The mean value theorems (MVT) for integrals are useful tools of mathematical analysis. However, the second MVT for integrals still lacks geometric interpretation and there is also no version yet in high dimensions in textbooks for undergraduate students. In this note, I attempt to give it a geometric interpretation. Then, by the inspiration of the geometric interpretation, an extended version in high dimensions of the second MVT is proposed and proved. This extension is useful in practice because many phenomena in the real world are related to many factors and parameters.
The trigonometric version of the well-known Ceva's Theorem for triangles is generalised to polygons, with an even number of sides and concurrent main diagonals. But one must note that the converse does not generalise. The same holds for another theorem, called Cartensen's Theorem, which in one direction can be generalised, but not in the converse. For both theorems, mathematical and educational items are discussed in this paper.
I obtain an integral representation of the exponential function exp(-gamma) using an elementary reduction to the Gaussian integral in a lively and engaging manner.
Mathematical fluency is critical to upward mobility in many STEM professions, yet mathematics is often viewed as a gatekeeper that filters students out of STEM or hinders degree completion. Prior research has examined the mathematical self-efficacy and mathematical beliefs of K-12 students, and in some cases undergraduates. Here, we present and analyse data from five undergraduate STEM majors engaged in organising and leading K-12 outreach. Reflective journals and semi-structured interviews provided insights into their mathematical beliefs and mathematical self-efficacy. In all cases, even those students majoring in mathematics-rich disciplines, we observed gaps in mathematical self-efficacy, unfavourable views of mathematics, or both. Each undergraduate in our sample could reasonably be classified as 'successful', in that they were making steady progress through their degree programmes and were engaged in their studies. Additionally, their participation in this programme can be seen as a proxy for engagement in their pursuit of a STEM degree. Our observations therefore suggest that a deeper investigation is necessary of how engaged undergraduate STEM majors, including those moving steadily towards graduation, feel about mathematics and their abilities as mathematics learners. To that end, we conclude with a discussion of potential next steps and implications for mathematics educators.
The Monte-Carlo method is a powerful tool for analysing a wide range of problems in mechanical engineering and physics. This paper considers how to introduce the Monte-Carlo method to undergrad-uate engineering students. It is proposed to use the Monte-Carlo method to evaluate the mass moment of inertia as an example application. This is an ideal application area as there is a hierarchy of complexity of implementation, starting with a one-dimensional shape, followed by a three-dimensional shape and finally a com-posite shape. A number of variants of the Monte-Carlo method are considered, with different complexities of implementation and numerical accuracy. The Monte-Carlo method that uses the paral-lel axis theorem as part of its basis is the most efficient method, with a maximum speed-up of 58,500 compared to the Monte-Carlo method that is the easiest to implement when reduced run-time is factored into the analysis. If the parallel axis theorem is not part of the Monte-Carlo method basis, then the maximum speed-up parameter is reduced to 79.4. The Monte-Carlo method that uses the parallel axis theorem uses proportionate stratified sampling to allocate function evaluations to the shapes that make up the composite shape.
This study examined the impact of analysing mathematical events on prospective teachers' conceptualisations of inclusion relationships among quadrilaterals. Grounded in the theoretical framework that emphasises the transformative potential of mathematical events for developing content knowledge, the study employed a mixed methods design with 20 prospective teachers. The intervention involved engaging participants in the critical analysis of three mathematical events, with data collected via pre- and post-tests and transcribed classroom observations. Initially, most participants struggled to identify these relationships, relying on non-critical attributes exclusive to prototypical examples rather than shape definitions. After the intervention, participants showed significant improvement in recognising inclusion relationships, using geometrical definitions and critical attributes to explain their responses. The observed improvements were related not only to the inclusion relationships discussed in the events, but also to inclusion relationships not discussed in the events, in the context of both verbal and visual tasks. The intervention enhanced the participants' ability to extend concept images to non-prototypical examples.
Many students enter their first university-level mathematics course feeling unprepared and overwhelmed. While some arrive with a strong foundation from the national curriculum, others, especially international students or those from alternative academic paths, face a different reality. The gap in mathematical background can quickly lead to confusion, discouragement and a sense of exclusion. To help bridge this gap, we introduced a set of structured, low-stakes support measures. All first-year students began the term with a short diagnostic test. Those who struggled were invited to complete ten focused learning modules designed to reinforce core skills, each comprising two self-assessment quiz attempts. Analysis of data from 258 students showed that those who completed the modules exhibited greater variability in final course grades, reflecting a wider dispersion of outcomes compared to students who met the diagnostic threshold. Despite this variability, a weak but statistically significant positive correlation was observed between quiz engagement and final examination performance, suggesting that structured, repeated practice can support academic improvement. We also saw significant performance variation across student groups, highlighting the value of tailored interventions. Our findings suggest that early diagnostic assessment combined with structured learning support can help transform a difficult start into a pathway toward success.
Assessment self-efficacy, the beliefs students have in their abilities to prepare for and take a specific assessment, is an under-explored predictor of achievement. In undergraduate mathematics, research is needed to address how it manifests, develops and interacts with assessment, particularly concerning the ubiquitous final examination. This study longitudinally collected measures of achievement and self-efficacy on low-stakes quizzes and a high-stakes examination in a New Zealand undergraduate mathematics course (N = 277). We present cross-lagged panel analyses examining the directions of influence between quiz and examination self-efficacy, together with how these are mediated by relevant achievement measures. We found quiz self-efficacy can influence students' beliefs about their ability to emotionally regulate on an examination; however, high-stakes assessment during the semester can overwhelm these effects. Findings inform sources of assessment self-efficacy, encourage adopting a comprehensive view to assessment design, and suggest the potential benefits of varying assessment to disrupt proceduralised beliefs.
R and its relevant packages including examinations are powerful tools to create randomised online questions in batch to be used on learning management systems such as Canvas and Blackboard, which significantly reduce faculty's workload for writing and grading traditional homework assignments and examinations on paper. We illustrate the complete process of using R to create question banks for mathematics courses. We also showcase the R package Tex4exams that we have developed to convert the output of some R functions to LATEX code for displaying mathematical formulas.
Students experience ordinary differential equations through examples that are mostly drawn from physics and engineering, with fewer opportunities to engage with applications in the social sciences including economics. This teaching module addresses that gap by presenting a nonlinear system of differential equations that models economic growth via the Cobb-Douglas production function, P(K, L) = AK(alpha)L(beta), where capital K and labour L dictate the production P based on productivity coefficient A and Cobb-Douglas exponents alpha,beta. Introduced earlier in our Calculus I and Calculus II courses, the Cobb-Douglas function provides a familiar yet rich context for students to extend into their Differential Equations course. Activities in this module include dimensional analysis, non-dimensionalisation process, equilibrium analysis, and linear stability analysis. MATLAB simulations allow students to visualise system behaviour and investigate how parameter choices and initial conditions influence longterm production outcomes. This work is part of a broader initiative to align our Calculus I, Calculus II, and Differential Equations curriculum with data-driven, interdisciplinary learning goals at a federal service academy.
Recent advances in digital technologies have given rise to the need to democratise advanced computer skills, such as programming and robotics. Among the digital skills expected of workers or citizens, those related to computer programming are increasingly at the heart of educational debates. In some countries, however, the integration of computer science into the school environment remains a challenge. As part of a research project, we propose a review of existing works that examines the links between computer programming and the teaching of mathematics at the end of primary and secondary school. Two major questions emerge from this study: one is the importance of jointly developing algorithmic knowledge and technical programming training, and the other is the question of teacher training. Without basic computer literacy, it is challenging to use computer programming to support mathematical knowledge and reasoning. However, teacher training must cover not only the learning of programming but also the intervention in the classroom and the design of activities that support the development of programming perspectives and practices. In the face of these challenges, it seems essential to strengthen the dialogue between the didactics of mathematics and algorithms, as initiated in the precursor research.
Fraction arithmetic remains a persistent challenge in elementary mathematics education, largely due to fragmented instruction that treats operations as disconnected procedures. This study develops and tests a unified didactic design integrating addition, subtraction, multiplication and division of fractions through a magnitude-based approach using student-constructed ribbons on a number line. Employing Didactical Design Research (DDR), this qualitative phenomenological study involved 23 Indonesian elementary students (grades 4-6) across five instructional sessions. The design and analysis are informed by the Anthropological Theory of the Didactic (ATD), the Theory of Didactical Situations (TDS) and the Concrete-Pictorial-Abstract (CPA) progression. Data from video-recorded dialogues, student worksheets and field observations were systematically coded to trace learning trajectories from concrete ribbon manipulation to abstract symbolic reasoning. Results show the development of three key knowledge types: magnitude understanding, equivalence via common denominators, and scaling for multiplication and division. Active ribbon construction fostered embodied understanding of fractions as actions rather than static objects. Retrospective analysis revealed both anticipated learning pathways and unexpected phenomena, including spontaneous discovery of equivalent fractions and difficulties distinguishing additive and multiplicative inverses. Overall, the approach reduced procedural fragmentation by unifying fraction operations within a coherent magnitude framework. Limitations and future research directions are discussed.
The goal of this research is to investigate the role of visual mediators in university students' solving processes. Specifically, this study aims to explore the relationship between resistance, understood as the tension students experience when encountering something new, and visual mediators, which are elements of mathematical discourse that support communication. The research is framed by commognition theory, locally integrated with the concept of resistance, which guided both the design and analysis. The participants, consisting of engineering and mathematics university students, worked in pairs to solve tasks during task-based interviews, with the entire problem-solving process video-recorded. The results suggest a potential connection between the use of different visual mediators and the emergence of moments of resistance. In particular, resistance appears to arise when students obtain conflicting results from different mediators and struggle to integrate them. These moments tend to be overcome when students construct new narratives that allow them to connect the different realisations of the mathematical objects involved.
This study examines the potential impact of using GeoGebra's dynamic software on the ability of high school students to define the inflection point concept and the development of their mathematical conceptual understanding. By engaging eleventh-grade students from two classes and using qualitative methods, our findings suggest that digital tools like GeoGebra aid students to construct a clear definition of the inflection point concept and further develop their concept definitions and concept images in a way that extends beyond merely identifying prototypical examples. Therefore, this research study shows that the functionalities found within GeoGebra play a crucial role in shaping students' concept definitions, concept images and more particularly, their ability to identify specific inflection points on function graphs. Uniquely, this study attempts to fill a gap in mathematics education research by investigating how multi-representational dynamic applets facilitate the transition from prototypical understandings to a comprehensive conceptualisation of non-prototypical inflection points. GeoGebra applets can contribute to creating a better integration of different mathematical representations: mental representations, computational representations and semiotic systems. These representations jointly construct a more comprehensive concept image of inflection points.
This paper proposes using the concept of triangulation with probabilistic models as a means to enhance theoretical inversion for deepening students' understanding of the nature of probability in real-world contexts. Triangulation refers to the combined application of multiple methodologies to investigate the same phenomenon, particularly in the social sciences. Theoretical inversion refers to a shift in focus from surprising outcomes to the theoretical foundations of probability. The paper introduces three types of problem-solving tasks designed to enhance one of four types of triangulations: theory triangulation. Theoretical inversion is expected to emerge through engaging in these tasks. The characteristics of the problems are as follows. Problem 1 promotes students to compare different probabilistic models of events under similar procedures. Problem 2 provides students with an opportunity to simplify an experiment by omitting steps that add no new information. Problem 3 enhances students' ability to recognise how subtle differences in the experimental setup can affect the resulting probability. These tasks are designed to encourage students to view probabilistic reasoning as a form of modelling and to appreciate the importance of assumptions, definitions of elementary events, and clarity in procedural descriptions.
Parallel transport is a fundamental concept in differential geometry and underpins the mathematical structure of spacetime in general relativity. However, its abstract formalism presents substantial cognitive challenges for students encountering relativity for the first time. This work introduces a geometric-computational framework for teaching parallel transport, complemented by a practical approach that enables systematic analysis and interactive visualisation in an educational setting. The proposed method generalises the intuitive notion of transport along geodesics to arbitrary curves by discretizing them into successive geodesic segments. Numerical simulations on spherical manifolds demonstrate that the approximation converges rapidly to the exact solution, with angular deviations becoming negligible even for moderate discretizations. By translating abstract geometric structures into interactive visualisations and computational exercises, the framework strengthens students' intuition, integrates digital tools into the learning of geometry and physics, and broadens access to advanced concepts - fostering a deeper engagement with the geometric essence of General Relativity. The article concludes with a detailed discussion of potential implications for physics and mathematics teaching, curriculum design, and future research, highlighting the method's broader relevance for both physics education and the practice of numerical modelling in curved spaces.
This study explores the impact of the Compare and Discuss Multiple Strategies (CDMS) framework for teaching multiple solution methods for factoring trinomials, a foundational algebraic skill serving as a gateway to college-level STEM coursework and frequently identified as a conceptual hurdle for students transitioning from arithmetic to algebraic thinking. Using a mixed-methods pre-post quasi-experimental design, 39 undergraduate students enrolled in developmental mathematics participated in a CDMS-based intervention focusing on multiple solution methods for factoring trinomials. The research investigated performance changes across trinomial types, method usage patterns, and student experiences. Results revealed large effect sizes significantly exceeding typical educational interventions, with substantial improvements across all trinomial types. Method diversity increased from two to four approaches, with students shifting from single-method dependency to flexible use of multiple strategies, particularly visual methods like Box and Tic-Tac-Toe methods. Qualitative findings indicated increased student confidence and strategic thinking, with nearly half expressing appreciation for multiple solution methods. The intervention was consistently effective across different algebraic structures and complexity levels. These findings provide the first empirical evidence for implementing CDMS in developmental mathematics at the college level, demonstrating that multiple solution methods can produce meaningful learning gains among undergraduate students who traditionally struggle with algebraic concepts.
