How Do Middle School Mathematics Teachers Conceptualize Open-Ended Questions?

Erhan Bingolbali, Hilmi Furkan Cevik

Abstract


The study aims at examining middle school mathematics teachers' conceptions of open-ended questions. A questionnaire consisting of open-ended items was applied to 40 mathematics teachers. The teachers were asked to define the open-ended question in general and the mathematical open-ended question in particular and provide examples to exemplify their definitions. This study employs phenomenographic study aiming at revealing middle school mathematics teachers' conceptions and experience regarding open-ended questions in an exploratory manner. The findings show that the teachers explained the open-ended question through its form (appearance), the number of outputs, the process/method required, and its functionality. In addition, teachers defined the open-ended question mostly using non-mathematical terms, and they had particular difficulties defining the mathematical open-ended question. The teachers regarded questions with variable correct answers as open-ended, could not give examples of open-ended questions with infinitely correct answers, and some deemed closed-ended questions as open-ended. Although the participants were mathematics teachers, the examples they presented for the open-ended question were mostly from outside the field of mathematics. This study points out the fact that teachers need a guiding conceptualization of open-ended questions.


Keywords


open-ended questions; mathematical open-ended questions; teachers’ conceptions; teachers’ difficulties; mathematics teachers

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References


Aziza, M. (2021). A Teacher Questioning Activity: The Use of Oral Open-ended Questions in Mathematics Classroom. Qualitative Research in Education, 10(1), 31–61. https://doi.org/10.17583/qre.2021.6475

Azungah, T. (2018). Qualitative research: deductive and inductive approaches to data analysis. Qualitative Research Journal, 18(4), 383–400. https://doi.org/10.1108/QRJ-D-18-00035

Becker, J. P., & Shimada, S. (1997). The Open Ended Approach. National Council of Teachers of Mathematics.

Bennevall, M. (2016). Cultivating Creativity in the Mathematics Classroom using Open-ended Tasks : A Systematic Review TT - Utvecklande av kreativitet i matematikklassrum med hjälp av öppna problem : En systematisk genomgång A Systematic Review (swe): Vol. Independen. http://liu.diva-portal.org/smash/get/diva2:909145/FULLTEXT01.pdf

Bingolbali, E. (2020a). An analysis of questions with multiple solution methods and multiple outcomes in mathematics textbooks. International Journal of Mathematical Education in Science and Technology, 51(5), 669–687. https://doi.org/10.1080/0020739X.2019.1606949

Bingolbali, E. (2020b). Çok Doğru Cevaplı ve Çok Çözüm Metotlu Etkinliklerin Ortaokul Matematik Ders Kitaplarındaki Yeri [An Examination of Tasks in Elementary School Mathematics Textbooks in Terms of Multiple Outcomes and Multiple Solution Methods]. International Journal of Educational Studies in Mathematics, 7(4), 214–235. https://doi.org/10.1080/0020739X.2019.1606949

Bingölbali, E., & Bingölbali, F. (2020). Divergent Thinking and Convergent Thinking: Are They Promoted in Mathematics Textbooks? International Journal of Contemporary Educational Research. https://doi.org/10.33200/ijcer.689555

Bingölbali, E., & Bingölbali, F. (2021). An Examination of Open-Ended Mathematics Questions’ Affordances. International Journal of Progressive Education, 17(4), 1–16. https://doi.org/10.29329/ijpe.2021.366.1

Boulton-Lewis, G. M., Smith, D. J. H., McCrindle, A. R., Burnett, P. C., & Campbell, K. J. (2001). Secondary teachers’ conceptions of teaching and learning. Learning and Instruction, 11(1), 35–51. https://doi.org/10.1016/S0959-4752(00)00014-1

Bragg, L., & Nicol, C. (2008). Designing open-ended problems to challenge preservice teachers’ views on mathematics and pedagogy. PME 32: Mathematical Ideas: History, Education and Cognition: Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education, 201–208.

Cai, J. (1995). A Cognitive Analysis of U. S. and Chinese Students’ Mathematical Performance on Tasks Involving Computation, Simple Problem Solving, and Complex Problem Solving. Journal for Research in Mathematics Education. Monograph, 7, i–151. https://doi.org/10.2307/749940

Cai, J. (2000). Mathematical Thinking Involved in U.S. and Chinese Students’ Solving of Process-Constrained and Process-Open Problems. Mathematical Thinking and Learning, 2(4), 309–340. https://doi.org/10.1207/S15327833MTL0204_4

Çekmez, E., Yildiz, C., & Bütüner, S. Ö. (2012). Fenomenografik Araştırma Yöntemi [Phenomenographic Research Method]. Necatibey Faculty of Education, Electronic Journal of Science and Mathematics Education, 6(2), 77.

Clarke, D. J., Sullivan, P., & Spandel, U. (1992). Student response characteristics to open-ended tasks in mathematical and other academic contexts (P. Sullivan, U. Spandel, & Australian Catholic University. Mathematics Teaching and Learning Centre (eds.)). The Centre.

Gracin, G. D. (2018). Requirements in mathematics textbooks: a five-dimensional analysis of textbook exercises and examples. International Journal of Mathematical Education in Science and Technology, 49(7), 1003–1024. https://doi.org/10.1080/0020739X.2018.1431849

Han, S., Rosli, R., Capraro, R. M., & Capraro, M. M. (2011). The Textbook Analysis on Probability: The Case of Korea, Malaysia and U.S. Textbooks. Journal of the Korean Society of Mathematical Education Series, 15(2), 127–140.

Hogan, T. P., & Murphy, G. (2007). Recommendations for Preparing and Scoring Constructed-Response Items: What the Experts Say. Applied Measurement in Education, 20(4), 427–441. https://doi.org/10.1080/08957340701580736

İnceçam, B., Demir, E., & Demir, E. (2018). Ortaokul Öğretmenlerinin Sınıf İçi Ölçme ve Değerlendirmelerde Yazılı Yoklamalarda Kullandıkları Açık Uçlu Maddeleri Hazırlama Yeterlikleri [Competencies of Middle School Teachers to Prepare Open-Ended Items Used in Open-Ended Test for In-Classroom Assess. İlköğretim Online, 17(4), 1912–1927. https://doi.org/10.17051/ilkonline.2019.506900

Kasar, N. (2013). Matematik Derslerinde Alternatif Çözüm Yollarına ve Farklı Soru Türlerine Ne Ölçüde Yer Verilmektedir?: Sınıf İçi Uygulamalardan Örnekler [To What Extent Alternative Solution Methods and Different Question Types Are Given Place in Mathematics Teaching?: E. University of Gaziantep.

Klavir, R., & Hershkovitz, S. (2008). Teaching and Evaluating ‘Open-Ended’ Problems. International Journal for Mathematics Teaching and Learning, 20(5), 1–24. https://www.cimt.org.uk/journal/klavir.pdf

Kwon, O. N., Park, J. H., & Park, J. S. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7(1), 51–61. https://doi.org/10.1007/BF03036784

Leung, S. S. (1997). On the open-ended nature in mathematical problem posing on the open-ended nature in mathematical problem posing. In E. Pehkonen (Ed.), Use of Open-ended Problems in Mathematics Classroom (pp. 26–33). Department of Teacher Education, University of Helsinki. http://coreylee.me/en/publications/2001_self-efficacy_change.pdf%5Cnhttp://files.eric.ed.gov/fulltext/ED419714.pdf

Lin, C.-Y., Becker, J., Ko, Y.-Y., & Byun, M.-R. (2013). Enhancing pre-service teachers’ fraction knowledge through open approach instruction. The Journal of Mathematical Behavior, 32(3), 309–330. https://doi.org/10.1016/j.jmathb.2013.03.004

Liz, Bills, Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. 30th Conference of the International Group for the Psychology of Mathematics Education, Leinhardt 2001, 126–154. http://users.mct.open.ac.uk/jhm3/PME30RF/PME30RFPaper.pdf

McMillan, J. H. (2017). Classroom Assessment: Principles and Practice that Enhance Student Learning and Motivation. Pearson Education.

Merriam, S. B., & Tisdell, E. J. (2016). Qualitative Research: A Guide to Design and Implementation. John Wiley & Sons.

MoNE. (2017). 8. Sınıf Merkezî Ortak Sınavlar Matematik Dersi Açık Uçlu Sorular Yapılandırılmış Cevapanahtarı Örnekleri [Open-Ended Questions for Eight-Grade Central Common Mathematics Exams and the Answer Key Examples]. http://odsgm.meb.gov.tr/meb_iys_dosyalar/2017_09/15135732_Mat_acik_uclu.pdf.

Nieminen, J. H., Chan, M. C. E., & Clarke, D. (2022). What affordances do open-ended real-life tasks offer for sharing student agency in collaborative problem-solving? Educational Studies in Mathematics, 109(1), 115–136. https://doi.org/10.1007/s10649-021-10074-9

Nohda, N. (2000). Teaching by open-approach method in Japanese mathematics classroom. Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education, 1, 39–53.

OECD. (2017). PISA 2015 (Assessment and Analytical Framework : Science, Reading, Mathematic, Financial Literacy and Collaborative Problem Solving). In Ming Pao Daily News. OECD Publishing.

ÖSYM. (2017). ÖSYS: Öğrenci Seçme ve Yerleştirme Sistemi [ÖSYS: Student Selection and Placement System]. https://www.osym.gov.tr/TR,12909/2017-lisans-yerlestirme-sinavlari-2017-lys-acik-uclu-sorular-hakkinda-bilgilendirme-ve-acik-uclu-soru-ornekleri-05012017.html

Pehkonen, E. (1997). Introduction to the concept “open-ended problem.” In E. Pehkonen (Ed.), Use of Open-ended Problems in Mathematics Classroom (pp. 7–11). Department of Teacher Education, University of Helsinki.

Pehkonen, E. (1999). In-service teachers’ conceptions on open tasks. In G. N. Philippou (Ed.), MAVI-8 Proceedings, Research on Mathematical Beliefs (pp. 87–95). University of Cyprus.

Reitman, W. R. (1966). Cognition and Thought: An Information Processing Approach. Wiley.

Shipman, M. D. (2014). The Limitations of Social Research. Taylor & Francis.

Silver, E. A. (1992). Assessment and mathematics education reform in the united states. International Journal of Educational Research, 17(5), 489–502. https://doi.org/10.1016/S0883-0355(05)80007-2

Silver, E. A. (1995). The Nature and Use of Open Problems in Mathematics Education: Mathematical and Pedagogical Perspectives.

Zentralblatt Fur Didaktik Der Mathematik/International Reviews on Mathematical Education, 27(2), 67–72.

Sullivan, P., & Clarke, D. (1992). Problem solving with conventional mathematics content: Responses of pupils to open mathematical tasks. Mathematics Education Research Journal, 4(1), 42–60. https://doi.org/10.1007/BF03217231

Sullivan, P., Warren, E., & White, P. (2000). Students’ responses to content specific open-ended mathematical tasks. Mathematics Education Research Journal, 12(1), 2–17. https://doi.org/10.1007/BF03217071

Sullivan, P., Warren, E., White, P., & Suwarsono, S. (1998). Different forms of mathematical questions for different purposes: Comparing student responses to similar closed and open-ended questions. In C. Kanes, E. Warren, & M. Goos (Eds.), Teaching Mathematics in New Times. MERGA.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619

Watson, A., & Mason, J. (2002). Extending example spaces as a learning/teaching strategy in mathematics. Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 4, 4–377.

Yang, D. C., Tseng, Y. K., & Wang, T. L. (2017). A comparison of geometry problems in middle-grade mathematics textbooks from Taiwan, Singapore, Finland, and the United States. Eurasia Journal of Mathematics, Science and Technology Education, 13(7), 2841–2857. https://doi.org/10.12973/eurasia.2017.00721a

Yıldırım, A., & Şimşek, H. (2011). Sosyal bilimlerde nitel araştırma yöntemleri [Qualitative research methods in the social sciences]. Seçkin.

Zaslavsky, O. (1995). Open-ended tasks as a trigger for mathematics teachers’ professional development. For the Learning of Mathematics, 15(3), 15–20. http://www.jstor.org/stable/40248183

Zhu, Y., & Fan, L. (2006). Focus on the Representation of Problem Types in Intended Curriculum: A Comparison of Selected Mathematics Textbooks from Mainland China and the United States. International Journal of Science and Mathematics Education, 4(4), 609–626. https://doi.org/10.1007/s10763-006-9036-9




DOI: https://doi.org/10.53400/mimbar-sd.v9i1.43742

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