Floridona Tetaj
email: floridona.tetaj@uia.no

Olov Viirman
email: olov.viirman@hig.se

A case study on mathematical routines in undergraduate biology students' group-work

Keywords: Commognitive perspective, mathematical discourse, mathematics in biology, routines, rituals and explorations

In this paper, we investigate the mathematical discourse of undergraduate biology students when working on biology tasks. Our data consists of students' and the lecturer's discussion when working on two tasks in an Evolutionary Biology course. In our analysis we make use of the commognitive framework and focus on the use of mathematical routines. We observed that although the overall aim of students' engagement with routines was exploratory, the way they engaged with mathematical routines when working on biology tasks was ritualized. However, students were aware of the need of using construction routines when trying to mathematize a biological phenomenon although the lack of familiarity with relevant construction routines constrained their ability to deal with certain task situations.

Introduction

In the last four decades, the role of mathematics in the science of Biology has greatly increased. This has influenced undergraduate and graduate biology students who need to be mathematically capable to meet the demands of this science (i.e. [START_REF] Labov | Integrated biology and undergraduate science education: A new biology education for twenty-first century?[END_REF]. Students need to understand and work with contemporary mathematical models and frameworks that are applicable in analysing the overwhelming flow of biological data. However, various studies have shown that biology students face many challenges and difficulties in their biology courses when mathematics is used as an analytical tool (i.e., [START_REF] Speth | 3,4: Infusing Quantitative Literacy Into Introductory Biology[END_REF][START_REF] Brewer | Vision and change in undergraduate biology education: A call to action[END_REF]. These difficulties can be partly explained by a prevalent epistemological perspective on the connection between mathematics and other disciplines that [START_REF] Barquero | Applicationism' as the dominant epistemology at university level[END_REF] have called "applicationism". According to this view, common at the university level, "first mathematical tools are built within the field of mathematics and then they are 'applied' to solve problematic questions from other disciplines, but this application does not cause any relevant change, neither in mathematics nor in the rest of disciplines where the questions to study appeared" (ibid, p. 1940). Barquero, Bosch and Gascón argue that such a perspective has problematic effects particularly on mathematical modelling practices, since these require a solid understanding of the connection between the mathematical tools and the context in which they will be used. This paper, which is based on the pilot study of a doctoral research project aimed at exploring and characterizing the mathematical discourses of undergraduate and graduate biology students, investigates students' group-work in a biology course in a Norwegian university when engaging with biology tasks that use mathematics as an analytical tool. Taking a discursive approach, we analyse students' and the lecturer's mathematical discourse when working on two tasks aiming to explore the routines that students engage with and the character of this engagement, and to provide insights into the kind of mathematical discourse that the lecturer expected students to engage with.

Commognitive framework

This study is grounded in the commognitive framework [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF]. In this framework, mathematical knowledge is conceived through a community's established modes of communication called discourses. A discourse is defined as a "specific type of communication made distinct by its repertoire of admissible actions and the way these actions are paired with reactions" (ibid., p. 297). In other words, discourses are different types of communication set apart by certain characteristic features. In particular, mathematical discourse is described by four features: word use, visual mediators, endorsed narratives and routines.

Our focus in this paper is on the students' use of routines. [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF] originally defined routines as "repetitive patterns characteristic of the given discourse" (ibid., p. 134). However, recent work by, for instance, [START_REF] Lavie | Routines we live by: from ritual to exploration[END_REF] elaborates further of the notion of routine. They define routines using the notions of task situation, task and procedure. First, task situation is understood as a setting in which "a person considers herself bound to actto do something" (ibid., p. 7). Second, a task is "the set of all the characteristics of the precedent events that she (the person) considers as requiring replication" (ibid., p. 9). The task refers to a person's interpretation of a given task situation; and, by precedent event, the authors mean all that happened in the precedent task situation. Third, a procedure is "the prescription for action that fits both the present performance and those on which it was modelled" (ibid., p. 9). The procedure is implemented by the task performer in response to a given task situation. Lastly, a routine which is performed in a given task situation by a given person is the task, as seen by the performer, together with the procedure the person executes to perform the task. Routines are distinguished as practical if the person interprets the task situation as requiring a change, reorganizing or re-positioning of objects; or discursive if the person interprets the task situation as requiring a communicational action [START_REF] Lavie | Routines we live by: from ritual to exploration[END_REF]. Moreover, a routine is characterized as exploration if it is oriented towards the outcome; and, as ritual if it is a processoriented performance. Thus, while an explorative task aims at producing a new "historical" fact about mathematical objects, a ritual is appreciated for its performance and not for its product. Explorations according to [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF] are divided into three categories: construction (a process resulting in a new endorsable narrative), substantiation (deciding whether to endorse previously constructed narratives) and recall (the process of citing a narrative that was endorsed in the past). According to [START_REF] Lavie | Routines we live by: from ritual to exploration[END_REF], students participate in unfamiliar discourse in a ritualized way. However, in further learning, their routines are expected to "undergo gradual deritualization until they eventually turn into full-fledged explorations" (ibid., p. 2). A person performs a certain mathematical routine in the present task situation because of its "precedentsto past situations which she (the person) interpret as sufficiently similar to the present one to justify repeating what was done then, whether it was done by herself or by another person" (ibid., p. 8). The precedents are chosen with the help of "precedent identifiersto those features of the current task situation that a person considers as sufficient to view a task situation from the past as a precedent" (ibid., p.8). The analysis in this paper examines this relation focusing particularly on the following research questions: "What characterizes students' mathematical discourse when engaging with biology tasks?" and "How does students' mathematical discourse relate to the lecturer's intended solutions of the tasks?"

Method

This research is a case study of the course Evolutionary Biology, an undergraduate biology course at a large Norwegian university. The study was conducted during the spring semester of 2018. All students enrolled in the course were required to have taken at least one course in Calculus and one in Statistics. The aim of this course is to provide students with a deeper insight into the evolutionary processes that can explain the genetic composition of populations, form, behaviour and distribution of organisms, and to acquaint students with the basic methods of analysing evolutionary relationships between species. Moreover, students are introduced to different mathematical models of evolution. During the course, three sessions are dedicated to students' group-work. In these sessions, students are asked to work on different questions related to the topics that have been presented by the lecturer in the lecturing sessions. In this paper, we present data from one of these group-work sessions. In this particular session, students were asked to work in groups on a quiz with multi-choice questions. Students had 30 minutes available to finish the quiz. They were first to work individually for 10 minutes, then discuss their answers in small groups. The lecturer also handed out an answer sheet, enabling the students to check if the proposed alternative was the right one. After the small group discussion, the group had to agree upon a correct answer, which was then evaluated using the answer sheet. As students were working on the quiz, the lecturer circulated among the groups, providing help if needed. At the end of students' group-work, the lecturer discussed each question of the quiz in front of the class.

In our analysis, we focus on two questions: Heritability Equation and Genetic Correlation. The first question (see Figure 1) requires students to find the correct formula of heritability ( ) which is given as a proportion of response (R) to selection (S).1 This formula derives from the definition of heritability as the degree to which a certain trait in individuals is genetically (rather than environmentally) determined when the genetic construction of an individual and the environmental factors have a normal distribution. This equation had been introduced in a previous lecture. The second question (see Figure 2) requires students to reason about negative and positive correlation between variables. Although this specific question had not been previously discussed, the lecturer had previously introduced the theoretic background on genetic correlation between variables.

For the purpose of this study, we video and audio-recorded two groups of five students each (in total there were 32 students in the class, divided into groups of five or six) and the lecturer's discussion afterwards. The two groups of students were chosen randomly. All collected data was transcribed. Students' and lecturer's discussions were conducted in English. When analysing the data, we focused on the identification of students' and lecturer's mathematical routines. We analysed separately the transcripts of each students' group-work and the lecturer's discussion, looking for instances where they were engaged in mathematical routines. Then we compared the findings from each students' group work with the lecturer's discourse. In the process of data analysis, we also looked for signs of exploratory or ritualized engagement, for instance, whether students' routine use was aimed at producing or justifying mathematical claims, or if they appeared to be engaging in routines without a regard for their relevance for the task at hand.

Results

In this section, we will analyse each question separately, considering both the students' groupwork and the lecturer discussion where he presented his expected solutions. Analyzing students' work on the Heritability Equation task, we observed that as the first step toward finding the correct response, students would engage with recall routinestrying to recall the equation given during the lectures or written in their textbook. Of the two groups of students that we video-recorded, only one student was able to remember the correct form of the equation, and she still expressed some doubt concerning her choice: "I think the right answer is . I'm just trying to remember what was written in the book but I'm not sure about it". After failing to recall the correct equation, students tried engaging in construction routines. They started discussing what the variables in the equation stood for and how they were defined. Although students were able to identify them biologically, they could not recall the meaning behind these variables: 'I think is everything that is heritable, and R is of course response and S the selection, but I don't remember what they mean or how you actually use them'. Being unable to explain what these variables represent biologically urged students to look for other solutions. They tried reasoning mathematically using the properties of ratio and squared numbers. As shown in the dialogue below, through the use of these mathematical routines, the students tried to construct a new endorsable narrative (the equation of heritability).

Student1:

I'm guessing b) or d) … because that would be response squared divided by selection, so the higher the response, the higher is the heritability; the higher the selection, less heritability.

Student2:

Yeah, but you can also see that there is only one answer where is the opposite which leads to one of these to be right.

Student1:

Yeah, but the question is which one should we square?

Student2: They can both be squared.

………………..

Student3:

Why would you have it squared?

Student2:

To negate the possibility of a negative answer. Just square something and it cannot be negative.

Despite engaging in construction routines, the students in the first group were unable to come up with an answer, since these routines were not helpful enough for justifying any of the proposed alternatives. Thus, they decided to postpone the question. They had a further reason for this: since the following question was directly asking for the definition of heritability, they believed that the alternatives given there would help them to find the expected definition of heritability.

After students had spent some minutes working on the quiz, the lecturer asked them if they needed help with the questions. He suggested using another definition of heritability given during lectures, but the group still felt lost and, in the end, the group gave up and just skipped the question. Meanwhile, the second group, after some discussion, agreed upon alternative ( , arguing that if you have a higher selection in the numerator then the response is higher in the denominator, although without explaining why that would make sense biologically. Checking the answer sheet and realizing that they had selected the wrong answer, they merely exchanged this for the correct answer. Thus, we can observe that in this exercise, although students used construction routines with the overall aim of formulating a new endorsable narrative, they engaged with the mathematical routines in a ritualized wayemploying the routines without much regard to their relevance for the problem at hand, and consequently failing to come up with a meaningful narrative.

In the follow-up discussion of the first question, the lecturer began by saying: "I told you, you have to learn these equations and some of you didn't". Although the lecturer was laughing as he said this, suggesting that he was perhaps not entirely serious, it still implies that he was not really expecting students to come up with the correct alternative from scratch. Rather, he just expected them to remember it. In other words, he expected them to use recalling routines rather than construction routines. However, he also explained how the model could be constructed and how the form of the equation could be justified using the graph of normal distribution which describes the relation of differential selection and response. Hence, during his explanation of the task, he used construction routines that differed from the ones the students used in their solution attempts. Neither normal distribution nor the alternative definition suggested by the lecturer to the first group, were mentioned by any of the students.

The second question (see Figure 2) focused on genetic correlations of three different variables. This question, unlike the first one, could not be solved using only recalling routines. Since the task required actions for deciding whether to endorse previously constructed narratives, the solution will involve substantiation routines. Working on this task, both groups struggled with the same issue as in the first question: converting variables into their biological context. However, in this case, students were able to come up with the meaning of genetic correlation, but as shown in the Student2: But if there is a negative ratio between fat and days to market, that means that if fat increases then days to market decrease, right? … Then, that would mean that less fat, more days to market? And if we apply the same logic to the first one, then fat decreases, then we need less food.

Student3:

Yeah, I thought that when I read it. I thought it was the amount of fat correlates to the Days to Market so that means that the percentage of fat…the amount of days at the market would lower the percentage of fat by a factor of .

Student2:

Which does not make sense.

………………..

Student4:

I understand your point. I just guessed something because I was thinking about what is the effect of selecting leaner pork on feeding costs. If you think logically, then it would take shorter time to get a leaner pork than a fat pork which means lower feeding costs, right? So, I just kind of started thinking without the mathematics.

After some discussion, both groups came up with the right answer. Still, the second group first tried drawing a conclusion through reasoning biologically, rather than considering the quantitative relations between variables. However, due to a mistake in their biological reasoning, they ended up with the wrong answer. Realizing that, they started re-examining the correlation between the variables, noticing that just considering the relation between variables would lead to the right answer. In his discussion of the solution to this question, the lecturer suggested that it should have been based on the meaning of genetic (negative or positive) correlation, which fits with what the students did. Thus, his expectation on the kind of routines that students should have used during the exercise were fulfilled (despite the fact that the second group failed in their first attempt).

Discussion

In this paper we have analysed students' and the lecturer's discourse when working on two biology tasks involving mathematical elements: Heritability Equation and Genetic Correlation. These two tasks set by the lecturer in a group-work session differ from each other concerning the expected routines that students were required to use when solving them. In the second question, the mathematization of the problem had already been done, meaning that the relation between variables was given, and the students just needed to use that relation. Thus, they were expected to use substantiation routines in order to endorse previously constructed narratives. In the first question, on the other hand, the mathematization of the problem was not given. Students had two possibilities in this caseuse recall routines to remember the correct form of the equation from previous lectures or use construction routines to construct a mathematical model for the problem.

Working on Heritability Equation, when the students were unable to remember the correct equation, they used construction routines. However, considering that they did not remember the biological meaning of any of the variables, students were not left with many choices other than identifying other features of their current task situation that would allow them to repeat precedents from their past situations. The choice to use properties of ratio and squared numbers as a step to overcome the constraints of the task, shows that they were trying to apply familiar routines in unfamiliar task situations aiming at innovation. However, we observed that students employed these mathematical routinesby identifying precedents that allowed them to use properties of ratio and squared numberswithout any apparent regard for their relevance for their task situation. Rather, they appeared to engage in familiar routines in a purely processoriented manner and not towards a specific goal. This is in accordance with the findings of [START_REF] Viirman | Negotiating different disciplinary discourses: Biology students' ritualized and exploratory participation in Mathematical Modelling activities[END_REF] where biology students were seen to employ mathematical routines in a ritualized way, even when their engagement with the biological content of the tasks was exploratory.

On the other hand, from the lecturer's discourse we can observe that his expected solutions to the task build primarily on recall routines. This fits with the aims of the course, which focused on learning how to use mathematical models rather than constructing them (although he did show the construction of the models during his discussion). Still, we do not exclude the possibility that the lecturer expected students to explore the exercise mathematically using construction routines since he gave a hint on how the problem can be modelled using the alternative definition of heritability. We note that when the lecturer gave the hint to the first group, he assumed that students knew the meaning of variables ( and ), which was not the case. However, as [START_REF] Lavie | Routines we live by: from ritual to exploration[END_REF] suggest, although the lecturer's own mathematical discourse can be a model for the students to follow, it is necessary to put a conscious effort into establishing whether the explorative nature of the task gets through to the students clearly enough (ibid., p. 20). We suggest that helping students with the biological meaning of the variables might have supported their engagement with construction routines. This being said, the lecturer was unaware of the students' difficulties since they did not mention that they were struggling with understanding the variables. Concerning Genetic Correlation, despite some struggles in the beginning, students were able to use substantiation routines in accordance with the lecturer's expectations. It is worth emphasizing that, contrary to the second task, the nature of the Heritability equation encouraged students to engage in explorative participation. The mathematization of the first task required engagement with an unfamiliar task situation, as suggested by the students' inability to use the hint that the lecturer gave them during group-work, and by the absence in the students' discourse of any of the routines used by the lecturer in his construction of the heritability model.

Figure

  Figure 1: Heritability equation

Figure 2 :

 2 Figure 2: Genetic Correlation dialogue below, they struggled with making sense of the biological interpretation of their result.

-is the response to selection for certain trait, -is the selection differential, defined as the mean phenotypic difference between selected individuals and the population mean (see,[START_REF] Falconer | Introduction to Quantitative Genetics[END_REF].

We are aware of the limitations of our study, and that the limited amount of data does not allow us to draw any general conclusions. Still, we find it noteworthy how the students' engagement with the tasks corresponded well with the lecturer's expectations, and that difficulties mainly occurred when they failed in invoking the kind of routines he had expected. We claim that relying purely on recall routines can be potentially problematic for students. We suggest that providing students with some opportunities for working with construction routines can be helpful for their learning. In the continuation of this doctoral project, it is our intention to observe a greater number of biology students' over a longer period of time, investigating what happens to their mathematical discourse as they participate in a course more closely connecting mathematical modelling with biology.


Athina Thoma
email: a.thoma@uea.ac.uk

Elena Nardi

Recall and substantiation routines in exam scripts: injective and surjective functions

Keywords: Substantiation, recall, routines, injective, surjective function

In this paper, we focus on first-year university students' engagement with university mathematics discourse in the context of a final examination question on injective and surjective functions. The data consists of twenty-two responses as well as excerpts from an interview with the exam-setting lecturer. Our commognitive analysis focuses on student engagement with recall and substantiation routines: how they recall and provide relevant definitions, and how they substantiate whether a function is injective or surjective. We identify three issues that are exemplified with samples from the data: ambivalent word use, and visual mediation, relating to equivalence; partial justifications regarding the injectivity of h(n) = 3n; and, conflation of justifications for a function's surjectivity in ℤ with those used for functions in ℝ. We conclude by discussing the notion of precedent events as evident in students' engagement with routines in a range of mathematical topics.

From secondary to university mathematics

Transition from secondary school to university mathematics is a topic of growing interest [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF]. While differences between secondary school and university mathematics vary across countries, there are aspects of this transition which seem to be common in different contexts. In the larger study from which this paper stems [START_REF] Thoma | Transition to university mathematical discourses: A commognitive analysis of first year examination tasks, lecturers' perspectives on assessment and students' examination scripts[END_REF], we explore this transition through analyses of Year 1 examination questions, lecturers' perspectives on designing and using these and students' responses. Functions, one of the topics examined in these questions, have been identified as a key mathematical topic in the transition from secondary to university mathematics [START_REF] Winsløw | Research on university mathematics education[END_REF] and research has reported issues around students' learning about functions for a long time [START_REF] Bansilal | An APOS study on pre-service teachers' understanding of injections and surjections[END_REF]. Analogous attention has been given by researchers to students' deployment of formal definitions and proofs [START_REF] Selden | Transitions and proof and proving at tertiary level[END_REF], their use of mathematical notation (e.g. [START_REF] Mamolo | Polysemy of symbols: Signs of ambiguity[END_REF] and their difficulties with mathematical objects that have different meanings in various mathematical areas [START_REF] Kontorovich | Why Johnny struggles when familiar concepts are taken to a new mathematical domain: towards a polysemous approach[END_REF]. Here, we focus on one aspect of learning about functions that has received relatively little attention: what students think a surjective and an injective function is and how they decide whether a function is injective or surjective.

In this paper1 , we focus on twenty-two students' responses to a Year 1 examination question on injective and surjective functions. We take a discursive approach, the theory of commognition [START_REF] Sfard | Thinking as communicating: Human development, development of discourses, and mathematizing[END_REF], in analysing the data, and we aim to characterise the routines that students engage in when answering the questions. Building on previously reported work (Thoma & Nardi, 2018a;[START_REF] Thoma | Conflating discourses: University students' exam responses on injective and surjective functions[END_REF], we examine the relevant definitions students provide and the procedures they use when substantiating that a given function is (or is not) injective or surjective.

Commognitive routines: recall and substantiation

A discursive perspective which is being increasingly used in university mathematics education research is [START_REF] Sfard | Thinking as communicating: Human development, development of discourses, and mathematizing[END_REF] theory of commognition [START_REF] Nardi | Commognitive analyses of the learning and teaching of mathematics at university level: the case of discursive shifts in the study of Calculus[END_REF]. In this perspective, mathematics is defined as a discourse [START_REF] Sfard | Thinking as communicating: Human development, development of discourses, and mathematizing[END_REF] which can be described in terms of the following four characteristics: word use (e.g. injective), visual mediators (e.g. algebraic and logical symbols), endorsed narratives (e.g. definitions) and routines (e.g. proving). [START_REF] Sfard | Thinking as communicating: Human development, development of discourses, and mathematizing[END_REF] describes three types of routines: deeds ("an action resulting in a physical change in objects", p. 236), rituals ("creating and sustaining a bond with other people", p. 241) and explorations ("producing endorsed narratives", p. 259) with the exploration routines further categorised as recall (e.g. recalling the definition of an injective function), substantiation (examining whether a given function is injective) and construction (e.g. constructing a function that is injective). Of relevance to our analysis here, are exploration routines, which often develop from rituals. This process has been described through the notion of task-situation in which an individual feels a need to act and which is taken as "the set of all the characteristics of the precedent events that she considers as requiring replication" (Lavie, Steiner & Sfard, 2018, p. 9). Thus, explorations can emerge as changes in the way learners search for precedents in task situations. In our study, we deploy task to mean a part of an examination question which the students are asked to engage with. These task situations are interpreted by the students, who draw on precedent events, which include previous examples or exercises each student has engaged with before and which they see as relevant to the present task. [START_REF] Sfard | Thinking as communicating: Human development, development of discourses, and mathematizing[END_REF] describes the procedure (or course of action) and the when of a routine, with the latter further described in terms of its applicability and closing conditions (pp. 208-209). The applicability conditions are "circumstances in which the routine course of action is likely to be evoked by the person" (p. 209, ibid) and closing conditions "[signal] a successful completion of performance " (p. 209, ibid).

Here, we explore how students recall the definitions of an injective and surjective function and how they engage in substantiation routines, in order to examine whether functions defined on the integers are (or are not) injective or surjective. Previous research has documented "how lack of flexibility in working across different representations influences students' encounter with the concept of function" (Nardi, 2008, p. 161) and has posited that students' prior experience with functions and sets may have contributed to this lack of flexibility [START_REF] Bansilal | An APOS study on pre-service teachers' understanding of injections and surjections[END_REF]. However, there is little insight from prior research into students' engagement with injective and surjective functions in various numerical domains (e.g. reals and integers). In the UK educational context of the study we report in this paper, engagement with various numerical domains is different in school and university mathematics. In school mathematics, students are initially introduced to the domain of integers. Later, they are introduced to real numbers and discursive activity with integers is subsumed within that of the reals. The tasks they engage with here are in the context of real numbers. At university, students are expected to engage with various numerical domains, often within the same task. Being aware of this variety and of the various restrictions within each domain is therefore crucial.

Our study aims to provide some insight into this through exploring the following research question:

What engagement in mathematical routines can we observe in students' scripts when they examine whether given functions are injective or surjective?

To answer this research question, we examine students' use of mathematical terminology and notation which may be incompatible with the mathematical discourses they are expected to engage with when responding to the task at hand. We also examine the closing and applicability conditions of the routines, in which the students are required to engage in when solving the question within the various mathematics discourses present in the question (e.g. discourse of functions, integers, reals). We examine the written scripts of the students which capture the output of students' engagement with these mathematical discourses.

The exam question, the participating students and the exam-setter's intentions

The data on which we focus are examination scripts from 22 students taking an exam on a Sets, Numbers and Probability Year 1 module. In this module, the focus is on Sets, Numbers and Theory in the first semester and on Probability in the second semester. The final examination consists of six questions, three coming from each part of the module, one compulsory and two optional from each of the two parts of the module. Here, we focus on one sub question of one of the optional questions from Sets, Numbers and Theory, on surjective and injective functions. During the year, the students were asked to engage with the properties of injectivity and surjectivity in exercise sheets and coursework with functions defined in different numerical domains. In this paper, we discuss students' responses to this examination sub question: Suppose and are sets and is a function. Define what is meant by being surjective and what is meant by being injective. For each of the following functions decide whether it is injective, surjective (or both, or neither). Give brief reasons for your answers.

(a)

ℝ ℝ where for ℝ.

(b) ℤ ℤ where for ℤ.

Fifty-four students took part in the final examination and the marks of their responses to the whole question (the first sub question is given above and the second dealt with modular arithmetic) ranged from 0 to 20 marks with an average of 14.31. The scripts of 22 students were selected by the first author to represent a variety of marks (for more information on the selection process, see [START_REF] Thoma | Transition to university mathematical discourses: A commognitive analysis of first year examination tasks, lecturers' perspectives on assessment and students' examination scripts[END_REF]Nardi, 2017, Fig. 3 on p. 2269). We analyse students' scripts according to the definitions they recall for injective and surjective functions, and their substantiations that the given functions are (or are not) surjective or injective. In our discussion of the students' scripts, we bear in mind also the following selected quote from an interview, conducted soon after the exam, with the lecturer who taught the module and set the examination task. This quote illustrates the lecturer's concern with a perceived discrepancy between "knowing how to write down" a definition for injectivity and knowing what injectivity "means". We return to this quote towards the end of paper.

"

Recall and substantiation routines in students' scripts

In the sub question under consideration, students are first asked to provide the definition of injective and surjective function (recall); then, to determine whether two functions are injective or surjective and provide brief explanations for their choices (substantiation). Here, we highlight three issues that emerged from the analysis of the scripts: ambivalent word use, and visual mediation, relating to the object of equivalence () instead of implication (⇒) in the definition of injective function; partial justification of the injectivity of h(n) = 3n; and, conflation of justifications for a function's surjectivity in ℤ with those used for functions in ℝ. We discuss each case providing also examples from students' scripts.

Ambivalent word-use and visual mediation relating to the object of equivalence

In the scripts of four students, there is ambivalent word use and visual mediation relating to equivalence (either using the logical symbol  or phrases that signal equivalence) in the definition of injective function, illustrating confusion between the definition of injective function and the definition of a function. In the definition for an injective function, student [02] writes "one to one relationship" and comments on the relationship between the elements of the domain and the codomain of the function (Figure 1). However, later, the student, in trying to clarify what this "one to one relationship" is, writes "f(a)=b and vice versa". This phrase is not further explained and is also used when the student examines whether the given function h is injective, signaling a confusion between the definition of a function (which must be single valued) and the definition of an injective function. Additionally, we note that the logical quantifiers "for every b" and "there exists a" in the definitions of injectivity and surjectivity are missing from student [02]'s response. This absence of logical quantifiers and the ambivalent use of logical phrases ("vice versa") highlights a difficulty with the logical connections between the various mathematical objects that are part of the definitions. Furthermore, we observe the difficulty to state the definition formally. The phrases used by the student (e.g. "only one value") are ambiguous and are not clearly showcasing whether the student discusses injectivity, which would be "at most one value" or bijectivity ("there exists one and only one value"). The ambiguity between the single valued function and the injective function is also illustrated in definitions given by student [06] (Figure 2) and two This use of equivalence (either in the form of the symbol (Figure 2) or using the phrase "vice versa" (Figure 1)) in the definition of injective function highlights difficulties in students' engagement when recalling and writing the definitions. This may lead to difficulties in the substantiation part of the question, illustrating the close connection between definitions of the properties of the functions and the substantiation routines. This ambivalence between definitions and substantiation routines could be explained by considering students' precedent events. The focus in secondary school is on the properties of a mathematical object. However, the students are rarely required to provide a set of conditions to serve as a definition of a mathematical object. In that occasion providing more conditions and properties is not considered incorrect. This is not the case at university level. The expectation at university mathematics is to provide a minimal set of conditions which suffices for the function to be injective without reiterating the definition of function. The examples from the students' scripts discussed above, highlight the need to alert the students to the characteristics of the routine in the context of university mathematics.

We now turn to students' engagement with substantiation routines in examining whether the function which is defined in integers is injective or surjective.

Partial justifications regarding injectivity of h(n) = 3n

In the scripts of ten students, we have partial justifications regarding the injectivity of the function h defined in ℤ . One of these scripts is in Figure 3. Student [22] claims that h(n) is not surjective by providing a counterexample and then tries to prove that h(n) is injective. We note here that [22] writes "surjective" instead of "injective" in the second part of the script. When discussing why the function is injective s/he says that "no two numbers which are also integers can be made by 3n" and provides a graph to support this claim. In the graph produced by the student, and from their answer regarding surjectivity, we can see that [22] sees h(n) as a discrete function. However, the argument accompanying the graph ("h is surjective because … can be made by 3n") is not sufficient. The relationship between two elements of the codomain and the corresponding elements of the domain is not clear. We note, that the difficulty here is in the closing conditions of the routine.

Conflation of justifications for a functions' surjectivity in ℤ and in ℝ

In seven student scripts, the procedure of substantiation that an integer function is or is not surjective is conflated with procedures that could be used when a function is defined in ℝ, not in ℤ .

One such example is shown in Figure 4. When substantiating, the student does not use the definition of the injective function but relies on other procedures. S/he talks about "turning points on this continuous function" and "continuously increasing". These properties of the function are describing functions which are continuous and defined in ℝ. This conflation of functions in ℝ and ℤ is also illustrated in the graph produced by student [11]. We observe that the arguments are based on the graph of the function, even though there is a symbol n at the x-axis, the values in the x-axis, y-axis and the line showing the function is a straight line without gaps. This engagement is ritualistic as the students seem to be drawing on familiar precedents: in the past, the majority of the tasks the students engaged with were dealing with functions in ℝ. The students are examining the function but do not take into account the domain as a significant precedent identifier regarding the routine that they should follow. This suggests that the student does not consider the integers as the domain of this function but thinks of the function as a function in the reals, highlighting that the applicability conditions of the substantiation routine are not being examined and thus a routine being used typically for a continuous function is used here for a discrete one. We should note that there is a relationship between properties of functions in ℝ which are restricted in ℤ . However, if this relationship is used, then this relationship between the real function and its restriction to integers should be examined as part of the justification provided by the student.

Returning to the lecturer's quote, it seems that the phrase "they know what it means" indicates the students' ability to decide whether a function is injective or surjective. This routine of deciding whether an object has certain properties has precedents in secondary school mathematics. However, in university mathematics, the substantiation routine is at least as important as deciding on the properties of a mathematical object, and the lecturer would like to see the substantiation of students' decisions relying on an accurately recalled definition, which often was not the case. This new routine of substantiation is challenging for students for many reasons; in the precedent space acquired in secondary school, decision tasks do not typically require the recall or reconstruction of definitions, nor do they require substantiation based on a definition. At university level, the students must become familiar with recalling and providing definitions, focusing on a minimal set of conditions rather than providing all the properties that they can recall relating to the mathematical object in question. In the definition of injective function, they must identify the domain and the codomain of the function, engage with different elements in the domain and with logical quantifiers that connect statements with elements from the domain and the codomain, and use all these in their substantiation.

The potency of a commognitive lens: Routines and precedent events

Our analysis suggests that, in these exam scripts, students face difficulties when they recall the definition of an injective function, particularly in relation to word use and visual mediation relating to the object of equivalence pertaining to the definition of injectivity and to conflating injective functions and the fact that a function is single valued. Furthermore, the closing conditions of the substantiation routines are not met. This may relate to differences between secondary and university mathematics in the UK context of our study. While in the context of secondary mathematics, students have engaged with examining whether a mathematical object has a property or not; they have not necessarily done so with the explicit requirements for rigour and logical connectedness expected in a Year 1 examination at university. Finally, students' ways of exploring whether function h(n) is surjective indicate that they base their responses on their prior experiences which are usually with functions defined in the reals. However, these procedures are not necessarily applicable in the context of the integers and this results in conflating the discourses on the two. We see the students' responses as underlain by a commognitive conflict: integers are seen as a subset of reals and not as a domain in which there is no closure for the operation of division.

The analysed students' scripts and the comment from the lecturer illustrate the importance of the precedent events. In aiming to deritualise students' engagement with these routines, a more nuanced search for the precedent events is neededand one which focuses clearly on the identifiers of the precedent events. This is illustrated in the comments of the lecturer as well as those student scripts which showcase ambivalent word use and visual mediation. Both of these cases highlight the difference with the precedent events in secondary school regarding the definition of a mathematical object. Then, the scripts illustrating partial justifications are also showing that students' engagement with precedent events at secondary school, where the justification whether an object satisfies a given identity does not necessarily require the depth of justification needed at university level. Finally, in the last category, the importance of the identifiers regarding the precedent events is further elaborated, as it showcases that the focus should not only be on the object (the function) but also on the domain of the function which is a substantial precedent identifier in terms of the routine required.

Our results offer further insight into students' engagement with routines of proving and recalling definitions, and the relationship of the two in the context of a question on injective and surjective functions. While previous research regards students' prior experience with functions and sets [START_REF] Bansilal | An APOS study on pre-service teachers' understanding of injections and surjections[END_REF], our results highlight the importance of examining the applicability conditions of substantiation routines in different mathematical contexts (e.g. integers and reals). These results illuminate instances where the lecturers could highlight more explicitly the differences between university and secondary school discourses; and, discourses of reals and integers. Similar results to the ones reported in the paper with students' engagement with recall and substantiation routines are visible to students' responses to other examination questions [START_REF] Thoma | Transition to university mathematical discourses: A commognitive analysis of first year examination tasks, lecturers' perspectives on assessment and students' examination scripts[END_REF] relating to: the definitions of reflexive, symmetric, and transitive relations and the substantiation of these properties; and, recalling Fermat's Little Theorem and applying it to find the remainder of a power (e.g., 27 313 ) divided by a prime number. We credit the commognitive lens for the insights into the student scripts that our analysis allows, and particularly the recent efforts [START_REF] Lavie | Routines we live by: from ritual to exploration[END_REF] to associate task situations, precedent events and the ways students engage with mathematical routines.

Figure

  Figure 1: Student [02]'s definitions

Figure 3 :

 3 Figure 3: Student [22]'s response to (b)

Figure 4 :

 4 Figure 4: Student [11]'s script to (b)

  They need to see this transition between (…) the symbols and the meaning and the logic of things and it's one of the most important things and it is one of the hardest things to teach"

somehow they know what injective means, they just don't know how to write down the definition (...) it is a very strange experience to see that a student knows what injective means but can't write down what it means, it's something about maybe not even about mathematics, it's about language and about logic (…)

An earlier version of this paper was presented as a PME42 Oral Communication(Thoma & Nardi,

2018b).


Stephanie Treffert-Thomas

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

Dave Hewitt
email: d.hewitt@lboro.ac.uk

Nikolaos Vlaseros
email: n.vlaseros@lboro.ac.uk

Marinos Anastasakis
email: m.anastasakis@lboro.ac.uk

Students as Partners in Complex Number Task Design

Keywords: Student-partner, computer-based task design, mediation and tool use, collaborative research, developmental research

We report on a collaborative project at university level involving students as partners in task design for a bridging mathematics module (known in the UK as a Foundation module) which is part of gaining access to first year degree studies. Three teacher-researchers met regularly with four student partners who developed a set of tasks on matrices and on complex numbers which were trialled with students on this Foundation module. We show the mediational processes by which the tasks developed from 'static' designs to more 'dynamic' designs using the software Autograph. Our analyses highlighted various tools in the mediation of the learning of mathematics, in the mediation of task development and in the mediation of the engagement of all team members in collaboration.

Background and literature

In 2011/12 findings from the ESUM project [START_REF] Jaworski | Developing teaching of mathematics to first year engineering students[END_REF][START_REF] Jaworski | An activity theory analysis of teaching goals versus student epistemological positions[END_REF] pointed to a problematic difference between teacher culture and expectations and those of students when researching students' use of mathematical tasks in a computer environmentwhen those tasks were designed for students by their teacher. In the current project we are trying to address this difference by engaging students in the design process and studying the emergent learning for the students and the researchers. The project was designed by three teacher-researchers (TRs, authors of this paper) to engage four students as partners in the design of mathematical tasks in a computer environment. Thus, the project is a collaboration between students (we refer to them as student-partners, SPs) and teacher-researchers (TRs) in the design of tasks. Several aspects were explored within this project: the development of mathematical tasks in a computer environment, SPs' perspectives on task design and participation in a research team, and the use of the tasks with current students studying on a Foundation module.

While there is research into the engagement of students as partners in course design at university level (see [START_REF] Mercer-Mapstone | A systematic literature review of students as partners in higher education[END_REF], few of these are within mathematics education (see however, [START_REF] Duah | Can peer assisted learning be effective in undergraduate mathematics[END_REF][START_REF] Fayowski | An evaluation of the Supplemental Instruction programme in a first-year calculus course[END_REF]. Our project builds on the work of [START_REF] Duah | Students as Partners in Mathematics Course Design[END_REF] who involved student interns in the design of resources for two 2 nd year mathematics modules experienced as difficult by students. Both the lecturer of the module and student interns learned considerably from this collaboration [START_REF] Duah | Students as partners and students as change agents in the context of university mathematics[END_REF].

Our aim for initiating this project and recruiting SPs was to foster and study a deeper understanding of mathematics in a Foundation mathematics module, specifically the two topics of complex numbers and matrices. The Foundation Studies Programme is a one-year course intended for students who wish to study for a STEM subject at our university but do not satisfy the entry requirements for their chosen degree. All must take a Foundation mathematics module as part of their programme.

We focus on complex number tasks for this paper. The decision to use dynamic software in task design was taken by the TRs in order to connect algebraic and geometric representations of the arithmetic operations on complex numbers. We consider geometry and algebra as fundamental to all mathematics [START_REF] Atiyah | Mathematics in the 20 th century: Geometry versus algebra[END_REF] and hence to its teaching and learning. While we are aware that providing geometric insights can potentially be problematic (it can both help and hinder, see [START_REF] Gueudet-Chartier | Should we teach linear algebra through geometry?[END_REF], we align ourselves with those who value these as a means of connecting with students' more intuitive notions of a concept [START_REF] Stewart | A framework for mathematical thinking: The case of linear algebra[END_REF][START_REF] Uhlig | Author's response to the comments on 'The Role of Proof in Comprehending and Teaching Elementary Linear Algebra[END_REF].

Theoretical and Methodological Perspectives

We take a socio-cultural perspective on teaching and learning. Central are the Vygotskian notions of mediation and tool usethe idea that attaining an object of activity is achieved better through the mediation of some tool, artefact, person or process [START_REF] Vygotsky | Mind in society[END_REF][START_REF] Wertsch | Voices of the mind: a socio-cultural approach to mediated action[END_REF]. All members of the project were learners: the SPs learned how to design mathematical tasks and work with the other members in a research environment; the TRs learned about student perspectives and the process of involving students in task development. Mediation was similarly diverse: all participants mediated the learning of others through the collaborative process. There were various tools: computer-based resources in the form of Autograph1 files that were accessed by Foundation students in the tutorials; paper-based resources such as lecture notes and problem sheets, additional instructions on the use of Autograph, and questions in relation to the Autograph files. We return to mediation and tools later.

The research team consisted of nine people: three teacher-researchers (TR1, TR2 and TR3, the first three authors), two doctoral students (the last two authors who helped with data collection and analyses) and four SPs. All were engaged in developmental research within a community of inquiry, seeking to create knowledge, improve practice and the mathematical learning experience of students (see [START_REF] Jaworski | Theory and practice in mathematics teaching development: critical inquiry as a mode of learning in teaching[END_REF]. All participants took part in design meetings and were involved in an iterative design-research approach where participants inquire into the processes in which they engage. The SPs learned mathematics through inquiry into task design. They particularly learned task design through several iterations of the design process (inherently an inquiry process) in which drafts of their tasks were critiqued by other members of the team.

Our approach used ethnographic methods with data collected that could help answer our research question. Data collected included (1) audio-recordings of project design meetings, (2) written reports by SPs reflecting on their experience of task design, (3) interviews with SPs, (4) computerbased tasks (Autograph files), (5) various field notes. We took a grounded approach to data analyses. As part of this process data reductions were made of all design meetings. For a data reduction one of the researchers listened to the audio-recording while simultaneously making timerelated factual notes on the content. This resulted in a factual summary in tabular form with interpretative comments written in a separate column alongside suggestions for areas of transcription. With these summaries we aimed to capture key points of the task design in order to chart their development, written reports were read and used to support the analysis in respect of the recordings; interview data were partially transcribed and key points summarised; computer-based tasks were recorded as screenshots; field notes were read and integrated into summaries.

In this paper, we focus on the SPs' design of tasks in complex numbers and specifically on their transition from thinking in terms of the nature of tasks with which they were familiar from their past mathematical experience, to developing more dynamic forms of the tasks. We ask: What was the process by which SPs' generation of tasks developed?

The development of the complex number tasks

The tasks were developed for use with students taking a mathematics module in their Foundation Studies Programme. The SPs were recruited from former Foundation students via an interview process. The four students, chosen from the sixteen who applied, were those who had achieved a good grade in their Foundation mathematics module and who showed interest and initiative in thinking about their potential involvement. All had begun their first year of an Engineering or Science degree (Chemistry, Physics, Mechanical and Chemical Engineering).

The SPs' first task in the project was to review their notes from the Foundation module on two topics, matrices and complex numbers. To initiate the project, the whole team met for an introduction to the computer environment (Autograph) by an expert in its design and use. The SPs knew TR2, the lecturer of the Foundation mathematics module and were introduced to the other project members. Audio data from the meeting shows that SPs participated in a discussion on the use of Autograph and expressed views on experiences from their Foundation studies. These were of particular value and interest to the TRs, who gained insight into student perspectives on teaching and mathematical learning. Following this initial meeting, SPs engaged in designing tasks and bringing their current designs to successive 'design meetings' (DMs) where designs were discussed and modifications suggested.

The computer environment, Autograph, can be seen as a tool in two ways: first, it was to be a tool for the design of tasks as we discuss below. Second, and more immediate in the initial meeting, it became a catalyst for drawing the SPs into a discussion with the team in a non-threatening way. The team was relatively new to Autograph and so all could ask questions and talk about the use of the software. TRs engaged in a mediational process of drawing the SPs into the team via questions to the expert about how to use Autograph and the dynamic nature of its commands. The SPs were asked to use Autograph to produce first (and successive) drafts of tasks to be used with the new cohort of Foundation students. Gradually, roles and relationships were established, with the researchers taking the lead initially and the SPs showing a growing confidence in participation in the project.

Design of tasks and the Design Meetingsan iterative process

The first design meeting (DM1) was a significant staging post in the developmental process. It involved the team and the expert in a discussion of the tasks which the SPs had brought to the meeting. Each task had a mathematical element and an Autograph element. The SPs presented their initial ideas, one of which involved a multiple-choice style task on addition of complex numbers based on the type of questions that they were used to on problem sheets, such as 'If z 1 =2+5i and z 2 =3-6i, what is z 1 +z 2 ?' The Autograph file displayed the complex numbers as vectors including also the correct answer (one variation was to include some incorrect answers with Foundation students asked to choose the correct one). The initial task made use of the commands within Autograph such as 'hiding' and 'unhiding' lines and objects and 'dragging' a complex number z to see what happened to a linked object such as z 1 +z 2 , z 1 z 2 or z 3 .

In SPs' initial designs, the mathematical element came first and the Autograph element followed. The SPs' expectation was for Foundation students to work out solutions on paper first and then use the software to view complex numbers in an Argand diagram and verify their results. Thus, the Autograph element was a tool to illustrate the mathematical relationship but it was left up to Foundation students to make the link. In the meetings the whole team discussed the tasks. In DM2, one member (TR1) commented that a task seemed rather "static". Discussion followed as to what aspects contributed to the task being static, and what a task that was more dynamic and exploratory might look like. TR1 wrote in a reflection afterwards: "So far, we are seeing quite 'static' taskstasks in which there is something to find, with a right answer." In this sense, the initial tasks mirrored typical questions on problem sheets that the SPs had been used to. TR1 continued writing, "A challenge now is to use the power of the software to offer more interactive, open-ended, exploratory tasks".

The word "static" acted as a tool in the mediational process. It was used briefly in DM2 with TR1 commenting that tasks might follow a "sequence from straightforward 'How do you work out this…?', straightforward questions like that, … but work towards something that is more openended and more exploratory, so that they [the Foundation students] have to actually do something themselves, to explore a situation" (DM2, 40:08). Discussions in the subsequent design meetings ensued. As team members expressed their view on a 'dynamic' task, a sense emerged of the mathematics of the task becoming more integrated with its representation(s) in Autograph and Foundation students engaging with these representations and the computer environment. In an endof-project interview when reflecting on the early design process, one of the SPs (SP1) recalled how the word "static" had been a catalyst for new ways of thinking about the tasks. He said that, as a Foundation student, he had been familiar with procedural tasks on the problem sheets. He and his partner (SP2) started from such a point of view to represent the tasks in Autograph with lecture notes and associated problem sheets acting as mediational tools, guiding their initial perceptions.

Working on the complex number tasks and communicating electronically, the two SPs had shared potential examples of dynamic tasks. As they exchanged ideas, they said, a clearer sense of possibilities emerged. These exchanges were themselves mediational; we interpret the oral and written words of the SPs as saying that different versions of the tasks acted as tools to promote new thinking and subsequent modifications. Discussion of static versus dynamic tasks in design meetings opened up new ideas and dimensions, again a mediational process. For example, the first iteration of the design process produced procedural tasks with expected answers while subsequent iterations shifted the nature of tasks to involve SPs in inquiry processes as they explored the questions asked. Team members expressed their ideas in different ways, encouraging a group perception of the nature of dynamical tasks (exemplified below). We see here a significantly new mode of mediation. The discussion of tasks allowed team members to question and share their own views with the SPs contributing alongside the other members of the team.

Finalising the tasks

The SPs developed six complex numbers tasks: adding and subtracting two complex numbers, multiplying together two complex numbers, multiplying a complex number and its complex conjugate and raising a complex number to a power. All final designs had their genesis in the ideas presented in DM1. SPs undertook major revisions of the tasks between the first and the second design meeting. Critiquing the tasks in the second design meeting, team members spent a lot of time discussing mathematical ideas, ways of viewing complex numbers (as a number, a point or a vector) and ways of solving. Thus in discussions involving all member of the team, the tasks were tools that mediated SPs' mathematical learning as part of the iterative design methodology and the mode of inquiry in our project. SPs offered Autograph files and mathematical explanations that gave insights into their development of mathematical sophisticationfor example, working fluently with vector representations in constructing complex number addition geometricallywith the teacherresearchers learning from this activity about student expectations and culture. TRs saw the tasks as designed to mediate Foundation students' learning of mathematics in tutorials and, to this end, constructed written instructions to accompany the Autograph files for the Foundation students.

In subsequent design meetings the team focussed almost entirely on technical aspects for improving the Autograph files. All tasks presented after DM2 had a 'dynamic' element such as 'dragging' a complex number around the screen until 'it fitted' a desired location, or exploring the relationship between the complex numbers by varying one or more of the linked objects. We provide examples of two tasks in more detail below. The first task centred on the addition of complex numbers. SPs designed the initial addition task by following closely the type of question they had encountered on problem sheets: Given z 1 and z 2 , find z 1 +z 2 and (making explicit the Autograph element) use Autograph to check that your answer is correct. After receiving feedback in DM1 the SPs set about 'reverse-engineering' the questions which SP1 described as looking at the answer and working backwards in order to design the task with the visualisation (Autograph) providing the information. Hence, for addition of complex numbers, the final task was set out to display three complex numbers, z 1 , z 2 and z where z was equal to z 1 +z 2 (see Figure 1). The instruction for Foundation students was to keep z 1 fixed and to move z 2 until z reached the position 6+5i (see Figure 2). At this point several lines on the screen moved as the linked object z 1 +z 2 moved dynamically with the movement of z 2 . The task was to find z 2 and determine the (arithmetic) relationship between z 1 and z 2 (addition) and its geometric representation (parallelogram law). Foundation students followed instructions on a separate hand-out on how to proceed with the Autograph files and could 'check' final answers by using the 'Unhide All' command in Autograph. The task in our second example centred on the multiplication of a complex number by its complex conjugate. Presenting their ideas in DM2 one of the SPs (SP2) said that "there are some [of our tasks] that are quite similar.

[But] we purposely put some in to try and be tricky" (DM2, 54:07). He was referring to the complex conjugate task. The idea originating from the design meeting was to display a complex number and its complex conjugate together on one screen as well as their product. One of the researchers (TR2) commented how nice this task was exclaiming "Oooh, I like that!" (DM2, 55:10) because you could see the angles cancelling outone angle taking a positive value while the other (while equal in size) was negative. Foundation students were asked to explore and comment on this task with TR1 suggesting to use the polar grid representation because "it would be quite nice to show the angles there" (DM2, 55:32). Hence, in the final design there were three complex numbers, z 1 , z 2 and z where z was equal to z 1 z 2 (see Figure 3). The task was to notice what happened to z when z 2 was moved into the position of the complex conjugate of z 1 (see Figure 4). In this task design the SPs made no reference to any numerical value for z 1 or for z 2 , or their product. This task has a very different character and is more general, inviting exploration.

In the end-of-project interview, one of the SPs (SP1) suggested that "these tasks invited a deeper understanding than standard [problem] sheets" and that while many students may know that the multiplication of a complex number and its complex conjugate results in a real number, they, as partners in the project -knew 'why' that was the case.

All six tasks were trialled with Foundation students over a two-year period with two different cohorts of students. Tasks were integral to the syllabus and formed part of the Foundation students' undergraduate learning experience. Analyses of these data are ongoing.

Summary of the project collaboration

The project involved students as partners in the design of tasks in complex numbers (as well as matricesnot reported here). Few examples in the literature exist but those that do point to the benefit of such a collaboration for both staff and students (see [START_REF] Bovill | Students as Co-Creators of Teaching Approaches, Course Design, and Curricula: Implications for Academic Developers[END_REF]. [START_REF] Duah | Students as Partners in Mathematics Course Design[END_REF] found as we did that the SPs gained a deeper understanding of the mathematics studied, in our case the mathematics of complex numbers. For example, SP1 recalled that "When I came across complex numbers in my third year [of degree study]…, it was immediately clear to me why the solutions appeared as complex conjugate pairs while many students had to spend time revising the principal." In our analyses we see multi-layered mediational actions. One layer relates to the mathematical learning of the SPs mediated by tools. These are the tasks themselves, the software used to create the tasks, the SPs' discussions when working together as a pair, and the inquiry mode in design meetings when all members of the team discussed drafts of the tasks.

A second layer relates to the work of the research team as a community of inquiry where we inquired into the design of computer-based tasks and into the learning processes of SPs and other team members. The SPs were drawn into the inquiry-based activity of designing dynamic tasks through the mediational nature of becoming familiar with Autograph and discussing its use together with other members of the team. Autograph acted as a tool for stimulating thinking and sharing of mathematical ideas and possibilities; it also acted as a tool for integrating SPs as partners in the task design process within the team.

We observed how the response of a team member using the word "static" was mediational in exerting a shift in SPs' perception of mathematics tasks: from static forms towards consideration of more dynamic forms of tasks using Autograph. Thus 'static' itself acted as a tool to promote a new perception of the nature of tasks. The word "static" was not planned. It emerged through the interactivity, and resulted in a stimulus for the SPs towards more dynamic forms of tasks.

At a third level, our engagement in research led to additional data being collected outside the design meetings. Interviews with SPs, where they were encouraged to reflect on their activity and learning at different stages in the project were revealing for the researchers. The interview process itself was mediational in allowing the researchers to perceive the stages in development of the SPs' thinking and perceptionfrom their experiences with questions on problem sheets towards their design of dynamic tasks to engage Foundation students in mathematical inquiry.

In summary, we have characterised learning as a mediational process involving a variety of tools.

Our analyses contributed to our understanding of the task design process where we see a new way of mediation that drew SPs into the collaboration as partners. Our project and analyses are contributing new knowledge in the field of students as partners in curriculum and course design and to the scarce body of literature in this area.

Figure

  Figure 1: Task 1 Figure 2: Task 1 with z 2 moved

Figure

  Figure 3: Task 4 Figure 4: Task 4 with z 2 moved

Autograph https://www.autograph-maths.com is a piece of software which dynamically links 'objects'. This allows the user to 'see' the result of moving/changing one objectother objects are linked and move/change with the first object.

Acknowledgements

This project is supported by HEFCE (Higher Education Funding Council for England) Catalyst Fund: Innovations in learning and teaching, and addressing barriers to student success A: Smallscale, 'experimental' innovation in learning and teaching. Project code: PK20.


Amanjot Toor
email: atoor@brocku.ca

Joyce Mgombelo
email: jmgombelo@brocku.ca

Chantal Buteau
email: cbuteau@brocku.ca

Post-secondary students' enactment of identity in a programming and mathematics learning environment

Keywords: post-secondary, identity, appropriation, community of practice

This paper draws from year one of a 5-year research study that seeks to examine how postsecondary mathematics students learn to use programming as a computational thinking instrument for mathematics. It focuses on how post-secondary mathematics students' identities as mathematics learners are enacted as they engage in a programming-based mathematical investigations and applications learning environment. Specifically, the paper offers a discussion of a case of one student's enactment of his identity while simultaneously learning to program and to use it for this kind of mathematical work. This paper highlights the importance of identity in learning mathematics and its role in the development of productive dispositions in learning to program for mathematics investigation and modeling.

Introduction

The Canadian federal government predicted that jobs in STEM fields will grow by 12% between 2013 and 2022, and that 35% of such jobs will be computer science-related ("Back to School", 2015). Few academic institutions attempt to prepare their graduates for the STEM workforce by integrating "Computational Thinking" (CT) in their programs. This integration of CT in STEM studies points to the need for research on what [START_REF] Healy | Charting the microworld territory over time: Design and construction in mathematics education[END_REF] describe as, "the complexities associated with the appropriation by the user of tools [such as programming]" (p. 66). This paper draws from year one of a five-year ongoing research study, funded by the Canadian Social Sciences and Humanities Research Council (SSHRC), entitled "Educating for the 21st century: Post-secondary students' learning to use computer programming for mathematical investigation, simulation, and real-world modeling" (referred to onward as 'progmatics'). The research addresses the need to better understand how post-secondary mathematics students come to appropriate programming as an instrument for mathematics investigations and applications. It is a naturalistic study that takes place in a sequence of three programming-based mathematics courses called Mathematics Integrated with Computers and Applications (MICA I-III), implemented in the mathematics department at a Canadian university in Ontario since 2001. In these courses, future mathematics teachers and undergraduate students majoring in mathematics learn to design, program, and use interactive computer environments to investigate mathematics conjectures, concepts, theorems, or real-world applications [START_REF] Muller | Learning mathematics through the design and implementation of exploratory and learning objects[END_REF]. The research project is framed by interrelated theories and concepts in mathematics education that articulate the complex mutual shaping of the learner (user) and artefact (programming) [START_REF] Buteau | Computational thinking in university mathematics education: A theoretical framework[END_REF], including Lave and Wenger's social theory of learning and theory of situated learning. [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF] analysis of situated learning is embodied in their productive concept of "legitimate peripheral participation." Legitimate peripheral participation is supported by systems of relationship in community between newcomers and old-timers, as well as relationships with outside communities and with other newcomers. Legitimate peripheral participation offers a two-way bridge: between the development of knowledgeable skills and identity-the production of persons-and the production and reproduction of communities of practice. In this sense, the newcomers become old-timers through a social process of increasingly centripetal participation, which depends on legitimate access to ongoing community practice. This paper focuses on identity. We are interested in exploring how post-secondary students (newcomers) enact their identities during their process of appropriation of programming as a mathematical instrument as mathematicians (old-timers) would do, as they engage (i.e., legitimately peripherally participate) in 'progmatics' tasks. Specifically, the paper offers a preliminary discussion of student enactment of identity while learning 'progmatics' in a first year MICA course through a case of one student, Roy (pseudonym).

Conceptual Framework

In this paper, the concept of identity is further articulated from three perspectives, namely selfefficacy, environment, and four faces of learner's identity, developed in our previous work [START_REF] Toor | Undergraduate students' perceptions of themselves as capable mathematics learners[END_REF]; see Figure 1. Researchers in the field believe that self-efficacy is learned, and that self-efficacy expectations are acquired through various sources including accomplishment, vicarious learning, verbal persuasion, and emotional arousal [START_REF] Bandura | Self-efficacy: Toward a unifying theory of behavioral change[END_REF]. Accomplishment as a source of self-efficacy refers to the way in which one's successful or unsuccessful experience on a given task increases or decreases the self-efficacy connected to that task. Another source, vicarious learning, can affect one's self-efficacy where one sees others-peers and classmates-succeed or fail on a given task, assessment, or course. Other sources of self-efficacy are verbal persuasion and emotional arousal. In verbal persuasion, beliefs about one's self are influenced by the messages conveyed by others. Emotional arousal refers to the stress and anxiety in a given task and its effect on self-efficacy. Identity is influenced by one's environment. In other words, identity is greatly formed by individuals' relationships with others from the past to present, stretching into the future [START_REF] Wenger | Communities of practice: Learning, meaning, and identity[END_REF]. As individuals progress through the post-secondary level, they develop a stronger sense of who they are as mathematics learners through their mathematics experiences, such as in lectures, classrooms, and seminars, interactions with teachers and peers, and in relation to their anticipated future (Sfard & Prusack, 2005). In addition to self-efficacy and environment, mathematical identity can also be explored from [START_REF] Anderson | Being a mathematics learner: Four faces of identity[END_REF] four faces of learning mathematics-engagement, relativity/imagination, alignment, and nature. Engagement looks at one's direct experience and active involvement of others within their environment and/or with the world around them. Relativity/imagination focuses on the images one has of him/herself and of how mathematics fits into the broader experience of life. The alignment face of learning mathematics refers to how one aligns their energies within given boundaries and requirements in response to their imagination face. Nature looks at the connection one makes of their natural characteristics. These characteristics refer to what nature provided one with from birth and over which one has no control.

Methodology

A mixed methodology approach is being used throughout the five-year ongoing study. The study uses an iterative design to refine and develop the research tools for yearly data collection and analysis. This paper draws from data that was collected in year one of the study, where six participants were recruited from the MICA I course. Data collected included each of the participants' four 'progmatics' project assignments (referred to as Exploratory Objects or EO), and reports, and semi-structured interviews with each of the participants after completing each of the four assignments (referred to as A1-A4). In addition, data collected included post-laboratory session reflections and a questionnaire. After each of the 10 weekly two-hour MICA lab sessions, participants recorded online their reflections on their learning during the lab session, L1-L10 (guiding questions were provided). All participants filled an online questionnaire (Q0) before the beginning of the MICA I course. This was followed by interviews where participants were asked to elaborate upon their questionnaire responses. The purpose of this questionnaire (and the first interview) is to serve as baseline information about participants' background of mathematics learning with technology.

Data analysis followed Cresswell's ( 2008) general principles of qualitative data analysis: preparing and organizing data, exploring data, describing and developing themes from the data. To begin the data analysis, codes were developed according to categories informed by our theoretical framework [START_REF] Buteau | Computational thinking in university mathematics education: A theoretical framework[END_REF]. Each participant's qualitative data were coded individually by two researchers, who then jointly completed a thematic analysis of the data. Themes were consolidated among six participants' analyses, leading to the development of sixteen overall themes. These themes were further regrouped into five meta-themes, one of which was identity. The first author was assigned to analyze one of the six participants' data, which was later further analyzed in order to explore the identity meta-theme specifically through the lenses of self-efficacy, environment, and four faces (see Figure 1) of learners. In this paper, we present and discuss the findings of the case of this participant, Roy's, enactment of identity in the MICA I course.

Findings and Discussion

Description of the case: Roy

Roy was in his first year of the Bachelor of Science in Mathematics program when he enrolled in MICA I. Prior to MICA I, Roy's experience with programming included programming in Julia (just the basics), R (just the basics), Maple, Wolfram, and Python, as well as HTML. In the first questionnaire (Q0), Roy noted that he did not have any prior knowledge of Vb.net (programming language used in MICA I), as he had, "never really heard about it" (Q0, 4). When asked to complete a statement about his feelings regarding the fact that the significant component in MICA I is computer programming, Roy responded "Very confident" (Q0, 5). When asked for his reason for his response, Roy said, "[m]ost languages have the same components just different syntax" (Q0, 5). Roy indicated that learning to program for mathematical activities is very useful because "it can be used in almost every area of science or just for figuring out proof that would take years for a human mind to do" (Q0, 6). According to Roy, students learn mathematics by learning how to derive the "equations of the proof and learning how to use the equations". He personally learns mathematics by, "teaching it to someone else" (Q0, 9). When asked, "What does 'doing mathematics' mean to you?", he responded, "It means solving a problem in the most logical way" (Q0, 10).

As part of the first MICA I assignment (A1), Roy was asked to select or state a conjecture, and to create and use his EO to explore it. When asked to describe his conjecture, Roy said:

Opperman's conjecture, [for every integer x > 1, there is at least one prime number between x (x -1) and x 2 , and at least another prime between x 2 and x (x + 1)] I did and that has to do with every real number greater than two…if you square it and then you minus that number and plus that number, in between those two zones there is at least one prime. (A1, 3)

In the second assignment, Roy implemented the RSA encryption algorithm, including encoding, decoding, and randomly generating keys. In the third assignment, Roy created an EO to explore the dynamical system based on a cubic (two parameters involved) and describe its behaviour. For the fourth EO, students choose a topic of their interest and work individually or in groups of two or three. Roy worked with a partner on the idea of trying to predict human population.

Roy's enactment of identity

Analysis of data from Roy indicates two ways in which Roy enacted his identity while learning 'progmatics' in MICA I: through ideal images and through his direct personal experiences [START_REF] Toor | Undergraduate students' perceptions of themselves as capable mathematics learners[END_REF]. Roy describes his perception of an ideal image of a mathematics learner in MICA I by describing his experiences in other non-MICA courses. When asked about his plan regarding taking the MICA II course, he responded that he was not planning to continue with MICA II as he did not find it engaging. When asked about the reason why he thought the course was not engaging, he responded:

With my straight math course, there were some things…Like being able to prove the volume of a sphere just through calculus. I find that very interesting cause then it's just like something you show people or… I found that interesting, whereas I didn't have that feeling with MICA I. (A4, 31)

It is clear from above that Roy's idea of a learner in MICA I is based on an ideal image of a learner in other non-MICA traditional courses, such as calculus. This ideal image of a learner in traditional courses manifests in a tension Roy experiences between how MICA courses are designed to engage students and how he perceives an ideal image of a learner in non-MICA courses. In Lab 10 of the course, students were asked to use their programs to explore the dynamical system based on the logistic function. When asked to reflect on his experience in the lab session, he noted:

The last class covered all of what I needed to start my project. This class didn't do much for me. This class should have been more about the assignment. I got some help with my project that I had almost finished but at times they refused to help since they were supposed to focus on the program we had to make to get 5% on the assignment. I understood the concept of a dynamical system but I never got why it was important. (L8-10, 10)

It is interesting to note that Roy did not take the opportunity to learn through inquiry. He struggles with the learning experience in the lab session because his ideal image informs him that, as a learner, the important thing is to complete the assignment and get a good grade.

Roy's self-efficacy

Roy's self-efficacy in terms of accomplishment in the MICA I course is influenced by his perception of his mathematics and programming ability. In terms of mathematics ability, Roy seems to perceive himself as someone who is better than his peers: "I wouldn't ask my peers, normally..., [I'm] ahead of them" (A3, 5.3). This perception seems to show Roy's hesitance to work or collaborate with peers. In addition, Roy's perception of his ability in mathematics separates his ability in mathematics and his ability in programming when he encounters a difficulty in a 'progmatics' task. For example, he states:

The first two codes we had to do were easy, the third left out too much information. The lab was to help us understand RSA and by the end I still didn't have a strong understanding of it. I didn't like that we were told to get into groups to test out our program...The most challenging part was understanding the RSA encryption. When the math is broken into parts none of it was very challenging or hard to understand. It was the major concept I still didn't get at the end. (L5-7, 6)

The experience of mastery influences one's perception of one's ability, where success and/or failure on a task have a direct impact [START_REF] Bandura | Self-efficacy: Toward a unifying theory of behavioral change[END_REF]. Roy's failure to deal with the third part of the lab due to the challenge it presented may influence his perception of his ability in programming. This part of Roy's self-efficacy in terms of accomplishment is also connected with his self-efficacy in terms of vicarious learning. On one hand, Roy perceives his mathematics ability as ahead of his peers, but on the other hand he perceives others' programming ability as more advanced than his own. In the interview, he said: No, it only worked…mine only works with numbers. Um, uh, the girl that I was sitting next to the previous one she did uh numbers she's like super advanced in computer science though (A2, 30). In other words, Roy's vicarious learning, a part of his selfefficacy, consists of him comparing his performance with others' and reflecting on his ability in mathematics and programming. Roy's selfefficacy in terms of emotional arousal could be seen in the choices that he made, the effort he expended, the perseverance he exerted in the face of difficulties, and the thought patterns and emotional reactions he experienced in MICA I tasks. When he was asked about why he was not planning to take MICA II, Roy said:

I like the in-class learning about the RSA and how different number theories, chaos theory and that kind of stuff. I didn't like doing the programming. Whenever I got an assignment, I did not find it fun or engaging to do, which I found in my math class. I really like doing the equations in the programming or in chemistry. I like doing that stuff but in this I didn't have much enjoyment in doing the programming. Whenever I got a problem…it just didn't…it just sucks. (A4, 29)

When Roy was given the freedom to select the topic for a task in the final assignment, he displayed excitement and satisfaction, which in turn allowed him to invest himself emotionally with the learning goal of the task. When asked about the outcomes of his final project, Roy said: "Really awesome. It was cool that we actually got the same result [as the World Infant Mortality Rate], I thought."

Roy's environment

One's environment is an essential element in building the identity of an individual as a member of the environment. [START_REF] Solomon | Safety in numbers: Mathematics support centres and their derivatives as social learning spaces[END_REF] claim that peer-group relations play an important role in the educational success of the individual at the post-secondary mathematics level.

As we have previously discussed, Roy seems to not like to ask his peers for help. This could be due to the fact that he views himself as "ahead" of others. Interestingly, Roy seems to be okay supporting others: "I did help another person with his conjecture by letting him see my code" (A1, 7). Also, he seemed to find it frustrating to work collaboratively with his partner in the final assignment. When asked about what was frustrating about working on his project, Roy answered:

Working with my partner, I guess. He wasn't a very independent partner, so he wanted us to sit beside each other while we do the project. I was busy doing other stuff, so I ended up doing the first part of the project but then he still wants to do the write up, which the write up only took two hours and I ended up doing the write up myself as well. (A4, 21)

Roy's relationship with faculty consisted of getting help from his professor and/or TA whenever he got stuck with a task.

Roy's four faces of learning mathematics

Identity in terms of each of the four faces of leaning mathematics exists as a way in which one comes to understand their practices and membership within the community of mathematics learners [START_REF] Anderson | Being a mathematics learner: Four faces of identity[END_REF]. Roy's enactment of identity in the MICA I course seems to be strongly influenced by his engagement. [START_REF] Anderson | Being a mathematics learner: Four faces of identity[END_REF] states; "much of what students know about learning mathematics comes from their engagement in mathematics classrooms" (p. 8). This enactment of Roy's identity in terms of the engagement face could be seen in the tension that Roy experienced in terms of the timing of when a task was assigned, which did not align with the timing of when some of the concepts needed to complete the task were taught in the class (even though they were eventually taught before the deadline): I had to do some research because in class we didn't, there was some of the stuff that were key concepts that we didn't actually, finish [before reading week] … one of the last concepts before the due date, was then given it was very key. (A2, 2). It seems that Roy has developed this identity from his experience of engagement in many mathematics classrooms where assignments are normally given after the required concepts or skills are covered. Another way Roy's identity in terms of his engagement with the MICA I course could be seen is in his perception of the computer language used in the tasks:

The program language itself isn't very good.

[Visual] Basic [is] pretty outdated-more or less obsolete. The only thing that [Visual] basic is used for is functions in Excel, like doing macros, which it's useful but it is not. It's getting outdated so I think the only part that was useful for me learning was the math that was taught in class and a little bit of how to implement that math into programming. But it's not going to be super useful in a job because I am not going to use [Visual] basic after I learn an entirely different programming language. (A4, 25)

This perception of the programming language seems to lead Roy to not identify with the MICA I environment and come to see himself as only marginally part of the MICA I community. In fact, when he was asked if he would take MICA II, he said no. Additionally, Roy's engagement with the programming language displays his identity in terms of his relativity/imagination face when he envisions how the computer language fits in with his future career goals. Further, when asked if he thought that doing mathematics in a programming setting was beneficial, Roy responded:

I think it would have to be more complex programming like when we were using Maple because that already has built-in functions, whereas this one, it just all, you have to do every bit of math yourself. When even today's time programming, you have to use other people's work to do more complex work. I think that eventually you would have to get to it. I think it's good for teachers but like for pure math students, that I don't think I got much out of it. (A4, 24)

This shows Roy's identity in terms of the alignment face of learning mathematics, where he chooses to align his energy toward a "progmatics" task that displays "more complex work" based on his perception of himself as a pure mathematics student. It is interesting to note that Roy's identity in terms of the nature face does not come forward in any of his interviews.

Conclusion

This paper draws from research that addresses the need to better understand how post-secondary mathematics students come to appropriate programming as an instrument for mathematics investigations and applications. [START_REF] Papert | Mindstorms: Children, computers, and powerful ideas[END_REF] proposes that every learner should program computers for personalizing learning and knowledge. He notes that at the heart of learning is 'appropriation' in the sense of making it 'yours.' This idea of taking the learning, and integrating it into one's way of being, thinking, and seeing necessitates an exploration of how one's identity plays a role in students' appropriation of programming (in the sense of transforming it as an instrument) for mathematics inquiry. This paper provides a beginning of this kind of exploration through a preliminary analysis of a case of one participant, Roy's enactment of identity in a first year MICA course. Roy came into MICA I with a perception of a learner in MICA I course based on his ideal image of a learner in traditional mathematics courses. Consequently, Roy's perception of this ideal image created a tension between the creative, inquiry-based expectations in MICA and his expectations of what the course should be. His perception of learning mathematics created a tension between his learning goals and the learning goals of MICA I-which are to learn to design, program, and use interactive computer environments to investigate mathematics conjectures, concepts, theorems, or real-world applications. This paper highlights the idea that learning to program for mathematics investigation and modeling requires not just developing skills and knowledge, but also developing an identity of a learner who engages in the inquiry-based tasks (peripherally) as mathematicians would do.

Figure 1 :

 1 Figure 1: Conceptual framework: Mathematical identity as capable mathematics learners (Toor & Mgombelo, 2013, p. 2458)