This study presents a critical reflection on the use of the flipped classroom (FC) method in teaching mathematics and statistics to engineers, focusing on evaluating various aspects of student engagement. We based our analysis on the approach implemented by Lo and Hew [1] in their literature review on student engagement in mathematics flipped classrooms. This conceptual framework is rooted in the multifaceted nature of engagement, including behavioural, emotional, and cognitive aspects [2].

We applied this framework to two different FC formats. The first format follows a traditional approach, where students are provided access to the learning material in advance. The students go through it independently and then participate in learning activities offered in class. This traditional approach has been widely used and evaluated, presenting its own benefits and challenges. The second format is a modernised version of the traditional setting. In line with the traditional approach, students go through the learning material themselves; however, the lecturer summarises a topic in a weekly lecture and goes through examples presented in pre-recorded lectures on the board.

While the FC approach might be considered a well-known method [e.g., 3], it has not been widely used in teaching mathematics until recently. The available analysis of the effectiveness of this method in relation to student engagement doesn’t show consistent results [e.g., 1, 4]. What affects students engagement? What new ideas in implementing FC would be worth developing further?

During the Covid-19 pandemic, the higher education sector had to radically change the ways the majority of courses were taught to accommodate new realities. The vast number of courses went online creating favourable conditions for implementing and advancing the FC teaching method in a variety of subject disciplines, including mathematics-based courses.

Grounded in constructivist learning theories [e.g., 5,6], the FC represents one of the student-centred instructional models. In the FC environment, students are the builders of their knowledge. Initially, students familiarise themselves with new material outside the classroom and then build upon this foundation through adaptation and application of knowledge in in-class discussions, problem-based and project-based learning, and peer learning.

The objectives of this critical reflection are as follows:

• To evaluate the three types of student engagement in studying mathematics and statistics in the FC environment in two settings.
• To analyse the benefits and challenges academic staff face when using the FC in two settings.
• To propose recommendations on using the FC method in teaching mathematics and statistics.

This paper presents the outcomes of the on-going collaboration between Lancaster University, UK, and University West, Sweden, focusing on the implementation of non-traditional teaching methods in mathematics and statistics [7-10]. It evaluates the outcomes of using the FC approach in two courses at University West, Sweden, during the first semester of the 2023-24 academic year. The first course, ‘Statistics,’ was offered to second-year electrical engineering students, employing the traditional FC setting. Meanwhile, the second course, ‘Algebra and Trigonometry,’ targeted first-year engineering students. A total of 42 students enrolled in the ‘Statistics’ course, while 190 students were enrolled in the ‘Algebra and Trigonometry’ course. In both cases, the students were provided with new material on a weekly basis to independently prepare for the upcoming seminars. Seminars were conducted twice per week, spanning two hours each session, over an eight-week period. First-year students were divided into four seminar groups. To evaluate student engagement, participants were required to complete a questionnaire comprising three parts. Each section featured a set of Likert-type questions designed to assess behavioural, emotional, and cognitive engagement. The first set of questions delved into students’ participation, effort, and preparation for class activities. The second set aimed to gauge satisfaction with learning and motivation levels, while the final set focused on students' investment in learning, confidence development, and deeper understanding. Additionally, discussions were held with the two course convenors to gain insights into their experiences with the FC methodology. These discussions aimed to provide supplementary qualitative data to complement the quantitative findings from the questionnaires.

In the study, 19 out of 42 students enrolled in the 'Statistics' module returned the questionnaire, highlighting strong engagement within the traditional setting. Students expressed significant learning through peer collaboration, effective communication with the lecturer, use of resources, and satisfaction with course organisation. Among 140 first-year students, 68 responses revealed less coherence. While 80% expressed emotional satisfaction with course delivery, 80% were uncertain about cognitive engagement with the FC method allowing pacing of their own learning. Additionally, 30% were unsure about ease of communication with the lecturer. Discussions with lecturers showed excellent student attendance. In 'statistics' course, the lecture used less group activities in class this time and focused more on going through solutions on the board. This was different from the previous year were the students were working in groups on solving problems. This might potentially affected student engagement and exam performance as the students were exposed more to passive learning. In the case of first-year students attending the summary lectures, lecture attendance was notably high, however, seminar attendance was comparatively low. Typically, students engage in problem-solving activities either individually or in groups during seminars, seeking guidance from tutors when needed. The lower seminar attendance suggests that students may have grasped the material well enough without collaboration with peers. Reflecting on these findings, repeating examples may enhance understanding and application of new concepts, while group work fosters active engagement, positively impacting exam performance. It's evident that using a variation of learning activities in class could positively impact different types of student engagement. The pandemic has accelerated changes in the way we teach our students. Transitioning to more flexible, mixed modes of teaching practices will provide opportunities to create a more engaging and motivating learning environment that reflects the rapidly changing world we live in.

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