Svitlana Rogovchenko
email: svitlana.rogovchenko@uia.no

Stephanie Treffert-Thomas

Yuriy Rogovchenko

Non-standard Problems in an Ordinary Differential Equations Course

Keywords: existence and uniqueness theorems, design research, non-standard problems, commognitive theory, mathematical discourse

We report first results from a teaching intervention in an ordinary differential equations (ODEs) course for engineering students. Our aim is to challenge traditional approaches to teaching of Existence and Uniqueness Theorems (EUTs) through the design of problems that students cannot solve by applying well-rehearsed techniques or familiar methods. We analyse how the use of non-standard problems contributes to the development of students' conceptual understanding of EUTs and ODEs.

INTRODUCTION AND BACKGROUND TO THE STUDY

Although ODEs are an important topic in the engineering curriculum, students experience difficulties with mastering ODEs and with the very concept of a differential equation [START_REF] Arslan | Do students really understand what an ordinary differential equation is[END_REF]. In our study, the lecturer, a mathematician, devised a set of non-standard problems (see below, Problem 1 of 6) to challenge students' conceptual understanding of the EUTs. These problems formed an assessed piece of coursework.

Figure 1. One of the problems in the study

We analysed how solutions changed and developed as students worked on the problems. Students' discussions in small groups were audio-recorded, transcribed, and then analysed using constructs from commognitive theory [START_REF] Sfard | Thinking as communicating: human development, the growth of discourses, and mathematizing[END_REF]. We are currently in the initial phase of the data analysis aimed at answering the following question: How do non-standard problems contribute to the development of students' mathematical discourse and further their conceptual understanding of fundamental notions and results in an ODE course?

RESULTS

For Problem 1a (P1a), students could use one of two solution methods: M1 (substitution) and M2 (integration). Working on the problems, several students changed their approach. In the final script, only one student produced a correct and complete solution (M2) while 14 (of 19) students used M1 verifying that a given function is a solution (which is sufficient for the particular solution), but failed to explain why this solution is the general one (hence incomplete M1). We conducted similar analyses for Problem 1b (P1b).

We present one extract (for P1b) as an example of our analyses of students' group discussions using commognitive constructs -narrative, routine, ritual, substantiation. S12. The first idea was just to try to solve for C and I got the same constant, so that's OK. And I checked for asymptotes and I got one on x=0, so I noted that the equation is split to get two curves, at least, according to calculator we got it split about zero. S11. So it's undefined at zero. S12. Undefined at zero, so we get two different curves and both solutions work. We do not have a continuous curve which happens to intersect at these two points […] S14. It's not continuous for 𝑥 = 0? S12. No. So if we take an interval from -3 to 1, it's discontinuous in this interval, so it's not a curve that happens to just hit these two points, it is two individual curves that have the same solution. So it's correct in just a tiny area. S11. That was my argument as well. As the theorem states, there is a continuous interval but here it is split into two which contain two different t 0 's. S13. The theorem says, that there is a unique solution for every interval where the function is continuous. Since there are two intervals and there are two solutions, it does not conflict with the theorem.

[…]

Note that S12 is using two different visual realizations of solutions, first the algebraic representation and then the graph plotted by calculator. He shows that the realizations are not equivalent, they do not produce the same result. We see how the student demonstrates the ability to solve the problem by developing the realization tree and employing the mathematical object of "continuous solution" (discursive object). S11 is not so sure at the beginning, he is guided by S12 (considered "more experienced" interlocutor) and adopts the narrative offered by S12. S13 concludes by reformulating the expression "does not violate the theorem" as "does not conflict with the theorem".

We see how students worked to substantiate the narrative. This routine can be characterized as the exploration. Students gradually improved their abilities in developing and endorsing the EUTs narratives while working on all six tasks during the group discussions.


Stacy Brown
email: brown@cpp.edu

Bartolini Mariotti

Bussi

Ferri & Boero

Garuti

A Study of Students' Reasoning About "There exists no …"

Keywords:

In this paper, we report findings from two studies of students' engagement in metatheoretical tasks drawn from a model of the reasoning requirements of a proof by contradiction. The studies aimed to explore students' engagement in the tasks, the extent to which they were successful, and the similarities and/or differences between students' and mathematicians' approaches. Findings indicate students tend towards syntactic, logical theory approaches while mathematicians gravitate towards semantic, mathematical theory approaches. Drawing on interview data, it is shown that students may use symbols to avoid employing fragile content knowledge, yet encounter further difficulties by viewing quantifiers as appended symbols.

interest, for it raises many questions: To what extent are novices successful when engaging in metatheoretical tasks? What approaches do they employ? Do their approaches differ from mathematicians'? The purpose of this paper is to take a preliminary step towards answering these questions. Specifically, we report on two studies of participants' responses to metatheoretical tasks drawn from a model of the reasoning requirements of a proof by contradiction, which is described below.

PROOF BY CONTRADICTION AND ITS REASONING REQUIREMENTS

In this section, our aim is to model the reasoning requirements of a proof by contradiction of a universally quantified conditional statement. To aid our discussion, we consider a specific example: Theorem 5. For all positive integers n, if n mod(3) º 2 then n is not a perfect square. To prove the theorem by contradiction, one must correctly negate the universally quantified conditional statement and take the resulting statement as one's primary assumption. Such actions require one accept (at least at an intuitive level) that for a conditional statement to be true universally, it must not be the case that there is some element in the universe of discourse for which the premise is true (has a truth-value of true) and the conclusion is false (has a truthvalue of false). In the case of our example, we assume "There exists a positive integer n, such that n mod(3) º 2 and n is a perfect square." As shown by Wu Yu, [START_REF] Wu Yu | Students' understanding of proof by contradiction[END_REF], this task is far from trivia for students 17-20 years of age. Moreover, as [START_REF] Antonini | Indirect Proof: What is specific to this way of proving?[END_REF] note, the validity of the work is determined by theorems that reside within the logical theory (i.e., the metatheory).

Having assumed the negation of the statement-to-prove, one must now explore the consequences of this assumption and identify a contradiction. Three aspects of this work are important. First, to carry out this work one must move back to the mathematical theory, for it is here that the contradiction will reside. Second, one's goal is open-ended, for one does not know in advance where one will find the contradiction. In fact, there may be many. Third, one must know one's commitments with regard to the mathematical theory. Otherwise, one will not have the means to recognize a contradiction. This point was made by [START_REF] Sierpinska | I need the teacher to tell me if I am right or wrong[END_REF] who argued, "sensitivity to contradictions in mathematics requires theoretical thinking … (thinking) concerned with internal coherence of conceptual systems" (p. 1-54). Once the contradiction is identified, one's work is not done. One must make sense of it.

In our example, we claimed that an integer existed but having produced a contradiction we now know that such a number cannot exist. Hence, one must conclude, i there exists no integer n, such that n mod(3) º 2 and n is a perfect square.

And it is at this point that one is faced with the very requirement that [START_REF] Antonini | Indirect Proof: What is specific to this way of proving?[END_REF] Fabio rejected; namely, seeing the proof of this statement as a proof of the statement-to-prove. In other words, having shown S*: There exists no integer n, such that n mod(3) º 2 and n is a perfect square, one must recognize (from a logical standpoint) that one has proven S: For all positive integers n, if n mod(3) º 2 then n is not a perfect square; that is, we must recognize S* implies S since S* is a non-identical but logically equivalent form of S. As [START_REF] Antonini | Indirect Proof: What is specific to this way of proving?[END_REF] note, this work relies on theorems in the logical theory rather than the mathematical theory; that is, it is metatheoretical. Thus, a proof by contradiction imposes two unique metatheoretical requirements. First, at the beginning, when one must produce the negation of a statement. Second, at the conclusion, when one must recognize that S* implies S. And, it is the latter requirement that is the focus of the reported studies.

AN OVERVIEW OF THE STUDIES

The reported studies examined students' engagement and extent of success in the metatheoretical reasoning requirements that arise at the conclusion of a proof by contradiction. All studies were conducted at a minority-serving university, where the majority of students qualify for need-based financial assistance and are firstgeneration university students. Study 1 explored the extent to which novices (i.e., students without prior logical training or who have limited training) are successful evaluating claims of the form S*implies S. Study 2 explored students' and mathematicians' approaches to and success with metatheoretical tasks. The aim of the combined studies was to explore the reasoning practices that may inhibit or support students' metatheoretical work and consequently, play a role in the extent to which students reach or fail to achieve cognitive unity in relation to indirect proofs.

Study 1 Methods and Findings

To explore novices' success with metatheoretical reasoning tasks prior to instruction, 46 university students were surveyed. The surveys were administered on the first day of a "Basic Set Theory and Logic" course that served as the universities' first logic course and their "Introduction to Proof" course. Prior to the course, students would have been enrolled in computation-focused courses on calculus and differential equations. Included on the survey were two tasks that asked students to compare a pair of statements and determine, "Can you prove Statement A by proving Statement B?" (Figure 1.) Task 1 involved a universally quantified statement and an incorrect alternative. Task 2 involved the same Statement A and a correct alternative. Survey results indicated that of the 46 students surveyed, 50% were successful at Task 1, 47.8% were successful at Task 2, and 24% were successful at both tasks.

Study 1 Discussion

The findings of Study 1 demonstrate that most of the students did not enter the Basic Set Theory and Logic course reasoning in ways aligned with the metatheoretical requirements of indirect proofs, as the rates were at or below guessing and less than a quarter successfully answered both questions. While the findings are not startling, they provide a warrant for further research. Indeed, prior to 1 there were no studies of novices' responses to such tasks prior to instruction. Thus, the findings warrant the following questions: Do novices' difficulties persist after instruction? Do students' and mathematicians' approaches differ?

Study 2 Methods

Study 2 aimed to explore university students' and mathematicians' extent of success and approaches to the metatheoretical task in Figure 2. Participants were 21 students drawn from the same student population as Study 1 and 6 mathematicians. However, the Study 2 students had completed the Basic Set Theory and Logic course. As the course focused on set theory and logic in the service of proof writing, the instruction on set theory and logic was limited to basic properties, terms, and definitions, as well as symbolizing practices, and then on specific proof techniques and/or strategies. All participants took part in video-recorded interviews during which the task was presented on a large piece of paper. The participants were given as much time as requested and then asked to explain their answer to the stated question.

Question: Can you prove Theorem 5 by proving Statement A? Theorem 5. For all positive integers n, if n mod(3) º 2 then n is not a perfect square. Statement A. There exists no positive integer n such that n mod(3) º 2 and n is a perfect square.

Figure 2. Study 2, Interview Task

To identify approaches the analysis focused on which theory (mathematical or logical) the participant worked in and how they engaged in that theory. Responses were considered mathematical theory approaches (MTA) if the participant was observed: (1) explicitly exploring mathematical statements, definitions and/or terms; and/or (2) constructing a proof of either statement. Responses were considered logical theory approaches (LTA) if the participant was observed: (1) posing explicit questions of equivalence; (2) constructing truth-tables and/or working with symbolic logic; and/or (3) citing logical theorems or practices. In addition to the approach, participants' responses were analysed for the form of engagement. Specifically, coding noted participants' use of syntactic and/or semantic reasoning, with semantic referring to reasoning that employs meanings and multiple representational systems and syntactic referring to rule-based reasoning within a representational system. ii

Study 2 Findings

In Table 2, we report the percentage of correct responses. The reader will notice that among the 21 students five types of responses were observed: yes, yes-no-yes, no-yesno, no, and don't know. Yes refers to students who, after a period of exploration, decided without hesitation that one can prove Theorem 5 by proving Statement A. Yes-no-yes refers to students who repeatedly switched answers, expressed hesitation and doubt, but ultimately choose "yes." No-yes-no were similar to yes-no-yes but were students who repeatedly switched answers and ultimately choose no. No refers to students who reached, with evident certainty, the decision you cannot prove Theorem 5 by proving Statement A. Uncertain refers to students who, after deliberation, responded to the prompt by remarking they "didn't know." Since findings that indicate the prevalence of difficulties are of little use without information on the nature of students' engagement, we turn to the analysis of participants' approaches. This analysis focused on the question of which theories the participant engaged with and their form of engagement (see Table 3). Looking at Table 3, the reader will notice that the majority of the students (18 of 21; 85.7%) engaged in a logical theory approach (LTA). For many this work occurred symbolically, with 15 of the 18 (LTA) students replacing the open sentences with the symbols (e.g., P, Q, ~P or ~Q or P(n), Q(n), etc.) and the phrases for all and there exists no with " and ∄, respectively. iii Indeed, except for one LTA-semantic (Don't know) and two LTA-syntactic (No-yes-no), the students worked symbolically. When asked about the use of symbols many students noted their discomfort with the content, "mod is really rough in my memory right now", and that "it's easier to work with symbols." Thus, the symbolic approaches enabled the students to avoid content for which they lacked confidence in their mathematical understandings.

For 3 of the LTA students their symbolic approach led to a quick and definitive yes, as shown Figure 3. The reader will notice the student initially focuses on the relationship between the quantifiers and then on how translating from ∄ to " requires one to act on the open sentences by negating a sentence of the form (PÙ ~Q). In contrast to those who readily replied yes, nearly half of the LTA students experienced a significant amount of hesitancy and doubt (Yes-no-yes; No-yes-no). Many of these students articulated difficulties with the phrase "there exists no" while, at the same time expressing certainty regarding the logical relationship between for all and there exists (i.e., they asserted the negation of one quantifier produced the other). Among these students, it was not uncommon for them to argue that there exists no means nothing and that, "nothing is the opposite of everything," a point which left many confused having already noted for all and there exist were "opposites" in logic. For nearly a third of the students (6 total) recognizing there exists no as the opposite of for all and the open sentence "n mod(3) º 2 and n is a perfect square" as the negation of "if n mod(3) º 2 then n is not a perfect square" led to the conclusion Statement A is the negation of Theorem 5, as illustrated in Figure 4.

Two aspects of this approach are important to note. First, the student compares the quantifiers (for all and none) and then compares the open sentences. Hence, the quantifiers are not seen as variable-binding operators that act on open sentences but rather as appended symbols. Such reasoning enables the student to translate ∄ into its "opposite" " independently of translating PÙ ~Q into its "opposite" PÞQ.

Second, such reasoning relies on the student incorrectly viewing ∄ (PÙ ~Q) as being of the form (~$) (PÙ~Q) rather than as of the form ~[($) (PÙ~Q)]. As seen in Table 4, the mathematicians' responses were quite different, with all but one engaging in an MTA. Though not shown, it is important to note that in three of the five MTA-semantic responses, the mathematician spent the majority of the time proving (or considering how they would prove) Theorem 5.

Faculty Response Category (n = 6)

Mathematical Theory Approach (MTA) Logical Theory Approach (LTA) Response Type Semantic Syntactic Semantic Syntactic Yes 5 1

Table 5: Faculty Response by Type and Form

This work lead all three to realized they would use a proof by contradiction and in so doing, prove Statement A to prove Theorem 5. In the other two MTA-semantic responses, the mathematicians repeatedly rephrased the statements, while explicitly noting the everyday meanings of the words, until they had convinced themselves that the statements were "essentially the same.' This work was often well-situated in the mathematical theory, as seen in the transcript below where the mathematician speaks of "turning around" Statement A and "running through" sets of numbers.

Mathematician a: I tend to take statements like that [Statement A] and try to rephrase them, so … for me, I would say, what does that actually say? It says that, umm, whenever, umm, a positive integer n is congruent to, umm, is congruent to 2 mod 3 then n cannot be a perfect square … like I … I try to turn it around … I'm sitting here almost hesitant about whether or not I've even done it correctly. But let me think … so, umm, let's see, so there exists no positive integer n such that these two things are true … so that's … what is that the same thing as saying, it's saying that, umm, if you ran over the positive integers n which were congruent to 2 modulo 3 you are never going to hit a perfect square but then that's what this is saying (point to Theorem 5), umm, if I think of for all positive integers n and this part is true, that n is congruent to two modulo three, then I am never going to hit a perfect square. So, … umm, actually, I think, umm, I would almost rephrase these things as being equivalent but I am feeling a little bit hesitant about that.

Here, it is important to note that in addition to Mathematician a, three other mathematicians expressed hesitancy with regard to their own reasoning; e.g., Dr. b remarked "just doubting myself for some reason." In each case, the mathematician was asked "Do you have some doubts about your answer?" and all responded "No." Thus, the participants appeared to be applying inferences with a high degree of (perhaps intuitive) certainty, while also doubting their own judgements of those inferences. Finally, two other observations are of note. First, like the students, one mathematician translated the statements into symbols. However, they immediately pushed the paper away saying, "I am not going to do that." Second, in the case of the LTA-semantic approach, the mathematician translated both statements into Venn diagrams (Figure 5) and then, by comparing the diagrams, reasoned through the task.

Dr. b: 3) is 2 expressed then … this will be expressed as a containment and ….it's not a perfect square … perfect square is on the outside …….. and, umm, let's see, if-then means that ….(long pause) that that set is inside that set or is it the other way … that implies that …(long pause). Yeah, it looks like it (laughs quietly). Study 2 aimed to explore students' and mathematicians' success with and approaches to a metatheoretical task. The data demonstrate that post-instruction, novices continued to struggle with the metatheoretical requirements of proof by contradiction and gravitated towards syntactic-LTA approaches, while the mathematicians tended towards semantic-MTA approaches. Furthermore, students' remarks indicated their use of syntactic-LTA approaches enabled them to avoid perceived content knowledge weaknesses, whereas the mathematicians drew heavily on this knowledge to produce proofs and explore concepts. The study also revealed a tendency among students who struggled with the tasks; namely, a tendency to view quantifiers as appended symbols rather than as variable-binding operators that act on open-sentences. Though far from providing definitive evidence, the study contributes to the literature by highlighting the logical complexities novices may encounter when producing or comprehending proofs by contradiction, given the approaches they gravitate towards.

CONCLUDING REMARKS

One question raised by the studies is, why didn't the students' reasoning progress, even after completing the Basic Set Theory and Logic course? Certainly, the lack of progress may be due to poor instruction, an insufficient curricular treatment, or the cognitive demand of the tasks. Turning to the curricular materials used, Chartrand, Polimeni, and Zhang's (2008) Mathematical Proofs: A Transition to Advanced Mathematics, one finds little in the ways of support for the metatheoretical tasks studied. This text includes an introductory chapter on logic with two subsections on quantifiers. In these subsections, the quantifiers for all and there exists are defined and discussed with regard to the variations of these phrases used in mathematics (e.g., for some; at least one, etc.). Neither are quantifiers discussed as variable-binding operators nor is the phrase "there exists no" or the symbol ∄ mentioned. The same is true in a latter chapter focused on proof by contradiction, where emphasis is placed on moving from "for all" to "there exists" when proving by contradiction without mention of what one must do once one determines something "does not exist." Thus, the lack of progress may be tied to an insufficient curricular treatment of the topic.

Turning to the mathematicians' responses an alternative rationale for students' persistent difficulties becomes evident. As discussed, most of the mathematicians in Study 2 expressed a lack of confidence in their own reasoning, while none wished to change their answer due to an intuition (i.e., "gut feeling"). Hence, it seems reasonable to conclude that the task was cognitively demanding. Consequently, even with instruction, we might expect low success rates among undergraduates, who are at the early stages of the education and lack the content knowledge experts employed.

Lastly, in an interesting study of effective proof comprehension strategies, [START_REF] Weber | Effective proof reading strategies for comprehending mathematical proofs[END_REF] found mathematicians preferred students, "rephrase theorems in their own words" and that students not use the strategy, "rewrite the theorem in first-order logic." These views reflect the practices of the mathematicians in Study 2, for none used the latter strategy, while nearly all used the former. However, their rephrasing of the statements relied on their extensive content knowledge; namely, as a tool for inferring meanings. Hence, the findings raise questions regarding whether or not the mathematicians would use these approaches if they were working with unfamiliar (or difficult) content. Indeed, it seems that we must be careful inferring instructional recommendations from the mathematicians' practices. Many appeared to generate inferences automatically -a practice that seemed to inhibit them from rationalizing their judgements; as illustrated by the mathematicians' repeated expressions of hesitancy.

Figure 1 .

 1 Figure 1. Study 1's Task 1 and Task 2

  Student A: Yes, you can prove Theorem 5 by proving Statement A … because when you say there exists no that implies ... well that's a for all statement and then you have to negate the umm … the umm … (writes PÙ ~Q).

Figure 3 :

 3 Figure 3: Student A's (Correct) Syntactic-LTA Response

  Student B: they're opposites […] this (Statement A) is the negation of Theorem 5 …it's saying for all of them, it's saying none of them [...] Yeah, (writes "(PÞQ)) and (writes ~ symbol before "(PÞQ))) is (writes ∄ (PÙ ~Q)).

Figure 4 :

 4 Figure 4: Student B's (Incorrect) Syntactic-LTA ResponseIn addition to the LTA responses, two MTA responses were observed. In the MTAsemantic response, the student spent his time considering numbers that satisfy n(mod 3)º2 and trying to understand the structure of a number that would disprove Theorem 5 or Statement A. Eventually, this student decided Statement A was false and, therefore, could not be used to prove Theorem 5. In the MTA-syntactic response, the student immediately remarked, "it's by contradiction." The student then proceeded to determine if Statement A provided the needed claims for such a proof: Student C: by contradiction […] he's claiming that there is no positive integer n, … such that (points to Statement A's open sentences) […] so, he's saying there is no positive integer n here so you can use that argument (points to open sentences again) and … so, yeah, you can put those together and prove it.

I

  'm going to draw some sets. … Statement A says to me the sets of n mod(3) … congruent to 2 and perfect squares … (long pause)… are disjoint. Right. There exists no positive integer … so this says for all positive integers if n mod(

Figure

  Figure 4: A Mathematician's (Correct) Semantic-LTA Response Study 2 Discussion

  For every integer n, if n is a perfect square, then n has an even number of factors. Statement B. There exists no integer n such that n has an even number of factors and n is not a perfect square. ☐ Yes, you can prove Statement A by proving Statement B.

	Task 1.
	Statement A.

☐ No, you cannot prove Statement A by proving Statement B. Task 2. Statement A. For every integer n, if n is a perfect square, then n has an even number of factors. Statement B. There exists no integer n such that n is a perfect square and n has an odd number of factors. ☐ Yes, you can prove Statement A by proving Statement B. ☐ No, you cannot prove Statement A by proving Statement B.

Table 2 : Student and Faculty Response by Category

 2

As seen in Table

2

, less than one-third of the students (28.6%) who had completed the Basic Set Theory and Logic course stated with certainty, yes one could prove Theorem 5 by proving Statement A. And, nearly as many (23.8%) reached this conclusion with significant hesitation (Yes-no-yes). Furthermore, 42.8% argued either with certainty (No) or with hesitation (No-yes-no) that you cannot prove Theorem 5 by proving Statement A. These findings indicate instruction had little impact on the students' success with the metatheoretical requirements of a proof by contradiction. In contrast, (without surprise) all of the mathematicians replied yes.

Table 3 : Student and Faculty Response by Type and Form *(NE is no evidence)

 3

	Student Response By Type and Form


Thomas Gjesteland

Pauline Vos
email: pauline.vos@uia.no

Margrethe Wold

Mathematical modelling and activationa study on a large class, a project-based task and students' flow

Keywords: flow, large class, mathematical modelling, mathematics for engineers, novel approaches to teaching, project-based tasks

) tracker software for capturing the motion. Through a poster, they had to report their video analysis. We framed activation through the concept of flow, which is a state of being fully absorbed by an activity. We administered a web-based questionnaire (response rate 69%). The results show that such a project-based task is feasible with >300 students and activated them: three out of five experienced flow. Also, we validated the theory that for experiencing flow, a task must be perceived as challenging and that one's skills should match that challenge.

INTRODUCTION

 [START_REF] Harris | Mathematics and its value for engineering students: what are the implications for teaching[END_REF]studied engineering students' values regarding mathematics finding that not many first-year engineering students have a positive stance towards mathematics. The students see mathematics as a hurdle in their studies, and they are disappointed by the mathematical demands in the first year of their studies. Some even indicate that they wouldn't have chosen the engineering direction if they had known about the mathematics demands before. Nevertheless, mathematics needs to be part of engineering studies, because alumni from engineering studies, such as engineers, managers, researchers, etc., need mathematical modelling competencies to describe, analyse, and predict phenomena to solve problems at the workplace [START_REF] Alpers | A Framework for Mathematics Curricula in Engineering Education[END_REF]. This means, that in particular mathematical modelling needs to be included in engineering studies. It can be integral part of the mathematics curriculum, but the learning of mathematical modelling can also take place in other disciplines, such as physics, where mathematical models are used to describe and analyse physical phenomena. The study described in this paper centres on a mathematical modelling task situated within kinematics (the physics of movement).

In university first-year studies, engineering students often attend large-scale lectures and have tutorial sessions to practice examination-like exercises. However, research has demonstrated the advantages of activating, inquiry-based tasks over these traditional instruction methods [START_REF] De | Physical and virtual laboratories in science and engineering education[END_REF][START_REF] Freeman | Active learning increases student performance in science, engineering, and mathematics[END_REF]. This means that we need research in engineering education into what mathematical modelling tasks can be activating, how these can be organised, how students experience these tasks, what task characteristics create challenges, etc. Moreover, studies with large groups are scarce; the review by [START_REF] Freeman | Active learning increases student performance in science, engineering, and mathematics[END_REF] shows that most studies on students' activation are carried out with small or medium size classes (up to 110 students). With more than 300 first-year students, we can add to the research on how engineering students in large classes can be activated. Sullivan et al. (2011) describe challenging tasks as requiring students to: plan their approach, especially sequencing more than one step; process multiple pieces of information, with an expectation that they make connections between those pieces, and see concepts in new ways; choose their own strategies, goals, and level of accessing the task; spend time on the task and record their reasoning; explain their strategies and justify their thinking to the teacher and other students. We used a task format that fits this description: a project-based task, which is a task that cannot be completed within limited time, which has a clear, but not straight-forward goal, there are various approaches to tackle it, and results must be presented through a product, such as a written report or an oral presentation [START_REF] Blomhøj | Teaching mathematical modelling through project work[END_REF]. [START_REF] Domínguez | Models and modelling in an integrated physics and mathematics course[END_REF] did research with a group of 20 engineering students and asked them: a child is throwing a candy to another; make a mathematical model of this movement. This modelling task is an open-ended task with characteristics of 'a challenging' task (Sullivan et al., 2011): students need to sequence more than one step; process multiple pieces of information and connect the throwing and the model; choose their own strategies, goals, and level of accessing the task; spend time on the task. We adapted the task in the following way: (1) students could choose whatever movement of whatever object: throwing a ball, jumping their skate board, etc.; (2) students were asked to use their smart phones for filming, as nearly all students nowadays have smart phones with high quality cameras; (3) students were asked to download tracker software ( http://physlets.org/tracker/ ), which captures motion in videos based on contrasts and yields a table of time and position coordinates (measurements). We made a tutorial video on the use of Tracker. The measurements were to be mathematically modelled (i.e. create a formula that approximates the movement). The required, final product was a poster, in which students presented their reasoninganother characteristic of a 'challenging task' (Sullivan et al., 2011). The poster had to contain the video analysis, including a discussion of the accuracy of their mathematical model in comparison to the measurements. The task had to be done in groups of two or three. Collaboration was convenient, because one student alone cannot throw and film simultaneously. In our communication with the students we indicated the task as the Tracker Project.

THE TRACKER PROJECT TASK

It was our first time to implement a project-based mathematical modelling task with such a large group. Unlike earlier studies (e.g. [START_REF] Domínguez | Models and modelling in an integrated physics and mathematics course[END_REF] we did neither have a group of 20 students, nor uniform equipment, nor sufficient staff. We couldn't learn from earlier experiences, asto our knowledgethere are no reports of similar studies carried out with more than 300 students. The few studies on the activation of students in large classes centre on using clickers in lectures [START_REF] Freeman | Active learning increases student performance in science, engineering, and mathematics[END_REF]. Thus, we didn't immediately want to focus on students' learning, but instead, first study the feasibility of such a task with such a large group, with the variation of cameras, and with students who have little experience with open-ended tasks. We felt that weas lecturersshould first take the opportunity to learn how it worked in practice, whether students liked the task and how they engaged with it.

In this paper, we report on our research into the extent to which students' were activated by the modelling task. By activation, we meanfor the time beingthat the task grasped them and that they liked working on it. Thus, our study is on students' attitudes, which is an aspect of their affect. Based on [START_REF] Harris | Mathematics and its value for engineering students: what are the implications for teaching[END_REF], we expected the engineering students to have preconceived beliefs about mathematics, and we wanted to avoid that our research would be contaminated by their biases. Therefore, we undertook our research by limiting the use of the word mathematics in our communication with students. Abundant use of the term mathematics could trigger memories and bias of traditional mathematics education, which could interfere with their evaluation of the Tracker Project Task.

THEORETICAL FRAME

Recent research in the field of mathematics education and affect conceptualize the latter in terms of complex, dynamic systems and participatory environments (Pepin & Roesken-Winter, 2015). However, while distinguishing between aspects of affect (values, emotions, beliefs, attitudes, etc.), these researchers don't differentiate between aspects of mathematics education. Yet, mathematics education contains many aspects, such as instruction formats, teacher attitudes, tasks, etc. These become invisible when researchers address mathematics holistically and ask students to mark their (dis-)agreement to statements such as: 'mathematics is my favourite subject'. (Dis-)agreement to such a statement gives little room for nuances and contexts. A student partly agreeing with this item might rather have said: "mathematics with this particular teacher is my favourite subject, but last year it was the opposite" or "mathematics could be favourite, if it had relevance for my future".

We wanted to study students' affect through an activity that differed from standard activities within traditional mathematics education. Thus, we sought an activity-based conceptualisation of affect. An activity-based perspective in mathematics education aligns with a socio-cultural perspective. One of its promoters, [START_REF] Lerman | The social turn in mathematics education research[END_REF], describes mathematics as a socio-cultural practice embedded within a community. Within a school institution, mathematics is a practice embedded in a community of a teacher and a group of students, its rules, language, etc. The activities consist, among others, of explanations by the teacher, and work on tasks by students. This practice differs markedly from mathematics as a practice within a research community, whereby the actors organize mathematical patterns, solve creatively a non-routine problem by using mathematics, and actors may reach different answers. Describing mathematics socio-culturally as a practice embedded within a community entails focusing on the activities undertaken by the actors, which are mediated by language, tools, etc. Using an activity-based conceptualisation of mathematics enabled us to relate affect to distinct activities and not to mathematics holistically, whereby we could distinguish mathematical activities as having different contexts. In our study the activity was guided by the Tracker Project Task and students had to use mathematics within a kinematics context.

We sought an activity-based conceptualisation of students' affect with respect to them being activated. Activation is an aspect of attitude, just like boredom or anxiety (Pepin & Roesken-Winter, 2015). For this conceptualisation, we turned to a concept, which describes "a state in which people are so involved in an activity that nothing else seems to matter; the experience is so enjoyable that people will continue to do it even at great cost, for the sheer sake of doing it" (Csíkszentmihályi, 1990, p.4). Nakamura and Csikszentmihalyi (2009) describe how they observed rock climbers, gamers, painters and researchers during their challenge, and how these people got absorbed in their activities, felt happiness, forgot about time and basic needs (eating, resting), and were intrinsically motivated (motivated by the activity itself, not by an external incentive). They coined this state: flow.

Flow is an activity-based concept: without activity, there cannot be an experience of flow. Flow is an experience of an individual, yet, the activity is culturally embedded (e.g. gamers play a game created by others, painters expose their work). In fact, social activities can intensify flow through group cohesion (group flow). We will use students' self-reported experience of flow as an operationalisation of their activation through the Tracker Project task. Flow has also been studied in mathematics education (a.o. [START_REF] Armstrong | The fragility of group flow[END_REF][START_REF] Drakes | Mathematical Modelling: from Novice to Expert[END_REF][START_REF] Liljedahl | Flow: A framework for discussing teaching[END_REF], observing that many students in traditional mathematics classes is to not experience flow at all.

Figure 1 (left) illustrates how flow depends on the perceived challenge of a task and perceived skills of a person engaging in the task (Nakamura & Csikszentmihalyi, 2009). If the activity is too challenging for the skills, then the task may cause anxiety. If the activity is too easy for the skills, then the task may cause boredom. When challenge and skills match, a person engaging in a task may experience flow. In later work, Csíkszentmihályi and colleagues adapted the diagram, adding more affective states, and stating that flow can be only experienced when a participant perceives the task as more than averagely challenging, and that he/she thinks to have the skills that match this challenge, see Figure 1 (right). The older diagram still appears in recent studies (e.g. [START_REF] Liljedahl | Flow: A framework for discussing teaching[END_REF]. Therefore, we opted to use our study to empirically validate the old versus the new theory and see whether flow occurs only when the actor perceives a more than average challenge.

METHODS

In the Spring of 2017 we offered the Tracker Project task to all first-year students in engineering at our university (Mechatronics, Electrical Eng., Data Eng., Renewable Energy, ICT). There were 346 students for whom the task was mandatory.

The research design for studying students' activation in terms of flow was a survey. We collected data through a digital questionnaire. Participation in the survey was voluntary, but encouraged with prizes of NOK 500 (approx $50) for three randomly drawn participants. After removing seven participants (four had constantly chosen a 3 as answer, three were 2 nd -year students for whom the task wasn't mandatory), we had n=239 students. This response rate of 69% is very high [START_REF] Bryman | Social research methods[END_REF].

Based on instruments from earlier research [START_REF] Armstrong | The fragility of group flow[END_REF][START_REF] Egbert | A study of flow theory in the foreign language classroom[END_REF], we developed 15 items in alignment with the task. Each item consisted of a statement, asking students for their (dis-)agreement on a 5-point Likert scale, from 1 (strongly disagree) to 5 (strongly agree), see the Appendix. Five items were designed to measure students' perception of flow. For this, they could indicate, for example, whether they forgot about the time, and whether they even would do the task if it wasn't obligatory. By having several items related to flow, a participant's score is indicator of the extent to which he/she had experienced flow. Five other items were designed to measure students' self-perceived skills (e.g. 'the Tracker technology was easy to use' or (inverted) 'It was complicated to find the right formula of the model'). And a further five items were designed to measure students' perception of the task's challenge (e.g. 'during this task I started thinking about other movements (what if..?)' and (inverted) 'this task was more for secondary schools').

We make a difference between flow as a concept (written in italics), and the scale of Flow (with a capital letter). The concept of flow is a psychological state of a person, and therefore it cannot be measured. However, we assume that it can be approximated by a score on the scale of Flow. A student's score on this scale results from his/her answers to the five questions in our questionnaire. The score on the Flow scale is calculated by adding the scores on the five questions. As the score on one question ranges from 1-5, the score on the Flow scale ranges from 5-25. Likewise for respectively, challenge and the Challenge scale, and skills and the Skills scale. To increase reliability, within each scale one or two questions were inversely posed, and the scoring was inverted, too. As measure of reliability (internal consistency), we calculated Cronbach's Alpha: the Skill scale yielded 0.55, the Challenge scale yielded 0.73, and the Flow scale yielded 0.63. A scale is considered unreliable if Cronbach's Alpha is less than 0.5 [START_REF] Bryman | Social research methods[END_REF]. Thus, the three scales can be considered as being reliable.

RESULTS

We observed students everywhere on campus, flying paper helicopters, riding skateboards, or throwing apples, cats or balls. We received more than 100 posters in our Virtual Learning System. As explained before, in this study we didn't want to focus on students' performance (the precision of their measurements, their understanding of modelling, the depth of their analysis, etc.). Instead, we focused on the feasibility of an activating tasks for massive students groups, which would show in their activation in terms of flow as measured through the questionnaire. Second, we aimed at seeing whether the measurement reproduced one of the two flow diagrams. The Appendix shows mean scores on all items.

The mean score on four items in the Flow scale is higher than 3.5, being well on the positive side. This indicates that a majority of the students experienced a state of flow to quite an extent, in particular with respect to losing track of time, and not being easily distracted. Only item 14 was answered below the middle range. This item focuses on doing the task even if at some costs [START_REF] Csíkszentmihályi | Flow: The psychology of optimal experience[END_REF], which translates in our study to: one out of four would even do the task voluntarily. students (59%) scored 17 points or higher. When we take 17 points as a threshold, then three out of five students experienced flow to quite an extent. The table in Figure 2 presents mean scores on the scales for Skills, Challenge and Flow (minimal score = 5, middle score range = 14-16, maximal score = 25). The scores on Skills are highest: generally, students perceived themselves as highly skilled; the low standard deviation indicates a high agreement among students. The scores on Challenge are around the middle range; these scores are most "normal" (making a Gauss curve).

To validate the Csíkszentmihályi diagrams (Figure 1), we created a scatter diagram. Each student was represented by a dot defined by his/her Skills score on the x-axis and his/her Challenge score on the y-axis, see Figure 2. The resulting diagram shows a scattered distribution, which means that there is no correlation at all between the scales Challenge and Skills (r = 0.097). In this diagram, we added the third scale, the one for Flow, by colouring the dots depending on the student's Flow scores. These scores range from red to orange (13 or lower), via yellow (middle range, 14-16) to green (17 or higher). Roughly, one can discern overlapping red, yellow and green areas. The red area is more visible at the bottom showing the students who experienced little flow (13% of the students). These students indicated that the task posed little challenge, independently of their perceived skills. The yellow area runs from bottom right to the centre showing the students who experienced medium flow (28% of the students). These students either indicated low challenge and high skills, or medium challenge and medium skills. The green area is the largest with the majority of students (59%). It is in the top-right, fading towards the centre, showing the students who experienced flow to quite an extent. These students indicated that they perceived the task as challenging, and they perceived themselves skilled. In light of the different regions in the later Csíkszentmihályi diagram, we also see many students who may fit into the affective states of 'control', 'arousal' and 'relaxation'. Only few students may fit the more negative affective states of 'apathy', 'boredom', 'anxiety' and 'worry'.

CONCLUSION, DISCUSSION, RECOMMENDATIONS

We studied whether a project-based task was feasible with a class of more than 300 students, that is: whether the task activated individual students. The first research question asked: to what extent does the Tracker Project Task make students experience flow? The results from the survey showed that a majority of the students (59%) experienced flow to quite an extent, forgot about time and wanted more of such activities. This result was confirmed by anecdotal evidence of their boasting stories in the tutorials of them throwing objects, and the high response rate to the survey. This means that the Tracker Project Task activated a majority of the students and that they had positive attitudes towards it. Thus, an activating mathematics task can be feasible with a large class of engineering students, even if they are known to have a negative stance towards mathematics [START_REF] Harris | Mathematics and its value for engineering students: what are the implications for teaching[END_REF].

The Tracker Project Task was designed to be challenging with characteristics such as: expecting students to process multiple pieces of information, that they make connections between those pieces, choose their own strategies, and explain their strategies to others [START_REF] Blomhøj | Teaching mathematical modelling through project work[END_REF]Sullivan et al., 2001). We observed other characteristics in the Tracker Project Task that activated the students. First, the task had a clear goal, which was understandable to all students. We observed this through the few questions that we got from the students on how to carry out the task. Thus, the task was easily accessible, also known as having a low floor. Second, the better students were able to challenge themselves further, allowing for a high ceiling.

Third, the use of readily-available technology (cameras in smart phones, tracker software) may have captivated the engineering students, who are known to be technology minded. Fourth, the task was a mathematical modelling task embedded in engineering practices, whereby mathematics served non-mathematical purposes; this showed students the relevance of mathematics to their studies, and contrasted with bare mathematics tasks that alienate and demotivate students.

Our second research question pertained to the theory of flow and how it can be conceptualized in a diagram (Nakamura & Csíkszentmihályi, 2009). Our data reject the earlier theory that flow depends on the alignment of skills and challenge. Instead, our data support the later theory that flow occurs when the participants perceive the task more than average challenging, and that their skills should match this challenge.

Furthermore, we take from our study that the concept of flow proved useful for activity-based research on affect in mathematics education. The Tracker technology was easy to use.

(Inv) It was complicated to find the right formula of the model.

The aims of the task were clear to me.

During this task I had full control over what we did.

Filming the movement of an object was easy.

3.89 (0.88)

Figure 1 :

 1 Figure1(left) illustrates how flow depends on the perceived challenge of a task and perceived skills of a person engaging in the task(Nakamura & Csikszentmihalyi, 2009). If the activity is too challenging for the skills, then the task may cause anxiety. If the activity is too easy for the skills, then the task may cause boredom. When challenge and skills match, a person engaging in a task may experience flow. In later work, Csíkszentmihályi and colleagues adapted the diagram, adding more affective states, and stating that flow can be only experienced when a participant perceives the task as more than averagely challenging, and that he/she thinks to have the skills that match this challenge, see Figure1(right). The older diagram still appears in recent studies (e.g.[START_REF] Liljedahl | Flow: A framework for discussing teaching[END_REF]. Therefore, we opted to use our study to empirically validate the old versus the new theory and see whether flow occurs only when the actor perceives a more than average challenge. Our research was guided by two questions. The first was empirical: To what extent did the Tracker Project Task make students experience flow? The second was about the choice of flow diagram: Can

Figure 2 :

 2 Figure 2: Scores to Skills, Challenge, and Flow scales (n=239)When adding the students' scores on the five questions, we obtain their score on the scale Flow. See Figure2(right) for a bar graph. This graph shows the frequencies of scores (number of students with certain scores). The green bars of the Flow scale show a skewed distribution. On this Flow scale, 31 students (13%) scored 13 points or lower, 67 students (28%) scored in the middle range of 14-16 points, and 141

Figure 3 :

 3 Figure 3: Flow score indicated by color, as depending on Skills and Challenge This colour distribution of Flow does not confirm the earlier Csíkszentmihályi diagram (Fig 1, left), as the green dots do not centre on the diagonal. Instead, the

  Nakamura, J., & Csikszentmihalyi, M. (2009). Flow theory and research. In S. J.Lopez & C.R. Snyder (Eds.), Oxford Handbook of Positive. Oxford, UK: Oxford University Press.Sullivan, P. et al. (2011). Challenging mathematics tasks: What they are and how to use them. In L. Bragg (Ed.), Maths is multi-dimensional (pp. 33-46). Melbourne: Mathematical Association of Victoria. This Tracker task took too much of my time Time was flying when we worked in this task.(Inv) I was easily distracted when we worked on this task.I would do this task even if it wasn't obligatory.I would like to have more of such practical tasks.

	APPENDIX
	Mean scores on all items (1=lowest, 3=middle, 5=highest).
	Flow questions	mean (std dev)
	(Inv) 3.67 (0.88)
		3.40 (0.92)
		3.55 (0.91)
		2.60 (1.13)
		3.70 (1.02)
	Skills questions	mean (std dev)

  This "Modelling med Tracker" task made me curious.Making a poster made me feel like a "real scientist".(Inv) This task is more suitable for secondary schools. This task helped me to better understand the theory.

		3.38 (0.92)
		3.96 (0.91)
		3.77 (1.13)
		3.76 (1.02)
	Challenge questions	mean (std dev)
		3.61 (0.75)
		2.52 (1.03)
		2.58 (0.95)
		3.39 (0.88)
	During this task I started thinking about modelling other	3.31 (1.12)
	movements (what if..?).


Rosa Delgado

Diana Zakaryan

Knowledge of the Practice in Mathematics in University Teachers

Keywords: teachers' and students' practices at university level, teaching and learning of analysis and calculus, university teachers' knowledge, knowledge of the practices in mathematics, Mathematics

This research is part of the study of university teachers' knowledge that has emerged as a new line of research, which aims to understand the components of this knowledge, its development, and how it is reflected in university teachers' teaching practice. This study, using the Mathematics Teacher's Specialized Knowledge model, seeks to characterize a university teacher's knowledge of practice in mathematics in the content area of mathematical analysis. Based on an instrumental case study, through classroom observation, we provide indicators of the teacher's knowledge of ways of reasoning, validating, and proceeding in mathematics, contributing to the understanding of the nature of this teacher knowledge.

INTRODUCTION

Research on mathematics teachers' knowledge began to be carried out in the nineties (e.g. [START_REF] Fennema | Teachers' knowledge and its impact[END_REF]Broome, 1994) and currently continues to be developed with great force. Studies in this line of research have focused on teachers' knowledge of various concepts, such as fractions [START_REF] Llinares | The knowledge about unity in fractions tasks of prospective elementary teachers[END_REF] and functions [START_REF] Even | Teachers' pedagogical knowledge: the case of functions[END_REF], and in different mathematical domains, such as algebra [START_REF] Mccrory | Knowledge of algebra for teaching: A framework of knowledge and practices[END_REF] and geometry [START_REF] Herbst | Mathematical knowledge for teaching high school geometry[END_REF]. We also find proposals of models of teacher knowledge such as the Knowledge Quartet [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF], Mathematical Knowledge for Teaching [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF] and, more recently, Mathematics Teacher's Specialized Knowledge (Carrillo, Climent, Contreras, & Muñoz-Catalan, 2013). These models have been used for studying mathematics teachers' knowledge, principally in primary and secondary education, with scarce accounts of studies of university teachers, as noted by [START_REF] Speer | Definitions of mathematical knowledge for teaching: using these constructs in research on secondary and college mathematics teachers[END_REF].

Currently, it can be said that research in higher education has gone from being centered on students to having a more balanced interest in both, students and teachers [START_REF] Artigue | Mathematics Education Research at University Level: Achievements and Challenges[END_REF], to the point that research on university teachers' knowledge has emerged as a new line of research from which it is asked what is understood as knowledge, how this knowledge is developed, and how it is reflected in the teaching practice of university teachers [START_REF] Biza | Research on Teaching and Learning Mathematics at the Tertiary Level: State-of-the-art and Looking Ahead[END_REF].

In agreement with the above and taking into account that the Mathematics Teacher's Specialized Knowledge model (MTSK) has been shown to be useful for studying university teachers' knowledge (e.g. [START_REF] Vasco | Conocimiento especializado del profesor de álgebra lineal: un estudio de casos en el nivel universitario[END_REF], we carry out our research based on the MTSK with the aim of characterizing the knowledge of a university mathematics teacher who teaches content in the area of mathematical analysis. In this text, some results are presented in relation to knowledge of the practice in mathematics, one of the sub-domains of knowledge considered in the model.

MATHEMATICS TEACHER'S SPECIALIZED KNOWLEDGE

Mathematics Teacher's Specialized Knowledge (MTSK) is an analytical model for understanding mathematics teachers' knowledge and at the same time a methodological tool that allows analyzing teachers' teaching practices [START_REF] Carrillo | Determining Specialised Knowledge for Mathematics Teaching[END_REF]. The model was developed based on a theoretical, empirical, and reflective work proposed by [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] regarding foundational knowledge for teaching and the refining of Mathematical Knowledge for Teaching [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF].

In MTSK, two domains of teacher knowledge are distinguished, mathematical knowledge (MK) and pedagogical content knowledge (PCK), considering that all of this knowledge is specialized, that is, it derives from the teaching profession and is conditioned by mathematics itself. Consequently, MTSK does not include, for example, general psychopedagogical knowledge included in Shulman's works. Also, the model takes into account that teacher's beliefs and conceptions about mathematics, its teaching, and its learning permeate the organization and the use of knowledge [START_REF] Carrillo | Determining Specialised Knowledge for Mathematics Teaching[END_REF]. Mathematical Knowledge comprises three sub-domains, knowledge of topics (KoT), knowledge of the structure of mathematics (KSM), and knowledge of the practice in mathematics (KPM).

KoT includes knowledge of concepts, properties, procedures, classifications, formulas, and algorithms, with their respective meanings and foundations. For example, knowing the topological property, density of the rational numbers in R, lies in this sub-domain. KSM includes knowledge of the interconceptual connections that can be established among mathematical concepts. So, knowledge of relationships between infinity and the Archimedean property of the real numbers belongs to this sub-domain. KPM includes knowledge of how to proceed, reason, and establish validity in mathematics. Knowledge of how proofs are made using different methods is part of teachers' KPM.

Pedagogical Content Knowledge contains three sub-domains: knowledge of mathematics teaching (KMT), knowledge of features of learning mathematics (KFLM), and knowledge of mathematics learning standards (KMLS).

KMT includes knowledge about didactic resources, strategies, tasks, and examples for making mathematical contents understandable. A teacher using an example to illustrate the meaning of a necessary condition forms part of his or her KMT. KFLM addresses the teacher's knowledge about mathematical contents as an object of learning, for example, teacher's knowledge about students' difficulties in understanding proofs belongs to this sub-domain. KMLS describes what students should achieve in a given course, conceptual and procedural capacities and mathematical reasoning that are promoted in given educational moments. The sequencing of the topics completeness theorem, characterization of the greatest element, and the Archimedean property of the real numbers is an example of KMLS.

KNOWLEDGE OF THE PRACTICE IN MATHEMATICS

The idea of knowledge of the practice in mathematics (KPM) comes from the works of [START_REF] Schwab | Education and the structure of the disciplines[END_REF], [START_REF] Ball | The mathematical understandings that prospective teachers bring to teacher education[END_REF][START_REF] Ball | The mathematical understandings that prospective teachers bring to teacher education[END_REF][START_REF] Ball | With an eye on the mathematical horizon: Knowing mathematics for teaching to learners' mathematical futures[END_REF] regarding syntactic knowledge of mathematics, which implies that the teacher should know how to reason mathematically, know different kinds of reasoning, and know in which mathematical contexts a particular kind of reasoning is more adequate than others. In this regard, within KPM, the importance of the teacher not only knowing established mathematical results, but also how to proceed and think in mathematics to arrive at these results is highlighted.

Knowledge that makes up part of this sub-domain is propositional logic, mathematical language and its precision, how definitions are made and used in the construction of mathematical knowledge, knowledge about different kinds of proof and their internal logics, the role that examples and counterexamples play in proofs, different kinds of heuristic reasoning, how knowledge is created in mathematics, how it is validated, reasoned, and generalized, and the role of mathematical conventions and symbols.

Although this knowledge is important, research carried out in MTSK with teachers at different educational levels reports scarce evidence of KPM in their mathematics classes (e.g. [START_REF] Montes | Conocimiento especializado del profesor de matemáticas acerca del infinito. Un estudio de caso[END_REF][START_REF] Vasco | Conocimiento especializado del profesor de álgebra lineal: un estudio de casos en el nivel universitario[END_REF]. In this regard, this investigation contributes to the understanding of the nature of this knowledge in university teachers.

METHODOLOGY

In this investigation, based on an interpretive paradigm and a qualitative methodology (e.g. Denzin & Lincoln, 2000), we carry out an exploratory study such that our results correspond to a first approximation for characterizing a university teacher's knowledge of the practice in mathematics (KPM). Our work is supported by an instrumental case study [START_REF] Stake | The art of case study research[END_REF] for which we chose as an informant a teacher who taught a real analysis course for prospective teachers of mathematics during one semester in a Chilean university.

The teacher, who we will call Diego, is a mathematics researcher with a Ph.D., with more than 20 years of teaching experience, and this is the sixth time in recent years that he has taught the real analysis course. These academic characteristics of Diego make it likely that he possesses plentiful knowledge in elements of KPM.

The data was obtained through video recording while Diego taught the system of real numbers. The video recordings were transcribed and later subjected to content analysis (Bardin, 1997), identifying the units of analysis associated with the KPM sub-domain and considering the differentiation between evidence and indication of knowledge [START_REF] Moriel-Junior | Explorando indícios de conhecimento especializado para ensinar matemática com o modelo MTSK[END_REF]). An evidence is an element that supports the presence of teacher knowledge, while an indication provides suspicion of the existence of knowledge but requires additional information in order to be confirmed as evidence.

RESULTS

In this section, we present episodes from one of Diego's classes on the properties of real numbers that allow us to observe his knowledge of the practice in mathematics (KPM).

Diego begins the class enunciating the property of the density of Q and R\Q in R. For the proof of this property, Diego takes an interval [𝑎, 𝑏] with 𝑎 and 𝑏 in R and considers the case 𝑎 = 0, then he generalizes it for any positive a. In fact, he takes 𝑎 and 𝑏 positive numbers and indicates that with this supposition there is no loss of generality.

Teacher:

We are going to assume that 𝑎 and 𝑏 are two positive numbers, which is not a great assumption. If they were negative, for example, I work with -𝑎 and -𝑏.

If I have 𝑎 and 𝑏 here [indicating on the number line], If I take -𝑎 and -𝑏 then they are on the other side of the zero.

And if I find a rational number here [indicating on the number line between 𝑎 and 𝑏], then its opposite, will also be rational.

And if I find an irrational number, its opposite will be irrational.

And if one number is positive and the other is negative, then zero is rational, and it is between them, and the irrational will come from a proposition.

In the previous episode, Diego's knowledge of the different cases that should be taken into account to make a proof whose hypothesis possesses an implicit or explicit disjunction is observed, as is his that a particular case can be sufficient for showing the behavior of other cases in the proposition. Following this, the teacher's knowledge about the consideration of cases to particularize or generalize, is a way of proceeding in mathematics and can be considered as an indicator of KPM. The teacher's knowledge that the proof goes beyond the example of a concrete case and addresses all the possible cases the statement can include (e.g. [START_REF] Brodie | Teaching mathematical reasoning in secondary school classrooms[END_REF][START_REF] Montes | Conocimiento especializado del profesor de matemáticas acerca del infinito. Un estudio de caso[END_REF] is part of his KPM.

Additionally, Diego emphasizes that in mathematics for a fact to be considered valid it must be proven:

Teacher: Now, a question, do I know of any irrational number that is less than 1 and greater than 0? Student:

The square root of square root of 2. Teacher:

Ah! Why is it less than 1? We have to prove that it exists, and, wait, we have to prove that the root exists and that it is a number less than 1. And prove something else, because they didn't ask us for a number between 0 and 1, but rather for an irrational number between 0 and 1. So, you can take a real number, prove that its root exits, but you still don't know if it is irrational or not.

In the prior episode, Diego's knowledge of proof as a way of validating in mathematics is shown. As [START_REF] Brodie | Teaching mathematical reasoning in secondary school classrooms[END_REF] maintains, in addition to knowing that a kind of proof exists that confirms the truth of a statement, it is necessary to know how this type of proof works. Furthermore, Diego points out the importance of this role of proofs (de Villiers, 1990), as he is in front of a course for prospective mathematics teachers and after proving the density property he says:

Teacher: So, now you can say to your students with certainty that between any two rational numbers there is always an irrational number.

The comment above shows as indicator of KPM, the teacher´s knowledge of the necessity and importance of proofs as a way of validating in mathematics (e.g. [START_REF] Balacheff | Procesos de prueba en los alumnos de matemáticas [Proof process in mathematics students[END_REF].

Continuing with the density proof, Diego discusses with the students about taking the maximum or minimum of a set of natural numbers. In the following episode, we observe Diego's knowledge about the convenience of selecting a certain element to develop an argument in a proof.

Teacher:

The issue is that if I take either of the two [the maximum or minimum] I can argue that there is a rational number in between them.

What I have to guarantee before choosing one of the two is that in fact there is something to choose, so, which is easier to guarantee?

The smallest or the largest? Student:

The smallest. Teacher:

Why? Student:

Because of the well-ordered principle. Teacher:

Of course, the minimum exists by the well-order principle.

Nevertheless, on occasions, rather than choose, is necessary construct an element that allows developing an argument. Diego also gives evidence of this knowledge when discusses with the students about irrationality of a number that belongs to the interval they are working on. He uses √2 to construct √2 𝑚 ⁄ as shown below.

Teacher: I know that √2 does not belong to the rational numbers. We proved it.

It is real, it exists, and it is not rational.

If I now divide this number by a whole number 𝑚, will it continue to be not rational? will it become a rational number? That is, this is not rational, but is this [pointing to √2 𝑚 ⁄ ] not rational either?

According to above episodes, an indicator of KPM is the teacher´s knowledge of the construction or selection of elements for developing an argument in a proof as a way of proceeding in mathematics.

Regarding to √2 𝑚 ⁄ , Diego prove its irrationality and establishes this affirmation as a lemma that he uses in different moments in the proof of the density property:

Teacher:

So, if I take the Archimedean property for 𝛿 = 𝜀 √2 ⁄ , then exists an 𝑚 ∈ 𝑁 such that 0 < 1 𝑚 ⁄ < 𝛿, and this implies that, √2 𝑚 ⁄ is smaller than 𝜀 and does not belong to the rational numbers. So, between 0 and 𝜀, no matter how small 𝜀 is, there is always a rational number, and there is always an irrational number. Ok, so this affirmation that I just wrote, we're going to write it as a lemma.

This episode gives us evidence that the teacher knows that establishing preliminary results is a way of proceeding in mathematics that facilitates the development and the communication of a long and/or complex proof.

On the other hand, regarding the existential quantifier present in the Archimedean property and the well-ordered principle, Diego expresses that the existential quantifier only gives information about the characteristics or properties of an element, but does not say what the element is like or which element it is, only that it exists Teacher: When the for all quantifier is given, I can always choose an element that works best for me, but when it is existence, it's "whatever you've gotten", no more. Then as exists gives you "whatever you've gotten", you have to do some work so that what you get, is what you need.

An indicator of KPM in this episode is the teacher's knowledge of the role and the meaning of the quantifiers when they are found in the hypothesis or in the conclusion of a proposition.

Summarizing, indicators of the teacher's knowledge of the practice in mathematics related to ways of reasoning, validating, and proceeding in mathematics has been evidenced. In the exposed episodes, the teacher teaches mathematical reasoning in order to give meaning to the mathematical activity (e.g. [START_REF] Brodie | Teaching mathematical reasoning in secondary school classrooms[END_REF], not only for students to understand and acquire sensibility regarding how to establish truth in mathematics, but also, they find meaning in the need to do so [START_REF] Montes | Conocimiento especializado del profesor de matemáticas acerca del infinito. Un estudio de caso[END_REF].

CONCLUSIONS

A mathematics teacher is a professional whose knowledge of the discipline he or she teaches has a level of deepening, organization, and structure that is greater than what the students are going to receive [START_REF] Ma | Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States[END_REF]. In this regard, KPM is necessary knowledge for the teacher, as it provides logical thinking structures that help to understand the function of diverse aspects of mathematics (Flores-Medrano, Escudero-Avila, [START_REF] Flores-Medrano | Nuestra Modelación del Conocimiento Especializado del Profesor de Matemáticas, el MTSK[END_REF]. As observed in the case studied, the teacher's knowledge of processes of particularization-generalization, of the necessity and importance of proof for validation, and the knowledge of different ways of proceeding in mathematics are closely linked to the transition to advanced knowledge (e.g. [START_REF] Pino-Fan | Conocimiento didácticomatemático de profesores en formación: Explorando el conocimiento especializado sobre la derivada. [Prospective teachers' didactic-mathematics knowledge: Exploring the specialized knowledge about the derivative[END_REF] and are related to the particular way of mathematical work.

In line with the ideas above, we consider that KPM allows the teacher to promote in students the construction of mathematical knowledge and the acquisition of abilities for reasoning, proof, and problem solving that are considered important for learning mathematics at all levels of education (e.g. [START_REF] Flores-Medrano | Nuestra Modelación del Conocimiento Especializado del Profesor de Matemáticas, el MTSK[END_REF].

With reference to the indicators of KPM obtained in this investigation regarding ways of reasoning, validating, and proceeding in mathematics, we agree with Sosa, Flores-Medrano, & [START_REF] Sosa | Conocimiento del profesor acerca de las características de aprendizaje del álgebra en bachillerato. [Teacher knowledge about the features of learning algebra in high school[END_REF] that indicators shown directly in empirical data can contribute to identifying, understanding, and analyzing teachers' knowledge in their discipline.

Given that teacher knowledge is a complex and multidimensional construct, more studies are necessary to deepen the understanding of its different components, in order to advance in interrogations proposed regarding how this knowledge is developed and how it is reflected in the teaching practice of university teachers (e.g. [START_REF] Biza | Research on Teaching and Learning Mathematics at the Tertiary Level: State-of-the-art and Looking Ahead[END_REF].

Figure 1 :

 1 Figure 1: Mathematics Teacher's Specialized Knowledge (Carrillo et al., 2013).


Ghislaine Gueudet
email: ghislaine.gueudet@espe-bretagne.fr

Pierre-Vincent Quéré

"Making connections" in the mathematics courses for engineers: the example of online resources for trigonometry

Keywords: Connectivity, Teaching and learning of specific topics in university mathematics, The role of digital and other resources in university mathematics education, Teaching and learning of mathematics in other fields, Mathematics

This paper concerns the teaching of mathematics for future engineers, focusing on the theme of trigonometry. We claim that the use of trigonometry in engineering courses requires different kinds of connections: connecting different domains, different concepts, frames and registers. We use here the concept of connectivity, developed in the frame of e-textbooks analysis, to analyse online courses for future engineers in France. We evidence that these courses propose some connections; but their connectivity is not developed enough to meet the requirements of engineering courses.

INTRODUCTION

How can a teaching of mathematics answer the needs of engineering courses, i.e. can provide the mathematics needed to understand the course and solve the problems proposed? Which should be the features of such a teaching, and do existing courses present such features? This is the general theme of the work presented here. Previous works addressing this theme observe a gap between the mathematics taught in "mathematics courses" and the use of them to solve problems in engineering courses (e.g. [START_REF] Redish | Problem solving and the use of math in physics courses[END_REF][START_REF] Biehler | Conceptualizing and Studying Students' Processes of solving typical problems in introductory engineering courses requiring mathematical competences[END_REF]. [START_REF] Winsløw | Research on University Mathematics Education[END_REF] note that several works presented at CERME conferences identify "a lack of connectedness of curricula integrating mathematics and other disciplines". Interviewing French engineers about their mathematical needs in the workplace [START_REF] Quéré | French engineers' training and their mathematical needs in the workplace: interlinking tools and reasoning[END_REF], and how the mathematics courses they followed as students answered or not these needs, we noticed that several of them declared that the mathematics courses "did not make enough connections". These connections, according to the engineers, can be of different kinds: links between mathematics and the real world, between different mathematical contents, between different representations etc. We consider that this is an important issue for understanding mathematics applied to engineering and their teaching. Which kinds of connections should propose a teaching of mathematics for engineering? Moreover, in the frame of another starting study, we are interested by the possible design of innovative curriculum resources for the teaching of mathematics for STEMSS (Science, Technology, Engineering, Medicine and Social Sciences) courses. Before starting the design of such resources, we have investigated existing resources, trying in particular to observe whether they tried to build the kind of connections evoked by the engineers. This investigation led us to choose a focus on trigonometry. Indeed, trigonometry appeared as extensively used in different kinds of engineering courses, and offering many possibilities of connections of all the kinds evoked above. In what follows, we firstly introduce our theoretical tools and research questions; then we present related works on the teaching of trigonometry. To exemplify the mathematical needs, we analyse the use of trigonometry in an electricity course for first year students; then we consider the trigonometry content in two online mathematics courses for engineering students.

THEORETICAL PERSPECTIVE AND RESEARCH QUESTIONS

The overarching perspective guiding our work is an institutional perspective [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF]. We consider that the mathematics taught are shaped by the institution where they are taught. Engineering studies at university or in engineers' schools constitute an institution, different from mathematics courses for maths majors. Within engineering studies, engineering courses also constitute an institution different from mathematics courses. Our aim here is to compare how these two institutions shape the mathematics contents, more precisely the trigonometry contents.

Our central focus is on "making connections". Students' understanding in terms of mathematical reasoning and problem-solving has been linked by several authors to "making connections" and "connectivity" (e.g. [START_REF] Hiebert | Learning and teaching with understanding[END_REF]. Drawing on these works, we have chosen to look for: "connections in, between, and across individuals' cognitive/learning tasks and activities, and how e-textbooks may support those (micro level); as well as for 'connected learning' between and across groups of individuals, teachers or students (macro level)." (Gueudet, Pepin, Sabra, Restrepo & Trouche to appear) We have therefore proposed a concept of "connectivity" to analyse e-textbooks (encompassing various kinds of digital curriculum), with the intention to evaluate their potential for the building of connections for the students.

Thus, connectivity has two components: "macro-level connectivity", which considers the e-textbook as a whole; and "micro-level connectivity", where the focus is on a particular mathematical topic (trigonometry here). In this paper we only use "microlevel connectivity"; more precisely, we observe in curriculum resources available online the presence of:

-Connections between different topic areas or frames; -Connections between different semiotic representations (e.g. text, figures, static and dynamic); […] -Connections between different concepts […]" (Gueudet et al. to appear).

Here we want to compare the connections concerning trigonometry when it is used in engineering courses and the connections concerning trigonometry in mathematics courses for engineers, more precisely the connectivity of online courses for engineers. Our purpose is not to discuss whether trigonometry should be introduced in mathematics courses or engineering courses, but to compare how it is introduced/used in these institutions. Hence the research questions we study here can be formulated as:

-Which connections concerning trigonometry appear in non-mathematical engineering courses?

-Which connectivity, concerning trigonometry, can be observed in online resources for mathematics courses for engineers, and how does this compare with the connections in non-mathematical courses?

In terms of methods, we have searched for curriculum resources and online courses on three major websites used in France: Unisciel1 , meaning "online science university", gathering many online courses; IUTenligne2 , a website for technicians institutes within universities; and FUN3 , meaning France Digital University, the national platform offering MOOCs. We have selected all the resources corresponding to mathematical courses for engineers or technicians on trigonometry, and have looked at the same time for resources on non-mathematical subjects using trigonometry. We have eventually chosen the theme of electrical engineering, because we have identified in it specific mathematical needs in trigonometry. Before analysing these resources, we now consider some works about trigonometry in mathematics education, and how they enlighten the connections issue.

CONTEXT AND RELATED WORKS

In France, trigonometry is firstly introduced in grade 8 or grade 9 through the definitions of cos, sin and tan as quotients of lengths in a right-angled triangle. The angles are measured in degrees. In grade 10, the unit circle and the radian are introduced, together with a new frame for cos and sin, which are now the coordinates of a point on the unit circle. In grade 11, cos and sin are studied as functions. Hence connections between frames and registers are extensively present in this teaching, and these connections can raise difficulties for the students [START_REF] Berté | Aide apportée aux enseignants par la recherche en didactique. Un exemple : le cosinus en quatrième[END_REF]. For example, grade 9 students already use for solving geometry exercises the "cos -1 " key on their calculator; nevertheless, this key refers to a reciprocal trigonometric function, which is only presented at university, and moreover belongs to the functional frame (Bueno-Ravel & Gueudet 2010).

The international research on the teaching and learning of trigonometry acknowledge the existence of all these different registers and representations and investigate their consequences. [START_REF] Kendal | Teaching trigonometry[END_REF] compare two teachings in grade 10 in Australia, using respectively ratios and the unit circle to introduce sin, cos and tan; they conclude that the ratio method appears as a better choice. Also at the university level, trigonometry remains a difficult subject for the students [START_REF] Weber | Students' understanding of trigonometric functions[END_REF]. [START_REF] Mesa | Conception of Angles, trigonometric functions, and inverses trigonometric functions in college textbooks[END_REF], studying the presentation of trigonometry in college textbooks, have evidenced that these textbooks propose different conceptions of angles, trigonometric and inverse trigonometric functions; depending on these conceptions, some problems can be delicate to tackle for the students. The textbooks do not try to link different conceptions, and do not highlight which one is more relevant for a given problem.

Trigonometry clearly requires many connections between frames and registers. It is moreover linked with many different mathematical subjects (geometry, functions, but also complex numbers); and is extensively used in physics. Several researchers have also studied trigonometry within physics courses. [START_REF] Chiu | The Challenge of Learning Physics Before Mathematics: A Case Study of Curriculum Change in Taiwan[END_REF] studied the impact of a new curriculum in Taiwan, where contents of physics requiring trigonometry are taught before the corresponding mathematics courses. She observes that, while curriculum designers are positive on the possible consequences of a teaching of trigonometry by the physics and then by the mathematics teachers, the students and the teachers are mostly negative about this experience. A teaching of trigonometry in mathematics courses seems necessary before using it in physics.

In his comparative study between France and Vietnam, Nguyen Thi (2013) shows that in both countries, trigonometry is present in physics courses with mathematical models for periodical phenomena, under two forms: uniform circular motion (represented by a point moving on a circle, in a graphic or algebraic register) and harmonic oscillation (represented by functions in a graphic, algebraic or vectorial register). Nevertheless only a few exercises propose modelling activities (in physics as well as in mathematics); and the two models are almost never connected.

TRIGONOMETRY IN ELECTRICAL ENGINEERING COURSES

In this section we draw on the content of several courses for future engineers or technicians, available on French websites (e.g. [START_REF] Piou | Electricité. Analyse des circuits et des signaux électriques[END_REF]). In electrical engineering, one of the major subjects is the study of the "alternating sinusoidal regime". In circuits with such a regime, the different signals (current and voltage in particular) are of the form: s(t) = A2sin(t+).

In a circuit where reactive loads are present (like capacitors or inductors) energy storage in the loads results in a phase difference between the voltage (u) and the current (i) waveforms. This difference is firstly introduced in the context of functions, with formulas like: u(t)=U2sin(t); i(t)=I2sin(t-), and associated with a graphical representation as two curves on the time axis (Figure 1, left part).

A connection is immediately established with a geometrical representation of these signals, through "Fresnel vectors". A signal defined by s(t) = A2sin(t+) (where A is positive) can be represented by a vector of length A, and a direction forming an angle  with the horizontal direction. Hence u(t) can be represented as a horizontal vector, and i(t) as a vector forming an angle (-) with it (figure 1, right part). Some courses also propose an interpretation in terms of complex numbers; for the sake of brevity, we do not present it here. We argue that the students in this case have to master connections between these two representations of signals: as two curves with a gap of  on the horizontal axis; or as two vectors forming an angle . Some courses propose animated pictures or exercises to work explicitly on this connection (see figure 2 for an example extracted from a teacher's website, http://fisik.free.fr/.)

Figure 2. Connection between a rotating vector and the corresponding curve, extract of an animated picture. The blue points on the right appear when the vector rotates.

In terms of micro-level connectivity, electrical engineering courses naturally connect trigonometry with electrical engineering; they also connect different concepts: in the example we considered here, functions and vectors, and in other courses also complex numbers. Since vectors are represented as arrows, and functions represented by their graphs, these two kinds of representations are also connected in the text of the course.

TRIGONOMETRY IN MATHEMATICS ONLINE COURSES FOR ENGINEERS A MOOC presenting the basics of mathematics for future engineers

The MOOC "Basics of mathematics", available on the FUN platform (freely available after inscription) presents its objective on its first page: "This MOOC aims at revising the basic notions of mathematics, needed to start engineering studies". The MOOC lasts 12 weeks, corresponding to 12 chapters. Chapter 2 is entitled "trigonometry". It comprises 7 course videos (from 5 to 12 minutes), interactive multiple-choice questions, and a final assessment. Most of the notions presented in chapter 2 of the MOOC are taught in France at secondary school; nevertheless the reciprocal functions (like sin -1 ) are only presented during the first year of university.

The course proposes almost no connection with engineering activities. The first video says that "Trigonometry appears in many domains, like drawing plans, navigating or mechanics"; the last video mentions the task of "triangulation", without definition. Three exercises are associated with each video. Most of these exercises are situated in the geometrical frame; some of them are in the frame of trigonometric equations. There is only one exercise in the whole chapter with a concrete context: computing the length of a cable joining the top of a pole to the ground, over a mountain with slope of 15%. The final evaluation is composed of two problems; the first one has a concrete context, measuring lengths, and the second one concerns trigonometric equations.

The definitions of cos, sin, tan are introduced in the geometrical frame in the right triangle (video 1). In video 2, a software (Maple reader) is used to establish a connection between cos, sin and tan in the unit circle and their graphs as functions (figure 3). The animated picture supports the discourse of the teacher explaining this link. This picture associates in fact the unit circle with the graphs of the functions, but also with the triangle. In fact the radius of the circle can be changed; it is not only the unit circle, but any circle; and the words "Adjacent" and "Hypotenuse" on the left of the screen link the circle and the other geometrical view as a triangle. This dynamic image is clearly used to connect the geometrical frames (triangle and unit circle) and the analytical frame; nevertheless, it might be very difficult to understand for participants who do not remember their school courses.

As a summary, we retain that micro-level connectivity in this MOOC comprises some connections between concepts and representations, including dynamic representations; but almost no connections with other domains or engineering contexts. Moreover the connections with dynamic representations can remain unclear for students because of a lack of explanations.

"Mathematical tools for physics", an online course "Mathematical tools for physics"4 is an online course addressed to first year students in physics, freely accessible (without inscription). It belongs to a complete first year course, which is always organised in three sections: "learning" (course) "practice" (exercises) and "self-assessment". It comprises 11 chapters; chapter 6 is entitled "Circular trigonometry -Hyperbolic trigonometry". This chapter comprises 8 subsections; here we are only interested in the 4 subsections concerning circular trigonometry.

After recalling the definition of an angle, the first subsection defines sin, cos and tan in the frame of the unit circle. Nevertheless, an animated picture proposes a link with the frame of the right-angled triangle. A subsection about formulas associates a functional frame: specific values of sin, cos and tan and register of the unit circle.

Then two sections are dedicated to the properties of the "direct circular functions" and of the "reciprocal circular functions", and only mention the functional frame. This course makes no link with physics or any real-life context.

There is only one problem, divided in three questions, in the "practice" section. Interestingly, it is a problem of physics: "the Compton effect" (scattering of a photon by an electron). To solve this problem, the students must master contents of physics: the law of conservation of energy and quantity of movement (and the associated formula). The initial model is in the frame of vectors; these vectors are projected on the two axes, and the students have then to use trigonometric formula, and finally to work in the frame of functions. The difference between the wavelengths before and after the scattering is indeed of the form coswhere is the angle characterizing the direction of the photon after the scattering; the students must observe that this function is increasing over [0, ]: a larger angle corresponds to a larger change in the wavelength.

The "assessment" section comprises five parts: 2 on circular trigonometry, 2 on hyperbolic trigonometry and one entitled "composition of vibratory motions". There is here again a connection between trigonometry and physics. Nevertheless its mathematical part remains entirely in the functions' frame.

Finally, concerning the micro-level connectivity of this online course, we retain that it proposes some connections with physics in the exercise and assessment part (but no such connection in the course part). The connections between concepts and between registers are restricted to the case of a single problem.

CONCLUSION

Trigonometry is a domain of mathematics where many different concepts (angles, vectors in geometry; functions) and semiotic registers (triangles, arrows, circle in geometrical register; curves in a graphical register; equations etc.) can be connected.

It is recognised as a difficult domain for students. Nevertheless, within mathematics some exercises are limited to a single register: the study of trigonometrical functions for example does not always require thinking in terms of angles. Using trigonometry in engineering courses, on the opposite, always requires such connections. The students must be able to associate an expression like s(t) = A2sin(t+) both with a function and its graph; and a vector represented by an arrow. Engineering courses have a high degree of micro-level connectivity, for trigonometry. Our analyses of two online courses of trigonometry for engineers in France lead us to observe that they have a reduced level of micro-level connectivity: limited to connections between concepts and semiotic registers for the first one, while the second one on the opposite offers more connections between trigonometry and physics, but reduced connections between concepts and semiotic registers.

Our exploratory work leads to formulate recommendations for mathematics courses for future engineers, concerning trigonometry (and possibly other topics). Our study of engineering courses confirms that developing the ability of these students to make connections is an important aim. Connections between frames, between registers, but also connections with engineering are possible, as evidenced by Nguyen Thi (2013).

Online resources could propose such connections, and moreover could draw on the possibilities offered by dynamic representations and various kinds of software. Contributing to the development of such resources is an important aim for mathematics education research, to address the need for students to make a relevant use of the mathematics they learn at university in and for their engineering courses.

Figure 1 .

 1 Figure 1. Signals in the alternating sinusoidal regime and phase difference. On the left: functional frame; on the right, Fresnel vectors in the vectorial frame.

Figure 3 .

 3 Figure 3. A dynamic representation making connections in the Mooc, case of cos.

http://uel.unisciel.fr

http://www.iutenligne.net/

https://www.fun-mooc.fr/

http://uel.unisciel.fr/physique/outils_nancy/outils_nancy/co/outils_nancy.html

Acknowledgments

The authors warmly thank Alain Jameau and Damien Grenier for their contribution on electrical engineering.