VISUAL AND COMPUTER-BASED MODELS IN INTEGRATED MATHEMATICS EDUCATION

Abstract

The article substantiates the pedagogical value of visual and computer-based modelling in secondary mathematics (Grades 7–11) and proposes an integration framework that combines GeoGebra, Desmos, and Python (turtle, matplotlib) to cultivate spatial and computational thinking. Conceptual-methodological reasoning and the authors’ synthesis indicate that dynamic geometry systems (DGS) enable students to reason with invariants, parameters, and transformations through direct manipulation, while Python translates visual insight into programmatic verification via a series of experiments, automated constructions, and data-driven checks. We formulate a didactic workflow-visualisation → analysis → programmatic verification → generalization – and show how it can be enacted within a two-period lesson through small-group roles (constructor, analyst, programmer, presenter) that support agency, collaboration, and reflective discourse. The paper presents two summary tables that map integration opportunities across major curriculum strands (geometry, functions, analytic geometry, transformations, statistics/probability, combinatorics, and STEM projects) and align tools to their pedagogical affordances. A companion table offers lesson fragments in which learners independently discover patterns (e.g., relations among triangle elements, parameter effects on a quadratic, linear dependence, law of large numbers, rigid motions, and fractals) and articulate conclusions grounded in both dynamic visuals and code outputs. Practical guidance is provided on low-barrier task design (short, readable code; prepared sliders and templates), process-oriented assessment (quality of hypotheses, parameter variation, algorithmic correctness, interpretation of error, teamwork), and differentiation («common core + extensions» for mixed-ability classes). We also outline organisational conditions and risk-mitigation strategies (offline GeoGebra, portable Python, cached applets, print-ready snapshots) to ensure continuity in constrained settings. The contribution is methodological rather than empirical; nonetheless, it delineates mechanisms by which modelling can impact learning: visual variability fosters spatial intuition; algorithmic iteration stabilises conjectures; and immediate feedback sustains motivation by shifting learners from observers to creators. The article concludes with prospects for future work – quasi-experimental evaluation of impacts on spatial/computational thinking, tighter Python–DGS bridges, and refined task taxonomies – positioning integrated modelling as a coherent pathway from intuition to formal mathematical explanation in contemporary classrooms. The perspectives for further research are related to the empirical evaluation of the impact of modelling on students’ spatial and computational thinking, as well as the exploration of task differentiation for different levels of learner preparedness.