# III: Teachers’ and students’ practices at university level

## Transitions to, across and beyond university

**Reinhard Hochmuth, Laura Broley and Elena Nardi**

**Introduction**

This chapter is about papers from INDRUM 2016 and 2018 considering transition issues. The aim is to give a systematic review of the themes discussed and the major theoretical lenses used. Transition to and across university mathematics has been attracting increasing research effort (e.g. Biza, Giraldo, Hochmuth, Khakbaz, & Rasmussen, C., 2016). Since all learning is somehow a transition from one state to another, something which is also noted in the introduction of the survey about transition by Gueudet, Bosch, diSessa, Kwon & Verschaffel (2016), nearly all papers presented at the INDRUM conferences could be seen as relevant to this chapter. To narrow our focus, we decided to start from the papers assigned to the transition working group at INDRUM 2018 and added further papers from both INDRUM 2016 and 2018. In total, we identified 32 papers that substantially referred to transition issues. Finally, we connected the INDRUM papers to other, pertinent works in the field and structured the chapter in accordance with three types of transition, a classification that follows a timeline across educational levels and the workplace: Type I: From school to university; Type II: Within and across university courses; and Type III: From university to the workplace.

In comparison with the survey by Gueudet et al. (2016), our chapter is limited to the transitions involving university mathematics education, and does not cover, for example, the pre-school to school or the primary-secondary transitions. Our choice regarding transition types resonates with the ICMI Pipeline Project (Hodgson, 2015), which focused on the following four transition points: A: School to undergraduate courses; B: Undergraduate courses to postgraduate courses; C: University into employment; and D: University into teaching. (A) is included in our (I), (B) is part of (II) and (C) and (D) are linked to (III). Our (II) might also include horizontal transitions between topics and transitions between service courses and non-mathematics major courses. Our (III) covers workplaces outside school and university to the limited extent allowed by INDRUM papers. (Dur stronger focus on research efforts with regard to (II), and the sparsity of works on (III), can be attributed to the backgrounds and priorities of INDRUM participants. Overall, we note that we faced a challenge with the general issue of typification: for example, what counts as school and university mathematics varies significantly across countries. We have been careful to attend to this variability by including notes on context where appropriate.

To illustrate this issue, and briefly introduce each of the transition types on which our chapter is based, we offer some prototypical examples. For instance, an often-investigated issue of transition type I is the mathematical topic of limits of sequences that has turned out to cause major problems for many students. In school, limits are introduced in an informal way and are algebraically calculated for rational terms. If a definition is presented in school, there are typically no tasks questioning or requiring this definition. In university analysis courses, the formal definitions of limits in R, C, metric- or Banach-spaces, which are axiomatically introduced, are fundamental for solving tasks and understanding continuity and differentiability. We note that the transition from an informal to a definition-based use of limits does not always occur in the transition from school to university. In countries where calculation-oriented calculus courses are at the beginning of university studies, the limit-issue shifts from transition type I to type II.

With type II in view, a mathematical notion that is treated in both first semester and advanced university mathematics courses is “continuity”. In beginner courses, the continuity of functions is essentially based on the definition of a limit. In topology courses, continuity is introduced more generally for functions between topological spaces. Besides such vertical transitions, in which areas of knowledge and courses build on one another in terms of content and follow one another in time, there are also horizontal transitions, in which areas of knowledge can be taught largely independently of each other and therefore courses do not have to follow each other in time. An example is the use, and further development of, the Riemann - or Lebesgue — integral in courses about partial differential equations or stochastics. A mixture of horizontal and vertical mathematical interrelations often appears in the transition from mathematics service courses to non-mathematics major courses. For example, Fourier series are typically considered in mathematics service courses for engineers, but also in electrical engineering signal theory courses.

As an example of the transition from university to the workplace (III), let us briefly consider the issue of teaching mathematics. Student teachers, lecturers of introductory and advanced mathematics courses and lecturers of service courses have the qualifications that suggest they know the mathematics they teach well. But knowing something and knowing how to teach something is not the same. The latter requires one to address students’ learning needs, to choose appropriate ways of introducing students to new content, to design and carry out assessment and to be aware of how mathematics is changed when it is transposed into educational institutions. In resonance with Klein’s (1908/1939) poignant points on the double discontinuity between mathematics in school, university and the workplace (school and elsewhere), teaching also necessitates an awareness of the various forms of mathematics needed in various workplaces (engineers’, economists’, psychologists’, etc.).

Since INDRUM is a network of university mathematics educators, typically lecturers doing research in mathematics and/or mathematics education, we see it as helpful to begin by identifying and discussing issues that our research aims to address mainly in *mathematical* terms, as we have done above. This said, and as will be shown throughout the rest of our chapter, the INDRUM papers on transition also highlight epistemological, historical, pedagogical, educational, cultural, sociocultural and affective issues. The studies we refer to have a variety of goals: the analysis of transition processes; development, implementation and evaluation of measures that intend to support students coping with major transition steps; and, integrative theoretical efforts. As a whole, they make a broad contribution to research on transition in university mathematics education and point to some areas that require more attention from the INDRUM community. In the following section, we present a short overview of the theoretical frameworks and constructs that underpin these papers. Then, we discuss the papers relating to the three transition types I, II and III. We conclude with a summary, noticing recent trends and major challenges.

**Theoretical lenses that capture transition**

Artigue (Chapter 1 in this book), Gueudet et al. (2016) and Nardi (2017) all highlight a first period of research in university mathematics education and transitions espousing cognitive approaches, e.g. identifying the necessities of conceptual changes. Under the umbrella term “advanced mathematical thinking” (Tall, 1991), works covered major differences between school and university mathematics in terms of oppositions such as informal vs. formal, concrete vs. abstract, calculation-vs. structure-oriented, etc. There are also Tail’s (2013) three worlds (embodied, symbolic and formal), which extend to the transition from primary to secondary school, and Fischbein’s (1989) tacit models, which theorize more generally transitions from previous to new knowledge (e.g. Kidron, 2018, referenced in subsection I). All these theoretical constructs and frameworks work well for analysing mathematical differences and related cognitive phenomena with respect to type I transitions. This is not necessarily the case for analysing transitions of type II or III which, as noted by Artigue (2016), Gueudet et al. (2016) and Thomas et al. (2015), may benefit from a general move in mathematics education research that aims to include a more systematic consideration of social, interactional and intersubjective issues (Lerman, 2000). This does not mean that the initial theories are out of use today. As Artigue (2016) noted in her INDRUM plenary, there has been “a reconstruction of their main outcomes, thus making possible some form of incorporation of these outcomes in the new paradigms” (p.19). We now outline theoretical approaches that are frequently used in INDRUM transition papers.

*Concept image / concept definition and APOS*

The concept image / concept definition distinction appeared in the work of Shlomo Vinner in the 1970s (e.g. 1976). It became a quintessential part of cognitively oriented, developmental analysis of mathematical learning at university level and elsewhere through the much-cited work of Tall & Vinner (1981). In research on transition phenomena, the distinction emphasizes that unique formal definitions and proofs of theorems based on definitions are an important feature of university mathematics and indicates major differences with school mathematics. A more layered cognitive theoretical framework is APOS (action-process-object-schema) (Dubinsky, 1991; Amon et al., 2013), which is based on Piaget’s genetic epistemology and, in particular, his notion of reflective abstraction. APOS theory - given that the advanced phases of “object” and “cognitive schema” extend naturally to university mathematics — is used particularly in investigations of type I transitions (e.g. Syamsuri, 2019, where cognitive processes in proving activities were studied). Regarding transition type II, one INDRUM paper that deploys the concept image / concept definition construct is by Hamza and O’Shea (2016) who used it to analyse interview data concerning open sets in metric spaces (see also subsection II).

*Anthropological Theory of the Didactic (ATD)*

ATD (Chevallard 1999) considers knowledge as something that lives in institutions (Bosch & Gascon, 2014) and regards the institutionalisation of knowledge as the result of complex transformation processes, referred to as external didactic transpositions (Bosch, Hausberger, Hochmuth & Winslow, 2019). Moreover, knowledge is regarded as co-detennined by an ecological and hierarchical scale of so-called “levels of codetermination” (Chevallard, 2002) - civilisation, society, school, pedagogy, discipline, domain, etc. — which also indicates that teaching and learning are shaped by constraints located at different levels of specificity. At the centre of the ATD lies the 4T-model of a “praxeology” (i.e., task, technique, technology, theory), which enables researchers to build reference models as a basis for further analyses of taught and learned knowledge. In INDRUM papers, transition can be seen as taking place between institutions (e.g. the transition from the institution of calculus to the institution of probability theory considered in subsection I, or from single to multivariable calculus courses considered in subsection II). Considering type I transitions, there is a change in tasks and techniques and, in particular, the formation and relevance of technological and theoretical blocks, which constitute the discourse necessary for interpreting and justifying the practical block (i.e., task and related techniques), including explanations and justifications of the techniques used, especially in terms of theorems and proofs. Type II transitions within mathematics involve further changes from procedural to theoretical practices. One might even identify phases of transitions within university with fundamental changes of the roles of those blocks; for example, theoretical blocks become techniques of new tasks in advanced courses, accompanied by the introduction of new theoretical blocks (see subsection II). Regarding mathematics service courses, ATD allows formalizing the disconnections between the mathematics taught in these courses and its use in the serviced disciplines. Type III transitions also fit very well with the institutional idea (see our examples regarding Felix Klein’s double discontinuity in subsection III). In recent years, ATD, in particular its “study and research paths” construct, a variation of inquiry-oriented teaching efforts (Barquero, Serrano & Ruiz-Munzón, 2016; Florensa, Bosch, Cuadros & Gascón, 2018), has also been used in professionalization courses.

*Theory of Commognition*

In the transition papers utilizing Sfard’s (2008) communicational perspective, the focus is on the shifts of mathematical discourses and rules within and between different groups. Commognition considers knowledge as something that comes into the world through communication, which is regarded not as a secondary aspect of teaching and learning, but as intrinsically related to individuals’ cognition. Commognitive notions present in INDRUM papers include word use, visual mediators, endorsed narratives and routines, which are further distinguished into rituals and explorations. A key incentive for developing commognition theory lies in the observation that a cognitive take on conceptual development is often binary (e.g. procedural vs. conceptual), whereas a commognitive take resonates with a spectrum-view of learning and teaching. Commognitive constructs enable descriptions and investigations of all three transition types. Regarding type I, for example, a word like “limit” is understood and used in different ways by lecturers and by first semester students arriving from school. Regarding type II, within mathematics, endorsed narratives change, for example they may become more general and abstract from one domain to another. This is also true of transitions of type III: there are changes of routines and endorsed narratives in the transition from pure mathematics to, say, engineering. Furthermore, the theory might be very helpful in sensitizing school teachers and university lecturers to commognitive conflicts inherent in or emerging from teaching-learning situations.

The referenced approaches facilitate not only the identification of mathematical learning obstacles in transitions, but also the development of well-founded teaching interventions. Some specific approaches in mathematics education research for developing, researching and evaluating interventions have also been applied in transition contexts considered in INDRUM papers: for instance, the theory of didactic situations (TDS: Brousseau, 1997), with its a priori and a posteriori analyses, Realistic Mathematics Education (van den Heuvel-Panhuizen & Drijvers, 2014), as well as guided intervention studies and design-based research as a rather general iterative approach (Prediger, Gravemeijer & Confrey, 2015). Variable psychological approaches adapted to mathematical contexts are also used in INDRUM’s transition research. Large-scale questionnaires, for example, are applied in type I transition studies evaluating measures that intend to support students regarding anxiety of mathematics, motivation and interest, mathematical self-efficacy, learning strategies, etc. (see, e.g., in subsection I, Biehler et al., 2018, and Kuklinski et al., 2018).

Last but not least, recent success in networking efforts in mathematics education research (Bikner-Ahsbahs & Prediger, 2014) point to the increasing need for more systematic inclusion of societal issues. One INDRUM example is (Hochmuth, 2018), which networks ATD and the subject scientific approach by Holzkamp (1985).

**Three types of transition: school-university, within university, from university to the workplace**

*/. School-university*

The transition from secondary to tertiary education is an exciting and often complex experience for young people — and with many facets: mathematical, curricular, academic and social (Nardi et al., 2017). University, as an institution, and university mathematics, can be seen in this sense as a new world governed by new rules (Gueudet et al., 2016), a new discourse that may intrigue as well as alienate newcomers (Sfard, 2014). Within mathematics, despite obvious contextual differences, there are several key features of this transition that are common across educational systems (e.g. Engelbrecht & Harding, 2008): a “gap” or a curricular discontinuity between what is defined in different contexts as “school” and “university” mathematics (Kajander & Lovric, 2005) and academic and social differences in learning environment (Cherif & Wideen, 1992). Recent studies have also been turning to social aspects of the transition such as changes in students’ living arrangements, location, friendship groups, as well as relationships with peers and staff (Bampili, Zachariades, & Sakonidis, 2017).

Most emphatically though, the transition from school to university has been seen as requiring a shift in thinking mathematically (Tall, 1992), particularly in relation to increased requirements for rigour and precision when engaging with the formalism and abstraction of university mathematics (Nardi 1996), for moving from instrumental to more relational approaches to doing mathematics (Clark & Lovric, 2009) and for developing more learner autonomy (Solomon, 2006). There is now a non-negligible body of work that discusses students’ arrival at university in terms of mathematical under-preparedness (e.g. Gill, O’Donoghue, Faulkner & Hannigan, 2010) with some effect, in some contexts, on student retention and success (e.g. Anthony, 2000). However, a deficit discourse on this issue is not universal (Engelbrecht & Harding, 2008) and a survey of the literature by Thomas et al. (2015) found a mixed picture in this respect. INDRUM papers minor this diverse landscape. In what follows, we sketch this landscape in three parts; each with a focus on studies that foreground the mathematics, students’ approaches to study and institutional support.

*Mathematics.* Lecorre (2016) explores pragmatic, empirical and theoretical aspects of rationality in relation to the concept of limit, especially at the transition from informal to formal calculus between school and university. With a focus on what means of justification students bring to their mathematical work, he creates a situation that aims to encourage a shift from dynamic to static perspectives on limit, and vice versa, which he sees as a great platform to develop aforementioned types of rationality. The situation includes forms of scientific debate around the conjectures being put forward by students and is trialled in the context of an upper secondary class in France. He observes students’ arguments gradually becoming more robust, until a collectively generated proof is finally endorsed by the class. Ghedamsi and Lecorre (2018), combining TDS and APOS, take this work further: eight tasks were trialled in upper secondary classrooms with reported success particularly with regard to creating a need for formalisation, a need that the secondary students had little prior experience of.

Schiiler-Meyer (2018) also works in the context of limiting processes. Defining the convergence of sequences is studied from a commognitive perspective as a discursive practice in transition. Upper secondary students reinvent the definition in a guided intervention process that aims to facilitate their transition from experiential to abstract approaches. Prior experiences of convergence from school are activated and secondary school discourses unlock objectified narratives.

Also with a focus on what students may bring with them from school mathematics, Kidron (2018) explores newly arriving students’ understanding of irrational numbers through the ways in which they choose to represent non-repeating decimals. Drawing on developmental constructs such as Fischbein’s (1987) mental model, she offers a quantitative analysis and classification of secondary students’ concept images of decimals and irrationals as evident in questionnaire responses of 91 Year 10 students and 97 Year 11 and 12 students in Israel. She identifies misconceptions about decimals and rational/irrational numbers. There is a strong parallel in this work with Durand-Guerrier and Tanguay (2016) who offer an epistemological analysis of the transition from upper secondary to university mathematics for real numbers and map didactic horizons that may emerge from such an analysis.

Bloch and Gibel (2016) deploy TDS constructs (particularly that of adidactic situations) as well as semiotic approaches to explore complications in students’ learning of calculus at the start of university studies. Their focus is on the increased complexity of signs and reasoning processes at this level. They identify challenges evident in responses from 14 university students in a French university to final exams questions about parametric curves and differential equations. Amongst these challenges are: the learning of new techniques; and, selecting, performing and interpreting the results of calculations. Students’ relationship with diagrammatic reasoning also emerges as potentially problematic. These challenges are seen as relating to transition, particularly as the students are in the first year of university studies.

Finally, Derouet, Planchón, Hausberger, and Hochmuth (2018) focus on the challenges posed by the structures of school and university mathematics programmes, such as the compartmentalisation of the curriculum both at secondary and tertiary levels. Drawing on ATD constructs, Derouet et al. analyse the prax-eological repertoire that students bring from school into university in probability

(continuous distributions) and calculus (integrals) - two topics that students are to some extent familiar with from school and that problematize the compartmentalization of secondary school curricula. Analysis of questionnaire data from 82 students with different backgrounds revealed almost no transfer of applicable techniques across the two topics.

*Approaches to study.* A small but distinct set of works presented at INDRUM revolves around how students’ independent work changes as they enter university mathematics studies (Quéré, 2016), in particular with regard to their use of resources (Gueudet & Pepin, 2016; Kock & Pepin, 2018). Quéré (2016) notes the concern, and sometimes hesitation, of engineering students about how to deploy the range of means and resources available to them as well as a deficit of strategy in their approaches to study for doing so. Gueudet and Pepin (2016) explore this potential deficit through comparing how university mathematics teachers expect students to use resources and how students actually use them. In their UK datasets, they survey students, interview students and lecturers, observe lectures and study university documents. In their France dataset (Year 1, Number Theory module), they survey students, interview one lecturer and explore the impact of coursework, also in follow-up interviews. The lecturers seem to expect students to use resources to understand mathematics (epistemic mediation) while students seem to use resources to find examples fully worked by their lecturers that they can emulate in examinations or other forms of assessment (pragmatic mediation). While pragmatic mediation does not preclude the emergence of epistemic mediation (for instance, a student might expand their understanding of mathematics while adapting a worked example), their analysis, which draws on instrumentation approaches (Rabardel & Bounnaud, 2003), stresses the discrepancy between the two.

Analogous findings, but in the context of calculus and linear algebra, are proposed by Kock and Pepin (2018) who juxtapose students’ use of resources at school and university. Their group and individual interviews with students, analysis of resources and interviews with lecturers and tutors highlight the greater variety of resources available to students at university level, the diminished role of the teacher as the main resource and the strong association between students’ use of resources and course organisation. In some courses, students spend less time interacting individually with a teacher and there is loose, even poor, alignment between how mathematical content is presented in textbooks and in lectures. This implies that students are expected to establish links between the two and thus invest substantial effort in organising their resource systems.

*Institutional support.* Overall, the work presented at INDRUM sees the transition to the study of mathematics at university as a multifaceted process that requires a shift in the way students think mathematically. We stress that this shift materialises in a new social and academic environment and according to the norms of a new learning community. This shift is also experienced differently by students with different socioeconomic and educational backgrounds and is part of broader and deeper identity formation processes they live concurrently. How universities support this multifaceted process is essential: student learning support systems, personal tutoring and peer-support systems are amongst the prevalent modes of support.

There are several studies presented at INDRUM that relate to attracting or retaining students and to supporting a successful transition. These focus on: evaluating programmes that aim to raise awareness and appreciation of university mathematics for high-achieving secondary students on the brink of deciding whether they will undertake university studies with a STEM focus and high mathematical component (Bracke et al., 2016); identifying newly-arriving students’ interest in different domains and aspects of mathematics such as real-world problems, calculations and proofs, and exploring who may be at risk of dropping out of university mathematics (Geisler & Rolka, 2018; Fuller et al., 2016; Rach et al., 2018; Gradwohl & Eichler, 2018); and, evaluating tailored support for students either on the cusp of or just arriving at university (Biehler et al., 2018; Kuklinski et al., 2018; Landgards, 2018). For example, Biehler et al. (2018) evaluate pre-university bridging courses that aim to clarify the goals and expectations of study at university level for incoming students: goals of lecturers and courses are juxtaposed and compared to perceived goals by students as well as students’ beliefs about how these goals can be achieved. Within the same larger programme (WiGeMath), Kuklinski et al. (2018) evaluate a first-semester course that aims to facilitate the transition to university study. The evaluation includes an analysis of some affective variables such as interest, mathematical selfconcept, basic needs and self-efficacy and goal-fulfilment and traces a decline in interest and some shifts in student beliefs e.g. about the “toolbox” nature of mathematics (Grigutsch & Tomer, 1998).

Also with a focus on the affective domain. Fuller et al. (2016) explore how newly-arriving students’ completion rates may be influenced by anxiety and personality factors. This empirical study explores psychological traits, uses scales such as the Abbreviated Mathematics Anxiety Rating Scale (MARS) as well as elements of the “Big Five” inventory (extraversion, agreeableness, conscientiousness, neuroticism and openness: Poropat, 2009) and combines extensive demographic data and results. 404 students’ mathematical behaviour is evidenced in task completion and mapped onto data concerning reported levels of anxiety, personality traits and course completion. Exam anxiety emerges as a significant influence on non-completion, with Big Five’s conscientiousness emerging as key in successful completion. In a similar vein, Gradwohl and Eichler (2018) explore predictors of performance in engineering mathematics with a focus on psychological constructs like motivation and learning strategies. Individual characteristics of students such as prior perfonnance, motivation and engagement are studied for correlation and for their impact on study success. The focus is on 182 engineering students and the aim is to identify domain-specific predictors of study success. The questionnaires include items on goal-orientation, interests and self-reporting on learning strategies (and prior perfonnance, e.g. grades). Strong correlations with mathematics self-concept resonate with other studies of affect and, unsurprisingly, the study identifies an impact of school grades and performance in earlier exams at university. With more of a focus on identifying predictors of dropping out as well as success in examinations, Geisler and Rolka (2018) collected questionnaire responses from 209 students reporting their mathematical self-concept, interest in mathematics, beliefs about the nature of mathematics, basic learning needs and general self-efficacy. Data concerning achievement during one semester and exam results were also collected. Students’ achievement during the semester emerged as the best predictor for their exam outcomes. Unsurprisingly, the best predictor for lecture and seminar attendance was their interest in a particular topic in mathematics.

Beyond aforementioned valuable systems of support and studies of affective factors, we see two further modes of supporting students’ transition from school to university - that do require, we note, deeper systemic shifts — as essential: (1) reconfiguration of teaching practices at university level that accommodate directly the learning needs of incoming students; and, (2) a stronger synergy between secondary and tertiary learning/teaching communities (e.g. secondary mathematics teachers and university mathematics lecturers).

One study presented at INDRUM that begins to address (1) is by Petropoulou et al. (2016). Drawing on Jaworski’s (Potari & Jaworski, 2002) teaching triad, as well as a combination of data grounded theory (Charmaz, 2006) and activity theory (Leont’ev, 1978), the study investigates teaching practice, in particular in relation to goals for supporting students in transition. Analyses of lecture observations, field notes, post-lecture interviews and informal discussions with students reveal various forms that sensitivity to students, both cognitive and affective, can take, even in the context of modes of expository teaching that are largely seen as impersonal, such as a lecture to a large cohort.

There is hardly any work presented at INDRUM that accommodates (2): we note this gap and ask for more work with the learning needs of incoming students in mind, especially work that brings together teaching communities from across the secondary/tertiary divide. Where this may become the case (Winslow, Biehler, Ronning, Jaworski & Wawro, 2020) is within the rapidly changing landscape of university teachers’ professional development.

In addition to experiencing an often far from seamless transition from school to university mathematics, students are typically expected to navigate across mathematical domains, and across other disciplines, when at university. Their lecturers also navigate between mathematical research and teaching cohorts of students with diverse needs. We review studies of these navigations, for students and their lecturers, in the sections that follow.

*//. Within university*

Beyond their first year, students typically do not have to deal with radical changes in their living and learning environments. Furthermore, as they study material more closely linked to their career goals, they may be less likely to experience a lack of interest or anxiety. This is perhaps why the studies summarized in this section focus on cognitive and epistemological aspects of “within university” transitions. Two kinds of transitions were highlighted by INDRUM. First, the transitions between mathematics courses that occur in succession (i.e., one is the prerequisite of the other); in particular, courses within the Analysis Path (i.e., single variable Calculus, multivariable Calculus, Real Analysis, Metric Spaces, ...). Second, the transitions from foundational mathematics courses to specialist courses for disciplines that use mathematics; in particular, the discipline of engineering.

*From one course to the next in the Analysis Path*

Transitions can vary internationally due to the different kinds of educational institutions that exist within different countries. In North America, for example, a student’s first encounter with Calculus can occur at the end of secondary school; nevertheless, most university programmes in STEM-related disciplines require their students to take single variable Calculus courses (again) in their first year^{1}. The transition from Calculus to Analysis hence occurs “within university”.

Two INDRUM papers (Brandes & Hardy, 2018; Broley & Hardy, 2018) relate to this North American context. They both refer to a model by Winslow (2006), which uses the ATD’s notion of praxeology to formalize the observation that, in the Calculus to Analysis transition, students’ practices become less procedural and more theoretical. As depicted in Figure 10.1, students’ work in introductory Calculus is principally practical (represented by FI|): that is, students are required to master techniques for solving different types of tasks. In later courses, two transitions may occur. First, students may be expected to gain awareness of the theoretical block (AJ missing from their practical work (e.g. they not only learn how to calculate the derivative of sin(.v) + *x ^{2},* but also to describe and prove the differentiation rules that justify their calculations). Second, students are invited to engage in new types of tasks (FL) involving the more abstract objects from the theoretical block A] (i.e., they learn how to develop proofs in a more general sense).

Brandes and Hardy (2018) extend the body of literature on single variable Calculus by exploring the nature of students’ work in a subsequent course, in multivariable Calculus. For one such course, they studied curricular materials and final examinations to make praxeological models of (a) the knowledge to be taught and

**Transition 1**

**Transition 2**

**FIGURE 10.1 **Transitions in mathematics coursework (adapted from Winslow, 2006)

(b) the minimal core of the knowledge students must learn to receive a passing grade. They found that, in moving from (a) to (b), the practical blocks were reduced, and the theoretical blocks were transformed into superficial versions. In short, the students were not yet required to go through Transition 1 as depicted in Figure 10.1. Broley and Hardy (2018) concerns what happens in a later course, in Real Analysis, when students are typically invited to start working with and on mathematical theory. The researchers describe a plan to model the knowledge students are expected to learn in the course (according to assignments and past exams), as well as the knowledge some students actually learn (according to task-based interviews), in hopes of gaining insight into how students’ practices are evolving under the conditions of the course. They question when (if ever) students’ practices reflect the second stage in Figure 10.1 (i.e., (П|, Л|). One prediction is that students experience the “transition” from Calculus to Analysis more like a “jump” from the first stage (i.e.. Пі) to the third stage (i.e., (Пз, Аз)). Winslow (2016) investigates this in relation to angles and trigonometry. He notes that the textbooks used in Analysis courses have the potential to assist students in building the theoretical blocks missing from their previously developed practical blocks; however, an interview with a master’s student suggests that most students do not (and possibly cannot) engage in this building on their own (i.e., in the absence of supportive didactic activities).

Another ATD-inspired INDRUM paper, by Berge (2016), adds to the discussion of the importance of activities given to students in their independent work. While studying a sequence of courses at an Argentinian university (Courses I and II, which concern Calculus; and Courses III and IV, which concern Analysis), Berge (2008) noticed that Courses II, III and IV all included an axiomatic presentation of R and deduction of related theorems by the professor; but the tasks completed by students varied greatly. For instance, Course II required students to find suprema of concrete sets, while Course IV invited them to use the supremum as a tool in proof tasks. The results of a questionnaire led to expected results: the majority of students in Course II saw the supremum either as not being useful, or as an ordinary upper bound; by Course IV, the majority of students could name situations where the supremum functions as a tool. Based on a historical analysis, Berge (2016) suggests there may be a better approach for introducing students to the “raison d’etre” of completeness.

Hamza and O’Shea (2016) use McGowen and Tail’s (2010) notion of “met-before” and Fischbein’s (1989) concept of “tacit model” to theorize that a transition into any course involves meeting new knowledge based on previously developed mental images, whose influence (positive or negative) may or may not be known to the learner. Task-based interviews with ten students at the end of a Metric Spaces course in Ireland showed that the students held a variety of images of open sets. As theorized, some students seemed to have tacit models based on previous experience with R", which led them to visualize an open ball as a circle or a disc when reasoning about more general spaces. This may have helped students build intuition; but it may also cause difficulties. For instance, the students who routinely based their reasoning on the formal definition of open sets were moresuccessful in task solving. Echoing results about the Calculus to Analysis transition (e.g. Tall, 1992), it seems the transition from Real Analysis to Metric Spaces also requires significant cognitive shifts, including an increased level of abstraction and appreciation of the role of mathematical definitions. Hamza and O’Shea (2016) note that including a range of examples of metric spaces other than R", as was done by the lecturer in their study, is not sufficient for assisting students in making the transition.

*From introductory mathematics courses to engineering courses*

An engineering programme usually starts with foundational courses, which are meant to provide skills needed for later, more specialized courses. Foundational courses typically include mathematics taught by mathematics faculty to large groups of students studying in different programmes (e.g. engineering, computer science, chemistry, biology and physics). Such courses have been found to pose problems for engineering students (Ellis, Kelton & Rasmussen, 2014), leading some to abandon their programme, and others to experience a bumpy transition to the specialized engineering courses. A current interest of mathematics education research, as evidenced by INDRUM, is to identify what kinds of “bumps” may occur and why.

Gonzalez-Martin and Hernandes-Gomes (2018) and Gueudet and Quéré (2018) adopt an institutional perspective based on the ATD (Chevallard, 1999), which offers one explanation for the bumps. The institution of engineering studies operates according to different ecological conditions when compared to the institution of mathematics studies for mathematics majors; in the least, they correspond to two different Scholarly institutions - Scholarly Mathematics and Scholarly Engineering -where, according to the theory of institutional transposition (see, e.g., Castela, 2016), mathematical knowledge exists in different ways. Mathematics faculty are often educated in the mathematics studies and Scholarly Mathematics institutions, and have little to no knowledge of specialized engineering studies or Scholarly Engineering. When they are responsible for teaching the foundational mathematics courses for engineering students, Schmidt and Winslow (2018) suggest that a separation of external and internal didactic transposition (in the sense of Chevallard, 1991) may take place: the initial selection of the mathematical contents to be taught (external transposition) is based on the needs of engineering disciplines, but the actual teaching is carried out (internal transposition) according to the standards of teaching mathematics.

Gonzalez-Martin and Hernandes-Gomes (2018) study (dis)connections that may arise in relation to Calculus. Assuming that post-secondary instructors rely on textbooks to organize their teaching, the researchers performed a comparative analysis of how integrals are presented in the textbooks used in a first-year Calculus course and a second-year course (Strength and Materials) at a university in Brazil. Since many civil engineering notions are defined as integrals, programmes often dictate that engineers learn about them. However, Gonzalez-Martin and Hemandes-Gomes (2018) found that the knowledge learned in Calculus (e.g. techniques for calculating integrals), is not needed to solve the tasks in Strength and Materials. It is therefore not surprising that engineering students question the need to leant integrals in Calculus. The researchers emphasize the importance of mathematics lecturers becoming more aware of how the content they teach is used in professional courses.

In a similar vein, Gueudet and Quere (2018) compare the content of existing courses in mathematics and engineering; the difference is that they study online courses (in France), which may offer more flexibility in tailoring contents to the intended audience (i.e., future engineers). These researchers are also interested in connections in multiple senses, not only between mathematics and engineering, but also between mathematical concepts and semiotic registers. They focus on trigonometry since it is rich in such connections and is extensively used in engineering courses. An analysis of course materials showed that electrical engineering courses require students to master connections between concepts and registers (e.g. students must learn to associate an algebraic expression such as s(t) = *A^2* sin(<0f + ф) with a function, a graph and a vector representation). In comparison, the mathematics courses lacked connections, either between concepts and registers or between mathematics and engineering. The researchers conclude that the development of online bridging resources is an important aim for mathematics education research.

*///. From university to the workplace*

The INDRUM papers related to the third transition type focus mainly on one kind of workplace: where the expectation is to teach mathematics. This is perhaps not surprising given the focus of mathematics education research on the training of future primary and secondary teachers, as provided by universities. This said, the INDRUM community has not only continued to deepen the discussion about “old” issues (i.e., Klein’s “double discontinuity”); it has also contributed to the exploration of relatively “new” lines of research, such as the transitions faced by post-secondary teachers (e.g. graduate teaching assistants, college instructors and university professors).

*To primary or secondary school teaching*

Felix Klein (1908/1939) coined the term “double discontinuity” to represent the disconnections between what education students do at university and what they previously did (as a student) and eventually will do (as a teacher) in schools. Teacher education programmes are often criticized if they do not (a) offer future teachers adequate opportunities to deepen both their content knowledge and pedagogical content knowledge in relation to school mathematics (e.g. when future secondary mathematics teachers follow almost the same curriculum as mathematics majors); and (b) make strong enough connections to actual teaching practices (Gueudet et al., 2016). INDRUM contributions point to several approaches for bridging the gaps.

Biza and Nardi (2016) highlight an approach taken in education courses: inviting teacher students to engage with fictive, albeit realistic classroom situations. The researchers exemplify the potential of asking the students to compose their own situations based on their school placements. Their analysis of the situations and ensuing group discussions is structured around four characteristics -consistency (between beliefs and practice), specificity (to the situation), reification of pedagogical discourse and reification of mathematical discourse — which allow them to study how the content addressed in the education programme interacts with the teacher students’ early teaching experiences. Since at this stage in the transition, teacher students have one foot in both university and workplace, there is the possibility not only to analyse the transition process, but also to facilitate it (e.g. by immediately addressing any challenges faced and informing better design of future courses).

Two INDRUM papers focus on bridging gaps that might be experienced in advanced mathematics courses. Hanke and Schafer (2018) observed that the future teachers at their university often feel they just need to get through these courses and do not naturally gain the “higher standpoint” on school mathematics that is intended by curriculum developers. These researchers are therefore redesigning the courses “in the shape of a Y”: At the beginning, all students learn foundational content together, and about two-thirds through the semester, there is a split into two groups based on future professions (i.e., teachers and mathematicians). Mathematics students learn advanced mathematics topics, while teacher students identify notions relevant for secondary school mathematics and create related exploratory learning environments that they test with school pupils. Initial results in a Complex Analysis course suggest that the profession-specific part is more engaging and memorable for the teacher students, though more explicit guidance may be needed to help them connect their pedagogical knowledge (learned in their education courses) and the specialized content knowledge (learned in Complex Analysis).

Stender and Stuhlmann (2018) leverage a theoretical distinction between content and methods (i.e., the ways in which content is created or investigated) to argue that there is a more natural connection between university and school mathematics: if the content seems drastically different, the methods are the same. Moreover, teacher students should eventually be able to make the methods explicit to students when they are teaching. Stender and Stuhlmann (2018) focus on a subset of methods called “heuristic strategies”, which are general problem-solving tactics: e.g. understand the problem, change the representation, consider special cases, break the problem into sub-problems and so on (the paper has many more examples). The researchers are designing voluntary seminars (an alternative to the redesign discussed above), which aim to assist teacher students in understanding lecture material (in Linear Algebra and Analysis) and introduce heuristic strategies relevant to that material. Interviews with five students showed some positive results, though it was unclear if the students had learned to explain their strategies to others. While implementing heuristic strategies in lectures is a goal, Stender and Stuhlmann (2018) mention the potential challenge of convincing lecturers to change their habits.

*To post-secondary teaching*

The separation of teacher students and mathematics majors in the approaches outlined above makes sense: teacher education programmes aim to educate future teachers, while mathematics programmes aim principally to educate future mathematicians. Yet, a large portion of mathematics majors will eventually become teachers at the post-secondary level. One INDRUM paper, by Mathieu-Soucy, Corriveau and Hardy (2018), was driven by the lack of research on this transition and the challenge of transferring results about schoolteachers to the post-secondary level. The researchers frame the transition in terms of changes in a person’s relationship with (i.e., visions, opinions, beliefs, attitudes and feelings about) mathematics and its teaching and learning (RWMTL). Interviews with three new post-secondary^{2} teachers throughout a semester enabled an identification of pertinent themes. Surprisingly, the teachers did not spontaneously talk about mathematics when asked about their daily life; rather, they spoke of more general issues related to teaching (e.g. classroom methods, assessment and student expectations). Moreover, the teachers’ reflections on recent events were not mature enough to have led to significant changes in their RWMTL. The researchers conclude that a different methodological approach is needed if one wishes to document changes in RWMTL. They conjecture, for instance, that inviting new teachers to share significant past experiences in relation to specific themes would lead them to reveal deeper reflections that give more insight into the factors shaping their RWMTL as they transition into their new roles.

Other INDRUM contributions follow the trend of designing and evaluating attempts at easing the transition. Florensa et al. (2018) discuss courses for new (and in-service) university lecturers. At the beginning of four courses, they collected 143 questions that participants indicated they would have (or have had) during their teaching. Like Mathieu-Soucy et al. (2018), they found that the lecturers were concerned mostly with general issues. Florensa et al. (2018) offer possible explanations; notably, the phenomenon of “pedagogical generalisin’’, which separates instructional processes from their content and leads lecturers to accept without question the content they teach. This can be perpetuated by teaching support programmes based solely on pedagogy. Florensa et al. (2018) propose that such programmes introduce certain didactic theories (e.g. from the ATD) to help participants formulate more pertinent questions about their teaching. How such theories are received and used by those who are principally mathematically trained seems to remain an open question.

Huget (2018) reminds us that the transition to post-secondary teaching can start early, when graduate students are asked to engage in teaching-related work. There is increasing interest in the training offered to these students since they not only tend to lack knowledge of teaching, but they may also play a significant role in shaping undergraduate mathematics education, both immediately (for the students in their courses) and in the long run (since they are potential future professors) (Speer, Gutmann & Murphy, 2005). Huget (2018) describes one program for new tutors involving three mandatory modules throughout a term (“introduction to tutoring”, “interaction and activation” and “problem solving”) and the opportunity to be observed and receive feedback. The program’s effects on both tutors and students taking tutorials were under evaluation at the time of INDRUM 2018.

INDRUM has engaged strongly with research about the transition from being a student to teaching mathematics at any level of education. Nonetheless, there are a few papers that touch on the transition from university to other professions. Broley (2016) presents a study of how 14 research mathematicians use computer programming in their research and teaching, and describes the potential gaps between how university mathematics is traditionally taught and the increasingly computational nature of some mathematical research practices. Schmidt and Winslow (2018) discuss efforts to bridge gaps between engineering students’ training (in mathematics) and their future work (as engineers). Their novel approach includes creating authentic engineering problems and integrating them in early mathematics courses. We now conclude with mentioning populations of students (and their transitions to work) that have not yet been significantly represented in INDRUM but are starting to (e.g. economics students) in work presented at a more recent conference on *Calculus in upper secondai-)' and beginning university mathematics* (e.g. Monaghan, Nardi & Dreyfus, 2019).

**Summary, conclusion, ways forward**

Our overview of the contributions about transition issues demonstrates that this is a vibrant and broadening research area that was well represented at the INDRUM conferences. The internationality of the contributions especially points to the importance of describing the contexts within which transitions take place, since educational institutions vary from country to country. There are still areas that attract little research, even though their importance is recognized in our community.

Research on transition issues within and across university, as well as from university to the workplace is rather new. It is therefore not surprising that there are many unexplored areas. While some contributions go beyond Calculus and Real Analysis, there is still a lack of research on advanced mathematics (beyond Year 1 or 2 of university studies). There are other paths of courses (e.g. in Algebra or Statistics) and other horizontal transitions (i.e., between courses that occur concurrently) that could be considered in future research and at INDRUM conferences. There is also less consideration of the use of mathematics within fields (e.g. psychology, economics, computer science) outside of engineering, all of which could refer important insights back to mathematics service courses. Lastly, there are other populations of students missing from the studies reported at INDRUM: for instance, social sciences and humanities students who are required to take mathematics or statistics courses at university. One might question, for example, how such courses relate to the workforce such students will enter and the mathematical skills (e.g. in numeracy, computational thinking, logic, statistical thinking) that would benefit them most.

For each topic noted, numerous pertinent research questions can be formulated from the point of view of the various theoretical approaches we discussed. For example, from an ATI) perspective, the transition model depicted in Figure 10.1 and related issues have mainly been investigated with respect to Calculus and Analysis. But what is the situation for other fields? Or: What does the observed jump from the first to the third stage of the model mean in a longer-term perspective, e.g. taking into account what students (are expected to) leam in more advanced mathematics studies? From a commognitive perspective, transitions are currently studied mostly in terms of changes in meta-discursive rules, changes in word use and emerging commognitive conflicts in the interactions between students and teachers.

There is ample space for data collection and analyses of a different grain size. For example, studies can examine how changes in the curriculum, and in other institutional factors, play out in the way students’ discursive activity changes from one educational level to another. Or, they can examine how such changes may play out in different courses as well as in the transition from the world of study to the world of work.

Several papers highlight the need for constructing and implementing interventions that assist students in overcoming difficulties, developing certain kinds of understandings or knowledge, and, ultimately, making smooth transitions. This has already been realised in many countries for the transition from school to university, which has been researched for a long time and could be seen as a classical research topic in university mathematics education. However, there is still not so much research that evaluates implemented interventions or discusses sustainability issues. Regarding all transition types, there is a need to consider more strongly the impact of socially institutionalized contexts (including, e.g., the number of students in courses, exam requirements, or the general role of education in society).

Many of the measures for supporting transitions have appeared in the form of “add-ons” to existing systems (e.g. additional courses or seminars). We note the importance of complementing such efforts with reflections on, and attempts at, deeper systemic changes. This could require more collaboration between communities of teachers and professionals from across transition divides (e.g. school teachers and university lecturers; lecturers of mathematics and specialists in other disciplines; university lecturers and professionals from the workplaces where mathematics is put to use). The INDRUM papers have just begun to hint at both the challenges and possibilities in this regard. University teachers’ training and professional development - discussed in chapter 4 of this book (Winslow et al., 2020) — may be one place where these much-needed synergies can materialise.

**Notes**

- 1 For a description of such courses: https://www.niaa.org/sites/default/files/pdf/cspcc/ InsightsandReconinicndations.pdf
- 2 The teaching context is described thoroughly in the paper. It is within college institutions (so-called CEGEPs) that serve as intermediary' education between secondary schools and universities in the province of Quebec, Canada.

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