21st century mathematics teacher candidates' use of representations and critical thinking skills

One of the important characteristics expected in individuals during the problem-solving processes that play a significant role in mathematics education is the ability to use representations, while another is critical thinking skills. Based on this, the aim of the research is to identify the representations and critical thinking skills used by 21st-century mathematics teacher candidates. In line with this objective, the study examines the representations used by prospective mathematics teachers according to the representation classifications developed by Janvier (1987) and Lesh, Behr, and Post (1987) (Verbal representation: expressing the problem and its solution and reasoning about the problem during the problem-solving process. Algebraic representation: the use of mathematical symbols or variables in the problem-solving process. Graphical representation: the use of number lines, pictures, diagrams, or charts in the problem-solving process. Numerical representation: the use of tables or matrices in the problem-solving process) and, as a result of the requirements of the current era, their views on critical thinking skills, including digital representations, were determined. The research is a qualitative descriptive survey. The data for the study were collected using interview forms (IF) developed by the researchers and evaluated using content analysis methods. In the study, interview forms were administered to teacher candidates selected according to an appropriate sampling method from among mathematics teacher candidates enrolled at the Faculty of Education at İnönü and Marmara Universities in the Fall Semester of 2025-2026. The research data were evaluated using the MAXQDA qualitative data analysis program and visualized with the help of code maps. While analyzing the findings, categories were shaped with the help of codes, and themes were shaped with the help of categories. The results show that prospective mathematics teachers focus on symbolic representations in operational and measurement terms, visual representations in two- and three-dimensional spaces, numerical representations in statistical and quantitative terms, and verbal representations in external and internal domains. Digital domain representations are concentrated on dynamic software and virtual applications, while artificial intelligence representations are concentrated on tools and platforms, and representations related to virtual manipulatives are concentrated on visual and numerical domains. Representations of daily life are concentrated according to decision and situation conditions, and multiple representations are determined as binary, ternary, and quaternary representations. Another condition identified in the study is the advantages and disadvantages of representations in relation to the problem-solving process. Representations provide advantages in terms of meaning/facilitation, concretization, different solution paths/strategies, permanence, mental visualization, attention, acceleration, and association, while they are disadvantageous in terms of confusion, loss of time, dependency, misconception, misunderstanding, and distraction. In the study, the mental components of critical thinking skills were evaluated in terms of the problem-solving process, specifically in identifying and diagnosing the problem, collecting and selecting data, identifying structured and unstructured assumptions, selecting and formulating assumptions that lead to conclusions, drawing valid conclusions, and discussing the validity of inferences. Furthermore, the digital components of critical thinking skills were identified in terms of applications for digital platforms and artificial intelligence. Furthermore, digital resources that facilitate the development of critical thinking skills, such as digital platforms like GeoGebra, Desmos, YouTube, PhET, TinkerPlots, EBA, and Wordwall, and artificial intelligence tools like ChatGPT, Khan Academy, Google Gemini, and Matific, have come to the fore.