Maria Teresa Sanz

Fractions are essential for students’ success in higher mathematics, yet many learners struggle to develop robust conceptual understanding in this domain. This study investigates fraction-related “mental objects” among a convenience sample of 292 first- (15–16 years old) and second-year (16–17 years old) higher secondary students from five public schools in Mexico City. Drawing on Freudenthal’s phenomenological ideas and more recent frameworks for fraction understanding (e.g., part-whole, operator), we developed a pencil-and-paper test comprising six tasks: four involving graphic or pictorial representations (discrete and continuous), one focusing on fraction arithmetic, and one requiring problem-solving with fractions. Results indicate that while students in both grades displayed basic competence with standard fraction operations and certain representations, they had significant difficulties with tasks requiring redefinition of the whole (“parts of parts”) or “what is left” scenarios. Notably, second-year students did not consistently outperform first-year students on fraction arithmetic, suggesting a reliance on memorized rules that may erode over time. Although these findings cannot be generalized to the entire Mexican education system, they highlight areas in which fraction teaching—particularly with varied representations and conceptual connections—might be strengthened. Future research with larger, more diverse samples could further clarify the implications for curriculum and instructional practices at the national level.

Teachers’ understanding of fractions—especially the fraction-as-operator interpretation—is essential for students’ learning of rational numbers, yet research shows persistent weaknesses in this area. This study presents an empirical analysis of Spanish preservice primary teachers’ pedagogical content knowledge (PCK) regarding the fraction-as-operator, using a content representation aligned with the mathematical knowledge for teaching framework. A total of 263 third- and fourth-year students of different specialization completed six fraction-focused items. Through directed content analysis, 53 indicators linked to PCK subdomains were generated. The results show low PCK, with limited identification of teaching and learning difficulties. The operator interpretation appeared in fewer than 15% of responses, despite being explicitly included in the task. Differences between academic years and specializations were minimal, indicating limited PCK development. These findings point to program-level challenges in initial teacher education within a European generalist-teacher model and highlight the need to strengthen opportunities for developing deeper, more diagnostically oriented PCK.

This study presents “Aula de Apoyo”, a Service-Learning programme in which undergraduate education students provide extracurricular mathematics support to low-achieving Primary learners, particularly those from disadvantaged backgrounds. By applying knowledge from Mathematics Didactics, undergraduates strengthen their teaching skills while helping to reduce educational gaps. A mixed-methods design was used with eight tutor–student pairs. Tutors evaluated their training positively, highlighting the usefulness of the Concrete–Pictorial–Abstract approach, increased responsibility, and the desire for a longer intervention. The findings indicate that Service-Learning supports subject-matter understanding, fosters social engagement, and develops key transversal skills such as empathy and adaptability, thereby contributing to more equitable learning opportunities and aligning with European Higher Education priorities. This study adds to an underexplored line of research by integrating structured extracurricular mathematics support within a Service-Learning framework in initial teacher education. Moreover, the programme helps prepare future teachers to face the challenges of a rapidly changing liquid society, where adaptability and social responsibility are increasingly essential.

Se presenta un estudio experimental y exploratorio abordando la dificultad de los estudiantes para comprender el significado operatorio de la fracción, especialmente en la resolución de problemas y ejercicios algorítmicos. Dada la escasa presencia de enfoques visuales en la enseñanza tradicional, se analiza la importancia de las representaciones pictóricas en el aprendizaje de la fracción como operador. La investigación se realizó con estudiantes de secundaria de 14 a 15 años, organizados en un grupo experimental (42 estudiantes) y un grupo control (19 estudiantes). El experimental trabajó con modelos lineales y de área para apoyar la resolución de problemas, mientras que el control recibió únicamente explicaciones aritméticas. Se aplicaron un pretest y postest con elementos isomorfos, se analizaron cualitativa y cuantitativamente para evaluar conocimientos procedimentales y conceptuales. Aunque el grupo control presentaba mayores habilidades iniciales, el experimental obtuvo mejores resultados, mostrando que la representación pictórica favorece la comprensión y el aprendizaje. Revolutionizing fraction as operator learning revolutionizing fraction as operator learning: comparing pictorial representations vs. memoristic approaches in secondary education Abstract An experimental and exploratory study is presented addressing students’ difficulty in understanding the operator meaning of fractions, especially in problem solving and algorithmic exercises. Given the limited presence of visual approaches in traditional instruction, the study examines the importance of pictorial representations in learning fractions as operators. The research was conducted with secondary students aged 14 to 15, organized into an experimental group (42 students) and a control group (19 students). The experimental group worked with linear and area models to support problem solving, while the control group received only arithmetic explanations. A pretest and posttest with isomorphic items were administered, and qualitative and quantitative analyses were carried out to evaluate procedural and conceptual knowledge. Although the control group initially showed higher skills, the experimental group achieved better results, indicating that pictorial representation enhances understanding and learning.

This study presents the design and validation of the P-MATHEX questionnaire, an instrument aimed at assessing preservice primary teachers’ perceptions regarding pedagogical knowledge in mathematics and experimental science education. Initially, a questionnaire comprising 50 Likert-scale items and 10 demographic questions was developed and subjected to a cognitive pre-test with 50 preservice teachers from a Spanish university. The instrument was then validated using a larger training sample of 264 participants. Psychometric validation involved exploratory factor analysis (EFA) and confirmatory factor analysis, item discrimination indices, and internal consistency reliability assessed by McDonald’s omega coefficient. EFA results identified five distinct factors: teacher training improvement, methodological shortcomings, applied teaching methods, trial-and-error approach, and student problem-solving difficulties. The revised questionnaire was further validated with an additional sample of 166 preservice teachers. Results confirmed excellent internal consistency (global ω = 0.95) and robust factorial structure (CFI = 0.986; TLI = 0.985; RMSEA = 0.061). The P-MATHEX questionnaire effectively highlighted preservice teachers’ perceived strengths in practical teaching methods and identified areas requiring further training, particularly inclusive education practices and integration of information and communication technologies skills. This validated instrument provides educators and policymakers a reliable and actionable tool to evaluate and enhance teacher education programmes in mathematics and experimental sciences, ultimately aiming to improve pedagogical practices in primary education classrooms.

Root canal access is essential for successful root canal treatment, yet it poses significant risks in teeth with calcified or constricted canals, such as root perforation or excessive loss of healthy dentin. The aim of this study was to develop a predictive model that could guide the design of a conservative, accurate endodontic access in maxillary central incisors using cone beam computed tomography (CBCT). In this retrospective cross‐sectional study, CBCT scans from 135 maxillary central incisors were analyzed to obtain anatomical and demographic data. Twenty‐four variables significantly correlated with three key aspects of access design—access starting point, depth to the pulp horn, and access angle (target variables). Mathematical functions were formulated using non‐linear regression, and the resultant model was validated for the three target variables with a new set of 18 maxillary central incisors ( R ² > 0.68, W > 0.90). The results showed that age, tooth length, and specific CBCT‐derived parameters, such as starting point, angle, and depth, which are related to the tooth's access opening, strongly influenced the predicted access cavity parameters. This predictive model has the potential to be integrated into dynamic navigation software, optimizing endodontic access and reducing iatrogenic errors for practitioners.