DESIGNING WORKED EXAMPLES TO SUPPORT MATHEMATICAL REASONING IN LEARNING CIRCLE CONCEPTS

Fitria Nengsi Puasa(1*), Kuswari Hernawati(2),

(1) Universitas Negeri Yogyakarta
(2) Universitas Negeri Yogyakarta
(*) Corresponding Author

Abstract


Students’ reliance on routine procedures and difficulties in understanding abstract geometric concepts highlight the need for instructional designs that support mathematical reasoning. This study aims to construct a worked example design framework that facilitates students’ transition from procedural understanding to mathematical reasoning in learning circles. The study employs a design and development research approach by adapting the Analysis, Design, and Development phases of the ADDIE model. the proposed worked example prototype was validated by two mathematics education experts using an expert validation sheet covering the aspects of material, construction, and language. The design integrates principles of Cognitive Load Theory and indicators of mathematical reasoning to produce a worked example prototype. The development process resulted in a worked example prototype that was validated by two mathematics education experts and deemed feasible for use after revisions. The prototype integrates Cognitive Load Theory principles and mathematical reasoning indicators to support students in recognizing patterns, formulating conjectures, and constructing logical arguments. This is achieved through an integrated presentation that minimizes extraneous cognitive load and the use of self-explanation prompts to encourage active cognitive processing. In addition, structured solution steps and paired problems are used as fading guidance to support the transition from guided learning to independent problem solving. Therefore, this study provides a worked example prototype that can serve as a reference for designing instructional materials to support mathematical reasoning in geometry learning.


Keywords


Circle concept, Cognitive load theory, Geometry learning, Mathematical reasoning, Worked example

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References


Almubarak, M., Maat, S. M., & Mahmud, M. S. (2025). Evolving three decades of geometry learning strategies: A combination of bibliometric analysis and systematic review. In Eurasia Journal of Mathematics, Science and Technology Education (Vol. 21, Number 6). Modestum LTD. https://doi.org/10.29333/ejmste/16515

Baars, M., van Gog, T., de Bruin, A., & Paas, F. (2014). Effects of problem solving after worked example study on primary school children’s monitoring accuracy. Applied Cognitive Psychology, 28(3), 382–391. https://doi.org/10.1002/acp.3008

Barbieri, C. A., & Booth, J. L. (2020). Mistakes on display: Incorrect examples refine equation solving and algebraic feature knowledge. Applied Cognitive Psychology, 34(4), 862–878. https://doi.org/10.1002/acp.3663

Barbieri, C. A., Miller-Cotto, D., Clerjuste, S. N., & Chawla, K. (2023). A Meta-analysis of the Worked Examples Effect on Mathematics Performance. Educational Psychology Review, 35(1). https://doi.org/10.1007/s10648-023-09745-1

Bentley, B., & Yates, G. C. R. (2017). Facilitating proportional reasoning through worked examples: Two classroom-based experiments. Cogent Education, 4(1). https://doi.org/10.1080/2331186X.2017.1297213

Bokosmaty, S., Sweller, J., & Kalyuga, S. (2015). Learning Geometry Problem Solving by Studying Worked Examples: Effects of Learner Guidance and Expertise. American Educational Research Journal, 52(2), 307–333. https://doi.org/10.3102/0002831214549450

Borthwick, A., & Cross, A. (2018). Reasons to Reason in Primary Maths and Science. Learning Matters.

Chen, O., Retnowati, E., Chan, B. B. K. Y., & Kalyuga, S. (2023). The effect of worked examples on learning solution steps and knowledge transfer. Educational Psychology, 43(8), 914–928. https://doi.org/10.1080/01443410.2023.2273762

Crompton, H., & Ferguson, S. (2024). An analysis of the essential understandings in elementary geometry and a comparison to the common core standards with teaching implications. In European Journal of Science and Mathematics Education (Vol. 12, Number 2, pp. 258–275). Bastas. https://doi.org/10.30935/scimath/14361

Demir, M., Zengin, Y., Özcan, ?., Urhan, S., & Aksu, N. (2023). Students’ mathematical reasoning on the area of the circle: 5E-based flipped classroom approach. International Journal of Mathematical Education in Science and Technology, 54(1), 99–123. https://doi.org/10.1080/0020739X.2022.2101955

Gregersen, R. M. (2024). Analysing Instrumented Justification: Unveiling Student’s Tool Use and Conceptual Understanding in the Prediction and Justification of Dynamic Behaviours. Digital Experiences in Mathematics Education, 10(1), 47–75. https://doi.org/10.1007/s40751-024-00134-z

Hänze, M., & Leiss, D. (2022). Using heuristic worked examples to promote solving of reality-based tasks in mathematics in lower secondary school. Instructional Science, 50(4), 529–549. https://doi.org/10.1007/s11251-022-09583-8

Hjelte, A., Schindler, M., & Nilsson, P. (2020). Kinds of mathematical reasoning addressed in empirical research in mathematics education: A systematic review. Education Sciences, 10(10), 1–15. https://doi.org/10.3390/educsci10100289

Jablonski, S., & Ludwig, M. (2023). Teaching and Learning of Geometry—A Literature Review on Current Developments in Theory and Practice. In Education Sciences (Vol. 13, Number 7). Multidisciplinary Digital Publishing Institute (MDPI). https://doi.org/10.3390/educsci13070682

Kamalodeen, V. J., Ramsawak-Jodha, N., Figaro-Henry, S., Jaggernauth, S. J., & Dedovets, Z. (2021). Designing gamification for geometry in elementary schools: insights from the designers. Smart Learning Environments, 8(1). https://doi.org/10.1186/s40561-021-00181-8

Kementerian Pendidikan dan Kebudayaan. (2017). Matematika SMP/MTs Kelas VIII Semester 2 (Edisi revisi). Jakarta: Kementerian Pendidikan dan Kebudayaan.

Lannin, J. K. ., Ellis, A. B. ., & Elliott, Rebekah. (2011). Developing essential understanding of mathematical reasoning for teaching mathematics in prekindergarten-grade 8. National Council of Teachers of Mathematics.

Lee, C.-Y., & Chen, M.-J. (2015). Effects of Worked Examples Using Manipulatives on Fifth Graders’ Learning Performance and Attitude toward Mathematics. In Journal of Educational Technology (Vol. 18, Number 1).

Lehmann, T. H. (2024a). Learning to measure the area of circles. Research in Mathematics Education. https://doi.org/10.1080/14794802.2024.2304329

Lithner, J. (2017). Principles for designing mathematical tasks that enhance imitative and creative reasoning. ZDM - Mathematics Education, 49(6), 937–949. https://doi.org/10.1007/s11858-017-0867-3

Michael–Chrysanthou, P., Panaoura, A., Gagatsis, A., & Elia, I. (2024). Exploring secondary school students’ geometrical figure apprehension: cognitive structure and levels of geometrical ability. Educational Studies in Mathematics, 117(1), 23–42. https://doi.org/10.1007/s10649-024-10317-5

Mukuka, A., Balimuttajjo, S., & Mutarutinya, V. (2023). Teacher efforts towards the development of students’ mathematical reasoning skills. Heliyon, 9(4). https://doi.org/10.1016/j.heliyon.2023.e14789

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM.

Nickerson, R. S. (2010). Mathematical reasoning: Patterns, problems, conjectures, and proofs. Psychology Press.

Öz, T., & I??k, A. (2024). Exploring mathematical reasoning skills of middle school students. Thinking Skills and Creativity, 53. https://doi.org/10.1016/j.tsc.2024.101612

Paas, F., & van Merriënboer, J. J. G. (2020). Cognitive-Load Theory: Methods to Manage Working Memory Load in the Learning of Complex Tasks. Current Directions in Psychological Science, 29(4), 394–398. https://doi.org/10.1177/0963721420922183

Pachman, M., Sweller, J., & Kalyuga, S. (2014). Effectiveness of combining worked examples and deliberate practice for high school geometry. Applied Cognitive Psychology, 28(5), 685–692. https://doi.org/10.1002/acp.3054

Putu Ade Andre Payadnya, I., Prahmana, R. C. I., Lo, J. J., Noviyanti, P. L., & Made Dharma Atmaja, I. (2023). Designing area of circle learning trajectory based on “what-if” questions to support students’ higher-order thinking skills. Journal on Mathematics Education, 14(4), 757–780. https://doi.org/10.22342/jme.v14i4.pp757-780

Renkl, A. (2014b). Toward an instructionally oriented theory of example-based learning. Cognitive Science, 38(1), 1–37. https://doi.org/10.1111/cogs.12086

Sundstrom, T. (2016). Mathematical Reasoning Writing and Proof (Version 2.1).Grand Valley State University.

Sweller, J. (2020). Cognitive load theory and educational technology. Educational Technology Research and Development, 68(1), 1–16. https://doi.org/10.1007/s11423-019-09701-3

Sweller, J., van Merriënboer, J. J. G., & Paas, F. (2019). Cognitive Architecture and Instructional Design: 20 Years Later. In Educational Psychology Review (Vol. 31, Number 2, pp. 261–292). Springer New York LLC. https://doi.org/10.1007/s10648-019-09465-5

Weigand, H. G., Hollebrands, K., & Maschietto, M. (2025). Geometry education at secondary level – a systematic literature review. ZDM - Mathematics Education, 57(4), 829–843. https://doi.org/10.1007/s11858-025-01703-1

Weingarden, M., & Buchbinder, O. (2023). Teacher learning to teach mathematics via reasoning and proving: a discursive analysis of lesson plans modifications. Frontiers in Education, 8. https://doi.org/10.3389/feduc.2023.1154531




DOI: https://doi.org/10.31327/jme.v11i2.2827

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