Julia Gradwohl

Andreas Eichler²
email: eichler@mathematik.uni-kassel.de

Predictors of performance in engineering mathematics

Keywords: Teaching and learning of specific topics in university mathematics, Mathematics for engineers, motivational variables, students' profiles, students' achievement

Mathematics in university courses was identified as a main obstacle for engineering students in the beginning of their study. Since difficulties with mathematics could lead to a dropout, our research aims to analyse students' profiles referring to individual characteristics that allow identifying possible risks for students' achievement or success in the first year of study. As a first step to identify possibly risky profiles, we started to find possible predictors of students' performance. For this, we give a short overview of the research state and our derived research interest. We discuss theoretical constructs that are possibly crucial characteristics of students with respect to encountering mathematics as an obstacle. Further, we describe the method for measuring different variables of 182 engineering students. Finally, we present results referring to predictors of performance in engineering mathematics and discuss further steps of our research.

INTRODUCTION

Besides the technical disciplines, mathematics is a crucial part of higher engineering education [START_REF] Sefi | A framework for mathematics curricula in engineering education[END_REF]. Especially in the first year, mathematics is usually taught without considering practical applications. In lectures and tutorials mathematical basics are provided for subsequent technical courses. However, engineering students "encounter epistemological/ cognitive, sociological/ cultural and didactical obstacles" [START_REF] Gómez-Chacón | Engineering students in Spain and Germanyvarying and uniform learning strategies[END_REF], p. 2117) with mathematics struggling with the transition from school mathematics to university mathematics [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF]. Considering mathematical school skills, students show remarkable deficits at the beginning of their study (e.g. [START_REF] Knospe | Zehn Jahre Eingangstest Mathematik an Fachhochschulen in Nordrhein-Westfalen[END_REF][START_REF] Thomas | Survey team 4: key mathematical concepts in the transition from secondary to university[END_REF]. Empirical studies show the importance of cognitive variables, since school grades and domain-specific previous knowledge are identified as important predictors of academic achievement (e.g. [START_REF] Hailikari | Academic self-beliefs and prior knowledge of student achievement in mathematics: a structural model[END_REF]. Moreover, there is also some evidence that mathematics plays a crucial role when dropouts from engineering studies are regarded. [START_REF] Heublein | Student Drop-out from German Higher Education Institutions[END_REF] stated the highest dropout rates for mathematics laden studies that were partly caused by a low motivation and partly by excessive demands in the first part of the studies. Also, an international review study mentioned mathematical competencies as part of reasons for dropping out at universities [START_REF] Søgaard Larsen | Dropout phenomena at universities: what is dropout? Why does dropout occur? What can be done by the universities to prevent or reduce it? A systematic review[END_REF].

Apart from cognitive aspects, further individual characteristics are relevant in the context of learning and study success. The importance of these different aspects, e.g. socio-demographic, motivational, emotional and social aspects, are explained by different theoretical models (e.g. utilization of learning opportunities model: [START_REF] Schrader | School achievement: motivational determinants and processes[END_REF]models of dropout: Heublein et al., 2010) and proved by empirical findings. The meta-analysis by [START_REF] Hattie | Visible learning. A synthesis of over 800 meta-analyses relating to achievement[END_REF] summarizes the results of over 800 studies and provides an overview of factors influencing learning success in school. Moreover, there is evidence for the impact of self-efficacy beliefs (e.g. [START_REF] Fellenberg | Kaum begonnen, schon zeronnen? Psychologische Ursachenfaktoren für die Neigung von Studienanfängern, das Studium abzubrechen oder das Fach zu wechseln[END_REF], academic self-concept (e.g. [START_REF] Hattie | Visible learning. A synthesis of over 800 meta-analyses relating to achievement[END_REF] and interest (e.g. Schiefele et al., 1993a) on performance.

RESEARCH QUESTION

In view of the research state, empirical findings show insufficient mathematical skills, high dropout rates and difficulties with mathematics at the beginning of engineering study. Research concerning higher engineering education mostly deals with the improvement of mathematics teaching through developing and evaluating interventions (e.g. through integrating mathematical and technical disciplines: [START_REF] Rooch | Brauchen Ingenieure Mathematik? -Wie Praxisbezug die Ansichten über das Pflichtfach Mathematik verändert[END_REF]. According to the utilization of learning opportunities model, learning success does not only depend on the teaching offer but also on its utilization by students. Therefore, in this project engineering students should be explored in more detail, especially with respect to mathematics and individual characteristics. Moreover, the study of [START_REF] Fellenberg | Kaum begonnen, schon zeronnen? Psychologische Ursachenfaktoren für die Neigung von Studienanfängern, das Studium abzubrechen oder das Fach zu wechseln[END_REF] gives hints that a domainspecific investigation is also empirically meaningful. Concerning the time frame, our project focuses on the first year of engineering study because the secondary-tertiary transition and dropout surveys indicate serious problems at the beginning of study. In particular, more students decide to abandon one's studies within the first two semesters [START_REF] Heublein | Ursachen des Studienabbruchs in Bachelor-und in herkömmlichen Studiengängen[END_REF].

Empirical basis for the relevance of individual characteristics in learning processes exists. However, most of the studies were conducted in the context of school. Since the subject matter and learning environment changes with the transition from school to university, these results cannot be transferred immediately. In contrast, studies of higher education with a special view of mathematics are rather rare. In particular, most studies concentrate on single aspects and not on an overview of different impacting variables (e.g. [START_REF] Schiefele | The influence of learning motivation and learning strategies on collage achievement: Results of a longitudinal analysis[END_REF]. Therefore, our project draws on previous findings to explore a multitude of impacts of individual characteristics in learning processes of higher education with the main aim of developing mathematics related profiles of engineering students. As a student's profile we understand the characteristic of different individual variables, e.g. performance, motivational variables like interest [START_REF] Gómez-Chacón | Engineering students in Spain and Germanyvarying and uniform learning strategies[END_REF], or engagement [START_REF] Rach | Welche Studierenden sind im ersten Semester erfolgreich[END_REF], and further the relationships among these variables. Students' profiles should allow identifying possible risks for study success in the first year of the study. The identification of students' profiles is useful because it allows the development of goal-oriented and adequate support services for students who encounter difficulties.

Since insufficient motivation and excessive demands for achievement are primary reasons for dropping out, we focus on motivational and cognitive variables as a first step to developing students' profiles. Moreover, students leaving their course of study are difficult to access, so we focus on available data like students' performance and individual characteristics with respect to the following research question: Which domain-specific predictors of study success or failure can be determined in the first year of engineering mathematics?

THEORETICAL CONSTRUCTS

As a main construct that impacts on students' achievement, we refer to the construct of learning motivation as an umbrella term for different motivational variables [START_REF] Spinath | Lernmotivation[END_REF]. Firstly, we choose all of the motivational variables that are summarized by the term learning motivation in order to identify the crucial impacting factors. Furthermore, all of the constructs are connected to the subject matter, so they might play a crucial role in the transition from school to university mathematics, a situation characterised by a changing subject matter and learning environment.

A first and main part of motivation is an individual's goal orientation [START_REF] Dweck | Motivational processes affecting learning[END_REF]. This dispositional variable involves individual's beliefs about appropriate goals as a trait referring to specific and, thus, context-related tasks [START_REF] Elliot | Achievement goals, study strategies, and exam performance: A mediational analysis[END_REF]. A further dispositional and motivational variable is interest which is differentiated into three components: feeling-related valences, value-related valences and an intrinsic character. Interest could be understood as an individual's development of an appreciation for a specific subject like mathematics (Wild & Möller, 2009). This definition involves the necessity to regard interest context-specific. One aspect of the construct of interest, i.e. the feeling-related valences is also a part of the expectancyvalue-theory of [START_REF] Wigfield | Individuelle Lernstrategien von Studierenden. Konsequenzen für die Hochschuldidaktik und die Hochschullehre[END_REF]. They derive an individual's motivation for doing a task from the individual's expectancy of the success on a specific task and the incentive value of this task. Besides the intrinsic value, which is similar to the feeling-related valences of interest, further variables, i.e. attainment value, utility value and costs are part of achievement-related values. The expectancy of a success referring to specific tasks could be understood as individuals' self-efficacy beliefs [START_REF] Wigfield | Individuelle Lernstrategien von Studierenden. Konsequenzen für die Hochschuldidaktik und die Hochschullehre[END_REF] that are close to the construct of self-concept [START_REF] Shavelson | Self-Concept: Validation of Construct Interpretations[END_REF].

Learning strategies is a further umbrella term that includes variables which also could have an impact on students' achievement (Wild, 2005). Learning strategies include cognitive learning strategies like strategies for elaborating a specific issue, meta-cognitive strategies like planning or monitoring the process of learning, or strategies to use resources like a specific learning environment. Finally, although students' achievement or success is hardly to define [START_REF] Heublein | Student Drop-out from German Higher Education Institutions[END_REF], it could be understood as the achievement in exams referring to a specific subject like mathematics or to proceed in a field of study like engineering despite encountered difficulties.

METHODOLOGY Sample

Our first research step involved 182 engineering students at the University of Kassel enrolled in a calculus course in the summer semester 2017. Among the participants were 158 men and 24 women. Most of them started their second semester (67 %), though a small group of beginners is integrated (14 %). In the beginning of the semester, students of the calculus course were given a questionnaire concerning sociodemographic factors and motivational orientation towards mathematics. To achieve a high response rate, the students had two weeks to answer the questionnaire and received additional points for their permission to the final exam that could be achieved by solving weekly exercises. Students were also assured of the anonymity of their responses.

Instruments

The so called SELLMO instrument [START_REF] Spinath | SELLMO. Skalen zur Erfassung der Lern-und Leistungsmotivation[END_REF] was used to measure students' goal orientation towards mathematics. Twenty out of thirty-one items were chosen and especially referred to mathematics courses at university. Goal orientation is divided in four sub-scales with each five items concerning approaching and avoiding achievement goals, work avoidance and learning goals. One example for an item referring to learning goals is: "My aim for mathematics courses at university is to gain a deep understanding for the content."

For interest, we used a scale with nine items of Schiefele et al. (1993b) and adapted it to mathematics. One example for a negative formulated item of this scale is: "To be honest, I less care for mathematics." Referring to the expectancy-value-theory, we measured values with a scale involving six items that we developed according to [START_REF] Wigfield | Individuelle Lernstrategien von Studierenden. Konsequenzen für die Hochschuldidaktik und die Hochschullehre[END_REF]. One example for an item of this scale is: "Mathematical skills will be crucial for my future professional career." Further, we adapted each three items from the PISA study [START_REF] Kunter | PISA 2000: Dokumentation der Erhebungsinstrumente[END_REF] to measure students' selfconcept and self-efficacy with respect to mathematics.

To measure the students' learning strategies, we focussed in the first step of our research on students' self-reports about the use of resources like lectures, tutorials and special exercises. Whereas in lectures different mathematics topics are taught, students practice in tutorials by means of additional exercises. In the special exercises they discuss their homework and resolve open questions. Finally, students' success was measured by the self-reported grade in the final exam of the linear algebra course (see Fig. 1). By contrast, further grades of final exams in the abovementioned calculus course was directly given but is not analysed yet. Finally, we collected sociodemographic variables, e.g. the type of matriculation standard, according to a questionnaire used for dropout studies [START_REF] Heublein | Ursachen des Studienabbruchs in Bachelor-und in herkömmlichen Studiengängen[END_REF].

RESULTS

Results in the first step of our research firstly refer to an evaluation of the instruments concerning the quality of scales. We further proved the predicting power of different variables on the students' performance measured by the self-reported grade in the final exam of the linear algebra course that students have taken the previous semester.

Evaluation of the instruments

In the first analyses, Cronbach's alpha estimates of reliability were determined for the scales from each instrument (see Tab. 1). Measures are adequately reliable, with values ranging from .552 to .844. Most of the values are appropriate, the lowest value (.552) was found for a scale with only three items.

Construct

Possible predictors of performance in engineering: correlations

In each of the following analyses, we defined students' achievement as the selfreported exam grade of the linear algebra course that students have taken the previous semester. We firstly assessed the relationship between students' individual characteristics referring to grades achieved in school. As seen in We further proved the correlations between the exam grade and variables constituting the expectancy-value-theory. Except for interest, the correlations between the motivational variables and the students' achievement are significant.

Particularly, there is a considerable relationship between the mathematics selfconcept and the students' achievement. Using correlation analysis, we finally assessed the relationships between students' achievement and the students' engagement referring to external resources given by the attendance rate of lectures, tutorials and special exercises. However, the attendance rates seem to be independent of the students' achievement.

Lecture Tutorials Special exercise exam grade .133 .074 .120

Table 5: Pearson's correlation coefficients between the exam grade and students' engagement

Possible predictors of performance in engineering: group comparison

To identify other possible predictors of the performance in engineering mathematics, we compared between distinct groups using t-tests. Firstly, we compared students that were enrolled in an advanced math course and students that were enrolled in usual math courses. Students in advanced math courses get more math lessons in a week and, thus, examine mathematics in a greater extent than students of usual math courses. As expected, students attending a math advanced course have significantly better grades in the exam of the linear algebra course (see Tab. 6). Further, we compared the group of students who were at a technical secondary school in which the extent of mathematics is lesser than in usual secondary schools. As expected, on average, students who had attended a technical secondary school obtained in the exam of the linear algebra course a grade of 4.0, whereas the corresponding grade for students who had not attended a technical secondary school was 3.5. Thus, students from a technical secondary school significantly perform worse in the linear algebra course than those who attended another type of school (see Tab.

DISCUSSION AND CONCLUSION

The development of engineering students' profiles referring to cognitive and motivational variables could potentially result in identifying students' risks for an undesirable low success or a dropout. For identifying possibly risky profiles, we started to find possible predictors of the students' achievement. Final school grades, maths self-beliefs, values of maths, the type of matriculation standard and the choice of advanced courses in school are meaningful predictors of performance in engineering mathematics. Hence, the results show that several variables determine the maths performance which should be considered seriously for the development of support services. However, the results reveal new questions leading to further possible research steps.

In conformity with the research state, the mathematical achievement as well as the amount of mathematics in school seem to be a predictor of the success in a final exam of the first semester. As a subsequent issue it is a crucial question if the impact of the former school time on the students' success in mathematics courses at university decreases or disappears. Moreover, it is interesting whether previous knowledge measured by a skills test has a greater effect compared to the school grades. Perhaps, domain-specific skills can be recognised as special predictors of mathematics courses in engineering. In addition, the results show that the highest correlation exists between the final school grade and students' performance. This implies further influencing variables developed in the students' school time that impact on the students' achievement at university, especially learning activities and strategies. The fact that the attendance rates seem to be independent of the students' achievement strengthens this perspective.

Concerning the motivational variables, it is interesting that not all of them have an impact on the students' achievement. The students' mathematics self-concept that is also developed in the time of learning mathematics in school shows the highest correlation to the students' achievement. Therefore, support services should not only focus on the deficits in mathematics skills but also on the assistance of students' selfbeliefs. Mathematics interest shows no impact on the mathematics performance. This result could approve findings like [START_REF] Eilerts | Kompetenzorientierung in der Mathematik-Lehrerausbildung: Empirische Untersuchung zu ihrer Implementierung[END_REF]. Since engineering students do not choose mathematics voluntarily, mathematics interest might have no predicting power in this context. In this respect, it could be also interesting to differentiate in further analyses specific groups of students, e.g. concerning gender, the school form or other variables and to investigate if different groups show different relationships between motivational variables and the students' achievement.

Regarding the method, proven scales have been used and adapted to mathematics.

Only scales with a low number of items can be extended to improve the reliability. All analyses base on simple correlations. Results can be improved and deepened by using further methods like regression analyses or structural equation models, so that an investigation of indirect effects is facilitated.

To conclude, in further steps of our research, the observation of motivational variables of engineering beginners should be continued and extended to the investigation of their development. Additionally, engineering students should be surveyed in respect of learning activities and strategies that we involved in this first study only by collecting data to the use of attendance rates (external resources). A detailed investigation of students' motives for non-attendance would give more information about engineering students' learning behaviour. Thus, building upon the first results of our research, we expect to deepen the desirable insight into students' profiles in the next steps of our research.

Figure 1 :

 1 Figure 1: Time of data collection

Table 1 : Sample constructs and Cronbach's Alphas

 1

		Number of Items	Cronbach's Alpha
	Mathematics Interest	9	.844
	Goal orientation:
	Approach achievement goals	5	.722
	Avoidance achievement goals	5	.827
	Word avoidance	5	.727
	Learning goals	5	.739
	Expectancy-value-theory
	Mathematics self-concept	3	.704
	Mathematics self-efficacy	3	.552
	Value of mathematics	6	.690

Table 2 : Pearson's correlation coefficients between the exam grade and students' achievement in school (*p < 0.05; **p < 0.01)

 2 Table 2, the school grades and exam grades of the linear algebra course are significantly and positively correlated.

		Math school grade	Final school grade
	exam grade	.321**	.393**

Table 3 : Pearson's correlation coefficients between the exam grade and students' individual characteristics (expectancy-value-theory)

 3 By contrast, the correlations between the students' achievement and the students' individual characteristics referring to the construct of goal orientation are weak and except of the working avoidance, not significant.

		self-concept	self-efficacy	values	interest
	exam grade	.554**	.363**	.385**	.166
		AAG1	AGG2	WA	LG
	exam grade	.000	.177	.207*	.109

Table 4 : Pearson's correlation coefficients between the exam grade and students' individual characteristics referring to AAG1 (Approach achievement goals), AAG2 (Avoidance achievement goals), WA (Work Avoidance), LG (Learning goals)

 4

Table 6 : Exam grade of the linear algebra course depending on different subgroups (1 = attended; 0 = not attended)

 6


Josep Gascón

Pedro Nicolás²

Inquiry-based learning and pre-service teachers education

Keywords: inquiry-based learning, teacher's and students' practices, novel approaches to teaching

We use Hintikka's analysis of the notion of inquiry to point out some obstacles that the implementation of inquiry-based learning might find, and to suggest a way to overcome these difficulties via the use of the Reference Epistemological Models developed in the framework of the Anthropological Theory of the Didactic.

INQUIRY-BASED LEARNING

Inquiry-based learning (IBL), roughly conceived as a teaching method in which students are invited to learn in a way similar to that of a scientist or a mathematician, has been suggested, with different levels of precision, by several researchers, institutions and pedagogical and didactic approaches. [START_REF] Artigue | Conceptualizing inquiry-based education in mathematics[END_REF] include as instances of IBL the teaching proposals made from the following approaches in Mathematics Education: problem-solving, theory of didactical situations, realistic mathematics education, modelling perspectives, anthropological theory of the didactic, and dialogical and critical approaches. For its part, European Union, through reports prepared by experts (see, for instance, [START_REF] Rocard | L'enseignement scientifique aujourd'hui: une pédagogie renouvelée pour l'avenir de l'Europe[END_REF]) and projects (see, for instance, PRIMAS, http:// www.primas-project.eu), has supported also the implementation of IBL in educative european institutions. It would be incorrect to believe that these IBL proposals have been suggested as a means to reach the very same educational end. Let us mention some proposal aiming different ends. The Theory of Didactical Situations proposes the IBL, embodied in the notion of situation, as a means to achieve a real knowledge of mathematics (Brousseau, 1997, p. 22). For its part, the Anthropological Theory of the Didactic proposes the IBL, via the notion of Study and Research Path, as a means to transform the cognitive ethos of our society, that is to say, to provide new attitudes and habits with respect to the acquisition of knowledge [START_REF] Chevallard | Teaching Mathematics in Tomorrow's Society: A Case for an Oncoming Counter Paradigm[END_REF]. European Union, in reports and projects like the ones mentioned above, stands up for IBL as a means to remedy the declining interest of youth in science, confirmed by some OECD reports, and the subsequent lack of technological innovation in Europe. According to this point of view, IBL would contribute to change science and mathematics learning into a motivating activity. On the other hand, Europen Union also promotes IBL, for instance in the PRIMAS project, as a means to prepare students for a future in which "it is no longer sufficient (…) to learn facts" but "to be able to solve non-routine problems, to analyse data, to discuss with colleagues, to communicate their result and to work autonomously" (Maaß & Reitz-Koncebovski, 2013, p. 10). Therefore, the ends to which different proposers aim to arrive through IBL are as varied as the very ends of regulated education: the acquisition of knowledge, habits, attitudes, values, etc.

KINDS OF KNOWLEDGE

To analyse possible uses of IBL in the acquisition of knowledge we have to wonder: are there different kinds of knowledge? In Epistemology it is customary to distinguish between three kinds of knowledge [START_REF] Ichikawa | The Analysis of Knowledge, The Stanford Encyclopedia of Philosophy (Fall 2017 Edition)[END_REF][START_REF] Fantl | Knowledge How, The Stanford Encyclopedia of Philosophy (Fall 2017 Edition)[END_REF]: knowledge by acquaintance, knowledge how and propositional knowledge. Knowledge by acquaintance is the kind of knowledge you have when you can identify something or someone (for instance, the label criterion of divisibility by 3) and the corresponding name, description, formulation, etc. (in this case: a natural number is divisible by 3 if and only if the result of adding all its digits is divisible by 3). Knowledge-how is the kind of knowledge you have when you carry out a series of intentional actions (for example, to apply the criterion of divisibility by 3) towards the attainment of an end (to know whether a given number is divisible by 3, in this case). Finally, propositional knowledge is the kind of knowledge you have when you know why a certain proposition is true (for example, when you know why it is true the criterion of divisibilty by 3). Notice that in the knowledge by acquaintance you also seem to know the truth of a proposition (for instance, the proposition asserted by the statement "The criterion of divisibility by 3 is the statement A natural number is divisible by 3 if and only if the result of adding all its digits is divisible by 3"). But in this case this is just a contingent truth, existing by convention.

INQUIRY AND PROPOSITIONAL KNOWLEDGE

In order to analyse possible obstacles to the implementation of IBL, we shall use a theoretical model of the notion of inquiry, namely, the one developed by the logician and philosopher Jakko Hintikka and collaborators in several works. Of course, one can also find other theoretical models of the notion of inquiry in some approaches to Mathematics Education. Anyway, here we choose the one provided by Hintikka due to its enlightening use of Logic, which seems to be an essential ingredient for a true comprehension of an inquiry process.

Interrogative Model of Inquiry

To get a better understanding of the concept inquiry, as claimed by [START_REF] Hintikka | A dialogical model of teaching[END_REF], is not enough to study individual acts instantiating (…) whatever the concept in question may be. We also have to study the more complicated rule-governed behavioral complexes in which their "logical home" is.

For this, Hintikka has developed a game-theoretical model, the so-called Interrogative Model of Inquiry (IMI), which presents a formal approach to inquiry. A good reference for this model, which has been explained here and there, is (Hintikka, Halonen, & Mutanenet, 2002). In few words, inquiry is regarded as a game with two players: Inquirer and Nature. The game starts with a pair, (T, Q), where T is a given theoretical premise, and Q is a question. The game finishes when the Inquirer finds an answer to Q. Along the game the Inquirer is allowed to make two kinds of moves: questions to Nature, and deductions. The only moves of Nature are answers to Inquirer's questions. We will not explain here all the details of Hintikka's IMI, but at least we will point out two of its main features:

1) The principal goal of an inquiry is the acquisition of propositional knowledge.

2) Inquiry is a process in which a complex dialectics between deductions and questions takes place.

We emphasize here these two aspects because, as we hope to show below, they seem to be crucial for practical didactic considerations about IBL.

The goal of inquiry is to acquire propositional knowledge

The inquiry finishes if we can deduce (based on the premise and on Nature's answers to our questions) an answer A to the initial question Q (Hintikka et al., 2002, Theorem 1). Of course, the deduction procedure, which is carried out through a series of familiar rules (Hintikka et al., 2002, § 2), has to do with semantics. Indeed, these rules are such that if the premise T and Nature's answers are true in a given model M, then the answer A is also true in this model M. This is why we can read in (Hintikka, 1996, p. 38) that what the winning of an "interrogative game" shows is "knowledge of truths".

Later, we will speak of the implications of the presence of the model M in the use of IBL for teaching. For the moment, I would like to stress the fact that, according to Hintikka's IMI, the result of an inquiry is a proposition. Inquiry is a quest for propositional knowledge, which is the kind of knowledge linked to non-contingent truth and falsehood. This does not prevent, in the course of an inquiry, the acquisition of knowledge-how and knowledge by acquaintance. But we would not consider an inquiry finished if, along this inquiry, we would have found a successful technique (knowledge-how), whose success remains unexplained. In other words, in the course of an inquiry we can get knowledge-how, but we then would pursue the corresponding propositional knowledge able to explain the success (and even its limitations, portability to other contexts, etc.) of this knowledge-how.

Dialectics between deductions and questions

In a game of inquiry not all the questions can be asked at any moment. On the contrary, one is forced to pay attention to the presuppositions of that questions. As explained by Cross and Roelofsen ( 2016), there are different logical kinds of questions: whether-questions, which-questions, why-questions, etc. Each kind of questions determines its own kind of presuppositions. For example, the presuppositions of whichquestions (e.g., "Which is the smallest prime number bigger than 7?") are existential statements (e.g., "There exists a smallest prime number bigger than 7").

According to [START_REF] Genot | The interrogative model of inquiry and inquiry learning[END_REF], based in turn on [START_REF] Hintikka | Reasoning about Knowledge in Philosophy: The Paradigm of Epistemic Logic[END_REF], the Inquirer's range of attention, at a given moment of inquiry, is the set of questions such that:

-The corresponding presuppositions are already available or can be obtained by a deduction move.

-Inquirer is aware of those presuppositions.

The main problem of interrogative inquiry is the mismatch between Inquirer's range of attention and available Nature's answers, since sometimes Inquirer might ask a question which has no answer, or fails to ask a relevant question whose answer is available.

In the literature on IBL it is usual to distinguish between several kinds of inquiries depending on two parameters: how open is the initial question Q, and how guided is the inquiry (PRIMAS, 2011, pp. 11-12).

Concerning the first parameter, a serious study of types of questions and degrees of openness in relation to the development of the subsequent inquiry is still to be done. Our point is that IMI provides a suitable theoretical framework to carry out such a study.

Concerning the second parameter, we refer to [START_REF] Hintikka | A dialogical model of teaching[END_REF] in order to learn about all the possible moves the teacher can do in a game broader than an inquiry-game, in which the Student-Inquirer and Nature are but two players among many others (the Teacher being one of them). In particular, a kind of move the Teacher can do in order to guide the inquiry is to manipulate Inquirer's range of attention so that to avoid the mentioned mismatch. Among the possible kinds of manipulation one finds: to ask whether we can deduce a certain proposition from the available information (in order to use it further as a presupposition of a question), to attract attention to a certain (already proved) proposition and to incite to consider it as a presupposition of a question, etc. One might wonder to what extent the Teacher can avoid this manipulation in a IBL process channeled to the acquisition of some propositional knowledge. Concerning this, [START_REF] Genot | The interrogative model of inquiry and inquiry learning[END_REF] proved that "a trade-off between success [of the inquiry] and autonomy is unavoidable" (p. 1). Indeed, in one hand, with no manipulation of Inquirer's range of attention success might not be achieved. On the other hand

The IMI [Interrogative Model of Inquiry] does however warrant the following conclusion: a guaranty that an inquiry learner will be able to solve interrogatively a problem can always be obtained by manipulating the learner's range of attention. (Genot & Gulz, 2015, p. 18)

OBSTACLES TO IBL

Admittidly, to plan for and support IBL is difficult due to the presence of obstacles of different nature: political, cultural, concerning teacher's view of her own profession, concerning teacher's training, epistemological, etc. Let us use the IMI as a microscope to inspect here two of them.

Propositional knowledge and models

According to (PRIMAS, 2011, p. 22):

(…) many teachers find that IBL comes into conflict with the way they learnt science and mathematics in school and at the university, and even with the way they have been teaching science and mathematics for many years. That is, with their beliefs about the nature of mathematics/science and/or their beliefs about teaching of mathematics/science. Probably this will be one of major obstacle you will find.

This is strongly related to the fact that content taught in formal education is, typically, knowledge by acquaintance and knowledge-how, with a bleak lack of genuine propositional knowledge. Therefore, in agreement with the analysis of inquiry previously exposed, this content, not being propositional knowledge, can not be learnt through a proper inquiry. This is the case of the knowledge by acquaintance consisting in knowing the statement of theorems (Pythagorean theorem, Thales theorem, etc.) without proof. It is also the case of the following examples of knowledge-how, which lives just in the realm of syntax, of manipulation of symbols, without being supported by meanings: all kind of algorithms of addition, subtraction, multiplication and division of several kind of numbers, algorithms to calculate the greatest common divisor and least common multiple, divisibility criteria for natural numbers, etc. Summarizing, for a piece of mathematics to be learnt in an inquiry-based fashion, it must be presented as propositional knowledge and, insofar as this kind of knowledge is concerned with true propositions, the presence of models fixing meanings and guiding the inquiry is unavoidable. In other words, without models there are no meanings, without meanings there are no truths, without truths there is no propositional knowledge, and so there is no inquiry.

One might argue that, after all, these mathematical objects do have a meaning. For instance, the fraction 2/3, applied to an object, refers to any portion of this object equivalent (with respect to a certain, previously fixed, magnitude: volume, mass, area, etc.) to the one obtained after performing the following steps: i) Split this object into three parts so that they were equivalent with respect to the fixed magnitude.

ii) Take any two of these three parts.

We would eventually agree that some objects, like natural numbers, positive fractions, etc., are attached to a meaning in regular teaching. But I claim that this meaning is not used later to support techniques. This makes the difference: whereas an unexplained technique (for example, to multiply fractions) is just mechanical knowledge-how, a justified one becomes propositional knowledge. Let us illustrate my claim with the case of the product of fractions.

Typically, we say that the product of two fractions, a/b and c/d, is the fraction (a•c)/ (b•d), whose numerator (respectively, denominator) is the product of the two numerators (respectively, denominators). Notice that this mirrors the common definition in formal mathematics, where a fraction is defined as an ordered pair (a, b) of integers, with b different from zero, and the product of two fractions, (a, b) and (c, d), is (a•c, b•d). This formal definition, taking part in the praiseworthy human enterprise of founding Mathematics in Set Theory, does not need any further proof. Actually, being a definition, it can not be proved. But, for the existence of a proof, there is a more important obstacle other than the fact that it is a definition: there is no model-interpretation neither of the term fraction nor of the term product referring to fractions.

Concerning this we would like to underly two points:

-Once you have a model-interpretation of these terms you can (at least) try 'to prove your definition', namely, you can try to ask the question "Is it true that, according to the fixed meaning of the terms fraction and product, the product of two fractions is calculated by following the former procedure?"

-The way you answer the question strongly relies on the model-interpretation of the terms.

Imagine, for instance, that your interpretation of the term fraction is the one above: the denominator indicates the number of equivalent (according to a fixed magnitude) parts into which a given object has been split, and the numerator indicates the number of these parts you are considering. You still have to give an interpretation of the term product. This is a hard task. To beging with, we can say that a product is the result of a multiplication. Now, what is a multiplication? In natural numbers, a multiplication is what we do to calculate an amount of magnitude which has been expressed as a whole amount of a whole amount, for instance, to calculate the cardinal of a set which results from the union of 27 sets, each of which has 63 elements. Similarly, we can say that the multiplication of fractions is what we do to calculate an amount of magnitude which has been expressed as a fraction amount of a fraction amount. For instance, to know which is the total fraction we are considering when we calculate 2/3 of 4/5 of some amount of a given magnitude? I would know how to answer it if I knew how to answer in the case of 1/3 of 4/5. Similarly, this will not be a problem if I knew how to calculate 1/3 of 1/5. But it is not difficult to calculate that if each of the 5 fifths were divided into 3 parts, then the initial amount would be divided into 15 parts. Thus, 1/3 of 1/5 is 1/15. Now, since 1/3 of 4/5 is 1/3 of 4 times 1/5, we get 4 times 1/15, which is 4/15. And 2/3 of 4/5 is 2 times 1/3 of 4/5, that is to say, 2 times 4/15, which is 8/15. One can see, in this and other examples, that the numerator (respectively, the denominator) of the final fraction can be directly obtained from the first two fractions just by multiplying their numerators (respectively, denominators).

These considerations show that only after having an interpretation of the term product (again, it is the result of a multiplication, and a multiplication is what you do to calculate an amount of magnitude which has been expressed as a fraction amount of a fraction amount) you can prove the truth of the following proposition: "the product of the fractions a/b and c/d is (a•c)/(b•d)". The good news is that models allow to go from knowledge-how to propositional knowledge. The bad news is that the acquisition of this propositional knowledge strongly relies on the chosen model. This is not a minor issue. What to do if we have many possible interpretations of our theoretical terms? Which of them should be considered? Is there any didactic criterion (for ins-tance, to overcome some learning obstacle) to choose among the different possible interpretations? For instance, the interpretation given above to the term fraction is the most usual, but it is not the only one. Should we also consider the others interpretations of the term fraction? If so, how to prove the multiplication formula for these interpretations? We face the problem to choose criteria for attaching meanings such that: 1) they help us to fight against undesirable didactic phenomena specific to the usual teaching of fractions; 2) they are compatible with the meanings attached to other numerical fields, like negative or real numbers.

How to plan and support inquiries for students?

Even if all the questions raised in the previous paragraph about the implementation of semantics in mathematics syntax are answered, we still have to deal with further difficult problems.

A very first one is: what could be the initial question Q of the inquiry? This question seems to be very difficult to find as it is intended to initiate an inquiry through which many kinds of mathematic entities would appear. Among them, notably, concepts. In this direction, we find the following teacher's claim which figures in (Maaß & Reitz-Koncebovski, 2013, p. 12):

What about conceptual knowledge -surely students cannot be expected to reinvent mathematical or scientific concepts for themselves?

In terms of Hintikka's IMI, to search for Q amounts to looking for a question such that, together with the premises T (student's previous knowledge) and Nature's answers (derived from mathematical examples possibly examined by students), allows to deduce (as a final product but also as something obtained in the course of inquiry) the aimed mathematical propositions.

But still, even if Q is already clear, as it is said in Anderson's study (as cited in PRI-MAS, 2011, p. 20), teachers have difficulties with managing "the challenges of new teacher roles and new student roles". According to Walker's work (cited in PRIMAS, 2011, pp. 20 -22),

Teacher loses control: although it depends on the degree of freedom teacher gives to students, it is clear that in IBL students should take control of the lesson.

Also, although the next quotation refers to IBL of science, it is perfectly translatable to mathematics:

Inquiry based lessons might not "work": there is the risk that experiments do not work, that students collect wrong data and that they will get a wrong idea. In the classical use of experiments, these are carefully planned so that they always work and offer the right exemplification of the phenomena that is at stake.

Therefore, teachers have questions concerning their role in and control of students' inquiries. In terms of Hintikka's IMI, the question is: which would be the dialectics between deductions and questions in the expected inquiry? To have this dialectics relatively planned contributes to prepare the teacher in her duty of manipulating stu-dent's range of attention, which, in turn, grants the success of the inquiry in order to provide the acquisition of the aimed propositional knowledge.

PROPOSAL FOR TEACHING IBL TO PRE-SERVICE TEACHERS

In order to solve the problem of finding, given a piece of mathematics to be taught, a good model for the theoretical terms of this content, a good initial question Q and a way to handle the corresponding inquiry, PRIMAS has the so-called professional development modules [1]. There is, for instance, a module with examples of questions to initiate an inquiry. There is a module with strategies to promote students' questioning. There is, also, a module with examples of ways of acquiring concepts. But still there is a need of complete examples, including all together an initial question, the relevant moves students should do in the corresponding inquiry game, and moments in which the teacher could manipulate well enough student's range of attention in this very game.

As we said at the beginning of this work, the Anthropological Theory of the Didactic (ATD) is one of the proposers of IBL. ATD suggests the implementation of the so-called paradigm of questioning the world [START_REF] Chevallard | Teaching Mathematics in Tomorrow's Society: A Case for an Oncoming Counter Paradigm[END_REF]. According to it, formal education would be carried out by means of study and research paths (SRP). Succinctly, a SRP is the process you follow to find the answer A to a question Q. ATD emphasizes that, along this process, you are involved in different kinds of activities: studying possible (perhaps partial) answers to Q, formulating new auxiliar questions, etc. Although less philosophically informed, ATD's analysis of the notion of inquiry, via the notion of SRP, runs almost parallel with that of Hintikka, via the IMI.

There is a continuous spectrum of types of SRP, the extremes of which are what we could call open and closed SRP. The open ones are those in which the teacher is not specially interested in leading the students towards a particular piece of knowledge O. In contrast, the closed ones are those in which the question has been selected with the intention of leading to the natural emergence, in the course of the SRP, of a certain piece of knowledge O previously selected.

In formal mathematics this knowledge O is not expressed as the output of an inquiry, but as a series of axioms, definitions, theorems, examples and standard techniques. Therefore, closed SRP demand, at least, to reorganise the piece of knowledge O to be found along the inquiry. The corresponding reorganisations are what ATD calls Reference Epistemological Models (REM) (see, for instance, [START_REF] Sierra | Lo matemático en el diseño y análisis de organizaciones didácticas[END_REF].

Typically a REM is expressed in terms of praxeologies [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF], that is to say, in terms of: types of tasks, techniques devoted to face these types of tasks, a techno-logical considerations about each technique (a detailed description, a justification, a study of its scope and reliability, possible enhacements) and, possibly, also some theoretical considerations about the situations under study (our metaphysical description of them: basic entities, basic properties, etc.). It is a key feature of a REM that essentially everything in it appears motivated by the study of the types of tasks.

It is worth mentioning that normally the construction of a REM on a piece of know-ledge O is not only guided by the aim of using it as a basis of a SRP, but also by the intention of counteracting some undesirable didactic phenomena specific to the usual teaching of O (Gascón, Nicolás, in press).

It is still an open question, but our point is that each REM [2] implicitly provides a complete example of a possible inquiry, and so a solution to the obstacles to the implemenation of the IBL previously mentioned, namely, the need of interpretationsmeanings, the need of a good initial question and the need of knowledge about when and how to guide the students in their inquiry. More precisely, we think each REM implicitly proposes: a model for the theoretical terms and syntax appearing in the mathematical content O to be studied, a question Q to initiate an inquiry, a proof (based on Logic and Game Theory) of the fact that the corresponding inquiry would fully cover O, and a set of moments of the inquiry at which teacher should evaluate, and possibly manipulate, Inquirer's range of attention. In future works we will try to provide evidences for this point via a logical analysis (IMI-like, in terms of deductions, questions and answers) of some of the published REM.

NOTES

1. Available at http://www.primas-project.eu/artikel/en/1221/Professional+development+modules/ view.do

For the moment, there are, among others, published REM on natural numbers, integer numbers, decimal numbers, proportionality, algebra and differential calculus. See http://www.atd-tad.org/ grupo-tad/


Rolf Biehler
email: biehler@math.upb.de

Elisa Lankeit
email: elankeit@math.upb.de

Silke Neuhaus

Reinhard Hochmuth

Christiane Kuklinski

Elena Leis

Michael Liebendörfer

Niclas Schaper

Mirko Schürmann

Different goals for pre-university mathematical bridging courses -Comparative evaluations, instruments and selected results

Keywords: mathematical bridging courses, instruments for evaluation, transition to and across university mathematics, teachers' and students' practices at university level

To ease difficulties in the transition from school to university, bridging courses are implemented at many German universities. In this paper, we present instruments we have developed for evaluating those bridging courses. We also show selected results from six bridging courses at five German universities, comparing their different goals and achievements.

INTRODUCTION

The transition from school to university is a big challenge for many students, especially in mathematics [START_REF] Biehler | Transition von Schule zu Hochschule in der Mathematik: Probleme und Lösungsansätze Münster[END_REF][START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF]. Several supportive measures such as pre-university bridging courses or mathematical support centres are implemented at German universities to ease students' difficulties in the transition phase [START_REF] Hoppenbrock | Lehren und Lernen von Mathematik in der Studieneingangsphase[END_REF]. But we often do not know how effective these supportive measures are as detailed studies on the effects and success conditions are missing.

THE WIGEMATH-PROJECT

At this point, the ongoing WiGeMath project (Effects and success conditions of mathematics learning support in the introductory study phase), a joint project of the universities of Hannover and Paderborn [START_REF] Liebendörfer | A framework for goal dimensions of mathematics learning support in universities[END_REF] in collaboration with 14 universities, comes in. We distinguish four types of support: pre-university bridging courses, mathematics support centres, newly designed bridging lectures in the first semester, and support systems accompanying traditional lectures such as elearning material or extra tutorials. The WiGeMath project's goals are developing a theoretical framework in order to be able to describe, analyse and compare support measures, investigating effects and success conditions and elaborating recommendations for effective designs for mathematical support measures in the introductory study phase. The theoretical framework for the examinations is the 3P model of [START_REF] Thumser-Dauth | Evaluation hochschuldidaktischer Weiterbildung: Entwicklung, Bewertung und Umsetzung des 3P-Modells[END_REF]. It describes a programme evaluation for higher education measures based on Chen's theory-driven evaluation approach [START_REF] Chen | Theory-driven evaluations[END_REF]. Based on transition literature from mathematics education, we refined this framework to make it content-specific. Interviews with our collaborating universities and document analysis were used to locate the specific measures in the framework [START_REF] Liebendörfer | A framework for goal dimensions of mathematics learning support in universities[END_REF]. The reconstructed programme theories contain goals, procedures, circumstances and expected effects of the measures. Based on the theoretical framework, instruments were developed for evaluating the success of the measures.

Bridging courses in Germany

Most German universities provide bridging courses in mathematics for various kinds of beginning students shortly before the first semester. They differ in length, structure, amount of e-learning, content, audience, and goals. Some courses focus on the repetition of school mathematics while others aim at introducing students to university mathematics content and working methods [START_REF] Bausch | Mathematische Vor-und Brückenkurse. Konzepte, Probleme und Perspektiven[END_REF]Biehler & Hochmuth, 2017).

One main aim is to evaluate the success of bridging courses by assessing short term and medium term effects on attitudes and mathematical knowledge of the students. Therefore, we questioned the students at the beginning of the course, immediately after the course and after two months in the first semester.

Sample: Selected Bridging courses in the WiGeMath study

The following six bridging courses at five German universities are included in the analysis.

University Online (O) or

Attendance (A)

RESEARCH QUESTIONS

In this paper, we will focus on the post-test directly after the course. Apart from evaluating different instruments used in the post-test questionnaire the research questions are:

1. Which goals do lecturers of bridging courses set for their courses? How can the profiles of the courses be compared and located in the WiGEMath framework?

2. To which extend do students think they achieved explicit or implicit goals of their bridging course?

3. How much do the results of two different instruments measuring to which extend the students think they achieved different goals in the bridging course differ?

4. How do the (theoretical) profiles set up by the lecturer differ from the empirical profiles of the course?

METHODS AND INSTRUMENTS FOR EVALUATING BRIDGING COURSES

Instruments based on the WiGeMath Framework

The 13-pages questionnaire for the post-test contains about 205 Itemsusually 6level Likert-scale from "strongly agree" to "strongly disagree". The following tables illustrate the scales with exemplary items. Most of our scales had a reliability above 0.6 in the majority of cases at all locations. Additionally, we asked the participants about some affective characteristics (such as mathematical fear and self-regulation), these items are not used in the analysis for this paper.

Category of goals

Instruments adapted to the explicit goals of the course

The WiGeMath instruments are based on a comprehensive framework of potential goals of a bridging course. As a supplement we used a learning outcome oriented evaluation system, called BiLOE, proposed by [START_REF] Frank | Lehrveranstaltungsevaluation als Ausgangspunkt für Reflexion und Veränderung. Teaching Analysis Poll (TAP) und Bielefelder Lernzielorientierte Evaluation (BiLOE)[END_REF]. For the BiLOE, each lecturer is asked to specify her/his three to six major learning goals in his/her own words. Additionally, the lecturers had to specify up to seven study activities that should help the students to achieve these goals. Students are asked to evaluate these goals and activities. An important further element of the BiLOE is that the students have to state their personal goals for the course and are asked to which extend they think they achieved them. Those students who did not believe they achieved a learning goal were asked to give reasons for this at the end of the questionnaire. The BiLOE also requests the students to evaluate the relevance of the lecturer's goals and how much certain activities helped them to achieve those goals.

SELECTED RESULTS

RQ 1: The different profiles of the investigated bridging courses

We categorized the major goals provided by the lecturers in the BiLOE from the perspective of the WiGeMath framework. The results can be found in table 3. Quite different profiles become visible.

Category A B C D A D O E

School math. knowledge and competencies 1 1 0 0 1 0 University math. knowledge and competencies 1 1 1 1 1 3 Mathematical terminology 1 0 0 0 0 1 Mathematical modes of working 0 0 0 0 0 2 University modes of working 0 0 1 0 1 0 Learning strategies 0 0 0 0 0 0 Learning and working behaviour 0 2 1 0 2 0 Social contacts 1 0 0 1 0 0

Making university demands transparent 0 0 0 2 0 0 It is striking that no attitudinal goals were mentioned among the major goals, neither beliefs, nor affective features, nor mathematical enculturation. Likewise, none of the lecturers mentioned teaching learning strategies as a goal of their bridging course.

RQ 2: Goal achievements

The results of the WiGeMath and the BiLOE instruments provide valuable information for every single lecturer. The broader spectrum of the WiGeMath results will moreover provide information on the effects of the course from the perspective of its participants that the course lecturer may not have explicitly thought of in the selected major goals. This analysis provides empirical profiles and assesses the success of the various bridging courses.

In all cases, we calculated the percentage of students who rather agree up to fully agree (meaning greater than 3 in Likert scales with 4 steps or greater than 4 in Likert scales with 6 steps, respectively).

The following tables show the percentage of participants agreeing to the WiGeMath scales concerning the respective categories.

Category of goals / Scale

RQ 3: Differences between the two evaluation tools

To compare BiLOE and WiGeMath data, we first matched the learning goals given by the lecturers with the framework categories. The BiLOE results are mostly similar to the WiGeMath results. There are only six cases with differences of more than 15 percentage points. We reported back the interesting differences to the respective lecturers but these are relevant only for the individual and provide the general insight that the WiGeMath framework is sufficient for the evaluation.

RQ 4: Comparison of theoretical and empirical profiles

With these empirical results, a re-evaluation of the profiles based on the formulated learning goals of the respective bridging courses is possible. We will evaluate in which categories the percentage of agreeing participants are high or low and compare these results to the theoretical profiles. Here, "high" means an agreement to the WiGeMath scales of more than 80% of the participants and "low" is an agreement of less than 40%.

Course A. Based on the formulated learning goals, bridging course A has many goals. School mathematics, university mathematics, mathematical modes of operation, and social contacts are aimed at equally strongly. The empirical results differ:

The only category with high agreement is school mathematics. There are some categories with low agreement, including social contacts, which was originally formulated as a goal by the lecturer.

Course B. Goals in various categories were stated as well. The empirical profile is similar but even broader: high agreement is reached in school mathematics, university mathematics, mathematical terminology, social contacts, and gaining insight in university learning and teaching methods. No goals concerning the last three categories were stated by the lecturer.

Course C formulated various goals. The empirical results show that there is high agreement in the categories university mathematics and study groups. There is also high agreement in the categories mathematical terminology, meta knowledge and beliefs towards high mathematics, social contacts and making university demands transparent.

Course D A 's empirical results also fit the theoretical classification very well. Additionally, high agreement is reached for mathematical terminology and meta knowledge and beliefs towards high mathematics. The only category with a low percentage is school mathematics, which was not an explicit learning goal, however.

Course D O . The empirical profile of this bridging course differs significantly from the profile based on learning goals. The only category with a high percentage of agreement is school mathematics. Therefore, the focus of the course seems to be more on school mathematics than on university mathematics. Based on the learning goals, both could have been seen as equally strong.

Course E was the only one with a clear profile based on the formulated learning goals which was on university mathematics (including mathematical terminology). This is reflected in the empirical results, which however show a broader spectrum. Additionally, there is also a high percentage in the categories studying in study groups, meta knowledge and beliefs towards high mathematics, social contacts, and gaining insight in university learning and teaching methods. Low agreement was found concerning school mathematics, which was not a formulated learning goal.

SUMMARY AND DISCUSSION

The presented results are an intermediate step in communicating back to those who were responsible for the respective bridging courses with two goals. The immediate goal is to give feedback in order to improve and change the profile of the courseif desired. The second goal is to redesign our instruments so that the future instrument combines scales from the WiGeMath framework and more specific goals of the lecturers. The lecturers' goals given are quite diverse but all goals could be classified into the theoretical framework of the WiGeMath project. Some WiGeMath categories remained empty, however, e. g. learning strategies. We asked for the most important 5 goals, so it may be the case that our lecturers regarded them as minor ones.

Additionally, some lecturers stated more specific goals, while other stated general ones. This may be due to the lack of experience with formulating learning goals as most of the lecturers do not work in the field of didactics. It seems necessary and valuable to extend the phase of specifying BiLOE goals by informing the lecturer in more depth about the WiGeMath framework as a supportive frame for specifying their own goals.

The students in the different courses differ when referring to their achievement of certain goals and categories. This is no surprise. For example, an online bridging course will not provide as much social contact to other students as an attendance based bridging course. It is important to mention that all answers are based on the students' self-assessment. The instruments developed (termed WiGeMath scales) and the BiLOE mostly yield similar results, sometimes the results differ. That can be explained by the BiLOE items being more specific or some of the learning goals only having a corresponding category but no perfectly fitting scale was found, as the questionnaire was already up to 13 pages long. The BiLOE is limited by the number of goals a lecturer can state, while the developed instruments of the WiGeMath projects allow providing a general survey of the bridging course. Additionally, the (theoretical) profiles set up by the lecturer differ from the empirical profiles of the course. For example, no lecturer mentioned an attitudinal goal. Nevertheless, the WiGeMath scales show that there is very high agreement in this category in relation to courses for students in mathematics and mathematics teacher education. In these courses the focus explicitly lies on university mathematics and not on the recapitulation of school mathematics.

Table 1 : Overview over the investigated bridging courses

 1

Scale name Example item

	School math. knowledge	Identifying and overcoming	"I got to know my indi-
	and competencies	deficiencies in school	vidual deficiencies in
		mathematics.	school mathematics."
		Recapitulating and elaborat-	"School mathematical
		ing school mathematics	topics were repeated."
	University math. knowledge	University mathematics	"I learned new mathe-
	and competencies	knowledge and competen-	matical topics."
		cies
	Mathematical terminology	Mathematical terminology	"I have learned new
			mathematical symbols"

Table 2 .1 Knowledge goals Category of goals Scale name Example item (of 2 to 4 per scale)

 2

	Mathematical	Process-related com-	"I have learned how to read mathemati-
	modes of working	petences concerning	cal texts."
		math. texts
		Metaknowl. for math.	"I know how to recapitulate a mathe-
		modes of working

Table 2 .2 Behavioural (action-oriented) goals. (*) only one item Category of goals Scale name Example item (of 3 to 7 per scale)

 2

	Beliefs	Metaknowl. and beliefs	"In the course, I recognised the role
		concerning higher maths.	of proofs in higher mathematics."
	Relevance of	Estimating how relevant	"In the course, I became aware that
	school maths. for	school mathematics are	school mathematics provides a basis
	future studies	for future studies and	for my further studies."
		later profession

Table 2 .3 Attitudinal goals Category of goals Scale name Example item (of 2 to 6 per scale)

 2

	Social contacts Social contacts between students "I met fellow students."
		Perceived social integration	"I think the other students of the
		(Rakoczy, Buff, & Lipowsky,	course would help me, if neces-
		2005)	sary."
		Studying together with fellow	"If I have an idea for a solution,
		students (Liebendörfer et al.,	I will discuss it with other stu-
		2014)	dents."
	Making uni-	Gaining insight into university	"I gained insight into higher

Table 2 .4 System-related goals

 2

Table 3 : Number of learning objectives in the respective category mentioned by the lecturers

 3

Table 4 : Results knowledge goals: Rounded percentage of participants agreeing to the WiGeMath scales (n=651)

 4

	A B C	D A D O E

Table 5 : Results action-related goals: Rounded percentage of participants agreeing to the WiGeMath scales (n=651)

 5

	Category of goals / Scale	A B C D A D O E
	Beliefs
	Meta knowledge and beliefs towards higher mathematics	50 70 94 90 59 90
	Relevance for eventual profession and for subsequent studies
	Estimating how relevant school mathematic is for university and pro-	63 74 63 54 71 44
	fession

Table 6 : Results attitudinal goals: Rounded percentage of participants agreeing to the WiGeMath scales (n=651)

 6

	Category of goals / Scale

Table 7 : Results system-related goals: Rounded percentage of participants agreeing to the WiGeMath scales (n=651)

 7

ACKNOWLEDGEMENTS

The WiGeMath project is supported by the German Federal Minister for Science and Research (BMBF) (FKZ 01PB14015B and 01PB14015A).


Ida Landgärds
email: ida.landgards@uia.no

Mathematics Teaching for Economics Students, But How?

The research presented in this poster addresses the poor performance of many economics students in their first mathematics course at university level. Two different universities are involved in the research,

How to structure teaching to improve economics students' mathematical competences? Research papers within economics are often heavily mathematical. Mathematics is an indispensable tool in studying economics and in the economist's working day. The Norwegian Association of Higher Education Institutions (2011) therefore stipulates that mathematics should be a useful tool for students in the learning of other subject areas within economics.

Research has proved that mathematical knowledge in algebra and arithmetic is a good indicator for performance in introductory economics courses (e.g. [START_REF] Ballard | Basic Math Skills and Performance in an Introductory Economics Class[END_REF]. However, many universities are struggling to find satisfactory formats for teaching mathematics as a service discipline, particularly in their economics undergraduate degree courses.

At the University of Agder there is a first year, first semester course in mathematics for economics students. The proportion of students failing this course, has for several years been about 40%. This is an alarming issue and I have experienced similar problems at Åbo Akademi University in Finland, the university where I studied for my master's degree. Currently these two universities have taken opposite directions regarding the teaching of mathematics to economics students.

Both universities face the issue of students' diverse mathematical backgrounds. At the Åbo Akademi University the course in mathematics for economics students has been moved to the first semester, to more naturally be a continuation of school mathematics. At the University of Agder, a diagnostic test, compulsory for all economics students, is being implemented. The test is followed by a preparatory mathematics module; although not compulsory it is recommended that students with weak mathematical background take it, prior to the main course in mathematics. The preparatory module will consist of online, self-study material with a clear structure, implemented in the Canvas digital learning environment with 3-4 workshops.

The overall goal of the research is to improve the teaching of mathematics for economics students and to optimize their learning. This poster is about the first stage in the research process and considers productive opportunities for research on the diagnostic test and the online preparatory mathematics module. Students studying and participating in activities online produce a huge amount of data which gives the opportunity to use Learning Analytics (LA) as a research method to better understand the mathematical needs of economics students and their use of mathematical resources. e a-Ayala (2017, p. 6) writes: "LA in the context of higher education is an appropriate tool for reflecting the learning behaviour of students and provide suitable assistance from teachers or tutors." I thus want to find out what factors in the preparatory mathematics module contribute or do not contribute to the students' progress in learning mathematics. Because of the preparatory module being optional there is also the possibility to find out amongst students recommended to make use of the preparatory mathematics module, did those who took part in the module perform better in the main mathematics course examination than those who did not take part? At this stage, there are no concrete results to report. Data will be collected and analysed in autumn 2018. The theoretical framework to be used is under consideration. The theoretical framework will underpin the formulation of research questions. [START_REF] Clow | An overview of learning analytics[END_REF] argues for Learning Analytics being atheoretical. There are limited studies linking learning theory to learning analytics, but for example [START_REF] Macfadyen | Mining LMS data to develop an "early warning system" for educators: A proof of concept[END_REF] acknowledge the importance of effective student centred learning and mention, as an example of the social aspect for learning, the possibility of learner-to-learner communication within digital learning environments, and thus they look into the socio-constructivist paradigm.

At the time of the conference I hope to be more knowledgeable about what theoretical framework could guide the proposed research. It will be a valuable opportunity to discuss the theoretical framework with more experienced scholars in the field during the conference.

As e a- Ayala (2017, p. 68) writes: "LA aims at developing models, methods, and tools that can be widely used, whose deliverables are reliable and valid at a scale beyond a course or cohort to provide benefits for learners and educators without distracting or misleading them" I hope the proposed research will provide new knowledge about teaching of mathematics to economics students.


Eivind Hillesund
email: eivind.hillesund@uia.no

Using schematic representation of resource systems to examine how first year engineering students use resources in their studies of mathematics

Keywords: Students' practices at university level, the role of digital and other resources in mathematics education, mathematics for engineers, documentational approach, hierarchical focus interviews, schematic representation of resource systems

Very little research has been done on how students use resources when studying mathematics. My project aims to examine this both quantitatively and qualitatively. The qualitative data collection includes hierarchical focus interviews, with schematic representation of resource systems as a supplement. The intent is to reduce the degree of co-producing answers and imposing terminology on the students.

INTRODUCTION

The focus of my PhD-project is engineering students' use of resources for learning mathematics. Data collection takes place at three Norwegian universities. I focus on which resources they use, to what extent and in what situations, as well as their rationale for how they use resources.

THEORETICAL FRAMEWORK

I use the documentational approach [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF] to examine those questions. Relevant to the approach is the term 'document' which is the joint entity of a set of resources and schemes for utilizing them in certain situations to achieve certain goals. Also relevant is the term 'resource system' for the set of all resources an individual is using, structurally organized. For instance, what resources to use in what situation can be part of the structure. The framework is designed to examine teachers' practices and professional development, but can be adapted to examine students' use of resources as well. One of the focuses of the framework is how students' documents develop (called 'documentational genesis'). I look at students' practices during their first year of university, when I expected a lot of development to occur.

METHODOLOGY

I use both qualitative and quantitative means for data collection. I do so because I want to study resource with some depth, while also getting an indication of the variety of uses. Here, I will focus on qualitative data collection. I used hierarchical focus interviews [START_REF] Tomlinson | Having it both ways: hierarchical focusing as research interview method[END_REF]. These contain strategies to reduce the degree to which the interviewer co-produces the answers. Prior to the first interview (of three), I asked each student to draw a 'mind map' about their use of resources in mathematics. I used their mind maps to create schematic representations of their resource systems (SRRS), inspired by [START_REF] Pepin | Developing a deeper understanding of mathematics teaching expertise: an examination of three Chinese mathematics teachers' resource systems as windows into their work and expertise[END_REF]. From a pilot interview in spring 2017, I theorized that the construction of a mind map helped students structure their thoughts about using resources prior to the interview.

RESULTS

Qualitative data collection spanned the fall semester of 2017, with nine students from three different universities. All students created a mind map during the first interview. In the other 18 interviews, they made changes to their mind maps a total of eight times. The students seemed comfortable talking about their use of resources after constructing a mind map. Their descriptions of their mind maps also yielded interesting insight into how they perceived their use of resources.

The students structured their mind map in several ways. They all contained resources, but five students also had categories in their mind maps, three had situations, five had what purpose they used certain resources for and one had features they appreciated about certain resources. The resources that the most students put in their mind maps were the textbook (nine), fellow students (seven), lectures (six), exercises (six), lecturer (five), pencil and paper (five), Wolfram Alpha (four), calculator (three) and google (three).

CONCLUSION

It is difficult to discern whether differences in students' mind map structure only represent stylistic choices, or meaningful differences in how they perceive their use of resources. While a student's mind map can say much about their documents, analysing the mind map in a vacuum is insufficient. However, when mind map construction is combined with an interview, the two forms of data may shed some light on one another. The interviews may also benefit from the construction of the mind map, as it gives students some time to consider their use of resources prior to the interview.