Abstract The secondary–tertiary mathematics transition remains a persistent challenge across educational systems. Despite decades of research, reforms often oscillate between deficit narratives (‘students are underprepared’) and narrow curriculum fixes (‘more algebra, more rigour’). In this paper, I argue for a different framing: the transition is a shared design problem distributed across institutions, curricula, pedagogies and assessments. I synthesize key lines of work: early studies on proof, formalization and advanced mathematical thinking; institutional analyses emphasizing didactical contract and discontinuities in mathematical organizations; and more recent socio-cultural, socio-political and affect-oriented research examining belonging, identity, norms and equity. I identify unresolved issues, including theory fragmentation, limited comparative curricular evidence, weak causal claims about interventions at scale, and insufficient attention to diverse student pathways and service mathematics. I propose a future research agenda structured around (i) theory networking, (ii) curriculum–assessment coherence, (iii) design-based and programmatic intervention research, (iv) equitable participation and identity-safe teaching, (v) cross-institutional teacher education, (vi) data infrastructures for transition monitoring and (vii) international and intra-national comparative studies. The argument is intentionally visionary: we need to move from diagnosing transition ‘problems’ to designing transition ‘handovers’ with shared responsibility and stronger evidence.
Abstract Nursing students often perceive little value in learning mathematics beyond simple calculations during their undergraduate academic preparation and have difficulty transferring their mathematics learning to the nursing profession. Expansive framing (EF) is a theory and instructional approach that supports connections across disciplines through teachers’ framing of content and learning contexts. In this study, we used EF principles in an undergraduate College Algebra course for nursing students in the United States. The purpose of the study was to investigate nursing students’ perceptions of College Algebra’s value and transferability after participating in three expansively framed webinars. We explored whether creating intercontextuality through broad framing in their webinar instruction improved students’ perceptions of mathematics value and transferability to the medical field. Data included Value Beliefs survey responses and semi-structured interviews from 11 participants. The findings highlight themes related to the instructor’s role in creating intercontextuality, student perceptions of valuable mathematics and specific classroom activities that contributed to changing perceptions. Overall, intercontextuality was a motivator of changes in nursing students’ perceptions of mathematics value and transferability, and students’ perceptions of transferability often led to improved perceptions of value of mathematics, which is a new contribution to the theory of EF.
Classroom teaching practices shape students' motivational and emotional experiences in mathematics, yet less is known about how supportive and creative contexts influence proactive learning. Using nationally representative data from the 2022 Programme for International Student Assessment in the Philippines (N = 6856), this study tested a socio-cognitive-emotional model linking mathematics teacher support (TS) and creative classroom environment (CE) to mathematics self-efficacy (SE), mathematics anxiety (MA) and proactive study behaviour (PB). Multi-group structural equation modelling showed that CE was the strongest predictor in the model, substantially increasing SE and directly promoting PB while slightly reducing MA. TS primarily predicted PB directly. SE and MA were positively related, suggesting that confidence can co-exist with activating tension in demanding classrooms. Gender differences in specific paths were statistically significant but small, and the overall structure was largely invariant. Findings and their implications for teacher education are discussed.
We report on an instructional sequence where prospective mathematics teachers discovered binomial identities-including Vandermonde's sum-of-squares formula-through systematic engagement with a path-counting model of Pascal's triangle. Students progressed from proving known identities to independently formulating new ones within 2 weeks. The cognitive architecture enabling this discovery aligns with Minsky's uniframing theory: students constructed mental structures linking spatial paths, symbolic expressions and combinatorial interpretations. When one student called Vandermonde's identity 'beautiful' and immediately asked about generalizations, we knew the approach had succeeded. Our findings demonstrate that mathematical creativity emerges not from exceptional talent but through carefully sequenced experiences that position outcome sets as primary objects of investigation, invert traditional formula-first approaches and make 'mysterious' algebraic results appear as inevitable consequences of visual structure.
Mathematics education is at a crossroads in this era, as students need transdisciplinary knowledge as well as practical application. Traditional teaching approaches focusing on memorizing abstract mathematical knowledge detached from students' lives often decrease their motivation and engagement in learning mathematics. To address this challenge, this study investigates the influence of incorporating inquiry-based learning (IBL) and the 5E framework in secondary mathematics instruction on student motivation and engagement. The results of this study reveal that this method greatly raises students' intrinsic motivation as evidenced by the increase in their engagement, self-efficacy and the increased interest in mathematics. Significant improvements were also seen in the levels of social, cognitive and emotional engagement, highlighting the effectiveness of IBL and the 5E framework in creating more meaningful and engaging learning experiences. The research concludes that a wider implementation of these approaches promotes a more dynamic and practical mathematics education, despite limits in sample size and scope.
Generative artificial intelligence (GenAI) has recently permeated many facets of society, including higher education. Answer engines, which could provide students with solutions for complex computations in mathematics, have been freely available since 2009. However, prior research on these and other emerging technologies has shown that availability of a mathematical technological resource does not necessarily translate into widespread usage of this resource. Therefore, it is prudent to investigate usage levels of GenAI in relation to mathematics in higher education, and to discover if this differs from usage levels in other areas, as well as what tools students are using. In this study, we conducted a survey of almost 700 undergraduate students in an Irish university who were taking at least one mathematics module. The anonymous survey was undertaken online in March 2025. The results showed that 91% of respondents used GenAI for mathematics purposes and 89% for non-mathematics purposes. Of the former, 23% had used GenAI for mathematics for the first time in the previous month, compared with 9% of the latter using GenAI for non-mathematics purposes. The most commonly used GenAI tool for both mathematics and non-mathematics purposes was ChatGPT, by 90% and 95% of the sample of users, respectively.
It has long been understood that the transition from school to university is challenging for many undergraduate mathematics students. This is a continual cause for concern for university staff and students, and may lead to student dissatisfaction, lost learning and students changing courses or leaving higher education. In our study, we conducted a survey of first-year mathematics students at seven UK universities with different entry requirements, curricula and support provision. Insights was gained into student motivations, attitudes and the barriers they face during their transition to university. Using survey responses from 289 students, we explore how students' perceptions and expectations of studying mathematics, as well as their utilization of support provision, may change after their secondary-tertiary transition. Key findings indicate that students much prefer the style of teaching found in secondary education compared to university, where they believe there is a lack of support and they are reluctant to ask their teacher for help.
The secondary-tertiary transition (STT) is a significant aspect of mathematics education research as the transition from post-primary to higher education instils many feelings of anxiety in incoming students into university, even among those considered high-achieving. Following research in the 1980s and 1990s, the issues underlining the STT were identified to fall under three main themes: cognitive/epistemological, socio-cultural and didactical. There exists a wealth of literature on the STT; however, the research predominantly focuses on students entering their first year of university. This paper is a follow-up and deeper analysis of previously published conference proceedings albeit with differences in the theoretical grounding, focus and findings. In this follow-up paper, we discuss a pilot study of first-, second- and third-year undergraduate engineering students (n = 100) enrolled at an Irish university. Through survey responses, we investigate how the students' perception of mathematics and their relationship with mathematics has evolved from their final year of secondary school, into their first year of university and throughout their university experience. We provide preliminary results indicating that issues common to the STT affect students beyond the first year of university. Moreover, we investigate the strategies students use to cope with these issues and discuss whether these coping strategies are appropriate and effective.
Mathematics self-efficacy affects students' perseverance, college major choice, self-regulation and academic performance. For students enrolled in university (tertiary) discrete mathematics, high mathematics self-efficacy can promote their learning by supporting their studying, self-regulation, engagement and ultimate success in the class. However, discrete mathematics is a course that features introduction to proof and axiomatic-formal mathematics and new topics such as graph theory and combinatorics. Researchers have hypothesized that some of these topics (e.g., graph theory) may be more accessible to students than previous mathematics courses such as pre-calculus while other research shows other discrete mathematics topics (e.g., proof) may be less accessible to students. These differences in topics compared to previous mathematics courses could increase or decrease students' self-efficacy. Nevertheless, little previous research has examined changes to mathematics self-efficacy in collegiate discrete mathematics. We present results of 14 collegiate discrete mathematics students, interviewed across two semesters, examining their global mathematics self-efficacy as well as reported changes to their self-efficacy. We qualitatively describe the cases of two students who reported an increase in global mathematics self-efficacy, four who reported a decrease in self-efficacy, two who reported their self-efficacy remained unchanged but felt lower mathematics self-efficacy for discrete mathematics compared with other mathematics content, and six students who reported no change to their self-efficacy. Results have implications for university instructors as well as for researchers' knowledge about how mathematics self-efficacy changes.
Mathematics is a central component of engineering education, yet many students struggle to succeed in first-year university mathematics courses, resulting in low pass rates and high attrition. Previous research highlights the importance of fostering continuous engagement with mathematical content, but how to achieve this remains a challenge. This study investigates the potential of weekly online quizzes to support engineering students' success in mathematics. Weekly quizzes were introduced in two calculus courses for first-year engineering students at a Swedish university. Quantitative analyses compared exam results in calculus to students' prior performance in linear algebra, while three surveys captured students' perceptions of the quizzes. While effect sizes were moderate, the findings suggest that quizzes can support students' understanding and retention when implemented regularly and in a low-stakes format. Survey data further revealed that most students perceived the quizzes as helpful. However, quizzes were not ranked as the most useful learning resource, with lectures consistently valued more highly, and students indicated that bonus points for completion of quizzes were crucial for their participation. Moreover, the quizzes primarily reached students already engaged in the course. The study concludes that weekly quizzes can promote continuous learning and complement traditional instruction, but they are unlikely to address low pass rates on their own. Their effectiveness depends on thoughtful integration with other teaching practices, as well as strategies to engage students who are less active with their studies.
Testing for equality is a key feature in computer aided assessment of mathematical sciences. However, it has its problems when dealing with partially correct answers and feedback generation. There are also cases where some types of equality might be overly sensitive or permissive. This paper explores non-equality based classification of answers, based on feature extraction, and presents a test case question requiring it. This test case can be used to test the capabilities of both automated assessment systems and authors using them.
The use of e-assessment to assess students is increasingly widespread in UK Higher Education. However, it is less common to see students writing the questions themselves. As part of their final-year dissertations, we taught students to write STACK questions and asked them to develop an e-assessment for one of their other mathematical modules. The students learned about the pedagogical theory behind different elements of assessment design, as well as the technical considerations necessary to write good questions; they also became more effective learners and gained new insight into their other modules.
Bhutan's education sector has attracted international attention due to reforms driven by the national development philosophy of Gross National Happiness (GNH), which aspires to balance change with the Bhutanese cultural values that form the backbone of the GNH philosophy. However, GNH values are not explicitly integrated into academic teaching despite the training provided to teachers. In particular, there is a limited understanding of the factors that impede the integration of GNH values into Mathematics teaching. Addressing this gap, the study aims to provide an empirical baseline on the factors influencing the infusion of GNH values in Mathematics teaching, based on teachers' perceptions, understandings and classroom practices in Bhutan. Guided by an interpretivist paradigm, the study employed a qualitative, multiple-embedded case study approach to explore the challenges. The study involved 10 Mathematics teachers teaching grades XI and XII students (ages 16 to 18) across five purposively selected schools in three districts of Eastern Bhutan. Data were collected through semi-structured interviews, classroom observations and field notes, which were analyzed thematically. The findings revealed that an emphasis on academic performance, societal pressures and traditional teaching methods limits opportunities for the meaningful infusion of GNH values. The knowledge generated from this study can serve as an empirical baseline for curriculum developers, educators, parents, students and future researchers in the field of GNH and Mathematics education.
Online self-assessments, a specific type of e-assessment, are gaining increased importance as they can facilitate the transition from school to higher education. However, many aspects of online self-assessments and e-assessments are currently under-researched. This also applies to automated randomization, i.e., the automated creation of task variants of equivalent difficulty, which is crucial to realize the full potential of e-assessments, for example, by helping to avoid plagiarism in summative assessments or by enabling students to answer different variants of a task in formative assessments. In particular, a systematic analysis of aspects of mathematical tasks that can be varied, from a content-general perspective, and of how the variation of these aspects relates to task difficulty is still missing. The present paper addresses this gap by proposing a cross-content model for the difficulty of variants of mathematical tasks. The model includes two key elements of mathematical tasks: the solution path and the calculation effort, which are relevant to task difficulty across various mathematical content areas. Both can be used to generate informed predictions about differences in expected task difficulty. This paper provides details on the proposed model, along with empirical evidence from a study involving 105 mathematics freshmen, to support the model and its validity. In this study, we systematically created task variants and used generalized linear mixed-effects models to examine the impact of variations in the solution path and calculation effort on the solution rates of these task variants. Our findings substantiate our hypotheses on how solution path and calculation effort can be used to influence task difficulty. We discuss implications for future research as well as practical implications of how the proposed model may serve as a useful tool for addressing task variants in general and automated, randomized task variants in e-assessments in particular.
This paper investigates demographic predictors of engagement with mathematics and statistics drop-in support at Coventry University, which has an extensive and well-used mathematics and statistics support (MSS) provision. The analysis focuses on students from a set of 12 disciplines where there is clear mathematics and/or statistics content. It explores the effect of gender, entry requirements, course stage, ethnicity, age, nationality and disability. Two dimensions of engagement with MSS are examined: the binary dimension of engaged (at least once) or not and, for those who did engage, the number of times they did. These two dimensions are modelled through a two-stage hurdle model using a binomial logistic regression model to predict engagement and a negative binomial distribution model to predict the number of visits made by a student who does engage. These results provide valuable insights into how higher education institutions can tailor their support depending on the demographic population and engagement at their universities.
This article presents the findings of a mixed-methods research study that investigated students' ocular activity while they engaged with a video-based mathematical task focused on infinite iterative processes. The objective of this study was to examine potential correlations between ocular activity metrics captured via the eye-tracking technique and varying levels of task difficulty associated with unconscious cognitive processing during learning. The difficulty level categories were established based on two types of criteria: a subjective one, through an evaluation carried out by the subjects, and a behavioral one, related to obtaining the correct solution. Correlations of these categories with ocular activity parameters, which are considered indicators of mental effort and the cognitive load index, were identified. Fixation-related parameters correlated with perceived task difficulty levels, supporting the role of these ocular activity parameters as indicators of conscious information processing, highlighting the role of students' prior knowledge in shaping the relationship between students' difficulties in solving a mathematical task and the task's difficulty levels, and showing their impact on their ability to manage unconscious cognitive processes. The results suggest that gaze patterns and students' ocular activity reflect the difficulty of tasks involving abstract mathematical concepts, such as infinite iterative processes, while also providing insight into the unconscious cognitive mechanisms and the difficulties involved in the learning processes.
The traditional lecture-based model of teaching in the mathematical sciences has been increasingly challenged by the adoption of flipped learning, a pedagogical approach where students engage with instructional content independently before class and participate in interactive, problem-solving activities during class sessions. By considering published meta-analyses, this paper reviews the current state of flipped learning in tertiary mathematical education focusing on its effects on student achievement, perceptions and engagement. While the literature generally supports the benefits of flipped learning, including improved performance and satisfaction, these outcomes can be inconsistent due to the varied implementation methods and diverse student cohorts. The paper concludes with recommendations for implementing flipped learning at tertiary level in the mathematical sciences along with opportunities for future research.
In this paper we analyse an online computer-graded module designed to teach first year engineers the method of integration using partial fractions. Our focus lies on both analysing the increasing complexity from question to question for online learning purposes and studying the effect of the variants within the questions for possible use in exams.
We describe a failed pilot that involved using the automated grading software M & ouml;bius in place of graduate student markers for three undergraduate courses delivered in the School of Mathematics and Physics in Queen's University Belfast. We analyze the effects of this change on student engagement and performance. Our evidence suggests that students are more likely to engage with formative assessment activities when they are marked with M & ouml;bius. Students also perform better in summative assessments when they have had M & ouml;bius assignments to complete-with one module having a stark reduction in failure rate from 32% to 5%. When we surveyed the students who had the opportunity to engage with M & ouml;bius, we did not find that they had much enthusiasm for the software. However, we found that students also lacked enthusiasm for the systems for assessment and feedback that M & ouml;bius had replaced. Their responses to our survey instead indicating that students may not fully understand the distinction between formative and summative assessment. As we discuss in the conclusion, this project failed because, in spite of this apparent success, we could not drum up the support for M & ouml;bius from students and colleagues that justified the expense associated with purchasing software licenses each year. To introduce automated grading in our context we need a system that has zero or negligible associated cost as it will likely only ever be used by a small number of staff.
We reflect on over ten years' experience of using the STACK computer-algebra-aided online assessment system to support the learning of large numbers of distance learning students at the UK Open University. The motivation for the use of, in particular, computer-algebra-aided assessment is discussed together with the use of formative practice quizzes to enhance learning and summative tests for assessment. We consider the challenges faced in the adoption and use of the system, as well as the successes achieved.
