Development of a diagnostic assessment for measuring students' conceptual understanding of real numbers
DOI:
https://doi.org/10.29408/jel.v12i3.34250Keywords:
diagnostic assessment, mathematical concepts, real numbers, research and developmentAbstract
This research was motivated by the limitations of assessment instruments capable of measuring students' initial conceptual understanding of mathematics related to real numbers, which is the basis of calculus. This study aimed to develop a valid, practical, and effective diagnostic assessment to measure students' conceptual understanding of mathematics. This research is developmental research with the Plomp model, which includes the stages of preliminary research, development or prototyping, and assessment of the product. Data were collected through interviews, observations, questionnaires, and tests and analyzed using quantitative descriptive analysis. The results of the validator assessment on the validation sheet indicate that the developed diagnostic assessment has a validity level of 91.8% (content aspect), 87.6% (construct aspect), and 92.6% (linguistic aspects). The practicality level was 70.85% (practical), as obtained from the questionnaire scores completed by the students. In terms of effectiveness, the developed assessment met the effectiveness criteria because it could identify areas of conceptual difficulty, differentiate levels of conceptual understanding, and provide meaningful information for decision-making. Its uniqueness lies in the development of a diagnostic assessment item design framework based on specific conceptual understanding indicators for real numbers in higher education and its assessment method.
References
Alfageh, D. H., York, C. S., Hodge-Zickerman, A., & Xie, Y. (2024). Elementary teachers' use of adaptive diagnostic assessment to improve mathematics teaching and learning: A case study. International Electronic Journal of Mathematics Education, 19(1), Article em0768. https://doi.org/10.29333/iejme/14190
Aristiawan, A., Wilujeng, H., Erwanto, M. R. Y., & Murtafiah, W. (2025). Development of a four-tier diagnostic test to detect middle school students' misconceptions in algebraic thinking. Prima: Jurnal Pendidikan Matematika, 9(2), 256–268. https://doi.org/10.31000/prima.v9i2.12700
Aziz, I. R., Harahap, N. S. R., & Siregar, R. N. (2025). Analysis of students' difficulties on understanding integral concepts. Journal of Innovative Mathematics Learning, 8(4), 891–904. https://doi.org/10.22460/jiml.v8i4.30135
Berger, S., Verschoor, A. J., Eggen, T. J. H. M., & Moser, U. (2019). Development and Validation of a Vertical Scale for Formative Assessment in Mathematics. Frontiers in Education, 4, 103. https://doi.org/10.3389/feduc.2019.00103
Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 21(1), 5–31. https://doi.org/10.1007/s11092-008-9068-5.
Caleon, I. S., & Subramaniam, R. (2010). Development and application of a three-tier diagnostic test to assess secondary students’ understanding of waves. International Journal of Science Education, 32(7), 939–961. https://doi.org/10.1080/09500690902890130
Ebel, R. L., & Frisbie, D. A. (1991). Essentials of educational measurement (5th ed.). Englewood Cliffs, NJ: Prentice Hall.
Ekawati, A., Siswono, T. Y. E., & Lukito, A. (2025). Psychometric analysis of high school mathematics ability test. Journal of Education Culture and Society, 16(1), 503–519. https://doi.org/10.15503/jecs2025.2.503.519
Fatih, J. G. (2026). Enhancing mathematical conceptual understanding through formative assessment with conceptual scaffolding. Journal of Applied Mathematics and Data Science, 1(1), 38–49. https://doi.org/10.64845/jamds.v1i1.291
Guskey, T. R. (2018). On your mark: Challenging the conventions of grading and reporting. Solution Tree Press.
Herliana, H., Maison, M., & Syaiful, S. (2024). Development and implementation of a five-tier diagnostic test to identify student misconceptions on fractions: A significant step towards improving mathematics education. Jurnal Ilmiah Ilmu Terapan Universitas Jambi, 8(2), 563–576. https://doi.org/10.22437/jiituj.v8i2.34159
Hidayat, W., & Prabawanto, S. (2018). Students’ difficulties in understanding irrational numbers. Journal on Mathematics Education, 9(1), 23–34. https://doi.org/10.22342/jme.9.1.5072.23-34
Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Information Age Publishing.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Lawrence Erlbaum Associates.
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., & Wearne, D. (1997). Making sense: Teaching and learning mathematics with understanding. Heinemann.
Jiang, B., Li, X., Yang, S., Kong, Y., Cheng, W., Hao, C., & Lin, Q. (2022). Data-driven personalized learning path planning based on cognitive diagnostic assessments in MOOCs. Applied Sciences, 12(8), Article 3982. https://doi.org/10.3390/app12083982
Kidron, I. (2018). Students’ conceptions of irrational numbers. International Journal of Research in Undergraduate Mathematics Education, 4(1), 94–118. https://doi.org/10.1007/s40753-018-0071-z
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. National Academies Press.
Metsämuuronen, J. (2026). Dimension-corrected discrimination index. Behaviormetrika, 53(1), 163–203. https://doi.org/10.1007/s41237-025-00279-0
NCTM. (2014). Principles to actions: Ensuring mathematical success for all. National Council of Teachers of Mathematics.
Nurhandayani, E. F., Mulyono, D., & Yanto, Y. (2022). Pengembangan e-modul matematika materi barisan dan deret dengan pendekatan problem-based learning (PBL) kelas XI SMA [Development of a mathematics e-module on sequences and series using a problem-based learning (PBL) approach for Grade 11 Senior High School Students]. Jurnal Pendidikan Matematika (Judika Education), 5(2), 126–137. https://doi.org/10.31539/judika.v5i2.4588
OECD. (2019). PISA 2018 results (Volume I): What students know and can do. OECD Publishing. https://doi.org/10.1080/09500690902890130
Parsinia, M., & Rott, B. (2026). Constructing and validating a test to assess pre-service teachers' knowledge of real numbers. Frontiers in Education, 11, 1776551. https://doi.org/10.3389/feduc.2026.1776551
Plomp, T. (1997). Educational design: Introduction. In T. Plomp & N. Nieveen (Eds.), Educational design research (pp. 9–20). Netherlands Institute for Curriculum Development (SLO).
Plomp, T., & Nieveen, N. (Eds.). (2013). Educational design research: Part A: An Introduction. Enschede, The Netherlands: SLO.
Pratiwi, D., Suryadi, D., & Sabandar, J. (2022). The role of diagnostic assessment in improving students’ mathematical understanding. Journal of Physics: Conference Series, 1987(1), 012034. https://doi.org/10.1080/09500690902890130
Rizos, I., & Adam, M. (2022). Mathematics students' conceptions and reactions to questions concerning the nature of rational and irrational numbers. International Electronic Journal of Mathematics Education, 17(3), Article em0686. https://doi.org/10.29333/iejme/11977
Ruamba, M. Y. (2025). Persistent hurdles: A systematic review of limit concept misconceptions in undergraduate calculus. Plusminus: Jurnal Pendidikan Matematika, 5(3, 495-508 https://doi.org/10.31980/plusminus.v5i3.3291
Sahin, M., Ozdemir, G., & Yilmaz, R. M. (2024). Mapping design stages and methodologies for developing STEM concept inventories: A scoping review. Frontiers in Education, 9. https://doi.org/10.3389/feduc.2024.1442833
Skemp, R. R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher, 26(3), 9–15.
Stewart, J., Redlin, L., & Watson, S. (2016). Calculus: Early transcendentals (8th ed.). Cengage Learning.
Tall, D. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619
Treagust, D. F. (1988). Development and use of diagnostic tests to evaluate students’ misconceptions in science. International Journal of Science Education, 10(2), 159–169. https://doi.org/10.1080/0950069880100204
Tessmer, M. (1993). Planning and conducting formative evaluations. London: Kogan Page.
Usman, Aiyub, & Hasbi. (2025). Students' conceptual understanding of limit of functions reviewed from mathematical beliefs. Jurnal Elemen, 11(3), 515-532. https://doi.org/10.29408/jel.v11i3.29144
Wiliam, D. (2011). Embedded formative assessment. Solution Tree Press.
Zhan, P., Li, F., & Jiao, H. (2021). Editorial: Cognitive diagnostic assessment for learning. Frontiers in Psychology, 12, 806636. https://doi.org/10.3389/fpsyg.2021.806636
Zazkis, R. (2005). Representations of irrational numbers: A case of conceptual understanding. Journal for Research in Mathematics Education, 36(5), 406–426. https://doi.org/10.1080/00207390412331316951
Zazkis, R., & Sirotic, N. (2010). Representing and defining irrational numbers: Exposing the missing link. CBMS Issues in Mathematics Education, 16, 1–27.
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