text:
Olov Lui

Elena Nardi
email: e.nardi@uea.ac.uk

From ritual to exploration: The evolution of biology students' mathematical discourse through mathematical modelling activities

Keywords: Theory of commognition, mathematical modelling in biology, mathematical discourse, routines, rituals and explorations

We report analyses from a collaborative, developmental research project between two Norwegian centres of excellence in higher education (MatRIC and bioCEED) in which biology-related mathematical modelling (MM) activities are introduced to biology students as a means to motivate their appreciation for, and competence in, mathematics. This phase of the project involved four sessions with 11 first-semester students. We report data and analyses from two activities: Yeast Growth and Digoxin. Our commognitive analyses trace the evolution of the students' mathematical discourse in two episodes, revealing a scaffolding story about the gradual transition from ritualized to exploratory engagement with MM, and pointing to the crucial role played by the teacher in this process. We conclude with discussing some implications of our analysis for the design and use of MM activities for students of Biology, and other non-mathematics specialists.

Teaching mathematics to biology students through mathematical modelling

Research into the mathematical needs of non-mathematics specialists is by no means new (e.g. [START_REF] Kent | Mathematics in the university education of engineers[END_REF]. Participants in many university-level studies are often non-mathematics specialists (e.g. engineers or pre-service teachers), but their specialism often remains a mere part of the study's backdrop [START_REF] Biza | Research on teaching and learning mathematics at the tertiary level: State-of-the-art and looking ahead[END_REF]. The relatively small but growing number of studies in this area (e.g., [START_REF] Gould | Mathematical Modelling Handbook[END_REF] have touched on issues such as: the double discontinuity between school, university and workplace mathematics; the challenges of teaching mathematical modelling at school and university levels; issues of confidence in and appreciation for mathematics; and, embeddedness of mathematics into other disciplines.

Within biology, mathematics is becoming increasingly important, placing new demands on the education of future biologists. In the US, for example, the recognition of these demands has led to two national projects focusing on developing undergraduate biology education (Brewer & Smith, 2011;[START_REF] Steen | Math & Bio 2010: Linking undergraduate disciplines[END_REF]. A potential problem with placing greater emphasis on mathematics in biology education is that "biology education is burdened by habits from a past where biology was seen as a safe harbour for math-averse science students" (Steen, 2005, p. 14). The project that we draw on in this paper aims to improve student appreciation for mathematics through helping them experience the relevance of mathematics to their field of study. It does so through exploring the suggestion made by several authors (e.g. Brewer & Smith, 2011;[START_REF] Steen | Math & Bio 2010: Linking undergraduate disciplines[END_REF] for greater integration of mathematics and biology in the curriculum. MM, as Brewer and Smith (2011) point out, is a basic skill within the 'core competencies and disciplinary practices' (p. 17) of biologyand a vehicle for improving student appreciation for the role that mathematics can play in scientific research.

Studies which have investigated the use of MM in university biology education (e.g. [START_REF] Chiel | From Biology to Mathematical Models and Back: Teaching Modeling to Biology Students, and Biology to Math and Engineering Students[END_REF] indicate that engagement with MM activities can contribute to more positive attitudes towards, and self-perceived competence in, both biology and mathematics. Concerning an integrated approach to mathematics and biology, [START_REF] Madlung | A study assessing the potential of negative effects in interdisciplinary math-biology instruction[END_REF] investigated whether such an approach might have adverse effects, such as breadth at the expense of depth, or mathematics anxiety problems. Two versions of a bioscience module, one of which contained a computational statistical element, were developed and offered to an introductory and an advanced biology class. Results showed no detrimental effects of an integrated approach but indicated that advanced level students were more able to benefit from it.

To examine the evolution of biology students' appreciation for, and competence in, mathematics as they engage with MM activities we espouse a discursive perspectiveparticularly that of the theory of commognition [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF]Nardi, Ryve, Stadler & Viirman, 2014, p. 183-5)according to which learning is change in one's participation in well-defined forms of activity (discourse). In what follows, we introduce those components of the commognitive perspective pertinent to the data analysis we present in this paper; we then present a sample of our data and analysis (two episodes from students' engagement with two MM activities, Yeast Growth and Digoxin).

The commognitive construct of routines: Explorations, deeds and rituals

According to the commognitive perspective, 'it is by reproducing familiar communicational moves in appropriate new situations that we become skillful discursants, and develop a sense of meaningfulness of our actions' (Sfard, 2008, p. 195). A routine is a set of meta-rules that describe a repetitive discursive action. Sfard defines three types of mathematical routines: explorations, deeds and rituals, with deeds and rituals presented as predecessors of explorations. A routine is called an exploration 'if its implementation contributes to a mathematical theory' (p. 224) (e.g. equation solving, defining and proving). Explorations involve the construction, substantiation or recall of narratives about mathematical objects. Routines that involve practical action (action resulting in change in objects, either primary or discursive, p. 241) are called deeds. Deeds are therefore different from explorations, which aim to effect change on narratives. Often, however, there are routines that "begin their life as neither deeds nor explorations but as rituals, that is, as sequences of discursive actions whose primary goal […] is neither the production of an endorsed narrative nor a change in objects, but creating and sustaining a bond with other people' (p. 241).

Sfard claims that rituals are a 'natural, mostly inevitable, stage in this development process ' (p. 245) and that the road to exploration often leads through ritual. The data and analysis sample we present in this paper examines this claim with a particular focus on the following research question: "What characterizes the use of routines by Y1 Biology students as they engage in MM activities?"

Aims, methods, data and participants of the study

The research design of our study comprises cycles of developmental activity (planning, implementation, reflection, feedback) which are theoretically informed, contribute to the emergence of theory and take place in a partnership between teachers (in this case, a university mathematician) and didacticians [START_REF] Goodchild | Critical alignment in inquiry-based practice in developing mathematics teaching[END_REF]. This ongoing project is a collaboration between two Norwegian centres of excellence in higher educationthe Centre for Research, Innovation and Coordination of Mathematics Teaching (MatRIC) and the Centre for Excellence in Biology Education (bioCEED). The aim of the project is to improve biology students' motivation for, interest in, and perceived relevance of mathematics in biological studies through the use of MM. The teaching took place at a well-regarded Norwegian university where biology students take one compulsory mathematics course, taught in the first semester, designed not specifically for the biology undergraduate programme but for students from about twenty different natural science programmes. Typically, in this university, there is little collaboration between the mathematics and biology departments, and few opportunities for focusing on issues specific to biology in the mathematics course. The data for this paper originate in four three-hour sessions with twelve volunteer students, nine female and three male. Activity during the sessions was video and audio recorded, both from whole-class and small-group work. Also, all written material produced by the students was collected. The teaching was conducted in English, but all student group work and most student contributions to group discussions were in Norwegian. The first session began with an introduction to the basic ideas of MM and to the modelling cycle. Students were then asked to work in smaller groups on modelling problems of varying complexity, but requiring only pre-calculus mathematics. The structure of the three remaining sessions was similar, but the initial exposition instead introduced specific types of models relevant to the problems given in that session. The data we draw on in this paper are taken from sessions two and three, and concern one group of four female students as they work on two different, but related, tasks, Yeast Growth and Digoxin. In the analysis, we examined the discourse of the students looking for recurring patterns that could be described as routines, for instance graph construction. Furthermore, we looked for signs indicating the type of routine use. For instance, we aimed to discern the motives (if any) students provide for their activity. Since the changes in discourse that we aim at charting in this paper take place gradually and over extended periods of time, they are difficult to exemplify through data excerpts within this short paper. Hence, in presenting the data analysis we have opted for offering instead a condensed, selectively detailed narrative account of key incidents illustrating these changes.

Mathematical modelling for biology students: Yeast Growth and Digoxin tasks

A large part of the first session was spent on a very open task where the students were asked to estimate the density of a rabbit population based on the number of roadkill rabbits along a stretch of highway. Reflecting on the session, the lecturer felt that the students had not been able to work productively enough on this task, and he decided to make the second session more structured. The first 45 minutes of that session were spent first on a follow-up of a homework task given at the end of the previous session, followed by a brief lecture on "steady-state box models" and, related to this, a very short task on pollution in a lake. Then the focus shifted to modelling change, introducing a task concerning the growth of a yeast culture in a petri dish (Yeast Growth). Contrary to the first session, however, the task was broken into subtasks that the students worked on for 10-15 minutes each, with whole-class summaries in between.

For the first Yeast Growth subtask, the students were given a first part of a table of data, taken from an old research paper [START_REF] Pearl | The growth of population[END_REF], with three columns (time, amount of biomass, change in biomass) describing the growth of a yeast culture. The students were asked to: analyze the numerical data in the table; plot the data and analyze the graph; suggest a simple model based on a difference equation of the form

n n p k p 1  
, where n p is the size of the yeast biomass after n hours, and, explain what their expectations would be regarding the predictive power of the model they constructed. The initial plan for the second subtask was to give students the second part of the table and ask them to: analyze this new data (noting the change in population per hour becomes smaller as the resources become more limited); plot the population against time, explore the shape of the graph and state what they would expect in the long run; and, calculate the expected value for "carrying capacity" in this case (noting that, based on the graph, the population appears to be approaching a limiting value, known in biology as "carrying capacity"). However, in the actual session (due to limitations of time) the students were instead given a non-linear model based on incorporating the carrying capacity: "We may estimate carrying capacity to be 665 (this value is not precise and your value may differ a bit k is a positive constant. What is your choice of 3 k ? (b) Now our objective is to consider the decay of digoxin in the blood stream to prescribe a dosage that keeps the concentration between acceptable levels so that it is both safe and effective. Design a simple linear model describing the following scenario: we prescribe a daily drug dosage of 0.1mg and know that half the digoxin remains in the system in the end of each dosage period. (c) Consider three different options where the initial one-time dose of medicine received by the patient is 0 a = 0.1mg, 0.2mg or 0.3mg. What are your conclusions? What would you recommend if you were this patient's GP?"

In Yeast Growth, the students were expected to find an approximately linear relation between the change and the amount of biomass, estimate the proportionality constant, and conclude that this rate of growth cannot continue indefinitely. With the additional data then provided, they were then expected to conclude that the growth decreases and the amount of biomass stabilizes at the carrying capacity of the petri dish, in this case 665. The students were then given a suggested non-linear model and were expected to check the validity of the model by finding the proportionality constant. Finally, they were expected to use the model to generate values that could be compared with the actual data. To do this, they needed to solve the equation ) 665 (

2 1 n n n n n p p k p p p       for 1  n p .
In Digoxin, in part (a) the students were expected to find a linear relationship between the change and the amount of digoxin remaining, and estimate the proportionality constant from the graph. In part (b) they were expected to construct a model of the form

1 . 0 5 . 0 1    n n a a
, and then, in part (c), use this model with the different initial conditions to realize that, in all cases, an equilibrium of 0.2mg will eventually be reached, leading to a recommended initial dose of 0.2mg.

In what follows we highlight two critical incidents, one from Yeast Growth and one from Digoxin.

Yeast Growth: Ritualized engagement with mathematical modelling

The group ignores the first question in the subtask, about analyzing the data in the table. Instead, their initial efforts concern the practical details around graph construction and data plotting: choosing the right scale for the axes, and the like. They do all work in parallel, constructing one graph each, on millimetre grid paper, but they still work collaboratively, discussing their work at every turn. The routines they are using seem familiar to them, but there is no evidence of any reflection concerning the purpose of the activity they are engaging in. The task requests of them to plot the data, and since this is something they know how to do, they do it. We see this as suggestive of ritualized routine use. After about ten minutes, however, they seem confused about how to interpret the data in the table: what does n p  actually mean? They start discussing how to fit a straight line to the data, but the relative inefficiency of their working methodputting a lot of effort into the design of the graph and all drawing their own copymeans that, in the end, they do not have the time to do this, let alone find the proportionality constant. In the first whole-class followup, the students quickly agree that the problem concerns exponential growth, but none of the groups have succeeded in finding the constant 1 k . It turns out that that they have constructed the wrong graph: plotting change against time, not against amount. We see this as evidence of ritualized routine use. Had the students engaged with the first question in the subtask, and reflected about the interpretation of the data, this mistake might have been avoided. Instead, the students resorted to a well-established routine for data plotting, using time as the independent variable. After this mistake has been clarified, the students are given additional data, and start discussing the validity of the model: is unlimited growth reasonable? The need for a revised model is established.

The work on the second subtask still mostly revolves around plotting the data, but now the group only constructs one plot. There is, however, some remaining confusion regarding the nature of the data: does n p  represent change or the actual amount? One of the students interprets the decrease in n p  as evidence of a population crash (a catastrophic decline in population), but the other group members point out that the decrease is in change, not actual amount: "But this is just the change, this is not the number of living cells." Thus, when engaged in biological discourse, they are able to reason in a meaningful manner about the interpretation of the mathematical symbols. However, the formulation of the task creates additional confusion. It explicitly mentions a nonlinear model, but at the same time asks for proportionality. Finding proportionality between the more complexly presented quantities in this task seems unfamiliar to the studentsand, since this is something not normally done in school, it probably is. Following the recent whole-class discussion, but contrary to what is written in the formulation of the subtask, the students do what they were expected to do in the first subtask, plotting the change n p  against n p instead of against ) 665 ( n n p p  . They thus struggle with fitting a straight line to the data, since their plot does not describe a linear relationship.

In the whole-class follow-up, it turns out that, yet again, none of the groups have been able to compute the constant 2 k , and, in the end, the lecturer provides the students with an estimated value and asks them to use the model they now have to compute a number of values of n p and to check the predictive value of the model. This turns out to be very confusing for our group, who are at a loss as to how to proceed: "I don't have a clue. I feel so stupid." The work they have been doing in both sessions so far has been geared towards constructing models, not validating them, leaving them unprepared for this way of using models. Furthermore, the routines they have been using have all concerned graph construction and plotting, and now they are supposed to compute values. After some initial confusion, they start doing computational work, but their nervous laughter and exclamations of surprise suggest that they have little faith in that what they are doing makes sense. Indeed, the different numbers they are juggling around suggest that they are making various computational errors. Also, they spend quite some time plotting the values that they obtain. We see this as indication that their routine use is still highly ritualized: they do certain things because they feel that it is expected of them, without having any clear rationale for why they are doing so.

Looking at the way the students engage with the Yeast Growth task, we conclude that what was intended by the lecturer as scaffoldingdividing the task into clearly delineated, smaller subtasksin fact amounted to restricting student agency. We propose that this restricted agency is connected to ritualized routine use. The formulations of the subtasks state explicitly what the students are supposed to do, and even suggest what specific routines to invoke (plot the data; estimate the constant). This decreases the need for reflection about what routines to use and why, thus inviting ritualized routine use. This interpretation is further supported by how they struggle when asked to perform a different set of routines, using a given model for substantiation purposes, rather than constructing a model from given data. This indicates to us that they are not yet using the construction routines in an exploratory manner.

Digoxin: Towards exploratory routine use

Although there seems to be a connection between the highly scaffolded format of the Yeast Growth task and students' ritualized routine use, we do not intend this to be seen merely as a cautionary tale. Indeed, looking at the students' work on the Digoxin task in session 3 four weeks later, there is evidence of progress towards making the discourse of growth model construction their own. The Digoxin task was presented as a whole, without the same amount of scaffolding as the Yeast Growth task. As in the second session, the group focuses their effort on constructing the graph, but has some problems interpreting the task because of unfamiliar terminology (e.g. difference equation).

Contrary to Yeast Growth, in Digoxin time is not included as a column in the table of data, thus minimizing the risk of students resorting to the "plotting against time" routine. Still, one of the students suggests using n as the independent variable, in an attempt to fall back on the familiar routine. After some discussion, they decide not to resort to the earlier default option of using time as the independent variable, and, using the graph and the table, they manage to find the proportionality accurately. This might be interpreted as an indication of what Sfard (2008, p.251) calls "thoughtful imitation". Having failed at constructing the requested plot in subtask 2 of Yeast Growth, and then being shown by the instructor what should have been done, they are now able to engage more fruitfully with this similar, but less complex, task. There is some additional confusion due to the formulation of the task (even though we are dealing with decay, the task still prescribes that 3 k should be positive). Here we see signs that the group have still not made the discourse fully their own, but rather are emulating the discourse of the teacher. Rather than trusting their own reasoning, they handle the problem in a manner familiar to many studentsthey adapt the answer to fit the teacher's expectations: "Let's just drop the minus sign." As for parts (b) and (c) of the task, they (as well as the other two groups) run out of time before managing to make much headway. Still, it appears as if the ritualized routine use when working on the Yeast Growth task has supported the students' pathway towards handling the Digoxin task in a more exploratory manner.

The path to exploration passes through ritual: Conclusions and ways forward

In this paper, we examine a case (Y1 Biology students' engagement with MM) of how new routines evolve, and particularly how discursants experience a step from ritualized to exploratory routines. The analysis points to the crucial role played by the teacher in facilitating this process. For instance, through the tasks presented to students, he influences their routine use, not only in the obvious way of suggesting what routines to use, but also in what way to engage with these routines. We have seen how a highly scaffolded task, which explicitly states what routines to invoke, might in fact invite ritualized routine use, whereas a less strongly scaffolded task might necessitate reflection about what routines to invoke and why, thus inviting a more exploratory engagement. At the same time, our analysis suggests that perhaps the ritualized routine use suggested by more scaffolded tasks might be a necessary step on the route towards exploratory routine use.

Per [START_REF] Sfard | Thinking as communicating. Human development, the growth of discourse, and mathematizing[END_REF], rituals are a 'natural, mostly inevitable, stage in this development process' (p. 245) and, recognizing this as so, recognizes fully the 'inherently social nature of human thinking and learning' (p. 245). Our claim here resonates with Sfard's: the road to exploration must sometimes pass through ritual. There is an inherent circularity in this evolutionary process: a learner 'could not possibly appreciate the value of the new routine until she was aware of its advantages; such appreciation, however, could only emerge from its use ' (p. 246). Furthermore, 'the deedenhancing mathematical explorations would sometimes involve new abstract objects, objects that can only emerge through implementation of this very routine' (p. 247) and this holds for the evolution of an individual's mathematical discourse as well as that of the field of mathematics as a whole. Discursive researchers -Sfard herself as well as Bakhtinposit that thoughtful imitation can be a transitory phase in transforming ritual into exploration (where imitation is meant as a nontrivial process that involves evaluation, assimilation, reworking and re-accentuation). Indeed, in the students' work on the Digoxin task, we have shown signs of such "thoughtful imitation".

Deritualization results in consolidated discourse, namely a 'well-developed network of interlacing, partially overlapping routines ' (p. 254). In this trajectory of growth there are at least two 'basic conditions for effective mediation': the principle of the continuity of discourse ('introducing a new discourse by transforming an existing one', p. 254); and, the principle of commognitive conflict ('the situation in which different discursants are acting according to different metarules' (p. 256)a potential source of discourse change, and thus of learning). In this paper, we sample evidence mostly of the former principle. Our scrutiny of the entire dataset is now gearing towards the identification of evidence of the latter. Further, we anticipate that rolling out more MM activities to a new cohort of Y1 Biology students will lend corroborative power to the conjectures we explore here. It may also provide an opportunity for a more extended testing out of using the commognitive framework towards analyses that inform pedagogical practice.

  of biomass between two measurements, and 1 k is a positive constant;

2k.

  What is your value?" For the third and final subtask, the students were asked to use the new model, with 2 k =0.00082, to compute values and compare them with the actual data ("Compute twelve values of n p using the formula and starting with the initial value Digoxin was the first task of session 3 and also concerned the modelling of change, in this case the decay in the body of Digoxin, a drug used in the treatment of heart disease: (a) For an initial dosage of 0.5mg in the bloodstream, the table shows the amount of digoxin n a remaining in the bloodstream of a particular patient after n days, together with the change the graph. Suggest a simple model based on a difference equation of the form

  ). As the number

										665		p	gets smaller and smaller as	p
										n	n
	approaches 665, we may adjust our simple linear model replacing it with a nonlinear model
	p 		k	p	665 (		p	)	or alike, if you have chosen 664 or 666. Test a new model by plotting
	n		2	n					n
	p  against	p	665 (		p	)	to check whether a reasonable proportionality is observed. Then, estimate
	n				n					n
	the proportionality constant

Proceedings of CERME10

Acknowledgements

We thank MatRIC for financial support, MatRIC colleagues Simon Goodchild, Yuriy Rogovchenko and Ioannis Liakos for their contributions to the study and bioCEED students for their participation.


domain: math shs

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

Elena Nardi

Discursive shifts from school to university mathematics and lecturer assessment practices: Commognitive conflict regarding variables

Keywords: Undergraduate examinations, assessment routines, commognitive conflict, variables

We report part of an ongoing study that aims to characterise lecturers' assessment discourse, especially on closed-book examinations. We focus particularly on lecturers' discourses that concern the transition from school to university mathematics, and we do so through highlighting one commognitive conflict regarding the use of variables in a task from an examination paper for a Year 1 module on Sets, Numbers and Probability taught in a UK mathematics department. We show evidence that the lecturer's assessment practices aim to facilitate students' avoidance of errors that are occurring because of said conflict. Here, we focus on students' scripts which illustrate that, nonetheless, students make errors and do not draw on the discourse of integers when deciding the domain of the variables used in the task. We conclude with a brief discussion of students' experience of commognitive conflict in the transition from school to university mathematics.

Introduction

Studies in mathematics education have focused on students' transition from secondary school to university (e.g. [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF]. Part of how students experience said transition is evidenced in their engagement with examinations during their first-year undergraduate modules. The nature of tasks in examinations has been studied using different theoretical frameworks (e.g. [START_REF] Tallman | A characterization of Calculus I final exams in US colleges and universities[END_REF]. Researchers have also examined lecturers' perspectives on examination tasks [START_REF] Bergqvist | University mathematics teachers' views on the required reasoning in Calculus exams[END_REF][START_REF] Tallman | A characterization of Calculus I final exams in US colleges and universities[END_REF]. In our study, we take a discursive approach in analysing examination tasks and lecturers' perspectives focusing on aspects of the transition from secondary school to university mathematics. This theoretical approach allows a characterisation of the mathematical discourse the students engage in when solving the tasks, and provides insight into lecturers' assessment practices and their expectations from students' responses.

In this paper, we analyse a task from a first-year module on Sets, Numbers and Probability offered in a UK mathematics department. We build on previously reported work from this study [START_REF] Thoma | A commognitive analysis of closed-book examination tasks and lecturers' perspectives[END_REF] in order to delve into lecturers' assessment practices facilitating students' transition to university mathematics in more detail. Specifically, we take the case of variables and the way these appear in a Number Theory task of the module's examination paper. In choosing this particular case, we take cue from previous works (e.g. [START_REF] Epp | Variables in mathematics education[END_REF] which note that variables have diverse uses in mathematics, some of which often create difficulties for students' transition to algebra and other advanced topics. Of particular relevance in this paper is the discussion by [START_REF] Biehler | Students' use of variables and examples in their transition from generic proof to formal proof[END_REF] about the difficulties with variables that students face.

In the part of our study reported here, we focus on a commognitive conflict concerning the number domains in the secondary school and university mathematics discourses. First-year mathematics undergraduate students' errors regarding variables, when engaging in a Number Theory task, provide evidence of this unresolved commognitive conflict. In what follows, we present briefly the theoretical framework of the study, the examination task and the study's participants. We then analyse the task and the interview data with the lecturer who posed the task. The interview data illustrate the lecturer's ways of assisting the students to avoid the errors. Finally, we highlight the errors evidenced in the student scripts despite this assistance, we present the case that these errors stem from aforementioned commognitive conflict and conclude with a discussion of findings and how these are embedded into the larger study.

Commognitive conflicts and assessment routines facilitating discursive shifts [START_REF] Sfard | Thinking as communicating: Human development, development of discourses, and mathematizing[END_REF] theory of commognition is a discursive approach that is being increasingly used in mathematics education [START_REF] Tabach | Communicational perspectives on learning and teaching mathematics: prologue[END_REF], as well as specifically in university mathematics education [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]. Mathematics in this approach is a discourse that can be described in terms of the following four characteristics: word use (e.g. divisor), visual mediators (e.g. algebraic symbols), endorsed narratives (e.g. definitions) and routines (e.g. proving). The routines are distinguished in deeds ("an action resulting in a physical change in objects"; Sfard, 2008, p. 236), rituals ("creating and sustaining a bond with other people", p. 241) and explorations ("producing endorsed narratives", p. 259) with the explorations further categorised in recall, substantiation and construction. Of particular relevance to our analysis here is the construct of commognitive conflict "the phenomenon that occurs when seemingly conflicting narratives are originating from different discoursesfrom discourses that differ in their use of words, in the rules of substantiation, and so forth." (p. 257). For example, a commognitive conflict may occur between the different relationships that the number domains have in school and university mathematics discourses. In school, number domains are introduced progressively. They are used for some time; and then subsumed in the next number domain. Positive integers are introduced first. Then, as the students learn about division, rational numbers follow. After a while, the discourse about unsigned rational numbers (which includes integers and rational numbers) together with negative numbers constitute the discourse on rational numbers (p. 121). The discourse of real numbers is introduced in later stages of the secondary school. In the university discourse, the number domains play a different role. They are presented as crucial abstract structures, the ring of integers and the fields of rationals, reals and complex numbers. In particular modules, the focus of study are those abstract structures and that is the case for Number Theory, where the domain of the variables is restricted in the discourse of integers.

In our study, we examine students' participation in the university mathematics discourse taking also into account the lecturers' perspectives, particularly their rationale for the choices of the examination tasks and the wording of the tasks. Our previous analysis of examination tasks and lecturers' assessment practices [START_REF] Thoma | A commognitive analysis of closed-book examination tasks and lecturers' perspectives[END_REF] highlighted the following assessment routines: giving directions to the students regarding the steps their response to a task may take; structuring the tasks and subtasks in ways that allowed students to secure and optimise marks as they progressed from one part of a task to another; and, providing guidance regarding expected justifications in the students' responses. Overall, these routines aim at assisting students' shifting from school to university mathematics discourse. Here, we aim to extend our previous analyses, also taking into account lecturers' assessment routines, which aim to avoid expected errors. We will make the case that unresolved commognitive conflicts are responsible for those errors. We are, therefore, starting to look in tandem at aspects of students' experience (here: commognitive conflicts relating to variables in a Number Theory examination task) and lecturers' perspectives onand intended practice relating to this experience. In the following, we outline the larger study our paper originates in; and, introduce the examination task and a brief commognitive analysis of it. We then offer an analysis of the lecturer's perspectives on the task, highlighting those assessment routines that aim to help students avoid errors relating to variables. Finally, we present the student examination scripts, which illustrate these errors and examine whether, and how, the unresolved commognitive conflict in the different relationships with number domains at school and university, may be seen as responsible for these errors.

The examination task and the participants of our study (lecturer and students)

The data of our study consists of examination tasks from different modules, lecturers' interviews on those tasks and students' scripts corresponding to these examination tasks. The focus of this paper is on one task from the module Sets, Numbers and Probability. This is a first-year module and has two parts: Sets, Numbers and Proofs taught in the autumn semester and Probability taught in the spring semester. The final examination includes six tasks: the first two are compulsory and the other four optional. One of the compulsory and two from the optional tasks are on Numbers, Sets and Proofs and the others on the Probability part of the module. At the final examination, the students have to solve both the compulsory tasks and three from the optional tasks. The total grade of the examination is 100 marks and the pass grade is 40 marks. This paper focuses on the compulsory task from the Sets, Numbers and Proofs part of the module (Figure 1). More specifically, in this part of the module, the topics covered are: Set Theory (notation, operations, cardinality and countability), Functions (introduction to functions, injection, surjection), Proofs (direct proof, proof by induction, proof by contradiction, proof by counterexample), Number theory (greatest common divisor, prime numbers, modular arithmetic) and Equivalence relations. The topic examined in this task is proof by induction and Number theory. Our analysis will focus on students' responses to the Number theory part of the task, task (ii). The model solution for part (ii) created by the lecturer for departmental use is in Figure 2. We note that this solution is not made available to the students. Fifty-four students took part in the final examination and the marks of their responses in this task ranged from 4 to 20, with the mean being 16.85 marks. The scripts of 22 students were selected by the first author to represent a variety of marks (Figure 3). The errors based on what we see as an unresolved commognitive conflict regarding variables were observed in 6 students' scripts. Here we report: first, analysis of the task and from the lecturer interview data; then, a sample from the analysis of the six students' scripts.

Task analysis and the lecturer interview

In part (i) of the task (Figure 1), the students are asked to engage in a substantiation routine (proof by induction). The wording of the task directs the students to this type of proof. In part (ii), the students are directed to engage first in a recall routine, giving the definition of a divisor, and then in a substantiation routine of a relationship describing the connection between the linear combination of a and b and the divisor d of a and b (iia). The students are then directed toward using the Euclidean Algorithm in (iib) and, in the last part (iic), they are expected to engage in a proof by contradiction (not explicitly mentioned in the wording of the task) in order to prove that the linear combination given is not divisible by 7see [START_REF] Thoma | A commognitive analysis of closed-book examination tasks and lecturers' perspectives[END_REF] for more detailed analysis of the task. For the purpose of this paper, we focus on the lecturer and student data corresponding to the second part of the task, (ii). During the interview, the lecturer said:

Lecturer: (…) my memory of school mathematics is that there was a lot of doing things but not necessarily a lot of formally defining things (…) And of course they came to university thinking that they knew what that [the definition of the divisor] meant but in this situation it really matters that they are restricting themselves to the-to the ring of integers (…) and all the symbols represent integers so what it means to divide is very different than if they were working with fractional numbers or something where they could write a over b and things like this.

Our commognitive analysis highlights the differences in the lecturer comments between the school discourse and the university discourse and, more specifically, with regard to the routine of defining and the importance of understanding that "all the symbols represent integers".

He highlights the differences between what the students are used to and what they are expected to do at university level. Our analysis sees this as the differences between the two discourses: on the one hand on the focus of the routines; on the other hand, on the constraints of the different discourses that exist within the mathematical discourse at university level. More specifically, the students, working with this definition have to restrict their work on integersand not on rational or real numbers. The lecturer, then, speaks about the nature of the symbols involved. We recognize this comment by the lecturer as foreseeing students' errors in the case that they treat the divisor as a rational instead of an integer, drawing on the discourse of rational numbers instead of the discourse of integers. Integer numbers are rational numbers, and making the distinction between the twoand then opting for working within the discourse of integersis not something that these students have been routinely working with in school. This shows that the lecturer expects students to use division in the way that they were taught in school, instead of considering the abstract structure that this task is asking them to restrict their activity in. In the excerpt that follows, the lecturer explains his assessment routines which aim to assist students with avoiding errors that are happening because of what we see as a commognitive conflict: subtask (ii) is gradually structured as first asking the definition, then, substantiating a narrative that draws on this definition, engaging with the Euclidean algorithm and, finally, combining all the above to engage in a proof by contradiction. He comments on the purpose of this gradual structure as follows:

Lecturer: (…) what's being tested here is their ability to write down something formally and correct. And I would worry that if I didn't prompt them to write down formally the definition of what it means for one integer to divide another in the exam, in the pressure of the exam and so on, then their answers could start looking very 'creative' at the second part and they might start writing down fractions.

Therefore, he aims that the gradual structure aids students towards achieving the expected solution.

This can be thought of as a way of helping students avoid experiencing errors stemming from what we labelled as commognitive conflict, where a and b, would be treated by students as rational numbers, instead of integers: this gradual structure serves as a reminder that they should restrict themselves in the discourse of integers. Additionally, the analysis shows that the lecturer stresses the routine of justification and the rigor of the university discourse compared to the school discourse, a further staple of the transition that these students are at the moment experiencing [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF].

Lecturer: (…) the only challenging part would be the last part, the part that requires some thought and they need to-to sort of understand or remember that somehow it relates to what happened up here [shows parts (iia) and (iib)] (…) to remind them that I want them to explain why they are answering what they are saying.

In the excerpt above, the lecturer comments on the challenge of the (iic) part of the task and the purpose of the prompt "Explain your answer carefully". In this part of the task, the students have to engage in a substantiation routine that is based on the endorsed narratives that they have created for parts (iia) and (iib). The lecturer suspects that the students may omit justifying their response regarding the substantiation of the given relationship and aims that this prompt will help them do so.

From the above, we see that the lecturer has identified students' difficulties with the nature of the variables being used in this task. Our commognitive analysis sees this as evidence that the lecturer appears alerted to this difficulty as a difference between the school and the university discourse. The students, during their school years, gradually moved from the discourse of the natural numbers, to the one of the integers, then to the rational numbers and finally to the reals. Now, in this task, they are asked to endorse the discourse of the integers, which is subsumed in the discourse of rational numbers, within which they have been performing division of numbers in school. We now turn to students' responses, which evidence that, despite aforementioned aid provided by the lecturer, errors The students' scripts Student [03] first communicates the relationship between the divisor d and a using written verbal visual mediators. In the second part of (iia) the student writes, using symbolic mediation, that d is a divisor of a and d is a divisor of b. However, in the symbolic realisation of the divisor, the student deploys fractions, with d being the numerator and a and b being the denominators. This way of writing that d divides a can be seen as a translation of the written verbal mediator into a symbolic mediator without taking into account that the fraction line means that the denominator divides the numerator. In the case of the task, this division would result in a non-integer number. This way of writing signals that the student may see m and n (written on the right hand of the script) as numbers and not as integers.

Then, the student writes the relationship between the symbolic mediators m, d and a and concludes that d is equal to the product of m and a. The student used all the symbols given in the wording of the task to produce a narrative that involves fractions. Fractions could be part of the discourse of rationals, and this task asks the students to restrict their activity within the discourse of integers. We see the appearance of fractions and the absence of the constraints regarding the variables as evidence of the commognitive conflict regarding the relationships between the number domains in school and university mathematics discourses. Unclear meaning making regarding the object of a divisor is evidenced as the student starts by explaining that d is a factor of a, then engages in the discourse of rationals concluding that d=ma but then saying that the product 2d has d as a divisor. Having concluded that the greatest common divisor is 3 (Figure 4) using long division -and not the Euclidean algorithm -the student writes 123m + 45n =3. Then s/he divides all the terms of the equality by 3 and takes different cases where the new equality is true.

In doing so though, [03] does not take into account the nature of the variable symbolic mediators and the variables become rational numbers. Also, there are multiple values in the rational numbers that satisfy this equality as can be seen in the response (Figure 4). Finally, in the last part of the task, the student responds affirmatively that there are "integers" s and t. However, the s and t s/he gives are

Symbolic visual mediation and the transition to university mathematics

Looking at the model solution produced by the lecturer (Figure 2) and the wording of the task (Figure 1), we can see that there are four different instances where the students have to define the symbolic mediators they use: first, in the definition of the divisor where an integer is introduced to illustrate the relationship between a and d; then, in the narrative which connects a, b and their divisor d; next, in the substantiation of the relationship between the linear combination of a and b and their divisor d; and, finally, in order to prove by contradiction that a linear combination of a and b is not divisible by 7. In the last instance, the symbolic mediators on both sides of the equality have to be integers and, as 7 is not divisible by 3, the contradiction occurs. The wording of the task, where the lecturer stresses that all the variables in this part of the task are integers, and the structure of the task aim to signal to the students that they have to be working within the domain of integers. Our analysis suggests that lecturers design the tasks being aware of the students' potential errors regarding the variables belonging to different number domains and thus adds to [START_REF] Bergqvist | University mathematics teachers' views on the required reasoning in Calculus exams[END_REF] results on what lecturers take into account when designing assessment tasks. The data from the students' scripts shows however, that, in six out of the twenty-two analysed responses, students' errors signal an unresolved commognitive conflict that relates to making a distinction about the nature of the variables. Engaging with this distinction is not a routine that students are typically engaged with, at least in the UK secondary classrooms where the students participating in this study have been educated in: number domains are introduced progressively and students are simply expected to always work within the latest introduced number domain. However, in the university, the numerical context of a task may differ from task to task and the students are expected to be able to swiftly identify the domain that is appropriate for each task and work within it. We approach this issue therefore as a non-negligible aspect of the students' transition from school to university mathematics. Our results resonate with those in [START_REF] Biehler | Students' use of variables and examples in their transition from generic proof to formal proof[END_REF] study. They found that, frequently, their participants would use symbols without providing information regarding the domain of the variable; in our case, not attending to such information results also in leaving parts of the task practically not answeredespecially in the cases where the explicit request for "integers s and t" in (iic) receives non-integer responses.

Figure 1 :

 1 Figure 1: Compulsory task on Sets, Numbers and Proofs

Figure 3 :

 3 Figure 3: Marks from all the students' scripts

  illustrating this commognitive conflict were not avoided. Of the twenty-two student scripts analysed, six contained said evidence ([01], [03], [06], [11], [16], [17]).

Figure 4 :

 4 Figure 4: Student [03]'s script to (ii)

Figure 5 :

 5 Figure 5: Student [16]'s to (iic)

Proceedings of CERME10


text:
Sepideh Stewart
email: sstewart@math.ou.edu

Clarissa Thompson

Noel Brady
email: nbrady@ou.edu

Navigating through the mathematical world: Uncovering a geometer's thought processes through his handouts and teaching journals

Keywords: Embodied, symbolic, formal, Tall's three worlds, Algebraic Topology

In this case study, we examined a mathematician's thought processes as he taught a course on Algebraic Topology. The mathematician shared his teaching-related journals over an entire semester and discussed them in depth during weekly meetings with the research team comprised of a mathematics educator, a cognitive psychologist, and a postdoctoral fellow in mathematics. Concurrently, one of his students took detailed journals on most lectures. The team employed Tall's three worlds of embodied, symbolic, and formal mathematical thinking as various lenses to gain insight into the mind of the working mathematician as he taught a course on Algebraic Topology. Although the analysis of the data from the instructor's journals and the in-depth discussion of the journals during the team meetings revealed his thought processes, the 35 handouts that he prepared, aligned with students' needs, provided the most insight into his way of thinking.

Introduction

Communicating advanced mathematical ideas to university students is a challenging endeavor. It is a common and accepted practice for many mathematicians to write definitions, theorems and proofs on the board and make comments as they introduce mathematical ideas to students. Thurston (1994, p. 162) asked the question: "How do mathematicians advance human understanding of mathematics?" In interviewing 70 research mathematicians, Burton (1999, p. 31) found that "intuition, insight, or instinct" was seen by most mathematicians as a necessary component for developing student knowledge. Although we have some literature on examining mathematicians teaching practices (e.g. [START_REF] Fukawa-Connelly | A case study of one instructor's lecture-based teaching of proof in abstract algebra[END_REF]Stewart, Schmidt, Cook & Pitale, 2015), research on what takes place in the minds of mathematicians and their students is still scarce [START_REF] Speer | Collegiate mathematics teaching: An unexamined practice[END_REF]. [START_REF] Dreyfus | Advanced mathematical thinking processes[END_REF] believed that, "one place to look for ideas on how to find ways to improve students' understandings is the mind of the working mathematician" (p. 29). In this study, we examined a mathematician and one of his students' daily thoughts on Algebraic Topology. The overarching goal of this research was to investigate the way mathematicians and students think about mathematics and the possible pedagogical challenges that they may face.

Theoretical framework

In this study, we employed [START_REF] Tall | How humans learn to think mathematically: Exploring the three worlds of mathematics[END_REF] three-world model of conceptual embodiment, operational symbolism, and axiomatic formalism in order to describe an expert geometer's ways of mathematical thinking. In Tall's view, the embodied world involves mental images, perceptions, and thought experiments; the symbolic world involves calculation and algebraic manipulations; the formal world involves mathematical definitions, theories and proofs. [START_REF] Tall | The transition to formal thinking in mathematics[END_REF] asserts that, "all humans go through a long-term development that builds through embodiment and symbolism to formalism " (p. 23). Bridging between the embodied and symbolic worlds is of critical importance. Tall emphasizes that "a curriculum that focuses on symbolism and not on related embodiments may limit the vision of the learner who may learn to perform a procedure, even conceive of it as an overall process, but fail to be able to imagine or 'encapsulate' the process as an 'object' (p. 12). [START_REF] Tall | The long-term cognitive development of different types of reasoning and proof, presented at the Conference on Explanation and Proof in Mathematics: Philosophical and Educational Perspectives[END_REF] 3) declared that the word 'world' is carefully chosen and has a 'special meaning' in order to represent "not a single register or group of registers, but the development of distinct ways of thinking that grow more sophisticated as individuals develop new conceptions and compress them into more subtle thinkable concepts". As Dreyfus (1991, p. 32) declares "One needs the possibility to switch from one representation to another one, whenever the other one is more efficient for the next step one wants to take… Teaching and learning this process of switching is not easy because the structure is a very complex one." [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF] claims that many students do not have the cognitive framework to perform the switch. The overarching goal of the first author's research program is to investigate the ways in which mathematicians move between modes of thought and facilitate their students' movements among these modes. Tall's theoretical framework accounts for movement between the worlds of mathematical thinking and is a suitable scaffold for this research. Through our collaborations, we are beginning to understand how the minds of working mathematicians operate. Thus, we hope to evolve Tall's theory and use it to analyze rich data from many mathematicians. We endeavored to investigate the following research questions: (a) How did the instructor and student move between the formal, symbolic, and embodied worlds? (b) How did the instructor use handouts in order to help students move between the worlds?

Viewing Homology Theory through three lenses

The mathematician appreciated the developmental aspect of Tall's framework in which one begins with a very embodied view of the world around them and then moves with increasing age and experience to a symbolic view as one matures. However, he took issue with the "formal" viewpoint as the ultimate destination of this progression, especially since formal from a math perspective (i.e., set theoretic axioms, definitions, and formal deductions from such a system) is not the way mathematicians think. One can program a computer to generate (i) statements and (ii) formal proofs of these statements within an axiomatic system. In what sense can we say that the computer is discovering a mathematical theory? Humans use a lot more when they discover/develop a mathematical theory. Among all the myriad of possible statements that could be true in this formal theory, mathematicians choose certain ones (usually as a result of intuition and metaphors possibly supported by symbolic computations to garner evidence for the particular statements) called conjectures, and they try to prove them. Instead, the mathematician made sense of Tall's worlds by thinking of them as three lenses that allowed him to view a mathematical reality/world. Figure 1 illustrates his views of Homology Theory through these lenses. The embodied lens allows the mathematician to see cycles as geometric objects, and similarly for chains and various topological spaces. The symbolic lens allows the mathematician to use symbolic computation tools such as the Mayer-Vietoris sequence and produce symbolic computations (e.g., the homology of the 2-torus). The formal lens allows the mathematician to work with the Eilenberg-Steenrod axioms and results which can be derived formally from these axioms. The geometric side of topology spans the embodied and symbolic lenses. Algebra, primarily in the form of Homological Algebra, spans the symbolic and formal lenses. We can think of similar lenses, for example, in medicine. One can look at a patient with one's eyes, take an x-ray or an MRI of the patient, view the patient through an infrared lens, listen to the patient's heart and lungs etc., talk to the patient about their symptoms, and draw blood and perform tests. These are different modes of gathering information to give a practitioner a more complete picture of the patient.

Method

The participants. Our qualitative narrative study investigated the ways an expert mathematician navigated among Tall's worlds of mathematical thinking. The research team consisted of four members: a mathematics education researcher; a geometer, Noel Brady (the course instructor); a cognitive psychologist; and a mathematics postdoc familiar with both Algebra and Topology.

The course. The Algebraic Topology course was the first in a two-semester sequence of graduate courses. There were eight graduate students enrolled in the course. During class meetings, Noel often passed out handouts to help students follow along with the topic of the day. He believed some topics covered in the chosen textbook [START_REF] Hatcher | Algebraic Topology[END_REF], needed to be handled in a more detailed fashion. "Hatcher is a bit fast and loose with all of this". Students actively solved problems together in groups, or individual students were called to the board to complete problems. Noel also helped to revive an extracurricular, student-led topology seminar.

Data and procedures.

In this study, we analyzed a geometer's thought processes and actions while he taught Algebraic Topology over the entire Fall 2014 semester. One source of data was a series of teaching journals that contained Noel's reflections on his preparations for class, what happened during class, as well as some descriptions of the events that took place during office hours and a student-led topology seminar. The research team read his daily journal entries and discussed them during weekly research meetings. During these meetings, we asked Noel further clarification questions, and he often drew additional pictures as he described the course content. These meetings were audio recorded and later transcribed and will be used as a source of data. Another source of data came from one of Noel's graduate students who also wrote daily journals. These student journals provided an additional perspective into the events that took place in class. In addition, further data came from 35 handouts that Noel provided.

Coding scheme. The data were analyzed thematically, meaning we mainly considered the key issues that emerged in this study. The main themes and their sub-categories were identified and coded (see Figure 2). In addition to assigning codes for the three worlds of mathematical thinking, we also created codes for movement between the worlds (e.g., embodied-symbolic). While coding Noel's journals, at times we assigned multiple codes for a particular instance. For example, an excerpt could be coded with both the "Teaching" and "Tall's Worlds and Movements" codes.

In the following section, we give a glimpse into the analysis of Noel's journals, as well as instances from the student's journals to illustrate how the student perceived movement among the worlds.

Results and discussion

Figure 2 shows the percentage of total qualitative codes that were applied to excerpts from Noel's teaching journals. The main theme of Tall's three worlds of mathematics comprised 25% of the total codes. Teaching was the main theme that was coded the most (46%) in Noel's journals. Reflections included 20% of codes, and codes pertaining to students involved 9% of the total codes. Analysis of the data revealed ample evidence that Noel repeatedly navigated between the three worlds of mathematical thinking. Below, we provide examples from our analysis of his teaching journals and a student's journals to illustrate movement between worlds.

Moving between embodied (intuition) and formal worlds

According to Noel, this may have been the type of movement that the students found the most challenging: "There were a lot of questions about how to pass from an intuition to a formal proof (many of these examples used techniques/results from quotient spaces)."

The analysis of the student's journals showed his concerns regarding the proofs. This excerpt was taken from one of his journals at the beginning of the semester: Dr. Brady's way of proving results that come from concepts we're already supposed to have come across before his class is nice, I think. He gives a detailed outline verbally, which is helped along visually by his pictures and hand gestures. For the most part I'll watch without writing almost anything, but I definitely get a lot out of reviewing concepts in this way. I'm a little worried, however, that when we get to brand new material Dr. Brady's way of proving results might remain in the same verbal/hand-waving/picture-drawing style and that this won't be enough for me to follow the proof right there and then. He tends to speak and write very quickly, which is fine when we're reviewing. But since I can either copy furiously what he writes on the board or listen to him, but not both, this could become a problem.

Noel refused to give students proofs that were pre-packaged. More specifically, he wanted to provide students with intuitions/pictures that would help them understand the conceptual nature of the proof and ultimately lead them to it. In one of the research meetings Noel said: I mean I can give verbatim proofs of things or give them more detailed proofs where Hatcher leaves stuff out, but that will just waste time and I'll reproduce a book and nobody will get anything out of it. So I've given them intuitions, enough of an intuition that they can tag that together with a formal proof.

Later in the course the student wrote: "I've seen van Kampen's theorem before, but Dr. Brady's fromthe-ground-up approach was very nice in that it showed us through comprehensive diagrams just where exactly the theorem comes from."

Movement between embodied and symbolic worlds

Noel discussed moving from embodied demonstrations (e.g., rope trick) to having students complete symbolic examples (e.g., right-angled Artin group (RAAG) complexes and the torus knot spine):

More of the same. I connected back to several examples from the first week and from the intro to 𝜋 1 . The pair of circle links in 𝑆 3 example (a.k.a. the rope trick) and the RAAGs. This seemed to go ok. Mentioned again that RAAGs are deceptively simple looking groups, but that their subgroup structure is surprisingly rich. In particular, Bestvina-Brady (1997) and Agol-Wise (2012) contain very surprising results about subgroups of RAAGs. Told them that the story is still ongoing. Left off with an example of a torus knot spine (Hatcher).

The handouts

Analysis of the 35 handouts that Noel created illuminated the motives behind some of his thought processes and movement between worlds. These handouts gave the team a more authentic glimpse into the mind of the mathematician than the teaching journals that Noel regarded as self-critical (selfaware). Figure 3 shows the first two pages of a handout Noel created on barycentric subdivision. The Thematic Working Group 14 start of the handout contains the formal definitions of "barycenter" and of "barycentric subdivision." These definitions build on a previous definition (and square bracket notation) of an n-simplex. The definition of "barycentric subdivision" is recursive (i.e., defined in terms of lower dimensional versions of itself). The rest of the two pages is devoted to building students' intuitions for these definitions. At the bottom of the first page, two embodied examples are provided which demonstrate how to unwrap the recursive definition to determine the barycentric subdivision of a 1-simplex (a line segment) and of a 2-simplex (a triangle). This is followed by an exercise which asks the student to add another layer of recursion and describe the barycentric subdivision of a 3-simplex (a triangularbased pyramid). This is a very embodied example. At this stage, Noel hoped that the student should be gaining confidence working with the recursive definition and should be developing an intuition that the symbolism will work in higher dimensions where one's embodied intuition fails. The second exercise asks the student to iterate the barycentric subdivision process for a 2-simplex. Again, this is very embodied and can be drawn easily in the plane. Noel pointed out that developing an intuition about iterated barycentric subdivisions is important since they will form the heart of the proof of the "locality result" and the proof of the "excision theorem" for singular homology later on in the course. The two Roman-numeral-labeled observations at the end of page 2 build on the student's embodied intuition of the behavior of iterated barycentric subdivisions in dimension 2 (obtained from doing exercise 2). They motivate the statement of the theorem that will be given and proven on subsequent pages of the handout. They also alert the student to the fact that some care will have to be given to the proofs on the subsequent pages. This is particularly so, since these proofs will hold in arbitrary dimensions.

Noel pointed out that, from a textbook perspective, one can skip straight from the definitions of barycenter and barycentric subdivision to the statement and proofs of the theorems about the behavior of the diameters of simplices under iterated barycentric subdivisions. Nothing in the logical progression and framework would be lost. However, students' intuitions would be lacking (save for the rare student or two who can do some mental exercise equivalent of the examples, exercises and observations of these two pages.). This handout is one of a sequence of three handouts. These handouts get increasingly symbolic and abstract. Eventually, the results contained in the last handout are just what are needed in the formal proof of the "locality theorem" (and the "excision theorem") of singular homology. At this stage, the proofs are very symbolic and far removed from geometry. It is good that students have developed an embodied intuition about iterated barycentric subdivisions, so that they have concrete models in their mind for how excision works on the geometric level of chains.

Concluding remarks

This study revealed that Noel viewed Algebraic Topology through all three mathematical lenses (embodied, symbolic, formal), and his handouts provided his students with opportunities to view the course material through these different lenses as well. In one of the research meetings, Noel mentioned:

When I think of the mathematical world of algebra I have examples in my mind, many of which are very embodied, and many of which are symbolic, I also know the axiomatic definitions of concepts in this world like "group," "ring," "field" etc. So, when I think of the world of algebra all three lenses (embodied, symbolic, formal kick into gear. Likewise, for the mathematical world of topology.

Our research team, comprised of a mathematician, a mathematics educator, and a cognitive psychologist, are working together to apply and evolve Tall's theoretical framework by analyzing the teaching journals of mathematicians and their students. We have come to realize that the embodied, symbolic, and formal worlds blend together as applied to Algebraic Topology; it is often not clear where one world starts and another world ends. In addition to thinking about problems from the ESF perspectives, mathematicians often translate a problem from one area of mathematics (e.g. Topology) to another (e.g. Algebra). This translation is achieved using mathematical constructs called functors. Noel used the analogy of a translator to describe the mathematical notion of a functor. When a statement of a problem is translated from one language to another, some of the details may get lost in the translation. Perhaps this loss of information has an unexpected benefit; the simpler formulation of the problem in the new language might allow for new insights or intuitions to be gained, and perhaps even for a solution to the original problem.

Noel talked about functors in his journals, and described how they are used to solve problems in topology by first translating then into algebra problems:

We introduced some other situations where Algebraic Topology functors might help solve topology problems, and mentioned that the homology functors would be introduced and studied in the course.

We are using analogies and metaphors to communicate with one another as we attempt to understand the pedagogical decisions of the working mathematician. As Thurston (1994, p. 168) asserted: "we mathematicians need to put far greater effort into communicating mathematical ideas. To accomplish this, we need to pay much more attention to communicating not just our definitions, theorems, and proofs, but also our ways of thinking...we need to appreciate the value of different ways of thinking about the same mathematical structure".

Figure 1 :

 1 Figure 1: The three-lens view of Homology Theory

Figure 2 :

 2 Figure 2: Qualitative coding scheme

Figure 3 :

 3 Figure 3: An excerpt from Noel's handout

Proceedings of CERME10


text:
Kirsten Schmitz
email: kschmitz@math.uni-bremen.de

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

Access to conceptual understanding -Summer courses for linear algebra and analysis after the first semester

Keywords: Mathematical concepts, advanced mathematical thinking, tertiary level mathematics

. In particular, we investigate the potential of this course design to contribute to the motivation of the students and to their ability to engage in concept construction. These qualitative preliminary results will be used in further research to quantitatively assess the effect of these courses as well.

Introduction

A recurring problem in undergraduate mathematical education is students' difficulty in coping with the transition from school mathematics to the higher paced, more demanding and more formal mathematics at university level. This can be seen in the high dropout rates of math students in the first semesters and the poor exam results. In particular, some students require more time to learn than the time span determined by the lecture course, and find it hard to catch up with their peers for the next course. The previous approach of the mathematics faculty at the University of Bremen had been to install weekly extra tutorials for the courses in Linear Algebra and Analysis, parallel to the corresponding lecture courses. Even though these were conducted by senior student tutors, they yielded poor results. Few students attended, and there was no perceived positive effect on motivation or performance with regard to concept building. Limited time during the semester was stated as the main obstacle to continuous attendance.

In response to this, a new approach consisting of summer courses in Linear Algebra and Analysis was developed and implemented by the authors. These were voluntary courses that took place within two weeks between the first and second semester, after the students had not only had the weekly feedback from their homework groups, but also some individual feedback on their progress by the end-of-semester test. In addition to the goal of enabling the students who had failed (or scored low in) the first test to pass the retake, the courses were intended to foster the students' motivation and help them create appropriate concept images [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] of the core concepts of each course. This is assumed to be of vital importance, in order for them to be able to profitably take part in the follow-up courses in the second semester. From the experience gathered in the first implementation of these courses, we intend to give a preliminary evaluation with respect to these two factors. This will be used to make some adaptions to the following implementations, which will then be evaluated more thoroughly.

Thematic Working Group 14

Proceedings of CERME10

In this paper, we will focus on the description and evaluation of the Linear Algebra additional course given by the first author. The analysis course was designed analogously by the second author and the results were similar. In figure 1 the structure of the whole Linear Algebra module in Bremen is displayed. The lectures extend over 14 weeks with 4 hours of lecture, 2 hours of exercise in groups and 2 hours of plenary exercises each week. The test at the end of the first semester takes place about a week after the lectures end and does not contribute to the final grade. The students nevertheless have to achieve a certain number of points in order to complete the module. The pass rate is set at a low level compared to the final exam, so the students know what to expect on the final, but are not impeded by having to retake the test, unless their score is very low.

Theoretical background

In their extensive study on teaching Linear Algebra in the first university semester, [START_REF] Dorier | The Obstacle of Formalism in Linear Algebra[END_REF] identified and described a main obstacle for students to learn the subject, which they called the obstacle of formalism. This obstacle is found to be a conglomerate of formal reasoning, abstraction and extracting ideas from concepts, due to the vast abstractions, simplifications, and unifications in the subject's history.

We tried to give students additional support to master this difficulty by a course design based on the Abstraction in Context framework, AiC, [START_REF] Dreyfus | Introduction to Abstraction in Context[END_REF]. This framework merges the idea of the vertical mathematization process by Freudenthal with Davydov's method of ascent to the concrete (ibd., p.87), in order to explain how abstraction processes can occur. Abstraction is assumed to happen in a "three-stage process: the need for a new construct, the emergence of the new construct, and the consolidation of that construct". The corresponding (observable) epistemic actions are recognizing (R) the relevance of certain known constructs in a given situation, building with (B) these constructs to achieve (local) solutions and constructing (C), i.e., integrating the previously known constructs into a new whole. This model is referred to as the RBC-model.

Since performance at challenging and creative tasks is dependent on intrinsic motivation, we make use of the self-determination theory of motivation (SDT), which postulates the existence of three basic psychological needs [START_REF] Deci | The 'What' and 'Why' of Goal Pursuits: Human Needs and the Self-Determination of Behavior[END_REF]. When these needs for autonomy, competence and sense of relatedness are fulfilled at a satisfactory level, intrinsic motivation is likely to occur. Low achieving students, in particular, may benefit from an environment that fosters positive experiences with regard to these needs in relation to mathematics [START_REF] Um | Motivation, autonomy support, and mathematics performance: A structural equation analysis[END_REF][START_REF] Rakoczy | Motivationsunterstützung im Mathematikunterricht Unterricht aus der Perspektive von Lernenden und Beobachtern[END_REF]. Hence, both in the course design and during the course hours themselves, special attention was paid to these factors of SDT.

As in the methodology of design research according to [START_REF] Gravemeijer | Design research from a learning design perspective[END_REF], we plan to develop the design of the additional courses in experiment-reflection-cycles. Gravemeijer and Cobb identify three main stages: (1) preparing for the concrete design experiment, (2) experimenting in the classroom (in this case the actual implementation of the course), and (3) analyzing the experiment retrospectively. At this point of our research, we consider ourselves to have obtained enough information from the initial design and implementation, as to be able to proceed to the first genuine design cycle within this framework.

Research questions

The main research questions we want to deal with are the following:

1.) How does the participation in the course change the perceived fulfillment of the basic psychological needs and, thus, intrinsic motivation? 2.) What is the effect on the epistemic actions with regard to the relevant concepts for the participants in the following course?

In addition, a minor question can be seen to be "Is there a beneficial short term effect on the retake test?", as this is the main motivation for the students to take part and also an institutional concern. We are, however, more interested in the long term effects.

Description of the course design and design principles with examples

The Linear Algebra course consisted of five days of activity spread out over two weeks (alternating with the corresponding Analysis course). Each day was divided into a morning and an afternoon session of three and two hours, respectively, with a lunch break in between. The sessions themselves were each devoted to a central topic of the Linear Algebra lecture, e.g., bases, linear maps or different interpretations of matrices, and were split into smaller working units of varying types and content, which were occasionally adjusted spontaneously according to the students' needs.

The instructional design of the course was based on the observation that low achieving students often do not have the mathematical resources to occupy themselves for a long time span with a given (in general more complex and open) task, but need to be guided to acquire these resources. Thus, the course proceeded from very short and clearly defined tasks of varying nature (but with immediate feedback) to longer and more open and self-determined ones. For example, the first task of the whole course consisted of a lecturer-guided group discussion of very small exercises concerning relevant geometric objects (lines, planes), while the very last task was a guided session, in which students were asked to create individual exercises to given topics by themselves and then solve each other's exercises, including negotiations on the wording of the task and different approaches and solutions. In this way, the possibility of increasing the levels of fulfillment of the basic psychological needs was provided.

With respect to the subject specific aims, on the one hand the focus was on developing the core ideas of Linear Algebra and important techniques (Gauss' algorithm, proofs relating to algebraic structures, etc.). On the other hand, great emphasis was placed on the creation and discussion of a zoo of examples and counter-examples to the relevant notions, as this is known to be very effective in the initial understanding of new concepts (see [START_REF] Dahlberg | Facilitating learning events through example generation[END_REF].

We illustrate how the AiC framework was used in the task design by the example of the concept of "linear independence" of vectors, which was comprehensively dealt with in the afternoon session of the first day and then later referred to throughout the whole course.

To aid the construction of viable concept images of "linear independence", the students were given geometric situations, where this concept is relevant, e.g. describing a plane in a three-dimensional space as the span of two vectors and characterizing the pairs of vectors where this description fails. Thus, the students had to recognize (R) that this particular knowledge of a linear relation between vectors is relevant to particular geometric problems (note that at this point the students were already aware of the existence of the definition of linear independence). Afterwards, the students were given tasks, where linear independence appeared in different situations and in relation to other concepts (such as basis, coordinate system, etc.) as to integrate the concept into their mathematical views and work out means to examine and apply the concept locally (B). Finally, by discussions among the students and with the whole class, the students were encouraged to express and evaluate different views on linear independence, which we hope has helped the students in building an (at least preliminary) concept image of "linear independence" (C).

Since the students had already encountered the definitions during the lecture and found themselves naturally confronted with extracting meaning from them, we did not incorporate a component of guided reinvention in the sense of Freudenthal into the course. Students were not encouraged to create models of the key concepts themselves as proposed in RME [START_REF] Gravemeijer | How Emergent Models May Foster the Constitution of Formal Mathematics[END_REF], but instead the "models-of" were already given (e.g. in the form of the definition of linear independence) and had to be realized as "models-for" in corresponding applications.

Field notes including students' actions and reactions during the course were collected by the first author.

Implementation and observations using AiC and SDT

We will illustrate the implementation of the course and relate it to the relevant theory by the help of two examples. In the first sessions of the course, the students were given explicit exercises of varying type and difficulty involving some recognizing of, but mainly building with, concepts. As a first example, we will therefore report on one specific exercise at this stage of the course. The exercise shown in figure 2 was given to students on the second day, after the concept of basis had been established and been related to the concepts of generating system and linear independence. It was part of a set of exercises, on which the students worked in small groups of two or three or individually according to their own choice. The students took to the exercises well, and most of them found access to part (a) quickly (possibly after short exchanges with other groups or with the lecturer). Part (b) was claimed to be impossible by some students, but this was resolved mainly among the students themselves without much intervention of the lecturer. The remaining parts were difficult. Part (c) was attacked by an attempt to prove that the vectors are linearly independent, and linear independence was claimed persistently. For time reasons, part (d) and (e) were only dealt with by some students, who all implicitly assumed K to be the real numbers, V to be the real plane and whose arguments were of a geometric nature. It is noteworthy, that the students refrained from trying to manipulate the expressions according to some formal rules they had not understood, but rather tried to give meaning to the statements, albeit not always succeeding. Similar effects could be observed throughout the course.

With regard to the epistemic actions in AiC, in this exercise the students were mainly building with the concept of basis, while recognition of the relevance and the role of linear independence was required. The students' reaction (i.e. the persistent claim of linear independence) to part (c) indicates that the concept of linear independence had not yet been consolidated, although it had been extensively covered before. The geometric approach to the tasks, however, seems to show that the students already had acquired a basic concept image and tried to use this, as opposed to merely manipulating the concept definition. There was no particular observation concerning the motivation for this exercise.

The second example to illustrate the implementation is taken from the last day of the course, where the students were given the task of creating exercises themselves. Each student was given an index card, which was labeled with one of the concepts that had been dealt within the course. He or she was then asked to make up an exercise together with a solution involving the corresponding topic and write up the exercise on the index card. Students who had quickly completed this task were given additional index cards, until everyone had made up at least one exercise. The cards were then redistributed randomly, and the students were asked to solve the exercise given to them. Once finished, they were told to check their solution with the creator and if both agreed, they repeated the process with another exercise.

There were very different approaches to this task by the students. Some students chose to model their exercises on the ones they had been given during the course with only slight changes (different numbers, different number of variables/equations, etc.), while others tried to produce an original task by combining things they knew. Both approaches were encouraged by the lecturer. During the creation of the exercises, the most striking observation was that the students suddenly felt the need for certain insights, which had been difficult to stir beforehand. E.g., one student wanted to create a linear system of equations, where a row of zeros would appear at some point in the process of Gaussian elimination, and was confronted with the need for a practical criterion to achieve this. Even though this issue had been dealt with during the course, she seemed to recognize the relevance of linear independence of the rows only at this point, when it was explicitly needed by her. Additionally, the participants were aware of the need to communicate their mathematical problem well enough for someone else to make sense of it and there seemed to be a genuine effort to achieve this. In the following session of solving each other's exercises, a dynamic of interaction was observed, where the randomly allocated pairs of creator and solver got together to sit down and negotiate the wording of the exercises and the validity of different approaches (in particular, students who had not closely worked together before). There was very little intervention from the lecturer.

Concerning the epistemic actions in AiC, there was much activity of recognizing and building with concepts. Moreover, the proposition that there has to be a need for a concept for an abstraction process to occur was confirmed in some cases, e.g., in the case of the student described above. We do not know, however, at this stage, whether this led to the construction and consolidation of the relevant concepts, which we consider to be a long term process and not measurable in this short time span. With respect to self-determination theory, after a short period of orientation a high level of motivation was observed during this task. The students seemed to feel a strong sense of autonomy, as they were given the freedom to create a problem of their choice with only the general topic prescribed. During the period of solving, many students displayed a boost in perceived competence, particularly when they were in the creator-role, and experienced their exercise and their comments on it to be of value for someone else. Furthermore, the students appeared to build new, largely positive, relationships with each other via the random and varying allocations of creatorsolver pairs. This interaction seemed to be of natural importance to the students (the extensive meetings of these pairs were their spontaneous creation and had not been suggested by the lecturer). Hence, the conditions for intrinsic motivation according to SDT were largely fulfilled, which might help to explain the unusually high motivation. We believe, however, that this exercise would not have worked out well if the participants had not been prepared for it in advance by the preceding days of the course.

Results

One of the institutional measures to evaluate the success of the course was the pass rate of the test and retake test, which was compared to the one of the previous year (as the style of the lecture and of the exams were largely the same, this comparison seems justified). In 2015, the pass rate for the regular test was about 65%, while the retake test was passed by 50% of the participants. In comparison, the regular test in 2016 was passed by 78% and the retake by 95%. Although there are of course various factors, which play a role in these results, an effect of the additional course seems likely.

In addition, the course was hoped to have a positive lasting effect both on the motivation as well as on the ability of the students to achieve concept construction by themselves with the means of the relevant epistemic actions. This has not been quantitatively assessed yet and will be the subject of further research. However, there are many indications that such an effect might indeed be observed.

During the sessions of the course, the students appeared to be (at least extrinsically) motivated and confident of benefiting from the course. Confidence and perceived competence seemed to increase, as the students advanced from the passivity of merely carrying out tasks imposed onto them by the instructor to more self-determined action. This seemed to be accompanied by a shift from mere extrinsic motivation (to pass the retake) to at least some intrinsic motivation, e.g. some students were observed to carry on discussing tasks during the break. The actual mathematical competence was seen to increase accordingly, as students gathered experience and perceived deeper mathematical insights, which they worked to develop. By the analysis of the two examples above, it Thematic Working Group 14 Proceedings of CERME10 can be inferred that in the framework of AiC and of SDT there have indeed been positive effects on motivation and the ability to perform epistemic actions with regard to the relevant concepts.

These positive effects reached far into the next semester. Most participants felt that they had acquired a basic knowledge and techniques, which were necessary for the following courses, and were repeatedly observed by the authors to use methods they had picked up on in the additional course during the exercise classes of the following course. In almost all cases this was bolstered by the fact that the retake had been passed. Many students expressed their conviction of having profited to a great extent from the additional courses, both immediately after the course and about half way into the next semester.

Conclusion and perspectives

Compared to the previous approaches of the faculty of mathematics and to other approaches described in the literature (e.g. remedial courses as discussed by Di Pietro (2014), which were shown to be largely ineffective) to help the students in the transition from school to university mathematics, the additional courses described here seemed to be more effective with regard to the research questions posed above. Various factors are assumed to be of importance for this.

As this was a voluntary course aimed at students having failed the end-of-term test, it was made clear beforehand that it should be seen as a chance to acquire skills not yet developed rather than an obligatory task (as in Di Pietro, 2014). The course was announced at the beginning of the semester, giving the students the opportunity to take it into account in their planning. In general, the allocated time was met with approval, both because it meant no conflict with the workload during the semester and because the end-of-semester test provided some feedback for the students whether attendance was advisable (due to other time restrictions regarding school and computer practica for the students, however, the course should not exceed two weeks). In addition, the common goal to prepare for the next semester and pass the retake seemed to ensure a sense of belonging to a peer group with a common aim. Hence, the general framework of the course proved to be more feasible than the previous ones, and will be kept for the time being.

The design of the course with sessions devoted to central topics, which were each split into smaller units of varying tasks, were also perceived to have a positive impact with regard to the above questions. In particular, the general progress from rigidly set tasks to more self-determined ones seemed to be appropriate in this setting. The exercises themselves were developed using the AiC framework, and their construction will be refined, based on the observation of how the different tasks affect the perceived concept building in the students. For example, in the exercise given in figure 2, part (c) has already been modified to ask for linear independence of the differences of the given vectors as opposed to their sums. This allows for the generalization of ideas from lower dimensions to higher ones, giving the students the possibility to use previously built knowledge more directly. Moreover, part (d) was simplified to ask only for real vector spaces, as the additional difficulty of an arbitrary field seemed to be ignored or actually hindered the building with in the previous course. This will be reflected on further, using the gathered experience, and systematically revised for the next implementations. After this first preparatory cycle, we hope to be able to contribute to the research regarding the mediation of students' problems in the first year of university by carefully designing and evaluating the next cycle of the described additional courses.

Figure 1 :

 1 Figure 1: Structure of the Linear Algebra module

Figure 2 :

 2 Figure 2: A problem from the course in linear algebra

Proceedings of CERME10

We are going to evaluate the two main questions in the next cycle as follows. To assess the effect with regard to SDT, we propose to use a pre-post-test according to the items presented in Rakoczy (2006) with the pre-test set at the end of the lectures of the first semester and the post-test in the middle of the second semester. We want to check for correlations of these results with the performance in the end-of-first-term test. With regard to concept building, we are going to evaluate some abilities concerning the relevant epistemic actions at the end of the additional course itself with a free form questionnaire, supplemented by two or three interviews with students to reach theoretical saturation and to clarify the data.