Main Article Content
Abstract
Keywords
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
- Abrahamson, D., & Cendak, M. R. (2006). The Odds of Understanding the Law of Large Numbers: A Design for Grounding Intuitive Probability in Combinatorial Analysis. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, 2, 1–8.
- 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
- Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Efect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181–199. https://doi.org/10.1023/A:1002954428327
- Busadee, N., & Laosinchai, P. (2013). Authentic Problems in High School Probability Lesson: Putting Research into Practice. Procedia - Social and Behavioral Sciences, 93, 2043–2047. https://doi.org/10.1016/j.sbspro.2013.10.162
- Cao, Y., Zhang, S., Chan, M. C. E., & Kang, Y. (2021). Post-pandemic reflections: lessons from Chinese mathematics teachers about online mathematics instruction. In Asia Pacific Education Review (Vol. 22, Issue 2, pp. 157–168). https://doi.org/10.1007/s12564-021-09694-w
- Carrillo-Yañez, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, Á., Ribeiro, M., & Muñoz-Catalán, M. C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model*. Research in Mathematics Education, 20(3), 236–253. https://doi.org/10.1080/14794802.2018.1479981
- Chick, H., & Beswick, K. (2018). Teaching teachers to teach Boris: a framework for mathematics teacher educator pedagogical content knowledge. Journal of Mathematics Teacher Education, 21(5), 475–499. https://doi.org/10.1007/s10857-016-9362-y
- Chotikarn, N., Kanhapong, A., Tupsai, J., & Yuenyong, C. (2021). Enhancing Grade 11 students ’ representation and connection in permutation and combination for their problem solving Enhancing Grade 11 students ’ representation and connection in permutation and combination for their problem solving. Jin Journal of Physics: Conference Series, 1835(1), 012023. https://doi.org/10.1088/1742-6596/1835/1/012023
- Eizenberg, M. M., & Zaslavsky, O. (2004). Students ’ Verification Strategies for Combinatorial Problems. Mathematical Thinking and Learning, 6(1), 15–36. https://doi.org/10.1207/s15327833mtl0601
- English, L. D. (2005). Combinatorics and the development of children’s combinatorial reasoning. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 121–141). Kluwer Academic Publishers.
- Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. Zentralblatt Für Didaktik Der Mathematik, 5, 193–198.
- Garfield, J., & Ahlgren, A. (1988). Difficulties in Learning Basic Concepts in Probability and Statistics: Implications for Research. Journal for Research in Mathematics Education, 19(1), 44–63.
- Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.
- Hsieh, H. F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288. https://doi.org/10.1177/1049732305276687
- Hubeňáková, V., & Semanišinová, I. (2020). Preservice teachers’ mathematical knowledge for teaching combinatorial thinking. Eleventh Congress of the European Society for Research in Mathematics Education (ERME), 0243042.
- Kemdikbud, Dirjen GTK. (2020). Pedoman Program Pendidikan Profesi Guru dalam Jabatan [In-service Teacher Professional Education Program Guidelines]. Jakarta: Direktorat Pendidikan Profesi dan Pembinaan Guru dan Tenaga Kependidikan
- Korkmaz, H. I., & Şahin, Ö. (2020). Preservice Preschool Teachers’ Pedagogical Content Knowledge on Geometric Shapes in Terms of Children’s Mistakes. Journal of Research in Childhood Education, 34(3), 385–405. https://doi.org/10.1080/02568543.2019.1701150
- Kuckartz, U. (2019). Qualitative Text Analysis: A Systematic Approach. In Compendium for early career researchers in mathematics education (pp. 181–197). Springer, Cham. https://doi.org/10.1007/978-3-030-15636-7_8
- Kuo, T. (2009). A New Method for Generating Permutations in Lexicographic Order. Journal of Science and Engineering Technology, 5(4), 21–29.
- Lamanna, L., Gea, M. M., & Batanero, C. (2022a). Do Secondary School Students’ Strategies in Solving Permutation and Combination Problems Change with Instruction? Canadian Journal of Science, Mathematics and Technology Education, 1–15. https://doi.org/10.1007/s42330-022-00228-z
- Lamanna, L., Gea, M. M., & Batanero, C. (2022b). Secondary school students’ strategies in solving combination problems. In Bozen-Bolzano (Ed.), Twelfth Congress of the European Society for Research in Mathematics Edu- cation (CERME12) (p. hal-03751834).
- Lapan, S. D., Quartaroli, M. T., & Riemer, J. F. (2012). Qualitative Research: An Introduction to Methods and Designs; Wiley:Weinheim.
- Lockwood, E. (2011). Student connections among counting problems: An exploration using actor-oriented transfer. Educational Studies in Mathematics, 78(3), 307–322. https://doi.org/10.1007/s10649-011-9320-7
- Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32(2), 251–265. https://doi.org/10.1016/j.jmathb.2013.02.008
- Lockwood, E., A. Swinyard, C., & S. Caughman, J. (2015). Patterns, Sets of Outcomes, and Combinatorial Justification: Two Students’ Reinvention of Counting Formulas. In International Journal of Research in Undergraduate Mathematics Education (Vol. 1, Issue 1). https://doi.org/10.1007/s40753-015-0001-2
- Lockwood, E., Reed, Z., & Caughman, J. S. (2017). An Analysis of Statements of the Multiplication Principle in Combinatorics, Discrete, and Finite Mathematics Textbooks. International Journal of Research in Undergraduate Mathematics Education, 3(3), 381–416. https://doi.org/10.1007/s40753-016-0045-y
- Loong, E. Y. K. (2014). Using the internet in high school mathematics. Journal on Mathematics Education, 5(2), 108–126. https://doi.org/10.22342/jme.5.2.1496.108-126
- Loughran, J., Berry, A., & Mulhall, P. (2012). Understanding and Developing Science Teachers Pedagogical Content Knowledge 2nd Edition. Sense Publishers.
- Loughran, J., Mulhall, P., & Berry, A. (2004). In Search of Pedagogical Content Knowledge in Science: Developing Ways of Articulating and Documenting Professional Practice. Journal of Research in Science Teaching, 41(4), 370–391. https://doi.org/10.1002/tea.20007
- Lune, H., & Berg, B. L. (2017). Qualitative Research Methods for the Social Sciences (9th Edition). In Pearson Education Limited (Ninth edit). Pearson Education Limited.
- Martin, G. (2001). Counting: The Art of Enumerative Combinatorics. Springer.
- Melusova, J., & Vidermanova, K. (2015). Upper-secondary Students’ Strategies for Solving Combinatorial Problems. Procedia - Social and Behavioral Sciences, 197(February), 1703–1709. https://doi.org/10.1016/j.sbspro.2015.07.223
- NCTM. (2000). Principles Standards and for School Mathematics. Reston, VA
- Permendikbud No. 37. (2018). Perubahan atas Peraturan Menteri Pendidikan dan Kebudayaan Nomor 24 Tahun 2016 tentang Kompetensi Inti dan Kompetensi Dasar Pelajaran pada Kurikulum 2013 pada Pendidikan Dasar dan Pendidikan Menengah. Available online at jdih.kemdikbud.go.id (accessed on 4 May 2022)
- Quinn, J. R., & Wiest, R. L. (1998). A Constructivist Approach to Teaching Permutations and Combinations. Teaching Statistic, 20(3), 75–77. https://doi.org/10.20305/it202202111142
- Rungtusanatham, M., Wallin, C., & Eckerd, S. (2011). The vignette in a scenario-based role-playing experiment. Journal of Supply Chain Management, 47(3), 9–16. https://doi.org/10.1111/j.1745-493X.2011.03232.x
- Salavatinejad, N., Alamolhodaei, H., & Radmehr, F. (2021). Toward a model for students’ combinatorial thinking. Journal of Mathematical Behavior, 61(100823), 1–19. https://doi.org/10.1016/j.jmathb.2020.100823
- Sandefur, J., Lockwood, E., Hart, E., & Greefrath, G. (2022). Teaching and Learning Discrete Mathematics. ZDM-Mathematics Education, https://do. https://doi.org/10.1007/978-94-007-4978-8_51
- Semanišinová, I. (2021). Multiple-solution tasks in pre-service teachers’ course on combinatorics. Mathematics, 9(18). https://doi.org/10.3390/math9182286
- Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2), 4–14.
- Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–23.
- Siegel, M. (1986). Report of The Committee on Discrete Mathematics in the First Two Years.
- Smith, C. E. (2007). On Students’ Conceptualizations of Combinatorics: A Multiple Case Study. University od Minnesota.
- Soto, O., Siy, K., & Harel, G. (2022). Promoting a set-oriented way of thinking in a U.S. High School discrete mathematics class: a case study. ZDM - Mathematics Education, 0123456789. https://doi.org/10.1007/s11858-022-01337-7
- Subanji. (2015). Teori Kesalahan Konstruksi Konsep dan Pemecahan Masalah Matematika [Error Theory of Concept Construction and Mathematical Problem Solving]. Universitas Negeri Malang: UM Press.
- Sukoriyanto, S., Nusantara, T., Subanji, S., & Candra, T. D. (2016a). Students’ Thinking Interference of Real Global Type in Solving Permutation and Combination Problems. IOSR Journal of Research & Method in Education, 6(5), 47–50. https://doi.org/10.9790/7388-0605034750
- Sukoriyanto, S., Nusantara, T., Subanji, S., & Chandra, T. D. (2016b). Students’ Errors in Solving the Permutation and Combination Problems Based on Problem Solving Steps of Polya. International Education Studies, 9(2), 11. https://doi.org/10.5539/ies.v9n2p11
- Tchoshanov, M. A. (2011). Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 76(2), 141–164. https://doi.org/10.1007/s10649-010-9269-y
- Turkdogan, A., Baki, A., & Cepni, S. (2009). The Anatomy of Mistakes: Categorizing Students ’ Mistakes in Mathematics within Learning Theories. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 1(1), 13–26. https://doi.org/10.16949/turcomat.09564
- Yetkin, E. (2003). Student Difficulties in Learning Elementary Student Difficulties in Learning Elementary. ERIC Digest., 1–6.
- Yunianto, W., Prahmana, R. C. I., & Crisan, C. (2021). Indonesian mathematics teachers’ knowledge of content and students of area and perimeter of rectangle. Journal on Mathematics Education, 12(2), 223–238. https://doi.org/10.22342/JME.12.2.13537.223-238
- Zhang, Y. (2015). Pedagogical content knowledge in early mathematics: What teachers know and how it associates with teaching and learning. Loyola University, Chicago, IL.
References
Abrahamson, D., & Cendak, M. R. (2006). The Odds of Understanding the Law of Large Numbers: A Design for Grounding Intuitive Probability in Combinatorial Analysis. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, 2, 1–8.
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
Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Efect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181–199. https://doi.org/10.1023/A:1002954428327
Busadee, N., & Laosinchai, P. (2013). Authentic Problems in High School Probability Lesson: Putting Research into Practice. Procedia - Social and Behavioral Sciences, 93, 2043–2047. https://doi.org/10.1016/j.sbspro.2013.10.162
Cao, Y., Zhang, S., Chan, M. C. E., & Kang, Y. (2021). Post-pandemic reflections: lessons from Chinese mathematics teachers about online mathematics instruction. In Asia Pacific Education Review (Vol. 22, Issue 2, pp. 157–168). https://doi.org/10.1007/s12564-021-09694-w
Carrillo-Yañez, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, Á., Ribeiro, M., & Muñoz-Catalán, M. C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model*. Research in Mathematics Education, 20(3), 236–253. https://doi.org/10.1080/14794802.2018.1479981
Chick, H., & Beswick, K. (2018). Teaching teachers to teach Boris: a framework for mathematics teacher educator pedagogical content knowledge. Journal of Mathematics Teacher Education, 21(5), 475–499. https://doi.org/10.1007/s10857-016-9362-y
Chotikarn, N., Kanhapong, A., Tupsai, J., & Yuenyong, C. (2021). Enhancing Grade 11 students ’ representation and connection in permutation and combination for their problem solving Enhancing Grade 11 students ’ representation and connection in permutation and combination for their problem solving. Jin Journal of Physics: Conference Series, 1835(1), 012023. https://doi.org/10.1088/1742-6596/1835/1/012023
Eizenberg, M. M., & Zaslavsky, O. (2004). Students ’ Verification Strategies for Combinatorial Problems. Mathematical Thinking and Learning, 6(1), 15–36. https://doi.org/10.1207/s15327833mtl0601
English, L. D. (2005). Combinatorics and the development of children’s combinatorial reasoning. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 121–141). Kluwer Academic Publishers.
Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. Zentralblatt Für Didaktik Der Mathematik, 5, 193–198.
Garfield, J., & Ahlgren, A. (1988). Difficulties in Learning Basic Concepts in Probability and Statistics: Implications for Research. Journal for Research in Mathematics Education, 19(1), 44–63.
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.
Hsieh, H. F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288. https://doi.org/10.1177/1049732305276687
Hubeňáková, V., & Semanišinová, I. (2020). Preservice teachers’ mathematical knowledge for teaching combinatorial thinking. Eleventh Congress of the European Society for Research in Mathematics Education (ERME), 0243042.
Kemdikbud, Dirjen GTK. (2020). Pedoman Program Pendidikan Profesi Guru dalam Jabatan [In-service Teacher Professional Education Program Guidelines]. Jakarta: Direktorat Pendidikan Profesi dan Pembinaan Guru dan Tenaga Kependidikan
Korkmaz, H. I., & Şahin, Ö. (2020). Preservice Preschool Teachers’ Pedagogical Content Knowledge on Geometric Shapes in Terms of Children’s Mistakes. Journal of Research in Childhood Education, 34(3), 385–405. https://doi.org/10.1080/02568543.2019.1701150
Kuckartz, U. (2019). Qualitative Text Analysis: A Systematic Approach. In Compendium for early career researchers in mathematics education (pp. 181–197). Springer, Cham. https://doi.org/10.1007/978-3-030-15636-7_8
Kuo, T. (2009). A New Method for Generating Permutations in Lexicographic Order. Journal of Science and Engineering Technology, 5(4), 21–29.
Lamanna, L., Gea, M. M., & Batanero, C. (2022a). Do Secondary School Students’ Strategies in Solving Permutation and Combination Problems Change with Instruction? Canadian Journal of Science, Mathematics and Technology Education, 1–15. https://doi.org/10.1007/s42330-022-00228-z
Lamanna, L., Gea, M. M., & Batanero, C. (2022b). Secondary school students’ strategies in solving combination problems. In Bozen-Bolzano (Ed.), Twelfth Congress of the European Society for Research in Mathematics Edu- cation (CERME12) (p. hal-03751834).
Lapan, S. D., Quartaroli, M. T., & Riemer, J. F. (2012). Qualitative Research: An Introduction to Methods and Designs; Wiley:Weinheim.
Lockwood, E. (2011). Student connections among counting problems: An exploration using actor-oriented transfer. Educational Studies in Mathematics, 78(3), 307–322. https://doi.org/10.1007/s10649-011-9320-7
Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32(2), 251–265. https://doi.org/10.1016/j.jmathb.2013.02.008
Lockwood, E., A. Swinyard, C., & S. Caughman, J. (2015). Patterns, Sets of Outcomes, and Combinatorial Justification: Two Students’ Reinvention of Counting Formulas. In International Journal of Research in Undergraduate Mathematics Education (Vol. 1, Issue 1). https://doi.org/10.1007/s40753-015-0001-2
Lockwood, E., Reed, Z., & Caughman, J. S. (2017). An Analysis of Statements of the Multiplication Principle in Combinatorics, Discrete, and Finite Mathematics Textbooks. International Journal of Research in Undergraduate Mathematics Education, 3(3), 381–416. https://doi.org/10.1007/s40753-016-0045-y
Loong, E. Y. K. (2014). Using the internet in high school mathematics. Journal on Mathematics Education, 5(2), 108–126. https://doi.org/10.22342/jme.5.2.1496.108-126
Loughran, J., Berry, A., & Mulhall, P. (2012). Understanding and Developing Science Teachers Pedagogical Content Knowledge 2nd Edition. Sense Publishers.
Loughran, J., Mulhall, P., & Berry, A. (2004). In Search of Pedagogical Content Knowledge in Science: Developing Ways of Articulating and Documenting Professional Practice. Journal of Research in Science Teaching, 41(4), 370–391. https://doi.org/10.1002/tea.20007
Lune, H., & Berg, B. L. (2017). Qualitative Research Methods for the Social Sciences (9th Edition). In Pearson Education Limited (Ninth edit). Pearson Education Limited.
Martin, G. (2001). Counting: The Art of Enumerative Combinatorics. Springer.
Melusova, J., & Vidermanova, K. (2015). Upper-secondary Students’ Strategies for Solving Combinatorial Problems. Procedia - Social and Behavioral Sciences, 197(February), 1703–1709. https://doi.org/10.1016/j.sbspro.2015.07.223
NCTM. (2000). Principles Standards and for School Mathematics. Reston, VA
Permendikbud No. 37. (2018). Perubahan atas Peraturan Menteri Pendidikan dan Kebudayaan Nomor 24 Tahun 2016 tentang Kompetensi Inti dan Kompetensi Dasar Pelajaran pada Kurikulum 2013 pada Pendidikan Dasar dan Pendidikan Menengah. Available online at jdih.kemdikbud.go.id (accessed on 4 May 2022)
Quinn, J. R., & Wiest, R. L. (1998). A Constructivist Approach to Teaching Permutations and Combinations. Teaching Statistic, 20(3), 75–77. https://doi.org/10.20305/it202202111142
Rungtusanatham, M., Wallin, C., & Eckerd, S. (2011). The vignette in a scenario-based role-playing experiment. Journal of Supply Chain Management, 47(3), 9–16. https://doi.org/10.1111/j.1745-493X.2011.03232.x
Salavatinejad, N., Alamolhodaei, H., & Radmehr, F. (2021). Toward a model for students’ combinatorial thinking. Journal of Mathematical Behavior, 61(100823), 1–19. https://doi.org/10.1016/j.jmathb.2020.100823
Sandefur, J., Lockwood, E., Hart, E., & Greefrath, G. (2022). Teaching and Learning Discrete Mathematics. ZDM-Mathematics Education, https://do. https://doi.org/10.1007/978-94-007-4978-8_51
Semanišinová, I. (2021). Multiple-solution tasks in pre-service teachers’ course on combinatorics. Mathematics, 9(18). https://doi.org/10.3390/math9182286
Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2), 4–14.
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–23.
Siegel, M. (1986). Report of The Committee on Discrete Mathematics in the First Two Years.
Smith, C. E. (2007). On Students’ Conceptualizations of Combinatorics: A Multiple Case Study. University od Minnesota.
Soto, O., Siy, K., & Harel, G. (2022). Promoting a set-oriented way of thinking in a U.S. High School discrete mathematics class: a case study. ZDM - Mathematics Education, 0123456789. https://doi.org/10.1007/s11858-022-01337-7
Subanji. (2015). Teori Kesalahan Konstruksi Konsep dan Pemecahan Masalah Matematika [Error Theory of Concept Construction and Mathematical Problem Solving]. Universitas Negeri Malang: UM Press.
Sukoriyanto, S., Nusantara, T., Subanji, S., & Candra, T. D. (2016a). Students’ Thinking Interference of Real Global Type in Solving Permutation and Combination Problems. IOSR Journal of Research & Method in Education, 6(5), 47–50. https://doi.org/10.9790/7388-0605034750
Sukoriyanto, S., Nusantara, T., Subanji, S., & Chandra, T. D. (2016b). Students’ Errors in Solving the Permutation and Combination Problems Based on Problem Solving Steps of Polya. International Education Studies, 9(2), 11. https://doi.org/10.5539/ies.v9n2p11
Tchoshanov, M. A. (2011). Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 76(2), 141–164. https://doi.org/10.1007/s10649-010-9269-y
Turkdogan, A., Baki, A., & Cepni, S. (2009). The Anatomy of Mistakes: Categorizing Students ’ Mistakes in Mathematics within Learning Theories. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 1(1), 13–26. https://doi.org/10.16949/turcomat.09564
Yetkin, E. (2003). Student Difficulties in Learning Elementary Student Difficulties in Learning Elementary. ERIC Digest., 1–6.
Yunianto, W., Prahmana, R. C. I., & Crisan, C. (2021). Indonesian mathematics teachers’ knowledge of content and students of area and perimeter of rectangle. Journal on Mathematics Education, 12(2), 223–238. https://doi.org/10.22342/JME.12.2.13537.223-238
Zhang, Y. (2015). Pedagogical content knowledge in early mathematics: What teachers know and how it associates with teaching and learning. Loyola University, Chicago, IL.