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Abstract
Mathematical Modeling is the cornerstone and central aspect of the PISA framework. However, students still encounter difficulties in dealing with problems related to mathematical modeling. This study aims to produce a learning trajectory of mathematical modeling on functions material using the context of an online taxibike. Mathematical modeling activities on functions material are designed using the mathematical modeling cycle in the PISA framework. The subjects in this study were students of class X Bina Ilmi IT Senior High School Palembang. The research method used is validation studies. The research consisted of three stages, namely the preparation stage, the experimental design stage consisting of two cycles (cycle I: pilot experiment and cycle II: teaching experiment) and the retrospective analysis stage. Data was collected through observation, interviews and written test (students' work on the activity sheet and the results of pre-test and post-test). The results showed that learning activities that have been designed can assist students in solving mathematical modeling problems on functions. It begins with the formulate, employed, interpret/evaluate process. Through these mathematical modeling processes, students have a better understanding of functions and become accustomed to using mathematical modeling processes in solving contextual problems related to function material.
DOI: https://doi.org/10.22342/jpm.16.3.17228.289-302
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Copyright (c) 2022 Nety Wahyu Saputri, Zulkardi, Darmawijoyo
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References
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- Hoboken, NJ: Jossey-Bass. Bardini, C., Pierce, R., Vincent, J. and King, D. (2014). Undergraduate mathematics students’ understanding of the concept of function. Journal of Mathematics Education, 5(2): 85–107. https://doi.org/10.22342/jme.5.2.1495.85-107
- Blanton, M. et al. (2015). “A learning trajectory in 6-year-olds” Thinking about generalizing functional relationships. Journal for Research in Mathematics Education. 46(5). 511–558. https://doi.org/10.5951/jresematheduc.46.5.0511
- Bliss, K., & Libertini, J. (2016). Guidelines for Assessment & Instruction in Mathematical Modeling Education: What is Mathematical Modeling?. USA: Comap, Inc & Siam.
- Burns-Childers, A., & Vidakovic, D. (2017). Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative. International Journal of Mathematical Education in Science and Technology, 49(5), 660–679. doi:10.1080/0020739x.2017.1409367
- Charmila, N., Zulkardi, & Darmawijoyo. (2016). Developing mathematics problem of PISA model using the context of Jambi. Jurnal Penelitian dan Evaluasi Pendidikan, 20(2), 198-207. https://doi.org/10.21831/pep.v20i2.7444
- Czocher, J. (2018). How Does Validating Activity Contribute to The Modelling Process?. Educational Studies in Mathematics. 99(2), 137–159. DOI:10.1007/s10649-018-9833-4
- Dasaprawira, M. N., Zulkardi, & Susanti, E. (2019). Developing mathematics questions of PISA type using Bangka context. Journal on Mathematics Education, 10(2), 303-314. https://doi.org/10.22342/jme.10.2.5366.303-314.
- Doerr, H.M., Arleback, J.B., Misfeldt, M. (2017). Representations of modelling in mathematics education. Dalam G.A. Stillman et al. (eds.), Mathematical Modelling and Applications, International Perspectives on the Teaching and Learning of Mathematical Modelling (pp, 71- 81) Cham: Springer. https://doi.org/10.1007/978-3-319-62968-1_6
- Doorman, M. & Drijvers, P. (2011). Algebra in Function. In P. Drijvers (Ed.), Secondary Algebra Education (pp. 119-135). Rotterdam, the Netherlands: Sense Publishers.
- Efriani, A., Putri, R. I. I., & Hapizah. (2019). Sailing context in PISA-like mathematics problems. Journal on Mathematics Education, 10(2), 265-276. https://doi.org/10.22342/jme.10.2.5245.265-276.
- Frejd, P., & Bergsten, C. (2016). Mathematical modelling as a professional task. Educational Studies in Mathematics, 91(1), 11–35. https://doi.org/10.1007/s10649-015-9654-7
- Fuson, K.C. & Bransford, J.D. (2015). Mathematical Understanding: An Introduction. Dalam M. Suzanne Donovan and John D. Bransford (Ed). How Students Learn: History, Mathematics, and Science in the Classroom. 217-256.
- Jannah, R. D., Putri, R. I. I., & Zulkardi. (2019). Soft tennis and volleyball context in asian games for pisa-like mathematics problems. Journal on Mathematics Education, 10(1), 157-170. https://doi.org/10.22342/jme.10.1.5248.157-170.
- Jupri, A. & Drijver, P. (2016). Student difficulties in mathematizing word problems in algebra. Eurasia Journal of Mathematics, Science & Technology Education, 12(9): 2481-2502. doi: 10.12973/eurasia.2016.1299a
- Kaiser, G. (2017). The Teaching and Learning of Mathematical Modeling. Dalam J. Cai (Ed.), Compendium for Research In Mathematics Education (pp. 267–291). Reston, VA: The National Council of Teachers of Mathematics, Inc..
- Kalchman, M. & Koedinger, K. R. (2005). Teaching and Learning Functions. Dalam M. Suzanne Donovan and John D. Bransford, Editors. How Students Learn: History, Mathematics, and Science in the Classroom. 351-393.
- Lee, K.H, Chong, Y., Na, G. & Park, J. (2020). Korean Mathematics Education Meets Dutch Didactics. Dalam van den Heuval-Panhuizen, M. (Ed.), International reflections on the Netherlands didactics of mathematics: Visions on and experiences with Realistic Mathematics Education, (pp. 279-296). ICME-13 Monographs. Cham, Switzerland: Springer.
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- NCTM, (2021). Principles and Standards for School Mathematics. [online] Retrieved from https://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Algebra/.
- Niss, M. (2015). Prescriptive Modelling – Challenges and Opportunities. Dalam G. A. Stillman,
- W. Blum, & M. Biembengut (Eds.), Mathematical Modelling in Education Research and Practice: Cultural, Social And Cognitive Influences (pp. 67–79). Cham: Springer.
- Niss, Ma. (2017). Obstacles Related to Structuring for Mathematization Encountered by Students When Solving Physics Problems. International Journal of Science and Mathematics Education, 15, 1441–1462. https://doi.org/10.1007/s10763-016-9754-6
- Niss, M. & Blum, W. (2020). The Learning and Teaching of Mathematical Modelling. New York: Routledge.
- Novita, R., & Putra, M. (2016). Using task like PISA's problem to support students' creativity in mathematics. Journal on Mathematics Education, 7(1), 33-44. https://doi.org/10.22342/jme.7.1.2815.31-42
- Nusantara, D. S., Zulkardi, & Putri, R. I. I. (2021). Designing PISA-like mathematics task using a COVID-19 context (PISAComat). Journal on Mathematics Education, 12(2), 349-364. http://doi.org/10.22342/jme.12.2.13181.349-364.
- OECD. (2018). PISA 2021 Mathematics Framework (Draft). OECD Publishing, Paris.
- OECD. (2019), “PISA 2018 Mathematics Framework”, in PISA 2018 Assessment and Analytical Framework, OECD Publishing, Paris. DOI: https://doi.org/10.1787/13c8a22c-en
- Prahmana, R.C.I., Zulkardi, & Hartono, Y. (2012). Learning multiplication using indonesian traditional game in third grade. Journal on Mathematics Education, 3(2): 115-132. https://doi.org/10.22342/jme.3.2.1931.115-132.
- Putri, R. I. I. (2015). The interaction effect of learning approach in the formative test form of the mathematics learning outcomes [in Bahasa]. Jurnal Pendidikan dan Pembelajaran, 22(1): 69- 75.
- Putri, R. I. I., & Zulkardi. (2020). Designing PISA-like mathematics task using Asian Games context. Journal on Mathematics Education, 11(1), 135-144. http://dx.doi.org/10.22342/jme.11.1.9786.135-144.
- Stillman, G. & Brown, J. (2014). Evidence of implemented anticipation in mathematising by beginning modellers. Mathematics Education Research Journal, 26, 763–789. https://doi.org/10.1007/s13394-014-0119-6
- Sadan, S. (2020). Teachers’ Handbook Volume 1: Mathematical Literacy. New Delhi: Central Board of Secondary Education.
- Saputri, N. W., & Zulkardi. (2020). Development of LKPD mathematical modeling for junior high school students using the context of online motorcycle taxis [in Bahasa]. Jurnal Pendidikan Matematika, 14(1), 1-14. https://doi.org/10.22342/jpm.14.1.6825.1-14.
- Saputri, N.W., Turidho, A., Zulkardi, Darmawijoyo, & Somakim. (2020). Designing PISA-like mathematics task Content of Uncertainty and Data using a COVID-19 context [in Bahasa]. EDU-MAT: Jurnal Pendidikan Matematika, 8(2), 106-117. DOI: 10.20527/edumat.v8i1.8564
- Turidho, A., Putri, R. I. I., Susanti, E., & Johan, M. (2021). Developing PISA-like math tasks on algebra using arabic contexts. Jurnal Pendidikan Matematika, 15(2), 203-216. https://doi.org/10.22342/jpm.15.2.14847.203-216
- Wang, Y., Barmby, P., and Bolden, D. (2015). Understanding linear function: A comparison of selected textbooks from england and shanghai. International Journal of Science and Mathematics Education, 15(1): 131–153. DOI 10.1007/s10763-015-9674-x.
- Warsitasari, W.D., (2015). Algebraic thinking in mathematical problem solving [in Bahasa]. Apotema, 1 (1): 1-17.
- Widiati, I. (2015). Developing mathematical representation ability of junior high school students through contextual learning [in Bahasa]. Jurnal Pengajaran MIPA, 20(2): 106-111. http://dx.doi.org/10.18269/jpmipa.v20i2.571.
- Wijaya, A., Van den Heuvel-Panhuizen, M., Doorman, M., & Veldhuis, M. (2018). Opportunity- tolearn to solve context-based mathematics tasks and students’ performance in solving these tasks – Lessons from Indonesia. EURASIA Journal of Mathematics, Science and Technology Education, 14(10), em1598; https://doi.org/10.29333/ejmste/93420.
- Yuniati, S., Nusantara, T., Subanji, & Sulandra, I.M. (2019). The Use of Multiple Representation in Functional Thinking. International Journal of Recent Technology and Engineering (IJRTE), 8(1C2), 672-678.
- Zbiek, Blume, Heid (2018). Facilitator’s Guidebook for Use of Mathematics Situations in Professional Learning. USA: NCTM & IAP, Inc.
- Zulkardi & Kohar, A.W. (2018). Designing PISA-like mathematics tasks in Indonesia: Experiences and challenges. Journal of Physics: Conference Series, 947(1), 012015. https://doi.org/10.1088/1742-6596/947/1/012015
References
Andreescu, T., Cordeiro, K., & Andreescu, A. (2020). Teaching Mathematics with Problem Solving.
Hoboken, NJ: Jossey-Bass. Bardini, C., Pierce, R., Vincent, J. and King, D. (2014). Undergraduate mathematics students’ understanding of the concept of function. Journal of Mathematics Education, 5(2): 85–107. https://doi.org/10.22342/jme.5.2.1495.85-107
Blanton, M. et al. (2015). “A learning trajectory in 6-year-olds” Thinking about generalizing functional relationships. Journal for Research in Mathematics Education. 46(5). 511–558. https://doi.org/10.5951/jresematheduc.46.5.0511
Bliss, K., & Libertini, J. (2016). Guidelines for Assessment & Instruction in Mathematical Modeling Education: What is Mathematical Modeling?. USA: Comap, Inc & Siam.
Burns-Childers, A., & Vidakovic, D. (2017). Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative. International Journal of Mathematical Education in Science and Technology, 49(5), 660–679. doi:10.1080/0020739x.2017.1409367
Charmila, N., Zulkardi, & Darmawijoyo. (2016). Developing mathematics problem of PISA model using the context of Jambi. Jurnal Penelitian dan Evaluasi Pendidikan, 20(2), 198-207. https://doi.org/10.21831/pep.v20i2.7444
Czocher, J. (2018). How Does Validating Activity Contribute to The Modelling Process?. Educational Studies in Mathematics. 99(2), 137–159. DOI:10.1007/s10649-018-9833-4
Dasaprawira, M. N., Zulkardi, & Susanti, E. (2019). Developing mathematics questions of PISA type using Bangka context. Journal on Mathematics Education, 10(2), 303-314. https://doi.org/10.22342/jme.10.2.5366.303-314.
Doerr, H.M., Arleback, J.B., Misfeldt, M. (2017). Representations of modelling in mathematics education. Dalam G.A. Stillman et al. (eds.), Mathematical Modelling and Applications, International Perspectives on the Teaching and Learning of Mathematical Modelling (pp, 71- 81) Cham: Springer. https://doi.org/10.1007/978-3-319-62968-1_6
Doorman, M. & Drijvers, P. (2011). Algebra in Function. In P. Drijvers (Ed.), Secondary Algebra Education (pp. 119-135). Rotterdam, the Netherlands: Sense Publishers.
Efriani, A., Putri, R. I. I., & Hapizah. (2019). Sailing context in PISA-like mathematics problems. Journal on Mathematics Education, 10(2), 265-276. https://doi.org/10.22342/jme.10.2.5245.265-276.
Frejd, P., & Bergsten, C. (2016). Mathematical modelling as a professional task. Educational Studies in Mathematics, 91(1), 11–35. https://doi.org/10.1007/s10649-015-9654-7
Fuson, K.C. & Bransford, J.D. (2015). Mathematical Understanding: An Introduction. Dalam M. Suzanne Donovan and John D. Bransford (Ed). How Students Learn: History, Mathematics, and Science in the Classroom. 217-256.
Jannah, R. D., Putri, R. I. I., & Zulkardi. (2019). Soft tennis and volleyball context in asian games for pisa-like mathematics problems. Journal on Mathematics Education, 10(1), 157-170. https://doi.org/10.22342/jme.10.1.5248.157-170.
Jupri, A. & Drijver, P. (2016). Student difficulties in mathematizing word problems in algebra. Eurasia Journal of Mathematics, Science & Technology Education, 12(9): 2481-2502. doi: 10.12973/eurasia.2016.1299a
Kaiser, G. (2017). The Teaching and Learning of Mathematical Modeling. Dalam J. Cai (Ed.), Compendium for Research In Mathematics Education (pp. 267–291). Reston, VA: The National Council of Teachers of Mathematics, Inc..
Kalchman, M. & Koedinger, K. R. (2005). Teaching and Learning Functions. Dalam M. Suzanne Donovan and John D. Bransford, Editors. How Students Learn: History, Mathematics, and Science in the Classroom. 351-393.
Lee, K.H, Chong, Y., Na, G. & Park, J. (2020). Korean Mathematics Education Meets Dutch Didactics. Dalam van den Heuval-Panhuizen, M. (Ed.), International reflections on the Netherlands didactics of mathematics: Visions on and experiences with Realistic Mathematics Education, (pp. 279-296). ICME-13 Monographs. Cham, Switzerland: Springer.
Liputan6.com. (2020). 5 Go-Jek services experience a surge in users during the Covid-19 pandemic {in Bahasa}. Retrieved from https://www.liputan6.com/tekno/read/4298473/5-layanan-gojek- ini-alami-lonjakan-pengguna-selama-pendemi-covid-19.
Meryansumayeka, Zulkardi, Putri, R. I. I., & Hiltrimartin, C. (2020). Secondary students’ higher- order thinking skills in solving PISA-like mathematical tasks. Journal of Physics Conference Series, 1480(1),012034. https://doi:10.1088/1742-6596/1480/1/012034
NCTM, (2021). Principles and Standards for School Mathematics. [online] Retrieved from https://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Algebra/.
Niss, M. (2015). Prescriptive Modelling – Challenges and Opportunities. Dalam G. A. Stillman,
W. Blum, & M. Biembengut (Eds.), Mathematical Modelling in Education Research and Practice: Cultural, Social And Cognitive Influences (pp. 67–79). Cham: Springer.
Niss, Ma. (2017). Obstacles Related to Structuring for Mathematization Encountered by Students When Solving Physics Problems. International Journal of Science and Mathematics Education, 15, 1441–1462. https://doi.org/10.1007/s10763-016-9754-6
Niss, M. & Blum, W. (2020). The Learning and Teaching of Mathematical Modelling. New York: Routledge.
Novita, R., & Putra, M. (2016). Using task like PISA's problem to support students' creativity in mathematics. Journal on Mathematics Education, 7(1), 33-44. https://doi.org/10.22342/jme.7.1.2815.31-42
Nusantara, D. S., Zulkardi, & Putri, R. I. I. (2021). Designing PISA-like mathematics task using a COVID-19 context (PISAComat). Journal on Mathematics Education, 12(2), 349-364. http://doi.org/10.22342/jme.12.2.13181.349-364.
OECD. (2018). PISA 2021 Mathematics Framework (Draft). OECD Publishing, Paris.
OECD. (2019), “PISA 2018 Mathematics Framework”, in PISA 2018 Assessment and Analytical Framework, OECD Publishing, Paris. DOI: https://doi.org/10.1787/13c8a22c-en
Prahmana, R.C.I., Zulkardi, & Hartono, Y. (2012). Learning multiplication using indonesian traditional game in third grade. Journal on Mathematics Education, 3(2): 115-132. https://doi.org/10.22342/jme.3.2.1931.115-132.
Putri, R. I. I. (2015). The interaction effect of learning approach in the formative test form of the mathematics learning outcomes [in Bahasa]. Jurnal Pendidikan dan Pembelajaran, 22(1): 69- 75.
Putri, R. I. I., & Zulkardi. (2020). Designing PISA-like mathematics task using Asian Games context. Journal on Mathematics Education, 11(1), 135-144. http://dx.doi.org/10.22342/jme.11.1.9786.135-144.
Stillman, G. & Brown, J. (2014). Evidence of implemented anticipation in mathematising by beginning modellers. Mathematics Education Research Journal, 26, 763–789. https://doi.org/10.1007/s13394-014-0119-6
Sadan, S. (2020). Teachers’ Handbook Volume 1: Mathematical Literacy. New Delhi: Central Board of Secondary Education.
Saputri, N. W., & Zulkardi. (2020). Development of LKPD mathematical modeling for junior high school students using the context of online motorcycle taxis [in Bahasa]. Jurnal Pendidikan Matematika, 14(1), 1-14. https://doi.org/10.22342/jpm.14.1.6825.1-14.
Saputri, N.W., Turidho, A., Zulkardi, Darmawijoyo, & Somakim. (2020). Designing PISA-like mathematics task Content of Uncertainty and Data using a COVID-19 context [in Bahasa]. EDU-MAT: Jurnal Pendidikan Matematika, 8(2), 106-117. DOI: 10.20527/edumat.v8i1.8564
Turidho, A., Putri, R. I. I., Susanti, E., & Johan, M. (2021). Developing PISA-like math tasks on algebra using arabic contexts. Jurnal Pendidikan Matematika, 15(2), 203-216. https://doi.org/10.22342/jpm.15.2.14847.203-216
Wang, Y., Barmby, P., and Bolden, D. (2015). Understanding linear function: A comparison of selected textbooks from england and shanghai. International Journal of Science and Mathematics Education, 15(1): 131–153. DOI 10.1007/s10763-015-9674-x.
Warsitasari, W.D., (2015). Algebraic thinking in mathematical problem solving [in Bahasa]. Apotema, 1 (1): 1-17.
Widiati, I. (2015). Developing mathematical representation ability of junior high school students through contextual learning [in Bahasa]. Jurnal Pengajaran MIPA, 20(2): 106-111. http://dx.doi.org/10.18269/jpmipa.v20i2.571.
Wijaya, A., Van den Heuvel-Panhuizen, M., Doorman, M., & Veldhuis, M. (2018). Opportunity- tolearn to solve context-based mathematics tasks and students’ performance in solving these tasks – Lessons from Indonesia. EURASIA Journal of Mathematics, Science and Technology Education, 14(10), em1598; https://doi.org/10.29333/ejmste/93420.
Yuniati, S., Nusantara, T., Subanji, & Sulandra, I.M. (2019). The Use of Multiple Representation in Functional Thinking. International Journal of Recent Technology and Engineering (IJRTE), 8(1C2), 672-678.
Zbiek, Blume, Heid (2018). Facilitator’s Guidebook for Use of Mathematics Situations in Professional Learning. USA: NCTM & IAP, Inc.
Zulkardi & Kohar, A.W. (2018). Designing PISA-like mathematics tasks in Indonesia: Experiences and challenges. Journal of Physics: Conference Series, 947(1), 012015. https://doi.org/10.1088/1742-6596/947/1/012015