In this section I will demonstrate some of the various theoretical approaches that have been used to investigate the secondary-tertiary transition. Consid- erable work was undertaken by the TRANSMATHS project (www.transmaths. org) which investigated, with the use of socio-cultural theories, the transi- tions to post-compulsory education, i.e. from school to college and from college to university. Their theoretical approaches include Activity Theory perspective on the concept of identity, a Marxist perspective on the concept of alienation, Bourdieu’s theory and also the creation of measures to under- stand students’ perceptions of the transition. Other scholars in the field have
which I will explore in the following. 2.1.2.1 Identity
Identity is a topic which has “enjoyed an explosion” (Darragh, 2016, p. 19) in the field of mathematics education during the last 20 years. Unsurpris- ingly, this explosion has had an impact upon research into the transition. Indeed, many scholars in recent years have used the concept of identity as a theoretical approach to investigate the issue. Hernandez-Martinez et al. (2011b) examine the ways in which the identities of students develop while they engage in different activities during the transition from school to college. They argue that students during this phase construct their identities through different social interactions and consequently they position themselves in the new institution in different ways. Interestingly, the troubles that these stu- dents encountered during their transition were seen as an opportunity for stepping-up and not as an obstacle.
In a later work, Black and Hernandez-Martinez (2016) emphasise the role of identity in accessing science capital while studying mathematically de- manding courses. The authors use the concept of science capital as developed by Archer, Dawson, Dewitt, Seakins, and Wong (2015) and this consists of: “scientific forms of cultural capital (scientific literacy; dispositions towards science, symbolic forms of knowledge about the transferability of science qualifications), science-related behaviours and practices (e.g. consumption of science media; visiting informal science learning environments, such as sci- ence museums), science-related forms of social capital (e.g. parental scientific knowledge; talking to others about science)” (p.3). Some students will access science capital for its exchange value, but others will recognise its use value. The choice is influenced by students’ identities and the difference in approach
will produce different forms of engagement with mathematics. Therefore, a promotion and alignment of the use value with the exchange value will be influential on students’ dispositions towards mathematics and will prevent a possible alienation from the subject (Black & Hernandez-Martinez, 2016).
Along similar lines, Holmegaard et al. (2013) suggest that in order to understand the transition we need to understand how people work on their identities while they move from one cultural context (the school) to another (the university). The authors investigate students’ transition as a process and a negotiation of identity. They focus on how students narrate the negotiation of their identities in the new context and how they develop an understanding of what it means to belong in this new context. The results suggest that the students renegotiate their narratives of why they chose this degree; what is it like to study for this degree; what kind of students they consider themselves to be in order to construct a new type of narrative which will include their experiences at university and their identity.
2.1.2.2 Alienation
Another concept that has been used in the transition research is the concept of alienation. Hernandez-Martinez (2016) uses the concept to explore drop- out of previously engaged students from mathematical study during their transition to university. Despite their considerable efforts to integrate and the support provided by the university (e.g. mathematics support centres) the students in this study were not able to negotiate their learning. The way they saw themselves as learners (for instance, engineering students who recognise themselves as more “practical”) prevented them from aligning with the academic practices employed at university.
plore the reasons why students disengage from university mathematics. The authors focus on the relationships with mathematics that students bring with them from school and on the changes that these relationships undergo at university. They argue that students who favoured more transmissionist teaching at school might feel less confident at university. At the same time, students who tried to build new relationships within the new context and invest in their participation in the community of mathematics, became frus- trated because they were not supported enough by the institution. Their results suggest that the majority of the students who participated in their study were in favour of the change in the nature of mathematics taught at university and they expressed a sense of ownership (which is the opposite of alienation) over their mathematical knowledge, as opposed to a number of students who “remained in ‘school mode’ ” (p.273).
2.1.2.3 Cooling-off phenomenon
The developing loss of interest in mathematics, also known as the “cooling- off” phenomenon, was also approached with a different focus (from the one we saw above, which uses the concept of alienation). Daskalogianni and Simp- son (2002) investigate students’ attitudes regarding mathematics before and while being at university during the first semester of their studies. They find that before entering university students’ attitudes are very positive and are mostly shaped by their beliefs about the nature of mathematics and the teaching-learning approaches employed at school. When they come to uni- versity for some of these students there is a mismatch between the previously shaped beliefs (about learning processes, nature of mathematics, assessment methods, etc.) and the actual university practices which can cause difficul- ties in the adjustment to the new setting. The students who experience such
a mismatch of their beliefs develop a negative attitude towards university mathematics which eventually entails a loss of interest in mathematics. The authors suggest that at this point two possible outcomes can happen; the stu- dents may recover and re-engage with their studies or they may lose interest in mathematics, lose interest in the course and then develop a “cooling-out” behaviour (some characteristic signs of this behaviour are no sign of interest in engaging with mathematical activities and denial of engaging because of the fear of failure).
2.1.2.4 Activity theory
Jooganah and Williams (2016) use activity theory to explain first year un- dergraduate students’ experiences in the learning of advanced mathematics. They focus on the contradictions that arise between the two activity sys- tems (school and university) and they claim that these contradictions play an important role in explaining the difficulties that students face while study- ing mathematics at this level. The results of the study suggest that when students move from one activity system to another, the contradictions aris- ing between the two systems can create conflicts in students’ mathemati- cal identities which influence the motivation and understanding of advanced mathematics.
2.1.2.5 Creation of measures
Statistical measures have been developed in an attempt to quantify the issue of the transition. Pampaka et al. (2012) created two measures of students’ perceptions of their transition to university, within the Rasch measurement framework, namely:
2. “degree of positive feeling about the transition”
These two measures allowed the researchers to track students’ disposition to complete their degree and their dispositions to study more mathematics. The authors suggest that the measures can be easily adopted by lecturers who want to get feedback regarding the practices they use during the tran- sitional period. The results of their study suggest that there are significant differences in the transitional experience between subgroups of students and that these contribute to the prediction of the development of positive dis- positions. Moreover, they argue that mathematics self-efficacy has a similar effect size to the “positive feeling about transition” (Pampaka et al., 2012, p. 1066).
2.1.2.6 Pedagogic discourse
In their study of first year engineering students at two Swedish universities Jablonka et al. (2016) investigate whether students were able to recognise the change in criteria regarding the mathematical rigour required at this level. The researchers presented to the students extracts from various mathematics textbooks and asked them which ones they considered more or less mathe- matical and why. Their theoretical framework was based on Bernstein’s (2000) work in pedagogic discourse, and more specifically on his ideas of knowledge classification and recognition rules, on features of Halliday and Hasan’s (1989) work on social semiotics, and on Eco’s (1979) concept of the model reader. The results of the study suggest that the students pay attention to a variety of aspects in the mathematics texts by which they “(mis)recognise” the precision of the mathematics pedagogic discourse. This (mis)recognition is eventually linked to their academic success.
2.1.2.7 Didactical Contract
Hourigan and O’Donoghue (2007) employ Brousseau’s concept of didacti- cal contract to study the pre-tertiary mathematics experience of entering students, in the Irish educational context. The authors investigate the di- dactical contracts of two distinct mathematics classrooms and identify many common features in the two. The “national obsession with the state ex- amination” (p.473) led the teachers to follow examination-driven practices. These practices narrowed students’ future potentials and did not promote skills that are necessary in tertiary level mathematics courses, such as prob- lem solving abilities, self-confidence in working with challenging problems and more flexible ways of thinking.
2.1.2.8 Anthropological Theory of Didactics (ATD)
In her review paper on the secondary-tertiary transition Gueudet (2008) gives an insight into studies in the French (Gueudet, 2006), Spanish (Bosch, Fon- seca, & Gasc´on, 2004) and Danish (Gronbaek & Winslow, 2007) educational context, which use the ATD (Chevallard, 1992) as an analytical tool for the investigation of the issue. According to the ATD a given institution proposes mathematical organisations (or praxeologies): “A mathematical organization [T, τ, θ, Θ] has four components: a type of tasks, T, associated techniques, τ , a technology, θ , which is a discourse explaining and justifying the tech- nique, and a theory, Θ , which is itself a discourse justifying the technology. These organizations can be observed in the official curriculum text, or in the textbooks provided by the institutions.” (Gueudet, 2008, p. 245, 246). The results of the aforementioned studies suggest that a focus on the math- ematical organisations can provide us with information regarding how an institution presents specific mathematical content and what can be the effect
of this particular choice on students’ transitional experience. 2.1.2.9 Rite of passage
A concept from anthropology - the rite of passage - is used by Clark and Lovric (2008) to build a theoretical model in order to explain the transition. The three phases of the rite of passage - separation (when a person encoun- ters a life crisis and isolates her/himself from the rest of the community), liminal phase (events that help the person to achieve the required changes and eventually bring her/him back to the community) and incorporation (at the end of the rite the person learns about the community that she/he will belong to and, supported by the members of the community, she/he will find eventually a place in this new community) - can all be identified as con- stituent stages of the transitional experience. The separation phase refers to students’ separation from the school context: “takes place while students are still in high school, and includes anticipation of forthcoming university life” (p.35). The liminal phase (from school to university) includes the end of school, the time between school and university and the beginning of uni- versity. Lastly, the incorporation phase refers to the first year at university. The exploration between the dynamics of these three stages offers a way to understand the difficulties arising during the transition.