text:
Jörg Kortemeyer
email: kortemey@math.upb.de

Rolf Biehler
email: biehler@math.upb.de

The interface between mathematics and engineeringproblem solving processes for an exercise on oscillating circuits using ordinary differential equations

Keywords: Engineering mathematics, competence, differential equations. Thematic Working Group 14 Thematic Working Group 14

In our project, we investigate the mathematical skills that are required in first-year courses of technical subjects of engineering bachelor courses, i.e., we do not look at the courses on pure math. We analyze four exercises from an exam in electrical engineering, which is compulsory for firstyears. To solve such exercises, students have to combine their knowledge in electrical engineering with their skills in mathematics. We introduce a theoretical approach consisting of three elements: a normative solution called "student-expert-solution, "low-inferent analyses" for qualitative studies with students, and categorizations of written solutions. We describe the newly developed tools and details on one of the exercises and present the results from the analysis of the exercise in reference to the three concepts mentioned above. This provides insight into the interface between mathematics and engineering courses in the first year at university.

Introduction

Engineering students at German universities are taught mathematical subjects as well as engineering subjects, which require some understanding of mathematical topics, at the same time. This leads to several challenges for the students: To begin with, the lectures on Mathematics for Engineering Students (MfES) and the Fundamentals of Engineering (FoE) are very often asynchronous. On the one hand, there is a deductive structure in the lectures on mathematics, which leads to a certain order of presentation of the different topics to assure understanding. On the other hand, there is also a standard way of presentation of the different engineering topics in the FoE-courses and because of that, mathematical topics are often needed earlier in the FoE-courses than they are presented in the MfES-courses. Moreover, there are different mathematical practices in MfES and FoE, for example in the use of vectors or differentials (see e. g. [START_REF] Alpers | Differences between the usage of mathematical concepts in engineering statics and engineering mathematics education[END_REF]. There is a mismatch between the mathematics in MfES-courses, the mathematics at school level, and the "contextual mathematics" required in engineering tasks (see e. g. [START_REF] Redish | Problem solving and the use of math in physics courses[END_REF].

At the beginning of the research, our central question was: how do engineering students solve tasks in basic engineering courses given this situation with two interconnected fields of competencesmathematics and engineering. We are interested in the modelling and assessing of explicit as well as implicit competences required and developed by students in this field. We investigate how students actually solve exercises in FoE-courses and which difficulties occur. Our focus is on four typical exercises of a FoE-exam after the first year at university. In this paper, we present a case study of these issues in the context of a single exercise on ordinary differential equations in the electrical engineering field of oscillating circuits. Our focus here is on the following research questions:

1. What are the expectations from students' solutions to an exercise on ordinary differential equations in a first-year electrical engineering course? 2. What are the characteristics of students' problem-solving processes (e. g. strategies, difficulties) in electrical engineering courses?

For the analysis of students' work we required a normative solution of the exercise, which was developed with engineering experts and which considers both fields of competences. This normative solution is based on relevant theoretical concepts that are presented in the next section.

Theoretical background

In this section, we present the theoretical tools that were used to develop the newly constructed methodology for our investigations. As a first step, the approaches deal with modelling processes using mathematical methods and mathematical problem solving. Both theoretical approaches are relevant, as they combine inner-and outer-mathematical solution parts and describe their connections. Next, we also consider actual solving processes that help us to supplement normative solutions with the steps students use when solving an exercise.

The first approach is the modelling cycle by [START_REF] Blum | How do students and teachers deal with modelling problems?[END_REF], which is used to describe idealized modelling processes of real world problems that can be solved using mathematics. In a broad outline, it divides the modelling processes into two distinct parts, the so-called "rest of the world" and "mathematics". Our second perspective is mathematical problem solving by [START_REF] Polya | How to solve it: A new aspect of mathematical method[END_REF], who intended to give advice to students on how to solve mathematical problems as well as applied problems referring to mathematics. He divides the solving processes into four phases: understanding the problem, devising a plan, carrying out the plan, and looking back.

For the analysis of actual solution processes of students, we use theoretical approaches developed by Redish and his working group, i.e., by [START_REF] Tuminaro | Elements of a cognitive model of physics problem solving: Epistemic games[END_REF] and Redish and Bing (2008), in addition to the normative solution. Their approaches discuss the role of mathematical resources and knowledge in solving processes by pairs of physics students. [START_REF] Tuminaro | Elements of a cognitive model of physics problem solving: Epistemic games[END_REF] distinguish three framings in qualitative solving processes: quantitative sense-making, qualitative sense-making, and rote equation chasing (without understanding the underlying physical situation). [START_REF] Bing | An epistemic framing analysis of upper level physics students' use of mathematics[END_REF] looked at mathematical justification strategies and found four distinct types of justifications: calculation (a correctly done algorithm gives a correct result), physical mapping (the physical behavior is described correctly by mathematical results), invoking authority (the result is consistent with the lecture) and math consistency (the same mathematical approach is used in a similar situation). The theoretical background is presented in more detail in [START_REF] Biehler | Conceptualizing and studying students' processes of solving typical problems in introductory engineering courses requiring mathematical competences[END_REF].

The newly developed methodology and its aims

In order to do research in this interface of two interconnected competence fields, new theoretical approaches had to be developed. This section presents the three main approaches that were developed on the basis of the theoretical approaches mentioned above. As shown in Figure 1, the central theoretical tool is the SES, which builds on expert interviews and the theoretical frameworks of the modelling cycle and mathematical problem solving. The SES is our tool to answer the first research question, i.e., it gives idealized solution processes which we can expect from first-years. It is used to analyze and structure the video-graphed solving processes, which were transcribed using LIAs, and the categorizations of written solutions. Details on the SES, the LIAs and the categorizations are presented in Figure 1: Initially we asked the task designer and the electrical engineering experts to solve the exercises from the perspective of students who understood the contents of the courses in the first year of studies well. Afterwards we interviewed them concerning their solution processes. The aim of the expert interviews was to identify the explicit and implicit competence expectations of instructors in electrical engineering courses. We conducted the interviews using the Precursor-Action-Result-Interpretation (PARI) method by [START_REF] Hall | A procedural guide to cognitive task analysis: The PARI Methodology[END_REF] which is a task-based interview technique. This solution was then subdivided using the language of the modelling cycle and mathematical problem solving, which in combination structured the normative solution of the exercise. The solving processes could be divided into three main phases: mathematization, mathengineering working, and validation and each main phase was subdivided by Polya's four phases. The expert interviews and the structure shown in Figure 1 were the basis of the student-expertsolution (SES), which was used to finally sharpen the theoretical description, and as a basis for the further analysis of students' work. SESs are represented by two columns: the first column provides a normative solution to the exercise in detail and is structured as mentioned above. The second column contains a division of the problem-solving process into phases, as well as remarks given by the experts on expected mistakes, alternative solutions, and learning goals for the different phases.

One of our main interests is to describe real problem-solving processes of students for the four exercises using both qualitative and quantitative methods. Those analyses are based on the SES. We conducted video studies of problem solving processes of three to four pairs of students per exercise, who were asked to solve the exercise and to think aloud during the solution processes. The videos were transcribed with additional remarks on the activities (especially gestures) performed. We analyzed the transcripts using our concept of the low-inference analyses (LIAs) with the aim of finding differences to ideal solutions (the SESs) and to identify students' difficulties. The LIAs consist of four parts: First, there is the connection of the phases in the SES and the phases in the problem-solving processes of the students. Second, the differences between the idealized solution parts in the SES and the actual solution paths of the students are described. The third part consists of commenting and interpreting of the differences, which forms the basis to conceptualize and describe problem-solving strategies, which are expected to be more general than just the process in the actual exercise alone. Finally, we connect the strategies we found with the strategies described by [START_REF] Tuminaro | Elements of a cognitive model of physics problem solving: Epistemic games[END_REF] in general, and [START_REF] Bing | An epistemic framing analysis of upper level physics students' use of mathematics[END_REF] in particular, in order to find typical strategies and challenges at the interface between math and engineering.

In addition to these qualitative studies, we also scanned 92 anonymized "solutions" of students from their written exams. In order to analyze the solutions, the phases in the SES were subdivided into the particular activities that students have to accomplish in order to solve an exercise. For example, the phase of the math-engineering work was subdivided into the forming and the evaluation of the formula. Each student's work was categorized using a partial credit system, i.e., they got two points if the activity was done correctly, or they got one point if it contains right parts (e. g. the solution would be correct if one multiplied it with a power of 10), or they got no points if the solution is completely wrong. The categorization "1" was subdivided into 1a, 1b, 1c etc. to distinguish different forms of mistakes. This provided quantitative results on the frequency of mistakes andby combination of activities in contingency tablesthe connection of successes in different activities. The results are used to confirm, refine, and enhance the results in the first two levels.

The SES for the analyzed exercise on ordinary differential equations

This section presents the first part of one of the exercises of the exam to exemplify our method and present exemplary results. It answers the first research question, which asks what we can expect from students in their first year at university. The exercise deals with oscillating circuits and transients. We present the problem setting as well as a short overview of the solution. This solution is enhanced by the remarks of the experts, which were elicited in the third phase of the PARIinterview. For the first time, this paper presents our total approach for an exercise using methods from MfES. In this exercise, the oscillating circuit contains a resistor R, an inductor L, a capacitor C and an ideal voltage source U0. In summary, the students have to read the sketchtaking into account conditions on the switches S1 and S2 -to be able to form an ordinary differential equation (ODE) and then to solve it. The exercise starts with the circuit diagram shown in Figure 2. It consists of eight subtasks. In this paper, we concentrate on the first five subtasks, which deal with the left part of Figure 2.

Initially, both switches are open, and the inductor and the capacitor are totally discharged. At the moment t=0 the switch S1 is closed, while S2 remains open. In subtask 1 and 2 the students are to give the values of uC(t), the voltage at the capacitor, iC(t), the electric current in the capacitor, and iL(t), the voltage at the inductor, before and after opening S1, i.e., before and after t=0. Solution: All three values are 0 before S1 is closed, because the components of the circuit are initially assumed to be discharged. After closing switch S1, uC(t) and iL(t) are still 0, as a voltage across a capacitor or an electric current through an inductor does not change discontinuously; a fact students learned in the lecture. iC(t)=U0/R directly after the switching of S1 due to Ohm's law and then declines due to the charging of the capacitor. In subtask 3 the students are to form an ordinary differential equation for uC (t) The ODE is to be solved in subtask 4. Solution: The solution can be done using either the separation of variables combined with a variation of constants, or alternatively, the solution can be found by superposition of the solution of the homogenized ODE, one particular solution of the inhomogeneous ODE and the using of the initial value uC(0)=0. The solution is uC(t)=U0(1-e t/(RC) ).

In subtask 5 the students are to sketch the voltage curve of uC(t). Solution: The graph of uC(t) starting at uC(t=0)=0 approaches an asymptote at uC(t)=U0, because e t/(RC) converges to 0 for t  .

Developing the SES for this exercise

At first sights, the solving process can be divided into three phases: mathematization (the given circuit diagram, subtask 1 to 3), math-engineering working (subtask 4), and validation, which is partly done in subtask 5, at least, if the students know the physical behavior of the setting.

As stated in [START_REF] Biehler | Conceptualizing and studying students' processes of solving typical problems in introductory engineering courses requiring mathematical competences[END_REF], there are differences to the modelling cycle in mathematization processes in exercises in basic courses of electrical engineering. Students do not construct a real model from a real situation-as suggested in the modelling cyclebut they need to have strategies to reconstruct the underlying real model -which was implicitly taught in the course (but usually not called model). This includes understanding conventionalized circuit diagrams. In contrast to the exercise on magnetic circuits that was presented in [START_REF] Biehler | Conceptualizing and studying students' processes of solving typical problems in introductory engineering courses requiring mathematical competences[END_REF], the equivalent circuit diagram does not have to be produced by the students; they can use the given diagram directly for their mathematization. In both cases, students use implicit idealizations and are not necessarily aware that they are idealizations. The students have to "read" the diagram and recall and use its physical background in the first two subtasks. The didactic motive of the first subtasks isas stated by the task designerto remind students of applying Ohm's law. Then in subtask 3, the mathematization consists of two independent competences: either recognizing certain components and translating them into their equations, or alternatively the translation of the experiment set-up into mesh and node equations using graph-theoretical arguments in an application of Kirchhoff's laws. The result are three equations: U0=uC(t)+uR(t), CuC'(t)=iC(t), and uR(t)=iC(t)R.

In the next step, there are similarities to the modelling cycle, except that physical quantities are used instead of numbers. The left part can be mathematized by the three equations mentioned and using them, an entering of the "world of mathematics of physical quantities" is possible. Students have to do equation management (see [START_REF] Biehler | Conceptualizing and studying students' processes of solving typical problems in introductory engineering courses requiring mathematical competences[END_REF] to combine the equations in order to get a formula, which also contains one unknown quantity (given by a function in this case), while all the quantities are given in the exercise or have already been calculated. The equation management includes equations with functions as objects and leads to an inhomogeneous ODE of order one. A further characteristic of the equation management is that, unlike in the solving of systems of linear equations, there are no methods to find out whether there are enough or too many equations to get a solvable ODE. Asked for typical mistakes the experts said that the students have some problems in applying mathematical methods to solve the ODE. He also said, that for some students, the application of Kirchhoff's laws is hard, as they do not obtain all the required equations.

Students have learned two different algorithms to solve such ODEs. In the MfES-courses, they solve the homogenous ODEs by separation of variables andusing the solution of the homogenous ODE they subsequently solve the inhomogeneous differential equation. In the FoE-course, they retain a solution by using superposition of the homogenous and the inhomogeneous solution. In the interview, the expert said that most students are able to set up the differential equation, the following solving of the ODE, however, is quite difficult for many students, especially finding the inhomogeneous solution. As the students work with functions instead of numbers or quantities, there is no difference in the use of the solving algorithm for ODEs, which was presented in the MfES-course. So, in this case, the solving process can be divided analogously to the modelling cycle, i.e., there is a "real" world (given by a conventionalized sketch), its translation using three equations and the solving in the mathematical world with quantities.

The solution of the inhomogeneous ODE, uC(t)=U0(1-e t/(RC) ), describes the behavior of the voltage in the capacitor in such a setting. Students know the qualitative behavior of this function from lab courses, which are obligatory in the first year at university. The didactic motive of the task designer was to make students see the connection between their solution of the ODE and the physical mechanisms they know from the lab courses, and use this as a validation strategy. Possible variations of exercises on this topic, which were suggested by the experts, can be either done by using further components (as in the right part of the sketch, which leads to a second order ODE) or by changing the setting of the circuit from a series connection to a parallel connection.

Analyzing the actual solution processes of the students

Selected results of the analyses of the videos in the LIAs. Three pairs of students worked on this exercise in our video studies. Each pair directly found the component equations using the concepts and the language of graph theory for applying Kirchhoff's laws was a bigger problem for two pairs: They were not sure whether one mesh equation would be enough to mathematize the setting, or if they also needed to have node equations, as there was a node between the two parts of the oscillating circuit. However, no pair started the equation management with an incorrect equation and they were also successful in combining them. In reference to solving the ODE, all three pairs used the superposition-method, i.e., they used the method presented in the FoE-course.

In subtask 5 the three pairs acted in different ways, which they described while thinking aloud. One pair found the solution of the ODE by inserting t=0 and realizing that the function converges to U0.

Another pair remembered the behavior they had seen in the lab courses, i.e., they knew that the graph should start at uC(0)=0 (also known from subtask 1 and 2) and would converge to the value of the ideal voltage source, so they applied their physical knowledge to get a mathematical representation of the result, i.e., they used the "mapping meaning to mathematics"-game (see [START_REF] Tuminaro | Elements of a cognitive model of physics problem solving: Epistemic games[END_REF]. The third pair used both arguments, i.e., they drew the solution of the ODE and validated it with the physical behavior, saying it confirms the result of the ODE.

Some results of the analyses of the written exams.

There is a connection between finding the component equations and the applying Kirchhoff's law: 84 of 92 students either did both types of equations right or both wrong. Here, 77 students were able to find the correct ODE; 56 of them solved the homogenous ODE correctly, i.e., for about 73% of the students solving the rightly formed ODE was no problem.

Summary and discussion of results

The solution processes of first-years (Research Question 1). The SES shows that this exercise has more similarities to the modelling cycle than the one presented in [START_REF] Biehler | Conceptualizing and studying students' processes of solving typical problems in introductory engineering courses requiring mathematical competences[END_REF]. Although the exercise uses quantities rather than numbers, it contains three distinct parts, which are analogous to the modelling cycle: mathematization, math-engineering and validation. The mathematization part consists of finding equations for the involved components and the experiment set-up by applying Kirchhoff's laws. In math-engineering these equations are combined in a purely mathematical way, they are solved using inner-mathematical algorithms. The validation part is attended to by a retranslation into the so-called real world by looking at the physical behavior.

The analysis of students' work (Research Question 2). In the mathematization part, most students were able to find both kinds of equations, and in the video-studies the biggest hurdle was, whether they had the right number of equations to get a solvable ODE. The component equations were cited from the FoE-lecture, i.e., the students invoked authority (see [START_REF] Bing | An epistemic framing analysis of upper level physics students' use of mathematics[END_REF]. The students in our studies could either find both the component equation and the equations by application of Kirchhoff's laws or none of them. In contrast to the remarks of the experts, the same holds for the solution processes in the math-engineering part, i.e., more than 90% of the students who solved the homogeneous part correctly also solved the inhomogeneous part. Moreover, some students only solved the inhomogeneous ODE using physical arguments. The question remains is whether students realize that they can also apply another method from the MfES. In the validation part, students used different strategies, involving both mathematical as well as physical arguments, i.e., some students did all steps of the modelling cycle, while others argued using inner-mathematical arguments. They showed different justification strategies, analogous to justifications like calculation and physical mapping, as defined in [START_REF] Bing | An epistemic framing analysis of upper level physics students' use of mathematics[END_REF]

Figure 1 :

 1 Figure 1: Diagram on the connection of the different elements of our analyses

Figure 2 :

 2 Figure 2: Sketch of an oscillating circuit containing the mentioned components

Table 1 : Connection between homogenous and inhomogeneous solutions

 1 Table1shows that all students who were able to solve the homogenous ODE could also solve the inhomogeneous ODE. Eight students only solved the inhomogeneous ODE correctly by finding one particular solution using physical arguments, i.e., they were able to solve at least one part of the task without applying any mathematical methods to solve ODEs, by looking instead at certain values of uC(t) that were known from the problem setting.

		Inhomogeneous solution:	Inhomogeneous	Total
		wrong resp. partly right	solution: right
	Homogenous solution: wrong	28	8	36
	Homogenous solution: right	0	56	56
	Total	28	64	92

Proceedings of CERME10

Acknowledgments

Project was funded by the German Federal Ministry of Education and Research (BMBF) under contract FKZ 01PK11021B.


domain: math shs

text:
Eivind Kaspersen
email: eivind.kaspersen@ntnu.no

Birgit Pepin

Svein Arne Sikko
email: svein.a.sikko@ntnu.no

The association between engineering students' self-reported mathematical identities and average grades in mathematics courses

Keywords: Mathematical identity, Rasch, ANOVA

Arguments have been made that one purpose of learning mathematics successfully is for students to develop mathematical identities. Thus, since students are frequently evaluated with grades in university mathematics courses, a relevant question is how mathematical identities are associated with average grades. This study has measured engineering students' mathematical identities and compared these measures with grades in university mathematics courses, and a Welch's ANOVA conclude that the mean average grade amongst students with high mathematical identities is significant, and about one grade higher than students with low mathematical identities. Moreover, the variance is greater amongst students with low mathematical identities, which indicates a strong association between mathematical identity and average grade only when mathematical identities are high.

Introduction

The transfer of mathematical knowledge from university to the world of work seems problematic. Specifically, evidence has been provided that "attainment" in university mathematics courses is poorly transferred. One example is an experiment that illustrated how 17 students and researchers all failed a mathematics examination they had previously passed, even the students who had recently passed the original exam with an "A" [START_REF] Rystad | Alt glemt på grunn av en ubrukelig eksamensform? En empirisk undersøkelse av Matematikk 2 eksamen ved NTH[END_REF]. Moreover, selected studies illustrate how the mathematics is often hidden in "black-boxes" (e.g. [START_REF] Williams | Black boxes in workplace mathematics[END_REF] in the world of work, and consequently, arguments have been made that the world of work seeks more general mathematical characteristics than what is typically assessed in standard exams (e.g. [START_REF] Hoyles | Mathematical skills in the workplace[END_REF]. On a general note of education, [START_REF] Wenger | Communities of practice: learning, meaning, and identity[END_REF] argued that learning is about developing identities in communities of practice. In general, over the last decades, there has been an increased attention towards the construct of identity, and mathematical identity in particular (e.g. [START_REF] Axelsson | Mathematical identity in women: The concept, its components and relationship to educative ability, achievement and family support[END_REF][START_REF] Black | Developing a 'leading identity': The relationship between students' mathematical identities and their career and higher education aspirations[END_REF][START_REF] Wenger | Communities of practice: learning, meaning, and identity[END_REF]. Thus, if the world of work seeks general characteristics of working mathematically, a relevant question is how mathematical attainment in university mathematics courses, as represented by average grades, is associated with mathematical identity. This paper addresses this question.

This study has examined the association between self-reported mathematical identities and average grades in university mathematics courses. From a Rasch calibrated instrument, previously validated in [START_REF] Kaspersen | Using the Rasch model to measure the extent to which students work conceptually with mathematics[END_REF], the students were categorised as having a "low," "medium," or "high" mathematical identity, and the paper will illustrate how the mean average grade of students with high mathematical identities was significant and about one grade higher than students with low mathematical identities. Moreover, the variance amongst students with low mathematical identities was higher than amongst students with high mathematical identities, although the difference was not significant (p=0.06). The paper concludes that high mathematical identities are associated with high average grades in university mathematics courses. However, the same conclusion is not true amongst students with lower mathematical identities.

Theoretical framework

The construct of identity suffers from a lack of consensus on general philosophical issues [START_REF] Cote | Identity, formation, agency, and culture: A social psychological synthesis[END_REF]. Specifically, identity is defined differently across different studies and paradigms, such as "a certain kind of person" (Gee, 2000, p. 99), "those narratives about individuals that are reifying, endorsable and significant" (Sfard & Prusak, 2005, p. 44), and "self-perceived mathematical knowledge, ability, motivation and anxiety" (Axelsson, 2009, p. 387).

This lack of consensus is typical in pre-paradigmatic fields [START_REF] Kuhn | The structure of scientific revolutions[END_REF]. Unlike firm paradigmatic fields where well-established theories tend to guide the analyses, research in pre-paradigmatic areas has a more dialectical relationship between data and theory [START_REF] Kuhn | The essential tension: Selected studies in scientific tradition and change[END_REF]. This description is a fair representation of how the theoretical perception in this study was chosen. That is, no ready-made theory was chosen on pure faith. Rather, a definition of identity was established that was consistent with measurement (i.e., consistent to conclude some persons to have stronger mathematical identities than others), yet, influences by fragments of multiple existing theories. The following theoretical perspective and a wider discussion on practical significance has been provided in more detail in [START_REF] Kaspersen | Measuring students' mathematical identities[END_REF].

On another note, we do not regard theories as mirrors of some true reality. Thus, we do not believe that some theories are true, and that others are false. When we propose the following theoretical perspective, therefore, we are not refusing other perspectives, for instance, a narrative view on identity. Rather, we claim that if we choose the following perspective, then the practical consequence is that mathematical identity can be measured.

The perspective of mathematical identity relies on two assumptions. First, we assume that identity (originated from the Latin idem) is about sameness and distinction. As such, the position in this study juxtaposes perspectives that consider persons to have their unique identity. That is, persons are indeed unique. However, they can be defined as identical with respect to a set of characteristics, just like mathematical objects can be identified by certain characteristics while remaining unique on others. Moreover, since there exists an infinite number of characteristics, identities have a varying degree of complexity. That is, mathematical identity can be binary, linear, or multidimensional, and we argue that there is no ontological limit to the number of dimensions. Consequently, there exists no set of criteria that dictates when researchers have arrived at the final dimension. Hence, the choice of complexity can be nothing but pragmatic, and in this study, we have chosen a one-dimensional perspective on mathematical identity, whereby persons are distinguished on a continuum from having a low to having a high mathematical identity within the engineering education context. Furthermore, if we accept that persons participate and contribute in multiple activities, a consequence is that each person has multiple identities, a position that is shared by many authors, for example Black and colleagues (2010) who, inspired by [START_REF] Leont'ev | Problems of the development of mind[END_REF], presented the idea of "leading identity." Since there is no limit to how many ways persons can be distinguished, we argue that there exists no limit to the number of identities, although the number of identities that individuals are consciously aware of is likely to be finite. Moreover, in this study, we take no definite position on the relationship between identities. Thus, when we later will conclude that selected persons have (more or less) the same mathematical identity, we do not make claims about how these are related to the multiplicity of identities-for instance, whether they are central/leading or peripheral identities.

Second, we assume that identity is relational by nature. That is, persons can be concluded to be identical relative to a set of characteristics, only if the structure of these characteristics is personindependent. Thus, in quantitative studies, we reject the assumption that persons with the same score on some test or questionnaire are identical unless statistical evidence is provided that the items stay invariant across relevant subgroups. Hence, there likely exist contexts that are so different that comparisons of identities across these contexts do not make sense. Consequently, we argue that the methods that are applied to capture identities should also capture the level of invariance.

In conclusion, we define mathematical identity to be where persons position themselves relative to the social structure of being mathematical within the activity in which they participate and contribute. From a one-dimensional perspective, "the social structure of being mathematical" is a personindependent set of characteristics and their internal structure (i.e., their relative distance) that distinguishes persons on a continuum from having a "low" to having a "high" mathematical identity. "Where persons position themselves" is persons' positions relative to the social structure.

Method

To test the relationship between engineering students' self-reported mathematical identities and average grade in mathematics courses, a convenience sample consisting of Norwegian engineering students (N=361) was selected. 47 students attended an "Introductory course in mathematics," 71 students attended a "Calculus 2" course, 113 attended a "Calculus 3" course, 11 a "Cryptography" course, and 119 were students from a variety of courses in their normalised final year of education. The participants responded to a Rasch-calibrated instrument [START_REF] Rasch | Probabilistic models for some intelligence and attainment tests[END_REF], previously validated in [START_REF] Kaspersen | Using the Rasch model to measure the extent to which students work conceptually with mathematics[END_REF], that measures persons on a continuum from having a low to having a high mathematical identity relative to 20 uni-dimensional characteristics. The items in the instrument were collected from three sources: the literature, other related instruments, and from persons contributing in mathematical activities (e.g., students and lecturers). The validation of the instrument will not be discussed in depth, as details can be found in [START_REF] Kaspersen | Using the Rasch model to measure the extent to which students work conceptually with mathematics[END_REF]. The person reliability, analogous to Cronbach's alpha, was 0.87. Moreover, from principal component analysis of residuals, the instrument was found to be sufficiently uni-dimensional for the purpose of measurement with a 1.99 unexplained variance (in Eigenvalue units) in a second contrast. Furthermore, the mean of the squared standardised residuals (outfit mnsq) and the information-weighted version (infit mnsq) (see e.g., Bond & Fox, 2003, p. 238 for a detailed description) indicated a sufficient data-model fit, with Item 6 and Item 15 as the most underfitting items (Table 1).

Rasch measurement requires additivity, uni-dimensionality, and invariance, and the probability of an observation is a function of the difference between a person's measure and a characteristic's measure (e.g. [START_REF] Wright | Best test design[END_REF]. Thus, most response strings follow a Guttman-like structure with most deviations around the measure of the person. Consequently, persons with approximately the same measures, except those with large misfit, have, not only the same measures but also approximately the same combination of self-reported characteristics (and thus concluded to be identical with respect to these characteristics).

After the validation of the instrument, the respondents were categorised as having either low (measures lower than -1), medium (measures between -1 and 1), or high (measures above +1) mathematical identities (all measures are in logit units). The distance from the "low"/"medium" to the "medium"/"high" thresholds was about the same distance as one response category. Consequently, persons with "high" mathematical identities were expected to respond at least one category higher on each characteristic than persons with "low" mathematical identities. Subsequently, a one-way ANOVA was conducted to compare the association between mathematical identity and the self-reported average grade in mathematics courses at the University (from grade F=1 to grade A=15). However, since the [START_REF] Levene | Robust tests for equality of variances[END_REF] test barely accepted the null hypothesis of homogeneity of variances (p=0.06), and the sample sizes across categories were unequal, the Welch's ANOVA was chosen since it is more robust to unequal sample size and variance.

Moreover, the assumption of normality was violated, and the grades were ordinal as opposed to interval measures. Since Welch's ANOVA assumes normal and interval measures, 10,000 simulations were made in R (R Core Team, 2015) to assess how these violations affected the robustness of the analysis. To ease this part of the analysis, we considered a transformed data set which had no difference in the mean across groups but was otherwise identical to ours-the assumptions of Welch's ANOVA were violated equally in the empirical study and the simulated studies. This transformation eased the interpretation since we could compare the results with the statistical ideal situation (perfectly normal interval data, equal sample size and variance). If our data set was as good as the ideal situation, we would expect the Welch's ANOVA to show a significant difference in about 5% of the simulations.

Specifically, from the empirical data frame, M, a new data frame, M', was made whereby each grade in the medium and high groups was shifted so that the mean of all three categories in M' were equal (i.e., keeping the sample sizes and distributions, but aligning the means). From M', 10,000 data frames, M1 -M10,000, were randomly sampled whereby the sample sizes in the three groups were equal to the original M. Subsequently, Welch's ANOVA was conducted on each simulated data frame. Since the result showed that 5.2% of the p-values in the simulations were less than .050, it was concluded to ignore violations of Welch's ANOVA's assumptions since they had only a trivial negative effect on the robustness.

Result Mathematical identities

Due to the Guttman-like response strings, a rough interpretation of Table 1 is that most students with low mathematical identities (measures lower than -1) agreed with characteristics much lower than -1, and disagreed with those much higher than -1. That is, students with low mathematical identities often keep trying when they get stuck, but they rarely study proofs until they make sense (to them), they rarely like to discuss mathematics, they rarely derive formulas, etc. Likewise, students with medium mathematical identities (measures between -1 and 1) frequently keep trying, connect new and existing knowledge, and can explain why their solutions are correct, but rarely take the initiative to learn more than expected, rarely take the time to find better methods, etc. Students with high mathematical identities (measures above +1) agree with most characteristics in the instrument. A more thorough discussion is discussed in [START_REF] Kaspersen | Measuring students' mathematical identities[END_REF]. Moreover, it is evident from Table 1 how the identities in this study were situated amongst the engineering student context. That is, persons with measures, say, around 0.5 in other contexts would be identical to engineering students with the same measures, only if the same set of characteristics were proven to be invariant (i.e., calibrated to have the same structure) in both contexts.

The relationship between self-reported mathematical identities and average grade

Figure 1 illustrates the relationship between self-reported mathematical identity and average grade in university mathematics courses. The Welch's ANOVA showed that the association between mathematical identity and self-reported average grade was significant, F(2, 110.79)=31.966, p=0.000. Moreover, the mean of the self-reported average grade amongst students with high mathematical identities was about one grade higher than those with low mathematical identities. The Games-Howell test showed that the difference was significant between all groups with low-medium as the least significant (p=0.001). The unequal variance is also illustrated in Figure 1. Specifically, the variances decreased with the increase of mathematical identity. That is, high mathematical identities are associated with high selfreported average grade. However, there seems to be no limit to how low mathematical identities students can have and still get high grades.

Conclusion and discussion

In this paper, we have argued that the average grade in university mathematics courses amongst students with high mathematical identities is about one grade higher than amongst students with low mathematical identities, and the difference is significant. Moreover, we have shown that the variance of self-reported average grades amongst students with low mathematical identities is higher than amongst students with high mathematical identities. That is, students with high identities get, for the most, high grades. However, the grades of students with lower identities are more uncertain.

We have in this study examined the association, and not the causal relationship, between self-reported mathematical identities and average grades, and therefore we argue that the significance of the result is that it points the direction for future research. Specifically, we suggest future research to address the following:

First, replicates of this study should seek more precise measures. That is, the precisions of the mathematical identity measures can be improved by including more response categories (as long as they are sufficiently validated) and more items, particularly near the "gaps" (e.g., between 0.5 and 1.2 logits). Moreover, the precision of the average grade would most likely be improved if selfreported average grades were substituted with actual average grades.

Second, future research should seek a more causal relationship between identities and grades. Specifically, this study does not conclude that an increase in mathematical identity infers an increase in attainment.

Third, future research could study the significance of mathematical identity versus the significance of attainment. For instance, students can be categorised as having "low identities and low grades," "low identities and high grades," or "high identities and high grades," and subsequently studied with respect to other variables, for example, in the transition from university to the world of work.

Fourth, we argue that future research can transfer the design of this study to other samples and forms of testing students' attainment. For example, relationships between mathematical identity and measures on international standardised tests, such as PISA and TIMSS, can be tested. Accordingly, we argue that future research can nuance the debate on the significance of these tests. If some districts/countries are "teaching to the test," then one might hypothesise that a relatively great proportion of students in these districts/countries are in the "top left corner"-that is, students with low mathematical identities, yet, high measures of attainment.

Figure 1 :

 1 Figure 1: The relationship between self-reported mathematical identity and average grade in university mathematics courses

Table 1 : Characteristics of mathematical identities amongst Norwegian Engineering students

 1

	Item statistics: Measure order
	Measure INFIT MNSQ OTFIT MNSQ	Item
	1.91	.81	.83	1. Takes time to find better methods
	1.58	1.08	.99	2. Takes the initiative to learn more
	1.24	.91	.86	3. Thinks of times when methods don't work
	.55	1.22	1.20	4. Struggles with putting problems aside
	.51	1.05	1.07	5. Derives formulas
	.45	1.36	1.37	6. (x) Likes to be told exactly what to do
	.41	.96	.95	7. New ideas lead to trains of thoughts
	.32	1.05	1.05	8. Likes to discuss math
	.20	1.07	1.07	9. Makes his/her own problems
	.05	.99	.99	10. Studies proofs until they make sense
	.04	.86	.88	11. Moves back and forth between strategies
	-.10	.87	.86	12. Tries to understand formulas/algorithms
	-.20	.72	.74	13. Considers different possible solutions
	-.26	1.03	1.05	14. Pauses and reflects
	-.38	1.32	1.31	15. Finding out why methods do not work
	-.47	.86	.86	16. Wants to learn more things
	-.77	1.20	1.20	17. Visualises problems
	-1.19	.71	.76	18. Can explain why solutions are correct
	-1.83	.83	.88	19. Connects new and existing knowledge
	-2.05	1.02	1.06	20. Keeps trying
	Note. Item 6 was negatively coded
	Items in their entirety in https://www.researchgate.net/publication/309740755_math_identity_questionnaire

Proceedings of CERME10


text:
Barbara Jaworski
email: b.jaworski@lboro.ac.uk

Despina Potari
email: dpotari@math.uoa.gr

Georgia Petropoulou

Theorising university mathematics teaching: The teaching triad within an activity theory perspective

Keywords: University mathematics learning and teaching; teaching triad, activity theory, didactical triangle and tetrahedron

We draw on our recent research to inspect again some of the theoretical perspectives we have been using to analyse data and to characterise teaching-learning in university settings. We focus particularly within a sociocultural perspective on Activity Theory (AT) and the construct 'the Teaching Triad (TT)', seeking to embed the TT within an AT perspective. To achieve this, we relate the Teaching Triad with aspects of the sociocultural setting both in and beyond direct interactions in face to face teaching. While this is mainly a theoretical paper, an example is taken from observations of teaching in university lectures in a Greek university to show how these theoretical perspectives have provided insights to the institutional and cultural complexities involved.

Introduction to University Mathematics Teaching (UMT)

By University Mathematics Teaching (UMT) we refer to any or all the teaching of mathematics which takes place at university level. In our own corpus of work we are particularly interested in face to face teaching in lectures and tutorials in which teachers design their teaching for the benefit of students who attend their sessions. We are interested in uncovering relationships between teaching and learning within the full sociocultural context of university life. This includes the institutional setting as well as the cultures from which teachers and students make sense of the interactions in which they engage. In particular, we seek to know more about "what teachers do and think daily, in class and out, as they perform their teaching work" (Speer, Smith & Horvath, 2010, p. 99). Our research addresses:

What is it that mathematics teachers do and think as they perform their teaching work in a university setting, and how does this relate to the mathematical meaning making of their students? [START_REF] Jaworski | Critical theorizing from studies of undergraduate mathematics teaching for students' meaning making in mathematics[END_REF] This question takes us into the didactical thinking of teachers who consider how best to enable students to think mathematically and develop understandings of mathematical topics; it includes teachers' pedagogic thinking in the ways in which they interact with students and use resources to promote students' engagement with mathematics; it includes also the ways in which teachers work within university affordances and constraints, the norms and expectations of university culture and their own educational histories, their views of mathematics and of what it means for students to learn mathematics and so on.

In our work to date we have used a number of theoretical perspectives to analyse data from teacherstudent interactions in university mathematics teaching. Largely we have taken a broad sociocultural perspective in which we aim to address both micro and macro aspects of teaching. In

Introduction to the Teaching Triad (TT)

The Teaching Triad (TT) is a theoretical construct developed from earlier research into the teaching of mathematics at secondary school level. It offers a way of characterizing mathematics teaching by acting as a tool for analyzing teaching data from classroom situations; it has also been used by teachers as a developmental tool [START_REF] Jaworski | Investigating mathematics teaching: A constructivist enquiry[END_REF][START_REF] Potari | Tackling Complexity in Mathematics Teaching Development: Using the Teaching Triad as a Tool for Reflection and Analysis[END_REF]. More recently it has been used to characterize mathematics teaching at university level and as an analytical tool at this level [START_REF] Jaworski | Sensitivity and challenge in University Mathematics Teaching[END_REF][START_REF] Jaworski | Critical theorizing from studies of undergraduate mathematics teaching for students' meaning making in mathematics[END_REF].

Although NOT a triangle, the TT comprises three inter-related elements or domains of teaching: Management of Learning (ML); Sensitivity to Students (SS) and Mathematical Challenge (MC). These have been interpreted in terms of the interactions that take place within a classroom setting and, as such, focus on the micro aspects of teaching, without overt focus on the broader situational and cultural focuses, the macro. Briefly, Management of Learning describes the teacher's role in the constitution of the classroom learning environment by the teacher and students. It includes classroom groupings; planning of tasks and activity; use of textbooks and other resources, setting of norms and so on. Sensitivity to Students describes the teacher's knowledge of students and attention to their needs, affective, cognitive and social; the ways in which the teacher interacts with individuals and guides.

group interactions. Mathematical Challenge describes the challenges offered to students to engender mathematical thinking and activity; this includes tasks set, questions posed and emphasis on metacognitive processing. These domains are closely interlinked and interdependent [START_REF] Jaworski | Investigating mathematics teaching: A constructivist enquiry[END_REF]. Research has shown that a good balance between SS and MC is needed for effective teaching: a lot of SS, but little MC can lead to good teacher-student relations but low mathematical progress; a lot of MC but little SS can result in students feeling stressed or unable to succeed. When challenge and sensitivity are well balanced, the result is "harmony"students are suitably challenged and stimulated while supported to achieve [START_REF] Potari | Tackling Complexity in Mathematics Teaching Development: Using the Teaching Triad as a Tool for Reflection and Analysis[END_REF].

The TT is associated with another familiar construct, the Didactic Triangle (DT) which links Teacher (Τ), Students (S) and Mathematics (M) and draws attention to relationships TeacherStudent; TeacherMathematics; StudentMathematics and links between these pairs (e.g., [START_REF] Rezat | From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation[END_REF]. The TT expands the "Teacher" node of the DT, illuminating the links TeacherStudent and TeacherMathematics through the constructs SS and MC respectively while extending the DT to the wider classroom context through the construct ML. This wider context includes the resources a teacher uses in mediating between students and mathematics as expressed in the idea of a Didactic Tetrahedron in which there are 4 planes: the original DT linking TSM and the planes linking TSR, TRM and SRM (R=Resources/artifact; see Figure 2). . Embedding the TT into the sociocultural perspective

In this paper we re-examine the TT as a construct used within a sociocultural perspective and particularly its relationships to and within an Activity Theory analysis of teaching data. As a backdrop to AT we take Vygotskian perspectives involving particularly mediation, tool use, scientific concepts and the zone of proximal development (ZPD). Briefly, we see teaching as a process of mediation between teacher, students and mathematics (relationships are expressed simply in the DT and expanded in the TT). Teaching can be seen as mediating between student and mathematics: this is not a simplistic relationship but one with several dimensions which the TT serves to accentuate. The resources that a teacher brings to teaching (examples include mathematical symbolism, dynamic software, display media) are tools used in the teaching process; tools to facilitate learning (indicated by the extension of the DT to the DTetra). Scientific concepts are those distinguished by Vygotsky as involving theoretical learning in contrast with spontaneous concepts which arise from empirical learning (examples are mathematical concepts which need to be introduced by someonethey are not naturally occurring in everyday interactions). Daniels (2008, p. 314) cites Hedegaard (1998, p. 120) to suggest that "the teacher guides the learning activity both from the perspective of general concepts and from the perspective of engaging students in 'situated' problems that are meaningful in relation to their developmental stage and life situations". These words capture importantly the basic ideas of ML and SS in the TT of which we say more below. Daniels emphasizes the important relationship between the idea of scientific concepts and the ZPD as involving a teacher in bringing general theoretical knowledge to her interactions with students, while engaging students in concrete tasks from which scientific concepts can be abstracted. This suggests important relationships between a teacher's didactics and pedagogy expressed simply, the former involving the transformation of mathematical concepts into tasks and activity for students and the latter involving the organization of the social setting to enable students' engagement with mathematics (together these form the basis of ML in the TT). Within the ZPD, student engagement with a teacher's theoretical input can achieve better learning outcomes than would be achievable by a student's engagement with empirical tasks alone.

The concepts expressed extremely briefly above fit with the sociocultural perspective of A. N. Leont'ev, who makes the following point "in a society, humans do not simply find external conditions to which they must adapt their activity. Rather these social conditions bear with them the motives and goals of their activity, its means and modes." (A. N. Leont'ev, 1979, pp. 47-48). Here we focus particularly on Leont'ev's three layers of human action which constitute Activity. The outer, or top layer is labelled 'Activity' which according to [START_REF] Leont'ev | The problem of activity in psychology[END_REF] is always motivated, although the motive might not be explicit. Within Activity, the second layer consists of the 'actions' of humans engaging in Activity. Actions are goal-directed, such that the goals are always explicit or conscious. In the third layer, actions include 'operations', which depend on the 'conditions' within which actions take place. In earlier research we have used Leont'ev's layers to explain issues and tensions which have emerged from analyses between teachers' and students' perspectives on mathematics teaching and learning [START_REF] Jaworski | Bridging the macro-and micro-divide: using an activity theory model to capture sociocultural complexity in mathematics teaching and its development[END_REF][START_REF] Jaworski | An activity theory analysis of teaching goals versus student epistemological positions[END_REF].

If we think of the Activity of a university teacher teaching mathematics, within a university setting, subject to all the sociocultural forces within which the Activity takes place, we might think of the motive of this activity to be the mathematical learning of students participating within the complexities of this setting. Actions here are the teaching actions which take place as the teacher engages in the teaching process in relation to the mathematics which is the focus of teaching. Such actions are goal directed and relate to ways in which the teacher thinks about her teaching and acts in relation to her students. Thus, teacher intentions and theoretical perspectives form goals, and didactical and pedagogic processes form actions in this activity setting. The operations within this role, with which the teacher engages, are closely related to the practicalities of the role; for example, setting exams, creating VLE pages, assessing students' work. These operations must take place within the affordances and constraints of the university system which impose conditions on the operations.

Another model which is used very commonly to represent an Activity System, is the Expanded Mediational Triangle (EMT) from Engeström (e.g. 1998). This developed originally from Vygotsky's (simple) mediational triangle (the top part of the EMT) linking a subject with the object of her activity via the resources (tools, artefacts) employed in mediation. Engeström recognized the important mediational functions of other aspects of the sociocultural setting, such as 'rules', 'community' and 'division of labour' which expanded the roles of tools/artefacts, and which he added overtly in the EMT (see Figure 3). The 'rules' include university procedures and constraints, community includes both student and academic communities, and division of labour recognizes differences between student and teacher roles within the academic setting.

Example: Teaching in a lecture course in calculus with first year undergraduates

This example comes from the study of university lectures in first year calculus teaching in a fouryear mathematics programme in the mathematics department of a Greek University (see [START_REF] Petropoulou | How do research mathematicians teach Calculus[END_REF]. The lecturer is an established mathematician, with extensive teaching experience, who is very popular among the students in this department. The teaching takes place in an amphitheatre with more than 200 students. The course is compulsory and its focus is theoretical with an emphasis on proofs; in this example, the mathematical focus is the convergence of series. The approach is new for the students who have previously experienced calculus in high school as a set of methods and computations. Students' expressed opinions and the very low success rate in the course examination suggest that this course is experienced as one of the most difficult during the four year programme. A large number of students take more than four years to complete their studies (the average time is 6.5 years) and some of the students have part time jobs in order to support their studies financially.

The lecturer is aware of these sociocultural issues and takes them into account in his teaching, as our analyses show. For example, he says, "I do know that students get lost in their first year and that most find mathematics too difficult…if you don't pay attention as a teacher, the average duration of their studies could easily become 7 or 8 years". (In analysis we see here SS in cognitive, affective and social dimensions as we explain below).

The lecturer's teaching appears rather traditional as he is seen mostly standing at the front of the room, writing at the board, "telling" or "explaining" the mathematics with rare interaction with students. Nevertheless, we see many elements of sensitivity, taking into account students' learning needs. By scrutinizing his teaching actions and goals, we see that his main teaching goal is to make the content relevant to students, with associated actions providing comprehensive explanations, highlighting subtle points that cause students' difficulties, linking informal and formal representations, making connections with students' prior school experiences, emphasizing the importance of the specific content in mathematics, in the course exams and in other courses.

At the beginning of this episode the lecturer reminds students that they know they can add a finite number of terms of a sequence i.e. they know that the sum of a finite number of terms always exists. He points out that a central question is whether the sum of an infinite number of terms exists.

He establishes the importance of this question by saying that this is exactly what we mean when we say that if it exists then the series converges. He also highlights that the "big difference here" is that the number of terms is infinite. By relating the convergence of the series to the existence of a sum, he attempts to help students to make sense of the meaning of convergence (which may still be difficult however). This can also offer a mathematical challenge that is possibly not appropriate for the students to respond to at this stage. It acts more as a situated problem for introducing the relevant theorems about the convergence of a series.

He sympathizes with the students, through a personal story about a teacher he had at school for whom the convergence was of great importance. We might say that this story supports their comfort zone, offering affective sensitivity. He then formalizes a basic proposition related to the necessary condition for a series to converge: "If a series 1 k k a    converges, then the sequence ak → 0".

Here we see SS-cognitive in alternative expressions of the meaning of convergence, helping students to make sense of the concept of series convergence. We categorise the personal story as SS-affective/social, encouraging students in the lecture to have rapport with the lecturer and feel empathy with his approach to teaching them. These are pedagogic strategies which enable the lecturer to proceed to a more formal didactic stage in his explanation in which he acknowledges a problem they might find in a text book on the topic. "The books write 'consider the sequence Sn-1'. But what is the sequence Sn-1 if n=1? Is it S0? S0 is not defined! Ok?". His solution to this problem is to introduce a second sequence tn: "Now, I define a second sequence tn as follows -I am going to write down for you the terms of this sequence. First, I set something… let's say t1, to be equal to 0. Then I set the 2nd term of tn to be equal to S1, the 3rd to S2… Ok? … the 4th to S3 etc. Namely I set t1 to be 0 -you can set everything you want. So let tn be Sn-1, if n ≥2." He concludes this proof and then he offers a second proof based on the formal ε-δ definition. He compares the two ways by characterising the first way of proving 'the quick way' and the second ε-δ proof 'the slow way'. He provides all the details in both of these proofs highlighting the problem solving strategies that are usually used in proofs about series such as for example the use of partial sums.

These steps challenge students to engage with the mathematics of series in a more formal way. Perhaps this MC is scaffolded by the sensitivity observed in the earlier considerations. We see again the lecturer's drawing of students into his confidence in encouraging them to be critical of the text book, and in involving them in his reasoning for introducing the new sequence. We might see these careful steps on the part of the lecturer as his sensitivity in "paying attention as a teacher to his students' potential difficulties".

The lecturer subsequently takes the opportunity to remind students of the harmonic series

1 1 k k  
 the sequence of which tends to 0 but the series itself does not converge, and he uses this to justify that the inverse of the above proposition does not hold. He draws students' attention to the usefulness of this example in the forthcoming exams.

Further actions include providing resources and materials to students for their individual studying especially for those students who cannot attend the lectures, the structuring of the content, the teaching tools (board, supportive resources) and the traditional communication norms. In the analysis, Mathematical Challenge (MC) is often difficult to distinguish, appearing to be integrated into the SS. It is usually addressed through problem solving heuristics that are presented by him in explaining general mathematical strategies in specific cases of problems and theorems (e.g., the use of partial sums for proving the convergence of series) and by emphasizing metacognitive processes (e.g., comparing different solution strategies).

By referring to the EMT, we identify some links to the TT. SS is related to the lecturer's attention to the students' community (e.g., offering supportive resources for the students who do not attend the lectures, the delay for completing their studies). ML is related to the lecturer's attention to the university community (e.g., the tools that the lecturer uses and develops, the institutional rules such as examinations, large cohorts of students,) On the other hand MC is related to the community of mathematicians and to the mathematical practices that the lecturer brings into the classroom.

Discussion

In this example, the Activity is the sociocultural setting of teaching and learning. Seeing the teacher as subject (in the EMT) with the object of enabling students to make sense of the mathematical topic, mediators are the various artefacts/resources (such as the lecturer's board writing; his provision of on-line resources) as well as the cultures of students or teachers (student community. academic community), differing roles of students and teachers (division of labour), and the expectations of university lectures/tutorials and the four-year programme (rules). In Leont'ev's terms the Activity is the whole, the lecturing, with the motive of enabling the students to learn basic concepts and theorems of calculus by taking into account their learning needs. We see actions and goals particularly in the activity of the lecturer: what he does to achieve the main goal of making the content relevant to the students, such as explaining mathematics at the board to ensure that students are provided with clear accounts of mathematical concepts with which they can work further, providing on-line resources to help students who must work to support their studies.

The Teaching Triad cuts across the Activity Theory frameworks to interrogate the activity of teaching. It captures the teacher's actions as related to mathematics and to the students (MC and SS). Through ML, we see the teacher's use of artefacts: tasks and resources, pedagogic strategies to include and engage students, orchestration of the environment to facilitate learning. MC can be seen in the ways the teacher presents or provides access to mathematics, linking with what the students know and with what they are expected to do in the course exams. SS links the affective, cognitive and social elements of student engagement, rationalizing conventions and norms within the constraints and affordances of the institution. The triad presents a framework in which we see all the aspects of Activity through its three dimensions.

Elaborating further the elements of the TT discussed above, we see a close link between SS and ML in the teaching of this lecturer. SS has a strong social dimension apparent inside and outside the amphitheatre. For example, we see his concerns for providing clear explanations without interaction with the students as the institutional context and the affective constraints do not allow it. He says, "In an audience of 200 students, if you discuss with 2-3 of them, these probably will be the strongest students and the others will feel bad. … And finally nothing will remain on the board". He also takes into account students who cannot attend the lectures for various reasons (e.g., participating in social associations; socializing after the hard entry examinations; or having to get a job for financial reasons) by providing supportive online resources and materials. The lecturer teaches within the sociocultural setting described above. He engages with mathematics and with students: the fundamental relationships expressed in the DT. He uses a range of resources with which to engage students as expressed in the DTetra. The TT enables us to inspect these relationships in more depth, addressing the ways in which the lecturer engages the students and provides for their needs. We come to see that despite an approach that seems transmissive, he is nevertheless sensitive in social, affective and cognitive ways to what students need in order to make sense of the mathematics he offers. These needs relate strongly to elements of the sociocultural context in which the activity takes place including the number of students, the financial provision for their studies, and their struggles with mathematical formalism. We see lecturer's goals and actions, through which he demonstrates challenge and sensitivity, to relate fundamentally to his recognition of these contextual demands. While the AT frames (EMT and Leont'ev's layers) characterize teaching activity in its relation to context, the TT zooms in on the goals and actions to specify qualities of sensitivity and challenge and their management within the given context.

  Figure 1. The Teaching Triad (Jaworski, 1994).

Figure

  Figure 2 . The Didactical Tetrahedron (DTetra) (Rezat & Straesser, 2012)

Proceedings of CERME10


text:
Ninni Marie Hogstad
email: m.hogstad@uia.no

Ghislain Maurice

Norbert Isabwe

Using the theory of instrumental genesis to study students' work with a digital tool for applying integrals in a kinematic simulation

Keywords: Digital tool, engineering students, instrumental genesis, epistemic value, pragmatic value

Digital tools are increasingly becoming part of mathematics in Higher Education, some of which are used pedagogically. An example is Sim2Bil, a digital tool that offers mathematical tasks about a simulation of cars. Students can solve the tasks using integrals. Applying the theory of instrumental genesis, in which techniques are analyzed in light of epistemic and pragmatic value, we studied the tool when used by groups of engineering students. We observed the students applying techniques such as instrumented and pen & pencil for solving the tasks. The techniques required the combination of understanding integrals kinematically (as distance travelled), graphically (as area under a graph) and as a calculation with symbols (finding the anti-derivative). In fact, the mathematical tasks provided opportunity for students to address at least one task, and we had students solving the most demanding ones.

Introduction

Solving mathematical tasks involves using different tools. These tools might be compasses and rulers, but also language, symbols, gestures, and digital tools. From a didactical point of view, it is important to study how such tools function, how students work with these tools and how the tools can be incorporated into educational practices. Within education, tools not only serve to do mathematics, such as carrying out calculations quickly, but they can also be pedagogical instruments for learning mathematics [START_REF] Artigue | Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work[END_REF]. In our study, we focus on digital tools. Examples of digital tools for doing mathematics are Mathematica (www.wolfram.com/mathematica/) and MatLab (www.mathworks.com), which are numerical computing environments used in academia and industries. Examples of digital, pedagogical tools for mathematics are Geogebra (www.geogebra.org) and MIT Mathlets (http://mathlets.org).

Our research is positioned within the education for engineers. The integral is one of the mathematical concepts to be learned. It can be perceived in many different ways, for example as an object (as a function or as an area) or as a process (calculating the anti-derivative or taking the limit of a Riemann sum). Researchers have found that students have difficulties conceptualizing the integral (e.g. [START_REF] Jones | Understanding the integral: Students' symbolic forms[END_REF][START_REF] Swidan | Learning the indefinite integral in a dynamic and interactive technological environment[END_REF]. [START_REF] Derouet | La fonction de densité au carrefour entre probabilités et analyse en terminale S. Etude de la conception et de la mise en aeuvre de tâches d'introduction articulant lois à densité et calcul intégral[END_REF] has focused on the relation between integral, area and probability, in which the integrand is the probability density. We take an approach where we study the relation between integral, area and distance in which the integrand is the velocity. We are representing the integral as a mathematical model for making objects move under certain conditions.

There are several digital tools used for the learning of integrals. On the internet one can find applets, in which the integral is demonstrated with the Riemann Sum with an interactive slider for showing the limiting process. [START_REF] Berry | Promoting students' graphical understanding of the calculus[END_REF] used a different tool, namely one that can record displacement of an object and then graphically display both a displacement-time graph and a velocity-time graph. These kinematical graphs were meant to assist students in visually making a connection between the function and its anti-derivative. [START_REF] Yerushalmy | Signifying the accumulation graph in a dynamic and multirepresentation environment[END_REF] describe another tool that, given a function and its graph, generates graphically an accumulation point graph. Further on, [START_REF] Swidan | Learning the indefinite integral in a dynamic and interactive technological environment[END_REF], presented the Calculus Integral Sketcher, which allows students to construct and drag a primitive graph when a function graph is given. In all cases, students' learning is supported by graphical means. We want to extend the research on pedagogical, digital tools for learning the concept of integrals by combining graphical and kinematical approaches with dynamic animations. Therefore, we considered a tool aiming to engage students to (1) combine different topics of their curriculum, such as calculus and kinematics, (2) calculate and interpret graphs etc, and (3) collaborate.

We studied Sim2Bil which can be used for groups within higher education where integrals and kinematics are part of the curriculum (students in the natural sciences, engineering, etc.). The tool requires users to work with velocity functions, of which the integral represents the distance travelled. The tool's name comes from a Norwegian word for car, 'bil', derived from automobile. The tool consists of mathematical tasks connected to dynamic animations. In addition to studying this tool, we want to study collaborative work between students, because inter-personal mediation is considered an important aspect of future education [START_REF] Lowyck | Bridging Learning Theories and Technology-Enhanced Environments: A Critical Appraisal of Its History[END_REF].

Figure 1 shows the interface of Sim2Bil. The top left part shows an animation of two cars driving in a straight line from a starting line to a finish line. This is the simulation area. The lower left part shows two separate graphs for the velocity-time function of both cars. The areas under the graphs represent the distance travelled by the cars. This is the graph area. The bottom right part is the space for the velocity functions of the cars. A user can set in parameters for polynomial velocity functions. This is called the formula area. Also, within this area there are buttons to click on. 'Formula1' and 'Formula2' give the generalized kinematical formulae for average and instantaneous velocity, and displacement, which include the integral symbol. Other buttons can hide/show the cars and graphs. When pressing the Start button in the bottom right corner, an animation starts showing the cars run, and at the same time the areas under the graphs are animated: the grey areas increase with time.

The tasks in the top right area were especially designed for the animations. In connection with the animations of cars driving by taking over each other and finishing at the same time and the graphs, the tasks were about making these cars run under certain conditions. Without the tasks, the animations of moving cars will not be very meaningful, and vice versa.

In the study presented in this paper, we investigate how Sim2Bil is used by groups of students. We use the theory of instrumental genesis to analyze our data. Our research question is: what forms does the instrumental genesis of students working with Sim2Bil take?

The theory of instrumental genesis

The theory has assisted researchers to study students' activities in CAS environments (e.g. Drijvers, 2003;Guin & Trouche, 1999) and dynamic geometry environments (e.g. [START_REF] Alqahtani | Instrumental appropriation of a collaborative, dynamicgeometry environment and geometrical understanding[END_REF]. First, we will explain the constructs instrumentalisation, instrumentation and technique which play an important role in the theory. Further, we will explain how techniques can have epistemic or pragmatic value.

Any artifact, a physical object, will remain a bare artifact for a person if the person does not know what to use it for. However, an artifact might turn into an instrument if there exists a meaningful relation between the person and the artifact [START_REF] Rabardel | From artefact to instrument[END_REF]. [START_REF] Drijvers | From artifacts to instruments: A theoretical framework behind the orchestra metaphor[END_REF] give a good example of how a hammer, an artifact, is turned into an instrument if a person has skills and experience of how to use it properly. The distinction between artifacts and instruments does not lay in a physical change of the objects, but a transformation of the way a person thinks about and practically uses the object. This transformation process is a learning process and it is called instrumental genesis. The transformation process works in two directions as explained by [START_REF] Trouche | Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students' Command Process through Instrumental Orchestrations[END_REF]: towards the artifact and towards the user. The first direction involves the user learning to use the artifact. This embraces the activities in which an artifact becomes an instrument for a user, and about how the action influences the user's activity and knowledge. This direction is called instrumentalisation. The other direction concerns the user using the artifact meaningfully for tasks. This is called instrumentation. As [START_REF] Artigue | Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work[END_REF] points out, this process involves developing ways for solving tasks.

We will analyse students' work with Sim2Bil. Our intention is to understand how students are appropriating the tool and the transformation process of the tool becoming an instrument. Since there exists a dialectic relationship between an artifact and a user [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF], we will look into both directions described above for the investigation.

Epistemic and pragmatic value of techniques

While doing mathematics, students can use several techniques, which may involve doing calculations and making drawings. A technique within the theory of instrumental genesis means "a manner of solving a task" (Artigue, 2002, p. 248). Artigue distinguishes between instrumented and paper & pencil techniques. Techniques yield a result and, therefore, they have a value. The value can be epistemic or pragmatic. Epistemic value involves that the technique has a meaning for the students related to the mathematical objects involved. We observe this, for example, when students find solutions to symbolic equations by creating a graphical representation with a graphing tool to use the zeros of the graph instead of solving the equations algebraically or numerically.

A technique might also have pragmatic value. [START_REF] Artigue | Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work[END_REF] explains that we can observe pragmatic value when a technique is applied by students who are focused on the productive outcome of that technique (e.g. to have a quick answer). Further, she states that epistemic value might be less recognizable than pragmatic value since the latter concerns the appearance of immediate results. We observe this, for example when students are randomly guessing an equation.

Methods

To investigate students' work with Sim2Bil we needed to observe students interacting with this digital tool. Hence, we created an environment in which students use the tool (tasks, animations), and other resources such as calculators, pens and paper.

As a default setting, there are two velocity functions given in the interface so that a student can press the Start button, see the cars run and the areas under the graphs grow. The two default velocity functions make the cars finish simultaneously. In this way, the forthcoming tasks were framed, which all have requirements on how the cars should finish. With a total of four tasks, Sim2Bil integrates tasks and animations.

The first task requires students to press the Start button and explain what the shaded areas represent. This task gives a visual introduction to the tool offering an association to distance as represented by area under a graph. After Task 1, there are Tasks 2-4, in which students are asked to find velocity functions that fulfill different requirements for the running cars. In Task 2, students are asked to find other functions so that the cars run with different velocities and arrive at the finish line at the same time. This will require students to translate the kinematics into mathematics and find functions v(t) so that 4 0 ( ) 400 vt   . In Task 3, they are asked to make the green car be only half way when the red car reaches the finish line, thus 4 0 ( ) 200 vt   . In Task 4, they are asked to make one car have half the velocity of the other when they arrive at the finish line simultaneously.

Our study was carried out at two different universities, at each of which we worked with one group of three students. All were within their first year of engineering studies. Group 1 comprised three boys, while Group 2 included one boy and two girls. All the six participants were not familiar with Sim2Bil. In their lectures, the students of Group 1 might have seen a similar tool as Sim2Bil, including a button to press on screen for making an animation run. All volunteered to participate in this study outside regular lectures. The groups sat in a room with a table and one laptop with the tool in front of them. They were informed that we would study how they interact with the tool, and that they would not be assessed. Both groups were given an unlimited time for group work, and it turned out that Group 1 spent 45 minutes and Group 2 used one hour.

There are some differences in how the groups were treated, with regards to parts of the interface and how the tasks were given. In the interface, the top right area showed an unused menu for Group 1 and they received the tasks on paper. For Group 2, the Tasks 2-4 were given on screen. In the formula area, the students of Group 1 could write in parameters to make up to third degree polynomials (see Figure 1). Group 2 students could write in any expressions. Since Task 1 has a different nature than the others, as it does not ask for mathematical expressions, we gave this task orally to Group 2, at about three minutes into the session. In this way, we could see whether the Start button would be quickly found. The data collection consisted of video recordings of the students' group work. The first author was present with both groups and two cameras were used at different angles to capture students' writings, gestures, and screen activity. The video recordings were transcribed fully for Group 1 and largely for Group 2, and analyzed in light of instrumental genesis. In particular, techniques the students used to solve the tasks were identified by going through the videos and transcriptions, and analyzed in terms of epistemic and pragmatic value.

Results

By analyzing the students' group work regarding the process of instrumentalisation, we observed that Group 1 easily found the Start button at the beginning. They saw the cars driving to the finish line and arriving together, the growing graphs and the increasing shaded areas. Then they read Task 1. They related the areas under the graphs with the distance covered by the cars.

Group 2 was not asked to press the Start button at the beginning. They started by explaining to each other what they saw on the screen: Dana:

We are going to work with the relationship between velocity and time. (…) I'm thinking we are supposed to come up with functions like this (points at the formula area) related to the graphs (points at the graph area). Isn't it?

Jeff: Mhm… Okay, it looks like we have the formula for the velocity of the first car… and for the second car. So, uhm….

In the episode above the students related velocity functions to the graphs and velocity to the cars. After this, Jeff turned to the researcher and asked: "Excuse me, what are we supposed to find out here? We are supposed to…". The researcher asked them to press the Start button. Then, they pressed Start and saw the animated cars and graphs, and they related the areas under the graphs to distance covered by the moving cars. Thereafter, they started working on the remaining tasks.

Based on the observation of Group 2, we saw that guidance of finding the Start button was needed. At first sight, the screen offers much information, so this button can be overlooked. By pressing Start, they were introduced to the tool's functionalities and to the conditions of the tasks. The animation showed two cars running differently but finishing simultaneously. We interpret that once the Start button was found, it enabled the appropriation of (1) the operations of the animations and

(2) the conditions of the tasks.

In regards to the process of instrumentation, the students used several techniques. For example, they set parameters in the formula area and pressed the Start button and watched the cars run. Another instrumented technique was to insert parameters and notice how a graph of a function looked like, dismissing it when it went too low. These techniques were applied by both groups within each task. On some occasions, we observed some students set in parameters seemingly at random in the formula area. Then, the graphs were noticed or they saw the cars run. On other occasions, the instrumented techniques were done as a final check whether their paper and pencil solutions were correct (paper & pencil techniques are explained below). These instrumented techniques had pragmatic value, since it was a quick check to see whether inputs were correct. The students might have reasonably guessing parameters, and on some occasions, the students explained why the animations appeared the way they did connecting it with algebraic expressions of the functions. The techniques connect symbolic, graphical, and kinematical representations. Therefore, it can be argued that the techniques also had epistemic value.

We observed both groups use paper & pencil techniques for the tasks 2-4. They calculated the integral as anti-derivative, but knowing that they calculated the area under the graph and that it was the distance covered by the cars. In task 4, Group 1 also used a technique consisting of making rectangle and triangle drawings and making area calculations based on the fact that one area needs to be equal the other one. The techniques have a pragmatic value since the focus was on the productive potential (finding velocity functions) and it was a way of checking their answers. It can also be argued that these techniques, inherent symbols, had an epistemic value, since integration can be regarded as anti-derivation and an area can be regarded as distance travelled.

Both groups discussed how to solve each task. Also, one group occasionally used gestures to visualize the cars and graphs in a way they could not visualize through instrumented techniques (e.g. get a specific graph). Also, gestures were used to support their imagination in discussions on the requirements. When one group started on task 4, they used their hands to gesture the run of the cars, mimicking the cars take over each other and finish together. Especially, the use of gestures occurred when the students faced challenges in finding parameters to solve the tasks.

For creating polynomial velocity functions, the tool included a hint for the problem-solving process. We observed some students being "stuck" while doing calculations and clicking around on possible buttons for a clue. The formula buttons confirmed to them, that they needed an integral, but it did not help them in calculating the integral or in mathematizing the requirements of the tasks.

At the end, the one group related their answer (v1=100, v2=-25t+150) to different parts in the areas on screen:

Erik: Can we prove that the answer is correct? Yes, by the calculations again. (…) But we can also see it here that it stops on half of the other (point at the screen). So, it stops on 40. So, you can see that on the graph, actually.

Sam:

Oh, yes and you can see that it is a relation between the area under the graph…which is the integrated of the velocity.

This episode shows that the students were reasoning on how they can check whether their answer is correct. Erik mentioned the paper & pencil technique (calculating the triangle and rectangle, area), and the instrumented technique (notice the graphs on screen including the equal areas and the velocity of the red car is half the velocity of the green car). Sam repeated the relation between areas and distance.

Discussion and conclusion

Research has focused on different approaches for learning the concept of integral. In some studies, the integral was visualized as Riemann sum. In our contribution, we have taken another approach: we investigated how students work with Sim2Bil, which is a digital tool designed for university level engineering studies that include mathematics and kinematics. Two groups of students were offered tasks and animations to make cars fulfill different requirements. They needed to collaboratively understand the distance travelled as an area under a graph.

The theory of instrumental genesis allowed us to investigate students' activities in a technological environment as previous studies have shown. In particular, focusing on particular aspects of the theory such as instrumentation, instrumentalization and technique, we were able to analyse the ways students used Sim2Bil.

In the theory of instrumental genesis, a distinction is made between learning to use a tool, and using the tool for solving tasks. For learning to use the tool, the Start button played a dominant role. By pressing the Start button, students observe how the velocity functions make the graphs appear and the cars run. Thus, students were introduced to the functionalities of the tool and the conditions of the tasks. However, if students weren't told to press Start, it could take long before they discover the dynamic animation. For using a tool for solving tasks, also known as instrumentation, [START_REF] Artigue | Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work[END_REF] distinguishes between two types of techniques, instrumented and paper & pencil techniques.

In tasks 2-4, our students needed pencil and paper for calculating the integral as anti-derivative, and making drawings of areas which they also calculated. Their instrumented techniques consisted of generating cars running and generating graphs. In both groups, the instrumented techniques were always the final activity in each of the tasks 2-4. With this technique, they could check whether the functions they had found met the conditions of the task.

Several techniques were used to solve the tasks. The instrumented techniques in the groups played a role as a check whether solutions are correct. Thus, they had pragmatic value. At the same time, the instrumented techniques had epistemic value because integration can be regarded as anti-derivation, and an area can be regarded as distance travelled. With task 1, our students learned the operations of the animations and the conditions of the tasks within only a few minutes. The Start button was included to get the cars running. This framed the forthcoming tasks. The challenge laid in finding functions for solving the remaining tasks. To overcome these challenges, the students used several techniques as explained above supported by gestures and discussions.

The study has some limitations. Sim2Bil is different from other tools in which students can construct mathematical objects (drag triangles, construct families of parabolas, etc). It is more of a question generator, the animation "explains" the task, and the students can use the animations or the graphs to check their answer. Additionally, the findings of this study may not be generalizable to larger groups of students since we have only observed two groups of students using the digital tool.

Figure 1 :

 1 Figure 1: Interface of Sim2Bil.

Proceedings of CERME10