MIDK 2025

Integrating Programming into Combinatorics Education: A Pathway to Combinatorial Reasoning and Computational Thinking

Problem-solving is central to mathematical education, fostering essential skills for success. Combinatorics provides unique problem-solving experiences, encouraging students to explore multiple solution paths, justify reasoning, and compare strategies, fostering metacognitive growth. Combinatorial thinking combines logical reasoning with systematic planning, helping students analyze all possible configurations of elements. It supports transitions from intuitive listing to abstract reasoning using sum and product rules, permutations, arrangements, and combinations. As a gateway topic, combinatorics boosts struggling students' confidence in mathematics. Integrating programming enhances combinatorial and computational thinking. Programming automates processes, algorithmizes problem-solving, and visualizes complex structures. It enables students to explore broader problem spaces by analyzing, modifying, or creating algorithms, deepening their understanding of systematic listing and combinatorial logic. This lecture explores how programming enriches combinatorics education through combinatorial and computational thinking frameworks, focusing on automation and algorithmization. Practical examples will illustrate how programming verifies systematic listings, models structures, and solves problems algorithmically. Engaging with code—whether analyzing, modifying, or creating—enhances mathematical learning and fosters computational literacy.

The diversity of motivation - games and experiences from mathematics and IT lessons

The basic components of our competences are the elements of the knowledge system (knowledge, skills, etc.). The science of education has many theoretical and methodological results on their development. The content regulations of education generally define the knowledge elements to be acquired at each age – sometimes also the criteria for these elements.
Equally important are the motives that inspirate students to engage in activities that lead to the solution of tasks. Motives differ significantly from the elements of the knowledge system in their development, functioning and accessibility for the educational assessment. The level and durability of the knowledge elements are well perceived, with little deterioration over time when the knowledge elements are acquired at the appropriate level. The assessment of motives is more difficult: depending on time and circumstances, the strength of the individual motives that are activated can vary greatly, and this can show significant individual differences.
Going beyond the concept of motivation at the beginning of the lesson, conscious motivation development aims to allow as many different motives as possible to appear in the lessons. This method allows students to build their own unique system of motives from the successes and failures of different activities.
In this presentation, I will give an overview of the development of the knowledge system and motivation, illustrate the diversity of the motivational system with some types of motives, and then present learning situations and tasks to activate several types of motives. The examples are based on my own experience at secondary and college levels. Games with different embeddedness are presented as elements to support the maintenance of motivation. At the end of the presentation, I will show a game that provides a framework for students' individual practice.

A novel approach to teaching calculus using computer-supported collaborative learning and its improvement from 2012 to 2025

A novel approach to teaching and learning calculus, developed using Computer-Supported Collaborative Learning (CSCL), will be presented, highlighting its continuous improvement from 2012 to 2025. This method was initiated in response to the challenges faced by first-year students, particularly in understanding fundamental calculus concepts and applying derivatives to analyze functions and draw their graphs. In 2011, educators at the University of Novi Sad and Subotica Tech College of Applied Sciences (Djurdjica Takaci and Gordana Stankov), Serbia, used collaborative learning, and in 2012 they decided to begin integrating the GeoGebra application into collaborative learning practices. A comparison between two groups of students, one from 2011 (the control group) and another from 2012 (the experimental group), focusing on their learning methods, performance, and responses in questionnaires and interviews, demonstrated that GeoGebra facilitated the development of an effective learning environment for this topic. The GeoGebra application allows students to verify each step in the process of solving a task. Findings from our research indicate that GeoGebra supports students with insufficient prior knowledge, helping them enhance their understanding and skills necessary for solving such tasks. This teaching approach and the resulting learning outcomes were detailed in a paper published in 2015 (Dj. Takači, G. Stankov, I. Milanovic, Efficiency of learning environment using GeoGebra when calculus contents are learned in collaborative groups, Computers and Education, Vol. 82, 2015, 421-431). In the years leading up to the COVID-19 pandemic, students used their phones instead of computers and independently began incorporating other software tools into their collaborative learning activities, further evolving the methodology of this approach to collaborative learning. Building on the positive outcomes, in 2024 we extended this approach to teaching students how to calculate the area of geometric figures bounded by function graphs using definite integrals. This extension of our CSCL method has shown promising results, which will also be discussed in our presentation.