The protocol for solving a combinatorial problem as a boundary object
Introduction. In the context of the proliferation of digital educational environments and artificial intelligence systems, the problem of transmitting mathematical meanings in teaching becomes particularly relevant. The availability of ready-made solutions does not guarantee that students will develop their own reasoning methods, which is especially significant in teaching combinatorics, where constructing a model, choosing an enumeration method, and justifying the solution are required. Aim. To identify the conditions under which the Wise Tasks Combinatorics system can act as a boundary object in teaching combinatorics. Materials and Methods. The study is based on analyzing students’ work with a digital environment that supports solving, constructing, and verifying combinatorial problems. A comparison was made between independent work, work with mandatory protocoling of the solution process, and work with pedagogical support. Solution protocols, errors, number of attempts, and reasoning patterns were analyzed. Results. It was found that during independent work, students primarily focus on obtaining an answer. Mandatory protocoling makes the solution process more detailed but does not in itself provide meaningful deepening. It is shown that only in combination with pedagogical support does the digital environment begin to support exploratory research activity and the transmission of basic combinatorial ideas. It is substantiated that the boundary object is not the system itself, but the system embedded in a specially organized interaction. Conclusion. It is not the Wise Tasks Combinatorics system itself that acts as a boundary object, but rather the same system embedded in a specially organized interaction that includes pedagogical support and reflection on the solution process – which alone ensures the transfer of combinatorial meanings.
While nobody left any comments to this publication.
You can be first.
Adlai, S.F. and Pozdnyakov, S.N. (2020), “Digital representations of mathematical objects in the context of different forms of mathematical knowledge representation”, Kompyuternye instrumenty v obrazovanii, 1, 58-86. DOI: 10.32603/2071-2340-2020-1-58-86. (In Russian).
Bashmakov, M.I., Pozdnyakov, S.N. and Reznik, N.A. (1997), Informatsionnaya sreda obucheniya [Information learning environment], Svet, St. Petersburg, Russia.
Bogdanov, M.S. (2006), “Automation of problem solution checking based on formal description of its condition”, Kompyuternye instrumenty v obrazovanii, 4, 51-57. (In Russian).
Bogoslovsky, V.I., Izvozchikov, V.A. and Potemkin, M.N. (2000), “Information and educational space or information and educational chronotope”, Nauka i shkola, 5, 42-46. (In Russian)
Vygotsky, L.S. (1999), Myshlenie i rech [Thinking and Speech], Labirint, Moscow, Russia.
Vilenkin, N.Ya. (1969), Kombinatorika [Combinatorics], Nauka, Moscow, Russia.
Malyutin, E.V. (2026), “The use of digital technologies in teaching mathematics: from black-box systems to intelligent pedagogical environments”, Razvitie obrazovaniya, 9 (1), 36-46. DOI: 10.31483/r-152277. (In Russian).
Malyutin, E.V. (2026), “Wise Task: intelligent partnership in a digital educational environment”, Mezhdunarodnyy nauchno-issledovatelskiy zhurnal, 3 (165). DOI: 10.60797/IRJ.2026.165.47. (In Russia).
Naumov, I.S. and Vykhovanets, V.S. (2014), “Assessment of difficulty and complexity of learning tasks based on syntactic text analysis”, Upravlenie bolshimi sistemami, 48, 97-131. (In Russian).
Papert, S. (1989), Perevorot v soznanii: deti, kompyutery i plodotvornye idei [Mindstorms: Children, Computers, and Powerful Ideas], Pedagogika, Moscow, Russia.
Pozdnyakov, S.N. (2017), “Computer algebra system as a pedagogical problem”, Kompyuternye instrumenty v obrazovanii, 2, 25-41. (In Russian).
Rokhlin, V.A. (2004), “A lecture on teaching mathematics to non-mathematicians”, Matematicheskoe prosveshchenie, Series 3, 8, 21-36. (In Russian).
Abramovich, S., Malyutin, E. and Pozdniakov, S. (2025), “Mathematization through application and common sense: motivating intellectual activities of schoolchildren with digital tools”, Digital, 5 (3), Art. 41. DOI: 10.3390/digital5030041. (In Switzerland).
Bannon, L. and Bodker, S. (1997), “Constructing common information spaces”, Proceedings of the Fifth European Conference on Computer Supported Cooperative Work, Dordrecht, Netherlands, 81-96. DOI: 10.1007/978-94-015-7372-6_6. (In Netherlands).
Dahl, O.J. and Nygaard, K. (1966), “SIMULA - an ALGOL-based simulation language”, Communications of the ACM, 9 (9), 671-678. (In USA).
Dolgopolovas, V., Dagiene, V., Pozdniakov, S. and Liaptsev, A. (2022), “Developing computational thinking skills to foster student research: contemporary scientific education through modeling and simulations”, in Rezaei, N. (ed.), Integrated Education and Learning, Integrated Science, 13, 417-443. DOI: 10.1007/978-3-031-15963-3_23. (In Germany).
Holmes, W. et al. (2022), “Artificial Intelligence in Education”, Computers & Education: Artificial Intelligence, 3, Art. 100099. (In Netherlands).
Lavrenov, A. and Pozdniakov, S. (2025), “Interaction of students with an intelligent system for setting and solving combinatorial tasks”, Computers in the Schools, 42 (3), 257-275. DOI: 10.1080/07380569.2024.2410906. (In USA).
Morgan, C. and Kynigos, C. (2014), “Digital artefacts as representations: forging connections between a constructionist and a social semiotic perspective”, Educational Studies in Mathematics, 85, 357-379. DOI: 10.1007/s10649-013-9523-1. (In Netherlands).
Moroshkina, N.V., Savina, A.I., Ammalainen, A.V., Gershkovich, V.A., Zverev, I.V. and Lvova, O.V. (2022), “How difficult was it? Metacognitive judgments about problems and their solutions after the Aha moment”, Frontiers in Psychology, 13, Art. 911904. DOI: 10.3389/fpsyg.2022.911904. (In Switzerland).
Nguyen, D.T. and Pham, Q.V. (2025), “The evolving landscape of AI integration in mathematics education: a systematic review of trends (2015–2025)”, Eurasia Journal of Mathematics, Science and Technology Education, 21 (10), Art. em2714. DOI: 10.29333/ejmste/17078. (In UK).
Nievergelt, J. (1980), “A pragmatic introduction to courseware design”, IEEE Computer, September, 7-21. (In USA).
Son, L.N., Tuyen, N.T.T., Tinh, P.T., Tu, N.X., Dinh, N.T., Huyen, L.T.T. and Trang, N.H. (2025), “Artificial intelligence in mathematics
education: a bibliometric analysis in Scopus
Database (2000-2024)”, in Nghia, P.T. et al. (eds.), Advances in Information and Communication Technology, ICTA 2024, Lecture Notes in Networks and Systems, 1205, Springer, Cham, Germany, 403-410. DOI: 10.1007/978-3-031-80943-9_44. (In Germany).
Son, T. (2024), “Intelligent tutoring systems in mathematics education: a systematic literature review using the SAMR model”, Computers, 13 (10), 270. DOI: 10.3390/computers13100270. (In Switzerland).
Star, S.L. (1989), “The structure of ill-structured solutions: boundary objects and heterogeneous distributed problem solving”, in Gasser, L. and Huhns, M.N. (eds.), Distributed Artificial Intelligence, Pitman, London, UK.
Star, S.L. (2010), “This is not a boundary object: reflections on the origin of a concept”, Science, Technology, & Human Values, 35 (5), 601-617. (In USA).
Star, S.L. and Griesemer, J.R. (1989), “Institutional ecology, ‘translations’ and boundary objects: amateurs and professionals in Berkeley’s Museum of Vertebrate Zoology, 1907–39”, Social Studies of Science, 19 (3), 387-420. (In UK).
Wiley, D.A. (2000), “Connecting learning objects to instructional design theory: a definition, a metaphor, and a taxonomy”, in Wiley, D.A. (ed.), The Instructional Use of Learning Objects, Bloomington, USA.