Proof to Practice: a day of mathematics education research in calculus and linear algebra

Rigour as a Collective Practice: Designing Tutorials to Support Belonging in University

In first-year linear algebra courses, students often face challenges with rigorous reasoning at a time when they are still developing a sense of belonging in the university mathematics community. In our ongoing project, we investigate how tutorials can be designed to foster participation in the university mathematics community that supports students in engaging with rigorous reasoning. Rather than treating rigour as an individual cognitive hurdle, we approach it as a collective practice that can be encouraged through shared norms, collaborative tasks, and dialogic interaction. Our project involves the co-design of tutorials by mathematicians and mathematics educators, alongside the collection of observational and interview data to trace how students and tutors negotiate rigour. In this presentation, we will share preliminary insights and invite discussion on how community-building designs can further support students in approaching rigour as an opportunity for meaningful participation in the university mathematics community.

 Exploring Linear Algebra’s Student Behavioral Engagement Trajectories in Online Learning Platforms

Student behavioral engagement (BE) can be viewed as a complex dynamical system, with its evolution over time marked by variations both within and between students. Understanding the longitudinal trajectories of engagement, defined as the progression of similar engagement states over time, alongside distinct patterns, is crucial for effectively identifying different types of students and improving student learning. In this study, we specifically focus on Linear Algebra, a foundational course in many STEM programs that is often perceived as abstract and challenging for students. By analyzing data from 236 students enrolled in Linear Algebra courses, we aim to identify distinct BE states and trajectories, explore factors that may influence trajectory membership, and investigate potential relationships with course performance.

Balancing Abstract and Applied Linear Algebra: Pedagogical Challenges and Opportunities

Freudenthal argued that separating theory from application—teaching pure mathematics first and only afterwards showing its uses—places learning in the wrong order. This tension is particularly evident in linear algebra, where achieving a balance between abstract concepts and practical use remains a persistent challenge. Drawing on a recent scoping review and framed by Sierpinska’s model of practical and theoretical thinking, this presentation highlights some recent instructional and research trends in linear algebra education. We also report on ongoing work bridging abstract linear algebra and numerical approaches, aiming to enhance both conceptual mastery and practical competence. These insights underscore the need for teaching approaches that balance abstract and practical learning, equipping students with the skills to navigate both mathematical reasoning and its real-world applications.

Comparative Judgement as a means to authentic student engagement with proof writing in a specialist linear algebra course

In this talk I will describe a student-centred approach to engaging learners with proof writing and proof comprehension in a second university linear algebra course for mathematics specialists. This novel approach makes use of group-based activities, peer-to-peer feedback, and pairwise comparison of student-generated proofs. By creating structured opportunities to repeatedly engage in robust and meaningful mathematical discourse, we claim our novel approach has scope to add variety and substance to the often narrow array of assessment activities seen in similar classrooms. We present preliminary quantitative evidence for the construct validity of two tasks, alongside qualitative evidence suggesting that these tasks led students to engage in productive mathematical discourse. 

Acknowledgement: This is joint work with Dr Ben Davies from the University of Southampton in the UK.

Developing basic mental models of derivative and integral

Developing a solid understanding of the basics of differentiation and integration is considered a key prerequisite for a lasting understanding of calculus. This presentation systematizes basic mental models related to the derivative (local rate of change, slope of the tangent line, local linearity, amplification factor) and the integral (area, reconstruction, accumulation, mean value) and discusses their respective potential and limitations. Empirical studies with students show that the slope of the tangent (derivative) and the area (integral) are most frequently mentioned, while more conceptually demanding basic mental models such as rate of change or reconstruction are less common. This underscores the need to promote less intuitive ideas through appropriate contexts and dynamic visualizations. The results illustrate that basic mental models and conceptual knowledge represent different but complementary categories, both of which are indispensable for a deeper understanding of calculus.

TBA

Challenges in double integral construction and vector calculus problem solving

Multivariable and vector calculus are core components of many technical undergraduate programmes. Students experience many challenges, from fundamental construction of integral bounds to advanced skills such as effective problem solving. 

This presentation will report on two studies. The first study was on students’ construction of iterated double integrals to represent area of a lamina. Student responses were analysed, using a suitable theoretical framework, to identify key points of difficulty in constructing and developing conceptualisations. Findings indicate that the action of “reading the region” left to right or bottom to top is significantly indicated in almost all different types of errors. Furthermore, the role of the integrand and its relationship to the region of integration was a major source of difficulty. The second study investigated the role of “writing a plan” in solving vector calculus problems with multiple possible solution strategies. Findings suggest that writing a plan might decrease trial and error in problem solving, but has no effect (or even a deleterious effect) on making good choices. For both of these studies, implications for teaching will be discussed. 

Workshop 'Linear Algebra Tile Game: A Game-Based Learning Activity for Deepening Linear Algebra Knowledge'

During this hands-on session, participants will experience the Linear Algebra Tile Game, a game-based learning approach to teaching learning algebra concepts. We will introduce the game and its underlying principles, next the participants will play the game, and finally participants can share feedback and reflect.