be used for teaching in method courses, allowing PMTs' noticing skills for student thinking to be revealed and improved.

Keywords

Noticing skills, Probability, Prospective mathematics teachers, Student’s thinking

Ethical Statement

The Human Research Ethics Committee of Trakya University's report E-29563864-050.04.04-272275, dated 15.06.2022, and the Ministry of National Education's report E-87085441-44-68502300, dated 17.01.2023, both confirm that the research does not pose an ethical problem.

Supporting Institution

The Scientific and Technological Research Council of Türkiye (2209-A-Research Project Support Programme for Undergraduate Students)

Project Number

1919B012201189 (2209 Üniversite Öğrencileri Araştırma Projeleri)

Matematik öğretmen adaylarının olasılık öğrenme alanına ilişkin öğrenci düşünüşünü fark etme becerilerinin incelenmesi

Abstract

Bu çalışmanın amacı, matematik öğretmen adaylarının öğrenci düşünüşünü tanımlama ve yorumlama becerileri ile birlikte öğrencilerin düşünüşüne ilişkin öğretimsel önerilerini (karşılık verme becerisi) ortaya çıkarmak ve değerlendirmektir. Bu çalışma 29 matematik öğretmeni adayının katılımıyla olasılık bağlamında nitel bir araştırma ile yürütülmüştür. İlk olarak, altmış iki ortaokul 8.sınıf öğrencisine üç olasılık problemi sorulmuş ve öğrencilerin çözümleri adaylara görev oluşturmak için kullanılmıştır. Matematik öğretmen adayları görevlere ilişkin yanıtlarını yazılı bir rapor halinde vermişlerdir. Daha sonra, bir sınıf tartışması yapılmış ve adaylara ilk raporlarını gözden geçirme ve düzeltme fırsatı verilmiştir. Veri analizi için içerik analizi kullanılmıştır. Matematik öğretmen adayları, öğrencilerin stratejilerini tanımlama ve yorumlama konusunda kısmi ya da güçlü kanıtlar sunabilmiştir. Ancak, öğrencilerin çözümlerine ilişkin öneri vermekte zorlanmışlardır. Düzeltmelerde, matematik öğretmen adaylarının fark etme becerileri sınıf tartışmasının katkısıyla daha iyi hale gelmiştir. Bu çalışma, özel öğretim yöntemleri derslerinde kullanılabilecek bir yaklaşım örneği sunarak, matematik öğretmen adaylarının öğrenci düşünüşünü fark etme becerilerinin ortaya çıkarılmasına ve geliştirilmesine olanak tanımaktadır.

Keywords

Fark etme becerisi, Matematik öğretmeni adayları, Olasılık, Öğrenci düşünüşü

Ethical Statement

The Human Research Ethics Committee of Trakya University's report E-29563864-050.04.04-272275, dated 15.06.2022, and the Ministry of National Education's report E-87085441-44-68502300, dated 17.01.2023, both confirm that the research does not pose an ethical problem.

Supporting Institution

TÜBİTAK - Türkiye Bilimsel ve Teknolojik Araştırma Kurumu (2209-A - Üniversite Öğrencileri Araştırma Projeleri Destekleme Programı)

Project Number

1919B012201189 (2209 Üniversite Öğrencileri Araştırma Projeleri)

References

• Alsawaie, O. N., & Alghazo, I. M. (2010). The effect of video-based approach on prospective teachers’ ability to analyze mathematics teaching. Journal of Mathematics Teacher Education, 13(3), 223-241, https://doi.org/10.1007/s10857-009-9138-8
• Amador, J. M., Bragelman, J., & Castro-Superfine, A. (2021). Prospective teachers’ noticing: a literature review of methodological approaches to support and analyze noticing. Teaching and Teacher Education, 99, 103256. https://doi.org/10.1016/j.tate.2020.103256
• Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
• Barnhart, T., & van Es, E. (2015). Studying teacher noticing: examining the relationship among prospective science teachers’ ability to attend, analyze, and respond to student thinking. Teaching and Teacher Education, 45, 83-93. https://doi.org/10.1016/j.tate.2014.09.005
• Barnhart, T., Johnson, H. J., & Tekkumru-Kisa, M. (2025). Pre-service teachers notice student thinking: Then what?. Journal of Teacher Education, 76(1), 57-70. https://doi.org/10.1177/00224871231220604
• Batanero, C., & Álvarez-Arroyo, R. (2024). Teaching and learning of probability. ZDM Mathematics Education, 56, 5-17. https://doi.org/10.1007/s11858-023-01511-5
• Batanero, C., Arteaga, P., Serrano, L. & Ruiz, B. (2014). Prospective primary school teachers’ perception of randomness. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: Presenting plural perspectives (pp. 345–366). Springer.
• Bergman, A. M., Gallagher, K., & Zazkis, R. (2023). Prospective teachers’ responses to students’ dialogue on fractions: attribute substitution and heuristic approaches. Research in Mathematics Education, 25(1), 85-104. https://doi.org/10.1080/14794802.2021.2020155
• Bryant, P., & Nunes, T. (2012). Children’s understanding of probability: A literature review. Nuffield Foundation.
• Callaert, H. (2004). In search of the specificity and the identifiability of stochastic thinking and reasoning. In M. A. Mariotti (Ed.), Proceedings of the Third Conference of the European Society for Research in Mathematics Education. Retrieved June 7, 2022, from http://www.mathematik.unidortmund.de/~erme/CERME3/Groups/TG5/TG5_callaert_cerme3.pdf
• Callejo, M. L., & Zapatera, A. (2017). Prospective primary teachers’ noticing of students’ understanding of pattern generalization. Journal of Mathematics Teacher Education, 20(4), 309-333. https://doi.org/10.1007/s10857-016-9343-1
• Copur-Gencturk, Y., & Rodrigues, J. (2021). Content-specific noticing: A large-scale survey of mathematics teachers’ noticing. Teaching and Teacher Education, 101, 103320. https://doi.org/10.1016/j.tate.2021.103320
• Copur-Gencturk, Y., & Tolar, T. (2022). Mathematics teaching expertise: A study of the dimensionality of content knowledge, pedagogical content knowledge, and content-specific noticing skills. Teaching and Teacher Education, 114, 103696. https://doi.org/10.1016/j.tate.2022.103696
• Creswell, J. W. (2009). Research design: Qualitative and mixed-methods approaches. Sage Publications.
• Falk, R., Yudilevich-Assouline, P. & Elstein, A. (2012). Children’s concept of probability as inferred from their binary choices—revisited. Educational Studies in Mathematics, 81, 207–233. https://doi.org/10.1007/s10649-012-9402-1
• Fernández, C., Llinares, S., & Valls, J. (2013). Primary school teachers’ noticing of students’ mathematical thinking in problem solving. The Mathematics Enthusiast, 10(1), 441-468. https://doi.org/10.54870/1551-3440.1274
• Fernández, C., Moreno, M., & Sánchez-Matamoros, G. (2024). Prospective secondary teachers’ noticing of students’ thinking about the limit concept: Pathways of development. ZDM–Mathematics Education, 56, 1137–1151. https://doi.org/10.1007/s11858-024-01573-z
• Fischbein, E. & Schnarch, D. (1997). Brief report: The evolution with age of probabilistic, intuitively based misconceptions. Journal of Research in Mathematics Education, 28, 96-105.
• Gal, I. (2005). Towards “probability literacy” for all citizens: Building blocks and instructional dilemmas. In G. A. Jones (Ed.), Exploring probability in school (pp. 39–63). Springer. https://doi.org/10.5951/jresematheduc.28.1.0096
• General Directorate of Assessment, Evaluation and Examination Services. (n.d.). 8th-grade study questions. Retrieved October 2, 2022, from http://odsgm.meb.gov.tr/www/8-sinif-calisma-sorulari/icerik/739.
• Girit-Yildiz, D., Osmanoglu, A., & Gundogdu Alayli, F. (2023). Providing a video-case-based professional development environment for prospective mathematics teachers to notice students’ misconceptions in measurement. Journal of Mathematics Teacher Education, 26(2), 179-209. https://doi.org/10.1007/s10857-021-09525-0
• Guner, P., & Akyuz, D. (2020). Noticing student mathematical thinking within the context of lesson study. Journal of Teacher Education, 71(5), 568-583. https://doi.org/10.1177/0022487119892964
• Ivars, P., Fernández, C., & Llinares, S. (2020). A learning trajectory as a scaffold for prospective teachers’ noticing of students’ mathematical understanding. International Journal of Science and Mathematics Education, 18(3), 529-548. https://doi.org/10.1007/s10763-019-09973-4
• Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169-202. https://doi.org/10.5951/jresematheduc.41.2.0169
• Jacobs, V. R., Empson, S. B., Jessup, N. A., Dunning, A., Pynes, D., Krause, G., & Franke, T. M. (2022). Profiles of teachers’ expertise in professional noticing of children’s mathematical thinking. Journal of Mathematics Teacher Education, 27, 295-324. https://doi.org/10.1007/s10857-022-09558-z
• Kazak, S. (2012). Olasılık konusunda karşılaşılan zorluklar [Difficulties in Probability]. In E. Bilgölbali & M. F. Özmantar (Eds.), İlköğretimde karşılaşılan matematiksel zorluklar ve çözüm önerileri [Mathematical difficulties encountered in elementary education and solution suggestions] (pp. 217-239). Pegem Akademi.
• Kazak, S., & Pratt, D. (2021). Developing the role of modelling in the teaching and learning of probability. Research in Mathematics Education, 23(2), 113-133. https://doi.org/10.1080/14794802.2020.1802328
• Konold, C. & Miller, C. D. (2005). TinkerPlots: Dynamic Data Exploration. Key Curriculum Press.
• Krupa, E. E., Huey, M., Lesseig, K., Casey, S., & Monson, D. (2017). Investigating secondary prospective teacher noticing of students’ mathematical thinking. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 49-72). Springer.
• Lee, M.Y., & Lee, J. E. (2023). An analysis of elementary prospective teachers’ noticing of student pattern generalization strategies in mathematics. Journal of Mathematics Teacher Education, 26, 155-177. https://doi.org/10.1007/s10857-021-09520-5
• Mason, J. (2002). Researching your own practice: The discipline of noticing. Psychology Press.
• Merriam, S. B. (2009). Qualitative Research: A guide to design and implementation. Jossey-Bass.
• Miles, M. B. & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. Sage Publication.
• Park, J. & Kim, D. W. (2023). Facilitating factors and obstacles in the coordination of theoretical probability and relative frequency estimates of probability. Educational Studies in Mathematics, 114, 439-460. https://doi.org/10.1007/s10649-023-10253-w
• Park, M., & Lee, E. J. (2019). Korean prospective elementary teachers’ abilities to identify equiprobability bias and teaching strategies. International Journal of Science and Mathematics Education, 17, 1585-1603. https://doi.org/10.1007/s10763-018-9933-8
• Rotem, S.H., & Ayalon, M. (2023a). Constructing coherency levels to understand connections among the noticing skills of prospective mathematics teachers. Journal of Mathematics Teacher Education, 27, 579-605. https://doi.org/10.1007/s10857-023-09574-7
• Rotem, S. H., & Ayalon, M. (2023b). Changes in noticing multiple dimensions in classroom situations among pre-service mathematics teachers. Teaching and Teacher Education, 121, 103932. https://doi.org/10.1016/j.tate.2022.103932
• Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2019). Relationships among prospective secondary mathematics teachers’ skills of attending, interpreting and responding to students’ understanding. Educational Studies in Mathematics, 100(1), 83-99. https://doi.org/10.1007/s10649-018-9855-y
• Santagata, R., Zannoni, C., & Stigler, J. W. (2007). The role of lesson analysis in prospective teacher education: An empirical investigation of teacher learning from a virtual video-based field experience. Journal of Mathematics Teacher Education, 10(2), 123-140. https://doi.org/10.1007/s10857-007-9029-9
• Schlesinger, L., Jentsch, A., Kaiser, G., König, J., & Blömeke, S. (2018). Subject-specific characteristics of instructional quality in mathematics education. ZDM Mathematics Education, 50(3), 475-490. https://doi.org/10.1007/s11858-018-0917-5
• Schack, E. O., Fisher, M. H., Thomas, J. N., Eisenhardt, S., Tassell, J., & Yoder, M. (2013). Prospective elementary school teachers’ professional noticing of children’s early numeracy. Journal of Mathematics Teacher Education, 16(5), 379-397. https://doi.org/10.1007/s10857-013-9240-9
• Sherin, M. G., & Han, S. Y. (2004). Teacher learning in the context of a video club. Teaching and Teacher Education, 20(2), 163-183. https://doi.org/10.1016/j.tate.2003.08.001
• Sherin, M. G., & van Es, E. A. (2009). Effects of video club participation on teachers' professional vision. Journal of Teacher Education, 60(1), 20-37. https://doi.org/10.1177/0022487108328155
• Star, J. R., & Strickland, S. K. (2008). Learning to observe: Using video to improve prospective mathematics teachers’ ability to notice. Journal of Mathematics Teacher Education, 11(2), 107-125. https://doi.org/10.1007/s10857-007-9063-7
• Stockero, S. L., Leatham, K. R., Van Zoest, L. R., & Peterson, B. E. (2017). Noticing distinctions among and within instances of student mathematical thinking. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 467-480). Springer.
• Strauss, A., & Corbin, J. (1994). Grounded theory methodology: An overview. In N. K. Denzin & Y. Lincoln (Eds.), Handbook of qualitative research (pp. 273-285). Sage.
• Supply, AS., Vanluydt, E., Van Dooren, & W. et al. (2023). Out of proportion or out of context? Comparing 8- to 9-year-olds’ proportional reasoning abilities across fair-sharing, mixtures, and probability contexts. Educational Studies in Mathematics, 113, 371-388. https://doi.org/10.1007/s10649-023-10212-5
• Tarr, J. E. (2002). Confounding effects of the phrase “50-50 chance” in making conditional probability judgments. Focus on Learning Problems in Mathematics, 24(4), 35-53.
• Thomas, J., Dueber, D., Fisher, M. H., Jong, C., & Schack, E. O. (2022). Professional noticing coherence: exploring relationships between component processes. Mathematical Thinking and Learning, 25(4), 361-379. https://doi.org/10.1080/10986065.2021.1977086
• Turkish Ministry of National Education [TMoNE] (2018). Ortaokul matematik dersi 5-8 sınıf öğretim program [Middle school (5-8) curriculum]. Retrieved August 30, 2020, from https://mufredat.meb.gov.tr/Dosyalar/201813017165445-MATEMAT%C4%B0K%20%C3%96%C4%9ERET%C4%B0M%20PROGRAMI%202018v.pdf.
• Ulusoy, F. (2020). Prospective teachers’ skills of attending, interpreting and responding to content-specific characteristics of mathematics instruction in classroom videos. Teaching and Teacher Education, 94, 103103. https://doi.org/10.1016/j.tate.2020.103103
• Ulusoy, F., & Çakıroğlu, E. (2021). Exploring prospective teachers’ noticing of students' understanding through micro-case videos. Journal of Mathematics Teacher Education, 24(3), 253-282. https://doi.org/10.1007/s10857-020-09457-1
• van Es, E. A., Cashen, M., Barnhart, T., & Auger, A. (2017). Learning to notice mathematics instruction: Using video to develop prospective teachers’ vision of ambitious pedagogy. Cognition and Instruction, 35(3), 165-187. https://doi.org/10.1080/07370008.2017.1317125
• van Es, E.A. & Sherin, M.G. (2002). Learning to Notice: Scaffolding New Teachers’ Interpretations of Classroom Interactions. Journal of Technology and Teacher Education, 10(4), 571-596. Retrieved December 27, 2023, from https://www.learntechlib.org/primary/p/9171/.
• van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers' "learning to notice" in the context of a video club. Teaching and Teacher Education, 24, 244-276. https://doi.org/10.1016/j.tate.2006.11.005
• van Es, E. A., & Sherin, M. G. (2021). Expanding on prior conceptualizations of teacher noticing. ZDM Mathematics Education, 53, 17-27. https://doi.org/10.1007/s11858-020-01211-4
• Yin, R. K. (2009). Case study research: Design and methods (4th ed.). Sage Publications.
• Walkoe, J. (2015). Exploring teacher noticing of student algebraic thinking in a video club. Journal of Mathematics Teacher Education, 18(6), 523-550. https://doi.org/10.1007/s10857-014-9289-0
• Warshauer, H. K., Starkey, C., Herrera, C. A., & Smith, S. (2021). Developing prospective teachers’ noticing and notions of productive struggle with video analysis in a mathematics content course. Journal of Mathematics Teacher Education, 24(1), 89-121. https://doi.org/10.1007/s10857-019-09451-2
• Wilson, P. H., Mojica, G., & Confrey, J. (2013). Learning trajectories in teacher education: supporting teachers’ understanding of students’ mathematical thinking. Journal of Mathematical Behavior, 32, 103-121. https://doi.org/10.1016/j.jmathb.2012.12.003
• Zeeb, H., Ibach, A., Voss, T., & Renkl, A. (2023). How does teachers' noticing of students' fixed mindsets relate to teachers’ knowledge, beliefs, and experience? An exploratory study. Teaching and Teacher Education, 130, 104170. https://doi.org/10.1016/j.tate.2023.104170