Design and Application of Wise Tasks as a Technology of Productive Learning
Abstract
The article examines the pedagogical potential of digital educational resources such as Wise Tasks and their role in implementing productive learning technologies. The relevance of the study stems from the widespread practice of using digital tools primarily for organizing the learning process or knowledge assessment, which often leads to a passive role for students and limits the development of their research and creative activity. The paper analyzes the limitations of such approaches and substantiates the need to create educational environments in which students act not only as consumers of ready-made tasks and solutions, but also as active participants in their formulation and development. As a solution, the use of formalized means for representing subject area problems and digital tools that allow for the automatic verification of solutions to constructive tasks is proposed. This approach is implemented in resources like Wise Tasks, where the learner can experiment with different solution methods, test hypotheses, and formulate new problems. Using the example of developing an educational resource on graph theory, it is shown how such systems can support the “learning by teaching” technology, develop algorithmic and computational thinking, and engage students in the collaborative development of learning materials and software modules. The obtained results demonstrate that the use of such digital environments contributes to a deeper mastery of the discipline’s content and the formation of research skills.
References
V. A. Dalinger, “Educational and research activities of students in the study of mathematics,” Almanac of Modern Science and Education, no. 11 (42), part 1, pp. 36–39, 2010 (in Russian).
G. Polya and G. Szego, ¨Problems and Theorems in Analysis, Berlin: Springer Science & Business Media, 1998.
G. Polya, Mathematics and Plausible Reasoning, Princeton, NJ, USA: Princeton University Press, 1954.
G. Polya, Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving, New York, NY, USA: Wiley, 1981.
G. Polya, How to Solve It: A New Aspect of Mathematical Method, Princeton, NJ, USA: Princeton University Press, 2004.
C. Hoyles, “Transforming the mathematical practices of learners and teachers through digital technology,” Research in Mathematics Education, vol. 20, no. 3, pp. 209–228, 2018; doi:10.1080/ 14794802.2018.1484799
D. G. Zaykov and S. N. Pozdniakov, “Wise Tasks Graphs System,” in Proc. Int. Conf. Polynomial Com- puter Algebra, Saint Petersburg, Russia, 2022, pp. 180–185.
E. Malyutin, “Wise Tasks: Algorithms for Self-Checking Problems and Intelligent Tutoring Systems,” in Polynomial Computer Algebra 2025, St. Petersburg, Russia: Euler International Mathematical Institute, 2025.
L. Fiorella and R. E. Mayer, “The relative benefits of learning by teaching and teaching ex- pectancy,” Contemporary Educational Psychology, vol. 38, no. 4, pp. 281–288, 2013; doi:10.1016/ j.cedpsych.2013.06.001
N. M. Webb, “Peer interaction and learning in small groups,” International Journal of Educational Research, vol. 13, pp. 21–39, 1989.
S. Stollhans, “Learning by teaching: developing transferable skills,” in Employability for languages: a handbook, E. Corradini, K. Borthwick, and A. Gallagher-Brett, Eds. Research-publishing.net, 2016, pp. 161–164; doi:10.14705/rpnet.2016.cbg2016.478
D. Duran, “Learning-by-teaching. Evidence and implications as a pedagogical mechanism,” Innova- tions in Education and Teaching International, vol. 54, no. 5, pp. 476–484, 2017; doi:10.1080/14703297. 2016.1156011
G. Biswas,J. R. Segedy, and K.Bunchongchit, “From Design to Implementation to Practice a Learning by Teaching System: Betty’s Brain,” Int. J. Artif. Intell. Educ., vol. 26, pp. 350–364, 2016; doi:10.1007/s40593- 015-0057-9
E. Yarzebinski, A. Ogan, M. M. T. Rodrigo, and N. Matsuda, “Understanding Students’ Use of Code- Switching in a Learning by Teaching Technology,” in Artificial Intelligence in Education, pp. 504–513, 2015 2015; doi:10.1007/978-3-319-19773-9_50
L. S. Vygotsky and A. R. Luria, “Tool and symbol in child development,” in The Vygotsky Reader, R. Van der Veer and J. Valsiner, eds. Wiley-Blackwell, Oxford, UK: Blackwell, 1994, pp. 99–174.
C.-Y. Chou, T.-W. Chan, and C.-J. Lin, “Redefining the learning companion: the past, present, and future of educational agents,” Computers and Education, vol. 40, pp. 255–269, 2003.
E. Tolkacheva, S. Ivanov, and S. Pozdniakov, “Implementation of Horizontal Connections in the Course of Mathematics by Combining Pedagogical and Digital Technologies,” Mathematics, vol. 10, no. 13, p. 2352, 2022; doi:10.3390/math10132352
A. Burago, Mathematical Circle Diaries, Year 2: Complete Curriculum for Grades 6 to 8, (MSRI Mathematical Circles Library, vol. 20), Berkeley, CA, USA: MSRI, 2018.
V. Freiman, “Types of technology in mathematics education,” in Encyclopedia of mathematics education, pp. 869–879, 2020; doi:10.1007/978-3-030-15789-0_158
P. Leadbetter and P. Thomas, “A review of computer algebra and its educational implications in the teaching of mathematics,” Education and Computing, vol. 5, no. 4, pp. 243–259, 1989.
M. Artigue, “Learning mathematics in a CAS environment: The genesis of a reflection about instrumen- tation and the dialectics between technical and conceptual work,” International Journal of Computers for Mathematical Learning, vol. 7, no. 3, pp. 245–274, 2002; doi:10.1023/A:1022103903080
A. N. Arkhangelsky, V. N. Dubrovsky, M. Y. Lebedeva, et al., “Personality Extended by Digital Means,” Herald of the Russian Foundation for Basic Research, no. 113, pp. 38–51, 2022. (in Russian).
I. Lyublinskaya and V. Ryzhik, Geometry investigations with GeoGebra, St. Petersburg, Russia: SMIO- Press, 2020 (in Russian).
E. V. Shumara, “Computer Dialogue in the Education System,” in Problems of the theory and practice of teaching mathematics: Proc. Int. Sci. Conf. “The 73rd Herzen Readings”, St. Petersburg, Russia, 2020, pp. 7–11, (in Russian).
V. I. Ryzhik, “Geometry and computer,” Computer Tools in Education, no. 6, 2008. (in Russian).
A. S. Chukhnov, “Constructive Tasks as a Tool of Invasive and Non-invasive Assessment of Knowledge,” Computer tools in education, no. 3, pp. 96–104, 2019; doi:10.32603/2071-2340-2019-3-96-104
A. Chukhnov, A. Maytarrattanakhon, I. Posov, and S. Pozdniakov, “Constructive graph tasks in distant contests,” Informatics in Education, vol. 19, no. 3, pp. 343–359, 2020; doi:10.15388/infedu.2020.16
S. Abramovich, E. Malyutin, and S. Pozdniakov, “Mathematization Through Application and Common Sense: Motivating Intellectual Activities of Schoolchildren with Digital Tools,” Digital, vol. 5, no. 3, p. 41, 2025; doi:10.3390/digital5030041
E. V. Malyutin, “The use of digital technologies in teaching mathematics: from ‘black box’ systems to intelligent learning environments,” Development of Education, vol. 9, no. 1, 2026 (in Russian).
U. Kortenkamp et al., “Interoperable interactive geometry for Europe — first technological and edu- cational results and future challenges of the InterGeo project,” in Proc. 6th Conf. European Research in Mathematics Education (CERME 6), Lyon, France, 2009.
H. Poincare, “On the Foundations of Geometry,” The Monist, vol. 9, no. 1, pp. 1–43, 1898; doi:10.1093/ monist/9.1.1
H. Poincare,ˊ The Foundations of Science, New York, NY, USA: Science Press, 1902–1908 (reprinted in 1921).
S. Pozdnyakov and S. Ivanov, “Computers in productive teaching of mathematics or how information technologies can support intellectual freedom of the learner,” in The 10th Int. Congr. Math. Education, National Presentation: Russia, Selected Materials, Copenhagen, Denmark, 2004, pp. 115–124.
D. I. Mantserov, “Environment Verifer-KD: Verification of solutions of problems in mathematics,” Computer Tools in Education, no. 4, pp. 36–41, 2006 (in Russian).
Wise Task Graph. (2026) Accessed: Jan. 20, 2026. [Online]. Available: https://wisetask.ru/graph/library
M. S. Bogdanov, “Automation of checking the solution of problems according to the formal description of its conditions,” Computer Tools in Education, no. 1, pp. 24–32, 2006 (in Russian).
M. Bogdanov, S. Pozdnyakov, and A. Pukhov, “Multiplicity of the knowledge representation forms as a base of using a computer for the studying of the discrete mathematics,” Pedagogika, vol. 96, pp. 136– 142, 2009.
O. V. Perchenok, S. N. Pozdnyakov, and I. A. Posov, “Automation of verification of solving geometric problems based on the description of their conditions in a domain-specific language,” Computer Tools in Education, no. 1, pp. 31–38, 2012 (in Russian).
S. N. Pozdniakov, “Computer Algebra System as a Pedagogical Task,” Computer tools in education, no. 2, pp. 25–41, 2017. (in Russian).
S. N. Pozdnyakov, A. G. Lyamov, and N. Y. Prokopenko, “Computer tool in teaching mathematics,” Computer Tools in Education, no. 1, pp. 27–35, 2006 (in Russian).
M. I. Bashmakov, S. N. Pozdnyakov, and N. A. Reznik, Learning Information Environment, Saint Petersburg, Russia: Svet, 1997 (in Russian).
S. F. Adlaj and S. N. Pozdnyakov, “Digital Representations of Mathematical Objects in the Context of Various Forms of Representation of Mathematical Knowledge,” Computer tools in education, no. 1, pp. 58–86, 2020 (in Russian); doi:10.32603/2071-2340-2020-1-58
S. Papert, Mindstorms: Children, Computers, and Powerful Ideas, New York, NY, USA: Basic Books, Inc., 1980.
S. N. Pozdniakov, A. S. Chukhnov, and N. N. Pangina, “Analysis of the Understanding of the Material of Theoretical Informatics in Competitions and Olympiads in Informatics,” Computer tools in education, no. 2, pp. 55–67, 2018; doi:10.32603/2071-2340-2018-2-55-67
M. Wertheimer, Productive thinking, New York and London: Harper & brothers, 1945.
M. I. Bashmakov, ed., Theory and practice of productive teaching, Moscow: Narodnoye obrazovaniye, 2000 (in Russian).
I. Bohm and J. Schneider, Produktives Lernen—eine Bildungschance fur Jugendliche in Europa, Berlin, Germany / Milow, Germany: Schibri-Verlag, 1996.
S. Pozdniakov, “Computer in productive learning of mathematics,” in Proc. PME and Yandex Russian Conf.: Technology and Psychology for Mathematics Education, Moscow: National Research University Higher School of Economics, 2019, pp. 77–92.

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