From Misconception To Conceptual Understanding: Hypothetical Learning Trajectory Design In Similarity Learning

Vira  Pitriatunnazwa; Dedi Muhtadi; Sukirwan

doi:10.24239/koordinat.v7i1.217

Vira Pitriatunnazwa Universitas Siliwangi
Dedi Muhtadi Universitas Siliwangi
Sukirwan Universitas Siliwangi

DOI: https://doi.org/10.24239/koordinat.v7i1.217

Keywords: Geometric Similarity, Hypothetical Learning Trajectory, Learning Obstacles, Students’ Errors

Abstract

Similarity is a fundamental topic in geometry, yet many students continue to experience difficulties in understanding proportional relationships, identifying corresponding elements, and constructing geometric reasoning. This study aimed to synthesize empirical evidence on students’ difficulties in learning geometric similarity and to develop a Hypothetical Learning Trajectory (HLT) to support instruction. A Systematic Literature Review (SLR) was conducted following the PRISMA 2020 guidelines. Data were collected from six databases: Scopus, ScienceDirect, SpringerLink, Taylor & Francis Online, ERIC, and Google Scholar. From 142 identified records published between 2015 and 2025, 15 empirical studies met the inclusion criteria and were selected for analysis. Data were analyzed using thematic synthesis through open coding, axial coding, and selective coding. The findings revealed six major categories of students’ difficulties: conceptual errors, transformation errors, procedural errors, visual–spatial difficulties, metacognitive misconceptions, and reasoning-and-proof difficulties. These difficulties form a developmental pattern, progressing from visual and conceptual misunderstandings to higher-order reasoning challenges. Based on the synthesis, a five-phase Hypothetical Learning Trajectory consisting of visual recognition, correspondence identification, proportional reasoning, contextual application, and deductive justification was developed. The proposed HLT provides an evidence-based framework for improving geometry instruction and supporting students’ conceptual understanding of similarity

References

Al-baqie, A. F. A., & Budiarto, M. T. (2022). Kemampuan Membaca , Menulis , dan Membuktikan Siswa dalam Menyelesaikan Masalah Kesebangunan Dua Segitiga. 5(1), 110–126. https://doi.org/10.30762/f

Alonzo, A. C., & von Aufschnaiter, C. (2018). Moving Beyond Misconceptions: Learning Progressions as a Lens for Seeing Progress in Student Thinking. The Physics Teacher, 56(7), 470–473. https://doi.org/10.1119/1.5055332

Ani Daniyati, Ismy Bulqis Saputri, Ricken Wijaya, Siti Aqila Septiyani, & Usep Setiawan. (2023). Konsep Dasar Media Pembelajaran. Journal of Student Research, 1(1), 282–294. https://doi.org/10.55606/jsr.v1i1.993

Artigue, M. (2009). Didactical design in mathematics education. ZDM Mathematics Education, 41 (1-2), 13–23.

Audétat, M., Laurin, S., Dory, V., Charlin, B., Laurin, S., Dory, V., & Charlin, B. (2017). Diagnosis and management of clinical reasoning difficulties : Part I . Clinical reasoning supervision and educational diagnosis. Medical Teacher, 39(8), 792–796. https://doi.org/10.1080/0142159X.2017.1331033

Bakker, A., & Smit, J. (2018). Using hypothetical learning trajectories in design research. In Design Research in Education (pp. 255–271). Routledge.

Bapa, A. T., Susanto, H., & Malang, U. N. (2025). Symmetry | Pasundan Journal of Research in Mathematics Learning and Education. 10(2007), 31–43. https://doi.org/10.23969/symmetry.v10i1.21959

Battista, M. (2007). The development of geometric and spatial thinking. In The Cambridge Handbook of Mathematics Education Research.

Bermudez, H., & Graysay, D. (2025). Supporting Students in Exploring, Developing, and Formalizing Mathematical Arguments. PRIMUS, 35(1), 102–133. https://doi.org/10.1080/10511970.2024.2445517

Borasi, R. (1994). Capitalizing on Errors as “Springboards for Inquiry”: A Teaching Experiment. Journal for Research in Mathematics Education, 25(2), 166–208. https://doi.org/10.5951/jresematheduc.25.2.0166

Clements, D. H., & Battista, M. T. (1989). Geometry and Spatial Reasoning. https://doi.org/10.1108/978-1-60752-874-620251022

Didi, S., Muhammad, P., & Endang, M. (2019). Hypothetical Learning Trajectory Of Students On Learning Geometry In Junior High School Grade 7Th. Advances in Social Science, Education and Humanities Research, 287(Icesre 2018), 180–184. https://doi.org/10.2991/icesre-18.2019.38

Ditinjau, K., Siswa, D., Hakiim, S., & Hidayati, F. H. (2021). Problematika Pembelajaran Daring Materi Kesebangunan dan. 1, 44–52.

Dreyfus, T. (2015). Constructing abstract mathematical knowledge in context. In S. J. Cho (Ed.), Selected Regular Lectures from the 12th International Congress on Mathematical Education(pp. 115-133). Springer International Publishing., 115–133. https://doi.org/https://doi.org/10.1007/978-3-319-17187-6_7Dreyfus, T., Hershkowitz, R., & Schwarz, B. (2015).

Duval, R. (1998). Geometry from a cognitive point of view. Educatinal Studies in Mathematics, 31 (1-2), 1–25.

Fadilah, R., & Bernard, M. (2021). ANALISIS KESALAHAN SISWA DALAM MENYELESAIKAN MASALAH MATEMATIKA KONTEKSTUAL MATERI KEKONGRUENAN DAN KESEBANGUNAN. Jurnal Pembelajaran Matematika Inovatif, 4(4), 817–826. https://doi.org/10.22460/jpmi.v4i4.817-826

Firdaus, A., & Mutaqin, A. (2019). DESAIN DIDAKTIS MATEMATIS PROBLEM SOLVING PADA ( MATHEMATICAL DIDACTIC DESIGN OF PROBLEM SOLVING ON. 02(01), 16–27.

For, A. F., Gray, E., & Tall, D. (2006). RETIREMENT.

Ginting, A. B., Aulia, D. U., Anindya, I. B., & Fauzi, M. A. (2025). ANALISIS KESALAHAN SISWA DALAM MENYELESAIKAN SOAL GEOMETRI MATERI KEKONGRUENAN DAN KESEBANGUNAN BERDASARKAN TEORI NEWMAN. Jurnal Review Pendidikan Dan Pengajaran, 8(1), 3206–3214. https://doi.org/10.31004/jrpp.v8i1.38421

Goldberg, S. R. (2016). Examining Students ’ Metacognitive Awareness Through Analysis of Student- generated Learning Responses. ASEE Annual Conference & Exposition, 16. https://doi.org/10.18260/p.26795

Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition. Educational Studies in Mathematics, 49 (2), 195–223.

Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics, 49, 193–223.

Gravemeijer, K. & Cobb, P. (2006). Design Research from a Learning Design Perspective”. In J. Van den Akker, K. Gravemeijer, S. McKenney & N. Nieveen (Eds). Educational Design Research.

Gravemeijer, K., & Doorman, M. (1999). Educational Studies in Mathematics,. Educational Studies in Mathematics, 39 (1-3), 111–129.

Gutiérrez, A., & Jaime, A. (2021). Revisiting Van Hiele’s model in light of current research on geometry thinking. International Journal of Science and Mathematics Education, 19 (16), 1021–1040. https://doi.org/https://doi.org/10.1007/s10763-021-10194-6

Hamzah, E., T, A. Y., & Nursangaji, A. (2019). ANALISIS KESALAHAN KONSEP DALAM MENJAWAB SOAL-SOAL PADA MATERI KESEBANGUNAN MENGGUNAKAN CERTAINTY OF RESPONSE INDEX. Jurnal Pendidikan Dan Pembelajaran Khatulistiwa (JPPK), 2003, 1–9. https://doi.org/https://doi.org/10.26418/jppk.v8i11.37595

Hiebert, J., Morris, A. K., & Spitzer, S. M. (2018). Diagnosing Learning Goals: An Often-Overlooked Teaching Competency. In Diagnostic Competence of Mathematics Teachers (pp. 193–206). https://doi.org/10.1007/978-3-319-66327-2_10

Hiele, V. (1986). Structure and Insight: A Theory of Mathematics Education. In Academic Press.

Hohenwarter, M., Hohenwarter, J., & Kreis, Y. (20007). Teaching and calculus with free dynamic mathematics software GeoGebra GeoGebra and Open Source. In 11th International Congress on Mathematical Education (ICME 11).

Inhelder, B., & Piaget, J. (1958). The growth of logical thinking: From childhood to adolescence. https://doi.org/https://doi.org/10.1037/10034-000

Irawan, D., Marpaung, M. T. F., Dewi, I., & Surya, E. (2025). Paradigms of Mathematics Education Research: A Theoretical Review and Implications for Learning Practice. International Journal Of Humanities Education and Social Sciences (IJHESS), 5(1). https://doi.org/10.55227/ijhess.v5i1.1898

Jones, K., & Tzekaki, M. (2016). 4 . RESEARCH ON THE TEACHING AND LEARNING OF GEOMETRY.

Laila, F. M., Helina Qatrunnada Nasution, Lisda Karolin Br Barus, & Kms Muhammad Amin Fauzi. (2025). Analisis Kesalahan Mahasiswa Dalam Menyelesaikan Soal Kesebangunan Dan Kekongruenan Di Jurusan Matematika Universitas Negeri Medan. Jurnal Pedagogy.

Mariotti, M. A., & Fischbein, E. (1997). DEFINING IN CLASSROOM ACTIVITIES. Educational Studies in Math.Ematics, 34, 219–248. file:///C:/Users/ASUS/Downloads/defESMcopia.pdf

Mosher, F. A. (2022). Learning Progressions. In Learning Progressions. https://doi.org/10.4324/9781138609877-REE115-1

Murio, N. (2022). Nazom Murio: Mahasiswa FKIP Universitas Jambi Page 1. 1–8.

Nafiin, M. W. (2020). ANALISIS KESALAHAN PESERTA DIDIK DALAM MENYELESAIKAN SOAL KESEBANGUNAN. Jurnal Ilmiah Pendidikan Matematika, 9(2), 21–29. https://doi.org/https://doi.org/10.26740/mathedunesa.v7n2.p485-492

Newman, M. A. (1977). An analysis of sixth-grade pupils’ errors on written mathematical tasks’, in M. A. Clements, and J. Foyster (eds.),. Research in Mathematics Education in Australia, 1977, 1, 239–258.

Nogueira, V. B., Teixeira, D. G., de Lima, I. A. C. N., Moreira, M. V. C., de Oliveira, B. S. C., Pedrosa, I. M. B., de Queiroz, J. W., &, & Jeronimo, S. M. B. (2021). Towards an inclusive digital literacy: An experimental intervention study in a rural area of Brazil. Education and Information Technologies, 27 (2), 2807–2834.

Nurmala, S., & Jupri, A. (2023). Common errors and cognitive obstacles in learning similarity. Journal on Mathematics Education, 14 (2), 155–172. https://doi.org/https://doi.org/10.22342/jme.14.2.17791.155-172

OECD. (2024). PISA 2022 Results Volume III : Creative Minds , Creative Schools. III.

Osibodu, O., & Sunghwan Byun. (2021). A Participatory Turn in Mathematics Education Research: Possibilities and Tensions. National Council of Teachers of Mathematics, 54(3). https://doi.org/https://doi.org/10.5951/jresematheduc-2021-0147

Osta, I. (2014). Mathematics Curriculum Evaluation. In Encyclopedia of Mathematics Education (pp. 417–423). Springer Netherlands. https://doi.org/10.1007/978-94-007-4978-8_168

Ouzzani, M. (2016). Rayyan — a web and mobile app for systematic reviews. Systematic Reviews, 2016, 1–10. https://doi.org/10.1186/s13643-016-0384-4

Page, M. J., Mckenzie, J. E., Bossuyt, P. M., Boutron, I., Hoffmann, C., Mulrow, C. D., Shamseer, L., Tetzlaff, J. M., Akl, E. A., Brennan, S. E., Chou, R., Glanville, J., Grimshaw, J. M., Hróbjartsson, A., Lalu, M. M., Li, T., Loder, E. W., Mayo-wilson, E., Mcdonald, S., … Moher, D. (2021). The PRISMA 2020 statement : an updated guideline for reporting systematic reviews Systematic reviews and Meta-Analyses. https://doi.org/10.1136/bmj.n71

Patton, M. (2015). Qualitative Research and Evaluation Methods. 4th Edition, Sage, Thousand Oaks.

Piaget. (1970a). Cognitive Developmental Standpoint.

Piaget, J. (1970b). Science of Education and the Psychology of the Child.

Polya, G., To, H., It, S., Principle, F., & Principle, S. (1945). Polya ’ s Problem Solving Techniques. 1–4.

Riyan, H., Mohd, S. M. R., Normala, I., & Noor, M. A. M. (2023). Systematic Review of Spatial Thinking in Mathematics Education. Educational Studies: Conference Series, 3(1), 47–60.

Rohmah, D. I., & Rosyidi, A. H. (2022). MATHE dunesa. 11(3).

Saldaña, Matthew B.Π, M., Johnny, & Huberman, A. M. (2014). Qualitative Data Analysis A Methods Sourcebook Edition 3. SAGE Publications. https://www.metodos.work/wp-content/uploads/2024/01/Qualitative-Data-Analysis.pdf

Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press.

Silmi, U., & L, D. A. M. (2022). Systematic Literature Review:Teori Van Hiele dalam Meningkatkan Kemampuan Berpikir Geometris Siswa Sekolah Dasar. PEDADIDAKTIKA: Jurnal Ilmiah Pendidikan Guru Sekolah Dasar, 9(2), 327–338. https://doi.org/10.17509/pedadidaktika.v9i2.53166

Simon, M. A. (1995). Reconstructing Mathematics Pedagogy from a Constructivist Perspective. Journal for Research in Mathematics Education, 26(2), 114–145.

Sinclair, N., & Bruce, C. D. (2015). New opportunities in geometry education at the primary school. ZDM Mathematics Education, 47(3), 319–329. https://doi.org/10.1007/s11858-015-0693-4

Snyder, H. (2019). Literature review as a research methodology: An overview and guidelines. Journal of Business Research, 104, 333–339. https://doi.org/10.1016/j.jbusres.2019.07.039

Solihah, S., & Muhtadi, D. (2025). Kesulitan konseptual dan prosedural siswa smp dalam materi kesebangunan Pendahuluan. 04(01), 123–140.

Sowder, L. (2007). Second Handbook of Research on Mathematics Teaching and Learning,. In Nantional Council of Teachers of Mathematics (Vol. 6179, Issue 619, pp. 1–60).

Stark Guss, S., Clements, D., & Sarama, J. (2022). High-quality early math: Learning and teaching with trajectories and technologies. In Handbook of Research on Innovative Approaches to Early Childhood Development and School Readiness. https://doi.org/https://doi.org/10.4018/978-1-7998-8649-5.ch015

Stylianides, G. J., Stylianides, A. J., & Moutsios, A. (2024). Proof and proving in school and university mathematics education research : a systematic review. ZDM – Mathematics Education, 56(1), 47–59. https://doi.org/10.1007/s11858-023-01518-y

Sulastri, R., Meutia, H., Johar, R., & Ikhsan, M. (2017). PROSES PEMBELAJARAN MATEMATIKA. SEMINAR NASIONAL KEMARITIMAN ACEH, 1, 285–288.

Tall, D. (2004). Introducing Three Worlds of Mathematics. For the Learning of Mathematics, 23, 29–33.

Tall, D. (2011). Crystalline Concepts in Long-Term Mathematical Invention and Discovery. For the Learning of Mathematics, 31 (1), 3–8. https://doi.org/https://eric.ed.gov/?id=EJ921149

Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. University of Chicago.

Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological Processes. Cambridge, MA: Harvard University Press.