Factors Contributing To Students’ Difficulties In Number Pattern Generalization: A Thematic Qualitative Meta-Analysis Within The Framework Of Early Algebraic Thinking

Nazwa Wulandari Azis; Dedi Muhtadi; Sukirwan

doi:10.24239/koordinat.v6i2.183

Nazwa Wulandari Azis Siliwangi University
Dedi Muhtadi Siliwangi University
Sukirwan Siliwangi University

DOI: https://doi.org/10.24239/koordinat.v6i2.183

Keywords: Early algebraic thinking, Mathematical difficulties, Pattern generalization, Qualitative meta-analysis, Scaffolding

Abstract

This study examines students’ difficulties in generalizing number patterns and their multidimensional interactions within early algebraic thinking. A thematic qualitative meta-analysis of 15 Scopus-indexed articles (2019–2024) was conducted under the PRISMA protocol using open coding and thematic clustering (κ = 0.82). Five categories of difficulty were identified: reasoning–generalization (80%), conceptual (73%), procedural–operational (60%), strategic (47%), and didactic (40%). Reasoning–generalization difficulties involved students’ inability to identify relationships or formulate general rules, while conceptual and procedural weaknesses reflected misconceptions of pattern structures and errors in extending sequences. Strategic and didactic challenges were linked to limited problem-solving approaches and insufficient scaffolding. The findings highlight dynamic interactions—especially between conceptual and procedural factors—and emphasize teachers’ mediating role in fostering the transition from arithmetic to algebraic symbolization

References

Aba‐Oli, Z., Koyas, K., & Husen, A. (2025). Higher‐order thinking skills‐oriented problem‐based learning interventions in mathematics: A systematic literature review. School Science and Mathematics, 125(3), 214–231. https://doi.org/10.1111/ssm.12676

Acosta, Y., Alsina, Á., & Pincheira, N. (2024). Computational thinking and repetition patterns in early childhood education: Longitudinal analysis of representation and justification. Education and Information Technologies, 29(6), 7633–7658. https://doi.org/10.1007/s10639-023-12051-6

Adu‐Gyamfi, K., Chandler, K., & Thompson, A. (2025a). Algebra story problem: The nature of students’ obstacles. School Science and Mathematics, 125(2), 140–153. https://doi.org/10.1111/ssm.12659

Adu‐Gyamfi, K., Chandler, K., & Thompson, A. (2025b). Algebra story problem: The nature of students’ obstacles. School Science and Mathematics, 125(2), 140–153. https://doi.org/10.1111/ssm.12659

Amin, M. E. K., Nørgaard, L. S., Cavaco, A. M., Witry, M. J., Hillman, L., Cernasev, A., & Desselle, S. P. (2020). Establishing trustworthiness and authenticity in qualitative pharmacy research. Research in Social and Administrative Pharmacy, 16(10), 1472–1482. https://doi.org/10.1016/j.sapharm.2020.02.005

Baumanns, L., Pitta-Pantazi, D., Demosthenous, E., Lilienthal, A. J., Christou, C., & Schindler, M. (2024). Pattern-Recognition Processes of First-Grade Students: An Explorative Eye-Tracking Study. International Journal of Science and Mathematics Education, 22(8), 1663–1682. https://doi.org/10.1007/s10763-024-10441-x

Belotto, M. (2018). Data Analysis Methods for Qualitative Research: Managing the Challenges of Coding, Interrater Reliability, and Thematic Analysis. The Qualitative Report. https://doi.org/10.46743/2160-3715/2018.3492

Blanton, M., Gardiner, A. M., Ristroph, I., Stephens, A., Knuth, E., & Stroud, R. (2024). Progressions in young learners’ understandings of parity arguments. Mathematical Thinking and Learning, 26(1), 90–121. https://doi.org/10.1080/10986065.2022.2053775

Blažun Vošner, H., Železnik, D., Kokol, P., Vošner, J., & Završnik, J. (2017). Trends in nursing ethics research: Mapping the literature production. Nursing Ethics, 24(8), 892–907. https://doi.org/10.1177/0969733016654314

Bramer, W. M., Rethlefsen, M. L., Kleijnen, J., & Franco, O. H. (2017). Optimal database combinations for literature searches in systematic reviews: a prospective exploratory study. Systematic Reviews, 6(1), 245. https://doi.org/10.1186/s13643-017-0644-y

Brod, G. (2021). Generative Learning: Which Strategies for What Age? Educational Psychology Review, 33(4), 1295–1318. https://doi.org/10.1007/s10648-020-09571-9

Brown, J. L., Ortiz-Padilla, M., & Soto-Varela, R. (2020). Does Mathematical Anxiety Differ Cross-Culturally? Journal of New Approaches in Educational Research, 9(1), 126–136. https://doi.org/10.7821/naer.2020.1.464

Byram, J. (2022). Do you trust me? Rigor and Information Power in Qualitative Research. The FASEB Journal, 36(S1). https://doi.org/10.1096/fasebj.2022.36.S1.0I620

Cook, J. P., Richardson, A., Strand, S., Reed, Z., & Melhuish, K. (2023). Examining the concept of inverse: Theory-building via a standalone literature review. The Journal of Mathematical Behavior, 72, 101100. https://doi.org/10.1016/j.jmathb.2023.101100

Czocher, J. A., & Melhuish, K. (2024). Attending to Coherence Among Research Questions, Methods, and Claims in Coding Studies. Journal for Research in Mathematics Education, 55(3), 148–155. https://doi.org/10.5951/jresematheduc-2022-0037

De Smedt, B. (2022). Individual differences in mathematical cognition: a Bert’s eye view. Current Opinion in Behavioral Sciences, 46, 101175. https://doi.org/10.1016/j.cobeha.2022.101175

Dennis, M. S., Sorrells, A. M., Chovanes, J., & Kiru, E. W. (2022). Ecological and Population Validity of Mathematics Interventions for Diverse Students With Low Mathematics Achievement: A Meta-Analysis. Learning Disability Quarterly, 45(1), 32–45. https://doi.org/10.1177/07319487211010342

Dong, L., & Kang, Y. (2022). Cultural differences in mindset and beliefs regarding mathematics learning. Current Opinion in Behavioral Sciences, 46, 101159. https://doi.org/10.1016/j.cobeha.2022.101159

Durkin, K., Rittle-Johnson, B., Star, J. R., & Loehr, A. (2023). Comparing and Discussing Multiple Strategies: An Approach to Improving Algebra Instruction. The Journal of Experimental Education, 91(1), 1–19. https://doi.org/10.1080/00220973.2021.1903377

Ellis, A. B., Lockwood, E., Tillema, E., & Moore, K. (2022). Generalization Across Multiple Mathematical Domains: Relating, Forming, and Extending. Cognition and Instruction, 40(3), 351–384. https://doi.org/10.1080/07370008.2021.2000989

Ellis, A. B., Waswa, A., Tasova, H. I., Hamilton, M., Moore, K. C., & Çelik, A. (2024). Classroom Supports for Generalizing. Journal for Research in Mathematics Education, 55(1), 7–30. https://doi.org/10.5951/jresematheduc-2022-0140

Fabry, R. E., & Pantsar, M. (2021). A fresh look at research strategies in computational cognitive science: The case of enculturated mathematical problem solving. Synthese, 198(4), 3221–3263. https://doi.org/10.1007/s11229-019-02276-9

González-Forte, J. M., Fernández, C., Van Hoof, J., & Van Dooren, W. (2023). Incorrect Ways of Thinking About the Size of Fractions. International Journal of Science and Mathematics Education, 21(7), 2005–2025. https://doi.org/10.1007/s10763-022-10338-7

Guetterman, T. C., & Fetters, M. D. (2018). Two Methodological Approaches to Integrating Mixed Methods and Case Study Designs: A Systematic Review. American Behavioral Scientist, 62(7), 900–918. https://doi.org/10.1177/0002764218772641

Hunter, J., & Miller, J. (2022). Using a Culturally Responsive Approach to Develop Early Algebraic Reasoning with Young Diverse Learners. International Journal of Science and Mathematics Education, 20(1), 111–131. https://doi.org/10.1007/s10763-020-10135-0

Janis, I. (2022). Strategies for Establishing Dependability between Two Qualitative Intrinsic Case Studies: A Reflexive Thematic Analysis. Field Methods, 34(3), 240–255. https://doi.org/10.1177/1525822X211069636

Jordan, N. C., Devlin, B. L., & Botello, M. (2022). Core foundations of early mathematics: refining the number sense framework. Current Opinion in Behavioral Sciences, 46, 101181. https://doi.org/10.1016/j.cobeha.2022.101181

Junker, A., Nortvedt, G. A., & Farsani, D. (2025). Patterning strategies in grade 1 students with low and high number sense proficiency. Educational Studies in Mathematics, 118(1), 29–51. https://doi.org/10.1007/s10649-024-10341-5

Kakos, M., & Fritzsche, B. (2017). Meta-ethnography E& E. Ethnography and Education, 12(2), 129–133. https://doi.org/10.1080/17457823.2017.1278578

Kärki, T., McMullen, J., & Lehtinen, E. (2022). Improving rational number knowledge using the NanoRoboMath digital game. Educational Studies in Mathematics, 110(1), 101–123. https://doi.org/10.1007/s10649-021-10120-6

Kaya, Z., Kaya, O. N., Aydemir, S., & Ebenezer, J. (2022). Knowledge of Student Learning Difficulties as a Plausible Conceptual Change Pathway Between Content Knowledge and Pedagogical Content Knowledge. Research in Science Education, 52(2), 691–723. https://doi.org/10.1007/s11165-020-09971-5

Kerig, P. K. (2020). Open Science Practices at the Journal of Traumatic Stress. Journal of Traumatic Stress, 33(2), 133–136. https://doi.org/10.1002/jts.22489

Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: background, overarching dimensions, and new directions. ZDM – Mathematics Education, 54(6), 1131–1150. https://doi.org/10.1007/s11858-022-01435-6

Kim, S. A., Bryant, D. P., Bryant, B. R., Shin, M., & Ok, M. W. (2023). A Multilevel Meta-Analysis of Whole Number Computation Interventions for Students With Learning Disabilities. Remedial and Special Education, 44(4), 332–347. https://doi.org/10.1177/07419325221117293

Lambert, R., & Tan, P. (2020). Does disability matter in mathematics educational research? A critical comparison of research on students with and without disabilities. Mathematics Education Research Journal, 32(1), 5–35. https://doi.org/10.1007/s13394-019-00299-6

Lariviere, D. O., Arsenault, T. L., & Payne, S. B. (2025). A literature review: Mathematics vocabulary intervention for students with mathematics difficulty. School Science and Mathematics, 125(1), 6–17. https://doi.org/10.1111/ssm.12684

Leary, H., & Walker, A. (2018). Meta-Analysis and Meta-Synthesis Methodologies: Rigorously Piecing Together Research. TechTrends, 62(5), 525–534. https://doi.org/10.1007/s11528-018-0312-7

Lee, J. (2025). Concept-Focused and Procedure-Focused Instruction on the Algebra Performance of Grade 9 Students With and Without Mathematics Difficulty. Journal of Learning Disabilities, 58(3), 192–209. https://doi.org/10.1177/00222194241249960

Lenz, K., Dreher, A., Holzäpfel, L., & Wittmann, G. (2020). Are conceptual knowledge and procedural knowledge empirically separable? The case of fractions. British Journal of Educational Psychology, 90(3), 809–829. https://doi.org/10.1111/bjep.12333

Lochmiller, C. (2021). Conducting Thematic Analysis with Qualitative Data. The Qualitative Report. https://doi.org/10.46743/2160-3715/2021.5008

Lopez, D. A., Cardenas-Iniguez, C., Subramaniam, P., Adise, S., Bottenhorn, K. L., Badilla, P., Mukwekwerere, E., Tally, L., Ahanmisi, O., Bedichek, I. L., Matera, S. D., Perez-Tamayo, G. M., Sissons, N., Winters, O., Harkness, A., Nakiyingi, E., Encizo, J., Xiang, Z., Wilson, I. G., … Huber, R. S. (2024). Transparency and reproducibility in the Adolescent Brain Cognitive Development (ABCD) study. Developmental Cognitive Neuroscience, 68, 101408. https://doi.org/10.1016/j.dcn.2024.101408

Margolis, A. A. (2020). Zone of Proximal Development, Scaffolding and Teaching Practice. Cultural-Historical Psychology, 16(3), 15–26. https://doi.org/10.17759/chp.2020160303

Mehta, P., Zimba, O., Gasparyan, A. Y., Seiil, B., & Yessirkepov, M. (2023). Ethics Committees: Structure, Roles, and Issues. Journal of Korean Medical Science, 38(25). https://doi.org/10.3346/jkms.2023.38.e198

Melhuish, K., Thanheiser, E., & Guyot, L. (2020). Elementary school teachers notice essential mathematical reasoning forms: justification and generalization. Journal of Mathematics Teacher Education, 23(1), 35–67. https://doi.org/10.1007/s10857-018-9408-4

Morse, J. M. (2015). Critical Analysis of Strategies for Determining Rigor in Qualitative Inquiry. Qualitative Health Research, 25(9), 1212–1222. https://doi.org/10.1177/1049732315588501

OECD. (2023). PISA 2022 Results (Volume I). OECD. https://doi.org/10.1787/53f23881-en

Parey, B., & Kutscher, E. (2024). Operationalizing and Expanding the Rigorous Mixed Methods Framework for Systematic Reviews: The Case of Inclusive Education. Journal of Mixed Methods Research, 18(2), 147–164. https://doi.org/10.1177/15586898231175192

Pinto, E., & Cañadas, M. C. (2021a). Generalizations of third and fifth-graders within a functional approach to early algebra. Mathematics Education Research Journal, 33(1), 113–134. https://doi.org/10.1007/s13394-019-00300-2

Pittalis, M. (2025). An empirically validated rational number sense framework. Mathematics Education Research Journal, 37(1), 125–160. https://doi.org/10.1007/s13394-024-00484-2

Post, M., & Prediger, S. (2024). Teaching practices for unfolding information and connecting multiple representations: the case of conditional probability information. Mathematics Education Research Journal, 36(1), 97–129. https://doi.org/10.1007/s13394-022-00431-z

Psyridou, M., Torppa, M., Tolvanen, A., Poikkeus, A.-M., Lerkkanen, M.-K., & Koponen, T. (2023). Developmental profiles of arithmetic fluency skills from grades 1 to 9 and their early identification. Developmental Psychology, 59(12), 2379–2396. https://doi.org/10.1037/dev0001622

Pulverer, B. (2014). Transparent, Reproducible Data. The EMBO Journal, 33(22), 2597–2597. https://doi.org/10.15252/embj.201490278

Radford, L. (2018). The Emergence of Symbolic Algebraic Thinking in Primary School (pp. 3–25). https://doi.org/10.1007/978-3-319-68351-5_1

Rea, S. D., Wang, L., Muenks, K., & Yan, V. X. (2022). Students Can (Mostly) Recognize Effective Learning, So Why Do They Not Do It? Journal of Intelligence, 10(4), 127. https://doi.org/10.3390/jintelligence10040127

Reid O’Connor, B. (2024). Methodologies to reveal young Australian Indigenous students’ mathematical proficiency. Mathematics Education Research Journal, 36(2), 311–338. https://doi.org/10.1007/s13394-023-00447-z

Rexigel, E., Kuhn, J., Becker, S., & Malone, S. (2024). The More the Better? A Systematic Review and Meta-Analysis of the Benefits of More than Two External Representations in STEM Education. Educational Psychology Review, 36(4), 124. https://doi.org/10.1007/s10648-024-09958-y

Rojo, M., King, S., Gersib, J., & Bryant, D. P. (2023). Rational Number Interventions for Students With Mathematics Difficulties: A Meta-Analysis. Remedial and Special Education, 44(3), 225–238. https://doi.org/10.1177/07419325221105520

Rudaizky, D., Mărcuș, O., Visu-Petra, L., & MacLeod, C. (2023). Anxiety-linked impairment in the ability to recode stimuli. Emotion, 23(3), 814–824. https://doi.org/10.1037/emo0001093

Sandelowski, M., Barroso, J., & Voils, C. I. (2020). Using qualitative metasummary to synthesize qualitative and quantitative descriptive findings. Research in Nursing & Health, 30(1), 99–111. https://doi.org/10.1002/nur.20176

Scheibling-Sève, C., Pasquinelli, E., & Sander, E. (2020). Assessing conceptual knowledge through solving arithmetic word problems. Educational Studies in Mathematics, 103(3), 293–311. https://doi.org/10.1007/s10649-020-09938-3

Spiers, J., Morse, J. M., Olson, K., Mayan, M., & Barrett, M. (2018). Reflection/Commentary on a Past Article: “Verification Strategies for Establishing Reliability and Validity in Qualitative Research.” International Journal of Qualitative Methods, 17(1). https://doi.org/10.1177/1609406918788237

Star, J. R., Tuomela, D., Joglar-Prieto, N., Hästö, P., Palkki, R., Abánades, M. Á., Pejlare, J., Jiang, R. H., Li, L., & Liu, R.-D. (2022). Exploring students’ procedural flexibility in three countries. International Journal of STEM Education, 9(1), 4. https://doi.org/10.1186/s40594-021-00322-y

Sterner, H. (2024). Using Graphical Representations to Develop Students’ Correspondence Relationships and Covariational Thinking in Pattern Generalizations in Primary School. International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-024-10520-z

Sun, S., Sun, D., & Xu, T. (2023a). The Developmental Progression of Early Algebraic Thinking of Elementary School Students. Journal of Intelligence, 11(12), 222. https://doi.org/10.3390/jintelligence11120222

Sun, S., Sun, D., & Xu, T. (2023b). The Developmental Progression of Early Algebraic Thinking of Elementary School Students. Journal of Intelligence, 11(12), 222. https://doi.org/10.3390/jintelligence11120222

Teare, M. D. (2016). Transparent reporting of research results in eLife. ELife, 5. https://doi.org/10.7554/eLife.21070

Thomas, J., & Harden, A. (2008). Methods for the thematic synthesis of qualitative research in systematic reviews. BMC Medical Research Methodology, 8(1), 45. https://doi.org/10.1186/1471-2288-8-45

Thompson, N. (2023). Vygotskian scaffolding techniques as motivational pedagogy for gifted mathematicians in further education: a diary-interview study. Journal of Further and Higher Education, 47(4), 492–512. https://doi.org/10.1080/0309877X.2022.2142103

Thurtle, V., Leather, A. J., Wurie, H., Foday, E., Samai, M., & Parmar, D. (2021). Strengthening ethics committees for health-related research in sub-Saharan Africa: a scoping review protocol. BMJ Open, 11(8), e046546. https://doi.org/10.1136/bmjopen-2020-046546

Van Wee, B., & Banister, D. (2024). Literature review papers: the search and selection process. Journal of Decision Systems, 33(4), 559–565. https://doi.org/10.1080/12460125.2023.2197703

Wang, Y., & Lewis, S. E. (2020). Analytical chemistry students’ explanatory statements in the context of their corresponding lecture. Chemistry Education Research and Practice, 21(4), 1183–1198. https://doi.org/10.1039/D0RP00063A

Wilkie, K. J., & Hopkins, S. (2024). Primary students’ relational thinking and computation strategies with concrete-to-symbolic representations of subtraction as difference. The Journal of Mathematical Behavior, 73, 101121. https://doi.org/10.1016/j.jmathb.2023.101121

Winker, M. A., Bloom, T., Onie, S., & Tumwine, J. (2023). Equity, transparency, and accountability: open science for the 21st century. The Lancet, 402(10409), 1206–1209. https://doi.org/10.1016/S0140-6736(23)01575-1

Yıldırım, D., Kırşan, M., Kıray, S., Korhan, E. A., Üstün, Ç., & Aykar, F. Ş. (2019). Nurse Academicians’ Attitudes Related to Academic Ethical Values and Related Factors. Journal of Academic Ethics, 17(4), 363–373. https://doi.org/10.1007/s10805-019-09339-z

Zwanch, K. (2022a). Using number sequences to account for differences in generalizations. School Science and Mathematics, 122(2), 86–99. https://doi.org/10.1111/ssm.12516

Zwanch, K. (2022b). Using number sequences to account for differences in generalizations. School Science and Mathematics, 122(2), 86–99. https://doi.org/10.1111/ssm.12516