In this study we explore (1) the curriculum resources engineering students use who are studying in their first year university mathematics (Calculus and Linear Algebra (LA)); and (2) how they use/orchstrate them, in particular how they develop their own learning paths with the support of these resources.

In most western (engineering) universities students now have access to a plethora of (1) digital/online resources, and of course to (2) ‘traditional’ curriculum resources, such as textbooks, readers, worksheets, etc., provided by the universities and by lecturers. In particular in large first year courses (e.g. calculus), students are typically expected to use and blend the available resources according to their individual needs, to support their learning. The rationale for our study is that, in order to be able to understand (and develop) ‘blended’ mathematics learning, one has to know, which resources are used by students (from the ones on offer), and how they use them for their learning of mathematics. 

The use of particular curriculum resources by teachers and students in higher education mathematics has been subject of current research. In a recent review study the opportunities afforded by introductory university mathematics textbooks have been described, and the actual use made of these curriculum materials by students (Biza, Giraldo, Hochmuth, Khakbaz, & Rasmussen, 2016). Anastasakis, Robinson, and Lerman (2017) investigated the different types of tools (‘external’ (to the university) and internally provided resources) that a cohort of second year engineering undergraduates used. Their results showed that although to some extent students used resources external to their university, their practices were dominated by tools that their institution provided. The students in their sample choose certain tools mainly because these enabled them to pursue their exam-driven goals. The use of visual resources (online lectures and mathematics learning support centers) has been studied by Inglis, Palipana and Ward (2011). Using the Documentational Approach to Didactics Gueudet (2017) investigated mathematics teachers’ interactions with resources at university, and Gueudet and Pepin (under review) explored how Brousseau’s Didactic Contract can be interpreted seen through the lens of (the use of) curriculum resources.  In terms of LA, Grenier-Boley (2014) investigated LA tutorials, and he noticed that the teacher’s interventions and the student activity in class were linked with the kind of mathematical tasks chosen by the teacher from a list of exercises. 

However, we note here that relatively little research is available on the broad range of resources available to first year university students to learn mathematics, and moreover how students actually orchestrate the resources in order to learn the mathematics. Typically, studies include the curriculum resources made available or recommended as part of mathematics courses, but there are also ‘human resources’ (e.g. lecturers, tutors, peers) that students tap into, and digital and other resources mobilized by students themselves. In addition, we know little about the roles these various resources play in the intention of the lecturers (as compared to the students’). Inglis et al. (2011) suggest that students might need explicit guidance on how to combine the use of various resources into an effective learning strategy. Before this guidance can be given, or can be reified in a blended learning environment, more in-depth information on their actual use is needed. Hence, we ask the following research questions.

(1) Which curriculum resources do first year engineering students use, and how do they use and orchestrate these resources to learn mathematics (in particular Calculus and LA) at university? 

(2) Using the lens of resources, which learning paths do students develop (or follow), using available resources, to learn Calculus and LA? 

Using a case study approach, we explore two first year mathematics courses in a Dutch engineering university, LA and Calculus, as our cases. LA is taught to one group of approximately 150 students, mainly mathematics and physics students. Calculus is taught in 6-7 groups of 300 students each (all types of engineering); which are streamed into Calculus A, B and C (according to perceived mathematical ability). In order to find out which resources students use, and how they use them, we have used the following data collection strategies for the exploratory study (2016/17): - individual and focus group interviews with students (23) - selected observations - interviews with lecturers and instructors (5) - calculus students were asked to draw Schematic Representation of Resource System (SRRS - see Pepin, Xu, Trouche, & Wang, 2017), and during the interviews students were asked to explain their resource use (based on their SRRSs) - selected students were asked to annotate mathematical tasks with the curriculum materials they used to solve those tasks. In terms of analysis, the interviews were transcribed and student interviews were analyzed with the help of the ATLAS-ti software. Interview quotations were coded with codes based on the themes from the literature with reference to the use of different resources, and to student approaches to learning mathematics, and then ‘constantly compared’ with those themes mentioned by students and lecturers. In the next step of the analysis the quotations with the same codes were examined to identify patterns regarding student learning trajectories, and their use of resources in the two courses.

Based on our results we claim that it is not sufficient to provide a plethora of curriculum resources, may they be digital, traditional text or human resources, but that serious consideration should be given to how students might combine these resources, orchestrate them into their preferred learning paths. In addition, it is advisable to help, perhaps even to train students to develop such learning paths, and these might be different from one subject to another (even from one mathematics course to another). This, we claim, is the responsibility of the lecturer/teacher/course developer. Such course development would involve purposeful design, including the development/envisaging of particular (intended) learning paths, and the design of particular resources supporting such paths. Simply providing access to curriculum resources does not seem to help students to ‘blend their learning’ and orchestrate the resources on offer, but may rather confuse and overwhelm them (due to the immensity of resources on offer), and drive them towards “learning for the test”. As Anastasakis, Robinson, and Lerman (2017) claim, under such conditions “students use the most popular resources [and] they aim mostly for exam-related goals” and use “certain tools because these enable them to pursue their exam-driven goals” (p. 67).

Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224. Anastasakis, Robinson, and Lerman (2017) Links between students’ goals and their choice of educational resources in undergraduate mathematics. Teaching Mathematics and Its Applications, 36, 67-80 doi:10.1093/teamat/hrx003 Biza, I., Giraldo, V., Hochmuth, R. Khakbaz, A.S., & Rasmussen, C. (2016) Research on teaching and learning mathematics at the tertiary Level. Springer Open. Grenier-Boley, N. (2014). Some issues about the introduction of first concepts in linear algebra during tutorial sessions at the beginning of university. Educational Studies in Mathematics, 87(3), 439–461. Gueudet (2017). University teachers’ resources systems and documents. International Journal of Research in Undergraduate Mathematics Education, 3(1), 198–224. Gueudet, G., & Pepin, B. (Under review). Didactic contract at the beginning of university: A focus on resources and their use. International Journal of Research in Undergraduate Mathematics Education. Inglis, M., Palipana, A., Trenholm, S., & Ward, J. (2011). Individual differences in students’ use of optional learning resources. Journal of Computer Assisted Learning, 27, 490-502. Pepin, B. (2014) Using the construct of Didactic Contract to understand student transition into university mathematics education. Policy Futures in Education, 12 (5), 646 - 657. Pepin, B., Choppin, J., Ruthven, K., & Sinclair, N. (2017) Digital curriculum resources in mathematics education: foundations for change. ZDM- Mathematics Education, 49(5), 645- 661. Pepin, B., & Gueudet, G. (2014). Curricular resources and textbooks. In S. Lerman (Ed.), Encyclopedia of mathematics education. Berlin, Heidelberg: Springer. Pepin, B., Xu, B., Trouche, L., & Wang, C. (2017). Developing a deeper understanding of mathematics teaching expertise: an examination of three Chinese mathematics teachers’ resource systems as windows into their work and expertise. Educational Studies in Mathematics, 94(3), 257-274. doi:Doi:10.1007/s10649-016-9727-2