text:
Erik Hanke
email: erik.hanke@uni-bremen.de

Ingolf Schäfer
email: ingolf.schaefer@uni-bremen.de

Students' view of continuity: An empirical analysis of mental images and their usage

Keywords: Continuity, mental images, Grundvorstellungen, concept image, acceptance

.

Introduction

More than 20 years ago [START_REF] Moore | Making the transition to formal proof[END_REF] did an empirical analysis on the difficulties students face when they are required to give formal proofs. He identified among other factors that the students had little intuitive understanding of the concepts and their concept images were not adequate to perform certain proofs. Moreover, [START_REF] Selden | Proof and problem solving at university level[END_REF] argue that the ability to choose the right conceptual representation is a vital part in proving and generally in problem-solving activities.

In the context of analysis the concept of continuity is one of the most fundamental notions needed to do rigorous proofs. It is well known that students have difficulties with the notion [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF]. This paper focuses on the mental images that future math teachers, pure math and applied math students have, which mental images they find acceptable and which they use to solve tasks. For the notion of continuity from the point of view of mathematics as well as mathematics education we refer to [START_REF] Tall | How humans learn to think mathematically: Exploring the three worlds of mathematics[END_REF].

Theoretical background

In the German tradition of subject-matter didactics the notion of Grundvorstellungen (regularly translated as "basic ideas") has gained much attention: The idea of Grundvorstellungen was extracted by vom Hofe (1995) after an analysis of related ideas of didactics of arithmetics and collegepreparatory didactics by pointing out the importance of creating internal representations of mathematical notions in the learners' minds [START_REF] Vom Hofe | Grundvorstellungen" as a category of subject-matter didactics[END_REF]. According to [START_REF] Kleine | With a focus on 'Grundvorstellungen' Part 1: a theoretical integration into current concepts[END_REF] Grundvorstellungen link mathematics and reality by pointing out that modeling is a central mathematical process which fulfills requirements of mathematical literacy (application, structure and problem orientation): The authors argue that this is only possible after having acquired internal representations of mathematical concepts, so called Grundvorstellungen, which connect learners' experiences and mathematical knowledge with real life. Primary Grundvorstellungen are directly related to concrete objects and actions in the environment of the learners whereas secondary Grundvorstellungen consist of imaginative actions with mental representations [START_REF] Vom Hofe | Grundvorstellungen" as a category of subject-matter didactics[END_REF]. The latter are particularly relevant for the notion of function and special classes thereof such as continuous real-valued functions.

We prefer to regard the essence of Grundvorstellung, using a subject-matter-didactical analysis, as a predominantly normative (or even prescriptive) approach to find internal representations learners should acquire in order to be able to recognize and use a mathematical notion in inner-mathematical or applied fields. But the idea of Grundvorstellungen is complemented by the wish of mathematical didactics specialists to observe actual mental models or images, respectively, that learners really develop [START_REF] Vom Hofe | Grundvorstellungen mathematischer Inhalte[END_REF][START_REF] Vom Hofe | Grundvorstellungen" as a category of subject-matter didactics[END_REF][START_REF] Kleine | With a focus on 'Grundvorstellungen' Part 1: a theoretical integration into current concepts[END_REF].

The notions of concept image and concept definition by [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] have been foundational for the existing literature on university students' conceptions of elements of analysis such as differentiation, integration but also limits and continuity. The concept image comprises "the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes" (Tall & Vinner, 1981, p. 152). Besides, the concept definition is a form of words to specify the concept and to communicate it. It can be personal or formal, thus individually shaped or widely accepted by the mathematical community [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF]. A formal concept definition rather reflects a normative viewpoint on what is actually forced to belong to the concept. We argue that concept definitions are part of the overall of concept images of a mathematical notion. Contrariwise, if learners are confronted with an existing concept definition they develop concept definition images, a part of their concept images that expressly comprises their associations with the definition. Additionally, it is understood that learners enter their acquisition process of a newly introduced concept with preexisting concept images [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF], and that teaching persons and environments can influence the acquisition of concept images [START_REF] Bingolbali | Concept image revisited[END_REF].

Since both the idea of Grundvorstellungen and concept images lack a distinctive description of what actually counts as internal representations, a conceptualization of mental images (Vorstellungen) was developed in [START_REF] Hanke | Ausprägung und Akzeptanz von Vorstellungen zur Stetigkeit bei Mathematikstudierenden[END_REF] which seems more appropriate to address the subtleties of precise and distinguished research questions in the scope of mental images. Mental images are substantiated as individual constructions and therefore reconstructions of all kinds of mathematical notions. They are of singular, regular or epistemological nature, can be subdivided into mental pictures (Vorstellungsbilder) and mental actions (Vorstellungshandlungen) [START_REF] Weber | Mathematische Vorstellungen bilden. Praxis und Theorie von Vorstellungsübungen im Mathematikunterricht der Sekundarstufe II[END_REF]. Due to the premise of being able to be communicated, mental images can be shared as well as accepted, rejected or even imposed on somebody.

The most important idea for our study-and in general empirical research-is the fact that mental images cannot be observed. Thus, the only way to do empirical research about mental images is to study their communicative simulacra, the transformation of a potential inner world of a learner into observable entities such as spoken words, written solutions to exercises and so on [START_REF] Hanke | Ausprägung und Akzeptanz von Vorstellungen zur Stetigkeit bei Mathematikstudierenden[END_REF]. In particular, communicative simulacra do not reflect normative assumptions on a notion as it is the case with Grundvorstellungen. In case of answers to a direct question on mental images (e.g. "What is your intuitive meaning of continuity?") we will speak of exclamatory simulacra. With this terminology we emphasize the fact that what is actually communicated by a learner depends on the occasion of communication and does not necessarily reflect the full entity of associations of the learner. We cannot even be sure that the learner is aware of the intentions of the researcher when asked about mental images. Rather, we find blurrings of the actual mental images of learners that could potentially be sharpened by further qualitative analysis. In particular, exclamatory simulacra are shaped by the learner's understanding of the concept in question and are only a subset of communicative simulacra which, in turn, can be expressed by different forms of communication.

Here we concentrate on descriptions of communicative simulacra.

Based on [START_REF] Moore | Making the transition to formal proof[END_REF] and Selden's and Selden's (2013) conclusions, we believe that the mere knowledge of definitions, the ability to reproduce them or the setup of mental images for a mathematical concept do not necessarily mean that the students are able to use the concept. Also, we believe that the more mental images students have the more they are able to apply at least some of these in inner-mathematical situations or in contexts. [START_REF] Moore | Making the transition to formal proof[END_REF] term concept usage is related to our idea of distinguishing between exclamatory simulacra of mental images and the usage of (probably different) mental images as required in the third section of our questionnaire. The second part of the questionnaire provides insight in the acceptance of attitude (Einstellungsakzeptanz) and acceptance of usage (Nutzungsakzeptanz) [START_REF] Weber | Mathematische Vorstellungen bilden. Praxis und Theorie von Vorstellungsübungen im Mathematikunterricht der Sekundarstufe II[END_REF] (see next section).

The review of central papers [START_REF] Bezuidenhout | Limits and continuity: some conceptions of first-year students[END_REF][START_REF] Núñez | What did Weierstrass really define? The cognitive structure of natural and ε-δ continuity[END_REF][START_REF] Schäfer | Vorstellung von Mathematiklehramtsstudieren[!] zur Stetigkeit[END_REF][START_REF] Takači | On the continuity of functions[END_REF][START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] on concept images and related results on students' conceptions of continuity lead to the classification in Table 1 of eight possible mental images that are reported in the literature following [START_REF] Mayring | Qualitative Inhaltsanalyse: Grundlagen und Techniken[END_REF] methodology of qualitative content analysis. We emphasize that these categories are representatives of communicative simulacra identified in the literature and we do not intend to judge about their formal or normative correctness.

Connections of continuity to the concept of integration is hardly ever noticed explicitly and therefore in case of appearance subsumed under miscellaneous. Likewise, the concept image of "pulling flat" the graph of a real-valued continuous function (Tall, 2009, p. 487) could not be found in any of the students' responses. It seems to be related to the rubber band metaphor often used in topology and usually is not part of standard German textbooks or lectures on analysis in one variable. Additionally, [START_REF] Schäfer | Vorstellung von Mathematiklehramtsstudieren[!] zur Stetigkeit[END_REF] Grundvorstellungen for real-valued functions (controlled stability while wiggling at a point, possibility of approximation at a point and connectedness of the graph) are subsumed in the categories of Table 1.

# Category Example

I

Look of the graph of the function "A graph of a continuous function must be connected" II Limits and approximation "The left hand side and right hand side limit at each point must be equal" III Controlled wiggling "If you wiggle a bit in x, the values will only wiggle a bit, too"

IV Connection to differentiability "Each continuous function is differentiable" V General properties of functions "A continuous function is given by one term and not defined piecewise" VI Everyday language "The function continues at each point and does not stop"

VII Reference to a formal definition "I have to check whether the definition of continuity applies at each point"

VIII Miscellaneous

Table 1: Categories for mental images of continuity

Thematic Working Group 14

Setup of the study and methodology

Our research questions have been:

1.) What mental images do students express by exclamatory simulacra?

2.) What mental images do students accept and make use of in argumentation?

3.) Is there a difference between students who want to become teachers and those studying pure and applied mathematics with regard to mental images or concept usage?

We distributed a questionnaire to 54 Bachelor students (first-year pure and applied mathematics and second-year mathematics teacher students) in Bremen after the completion of a lecture with exercise classes on Analysis I [START_REF] Hanke | Ausprägung und Akzeptanz von Vorstellungen zur Stetigkeit bei Mathematikstudierenden[END_REF]. The course covered approximately the content of [START_REF] Binmore | Mathematical analysis. A straightforward approach[END_REF]. The chosen methodology of the questionnaires is very similar to the one used often to investigate concept images (e.g. [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF][START_REF] Bezuidenhout | Limits and continuity: some conceptions of first-year students[END_REF][START_REF] Nordlander | On the concept image of complex numbers[END_REF]. Our questionnaire, described in detail below, is an extended version of those described in [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] and in particular [START_REF] Schäfer | Vorstellung von Mathematiklehramtsstudieren[!] zur Stetigkeit[END_REF]. No questions concerning applications are given in order to identify students' conceptions of continuity solely related to the mathematics itself. New is the differentiation as described in the taxonomy of communicative simulacra and the comparative approach of acceptance of attitude and acceptance of usage, i.e. if students accept certain concept images and if they can apply those different images to some example functions.

To identify the different types of simulacra, the types of questions we provided different stimuli. In the first part of the questionnaire the students were asked to freely verbalize what the intuitive meaning of continuity from their point of view is. In the second part we probed the acceptance of attitude of the verbalizations of mental images presented to the participants in fictive statements on a 6-point Likert scale (totally decline (0), …, totally accept ( 5)) (Table 2). Furthermore, the third part of the questionnaire focused on acceptance of usage of mental images since we asked to give arguments for whether the following functions in Table 3 are continuous at the respective locations with multiple mental images. We have chosen the functions because they have discontinuities of different kind: the right and left hand side limit of g(x) exist and do not coincide as x approaches 0, and the limits of f(x), and h(x) respectively, do not exist as x approaches 1, or 0 respectively, while the graph of f is disconnected in every neighborhood of 1, whereas the graph of h is connected in every neighborhood of 0. Summarizing, we are interested in the threefold of mental images through communicative simulacra: exclamatory simulacra and the usage and acceptance of mental images observable in communicated outcomes. Due to page restrictions, we limit ourselves on an overview and provide some statistics.

For instance, the flexibility of usage and acceptance of individuals will be preserved for an upcoming paper.

Results and discussion

All answers to our questionnaires were categorized according to Table 1. Multiple responses of the students were not only possible but desired and multiple categorization of a single answer into the categories was also possible. The categorization of the answers to the first question "What is the intuitive meaning of continuity from your point of view?"-i.e. the exclamatory simulacra of mental images of continuity-led to the following: five students did not give an answer, 37 answers fell into only one category, eleven in two and the one remaining answer in three categories. Around 70% of the overall codes were found in "Look of the graph" (I) and all the other categories appeared in no more than 10% of the cases each.

The "Look of the graph" (I) is the dominant mental image among students when asked to give one. Nevertheless, some students are able to accept other mental images as well. Figure 1 illustrates this. While the items "Graph has no holes" (4) and "Graph has no jumps" (7) are accepted by the majority, items close to the limit (2) or Weierstraß definition (3) of continuity have 40% to 50% acceptance. The fact that the majority wrongly connects differentiability as a necessary condition for continuity may be a problem of the item in the questionnaire which included two negations. Based on the very high rejection rates for the items "Having minima and maxima is characteristic for continuity" (1) and "Graph does not swing too much back and forth" (8) in the second part of the questionnaire, it seems to be certain that these are not common misconceptions about continuous functions.

The functions f, g and h in Table 3 were all known to the students and had been part of the course in analysis and also of the exercises. In contrast to [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] we did not give a picture of the graphs. The functions f and g seem familiar to the majority of students so about 50% are able to give a correct answer. The function h seems more complicated and most students do not answer the question at all and about half of the answers are false. This item is one where there is a real difference between those who study to become teacher and those who want to work as mathematicians. In the latter group the percentage of a correct answers is about twice as high [START_REF] Hanke | Ausprägung und Akzeptanz von Vorstellungen zur Stetigkeit bei Mathematikstudierenden[END_REF]. To identify group differences between the different study groups (pure and applied math students vs. future teachers), we counted the occurrences of every category in Table 1 in the answers of the students for each of the questions. Concerning the overall usage of certain mental images measured with the coding of all answers to all functions of the third part of the questionnaire Fisher's test on the resulting contingency tables did not yield a significant result. We interpret that there are no observable differences in the acceptance of usage of the different study groups. Using the Kruskal-Wallis-Test, we could not find statistically significant differences except for the acceptance of the limit definition (2) in part two of the questionnaire (p < 0,03) where teachers students tended to express their acceptance with higher values on the Likert scale than the others.

Comparing the results with [START_REF] Schäfer | Vorstellung von Mathematiklehramtsstudieren[!] zur Stetigkeit[END_REF] we identified more detailed mental images via their exclamatory simulacra (categories 4 to 7). While the concept image of "look of the graph" (I) was dominant here as well, it was not so dominant in Schäfer's study (2011). We see a more diverse pattern in the argumentation for the three functions instead.

The most interesting part of the empirical results is that the same mental image is used by the students either to justify a wrong or a correct answer (based on their judgment whether the given function is continuous or not; cf. Figure 2): The look of the graph (I) was used most frequently for a correct but also for a wrong answer. For example, the graph of function h is connected (i.e. has no jumps) but the function is discontinuous at the origin. Among the answers to this function we found e.g. "Yes [h is continuous], since [it is] going through, without gaps or jumps," or "The function looks continuous, since it does not 'jump'." General properties of functions (V) were even used more often for a wrong than for a correct judgment of continuity. Again, for the function h, some students argued it is "discontinuous because of a pole," but also h is "continuous at x=0, since it lies in the domain of the function." For this function we could also find various different justifications for (dis-) continuity like "it gets area-like at the origin" or "it wiggles too much." This is also the function where more wrong than correct answers were justified with mental images.

Outlook

In this study we provided a taxonomy that guides empirical research related to mental images of mathematics students in several directions. We pointed out that communicative simulacra of mental images of real-valued continuous functions depend on the context in which mental images are used: Figures 1 and2 show that the spectrum of mental images used or accepted by the students we investigated is broader than the spectrum of explicit exclamatory simulacra. We also found out that in the overall of justifications of (dis-) continuity a single mental image does not exclusively help or misguide. A first step into mental images of metric space-valued continuity is also given in [START_REF] Hanke | Ausprägung und Akzeptanz von Vorstellungen zur Stetigkeit bei Mathematikstudierenden[END_REF].

We believe that future research on teachers', doctoral students', tutors' or university lecturers' conceptions of continuity will provide insight into similarities and differences between social groups in the overall process of teaching and learning of a particular mathematical notion. This will be of particular importance for the teaching of real-valued continuity in today's university classrooms. Since continuity is disappearing from the curricula in secondary schools in Germany, it would be interesting to find out how the teaching of continuity in secondary schools implicitly or explicitly influences the mental images of beginning university students. Particularly, the question on the stability of mental images arises. Focusing on the analysis courses taught at secondary schools, an upcoming research area are teachers' judgments of the adequacy of teaching continuity in schools as a prerequisite for important facts on differentiability and integration such as the fundamental theorem of calculus.

Figure

  Figure 1: Acceptance of item on Likert scale from 0 to 5: Reject (0, 1), Neutral (2, 3) and Accept (4, 5) in percent of answers

Figure 2 :

 2 Figure 2: Numbers of correct and incorrect answers per category summed up for all four functions

Table 2 : Mental images of continuity probed in the second part of the questionnaire

 2

	Thematic Working Group 14

Table 3 : Different functions in the questionnaire

 3

Proceedings of CERME10

Acknowledgment

We would like to thank the Bundesministerium für Bildung und Forschung for its support in the subproject "Spotlights Lehre" within the project "Schnittstellen gestalten."


domain: math shs

text:
Helge Fredriksen
email: helge.fredriksen@uia.no

Said Hadjerrouit
email: said.hadjerrouit@uia.no

John Monaghan
email: john.monaghan@uia.no

Ragnhild J Rensaa
email: ragnhild.rensaa@uit.no

Exploring tensions in a mathematical course for engineers utilizing a flipped classroom approach

Keywords: Activity theory, flipped classroom, tensions

Flipped Classroom approaches to teaching are becoming increasingly popular in higher education, but there is a lack of empirical research. We present here a study performed during an engineering course for 20 students at a Norwegian university, on student appropriation towards Flipped Classroom through interviews, questionnaire, video/quiz usage statistics and classroom filming. We approach this research through an activity theoretical framework, focusing on tensions experienced as the students try to tackle the demand of video preparation and active learning in class. In line with much of the recent research on the topic, we find that most students seem to appreciate more collaboration with peers and teacher. However, there is also evidence that the new form of teaching creates various tensions; a minor part of the cohort demonstrates conflicting beliefs about mathematics learning, resisting the active learning part of Flipped Classroom.

Introduction

Flipped Classroom (FC) is most commonly known as a method that arranges the lecturing part of the teaching as homework through videos. This is considered the out-of-class part of the FC. When students come to class, the stage is set for learning in a student-centered manner, using various problem-solving activities [START_REF] Bergmann | Flip Your Classroom, Reach Every Student in Every Class Every Day[END_REF]. This is considered as the in-class part of FC. Both are vitally important for the FC learning model to work. The out-of-class video learning "primes" the students for the crucial in-class active phase (Seyfedine, [START_REF] Kadry | Flipped classroom model in calculus II[END_REF], where hopefully the active "learning-by-doing" understanding and adaptation takes place. The idea is that through well-designed activity sets in class, the teacher has the opportunity to challenge the students at both a collaborative and conceptual level in this phase [START_REF] Wan | Technology Integration and the Flipped Classroom[END_REF].

In this paper, we describe a study that was conducted in the spring term of 2016 at a Norwegian university, where students in their first year of engineering studies were exposed to several interventions of the FC way of teaching. The teaching setup in this university is well suited for flipped teaching. It is a small campus, with only 20-30 students per-year in a 3-year long bachelor study in computer engineering, allowing for a tight integration between the students and the teacher.

Many studies on the implementation of FC seem to indicate that motivation might increase among students in mathematics [START_REF] Franqueira | To Flip or Not to Flip: A Critical Interpretive Synthesis of Flipped Teaching[END_REF][START_REF] Kadry | Flipped classroom model in calculus II[END_REF][START_REF] Roshan | The Flipped Classroom: Touch Enabled, Academically Proven[END_REF]. However, there are also research studies that indicate the opposite. Wasserman, Quint, Norris, and Carr (2015) found that students in a flipped calculus III class were critical to the use of class time for group work. Strayer (2015) reports that students felt "lost" and disengaged with the material sooner than students in the traditional classroom did. [START_REF] Ramaglia | The flipped mathematics classroom: a mixed methods study examining achievement, active learning, and perception[END_REF] did a comparative study between Thematic Working Group 14 Proceedings of CERME10 flipped and non-flipped high school and middle school mathematics classes in her PhD thesis, but failed to find consistently increased peer-to-peer activity among students. Referring to these mixed results from other studies, it seems interesting to gain more insight into what kind of tensions, strains and possible resolutions of these can be observed in a FC realization. Based on this background, our research questions are formulated as follows:

Research Question: What are the tensions that emerge from students' attempts to appropriate change towards FC facilitated by videos and quizzes?

Theory

We believe that learning can best be understood when considered as a common enterprise among students and teacher, emerging as culturally negotiated in an environment based on constructive criticism. Turning to Cultural Historical Activity Theory (CHAT) (Engström, 1994), we have a rich theoretical basis to put FC in a broader perspective. The primary part of the activity system in this study is the student as a subject in her object-oriented activity to learn engineering mathematics. In attempting this, she uses various instruments. The most important ones are Campus Inkrement (a virtual learning environment used for distributing videos and quizzes), classroom discourse and curriculum literature. The dominant new rule governing FC compared to "traditional teaching", is the video preparation part, forming the out-of-class component of FC. The mathematics in the video should form a common ground of knowledge for the community consisting of students and teacher. For the division of labor, we consider how students attain various roles in their collaboration to solve tasks in-class.

In any activity system there exist tensions and contradictions. Engström (1994) summarizes activity theory in five principles, and among these, he mentions contradictions as one of the leading sources for change and development. Basically, contradictions can be defined as a misfit within elements of an activity system, between them, and between different activity systems. [START_REF] Engeström | Learning by expanding: An activity-theoretical approach to developmental research[END_REF] argues that four levels of contradictions are present in an activity system, and identified tensions in interactions within and between activity systems. The contradictions can be identified at four levels: primary, secondary, tertiary, and quaternary. If we apply this model in the context of FC, we can describe the contradictions as follows:

1. The primary contradictions occur within the elements or components of FC as an activity system, e.g. within the community of students and teacher.

2. Secondary contradictions arise between the elements of FC, or when two or more elements of FC conflict with one another, e.g. between the community and subject (for example between the class and the individual student), between the object and the community, or between the rules and the community, etc. 3. Tertiary contradictions arise when a new and advanced method or artefact is used to achieve an objective, e.g. when videos are introduced as a new artefact to teach mathematics. 4. Quaternary contradictions occur between FC (as activity system) and another activity system.

Methodology

We performed two separate periods of FC teaching during the second semester of study year 2015/2016. We performed data collection by issuing an anonymous questionnaire, doing three semistructured interviews and two rounds of classroom filming. In addition, students' usage statistics of video and quizzes were collected through the Campus Inkrement software. As our theoretical stance is in the socio-cultural field, an interpretative research paradigm was chosen. The questionnaire and the interviews were performed after the students' first encounter of FC teaching, informing us on student impressions on pedagogical and technical impressions with the learning platform chosen for distributing video/quizzes, the in-class group work activities and the quality of interaction with the teacher and the other students. Episodes relevant for the enlightenment of the research questions of the paper, tensions and student appropriation towards FC, are highlighted in the results section.

Campus Inkrement (CI) as a mediating artefact

Preceding each in-class session, a corresponding out-of-class session of videos and quizzes was presented to the students in CI, which is a web-application fulfilling the role of the out-of-class component of FC. Built from the ground up to be consistent with the FC teaching design, the teacher/researcher also has the capability to highlight video watching statistics and quiz results for the individual student. From a student perspective, CI brings the opportunity to give feedback on how well the student understood the current topic on a scale 1-5. In addition, self-perceived effort can be reported on a similar scale. The student also has the opportunity to ask for further guidance from the teacher on specific topics. This opens up for an out-of-class possibility for students to prompt the teacher for assistance without revealing their uncertainty to peers in-class.

FC implementation

In this class, there were 20 students following the course. Before attending the spring term, these students had all background from a 10 ECTS (European Credits) calculus based Math-1 course with traditional lecture-based teaching. The course in the spring term that was subject for FC teaching was labelled Math-2, consisting of 10 ECTS containing series, Fourier and Laplace transform, recursion equations, proofs and optimization on functions with two variables.

After having informed the students thoroughly about the new form of teaching in the beginning of the term, we started out the term with one month of FC teaching in January. The topic for this first round of FC teaching was sequences and series, studying criteria for convergence, and in the end Taylor expansions and Maclaurin series. Although we did not influence the curricula, obligatory assignments and exam, we could plan and implement FC as we saw fit, including the teaching performed in-class. The teaching consisted of two or three 90-minutes sessions each week. To prepare each in-class session, 3-4 videos each of 8 to 15 minutes in length were available for the students. In between the videos, quizzes directly related to video contents were given. The videos presented the mathematics in a chalk-and-talk fashion, screen-capturing teachers writing using a tablet, including some demonstrations made in geogebra. We produced 12 of the 36 videos, the rest were collected from online resources mainly from Khan Academy (https://www.khanacademy.org). The videos were procedural in content, in the sense that there was little time to go into proofs or elaborate on deeper concepts. This choice was intended to make the video homework manageable in length for the limited out-of-class time. In line with FC ideas, in-depth understanding should be elaborated in an in-class setting.

After this first attempt at FC teaching, we the middle of the course teaching traditionally with other teachers involved. The reason for this shift was the necessity for collecting feedback through interviews with a representative selection of students, in addition to an anonymous questionnaire. This to inform us on potentially needed adjustments in the second phase of FC. At the end of the term, we ran two more FC teaching weeks on the introduction of functions with several variables, linearization of these, partial derivatives and optimization. On most occasions, specially adopted task sheets were prepared for in-class active learning to provoke discussion and in-depth conceptual reflection about the mathematics, the purpose being to raise the abstraction level.

Results

These three sources of data, the questionnaire, the interviews and the filming, provide the possibility for us to triangulate findings. As this is a paper investigating tensions in the CHAT sense, we have been actively looking for excerpts where such qualities are prominent.

Questionnaire

At the beginning of March, we invited all students participating in the class to answer an anonymous questionnaire.

Here we asked the participants to agree or not on fifteen statements, in a 5-point Likert scale fashion, about various features of our FC implementation. The purpose of this was primarily to inform us towards the next iteration of FC. Additionally, the questionnaire contained three openended questions, prompting the students to express their opinions about the method with their own words. n=15 out of N=20 students responded.

We have chosen to highlight three responses to the open-ended question: "What did you feel was most inconvenient with this method of teaching and learning mathematics?" While other questions highlighted the positive sides of FC, the three statements below are representative for most of the answers to this question, and are important for the analysis of tensions:

"Personally, it works better for me when I spend time on my own with the tasks. Thus, the session in the classroom became wasted for me. I believe I should learn new things in the class, and then work on my own with the topic afterwards, and then turn to the videos for assistance."

"I got "pushed away" from the classroom using this method, since I do not like to work on tasks in groups. I feel that group work is difficult since many do not understand the topic 100%, which means that many just do not participate in the discussions."

"Group work was unsuitable, since mathematics is a more "individualistic" subject."

Interviews

In addition to the questionnaire, we performed interviews with a representative set of students in a semi-structured fashion. Due to time constraints, we had to limit the sample to three persons. This group of students was chosen as a representative sample according to gender and age, but also due to observed willingness to make critical remarks about the teaching. The interview tried to dig a bit deeper into topics of engagements, impressions about videos, group work and interaction with the other students and the teacher and lasted for about 30-40 minutes. With respect to our consideration of tensions, we present interview excerpts from students with positive and negative views on FC.

The first interviewee was an engaged student in mathematics, with almost 100% attendance in class.

He favored learning by videos over traditional learning, and liked the fact that the teacher was more available for questions than traditional lecture-based teaching. As the problematic part of all the group work, he pointed at troubles with fluency in using new mathematical vocabulary. However, he noted that by trying to communicate verbally the task with the others in the group, it became easier to understand how to solve it for himself.

The second interviewee had most of his career from offshore industry but turned to engineering studies for health reasons. He had been away from mathematics for a long time and sometimes struggled to keep up with the pace in the group work Student: I did not like the specially adopted tasks we got for the class session, and the way we worked in the groups was very inefficient for me. Because many in the class are above me, I am stuck behind the rest during the work.

Interviewer: Ok, but you liked to prepare using the videos?

Student: Yes, I liked that very much.

Interviewer: But you think it would be easier for you to find the answer to the tasks if you were all by your own solving them?

Student:

Not easier, but it would have been a better way for me to understand them, since I would be alone to think it over, instead of the others in the group just working fast through them.

Interviewer: So you weren't able to engage in the conversation and participate with your own thoughts?

Student: Not to the degree I wanted.

Both interviewees 1 and 3 expressed concerns about using specially adopted tasks for the in-class work. They worried about the tasks not having sufficient relevance for the final exam, and would rather spend time solving tasks from the textbook. I chose to not include excerpts from the third person being interviewed, since there were little indications of tensions in this interview.

Classroom filming

During the second FC intervention period, we filmed two in-class sessions. We filmed several of the groups, primarily motivated by how the out-of-class teaching affected in-class group work. Two or three students worked together solving problems related to the videos, but on a slightly higher level

than the examples used in the videos. One episode in particular caught our attention. One student in a pair (let us call her Silvia) attended class seemingly well prepared and brought notes with her that she had taken from the videos. The other student in this pair (let us call him Nick) seem to be quite unprepared. A study of CI usage statistics confirms this impression. He did not bring notes, and barely spoke during the beginning of the episode. Silvia on the other hand expressed interest in how the formula of linear approximation for functions in two variables came about. The formula referred to is the well-known linearization

∆𝑓 = 𝜕𝑓 𝜕𝑥 ∆𝑥 + 𝜕𝑦 𝜕𝑦 ∆𝑦
The first author hinted that this formula was based on an extension of the single variable case that was derived in the video, but Silvia was clearly not satisfied with this, wanting to know more. In addition, she was the dominant speaker in the group: In the 22 minutes that the episode lasted, we counted 488 words spoken by Silvia, whereas Nick spoke 236. He did catch up in the last third of the episode though, speaking almost as much as Silvia does then. We observe from the videos that this occurs after he had listened carefully to her struggles with the problems and the conversation that she had with the teacher in connection with this.

It was evident from the CI user statistics that many students had not prepared in the last period of FC teaching. This was influencing their progression in-class, although many seemed to use other means of catching up with the topic. They used other resources such as the curricula book, discussions with more knowledgeable peers like prepared students and the teacher, and even to some extent looking at the videos in-class on their own laptop.

Discussion and summary

As previously discussed, activity theory can be used to depict tensions in the FC teaching. Studying the activity triangle in Figure 1, one of the most prominent changes in FC compared to traditional teaching are the rules. These undergo a radical change in FC, enforcing video preparation for in-class active learning.

There are two important observations we would like to highlight. Firstly, the second statement from the questionnaire excerpts hints towards a lack of understanding among several group members (the community) about the mathematical topic at hand. It seems that many members in the group had not grasped the mathematics in the videos, or they simply had not watched them, leading to a breach in the quality of student group collaboration.

The filmed episode of Silvia and Nick confirms this impression. Nick appeared to struggle to follow the arguments of Silvia, although there were evidence that he somehow changed from being an 'eavesdropper' of her struggling and collaboration with the teacher to becoming an active participant. However, Nick was not playing as the part of a collaborating peer, and thus failing to support Silvia in the discourse. We believe that this was due to his lack of preparation using the videos.

Classroom discussion is considered a vital instrument of learning in FC, and it constitutes a major tension if this is not taking place inside a group. We consider this as a secondary contradiction between rules and community. As previously explained, the major rule to consider in the student FC activity is the necessity to arrive at the in-class session being 'primed' by the out-of-class session. If a major part of the group has failed to do this, the in-class discourse, considered a CHAT instrument, is hampered. Thus, this contradiction could also be seen as a secondary contradiction between rules and instrument.

Considering the data excerpts, we can also mention the tensions below, even though not substantiated through triangulation as the one already mentioned:

1. Tension in expectations/beliefs/rules: Students expect to be "taught" by the teacher, but FC rules and division of labour directs students towards learning through collaboration with peers (subjectdivision of labour tension), (secondary contradiction). 2. Students disagree with the new rule that tasks should be solved during class time. Preference towards solving them in solitude (objectrules tension, students feel this is not the best way to learn math), (secondary contradiction). As we discussed earlier on, in the filmed classroom episode with Silvia and Nick there is indication that students who have prepared by engaging in the out-of-class work seem to express themselves fluently in the mathematical problems, and in addition seem eager to learn more about the concepts behind the procedural mathematics shown in the videos. This provides empirical evidence (though only a single case) of the potential of FC to motivate students to strive towards a higher level of abstraction.

Validity and reliability issues

This paper must be viewed in the context of a report on a pilot study. More elaborate studies will be carried out during 2016/2017 and 2017/2018 with engineering students in the same institution. Thus, there is little rigid design according to how data have to be collected to obtain optimal analysis and results. Handpicked excerpts from the data material were chosen to highlight the findings. There is also the issue of the researcher being present as the teacher, a classical objectivity dilemma found in many small-scale educational research settings. However, as we are presenting the data using several methods, both quantitatively and qualitatively, we can to some degree state that we have made valid triangulation of the findings.

Conclusion

Our analysis of the data collected in this study, shows evidence that there exist several tensions in FC; some of these could be expected from the outset, while others are surprising. Data seem to point towards various aspects of the active learning being the most problematic part for many students towards a FC realization. Considering the activity system of the student, a secondary contradiction or tension materializes between the rules and the community, since many students was not adhering properly to the out-of-class part of FC. This is also seen to hamper the in-class discourse, considered to be an important instrument of learning.

3 .

 3 Students need to adopt to a new paradigm of work: Preparation through video lessons requires discipline, which results in tension, especially when a heavy workload is expected in courses taken in parallel (subjectrules tension), (tertiary contradiction). 4. Fluency in discourse. Problems expressing the mathematical problems verbally to other students. (subjectinstrument (discourse) tension), (tertiary contradiction). 5. Students failing to keep up with the others during group-work (subjectcommunity tension), (secondary contradiction).

The CHAT triangle adopted to the Flipped Classroom from a student's perspective

						Campus Inkrement
						Classroom discourse
		Instruments				Curriculum literature
	Subject		Object	Outcome	Student	Learn Math	Become engineer
	Rules	Community	Division of labour	Prepare using videos	The class	Roles during group work
	Figure 1:

Proceedings of CERME10


text:
Frank Feudel
email: feudel@math.upb.de

Students' interpretation of the derivative in an economic context

Keywords: Derivative, economics, concept image, economic interpretation, marginal cost

The concept of derivative plays a major role in economics. One important competence for students of economics is to interpret values of the derivative in an economic context. In the study presented in this paper it was investigated how students of economics interpreted values of the derivative in an economic context before and after their Calculus course at university. Before the course only very few students were able to interpret these values adequately. After the course about half of the students were able to state an adequate economic interpretation of values of the derivative, mostly as amount of change of the function while increasing the output by one (marginal) unit, which is a common interpretation in economics. However, the data indicates that many of these students just identified the derivative with that amount of change without understanding the differences and the connection between these two different mathematical objects.

Introduction and embedding of the research

In order to maximize their profit firms have to make many profound decisions, for example about possible investments. An important tool that aids economists to make optimal decisions is marginal analysis. [START_REF] Baumol | Microeconomics: Principles and policy[END_REF] listed marginal analysis even as one of the most important ideas of economy. In marginal analysis the consequences of making relative small changes from the current situation are examined [START_REF] Ruffin | Principles of economics[END_REF]. Typical problems are the effects on cost or revenue if the output is increased by a small amount (normally one unit), or the effects on the demand of a product if the price is increased by a small amount. Marginal analysis is closely connected with the mathematical concept of derivative, which serves as a tool to measure the effects of these small changes on cost or revenue. Therefore, students of economics should have a proper understanding of the mathematical concept of derivative and its use in marginal analysis, which is currently investigated in my PhD-Thesis (supervised by Prof. Dr. Rolf Biehler).

Unlike for students of engineering or physical science, whose understanding of the derivative has been examined more extensively [START_REF] Bezuidenhout | First-year university students' understanding of rate of change[END_REF][START_REF] Bingolbali | Concept image revisited[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] Maull | A questionnaire to elicit the mathematical concept images of engineering students[END_REF], only little research about the understanding students of economics have of this concept exists. Only two studies are known to the author dealing with the understanding of rate of change by students of economics (Mkhatswa & Doerr, 2015;[START_REF] Wilhelm | Projecting rate of change in the context of motion onto the context of money[END_REF], and none of these involves the concept of derivative explicitly. To use the derivative in marginal analysis the students have to interpret calculated values of the derivative in an economic context. Interpreting values of the derivative in contexts is not easy for students, as shown by [START_REF] Bezuidenhout | First-year university students' understanding of rate of change[END_REF] for students of engineering. He showed, for example, that many of these students could not interpret values of the derivative of a stopping distance function S(v) (v is the velocity of a vehicle) adequately. They overgeneralized that the derivative is the acceleration or the velocity, or they had problems with the unit, which shows that they did not have a profound understanding of the derivative itself as rate of change. The economic interpretation of the derivative has the additional difficulty that it does not directly correspond to any of the usual mathematical representations of the derivative, which is explained in detail in the next section. Therefore, it can be expected that students of economics have even more difficulties in understanding this interpretation. To address this conjecture, the study presented in this paper is guided by the following research question: "How do students of economics interpret the derivative in an economic context before and after their Calculus course?" The results will extend the knowledge about students' difficulties in the understanding of the derivative with a focus on students of economics, who have been rarely considered until now.

Theoretical background of the study

The economic interpretation of the derivative

In physical contexts, the derivative is often interpreted as rate of change (for example as speed), which directly corresponds to other representations of the derivative like the slope or the limit of the difference quotient. The common economic interpretation, however, represents a different mathematical object, which is now explained for cost functions. If :[0; [ [0; [ C    is a cost function and C(x) represents the cost of a given output x, '( ) Cx(called marginal cost) is often interpreted as the additional cost while increasing the output by one unit [START_REF] Schierenbeck | Grundzüge der Betriebswirtschaftslehre[END_REF]. This additional cost, exactly calculated by ( 1) ( )

C x C x



, differs from the derivative C'(x) in its numerical value and in the unit. Hence, this interpretation has to be connected to the students' other knowledge of the derivative and needs to be justified. A typical justification is via the approximation formula ( ) ( ) ( )

C x h C x C x h     
for h close to 0 with the additional argument that 1 h  is really small in economic contexts. A more detailed description of the connection between the derivative and its economic interpretation as additional cost 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]. Some books of economics add the word "marginal" to the unit when describing marginal cost. The marginal cost is then the cost that arises if the output is increased by a marginal unit and is written as ' dC C dx  (e.g. in [START_REF] Reiß | Mikroökonomische Theorie: historisch fundierte Einführung[END_REF]). In economic literature the term "marginal unit" is often used as a synonym for a very small, but finite, unit. In [START_REF] Dyckhoff | Grundzüge der Produktionswirtschaft[END_REF], for example, the term is used to emphasize that one unit is small enough and that the derivative C'(x) can be used as approximation for the difference quotient

C x  
for Δx = 1. [START_REF] Reiß | Mikroökonomische Theorie: historisch fundierte Einführung[END_REF] emphasizes furthermore that it depends on the context whether a unit can be considered as marginal or not (example of the book: if water utility is measured in cubic meters a marginal unit might be a cubic millimeter).

Theoretical tools for the study

An adequate economic interpretation should be part of the students' conceptual knowledge of the derivative, which can be described with the term concept image by [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF]. The concept image describes the total cognitive structure that is associated with the concept. This includes are all mental pictures, associated properties and processes. Concerning the derivative the students' concept image should contain the different representations of it, the differentiation rules, connections to other mathematical concepts like monotonicity, and in the case of students of economics also an adequate economic interpretation of the concept. To understand this interpretation properly students should connect it to other knowledge of the derivative they already have from school. The process of making these connections, called synthesizing by [START_REF] Dreyfus | Advanced mathematical thinking processes Advanced mathematical thinking[END_REF], is a special challenge in the case of the common economic interpretation of the derivative as additional cost because this interpretation does not directly correspond to any of the other representations of the derivative, and, as described above, justifying it needs some argumentation.

Knowledge about the derivative covered in the students' Calculus course

In the course the study took place in (University of Paderborn, Germany), the derivative concept itself was covered in two lectures. In the first lecture the definition of the derivative as limit of the difference quotient, which was also visualized by secant lines "converging" to the tangent line, and the differentiation rules were presented. Afterwards the unit of the derivative in comparison with the unit of the original function was discussed in the case of a cost function C. In the second lecture the economic interpretation was presented and justified. Two possibilities were given in the course:

1. Approximation of the additional cost of the next unit

This interpretation was justified in the course via the above mentioned approximation formula ∆C ≈ C'(x)•∆x that was derived by deencapsulating the limit in the definition of the derivative and using the limit as an approximation of the difference quotient.

Additional cost of the next marginal unit

It was illustrated in the lecture with the help of the tangent line in the case of a convex function that the error between ΔC and C'(x)•Δx in the approximation '( )

C C x x  
  becomes smaller, the smaller Δx is. It was explained that the limiting process ∆x→0 results in a fictional equation '( ) dC C x dx  , in which dx was called a marginal unit.

Some lectures later, after the introduction of the concepts of monotonicity and convexity, the connection between the derivative and these two concepts was discussed. In a last step the derivative was used as a tool to solve optimization problems.

Besides the lectures the students had to solve problems referring to the content of the lectures. The problems were solved by the students in small groups. One week later the solutions were presented on the board in the lecture hall. Relevant for this study is that the 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.

Methodology of the study Data Collection

Students of economics at the University of Paderborn were administered a pre-test addressing their previous knowledge of the derivative concept in September 2015 in a voluntary bridging course before their math courses. The pretest contained the following task, to check if an adequate economic interpretation was part of the students' concept image:

A company produces pens. The cost (in euro) for the production of a number of x pens can be described with a cost function with the following equation: (GE = units of money, ME= units of quantity).

State an economic interpretation of this value.

As described in the previous section, the students were familiar with that type of task.

Data analysis

The answers to the two tasks were categorized by the author with quantitative content analysis.

Besides two categories given by the interpretations "additional cost of the next unit" and "additional cost of the next marginal unit", which were derived from literature in economics (see "Theoretical background"), the categories were created inductively because it was not clear what answers the students might state. This led to a system of 10 categories for the task in the pretest (other answers, which were all wrong, were given by less than 3% of the students). The answers in the first three categories were adequate economic interpretations (detailed descriptions in Figure 1). The answers in the categories 4-10 were not adequate economic interpretations (detailed description in Figure 2): the answers in categories 4-5 contained at least a correct idea, the answers in categories 6-8 were wrong economic interpretations, and the answers in categories 9-10 were no economic interpretations and therefore were wrong answers to the task as well. After the development of the category system the data was re-coded by a student to check inter-rater reliability. The reliability coefficient Cohen's Kappa was κ = 0.82, which is good.

The 10 categories mentioned above were also used to categorize the answers to the task in the exam to interpret '(73) 0.2 P  for a profit function P. However, three additional categories that contained more than 3% of the answers had to be added (Figure 3). In addition, the category "cost doubles or halves" was adapted to "Increase of profit by 20%".

Results

The students' answers to the task to interpret '(200) As can be seen in Figure 4, the economic interpretation of the derivative of a cost function C as the additional cost of the next unit was not known by the students from school, and is not as intuitive that the students stated it spontaneously. Instead, some students tried to use their knowledge about the derivative being the slope at a point (category: Gradient of cost at the point) or the rate of change (categories: Growth rate at of cost or Cost per unit). Others had misconceptions.

The students' answers to the task to interpret '(73) 0.2 P  for a profit function P in an economic context in the exam after their Calculus course are shown in Figure 5. As can be seen in Figure 5, more than half of the students knew the common economic interpretation of the derivative of a profit function as additional profit of the next (marginal) unit after the course (but still various misconceptions occurred). Thus, an adequate economic interpretation of the derivative was part of their concept image for about half of the students. However, the data indicates that many of these students did not really integrate their economic interpretation of the derivative with the rest of their concept image, i.e. they did not fully understand the differences between the derivative '( )

Px and its economic interpretation as profit of the next unit (differences in the numerical values and in the units). Of the 422 students who interpreted the derivative as additional profit, 100 students (23.7%) mentioned "unit of money per unit of quantity" as corresponding unit, which is the right unit of the derivative but not of its interpretation as additional profit. Furthermore, of the 103 students who interpreted the derivative as the additional profit of the next unit, only 20 students (19.4%) mentioned in their interpretation that the value of the derivative '( ) Px is just an approximation of the additional profit ( 1) ( )

P x P x 
, although the lecturer had emphasized that this has to be stated explicitly in the interpretation (and had shown graphically the error on the board). This indicates that many students probably did not understand the differences between the derivative '( ) Pxand its economic interpretation. They just identified both objects, which leads to an incoherent concept image of the derivative [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF].

Limitations of the study

Due to organizational problems it was not possible to match the students in the pretest and the exam. So it is not possible to investigate the progress of individual students with the data. It is just possible to compare the amount of students giving certain answers before and after the course.

Discussion and conclusions for further research

The study shows that an adequate economic interpretation was not part of the students' concept image of the derivative (with few exceptions) when entering university. Hence, this interpretation should be introduced in the course with caution and connected to the rest of their concept image of the derivative concept. After the course an adequate economic interpretation of the derivative as the additional cost/profit of the next (marginal) unit was part of their concept image for about half of the students. However, the data indicates that many of these students did not understand the differences between the derivative as a mathematical concept and its economic interpretation. For example, one quarter of the students who interpreted the derivative as additional profit gave the wrong unit (a unit of rate). They did not distinguish between the derivative being a rate of change and its economic interpretation being an amount of change, which is also documented in literature [START_REF] Mkhatshwa | Students' understanding of marginal change in the context of cost, revenue, and profit[END_REF]. Furthermore, the data indicates that many students were not aware that the value of the derivative is just an approximation for the additional cost/profit of the next unit. Students having these problems probably did not connect the economic interpretation of the derivative to their concept image of the derivative as a pure mathematical concept properly.

To find out to what extent the students really integrated the economic interpretation of the derivative as additional cost/profit into their concept image as intended in their Calculus course (i.e. that they know the differences and the connection between these two different mathematical objects, and can justify the identification of them), further research is necessary. Therefore, in spring 2015, an interview study addressing this identification directly was conducted to find out to what extent the students really integrated the economic interpretation of the derivative into their concept image of the derivative concept (and not just memorized it) and what cognitive obstacles occur during the justification of the economic interpretation of the derivative in the way it was done in the course.

The above mentioned research, however, only takes a cognitive perspective into account. There are more factors influencing the way students interpret the derivative in an economic context like institutional practices of the two institutes involved in their education (institutes of mathematics and economics). Taking these institutional influences into account, which has great potential for further research, would, however, require other theoretical perspectives like ATD [START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF].

Figure 1 :

 1 Figure 1: Adequate response categories for the task to interpret '(200) 2 C  for a cost function C in an economic context

Figure 2 :

 2 Figure 2: Not adequate response categories for the task to interpret '(200) 2 C  for a cost function C in an economic context

Figure 3 :

 3 Figure 3: Additional response categories for the task to interpret '(73) 0.2 P  for a profit function P in an economic context

2 C

 2 for a cost function C in an economic context before their Calculus course are shown in Figure 4.

Figure 4 :

 4 Figure 4: Students' answers to the task to interpret '(200) 2 C  for a cost function C in an economic context before their Calculus course (N = 143)

Figure 5 :

 5 Figure 5: Students' answers to the task to interpret '(73) 0.2 P  for a profit function P in an economic context in the exam at the end of their Calculus Course (N = 821)

  After their Calculus course in February 2016 the students had to take an exam to finish the course successfully. In the exam the students also had to answer a similar task: R be a profit function of a company, which manufactures a product in an unlimited and indivisible amount. The profit is measured in units of money, the output measured in units of quantity. It is known that the derivative function P' is called marginal profit. You get

	Let	P	:[0; [
	to know that	'(73) 0.2 	GE
	( ) C x		1 30000	x	3		1 100	x	2		2 , x x	0  . It can be determined that '(200) 2 C  . Interpret this
	result in the above context.
				P					ME

Proceedings of CERME10


text:
Faïza Chellougui
email: chellouguifaiza@yahoo.fr

Proof and formalism: The role of letters' logical status

Keywords: Formalism, proof, letters' logical status, quantifiers, natural deduction

In mathematical activity, especially in proof, it is a fairly common practice to change letters' logical status without giving any indication of these changes. Nevertheless, in some cases this practice is likely to hidden invalid steps in the proving process. In this paper, I show on an example that Copi's system for natural deduction provides a methodological tool that allows us both to anticipate where such invalid steps could appear and to analyse students' proof productions.

Introduction

Difficulties met by university students in the use of logical formalism are well documented in the literature (e.g. [START_REF] Selden | Unpacking the logic of mathematical statements[END_REF][START_REF] Dubinsky | On students understanding of AE and EA quantification[END_REF][START_REF] Chellougui | L'utilisation des quantificateurs universel et existentiel, entre l'explicite et l'implicite[END_REF]). An important issue on the development of formalism in university mathematics education is to reduce ambiguities conveyed by natural language, in order to foster the understanding of mathematical statements and to improve the development of proving skills. However, Durand-Guerrier and Arsac ( 2005) have underlined a rather common practise in mathematics textbooks of changing letters' logical status in a proof without giving any indication of these changes. This practice introduces additional ambiguities in the proving procedure. The aim in this paper is to address the conjecture that university students who are not able to cope with these ambiguities face difficulties with proofs.

In this paper I use the system for natural deduction (i.e. a formalisation of mathematical reasoning) suggested by [START_REF] Copi | Symbolic Logic[END_REF] in predicate calculus (i.e. an extension of propositional calculus that deals with the internal structure of propositions, with symbols for properties, relations, quantifiers and individuals) as a tool for checking the validity of proof. A main interest of this system is to make explicit the difference between bound variables (i.e variables in the scope of a quantifier) and generic elements (i.e any individual element of the domain of quantification at stake)1 . In the first part, I present the Copi system which has been used for both a priori and a posteriori analysis in my research. The second part focuses on the letters' status in a proof from the logic and didactic point of view. To this purpose, I present results demonstrating student's difficulties with letters' status at the beginning of their university studies. In the third part, I present results from first year university student's responses in a proof where the use of letters challenges the validity of the proof [START_REF] Chellougui | L'utilisation des quantificateurs universel et existentiel, entre l'explicite et l'implicite[END_REF].

Natural deduction system

I first introduce the natural deduction system proposed by [START_REF] Copi | Symbolic Logic[END_REF] and show that this system provides tools to detect invalid steps in a proof by remaining as close as possible of the usual modes of reasoning of mathematicians [START_REF] Durand-Guerrier | An epistemological and didactic study of a specific calculus reasoning rule[END_REF]. In my work, I use this system in the frame of predicate calculus, where four rules for introduction and elimination of quantifiers are introduced. In Figure 1, I summarize a presentation offered by [START_REF] Durand-Guerrier | Natural deduction in Predicate Calculus. A tool for analysing proof in a didactic perspective[END_REF] of these rules completed by some specific restrictions needed for preserving validity. This system can be used to control locally the validity of mathematical proofs. It can act as an intermediate between the usual practice and the completely formalized system. The rules of introduction and elimination of quantifiers from Copi address the semantic dimension, because there is an introduction of letters referring to generic elements. In ordinary textbooks, most often, both operations of elimination and introduction of quantifiers are either absent or partial [START_REF] Durand-Guerrier | An epistemological and didactic study of a specific calculus reasoning rule[END_REF][START_REF] Chellougui | L'utilisation des quantificateurs universel et existentiel, entre l'explicite et l'implicite[END_REF]. By making explicit the rules of introduction and elimination of quantifiers, Copi's natural deduction system allows the identification of implicit steps, especially in cases of proof that use multi quantified statements where there is, at least once, each of both universal and existential quantifiers. In my research, I use Copi's natural deduction as a tool for a priori analysis in order to anticipate possible invalid steps in the proving process of a given statement and as a tool for a posteriori analysis of the proofs offered by undergraduate students. In a following section, I present an example of such analysis.

The logical structure of mathematical statements: Changing letters' status [START_REF] Durand-Guerrier | An epistemological and didactic study of a specific calculus reasoning rule[END_REF] have investigated the letters' logical status in mathematics teaching. In their work, they refer to the predicate calculus in order to analyse quantified statements with a focus on the variables' dependence. They analyse a specific mistake which appears in proofs where one applies twice or more a statement of the kind "for all X, there exists Y such that R(X,Y)", abbreviated to AE statements, and a student may ignore that in that case, a priori, "Y depends on X". The misuse of AE statements in calculus have been demonstrated in an invalid proof of Cauchy's mean value theorem (Figure 2, Durand-Guerrier and Arsac, pp. 151-152): We can read: "A proof rather often provided by first year science students consists of the following deduction of Theorem 2 from Theorem 1: Function f satisfies the conditions for applying Theorem 1; hence there is a number c in ]a;b[,

such that f'(c)(b-a)=f(b)-f(a)
. Also g satisfies the conditions for applying Theorem 1; hence

there is a number c in ]a;b[, such that g'(c)(b-a)=g(b)-g(a). As g' is never equal to zero on ] a ; b [ , then g'(c)0; hence g(b)-g(a)0.
The result follows from the quotient of the above two equalities. This proof is invalid; one can prove it by considering two functions for which it is not possible to choose the same number c." (Durand-Guerrier et Arsac, 2005, p.152).

According to the authors, the error from a logical point of view is the following: since c is a bound variable following an existential quantifier, it cannot denote a particular real number. However, the existential elimination that must be applied here allows to consider a real number r such that:

f'(r)(b-a)=f(b)-f(a).
When thus applying the same rule to g, it is necessary to consider a real number s, that may or not be equal to r, such as g'(s)(b-a)=g(b)-g(a). It is important to notice that this logical analysis depends only on the logical structure of Theorem 1, and not on the mathematical meaning of the letters f, c, r, etc. The same reasoning allows us then to derive the quotients' equality: but this does not provide a proof of Cauchy's mean value theorem [START_REF] Durand-Guerrier | An epistemological and didactic study of a specific calculus reasoning rule[END_REF].

This example illustrates the difficulties linked with the logical status of letter in proof and proving. In the next section, I illustrate one example in Algebra how the use of Copi's natural deduction allow us to anticipate student's difficulties and to analyse their proofs.

An example in elementary set theory

In the context of my PhD (Chellougui, 2004) conducted in Tunisia, I distributed a questionnaire to ninety-six mathematics students arriving at university in November 2001 (details on the analysis and main results can be found in [START_REF] Chellougui | L'utilisation des quantificateurs universel et existentiel, entre l'explicite et l'implicite[END_REF]. In this paper, I focus on a specific example that I analysed in detail in order to highlight the methodological relevance of Copi's natural deduction for a priori and a posteriori analysis of proofs. In the Tunisian university, the first elements of elementary set theory including equivalence relation, order relation and binary relation, are taught at the beginning of the first academic year. The example I discuss here regards the proof that a given binary relation  is an order relation (Figure 3). My main objective was to identify precisely students' difficulties in the use of multi quantified statements in proof and proving.

We consider the set IN*2 endowed with the relation  defined by: (p,q) IN*x IN*(pqn IN*; p n =q). Show that  is an order relation. I hypothesised that the students would be able to recall each of the three properties that an order relation checked: reflexivity, antisymmetry and transitivity, because they have met this type of questions in the course and in the series of exercises, although the formalisation of an order relation was new to them. The three definitions of the properties above were given to the students in the general case of a binary relation  as follows:

Reflexivity: p pp. Formulation containing one universal quantifier.

Antisymmetry: pq(pqqp p=q). Formulation containing two universal quantifiers.

Transitivity: pqs(pqqs ps). Formulation containing three universal quantifiers.

In this paper, I focus on the proof of antisymmetry for a binary relation whose definition involves an existential quantifier, leading to a rather complex logical structure as will be shown in the a priori analysis. I first present some elements of a priori analysis; then I present results from students' responses.

Mathematical and logical analyses of proof of antisymmetry

The definition of the binary relation  involves two universal quantifiers in the beginning of the formula and an existential quantifier in the second part of the equivalence.

In order to anticipate the difficulties that the students could meet in proving that the given binary relation owns the property of antisymmetry and to make explicit the steps needed for a complete proof, I provide (Figure 4) a mathematical and logical analysis, using the Copi's system for natural deduction (see Figure 1) with a specific focus on introduction and elimination of quantifiers This formalized demonstration starts with a universal premise followed by four successive universal instantiations; twice in (2) with two different letters a and b, and twice in (3) with the two same letters a and b; so, one works then with two generic elements. In (4) an auxiliary premise is introduced to express the antecedent of the property of the antisymmetry on the generic elements (this is a standard way to prove a conditional statement in the frame of Copi's natural deduction: proving B under hypothesise A provides a proof of  The two existential statements ( 6) and ( 8) are followed by two existential instantiations with two different letters: m in (9) and k in (10).

3 A proposition upon which an argument is based or from which a conclusion is drawn.

The mathematical argument is developed in ( 11) to ( 14) using mathematical properties without the quantifiers. The passage from ( 13) to ( 14) does not appear in this demonstration. It can be expressed in the following way:

(13): a=1 or mk=1; First case: a=1 then a=b=1; Second case: if mk=1, with mIN* and kIN*, then m=k=1; finally, in both cases, a=b (

Framing the proof in Copi's system allows us to anticipate potential flows likely to appear in students' responses from ( 6) and ( 8) to ( 9) and ( 10), respectively, in case the restriction rule for two successive existential introductions is not applied.

I provide an a priori classification of the answers that I will use for a posteriori analysis.

Category 1: answers for which two different letters are quantified existentially.

Category 2: answers for which the same letter is quantified existentially.

Category 3: answers for which the same letter is used without existential quantifier, Category 4: answers where two different letters are used without existential quantifier.

Classification of students' answers

Among the ninety-six students that answered the questionnaire 80 students have produced a proof of the antisymmetry property for the given binary relation.

1) There are 17 copies in category 1 (about 21%) with representative examples that illustrate this category (Figure 5). In the solution of student 1, the mathematical argument for step (11) to ( 14) is fully developed. This is not the case of student 2: the mathematical argument is absent, there is no mathematical justification for the equality of both natural numbers x and y.

Student 3 used two different letters in the existential statements, however student 3 declared them to be equal. In this case we can make the hypothesis that the student attributed the same value to m and n in order to satisfy the validity of the statement. Another hypothesis could be that student 3 thinks that the equality is necessary.

2) There are 27 copies in category 2, that I have subdivided in two cases:

Thematic Working Group 14

(a) The existential quantifier is present twice (21 copies), see Figure 6. In the production of student 4, there is a presence of both quantifiers, the student tries to construct the object n existentially introduced in such a way as to have the conclusion. So, to verify the property of antisymmetry and have the equality p=q, the student takes, for the natural number n, the value 1 which is a solution of the equation p n =q of unknown n. Also for the student 5, where there is absence of universal quantification, the construction of the object n is implicit and the conclusion is immediate for the student.

(b) The existential quantifier is present only once (6 copies), see Figure 7. I notice with the student 6, that the existential introduction of n for the relation pq is also considered for qp. Let us note here that, on one hand, both variables p and q are not introduced and that, on the other hand, the elimination of the number n is not declared. This illustrates the fact that there are implicit arguments in the use of variables and in the steps required to prove the antisymmetry property.

3) There are 36 copies classified in category 3. I have considered in this category copies where the letter is used for both statement and is not in the scope of a quantifier. I have consider it as an implicit existential introduction without taking in account the restriction rule. The example in Figure 8 is typical of answers in this category. In the production of student 7, the variable n is the same in the two equivalences, and there is no mathematical argument supporting the conclusion. It is possible that the student wrote directly the conclusion p=q to fulfill the antisymmetry property; another possibility is that he considered that it was obvious that both equations p n =q and q n =p allow to conclude that n is 1 and to deduct the equality.

4) There is no copy in category 4, provided that we consider that the existential introduction may remain implicit.

These results confirm my hypothesis that the complexity of the logical structure on the side of quantifiers is likely to create an obstacle for the students to provide a correct mathematical argument. In particular, it is noticeable that the only students that provide sound mathematical arguments are those in category 1, i.e. those who take in account the restriction rule for existential introduction.

These results also highlight the fact that in case of two successive applications of an existential definition that a priori requires the use of different letters, many students use the same letter. In other words, they do not respect the restrictions on the names of objects associated with the rule of existential instantiation. We could suppose that the symmetry in pq and qp triggers the choice of the same letter; however in line with other results (e.g. Durand-Guerrier et Arsac 2005) we would say that this students' practice could be found in various other contexts.

Conclusion

In this paper, I aimed to illustrate the relevance of our methodology relying on Copi's natural deduction that allowed detailed a priori analysis of proofs and a posteriori analysis of students' productions. From the a priori analysis of the proof of the antisymmetry property of the binary relation at stake, I identified possible invalid steps. The a posteriori analysis of the proofs provided by students has shown that such invalid steps appeared in many answers and that in some cases, this invalid steps prevent the students from identifying the mathematical property required for providing a valid proof.

Figure 1 :

 1 Figure 1: Copi's rules (Durand-Guerrier, 2005, pp. 413-414)

Figure 2 :

 2 Figure 2: Two theorems: mean-value theorem and Cauchy's mean-value theorem.

Figure 3 :

 3 Figure 3: The exercise submitted to students

(

Figure 4 :

 4 Figure 4: Formalisation of the proof in the frame of Copi's natural deduction

Figure 5 :

 5 Figure 5: Responses of Student 1, Student 2 and Student 3

Figure 6 :

 6 Figure 6: Responses of Student 4 and Student 5

Figure 7 :

 7 Figure 7: Response of Student 6

Figure 8 :

 8 Figure 8: Response of Student 7

"[…] the proof of a universal statement, apart from the case of proof by induction, is always done by the method of the generic element: in order to prove a statement of the kind "for all x ∈ E, P(x)", you prove P(a) for a generic element a ∈ E, then after verifying that only properties of a that are common to all elements in E were used, you conclude that P(x) is true for every x in E."(Durand-Guerrier and Arsac,

2005, p.153)

Proceedings of CERME10

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