Alejandro S González-Martin
email: gonzalez-martin@umontreal.ca

Irene Biza
email: i.biza@uea.ac.uk

Jason Cooper
email: jason.cooper@weizmann.ac.il

Imène Ghedamsi
email: ighedamsi@yahoo.fr

Ghislaine Gueudet
email: ghislaine.gueudet@espe-bretagne.fr

Vilma Mesa
email: vmesa@umich.edu

Alon Pinto
email: alon.pinto@weizmann.ac.il

Olov Viirman
email: olov.viirman@hig.se

Alejandro S González-Martín

Keywords: University mathematics education, transition to and from university, resources in university mathematics, mathematics in non-mathematics disciplines, university

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction to the papers of TWG14: University Mathematics Education Introduction

The TWG14, University Mathematics Education (UME), was launched in CERME7 [START_REF] Nardi | University Mathematics Education[END_REF] showcasing the fast growth of research in UME and also the specificity of the research context. In particular, the abstract, formal nature of a significant portion of the mathematical content; the absence of national curriculum guidelines, and therefore the great variations in organization and practices across institutions; the general lack of professional development or of systematic preparation for teaching; and, the volume of content to learn in a short period of time, and the degree of autonomy expected from the students and faculty. The consolidation of this research area, both outside and inside CERME, was recognized by the ERME community in inviting the first, three-time leader of this TWG to present a summary of UME research as a CERME10 plenary lecture [START_REF] Nardi | From advanced mathematical thinking to university mathematics education: a story of emancipation and enrichment[END_REF] (Durand-Guerrier, Hochmuth, Nardi, & Winsløw, in preparation).

The number of papers submitted to TWG14 has been increasing since its inception. This year, we received 52 paper and 13 poster submissions, with 35 papers and 15 posters presented at the conference and published in the proceedings. Having received the largest number of papers in CERME10 and the second largest in CERME11, the number of papers led to the decision to split TWG14 into two isomorphic groups, TWG14A (18 accepted papers) and TWG14B (17 accepted papers) that ran in parallel; we held some common sessions during the conference. Each of the groups covered the same range of areas of research pertaining to UME, therefore the isomorphic label. The decision was prompted because splitting thematically would have resulted in losing perspective on the complexity of the issues that participants wanted to address. Furthermore, because the call for papers was open to a range of themes it would be unfair to assign contributions to a new group that might not align with the authors' original aim. This introductory paper summarizes the works presented in both groups organized according to the discussed themes, as well as the common discussions.

Some themes continue work from CERME10. For instance, we received a large number of papers focusing on students' learning of specific topics (10 papers), such as calculus, reasoning, and proof. We accepted seven papers proposing interventions, whereas in CERME10 we received five papers; however, relative to CERME10, the number of papers about teachers and their practices and about mathematics for non-specialists decreased from ten to four and from seven to two, respectively. We had two new themes: one on curriculum and resources and one on transition. In the next section, we briefly present the seven themes used to organize the work this year, with examples from paper contributions. It is important to note that many papers could fit in more than just one theme; this classification helped us to structure the presentations and the discussions during the sessions.

Themes and paper contributions

Students' learning of specific topics Some of the papers studied the relation of university mathematics instructors with resources. For instance, Viirman and Jacobsson put forward the role of the mathematical content in the didactical choices made by university mathematics lecturers in course design; they highlighted how differences in the epistemological character of the topics to be taught may have affected the course design. Similarly, Sabra showed how different teachers with different training may focus on different aspects of the content to teach using some resources.

Finally, Bosch et al. presented a study about the didactic transposition processes that lead to the creation of undergraduate mathematics programmes, which fills a gap on the rationale behind the creation of such programmes. Overall, the papers discussed in this theme show how the use of resources is related to learning and teaching practices, a question that was discussed in the other themes as well.

Transition

Four papers were discussed in this theme: Bampili, Zachariades, & Sakonidis; Bergsten & Jablonka; Doukhan & Gueudet; and, Pinkernell.

Different aspects of transition were present in these papers. For instance, Doukhan and Gueudet studied praxeologies in pre-university and university courses regarding the notion of random variable; they showed that the content learned in secondary school may not prepare students for tertiary courses. Regarding institutional support, Bampili et al. showed how institutional support (or the lack of it) at university can influence the trajectories of novice students. Furthermore, Pinkernell reported on a frame (WiGORA), founded on five characteristics of understanding mathematics, for supporting students' transition towards university mathematics.

Transition towards and within university courses has been investigated in the last years through several lenses. Bergsten and Jablonka presented a review of literature concerning the secondarytertiary transition, referring to five theories. Their review evidenced that, while there is a move towards socio-cultural theorizing, some dimensions pertaining to social issues still appear underresearched.

Literature on transition addresses, in many cases, students' challenges. This calls for intervention, and some specific examples were discussed in the following theme. showing how the use of some didactic constructs may be useful for the mathematician to organize content and to reflect on his practice.

Interventions

Use of mathematics by non-mathematicians

Two papers were discussed in this theme: González-Martín & Hernandes-Gomes and Tetaj & Viirman.

Artigue, Batanero, and Kent (2007) stated ten years ago that the study of issues related to mathematics as a service course would certainly need the use of sociocultural approaches. This is the case of the two papers discussed in this theme. For instance, González-Martín and Hernandes-Gomes studied the teaching of mathematics to future engineers with an institutional perspective based on ATD. Using textbook analysis, they compared how the integral is taught in a calculus course and in a mechanics of solids course for engineers. The authors identified a rupture in the study of a similar task (the sketching of the graph of an antiderivative), likely to prevent students from using their calculus knowledge in professional courses. Tetaj and Viirman used the commognitive approach to analyze the mathematical discourse of undergraduate biology students.

Their results showed students' tendency to develop a ritualized approach, as well as their difficulties to mathematize a biological phenomenon; their lack of familiarity with relevant construction routines may prevent them from developing abilities to deal with certain problems.

Transversal issues addressed in plenary discussions

Resources in university mathematics education

Research on resources and their use in UME is growing, with works investigating the use of textbooks (e.g., González-Martín, [START_REF] González-Martín | From resource to document: Scaffolding content and organising student learning in teachers' documentation work on the teaching of series[END_REF][START_REF] Mesa | Textbook mediation of teaching: An example from tertiary mathematics instructors[END_REF] and technology (e.g., [START_REF] Lavicza | Integrating technology into mathematics teaching at the university level, ZDM[END_REF]. Some authors consider a general concept of resource, as anything likely to re-source the activity of a teacher (e.g., [START_REF] Gueudet | University teachers' resources systems and documents[END_REF] or a student (e.g., [START_REF] Gueudet | Didactic contract at university: a focus on resources and their use[END_REF]. Since CERME10, TWG22 has been dedicated to studies related to the use of resources in mathematics education, but, in TWG14, some papers explicitly or implicitly addressed resource use and analysis of resources. For instance, the design of interventions is linked to the design of specific resources and the teaching of mathematics for non-specialists uses particular resources. The discussions on this topic in TWG14 raised several questions. Choosing an appropriate definition of resource is a first complex step. Should this definition encompass mathematical knowledge, professional knowledge (in the case of teachers), or affect? Different definitions may be needed depending on the question studied. The question of the use of resources for the research activities of the lecturers, as well as lecturers' interactions (or absence of interactions) with resources for teaching also requires further study.

University mathematics: teachers and teaching

Relative to school settings, research on teachers and teaching is less common in UME (e.g., [START_REF] Speer | Collegiate mathematics teaching: An unexamined practice[END_REF]. In contrast, CERME has three groups dedicated to teachers and teaching (TWG18, TWG19, and TWG20), four if one counts assessment (TWG21). The growing interest of UME in this area is evidenced in our TWG14, which discussed some of the challenges of working at the intersection of university mathematics teaching and research in mathematics education (RME). To begin with, the conception of what it means to know mathematics at university may be quite different from conceptions that are common in pre-tertiary RME. As a result, the models of university-level teaching may not be consonant with theoretical models and frameworks that typically guide RME. This state of affairs can be particularly challenging for researchers who set out to investigate their own mathematics lecturing. Another type of challenge has to do with relationships between the two communities involved: university lecturers do not always value RME as a scientific field, while educational researchers are often critical of lecturers' approach to teaching. Thus, to support research-based professional development for university lecturers there is a need for teacher educators with expertise in both content and teaching, credentials that may be scarce. These challenges and others were addressed in TWG14 contributions, as summarized in the Teachers and Teaching section. Finally, many research works involving the study of practices (e.g., teaching, resource use, planning, and assessment) of mathematics lecturers and also mathematicians call for the development of fruitful collaborations.

Collaboration with mathematics lecturers, mathematicians, and other professionals in research and development in UME

Collaboration and relationships between mathematics lecturers1 (ML) and mathematics education researchers have been addressed in UME research studies (e.g., [START_REF] Nardi | Among mathematicians. Teaching and learning mathematics at university level[END_REF]. This collaboration can be seen from (at least) two different perspectives: (1) through the roles of each group in such collaboration (e.g., UME researchers researching the practice of ML; ML researching their own practice in collaboration with UME researchers; or UME researchers and ML collaborating in UME research and development); and (2) through the contribution of research in UME to developmental projects in industry and practice (e.g., what is the 'utility' of research to practice?) and to mathematics teaching and curriculum development (e.g., how a research finding is transformed in order to be used as a curricular 'product'?). Collaborations and tensions in such collaborations were addressed in contributions to this conference as we have exemplified in the themes above. The discussion on this topic had two directions: (a) theoretical and methodological tools that will help to study such collaborations and make them fruitful; and (b) the role and impact of theory in such collaborations, from the perspective of both UME researchers and ML. Indicative points raised in the discussion concerned the observation that collaborations engage communities with different practices, as a result we should work on the: need of theoretical tools to study collaborations; need for tools to study boundary crossing, even though the boundaries between the communities are not clear (e.g., ML may also be UME researchers); need of ways to identify the goals and expectations of different professionals (e.g., mathematicians, chemists, biologists, engineers, etc.) who are involved in such collaborations; and, a need for better understanding of institutional factors (e.g., institutions may or may not facilitate such collaborations).

Reflection and ways forward

We initiated the discussion in the concluding session of our work at the conference summarizing some of the main points of the chapter Research on university mathematics education [START_REF] Winsløw | Research on university mathematics education[END_REF], published in the book celebrating twenty years of CERME. In our discussion in the concluding session we reflected on these areas as well as on the studies we discussed at the sessions and we suggested the following ways forward: 1) because many areas and topics of UME require the establishment of varied collaborations (with ML, mathematicians, engineers, school mathematics researchers) we need to develop theoretical tools to study these collaborations; 2) some papers presented an articulation or networking of different perspectives which could be a way to study complex phenomena (there were papers using "external" theories, e.g., sociology); 3) large-scale studies are needed to consolidate some results; 4) innovative methodologies are also being developed, such as those to study the use of digital textbooks; 5) most of the current studies in UME are local, and therefore, we need to develop studies and tools that help to transfer findings to other educational contexts; and 6) new topics appeared in this conference: the use of history of mathematics, curriculum design and its origin, the role of course coordinators. We predict more new topics will appear in the years to come. We anticipate that the coming CERME conferences will allow us to pursue research on the areas and questions discussed this year and to bring research on some of these areas forward.

  . Moreover, an ERME Topic Conference, International Network for Didactic Research in University Mathematics (INDRUM) was launched in 2015 with two successful conferences (INDRUM2016 and INDRUM2018) since then, with the first one leading to a special issue in the International Journal of Research in Undergraduate Mathematics Education, the new journal in the UME field. Future developments include a forthcoming book on Research and development in University Mathematics Education

  study of mathematics. They found that students' type and use of resources varied in relation to the type of course and the lecturer's expressed expectations. In relation to assessment, Howard et al. discussed student engagement with resources and continuous assessment in a mathematics course for business students.

	Resources and curriculum
	Five papers were discussed in this theme: Bosch, Hausberger, Hochmuth, & Winsløw; Howard,
	Meehan, & Parnell; Pepin & Kock; Sabra; and, Viirman & Jacobsson.
	From a student perspective, but also in relation to lecturers' intentions, Pepin and Kock explored
	what type of resources first year engineering students use and how they use/orchestrate these
	resources for their
	Ten papers were classified under this theme: Aaten, Roorda, Deprez, & Goedhard; Gabel &
	Dreyfus; Herrera Alva, Figueroa, & Aguirre de la Luz; Kondratieva; Kontorovich; Lankeit &
	Biehler; Malaspina & Torres; Bašić & Milin Šipuš; Mkhatshwa; and, Pinto & Cooper.
	Calculus was prominent in several papers (Aaten et al.; Herrera Alva et al.; Lankeit & Biehler;
	Malaspina & Torres; Mkhatshwa), with an increasing interest in two-variable calculus (Malaspina
	& Torres; Mkhatshwa) and analysis (Kondratieva). Under-researched topics were also present:
	differential geometry and its connections with two-variable calculus (Bašić & Milin Šipuš) and the
	use of conventions (Kontorovich). We also received papers addressing reasoning and proof (Gabel
	& Dreyfus; Pinto & Cooper).
	Some of the papers discussed in this theme also concerned questions related with teachers and
	teaching. For instance, an intervention to improve in-service teachers' comprehension of
	discontinuity in one or two variables was also presented (Malaspina & Torres), as well as studies
	about the challenges in providing feedback that is actionable by students when revising a proof
	(Pinto & Cooper) or when teaching proofs (Gabel & Dreyfus). These last two studies raised
	questions in relation to the teaching of, and communication about, proof relevant to other themes
	we discuss below.

  Seven papers were discussed in this theme: Feudel & Dietz; Florensa,Barquero, Bosch, & Gascón; Ghedamsi & Lecorre; Hochmuth, Schaub, Seifert, Bruder, & Biehler; Kuklinski et al.; Thomas, Jaworski, Hewitt, Vlaseros, & Anastasakis; and, Uysal & Clark. A relatively recent approach to constructing interventions-framed by the Anthropological Theory of the Didactic (ATD)-is the study and research paths (SRPs); SRP is a specific form of inquirybased intervention. Florensa et al. discussed ten years of SRPs implemented at university level, showing their utility as a methodological tool for the systematic design of interventions. Other interventions included the implementation of a coaching program to support conceptual learning in a first-year mathematics course(Feudel & Dietz) or innovative lectures that foster more social interactions and student engagement than traditional ones(Kuklinski et al.). These studies investigated the impact of the interventions on student engagement. Finally, Uysal and Clark reported on the use of history in learning mathematics and its potential to support mathematical and emotional aspects of students' transition from high school to university.On a more theoretical level, Ghedamsi and Lecorre compared and contrasted the Theory of Didactical Situations (TSD) and ATD with the aim of combining them in the design of a teaching intervention on real numbers and sequences. Many papers presented in this theme were related to other themes. Hochmuth et al. presented the structure of a diagnostic and evaluative test that helps students decide about what bridging courses are more appropriate for them; this paper has connections to the transition theme as well. Thomas et al. presented a project involving a bridging module, and showed that student-partners in task design furthered their own understanding of the topics (complex numbers) and developed didactical insights when designing mathematical tasks for others; this paper has connections to the transversal issue of collaborations in UME.Students' experiences were seen by Thoma and Nardi in relation to their participation in the mathematical discourse. They used the theory of commognition to study students' substantiation of claims in exam scripts, in a first-year course called "Sets, Numbers and Theory." They demonstrated that students' participation in mathematical discourse, at least in the exam, was mostly ritualized. On the other hand, Toor et al. studied the experiences of post-secondary mathematics students learning to use programming as a computational thinking instrument for mathematics. They argued that identity is essential to the development of productive dispositions in learning to program for mathematics investigation and modeling. The notion of identity was central in the work of Voigt et al. as well. Adopting a sociocultural perspective and the notion of figured world, they showed how three different variations in a calculus course can have an impact on a student's trajectory and relationship with mathematics. Four papers were discussed in this theme: Costa Neto, Giraldo, & Nardi; Mali, Cawley, Duranczyk, Mesa, Ström, & Watkins; Mesa, Duranczyk, Watkins, & AI@CC Research group; and, Stewart, Epstein, Troup, & McKnight. The involvement of and the tension between different communities in relation to the teaching of mathematics was addressed by Costa Neto et al. Using a re-storying methodology, their study examined disputes between mathematicians and mathematics educators in Brazil over the curriculum of a pre-service teacher education program. Teaching at the post-secondary level is the focus of Mesa et al. who describe the development of an instrument for the evaluation of community college algebra instruction, Evaluating the Quality of Instruction in Postsecondary Mathematics (EQIPM), through the analysis of video recorded data that include codes for studentcontent, instructor-content and instructor-student interactions. Specifically, Mali et al presented two of the twelve characteristics of instruction included in EQIPM: Instructors Making Sense of Procedures, and Student Mathematical Reasoning and Sense Making. Although these characteristics can contribute to the quality of instruction, the authors showed that exemplary instances of instruction with these characteristics were relatively rare. Finally, Stewart et al. addressed the work of a mathematician preparing his course on eigenvalues and eigenvectors,

	Teachers and teaching

Students' identity and experience

Three papers were discussed in this theme:

Thoma & Nardi; Toor, Mgombelo, & Buteau; and, Voigt, Rasmussen, & Martínez.

  [START_REF] Winsløw | Research on university mathematics education[END_REF] synthetize the main contributions of CERME to research in UME in terms of:(a) what is it?, namely research into current practices of UME (with no direct intervention), such as: mathematical content; methods and resources; transition phenomena; student experiences; and, teaching non-mathematics specialists; and (b) what could it be?, namely developmental or experimental research, that includes an intervention design as part of the research project (e.g., research on, and for innovation in UME; i.e., interventions in specific courses or programmes) and professionalization of UME practice (preparation of ML).

We use "mathematics lecturer" or "mathematics teacher" to refer to a person who teaches mathematics at university. This person may be active in research in mathematics or not.


Aaltje Berendina Aaten
email: aldine.aaten@kuleuven.be

Gerrit Roorda
email: g.roorda@rug.nl

Johan Deprez
email: johan.deprez@kuleuven.be

Martin Goedhart
email: m.j.goedhart@rug.nl

How

How Lisa's mathematical reasoning evolved at undergraduate level -on the role of metacognition and mathematical foundation

Keywords: Mathematical reasoning, undergraduate students, calculus, metacognition

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Since [START_REF] Lithner | Students' mathematical reasoning in university textbook exercises[END_REF][START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] introduced his theoretical framework on mathematical reasoning, many studies have used his distinction between creative mathematically founded reasoning and imitative reasoning. [START_REF] Lithner | Students' mathematical reasoning in university textbook exercises[END_REF] showed how undergraduates' reasoning is often not mathematically founded, but instead relies on imitation of exemplary or recalled solution strategies. [START_REF] Jonsson | Learning mathematics through algorithmic and creative reasoning[END_REF] have investigated learning effects of tasks requiring creative mathematically founded reasoning and found that it enables students to solve the same task at a later point in time. For moderately non-routine tasks however, undergraduate mathematics students have difficulties to think of suitable strategies [START_REF] Selden | Why can't calculus students access their knowledge to solve non-routine problems?[END_REF]. To gain more insight into undergraduates' mathematical reasoning, also on moderately non-routine tasks, we investigated its development at undergraduate level at two research universities and used Lithner's ideas concerning creative mathematically founded reasoning as a starting point.

Framework to analyze mathematical reasoning

For our research, we used [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] concepts in a different manner than in the original framework, because of its limitations for characterizing any reasoning observed in students' task solving [START_REF] Aaten | Undergraduates reasoning' while solving integration tasks: discussion of a research framework[END_REF][START_REF] Mac An Bhaird | A study of creative reasoning opportunities in assessments in undergraduate calculus courses[END_REF]. From the single category of creative mathematically founded reasoning, we disentangled two dimensions: reasoning foundation and novelty. Furthermore, we added a third dimension: metacognition level. Building upon [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF], we define mathematically founded reasoning as reasoning in which decisions are based on mathematical properties that are relevant to the decision. Superficially founded reasoning is reasoning that is not mathematically founded. Lithner (2008, p. 266) defined novel reasoning in situations when "a new (to the reasoner) reasoning sequence is created, or a forgotten one is re-created". Since we encountered difficulties in discerning whether a reasoning sequence is new or not, we choose to see novel reasoning as the opposite of recall, where recall means that a strategy is selected because the strategy was used before to solve similar tasks.

Metacognition levels

Since metacognition plays a central role in learning mathematics (Desoete & Veenman, 2006;[START_REF] Schneider | Metacognition and mathematics education[END_REF] it is important to include this aspect into our analysis. Several studies indicate that undergraduates' use of metacognition is limited [START_REF] De Backer | Exploring the potential impact of reciprocal peer tutoring on higher education students' metacognitive knowledge and regulation[END_REF][START_REF] Radmehr | Exploring students' mathematical performance, metacognitive experiences and skills in relation to fundamental theorem of calculus[END_REF]. However, not much research has been done on the evolution of metacognition during undergraduate mathematics courses. Since [START_REF] Flavell | Metacognition and cognitive monitoring: A new area of cognitivedevelopmental inquiry[END_REF] introduced the notion of metacognition, various conceptions have been developed. In this study we build upon research that distinguishes levels of metacognition explicitly, or implicitly by discerning types of metacognition or metacognitive behavior. [START_REF] Wilson | Towards the modelling of mathematical metacognition[END_REF] consider three aspects of metacognition: awareness, evaluation, and regulation, where awareness concerns the connection between the task at hand and past experience, evaluation is about the task solving actions just undertaken, and regulation anticipates specific task solving actions. In analyzing students' solutions, they discern regulation that builds upon awareness (e.g. retrieval of procedures) from regulation that builds upon evaluation of awareness, for instance, regulation as a "consequence of the need for a decision [...] as to how best to proceed" (Wilson & Clarke, 2004, p. 38). This distinction is similar to [START_REF] Koichu | On the relationships between (relatively) advanced mathematical knowledge and (relatively) advanced problem-solving behaviours[END_REF] distinction between circular heuristic behavior and spiral heuristic behavior. The former concerns employing strategies without building upon the intermediary outcomes, while the latter does build upon intermediary outcomes. [START_REF] Nunokawa | Mathematical problem solving and learning mathematics: What we expect students to obtain[END_REF] similarly distinguishes initial problem analysis and knowledge application from problem solving by building upon obtained information, e.g. transforming the problem or finding new perspectives.

Based upon these models, we define two metacognition levels. High-level metacognition builds upon evaluation of components of the task solving situation. By task-solving components we mean the task itself, strategies that are used or considered, intermediary outcomes of strategies, and subgoals formulated during task-solving. Low-level metacognition lacks such evaluation, for instance, if a student takes the decision to use a different strategy without reflecting upon what could be learned from the steps taken so far.

Research question

In the light of the above, we aim to answer the question: in what way do undergraduates' reasoning characteristics evolve over the course of bachelor studies in mathematics and what is the role of mathematical foundations and metacognition? In our larger study on undergraduates' mathematical reasoning when solving calculus tasks, we noticed how some students' reasoning barely evolved over the years, while the reasoning of other students did develop, even though the courses taken by the students, and the way the courses were taught (lectures and tutorials, a little project work), were similar. To answer the research question of this study, we focus on a student whose reasoning did evolve over the years: Lisa 1 , who started off unsuccessfully but improved over the years is a bachelor student in mathematics but has taken courses in econometrics before. In the first year of her mathematics studies, Lisa's average exam scores were low but sufficient to pass most exams.

Method

To investigate students' reasoning, the first author administered task-based think-aloud interviews in three different years. In each interview, Lisa solved three tasks, see Table 1. In post-interviews after each task, she explained her solutions and answered questions such as: 'How did you come to think of using this strategy?', 'How certain were you that this strategy would help you solve the task, and why?', 'Have you seen this type of task before?'. The interviews were video and audio recorded and transcribed.

The tasks used are indefinite integrals that can be solved with methods from the first year course on integral calculus. We chose these tasks, because students had learned a large number of integration strategies they could choose from, and because we expected that these specific tasks would be difficult to many students. This would give rise to longer and more elaborate solutions compared to solutions of routine tasks, therewith giving more insight into students' reasoning. To be able to compare reasoning over the years, some tasks have been used each year. In order to be able to correct for a possible test-retest effect, we alternately used task A and B. Although Lisa was given task A both in year 1 and in year 3, Lisa did not seem to recognize the task.

Analysis

To analyze the interview transcripts, we used the coding scheme in Table 2. For each dimension, one code is given per task solution as a whole, based upon Lisa's on-task utterances, and upon explanations and answers during the post-interview. We made use of Lisa's own analysis of her reasoning (e.g., that she recalled a strategy) and of our impression of her reasoning inferred from Lisa's statements and actions. These various angles served to complement each other and where possible to triangulate our findings. This analysis led to nine codes for each dimension, which together give the global picture of Lisa's reasoning evolution, or more specifically, the evolution of each of the dimensions.

To determine one code per dimension per task, we chose to focus on the phase of strategy selection, since that phase is especially essential in solving difficult tasks [START_REF] Selden | Why can't calculus students access their knowledge to solve non-routine problems?[END_REF]. This implied in general that the initial orientation phase is left out of the analysis, just like the execution phase. While one solution often contains various strategy selection phases, we focus on the phases in which the solution progressed, or was close to making progress. Considering these progressing task selection phases, we determined what type of reasoning was prominent.

When coding the reasoning characteristics, we deliberately ignored mathematical correctness of calculations and arguments, but we did evaluate the success of the solution as a separate category. This code is given somewhat indulgently: if a (lengthy) task was solved correctly almost completely, and only a minor error or lack of time appears to have been the cause of not finding the correct answer, the solution is coded as successful.

Codes are visualized in the scheme in Figure 1. Coding of the transcripts using the framework was executed by two of the authors independently. Discussion until agreement led to minor adjustments of the coding only. In the results section we include fragments from the interview transcripts to illustrate our coding.

Results

We first describe how Lisa's reasoning on task A evolved over time. Due to space limitations, Lisa's reasoning on the other tasks is not discussed in detail, but the codes are included in the overview.

Lisa's reasoning on task A:

We briefly describe and illustrate Lisa's solutions. In year 1, Lisa used only one strategy to solve this task: she calculated the derivative of the integrand, then considered what adjustments should be made to the integrand to obtain the anti-derivative. We call this strategy derive-and-correct.

Lisa:

Eh, what sometimes helps is taking the derivative of that part. I am not sure whether that will help this time. Sometimes I see a pattern that I can apply to the integral. I will try that now.

[…]

She made an error here, in applying the quotient rule, but did not notice.

Lisa:

Well, the only similarity I see is that both have a fraction , but this is twice

with exemplary coding) and legend

There was no progress in solution, so we considered all strategy selection phases. Lisa did not explicate relevant mathematically founded reasons for her strategic decisions: her reasons are non-mathematical ("I can always try") or merely superficially related to the task ("this is twice that fraction" to decide to take the square root). This latter quote illustrates how Lisa, although superficially and incorrectly, did evaluate intermediary outcomes for how to proceed, which is coded as high-level metacognition. Lisa used only one strategy, which she also used when solving the other tasks in the interview. She chose it because it 'sometimes helps', which we coded as recall. See figure 2.

In year 3, Lisa employed various strategies to solve task A: Lisa:

Eh, I try to solve this integral by substitution. Let's see whether that works. I take , so this is equal to . Ah wait, this is getting complicated. […] Ehm, if I take , then => . Ehm. This is not correct.

Then Lisa attempted substituting but stopped. She then decided to derive the integrand.

Lisa: I don't see it. Eh. I'll check what the derivative becomes. Perhaps that will help.

.

(silence) […]

If I take the derivative of this, , then I have , so that is not possible either. Hmm.

Next, Lisa modified the integrand.

Lisa: And… . If I take this , take , .

Ehm. Then it becomes . Does this help?

After some more thinking, Lisa thought of partial fractions, and solved the task correctly.

The episode in which the integrand is modified after which substitution is implemented, is the progressing phase in the solution. Here, the task is manipulated in such a way that the task is simplified. In the post-interview, Lisa explained how she came to think of this: In this last phase, Lisa evaluated how she could use the intermediary outcome obtained by deriving the integrand and combined it with the requirements of the previously attempted substitution , which indicates high-level metacognition. She did give relevant mathematical foundations for these strategic choices: "if you have an extra , then you have and that can be replaced by ". Lisa did not make any statements that indicate she recalled the strategy to use for this type of task, but instead she considered various substitutions and task modifications, which indicates novel reasoning. The coding is visualized in Figure 3.

Lisa's reasoning evolution

We analyzed Lisa's reasoning on nine tasks over three years. The codes for each dimension for each task in each year are put together in Figure 4 and summed in Figure 5. We summarize the coding below and frame them based upon observations in the data. Since Lisa was given task B only once, her reasoning on task B is excluded from the analysis of reasoning evolution. However, since task A and B require the same integration strategies, we did consider them sufficiently comparable for the global analysis of Lisa's reasoning evolution.

On task A, we see a change from superficially founded reasoning to mathematically founded reasoning, and from recall to novel. Level of metacognition is high in both years. A factor that was apparent from the data but not visible in these codes is that in year 3 Lisa used a much larger variation of strategies.

On task C, we see a change from low-level metacognition to high-level, from superficially founded reasoning to mathematically founded reasoning, and from novel to recall. In addition to these codes we remark that Lisa's solution in year 1 was extremely brief, without any real solving attempt. After the interview in year 2, Lisa had looked up the solution to this task, which she recalled in year 3.

On task D, Lisa's reasoning was consistently superficially founded, changed from recall to novel reasoning, and with varied level of metacognition. We add that in year 2 she used various strategies, and made several correct manipulations. In year 3, Lisa built upon her successful solution strategy used on the related task C. She modified that strategy to fit task D, which appeared somewhat mathematically founded, but her explanations were not explicit enough to code her reasoning as mathematically founded.

For each year, the number of tasks characterized by the codes of mathematically founded reasoning, novel reasoning, high-level metacognition, and low-level metacognition high-level metacognition task not successfully solved success are included in Figure 5. These numbers give an overview of how Lisa's reasoning evolved over time. The overall impression is in line with the examples described and the coding per task: at the beginning of her bachelor in mathematics, Lisa used high-level metacognition, no mathematical foundations, little novel reasoning and was not successful in solving the tasks. Over time, Lisa consistently used high-level metacognition, used more mathematically founded, some novel reasoning, and was more successful in solving the tasks.

Conclusion and discussion

The analysis of Lisa's reasoning using the framework shows how her reasoning evolved during three years of bachelor studies in mathematics, which is especially interesting given the many worrying characteristics in the first year: Lisa's reasoning was superficially founded (like often observed in undergraduate students), the solutions were full of errors and not successful. In this first year, Lisa appeared to have limited access to strategies and to be unaware of what mathematical properties are relevant in selecting a strategy. Given these rather weak characteristics, coding Lisa's reasoning as using high-level metacognition even seemed somewhat out-of-place. However, in comparison to other students who reasoned superficially founded, Lisa's metacognition level was clearly higher. The finding that Lisa's reasoning evolved towards mathematically founded (while other students' reasoning remained superficially founded) suggests that high-level metacognition may play a supportive role in developing mathematically founded reasoning.

The variety of strategies used by Lisa increased, which is captured by the 'novel' code. Thinking of suitable strategies to solve moderately non-routine tasks is difficult [START_REF] Selden | Why can't calculus students access their knowledge to solve non-routine problems?[END_REF] and requires insight into what mathematical characteristics are relevant for choosing strategies [START_REF] Mason | Having good ideas come-to-mind: Contemporary Pólya based advice for students of mathematics[END_REF], which is needed to make mathematically founded strategic decisions. In Lisa's reasoning, we see that her use of high level metacognition preceded mathematically founded strategy choices. This observation can be used as a starting point to stress the use of high-level metacognition in mathematics teaching.

Limitations

Our choice to use integration tasks has influenced the reasoning observed in this study, so we should be careful about extrapolating the findings to other contexts. However, the tasks have proven to arouse extensive reasoning of various kinds and, by using the same/similar tasks over the years, to bring to light an evolution in the way Lisa reasoned about challenging integration tasks.

  that fraction. . Ehm, to find the anti-derivative, I would, in one way or another , the fraction, I think. […] Let's see if I get the same. I'm afraid I won't, but I can always try. (Then she calculated the derivative of ) Here, Lisa again made an error, interchanging taking-power and taking-derivative, which are not commutative. She continued for some time, observing intermediary outcomes, considering what to infer to adjust the presupposed antiderivative, et cetera. She did not find the correct answer.

Figure 1 :

 1 Figure 1: Framework visualization (with exemplary coding) and legend

  derivative to see whether there is a pattern […] to see what step you can use to find the antiderivative […] Sometimes you obtain an , you see what to do to find the anti-derivative. So in the end it did help for this step. So I thought, wait, if I multiply with , that is simply times 1. Then I used again for substitution. […] And I thought, what if I put an here, multiply the antiderivative 2 with . […] Because I still had this substitution ( ) in mind, and then, if you have an extra , then you have and that can be replaced by .

Figure

  Figure 2: Coding of Lisa's reasoning, task A , year 1

Figure

  Figure 3: Coding of Lisa's reasoning, task A, year 3

Table 1 : Overview of tasks given to Lisa.

 1

Table 2 : Coding scheme

 2

	CODE:	CODE ASSIGNED IF:
	Mathematically	student explicitly gives relevant mathematical reasons
	founded reasoning	for why the strategy selected will help to solve the task
	Superficially	reasoning does not comply with the criterion for
	founded reasoning	mathematically founded reasoning
	Recall	student selects strategies because they have been used before to solve similar tasks
	Novel reasoning	reasoning does not comply with the criterion for recall
	High-level	student evaluates components of the task solving
	metacognition	situation when selecting a strategy
	Low-level metacognition	little to no evaluation takes place for selecting strategies

Pseudonym

Probably Lisa meant: integrand

Acknowledgment

We would like to thank Anke Haaksma for her help in transcribing the interviews.


Eszter Kovács-Kószó

Katalin Gosztonyi
email: kgosztony@caesar.elte.hu

József Kosztolányi

How can we help teachers using guided discovery method, who have not used it before

Keywords: Guided discovery, group working, series of problems

Introduction

In Hungary, there is a tradition of mathematics education focusing on problems and on mathematical discovery that we call Guided Discovery approach. This method was elaborated in the 20st century, first of all for the education of gifted students. Later the reform movement led by Tamás Varga in the 1960s and 70s developed a coherent general educational program for primary and lower secondary schools based on this approach (Pálfalvy, 2000). Although Hungarian specialists of mathematics education consider the approach actual and relevant until today, only a narrow circle of teachers follow it, and its large-scale dissemination remains problematic.

In my PhD research project, launched in September 2018, I would like to contribute to the elaboration of a more successful dissemination processes. In my analyses, I want to focus on one significant aspect of teaching by Guided Discovery: on the interactions between teacher and pupils, and amongst students. I plan to realize experimentations with teachers are willing to this approach and who want to discover it to improve their teaching practice. My purpose is to give an aid to teachers which could improve the number of "rich interactions" in their classroom.

Experiments

I plan to follow two directions. The first one is based on my previous project held during my master studies, when I have tried to develop teaching projects based on group work. The second direction will be realized in collaboration with the members of the MTA-ELTE Complex Mathematics Education Research Group. A subgroup of this research project focuses on one of the specificities of Varga's Guided Discovery approach: the planning of teaching trajectories in form of series of problems [START_REF] Gosztonyi | Teachers documentary work with series of problems[END_REF]. They examine the work with series of problems of expert teachers following the guided discovery approach, and they elaborate a commented collection of examples of series of problems to initiate other teachers into the approach.

In my experimentations first, I will analyze some lessons focusing on the interactions in both cases. Then I will try to help to the teacher to use one of the methods mentioned above and check the following lessons in the view of how the selected method could improve the quality of the teaching. Especially I want to answer the following question: Does the number of 'rich interactions' rise as an influence of the new way of teaching? If the answer will be 'yes, it does', I believe the quality of the teaching will rise also.

The project above has five parts. The first and most important element is the definition of 'rich interactions'. If the number of 'rich interactions' and its ratio with all interaction is rising, it shows at least three things: students become more active and thinking more intensively about the mathematical content, realize more effectively what they have not understand yet and become braver about asking from the teacher or from another student.

Secondly, I will try to help the teacher to familiarize with the 'new' method. This will be different in the two threads of the project. In the first one, the teacher will get a complete plan for a period of teaching which will take about five to ten lessons. During this time group learning will dominate on the lessons which is unusual in Hungarian practice so the teacher will get a short, practical summary about how he or she should organize and control work.

In the second project the teacher will get a commented series of problems issued from the abovementioned collection of the MTA-ELTE project. It is not necessarily connected with the following topic of the lesson, it is just an example. Then he or she will get time to measure it. The purpose is to help the teacher be able to create his or her own series of problems for his or her own class for the next period. During this he or she will have the opportunity to consult with one of the expert teachers mentioned above.

The third part will be the classroom realization. I will film the lessons and let the teacher work alone with the class.

After this, in the fourth part I will take an interview with the educator about how useful or efficient the experiment was in his or her opinion and I would be also interested in what he or she sees the most critical points of the lessons.

The last part will be the analysis of the data collected. The whole experiment will be repeated for three times in case of both versions. We hope this experiment will help us to support enthusiastic teachers to improve their method.

Acknowledgment

Eszter Kovács-Kószó is grateful for the scholarships of New National Excellence Program of the Ministry of Human Capacities of Hungary (ÚNKP-18-3-II-SZTE-13).

The poster

The presented poster is available through the following link: https://drive.google.com/file/d/10ZRMvcth-snOqwTvG7e8dGqsyWOnhOqK/view?usp=sharing


Matija Bašić

Željka Milin Šipuš

Students' understanding of the interplay between geometry and algebra in multidimensional analysis: representations of curves and surfaces

Keywords: Transition across university mathematics, multivariable calculus, curve, representation, didactic contract

HAL is

Introduction

In this paper we investigate learning difficulties that students face in the transition from the first courses in linear algebra and multivariable calculus, to more advanced mathematical courses that involve mathematical analysis of functions of several variables. As described by Kondratieva and Winsløw (2018), requests for efficiency and economy of exposition at university level often lead to narrow and disconnected teaching modules. Our teaching experiences indicate similar disparities in the transition from courses in analytical geometry, linear algebra and calculus on one side, to mathematical analysis of functions of several variables and differential geometry on the other. We noticed misconceptions and knowledge gaps in dealing with basic objects such as curves and surfaces in three-dimensional space, in using their implicit and parametric representations, and in converting from one representation to the other. These notions are essential for calculating curve integrals, applying Green-Stokes theorem, or in introductory lessons in differential geometry, for instance, the introduction of curve and surface curvature.

A curve in three-dimensional space can be presented either by a parametrization, that is, as a trace of a moving particle, or by a system of equations in three variables, that is, as an intersection of surfaces. Transitions from one representation to the other and vice versa are theoretically underpinned in multivariable calculus by implicit and inverse function theorems. However, students' work in basic examples usually happens in the domain of algebra and geometrysolving systems of (not necessarily linear) equations and giving them geometrical meaning. We noticed many students' difficulties in this process, starting from a non-meaningful manipulation of equations in a way that just leads to new equations, understanding what a solution of a system of equations is, especially when a solution is not unique, and, finally, providing a geometrical interpretation of the obtained objects. Students' difficulties in the same field but with linear objects, in recognition of lines and planes given by their implicit (Cartesian) and parametric equation, and the presentation of these objects as sets (subsets in space satisfying the same conditions or equations) are well evidenced [START_REF] Artigue | The Teaching and Learning of Mathematics at the University Level. Crucial Questions for Contemporary Research in Education[END_REF][START_REF] Alves-Dias | Les problèmes d'articulation entre points de vue cartésien et paramétrique dans l'enseignement de l'algèbre linéaire[END_REF][START_REF] Alves-Dias | Les problèmes d'articulation entre points de vue cartésien et paramétrique dans l'enseignement de l'algèbre linéaire[END_REF][START_REF] Nihoul | Quelques difficultés d'étudiants universitaires à reconnaître les objects "droites" et "plans" dans l'espace: une étude de cas[END_REF]. However, students' work related to curved objects in space is an under-researched area.

Motivation

Our study was motivated by the following question posed as one of five questions in the midterm exam for the course Introduction to differential geometry in spring semester 2017:

The question could be answered by following the procedure taught during the exercise classes (for a somewhat more complicated curve), which requires a curve to be parametrized. From the parametrization, one calculates its torsion, which should vanish for a planar curve, or determines its binormal field, which should be constant. However, in the above example the tedious calculation of a torsion (which often results in a calculation error) could have been avoided by using the definition of the planar curve emphasized during lectures. By a simple manipulation of the given equations, it is possible to eliminate the square terms to get 2x + 2y + z = 1, which could be recognized as an equation of a plane in which the curve lies. Among 136 students who provided any answer to this question (of 172 who participated in the exam), only 7 % (10 students) solved the task using that approach, and 10 % more solved it using both approaches, that means, they obtained the equation 2x + 2y + z = 1, and then still parametrized the curve and applied the tedious procedure. Some 27 % of students obtained the plane equation without providing an interpretation, and 30 % of students solved the question by immediately parametrizing the curve and appling the procedure only. The remaining 26 % did not obtain the equation nor a correct parametrization. This high number of students who focused only on procedure and failed to apply the definition of a planar curve, encouraged us to implement a more though-out approach in the next year courses followed by a deeper analysis of students' work.

Theoretical framework and research questions

Mathematical concepts are identified and represented through various representations, which reflect possibly different features of a concept, but simultaneously complement each other. Treating a concept within a certain representation (representational mode or register) and successfully converting between different registers of representation, is considered as a prerequisite for conceptual understanding. Conversion processes entail recognition of the same mathematical objects through representations from different registers and therefore require coordination of registers. Many students' difficulties can be described and explained by the lack of coordination of different registers of representation [START_REF] Duval | Registres de représentation sémiotique et fonctionnement cognitif de la pensée[END_REF][START_REF] Duval | A cognitive analysis of problems of comprehension in the learning of mathematics[END_REF]. Research provides examples [START_REF] Artigue | The Teaching and Learning of Mathematics at the University Level. Crucial Questions for Contemporary Research in Education[END_REF][START_REF] Alves-Dias | Les problèmes d'articulation entre points de vue cartésien et paramétrique dans l'enseignement de l'algèbre linéaire[END_REF][START_REF] Nihoul | Quelques difficultés d'étudiants universitaires à reconnaître les objects "droites" et "plans" dans l'espace: une étude de cas[END_REF] of students' difficulties in using the algebraic representation of lines and planes in space, and vice-versa, in recognizing straight lines and planes in space from their equations, and in converting between parametric and implicit viewpoints.

Question. Prove that the space curve given by 2

2 2 2 , 2 2 1 x y z x y x y     
 is planar and determine its osculating plane.

In this study we tried to answer the following research question: What are students' learning difficulties in relation to different representations of curves and surfaces in 3-dimensional space?

As observed in the motivational example, it seems that students tend to use lengthy procedures as opposed to conceptual definitions. This indicates that the work of some students does not show their complete mathematical understanding and difficulties, but rather that their answers are governed by reasons of didactical nature. We use the notion of didactic contract from the Theory of Didactic Situations (TDS) by [START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques. 1970 -1990[END_REF], a framework useful to study phenomena at university level (González-Martín, Bloch, Durand-Guerrier, & Maschietto, 2014):

A core TDS conceptual tool is Brousseau's notion of the didactic contract, the implicit set of expectations that teacher and students have of each other regarding mathematical knowledge and regarding the distribution of responsibilities during the teaching and learning processes. … The didactic contract is linked to an institution and, in particular at university level, the terms of the didactic contract can be quite strong. (pp. 119, 131) Although our ultimate goal is to study the cognitive and epistemological dimensions for the concept of a curve in space, we first had to analyze the didactical constraints that govern students' answers. Hence, motivated by our students' work we wanted to trace the impact of the didactic contract on students' exam productions and investigate if students' understanding could be improved through some changes in this contract.

Methodology and mathematical context

The participants of our study in 2018 were in total 173 students at a department of mathematics in Croatia, while participating in elective course Introduction to differential geometry (students of undergraduate program in mathematics and in mathematics education). The course has 2 hours of lecture and 2 hours of exercise class per week for 13 weeks. It is naturally divided into two parts, with the first part that focuses on the use of differential calculus to study geometrical properties of curves in 2D and 3D (curvature and torsion). The second part focuses on the study of surfaces and special curves on surfaces. The emphasis of the course is on low dimensions to encourage students to use the geometrical register in interpreting and understanding the results. The second author is the lecturer, while the first author is one of three teaching assistants for the course.

In 2018, during the exercise classes, the teaching assistants led the discussion about the exam question from the year before. The aim of this discussion was to re-negotiate the didactic contract built during previous courses in which teachers assign tasks and students answer in the way they were taught. In this case, it means that students apply the lengthy procedure, not the definition of a planar curve in space, which is intuitive and, although presented during the course, not exclusively a part of differential geometry content. As learning can be accomplished, in many cases, through the ruptures of the didactic contract [START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques. 1970 -1990[END_REF], the assistants showed both strategies and emphasized that the aim of the course is to encourage students to use the geometrical interpretation.

In other words, it is the conceptual and not the lengthy approach that was expected and aimed for. It was important for the students to hear that all correct answers are acknowledged at the exam, not just those that follow procedures from the exercise classes, since their comment during the discussion on the "non-standard" approach was that "they did not know this was allowed". In addition, the teaching assistants addressed other students' difficulties observed by the preliminary insight into students' work in 2017, like false generalizations and analogies from plane to space geometry (2D to 3D) in interpretation of equations and non-adequate conversions between implicit and parametric viewpoints of a curve.

The source of data for the in-depth analysis were midterm exam solutions. Questions were formulated in the geometric register, where curves are given as intersections of surfaces. The properties of the intersection curve can be recognized algebraically (from its equation) or geometrically (by some geometrical properties, such as its curvature). In this paper we analyze the questions presented in Figure 1.

Figure 1: Exam questions analyzed in the study

In Question 1, the set of points satisfying both equations, the equation of a hyperboloid of one sheet and a special plane, is to be identified either from a system of transformed implicit equations, or from its obtained parametrization which enables determining the curvature. Simple calculation in the algebraic register gives the system of transformed implicit equations as x = y, z = ±2, therefore the intersection set consists of two parallel lines that are given as intersections of two pairs of nonparallel planes.

In Question 2, the expected solution is to state that y = x + 1 is an equation of a plane, so the curve is planar by definition. For the second part it is expected that the equation of the sphere is recognized, and that the curve is a circle since it is the intersection of the sphere and the plane. The radius R of the circle may be determined geometrically, so the curvature can be expressed as 1/R.

The projection onto xz-plane is given by the equation

22 1 2( ) 8, 2 xz 
  which is the equation of an ellipse. Another correct approach is to parametrize the obtained curve, and then to calculate its curvature by using the formulas from the lecture. Conclusion that the curve is a circle should follow from the fact that the curvature is a non-zero constant, and the projection curve could be recognized from the parametrization with y = 0.

Students' answers were processed in an iterative way. First, based on our mathematical analysis of the question, we formulated codes regarding the register in which the task was solved: algebraic (following the algebraic procedure given in the exercise class) or geometric (geometrically interpreting equations), as well as the success of the students (if the calculation was completed or the conclusion reached). Second, the codes were refined to track students' different strategies and Question 1. Determine the set of points in space given by the equations unexpected answers pointing to some difficulties. Next, the observed categories and special cases were discussed between the authors and an agreement was reached to cluster the codes with low frequency pointing to the same type of difficulties. As an example, in Table 1 we present the data for Question 2 (a).

Results

In Question 1, approximately half of the students (51 %) noted correctly that the intersection of a doubly-curved surface (an hyperboloid of one sheet) and a plane is a pair of parallel straight lines. Most of the students (76 %) who reasoned correctly relied on parametrizing the obtained set as c(t) = (t, t, ±2) and calculating its curvature, which turned out to be 0 in all of its points, and some (29 %) relied on recognizing the lines from the system of transformed equations, whereas 5 % of students used both approaches.

Among students who did not provide a complete solution, approximately half gave no answer at all, further 30 % succesfully obtained transformed equations or calculated the curvature, but did not interpret the result geometrically and approximately 20 % misinterpreted the obtained result. Most of the errors in the last group occcured when trying to interpret a transformed implicit equation, which was seen as the requested intersection curve, and not as a new surface on which the intersection curve lies (the intersection set described as a plane, a cone, or a hyperbolic paraboloid); many errors also come from the misinterpretation of a transformed implicit equation without zcoordinate as a curve (a hyperbola instead of a hyperbolic cylinder).

In Question 2, 46 out of 173 students (27 %) noted that the given equation is an equation of a plane and successfully concluded that the curve is planar without further calculation, and a smaller number (18 %) of them went still for calculation of the torsion. More students than in 2017 successfully concluded that a curve is planar by the first approach, but still more than 70 % did not use this approach. Among the students that did not provide a solution, 9 % stated that y = x + 1 is an equation of a straight line in space, which is a well-evidenced false generalization in transition from 2D to 3D in linear algebra [START_REF] Nihoul | Quelques difficultés d'étudiants universitaires à reconnaître les objects "droites" et "plans" dans l'espace: une étude de cas[END_REF]. Further students' false argumentation runs as follows "since this is a line, and lines belong to a certain plane, therefore the curve is planar". Furthermore, careless approach to ambient space in which a curve is given seems to generate another case of false transition from 2D to 3D, since 38 students chose the formula for curvature that is the formula for curves in IR 2 , not in IR 3 .

Question 2 (a) No answer 57 33% Recognizing the equation of the plane 46 27% Using torsion = 0 as criterion Showing that torsion equals 0 8 5% Only stating the criterion 23 13%

Wrong arguments y = x + 1 is a line, and lines lie in planes 15 9% x 2 + y 2 + z 2 =17/2 is a plane 7 4% Substituting and stating that the curve is an ellipse in yz-plane Among 122 students who tried to parametrize the curve, 80 students (65 %) used the local parametrization 22 ( ) ( , 1, 17 / 2

( 1) ) c t t t t t      of the intersection curve, which is not adequate to conclude that the (whole) curve is planar, contrary to the linear case, where the standard procedure provides a global parametrization. To determine the curvature of the curve, students predominantly (70 %) used parametrization for further calculation over geometric approach. Only 15 (9 %) of them used a geometric approach by stating that the plane intersection of a sphere is a circle, with only 4 of them even moving forward to determine geometrically the radius of a circle.

Discussion

In general, we may state that students' difficulties in converting algebraic equations into geometrical objects are persisting, even after the first courses in linear algebra and calculus. In the case of a planar curve identification, students predominantly focus on the known procedure, although it is burdened with calculation, over using the conceptual definition of a planar curve which involves recognizing a plane by its implicit equation. We may also say that we observed the influence of the didactic contract, as the students tried to ensure that their answers included newly learned procedures during the exercise classes. However, students' difficulties in recognizing a plane by its equation can be seen as a particular case of a more general lack of coordination between algebraic equations and spatial geometrical objects. When manipulating equations defining a spatial curve, students interpret the new implicit equation as that intersection curve, not as a new surface on which the curve lies.

The above general observations point out to several further remarks. The first remark concerns evidence on students' false generalizations and analogies in their passage from 2D to 3D. As already stated, this was particularly the case when they identified the equation of a special plane with a line, a circular cylinder with a circle, or a hyperbolic cylinder with a hyperbola. Similar difficulties are observed in linear algebra, e.g. in the case of a plane in space given by an equation [START_REF] Alves-Dias | Les problèmes d'articulation entre points de vue cartésien et paramétrique dans l'enseignement de l'algèbre linéaire[END_REF][START_REF] Nihoul | Quelques difficultés d'étudiants universitaires à reconnaître les objects "droites" et "plans" dans l'espace: une étude de cas[END_REF]. It is a clear example of the fact that a piece of existing knowledge (a single equation represents a curve) may undergo a hasty generalization, and therefore standing in the way of full comprehension. It is not irrelevant to notice that the missing insight of different ambient (2D and 3D) is also identified in students' inappropriate use of formulas for the curvature of curves.

ax + by = c (Alves
Our second remark concerns students' use of and conversion between the implicit and the parametric representation (viewpoints) of curves. Again, students' difficulties in this process in linear algebra are well evidenced [START_REF] Artigue | The Teaching and Learning of Mathematics at the University Level. Crucial Questions for Contemporary Research in Education[END_REF][START_REF] Alves-Dias | Les problèmes d'articulation entre points de vue cartésien et paramétrique dans l'enseignement de l'algèbre linéaire[END_REF][START_REF] Alves-Dias | Les problèmes d'articulation entre points de vue cartésien et paramétrique dans l'enseignement de l'algèbre linéaire[END_REF][START_REF] Nihoul | Quelques difficultés d'étudiants universitaires à reconnaître les objects "droites" et "plans" dans l'espace: une étude de cas[END_REF]. In our study with curved objects, we observed that students did not invoke the need for conversion to a parametric representation, nor is the conversion found as flexible as it is assumed, already discussed in the case of linear algebra [START_REF] Artigue | The Teaching and Learning of Mathematics at the University Level. Crucial Questions for Contemporary Research in Education[END_REF](Artigue, , p. 1384)):

Flexibility seems to be considered as automatically internalized once one has "understood" the notion, as if it were a simple question of homework that one can leave to the private work of the student.

Another remark is related to students' difficulty coming from the character of non-linear problems.

Conversion from implicit equations to parametrization and vice-versa, that relies on a procedure of "eliminating a parameter" or "substituting a variable by a parameter" from linear algebra has a local character in the non-linear case and cannot produce an instant answer as in the linear case. This conversion brings many different transformation approaches, usually non-routine ones (especially when trigonometric and hyperbolic functions are involved). For example, in Question 2, a local parametrization is not adequate to conclude that the curve is planar. A relevant explanation can be provided with the implicit and inverse function theorems, which although do not provide a recipe to change representation, should be discussed during the class to emphasize the subtleties in the nonlinear case. Students' focus on procedures suggests that their knowledge is an "amalgamation of practical blocks" (Brandes & Hardy, 2018, p. 506). Moreover, the observed absence of theoretical insight in students' knowledge might be seen as a missing brick in Transition of type 1 [START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF].

Conclusion

Different representations of curves and surfaces and their flexible conversions are essential for various mathematical courses that involve the analysis of functions of several variables. We also repeat that our study tries to see some of the students' bad performances as a result of the impact of the didactic contract. However, students' productions firstly point out to the existence of difficulties which were also observed in linear algebra in representations of lines and planes [START_REF] Artigue | The Teaching and Learning of Mathematics at the University Level. Crucial Questions for Contemporary Research in Education[END_REF][START_REF] Alves-Dias | Les problèmes d'articulation entre points de vue cartésien et paramétrique dans l'enseignement de l'algèbre linéaire[END_REF][START_REF] Nihoul | Quelques difficultés d'étudiants universitaires à reconnaître les objects "droites" et "plans" dans l'espace: une étude de cas[END_REF]. False generalizations from 2D to 3D appear (a line in space is represented by a single implicit linear equation, and vice-versa, single implicit linear equation with, for instance, no z-coordinate is an equation of a line in space; the same student's reasoning appears for e.g. a circle and a circular cylinder). This observation may also point to the presence of some learning obstacles in the sense introduced by [START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques. 1970 -1990[END_REF], which needs to be further explored. According to Brousseau, obstacles of various origins can stand in the way of an efficient learning of mathematics. Brousseau distinguishes between cognitive (due to the state of mental development), didactical (due to a certain way of teaching) and epistemological obstacles (due to the very nature of mathematical concepts). Clear distinction among them, due to the complex nature of human's knowledge acquisition, is not always possible. Obstacles become evident in students' errors which are not related to chance but persist, or in difficulties and problems that students encounter that slow down the learning process. When this becomes evident, it does not point out to the lack of knowledge, but often to the piece of knowledge that was adequate in a previous situation, but now leads to incorrect reasoning. Brousseau suggests that to overcome an obstacle, a sufficient flow of new situations challenging old knowledge is needed. Whether careful task design with different potentials for student learning would be beneficial as well [START_REF] Gravesen | Task Design for Students' Work with Basic Theory in Analysis: the Cases of Multidimensional Differentiability and Curve Integrals[END_REF], is another question to be explored in the future.

   Find its parametrization and determine its curvature. Which object does it represent geometrically? Question 2. The curve c is given by the equations a) Explain why c is planar. (b) Determine the curvature of c and decide what curve it is. (c) What curve is obtained by projecting c onto the xz-plane?

Table 1 : Codes and frequencies for Question 2(a)

 1

Acknowledgment

We dedicate this paper to the memory of our dear colleague doc.dr.sc. Stipe Vidak (1980Vidak ( -2018) ) and to our many fruitful joint discussions.