Theresia Tabchi
email: theresia.tabchi@etudiant.univ-reims.fr

University Teachers-Researchers' Practices: the Case of Teaching Discrete Mathematics

Keywords: Teachers' practices at university level, discrete mathematics, undergraduate education, impact of research, resources

Our work is part of a thesis which deals with the teaching practices of university mathematics professors. Practices of university professors concur specificities: an example is the articulation between teaching and research. We seek to characterize the research impact in the teaching practices at university. We choose discrete mathematics as an object of study. It is an area whose didactic transposition is not fully achieved while the links with other mathematical fields remain undefined. These elements make the choice of resources to be mobilized and conceived quite complex. Our exploratory study draws on interviews with university professors aiming to characterize the interaction with resources; which can help us clarify the research activities impact on teaching practices in the case of discrete mathematics.

INTRODUCTION

Our work is part of an ongoing thesis which deals with the practices of university mathematics professors. The study belongs to the growing body of research on university professors' practices [START_REF] Biza | Research on Teaching and Learning Mathematics[END_REF]. At this level of our study, we designate by "University professors" the university mathematics teachers who occupy teaching/research positions. While we are interested in teaching practices at university, we try to characterize the different factors that impact them (institutional, didactic and epistemological), in particular, the impact of the research activity on the teaching practices. Throughout the text, we will be relying on the following definitions: -"Teaching practices" and "university teachers' practices" to refer to the "teaching" aspect of the university professors' work; -"Research activity" to refer to the research aspect of the profession. Our study focuses on the place of discrete mathematics in undergraduate level. Discrete mathematics holds an epistemological importance, and it is a branch whose topics are not stabilized in mathematics curricula [START_REF] Heinze | Discrete mathematics and Proof in the High School[END_REF]Hart & Sandefur, in press). Hence, it is particularly interesting to study university mathematics professors' practices in the field of discrete mathematics, a choice that we will justify later in the document. In a study of ICME [1], [START_REF] Biza | Research on Teaching and Learning Mathematics[END_REF] conducted a survey of existing research on mathematics education at university. Drawing on journal publications and conference proceedings on university mathematics education published since 2014, Biza et al. identified the current trends and the most recent advances in the field. They classified the existing work according to the topic of research. Among these, the influence of teachers' research activity on their teaching approaches in particular contexts: the use of examples in mathematics tutorials and the use of graphic representations. The study highlights the emerging interest in taking into consideration resources in teaching at university, in particular, interactions with resources for mathematics education and their impact on teachers' professional development. The study points as well the need for further research on practices of university mathematics teachers and the possible impact of their research activity on their teaching practices.

The work that we present in this text is a contribution to the growing body of research in mathematics education on the practices of mathematics teachers at university level. We will present first the context of the study and the research questions. Then, we will describe the first steps in the construction of our theoretical framework and the collection of data. Finally, we will present the methodology we used in our study as well as some extracts, along with some preliminary results and perspectives.

CONTEXT AND RESEARCH QUESTIONS

The study of the practices of university professors is part of a field of research currently in development, but still largely unexplored. There are similarities in many aspects of the practices of secondary and university teachers: the preparation of the courses and tutorials, the design of instruction, the conception of evaluations, the classroom management, and the interactions with students. However, the practices of university professors conquer specificities, from an institutional point of view and given their training and their academic background.

University professors have some freedom in the design of their courses and the choice of contents to be taught, and in the selection and development of resources. Moreover, their research activity can impact (or not) different aspects of their teaching practices [START_REF] Biza | Research on Teaching and Learning Mathematics[END_REF][START_REF] Mali | Characterising university mathematics teaching[END_REF] such as the instructional approach, the form of interaction with students, and the choice of new knowledge to be taught.

University professors, in their teaching practices, maintain a constant interaction with various resources [START_REF] Gueudet | University Teachers' Resources Systems and Documents[END_REF] derived from their teaching practices as well as their research activity (old resources, software, online resources, videos and podcasts, manuals, research articles, etc.). The interaction with the resources can take place at different moments of their teaching practices: in the design of the sessions, the choice of contents to be taught, the teaching in the classroom, and the evaluation of learning. In the working environment of a university professor, the same resources can be used in his teaching practices and his research activity (numerical computation software for example). Other interactions between teaching and research can take place, but in a less tangible way. The study of the university professors' interactions with the resources could be a first step in clarifying the nature of the relationship between their teaching practices and their activity of research: in the choice of contents to be taught, in the teaching mode of different mathematical contents, and in the learning they wish to develop with students (reasoning, application of properties, etc.).

In mathematics education, teaching practices are not considered independently of the contents taught. The relationship between university teachers' practices and their research activity may depend on several parameters, including the mathematical branches addressed in teaching and research; discrete mathematics, in particular, offer an interesting experience. Discrete mathematics have a real epistemological importance as they unfold in different mathematical domains; the objects are easy to access and can contribute to the understanding of other branches of mathematics [START_REF] Grenier | Spécificités de la preuve et de la modélisation en mathématiques discrètes[END_REF][START_REF] Maurer | What is Discrete Mathematics? The many answers[END_REF]. They provide an introduction to modeling, optimization, operational research, and experimental mathematics [START_REF] Grenier | Spécificités de la preuve et de la modélisation en mathématiques discrètes[END_REF][START_REF] Maurer | What is Discrete Mathematics? The many answers[END_REF]. They promote the learning of proof and the development of heuristic processes in students [START_REF] Debellis | Discrete Mathematics in Primary and Secondary Schools in the United States[END_REF][START_REF] Goldin | Problem Solving Heuristics, Affect, and Discrete Mathematics[END_REF].

Discrete mathematics is a field in expansion, with significant achievements in society. Attempts to integrate discrete objects in the curricula appear in several countries at the international level. In France, the place of discrete mathematics in the curriculum is still not sufficiently stabilized and takes various forms depending on the educational context (Ouvrier-Buffet, 2014); which makes this branch very interesting to study the relationship between research activity and teaching practices of university professors. Hence the research questions can be formulated as follows:

-How do university professors interact with the resources in / for the teaching of discrete mathematics?

-How can we characterize the relationship between the teaching practices of discrete mathematics at the tertiary level and research activities in the same field?

Trying to answer these questions requires specific theoretical and methodological developments.

TOWARDS THE CONSTRUCTION OF A THEORETICAL FRAMEWORK

We conducted a first theoretical choice: the documentational approach [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF]. We will justify the choice in what follows. We note that the choice of the documentational approach constitutes a tool for a first exploration of the field. [START_REF] Gueudet | University Teachers' Resources Systems and Documents[END_REF] based her work on the documentational approach [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF] to analyze the interactions of university professors with resources derived from their teaching practices. In the documentational approach, there is a distinction between resources and documents. University professors select, mobilize and use various resources (contents to be taught in class, written evaluations, old resources, etc.). This interaction with the resources generates a document, which is the association of resources and a scheme of use of these resources. A scheme is used here as it was defined by [START_REF] Vergnaud | The theory of conceptual fields[END_REF] as the invariant organization of conduct for a set of situations having the same aim. According to [START_REF] Vergnaud | Towards a cognitive theory of practice[END_REF], a scheme is a dynamic that has four interacting components:

-An aim that can be easily identified and that indicates intentionality in the organization of the activity; -Rules of actions which are the ways of acting generated by the scheme in order to achieve a specific aim; -Operational invariants that influence the rules of action. They can be theorems-inaction (propositions considered as true by the subject, but may be true or false) or concepts-in-action (considered as relevant in a given situation); -Possibilities of inferences, that are the adaptations that the subject can bring to his activity in order to respond to the specificities of a situation corresponding to an aim.

A scheme developed by a subject is associated with a class of situations [START_REF] Vergnaud | The theory of conceptual fields[END_REF]. A class of situations includes all situations having the same aim. In her work, Gueudet (2017) considered classes of situations for specific aims (for example, preparing an assessment in linear algebra) and thus corresponding to a single document, and larger classes of situations that are independent of the mathematical content (for example, preparing an assessment). She made these choices in order to observe the organization of the resources of university mathematics professors globally, and to analyze the operational invariants and the rules of actions associated to mathematical contents in more restricted classes of situations. Although [START_REF] Gueudet | University Teachers' Resources Systems and Documents[END_REF] recognizes the impact that university professors' research activity can have on their teaching practices, she did not focus on this aspect in her study.

We rely on the methodology developed by [START_REF] Gueudet | University Teachers' Resources Systems and Documents[END_REF]. We seek to analyze the choices, of contents and resources, of university professors in their teaching of discrete mathematics. Furthermore, we are interested in the impact of the research activity on the teaching practices. Therefore, we will adapt and develop Gueudet's methodology (2017) in a way to be able to consider the particular field of discrete mathematics and to take into consideration the research activity's impact on teaching.

ELABORATION OF INTERVIEW GUIDELINES AND DATA COLLECTION

The methodology of our exploratory study is based on interviews. We consider university professors whose field of research is discrete mathematics in Lebanon and in France. The interviews, of approximately 90 minutes, took place in workplaces of the Lebanese professors in order to have access to their resources. For French professors, the interviews were conducted via Skype.

We have developed the interview guidelines in order to characterize the interactions of university professors with various resources, resulting from their teaching practice and activity of research. We asked questions such as "What are the main resources you use for your teaching? How do you select the contents of your courses?" these questions intended to give a panoramic view of the resources selected, mobilized and used in the different classes of situations (preparation of lectures and tutorials, evaluation); We believe that the type and nature of the resources (resources resulting from research or teaching) can inform us about the relations between the research activity and the teaching practices of university professors; furthermore, the choice of contents (lectures, tutorials and evaluations) can give us insight about the different rules of action and the operational invariants corresponding to the different classes of situations. Another question was "What are the conditions and constraints that guide your different choices?" Learning about the institutional conditions and constraints (related to the educational institution or to the research institution) can contribute in characterizing the research activity's impact on teaching practices. Other questions were about the links with other mathematical fields, the collective work, the experimentation in classrooms, etc.

We conducted pilot interviews with two discrete mathematics university professors, one in Lebanon and the other in France, to test the interview guidelines. We recorded and transcribed the interviews. To keep the anonymity of the professors interviewed, we will call them Michel (Lebanon) and Bertrand (France) throughout the document.

THE METHODOLOGY OF ANALYSIS

Our methodology of analysis consists of two steps which we will be presenting next.

The first step is analyzing the practices of university professors in terms of their interactions with resources. For that, we detected the aims, mentioned by the university professors in the interviews, related to their teaching practices. In other words, we identified possible classes of situations based on statements, such as "preparing a lecture" or "preparing an assessment". For each aim, we identified the associated resources (manuals, books, scientific articles, computers, online resources etc.) that were explicitly mentioned in the interview. For the rules of action, we relied on the university professors' declarations; they described the ways they behaved in order to achieve an aim, and the adaptations they brought to their actions in accordance with the peculiarities of each situation. These are the conditions and actions that can be expressed in statements of the form "if (condition) ... then (rules of action) ..." or "for (aim) … I (action) ...". The regular ways of acting in specific situations reveal the presence of operational invariants that university professors do not always express in their discourse. To detect them, we identified in the interviews, the statements justifying the rules of actions, the propositions held to be true by the university professors (theorems-in-action) and the logical reasoning underlying the choice of actions to be conducted according to the specificities of each situation.

We constructed a table for the documents of each university professor; horizontally, in each table, we come across a document associated with a given class of situations (the resources used and the scheme of use). Vertically, we find the sets of the resources used, the rules of action and the operational invariants.

The second step consists in studying the impact of the research activity on the teaching practices in classes of situations related to teaching. For each class of situations, first we choose to detect, in the set of resources used, the presence / absence of an impact of the research activity (for example, the choice of contents to be taught in class might be influenced by the research activity of the university professors or not). Then, we will define a typology of operational invariants; it would allow us to classify the reasons behind some rules of actions of the university professors and behind the "transfer" (use, selection and modification) of some resources from their research activity to their teaching: beliefs developed by the activity of research, gestures acquired with professional experience or institutional constraints and conditions. We will eventually search for additional tools for a deeper study of the impact of research activity on teaching practices, following the results of the pilot interviews.

ANALYSIS OF THE FIRST PILOT INTERVIEW

Our first interview was with Michel (Lebanon), whose research domain is graph theory. Michel teaches graph theory for students in masters' degree and discrete mathematics for students majoring in mathematics in year 2. We detected, in Michel's declarations, the resources produced and / or used in his teaching practice. the ideas in discrete mathematics.

Aims

Table 1-Extract of Michel's documents table from his teaching practices (aims, resources, rules of action and operational invariants)

The operational invariants that justify the actions of Michel can be classified as follows. From a didactical point of view, the importance of understanding and writing ideas, as well as illustration; and the importance of examples and applications in familiarizing students with the field of discrete mathematics. From an epistemological point of view, the characteristics of proofs in discrete mathematics such as the accessibility of discrete objects and the simplicity of definitions. From an institutional point of view, the alignment of contents to ensure a smooth progressivity.

In developing the contents of his courses, Michel uses examples and "technical proofs". He believes that students need time to get used to reasoning in discrete mathematics. He tries to convey the basic ideas in the theory of graphs such as the definitions and characteristics of objects; a condition that guides his choice of contents is the compliance with the instructions of the major.

In teaching, Michel stresses the importance of proof and illustration, and engages his students in reading and writing proofs. He teaches discrete mathematics in a theoretical way without real life applications; there is a lack of experimentation in his courses. He recognizes the importance of computers for research (for expanding the empirical space), but he does not use them for teaching in his classes.

The interview with Michel was set up to test the guidelines. We have identified areas for improvement and adapted the guideline before the second interview.

ANALYSIS OF THE SECOND PILOT INTERVIEW

The second interview was with Bertrand. Bertrand (France) is a researcher in graph theory and a research director at the National Center of Scientific Research. He teaches by choice: an optional course of combinatory games and mathematical reasoning to students in first and second year and a course of graph theory for master's degree students.

We are particularly interested in his teaching practices at university. Therefore, we will identify the different classes of situations related to his teaching practices. We present in Table 2 In discrete mathematics, it is important to experiment with material objects.

A computer can be interesting for research purposes, but does not contribute in the teaching of discrete mathematics.

Table 2 -Extract of Bertrand's documents table from his teaching practices (aims, resources, rules of action and operational invariants)

The operational invariants that justify Bertrand's actions can be classified as follows. From a didactical point of view, the role of discrete mathematics in developing reasoning skills, the importance of experiencing research moments for the students. From an epistemological point of view, the importance of proofs, the characteristics of problems in discrete mathematics (fun and accessible). From an institutional point of view, the freedom of a researcher in the choice of contents of his courses.

To develop a course in discrete mathematics, Bertrand selects research problems and problems arising from situations experienced in workshops. The course of discrete mathematics to students in first and second year of university is optional and students are not all in scientific majors. Bertrand has some freedom in the choice of contents. He is convinced that the course should develop students' reasoning and knowledge about proofs, as well as knowledge related to discrete mathematics; these convictions guide his choice of contents. He focuses in his teaching on the activity of proof.

Bertrand's classroom management is based on putting students in situations of research. He does not provide the answers, he believes that they must experience research situations like real scientists. He engages students in experimentations with games such as hunting the beast. He thinks that computers are not useful in class.

RESULTS AND DISCUSSION

First results at the interview grid and difficulties encountered in the analysis

In the analyses, we realized the impact of the level of education (Bachelor or Master) on the interactions of university professors with the resources in their teaching practices and on the relationship between teaching practice and research activity.

We relied on the methodology developed by [START_REF] Gueudet | University Teachers' Resources Systems and Documents[END_REF] to study the interactions of university professors with resources derived from their teaching practices. It seemed to us that the institutional context has a large impact on teaching practices in university (distinction between Bachelor/Master, progressivity in certain fields, training of non-scientists, etc.). Taking into account the institutional context implies new theoretical and methodological choices.

First results at the research questions

Our first research question is "How do university professors interact with the resources in / for the teaching of discrete mathematics?" There is a big difference between Michel and Bertrand. Michel's course of discrete mathematics in the second year of university aims to prepare the students for a Master's degree in graph theory; a constraint on the choice of contents is the compliance with the instructions of the major. On the other hand, Bertrand's course is optional, which allows him a larger degree of freedom in the selection of contents. Bertrand uses, in teaching, resources derived from his research activity, which may be due to his involvement in research.

Both professors insist on the importance of reasoning and proofs, therefore they select contents that contribute to develop students' ability to write proofs. Bertrand uses games and material objects for experimentation while Michel's approach focuses more on theories. Both professors state that the use of software cannot contribute to reach the teaching objectives in class, although it can be useful for research purposes.

The second research question is "How can we characterize the relationship between the teaching practices of discrete mathematics at the tertiary level and research activities in the same field?" Bertrand holds a position of research. He adopts a researcher's attitude in class; he engages students in research and adapts the plan of his course according to their results. On the other hand, Michel follows a stiffer plan in his teaching. Quoting Bertrand "A teacher has to have the knowledge and pass it on, a researcher does not know everything, seeking answers is part of the research".

The analysis of the pilot interviews shows that the choice to consider the interaction with the resources is revealing for the study of the relations between teaching practices and research activity. The operational invariants identified are based on experience in teaching as well as experience in research. It seems interesting to define, according to the interviews, a typology of operational invariants (for classes of teaching situations). We plan to complete the data collection with a questionnaire that will be disseminated in universities in Lebanon and France for an analysis of the institutional context of discrete mathematics education. The results of this questionnaire will give us a map that will allow us to better study the case.

[1] International Congress on Mathematical education

  , an extract from the table of documents of Bertrand.

	mathematical	beast	and	He makes choices	accessible.
	reasoning for students in first and second year of university	problems issued from situations proposed in workshops.	that emphasis in his can put teaching on proofs.	Discrete mathematics makes it possible to teach mathematical knowledge and knowledge related to the proof.
	Implement a	Research		He focuses more on	Doing mathematics means
	discrete		problems		proofs	than	on	working on proofs.
	mathematics	such	as	mathematical
	course	in	hunting the	objects.			It is important that students
	class		beast problems workshops. and proposed in situations issued from	experimentation. objects He uses material for researchers. He puts the students in the position of	experience research problems in mathematics.
					He does not use
					computers		in
					teaching.
	Aims		Resources		Rules of actions		Operational invariants
	Develop	a	Research		He selects problems	It is important that students
	course	of	problems		arising from research	experience research problems.
	combinatory games and	such hunting the as	situations.			Problems mathematics	in	are	discrete easily


Frank Feudel
email: feudel@mathematik.hu-berlin.de

C' (x) = C(x+1)-C(x)? -Students' connections between the derivative and its economic interpretation in the context of marginal cost

Keywords: derivative, students of economics, economic interpretation, concept image, marginal cost

    C' (x) = C(x+1)-C(x)? -Students' connections between the derivative and its economic interpretation in the context of marginal cost

INTRODUCTION

The derivative plays an important role in economics. It is used to solve optimization problems, to describe and characterize economic functions, and in marginal analysis, in which the impact of small changes from the current state is examined (example: the effect of small changes in the price of a product on the demand) in order to make optimal decisions. Hence, students of economics should have an adequate understanding of the derivative concept in order to be able to use it in economics in a reflective manner. The study presented here focuses on students' of economics understanding of the derivative after their Calculus course with special emphasis on its economic interpretation, which is essential for the ability to apply the concept in economics. It is part of a larger research project about the understanding of the derivative in mathematics for students of economics (my PhD-thesis, supervisor: Rolf Biehler) at the Centre for Higher Mathematics Education in Germany (khdm).

LITERATURE REVIEW AND EMBEDDING OF THE RESEARCH

There is a lot of research about students' understanding of the derivative. Concerning the interpretation of the slope and the derivative in contexts, different difficulties are documented. Typical mistakes are the slope/height confusion or the graph-as-picture error [START_REF] Beichner | Testing student interpretation of kinematics graphs[END_REF][START_REF] Çetin | The Ability of Students to Comprehend the Function-Derivative Relationship with Regard to Problems from Their Real Life[END_REF][START_REF] Carlson | The precalculus concept assessment: A tool for assessing students' reasoning abilities and understandings[END_REF]). An interesting study, which included a task to interpret the derivative in the context of motion explicitly, was conducted by [START_REF] Bezuidenhout | First-year university students' understanding of rate of change[END_REF]. He asked students to interpret '(80) 1.15 S  if () Svis the stopping distance of a vehicle in metres in dependence of the velocity in km/h. Many students overgeneralized that the derivative is the acceleration or the velocity itself. Many also had problems with the units. These students did not understand the derivative as rate of change of the given function S properly.

While there is some research of students' understanding of the derivative in physical contexts (examples mentioned above), there is little research related to economic contexts. [START_REF] Wilhelm | Projecting rate of change in the context of motion onto the context of money[END_REF] showed that students cannot automatically transfer their knowledge about rate of change from a physical context to the context of money. Hence, an economic interpretation of the derivative should be explicitly taught in a Calculus course for students of economics. But even if an economic interpretation of the derivative was covered in the students' Calculus course, problems occur. [START_REF] Mkhatshwa | Students' understanding of marginal change in the context of cost, revenue, and profit[END_REF] showed that many students talked about marginal cost (the derivative of a cost function) as amount of change when solving economic problems, although it was underlined in the Calculus course that the derivative is a rate. This indicates a rather superficial understanding of the connection between the derivative as a mathematical concept and its economic interpretation. A similar result is also found in [START_REF] Feudel | Students' interpretation of the derivative in an economic context[END_REF]. Students' answers in a task to interpret '(73) 0.2

GE P ME



(GE = units of money, ME = units of quantity) of a profit function P economically indicated that many students were not aware of the numerical differences and the differences in the unit between the derivative and its economic interpretation as additional profit. However, in these two studies the students were not obliged to reveal their ideas about the connection between the derivative as a mathematical concept and its economic interpretation explicitly. The study presented here directly focuses on this connection.

THEORETICAL BACKGOUND OF THE STUDY

The economic interpretation of the derivative

To be able to use the derivative in economics, students need to be able to interpret its values in economic contexts. However, understanding the interpretation of the derivative commonly used in economics is a special challenge for students because it does not directly correspond to any of the usual representations of the derivative as limit of the difference quotient, slope of the tangent line, local rate of change, or as instantaneous velocity. If :[0; ) [0; ) C    is a cost function (the variable x represents the output of a product), the derivative '( ) Cx, called marginal cost, is often interpreted as additional cost of the next unit. However, if one takes this interpretation literally it corresponds to the difference ( 1) ( )

C x C x 

, which differs from the derivative in its unit and in its numerical value. Since the students already have previous knowledge about the derivative from school, e.g. as slope of a function at one point, this might confuse them. Hence, the economic interpretation of the derivative should be carefully connected to the students' previous knowledge, and justified for economic contexts in the students' of economics Calculus course. A typical justification is via the approximation formula (

) ( ) '( ) C x h C x C x h     for 0 h  . Since 1
h  can be considered as small in economics, the numerical values of '( )

Cx and ( 1) ( )

C x C x 

are close to each other, and they can be identified (a more detailed explanation can be found in [START_REF] Feudel | Relevant knowledge concerning the derivative concept for students of economics -A normative point of view and students' perspectives[END_REF]).

The above mentioned perspective on marginal cost as being defined as derivative and interpreted as additional cost of the next unit coincides with what is taught in mathematics courses for students of economics (see e.g. (Sydsaeter and Hammond, 2013)) and with what is presented in some books of economics like [START_REF] Breyer | Mikroökonomik[END_REF]. However, marginal cost can also be defined as additional cost of the next unit like in [START_REF] Blum | Grundlagen der Volkswirtschaftslehre[END_REF]. In this case the derivative is viewed as method of calculation of the additional cost. Nevertheless, the problem to justify the identification of the two different mathematical objects in economic contexts also remains in this approach.

The notion of concept image to describe students' conceptual knowledge

The economic interpretation of the derivative and its connection to the pure mathematical concept, as it was explained above, should be part of students' of economics conceptual knowledge of the derivative concept. To describe students' conceptual knowledge I will refer to the notion of concept image by [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF], which describes the total cognitive structure associated to a concept. This includes all mental pictures, properties and associated processes. In the case of the derivative students' of economics concept image should contain its representations, the differentiation rules, its connection to the concepts of monotonicity and convexity, its use as a tool for optimization problems, and in particular an adequate economic interpretation of the derivative. Since the common economic interpretation of the derivative as amount of change while increasing the production by one unit is a different mathematical object, it should be in particular carefully connected to the rest of the students' concept image, called synthesizing in literature (Dreyfus, 2002).

Knowledge concerning the derivative covered in the students' Calculus course

In the Calculus course for students of economics in which the study took place (University of Paderborn 2015, Germany), the sessions involving the derivative began with the definition of the derivative as limit of the difference quotient. Alongside with the symbolic definition, its representations as slope of the tangent line (tangent line introduced as limit of secant lines) and as rate of change were introduced. Afterwards, the unit of the derivative in the case of a cost function C was discussed and justified via the symbolic definition of the derivative. In the second lecture the economic interpretation of the derivative in the context of marginal cost was introduced, which is essential for the study presented here. Two possible economic interpretations of the derivative were presented in the lecture:

Interpretation as approximation of the additional cost of the next unit

This interpretation was justified via the approximation formula '( )

C C x x  
  , which was derived from the definition of the derivative by using the approximation aspect of the limit. The terms C  and '( )

C x x
 were also visualized on the board with the help of the tangent line.

Interpretation as additional cost of the next marginal unit

It was visualized that the mistake between C  and '( )

C x x  becomes smaller if 0 x  . This results in the asymptotic equation '( ) dC C x dx 
in which the lecturer called the "fictive infinitely small quantities" dx and dC marginal units. Some lectures later the concepts of monotonicity and convexity and their connection to the derivative were discussed. The sessions finished with optimization problems.

All the topics covered in the lecture were also practised in small groups, in which the students had to solve problems. Relevant for the study presented here is, that these problems also included a task to interpret the value '( 5)

C of the cost function 2 ( ) 8 10 700 C x x x   
in an economic context (in the way presented above).

METHODOLOGY OF THE STUDY

The study aimed to find out to what extent students of economics can make an adequate connection between the derivative and its economic interpretation after their Calculus course. Hence, eight economics students who successfully completed their Calculus Course at the University of Paderborn were interviewed. Each interview lasted about 30 minutes. The interviews were structured by four tasks:

1. Consider the cost function C that is given by the following equation:

Is the derivative '( )

Cx the same like the additional cost while increasing the production from

x units of quantity by one unit? Justify yours answer. ).

4. Justify with the definition of the derivative why it can often be assumed in economics that the values of '( ) Cx and ( 1) ( )

C x C x  are almost identical.
The tasks were not always presented to the students in written explicitly, but were sometimes given by the interviewer verbally during the interview process.

Task 1 was an introductory task with the aim to find out which of the two mathematical objects ( '( ) Cx or ( 1) ( )

C x C x 

) the students associated first with the term "marginal cost". Task 2 was central in the interview. It aimed to find out to that extent students knew the differences between the derivative and the additional cost of the next unit. If the students claimed both objects to be equal they got the additional task to determine the additional cost of the 11 th unit of a function, which was only given graphically (see right graph of figure 1). Its aim was that the students could no longer use their algorithm to determine the derivative and would use the difference ( 1) ( )

C x C x 
instead. This should make the students' concept image of marginal cost incoherent and provoke a cognitive conflict to make them rethink their ideas about the identity of the derivative and the additional cost of the next unit, and to reorganize their concept image. With the interviewer's help the students were then led to the differences between the derivative and the additional cost of the next unit.

Task 3 had the aim to find out if the students could justify the identification of '( ) Cx and ( 1) ( )

C x C x 

within the graphical representation with the help of the tangent line (similar to the visualization presented in the lecture). Task 4 finally aimed to find out to what extent the students have internalized the justification of the identification of '( )Cx with ( 1) ( )

C x C x 
in economics symbolically via the approximation formula

( ) ( ) '( ) C x h C x C x h   
for 0 h  (as it was taught in their Calculus course). The results of tasks 3 and 4 are not discussed in the paper in detail due to limited space.

The interviews were recorded, transcribed, and interpreted. First, individual cases were interpreted line by line. Later the results between different individuals were compared. To ensure reliability, the author's interpretations were discussed with colleagues of the Centre of Higher Mathematics Education in Germany (khdm).

SOME DETAILED RESULTS

Altogether eight students were interviewed. Due to limited space, two students (Holger and Lisa) with different understandings of marginal cost were chosen, whose interview parts referring to the tasks 1 and 2 are presented here in detail.

Holger's understanding of marginal cost

Holger immediately solved the first task to determine the marginal cost at

C

. This means his first association of marginal cost was the derivative and not its economic interpretation. His result was '(100) 1 C  . He did not mention a unit himself. After having been asked for the unit by the interviewer explicitly he mentioned "Euro", but could not justify it. This shows that he had an incoherent concept image of marginal cost: he associated the derivative for calculations but did not think of marginal cost as a rate (otherwise the unit would have to be Euro per unit of quantity).

In the next part of the interview Holger was explicitly confronted with the question if a definition of marginal cost as additional cost of the next unit would represent the same mathematical object like the derivative. He started thinking about it and then agreed. Therefore, the interviewer tried to provoke a cognitive conflict by giving Holger the task to determine the additional cost while increasing the production from 10 units by one unit (see right graph of figure 1). Holger first wanted to use the derivative again, but then started to think about the task again: 32 Holger:

You would have to imagine the derivative. Then you would see the additional cost. The derivative is nothing else than the slope at a point. If we take any point. 33 Interviewer: Here is one explicitly given. We search for it at a particular point. 34 Holger:

Here is one given, 10 units of quantity. So we have

10 x 
where the total cost is 21. Now we need the cost if one more is produced. Well, but we do not need this because if we are at 11 the cost is 22 point something. So the additional cost has to be one point, yes 1.1.

Holger now used the difference ( 1) ( )

C x C x 

to solve the task. To the following question of the interviewer if the value would have been the same by using '( ) Cx, Holger agreed. He then determined the slope at 10 x  graphically and got the solution "round about one" (correct value: 1, see right graph in figure 1). After the interviewer emphasized that Holger just said "round about", Holger claimed that he cannot determine the value exactly by graphical means. Hence, the interviewer asked Holger afterwards to determine the cost difference for the cost function C from task 1 given by an equation. He now got the result 1.051 and justified the "error" compared to '(100) 1 C  as follows:

77 Holger: One nearly gets [1], but only nearly. This is probably due to rounding.

One can see that Holger was really convinced that the derivative and the additional cost of the next unit are exactly the same, even if the calculated values differed. The interviewer now emphasized that there was no rounding involved. He then pointed to the graph of the cost function (left graph in figure1) and underlined that one can see the error in the graph, too. After the interviewer had asked Holger again to determine the value of the derivative by graphical means, now for the cost function of task 1 (left graph in figure 1), Holger found for himself a resolution of the conflict:

Afterwards you try to make this distance as small as possible, as you could think of, but we cannot reach the one point, but in our mind we want to reach it. And I assume this the very small rounding mistake, no, not rounding mistake, but this small difference is due to the fact that you do not reach the point exactly.

Two misconceptions of the derivative occurred here. Holger did not imagine the derivative as limit of slopes of secants through 00 ( , ( ))

x f x and 00 ( , ( )) x h f x h  for 0 h  but as limit of secants through 00 ( , ( ))

x h f x h  and 00 ( , (
))

x h f x h  for 0 h  .
But the important misconception that now prevented him from questioning the identity of '( )Cx and ( 1) ( )

C x C x 

was his opinion that the "true slope" at a point 0 x was not reached by taking the limit. He imagined the derivative to be the slope of a secant through 00 ( , ( ))

x h f x h  and 00 ( , (
)) x h f x h  with a very small 0 h  (he repeated this several times later, even more explicitly than in the lines above).

To sum up, at the start of the interview, Holger identified the derivative '( ) Cx with its economic interpretation as additional cost of the next unit. During the interview a conflict occurred due to different numerical values of these two. But instead of questioning the identity between the derivative and the additional cost of the next unit he made his concept image coherent again by attributing this error to an error between the "true slope at a point" and the derivative as result of a limiting process.

Lisa's understanding of marginal cost

Similar to Holger, Lisa also immediately solved the task to determine the marginal cost at the output 100 x  by using the derivative. Unlike Holger she stated as unit "Euro per unit of quantity", a unit of a rate. So Lisa also associated the derivative with the notion of marginal cost first. When confronted with the definition of marginal cost as additional cost of the next unit she replied: 20 Lisa: Yes, I really thought about this last semester. In the economic subjects we really learn it this way. […] And I always had to say: If you increase x by one unit, y increases by these many units, eh? This is really the case. But I have, since I had mathematics last semester, always thought that you learn it differently in mathematics. In mathematics you say, if you increase

x by one marginal unit, y increases by these many marginal units.

Unlike Holger, Lisa did not identify the derivative with the cost of the next unit. She even felt a conflict between the knowledge about marginal cost she learned in her maths course and the actual use of marginal cost in econometrics. Her remarks point out that she was of the opinion that marginal cost is not the additional cost of the next unit but of the next marginal unit. In which way she understood the term "marginal unit" was not clear yet. Therefore, the interviewer asked about this term: The interviewer gave Lisa the graph of the function of task 1 (left graph in figure 1).

32 Lisa:

Well, one unit could be from 100 to 101 we said. So you increase x by one unit from 100, eh? From the actual output. Then I would be here [pointing on (101)

C

].

After the interviewer's comment to determine the accurate value by calculation she got (101) (100) 1.051

CC 

(compared to '(100) 1 C  ), and continued as follows:

48 Lisa: And a marginal unit I imagine very, very small. Here I would go a right very, very little bit to the right and then a very little, little bit upwards.

We see here that Lisa understood a marginal unit as a very small, but finite unit. This understanding is also found in books of economics (and differs from the way a marginal unit was taught in the maths course as a "fictive infinitely small quantity"). After Lisa's explanation of the "marginal unit", the interviewer tried to induce a cognitive conflict by asking Lisa if the additional cost of such a small unit should not be close to zero. She then explained the following: This shows that Lisa understood the additional cost of a marginal unit as '(100

) C dx 
with dx being a very small, but finite unit. This is in her opinion numerically the same like '(100) C because the slope stays the same at 100 dx  . The interviewer again asked if additional cost slope are the same whereat Lisa agreed.

To sum up, Lisa knew that the derivative is not the additional cost of the next unit. She remembered the interpretation of the derivative to be the additional cost of a marginal unit from her Calculus course. She imagined a marginal unit dx to be a very small, but finite unit (and not in the way it was taught in the course as a "fictive infinitely small quantity" in the asymptotic equation '( ) dC C x dx 

) and identified the additional cost of a marginal unit in her mind with '(100

) C dx 
, which is in her opinion the same like '(100) C . She did not recognize the different nature of the derivative being a rate of change and the additional cost being an amount of change.

SUMMARY AND DISCUSSION

Just like in the cases of Holger and Lisa presented in detail here, the study showed that the students had difficulties to make an adequate connection between the derivative as a mathematical concept and its economic interpretation as additional cost. The author considered a connection as adequate if the students were aware of the differences between '( ) Cx and the additional cost of the next unit, and were able to justify their identification in economics (it did not matter if they associated marginal cost with the derivative or the additional cost of the next unit first).

At the beginning of the interview no student could make an adequate connection between the derivative and the additional cost of the next unit. The majority just declared these objects to be exactly the same (like Holger).

During the interview, all students recognized the question concerning the identity of the derivative and the additional cost of the next unit as very relevant and became aware of differences between these two with the interviewer's help. However, in the process of leading them to these differences on the graphical level via the tangent line, several problems occurred (not discussed here in detail due to limited space):

1. Misconceptions concerning the derivative concept (like Holger's misconception about the derivative being not the "true slope" at the point)

2. Incomplete concept images (example: knowledge of the geometric representation of the derivative as slope at a point but no association of the tangent line)

3. Problems in determining the slope of a linear function Only one student could make a connection on the symbolic level via the formula Furthermore, most participants of the study had not thought about the differences and the connection between the derivative and its economic interpretation as additional cost before the interview, although these were presented in their Calculus course.

A solution to these problems in a traditional Calculus course for students of economics, in which the concept of derivative is taught first, an economic interpretation afterwards, could be to confront the students with the two different notions of marginal cost in the tutorials of the course directly, and to provoke a cognitive conflict, just like in the interview. Afterwards, one could let them try to connect the derivative and the additional cost of the next unit by themselves or in small groups, and help them individually if misconceptions or incomplete concept images of the derivative occur. Another solution could be to start with the concept of marginal cost as additional cost first, which can be approximated by the derivative as linear approximation. However, an understanding of the mathematical concept of the derivative as slope of the tangent line is also necessary in this approach.

Concerning future research, one has to emphasize that the study presented here relied on one perspective on the connection between the derivative as a mathematical concept and its use in economics: additional cost as interpretation of the derivative. This perspective is important because students of economics are often confronted with it their maths course and in courses about economic theory (it can be found in respective books of economics). But as explained in the theoretic part of the paper: it is not the only one. Other perspectives may further enrich the knowledge about students' of economics understanding of the derivative and its use in economics.

  output x is given in units per quantity, the cost() Cx is given in Euro. Determine the marginal cost for an output of 100 x  units of quantity. Determine the unit (of the marginal cost), too.

Figure 1 :

 1 Figure 1: Graphs of the cost function from task 1 and the cost function from the task, in which the additional cost for the 11 th unit had to be determined

21

  Interviewer: Now the question, what is a marginal unit? 22 Lisa:Well, a marg/ Shall I draw it?

h

   like presented in the Calculus course.

  52 Lisa:This has to do with the slope you have. For the derivative you calculate the slope of the tangent line. The slope of the tangent line is what you calculate, isn't it? 53 Interviewer: Right, the slope of the tangent line, yes. 54 Lisa: Yes, and this is also what I get if you increase x by a marginal unit, starting at 100. Ah, what do I get? No, if you increasex by one marginal unit, the marginal cost still increase by one. I think the slope still remains one, right?

Holger: Oh, the reason is, because it is not exact. The origin of the derivative was to determine the slope at a point. To achieve this you take one point left and one point right of it, which have the same distance, and the slope between.


Stefanie Rach
email: rach@math.uni-paderborn.de

Stefan Ufer²

Timo Kosiol²

Situational interest in university mathematics courses: similar for real-world problems, calculations, and proofs?

Keywords: Students' interest, Feeling-resp. value-related component, Mathematical practices, Transition to and across university mathematics, Teachers' and students' practices at university level

In the educational context where this study was conducted, the transition from school to university is associated with changes of the learning domain mathematics. In high school, solving real-world problems and performing calculations are dominating practices, whereas in university, the main focus is on proving. Successful learning processes are associated with appropriate prerequisites, e. g. interest. The question is which component of interest concerning which practice is important for successful processes. We developed an instrument to differentiate these facets of students' situational interest. A study with 339 first-semester students in mathematics partially confirms the expected factorial structure of this instrument. Precise information concerning learners' interest may help us to support students at this challenging transition.

INTRODUCTION

The transition from school into a university mathematics programme is a challenging phase for many students. Reports mentioning high dropout rates in academic study programmes with a focus on mathematics (OECD, 2010) illustrate this fact. Reasons for students' problems in the first year of university study are primarily attributed to the changes of the learning domain: while the school subject mathematics strongly relies on performing calculations and solving real-world problems, mathematics in university is presented as a scientific discipline with a focus on proving [START_REF] Rach | Welche mathematischen Anforderungen erwarten Studierende im ersten Semester des Mathematikstudiums? [Which Mathematical Demands do Students Expect in the First Semester of Their Study?[END_REF]. It is yet not clear which of the individual prerequisites, that students bring from school, support successful learning processes in university courses. Many researchers assume that subject-specific interest, in the sense of a person-object or a person-situation relationship, is an important learning prerequisite in general [START_REF] Wigfield | Students' achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes[END_REF]. Common instruments to survey interest in mathematics are questionnaires. Mostly, these use the word "mathematics" to describe the objects of interest. At the transition from school to university, however it is not clear if students refer to school or university mathematics when reporting their interest in this questionnaires. To obtain a more differentiated insight into students' interest with a stronger relation to concrete learning situations, we develop a taskbased instrument to measure students' interest concerning different mathematical practices. In this contribution, we present the conceptualisations of this instrument in detail and report results of an empirical study with 339 first-semester students that investigates the factorial structure of the instrument.

THEORETICAL BACKGROUND

In the first section, we give an overview about the motivational construct "interest" by addressing common definitions and results concerning its role in learning processes. In the second section, we summarize ideas to the objects of interest, mathematics.

Interest in mathematics learning processes

Researchers agree that interest is an important motivational variable. Its role as a learning prerequisite, a learning processes measure, and as a learning outcome has been put forward repeatedly [START_REF] Rotgans | The Role of Interest in Learning: Knowledge Acquisition at the Intersection of Situational and Individual Interest[END_REF][START_REF] Sonnert | The Impact of Instructor and Institutional Factors on Students' Attitudes[END_REF][START_REF] Wigfield | Students' achievement values, goal orientations, and interest: Definitions, development, and relations to achievement outcomes[END_REF]. Interest is defined as a specific relationship between a person (here: a student) and an object or a situation (here: mathematics resp. mathematical practices) [START_REF] Krapp | Structural and dynamic aspects of interest development: theoretical considerations from an ontogenetic perspective[END_REF][START_REF] Schiefele | Situational and individual interest[END_REF]. The objects of interests may be concrete objects, topics, or school subjects. Whereas Schiefele, [START_REF] Schiefele | Interest as a predictor of academic achievement: A meta-analysis of research[END_REF] conceptualize interest mainly as a relatively stable trait: "Individual interest is conceptualized as a relatively stable affective-evaluative orientation toward certain subject areas or objects" [START_REF] Schiefele | Situational and individual interest[END_REF], pp. 198), Linnenbrink-Garcia et al. (2010[START_REF] Linnenbrink-Garcia | Antecedents and consequences of situational interest[END_REF], in contrast, see interest as specific to a situation: "Situational interest is a temporary state aroused by specific features of a situation, task, or object (e. g., vividness of a text passage)" (Schiefele, 2009, pp. 197-198). Situational interest is limited at a point of time, e. g., in a certain learning situation and may or may not develop into individual interest [START_REF] Hidi | The Four-Phase Model of Interest Development[END_REF]. [START_REF] Schiefele | Situational and individual interest[END_REF] distinguishes between two components of interest: (1) a feeling-related component, related to fun or other positively experienced emotions, and (2) a value-related component, related to a high importance of the objects of interest for oneself (adapted from [START_REF] Linnenbrink-Garcia | Measuring Situational Interest in Academic Domains[END_REF][START_REF] Linnenbrink-Garcia | Antecedents and consequences of situational interest[END_REF]. Linnenbrink-Garcia and colleagues (2010,2013) could confirm in empirical studies the two-factor structure of (maintained) situational interest and separate the two components from each other. The separation of interest in a feeling-related and a value-related component is also in line with expectancy-value models that distinguish between an intrinsic and a utility value [START_REF] Eccles | Motivational beliefs, values, and goals[END_REF].

Several researchers assume that interest can trigger successful learning processes: interest can lead to more engagement and to an enhanced use of deeper learning strategies which then result in better learning achievement [START_REF] Hidi | The Four-Phase Model of Interest Development[END_REF]. However, empirical studies could not confirm this assumption for learning processes in undergraduate mathematics courses [START_REF] Rach | The Transition from School to University in Mathematics: Which Influence Do School-Related Variables Have[END_REF]. The conceptualisation and operationalisation of interest in mathematics may be one reason for these conflicting findings: firstly, the interest object "mathematics" changes its character at the transition from school to university. So the objects of reported interest may differ from the learning object in the first semester (see next section). Secondly, many of the used instruments measure the individual interest rather than the situational interest, although the situational interest is a more proximal variable to the specific learning situation.

Mathematical practices at the transition from school to university

In the literature [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF][START_REF] Engelbrecht | Adding structure to the transition process to advanced mathematical activity[END_REF], changes in two relevant aspects of the learning environment at this transition are described: a shift in the character of the learning domain, mathematics, and a change in the learning opportunities and their use. As interest is a person-object relationship resp. a person-situation relationship, the change of the learning domain is important for the conceptualisation of interest. The specific differences of mathematics at school and at university might vary between countries due to traditions concerning learning goals etc. The subsequent presentation refers to the situation in Germany which was analysed in empirical studies [START_REF] Rach | Welche mathematischen Anforderungen erwarten Studierende im ersten Semester des Mathematikstudiums? [Which Mathematical Demands do Students Expect in the First Semester of Their Study?[END_REF] and which is relevant for the empirical study we present below. Nevertheless, several of the described features might hold for other countries as well. One central goal of teaching and learning at school is to apply mathematics for solving real-world problems (CCSSI, 2010;OECD, 2016). Thus, describing realistic situations mathematically, performing computations, and applying mathematical procedures are central. On the contrary, university mathematics is usually taught as a scientific discipline based on formal definitions of concepts and formal-deductive proofs [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF][START_REF] Nardi | The novice mathematician's encounter with mathematical abstraction: Tensions in concept-image construction and formalisation[END_REF]. Mathematics at university is often presented in a DTP (Definition-Theorem-Proof) structure to emphasise its logical rigidity [START_REF] Engelbrecht | Adding structure to the transition process to advanced mathematical activity[END_REF][START_REF] Weber | Effective Proof Reading Strategies for Comprehending Mathematical Proofs[END_REF]. These conclusions are mainly based on observations of lectures [START_REF] Weber | Traditional instruction in advanced mathematics courses: a case study of one professor's lectures and proofs in an introductory real analysis course[END_REF] or of tutorials [START_REF] Nardi | The novice mathematician's encounter with mathematical abstraction: Tensions in concept-image construction and formalisation[END_REF], interviews with involved parties [START_REF] Nardi | Amongst Mathematicians: Teaching and Learning Mathematics at University Level[END_REF], or analyses of tasks [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF]. In general, mathematical tasks are often used to describe and examine learning situations and their cognitive demands (e. g., [START_REF] Stein | Instructional Tasks and the Development of Student Capacity to Think and Reason: An Analysis of the Relationship between Teaching and Learning in a Reform Mathematics Project[END_REF].

Theoretical analyses and anecdotal evidence from students and teaching staff have described a change of the learning domain (e.g., [START_REF] Engelbrecht | Adding structure to the transition process to advanced mathematical activity[END_REF][START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF][START_REF] Thomas | The school-tertiary interface in mathematics: teaching style and assessment practice[END_REF]. However, empirical studies supporting the role of this shift and its effects on student learning are scarce. In particular, reliable evidence on the role of students' motivational variables, especially of students' interest, is scarce.

The SISMa project

The goal of the SISMa project ("Self-concept and Interest when Studying Mathematics") is to contribute evidence concerning students' interest and selfconcept with regard to certain mathematical practices [START_REF] Ufer | Interest in mathematics = Interest in mathematics? What general measures of interest reflect when the object of interest changes[END_REF]. As a first step of the project, we developed interest and self-concept measures that focus central mathematical practices. Using these measures, the development of these variables during students' learning processes and their effect on student learning in undergraduate mathematics programmes are investigated.

In this contribution, we focus on the measurement of situational interest. We developed a questionnaire of situational interest that is based on the conceptualisation of interest as a person-object (situation) relationship, and on the assumption that interest concerning a specific task is an indicator of situational interest. Here, we follow the idea of Schukajlow and colleagues (2012) who operationalize the two components of situational interest with the constructs enjoyment and value (see also [START_REF] Schukajlow | Are Interest and Enjoyment important for students' performance?[END_REF]. In this questionnaire, mathematics tasks are presented to students and they are asked to imagine solving this task (not to solve the task), and then to state their anticipated enjoyment and value appraisals when working on the task. For our questionnaire, we designed mathematical tasks concerning the topic derivatives that each prototypically represent one of the three practices "solving realworld problems" (resp. applying mathematics, sample item: "Using metal, you should produce a cylindrical can with a prescribed volume. For which radius is the material consumption minimal?"), "performing complex calculations" (sample item: "Let f be . Calculate the extrema of the function f."), "and "proving" (sample item: "Let be a differentiable function. Show that f is continuous."). After a pilot study (see [START_REF] Rach | Welche mathematischen Anforderungen erwarten Studierende im ersten Semester des Mathematikstudiums? [Which Mathematical Demands do Students Expect in the First Semester of Their Study?[END_REF], we used 12 tasks. While the first and second practice are considered central in mathematical lessons at school, the last practice predominates in university courses. After reading each task, students rate their agreement to one statement concerning the feeling-related component (item: "It would be fun to me to work on this task.") and one statement concerning the valuerelated component of interest (item: "Even if the task is not part of an exam, it is important to be able to solve the task."). So in sum, situational interest is divided into six subscalesone for each component and each mathematical practice (see table 1).

RESEARCH QUESTIONS

The aim of this contribution is to investigate the structure of students' situational interest as measured with our instrument. To achieve this, we applied the instrument to students from a first semester mathematics course. The questions focused in this contribution address students' reported levels of interest and the empirical structure of the questionnaire:

1. What level of feeling-and value-related situational interest do students from a first semester mathematics course report concerning the three practices? Since prior research has shown similar trends, we expected that students would report lower interest levels for each component and practice after six weeks, compared to the first day of their studies. One reason might be that high demands in the first semester lead to a decrease in interest.

2. Is the theoretical structure of subscales, that guided their development, reflected in the factorial structure of the newly developed instruments?

We expected that the two components of situational interest, the feeling-and the value-related component, are reflected in the empirical data. Moreover, we expected that subscales concerning the three different practices can be separated from each other.

METHOD

Design and sample

We present data from two measurements, at the beginning of the first semester (T1) and in the middle of the first semester (T2). Our sample consists of 339 mathematics students (162 female) of one German university of the course "Analysis I" which is compulsory for first-semester mathematics students. In this course, mathematics is presented as a scientific discipline with a strong focus on formal concept definitions and deductive proofs. The students were enrolled in the bachelor's programmes "mathematics" (n = 90), "business mathematics" (n = 91), or a mathematics teacher education programme for the highest attaining secondary school track in Germany (n = 104)for the remaining students, we have no information about their study programme. The participation in the study was voluntary.

Instruments

The mathematical tasks were arranged in a fixed, random order. The questionnaire was submitted to the students with the following instructions: "Imagine how you solve these tasks. Do not solve the tasks, but report your agreement to the following statements". Students rated each statement on a four-point likert scale from 0 (disagree) to 3 (agree). The individual mean value of a single student on a scale was computed if this student had answered at least half of the items of the scale.

RESULTS

Table 2 shows the means, standard deviations, and Cronbach's Alpha of the six scales at the first and second measurement point. Concerning all 24 items (3 practices, 4 tasks, 2 components), there are no floor or ceiling effects. Reliability analyses underpin the internal consistency of the six subscales. For T1 (N = 323-331), Cronbach's alpha of each scale ranges from .73 to .88, for T2 (N = 224-237), from .76 to .86. Indeed, students reported higher value and feeling ratings concerning all practices in the first as compared to the second measurement (cf. table 2; t(194-197) = 5.42-6.83; p < .001; d = 0.39-0.49; measured with students who were present at both measurements). Levels of value-and feelingrelated interest were relatively similar between the three practices.

Correlation analyses (see table 3) show that, as expected, the feeling-related and value-related scales concerning each practice strongly relate to each other (Solving real-world problems: T1: r = .55, T2: r = .65, Performing complex calculations: T1: r = .49, T2: r = .53, Proving: T1: r = .53, T2: r = .60). Moreover, the three feelingrelated scales (r = .43-.60) resp. value-related scales (r = .59-.79) correlate strongly.

The correlations of feeling-related scales with value-related scales for different practices are moderate (T1: r = .28-.44, T2: r = .31-.39). As some of the situational interest scales correlate strongly, we investigated whether our expected scale structure would be replicated by exploratory factor analyses. The results of Principal Component Analysis with Varimax rotation for every measurement point partially support our expected structure of our scales. For T1, results indicate four factors that explain nearly 63% of the variance. Table 4 shows the factor loadings of the items on the four identified factors.

As expected, the feeling-related items concerning each of the three practices load strongly on one of the factors two to four each. Contrary to the theoretical construction of our questionnaire, the first factor includes the value items for all three practices. Some value-related items have cross-loadings on the feeling-related scales for the same practice. For T2, the factor analysis shows similar results, with the four factors explaining 65% of the variance. Thus, it also seems to be possible to combine all value items into one value scale. This value-related scale with twelve items has an excellent reliability of α = .93 (N = 317, T1) resp. α = .92 (N = 223, T2).

DISCUSSION

Students' interest is an important variable in successful learning processes, including the transition to university mathematics. Based on prior research, we argue that to measure students' interest validly, one requires situated, e.g. task-related, measures, to cover feeling-and value-related components of interest and to take into account the changing role of different mathematical practices during this transition [START_REF] Rach | The Transition from School to University in Mathematics: Which Influence Do School-Related Variables Have[END_REF]. Contrary to the findings of [START_REF] Schukajlow | Teaching methods for modelling problems and students' task-specific enjoyment, value, interest and self-efficacy expectations[END_REF], the results of our empirical study show that the relations between the feeling-related components for different practices are relatively low as compared to the respective correlations between the value-related components. This partially supports the expected six-factor model (c. f. [START_REF] Schiefele | Situational and individual interest[END_REF], and underpins the different roles the three practices play during the transition in the educational context in Germany. Beyond this, factor analyses indicate that the value-appraisals seem to be fairly consistent over all three practices and the exploratory factor analyses also allow a model assuming only one value factor over all practices. This observation applies to data from the first day of university study, but also to data collected after six weeks into the semester. After six weeks, students report across all scales lower approvals. Interestingly, even though students seem to differentiate their interest reports by practices, we find almost no differences in their mean levels of interest between the different interest scales.

Of course, the results of our study rely on students' self-reports about their anticipated situational interest in a set of specific tasks. Even though the differences between school and university mathematics have been described internationally, it might be interesting to replicate the studies in other educational systems. In future studies, the relation between the anticipated situational interest and the actual engagement in learning situations should be considered. However, [START_REF] Schukajlow | Are Interest and Enjoyment important for students' performance?[END_REF] found only slight differences in prospective and retrospective ratings of interest in working with mathematical tasks. In sum, the newly developed instrument may provide more differentiated insights into students' interest concerning different mathematical practices. Further research should investigate which facets of situational interest indeed go along with learning gain in the study entry phase. In particular, it is an open question if interest in practices that are typical for university mathematics, such as proving, are more important for learning gain than other interest facets. In the future, this instrument may help to evaluate support courses in the study entrance phase, to analyse the development of students' interest (c. f. [START_REF] Hidi | The Four-Phase Model of Interest Development[END_REF], and to investigate the impact of different interest facets on study success.

Table 1 : Two components of situational interest and three practices.

 1

		Solving real-world	Performing complex	Proving (4 tasks)
		problems, applying	calculations
		mathematics (4 tasks)	(4 tasks)
	Feeling related	Feel Apply	Feel Calc	Feel Proof
	Value related	Value Apply	Value Calc	Value Proof

Table 2 : Means, standard deviations, and Cronbach's Alpha of the interest scales. Likert scale from

 2

		Beginning of the first semester (T1)	Middle of the first semester (T2)
		N	M (SD)	α	N	M (SD)	α
	Feel Apply	323	2.22 (0.61)	.73	226	2.02 (0.67)	.76
	Feel Calc	323	2.16 (0.71)	.79	230	2.01 (0.74)	.80
	Feel Proof	331	2.18 (0.61)	.77	237	2.00 (0.61)	.82
	Value Apply	323	2.25 (0.64)	.82	224	2.04 (0.71)	.82
	Value Calc	325	2.26 (0.70)	.86	232	2.00 (0.76)	.86
	Value Proof	331	2.31 (0.61)	.88	237	2.06 (0.67)	.86

0 (disagree) to 3 (agree).

Table 3 : Correlations between the interest scales. Over the diagonal T1 (N = 334-339), under the diagonal T2 (N = 239-243). All correlations significant with p < .01.

 3

		Feel	Feel	Feel	Value	Value	Value
		Apply	Calc	Proof	Apply	Calc	Proof
	Feel Apply		.47	.43	.55	.31	.31
	Feel Calc	.47		.60	.28	.49	.32
	Feel Proof	.44	.54		.38	.44	.53
	Value Apply	.65	.31	.31		.66	.68
	Value Calc	.30	.53	.39	.59		.79
	Value Proof	.37	.37	.60	.66	.79

Table 4 : Factor loading on the four extracted factors (T1, N = 313). Loadings under .30 not shown. The expected loadings on a factor are bolded.

 4

		Factor 1	Factor 2	Factor 3	Factor 4
	Feel Apply 1		.35	.36	.42
	Feel Apply 2		.31		.72
	Feel Apply 3			.44	.64
	Feel Apply 4				.81
	Feel Calc 1		.50	.41
	Feel Calc 2		.38	.67
	Feel Calc 3			.78
	Feel Calc 4		.35	.74
	Feel Proof 1		.71
	Feel Proof 2		.62
	Feel Proof 3		.65
	Feel Proof 4		.70
	Value Apply 1	.70			.44
	Value Apply 2	.61			.50
	Value Apply 3	.60			.56
	Value Apply 4	.55
	Value Calc 1	.75
	Value Calc 2	.75		.32
	Value Calc 3	.69		.46
	Value Calc 4	.76		.35
	Value Proof 1	.73	.35
	Value Proof 2	.82
	Value Proof 3	.71	.32
	Value Proof 4	.76	.35