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Abstract

Mathematical representation plays a pivotal role in students’ understanding, reasoning, and problem-solving processes. Despite its centrality in mathematics education, systematic approaches to assessing representational behavior remain limited, particularly within diverse cultural and curricular contexts. Existing assessment practices often emphasize cognitive outcomes, overlooking affective, psychomotor, and meta-representational dimensions that shape students’ mathematical understanding. Addressing this gap, the present study developed and validated an analytical rubric designed to assess mathematical representation behavior comprehensively across these four domains. Grounded in the Educational Design Research (EDR) framework, the rubric was constructed through four iterative stages—reflection, recording, grouping and naming, and application. Six mathematics education experts from Indonesia and Malaysia participated in the validation process, while empirical data were collected from 42 undergraduate students who had completed a geometry course. The analysis revealed strong content validity, with Aiken’s V coefficients ranging from 0.78 to 0.93 and full expert agreement, confirming the rubric’s clarity and relevance in evaluating representational behaviors. The rubric categorized student performance into three levels—Eikasia, Dianoia, and Intellectus—providing a nuanced diagnostic framework for assessing students’ mathematical representation. This study contributes to the field by introducing a cross-culturally grounded assessment tool that integrates cognitive, affective, psychomotor, and meta-representational perspectives. The findings highlight the rubric’s potential as both a formative and diagnostic instrument, enhancing the precision of assessment and offering insights for improving mathematics instruction and future digital-based evaluation practices.

Keywords

Analytical Rubric Behaviour Developmental Research Mathematical Representation

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Harisman, Y., Asra, A., Hafizatunnisa, Elniati, S., & Adnan, M. (2025). Analytical rubrics for mathematical representation behaviour assessment: Development, validation, and cross-cultural application. Journal on Mathematics Education, 16(4), 1137–1166. https://doi.org/10.22342/jme.v16i4.pp1137-1166
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References

  1. Aiken, L. R. (1985). Three coefficients for analyzing the reliability and validity of ratings. Educational and psychological measurement, 45(1), 131-142. https://doi.org/10.1177/0013164485451012
  2. Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198. https://doi.org/10.1016/j.learninstruc.2006.03.001
  3. Apsari, R. A., Putri, R. I. I., Sariyasa, Abels, M., & Prayitno, S. (2020). Geometry representation to develop algebraic thinking: A recommendation for a pattern investigation in pre-algebra class. Journal on Mathematics Education, 11(1), 45–58. https://doi.org/10.22342/jme.11.1.9535.45-58
  4. Beswick, K., & Muir, T. (2004). Talking and writing about the problem solving process. Mathematics Education For the Third Millennium: …, 1999, 95–102. http://www.merga.net.au/publications/counter.php?pub=pub_conf&id=202
  5. Brookhart, S. M. (2013). How to create and use rubrics for formative assessment and grading. Ascd.
  6. Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182. https://doi.org/10.1016/0364-0213(89)90002-5
  7. Creswell, J. W., & Creswell, J. D. (2018). Research Defign: Qualitative, Quantitative, and Mixed M ethods Approaches. In Research Defign: Qualitative, Quantitative, and Mixed M ethods Approaches.
  8. Davidson, A., Herbert, S., & Bragg, L. (2018). Supporting Elementary Teachers’ Planning and Assessing of Mathematical Reasoning. International Journal of Science and Mathematics Education, 17. https://doi.org/10.1007/s10763-018-9904-0
  9. de Boer, I., de Vegt, F., Pluk, H., & Latijnhouwers, M. (2021). Rubrics-a tool for feedback and assessment viewed from different perspectives. Springer.
  10. Domu, I., Regar, V. E., Kumesan, S., Mangelep, N. O., & Manurung, O. (2023). Did the Teacher Ask the Right Questions? An Analysis of Teacher Asking Ability in Stimulating Students’ Mathematical Literacy. Journal of Higher Education Theory and Practice, 23(5), 248–256. https://doi.org/10.33423/jhetp.v23i5.5970
  11. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131. https://doi.org/10.1007/s10649-006-0400-z Fauzan, A., Harisman, Y., & Arini. (2019). Analysis of students’ strategies in solving multiplication problems. International Journal of Scientific and Technology Research, 8(10), 568–573.
  12. Gezie, A., Khaja, K., Chang, V. N., Adamek, M. E., & Johnsen, M. B. (2012). Rubrics as a Tool for Learning and Assessment: What Do Baccalaureate Students Think? Journal of Teaching in Social Work, 32(4), 421–437. https://doi.org/10.1080/08841233.2012.705240
  13. Goldin-Meadow, S., So, W. C., Özyürek, A., & Mylander, C. (2008). The natural order of events: How speakers of different languages represent events nonverbally. Proceedings of the National Academy of Sciences of the United States of America, 105(27), 9163–9168. https://doi.org/10.1073/pnas.0710060105
  14. Goldin, G. (2020). A Joint Perspective on the Idea of Representation in Learning and Doing Mathematics. Theories of Mathematical Learning, September, 409–442. https://doi.org/10.4324/9780203053126-30
  15. Gunawan, I., Darhim, D., & Kusnandi, K. (2019). Exploration of the behavior of understanding mathematical concepts of junior high school students. Journal of Physics: Conference Series, 1157(4). https://doi.org/10.1088/1742-6596/1157/4/042098
  16. Hanifah, Waluya, S. B., Rochmad, & Wardono. (2020). Mathematical Representation Ability and Self -Efficacy. Journal of Physics: Conference Series, 1613(1). https://doi.org/10.1088/1742-6596/1613/1/012062
  17. Hannula, M. S. (2006). Motivation in mathematics: Goals reflected in emotions. Educational Studies in Mathematics, 63(2), 165–178. https://doi.org/10.1007/s10649-005-9019-8
  18. Harisman, Y., Noto, M. S., & Hidayat, W. (2021). Investigation of Students’ Behavior in Mathematical Problem Solving. Infinity Journal, 10(2), 235–258. https://doi.org/10.22460/infinity.v10i2.p235-258
  19. Hegarty, M., & Kozhevnikov, M. (1999). Types of visual-spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689. https://doi.org/10.1037//0022-0663.91.4.684
  20. Hostetter, A. B., & Alibali, M. W. (2008). Visible embodiment: Gestures as simulated action. Psychonomic Bulletin and Review, 15(3), 495–514. https://doi.org/10.3758/PBR.15.3.495
  21. J. H., F. (1979). Metacognition and cognitive monitoring: A new area of cognitive–developmental inquiry. American Psychologist, 157(DEC14), 424. https://doi.org/10.1093/nq/CLVII.dec14.424-a
  22. Ikashaum, F., Mustika, J., Wulantina, E., & Cahyo, E. D. (2021). Analysis of Students' Symbolic Representation Errors in Plane Analytic Geometry Problems. Al-Khwarizmi : Jurnal Pendidikan Matematika Dan Ilmu Pengetahuan Alam, 9(1), 57–68. https://doi.org/10.24256/jpmipa.v9i1.1701
  23. Khalid, M. (2017). Fostering Problem Solving and Performance Assessment among Malaysian Mathematics Teachers. Sains Humanika, 9(1–2), 51–55. https://doi.org/10.11113/sh.v9n1-2.1098
  24. Karaca, H., Ertekin, E., & Cagiltay, K. (2025). Investigating middle school students’ eye movements on the mathematical representations: An eye-tracking study. Education and Information Technologies, 30(11), 16189–16210. https://doi.org/10.1007/s10639-025-13436-5
  25. Kohen, Z., & Gharra-Badran, Y. (2023). A rubric for assessing mathematical modelling problems in a scientific-engineering context. Teaching Mathematics and Its Applications, 42(3), 266–288. https://doi.org/10.1093/teamat/hrac018
  26. Krathwohl, D. R. (2002). A Revision of Bloom’s Taxonomy: An Overview. Theory into Practice, 41(4), 212–218. https://doi.org/10.1207/s15430421tip4104_2Khairunnisak, C., Johar, R., Morina Zubainur, C., & Sasalia, P. (2021). Learning Trajectory of Algebraic Expression: Supporting Students’ Mathematical Representation Ability. Mathematics Teaching Research Journal, 13(4), 27–41. https://commons.hostos.cuny.edu/mtrj/
  27. Kusumaningtyas, A. A. S., Kartono, & Asih, T. S. N. (2020). Error Analysis in Mathematical Representation Through the Think-Talk-Write Model with Verbal Feedback. PRISMA, Prosiding Seminar Nasional Matematika, 3, 518–520. https://journal.unnes.ac.id/sju/index.php/prisma/article/view/37574/15534
  28. Lesh, R., Post, T. R., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In Problems of representations in the teaching and learning of mathematics (pp. 33–40). Lawrence Erlbaum.
  29. Lestari, D., Usman, U., & Munzir, S. (2024). Mathematical Representation Abilities and Self-Confidence through Application of Discovery Learning Model with Geogebra-assisted. 98–106. https://doi.org/10.2991/978-2-38476-216-3_11
  30. Marnizam, F. I., & Ali, S. R. (2021). Pentaksiran Dalam Pendidikan. Jurnal Pendidikan Sains Dan Matematik Malaysia, 11(2), 81–94
  31. Martínez-Planell, R., & Cruz Delgado, A. (2016). The unit circle approach to the construction of the sine and cosine functions and their inverses: An application of APOS theory. Journal of Mathematical Behavior, 43, 111–133. https://doi.org/10.1016/j.jmathb.2016.06.002
  32. Midgett, C., & Eddins, S. (2001). NCTM’s Principles and Standards for School Mathematics: Implications for Administrators. Nassp Bulletin, 85, 35–42. https://doi.org/10.1177/019263650108562305
  33. Muir, T., Beswick, K., & Williamson, J. (2008). “I’am nor very goot at solving problems”: An exploration of students’ problem solving behaviours. The Journal of Mathematical Behavior, 27(3), 228-241. https://doi.org/10.1016/j.jmathb.2008.04.003
  34. Musdi, E., Syahputra, H., Arnellis, A., & Harisman, Y. (2024). Students ’ m athematics communication behavior : Assessment tools and their application. Journal on Mathematics Education, 15(1), 317-338.. https://doi.org/10.22342/jme.v15i1.pp317-338
  35. NCTM, N. C. of T. of M. (2000). Principles and standards for school mathematics. NCTM. (Vol. 17, p. 302).
  36. Ng, C. H., & Adnan, M. (2018). Integrating STEM education through Project-Based Inquiry Learning (PIL) in topic space among year one pupils. IOP Conference Series: Materials Science and Engineering, 296(1). https://doi.org/10.1088/1757-899X/296/1/012020
  37. Novick, L. R. (2004). Diagram Literacy in Preservice Math Teachers, Computer Science Majors, and Typical Undergraduates: The Case of Matrices, Networks, and Hierarchies. Mathematical Thinking and Learning, 6(3), 307–342. https://doi.org/10.1207/s15327833mtl0603_3
  38. Novotná, Jarmila, Eisenmann, P., Přibyl, J., Ondrušová, J., & Břehovský, J. (2012). Problem Solving in School Mathematics Based on. Journal on Efficiency and Responsibility in Education and Science, 7(1), 1–6. https://doi.org/10.7160/eriesj.2013.070101.Introduction
  39. OECD. (2023). PISA 2022 Assessment and Analytical Frameword. In Paris: Journal of OECH Publishing
  40. Pape, S. J., & Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical understanding. Theory into Practice, 40(2), 118–127. https://doi.org/10.1207/s15430421tip4002_6
  41. Post, M., & Prediger, S. (2024). Teaching practices for unfolding information and connecting multiple representations: the case of conditional probability information. In Mathematics Education Research Journal (Vol. 36, Issue 1). Springer Netherlands. https://doi.org/10.1007/s13394-022-00431-z
  42. Rahmawati, D., Purwantoa, P., Subanji, S., Hidayanto, E., & Anwar, R. B. (2021). Process of Mathematical Representation Translation from Verbal into Graphic. International Electronic Journal of Mathematics Education, 12(3), 367–381. https://doi.org/10.29333/iejme/618
  43. Rebolledo-Mendez, G., Huerta-Pacheco, N. S., Baker, R. S., & du Boulay, B. (2022). Meta-Affective Behaviour within an Intelligent Tutoring System for Mathematics. International Journal of Artificial Intelligence in Education, 32(1), 174–195. https://doi.org/10.1007/s40593-021-00247-1
  44. Rittle-Johnson, B. (2006). Promoting transfer: Effects of self-explanation and direct instruction. Child Development, 77(1), 1–15. https://doi.org/10.1111/j.1467-8624.2006.00852.x
  45. Rohati, R., Kusumah, Y. S., & Kusnandi, K. (2022). The development of analytical rubrics: An avenue to assess students’ mathematical reasoning behavior. Cypriot Journal of Educational Sciences, 17(8), 2553–2566. https://doi.org/10.18844/cjes.v17i8.7043
  46. Rohati, R., Kusumah, Y. S., & Kusnandi, K. (2023). Exploring Students’ Mathematical Reasoning Behavior in Junior High Schools: A Grounded Theory. Education Sciences, 13(3). https://doi.org/10.3390/educsci13030252
  47. Santia, I., Purwanto, Sutawidjadja, A., Sudirman, & Subanji. (2019). Ill-Structured Problems : the Case of Quadratic. Journal on Mathematics Education, 10(3), 365–378.
  48. Sari, D. P., Darhim, & Rosjanuardi, R. (2018). Errors of students learning with react strategy in solving the problems of mathematical representation ability. Journal on Mathematics Education, 9(1), 121–128. https://doi.org/10.22342/jme.9.1.4378.121-128
  49. Sari, R. H. N., & Wijaya, A. (2017). Mathematical literacy of senior high school students in Yogyakarta. Jurnal Riset Pendidikan Matematika, 4(1), 100–107. https://doi.org/10.21831/jrpm.v4i1.10649
  50. Shimakeleni, L. (2022). the Use of Smartphones and Visualisation Processes for Conceptual Understanding of Mensuration: a Case Study of the Mathcitymap Project in Namibia. January.
  51. Sofnidar, S., Kamid, K., Anwar, K., Basuki, F. R., & Kurniawan, D. A. (2019). Student ’ s Behavior Base Cognitive Style In Outdoor Learning-Mathematiacal Modelling. 8(10).
  52. Stemler, S. E. (2004). A comparison of consensus, consistency, and measurement approaches to estimating interrater reliability. Practical Assessment, Research and Evaluation, 9(4).
  53. Stevens, D. D. (2023). Introduction to rubrics: An assessment tool to save grading time, convey effective feedback, and promote student learning. Routledge.
  54. Stevens, D. D., & Leve, A. J. (2005). Introduction to Rubrics: An Assessment Tool to Save Grading Time, Convey Effective Feedback and Promote Student Learning. Stylus, Sterling, Virginia. (2nd ed.).
  55. Stylianou, D. A. (2002). On the interaction of visualization and analysis: The negotiation of a visual representation in expert problem solving. Journal of Mathematical Behavior, 21(3), 303–317. https://doi.org/10.1016/S0732-3123(02)00131-1
  56. SUNY Discipline Panel in Mathematics. (2005). Standards and Rubrics for Assessing General Education in Mathematics. The State University of New York, Ic, 1–9. https://system.suny.edu/media/suny/content-assets/documents/academic-affairs/assessment/SUNY-Math-Rubric.pdf
  57. Telaumbanua, Y. N., Syahputra, E., & Surya, E. (2024). Students’ Metacognitive Awareness in Mathematics Learning. Academy of Education Journal, 15(1), 1047–1055. https://doi.org/10.47200/aoej.v15i1.2359
  58. Tian, Y., Fang, Y., & Li, J. (2018). The effect of metacognitive knowledge on mathematics performance in self-regulated learning framework-multiple mediation of self-efficacy and motivation. Frontiers in Psychology, 9(DEC), 1–11. https://doi.org/10.3389/fpsyg.2018.02518
  59. Toalongo, X., Trelles, C., & Alsina, Á. (2022). Design, Construction and Validation of a Rubric to Evaluate Mathematical Modelling in School Education. Mathematics, 10(24), 1–19. https://doi.org/10.3390/math10244662
  60. van der Wal, N. J., Bakker, A., & Drijvers, P. (2023). Designing an instrument to measure the development of techno-mathematical literacies in an innovative mathematics course for future engineers in STEM education. ZDM - Mathematics Education, 55(7), 1243–1254. https://doi.org/10.1007/s11858-023-01507-1
  61. van Lieshout, E. C. D. M., & Xenidou-Dervou, I. (2018). Pictorial representations of simple arithmetic problems are not always helpful: a cognitive load perspective. Educational Studies in Mathematics, 98(1), 39–55. https://doi.org/10.1007/s10649-017-9802-3
  62. Wijaya, A. (2016). Students’ Information Literacy: A Perspective from Mathematical Literacy. Journal on Mathematics Education, 7(2), 73–82. https://doi.org/10.22342/jme.7.2.3532.73-82
  63. Wulandari, E. D., Hidayanto, E., Subanji, & Rahardi, R. (2019). Mathematical Representation of Cerebral Palsy Students in Constructing the Concept of Plane Geometry Based on APOS Theory. Journal of Physics: Conference Series, 1227(1). https://doi.org/10.1088/1742-6596/1227/1/012018
  64. Yelisieiev, V., Lutsenko, V., & Berkout, V. (2022). Mathematical model of porous solid impregnation based on multichannel representation of the pore system. IOP Conference Series: Earth and Environmental Science, 970(1). https://doi.org/10.1088/1755-1315/970/1/012023
  65. Yudhanegara, M. R., Lestari, K. E., Studi, P., Matematika, P., & Karawang, U. S. (2014). Analysis of Students' Mathematical Representation Abilities in Geometry Courses Based on Their Background in Learning Achievement in Transformation Geometry Courses. JP3M (Jurnal Penelitian Pendidikan Dan Pengajaran Matematika), 3(2), 83–88. http://jurnal.unsil.ac.id/index.php/jp3m/article/view/258
  66. Yusuf Sodhiqin, Haryanto, & Abdul Muin. (2024). Problem Solving Learning With Drawing A Diagram Strategy To Improve Mathematical Reflective Thinking Skills. Jurnal Pendidikan Dan Pengajaran, 57(3), 551–562. https://doi.org/10.23887/jpp.v57i3.80361