Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları

Muhammet Şahal; Ahmet Şükrü Özdemir

doi:10.9779/pauefd.1340106

TR EN

Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları

Öz

Matematiksel modelleme karmaşık gerçek yaşam problemlerinin çözümü için matematik eğitiminde öne çıkan araştırma alanlarından biridir. Çalışmanın amacı ortaokul matematik öğretmeni adaylarının hangi modelleme aşamalarını deneyimlediklerini, modelleme aşamalarında karşılaştıkları zorlukları ve modelleme döngülerinde ortaya çıkan rotaları incelemektir. Bu bağlamda durum çalışması ile yürütülen araştırmada yirmi bir ortaokul matematik öğretmeni adayı gruplar halinde üç modelleme problemi üzerinde çalışmıştır. Katılımcıların not defterlerinden, çözüm izleme şablonlarından, ses ve video kayıtlarından elde edilen veriler içerik analizi yöntemiyle analiz edilmiştir ve “modelleme rotalarını” görünür kılan modelleme döngüsüne aktarılmıştır. Gruplarda ortaya çıkan modelleme rotalarının çoğu “düzensiz” ve “tamamlanmış” kategorisinde değerlendirilmiştir. Modelleme rotalarında öğretmen adaylarının “durumun zihinsel temsili”, “gerçek model oluşturma” ve “matematiksel çözüm/sonuç” aşamalarında başarılı oldukları görülmüştür. Modelleme döngülerinde en fazla atlanan aşamaların “matematiksel model”, “gerçek sonuçları yorumlama” ve “doğrulama” aşamaları olduğu ortaya çıkmıştır. Ayrıca atlanan aşamalardan matematiksel “model aşamasına” dair öğretmen adaylarının zorluk yaşadıklarını belirttikleri görülürken; “gerçek sonuçları yorumlama” ve “doğrulama” aşamalarına dair zorluk yaşadıklarını belirten herhangi bir ifade görülmemiştir. Çalışmadan elde edilen sonuçların öğretmen adaylarının eğitiminde matematiksel modelleme uygulamalarına ışık tutacağı ve karşılaşılan zorluklara ilişkin literatüre katkıda bulunacağı düşünülmektedir.

Anahtar Kelimeler

matematiksel modelleme, modelleme döngüsü, modelleme rotaları, öğretmen adayları, matematiksel modelleme problemleri

Pre-service Mathematics Teachers’ Mathematical Modeling Experiences: Modeling Cycle Phases, Challenges, and Modeling Routes

Öz

Mathematical modeling is one of the prominent research areas in mathematics education for solving complex real-life problems. This study aims to reveal the mathematical modeling phases performed by pre-service mathematics teachers, the phases they have difficulties, and the modeling routes they follow in modeling cycles. In this case study, twenty-one pre-service mathematics teachers worked in groups on three modeling problems. We obtained the data from student notebooks, solution tracking templates, and audio and video recordings. We analyzed them using content analysis and transferred them to the modeling cycle, allowing us to see “modeling routes”. Most modeling routes were “irregular” and “completed.” In the modeling routes, pre-service mathematics teachers were successful in terms of “mental representation of the situation”, “creating the real model”, and “mathematical result(s)”. The most skipped phases in the modeling cycles were “creating a mathematical model,” “interpreting the real results,” and the “validation” phases. Furthermore, while they reported difficulties in “creating the mathematical model,” none of the pre-service mathematics teachers reported in the “interpretation” and “validation” phases. The study results are expected to shed light on mathematical modeling applications in the education of pre-service teachers and contribute to the literature on the difficulties encountered.

Anahtar Kelimeler

mathematical modeling, modeling cycle, modeling routes, pre-service math teacher, mathematical modeling problems

Kaynakça

Albayrak, H. B., & Tarim, K. (2022). Mathematical modelling competencies of pre-service primary school teachers’: The time at school. Journal of Theory and Practice in Education, 18(2), 95-112. https://doi.org/10.17244/eku.1163414
Alwast, A., & Vorhölter, K. (2022). Measuring pre-service teachers’ noticing competencies within a mathematical modeling context – an analysis of an instrument. Educational Studies in Mathematics 109, 263–285. https://doi.org/10.1007/s10649-021-10102-8
Anhalt, C.O., & Cortez, R. (2016). Developing understanding of mathematical modeling in secondary teacher preparation. J Math Teacher Educ, 19, 523–545 https://doi.org/10.1007/s10857-015-9309-8
Ärlebäck, J.B., & Doerr, H.M. (2018). Students’ interpretations and reasoning about phenomena with negative rates of change throughout a model development sequence. ZDM Mathematics Education 50, 187–200. https://doi.org/10.1007/s11858-017-0881-5
Aydin Güc, F., & Baki, A. (2019). Evaluation of the learning environment designed to develop student mathematics teachers’ mathematical modelling competencies. Teaching Mathematics and its Applications: An International Journal of the IMA, 38(4), 191–215. https://doi.org/10.1093/teamat/hry002
Berry, J. (2002). Developing mathematical modelling skills: the role of CAS. Zentralblatt für Didaktik der Mathematik-ZDM, 34(5), 212-220.
Berry, J. S., & Houston, S. K. (1995). Mathematical modelling. Edward Arnold.
Blum, W. (2002). ICMI Study 14: Applications and modelling in mathematics education-Discussion document. Educational Studies in Mathematics, 51, 149–171.
Blum, W. (2011). Can modelling be taught and learnt? Some answers from empirical research. G. Kaiser, W. Blum, R. Borromeo Ferri & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (pp. 15–30). Springer.
Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education (pp. 73–96). Springer.

Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of Mathematical Modelling and Application, 1(1) 45-58.
Blum, W., & Kaiser, G. (1997). Vergleichende empirische Untersuchungen zu mathematischen Anwendungsfähigkeiten von englischen und deutschen Lernenden. Unpublished application to Deutsche Forschungsgesellschaft.
Blum, W., & Leiß, D. (2007). How do students and teachers deal with modeling problems? C. Haines, P. Galbraith, W. Blum & S. Khan (Eds.), Mathematical modeling (ICTMA 12): Education, engineering and economics (pp. 222–231). Horwood Publishing.
Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. Zentralblatt für Didaktik der Mathematik-ZDM, 38(2), 86-95.
Borromeo Ferri, R. (2007). Personal experiences and extra-mathematical knowledge as an influence factor on modelling routes of pupils. Pitta-Pantazi, D & Philippou, G. (Ed.), CERME 5 – Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education, 2080-2089.
Borromeo Ferri, R. (2010). On the influence of mathematical thinking styles on learners’ modelling behavior. Journal für Mathematik-Didaktik, 31, 99–118. https://doi.org/10.1007/s13138-010-0009-8
Borromeo Ferri, R. (2011). Effective mathematical modelling without blockages - A commentary. G. Kaiser, W. Blum, R. B. Ferri ve G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling: The 14. ICMTA study içinde (pp. 181–185). Springer.
Borromeo Ferri, R. (2018). Learning how to teach mathematical modeling in school and teacher education. Springer.
Bukova Guzel, E. (2011). An examination of pre-service mathematics teachers’approaches to construct and solve mathematical modeling problems. Teaching Mathematics and Its Applications, 30(1), 19-36.
Chang, Y. P., Krawitz, J., Schukajlow, S., & Yang, K. L. (2019). Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions. Zentralblatt für Didaktik der Mathematik-ZDM, 52, 59-72. https://doi.org/10.1007/s11858-019-01090-4
Ciltas, A., & Isik, A. (2013). The effect of instruction through mathematical modelling on modelling skills of prospective elementary mathematics teachers. Educational Sciences: Theory & Practice, 13(2), 1187–1192.
Cohen, L., Manion, L., & Morrison, K., (2007). Research methods in education (Sixth Edition). Routledge.
Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods approaches (Third edition). Sage.
Creswell J. W., & Miller D. L. (2000). Determining validity in qualitative inquiry. Theory into Practice, 39, 124–130.
Czocher, J. A. (2016). Introducing modeling activity diagrams as a tool to connect mathematical modeling to mathematical thinking. Mathematical Thinking and Learning, 18(2), 77–106.
Czocher, J.A. (2018). How does validating activity contribute to the modeling process? Educational Studies in Mathematics, 99, 137–159. https://doi.org/10.1007/s10649-018-9833-4
Deniz, D., & Akgun, L. (2016). The sufficiency of high school mathematics teachers’ to design activities appropriate to model eliciting activities design principles. Karaelmas Journal of Educational Sciences, 4, 1-14.
Deniz, D., & Akgun, L. (2018). Investigation of prospective secondary mathematics teachers’ mathematical modellling skills. Mediterranean Journal of Educational Research, 12(24), 294-312. https://doi.org/10.29329/mjer.2018.147.16
Diefes-Dux, H. A., Zawojewski, J. S., Hjalmarson, M. A., & Cardella, M. E. (2012). A framework for analyzing feedback in a formative assessment system for mathematical modeling problems. Journal of Engineering Education, 101(2), 375–406. https://doi.org/10.1002/j.2168-9830.2012.tb00054.x
Doerr, H.M., Ärlebäck, J.B., & Misfeldt, M. (2017). Representations of modelling in mathematics education. In: Stillman, G., Blum, W., Kaiser, G. (eds) Mathematical Modelling and Applications. International Perspectives on the Teaching and Learning of Mathematical Modelling (pp. 71-81). Springer, Cham. https://doi.org/10.1007/978-3-319-62968-1_6
Doerr, H. M., & English, L. D. (2006). Middle grade teachers’ learning through students’ engagement with modeling tasks. Journal of Mathematics Teacher Education, 9(1), 5–32. https://doi.org/10.1007/s10857-006-9004-x
Duran, M., Doruk, M., & Kaplan, A. (2016). Mathematical Modeling Processes of Mathematics Teacher Candidates: The Example of Tortoise Paradox. Cumhuriyet International Journal of Education, 5(4), 55–71. https://doi.org/10.30703/cije.321415
English, L. D., Ärlebäck, J. B., & Mousoulides, N. (2016). Reflections on progress in mathematical modelling research. A. Gutierrez, G. Leder & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 383–413). Sense Publishers.
English, L. D., & Mousoulides, N. G. (2015). Bridging STEM in a real-world problem. Mathematics Teaching in the Middle School, 20(9), 532–539. https://doi.org/10.5951/mathteacmiddscho.20.9.0532
English, L., & Watters, J. (2004). Mathematical modeling in the early school years. Mathematics Education Research Journal, 16(3), 59–80.
Eraslan, A. (2012). Prospective elementary mathematics teachers’ thought processes on a model eliciting activity. Educational Sciences: Theory & Practice, 12(4), 12-16.
Etsey, Y. K. (1997). Teachers and administrators' perspectives and use of standardized achievement tests: A review of published research. Paper presented at the annual meeting of American Educational Research Center, Chicago, IL.
Ferrando, I., & Albarracín, L. (2019). Students from grade 2 to grade 10 solving a Fermi problem: Analysis of emerging models. Mathematics Education Research Journal, 33, 61-78. https://doi.org/10.1007/s13394-019-00292-z
Gibbs, G. R. (2007). Analyzing qualitative data. U. Flick (Eds.), The SAGE qualitative research kit (pp. 100–108). Sage.
Goos, M. (2002). Understanding metacognitive failure. Journal of Mathematical Behavior, 21(3), 283–302.
Hess, F. M. (2002). Reform, resistance, ... retreat? The predictable politics of accountability in Virginia. In D. Ravitch (Ed.), Brookings papers on education policy (pp. 69–122). Brookings Institution Press.
Hidiroglu, C. N., Ozaltun Celik, A., Kula Unver, S., & Bukova Guzel, E. (2018). Prospective mathematics teachers' actions in technology-aided mathematical modeling process: distance problem. Erzincan University Journal of Education Faculty, 20 (3), 782-809. https://doi.org/10.17556/erziefd.441732
Julie, C. (2020). Modelling competencies of school learners in the beginning and final year of secondary school mathematics. International Journal of Mathematical Education in Science and Technology, 51(8), 1181–1195. https://doi.org/10.1080/0020739X.2020.1725165
Kaiser, G. (2005). Mathematical modelling in school – Examples and experiences. Kaiser, G. & Henn, H.-W. (Eds.), Mathematikunterricht im Spannungsfeld von Evaluation und Evolution (pp. 99-108). Franzbecker.
Kaiser, G. (2017). The Teaching and Learning of Mathematical Modeling. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 267–291). National Council of Teachers of Mathematics.
Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. G. Stillman, W. Blum & M. S. Biembengut (Eds.), Mathematical modelling in education research and practice. Cultural, social and cognitive influences (pp. 129–149). Springer.
Kaiser, G., Schwarz, B., & Tiedemann, S. (2010). Future teachers’ professional knowledge on modeling. R. Lesh, P. L. Galbraith, C. R. Haines & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies, ICTMA 13 (pp. 433–444). Springer.
Karahan, M., & Ergene, O. (2023). Investigation of pre-service mathematics teachers’ modeling processes in the context of crop ınsurance model-eliciting activity. Sakarya University Journal of Education Faculty, 23(1), 1-22. https://doi.org/10.53629/sakaefd.1271618
Kaya, D. & Kesan, C. (2022). Mathematical modelling processes of elementary mathematics teacher candidates: An example of waste of water. Van Yüzüncü Yıl University Journal of Education, 19(3), 1068-1097. https://doi.org/10.33711/yyuefd.1177845
Kaygisiz, I., & Senel, E. A. (2023). Investigating mathematical modeling competencies of primary school students: Reflections from a model eliciting activity. Journal of Pedagogical Research, 7(1), 1-24. https://doi.org/10.33902/JPR.202317062
Leiss, D., Plath, J., & Schwippert, K. (2019). Language and mathematics - Key factors influencing the comprehension process in reality-based tasks. Mathematical Thinking and Learning, 21, 131–153. https://doi.org/10.1080/10986065.2019.1570835
Leong, R. K. E. (2012). Assessment of mathematical modeling. Journal of Mathematics Education at Teachers College, 3, 61–65.
Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modelling perspective on mathematics teaching, learning, and problem solving. R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: models and modelling perspectives on mathematics problem solving, learning and teaching (pp. 3–33). Lawrance Erlbaum Associates Publishers.
Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5, 109–130. https://doi.org/10.1080/10986065.2003.9679996
Maaß, K. (2006). What are modelling competencies? The International Journal on Mathematics Education, 38(2), 113–142.
Maaß, K. (2007). Modelling in class: What do we want the students to learn? C. Haines, P. Galbraith, W. Blum & S. Khan (Ed.), Mathematical modelling (ICTMA 12): Education, engineering, and economics: Proceedings from the twelfth International Conference on the Teaching of Mathematical Modelling and Applications (p. 63–78). Horwood.
Maaß, K. (2010). Classification scheme for modelling tasks. Journal für Mathematik-Didaktik, 31, 285-311. McMillan, J., & Schumacher, S. (2014). Research in education. Evidence-based inquiry (Seventh edition). Pearson.
Miles, M, B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded Sourcebook (2nd ed). Sage.
MoNE (Ministry of National Education) (2018). Mathematics Curriculum (Elementary and Secondary School Year 1, 2, 3, 4, 5, 6, 7, and 8.). MEB.
NCTM (2000). Principals and standards for school Mathematics. Reston, Va: National Council of Teachers of Mathematics Pub.
Niss, M. (2010). Modeling a crucial aspect of students’ mathematical modeling. R. Lesh, P. Galbraith, C. R. Haines & A. Hurford (Ed.), Modeling students’ mathematical competencies (pp. 43–59). Springer.
Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. London & New York: Routledge.
Patton, M. Q. (2002). Qualitative research & evaluation methods (3rd ed.). Sage.
Ramírez-Montes, G., Henriques, A., & Carreira, S. (2021). Undergraduate students’ learning of linear algebra through mathematical modelling routes. Canadian Journal of Science, Mathematics and Technology Education, 21, 357–377 (2021). https://doi.org/10.1007/s42330-021-00149-3
Schukajlow, S., Kaiser, G., & Stillman, G. (2021). Modeling from a cognitive perspective: theoretical considerations and empirical contributions. Mathematical Thinking and Learning. https://doi.org/10.1080/10986065.2021.2012631
Schukajlow, S., Kolter, J., & Blum, W. (2015). Scaffolding mathematical modelling with a solution plan. Zentralblatt für Didaktik der Mathematik-ZDM, 47(7), 1241–1254. https://doi.org/10.1007/s11858-015-0707-2
Schukajlow, S., Krug, A., & Rakoczy, K. (2015). Effects of prompting multiple solutions for modelling problems on students’ performance. Educational Studies in Mathematics, 89(3), 393–417. https://doi.org/10.1007/s10649-015-9608-0
Sevinc, S. ve Lesh, R. (2018). Training mathematics teachers for realistic math problems: A case of modeling-based teacher education courses. ZDM Mathematics Education, 50, 301–314. https://doi.org/10.1007/s11858-017-0898-9
Shahbari, J. A., & Tabach, M. (2020). Features of modeling processes that elicit mathematical models represented at different semiotic registers. Educational Studies in Mathematics, 105, 115–135. https://doi.org/10.1007/s10649-020-09971-2
Simon, L. H., & Cox, D. C. (2019). The role of prototyping in mathematical design thinking. The Journal of Mathematical Behavior, 56, 100724. https://doi.org/10.1016/j.jmathb.2019.100724
Sol, M., Giménez, J., & Rosich, N. (2011). Project modelling routes in 12–16-year-old pupils. G. Kaiser, W. Blum, R. B. Ferri &G. Stillman (Ed.), Trends in teaching and learning of mathematical modelling (ICTMA 14) (pp. 231–240). Springer.
Stecher, B. M. (2002). Consequences of large-scale, high stakes testing on school and classroom practice. L. S. Hamilton, B. M. Stecher& S. P. Klein (Eds.), Making sense of test-based accountability in education (pp. 79–100). RAND Corporation.
Stillman, G.A. (2015). Applications and modelling research in secondary classrooms: What have we learnt?. Cho, S. (Eds.), Selected Regular Lectures from the 12th International Congress on Mathematical Education (pp. 791–805). Springer. https://doi.org/10.1007/978-3-319-17187-6_44
Stillman, G., Brown, J., & Galbraith, P. (2010). Identifying challenges within transition phases of mathematical modelling activities at year 9. Lesh, P. L. Galbraith, C. R. Haines & A. Hurford (Eds.), Modelling students’ mathematical modelling competencies ICTMA 13 (pp. 385–395). Springer.
Stillman, G., Galbraith, P., Brown, J., & Edwards, I. (2007). A framework for success in implementing mathematical modelling in the secondary classroom. J. Watson & K. Beswick (Eds.). Mathematics: Essential research, essential practice (pp. 691–700). Merga.
Stillman, G. A., Kaiser, G., Blum, W., & Brown, J. P. (2013). Mathematical modelling: Connecting to teaching and research practices–The impact of globalisation. G. A. Stillman, G. Kaiser, W. Blum & J. P. Brown (Ed.), Teaching mathematical modelling: Connecting to research and practice (pp. 1–24). Springer.
Stohlmann, M., & Yang, Y. (2021). Investigating the alignment to mathematical modelling of teacher-created mathematical modelling activities available online. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2021.1961030
Sen Zeytun, A. (2013). An Investigatıon of Prospective Teachers’ Mathematical Modelling Processes And Their Views About Factors Affecting These Processes [Unpublished doctoral dissertation, Middle East Technical University]. National Theses Center.
Sen Zeytun, A., Cetinkaya, B., & Erbas, A. (2017). Understanding prospective teachers’ mathematical modeling processes in the context of a mathematical modeling course. Eurasia Journal of Mathematics, Science & Technology Education, 13(3), 691–722. https://doi.org/10.12973/eurasia.2017.00639a
Taşpınar Şener, Z. (2017). The Examination of Preservice Middle School Mathematics Teachers' Model Eliciting Activities and Their Opinions on the Use of These Activities in the Teaching Process [Unpublished doctoral dissertation, Gazi University]. National Theses Center.
Tekin Dede, A., & Yilmaz, S. (2013). Examination of primary mathematics student teachers’ modelling competencies. Turkish Journal of Computer and Mathematics Education, 4(3), 185-206.
Turker Biber, D. B., & Yetkin Ozdemir, I. (2021). Teacher’s noticing and noticing strategies about student's thinking in the context of mathematical modeling activities. Pamukkale University Journal of Education, (53), 521-554. https://doi.org/10.9779/pauefd.761629
Vorhölter, K. (2018). Conceptualization and measuring of metacognitive modeling competencies: Empirical verification of theoretical assumptions. ZDM Mathematics Education, 50, 343–354. https://doi.org/10.1007/s11858-017-0909-x
Wagner, T. (2008). Rigor redefined. Educational Leadership, 66(2), 20–24.
Weber, R. P. (1990). Basic content analysis. Sage.
Wessels, H. (2014). Levels of mathematical creativity in model-eliciting activities. Journal of Mathematical Modelling and Application, 1(9), 22–40.
Yenmez, A.A., & Erbas, A.K. (2022). Facilitating a Sustainable Transformation of Sociomathematical Norms Through Mathematical Modeling Activities. International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-022-10275-5
Yilmaz, S., & Tekin Dede, A. (2016). Mathematization competencies of pre-service elementary mathematics teachers in the mathematical modelling process. International Journal of Education in Mathematics, Science and Technology, 4(4), 284-298. https://doi.org/10.18404/ijemst.39145
Zawojewski, J. (2010). Problem solving versus modeling. R. Lesh, P. Galbraith, C. R. Haines & A. Hurford (Ed.), Modeling students’ mathematical modeling competencies: ICTMA 13 (pp. 237-244). Springer.
Zbiek, R., M., & Conner, A. (2006). Beyond motivation: Exploring mathematical modeling as a context for deepening students’ understandings of curricular mathematics. Educational Studies in Mathematics, 69, 89-112.

Ayrıntılar

Birincil Dil

Türkçe

Konular

Matematik Eğitimi

Bölüm

Araştırma Makalesi

Yazarlar

Muhammet Şahal ^*
0000-0003-3625-2456
Türkiye

Ahmet Şükrü Özdemir
0000-0002-0597-3093
Türkiye

Erken Görünüm Tarihi

9 Ekim 2024

Yayımlanma Tarihi

20 Ocak 2025

Gönderilme Tarihi

9 Ağustos 2023

Kabul Tarihi

9 Eylül 2024

Yayımlandığı Sayı

Yıl 2025 Sayı: 63

DOI

https://doi.org/10.9779/pauefd.1340106

IZ

https://izlik.org/JA89EK55UT

APA

Şahal, M., & Özdemir, A. Ş. (2025). Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 63, 1-38. https://doi.org/10.9779/pauefd.1340106

AMA

1.Şahal M, Özdemir AŞ. Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları. PAÜEFD. 2025;(63):1-38. doi:10.9779/pauefd.1340106

Chicago

Şahal, Muhammet, ve Ahmet Şükrü Özdemir. 2025. “Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları”. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, sy 63: 1-38. https://doi.org/10.9779/pauefd.1340106.

EndNote

Şahal M, Özdemir AŞ (01 Ocak 2025) Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi 63 1–38.

IEEE

[1]M. Şahal ve A. Ş. Özdemir, “Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları”, PAÜEFD, sy 63, ss. 1–38, Oca. 2025, doi: 10.9779/pauefd.1340106.

ISNAD

Şahal, Muhammet - Özdemir, Ahmet Şükrü. “Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları”. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi. 63 (01 Ocak 2025): 1-38. https://doi.org/10.9779/pauefd.1340106.

JAMA

1.Şahal M, Özdemir AŞ. Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları. PAÜEFD. 2025;:1–38.

MLA

Şahal, Muhammet, ve Ahmet Şükrü Özdemir. “Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları”. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, sy 63, Ocak 2025, ss. 1-38, doi:10.9779/pauefd.1340106.

Vancouver

1.Muhammet Şahal, Ahmet Şükrü Özdemir. Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları. PAÜEFD. 01 Ocak 2025;(63):1-38. doi:10.9779/pauefd.1340106

27767 https://creativecommons.org/licenses/by-nc-nd/4.0/

Matematik Öğretmen Adaylarının Matematiksel Modelleme Deneyimleri: Modelleme Döngüsü Aşamaları, Zorluklar ve Modelleme Rotaları

Öz

Anahtar Kelimeler

Pre-service Mathematics Teachers’ Mathematical Modeling Experiences: Modeling Cycle Phases, Challenges, and Modeling Routes

Öz

Anahtar Kelimeler

Kaynakça

Ayrıntılar

Birincil Dil

Konular

Bölüm

Yazarlar

Erken Görünüm Tarihi

Yayımlanma Tarihi

Gönderilme Tarihi

Kabul Tarihi

Yayımlandığı Sayı

DOI

IZ

Kaynak Göster