Contrasting undergraduate mathematics students’ approaches to learning and their interactions within two student-centred learning environments

ABSTRACT

The Students’ Approaches to Learning (SAL) tradition comprehends an approach to learning as a combination of students’ aims for learning and the processes used to achieve them. The tradition values a deep approach to learning; this is the case also in the context of university mathematics, where a deep approach is seen as a requirement for learning proof-based mathematics. However, supporting a deep approach through learning environments has turned out to be a complex endeavour. The present study addresses this complexity by qualitatively contrasting the same students’ approaches learning in two pedagogically different undergraduate mathematics learning environments. The study is a follow-up on a quantitative study and reports on 16 student interviews aiming to connect the students’ quantitatively determined approaches to learning with their aims for learning and the actualized learning processes within the learning environments. The results suggest that student subgroups with different approaches to learning are distinguished by their aims for learning, and based on the students’ own reporting, by the ability to reflect on learning and utilize student collaboration. We discuss the results and conclude that a sequence of courses with well-aligned instructional practices is needed to support the development of a deep approach to learning.

KEYWORDS:

• Student-centred
• teaching practices
• learning environments
• approaches to learning
• higher education
• qualitative research
• undergraduate mathematics education

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 97C30
• 97C70
• 97D10
• 97D40

1. Introduction

In higher education research, the Students’ Approach to Learning tradition (SAL) has reached a central role in understanding the processes related to students’ learning. The research tradition originates in 1976, when Marton and Säljö distinguished between two qualitatively different ways of learning, namely deep- and surface-level processing (Marton & Säljö, 1976). They later refocused the names as deep and surface approaches to learning, as the word ‘approach’ intentionally captures both students’ aims for learning and the processes used to achieve them (Marton & Säljö, 1984). Of the two approaches to learning, the deep approach refers to learning that aims at understanding and creating a holistic view of the studied content. In contrast, the surface approach refers to instrumental learning characterized by memorization and reproduction. There is also a third approach, namely organized studying (Biggs, 1987). However, it is not included in the present study as it emphasizes time and effort management and is therefore viewed as an approach to studying rather than an approach to learning (Biggs, 1987; Parpala & Lindblom-Ylänne, 2012). There are multiple studies linking the deep approach to higher, and the surface approach to lower academic achievement (see, e.g. Marton & Säljö, 1976; Trigwell & Prosser, 1991; in the context of mathematics Crawford et al., 1998; Maciejewski & Merchant, 2016; Murphy, 2017). Also, a deep approach is linked to students’ cohesive conception of mathematics (Crawford et al., 1998) and is characterized as essential for learning proof-based mathematics (Maciejewski & Merchant, 2016). Therefore, the undergraduate mathematics learning environments should support students to apply and develop a deep approach to learning.

The present study uses the concept of ‘learning environment’ as a set of instructional practices, including the teacher–student and peer relations. A student’s interaction with a learning environment refers to their learning processes and social relationships within the learning environment. Designing learning environments that support a deep approach to learning has turned out to be a rather complex endeavour. During the last decades, new student-centred higher education learning environments implying fostering students’ own activity, responsibility and independence of learning have emerged (Baeten et al., 2010). In the disciplinary context, previous literature reports positive results for example from inquiry-based mathematics education (Kogan & Laursen, 2014; Rasmussen & Kwon, 2007), flipped learning (Lesseig & Krouss, 2017), adaptive instruction (Konstantinou-Katzi et al., 2013) and long-term department-level change efforts (Rämö et al., 2019). Although these student-centred learning environments have been reported contributing positively to students’ learning, the literature does not provide a straightforward answer about the relationship between different learning environments and students’ approaches to learning, even in the broader higher education context (Asikainen & Gijbels, 2017; Baeten et al., 2010; Gijbels et al., 2008). As Fredriksen and Hadjerrouit (2020) conclude, previous research on university mathematics learning environments focuses on the teacher-centred vs. student-centred dichotomy and fail to address the more nuanced ways different student-centred teaching practices support students’ learning (see also Rasmussen et al., 2021). The present study approaches this research gap by contrasting student subgroups that apply different approaches to learning in two student-centred university mathematics learning environments. It builds on a previous quantitative study (Lahdenperä et al., 2019) that identified three student subgroups, (1) students applying a deep approach to learning in both courses, (2) students applying a surface approach to learning in both courses and (3) students applying a context-sensitive surface approach to learning, and now qualitatively explores how the student subgroups’ approaches to learning revealed in their interactions with the learning environments. The aim is to understand the reasons behind the student subgroups’ different course-level approaches to learning and, as some subgroups changed their dominant approach to learning and some did not, to shed light to both the stability and change of approaches to learning. Overall, the motivation is to identify elements of the student-centred learning environments that positively contribute to students’ approaches to learning and thus, to offer implications for practice that have the potential to promote more effective mathematics teaching for all students studying mathematics at university.

1.1. Approaches to learning and learning environments

Researchers have suggested that the research linking SAL to learning environments should focus on the students’ perceptions of the learning environments (see, e.g. Ramsden, 1987). In general, researchers agree that students’ approaches to learning are linked to their perceptions of the learning environment; students applying a deep approach perceive the learning environment more positively than students applying a surface approach (Baeten et al., 2010; Parpala et al., 2010; in the context of mathematics Lahdenperä et al.,  2019; Crawford et al., 1998; Mji, 2003). To continue, Alansari and Rubie-Davies (2020) illustrate in their literature review that positive perceptions of the learning environment also have other beneficial consequences, such as higher academic achievement.

Baeten and colleagues (2010) argue in their literature review that a deep approach to learning can be supported with student-centred learning environments. However, the relationship is not straightforward; sometimes student-centred learning environments can push students towards a surface approach (Baeten et al., 2010). The challenge of supporting a deep approach through learning environments derives from multiple sources. First, approaches to learning are a combination of both students’ aims for learning and the complementary processes used to achieve them (Marton & Säljö, 1984; Wilson & Fowler, 2005). This means that a students’ approach to learning is formed in two parts; the students enter a learning environment with tendencies towards a certain approach to learning, which is then – to a different degree – related to the actualized approaches to learning (Alansari & Rubie-Davies, 2020). Second, some previous studies fail to distinguish students’ predispositions from their actualized approaches to learning; this has produced seemingly contradictory results in the current literature regarding the relationship between approaches to learning and student-centred learning environments (Wierstra et al., 2003). Third, there is mixed evidence on the stability of the tendencies towards a certain approach to learning (Lindblom-Ylänne et al.,  2013). For example, Öhrstedt and Lindfors (2016) conclude that even small changes in the learning environment can induce changes in the students’ course-level approaches to learning. However, Wilson and Fowler (2005) suggest that students applying a deep approach are more consistent in their approaches across different learning environments, while students applying a surface approach can move towards a deep approach in a student-centred learning environment. Fourth, it is rare that a student applies a pure deep or a pure surface approach to learning. For example, Lindblom-Ylänne et al. (2018) identified five surface-approach student profiles; four of the profiles were dissonant profiles implying the inclusion of some elements of a deep approach. This is linked to the stability of the approaches to learning, as they argue that the dissonant profiles might imply that students are in a transition phase from a surface approach towards more favourable approaches to learning (Lindblom-Ylänne et al., 2018). In fact, Kember (2016) suggests that deep and surface approaches are not two distinct extremes but a continuum. Therefore, supporting the development of a deep approach to learning is not a single change but a process. This creates the final challenge; researchers hypothesize that course- or semester-long isolated student-centred interventions are not long enough for the process to take place (Baeten et al., 2010; Entwistle & Peterson, 2004; Wilson & Fowler, 2005). However, positive results are reported in a recent literature review from the field of health sciences arguing that curriculum-wide problem-based instruction supports the development of a deep approach to learning (Dolmans et al., 2016).

1.2. Aims and research questions

The present study is positioned in the intersection of teaching and learning of university mathematics; the aim is to shift the attention from the teacher-centred vs. student-centred dichotomy towards a more nuanced understanding of how different student-centred learning environments are related to the students’ learning. The study conceptualizes learning through students’ approaches to learning as they are widely used to understand learning in higher education in general and also seen as essential to proof-based mathematics education.

The study builds on a previous quantitative study (Lahdenperä et al.,  2019) and aims to qualitatively explore and identify the elements of the learning environments that positively contribute to students’ approaches to learning. As especially students applying a surface approach to learning have reported various aims for their learning in previous literature (Lindblom-Ylänne et al.,  2018), and to distinguish between the intention and process components of an approach to learning (Marton & Säljö, 1984; Wierstra et al., 2003), students’ aims for learning and their interactions with the learning environments are investigated separately. The research questions are:

1. What kind of aims do the student subgroups with different approaches to learning have for their learning? How do these aims differ between the subgroups and the learning environments?

2. How do the student subgroups with different approaches to learning interact with the two learning environments? How do these interactions differ between the subgroups and the learning environments?

3. What makes the context-sensitive student subgroup change their course-level approach to learning from a surface to a deep approach?

The investigation of the same students in two contexts reveals how a student perceives different student-centred learning environments and sheds light to the reasons behind the mixed evidence reported in the literature on the relationship between student-centred learning environments and students’ approaches to learning. Overall, the motivation of the present study is to identify elements of the learning environments that are linked to students’ approaches to learning and provide implications that facilitate the development of a deep approach to learning in the mathematics context.

2. Methods

The present study continues a quantitative study (Lahdenperä et al.,  2019) which identified three student subgroups applying different course-level approaches to learning in a university mathematics context. Here, we address the research questions of the present study by reporting on the interviews of students from all the three subgroups. The following subsections describe the research context and the procedures of data collection and analysis in greater detail.

2.1. Context

The research is conducted in a mathematics department of a research-intensive university in Finland. In Finland, university studies usually last for five years resulting in a master’s degree. Students are accepted into a specific degree programme with a focus on their chosen major subject from the very beginning. The Universities Act (2009) provides academic freedom for teachers to design and develop their own teaching. In this vein, the department serving as the context has been a platform for individual educational change efforts that have led to a systemic department-level improvement (see Rämö et al.,  2019).

The present study investigates the same students in two different student-centred courses. The two courses run parallel, are usually taken by students during the first semester of their university mathematics studies, and both courses are six-week, five-credit (ECTS) courses with approximately 200 students. Course A functions within a traditional lecture-based setting but includes student-centred elements, such as activating lectures and small group sessions that support students’ participation in mathematical discussions. Course XA uses the Extreme Apprenticeship approach (see Rämö et al.,  2019), a form of inquiry-based mathematics education (IBME; cf. Artigue & Blomhøj, 2013) that supports discussions and student collaboration in an open learning space and in lectures. In practice, the main differences between the course implementations centred on the design of the tasks (gradually increasing difficulty in XA), the form of support given to the students by the tutors (in scheduled small groups sessions in A vs. in an open learning space in XA), and on the role of lectures (lectures first in A vs. tasks first in XA). At the time of data collection, both instructional models had been implemented for over five years. The instructional practices of the courses are summarized in Table 1.

Table 1. Summary of the instructional practices of Courses A and XA (slightly modified from Lahdenperä et al., 2019).

	Course A	Course XA
Stage	First year	First year
No. of students enrolled	approx. 200	approx. 200
Content	Analysis course emphasizing exact definitions and proof construction	Linear algebra course emphasizing skills such as reading mathematical text, oral and written communication, and proof construction
Lecturer	Mathematician and mathematics educator acknowledged for their excellent teaching by both students and the university	Mathematician and mathematics educator acknowledged for their excellent teaching by both students and the university
Lectures	4 h a week; lectures focus on how to approach the main content of the course	3 h a week; lectures come after students have solved problems; the lectures are based on students’ discussions and focus on a holistic understanding of the main content of the course
Course material	A general Analysis book	Course book written by the lecturer
Task design	Students solve problems every week prior to and during small group sessions; tasks are designed by the lecturer	Students start a new topic every week with approachable problems that teach content and study skills; more challenging problems follow the lecture; tasks are designed by the lecturer
Form of support^a	Two-hour small group sessions (approx. 20 students) twice a week with a tutor	Open learning space on a drop-in basis with tutors
Tutors^a	Older mathematics students	Older mathematics students who attend weekly pedagogical meetings with the lecturer
Assessment	Final grade based on a written exam; bonus points from completed course tasks	Final grade based on a written exam; bonus points from completed course tasks

^a Later in the analysis process, form of support and tutors are combined as guidance.

2.2. The previous study

A previous study (Lahdenperä et al., 2019) investigated the same learning environments as the present study by quantitatively comparing students’ approaches to learning in the two contexts. The quantitative data was collected from both courses using the HowULearn instrument, modified from ALSI (Entwistle et al., 2003) and RLPQ-2F (Kember et al., 2004) and validated in the Finnish higher education context (Parpala & Lindblom-Ylänne, 2012). Based on the students’ course-level approaches to learning, the previous study identified three student subgroups: (1) students applying a deep approach to learning in both courses, (2) students applying a surface approach to learning in both courses and (3) students applying a context-sensitive surface approach to learning. In contrast to the first two clusters, students applying a context-sensitive surface approach changed their approaches to learning according to the course context: these students applied a surface approach on Course A but not on Course XA. The three student subgroups did not differ statistically significantly in any of the background variables, except for the surface approach subgroup scoring statistically significantly lower in the Course XA exam than the other two subgroups (Lahdenperä et al., 2019). These three quantitatively determined student subgroups serve as the starting point for the present study. The present study addresses the need to acquire a qualitative understanding of how these quantitative differences show in students’ learning practices by reporting on the interviews of students from all the three student subgroups.

2.3. Data collection and participants

All participants in the prior quantitative study were invited for an interview on a voluntary basis. In the present study, the data consist of interviews of 16 students who attended both courses and, in the interviews, reflect on their experiences of the two courses. The students gave their active consent to participate in the research. The interviews were semi-structured interviews with an average length of 1 h 13 min (Min. 49 min and Max. 1 h 51 min). In the interviews, the students were asked about their aims for and processes of learning by, for example, asking them to describe how they studied in the courses, what kind of aims they had for the courses, and what supported and hindered their learning in the courses; depending on the answers, the students were prompted with more detailed follow-up questions. These questions are aimed at capturing students’ interactions with the learning environments. The first author conducted the interviews in spring 2017 three to four months after the quantitative data collection. The interviews were audio recorded and then transcribed verbatim by the first author. The first author was not part of the courses’ teaching teams, and the students were informed that participating in the research does not affect their course assessment or future studies in anyways.

The students’ background information is given in Table 2. The students represent all the three student subgroups determined in the previous quantitative study. The students’ major subject is reported using three categories: ‘mathematics’ refers to students majoring in mathematics, ‘science’ refers to students majoring in physics, chemistry or computer science, and ‘other’ refers to students majoring in other subjects. Gender is not reported to protect students’ anonymity. However, all subgroups include both students who identify as male and students who identify as female.

Table 2. Students’ background information (N = 16).

Subgroup	ID	Major subject	Study year
Deep approach (n = 7)	6	Mathematics	1st
	7	Mathematics	1st
	8	Mathematics	1st
	11	Mathematics	1st
	18	Science	>5th
	19	Mathematics	1st
	24	Science	3rd
Surface approach (n = 4)	1	Other	≥5th
	3	Science	1st
	9	Other	1st
	17	Mathematics	1st
Context-sensitive surface approach (n = 5)	2	Other	3rd
	4	Science	3rd
	5	Mathematics	1st
	16	Mathematics	1st
	21	Other	≥5th

2.4. Data analysis

The aim of the data analysis was to investigate how the students’ course-level approaches to learning were revealed in their interactions with the two student-centred learning environments. The analysis was based on qualitative framework analysis (Ritchie & Spencer, 1994) as it is developed for applied qualitative research and it addresses the need for a comprehensive between- and within-subject comparison. The framework analysis process consists of five stages: (1) familiarization, (2) identifying a thematic framework, (3) indexing, (4) charting and (5) mapping and interpretation. The data was analysed by the first author using AtlasTI-software (stage 3) and Microsoft Excel (stages 4 and 5). A data-driven approach was chosen to answer the research questions, and the analysis process sought for students’ concrete actions in the two learning environments and their perceptions of, and explanations for, the actions.

The analysis process started with reading the interviews several times to become familiar with the data. A framework was developed based on a priori themes – the course elements (lectures, course material, working on tasks, guidance and assessment). Based on the previous literature, a framework was also developed for indexing students’ aims for learning. Then all interviews were indexed regarding each course element and students’ aims. The course elements were easily identified from the transcripts, although differentiating ‘working on tasks’ and ‘guidance’ needed special attention in cases where a student e.g. worked on tasks in the open learning space. The interview included a direct question about student’s aims for the courses, so the aims were also easily identified from the transcripts. The charting stage where the data was thematically rearranged was conducted for each course element and student’s aims individually. The research questions were addressed in the mapping and interpretation phase by identifying the main concepts in the data, and then contrasting and seeking patterns and explanations in students’ interactions with the learning environments. The analysis was conducted similarly for both learning environments; however, at the same time being aware that some themes may be identified in only one of the contexts. To increase the stability of the analysis process, the first author developed explicit written-out instructions for applying the frameworks for indexing the transcripts and discussed them with the second author. In addition, the stability was increased by the iterative nature of the analysis process; the indexing was checked in the charting process, and the charting stage was checked while conducting the mapping and interpretation process.

3. Results

The results are reported in four parts. First, an overview of the data is provided in the form of students’ aims and their ways of interacting with the learning environment elements. This section helps the reading of the following three sections, each addressing the unique characteristics of one student subgroup. All text in italics is a student quote; the longer student quotes are signed as ([subgroup], [student ID], [course context]).

3.1. Students’ aims for learning and their interactions with the learning environments

The students had set qualitatively different aims for the courses; students could aim either to understand the course content, to survive the course, or they could see the courses in a larger context, for example, as aiming to fulfil degree requirements. Almost without exceptions, the students reported having similar goals for both courses, so the aims remained the same although the learning environment changed.

In terms of the learning environment elements, students reported on three qualitatively different ways of taking part in lectures. First, students could attend lectures, find the lectures relevant, and participate actively during the lectures. Second, students could attend lectures, but they were not always active participants nor found the lectures very relevant. Third, students could find the lectures irrelevant and therefore did not attend them; however, if they occasionally did so, they participated in a passive manner.

Regarding the course materials, students reported on material clarity; students could find the materials either unclear or articulate. In addition to material clarity, students also reported on different ways of utilizing the material while studying. Students could utilize the course material by reading the material independently and while working on course tasks. In contrast, students who did not utilize the course material reported using, for example, their own notes and/or online resources.

All students reported that they worked actively on the course tasks in both course contexts; the students were unanimous in emphasizing the importance of solving the mathematical problems as they saw it the only way of learning mathematics. However, the students perceived the difficulty of the course tasks differently; students could perceive the level of difficulty as suitable but also too challenging and therefore experiencing a lot of frustration while working on them.

Regarding guidance, the students reported on ways of working on the tasks, relevance of the guidance, in general, and helpfulness of the tutors. Students could work on the weekly tasks either at home, in the open learning space at the department, or elsewhere at the campus; also, students could work alone or with peers. Although some students worked together, some students stayed home as they found it easier, or they missed peer students to work with. All students attended the small groups sessions in Course A, but not all students utilized the guidance offered in the open learning space, as they reported being able to complete the tasks on their own. Students in both contexts reported not getting enough help for completing the course tasks from the tutors. However, this was not the case for students who attended the guidance offered in the open learning space.

In terms of assessment, the students were unanimous in liking the bonus points from the completed course tasks; they reported that the points motivated them to work on the tasks and forced them to study. Still, when students reflected on the assessment, they mostly talked about the course exams. The students reported on two different experiences on the course exams; the students could find that there were no surprises in the exam and/or the exam was aligned with teaching, or that the exam was difficult, not aligned with teaching, and/or yielded in unexpected results.

3.2. Deep approach to learning subgroup

Distinctive from the other two subgroups, the students in the deep approach subgroup aimed at understanding in both courses. In terms of their interactions with the learning environment, there were three distinctive elements for this subgroup: lectures, course material, and guidance. The students attended more in Course A lectures, which they found more relevant, and during which they participated more in the social interaction. The students did not find the Course XA lectures so relevant because they had already understood the topic and therefore attended fewer. For example, a student who was active in Course A lectures and inactive in Course XA lectures, stated:

[The lectures] were not always so relevant. […] [A]s I had already done […] the weekly tasks, and in the lectures the same tasks are discussed in detail, [the lectures] were a little boring. […] So I thought I’ll read the course material, it’s very good, and I get everything from there. (DA, 19, XA)

Regarding the course material, the deep approach subgroup was unique in utilizing the course material in Course A. Like the other two subgroups, the students in this subgroup reported the material to be unclear:

You had to read the textbook much longer just to understand what it’s about.

Nevertheless, the students utilized the material while studying – the student continues:

Perhaps it was one of the goals [of the course] to learn to read the textbook, what the notations mean and, in general, to learn to get better insight into mathematical texts. […] I have just been reading the textbook, somehow I’ve understood that you need to understand the course material and then be able to apply and use it. (DA, 8, A)

For guidance, the students in the deep approach subgroup reported very positive experiences from the Course A small group sessions, where they worked together with peer students and reported having received the help needed from the tutor. This is the only student subgroup with a very positive experience from the Course A small groups emphasizing the possibility of working together with other students, getting peer support, and experiencing learning successes. This is elaborated below by a student in this subgroup:

You eventually had certain people with whom you went to sit at the same table […] to work on the tasks. […] [I]t of course made the atmosphere better […], you knew the people […]. Probably I was not so actively […] trying to approach anybody, it was more like a coincidence […]. Probably the small group sessions are arranged well as you can get to know random people.

Later, the same student continues:

Some people […] did not know how to get started with the problems, so then I could guide them to the right path. […] [S]o I gained quite good insight when I had to break it down to simpler terms. […] [I]n the sessions, […] I noticed that [solving the problems] is not super easy for anybody […], instead, the [problems] do require some thinking. (DA, 7, A)

Similar positive experiences did not occur in Course XA, as the students did not actively utilize the guidance offered in the open learning space, mostly because they reported that they did not need any support in completing the tasks. However, although finding the difficulty of the course tasks suitable and being rather independent in working on the tasks, some students reported not getting enough help for completing the course tasks. For example, a student stated:

I didn’t utilise the guidance, […] I probably should have […] but I was not used to doing so and I didn’t have the courage or something. (DA, 18, XA)

Some students also worked alone at home because they did not have peer students to work with in the open learning space:

It’s challenging because […] the others have their own groups already and then it’s very difficult to go there [to work together]. (DA, 24, XA)

3.3 Surface approach to learning subgroup

All students in the surface approach subgroup aimed at staying alive in both courses. Their distinctive interactions with the learning environments can be described in terms of lectures, guidance, and assessment. The students did not get much out of the lectures and often did not attend them; however, as the following quote shows, they found the lectures irrelevant mostly because they were unable to engage in the social interaction:

I attended the lectures whenever I could, about half of them. […] I took some kind of notes but […] they turned out to be such that I didn’t read […] them afterwards […]. I don’t know whether I got so much out of the lectures. I tried to listen and stay on track. There were some tasks like ‘think about these problems together’ but […] usually it was someone else who came up with the answer, so I was unable to participate so much in those [activities] either. (SA, 9, A)

The students reported varied experiences of the Course A small group sessions. Some of the students perceived the small groups very positively reporting that they were just the kind of contact teaching I wanted, when other students reported negative experiences for not getting enough help and complaining that the sessions could have been used more for teaching so that the tutor would have actually taught us. These expectations for teaching might have derived from the tasks being reportedly high in workload and such that one could barely make it, and not getting enough help for completing the course tasks prior to the sessions. This lack of support was highlighted by another challenge for the subgroup; when faced with challenges, the students had a narrow repertoire of actions to be taken to overcome these challenges. For example, a student stated:

I would always try to do [the tasks] based on my intuition and if I didn’t have any, then I didn’t get anywhere from there. (SA, 9, A)

In the Course XA context, the students in this subgroup perceived the open learning space positively as they received enough help and guidance; they reported on feelings of safety such as you don’t need to be alone and me and my fellow students, I think it had a great impact, all the support and safety. Also, the students reported accounts of individualized teaching like the tutors truly […] cater for you. Like all other students in general, the students in this subgroup found the difficulty of the Course XA tasks very suitable – they survived and were able to complete the tasks.

The surface approach subgroup experienced the Course A assessment in diverse ways. However, in Course XA, their experience of the assessment was solely negative. As demonstrated below, the negative experiences derived from the perceived dissonance with teaching and unexpectedly low course grades:

The exam questions were more difficult and maybe of a different type than the course tasks. […] [I] felt that the [exam] results did not reflect how much effort I had put into the course. (SA, 3, XA)

Indeed, this demonstrates a more general challenge for the surface approach students. During the course, they studied in a way that they thought was the most efficient, but in the interviews, they reported that now that I think retrospectively it might not be true and perhaps my way of going through things wasn’t that optimal in the end. One student went even further and asked that could it be […] that I used torpedoes against my own learning?

3.4. Context-sensitive surface approach to learning subgroup

The students in the context-sensitive surface approach subgroup can be described in terms of their simultaneous multiple aims for the courses and the very different overall experiences in the two learning environments. Besides understanding and surviving reported by the other two subgroups, the students in the context-sensitive surface subgroup reported such aims as completing course modules and fulfilling degree requirements. As the following quote demonstrates, a student in this subgroup could aim at both understanding and surviving, while aiming to complete the course:

I wanted to pass the course, […] and then […] it was very important for me […] to complete lots of study credits. […] But it was not all about achievement, because I also wanted to understand, it was also my aim. (CSSA, 21, A)

As an overall experience in Course A, the students reported multiple challenges in engaging in learning. However, their experiences were quite the opposite in Course XA, which showed especially in their interactions with lectures and guidance. In contrast to the other subgroups, the students were more active in Course XA compared to Course A. They attended lectures actively in both courses, but in Course A, their participation was passive, and they did not always find the lectures relevant. This changed in Course XA, where the students explicitly reflected on lectures, especially the successful social interaction, as supporting their learning:

I attended all the lectures because I found them very useful, you got much out of them […]. [T]here were so many different perspectives and you deepen your learning. […] I was very active as a speaker; we discussed a lot there. […] [T]he questions were easier to approach, […] because in [Course A], already the questions were such that you didn’t get a grasp on them so how could you discuss about it, but here [in Course XA] they were so well formulated that you were able to take a stand and reflect, so I was a very active learner […] in the lectures. You always left the [lecture hall] with a feeling that something had just opened up in a completely new way. (CSSA, 5, XA)

The students report contradictory experiences on the small group sessions in Course A. The students perceived working with peers very positively. However, they experienced frustration as they reported negative experiences on the general atmosphere or relevance of the sessions, such as not getting enough support from the tutor and having challenges in getting started with the tasks. A student stated:

The group is big and there is that one tutor and they don’t have time for you […] so there wasn’t at all similar kind of safety net as in [Course XA]. (CSSA, 21, A)

In Course XA, the students actively utilized the guidance offered in the open learning space while working with peers. Overall, they report on a culture of collaboration and a sense of community. The students specifically emphasized experiencing learning successes due to a successful social interaction with the other students and tutors, as demonstrated by the following:

In the [open learning space] we worked together [on the tasks], but […] here too it was the discussions that were the basis of everything. […] [L]ike how the other person had understood it and have you […] understood it – you also get to explain […]. And if no one […] has understood, support was offered. […] [I]t is much easier to ask for help in the [open learning space]. […] [A]nd when you’ve been sitting there a lot, the tutors started to recognise you. Sometimes I felt that oh no, the tutor remembers me and I’m asking this again, I still can’t do this, but despite that the tutor was like ‘hey, you remember, just like last time, like this’, it created cosiness. (CSSA, 2, XA)

Overall, it seems that engaging in social interaction was very central to this student subgroup. Even the one student in this group who thought it was easier to work alone hoped for more interaction between students and students and the tutors.

4. Discussion

This study investigated the same mathematics students in two different student-centred learning environments and built on the three student subgroups with different approaches to learning identified in a prior quantitative study (Lahdenperä et al., 2019). The study aimed at identifying elements of the mathematics learning environments that positively contribute to students’ approaches to learning. To distinguish between the intention and process components of an approach to learning (Marton & Säljö, 1984; Wierstra et al., 2003), students’ aims for learning and their interactions with the learning environments were investigated separately.

4.1. RQ1: the student subgroups’ aims for learning

In line with previous research (Lindblom-Ylänne et al., 2018; Marton & Säljö, 1984), the study identified three different aims for learning; students could either aim to understand the course content, aim to survive the course or they could see the courses in a larger context, for example, aiming to fulfil degree requirements (Lindblom-Ylänne et al., 2018). The aims were stable across the two learning environments, and the students set their aims in line with their course-level approach to learning: students applying a deep approach to learning aim to understand, students applying a surface approach to learning aim at surviving, and students applying a context-sensitive surface approach report all of the three aims for a course. This shows the benefits of Wierstra and colleagues’ (2003) call to distinguish between the aims and the processes; perhaps the diverse aims distinctive to the context-sensitive surface approach subgroup are the origin for their context-sensitivity and interactions that make the students’ ‘deep shifts’ (c.f. Wilson & Fowler, 2005) possible; as they have many aims, the context has a great impact on the aim they start to pursue.

4.2. RQ2: the student subgroups’ interactions with the learning environments

The students in the deep approach subgroup were independent and acted based on perceived relevance to their learning. In general, the deep approach to learning showed in students’ stable interactions with the learning environments. This is supported by previous research stating that the deep approach to learning is the most stable approach to learning (Wilson & Fowler, 2005) and that the students reported quite similar experiences of the two –learning environments in the prior quantitative study (Lahdenperä et al., 2019). However, the results revealed one major difference in the students’ experiences between the two courses. The most positively experienced element of the learning environment was the scheduled small group sessions in Course A – for providing possibilities for discussions and student collaboration. The students did not have similar access to discussion and collaboration in the more unstructured open learning space in Course XA. Although doing well and being independent in studying, perhaps these students require more structural support from the learning environment to participate in the yet desired student collaboration. This is important considering their future studies; although feeling competent now, the skills to seek help will be essential when facing challenges later in their study.

The surface approach to learning showed – in line with Lindblom-Ylänne et al. (2018) – in students’ unreflective interactions with the learning environments. The students in this subgroup expected direct individualized teaching; they reported that it was successfully realized in the open learning space in Course XA, and for some students, it also happened in the small group sessions in Course A. This might explain their more positive experiences of the XA learning environment reported in the previous study (Lahdenperä et al.,  2019). However, the students had challenges engaging in the social interaction within the learning environments. Unreflective interaction also showed in students’ negative experience of Course XA assessment. Course XA guidance and the gradually increasing level of difficulty supported the students to solve the majority of the course tasks. However, perhaps the students expected more success in the Course XA exam based on the number of tasks completed but did not reflect on the level of difficulty of the tasks completed. This kind of unreflective studying results in a fragmented knowledge base (Lindblom-Ylänne et al.,  2018), which showed in the statistically significantly lower exam points in the Course XA exam compared to the other two subgroups (Lahdenperä et al.,  2019). Overall, the students did not report many learning successes; as a student put it, I can’t say that I had any aha moments. To conclude, the reflection on the productivity of their learning processes came afterwards but it was not present in their accounts of what happened during the course. It is not that the students did not put effort into their studying, quite the contrary. However, these students need support to reflect on their (unproductive) learning processes not just afterwards but already during the courses.

The context-sensitive surface approach showed active engagement in student collaboration. In Course A, besides working with peers, the students reported a lot of frustration as the course structures did not provide enough support for their learning. Their experience was quite the opposite in Course XA, where they were provided with the possibility to engage in social interaction and collaborative construction of meaning. In the prior quantitative study, this showed the largest increase in positive experiences of the teaching–learning environment when moving from Course A to Course XA context; distinctive to this subgroup, the increase showed also in the factor measuring peer support (Lahdenperä et al.,  2019). As Wilson and Fowler (2005) state, ‘it may not be so much […] learner activity per se that contributes to “deep shifts”, but activity within a context which expects student responsibility and contribution’. In essence, the students benefit from a learning environment expecting and supporting students’ activity and collaboration.

4.3. RQ3: the change from a surface to a deep approach to learning

But what could further explain the change in the context-sensitive surface approach subgroups’ dominant approach to learning? The students in the deep approach subgroup and the students in the context-sensitive surface approach subgroup preferred different learning environments; the deep approach students enjoyed especially the Course A small groups, whereas the context-sensitive surface approach students enjoyed the Course XA open learning space. It seems that the deep and context-sensitive students had access to learning successes that supported their positive perceptions of themselves as mathematicians in different kinds of environments. This is demonstrated by a student from the deep approach subgroup describing their experience in the small group sessions:

It made me feel that now I do mathematics, […] I fill the blackboard with some real proof and it is a big experience of success, […] it is somehow so stereotypical mathematics thing and then it is just so fun to get to do it yourself. (DA, 6, A)

Similarly, a student from the context-sensitive surface approach subgroup described their experience in the open learning space:

I was supported by the experiences of success, […] the tutor gave me a small hint that I was missing but I got to do it by myself so that the joy of discovery was not wasted, […] that made me feel very mathematical. (CSSA, 5, XA)

Perhaps the students are situated differently in the surface–deep approach continuum (cf. Kember, 2016) and therefore are supported to apply a deep approach to learning in different ways. However, it seems that the deep approach students are doing well for the time being; therefore, open but guided learning environments such as in Course XA can be used to support especially the context-sensitive students that have the instant potential to change their dominant approach to learning.

4.4. Implications for practice

Overall, the students reported on having various aims for their learning and various ways of interacting with the learning environments. This forms challenges for the mathematics instructors planning their teaching practices. However, the results showed that of the learning environment elements, tasks and guidance were the most central for students’ learning as they were reported on the most often. This suggests that when (re)designing mathematics teaching, the tasks and the guidance offered for students to solve them are crucial elements for students’ learning experiences.

As the students were stable in their aims across the contexts, it seems that students enter a course with these fixed aims, creating a challenge to change their approaches to learning through instruction (cf. Asikainen & Gijbels, 2017; Baeten et al., 2010; Gijbels et al., 2008). However, the results showed that despite having a similar aim for the two courses, a student’s actualized learning processes can be different when the context changes. It is plausible that the learning processes in a course will influence the aim(s) of a subsequent course. Therefore, to facilitate students to aim for understanding, the learning environments need to support them in developing their learning processes. But what kinds of processes have a positive contribution? The results showed that all the students who participated in the study put effort into their studying. Therefore, it is not the amount but the quality of interaction within the learning environments that differentiated the student subgroups with different approaches to learning. The quality seems to be related to engaging in social interaction and student collaboration that induce experiences of learning success. Therefore, to advance a deep approach to learning, students could benefit from access to collaboration (deep approach and context-sensitive surface approach subgroups) and support in developing skills needed in collaboration (surface approach subgroup). These are provided in different ways to different student subgroups. Therefore, the learning environments within a degree programme need to be both flexible and diverse. Also, as the students’ aims for learning were more stable than the processes used to achieve them, we conclude – in line with previous literature (Baeten et al., 2010; Entwistle & Peterson, 2004; Wilson & Fowler, 2005) – that one course is not enough to support the development of deep approaches to learning; instead, we need a sequence of courses with well-aligned and versatile instructional practices.

4.5. Limitations of the study and future research

There are some limitations to this study that we want to address here. One of the limitations of the study is the small number of self-selected participants (N = 16). However, the interviews were an hour long in which students reflected on the two learning environments, creating a solid base for contrasting their experiences. In addition, the interviewees equally represented the three predetermined student subgroups. Another limitation was created by the unavailability of a second coder in the data analysis. To address this, the qualitative framework analysis was a fit choice: the analysis procedure is well defined, systematic, documented, and commonly accompanies quantitative studies (Ritchie & Spencer, 1994). Further, the analysis did not create much ambiguity as it sought for students’ concrete actions in the learning environments, and the qualitative results did support the quantitatively predetermined student subgroups. Finally, an additional limitation derives from the fact that the courses differed in mathematical content. However, the students did not reflect very much on the content. Mathematical content was one of the a priori themes in the analysis framework and was indexed in the transcripts. However, the number of instances where students reflected on the mathematical content in relation to their learning was minimal compared to the other themes (e.g. lectures, guidance, tasks). Therefore, mathematical content was not included in the proceeding analysis process.

As student collaboration played a central role in the subgroups’ interactions with the learning environments, the future research should incorporate SAL with self- and co-regulation of learning (see, e.g. Räisänen et al.,  2016). Also, studies incorporating the self-determination theory and SAL might prove fruitful, as demonstrated by Kyndt and colleagues (2011). By extending the theoretical lenses, it could be possible to even better capture the students actualized learning processes. Also, it would be interesting to conduct longitudinal studies on the development of the students’ approaches to learning (cf. Asikainen & Gijbels, 2017). For example, in this study, it was the participants’ first year of studying mathematics at university; with longitudinal research designs, we could investigate whether the emphasized need to engage in social interaction and the aim for surviving were only part of the first-year experience. Also, with parallel learning environments, it was possible to contrast the same students in two different contexts and draw conclusions on the pedagogical factors creating stability and change in the students’ approaches to learning. Now, it is possible to extend the research to the whole student population with a research design not involving parallel courses. We also call for research that directly addresses the students who, depending on the context, change their dominant approach to learning. However, in terms of educational development, we should aim high – for department-level research-based interventions (cf. Dolmans et al., 2016; Kember, 2016; Reinholz et al., 2020).

Authors’ contributions

Conceptualization: JL, JR, LP; Methodology: JL; Validation: JL, JR; Formal analysis: JL; Investigation; JL; Data Curation: JL; Writing – Original Draft: JL; Writing – Review & Editing: JL, JR, LP; Project administration: JL

Acknowledgements

The authors would like to express their gratitude to Juuso Henrik Nieminen for his valuable comments on the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability

Due to the nature of this research, participants of this study did not agree for their data to be shared publicly, so supporting data is not available.

Additional information

Funding

The research was supported by the Finnish Cultural Foundation under grants 00160520, 00172299 and 00182449.