text:
Ignasi Florensa
email: iflorensa@euss.es

Marianna Bosch
email: marianna.bosch@iqs.url.edu

Josep Gascón
email: gascon@mat.uab.cat

Noemí Ruiz-Munzon

Teaching didactics to lecturers: A challenging field

Keywords: Lecturer education, mathematical modelling, Anthropological Theory of the Didactic

Lecturers' professional activity is, at least, twofold: research and teaching. However, their professional development is generally mostly based on research achievements and little effort is made to empower lecturers overcome the difficulties experienced during their teaching activities. We postulate that didactics of mathematics can be a powerful tool to help lecturers question and reorganize the knowledge to be taught, and to make them aware of the conditions enabling and the constraints hindering new modalities of teaching mathematics, more based on its use as a modelling tool to approach open questions. We present in this paper a first edition of a professional development course for lecturers designed for and experienced in an Engineering School in Barcelona. The results obtained are then used for a subsequent course redesign to be conducted with lecturers of a similar university school.

Introduction

Traditionally, lecturers' development courses have not been considered relevant by research in teacher education. This is a normal phenomenon considering universities' criteria when hiring lecturers and evaluating those already lecturing: mainly research activities and merits are considered. In contrast, lecturers' didactic or pedagogical education is usually ignored or, at most, considered as a positive complement. The absence of regular lecturers' teaching training is a worldwide phenomenon with fewand not always successfulexceptions. In the United Kingdom, the Higher Education Academy (HEA), the UK Professional Standards Framework (UKPSF) and its accreditation process made a first attempt to incorporate lecturer training as a requirement to teach in UK universities (Department for Education and Skills, 2003). Nevertheless, this program that was thought to be central in lecturers' professional development has finished as a volunteer training and accreditation scheme for both individuals and institutions involved in teaching at higher education (The Higher Education Academy, 2011).

We consider that, as long as their activity has a clear twofold character based on research and teaching, in addition to the traditional training in research (Master's Degree and PhD program), lecturers also need an explicit pedagogical and didactic education. In fact, universities are among the sole existing teaching institutions where teachers are not required an explicit training course on teaching and learning processes. We consider that this crucial difference should not be accepted as a given: the conditions of existence of a university teacher education course have to be studied, especially with the possibility to base it on contents emerging from research in didactics.

In order to have a first set of empirical data to evaluate the conditions of existence of such a course for lecturers at university level we designed a course for 14 lecturers of an Engineering School in Barcelona (www.euss.es). Lecturers participating in the course teach Analysis (3), Strength of Materials (4), Physics (2), Electronical Technology (2) and Informatics (2). We took as starting point the frame of "study and research paths for teacher education" (SRP-TE) based on recent investigations in the Anthropological Theory of the Didactic (ATD) for pre-service and in-service secondary teachers. The lecturers' course was experienced in February 2016. We present the design principles and results of this first edition, as well as the subsequent re-design for new editions, to overcome the experienced difficulties and take advantage of its potential strengths.

University teacher education: A field to be explored in ATD

Courses for pre-and in-service lecturer professional development are an unexplored field in research. There exists very little literature regarding this subject and the few experiences reported involve only general pedagogical contents not taking into account the very nature of the knowledge involved in the teaching and learning processes. It is important to highlight that no paper on this field was presented at the last CERME9 (neither at TWG 14, University Mathematics Education; or at TWGs 18, 19 and 20, Teachers' Knowledge, Practices and Education), or at groups regarding teacher training or university teaching at the last ICME 13, except for a preliminary version of this paper [START_REF] Florensa | Lecturer Education: a course design[END_REF]. The structure of ICME13 Topic Study Groups about teacher education is especially revealing at this respect: there were four groups on teacher education, two (in and pre-service) centered on the elementary level and two on the secondary level, but none on the tertiary level. At the recent conferences on Mathematics Education in North America, only Ellis presented research on teacher assistants training (Ellis, 2014a[START_REF] Ellis | Preparing future professors: highlighting the importance of graduate student professional development programs in calculus instruction[END_REF].

Regarding the presence of papers in journals about lecturers' education we have found very little production: only two papers [START_REF] Guasch | University teacher competencies in a virtual teaching/learning environment: Analysis of a teacher training experience[END_REF][START_REF] Postareff | A follow-up study of the effect of pedagogical training on teaching in higher education[END_REF] and the Handbook on Teaching and Learning in Higher Education (Fry, Ketteridge & Marshall, 1999). We have developed a research from the initial year of publication to the end of 2015 in these journals: Educational Studies in Mathematics, Higher Education, Journal of Mathematics Teacher Education, Mathematical Thinking and Learning, Journal of Teacher Education, Recherches en Didactique des Mathématiques, REDIMAT, RELIME.

As said before, we consider that research in didactics can be used as the basis for courses on lecturer education regarding teaching and learning processes. Our starting hypothesis is that results emerging form secondary teacher education can be used at this level. Our results will be used to partially confirm this assumption. The Solid Findings in Mathematics Education on Teacher Knowledge (Education Committee of the EMS, 2012) state explicitly that "content knowledge" (CK) is necessary but not sufficient for teaching. The report of the Education Committee highlights as crucial notions to be developed in teacher education the "pedagogical content knowledge" (PCK) [START_REF] Shulman | Knowledge and teaching: foundations of the new reform[END_REF] and the different dimensions of the "mathematical knowledge for teaching" (MKT) [START_REF] Ball | Content Knowledge for Teaching: What Makes It Special[END_REF]. Both approaches clearly go further than the traditional conception of teaching as transmission of knowledge and consequently ask for changes in teacher education concerning the way mathematical knowledge should be approached.

We use the Anthropological Theory of the Didactic (ATD) as a main framework for the design, experience and analysis of the course. The last investigations on teacher education in ATD show that the use of notions such as PCK and MKT do not ensure researchers/educators to include a questioning of the nature, selection and organization of the contents to be taught [START_REF] Ruiz-Olarría | La formación matemático-didáctica del profesorado de secundaria[END_REF]. Under the ATD approach, the role of teacher education is not limited to enrich teachers' pedagogical performance, but also to provide them with tools to contest the so-called dominant epistemology and emancipate from it when designing study processes [START_REF] Gascón | Los modelos epistemológicos de referencia como instrumentos de emancipación de la didáctica y la historia de las matemáticas[END_REF]. This questioning and reorganization of the knowledge to be taught is not spontaneous for teachers (nor for lecturers) because they tend to assume the institutional dominant epistemology as their own. The way proposed by ATD research to locate it at the core of teacher educational processes has very much evolved in this last decade. It started with a first experience in secondary teacher education based on the "questions of the week" [START_REF] Cirade | Devenir professeur de mathématiques: entre problèmes de la profession et formation en IUFM[END_REF] and nowadays takes the form of an inquirybased device called "study and research path for teacher education" (SRP-TE), which starts from a problematic question appearing in the field of the teacher profession and leads to the search, development and analysis of alternative teaching proposals [START_REF] Barquero | A study and research path on mathematical modelling for teacher education[END_REF], 2016). The main idea of the SRP-TE is to generate a practical and theoretical questioning of the school activities linked to the teacher professional initial question. It is structured in five modules:  M0: Formulation and first exploration of the generating question Q0 of the SRP-TE, for instance one of the kind: "How to teach (a specific content)?" which is to be partially answered at the end of the process.

 M1: Living a "study and research path" (SRP) as a student. The main goal is to make teachers encounter an unfamiliar inquiry-based activity related to Q0 that could exist in a normal classroom of the considered educational level.

 M2: Adaptation of the lived SRP to be experienced in a real school situation. During this adaptation, many of the institutional restrictions teachers should face are expected to show up. They can thus be afterwards analyzed from an epistemological, didactic and ecological perspective (what can "live" and under what conditions in a given educational setting).

 M3: Experimentation, management and carrying out of in vivo and a posteriori analyses of the adapted teaching proposal.

 M4: Joint elaboration of a critical analysis of traditional teaching practices and the possibilities (and limitations) of introducing new proposals, as well as generation of a partial answer to Q0.

During the development of SRPs-TE for secondary school teachers, an epistemological tool has been adapted and developed to facilitate the analysis of the SRP and the questioning of school contents: what we call "question-answer maps". Following other authors, we consider these maps, which are used as a key tool in ATD research, as a powerful instrument for teacher education:

We hypothesize that such a representation is sufficiently close to teachers' concerns, and also captures such essential parts of a didactic design, that one could use it as a tool for collaboration and communication with and among teachers, regarding concrete teaching designs (Winsløw, Matheron, & Mercier, 2013, p. 281) Some preliminary and promising experiences exist in using these maps in teacher training courses to describe the dynamic and collective aspects of mathematical activity [START_REF] Barquero | An online course for inservice mathematics teachers at secondary level about mathematical modeling[END_REF]Florensa, Bosch, & Gascón, 2016a;[START_REF] Jessen | How can study and research paths contribute to the teaching of mathematics in an interdisciplinary setting?[END_REF]. The work with the maps seem to be useful for teachers in order to describe knowledge in inquiry activities and to act as a counterpoint of the official curricular organization of contents.

Research questions

The work presented in this paper is considered as an exploratory design [START_REF] Singh | Quantitative social research methods[END_REF] to obtain and analyze a first set of data from the first implemented course and to redesign it to be applied in another institution. The specific research questions that will be studied are: RQ1. The role played by question-answer maps in teacher education: Do they help lecturers describe, analyze and design inquiry and modelling processes and the involved knowledge? RQ2. Does the course empower lecturers to identify the dynamic and collective nature of the lived SRP in contrast to the static, individual and compartmentalized dominant conception of knowledge?

Course description

The engineering school where the course was implemented keeps a four-hour time slope with no teaching for all lecturers all Wednesdays: they use this time for professional development, meetings, pedagogical courses or activities. In fact, it is a Salesian university with a special concern about teaching and learning processes, as well as students' personal evolution. The course was structured in six two-hour sessions during three weeks, and the central question to be partially answered was: "Could modelling be the main motivation of my subject? Which conditions enable and which constraints hinder this modelling activity?"

Because of the time restriction, the five-module structure of the SRP-TE had to be adapted. The six sessions appeared to us (designers and course leaders) as a short course. However, they finally seemed to be enough for the work planned. Of course the true work is to be carried out afterwards, when lecturers decide to introduce some new proposals in their subject based in the work initiated at the course. During this application phase teachers implementing SRPs asked for help to the researchers-educators, thus extending the real duration. We planned the course as follows:

 1 st session: Explicitly state the professional question Q0 and shortly present the ATD framework including the notions of praxeology, Herbartian schema and media-milieu dialectics, topogenesis, mesogenesis and chronogenesis [START_REF] Barquero | Didactic Engineering as a Research Methodology: From Fundamental Situations to Study and Research Paths[END_REF]. They seemed to be well understood and some of them were mobilized during the 4 th session.

 2 nd and 3 rd sessions: A SRP was proposed to be carried out in groups of up to three lecturers. "Taking into account the incidence index of the last 9 months of a dengue outbreak: could you forecast the incidence index for the next 3 months (already known)?" (Figure 1)

 4 th session: Lecturers generated a question-answer map of the lived SRP including aspects such as media-milieu dialectics. One of the generated maps can be seen in Figure 2.

 5 th session: Lecturers are invited to create new small groups with the colleagues teaching the same subject. They are asked to design a SRP by choosing a generating question in their field trying to overcome some observed didactic facts such as the absence of raison d'être, the disconnections of topics or the poverty of the experimental work, among others.

 6 th session: Sharing some possible teaching proposals and conclusions of the course. In the introduction to the 5 th session, lecturers were invited to identify didactic facts that they would like to overcome through the new didactic proposal. The goal was not to implement the inquiry by itself, but to identify how the dominant epistemology in the institution is related to these problematic phenomena and roughly propose new possible epistemological and didactic organizations to face them. The question-answer maps were the tool provided to lecturers to carry out this work. During the implementation of the course, some of the contents that we initially considered as difficult had an easier reception than expected (especially the notion of media-milieu dialectics) and, on the contrary, some basic notions were difficult to share with the participants, for instance the description of contents in terms of questions instead of topics.

In order to obtain data to evaluate the course, all the questions-answer maps of all groups, both from the analysis of the modelling lived activity and from the a priori design of the SRP, were collected.

We have also obtained data from a final survey filled in by all lecturers attending the course. The survey was structured in three main blocks. The first block addressed general aspects of the course such as duration, balance between individual and team work, time structure, etc. The second block asked about content-related aspects of the course like the work developed with question-answer maps and with the media-milieu dialectics. Finally, the survey asked the lecturers about the possible consequences of the course on their teaching activities: changes in the conception of knowledge, dynamics and collective aspects of activities, and availability of new designing and evaluating tools.

Results and discussion

The question-answer maps regarding the dengue outbreak SRP shew up how the inquiry was capable to connect fields usually disconnected in the traditional curricular organization of contents. For example, the map of Figure 2 reveals that functions, differential equations, regression, average rate of change and epidemiological notions are deeply interrelated. An interesting fact emerged when analyzing different maps from different working groups: depending on their lecturing field, they approached the problem quite differently. For instance, Mathematics lecturers' work was centered on finding a mathematical model fitting the data, whereas Chemistry lecturers' work evolved around the epidemiological data, the notion of "incidence index" and searching scholar literature regarding other similar outbreaks. The different teaching fields of lecturers permitted to share different visions of the knowledge at stake in the proposed SRP. The use of the maps was a key factor to describe this connection of fields usually lacking in school institutions.

The second part of the survey about the content of the course reveals that the work developed by lecturers with the question-answer maps and the media-milieu dialectics was difficult for them (more than 70% of the teachers found it hard or very hard) but at the same time they identified this work as "easily applicable to design and manage new teaching and learning processes" (more than 70% of the lecturers found contents and tools of the course easy to use and to implement).

Regarding the consequences of the course on the lecturers' teaching practices, the survey showed that it helped (more than 90% totally agreed) to change their previous conception of knowledge towards a dynamical-collective conception in terms of modelling activities.

The third source of evidence are the maps generated by the lecturers as a priori analysis for an SRP to be experienced in their subjects. In total, six maps where generated by lecturers, all of them with a generating question and making explicit the didactic facts intended to be overcome. Two of these a priori SRP designs where experienced during the spring semester, starting just after the lecturers' course. These two emerging SRP have been experienced and managed only by lecturers that followed the course and did not have any other didactic experience or training. This fact is especially interesting because with the analysis of these experiences a first set of data can be collected regarding the conditions of existence of SRPs at the university level led by lecturers with almost no direct connection with research in didactics. This first experience in lecturer education seem to preliminary validate [START_REF] Winsløw | Study and research courses as an epistemological model for didactics[END_REF] hypothesis about the use of question-answer maps in teacher education and confirm [START_REF] Barquero | An online course for inservice mathematics teachers at secondary level about mathematical modeling[END_REF] results. Lecturers have worked with the maps and have used them to both model a lived study process and a priori analyze their own designed SRP. Moreover, the maps have been used to compare the knowledge mobilized during a specific SRP and the school knowledge. The Q-A duplets appearing in the map were used as the elements to contrast with curricular requirements.

The course also appears as a good tool to empower lecturers to question and put under vigilance the dominant epistemology at the university. It produced a discussion (and thus enabled a reflection) on what knowledge has to be taught at the university and how the modelling activity with its dynamics and collective aspects could be considered. Regarding the conditions of existence of a lecturer course based on the ATD, it seems that the described conditions make it viable and that some lecturers have taken it as an opportunity to redesign their teaching and learning activities. However, an important aspect to take into account is the fact that one of the leaders of the course is also a lecturer in the considered Engineer School, what certainly affected the good predisposition of the attendees due to his personal leadership in the institution. This particular condition has to be considered in new editions of the course and the question of its reproducibility remains open.

Figure 1 :

 1 Figure 1: Data used for the lived SRP

Proceedings of CERME10

Acknowledgments

Founded by the Spanish Ministry of Economía y Competitividad. Project EDU2015-69865-C3-1-R


domain: math shs

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

"Points", "slopes" and "derivatives": Substantiations of narratives about tangent line in university mathematics students' discourses

Keywords: Tangent line, mathematical discourse, derivative, narratives, routines. Thematic Working Group 14

 in order to identify characteristics of responses to a questionnaire in which 182 students were asked to explain in simple words the tangent line, describe its properties, provide its definition and identify if a drawn line is a tangent of a given curve. Findings suggest that students engage with analytical, geometrical and algebraic discourses in their substantiations about tangents, sometimes by engaging with more than one discourse in the same response or/and across different responses in the same script.

Introduction

Research reports students' difficulties with their meaning making of the tangent line to a function graph. These difficulties have been attributed to students' encounter with the tangent line in different mathematical contexts (e.g. Euclidean geometry, analytic geometry or analysis), disconnection between algebraic or analytical approaches (e.g. rate of change, slope, derivative or tangent line formula) and graphical approaches (e.g. limiting position of secants, visualisation of tangents) and differences between a global perspective (relation of the line and curve as a whole) and the local perspective (relation of the line to the curve at a specific point). Most challenging cases are: when the tangent has more than one common point with the graph (e.g. f(x)=sinx at π/4) or coincides with the graph or a part of it (e.g. when the curve is a straight line); tangency at inflection points (e.g. f(x)=x 3 at 0); and, points in which the limit of the difference quotient from the left and the right are different real numbers (e.g. f(x)=x at 0) or infinity (e.g. f(x)=x at 0) [START_REF] Biza | Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis[END_REF][START_REF] Castela | Apprendre avec et contre ses connaissances antérieures: Un example concret, celui de la tangente[END_REF][START_REF] Park | Is the derivative a function? If so, how do we teach it?[END_REF][START_REF] Vinner | The role of definitions in the teaching and learning of Mathematics[END_REF].

In this paper, I draw on my previous research on students' perspectives about tangent line [START_REF] Biza | Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis[END_REF][START_REF] Biza | First year mathematics undergraduates' settled images of tangent line[END_REF] by analysing not only what lines students recognise as tangent or not, but also by considering how they justify their choices. The conjecture I examine here is that students use a range of arguments from geometry, algebra and analysis to justify their choices that go beyond the correctness or not of these choices. With this analysis, my objective is to gain insight into how students make meaning of mathematical objectsin the case of this paper tangentsthrough their communication about them. To this aim, I analyse first year university mathematics students' responses to a questionnaire about tangent line by drawing on the commognitive approach [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF]. In what follows, I introduce the main tenets of the commognitive approach that I employed in the analysis, and the methodology of the study. Then, I present preliminary findings from the analysis and I discuss them also in relation to potential implications for teaching.

Theoretical underpinnings of the study

According to the commognitive approach [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF] communication about mathematics in written or verbal responses is not a window to thinking but an inseparable part of this thinking that makes sense only in the context in which this communication takes place. A mathematical discourse is defined by four characteristics: word use, visual mediators, narratives and routines. Word use includes the use of mathematical terms (e.g., in the context of this study, 'tangent', 'derivative' or 'direction coefficient') as well as everyday words with a specific meaning within mathematics (such as 'touch', 'region' or 'point'). Visual mediators include mediators of mathematical meaning (e.g., function graphs, diagrams, geometrical figures or symbols) as well as physical objects. Narratives include texts, written or spoken, which describe objects and processes as well as relationships among those (e.g., definitions, theorems or proofs), and are subject to endorsement, modification or rejection according to rules defined by the community (e.g., 'a tangent line is a line that has one common point with a curve' is an endorsed narrative for tangents in Euclidean geometry but not in analysis). Routines include regularly employed and well-defined practices that are used in distinct, characteristic ways by the community (such as defining, conjecturing, proving, estimating, generalising and abstracting). Sfard elaborates three kinds of routines: deeds, explorations and rituals where explorations are categorised into substantiations, recall or constructions (ibid, pp. 223-245). Recently, there has been increasing interest in discursive approaches and, especially, in university mathematics teaching and learning research, discursive approaches are gaining more momentum [START_REF] Nardi | Commognitive analyses of the learning and teaching of mathematics at university level: the case of discursive shifts in the study of Calculus[END_REF] in the investigation of university teachers' discursive practices (e.g. [START_REF] Park | Is the derivative a function? If so, how do we teach it?[END_REF][START_REF] Viirman | Explanation, motivation and question posing routines in university mathematics teachers' pedagogical discourse: A commognitive analysis[END_REF] and student learning (e.g. [START_REF] Güçler | Making implicit metalevel rules of the discourse on function explicit topics of reflection in the classroom to foster student learning[END_REF].

In this study, students' responses to a questionnaire are seen as acts of communication and thus part of their meaning making about tangent lines. Mathematical routines such as investigating if a line is a tangent (see question Q3 in Figure 1) can be explorations that include recall of previously endorsed narratives, substantiations of narratives about why a line is (or is not) tangent or constructions of new objects such as formulae and graphs. However, there are differences in the mathematical discourses about tangency in analysis, geometry and algebra. For example, in Euclidean geometry, whether a line is tangent or not depends on the number of common points and the relative position between the line and the curve (geometrical routines) because a tangent line to a circle has one common point and keeps the circle to one side (geometrical narratives). In analysis, tangency is checked locally (analytical routine) and is defined by the derivative at a point (analytical endorsed narratives) which is the slope of the line (algebraic narrative). In algebra, the tangent line will be justified through calculations (algebraic routine) of the slope and defined through its equation or the vector that gives its direction (algebraic narratives). Identifying how students' responses to a questionnaire engage with these discourses is the focus of this paper.

Methodology

Data reported in this paper were collected from a questionnaire administered to 182 first year university students (97 female) from mathematics departments in Greek universities. All participants had been taught about the tangent line in Euclidean and analytic geometry, and in elementary analysis courses in Years 10,11 and 12, but not yet at university as the questionnaire was administered at the beginning of their first year. The questionnaire included tasks (see a sample of questions from the questionnaire in Figure 1) in which the students were asked to explain in their own words the tangent line (Q1); to describe properties of it (Q2); to identify if a drawn line is a tangent line of a given curve (Q3); to construct the tangent line, if it exists, of a given curve through a specific point on the curve or outside the graph (Q4 and Q5); to provide definitions (Q6), to write the formula, and to apply the formula in specific cases (Q7 and Q8). In questions Q3, Q4 and Q5 only the graph was provided and no formula of the corresponding curve was given; students were asked to identify or construct the tangents based on the graphs and justify their choices. The proposed curves were chosen to reflect students' common difficulties with tangent lines identified by previous research [START_REF] Biza | Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis[END_REF][START_REF] Castela | Apprendre avec et contre ses connaissances antérieures: Un example concret, celui de la tangente[END_REF][START_REF] Vinner | The role of definitions in the teaching and learning of Mathematics[END_REF]. For example, the corresponding line: had more than one common point with the curve (e.g., in Figure 1 In earlier analysis [START_REF] Biza | First year mathematics undergraduates' settled images of tangent line[END_REF], student choices in questions Q3, Q4, Q5, Q7 and Q8 were characterised according to their correctness and analysed quantitatively. This analysis suggested a classification of students regarding their perspectives on tangent line and its relation with the corresponding curve into three groups with analytical local perspectives (closer to the tangent line in the context of analysis -25.8%); geometrical global perspectives (more relevant to the tangent line in the context of geometry -17.6%); intermediate perspectives between the analytical local and the geometrical global perspectives (56.6%). Although this classification indicated a spectrum of students' perspectives about tangency, it does not grasp the subtlety of these perspectives as they were evident in students' choices and justifications of these choices. To this aim, student responses to questions Q1, Q2 and Q6 and their justifications in questions Q3, Q4 and Q5 were analysed qualitatively. Part of this analysis focuses on the mathematical discourses students engaged in in their responses (analytical, geometrical and algebraic) with specific emphasis on the words used, routines and narratives, substantiation of these narratives, how this discourse is related to their choices (correct or not) in the questionnaire and the consistency of student responses across the questionnaire. This paper discusses preliminary findings from the 182 student responses to the items: Q1, Q2, Q3a-e and Q6 presented in Figure 1.

Student justifications on why the sketched line is a tangent or not

Justifications students offered in order to accept or reject a tangent line when it does not have any other common point and keeps the graph at the same semi-plane (Q3.a) or when it has other common points sketched (Q3.c) or not (Q3.b) are summarised in Table 1. "The rays ε1, ε2 which are tangents at A are opposite" Other "There is only one tangent at the point A" or "There is a limit which is the same from the left and the right side or "ε: it is tangent, the point A is defined and belongs to the domain of the graph"

Table 1: Student choice justifications to questions Q3.a, Q3.b and Q3.c

1 Data have been translated from Greek to English. In Greek, the noun tangent [line] (εφαπτομένη [ευθεία]) and verbs such as being tangent, abut, touch (εφάπτεται) have the same origin. In Greek, the excerpt: "No, it is not a tangent because although it touches the graph …" sounds contradictory ("Όχι, δεν είναι εφαπτομένη γιατί αν και εφάπτεται στη γρ. παράσταση …"), one explanation is that the noun "tangent" draws on the mathematical discourse, whereas the verb "touches" draws on the everyday discourse.

Justifications students offered in order to accept or reject a tangent line when the tangency point is an inflection point (Q3.d and Q3.e) are summarised in Table 2. Similarly, the word use includes verbal descriptions as well as terms and symbols from geometry, analysis and algebra. The relation of the line and the curve especially at the point A are described in a range of ways, not necessarily with consistent (in terms of the different discourses) meaning. For example, in questions Q3.d and Q3.e where point A is an inflection point the line intersects (τέμνει); pierces (τρυπάει); cuts (κόβει); crosses (διαπερνά) or bisects (διχοτομεί). The same word can be used with a range of meanings in different scripts or across the same script. For example, "intersection" may mean the common point regardless of the position of the curve in relation to the line, whereas in other cases (in the same or different scripts) it means the split of the curve into parts. The analysis aimed to identify evidence not only regarding the common points but also regarding the overall relation of the line and the curve and their relative position. These subtle differences are not always evident in student responses.

Furthermore, the analysis indicated the use of endorsed narratives from more than one discourse in the same response or/and across responses within the same script. For example, student S[149] who performed well in all questionnaire items, writes in Q2: Another student (S[261]), who also had problems with tangency at inflection points, writes in Q1: "Gradient (λ), the tangency point, the formula of the line, f( x

f΄(xA)=λ the direction coefficient. A A x x A x x x f x f x f A     ) ( ) ( lim ) ( ' [ analytical

Discussion

This paper reports on my first attempt to draw on the commognitive approach to analyse 182 first year university mathematics students' justifications about tangent line. My initial conjecture was that students engage with different discourses (geometrical, algebraic or analytical) even for the substantiation of similar choices regarding the tangent line. Although the findings presented in this paper cover only a small slice of the data, in terms of questionnaire items, I would say that there is evidence supporting my conjecture. Students engage with analytical, geometrical or algebraic discourses in terms of the endorsed narratives and routines I identified in their responses. Also, the word use includes verbal descriptions as well as terms and symbols from geometry, analysis and algebra. Additionally, the same justification may engage with more than one discourse or/and different discourses across the script. Also, it seems that in several responses there are arguments that use analytical endorsed narratives (derivative etc.) applied through geometrical routines (check the tangency globally). I note here that my analysis considers students' responses in relation to the discourses of the mathematical community in different mathematical areas and not in relation to their correctness. Use of analytical narratives, for example, do not necessarily 'secure' the correctness of the response and, the other way around, a correct choice does not necessarily draw on a coherent and consistent justification. Furthermore, the type of the task may also affect the type of discourse students engage in. For example, a graphical question (e.g. sketch the tangent line) may trigger geometrical discourses whereas an algebraic question (e.g. calculate the formula) may trigger analytical or algebraic ones. The reported findings are indicative and further investigation is in process also in relation to inconsistencies within a response with potential commognitive conflicts between geometric, algebraic and analytical discourses of tangent lines as well as to the extent student responses are mediated by the task formation.

This work contributes to our insight into what students bring with them when they join postsecondary mathematics courses and I credit to the commognitive approach the deepening of this insight in the case of students' meaning making of tangency. I envisage teaching implications of the outcomes of this analysis in calculus or analysis introductory courses. For example, the observed mismatch between lecturers' and students' discourses [START_REF] Park | Is the derivative a function? If so, how do we teach it?[END_REF] would be dealt by explicitly addressing commognitive conflicts with the use of appropriately selected examplessee (Biza,

  , Q3.b and Q3.c in comparison to Q3.a challenge the geometrical routine of checking the number of common points and the relative position between the line and the curve) or passed through an inflection point (e.g., in Figure 1, Q3.d and Q3.e challenge the geometrical routine of the relative position between the line and the curve)for more about the questionnaire design see Biza et al. (2008). Q3: Which of the lines that are drawn in the following figures are tangent lines of the corresponding graph at point A? Justify your answers. is the definition of the tangent line of a function graph at its point A? Q1: Explain, in simple words, what you are thinking when you hear the term "tangent line". Q2: Write as many properties as you can think of about the relationship between a curve and its tangent line at a point A.

Figure 1 :

 1 Figure 1: Questionnaire sample

  JustificationScript Example Rejection of the line as a tangent Common points between line and curve, global view "No [it is not a tangent], the line has 2 points in common with the function graph" Relation of the line and the curve, global view "No it is not a tangent, although it touches 1 the function graph at the point A, it cuts [the graph] at another point" Relative position of the line and the curve, global view "[The line] splits the curve in two semi-planes" Only local acceptance of the tangency "Not [a tangent] in general […] in a small interval (δ>0 (x-δ, x+δ) it is [tangent]" Derivative / differentiability "Although the function is differentiable at A and thus it has a tangent, the extension of the [line] ε that goes through A has another common point with the function and as a result it is not a tangent" Acceptance of the line as a tangent Common points between line and curve, local view "[It is a tangent, b]ecause if we consider a small region (κ, γ) around the point A where [the line] ε is tangent we can see that [the line] ε does not touch any other point" Relation of the line and the curve, local view "Yes [it is tangent] because it touches exactly at [the point] A and it does not cut it [the graph]" Relative position of the line and the curve, local view "The part of the function graph which is close to the point A is located at the same side of the line ε" Common points between line and curve, global view "It [the line] has one common point with the curve" Relative position of the line and the curve, global view "f(x)>(ε)" Slope of the line "Yes, the line ε is tangent at A, the slope equals to the direction coefficient of the line" 2 Derivative / differentiability "It is [tangent] because it has slope [equals to] the derivative of the function at this point" Opposite rays

  line as a tangent Common points between line and curve, global view "No [it is not a tangent], because the curve and the line cut each other in several points" Relation of the line and the curve, global view "It is not [a tangent] because [the line] penetrates the curve" Relative position of the line and the curve, global or local view "It [the line] intersects the function graph by going to its both sides" Inflection point / concavity change "It [the line] is not tangent because A is inflection point" Change of function formula "Because the formula of the graph changes" Derivative / differentiability "The graph does not have tangent at [the point] A because the graph is an image of a function f and A(x 0 ,f(x 0 )), f΄(x)=κ for x<x 0 and f΄(x)=λ for x>x 0 κ≠λ close to x 0" Solution of the corresponding system of simultaneous equations (line and curve) "[The line] ε is not tangent because the system linecurve has one solution and not a double solution" Other "more than 1 lines can be sketched through point A with at least one common point with the graph" or "if we consider the figure as two figures with A as the unique common point the line ε is a tangent of the two figures. If we consider the figure as a whole the [line] is not a tangent of this figure» Acceptance of the line as a tangent Common points between line and curve, local view "The ε has only point in common with Cf in the region (xΑ-κ, xΑ+κ), κ>0 and very small" Common points between line and curve, global view "The line ε is tangent because it has one common point with the curve and the concavity of the graph changes at this point" (these participants rejected the line when it had more than one points in common) Opposite rays "It is [tangent] but for the right part of the function after A, tangent is the right part of the tangent and respectively for the left [part]" Slope of the line "Yes, the line ε is tangent at A, the slope equals to the direction coefficient of the line" Derivative / differentiability "derivative equals to the slope of the tangent" Inflection point / concavity change "It is [tangent] and the [point] A is inflection point" Other "The ε is tangent at the point A especially internal"

  Figure 2: Student responses Student S[123], on the other hand, who had difficulties with accepting tangency at an inflection point writes in question Q1: "The tangent line is a line that touches a graph at a point A and in a small area around it [the point] it does not intersect the graph" [geometrical narratives applied locally]. Then, in question Q2 he writes: "The slope of the tangent at the point A=(x0, f(x0)) is equal to f΄(x0)" [analytical narrative]. In Q3.a and Q3.b he accepts the line and justifies the choice: "Because if we consider a small interval (κ, γ) around the point A where [the line] ε is tangent we can see that [the line] ε does not touch any other point" (Figure 2b) [geometrical narratives applied locally]. In Q3.d he responds "εit is not [tangent] because it bisects f" [geometrical narrative].

Table 2 : Student choice justifications to questions Q3.d and Q3.e

 2

	In both set of questions students engage with analytical, geometrical or algebraic discourses. They
	use narratives such as derivative, differentiability, intervals, regions close to the tangency point,
	inflection point or concavity (analytical discourse); common points, relative position of curves,
	same-plane or ray (geometrical discourse); and, slope or system of simultaneous equations
	(algebraic discourse). Routines include checking for common points or for the relative position
	between line and curve (geometrical discourse) or for derivatives (analytical discourse) or slopes
	(algebraic discourse). Routines are applied locally around the point A (analytical discourse) or
	globally for the whole figure (geometrical discourse). Indicatively, of the justifications offered in
	Q3.

c, 83.3% were geometrical (either global or local); 12.1% analytical; and, 4.6% a mixture of analytical and geometrical/algebraic. Whereas, in Q3.d, 70% were geometrical (either global or local); 2% algebraic; 22% analytical; and, 6% a mixture of geometrical and analytic/algebraic.

  It satisfies the equation of the tangent as well as of the figure. It is λ=tanω (the angle between the figure and x'x)" [algebraic narratives]. In Q3.a and Q3.b she accepts the line and justifies the choice: "(ε) is tangent at A [the line is] at the same side of the graph" [geometrical narratives applied globally]. In Q3.c she writes "(ε) is tangent at A only. There [the line is] at the same side of the graph" [geometrical narratives applied locally]. I have highlighted "there" in her response as an indication of the focus at the area around point A. In question Q3.d she responds: "(ε) not tangent. It intersects with the graph going through both its sides" [geometrical narratives applied locally]. Finally, in Q6 she responds by using mainly geometrical narratives with symbolisation from analysis:The tangent line of a function graph at a tangency pointA=(x0, f(x0), that belongs the function and the tangent, is a line that intersects the function without going from its one side to the other but remains at the same side of the function with only one common point the [point] A.[her emphasis]

)=λ=(y2-y1)/(x2-x1), M(x,y), y2-y1=f(x)(x2-x1)" [algebraic narratives]. Then in Q2 she responds: "The [point] A is a tangency point and belongs to the figure.

Proceedings of CERME10

In the Greek curriculum, the "direction coefficient" is the coefficient m in y=mx+b, that indicates the slope of a line.

Acknowledgements

I would like to thank Elena Nardi and Athina Thoma for the inspiring conversation about the analysis of these data from a commognitive perspective.

2011) about the role of examples in student meaning making of tangency.


text:
Chiara Andrà
email: chiara.andra@gmail.com

An analysis of freshmen engineering students' notes during a preparatory mathematics course

Keywords: Secondary-to-tertiary education transition, students' notes, narrative approach

I analyse the notes of a group of first year engineering students who attended a course in precalculus mathematics. Being interested in verbalisation skills at the beginning of university, I adopt a narrative lens to analyse the notes: I see a lecture as a story being told by the teacher and the students' notes as re-tellings of the teacher's story. I focus on the way the students condense the mathematical content in their written notes in two distinct teaching formats: a traditional frontal lesson and the concluding phase of a classroom discussion after a

Introduction

This paper is part of a wider research project on freshmen engineering students at the Polytechnic of Milan, aimed at understanding their difficulties during the first year of their studies. Recent research has found that mathematics at tertiary level is difficult for engineering students (see e.g. Gomez-Chacon, Griese, Rosken-Winter & Gonzales-Guillen, 2015). [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF] provides a detailed overview of mathematics education studies concerning the transition from school to university and she identifies the theme that is the core of my research, namely: the students' organisation of knowledge that is new to them at the beginning of university. [START_REF] Boesen | The relation between types of assessment tasks and the mathematical reasoning students use[END_REF] argue that the kind of task assigned to the students affects their learning: tasks with low levels of cognitive demand lead to rote-learning by students and, consequently, their inability to solve problems that are unfamiliar to them (for instance, the ones that require conceptual understanding). [START_REF] Breen | The use of unfamiliar tasks in first year calculus courses to aid the transition from school to university mathematics[END_REF] define an 'unfamiliar task' as a task "for which students have no algorithm, well-reharsed procedure or previously demonstrated process to follow" (p. 2318), and provide evidence that this kind of tasks raises an awareness about the need for more than procedural understanding of mathematics, thus easing the students' transitions to university math practices. Also, the teaching strategies employed in the class can influence the development of one type of knowledge more than another: teacher-centred methods would favour the development of procedural knowledge and student-centred methods would favour conceptual knowledge (see e.g. [START_REF] Garner | Retention of concepts and skills in traditional and reformed applied calculus[END_REF][START_REF] Allen | Students' retention of mathematical knowledge and skills in differential equations[END_REF]. Inspired by these studies, I investigate how the students organise the knowledge in their notes during a preparatory math course.

To take notes does not involve only mathematical ability. It involves also verbalisation skills. O'Neill, [START_REF] O'neill | Predictive relations between aspects of preschool children's narratives and performance on the Peabody Individualized Achievement Test -Revised: Evidence of a relation between early narrative and later mathematical ability[END_REF] found that there is a correlation between performance in generating narratives and mathematical ability as early as in primary school years. With [START_REF] Nardi | Driving noticing' yet 'risking precision': university mathematicians' pedagogical perspectives on verbalization in mathematics[END_REF], I recognise the centrality of the students' ability to use ordinary language to construct and convey mathematical meaning and I investigate undergraduate engineering students' notes in a preliminary math course at their first year at university. With [START_REF] Nardi | Driving noticing' yet 'risking precision': university mathematicians' pedagogical perspectives on verbalization in mathematics[END_REF], I value the students' attempts to mediate the mathematical meanings through words, symbols and diagrams and I maintain that at the basis of the students' difficulties in dealing with a discursive shift from secondary to university mathematics there are: (a) undervalued verbalization and (b) premature compression. According to the former, "the students undervalue, and often avoid entirely, expressing their mathematical thoughts verbally" [START_REF] Nardi | Driving noticing' yet 'risking precision': university mathematicians' pedagogical perspectives on verbalization in mathematics[END_REF](Nardi, , p.2056)); according to the latter, "students' mathematical writing is typically prematurely compressed, namely ridden with gaps, leaps and omissions" (ibid.). [START_REF] Andrà | How do students understand mathematical lectures? Note-taking as retelling of the teacher's story[END_REF] examines the relationships between a teacher's lecture and the students' notes by viewing the lecture as a kind of story that the teacher tells and the students' notes as retellings of the teacher's story. A mathematical lesson seen as a story can be analysed in terms of its components: its characters, setting, action, plot, and moral (see also [START_REF] Bal | Narratology: Introduction to the theory of narrative[END_REF]. Mathematical objects, in fact, can be considered the mathematical characters of a story (Dietiker, 2012). They can play a central or a peripheral role, have multiple names, and have properties that can be introduced and developed gradually. The setting is the space where characters are placed. Sometimes the setting is not obvious, as it refers to underlying assumptions and/or axioms. The setting may also involve different registers, such as algebra or the Cartesian coordinate system. The action is that which the actor performs. In mathematical stories, the result of an action can be a change in an object or in a setting, or both. According to Dietiker (2012), we see that, unlike in literary stories, mathematical ones can change actions into objects (through reification). [START_REF] Andrà | How do students understand mathematical lectures? Note-taking as retelling of the teacher's story[END_REF] observes that the students miss important (teacher's) mathematical actions in their notes and more in general [START_REF] Morgan | Writing mathematically. The discourse of investigation[END_REF] notices a relative absence of active verbs in mathematical writing. The moral can be seen as the intended message of the lesson. The plot is the sequence of actions and it involves the shaping of the story, which is linked to its aesthetic effects: for instance, the rhythm and the frequency of the story [START_REF] Bal | Narratology: Introduction to the theory of narrative[END_REF] may affect the students' focusing on the areas of emphasis of the story and foster his anticipative acts. Some moves might displace attention away from what the teacher wants to communicate: for example, repeating the name of a character often may lead the students to think that the actual name is important, or more important than its properties. Other moves may induce the students to believe that the setting is unimportant. These moves can be interpreted in terms of [START_REF] Rotman | Toward a semiotics of mathematics[END_REF] schema, which distinguishes between invitations for the reader to be a "thinker" and ones that prompt the reader to be a mere "scribbler". This distinction provokes a further distinction, namely: to think about note-taking as mere consumption of mathematical meaning or to think about it as active production of meaning. It is possible to interpret Rotman's schema with respect to the students' notes in this way: a scribbler is a student who reports mainly the mathematical characters of the story and misses the actions, so that the notes result to be compressed and ridden with gaps, leaps and omissions (see also [START_REF] Nardi | Driving noticing' yet 'risking precision': university mathematicians' pedagogical perspectives on verbalization in mathematics[END_REF]. A scribbler is also a student who avoids putting her thoughts in her notes, and limits herself to copying and/or reporting what the lecturer is saying/writing. The plot is the same plot as that of the story told by the lecturer. A thinker, instead, re-organises the content of the lesson, she (re)structures the plot so that it becomes accessible to herself even after the lesson ends. A thinker pays attention to the details and also records the mathematical actions so that her notes are not overly compressed and under-verbalised. In [START_REF] Andrà | How do students understand mathematical lectures? Note-taking as retelling of the teacher's story[END_REF] understanding of Rotman's schema, furthermore, some moves invite the students to be thinkers while others invite them to be scribblers. In view of the findings of [START_REF] Boesen | The relation between types of assessment tasks and the mathematical reasoning students use[END_REF] we consider two scenarios: in the first one, the lecturer proposes a group activity on a conceptual and unfamiliar task and the students' notes are taken during the classroom discussion that follow the group activity; in the second scenario, the lecturer assigns a procedural task and corrects it on the blackboard. The interest is to see how the students' notes change (if so) in the two different scenarios.

Theoretical framework

Methodology

The Polytechnic of Milan, like many universities all around the world, organizes some courses before the beginning of the first semester, which have the purpose to recapitulate the basic knowledge that is necessary for the students to successfully attend the courses at the first academic year. One of these preparatory courses is on pre-calculus mathematics. Since three years, the course is organised according to a flipped classroom pedagogy: the students (are supposed to) watch a series of videos in a MOOC and at university the lecturers of the preparatory course involve them in groupwork activities aimed at deepening their understanding of the basic math concepts and expose the students to frontal lessons with routine exercises. The mixed method of teaching serves the purpose of both exposing the students to a new, "conceptual" teaching and to make them feel comfortable with teaching practices that are more typical of secondary school. During the first lesson of the preparatory course, among the exercises given about polynomials, one had a conceptual nature, since it said "The polynomial p(x) is divisible by the polynomial q(x) if…". This is a kind of task that is unfamiliar for Italian students, since it asks for reflection about the definitions and the students do not have a wellestablished procedure to resort to. It was first dealt with in small groups, then discussed at classroom level. During this last phase, the teacher wrote the steps of the solution at the blackboard and the students took notes. Another task had a procedural nature and was not unfamiliar for the students: two polynomials were given, p(x) and q(x), and the students were asked to divide p(x) by q(x). It was solved at the blackboard by the teacher. I compare and contrast the students' notes in these two different situations: one familiar and procedural (i.e., linked to actions), one unfamiliar and conceptual (i.e., linked to characters). In this study, the lecturer under consideration is also the researcher and the author of this paper. At the end of the lesson, the students were informed about the study and invited to provide their notes for research purposes. I am aware of the potential issues concerning this method of data collection, and I followed ethical guidelines in order not to expose the students to risk. Ten students offered their notes; of those I selected four to be analysed in this paper since they are contrasting. The students are identified with four fictitious names: Angela, Filippo, Roberto and Vincenzo. Their notes are analysed through a narrative lens, identifying: the mathematical characters; their setting; the mathematical actions, understood in terms of operations made on/by the mathematical characters; the plot, or the organisation of the content on the sheet of paper; the moral. The four students are inferred to be scribblers or thinkers by looking at these elements. The research questions read as follows: (a) how do students organise their notes? (b) which elements of the teacher's "story" are recorded, and which ones are discarded? (c) in which cases do the students act as scribblers and in which ones as thinkers?

Data analysis

Figures 1-8 report the four students' notes regarding the two tasks. Since they are in Italian, a translation is provided in the caption of each figure.

Figure 3: Filippo's notes about the "conceptual task". On the first row, he writes "q(x) divisible by p(x)", then he draws an arrow and writes a ratio, from which he draws another arrow and writes "polynomial". On the second row, he writes "example" and in the third row he writes a formula. In the last row: "hence it means that there exist a polynomial g(x) such that I can write q(x) = a(x)p(x)".

Figure 4:

Roberto's notes about the "conceptual task". On the first row, he writes "q(x) divisible by p(x)", then he draws an arrow and writes "a polynomial a(x) | q(x) = a(x)p(x)".

Vincenzo's notes (Figure 1) report exactly what the lecturer wrote on the blackboard. From these notes, we can see that the story has: (A) two mathematical characters, p(x) and q(x) and there's a relationship between them, being the latter divisible by the former; (B) the mathematical action is a division; (C) the moral is that there exists a polynomial a(x) such that q(x) = a(x)p(x). Vincenzo's plot is linear: each row is put below the other one, with no connections. The plot of Angela's story (Figure 2) is also linear: (A) is followed by (B) that is followed by (C). But she also adds an arrow on the right side of (B) and she writes "result", then another arrow and then "polynomial". These On the first row, he writes "q(x) divisible by p(x)". On the second row, he writes a formula and in the last row: "polynomial a(x) s.t. q(x) = a(x)p(x)".

Figure 2: Angela's notes about the "conceptual task". On the first row, she writes "q(x) divisible by p(x)". On the second row she writes the ratio, then she adds an arrow and writes "result", then another arrow and "polynomial". On the third row, she writes "hence there exists a polynomial a(x) such that q(x) = a(x)p(x)".

details have been told by the lecturer orally. For Angela, it is worth noticing that the result of the mathematical action is a polynomial (which is a(x), a new character), namely a character that still has the same properties of the other two. While the setting was implicit in Vincenzo's notes, it emerges in Angela's ones: the setting is the set of polynomials. Instead of the mathematical symbol for "there exists", she writes in words and she also adds "hence" at the beginning of (C): the moral is made more explicit and the verbalisation is less condensed compared to Vincenzo. Filippo (Figure 3) organises the plot in a non-linear way: he writes (A) and on the same row he writes (B), to which he draws an arrow and writes "polynomial". Like Angela, Filippo also remarks about this detail. Vincenzo and Angela do not write down the example, while Filippo does. Hence, a story in the story is told: it's the story of the two characters that become two particular polynomials. At the end of the story, Filippo writes (C) in a fashion that is similar to Angela's.

Filippo's story is slightly less linear than the stories re-told by Angela and Vincenzo, but the student that writes a different plot is Roberto (Figure 4): he writes (A), an arrow, then (C) on the first row, namely he puts at the first line the characters and the moral, then on the second row he writes the action, which is (B), and the story in the story, namely the example. We can infer that Roberto is a thinker, since he re-organizes the knowledge, while Vincenzo is a scribbler, since he reports the story in a linear way. Angela and Filippo also act as scribblers: in a sense, we can say that they are more accurate than Vincenzo, since they report more details, but do not re-organise the content of the lesson as Roberto does. Roberto, in fact, does not only remark what is worth noticing, he establishes a hierarchy in the mathematical content: characters and moral on the same, first row, and the action plus the example on the same, second row.

Figure 5: Filippo's notes about the "procedural task". On the first row, he writes "Exercises on euclidean division and Ruffini's division". To the right of the first arrow he writes "Ruffini is used because a first order polynomial is present". To the right of the arrow pointing to -60, he writes "this division is a factorisation, because there's no reminder".

If we look at the "procedural task" in Filippo's notes (Figure 5), we notice that he employs a more linear structure compared to the "conceptual" task. He writes: (0) the title of the story ("Exercises on the Euclidean division and Ruffini"), then (1) he presents the characters P(x) and Q(x), then (2) the series of actions in the Ruffini's grid. The new character, (3) A(x), the result of the actions, is present to the right of the grid. At the right side of the paper he adds comments that are connected to the "story" by means of arrows: such comments better characterise, and justify, the actions that are made. Like in the conceptual case, we see them as details that are worth to be noticed by Filippo. Like in the conceptual case, we can say that Filippo is an accurate scribbler. Vincenzo's notes (Figure 6) also have a linear structure with no connections between (1) and ( 2), or between (2) and (3). Vincenzo also adds the comment "we use Ruffini when we divide by an order-1 polynomial", but this comment about the actions is put below the characters with no arrow. Angela's notes (Figure 7) reflect exactly what the lecturer wrote on the blackboard. The notes have no words, just symbols: we can say that there's only one register present, the symbolic one. She records the characters, i.e. (1), the actions, i.e. (2), and the new character that results from the action, i.e. (3). Differently from Filippo, whose notes have a rather linear structure, Angela's ones are even more linear and essential, as if she wants to record just the essential facts. Like Vincenzo, Angela is an (inaccurate) scribbler.

Roberto (Figure 8) records (1), then (2), then ( 5)-( 3)-( 4) on the same line. To the right of (1) he remarks "Ruffini, because x=5 is such that P(5)=0", hence noticing a detail that is different from the ones recorded by Filippo and Vincenzo and also less general than those: Ruffini's algorithm can be used for any value of x when q(x) is an order-1 polynomial, not only for those polynomials where the value of x is a zero. Since the lecturer has said something different (see Filippo's or Vincenzo's notes), we can infer that Roberto added a detail that generated from his own knowledge about polynomials. As for the conceptual task, Roberto reorganises the space of the sheet and we can infer that he acted as a thinker.

Discussion

I discuss the data analysis in terms of what the distinction between scribblers and thinkers can add to:

(1) our knowledge of the students' verbalisation skills (responding to the question which elements of the teacher's "story" are recorded, and which ones are discarded?); (2) our understanding of how the students organise their knowledge (how do students organise their notes?); (3) whether the teaching strategies have an influence on conceptual and procedural understanding of mathematics (when do the students act as scribblers and when as thinkers?). I would like to underline that I am not valuing "thinker" over "scribbler", on the contrary I am interested in seeing which elements of the lesson provoke either modality in students. Comparing Angela's and Vincenzo's notes, I tend to say that both them act as scribblers in both tasks. I can further infer that Angela is a scribbler because the course is recapitulating mathematical concepts that are familiar for her: she probably does not need to put so many details in her notes. Angela's notes of the procedural task report only what has been written on the blackboard. Looking at her notes for the conceptual task, furthermore, I commented that she had time to record details that are worth to be noticed and she didn't give us the impression that she was rushing to keep the pace of the lecturer. For Vincenzo, it is a completely different story: he records only what has been written on the blackboard during the conceptual task and during the procedural one he added "We use Ruffini when we divide by an order-1 polynomial" to his notes with respect to what has been written on the blackboard. I can see that Vincenzo is struggling to remark all that is relevant, since the lesson is difficult for him. A conclusion that can be drawn from these observations is that a student acts as a scribbler in two cases: either if the mathematical content is too easy for her, or if it is too hard.

Nardi (2011) pointed out that the students under-verbalise and hyper-condense the mathematical discourses. As regards the conceptual task, I can see that Vincenzo and Roberto condense the mathematical content more that Filippo and Angela, but Roberto does it in a completely different way compared to Vincenzo: Roberto compresseseorganises the content, to have the character and the moral on the same row, and the actions plus the example on the second row, while Vincenzo linearly puts the elements of the story one after the other. [START_REF] Andrà | Teaching styles in the classroom: how do students perceive them?[END_REF] analysed the teaching styles of university lecturers and concluded that in a blackboard modality (namely, when the lecturer is mostly writing on the blackboard) the students have to adjust the pace of their note-taking to the pace of the lecturer's writing. By comparing Vincenzo's and Roberto's notes, indeed, I can imagine the former making an effort in dealing with a pace that is too fast for him, to the point that he does not have time to record the details that Angela and Filippo remarked, while Roberto seems to stop and think (fast) where he wants to put what is told by the teacher. Angela and Filippo can be seen as accurate scribblers, and it seems that they tend not to hyper-condense the math content. As well, Angela and Filippo tend not to under-verbalise when they take notes on the conceptual task. Why are some students more accurate scribblers than others? [START_REF] Andrà | Teaching styles in the classroom: how do students perceive them?[END_REF] interpreted this difference in terms of each student's ability to keep the pace of the lesson at the blackboard, but I would also add that it depends on the student's views: for some students, it seems necessary to record all the possible details, while for others it seems a question of being brief. Looking closely at Angela's notes, and comparing her notes on the conceptual task and on the procedural one, I can see a difference: in the first case, she adds details and comments that she discards in the second case. Vincenzo does not add comments in neither case, and Filippo accurately adds details in both cases, hence Angela seems to be the student on which the procedural vs conceptual nature of the task provokes different modalities of taking notes, and actually the conceptual nature of the task invites her to remark more details. This seems to have an impact on her verbalisation skills and on conceptual reflection.

Figure 1 :

 1 Figure 1: Vincenzo's notes about the "conceptual task". On the first row, he writes "q(x) divisible by p(x)". On the second row, he writes a formula and in the last row: "polynomial a(x) s.t. q(x) = a(x)p(x)".

Figure 8 :

 8 Figure 8: Roberto's notes about the "procedural task".

Figure 6 :

 6 Figure 6: Vincenzo's notes about the "procedural task". The first sentence reads "We use Ruffini when we divide by an order-1 polynomial".

Figure 7 :

 7 Figure 7: Angela's notes about the "procedural task".

Proceedings of CERME10


text:
Marinos Anastasakis
email: m.anastasakis@lboro.ac.uk

Carol Robinson
email: c.l.robinson@lboro.ac.uk

Stephen Lerman
email: s.lerman@lboro.ac.uk

A categorisation of the resources used by undergraduates when studying mathematics

Keywords: Blended learning, educational resources, student centered learning, activity theory, Wartofsky

This paper presents results from a survey exploring the kind of resources that engineering undergraduates (N=201) use when studying for their mathematics modules. By using Factor analysis, we were able to produce a typology of these resources. The resulted typology was further analysed by combining Leontiev's version of Activity Theory and Wartofsky's hierarchy of artefacts. This helped us to draw links between the tools that students use and infer about their learning actions.

Introduction

Historical records suggest that using tools has always been inseparable from expressing and doing mathematics [START_REF] Roberts | From the Slate to the Web: Technology in the Mathematics Curriculum[END_REF] with numerous examples demonstrating the capacity of tools to influence the development of mathematics itself as a scientific discipline [START_REF] Laborde | Place and use of new technology in the teaching of mathematics: ICMI activities in the past 25 years[END_REF] 1 . Since the end of the nineteenth century many different types of tools have been used for the teaching and learning of mathematics [START_REF] Kidwell | Tools of American Mathematics Teaching, 1800-2000[END_REF]) however for many the term "technology" corresponds to electronic or digital in nature tools (e.g. calculators) something that manifests a kind of historical amnesia [START_REF] Roberts | From the Slate to the Web: Technology in the Mathematics Curriculum[END_REF]. Students nowadays have access to a plethora of digital/online resources that they could use alongside more "traditional" ones (e.g. textbooks, lecturers or their own notes) and thus blend their learning. As a matter of fact, [START_REF] Masie | The blended learning imperative[END_REF] asserts that this has been always the case as students were always combining resources in order to support their learning. The notion of blended learning (BL) has been introduced almost 30 years ago however, the term has not been clearly defined yet [START_REF] Bos | Profiling student behavior in a blended course: Closing the gap between blended teaching and blended learning[END_REF][START_REF] Graham | Emerging Practice and Research in Blended Learning[END_REF][START_REF] Torrisi-Steele | The literature landscape of blended learning in higher education: the need for better understanding of academic blended practice[END_REF]. The three main definitions of BL used in education are [START_REF] Sharma | Blended learning[END_REF]: BL as a combination of face-to-face and on-line teaching; BL as a combination of technologies and; BL as a combination of pedagogical methodologies. In this paper, we are mainly referring to BL as a combination of technologies or better as a combination of tools since each era has its own technologies not only digital ones. Despite what its name implies, BL has received criticisms and some authors argue that it should be rather called blended teaching because current views adopt a teacher-centred and not a student oriented perspective by focusing on the resources that instructors choose for their students [START_REF] De George Walker | Self-determined blended learning: a case study of blended learning design[END_REF][START_REF] Oliver | Can "Blended Learning" Be Redeemed? E-Learning and Digital Media[END_REF]. If students indeed blend their learning by mixing different resources not just digital or the ones provided to them by their institution, what kind of resources do they blend and how can these resources be classified? In terms of the resources that students use, we have found a lack of empirical studies exploring the kind of resources that undergraduates themselves choose and use, with previous studies focusing 1 Although many authors consider the terms "tools", "artefacts", "instruments" or "resources" to be different, for the purposes of this paper we treat them as having the same meaning.

Thematic Working Group 14

Proceedings of CERME10 mostly on digital/online or institutionally-led resources and thus neglecting the "blended" nature of their learning (for a short review see [START_REF] Anastasakis | Do you really know what resources your students use to learn mathematics[END_REF]. In effect, our sense of the field aligns with comments from authors suggesting that the student perspective has not been taken into account in the BL literature (e.g. [START_REF] Ituma | An evaluation of students' perceptions and engagement with e-learning components in a campus based university[END_REF][START_REF] López-Pérez | Blended learning in higher education: Students' perceptions and their relation to outcomes[END_REF]. Having previously identified the kind of resources that a sample of engineering undergraduates uses when studying mathematics [START_REF] Anastasakis | Do you really know what resources your students use to learn mathematics[END_REF], our aim in this paper was to propose an empirically based typology of these tools.

Theoretical framework

Long before the dominance of the world wide web and computers, many researchers had emphasised the significance of tools in our everyday activities; a well-developed theoretical account of human praxis which emphasises the role of physical tools is second generation Activity Theory (AT) [START_REF] Leontiev | The problem of activity in psychology[END_REF]. From an AT perspective, our relationship with the "objective" world is mediated by tools: the central role of tool mediation within this framework is due to the fact that tools shape the ways we interact with reality and they reflect past people's experiences and practices [START_REF] Kaptelinin | Acting with technology: Activity theory and interaction design[END_REF]. In AT, activities have a hierarchical structure and they consist of three layers (Ibid.): at the top is the activity itself directed towards the object of the activity (our overarching goal); in the middle lie actions (what we do) directed at goals (what we want to achieve) and; finally, at the lower layer we find operations (non-conscious, routine processes) which are directed to conditions (nonsubjective factors that affect our actions). In sum, AT asserts that tools are the means by which subjects are trying to achieve their goals and in this sense tools are bounded with a subject's practice. Despite the central position that tools hold in AT, little is written from this perspective in regards to how they can be categorised; among the little accounts found in the literature, is Wartofsky's typology of artefacts. [START_REF] Wartofsky | Perception, Representation, and the Forms of Action: Towards an Historical Epistemology[END_REF] considered tools as the genes of our cultural evolution and proposed that they can be classified into primary, secondary and tertiary artefacts. As primary are considered the tools themselves [START_REF] Engeström | When is a tool? Multiple meanings of artifacts in activity[END_REF]; secondary artefacts represent "modes of action using primary artefacts" (Cole, 1996, p. 121) and they "synthesise the ways and procedures of using instruments and materials" (Miettinen, 1999, p. 189). Finally, tertiary artefacts are those which emphasise creativity [START_REF] Kaptelinin | Acting with technology: Activity theory and interaction design[END_REF] and "transcend[s] the more immediate necessities of productive praxis" [START_REF] Wartofsky | Perception, Representation, and the Forms of Action: Towards an Historical Epistemology[END_REF]. As an example of what constitutes a primary and a secondary artefact, [START_REF] Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF] refer to the abacus: the abacus itself is a primary artefact and the ways of using abaci for counting, keeping records or making computations represents a secondary artefact. Engestrøm notes that Wartofsky's typology is closely related to Leontiev's levels of activity [START_REF] Petersen | The usability of everyday technology: emerging and fading opportunities[END_REF]: primary artefacts correspond to the level of operations/conditions, secondary to the level of actions/goals while tertiary to the level of the activity [START_REF] Engeström | When is a tool? Multiple meanings of artifacts in activity[END_REF][START_REF] Engeström | Learning by Expanding[END_REF]. Our focus in this paper are primary and secondary artefacts.

Method

This study is part of a doctoral project that aims to identify the kind of tools that undergraduates use when studying mathematics, how these tools are used and the reasons for using them. During the autumn term, a paper-based questionnaire was administered to four different groups of second year engineering students in Loughborough university and in total 201 completed it. Loughborough has one of the largest cohorts of engineering students (over 3000 undergraduates) in the UK and a well established provision of Mathematics Support (http://www.lboro.ac.uk/departments/mlsc). It has also led on significant projects producing high quality printed material (e.g. the HELM project: http://helm.lboro.ac.uk). The questionnaire consists of three main parts and its design was guided by Activity Theory (AT) [START_REF] Leontiev | The problem of activity in psychology[END_REF]. Here we report only on the part related to the resources (tools) that undergraduates use. In this, students were explicitly asked to identify how often they use a list of 14 resources on a 6-point semantic scale (1/Never, 2, 3, 4, 5, 6/Always) with two additional open ended items for other resources not listed in the questionnaire. The list was based on our literature review, five in depth interviews with undergraduates conducted in 2015 and the resources that Loughborough University offers to students e.g. the Learn website (university's VLE). The list of resources was carefully generated and encompasses a great variety of tools available to students; in this way it reflects -to a certain degree-students' reality as learners when it comes to the resources they use. Students were also asked to identify which five of these 14 resources they use the most (top-5) and rank them in a descending order (not reported here).

Analysis and results

Summary statistics

Results for the tools that students use are presented in Figure 1. These results have been already presented elsewhere [START_REF] Anastasakis | Do you really know what resources your students use to learn mathematics[END_REF] but we include them here for clarity. By using each tool's mean, we categorised them into three main groups: tools with a mean greater than or equal to 4.5 were characterised as high-use, those with a mean between 3 and 4.5 were assigned into the miduse group while resources with a mean between 1.5 and 3 were put into the low-use group.

Factor Analysis

Exploratory Factor Analysis (EFA) is a statistical method aiming at grouping variables which have something in common (i.e. they correlate with each other). This enables researchers to identify latent constructs in the data that cannot be measured otherwise directly. Each group (or cluster) of variables is then called a factor and the variables consisting each factor are thought to be measuring the same underlying/latent construct i.e. each factor represents one underlying construct. An initial EFA was performed on all the 14 variables for tools. We used an oblique rotation because the underlying constructs sought in our data were expected to be related (all variables are related to tool-use after all). Our sample's adequacy was measured by the Kaiser-Meyer-Olkin measure of sampling adequacy and found to be above the minimum value of .5 (KMO=.711). Bartlett's test of Sphericity also showed that the correlations between variables are not 0 i.e. the correlation matrix is not an identity matrix: this is true when the significance value for this test (p) is less than .05 and in our case, it was p<.001.

The scree plot was used as a criterion for determining how many factors should be kept. Two lines that best summarise the scree plot were drawn with the intersection of these lines (called point of inflection) indicating how many factors are present in our data, excluding the factor on the point of inflection (in our case 3 factors). We additionally examined both the pattern and structure matrices and decided to exclude the variables "own written lecture notes", "HELM workbooks", "Learn website" and "Wolfram Alpha" from our subsequent analysis. This was done because these variables were either having factor loadings below the cut-off value of .364 that we used based on our sample's size (see Stevens, 2002, p. 374) or because only one variable was present on a single factor (the goal of EFA is to group similar to something variables). Based on this analysis, we run a second EFA on the 10 remaining items with an oblique rotation (KMO = .711, p<.001) by requesting a 3 factor solution. Both the pattern and structure matrices were interpreted (Tables 1 and2). Factor loadings (numbers at structure and pattern matrices) can be thought as the correlations between each variable and the factors and they represent how well a variable "fits" into a factor. The final obtained factors included the following variables:

 Factor 1 (5 variables): "Mathematics Learning Support Centre", "other textbooks", "lecturers", "pre-university notes" and "staff at tutorials"  Factor 2 (3 variables): "other students", "instant messaging" and "social media"  Factor 3 (2 variables): "online videos" and "online encyclopaedias".

At this point, we decided to treat the 4 variables not loading on any Factor ("HELM workbooks", "own written lecture notes", "Learn website", "Wolfram Alpha") as latent constructs too. This decision was made for two reasons: first, these variables did not load on any Factor (i.e. not relating with other variables); and second, the nature of each resource is different and unique when compared with the other resources, thus they can be thought "measuring" something on their own. By adopting a descriptive approach [START_REF] Rummel | Interpretation[END_REF] we named the 7 identified types of resources as follows:

1. The "official" mathematical textbook: "HELM workbooks" 2. Students' lecture notes: "own written lecture notes" 3. University's VLE: "Learn website" 4. The calculator: "Wolfram Alpha" 5. Teaching staff: Factor 1 6. Peers and communication tools: Factor 2 7. External online tools: Factor 3 From the above types of resources, "teaching staff" (Factor 1) contains 5 variables which they seem not fitting together; is it reasonable to interpret together different in nature variables such as "preuniversity notes", "other textbooks" and "lecturers" for example? In our opinion, it makes good sense since they correspond to students' direct interactions with university's teaching staff ("Mathematics Learning Support Centre", "lecturers", "staff at tutorials") and ways that students interact indirectly with teaching staff; this includes the use of resources probably suggested by teaching stuff ("other textbooks") or resources which are the product of prior interactions with a person holding a teaching position e.g. A-levels tutor ("pre-university notes").

Discussion

Our aim for this paper was to produce a typology of the resources that engineering students in our sample reported using. By performing an Exploratory Factor Analysis on our data, we were able to identify 7 different types of resources that undergraduates in our sample reported using (Table 3, left column). From a Wartofskian point of view, all the tools used by undergraduates when examined separately are primary. On the other hand, secondary artefacts represent "modes of action using primary artefacts" [START_REF] Cole | Cultural psychology[END_REF] and they "synthesise the ways and procedures of using instruments and materials" (Miettinen, 1999, p.189, our emphasis); this means that our proposed typology corresponds to the different secondary artefacts that undergraduates use. When examined from an AT perspective, the typology of tools corresponds to thematically related actions that students undertake when studying mathematics. This is because actions in AT are the "...specific interactions that people have with artefacts and other people..." [START_REF] González | Ensembles: understanding the instantiation of activities[END_REF] i.e. as actions we account the processes of using a tool (types 1, 2, 3, 4 and 7) and/or interacting with other subjects (types 5 and 6). Our Wartofskian and AT-based interpretations are also consistent from a statistical point of view: students were asked how frequently they use a resource thus, the nature of each variable is related to using a tool or interacting with a person i.e. an action from an AT perspective (Ibid.) or the ways of using primary artefacts. Thus, from both a Wartofskian and AT perspective, the 7 different types of resources that undergraduates use, correspond to the following secondary artefacts/actions (

Conclusion

In this paper, we analysed survey data from a cohort of second year engineering students (N=201) about the kind of tools they use when studying for their mathematics modules. In contrast with common approaches found in the literature, we did not focus only on digital/online or institutionally provided resources but rather we incorporated a variety of resources that students have at their disposal. An Exploratory Factor Analysis of our data allowed us to identify 7 different types of tools that students in our sample reported using: the "official" mathematical textbook ("HELM workbooks"); their own written notes; university's VLE ("Learn website"); a sophisticated calculator ("Wolfram Alpha"); university staff (Factor 1); peers and social apps (Factor 2); and non-institutional online tools (Factor 3). By adopting Wartofsky's hierarchy of tools these dimensions represent secondary artefacts i.e. 7 different ways that students use primary artefacts or students' modes of action when studying mathematics: studying the mathematical textbook, taking notes during a lecture, accessing institutionally provided material, performing calculations, interacting with teaching staff, interacting with peers and searching online for additional/alternative sources of information. This interpretation is consistent with AT because these dimensions represent students' actions when studying mathematics. One important implication of our analysis is that although some resources are different (e.g. people, digital) they may be used by students in a similar way: this was the case of Factor 1 which contained different in nature resources. This result contradicts our common assumptions when categorising resources (e.g. people, digital, online etc.) and adds an empirical basis for the argument that the way we usually classify resources does not necessarily reflect the ways these resources are used. Because of the nature of our data, we could only infer about the nature of the 7 types of tools that undergraduates use by only examining the resources included in each type. However, our preliminary analysis of 14 interviews suggests that our interpretation aligns with these resources' actual use: for example, students who interviewed reported using Facebook for communicating with peers when having an issue with mathematics (either by using Messenger or by posting a question on Facebook groups created by undergraduates). Our intention for the future is to complement the survey data with data gathered with other methods (interviews and diaries). Finally, we are of the opinion that the results of our analysis (students' learning actions), highlight the temporal nature of all primary tools used in learning and suggest that our future research foci in mathematics education should be on the ways that these tools are used rather than the tools themselves.

Figure 1 :

 1 Figure 1: Tools and their grouping based on their mean (high-use: red, mid-use: green, low-use: blue)

Table 1 : Pattern matrix of the final EFA (please note that values below .3 are omitted) Table 2: Structure matrix of the final EFA (please note that values below .3 are omitted)

 1

Table 3

 3

	, right column):
	1. Studying the mathematical textbook
	2. Taking notes during a lecture
	3. Accessing institutionally provided material (online)
	4. Performing (complex) calculations
	5. Interacting with teaching staff
	6. Interacting with peers (in-person or virtually)
	7. Searching for external/alternative material online
	Typology of Tools	Secondary Tools (Wartofsky) -Actions (AT)
	(1) The "official" mathematical textbook	Studying the mathematical textbook
	(2) Students' lecture notes	Taking notes during a lecture
	(3) University's VLE	Accessing institutionally provided material (online)
	(4) The calculator	Performing (complex) calculations
	(5) Teaching staff	Interacting with teaching staff
	(6) Peers and communication tools	Interacting with peers
	(7) External online tools	Searching for additional/alternative resources of
		information

Table 3 : Proposed typology of tools (left) and their representations as secondary artefacts and actions

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Proceedings of CERME10