text:
Maria Meehan
email: maria.meehan@ucd.ie

Ann O'shea
email: ann.oshea@nuim.ie

Sinéad Breen
email: sinead.breen@dcu.ie

"What you see and don't see, shapes what you do and don't do": Noticing in first year mathematics lectures

Keywords: Teacher noticing, decision points, mathematics lecture, university mathematics

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Introduction

Today I was happy with my 'performance' from a teacher-centred perspective as the lecture evolved: I felt I was coherent, explained and connected ideas well, used multiple representations of concepts and built on students' prior knowledge, However, I realised more than halfway through the class that my lecture was just thatvery teacher-centred! I tried to rectify this but was not happy that my attempts were successful. (Lecturer C) Lecturer C speaks about a tension inherent in large-group mathematics lectures between teachercentered and student-centered methodologies, and this is probably familiar to anyone who has taught such a course. It also highlights the dilemma that faces a lecturer when it occurs to her mid-way through a lecture that she might deviate from her original plan. The change of plan can feel like a risky strategy, especially when working with very large groups of students; however research on mathematics teaching at school level indicates that rich learning opportunities can arise from decisions to change direction (Rowland and Zaskis, 2013). It is important then, that lecturers are aware of opportunities that present themselves, for the simple reason that the quote in the title, borrowed from Schoenfeld (2011, p. 228), suggests. It is also important that lecturers develop the skills to take advantage of these opportunities as they arise. In this paper, we explore the opportunities to make decisions that might arise in a mathematics lecture; we have named instances when a nontrivial choice between alternative courses of action could be taken 'decision points'.

The Discipline of Noticing developed by [START_REF] Mason | Researching Your Own Practice: The Discipline of Noticing[END_REF] consists of "a collection of practices which together can enhance sensitivity to notice opportunities to act freshly in the future" (p. 59). Mason notes that practitioners, of necessity, form habits in order to deal with issues that arise in their everyday practice. The practices described by Mason provide educators with techniques to assist them in developing the dual abilities to notice key events in the classroom, and secondly to have possible actions come to mind in-the-moment in order to respond to these in non-habitual ways.

Thematic Working Group 14 Proceedings of CERME10 Mason (2002, pp. 33-34) distinguishes between levels of noticing, from ordinary-noticing, through to marking, and finally recording. "Ordinary-noticing" is where a person's memory of something can be jogged if another person remarks upon it; "marking" is where someone has taken sufficient notice of something to "re-mark" upon it at a later stage; and, finally, "recording" is where one records or makes a note of something one has noticed, usually in writing. To do the latter, Mason (2002, p. 46) advocates the practice of writing "brief-but-vivid accounts". These are brief notes which give an "account-of" an incident, rather than an "account for" as Mason explains:

To account-for something is to offer interpretation, explanation, value-judgement, justification, or criticism. To give an account-of is to describe or define something in terms that others who were present (or who might have been present) can recognize. (p. 41)

The data for this paper comes from a set of brief-but-vivid accounts written by the three authors on incidents that occurred during teaching over a two-year period when they took part in a project aimed at using the Discipline of Noticing to study their own teaching (see [START_REF] Breen | A year of engaging with the Discipline of Noticing: five mathematics lecturers' reflections[END_REF] for more details).

In this paper we will present an analysis of the accounts which relate to large-group teaching of first year modules; in particular we will endeavour to answer the questions: What types of decision points can occur during a mathematics lecture? What triggers these decision points?

Literature review

Teacher noticing has been receiving attention in the research literature recently, see for example the research studies included in the book edited by Sherin, Jacobs & Philipp (2011). In presenting an overview of these studies, the editors observe that while the conceptualisation of teacher noticing may vary, it is generally considered as consisting of one or both of two main processes. The first process, "attending to particular events in an instructional setting" (p. 5), relates to where the teacher does (and does not) place her attention in the classroom setting. The second process they describe as "making sense of events in an instructional setting" (p. 5) and note that some researchers conceptualise this process only as "interpreting" (p. 9) what is noticed, while others view it as "both interpreting and deciding how to respond" (p. 9).

Instances when a choice presents itself during teaching are labeled "contingent moments" by [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] in their work on the Knowledge Quartet. At these moments teachers have to think on their feet and possibly deviate from the planned lesson. [START_REF] Rowland | Triggers of contingency in mathematics teaching[END_REF] identified three types of triggers of contingent moments in their study of mathematics teaching: responding to student ideas; teacher insight; and, responding to the availability of tools and resources. Teachers may deal with the first trigger in one of three ways: ignore; acknowledge but put aside; acknowledge and incorporate (Rowland and Zaskis, 2013). [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision-making and its educational applications[END_REF] has developed a theory to explain what influences the decisions an individual makes when carrying out a particular task. He has applied his theory in particular to decision-making in mathematics classrooms. He proposes that decisions a teacher makes while teaching are a function of her resources, orientations and goals. Although "resources" is a broad term, Schoenfeld classifies the knowledge that a teacher possesses as being a key component of the resources she brings to the classroom. He uses "orientations" to encompasses one 's "dispositions, beliefs, values, tastes, and preferences" (p. 29). And while the term "goal" is self-evident, Schoenfeld notes that an individual may pursue a particular goal "simply in the service of other goals" (p. 20).

Most of the research studies discussed above in relation to noticing, contingent moments, or decisionmaking in the classroom have been conducted at the school level. An exception is the work of [START_REF] Barton | A marriage of continuance: professional development for mathematics lecturers[END_REF] and colleagues in New Zealand who use [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision-making and its educational applications[END_REF] theory to discuss taped video excerpts from participating mathematicians' and mathematics educators' lectures as a means of engaging in professional development on teaching practice. However in terms of noticing and the occurrence of decision points in lectures, there is little research. Indeed to many it might seem like traditional lectures provide few opportunities for contingency. However [START_REF] Mcalpine | Monitoring Student Cues: Tracking Student Behaviour in Order to Improve Instruction in Higher Education[END_REF] report on a study of monitoring of student cues by university lecturers. The lecturers in this study were found to attend to four types of cues: student written, student verbal, student non-verbal, student state. McAlpine et al. (1999, p. 117) posit that the lecturers had a corridor of tolerance for these cues and a decision to change practice was only taken when the cue lay outside of this corridor. In contrast to the work of both [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision-making and its educational applications[END_REF] and [START_REF] Mcalpine | Monitoring Student Cues: Tracking Student Behaviour in Order to Improve Instruction in Higher Education[END_REF], we focus here on the opportunities for decisionmaking that arise in lectures rather than the process of decision-making itself.

Methodology

The authors are lecturers of mathematics at three different universities in Ireland. Each has a doctorate in mathematics or applied mathematics and has a minimum of fifteen years' experience of teaching mathematics at the tertiary level. Between them they have taught mathematics classes from first year undergraduate through to postgraduate level, and have experience of teaching students in class sizes ranging from single figures up to a few hundred students. In 2010/11, along with two other colleagues, they embarked on a project aimed at reflecting on their teaching using the ideas and philosophy described by [START_REF] Mason | Researching Your Own Practice: The Discipline of Noticing[END_REF]. As part of this process, over the course of two years, they engaged in writing brief-but-vivid accounts of incidents or moments that occurred in relation to their teaching. See [START_REF] Breen | A year of engaging with the Discipline of Noticing: five mathematics lecturers' reflections[END_REF] for further details.

The accounts of all five members of the group were collected and a general inductive approach (Thomas, 2006) was used to identify themes in a sample of the accounts. We noticed that many accounts described instances where the lecturer was faced with a decision about what to do next, or, instances where an opportunity to make a decision that might change the course of the lecture or discussion was implicit. We labeled these moments decision points (DPs). In order to focus specifically on these moments, all accounts that did not specifically deal with lecturing were removed. The first and second authors (AOS and MM) independently analysed all the accounts to both identify and code the DPs. After some discussion they agreed on the identification of DPs and the codes assigned. In addition, AOS identified the triggers. Then all five members worked through all the accounts to confirm their agreement or express disagreement with the DPs and triggers identified and the codes assigned. By the end of this process, the group had reached a consensus. AOS then grouped the codes into categories, and the third author (SB) examined these for consistency. In some of the accounts, the action taken by the lecturer as a result of a DP was recorded, however we will not discuss the identification or classification of these here.

In this paper we present findings on the DPs and triggers from accounts written by each of the three authors while lecturing a first year mathematics class. In total there were 135 accounts with 141 DPs identified, as some accounts contained more than one DP. These DPS were classified into 8 categories. In order to provide a context for the accounts, we note that Lecturer A taught Calculus to a group of approximately 200 students consisting of both mathematics and finance students in the first semester of both 2010/11 and 2011/12. Lecturer B taught mathematics to a group of over 200 first-year business students in the first semester of both 2011/12 and 2012/13, while Lecturer C taught Calculus to a first year class of approximately 50 mathematics students for the duration of the academic years 2010/11 and 2011/12. The format for each course consisted of either 2-3 lectures per week given by the lecturer, to which all students were required to attend. While all three lecturers engaged in what might be considered lecturing -that is, the lecturer speaks to the whole class, and perhaps writes on a projector or board while the class is expected to remain silent -they also engaged in initiatives aimed at increasing student participation. These fall into two categorieswhole class question or discussion and class activity. The former relates to where the lecturer asks the whole class a question or attempts to conduct a whole-class discussion, while by class activity we mean an activity that students are expected to engage in during class, usually in small groups. As a final part of the analysis MM classified each account containing a DP as occurring in either Setting 1 (S1)lecturing; Setting 2 (S2)whole-class question or discussion; and, Setting 3 (S3)class activity. We now present the findings.

Lecturer

Findings

In Table 1 we present the categories of DP with frequency by lecturer and setting. Lecturer A mainly wrote accounts about S1 and S3, Lecturer B about S1 and S2, while Lecturer C wrote about all three. This is perhaps not surprising as she had a much smaller class. Over a quarter of all DPs identified belongs to the category "How to deal with students' mathematical difficulties?" (DP6, n=36, 25.5%).

It is noteworthy that 31 of these occurred in Settings 2 and 3. The next largest category of DPs is "How to deal with disruption?" (DP4, n=24, 17.0%) with three-quarters of these attributed to Lecturer B in the lecture setting. The third largest category is "How to respond to students' questions, answers or comments?" (DP2, n=19, 13.4%) and while these DPs may be expected to occur in Settings 2 and 3, it is interesting to note that just under half are attributable to Lecturer A in the lecture setting. The category "What to do next in the lecture?" (DP7, n=17, 12.0%) contains DPs relating to opportunities for decisions that present themselves when moving from a whole-class question/discussion or class activity, back to the lecture setting. Four of the DPs did not seem to fit in any of the categories identified and were grouped as "other".

Each DP was found to an associated trigger and in The first five triggers listed in Table 2 (T1-T5) are as a consequence of the lecturer monitoring her aims/goals for the class, her practice, what the students were (not) doing, which students were not answering questions or contributing, and student behaviour. These account for 58.2% of triggers identified. In terms of the triggers identified when the lecturer monitored students (T3-T5) there are some similarities between the cues identified in [START_REF] Mcalpine | Monitoring Student Cues: Tracking Student Behaviour in Order to Improve Instruction in Higher Education[END_REF] and those described in the accounts relating to these triggers. The category labeled "Student question or comment or answer" (TP6, n=56, 39.7%) contains student-initiated triggers and is the largest of the trigger categories. The "Other" category relates to triggers that are neither lecturer-nor student-initiated and relate to issues such as a cold room or poor attendance due to bad weather. These findings are similar to those of [START_REF] Rowland | Triggers of contingency in mathematics teaching[END_REF] who in their study classified triggers of contingent moments as emanating from the teacher, the students, or resources and tools, with the latter category accounting for far fewer triggers than the first two. Exploring links between DPs and triggers, not surprisingly almost all DPs categorized as DP2 (18/19) and most of those as DP6 (29/36) arose from T6 triggers. Over half of DP3 and most of DP4 (20/24) resulted from T1 and T5 triggers respectively.

We now present some examples of accounts featuring DPs and associated triggers that occurred in the lecture setting. We note that the account by Lecturer C at the start of the paper is an example of a lecturer attempting to make her lecture more student-centred in order to engage students (DP1) as a consequence of monitoring her practice (T2). In the following account by Lecturer A, the trigger for the decision point about how to engage students (DP1) is the lecturer monitoring what the students are (not) doing (T3) while she is lecturing.

I continued the introduction to limits today. I was doing a lot of talking and I realized that people weren't taking anything down. I tried writing more explanations on the pictures I was drawing, so hopefully it will make more sense when they look at it again. (Lecturer A)

Similarly the account by Lecturer C describes how noticing students' expressions (T3) during a lecture, prompts her to explain what mathematicians do in an attempt to engage the students (DP1):

On noticing students exchanging glances when I asked them why they would attempt to solve a particular problem in a particular way, I was prompted to reiterate what "doing mathematics" (at this level) entails. (Lecturer C)

The next account illustrates how a student asking a question (T6) during a lecture results in a decision point for the lecturer about whether to review material already covered (DP2).

I found the limit of as when a mature student, who asks lots of questions, asked why you couldn't just cancel the sin's and the x's to get 2/3. We proceeded to discuss the meaning of the term and the difference between multiplication and composition. I had spoken about this before but felt talking about it here was useful. (Lecturer A)

When a lecturer monitors student behaviour (T5) during a lecture, decision points about how to deal with disruptions (DP4) may arise as illustrated in the following account by Lecturer B:

Shortly into the lecture I ask a group of four students to stop talking. Minutes later I tell two other students to stop talking. Some minutes later, I ask the first group to stop talking again. I look at the classmost of them are staring straight at me and not moving. I realize I am nagging and stopping the lecture for the sake of a few "talkers". As I continue to write and talk, I hear whispering coming from various parts of the theatre and my explanation falters. I put up an exercise for the students to work on. I remind them that the first part is revisionthey have to find the profit function. As I walk around the theatre a student asks: "What are the fixed costs?" I remind him that this example is different to the one I did earlier in class. Another asks: "Do I multiply this function by q to get total revenue?" "No, you are given total revenue, you don't have to find it", I reply. "What is q?" another asks. I feel deflatedthis is revision. I planned to sin(2x) / sin(3x) x ® 0 sin(2x) finish the exercise today, but instead, show them how to get the profit function and complete the exercise in the next lecture. (Lecturer B)

Discussion and conclusions

We wish to discuss three points in this section. The first concerns decision points that arise while lecturing. One might assume that in the traditional sense of a lecture, a lecturer delivers from a preprepared script and unless a student asks a question, the lecturer will not deviate from the script. However there is some evidence from the three lecturers' accounts on their practice that indicates that even while lecturing in the traditional sense, they monitor their aims for the class, their own practice, and how the student cohort are acting, and that this monitoring leads to the occurrence of many of the decision points. About one third of the triggers for decision points in our study were student initiated, (T6 in Table 2) in contrast to the findings of [START_REF] Mcalpine | Monitoring Student Cues: Tracking Student Behaviour in Order to Improve Instruction in Higher Education[END_REF], but in comparing one must be cognizant that our methodology differs from theirs. While we acknowledge that the fact that the three lecturers undertook a professional development project on using the Discipline of Noticing [START_REF] Mason | Researching Your Own Practice: The Discipline of Noticing[END_REF] to improve their practice may mean that they are not typical, it would be interesting to explore further what mathematics lecturers focus on while in a lecture as well as gathering more data to illuminate the relationship between triggers and decision points.

Secondly we observe from our findings that decision points relating to how to address students' mathematical difficulties usually emerged in the context of a whole class question or discussion or a class activity. This is perhaps not surprising but does highlight the importance of including such activities during a lecture in order to assess students' mathematical understanding. However it is also worth pointing out that over half of all the decision points identified in this study were as a consequence of such an activity, which may suggest an increased cognitive load for the lecturer who engages in such activities.

Our final point relates to the methodology used in this study. Our analysis of the settings seems to indicate the lecturers' individual preferences for pedagogical techniques and activities however we cannot use our data to draw conclusions about the proportion of class-time they each spent in the three settings, or make general claims about the relationship between the settings and the occurrence of decision points. This is because the lecturers in this study had complete autonomy over what incidents they chose to write about. They wrote about what mattered to them and many of the accounts relate to incidents where the lecturer felt unsure about what to do, or uncomfortable about a decision she had made. In any lecture, there may have been a multitude of more interesting moments worthy of noting, but either they did not notice them, or chose not to write about them. In this way there is a parallel with the findings of [START_REF] Barton | A marriage of continuance: professional development for mathematics lecturers[END_REF]. In choosing excerpts from a video-taped lecture for group discussion, the authors note that: "Counterintuitively, lecturers chose parts in which they felt less comfortable" and in group discussions "frequently chose to focus the group's attention on interludes in the lecture when unexpected decisions were made" (p. 152).

We return to the quote from [START_REF] Schoenfeld | Noticing Matters. A Lot. Not What[END_REF] (p. 228). We suggest that the use of the Discipline of Noticing [START_REF] Mason | Researching Your Own Practice: The Discipline of Noticing[END_REF] can help lecturers to identify opportunities for making (possibly different) decisions in their lectures. Individually the process also highlighted for each of us different aspects of our practice that we wanted to work onfor Lecturer A it was how to make her lectures more student-centered, for Lecturer B the issue was how to deal with disruptive behavior, for Lecturer C, how to address mathematical difficulties effectively and sensitively. We also recommend that lecturers, as a professional development exercise, write a selection of brief-but-vivid accounts and discuss them in a group setting using Schoenfeld's theory (2011) to frame the discussion.

  I decide I can't get any more annoyed. I put up a question and ask the class to work on it. (Lecturer B, T5, DP4) Unlike the accounts so far which occurred in the lecture setting, the following account takes place during a class activity. During the activity the lecturer experiences a decision point concerning how best to respond to student questions (T6 and DP2). But on completion of the activity another decision point arises about what to do next in the lecture (DP7).

Table 1 : Decision Points by Lecturer and Setting

 1

	A	Lecturer B	Lecturer C	Total

  Table 2 we present the categories of triggers with frequency by lecturer and setting.

			Lecturer A	Lecturer B	Lecturer C Total
		Triggers	S1 S2 S3 S1 S2 S3 S1 S2 S3
	T1 Lecturer monitors aims/goals	3	0	1	1	0	0	2	1	3	11
	T2 Lecturer monitors practice	6	0	1	2	1	0	3	1	0	14
	T3	Lecturer monitors student nonverbal	9	0	0	0	5	1	8	4	0	27
	T4	Lecturer monitors absence of student verbal	0	0	6	0	0	0	0	0	4	10
	T5	Lecturer monitors disruptive behavior	1	0	0	18 0	0	0	1	0	20
	T6	Student question or comment or answer	8	0	3	2 19	0	6	9	9	56
	T7 Other	1	0	0	2	0	0	0	0	0	3
		Total	28	0	11 25 25	1 19 16 16 141

Table 2 : Triggers by Lecturer and Setting

 2

  : "Noticing is consequential -what you see and don't see shapes what you do and don't do"

Proceedings of CERME10


domain: math shs

text:
Tobias Mai
email: tmai@math.upb.de

Frank Feudel
email: feudel@math.upb.de

Rolf Biehler
email: biehler@math.upb.de

A vector is a line segment between two points? -Students' concept definitions of a vector during the transition from school to university

Keywords: vector, concept definition, transition, textbook analysis

The vector concept is an important concept students are confronted with in their first year at university. To be able to build on students' previous knowledge it is important to find out what they have learned about vectors from school. This study aims at exploring university freshmen's personal concept definitions of a vector. We therefore analyzed common German school textbooks to find out how vectors are introduced and what conception of a vector students might have developed at school. In addition, we administered a short pretest in which students were asked what a vector is and to explain vector addition and its properties. We ascertained that freshmen stated a lot of individual concept definitions. The majority of students stated geometric ones, which were mostly not fully adequate, i.e. improperly formalized to be embedded into the theory of abstract vector spaces. Furthermore, various misconceptions were identified.

Introduction

Many students face problems connected with the transition from school mathematics to university mathematics. To reduce students' difficulties during this transition, the Ministry of Innovation, Science and Research initiated the Studifinder project. Part of this project are the studiVEMINT learning materials. We have developed these learning materials (which are an e-learning based bridging course for mathematics) at the University of Paderborn since 2014 [START_REF] Colberg | Studifinder: Developing e-learning materials for the transition from secondary school to university[END_REF]. The development was completed in summer 2016, although quality assurance still takes place. Students can use the course for several purposes: to fill gaps in their mathematical school knowledge, or to get used to elaborated forms of school content. In any case a focus lies on accurate language and notions and a mathematical discourse based on the definitions introduced, as it is expected at university. An important concept students are confronted with at school and university (at a more abstract level) is the concept of vectors. In early 2016 we started the development of a chapter on vectors for the studiVEMINT learning materials by looking into school textbooks to explore how the concept could be introduced. While looking into many of these textbooks we observed the following:

1. The formal definitionthat is either geometric as an infinite set of arrows with the same direction and length or symbolic as a triple of real numbersis often not referred to again in the chapters following the introduction of the vector concept.

2. During the mathematical discourse related to vectors, several models (in the sense of [START_REF] Dörfler | Means for meaning[END_REF]) of the concept are used without arguing about isomorphism.

3. The symbols labeled as vectors and used are not always conceptually coherent with the axiomatic definition of a vector space.

Thematic Working Group 14

Proceedings of CERME10

These observations led us to question which of the many representations of a vector that were labelled as vectors at school the students actually consider to be a vector. This question was examined by analyzing the common German school textbooks dealing with the vector concept and by administering a short test to university freshmen at the beginning of their university studies. The results are presented in this paper.

Theoretical background

Although every formalized mathematical concept has a precise definition, students need to give it a meaning by operating with the concept (maybe just mentally) in order to understand it. [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] use the term concept image to describe all associations students may have acquired by operating with it. These include examples, counterexamples, visualizations as well as properties of the concept. In order to specify the concept with words it has a concept definition. This can either be the formal definition accepted by the mathematical community, or student's reconstruction of a definition of the concept from their concept image (more precisely from the parts of the concept image that were activated during this reconstruction process, which [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] call evoked concept image). In the latter case [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] call it personal concept definition.

The formal definitions of the vector concept the students might have learned at school are the following ones, which were discovered when analyzing German school textbooks:

1. A vector is an infinite set of arrows with equal direction and length [START_REF] Bigalke | Mathematik Gymnasiale Oberstufe, Nordrhein Westphalen[END_REF][START_REF] Bossek | Lehrbuch Analytische Geometrie, gymnasiale Oberstufe[END_REF][START_REF] Brandt | Lambacher Schweizer[END_REF][START_REF] Weber | Mathematik Leistungskurs-Lehrbuch. Analysis, Analytische Geometrie und lineare Algebra[END_REF].

2. A vector is a triple of real numbers or a matrix with one row [START_REF] Alpers | Mathematik, Analytische Geometrie/Lineare Algebra[END_REF][START_REF] Artmann | Lineare Algebra und Geometrie. Grund-und Leistungskurs[END_REF][START_REF] Griesel | Elemente der Mathematik Qualifikationsphase, Grundund Leitungskurs[END_REF][START_REF] Griesel | Mathematik heute[END_REF].

However, students' personal concept definitions, which they reconstruct from their concept images, may differ depending on individual experiences with the concept.

In the following we present for each of the two formal definitions of the vector concept, how they are introduced in German school textbooks and what possible personal concept definitions university freshmen might have, assuming these introductions formed their concept image at school from which they reconstructed their personal concept definitions. Then we discuss how the formal definitions are referred to further in the books when operating with vectors and how this might again influence the students' concept definitions reconstructed from their evoked concept image.

Analysis of books using the geometric definition of a vector

The geometric definition as an infinite set of arrows with same length and direction is often motivated by translations [START_REF] Bigalke | Mathematik Gymnasiale Oberstufe, Nordrhein Westphalen[END_REF][START_REF] Weber | Mathematik Leistungskurs-Lehrbuch. Analysis, Analytische Geometrie und lineare Algebra[END_REF], and sometimes even defined by these [START_REF] Brandt | Lambacher Schweizer[END_REF]. The translations are then represented by arrows with the same length and direction. Afterwards, students are told that all of these arrows describe the same translation and can therefore be identified as the same object (e.g., see [START_REF] Weber | Mathematik Leistungskurs-Lehrbuch. Analysis, Analytische Geometrie und lineare Algebra[END_REF]). This path would lead to the adequate concept definition of a vector as an infinite set of arrows with equal length and direction (D1). However, this identification step is rather difficult as is denoted in the literature, and may result in the incomplete conception that a vector is considered as a single arrow (D2) [START_REF] Malle | Schwierigkeiten mit Vektoren[END_REF]. The motivation of the formal geometric definition as a set of arrows with equal length and direction may also lead students to think that a vector is a translation. While a definition of a vector as a translation mapping operation on the whole plane is consistent with its formal concept, literature shows that translations are often understood as the motion of an object [START_REF] Yanik | Prospective middle school mathematics teachers' preconceptions of geometric translations[END_REF]. So the students might think of a vector as a translation of an object or translation of a point (D3). Yanik (2011) also found out that the connection between a vector and a translation is often not understood, and that many teacher students thought that a vector only gives the direction of a translation. This might lead to the following misconception: vector as direction indicator (D4).

Besides using translations, some books also motivate arrows in space as a quantity characterized by length and direction in physical contexts like speed or force [START_REF] Bossek | Lehrbuch Analytische Geometrie, gymnasiale Oberstufe[END_REF][START_REF] Weber | Mathematik Leistungskurs-Lehrbuch. Analysis, Analytische Geometrie und lineare Algebra[END_REF]. The recognition that two of these arrows can be considered as the same, since only the magnitude and direction matter (e.g., for the resulting movement of an object) leads to the adequate concept definition of a free vector, which is a quantity characterized by length and direction and represented by a free movable arrow (D5) [START_REF] Watson | The relationship between physical embodiment and mathematical symbolism: The concept of vector[END_REF]. But since forces are normally considered as dependent also on the point of origin [START_REF] Watson | The relationship between physical embodiment and mathematical symbolism: The concept of vector[END_REF], this approach can again lead students to the consideration that a vector is a single arrow (D2).

The geometric definition requires not only a lot of effort in its introduction, it is also difficult to handle afterwards. In literature, this is denoted as a lack of operability of the definition [START_REF] Bills | Operable definitions in advanced mathematics: The case of the least upper bound[END_REF].

For example the geometric definition is difficult to handle when defining vector operations because for all operations the independence from the chosen representative of the vector has to be justified.

In some books, this problem is discussed [START_REF] Weber | Mathematik Leistungskurs-Lehrbuch. Analysis, Analytische Geometrie und lineare Algebra[END_REF], others ignore it, and vectors are simply identified with arrows when defining vector operations geometrically [START_REF] Bossek | Lehrbuch Analytische Geometrie, gymnasiale Oberstufe[END_REF]. This can again lead to the conception of a vector as a single arrow (D2). Another option to deal with these difficulties is highlighted in [START_REF] Bigalke | Mathematik Gymnasiale Oberstufe, Nordrhein Westphalen[END_REF]: the addition of vectors is defined via the addition of the components in the symbolic representation as an n-tuple (directly after its introduction) and from then on the geometric addition only serves as a visualization. This does not result in a misconception but becomes problematic when trying to embed the geometric vectors with operations defined between triples into the formal theory of vector spaces because in the formal theory the operations have to be defined on the set, whose elements will be the vectors if the axioms are satisfied. After the introduction of the vector operations and their properties, the definition as a set of arrows (or as a translation) is not referred to again [START_REF] Bigalke | Mathematik Gymnasiale Oberstufe, Nordrhein Westphalen[END_REF][START_REF] Brandt | Lambacher Schweizer[END_REF][START_REF] Weber | Mathematik Leistungskurs-Lehrbuch. Analysis, Analytische Geometrie und lineare Algebra[END_REF]. Instead, in the following chapters on analytical geometry, single arrows and their corresponding number triples are used to describe geometric objects. This can lead to a loose connection between the formal definition and students' concept image from which they might deduce their own personal concept definition [START_REF] Vinner | The role of definitions in the teaching and learning of mathematics Advanced mathematical thinking[END_REF]. The resulting personal concept definitions in this case would be: vector as a single arrow (D1) or vector as a number triple (D6).

In summary, if the vector concept was introduced geometrically as an infinite sets of arrows with equal length and direction, the following concept definitions can be expected: vector as an infinite set of arrows with the same length and direction (D1), vector as a single arrow (D2), vector as a translation of an object or translation of a point (D3), vector as direction indicator (D4), vector as a quantity characterized by length and direction (D5), or vector as a triple of numbers (D6). The personal concept definitions D1 and D6 correspond directly to possible formal definitions of the vector concept, D5 is also an adequate conception, in which the equivalence of arrows with equal length and direction is realized by independence from the space, D2 and D3 are incomplete concept definitions (D2 does not take into account that vectors are equivalence classes, D3 does not take into account that a translation is a mapping on the whole plane) and D4 is a misconception.

Analysis of books using the symbolic definition of a vector as n-tuples

The symbolic definition is often motivated geometrically by translations or arrows [START_REF] Alpers | Mathematik, Analytische Geometrie/Lineare Algebra[END_REF][START_REF] Griesel | Elemente der Mathematik Qualifikationsphase, Grundund Leitungskurs[END_REF] or as coordinates of the points in the space [START_REF] Griesel | Mathematik heute[END_REF]. Sometimes the symbolic definition is introduced earlier in connection with the theory of systems of linear equations [START_REF] Artmann | Lineare Algebra und Geometrie. Grund-und Leistungskurs[END_REF]. The symbolic definition of a vector has the advantage of allowing a flexible interpretation as a point or an arrow. This can avoid the discussion about the equivalence of arrows (e.g., see [START_REF] Alpers | Mathematik, Analytische Geometrie/Lineare Algebra[END_REF]). However, besides the already mentioned incomplete conception of a vector as a single arrow, this flexibility can lead to another inadequate conception: vector as a point (D7). The identification of vectors and points becomes problematic in higher mathematics, e.g., in the theory of affine spaces, in which they are considered different objects [START_REF] Henn | Analytische Geometrie Didaktik der Analytischen Geometrie und Linearen Algebra[END_REF].

The way the vector concept is introduced in [START_REF] Artmann | Lineare Algebra und Geometrie. Grund-und Leistungskurs[END_REF] can also lead to another adequate concept definition. [START_REF] Artmann | Lineare Algebra und Geometrie. Grund-und Leistungskurs[END_REF] restrict their visualizations of vectors on points and arrows starting at the origin. If students identify the number triples with these arrows starting at the origin, they might consider a vector as an arrow starting at the origin (D8). These arrows starting at the origin can serve as elements of a vector space (with suitable operations defined between them).Unlike the geometric definition of a vector as a set of arrows, the symbolic definition is operable when defining vector operations and justifying their properties like the commutative law. However, some books do not mention these properties explicitly [START_REF] Alpers | Mathematik, Analytische Geometrie/Lineare Algebra[END_REF][START_REF] Artmann | Lineare Algebra und Geometrie. Grund-und Leistungskurs[END_REF][START_REF] Griesel | Elemente der Mathematik Qualifikationsphase, Grundund Leitungskurs[END_REF]. One reason, which is also noted in literature, might be their selfevidence [START_REF] Harel | Three Principles of Learning and Teaching Mathematics[END_REF]. However, the symbolic definition can also be difficult to handle in the case of the definition of geometric concepts related to vectors such as the norm of a vector. Purely algebraic definitions of these concepts seem unnatural without further explanation (e.g., see [START_REF] Alpers | Mathematik, Analytische Geometrie/Lineare Algebra[END_REF]). Geometric definitions of these concepts on the other hand (e.g., see [START_REF] Griesel | Elemente der Mathematik Qualifikationsphase, Grundund Leitungskurs[END_REF]), have the danger that the vector defined as a triple is again identified with just a single arrow, which is an at least incomplete vector conception.

After the introduction of vector operations, the concept of a vector is mainly used in geometrical settings (describing lines and planes in the space). This might cause students to not identify vectors with the originally defined 'triple' but with its geometrical representations such as points (D4) or single arrows (D2) (students might reconstruct their personal concept definitions of vectors from these representations and not from the formal symbolic definitions).

In summary, the symbolic approach can lead to two further personal concept definitions besides the intended definition of a vector as a triple (D6), which have not been mentioned yet: vector as a point (D7) or vector as an arrow starting at the origin (D8). The identification of symbolic vectors with arrows starting at the origin is not problematic because the latter ones can truly serve as objects, which the vector operations can be defined upon. The identification of vectors with points, however, can cause conflicts later in the theory of affine spaces, where these two objects have to be distinguished.

Methodology of the empirical study

Research question

On entering university, what personal concept definition of the vector concept do students have?

Data Collection

In September 2016 a short test was administered to 103 university freshmen in a mathematics bridging course at the University of Paderborn. These students were either freshmen majoring in mathematics or in mathematics for teachers at grammar schools. The pretest consisted of three open questions:

1. What is a vector?

2. Explain how you add two vectors a and b .

3. Explain, why for all vectors a and b the following is valid:

a b b a    .
The first question was asked to identify what the students' personal concept definition of a vector in the sense of [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] is. We did not ask for a definition because we did not want the students to try to recall the formal definition they had learned at school, but rather to specify the concept in their own words. We also did not use the term "definition" because we suspected that many students might not be familiar with the term and therefore might get confused.

The other two questions were asked to further analyze if the students used the defined objects to explain vector operations and their properties. This is important for a proper embedding of the old vector concept into the abstract notion of a vector space, which is a set with operations defined on its elements. However, this problem will be investigated later.

Data Analysis

The answers to the first question "What is a vector?" were categorized by using possible personal concept definitions deduced from the analysis of the textbooks (see theoretical background, categories D1,…, D8). Furthermore, four additional categories have been added. The first one, a vector as an element of a vector space was added before the analysis because, although this generalization is not taught at school, it may happen that some students had heard about it (e.g., in mathematical clubs at school). The other categories depict inadequate personal concept definitions that often showed up during the analysis: a vector as a line segment, a vector as a line and a category containing other inadequate concept definitions not yet mentioned.

The whole typology of 12 categories is shown in Table 1. The first five categories can be considered adequate, which means that objects described in the definition can serve as concrete examples of vectors in a vector space (if suitable operations are defined on them) or if vectors are already considered as elements of vector spaces. Categories 6 and 7 contain incomplete concept definitions, categories 8 to l2 contain inadequate concept image definitions, which can be considered as misconceptions. Two of the authors separately coded the data from the questionnaire. The interrater-reliability coefficient, Cohen's Kappa, was κ=0.803, which is good. Afterwards, they discussed the answers they had coded differently and agreed on a categorization.

Results of the study

The students' personal concept definitions of the vector concept that were identified form the students' answers to the question "What is a vector?" are shown in Figure 2. The bars of adequate personal concept definitions (which correspond roughly to possible formal concept definitions of models of the vector concept) are green, not fully adequate concept (i.e. they cannot be properly formalized or embedded into the abstract theory of vector space) definitions are marked yellow, inadequate concept definitions are marked red. As can be seen in figure 2, the students had a variety of individual concept definitions of the vector concept when entering university. Most of them had a geometrical basis. However, in most cases these geometric concept definitions were either incomplete (the yellow bars, in which either the nature of a vector being an equivalence class was not mentioned or in which a vector was considered as a translation of points or objects and not as translations of the whole space) or inadequate (the red bars). Nevertheless, even the inadequate conceptions of a vector like "a direction", "a connection between two points or a line segment" or "a point" have some properties of the adequate conceptions of vectors (e.g. if a vector is considered as a line segment, it has the property "finite length", which is as basic property of the arrows, which represent a vector geometrically.

Conclusion and outlook on possible further research

Our study shows that the students have a variety of concept definitions of what a vector is when entering university. Thus we should keep in mind that freshmen do not come to university with a shared idea on what a vector is. The majority of students stated geometric definitions which were mostly inadequate definitions in the sense that they cannot be properly formalized or embedded into the common definition of a vector in mathematics. This indicates that it is difficult for students to fully grasp the concept of a vector. However, many students seemed to be familiar with the symbolic definition of a vector as an n-tuple and that it can be interpreted in manifold representations. This property of the n-tuple approach seems very appealing. Dealing consistently with equivalence classes including the independence from the chosen representative can be circumvented with this approach. Hence, we chose this approach for the studiVEMINT course. We utilized the connection between the symbolic and geometric representations as often as possible. However, we avoided using the geometric representations while introducing the mathematical discourse on vectors that we wanted to be consistent with the provided definition [START_REF] Sfard | Symbolizing mathematical reality into being-or how mathematical discourse and mathematical objects create each other[END_REF], similar to what is required from the students in their upcoming lectures about linear algebra.

For further research we will look into the students' answers to questions 2 and 3 more thoroughly with a more elaborated theoretical framework. We also intend to do a follow-up study to investigate the influence that the linear algebra course achieved on students at the end of the currently ongoing winter term 2016/17. Including a semiotic point of view and an analysis of textbooks from school as well as the introduction of vectors within the linear algebra course will improve the theoretical framework and provide further insights.

Figure 1 :

 1 Figure 1: Answer categories to the question "What is a vector?"

Figure 2 :

 2 Figure 2: Students' answers to the question "What is a vector?" (N=103)

Proceedings of CERME10


text:
Michael Liebendörfer
email: liebendoerfer@khdm.de

Reinhard Hochmuth
email: hochmuth@idmp.uni-hannover.de

Rolf Biehler
email: biehler@math.upb.de

Niclas Schaper
email: niclas.schaper@upb.de

Christiane Kuklinski
email: kuklinski@idmp.uni-hannover.de

Sarah Khellaf
email: khellaf@idmp.uni-hannover.de

Christoph Colberg
email: christoph.colberg@math.upb.de

Mirko Schürmann
email: mirko.schuermann@upb.de

Lukas Rothe
email: rothe@math.uni-hannover.de

A framework for goal dimensions of mathematics learning support in universities

Keywords: Educational objectives, Mathematics support, Taxonomy, Tertiary education

In university mathematics, support measures address diverse goals in terms of students' knowledge and abilities, motivation and beliefs, or institutional goals like the reduction of dropout rates. In order to facilitate the analysis of specific support measures' goals, their evaluation and comparisons with other forms of support, we aim at developing a taxonomy for these goals. To this end, we have analyzed documents of 44 innovative projects of mathematics support in Germany and conducted supplementary interviews with teaching staff. We present the method and intermediary results of this research and discuss its potential use for researchers, policy makers and teaching.

Background and aim of the paper

Mathematics support has become a common endeavor in many universities and both researchers and teaching staff are interested in identifying the "best way" of supporting their students. The research we present here is part of the ongoing WiGeMath project (Wirkung und Gelingensbedingungen von Unterstützungsmaßnahmen für mathematikbezogenes Lernen in der Studieneingangsphase; Effects and success conditions of mathematics learning support in the introductory study phase), which is a joint research project of the Universities of Hannover and Paderborn (Colberg et al., in press) led by Biehler, Hochmuth and Schaper. In a first phase, the research aims at identifying and structuring goals that have been pursued in selected Projects of Mathematics Learning Support (PMLS). This paper reports on the methodology and the research outcomes of this first phase, drawing upon the project proposal and the intermediate project report of the three project leaders.

In WiGeMath, we examine 44 PMLS in mathematics programs, teacher education and diverse engineering programs at 14 German universities. These projects all have in common that they are trying to help students acclimatize to university mathematics. We label them innovative as they aim at deviating from the formats of standard lectures or standard tutorials. The extent to which they can fulfill this deviation is part of the evaluation in WiGeMath. The PMLS are clustered into four types: bridging courses, mathematics support centers, redesigned lectures and support measures that parallel courses. In most German universities, bridging courses are offered shortly before the beginning of the first semester and aim to bridge the gap between school and university, dealing with school contents as well as university contents. Mathematics support centers are relatively new to German universities and have been implemented in rather few universities. They are designed as optional classes for students where they can go and seek help or work on their math problems under the supervision of experienced tutors. Redesigned lectures are also relatively new and are offered particularly for preservice secondary teachers in order to support their transition from school mathematics to more abstract mathematical content. They focus more on study techniques like problem solving or reading and writing mathematical texts than on mathematical content. Support measures that parallel courses are diverse and include formats that provide online learning material as well as tutorials for special study groups (e.g. students at risk of failing the course), with a special focus (e.g., applications) or for a special purpose (e.g., exam preparation). Using a program evaluation approach [START_REF] Chen | Theory-driven evaluations[END_REF], WiGeMath aims at evaluating different PMLS based on their own assumptions and comparing them. Therefore, we developed a taxonomy in the sense of a descriptive (non-normative), structured set of goals, features and conditions of the PMLS. In this paper, we focus on the taxonomy for the goals of PMLS. Here, the term "goal" is interpreted in the sense of any observable criterion that provides ground for the evaluation of a PMLS against its own initial conception. Existing taxonomies of educational objectives include cognitive and affective aspects of students' learning (e.g. [START_REF] Krathwohl | A Revision of Bloom's Taxonomy: An Overview[END_REF]. However, they only describe learning outcomes of the individual. For evaluating the PMLS' outcomes against their respective initial conception and comparing different PMLS, existing taxonomies have proved insufficient. We found, for example, that some PMLS have goals that cannot be assessed in terms of individual learning objectives, like the support of certain study groups.

Method

We constructed a new taxonomy using an empirical approach consisting of two steps. In the first step, a document analysis was conducted based on documents from the partners' PMLS. This step resulted in a first draft for the taxonomy. In a second step, this rough taxonomy guided interviews with teaching staff to check, refine and supplement the categories.

For the first step, we asked our partners to send us any document that might inform us about the projects, including self-descriptions and learning material. The kind of information greatly varied across the projects: some PMLS had been described in conference papers or posters; others sent us lecture notes or books, and in several cases, flyers or websites addressed at students were available. In the case of redesigned lectures, we also analyzed study regulations. Moreover, the depth of information varied as some projects had explicit descriptions of their goals, others only had learning resources and some PMLS were just in the making and could not give us any documents. We thus asked several partners to describe the goals of their projects either via email or in unstructured interviews and then included these emails or interview notes and transcripts in the analysis. In the following phase of the document analysis, we followed the typical steps of skimming the documents for relevant passages, reading them and interpreting them as an iterative process [START_REF] Bowen | Document Analysis as a Qualitative Research Method[END_REF]. Following the principles of inductive category formation in qualitative content analysis [START_REF] Mayring | Qualitative Content Analysis: Theoretical Background and Procedures[END_REF], we worked through the relevant passages line by line, either subsuming the goals under existing categories or constructing new ones. In this step, we also reconstructed goals that were not mentioned explicitly but were apparent in the documents. If, for example, learning material explicitly asked the students to consider mathematics as a process of trial and discovery, we assumed the change of students' mathematical beliefs to be a goal. Due to the high diversity of materials and goals, we worked through all the material before reorganizing the categories in a rough taxonomy. Here, distinctions from the literature were implicitly taken into account, like the distinction of cognitive and affective outcomes.

A limitation of document analyses may lie in insufficient details and a bias in the document selection [START_REF] Bowen | Document Analysis as a Qualitative Research Method[END_REF]. Specifically, the teaching staff could have had goals that were not reflected in the documents. Thus, in the second step the rough taxonomy was used to guide eight interviews. Two interviews were held for each of the investigated designs, i.e., bridging courses, mathematics support centers, redesigned lectures and support measures that parallel courses. The interview guide first asked for the goals of the PMLS in general and then used the subcategories for deeper inquiry of each specific aspect. In the interviews, we found that some staff members did not mention all goals immediately. In reaction to a general question, only some goals were named. Explicit questions mentioning specific goal categories, however, led to additional goals named by the staff members. These interviews, taped and transcribed, were coded using the rough taxonomy as a coding scheme in order to see if each goal that was mentioned by the interviewees fit into one of our categories. This led to the refinement or reformulation of subcategories, but generally, all goals were covered by the taxonomy.

Results

The goals we identified were split into two main categories of educational goals and system-related goals, each of which consists of several sub-categories. We proceed with a presentation and description of the emerged categories and sub-categories that are summarized in Table 1.

Educational goals

The first main category of educational goals comprises learning outcomes with a focus on objectives regarding the individual learner. These educational goals are subdivided into knowledge goals, action-oriented goals and attitudinal goals. As these goals are more or less covered in existing taxonomies (e.g. [START_REF] Krathwohl | A Revision of Bloom's Taxonomy: An Overview[END_REF] and space is limited, we omit a description and discussion of these goals. We only shortly mention the last category of learning and working conduct. It refers to learning rhythm (i.e., when do participants study), learning expenditure (i.e., how much do participants study), learning materials (i.e., which resources do participants use when studying), learning environment (i.e., where and with whom do participants study) and use of the PMLS' provision. This goal differs from the ones mentioned so far as it does not represent a final goal of studying but rather functions as a mediating partial objective that facilitates the fulfillment of other goals at a later point in the course of studies. Staff members explicitly mentioned that they wanted their students to work in specific ways they assumed to be most efficient.

System-related goals

In our study, we found goals which do not focus on students as individuals but rather take into account the university as a broader organization and therefore decided to label them system-related goals. We specify how we understand each of these categories and which aspects they include. It is important to note that the system-related goals and the educational goals are not necessarily disjoint. When using the taxonomy to categorize the goals of a PMLS, these goals may fall into one category under educational goals and, at the same time, into another category under system-related goals. The difference lies in the focus on the individual learner versus the institution as a broader organization which for example has to establish its own reputation, take into account questions of funding as well as maintain its societal position in providing studies that lead to certain certificates. As to the creation of prerequisites for knowledge/abilities, PMLS with this target would aim at qualifying students to participate successfully in subsequent university classes. This might be achieved via the improvement of school knowledge and abilities as a prerequisite for university studies, a category that we defined as a first sub-category. We refer to topics and methods that are not part of regular university lectures but should be familiar to students from their school background like doing arithmetic with fractions, sine, cosine, and solving systems of linear equations. The difference between this goal and the educational goal of fostering school mathematics knowledge and abilities is that the educational goal is only achieved when each and every student has gained the knowledge in focus, whereas the system-related goal is reached when the bigger part of students has gained this knowledge so that future teaching can take this knowledge as shared. In the latter case, it is inconsequential whether all students make use of the knowledge supply and integrate it into their own learning. It is well known, for example, that students entering university often have gaps in their knowledge while lecturers want to give their lectures as if this knowledge was present. In one bridging course, asked for knowledge as a prerequisite in future lectures, a staff confirmed this (all quotations are own translations):

Educational goals

Staff member: Due to the fact that we refresh the students' school knowledge, there is something they can build on later [in the lectures] and certainly will.

[…] If I look at a proof of continuity, I need absolute values and inequalities. If I do not know them, particularly now that they were removed from the school curriculum, then I have to learn them before.

Interviewer: That means, you try to compensate some deficits, in particular after changes through a school reform where some topics have been canceled from the curriculum?

Staff member: Yes, I would say.

Often, PMLS also aim at providing requirements for lectures that exceed school knowledge. In the innovative courses included in our study, examples may be seen in topics like groups, rings or fields that could be shifted from regular university courses, in which they would usually be discussed. Sometimes, bridging courses also cover topics that were considered school knowledge in the past but are no longer taught in schools today.

For some institutions, the improvement of formal study success also proved to be important. This relates to objectively measurable success criteria like dropout rates, defined as the number of students who withdraw from their studies, and passing rates/achievements, which can be measured via the final exam. For a support center, for example, this was central in the interview: "Clearly, I mean, the big credo was always to reduce the dropout rate." In German universities, dropout rates in mathematics and engineering programs exceed 30 % [START_REF] Heublein | Student Drop-out from German Higher Education Institutions[END_REF]. These criteria are of special interest as they may be used for an institution's quality evaluation. PMLS addressing dropout include tutorials for specific study groups with low success rates or students who already failed an exam twice and now make their final attempt.

The improvement of teaching quality is another goal of PMLS. The aim is to improve the teaching quality as perceived by the students and gain better evaluations from the participants. This may include changing the teaching styles and improving the communication with the students.

In our analysis, three more goals emerged, which all reflect the position that, the university on the one hand wants to provide an optimal environment for the students' learning, yet on the other hand does not take the responsibility for each individual student. The sub-category of improvement of feedback quality covers the aim to provide students with high-quality feedback on their state of studies that helps them improve their learning. Under promotion of social contacts and connections relevant for the studies, we subsume the support of social exchanges and conversations, technical aids, and stimulation to form study groups. For example, students may be offered specific hours for a specific lecture in a mathematics support center so that they may form learning groups on their own account. Also, the design of such centers may reflect this goal: "These workstations there have desks, where eight students may sit at a time. Students have increasingly come in larger groups. And it was clearly our goal, to foster this". A further category that some PMLS aimed at is making university study demands transparent. This includes giving insight into demands of university education, especially with future orientation concerning preconditions and requirements in the course of further studies. Together, these aspects enable the students to make well-informed decisions on how to study according to their personal preferences.

Finally, some PMLS were designed to help certain students more than others, for example, students with a different language background, females, students with a sideline job or students with children. These aims would be encoded under supporting of certain student groups. For one support center, for example, the focus was on low and medium-achieving students, as we could see in the interview: "Our priority is on everyone but the high-performing students. So, from lowachieving to average. […] If we have resources left, then the high-performers are also welcome". In contrast, a redesigned lecture addressed the average-achieving students but not the low-achieving ones: "We have, so to say the very good students, a broad midfield and then the lower fifth part or so, where most of them will probably dropout from their studies, because they are not suitable after all. But they take part at the beginning and want to try it, anyway. And my objective during the development and conduction of this course was to address the broad midfield and in addition provide challenges for the very good students." Another specific group that might be mentioned consists of students without "Abitur". The German school system is split into schools for lowerachieving students where students attend nine/ten years and the so-called "Gymnasium" that adds another three years of school. Generally, only the students of the Gymnasium are allowed to attend university after they have passed the final exam called Abitur. Yet, there are exceptions where students without Abitur may attend university, a scenario that occurs particularly frequently in engineering programs. As engineering was one of the study fields we focused on, this category seemed especially relevant. This goal may reflect a special profile of the university or study program or societal goals and may come along with special funding opportunities.

Discussion

Methodological discussion

As mentioned above, the taxonomy at hand was developed in cooperation with university staff involved in innovative PMLS. Contrary to our expectation that the goals of PMLS were thought through and decided upon in advance, we found, in the course of our work, that many of their goals had remained implicit until our inquiry. This observation raises the question of whether important goals of innovative PMLS can be fully accessed from documents and interviews. Similarly, goals of traditional lectures might also be implicit, since some lecturers may be reproducing these longestablished formats without further reflection. To be more specific, it seemed as if some PMLS aim at establishing a specific didactic contract [START_REF] Brousseau | The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics[END_REF]. For example, students are to be offered a good learning environment in hope of thereby increasing success rates but the university does not take responsibility for the success of each individual. However, this aspect was not included in the taxonomy as we lacked clear evidence. The lack of reference to the didactical contract in the materials and the staff interview might be because of its implicitness in PMLS goals and to the less familiarity of staff to such theoretical terms.

Discussion of the taxonomy

The educational goals in our taxonomy show similarities to the objectives of other taxonomies [START_REF] Krathwohl | A Revision of Bloom's Taxonomy: An Overview[END_REF]. Whereas these models focus on individual learning outcomes, the WiGeMath model aims at comparing and evaluating innovative measures as a whole, in particular following the program evaluation approach [START_REF] Chen | Theory-driven evaluations[END_REF]. This reflects in the new category of system-related goals. The system-related goals reflect three purposes: Some of the goals ensure the preservation of the institution. The creation of prerequisites for knowledge/abilities, improvement of formal study success, and improvement of teaching quality would fall into this category. Another set of systemrelated objectives consists of improvement of feedback quality, promotion of social contacts and connections relevant for the studies, and making university study demands transparent. These goals improve the environment for students' self-directed learning. Supporting certain groups of students is a goal of a third kind as it represents the societal goal of creating equal study conditions in higher education for a wide range of students. The taxonomy thus reflects the institutional framing, in particular goals related to the institutions' preservation as well as the preservation of the innovative PMLS, which mostly had no regular funding but were financed by federal grants for innovative support. Related notions can be found in classification systems for higher education institutions. However, these categories are intended to be strictly descriptive and can therefore not be reformulated as goals. An early example is provided by the Carnegie Classification of Institutions of Higher Education, which was created in the 70s for US institutions. Initially, it was mainly concerned with structural and organizational characteristics of institutions but has undergone major changes in 2000, strengthening the emphasis on teaching-related institutional characteristics [START_REF] Bartelse | The European Higher Education Classification: Objectives and Concepts[END_REF]. A shortcoming of the model is that it lacks a consistent theoretical framework. In following the program evaluation approach, we based its development on the individual language and paradigms of the PMLS and did not question them in their data. From a practical point of view, the model helps to highlight their similarities and specialties. From a theoretical point of view, a next step could be the consistent re-interpretation of their goals from a specific theoretical and epistemological stance, e.g. clarifying basic notions of knowing and learning. We should keep in mind, however, that such a taxonomy might not only be used for the non-normative exchange of ideas, but also be turned into a normative model.

Implications for research, policy and teaching

It seems obvious that a model like the one developed above is never complete and could be expanded not only to fit more goals of innovative measures but also in order to serve regular courses or classes in other fields of study. So far in the first study phase of university mathematics education, we focused on innovative PMLS. We may thus have missed goals that are related to studies beyond the introductory year or to innovative measures integrated in the regular classes. An example for a possible system-related goal that was not mentioned in our study is the qualification of future staff or PhD-students.

Since many practitioners had to reconstruct their goals during the interviews, we believe that our taxonomy may prove beneficial to both teaching staff and developers of courses and support measures as a heuristic tool, helping them reflecting on their goals and teaching practice. It may also highlight ethical questions, e.g. by pointing out that some PMLS were designed to specifically support some students but not others. This taxonomy could also improve the communication about goals between students, teachers and institutions. It provides a common language and a frame of reference. In research, this framework is ultimately intended for the evaluation of innovative measures and so far proved useful in doing so in the ongoing WiGeMath project. For policy-makers, a taxonomy may help in the evaluation of their decisions. In our study, we found that many practitioners had thought about various aspects of their work, but did not have a (shared) language to communicate these thoughts.

Table 1 : Categories of the WiGeMath taxonomy

 1

	Knowledge goals	-school mathematics knowledge and abilities
		-higher mathematics knowledge and abilities
		-the language of mathematics
	Action-oriented goals	-mathematical modes of operation
		-university modes of operation
		-learning strategies
	Attitudinal goals	-beliefs
		-affective features
		-perceived relevance for the future job
		-perceived relevance for future studies
		-mathematical enculturation
	Learning and working conduct
	System-related goals
	Creation of prerequisites for	-improvement of school knowledge and abilities as a
	knowledge/abilities	prerequisite for university studies
		-requirements for lectures that exceed school knowledge
	Improvement of formal study	-dropout rates
	success	-passing rates/achievements
	Improvement of teaching quality
	Improvement of feedback quality
	Promotion of social contacts and
	connections relevant for the studies
	Making university study demands
	transparent
	Supporting of certain student groups

Proceedings of CERME10

Acknowledgment

The WiGeMath project is funded by the German Federal Ministry of Education and Research (BMBF), grant identifiers 01PB14015A and 01PB14015B.


text:
Ronja Kürten
email: ronja.kuerten@uni-muenster.de

Self-efficacy of engineering students in the introductory phase of studies

Keywords: Self-efficacy, transitional programs, undergraduate students, higher education

In this paper we investigate the self-efficacy (SE) of engineering students in the introductory phase of studies. We focus on changes occurring in the first weeks, reasons for those changes, and the effects of a mathematics preliminary course on SE. Based on results of interviews with first year students in 2014, the preliminary course was adapted in the following year. In 2015 we interviewed first year students and collected questionnaires on SE. In the analysis we focus on mathematics, social, and general SE.

Introduction

The start of studies at tertiary education institutions marks a new stage of life for young people. Students often move to a new location, meet new people, learn new rules, and are required to become accustomed to new educational settings. These changes can influence the students' self-efficacy (SE) beliefs [START_REF] Schunk | Self-efficacy development in adolescence[END_REF], which can be defined as "beliefs in one's capabilities to organize and execute the courses of action required to produce given attainments" (Bandura, 1997, p. 3). In academic contexts these beliefs correlate with perseverance, persistence, and achievement [START_REF] Pajares | Self-efficacy beliefs in academic settings[END_REF][START_REF] Schunk | The Development of Academic Self-Efficacy[END_REF], and therefore mediate study-success [START_REF] Fellenberg | Kaum begonnen, schon zerronnen? Psychologische Ursachenfaktoren für die Neigung von Studienanfängern, das Studium abzubrechen oder das Fach zu wechseln[END_REF]. Depending on the given task, different SEs are of importance: Engineering students for example face mathematical, social, and course-specific tasks. For this reason mathematics, social, study-oriented, and course-related SEs may influence the study progress. In this paper we focus on mathematics and social SE, as it seems reasonable to assume that they might be influenced by a mathematics preliminary course. [START_REF] Van Dinther | Factors affecting students' self-efficacy in higher education[END_REF] reviewed 39 studies investigating factors that affect SE at higher education level, which were conducted between 1993 and 2010. None of these studies considered effects of preliminary courses that are common measures at the introductory phase of studies. [START_REF] Fischer | Mathematische Vorkurse im Blended Learning Format[END_REF] investigates the perceived change in SE occurring during a mathematics preliminary course in his dissertation and finds that students perceive only minor changes in SE caused by the preliminary course. This study, however, offers no reasons for existing or non-existing changes. Another educational transition, namely the transition from primary to secondary school, is well investigated concerning SE beliefs. "Young adults often experience declines in their competence and efficacy beliefs as they make the transition from elementary to middle school […] [that] may result from changes in the school environment" (Schunk & Meece, 2006, p. 80).

The results of interviews conducted with first year students at the University of Applied Sciences Münster in 2014 suggested that several study-related SEs might change during the first weeks at university. The students reported rising uncertainty concerning the new style of teaching and learning at university and the finding and joining of a study group. Furthermore they feared to fail in their mathematics studies whilst being confident about success in other subjects. When asked for mastery experiences in mathematics during the first weeks they didn't report any preliminary course Thematic Working Group 14 Proceedings of CERME10 experiences, and even improvement in the mathematics-tests didn't raise their confidence, as improvement was the least they had expected in a test that mostly differed only in numbers from the first one. As a consequence of these results, the preliminary course was adapted in 2015 to promote participation and cooperation of students.

In this paper the development of SE (mathematics and social) of engineering students at the start of studies is investigated. The study focusses on changes occurring during the first weeks at university, their reasons, and the role of the redesigned mathematics preliminary course regarding those changes.

Theoretical framework

SE is "competence-based, prospective, and action-related" (Luszczynska, Scholz & Schwarzer, 2005, p. 440), as shown in items such as "I am confident to solve the systems of equations with 𝑥 + 𝑦 = -7 and 𝑥 ⋅ 𝑦 = 30" [START_REF] Zimmermann | A questionnaire for surveying mathematics self-efficacy expectations of future teachers[END_REF], p. 2137) or "I am confident that I can start a conversation with someone I don't know very well" (Hermann, 2005, p. 107). Efficacy beliefs are multi-dimensional, varying in level, generality, and strength [START_REF] Bandura | Self-efficacy: The Exercise of Control[END_REF]. Generality of SE can range from task-specific SEs as used in the original definition of [START_REF] Bandura | Self-efficacy: The Exercise of Control[END_REF] to domainspecific SEs such as mathematics SE or social SE to general SE. "Mathematics self-efficacy expectations indicate the belief of a person in his/her own competence to solve mathematical problems and tasks successfully" [START_REF] Zimmermann | A questionnaire for surveying mathematics self-efficacy expectations of future teachers[END_REF](Zimmermann et al., , p. 2136)). Engineering students are confronted with a considerable amount of mathematics in their studies, although they usually don't choose their course of studies for that reason. Social SE can be defined as "an individual's confidence in her/his ability to engage in the social interactional tasks necessary to initiate and maintain interpersonal relationships" (Smith & Betz, 2000, p. 286). Students beginning their studies at university face these tasks daily. "General self-efficacy […] reflects a generalization across various domains of functioning in which people judge how efficacious they are" (Luszczynska et al., 2005, p. 440). It can be defined as "the belief in one's competence to cope with a broad range of stressful or challenging demands" (Luszczynska et al., 2005, p. 439). We use this rather stable SE to control our results for testing effects. SE beliefs influence goal-setting, motivation, or perseverance of people [START_REF] Bandura | Self-efficacy: The Exercise of Control[END_REF]. Empirical studies have shown that in academic contexts, low study-related SEs influence proneness to drop out of studies, whereas high SEs are supposed to promote study success [START_REF] Fellenberg | Kaum begonnen, schon zerronnen? Psychologische Ursachenfaktoren für die Neigung von Studienanfängern, das Studium abzubrechen oder das Fach zu wechseln[END_REF]. [START_REF] Pajares | Self-efficacy beliefs in academic settings[END_REF] discovered that SE influences performance independently from and as strong as ability, and [START_REF] Schunk | The Development of Academic Self-Efficacy[END_REF] showed that mathematics SE is a better predictor of achievement than self-concept, anxiety or prior experiences in mathematics. In general, SEs that slightly exceed actual skills are the most functional, as they "lead people to undertake realistically challenging tasks and provide motivation for progressive self-development of their capabilities" (Bandura, 1986, p. 394). SE can be developed through four main sources: enactive mastery experiences, vicarious experiences, verbal persuasion, and physiological and affective states. Enactive mastery experiences (failures or successes) are the most influential source, especially when they are attributed to personal effort. When no absolute measures of adequacy exist, social comparisons can function as source of SE. These vicarious experiences, i. e. observed experiences of models, are of greater influence when the observer perceives higher resemblance to the model. Verbal persuasion can be a third source of SE if significant others express faith or doubts in one's capabilities. This source is less influential than the first two. The last source mentioned by Bandura is that of physiological and affective states. Stress reactions or feelings of joy in the face of certain tasks may influence one's SE if no other information is available. The way these sources influence one's SE is dependent on the cognitive processing, and therefore SE cannot simply be interpreted as the sum of prior mastery experiences [START_REF] Bandura | Self-efficacy: The Exercise of Control[END_REF]. Of these sources, enactive mastery experiences appear to be the strongest at higher education level [START_REF] Van Dinther | Factors affecting students' self-efficacy in higher education[END_REF].

Research questions and methods

Based on the preliminary findings and open questions described above, this study aims to find answers concerning the following research questions:

1.

Do mathematic and social SEs change during the first months at university and if so, how do they change?

2. How does the mathematics preliminary course affect mathematic and social SEs?

At University of Applied Sciences Münster each year in September before the start of the semester, a mathematics preliminary course takes place. It consists of twelve modules: The first one -"How to study"addresses general differences between school and university and other study-related aspects, such as time-management or preparation for exams. The second module -"practicing mathematics" gives an introduction into reading and writing mathematics, set-theory, propositional logic and proofs. The remaining ten modules focus on contents of school mathematics from lower and upper secondary level [START_REF] Kürten | The Rechenbrücke -A project in the introductory phase of studies[END_REF]. For the quantitative part of the study, three data collection points were chosen: At the day before and at the first day of the preliminary course (pretest), in the first two weeks of the semester (posttest) and in the first two weeks of January (follow-up test, see Figure 1). Participation in the first test was mandatory for all students taking part in the preliminary course and voluntary for the rest. Participation in the posttest and follow-up test was voluntary because of organizational reasons. Each mathematics test consisted of 19 items associated with the contents of the mathematics modules of the preliminary course, with items that vary from test to test only in the used numbers or the given context. After completing the mathematics test, the participants submitted a questionnaire on statistical data, such as the type of school qualification and the elapsed time since the end of school, and a SE questionnaire. This questionnaire, in 2015, was composed of the German versions of three category systems: general SE [START_REF] Hinz | Bevölkerungsrepräsentative Normierung der Skala zur Allgemeinen Selbstwirksamkeitserwartung[END_REF], mathematics SE [START_REF] Zimmermann | A questionnaire for surveying mathematics self-efficacy expectations of future teachers[END_REF], and social SE [START_REF] Hermann | The influence of social self-efficacy, self-esteem, and personality differences on loneliness and depression[END_REF]. The general efficacy part uses a four level Likert scale, while the other two operate with a five level Likert scale. The results of the general SE test were used to complement the specific view of the other scales and provide an opportunity to detect testing-effects. The other scales were chosen due to the results of prior analysis (2014), which suggested changes in mathematics and social SEs. This specific mathematics SE scale was chosen, as it presents tasks that can be solved using school mathematics, the only mathematics most of the students taking part in our survey knew when filling in the pretest. The reasoning behind the choice of social SE scale lies in its specific design for students at tertiary education. It was translated into German, and the translation was validated by retranslating the items back into English and comparing the two English versions. These scales were chosen because they were already tested for validity, objectivity, and reliability with satisfying results (e.g. Cronbach's α between .80 and .94) [START_REF] Zimmermann | A questionnaire for surveying mathematics self-efficacy expectations of future teachers[END_REF], Blömker, 2016[START_REF] Smith | Development and validation of a scale of Perceived Social Self-Efficacy[END_REF].

For the qualitative part of the study we chose two data collection points: In the week before the preliminary course began (first interview) and during the third and fourth week of studies in October (second interview, see Figure 1). Participation in the interviews was voluntary. In the first interview 14 students were interviewed, and out of this group 8 were chosen for a second interview according to their department, their results in both mathematics tests, and their school-leaving qualification, to generate a heterogeneous sample and thus gather a large range of positions present in the population. The interviews were conducted using a semi-structured interview guide. The questions focused on mathematics and social SE as well as the students' motivation for studying and learning. In addition to the qualitative data gained in the interviews, the results of the SE scales were analyzed using t-tests to find changes in SE during the course, and semi-partial correlations to assess the relationship between attending the preliminary course and changes in mathematics SE. The interviews were transcribed and analyzed using qualitative content analysis according to [START_REF] Mayring | Qualitative content analysis: theoretical foundation, basic procedures and software solution[END_REF]. As the interviews should be used to understand the quantitative findings, our focus lay on mathematics and social SE. To find reasons for changes and the perceived level of SE, inductive category formation was used conducting a summarizing content analysis [START_REF] Mayring | Qualitative content analysis: theoretical foundation, basic procedures and software solution[END_REF].

Results

Self-efficacy scales

In 2015 the SE questionnaire was completed by 409 students in the pretest, by 243 students in the posttest and by 135 students in the follow-up test. The downturn in the number of participants from pre-to post-and follow-up test can be ascribed to the change from mandatory to voluntary participation. 167 students completed the questionnaires in pre-and posttest and 54 students completed all three questionnaires. Comparison of the results of students taking part in the pre-and posttest (n = 167) respective in all three tests (n = 54) shows no significant differences in general SE and significantly higher results in the post or follow-up test for social SE (small effect size: dz_post = 0.20, resp. dz_fu = 0.40) and mathematics SE (medium effect size: dz_post = 0.61, resp. dz_fu = 0.79). To measure the relationship between attendance in the course and changes in mathematics SE we compute a semi-partial correlation of preliminary course attendance (as stated by the students) and mathematics SE after three months. We assume that for students not taking part in the preliminary course, mathematics SE didn't change in the weeks of the course. As those students neither took part in the course nor in the associated e-learning, we suppose they didn't engage in mathematics during this time. As we did not have results from those students in the pretest, we compare the first result of each student (whether it is from the pre-or posttest) with their results in the follow-up test. We found a significant correlation of .26 (n = 112, p = .006) between preliminary course attendance and mathematics SE in the follow-up test, after partialing out mathematics SEresults of the first test.

Interviews

In 2015 eight students taking part in the preliminary course were interviewed twice. They were of age 18 to 22 and studied chemical engineering (2), informatics (1), and electrical engineering (5). The qualitative content analysis resulted in six main categories: mastery experiences, vicarious experiences, verbal persuasion, goal setting, temporal effects, and type of teaching. In the first interviews some students explained their perceived competence in mathematics by the time that has elapsed since their last studies of mathematics at school (Christoph, Leon) or by social comparison with students coming from more demanding schools (Dennis):

Christoph: Well, my mathematics are a bit rusty. I had in the advanced technical certificate/ I was quite good.

[…] And now I haven't had any mathematics in the last three years apprenticeship. Really not a bit. (1 st interview, translation by the author)

Dennis:

If I now/ If I consider, I've been at a main school (Hauptschule), so surely there is something missing in maths compared with those of a middle school (Realschule) or a grammar school (Gymnasium). (1 st interview, translation by the author)

In some of the answers the changes in the preliminary course are quoted as reasons for confidence in social or mathematical settings. The groupwork at the beginning of each tutorial offered possibilities to meet people and to solve harder tasks together. In the second interviews differences in schoolleaving certification were of lower importance than mastery experiences in social tasks (finding a study group) as well as in mathematical tasks, and in the latter case the mastery experiences were attributed to personal effort:

Dennis: It worked out well. I met new people right after the preliminary course. It opened up with the groupwork. You have to talk to other people there.

[…] And then groups formed right away and why should it worsen?! (2 nd interview, translation by the author)

Roman:

[…] I was first put off a bit by the exercises because I wasn't able to figure them out. But as I took the lecture notes, I understood it and I liked that a lot. Because I realized that I understand things I didn't understand before. And I could solve exercises I hadn't been able to solve and then I saw there is improvement. (2 nd interview, translation by the author)

Christoph: That shows me that I'll be able to solve hard problems that I couldn't solve at first if I study for them. And it shows me concerning my studies that a 2.0 might be possible. Depending on how much I will knuckle down and then practice, practice, practice. (2 nd interview, translation by the author)

The prediction of failure rates was mentioned by some students as a reason for their fear to fail in mathematics. These vicarious experiences didn't differ from those reported in 2014:

Interviewer: Okay. We get to the next statement: "I'm afraid to fail in my studies due to mathematics".

Leon:

Well, I would say four. That's more like the case, there is a certain fear of course.

Interviewer: And can you explain why you put the cross at the four?

Leon: Well, if you look at the rate of the last year, there are many failing. And there is of course/ you fear a bit that you won't be able to make the cut somehow. (2 nd interview, translation by the author)

Discussion and perspectives

In the study presented here we show that students' mathematics and social SE did rise during the preliminary course, and that this effect was stable over a period of at least three months.

As participation in the interviews and the posttests was voluntary, and participation in the pretest only mandatory for those taking part in the preliminary course, the sample is probably not representative of the population of first year students. For example, motivation for learning or studying might influence the students' decision whether or not to take part in the course, the interviews or the tests. This is especially important for the quantitative analysis of the SE scales. Although the qualitative analysis of the interviews doesn't need statistical representativeness, it should be considered that certain interesting types of students (for example those with low motivation) were not included in our sample. Besides this limitation of the study, the results presented show that even in the first weeks at university study-related (mathematics or social) self-efficacies change, while general SE remains relatively stable. For ethical reasons we had no control group of students wishing to take part in the preliminary course, and for organizational reasons we weren't able to collect pretest results of those students not attending to the preliminary course. Therefore, the quantitative data does not reveal whether the perceived changes in SE are caused by the preliminary course or not. At that point the qualitative data help interpret the quantitative results, as their analysis suggests that experiences in the preliminary course influence changes in SE. Those experiences resulted partly from the redesign of the preliminary course in 2015, for example, the forced group work at the beginning of each tutorial that fostered social SE.

We did assume that for students not attending the preliminary course, no changes in mathematics SE occurred during the time of the course. However, some students might have used other resources to prepare for their studies or just reflected on their competencies in comparison to the requirements of the University of Applied Sciences Münster. Again, the qualitative data help us justify our conclusions, as none of the students taking part in post-or follow-up tests stated other mathematical activities as reasons for not attending the preliminary course in the open ended questions of the survey. We feel therefore confident that our conclusion is valid for at least the bigger part of the students. Nevertheless, the correlation between course attendance and development of mathematics SE has to be treated with caution.

With regard to our first research questions we can say that in contrast to the findings of [START_REF] Fischer | Mathematische Vorkurse im Blended Learning Format[END_REF], SE beliefs did change in the first months at university at least in the part of the population that attended the preliminary course. Social SE rose slightly and mathematics SE rose moderately during the preliminary course. These results are in contrast to the findings described in the introduction that predict a decline of self-efficacies during transitions [START_REF] Schunk | Self-efficacy development in adolescence[END_REF]. The differences may be a result of different ages of the observed students, effects of the preliminary course, or other reasons. Another possible explanation is that the decline of SE takes place even before the preliminary course and is therefore missed by our survey. Further research could clarify this by assessing SE well before the move to tertiary education, as well as before and after a preliminary course. With regard to the second research question, we did find clues that there is a relationship between the changes in SE and the preliminary course attendance. Those students who reported mastery experiences (social or mathematical) in the second interview showed an increased value for the related SE in the second test as well. Further research should investigate changes in SE during a preliminary course in an experimental or quasi-experimental design to investigate whether the changes reported here are indeed caused by the preliminary course.

Of the students cited in this paper, those who had been out of school for a longer period showed lower mathematics SE than the others. As they also stated the elapsed time without mathematics training as a reason for their perceived mathematics ability, it might be interesting to evaluate the effect of time on mathematics SE.

Figure 1 :

 1 Figure 1: Time bar of the preliminary course and the accompanying surveys

Table 1 : Statistical data of the interviewees cited in the article with German grades ranging from

 1

	Pseudo-	School-leaving	Last grade	Test results	Mathematics	Social SE	Time
	nym	qualification	in mathe-			SE				since end
		Type	Grade	matics	Pre	Post	Pre	Post	Pre	Post	of school
	Dennis	ATC	1	1 81% 84%	80%	96% 59% 73%	0 years
	Roman	A-level	3.3	4 68% 74%	72%	75% 53% 47%	1 year
	Christoph	ATC	2.3	2 11% 61%	59%	72% 71% 80%	3 years
	Leon	A-level	2.6	2.7 42% 95%	69%	80% 70% 72%	3 years

1 (very good) to 6 (inadequate) and German school-leaving qualifications "Abitur" (A-level) and "Fachgebundene Hochschulreife" (advanced technical certificate [ATC])

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