Designing Feedback for Exploratory Mathematical Activities: A Computational Thinking Perspective
Computational Thinking Scaffolding for Exploratory Mathematical Activities
DOI:
https://doi.org/10.21240/constr/2025/78.XKeywords:
Exploratory mathematical activities, Feedback design, Computational thinkingAbstract
Computational thinking (CT), recognised as vital in mathematics education, closely aligns with problem-solving and is increasingly integrated into global curricula. In this context, designing and engaging learners in exploratory activities through programming and debugging is valuable. However, challenges remain in effectively incorporating CT into teaching, especially in designing and assessing such exploratory activities. This study explores these issues by developing and evaluating a prototype incorporating CT as a framework for feedback design in exploratory mathematical activities involving programming. The prototype employs the MaLT2 programming environment, inspired by Papert’s turtle geometry, within AuthELO, an authorable feedback design system. MaLT2 aids mathematical understanding through programming, while AuthELO offers real-time, context-aware feedback based on students’ interactions. We design a feedback mechanism that applies CT components to guide learners in open-ended tasks. Qualitative data from a semi-structured focus group with educators reveals that they appreciate CT-based feedback for its relevance to math and programming, though they struggle with abstraction. Participants indicated a need for specific and motivating feedback, emphasising the balance between guidance and exploratory learning. The study highlights the importance of authorable feedback systems facilitating iterative refinement of exploratory mathematics programming activities with CT as a scaffolding framework.References
Aho, A. V. (2012). Computation and computational thinking. The Computer Journal, 55(7), 832-835. DOI: 10.1093/comjnl/bxs074
Ainsworth, S., Major, N., Grimshaw, S., Hayes, M., Underwood, J., Williams, B., & Wood, D. (2003). Redeem: Simple intelligent tutoring systems from usable tools. In Authoring tools for advanced technology learning environments (pp. 205–232). Springer.
Aleven, V., McLaren, B. M., Sewall, J., & Koedinger, K. R. (2009). A new paradigm for intelligent tutoring systems: Example-tracing tutors. International Journal of Artificial Intelligence in Education, 19(2), 105–154.
Amershi, S., & Conati, C. (2006). Automatic recognition of learner groups in exploratory learning environments. In International Conference on Intelligent Tutoring Systems (pp. 463–472). Springer.
Amir, O., & Gal, Y. (2013). Plan recognition and visualisation in exploratory learning environments. ACM Transactions on Interactive Intelligent Systems (TiiS, 3)(3), 1–23.
Balacheff, N., & Kaput, J. J. (1996). Computer-based learning environments in mathematics. In International Handbook of Mathematics Education: Part 1 (pp. 469-501). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-009-1465-0_14
Bernardini, A., & Conati, C. (2010). Discovering and recognising student interaction patterns in exploratory learning environments. In International Conference on Intelligent Tutoring Systems (pp. 125–134). Springer.
Blessing, S., Gilbert, S., Ourada, S., & Ritter, S. (2007). Lowering the bar for creating model-tracing intelligent tutoring systems. Frontiers in Artificial Intelligence and Applications, 158, 443.
Boulden, D. C., Rachmatullah, A., Oliver, K. M., & Wiebe, E. (2021). Measuring in-service teacher self-efficacy for teaching computational thinking: Development and validation of the T-STEM CT. Education and Information Technologies, 26(4), 4663–4689. https://doi.org/10.1007/s10639-021-10487-2
Bråting, K., & Kilhamn, C. (2021). Exploring the intersection of algebraic and computational thinking. Mathematical Thinking and Learning, 23(2), 170–185. https://doi.org/10.1080/10986065.2020.1779012
Brennan, K., & Resnick, M. (2012, April). New frameworks for studying and assessing the development of computational thinking. In Proceedings of the 2012 annual meeting of the American Educational Research Association, Vancouver, Canada (Vol. 1, p. 25).
Brusilovsky, P. (2003). Developing adaptive educational hypermedia systems: From design models to authoring tools. In Authoring tools for advanced technology learning environments (pp. 377–409). Springer.
Bunt, A., Conati, C., & Muldner, K. (2004). Scaffolding self-explanation to improve learning in exploratory learning environments. In International Conference on Intelligent Tutoring Systems (pp. 656–667). Springer.
Chao, P. Y. (2016). Exploring students’ computational practice, design and performance of problem-solving through a visual programming environment. Computers & Education, 95, 202-215.
Chen, M. (1995). A methodology for characterising computer-based learning environments. Instructional Science, 23(1–3), 183–220.
Clements, D. H., & Meredith, J. S. (1993). Research on Logo: Effects and efficacy. Journal of Computing in Childhood Education, 4(4), 263-290.
Clements, D. H., Battista, M. T., & Sarama, J. (2001). Logo and geometry. Journal for research in mathematics education. Monograph, 10, i-177. https://doi.org/10.2307/749924
Cocea, M., & Magoulas, G. (2009a). Context-dependent personalised feedback prioritisation in exploratory learning for mathematical generalisation. In International Conference on User Modeling, Adaptation, and Personalization (pp. 271–282). Springer.
Cocea, M., & Magoulas, G. D. (2009b). Hybrid model for learner modeling and feedback prioritisation in exploratory learning. International Journal of Hybrid Intelligent Systems, 6(4), 211–230.
Cocea, M., & Magoulas, G. D. (2010). Group formation for collaboration in exploratory learning using group technology techniques. In International Conference on Knowledge-Based and Intelligent Information and Engineering Systems (pp. 103–113). Springer.
Cocea, M., Gutierrez-Santos, S., & Magoulas, G. (2008). Challenges for intelligent support in exploratory learning: The case of shapebuilder. In 1st International Workshop on Intelligent Support for Exploratory Environments. CEUR Workshop Proceedings.
Cui, Z., & Ng, O.-L. (2021). The interplay between mathematical and computational thinking in primary school students’ mathematical problem-solving within a programming environment. Journal of Educational Computing Research, 59(5), 988–1012. https://doi.org/10.1177/0735633120979930
Denning, P. J. (2017). Remaining trouble spots with computational thinking. Communications of the ACM, 60(6), 33-39. https://doi.org/10.1145/2998438
Feurzeig, W., Papert, S. A., & Lawler, B. (2011). Programming languages as a conceptual framework for teaching mathematics. Interactive Learning Environments, 19(5), 487-501. https://doi.org/10.1080/10494820903520040
Gal, Y., Yamangil, E., Shieber, S. M., Rubin, A., & Grosz, B. J. (2008). Towards collaborative intelligent tutors: Automated recognition of users’ strategies. In International Conference on Intelligent Tutoring Systems (pp. 162–172). Springer.
Geraniou, E., Mavrikis, M., Hoyles, C., & Noss, R. (2010). A learning environment to support mathematical generalisation in the classroom. Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education, 978-2-7342-1190-7. hal-02182374. https://hal.science/hal-02182374v1
Ginon, B., Jean-Daubias, S., Lefevre, M., Champin, P.-A., et al. (2014). Adding epiphytic assistance systems in learning applications using the sepia system. In European Conference on Technology Enhanced Learning (pp. 138–151). Springer.
Grover, S., & Pea, R. (2013). Computational thinking in K–12: A review of the state of the field. Educational Researcher, 42(1), 38-43. https://doi.org/10.3102/0013189X12463051
Gutierrez-Santos, S., Mavrikis, M., & Magoulas, G. D. (2012). A separation of concerns for engineering intelligent support for exploratory learning environments. Journal of Research and Practice in Information Technology, 44(3), 347.
Kale, U., Akcaoglu, M., Cullen, T., Goh, D., Devine, L., Calvert, N., & Grise, K. (2018). Computational what? Relating computational thinking to teaching. TechTrends, 62(6), 574–584. https://doi.org/10.1007/s11528-018-0290-9
Kaufmann, O. T. & Stenseth, B. (2021). Programming in mathematics education. International Journal of Mathematical Education in Science and Technology, 52(7), 1029-1048.
Karkalas, S., & Mavrikis, M. (2016a). Feedback authoring for exploratory learning objects: Authelo. In CSEDU 2016 – Proceedings of the 8th International Conference on Computer Supported Education (Vol. 1, pp. 144–153). Science and Technology Publications, Lda.
Karkalas, S., Mavrikis, M., Xenos, M., & Kynigos, C. (2016b). Feedback authoring for exploratory activities: The case of a logo-based 3D microworld. In International Conference on Computer Supported Education (pp. 259–278). Springer.
Kastner-Hauler, O., Tengler, K., Sabitzer, B., & Lavicza, Z. (2025). A learning environment to promote the computational thinker: A Bebras perspective evaluation. In Z. Pluhár & B. Gaál (Eds.), Informatics in schools. Innovative approaches to computer science teaching and learning (Vol. 15228, pp. 85–98). Springer Nature Switzerland. https://doi.org/10.1007/978-3-031-73474-8_7
Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.
Klahr, D., & Nigam, M. (2004). The equivalence of learning paths in early science instruction: Effects of direct instruction and discovery learning. Psychological Science, 15(10), 661–667.
Kynigos, C. & Grizioti, M. (2018). Programming Approaches to Computational Thinking: Integrating Turtle Geometry, Dynamic Manipulation and 3D Space. Informatics in Education, 17(2), 321–340. https://doi.org/10.15388/infedu.2018.17
Kynigos, C. (2007). Half-baked logo microworlds as boundary objects in integrated design. Informatics in Education-An International Journal, 6(2), 335-359. https://doi.org/10.1007/s10758-007-9114-2
Kynigos, C., & Karavakou, M. (2023). Coding dancing figural animations: mathematical meaning-making through transitions within and beyond a digital resource. Digital Experiences in Mathematics Education, 9(2), 283-314. https://doi.org/10.1007/s40751-022-00118-x
Lee, H.-Y., Wu, T.-T., Lin, C.-J., Wang, W.-S., & Huang, Y.-M. (2024). Integrating computational thinking into scaffolding learning: An innovative approach to enhance science, technology, engineering, and mathematics hands-on learning. Journal of Educational Computing Research, 62(2), 431–467. https://doi.org/10.1177/07356331231211916
Mavrikis, M., Geraniou, E., Noss, R., & Hoyles, C. (2008). Revisiting pedagogic strategies for supporting students’ learning in mathematical microworlds. In Proceedings of the International Workshop on Intelligent Support for Exploratory Environments at EC-TEL (Vol. 8, pp. 41-50).
Mavrikis, M., Geraniou, E., Gutierrez Santos, S. and Poulovassilis, A. (2019), Intelligent analysis and data visualisation for teacher assistance tools: The case of exploratory learning. British Journal of Educational Technology, 50: 2920-2942. https://doi.org/10.1111/bjet.12876
Mayer, R. E. (2004). Should there be a three-strikes rule against pure discovery learning? American Psychologist, 59(1), 14.
Mitchelmore, M. C., & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics, 41, 209-238. https://doi.org/10.1023/A:1003927811079
Morales Gamboa, R., & Gamboa, R. M. (2000). Exploring participative learner modelling and its effects on learner behaviour.
Mumcu, F., Kıdıman, E., & Özdinç, F. (2023a). Integrating computational thinking into mathematics education through an unplugged computer science activity. Journal of Pedagogical Research, 7(2), 72-92. https://doi.org/10.33902/JPR.202318528
Mumcu, F., Uslu, N. A., & Yıldız, B. (2023b). Teacher development in integrated STEM education: Design of lesson plans through the lens of computational thinking. Education and Information Technologies, 28, 3443–3474. https://doi.org/10.1007/s10639-022-11342-8
Mumcu, F., Andic, B., Maricic, M., Tejera, M., & Lavicza, Z. (2025). Unpacking teachers’ value beliefs about computational thinking and programming. Journal of Educational Technology & Online Learning, 8(1), 41-63. https://doi.org/10.31681/jetol.1497284
Ng, O.-L., & Cui, Z. (2021). Examining primary students’ mathematical problem-solving in a programming context: Towards computationally enhanced mathematics education. ZDM – Mathematics Education, 53(4), 847–860. https://doi.org/10.1007/s11858-020-01200-7
Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers (Vol. 17). Springer Science & Business Media.
Papert, S. (1980). Computers for children. Mindstorms: Children, computers, and powerful ideas. Basic Books.
Pawar, U. S., Pal, J., & Toyama, K. (2006). Multiple mice for computers in education in developing countries. In 2006 International Conference on Information and Communication Technologies and Development (pp. 64–71). IEEE.
Pearce-Lazard, D., Poulovassilis, A., & Geraniou, E. (2010). The design of teacher assistance tools in an exploratory learning environment for mathematics generalisation. In European Conference on Technology Enhanced Learning (pp. 260–275). Springer.
Polya, G. 1945. How to Solve It: A New Aspect of Mathematical Method. Princeton University Press: Princeton, New Jersey.
Refvik, K. A. S., & Bjerke, A. H. (2022). Computational thinking as a tool in primary and secondary mathematical problem solving: A literature review. NOMAD Nordic Studies in Mathematics Education, 27(3), Article 3. https://doi.org/10.7146/nomad.v27i3.149191
Rich, K. M., Spaepen, E., Strickland, C., & Moran, C. (2020). Synergies and differences in mathematical and computational thinking: Implications for integrated instruction. Interactive Learning Environments, 28(3), 272–283. https://doi.org/10.1080/10494820.2019.1612445
Turgut, M., Kohanová, I., & Gjøvik, Ø. (2024). Developing survey-based measures of mathematics teachers’ pedagogical technology knowledge: A focus on computational thinking and programming tools. Research in Mathematics Education, 1–24. https://doi.org/10.1080/14794802.2024.2401482
Van Joolingen, W. R., & Zacharia, Z. C. (2009). Developments in inquiry learning. In Technology-enhanced learning (pp. 21–37). Springer.
Voogt, J., Fisser, P., Good, J., Mishra, P., & Yadav, A. (2015). Computational thinking in compulsory education: Towards an agenda for research and practice. Education and Information Technologies, 20(4), 715–728. https://doi.org/10.1007/s10639-015-9412-6
Weintrop, D., Morehouse, S., & Subramaniam, M. (2021). Assessing computational thinking in libraries. Computer Science Education, 31(2), 290–311. https://doi.org/10.1080/08993408.2021.1874229
Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33–35. https://doi.org/10.1145/1118178.1118215
Yadav, A., Stephenson, C., & Hong, H. (2017). Computational thinking for teacher education. Communications of the ACM, 60(4), 55-62. https://doi.org/10.1145/2994591
Yadav, A., Zhou, N., Mayfield, C., Hambrusch, S., & Korb, J. T. (2011). Introducing computational thinking in education courses. In Proceedings of the 42nd ACM Technical Symposium on Computer Science Education (pp. 465–470). https://doi.org/10.1145/1953163.1953297
Downloads
Published
Conference Proceedings Volume
Section
License
Copyright (c) 2025 Filiz Mumcu, Manolis Mavrikis, Sokratis Karkalas, Christina Gkreka, Chara Korompli, Dimitris Diamantidis, Zsolt Lavicza
This work is licensed under a Creative Commons Attribution 4.0 International License.
