This study examines the calculus learning experiences of first-year university students in a multilingual classroom in the Kurdistan Region of Iraq (KRI), with a particular focus on students from diverse regional and educational backgrounds. A purposive sampling method was used to select 23 students enrolled in a foundational calculus course, including two students from central Iraq and 21 from the KRI. The study aimed to explore performance and behavioral differences in classroom engagement, particularly in response to abstract mathematical concepts such as limits. Data were collected using a structured classroom observation checklist and field notes maintained over twelve instructional sessions. The observation rubric included behavioral indicators such as participation, engagement, peer interaction, and visible confusion. A quantitative content analysis approach was employed, utilizing Likert-scale ratings to convert qualitative observations into numerical data for comparison. Descriptive and nonparametric statistical methods, including time series plots, were employed to visualize and analyze patterns in student behavior over time. The findings revealed stark differences in engagement and confusion levels between the two student groups, attributable to disparities in prior learning experiences, language proficiency, and cultural learning norms. These insights contribute to our understanding of how regional educational inequalities and linguistic diversity impact mathematics learning. The study recommends aligning the curriculum across regions and implementing bridging courses, as well as culturally responsive pedagogies, in multilingual settings. Limitations and directions for future research are also discussed.

This article analyzes the methodological potential of a multiple-representation approach based on visual and illustrative modeling in teaching the fundamental concepts of trigonometry. During the learning process, students frequently experience difficulties in understanding the intrinsic relationships among the unit circle, right triangles, and the graphs of trigonometric functions. To address these challenges, the study proposes an alternative model for constructing the trigonometric circle. The proposed model aims to reveal the logical origins of trigonometric concepts, support their spatial and intuitive perception, and establish a holistic connection among the unit circle, graphical representations, and geometric models. The article examines the solution of trigonometric inequalities as an instructional example by employing various visual and analytical methods and provides a rationale for their didactic effectiveness. The impact of this approach on the efficiency of learning trigonometry was evaluated with 94 final-year high school students from two selected public schools in Uzbekistan (45 in the control group and 49 in the experimental group). A mixed-methods research design was adopted, integrating quantitative data obtained from student assessments with qualitative data collected through interviews. The findings indicate that students’ abilities to analyze problems, justify solutions, and make generalizations were significantly improved.

This article focuses on assisting teachers and students who face challenges in transcendental equations, opening new horizons for them, and enhancing their perspective on mathematics through the positive influence of technology. In this context, this article presents an approach to addressing challenging mathematical concepts with the help of math apps such as Desmos. In this way, students can enhance their understanding by visualizing solutions that would be difficult to grasp algebraically and broaden their perspectives on challenging questions. Simultaneously, interactive and didactic materials can promote students’ and teachers’ enthusiasm for using digital technology in the educational process. A case study revealed that both teachers and students found the graphing method for solving transcendental equations on Desmos beneficial for enhancing cognitive skills, fostering creativity, and improving teaching quality; therefore, this technique should be more prominently incorporated into the high school curriculum. This study, designed as a case study employing a mixed-methods approach, was conducted during the 2024-2025 academic school year with the participation of 24 twelfth-grade students enrolled at Stirling Girls’ High School in Kirkuk, as well as four mathematics teachers working at the same school. Data were collected through the implementation of questionnaires in addition to classroom activities and interviews. The findings indicate that applying alternative approaches to the teaching of transcendental equations—topics included in the high school curriculum but often given limited emphasis—and supporting these approaches with technology resulted in a significant increase in students’ academic achievement.

This article primarily aims to introduce high school students to the mystery of Lobachevsky geometry, one of the cornerstones of non-Euclidean geometries. Lobachevsky geometry, often known as hyperbolic geometry, differs from Euclidean geometry in several basic ways. The concepts and figures of Lobachevsky geometry can appear in different plane models, such as the Klein and Poincaré disk models. It further examines students' general attitudes and behaviors toward non-Euclidean geometries. Lobachevsky's geometry has helped expand students' horizons and enriched their critical thinking skills by challenging traditional Euclidean paradigms. This study is supported by a mixed-method approach utilizing quantitative and qualitative data. The mock exam results obtained from students during the educational process were compared, and the study was further supported by the positive feedback received from the participating students. The intriguing lessons on Lobachevsky geometry were conducted over 4 weeks, with weekly 2-hour geometry classes involving 12th-grade students at Stirling Schools in Erbil. Throughout the study, we observed significant improvements in students' ability to adopt, understand, and apply advanced geometric concepts. This article also discusses findings and implications that address gaps in the literature and considers the potential for curriculum updates to enhance the future of geometry education.

We generalize some sequence spaces from single to double, we study some topological properties of these double sequence spaces by using ideal convergence, difference sequence spaces, and an Orlicz function in 2-normed spaces, and we give some results related to these sequence spaces.

This mixed-methods study investigates the perceptions of Grade 12 students enrolled in Advanced Placement (AP) Calculus regarding their least preferred topics, to identify instructional and cognitive challenges that hinder their engagement. Data were collected from 53 students across five schools in the United States and Iraq using a survey that included both quantitative and qualitative items. Quantitative analysis revealed that topics related to integration, particularly Applications of Integration and Accumulation of Change, were most frequently identified as difficult. Latent class analysis further distinguished two distinct student groups based on topic preferences. Thematic analysis of open-ended responses highlighted key challenges such as cognitive overload, lack of engagement, struggles early in the semester, and anxiety related to assessments. Students also expressed negative emotional responses, emphasizing the affective dimensions of learning advanced mathematics. Despite these difficulties, participants identified effective instructional methods, including concise teacher explanations, visual aids, guided practice, and the use of multimedia resources. The findings suggest that student-centered, scaffolded instruction can help alleviate conceptual barriers and improve learning outcomes in AP Calculus. This study adds to the growing body of literature advocating for responsive and differentiated teaching practices in high-stakes mathematics courses. Implications for instruction, curriculum design, and teacher training are also discussed.