Barbara Jaworski

Stephanie Treffert-Thomas

Dave Hewitt

Matt Feeney

Damma Shrish-Thapa

Dan Conniffe

Ayesha Dar

Nikolaos Vlaseros

Marinos Anastasakis

Student Partners in Task Design in a computer medium to promote Foundation students' learning of mathematics

Keywords: Teachers' and students' practices at university level, Novel approaches to teaching, The

A team consisting of three mathematics education teacher-researchers, four former Foundation students (called Student Partners, SPs), and two analytic assistants worked together to produce mathematical tasks in a computer medium for the mathematical learning of current Foundation students (FSs). We have explored the collaboration between the SPs and researchers, the processes and outcomes of task design, and the contribution of the collaboration to tutorial teaching of FSs. We seek insight into the learning of all concerned of mathematics, mathematics teaching, task design and personal-professional development. The project is ongoing. Here we introduce the project and present early findingsspecifically related to task design and the contribution of SPs.

INTRODUCTION TO THE CATALYST PROJECT

We report from an exciting new development project (2016-18: The Catalyst Project i ) in which mathematics teacher-educators collaborate with former Foundation students as partners in designing computer-based mathematical tasks for current Foundation students. Our research explores the collaboration of participants, the design of tasks, teaching of students in tutorials, use of computer software for teaching and learning mathematics and the learning of all concerned.

The mathematics learning of students in our university Foundation Studies programme (henceforth Foundation Studies students -FSs) is the focus of this developmental project. These are students who need a higher-level qualification than they hold currently in order to be able to enter the first year of their desired undergraduate programme (e.g., programmes in engineering or science). For such programmes, mathematics is an essential component. All FSs are required to pass their year-long module in mathematics. It has been observed that teaching and learning in this module in the past has been rather procedurally based: students have been introduced to and expected to learn the application of procedures to mathematical problems and have been examined on their procedural competency. A current aim is to make the module more conceptually based, and to explore the use of computer-based tasks for this purpose. Our previous research has shown that:

a) The involvement of peer learning-teaching in a mathematics module has resulted in participating students gaining higher marks, with associated experiences reported as positive to their understanding of concepts; (SYMBoL, e.g., [START_REF] Duah | Can peer assisted learning be effective in undergraduate mathematics[END_REF][START_REF] Solomon | Reshaping understandings of teaching-learning relationships in undergraduate mathematics: An activity theory analysis of the role and impact of student internships[END_REF];

b) The cultural differences between teacher-researchers designing an innovation in mathematics teaching-learning and students engaged in learning mathematics through the innovation have contributed positively to outcomes/higher marks. (ESUM, e.g., Jaworski, Robinson, Matthews, & Croft, 2012). ii Beyond this activity, we have found very little other research involving students' engagement through partnerships in mathematics teaching and learning. There is relevant work in Higher Education more generally involving Partnership Learning Communities (Healey, Flint and Harrington, 2014), but this does not include mathematics specifically. A recent special issue of NoMAD, the Nordic Journal of research in mathematics education, included several papers in which teachers were involved in exploring their own practice. The reports address three themes, one being 'innovative approaches to teaching and learning, with emphasis on student participation in the educational process' [START_REF] Goodchild | Developing practice through research into university mathematics education[END_REF]. One paper, in particular, reported positively on students' activity in oral presentations as a tool for promoting metacognitive regulation in Real Analysis (Naalsund & Skogholt, 2014). Student engagement and understanding were seen to improve through their participative activity. Searches to date have revealed no other relevant work.

Building on our experiences in SYMBoL and ESUM, we sought to design mathematical tasks which would challenge the FSs in new ways, engage them visually and actively and promote the beginnings of a new learning culture. Recognising the value of student design of resources and peer support as demonstrated in the SYMBoL Project we built both of these aspects into our Catalyst project. One of the Catalyst aims was: To promote collaboration between staff and students that results in higher degrees of confidence, motivation and learning in mathematics and a new culture in the teaching-learning of mathematics (e.g., HEA, 2014). We hoped to learn from the various elements of this project in ways that would have relevance beyond mathematics, particularly in the inclusion of former FSs as Student Partners (SPs) in course design and teaching. Our own learning from working with students in these ways was also a central aim, with the intention to bring staff and student cultures into focus through our joint activity. The reflections of the students on their activity, both SPs and FSs, were seen as an important outcome of the project. Thus, our innovation in the Catalyst Project has two main areas of inquiry:

 The design of computer-based tasks in matrices and complex numbers for the FSs using Autograph software.

 The involvement of former FSs (the SPs) in the design of teaching and in the tutorial teaching of the current FSs.

According to its specification, the Foundation Studies programme provides "An opportunity to make it onto a degree course at Loughborough University". Engineering, Physics and Chemistry. At the time of their recruitment, they were first year students in their current programmes. In addition, two doctoral students in Mathematics Education were recruited as "Analytical Assistants" to support data collection and analysis. Thus, nine participants have been involved in the project, with differing roles.

THEORETICAL BACKGROUND

We are concerned with learning at a number of levels.

 FSs learning of mathematics;

 SPs learning of mathematics, task design and participation with staff in preparing for undergraduate learning;

 Mathematics teachers and researchers learning about the design of teaching in partnership with students.

This learning is influenced by a wide range of factors which include the curriculum, and institutional settings within the broader sociocultural setting. Some of these factors we can seek to influence; others are less amenable to innovation. We take a fundamentally Vygotskian (e.g., [START_REF] Vygotsky | Mind in society[END_REF][START_REF] Wertsch | Voices of the mind[END_REF] perspective recognising particularly mediation by people and tools that support learning; goaldirected activity and action related to learning and teaching; scientific concepts that require pedagogic mediation; and the zone of proximal development in which mediation fosters learning and development. We engage particularly with digital tools, their design and use, and the ways in which they mediate the learning process through both support and challenge for making sense of mathematical concepts.

An important theoretical concept is that of "partnership" between staff and SPs (e.g., Healey, Flint and Harrington, 2014). Relationships within the partnership have resulted in the design of mathematical tasks and their use with the FSs. The nature of this partnership is central to project outcomes, in terms of the designed tasks and their use. We see ourselves as having formed a 'Learning Community' in which colearning is an important concept [START_REF] Wagner | The unavoidable intervention of educational research: A framework for reconsidering research-practitioner cooperation[END_REF], and which demonstrates tenets of a Community of Practice, such as mutual engagement and joint enterprise [START_REF] Wenger | Communities of practice[END_REF]) and a Community of Inquiry, such as critical alignment [START_REF] Jaworski | Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching[END_REF].

METHODOLOGY

We take a developmental research approach, consistent with our Vygotskian theory, which both studies project development and learning within the project and contributes to development and learning [START_REF] Goodchild | A quest for 'good' research[END_REF][START_REF] Jaworski | Research practice into/influencing mathematics teaching and learning development: towards a theoretical framework based on co-learning partnerships[END_REF]. Mediation and tool use, for example, can be seen in an interactive stance of reflection and negotiation in which participants engage together in activity and action with growth of mutual understanding and co-learning [START_REF] Wagner | The unavoidable intervention of educational research: A framework for reconsidering research-practitioner cooperation[END_REF].

Analysis begins in questioning of what is done and achieved and is formalised through scientific inquiry addressing a range of data through recognised methods.

Research questions and data

Our Research Questions relevant to this paper are as follows:

 How have SPs engaged with task design and what has been the outcomes and issues arising?

 How have FSs worked with the designed tasks?

 What have we learned about the FSs' learning of mathematics with the designed tasks? What issues arise?

Data, which are being analysed to address these questions include:

 The involvement of SPs and staff in the design process as shown through recordings of project meetings, SP reflections/reports, documents (collected at the design meetings and from the SPs' own work).

 The tasks, and their use as seen through observation, screen capture and discussion. The teacher's narratives from her reviewing of tutorial data.

Analytical approach to date

Reflection and negotiation have taken place through meetings, discussion, sharing and review of designed tasks leading to increased awareness of issues in design and communication. This collaborative co-learning has involved a bringing to collective consciousness of key elements of the design process and issues to be resolved.

Analysis of collected data according to research questions has been qualitative, focusing on data from the design meetings, and from tutorials with FSs in which designed tasks have been used. A process of data reduction has summarised and coded recorded data, allowing initial identification of key elements and issues relating to research questions. The process is cyclic with developing depth of inquiry and insight to significant issues expressed through analytical memos. The tasks themselves have also been a focus of scrutiny which is ongoing. These analyses are as yet in their very early stages, so what we report below is tentative and indicative.

Here, we discuss some emergent findings in task design and use of tasks with FSs.

EMERGENT FINDINGS

Task Design

The teacher/lecturer of the Foundation course provided course notes on the two topics, Complex Numbers (CN) and Matrices. An expert in Autograph gave the group an induction into its use and potential for mathematical representation and exploration. SPs were asked to review the notes and think about possible tasks using Autograph. Task design, in 2 SP pairs (one to each topic) took place over 6 weeks and across 4 meetingsfinding times for these meetings, from timetables of 9 partners, over a short time period was challenging. At the meetings, SPs' presented their ideas to the whole team, hesitantly in the beginning but with growing confidence in response to expressed appreciation and suggestions from the team. The pair working on tasks in complex numbers were quick to provide examples and to modify them according to suggestions in meetings. The pair who worked on matrices found it harder to get going. Tasks in complex numbers appeared to be more readily achievable than in matrices where concepts seemed less amenable to digital representation/questioningit became necessary both to identify the barriers and to find some resolution. Collaboration between the SP pair and the PLs, focusing on the mathematics of matrices and the learner difficulties suggested by SPs, resulted in a set of tasks on matrices. One of the PLs also designed a set of tasks in GeoGebra focusing on matrix arithmetic. The emerging 'raw' tasks, consisting of an Autograph (or GeoGebra) file with brief associated notes, were then 'prepared' by the FS lecturer to make them ready for FS use. We see two examples of the prepared tasks, one for each topic, in Figure 1. Certain characteristics, incorporated by the SPs into these tasks can be seen in the examples; FSs have to:

 use several display features of Autograph to "see" mathematical objects and relationships;

 undertake some associated calculation by hand;

 relate movements on their screen to the handwork and theory involved;

 reflect on specific results to develop a more general awareness of concepts.

Some of these demands turned out to be a challenge for many FSs as we see below.

Use of tasks with Foundation students

The tasks were used in timetabled tutorials with the FSs, who were asked also to comment on the tasks for the research. Those agreeing to participate were audiorecorded in conversation with the teacher and one researcher. From analysis to date we are gaining some insight into FS participation with the tasks. We have recorded instances of FS requesting help from the teacher, asking questions, explaining their solutions, revealing mathematical insight, surprise, or lack of understanding. In interactions with the FS, the teacher responds to students, asks questions, explains, and provides technical information. The dialogue in Figure 2 shows a teacher-student Question 7: Open the Autograph File Matrices 5 On this page you see two straight lines. Their equations are 4x -y = 14 and 7x + 4y = a (a) By hand, using matrices, calculate the value of a so that the solution to the simultaneous equations is 3 . 2 (b) When you have a solution, use the "constant controller" to vary the value of a until the point (3, 2) is clearly displayed. (c) Select both lines by holding down the "Shift" key. Both lines should have changed colour. Go to "Object" in the menu bar and choose "Solve f(x) = g(x)". A point is displayed. To see what the co-ordinates of this solution are, go to "View" in the menu bar and choose "Results Box". Does your solution make Autograph show the intersection of the lines to be x = 3 and y = 2? Question 2: Open the Autograph File Task 2 There are three complex numbers labelled z1, z2 and z. z1 is fixed while z2 and z can be moved. Select z2 and move it until z reaches the position 3 + i. (a) What complex number is z2? (b) Right click and "Unhide All" to check your answer. (c) What is the relationship between z1, z2 and z? (d) Explore subtraction of complex numbers in Autograph. (e) Now calculate by hand: With z1 = -1 -3i and z = 3 + i, find z2 such that z2 -z1 = z. (f) Draw (by hand) all three complex numbers on an Argand diagram. Give a geometric interpretation of the relationship between z and z1 and z2.

exchange, relating to Question 2 above, that reveals an issue in the student's engagement with the task that is typical of several such exchanges:

We see that the student had engaged with the task, moving z 2 as instructed. However, he did not know what is meant by "the relationship between z1, z2 and z" although he had written that "z 2 -z 1 gives you z". It seems a case of not understanding the meaning of the word "relationship", although he had written the relationship. We are learning here about language issues in working between the visual mode of Autograph and the symbolic mode of expressing complex relationships. Although the teacher had used Autograph, in demonstration mode, in her lectures, students had not developed a language to express what they could see visually. So, from what the student said here, it was difficult to gauge the contribution of Autograph to the student's understanding of complex subtraction. When pushed by the teacher to explain his process in getting an answer, he replied that he "worked out" the result. As the teacher interpreted, this might mean that he calculated the result. Whether this is as well as discerning it from the movements in Autograph or instead of this, is not clear. Thus, it might be that the student had used Autograph effectively and made links with the symbolic forms. Or it may be that he had sidestepped interpretation of the visual and instead had worked out the result symbolically (the latter perhaps being a more familiar task). The teacher's reflective narrative relating to the recordings from the second tutorial on CN reveals the following example [Teacher's written words are italicised]:

[Student] found the additive relationship by adding separately the real parts and then the imaginary parts. He says "is it bisecting the angle?" I reply "it's something to do with it". [She asks about her related lecture presentation and mentions a comparison with vectors.] The student correctly relates vector addition to "adding head to tail" and that this forms a "triangle". [She indicates that he does not seem to understand what is meant by a "geometric interpretation".] The teacher's reflection suggests a student engaging correctly with several concepts including complex addition and vector representation. Yet, again, we see a problem with language -the term "geometric interpretation" is unfamiliar to the student. One student, asked about his work on the Autograph tasks, replied that he found the questions "hard to read" and could not "understand the way they are worded".

The two examples quoted are typical of recorded exchanges. We see from these a) Students not understanding the words or expression of ideas in the written questions (e.g., "relationship", "geometric interpretation") b) Students not articulating explicit conclusions from what they see on screenrather using the familiar forms of calculation to answer questions.

These observations lead us to question both the Autograph tasks and the wording of the tasks. How might we have worded the tasks differently so that students would engage with what they could see on screen and discern the mathematical relationships that the task was designed to reveal? How might we wish to modify the task itself so that students engage visually rather than depending on calculation? The challenge for the team here is twofold: to design a teaching approach that introduces the language we want students to use and enables students to become familiar with its use; and to design tasks that are revealing of concepts in and of themselves, so that students can see visually what they familiarly work out in calculation. These seem to be important elements of the learning culture we are trying to foster.

We are aware that many FSs come to their university course from school or college where their mathematical enculturation towards success in national final examinations may have encouraged a procedural perspective on learning mathematics [START_REF] Minards | An exploration of high-achieving students' experiences of Alevel mathematics[END_REF] and that we see the results of this to some extent in their response to the designed tasks. As well as looking critically therefore at the design of the tasks, we have to consider the wider mathematical culture, the nature of teaching that seeks to interact with this culture and the ways in which the tasks can be incorporated within the teaching-learning interface.

CONCLUSION AND FURTHER RESEARCH

The developmental nature of the project can be seen through the development of mathematical tasks for Foundation students by SPs and PLs in partnership, the subsequent use of the tasks in tutorials with the FSs, and issues arising from task design and use revealed both in practice and in analysis of data from the various events. An aim of the project was to foster conceptual understanding of mathematics by the FSs. We see above some issues arising from the nature of the tasks and the ways in which they are written, and from the ways in which FSs' mathematical experience influences their engagement with the tasks.

Because analysis is in its very early stages, we are not yet in a position to report on many of the aspects of learning in the project (such as aspects of the learning of the SPs). However, already we can start to see indications of important learning and the feedback element of the developmental research. Co-learning has been demonstrated between the FSs and the teacher-researchers -FSs' learning of mathematics has raised issues for the intention and preparation of tasks which offer challenges to the researchers and for future work with the SPs. The tasks and their design have been mediational tools not only for the mathematical learning of the students but also for the awareness of the researchers about teaching-learning issues, not least the issue of language in which tasks are expressed. The project is ongoing, both in terms of teaching-learning development and of analysis of the data collected so far.

A major issue for the project has been the timescale as dictated by the funding body and university organisation of teaching. We had barely half a semester to recruit SPs, initiate the design process, hold 4 spaced meetings, prepare the tasks for FSs and hold the tutorials. The project end coincides with the time for the next cohort to reach the teaching of matrices and complex numbers, so we could not build this into the project. We expect to use the same tasks again with the new cohort and collect further data, informed by our experiences a year ago. Since our data is extensive, in depth analysis is ongoing from which more in-depth reporting should be possible. We expect to be reporting further on the many aspects of this project.

i Supported by HEFCE (Higher Education Funding Council for England) Catalyst Fund: Innovations in learning and teaching, and addressing barriers to student success A: Small-scale, 'experimental' innovation in learning and teaching.

ii SYMBoL -Second Year Mathematics Beyond Lectureswas a project designed to support teaching in two second year mathematics modules, Vector Calculus and Complex Analysis. Students who had experienced these modules were employed over a summer period to design resources in collaboration with mathematics staff. The resources were used in subsequent delivery of the modules and a peer support system was initiated in which third year undergraduates supported their second year counterparts.

ESUM -Engineering Students Understanding Mathematicswas a developmental research project involving an innovation in mathematics teaching seeking to engage students more conceptually with mathematics through inquiry-based activity, a computer-based learning environment, small group tasks and an assessed small group project.

Figure 1 :

 1 Figure 1: Examples of prepared tasks in complex numbers and matrices

Figure 2 :

 2 Figure 2: Example of dialogue between FS, working on Task 2, and the teacher

  What are you doing in Question 2? [She looks at what he has written] 3+i... Haa! What did you find out? What relationship? FS: I don't know what it means. T: You don't know what that means? Well they are connecting aren't they? You must be doing something with them. If they are moving together [z1, z2 and z are moving together] we do something to them and then you get the third. FS: I wrote that z 2 -z 1 gives you z. T: So you did write it down. For [part] c, that is the relationship. Yes, that's what we meant. FS: OK T: So the relationship is subtraction. You are subtracting two complex numbers. How did you know that though? Did you know that from the picture? Or did you do something else? FS: I worked it out. T: Ah ok. So you actually did the calculation. T=Teacher FS= Foundation Student

T:


Cross-linking mathsusing keynotes to structure a curriculum for future teachers

Tanja Hamann 1 and Barbara Schmidt-Thieme² 1 2 Universität Hildesheim, Institut für Mathematik und Angewandte Informatik, Germany, 1 hamann@imai.uni-hildesheim.de

The paper gives a frame for a concept for university maths teacher education, that is based on three keynotes: central scientific notions, history and language. Amongst other benefits, the keynotes serve as cross-links between the different courses the students go through in their studies.

Curricular and institutional issues concerning the teaching of mathematics at university level, Teaching and learning of specific topics in university mathematics, teacher education, history of mathematics, language education.

INITIAL CONDITIONS AND REQUIREMENTS

Subject-specific content knowledge is an essential component of professional teacher knowledge (for the concept of professional knowledge see for example [START_REF] Schwarz | Professionelle Kompetenz von Mathematiklehramtsstudierenden: Eine Analyse der strukturellen Zusammenhänge[END_REF]). It forms a basis for the ability to judge a specific topic's significance for an entire subject and thus forms an important requisite for didactical reflections. Considering this, mathematics teachers' content knowledge must comprise not only advanced insight into singular topics but also a netlike overview of mathematics as one, consisting of notions and relations between them as well as methods and basic principles. To achieve this, a university curriculum for future teachers demands cross-links between the single courses. Basic notions (such as sets, functions, algorithms…) that are found in diverse mathematical subdomains seem suitable for serving as cross-links between scientific lectures, both, in a horizontal and a vertical way. This would transfer [START_REF] Bruner | The process of education[END_REF] concept of fundamental ideas and a spiral curriculum onto teacher education.

Basic ideas and fundamental notions can be found when one looks into the history of a subject as they should appear throughout time [START_REF] Schweiger | Fundamentale Ideen" der Analysis und handlungsorientierter Unterricht[END_REF], though maybe in different disguises. Apart from that aspect, including the history of mathematics into teacher education has been suggested long-since, presumably bearing a whole lot of further benefits [START_REF] Schubring | History of mathematics for trainee teachers[END_REF][START_REF] Katz | Using history to teach mathematics: An international perspective[END_REF][START_REF] Jankvist | Mathematical Knowledge for Teaching Teachers: The case of history in mathematics education[END_REF], for example experiencing mathematics as a process conducted by human beings rather than as a mere product.

Generation, clarification and precision of mathematical concepts and notions require the use of language (e.g. [START_REF] Morgan | Language and communication in mathematics education: an overview of research in the field[END_REF]. Without language, definitions and propositions could not be worded and not be taught either, so language forms another essential part of content knowledge, concerning scientific as well as educational knowledge. Due to increasing heterogeneity in pupils' language abilities, it is necessary that future teachers learn to illustrate subject matters on differing language levels and that they become able to vary by those levels.

THE HILDESHEIM CONCEPT OF LEARNING ALONG KEYNOTES

We believe that integrating the three keynotesbasic notions, history and language of mathematics -into a teachers' curriculum and thus cross-linking the courses supports the construction of a professional teacher content knowledge as required. Therefore, they should become a constitutional part of all scientific lectures throughout studies. Our poster presents a yet draft concept of how the three might constitute a scaffolding for a spiral curriculum, where historical and language elements serve as enrichment as well as embedding of basic contentual and methodical concepts. The concept seems convenient for linking subdomains as well as different levels of challenge throughout a curriculum that is adapted to the specific needs of future teachers. A research frame for evaluating efficacy of the concept has not yet been developed.

Examples of implementation by emphasis on basic ideas in the lectures, integration of historical tasks in practices as well as inclusion of training of language skills (for more details on this see the poster presented by Schmidt-Thieme at INDRUM on the project SuM_MaSt) will be presented on the poster.


Getting language awareness: A curriculum for language and language teaching for pre-service studies for teachers of mathematics Barbara Schmidt-Thieme 1 1 University of Hildesheim, Germany, bst@imai.uni-hildesheim.de

As language is an important medium as well as an aim of mathematics lessons at school, teachers of mathematics have to achieve the required competencies in language and language teaching. Therefore, we designed a curriculum, which offers students opportunities to get these competencies throughout their academic studies. Some examples and first results will be shown.

Keywords: Teaching and learning of specific topics, Curricular issues, Language awareness, Special language; Explaining.

LANGUAGE AND LANGUAGE TEACHING COMPETENCIES AS PART OF TEACHER PROFESSIONAL KNOWLEDGE

Teaching and learning mathematics is based on the use of language: for introducing and defining new mathematical objects, discussing different ways of calculating, documenting the results of a proof or explaining how to handle with teaching materials, different representations are used, but almost always accompanied by language. Language as a medium and an aim of mathematics lessons at school is a well-known object of didactic research and laid down in several publications (see [START_REF] Morgan | Language and communication in mathematics education: an overview of research in the field[END_REF]. But in the last years one can observe a focus being laid on differentiations and transitions between different languages or registers of one language used in mathematics [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF]). On the other side, the professional knowledge of teachers, and questions on how this can be developed in pre-service and in-service lessons gained considerable interest [START_REF] Gniffka | Fachsprachenvermitltung im Unterricht[END_REF]Kunter et al., 2013). This leads to following questions: (1) Which languagerelated competencies a teacher has to have? How do they interact with other parts of his professional knowledge? (2) When, where and how can he achieve these competencies? For this in the last years several universities in Germany designed and developed exemplary trails in pre-service teacher-studies. In most of the case these are extra, but obligatory courses on "German as second language". Another approach is followed for example by a project called "Umbrüche gestalten" (http://www.sprachen-bilden-niedersachsen.de/-index.php/projekt.html).

A "LANGUAGE CURRICULUM" (SUM_MAST)

Following the approach of this project we developed a pre-service "language curriculum" for teachers of mathematics (based on experiences out of the implementation of language studies in teacher studies above mentioned) following four requirements for the learning opportunities (used to guide the curriculum): they should (a) be spread from the beginning to the end of the academic studies, like a vertical spiral curriculum; (b) be integrated -besides explicit language related courses -in mathematical lessons, for the technical language of mathematics is best learned by doing mathematics; (c) evoke an active and reflective handling with language in learning situations and be applied and tested in authentic situations; (d) include individual feedback and allow some comparative measurements. (And in addition: It should be transferable to other designs of academic studies for teachers at other universities.) Our suggestions is, that a curriculum following these leads to a connection of linguistic and mathematical knowledge as a theoretical base for teaching practise.

After some pre-studies the design based research project SuM_MaSt started fully 2016 at the University of Hildesheim. Besides schemas of linguistic requirements of mathematics lessons (in multilingual classrooms) and the language-related competencies of teachers of mathematics (as a part of PCK), the poster will give an overview over the types of "linguistic tasks" (theory/practice, real/simulated, ...) and the curriculum implemented at the University of Hildesheim. Four examples will clarify the tasks, the type of language competence, which can be achieved, and show some first results of the evaluation of the above mentioned suggestion :

(a) linguistic tasks in mathematical lectures (examples: geometry, arithmetic, number theory; evaluation (qual.): correctness of the mathematical content and adequacy of language), (b) input and tasks on "Explaining" in the lecture "arithmetic" (evaluation (qual.): describing the development of linguistic competencies), (c) contents and tasks of an explicit course "Language and Mathematics" (evaluation (qual.): analysis of the products during the lessons), (d) questionnaire for reflecting and self-assessment of the competence (evaluation (quant.): questionnaire).


Ibrahim Alzubaidi

Keith Jones²

A CASE STUDY OF A UNIVERSITY TEACHER OF CALCULUS 1

Keywords: teachers' practices, university level, teaching calculus

This case study of a university teacher of calculus1, based on data from a questionnaire, semi-structured interview, and observation, illustrates how the teacher used their knowledge of calculus teaching to sequence the building blocks of mathematical theories (BBMT) of the concepts of calculus.

INTRODUCTION

Calculus is important at the university level because understanding calculus is an essential step in understanding how the world works. It is a foundation on which other skills can be built and is often a requirement or prerequisite for STEM programmes. Nevertheless, it is a challenging area regardless of which educational institution is offering the instruction [START_REF] Petropoulou | Addressing large cohorts of first year mathematics students in lectures[END_REF]. As such, while there is much research on calculus learning and teaching, [START_REF] Rasmussen | Research on calculus: what do we know and where do we need to go?[END_REF] identify teacher knowledge as an under-developed topic. In this study our research questions are; how does a calculus teacher demonstrate their pedagogical content knowledge (PCK), to achieve their aims, to develop instructional strategies (ISs), and to assess students' understanding? Here we present elements of a single case study.

RESEARCH ON PCK OF UNIVERSITY TEACHERS

Numerous studies, from [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF] to Khakbaz (2015), have indicated that teachers require different types of knowledge in the classroom. PCK was proposed by [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF] as an essential component of teacher knowledge, defined as a "special amalgam of content and pedagogy that is … understanding" (p.8). Later work by, for example, Khakbaz (2015), confirms that it is necessary to go beyond the subject (e.g. Calculus) and examine how the teacher interprets the subject matter (e.g. the interpretation of calculus) and how this is linked to their role in facilitating learning in the classroom. Research on PCK of university teachers remains limited.

THEORICAL FRAMEWORK AND METHODOLOGY

For the purpose of this study, we adopted Lessing's (2016) PCK framework. Based on this, we take PCK as a combination of knowledge of purposes of teaching calculus, of the building blocks of mathematical theories (BBMT) and of ISs, while taking into account knowledge of learners' conceptions and difficulties.

A mixed methods multiple case study design was adopted for this research. Qualitative data was gathered by semi-structured interview and observation of eight taught sessions of the module from 15 April to 15 July 2017 to explore the ways in which calculus teachers implemented their PCK in the classroom setting. Quantitative data was gathered using a questionnaire (multiple choice) on their knowledge of ISs, learners' conceptions in teaching and learning calculus, and learning difficulties. The qualitative data was analysed by coding and categorising according to the theme in order to identify the ways that the participating teachers demonstrated their PCK during their calculus teaching.

THE CASE OF TEACHER JOHN

John is a mid-career mathematician with four years of experience of teaching calculus1 at a university. The calculus teaching strategies employed by John were a pattern of topic-specific ISs based on using the BBMT in teaching calculus. From our data, it was clear that John approached each calculus topic through the BBMT using axioms, definitions, theorems and proof. Therefore, the lecture structure chosen by John often aligned with the mathematical concept used in the topic. Here, John often started the lecture with a definition, then theorem, and then sometimes the proof and then examples illustrated with graphs. On one occasion, John asked the students to read a proof. In an interview, John explained that, in the lectures, students are sometimes asked to read a proof because John wants to understand the students' conceptions, and misconceptions, of proofs. On one occasion, in a part of the topic 'continuous functions on an interval', John gave some examples and asked students to give a definition. At other times, John used ISs such as diagnostic techniques, (through class discussion, etc.), reviewing previous lessons as a way to introduce subsequent lessons, and using various mathematical representations.

CONCLUSION

The analysis of John illustrates how knowledge of calculus teaching (i.e. calculus PCK) was used in sequencing the BBMT of the concepts of calculus. The next stage of the research is documenting other cases as part of the multiple case study.


Tommy Dreyfus
email: tommyd@tauex.tau.ac.il

Chris Rasmussen²

Naneh Apkarian²

Michal Tabach

The complexity of knowledge construction in a classroom setting

Keywords: Teachers' and students' practices at university level, teaching and learning of analysis and calculus, knowledge construction in classrooms, paradoxes

We study a class of mathematics education MA students in an introductory course on Chaos and Fractals, as they grapple with the Sierpinksi triangle, and in particular with the apparent paradox that its area equals 0, while its perimeter is infinitely long. For this purpose, we network an approach for investigating the construction of knowledge in small groups with one for examining how ideas and ways of reasoning function-as-if-shared in a classroom. Our results show complexities: (i) small group work and whole class discussions mutually influence each other; (ii) ideas may function-as-if-shared in the whole class even if the majority of students have not previously constructed them in their groups; (iii) knowledge constructed in the small groups may or may not later function-as-if-shared in the whole class.

INTRODUCTION

The research presented here deals with the construction of knowledge in a student centred, inquiry-based classroom, where small group work (SGW) alternates with whole class discussions (WCDs). Construction of knowledge is usually investigated by observing small groups (1 to 4 students) of students. The reason for this is that in larger groups, the density of information for each student is low and does not allow the researcher to interpret their utterances or actions. However, intentional learning more often than not takes place in classrooms with many more than 4 students. We therefore use different approaches for analysing the SGW and the WCDs. The aim of our research is to link the two analyses by following ideas from their emergence in SGW or WCD, via their flow between SGW and WCD settings, until they possibly function-as-if-shared in the class, even though they may not have been constructed by all students. We thus aim at tracing and describing the complexity of knowledge construction across several classroom settings.

THEORETICAL BACKGROUND

The perspective we adopt for analysing the construction of knowledge during SGW is Abstraction in Context (AiC), a theoretical framework for analysing processes of constructing abstract mathematical knowledge [START_REF] Dreyfus | The nested epistemic actions model for abstraction in context -Theory as methodological tool and methodological tool as theory[END_REF]. AiC methodology begins with an a priori task analysis identifying the new (to the learner) knowledge elements required or useful when solving the task. It then uses a model of three types of epistemic actionsactions pertaining to the knowing of the learnersto analyze their learning processes. The model suggests constructing as the central epistemic action of mathematical abstraction. Constructing consists of assembling, interweaving and integrating previous constructs to make a new construct emerge. It refers to the first time the new construct is expressed or used by the learner. Hence, while the term constructing refers to the process, the term construct refers to the outcome of the action.

The perspective we adopt for analysing WCD episodes is documenting collective activity (DCA). Collective activity of a class refers to the ways of reasoning that function-as-if-shared (FAIS) as students work together to solve problems, explain their thinking, represent their ideas, and so on [START_REF] Rasmussen | A methodology for documenting collective activity[END_REF]. These FAIS ways of reasoning can be used to describe the mathematical activity of a group and may or may not be appropriate descriptions of the characteristics of each individual student in the group. The empirical evidence that a way of reasoning is FAIS is obtained by using [START_REF] Toulmin | The uses of argument[END_REF] model of argumentation, the core of which consists of Data, Claim, and Warrant. Typically, the data consist of facts or procedures that lead to the claim that is made. To further improve the strength of the argument, speakers often provide more clarification, which serves as a warrant for connecting the data to the claim. Backings provide further support for the core of the argument. For examples, see the data analysis below, e.g., in WCD 9. The following three criteria are used to determine when a way of reasoning becomes normative: 1) When the backing and/or warrants for particular claim are initially present but then drop off; 2) when certain parts of an argument shift position within subsequent arguments (e.g., a claim shifts to data); or 3) when a particular idea is repeatedly used as either data or warrant for different claims across multiple arguments.

In earlier studies [START_REF] Tabach | Knowledge shifts in the classroom -A case study[END_REF][START_REF] Hershkowitz | Knowledge shifts in a probability classroom -A case study coordinating two methodologies[END_REF], we have shown how DCA and AiC combine to provide an in-depth analysis of knowledge shifts in the classroom and of the knowledge agents that initiate these shifts. In [START_REF] Tabach | Abstraction in context and documenting collective activity[END_REF], we articulate why and how the two approaches are theoretically compatible. In this paper, we analyse a lesson where students dealt with an apparent paradox because of its potential to bring to the fore the complex nature of knowledge constructing processes across social settings in a classroom. Specifically, the paradox is an infinite perimeter that delimits a shape with no area, a phenomenon occurring in fractals. While paradoxes are abundant in the study of infinity, we found only two studies relating to similar ones: [START_REF] Sacristán | Students' shifting conceptions of the infinite through computer explorations of fractals and other visual models[END_REF] examined how the coordination of visual and numerical representations supported a single student's resolution of this apparent paradox. [START_REF] Wijeratne | On painter's paradox: Contextual and mathematical approaches to infinity[END_REF] found that their students were hindered by contextual considerations when attempting to resolve a similar paradox of a solid of revolution with finite volume but infinite surface area. Neither of these studies focused on the construction of knowledge in a classroom community.

METHODOLOGY

The setting for the research was a course on Chaos and Fractals at a US university, which formed part of the mathematics requirement toward a master's degree in mathematics education. Participants were 11 students with an undergraduate degree in mathematics, the teacher, and an instructor/observer who occasionally intervened. The teacher and instructor were both part of the research team. Classes took place during one semester twice a week for 75-minutes each; typical class periods alternated between SGW and WCD. During SGW, students worked in four stable groups; they were invited to use huddle boards -one table sized white board per group -in order to promote group communication and to facilitate subsequent whole class presentation of their work. The teacher and instructor went from group to group, trying to understand student thinking and attempting to focus students' activity on what they saw as the main issues; they did this mainly by asking questions but did not otherwise intervene in the SGW. The four stable groups will be numbered 1 (Carmen, Jan and Joy); 2 (Kevin, Elise and Mia); 3 (Soo, Kay and Shani); and 4 (Curtis and Sam). All names are pseudonyms. Groups 1 and 2 were video-recorded during SGW; the class was video-recorded during WCDs. In WCDs, groups had the opportunity to use the huddle boards to share their thinking; there were also teacher led discussions and short lectures whose aim it was to facilitate reflection on issues having been discussed by some groups.

On Day 9 (out of 24), class work was based on an activity about the Sierpinski Triangle (ST). As shown in Figure 1, the ST may be produced by a recursive procedure: Draw an equilateral triangle; connect the midpoints of its sides; remove the middle triangle to get three equilateral triangles (of side ½ of the original one); repeat these steps (including the repetition) for each of the three smaller triangles. The ST is obtained by means of the (infinite) recursion. The activity was based on a three-part worksheet. In Part A, students were asked to carry out the recursive procedure six times, blackening the removed triangles on the huddle boards. In Part B, they were asked to imagine continuing the recursion forever and to discuss the figure they would obtain, in particular its area and its perimeter. In Part C, they were asked about properties of the resulting figure, especially comparing it to its parts.

The teacher had planned for the students to come up with enough properties in Part C to enable a definition of self-similarity, and as a by-product of learning about selfsimilarity, to realize that self-similar objects may have finite (or even zero) area and infinite perimeter. However, as will be seen below, the class had its own emergent goals, which led us to investigate student reasoning about area (A) and perimeter (P).

Our task analysis yielded 4 knowledge elements for each of A and P: nature, process, limit, and infinity, denoted A n , A p , A l , A ∞ , and P n , P p , P l , P ∞ . Nature refers to the nature and properties of the region at step k whose area/perimeter is being considered. Process refers to the process of removing the triangles and computing the relevant area/perimeter. Limit refers to the realization that the process is never-ending and the areas form a sequence converging to zero whereas the perimeters form a diverging sequence. Infinity refers to the awareness that the actual figure has zero area and an infinitely long perimeter. The difference between limit and infinity may be construed as the difference between potential and actual infinity. These knowledge elements have been formulated in precise language and operational criteria have been fixed to decide whether a student or group of students has constructed each knowledge element. As an example, P ∞ is defined as follows: Eventually, there actually is a figure whose perimeter is longer than any finite curve. Operationally, we will say that a student has constructed P ∞ if the student explicitly claims that the eventual shape or figure or region has an infinitely long perimeter.

The data used in this paper consisted of transcripts from Day 9, images of the groups' huddle boards, and researcher notes taken in the classroom. SGW was analysed using AiC. We present only constructing actions here. In most cases, these will be attributed to the groups rather than individual students; exceptions will be noted. The WCDs were analysed using DCA; for each argument (numbered as A1, A2, etc.), Claim, Data and Warrant were identified, so that the criteria for ideas that FAIS could be applied.

SMALL GROUP WORK AND WHOLE CLASS DISCUSSIONS

The class on Day 9 started by watching and discussing an excerpt from a video about fractals with real world examples including a cauliflower, mountains, a magnetic pendulum and the coast of Britain; this took about 25 minutes and included WCD 1, SGW 2, and WCD 3. Then students were then asked to start working in groups on the worksheet; they spent about 26 minutes on drawing according to Part A and discussing the meaning of repeating the repetition (SGW 4, WCD 5). They used terms such as "infinite loop" and "zooming in". The focus of this paper is their work on Part B of the activity during the remaining 24 minutes, split into three episodes of SGW (6, 8, 10) and three WCDs (7,9,11). The class did not reach Part C on Day 9. SGW 6 (area as process) -The teacher invited the students to develop a conjecture about area and perimeter. After a brief discussion about the perimeter showing confusion (Joy: "in one sense it's infinity, because you keep adding a little bit more. But it should approach a number, right?"), Group 1 focused on area, and attempted to compute the area after one repetition. A reminder by the teacher to produce a conjecture led to a seed of the idea of recursion (Carmen: "That's one-fourth of it, so each term maybe three-fourths of it") thus starting the construction of A p .

Group 2 quickly came up with a formula (Elise: "So it's three fourths to the n of our A 1 ?") and spent the reminder of the time discussing what n means and how to denote things (e.g., A 0 for the initial area). We interpret this as having constructed A p .

WCD 7 (computations) -The teacher asked whether the groups had come up with a conjecture, and the students reacted by presenting computational results. Kevin presented their result as the sequence (3/4) n and Joy added that they had just started in this direction after computing the area of the first triangle. Student arguments included only claims and hence were not analysed per the DCA approach.

SGW 8 (perimeter as process) -Group 2 focused on computing the perimeter (Kevin: "we have an additional a, we have three halves more a", and later Mia: "So, it's like, it's going by a scale of three over two, to the n"). The group also made attempts at seeing what happens in the long run (Mia: "the perimeter is just keeps getting bigger, and bigger and bigger"; Elise: "Or is there, like, a limit? That it stops?"). While this points in the direction of P l , our interpretation is that they have not constructed P l yet: In spite of them having identified what an expert might see as a diverging geometric sequence, they question whether it converges or not. We also note that some students may be thinking additively rather than multiplicatively.

Group 1 quickly constructed A p (Carmen: "And then three-fourths of our threefourths"; Jan: "It's alright, we got enough… to do the whole") and somewhat hesitatingly, A l (Carmen: "Maybe zero?"; Joy: "No no no, because this is like threefourths time three-fourths is nine-sixteenths, and after that would be… what? Twenty-seven over sixty-four?"; Carmen: "Is it approach… zero? I think it does"; Joy: "Okay, so you are right, it approaches zero"). There is evidence that they also constructed A ∞ (e.g., Joy: "If you keep filling it in, there's not going to be any white area"). We note that there was no discussion about A n . However, they then held a long discussion about P n (e.g., Joy: "So what counts as the perimeter?"; Carmen: "Is it cumulative perimeter?"). Our analysis resulted in the decision that while constructing P n was under way, it had not yet been achieved. Next, they mentioned aspects of P p (Joy: "So let's say the perimeter of this is three, we would add in… half of each. So, like, three… times the half") without completing a constructing process. They were reminded by the instructor of the area tending to zero, which brought tension with respect to the perimeter (e.g., Carmen: "if we keep zooming in, there's no area, so there can be no fence [perimeter]").

WCD 9 (the controversy) -This discussion in Group 1 prompted the instructor to ask for the teacher's permission to ask Carmen and Joy to present their controversy to the class. Carmen's argument (A2) used as data "there's no area" and claimed "there'd be no perimeter" with warrant "there's nothing to… nothing to put a fence around it". Joy, on the other hand, argued (A3) for the opposite, using as data "as you zoom in there's more and more to fence", supported by the warrant that one keeps putting in more fencing material. This brought about a suggestion that when one removed a triangle (i.e. colours it black), the perimeter of this black triangle is added to the existing perimeter. In argument (A4) the claim is that "the perimeter of the, the white is also the same as the perimeter of the, perimeter of the black part" (Curtis), and this is based on the data "When you shade it in, you're adding the perimeter of the black" (Kevin) with the warrant that "the fence is guarding both properties" (Carmen). When encouraged by the instructor to explain Carmen and Kevin's thinking, Soo built argument A5: Claim: "So you keep adding the numbers, right?"; Data: "So you have more areas"; Warrant: "You keep zooming in, you're going to get more triangles forming". Next, Mia argued (A6): Claim: "I see the perimeter increasing and then this, the unshaded area is what's left over, and that's constantly decreasing and going to zero"; Data: "You're going to have all these shaded triangles, with perimeters"; Warrant, upon Carmen's question "Is this a cumulative perimeter or a perimeter at a point in time?": "I see it the first way" (Mia). Several more arguments (A7, A8, A9) in this WCD focused on the area decreasing and tending to zero in an unending process of creation. The analysis of arguments in this WCD resulted in two ideas that FAIS:

FAIS A: Perimeter of white is also perimeter of black; this was a claim in argument A4 but a justification in argument A6; the justification was that the perimeters of the shaded (black) triangles cumulatively constitute the perimeter of the remaining, unshaded, white area. Hence this idea satisfies Criterion 2. FAIS B: The perimeter is cumulative; this was a claim in A5 and a warrant in A6 (we will see it serving as justification again in A13), and hence also satisfies Criterion 2.

SGW 10 (connections) -Group 1 had a discussion of all four aspects of perimeter, completing the construction of, at least, P n and P p . They built their thinking on the fact that they used more and more ink at each stage to draw the additionally generated bits of perimeter, and concluded (Joy): "I thought it went infinitely, because if you zoom in, there's more fencing to put in. And if you zoom in there's more fence to put in". We have no evidence that they constructed P l and P ∞ . In fact, this is unlikely since they only completed constructing P n and P p toward the very end of the SGW. Moreover, Carmen, while admitting that the perimeter tends to infinity, insisted that intuitively, no area implies no perimeter.

Group 2 attempted to combine what they knew about the unending processes of area and perimeter. For example, Mia: "There's nothing for the area, but you're still… you're counting the perimeter of what you're taking out" and Kevin: "…as soon as you say -as n approached infinity, that means you're going to computation. So, I think what we want is something general, like -the area is getting smaller, but the perimeter is getting larger, and just leave it at that general statement". Our interpretation is that they may have started constructing A ∞ and P ∞ but are still far from completing these constructions.

The constructing processes resulting from the AiC analysis are summarised in Table 1. The table lists only constructs that we have evidence for; in other words, the fact that, for example, A n does not appear does not mean they have not constructed A nit only means that A n has not been discussed during SGW in a manner that lets us as researchers conclude that A n has been constructed. WCD 11 (linking area and perimeter processes) -The instructor called on group after group to present their thinking, however tentative. Group 1 used an ink metaphore, the ink being used to draw the additional perimeter bits. In this way, they explained how cutting out further triangles reduces the area while using more inkthus increasing the perimeter. Carmen, however, added, that for her "that's not resonating". The researchers identified two arguments while this group was reporting (A10, A11) and one (A12) while another group was reporting. During their report, Group 2 connected the perimeter process to the area process like Group 1. Elise built the following argument (A13): Claim: "The area is getting smaller and the perimeter is getting bigger". Data: "You're adding smaller and smaller pieces". Warrant: "…but you're adding those pieces to what you already have". Backing: "The perimeter is, like, all of this, combined with all of this, combined with all of this, combined…". We note that the argument focuses almost completely on perimeter, although the claim equally relates to area. Finally, the report of Group 4 included arguments A14 and A15. We only describe A15, produced by Sam: "So the area would be… go to zero… There would be limited amount of areas, so we're going to have a limited number of… perimeters. So, we don't have infinite number of perimeters". Sam's claim of a "not infinite perimeter" is based on the data that the area goes to zero and hence there is a limited amount of area, with the "limited number of perimeters" serving as warrant. Based on these arguments, we identified two more ideas that FAIS:

FAIS C: Unending process of creation; this meets Criterion 3: Continued use of an idea (e.g., keep adding) across multiple arguments to describe the process that is being analysed. This idea is related to potential infinity.

FAIS D: Area going to zero; this was repeatedly a claim, including in arguments A6, A7, A8, A9, A13 and became data in argument A15, hence satisfying Criterion 2.

The relationship between SGWs and WCDs

As a preliminary, we note the richness and diversity of students' ways of reasoning about area and perimeter, which in a less student-centred classroom might have been quickly undermined with an infinite geometric sequence that is decreasing (r<1) and hence tending to zero for area and an infinite geometric sequence that is increasing (r>1) and hence tending to infinity for perimeter. We note that the term geometric was once mentioned briefly with respect to area by Sam during WCD 11 ("it's geometric, so it's going to converge. So, the area would be… go to zero") but this was rather toward the end of class and was not picked up by any of the other students. Generally, students seem to have been satisfied by arguments of the type "the sequence is infinite and decreases, hence it tends to 0" (as in the discussion of Group 1 in SGW 8) or "the sequence is infinite and increases" (e.g., Mia and Elise of Group 2 in SGW 8).

The diversity manifests itself, among others, in the metaphors the students used (fence, ink), in a tendency to use, at least initially, numerical considerations for area and perimeter, in the attempt to link the area process with the perimeter process, and the related discussion about the nature of the perimeter as separating the region that belongs to the ST from the one that doesn't. Little of this was initiated or suggested by the teacher or the worksheet.

This last issue appears as P n in the AiC analysis of the SGWs and as FAIS A in the DCA analysis of the WCDs. Similarly, there are relationships between the other FAIS ideas and knowledge elements. Table 2 shows these relationships.

FAIS A B C D Constructs P n P p A l , P l A l , A ∞

Table 2. Relationship between FAIS and constructs

While FAIS idea A (the perimeter of white is also perimeter of black) is related to construct P n , the relationship between the constructing process of P n and the arguments establishing A as FAIS is complex. We don't have evidence of P n having been constructed in either of the two observed groups before the relevant WCD 9; and the first argument establishing idea A as a claim (A4) was initiated by Curtis but immediately supported by Kevin and Carmen. Moreover, the second argument, in which idea A became a justification, A6, was presented by Mia. This may indicate that the beginning P n construction we identified in Group 1 was substantial, and maybe even that the discussion in Group 2, which on the face of it focused on P p caused Kevin and Mia to think about P n . Finally, we ask ourselves to what extent the constructing process of P n continued during WCD 9 for Kevin, Carmen and Mia. FAIS B (the perimeter is cumulative)is similar to FAIS C (and intimately related to it from the point of view of the mathematical content). While we hesitated to claim that P p has been constructed by Group 2, it is Mia from that group who produced A6 and Elise from that same group who produced A13, the two arguments where the element switched position to becoming a justification and thus allowed us, according to DCA to categorize this idea as FAIS.

FAIS C (unending process of creation) exhibits a case in which the two analyses connect rather smoothly. Group 1 constructed A p and P p , and Group 2 may be assumed to have implicitly constructed A p and to be progressing in the constructing process of P p . The frequent use of this knowledge element in many WCD arguments may indicate a similar situation in the other two groups. Soo from the Group 3 produced Argument A5; and Sam from the Group 4 produced Argument A9.

Finally, the relationship between FAIS D (area going to zero) and A l seems obvious and needs little comment. When this idea functions as if shared in the classroom, it is possible that some of the students relate to construct A ∞ (as shown above for Carmen and Joy) and others think in terms of A l or even in terms of A p only. While to the expert, thinking in terms of A l may be satisfactory at best, and thinking in terms of A p may be insufficient, such differences are tended to be glossed over in this classroom with respect to the rather basic construct of area, and we may speculate that similar situations pertain to more complex constructs in this and other classrooms.

It becomes obvious that there are many ways, in which SGW and WCD can interact. The relationship is by no means unidirectional from constructing an idea in SGW to this idea FAIS in WCDs. Rather, constructing processes may well be continued or even initiated in WCDs. On the other hand, ideas may FAIS without having been constructed by all or even by a majority of the students in class. For example, we have no evidence for P l having been constructed in either of the two observed groups, though Group 2 had started this constructing process; but C (unending process of creation) is anyway FAIS in relation to both area and perimeter. Of course, it could have been constructed in the groups we have no data on. This raises the question whether an idea can FAIS if it has not been constructed at least in some group. We can only say that we have no example for this having happened in these data.

On the other hand, notions can be constructed by some students, or in some groups, without ever functioning as if shared. In fact, some of the constructs may not have a chance to come up in any WCD. We have no unequivocal evidence for this happening but we do know that in the first few minutes of the next lesson (Day 10), which took place two days later, Kevin referred to the perimeter as an increasing and hence diverging geometric sequence. We could also point to the fact that A p has been constructed by both analysed groups but does not appear in the lower row of Table 2. However, this argument is weak since A p appears indirectly as a component of A l .

CONCLUSION

The complexity of knowledge flow in the classroom, even based on this one class session, is far greater than one might imagine. Inquiry-based instruction features students' deep engagement in mathematics and peer to peer interaction. As such instruction increases at the university level, the field is in need of theoretically grounded approaches for analysing individual and collective mathematical progress. This paper makes a contribution in that direction. A strength of AiC is that the approach allows researchers to gain insight into the ideas that individuals or small groups of individuals construct, as long as the number of students remains small. DCA provides a complementary approach that provides researchers insight into the ideas and ways of reasoning that characterize the collective progress of the classroom community. In [START_REF] Tabach | Abstraction in context and documenting collective activity[END_REF], we showed how the two approaches combine theoretically, and the present paper adds to this a detailed analysis of how an AiC analysis interacts with a DCA analysis to expresses the complexity of knowledge construction in the classroom. An open question, and one that we are currently pursuing with this data, is how to coordinate the small group and classroom level findings with individual interviews conducted shortly after such rich class sessions.

Finally, the analysis presented here also contributes to what we know about how students reason about area and perimeter in a paradoxical situation. In contrast to the findings of [START_REF] Wijeratne | On painter's paradox: Contextual and mathematical approaches to infinity[END_REF], the students in our classroom found ways to profitably use contextual considerations, metaphors, numerical computations, and figural reasoning to support their endeavour to understand the apparent paradox.

Figure 1 .

 1 Figure 1. The Sierpinski Triangle (shown after 6 iterations)

Table 1 . The constructs; parentheses mean (under construction)

 1

	Group SGW6	SGW8	SGW10

Acknowledgment

This work has been partially supported by Israel Science Foundation grant 438/15.


Peter Stender
email: peter.stender@uni-hamburg.de

Ann Sophie Stuhlmann

Fostering Heuristic Strategies in Mathematics Teacher Education

Keywords: Novel approaches to teaching, teaching and learning of analysis and linear algebra, mathematical methods, heuristic strategies, teacher education, Double discontinuity

The "double discontinuity" stated by Felix Klein 1908 is still relevant in the mathematics teacher education at German universities. We are developing a course approach, which is intended to bridge the double discontinuity in a didactic dimension. We offer additional learning opportunities for teacher students, which are characterized by clarifications of mathematical methods that are fundamental to the regular lecture contents. We focus especially on reflections on heuristic strategies, which are the core part of mathematical methods according to George Pólya. Feedback shows that students consider our approach as helpful in the transition from school to university.

INTRODUCTION

In Germany, the first year at university is often associated with great problems for mathematics teacher students. The dropout rates in the study entrance phase are extremely high in mathematics compared to other study programs. [START_REF] Dieter | Zahlen rund um das Mathematikstudium -Teil 3[END_REF]. Klein's "double discontinuity" is still an issue in the mathematics teacher education at German universities [START_REF] Hefendehl-Hebeker | Doppelte Diskontinuität oder die Chance der Brückenschläge[END_REF]). On the one hand, the double discontinuity can be understood as a discontinuity of content, but Klein also assigns a didactic dimension to it [START_REF] Allmendiger | Die Didaktik in Felix Kleins "Elementarmathematik vom höheren Standpunkt aus[END_REF]. While mathematics teaching at school can be described as intuitive and problem-based, mathematics at the university is characterized by a deductive structure (Klein, 1905). The ideas and strategies behind the findings of mathematics are rarely presented at the university. The mathematics teacher students take part in the same mathematics lectures as regular students who study mathematics as a major subject. Therefore, bridging the double discontinuity is only possible by offering additional learning opportunities for teacher students. A few years ago, we implemented additional seminars for teacher students where we showed connections between school mathematics and university mathematics. Feedback of such seminars showed that students did not consider our approach as helpful for bridging the discontinuity. They said that the regular lectures were challenging and time consuming enough so they did not want to think about additional issues in our seminars. For this reason, we have developed an approach for additional learning opportunities for teacher students, which is characterized by reflections on intuitions and strategies that are fundamental to the regular lecture contents. Our approach is intended to bridge the didactic discontinuity. On the one hand, we want to support teacher students in the transition from school to university, and on the other hand, knowledge about strategies and intuitions is important in the future teaching activity at school. In the following report, we describe first experiences we have made with our seminars.

THEORETICAL FRAMEWORK

Subjects and methods in mathematics

In the philosophy of science, it is generally accepted that each science has its specific subject and its specific method i . As mathematical subjects in university mathematics, we deem the mathematical definitions, theorems and their proofs including mathematical algorithms. The mathematical methods describe how the subjects of mathematics are created or investigated by a mathematician ii . Which methods does a mathematician use, when formulating a new definition or a theorem? Furthermore: how does the mathematician come to the ideas when developing a proof? Sometimes the steps of a proof are easy in the sense that a beginner could do them but the decision to do these steps needs a lot of experience. In fact, producing a proof that does not already exist is an act of problem solving according to the definition of Schoenfeld (1985, p.74) or the idea Pólya (1973, p. 3) describes by mentioning "problems to proof". This point of view corresponds also with the definition of the concept "problem" the German cognitive psychologist Dörner gave in 1976 (p. 10 iii , own translation):

What a problem is, is easy to define: an individual faces a problem if he or she is in an inner state that is unwanted by any reason, but the individual has not the means in the moment to transform the unwanted situation into the desired one. Three aspects characterize a problem:

1. An unwanted start situation s α .

2. A desired final situation s ω .

3. A barrier that prevents the transformation from s α to s ω at the moment.

"We distinguish problems from tasks. Tasks are intellectual challenges for which the methods iv are known. […]. Tasks only need reproductive thinking while in problem solving you have to produce something new." (1976, p. 10, own translation) Dörner (1976, S. 10, own translation) emphasised: "Whether a question is a problem or a task depends on the experience of the individual." If a student just learned induction "Prove the Bernoulli Inequality by induction!" as the first exercise might still be a problem but having done a series of similar proofs, it should be a task.

Based on this definition of a problem [START_REF] Dörner | Problemlösen als Informationsverarbeitung[END_REF] defines heuristic strategies as "the structure that organises and controls a problem-solving process" (Dörner, 1976, p. 273, own translation), which covers a very wide range of methods and is not bound to mathematics. Following this concept, heuristic strategies cover much of what is meant by the concept of "method" in the philosophy of science once applied to mathematics. [START_REF] Pólya | How to solve it. A new aspect of mathematical methods[END_REF] already underlined this as the subtitle of "How to solve it" is "A new aspect of mathematical methods".

The following list of heuristic strategies was formulated based on the work of Pólya and others relying on the broad definition by Dörner. These strategies are arranged in groups for a better overview: Examples and detailed descriptions for each strategy are shown in [START_REF] Stender | The use of heuristic strategies in modelling activities[END_REF]. These strategies do not cover all the methods in the work of mathematicians but a broad range due to our literature analysis. Many of these strategies are also applicable beyond the field of mathematics v and this way of importance as general problem solving strategies. We would call a list of heuristic strategies that are used only in mathematics as they rely on a formal language "proof strategies". This list contains for example mathematical induction, proof by contradiction, proof by exhaustion, the Invariance Principle (Engel, 1998, p. 1), Cantor's diagonal argument, the box principle (Engel, 1998, p. 59) but also very simple ideas like "use an adequate substitution" or "add zero in an appropriate way." We do not claim that any of these lists is complete. A third aspect is the use of mathematical language like reading a mathematical text appropriately (for example see [START_REF] Hodds | Self-Explanation Training Improves Proof Comprehension[END_REF]. We visualised these aspects in figure 1.



The importance of the methods in mathematics

The discussion on problem solving since [START_REF] Pólya | How to solve it. A new aspect of mathematical methods[END_REF] shows the importance of heuristic strategies in mathematics and thus (if one follows the argument above) the importance of mathematical methods in the process of learning mathematics. But mathematical methods are not emphasized often in (German) lectures in the mathematics departments. Therefore, we formulate an additional argument here.

We look at a special case of mathematics competence. Imagine a person that knows many mathematical definitions, theorems and even proofs by heart, but has hardly any mathematical methods available. This person would not be able to solve any mathematical problem.

The other extreme would be a person with little declarative knowledge of mathematical facts but competent in applying mathematical methods. He or she just has forgotten most of the mathematical content he or she learned. This person would be able to understand new mathematical content very fast and could fill gaps in the mathematical knowledge quickly if necessary.

This thought experiment shows that a lack of mathematical methods is a much more severe problem while doing mathematics than a lack of knowledge about mathematical content.

A mathematician often uses mathematical methods unconsciously. This is appropriate, as this is faster in the most cases than channelling everything through the restricted working space (see Miller, 1956) in the prefrontal cortex. For a future teacher it is much more important to be aware of his own methods as he or she shall later explain his actions to children. An explicit reference to his own methods is essential in these explanations.

Furthermore, the methods mentioned are the same in university mathematics and in school mathematics while the content shows little similarities as [START_REF] Klein | Elementary mathematics from an advanced standpoint[END_REF][START_REF] Klein | Elementarmathematik vom höheren Standpunkt[END_REF] pointed out. This way the methods are an important link between school mathematics and university mathematics and give studying abstract mathematics a meaning for students who want to become a teacher.

These considerations lead us to the following goal for our research: we want to implement and evaluate teaching and learning of heuristic strategies in the mathematics teacher education in a close connection to the mathematics content trying to show that learning abstract mathematics means learning mathematical methods which is crucial for teaching mathematics in school.

IMPLEMENTED INTERVENTIONS

The approach we have developed for teacher students includes different additional learning opportunities:

 We offer special tutorials for mathematics teacher students connected to the Linear Algebra and mathematical Analysis lectures.

 Within the scope of Linear Algebra and Analysis lectures students must do weekly exercises. We have implemented a weekly workshop where students get support in creating and formulating proofs.

 All students may choose a preparation lecture before beginning their studies. Part of this lecture is a tutorial. We offer special tutorials for mathematics teacher students too.

These learning opportunities have in common that they do not intervene in the regular lectures and are not obligatory. This way we are free to realise our teaching approach, but this also leads to smaller groups so a standard pre-test-post-test design to evaluate the tutorials is not appropriate. We offered the tutorial over four semesters with 20 participants. This number was quite stable while participants changed: some gave up studying mathematics, some new students joined during the semesters when they heard about the tutorial. Due to the individual timetable of students, some of them couldn't participate over all four semesters. The tutorial was open to students with a major in math too (parallel traditional tutorials exists that only focus on mathematical content) so overall five teacher students participated over the whole time.

As the tutorials fulfil different functions in the learning process, we offer the following aspects for the students:

Among other aspects as answering questions related to the lectures, we point out the appearance of heuristic strategies in the lecture content. Besides students solve small exercises so we can exemplify the application of heuristic strategies. We discuss the application of heuristic strategies in school by means of examples. Furthermore, we give a structured overview over the lecture content several times per semester to help students to network their knowledge. The overview is closely connected to some heuristic strategies as using analogies and generalise the situation.

An example for the use of heuristic strategies is the use of equivalence classes in linear algebra for example using quotient rings. Quotient rings were presented in the lecture using a formal definition. The lecturer gave ℤ /mℤ as an example but this structure was not familiar to the students. In the tutorial we showed the following ideas to the students:  Using a special example, we presented the ring ℤ /4ℤ .

 For this structure, we gave several different representationsa number line where the numbers of the same equivalence classes were coloured in the same way, the four classes as sets, a representation on a circle and a table that showed the calculations in this structure.

 The importance of the concept of equivalence classes was emphasised: a number of objects is chunked together to one new object according to Miller (1965) so an element of a quotient ring is what we call a supersign according to [START_REF] Kießwetter | Modellierung von Problemlöseprozessen[END_REF].

 We pointed out, that similar structures occur in school when dealing with rational numbers, which are classes of pairs of natural numbers (supersigns) and where several representations are necessary while working with fractions. Even if fractions are not implemented in this abstract description in school, students must deal with this mathematical structure.

This way similarities in school mathematics and university mathematics are brought into the focus of the students and showed that students in school dealing with fractions might experience the same difficulties as students in university dealing with quotient rings.

A second example from the tutorial is a short exercise we presented to the students:

In the tutorial matrices were used in vector spaces with bases. In the tutorial, we showed how the matrix describing a rotation (ℝ 2 →ℝ 2 , angle α) could be build. Then we asked the students to develop the matrix for a reflection through an axis that has the angel α with the x-axis. As this group was already used to heuristic strategies, we gave the following help:

 Change the representation (figure).

 Look for special cases (basis).

 Trace back new problems to familiar ones (use the knowledge from the rotation).

 Break down your problem into sub-problems (work on one element of the basis at a time).

 Use symmetry (to work on sin(2α-180°).

This example could also be situated in high school, so it breaks down university mathematics to school mathematics and shows in addition how heuristic strategies can be used supporting students work.

In the weekly workshops, students do exercises on their own. Mostly students have to prove small theorems related to the lecture content. They work in groups of three or four and a tutor supports them in developing and formulating proofs. In case students are not able to overcome barriers in proving processes by themselves, the tutor promotes the application of an appropriate heuristic strategy. Following a proving process, students reflect on the strategies they have applied in the process.

EVALUATION RESULTS

As said before the number of students participating in the additional learning opportunities is too small for quantitative designs. We are in the process of designbased research as our interventions are an early approach for the use of heuristic strategies in mathematics education. This means results of efficiency do not make sense in this phase of the research.

We have conducted interviews with five participants of the tutorials and the workshops at the end of each semester. We have analysed these interviews using qualitative content analysis according to [START_REF] Mayring | Qualitative Inhaltsanalyse: Grundlagen und Techniken[END_REF]. In the interviews, the students were asked what they remembered from the tutorial and what they think was helpful (not discriminating helpful for studying mathematics or helpful for the later profession). If whole aspects were not mentioned the interviewer stimulated an answer using a key word. The two kinds of answers (with or without stimulation) were separated in the analysis.

Over the four semesters of this process, a slightly increasing awareness of the heuristic strategies in mathematics can be observed in the responses of the students. Over the time, the students gave more examples of the use of heuristic strategies during the interviews. The students gave a positive feedback according to the heuristic strategies as a link between school and university. They said that it delivered a meaning to the fact that the content of the mathematical studies had hardly any connect to school mathematics. One student mentioned that he used heuristic strategies to explain mathematics when he was working in school as a substitute teacher. The students regard the workshops as very helpful for their learning progress. They emphasize the importance of developing proofs by their own to understand the lecture content. Therefore, they consider it beneficial that the tutor gave only a few strategic hints in case of barriers in the proving process rather than presenting solutions. Especially the heuristic strategy of changing the representation helps students to develop new ideas.

We think that it can be expected that the heuristic strategies integrated in the lecture as metacognition, added to the mathematical content would be far more effective for learning mathematical methods.

CONCLUSIONS AND LOOKING AHEAD

Overall, we can observe that there are effects of the metacognitive input according the heuristic strategies as mathematical methods, but the learning process is very slow. It needs many different examples for each strategy to be able to realise the importance of these strategies and even more to identify them in your own mathematical work or use them explicitly to solve problems. Pólya (1973, p. 208) quotes a fictional traditional professor "A method is a device which you use twice."

Obviously "twice" is far not enough in the learning processand not enough to make a device so important that it should be taught.

The teaching concept to support students by stressing the mathematical methods particularly by stressing the heuristic strategies seems to have a positive impact but is not realised easily. This corresponds with the difficulty of teaching problem solving and heuristic strategies (Schoenfeld1985, p. 95). The examples we gave to the students during our tutorials and workshops had an impact on realising heuristic strategies in mathematics on the long run. However, we do not know if they were not enough to let the student's use these strategies in their mathematical work explicitly.

We redesign the tutorial if necessary and expand the number of learning situations where we implement metacognition on the use heuristic strategies. A long-time goal is to include metacognition in the lectures itself but that means to convince mathematics lecturer to change their lectures.

Our next project is developing a tutor training for master students tutoring beginners.

In the tutor training, we will use former homework tasks and analyse the solutions according to the use of heuristic strategies. This will be presented to the students in few examples and then done on further tasks by the students working in small groups. In the next step, we will demonstrate how these strategies can be used explaining mathematics and supporting students to overcome barriers during problem solving. The students shall use their own results from the first step to practise this.

  organise your material / understand the problem: change the representation of the situation if useful, trial and error, use simulations with or without computers, discretize situations,  use the working memory effectively: combine complex items to supersigns, which represent the concept of 'chunks', use symmetry, break down your problem into sub-problems,  think big: do not think inside dispensable borders, generalise the situation,  use what you know: use analogies from other problems, trace back new problems to familiar ones, combine particular cases to solve the general case, use algorithms where possible,  functional aspects: analyse special cases or extreme cases, in order to optimise you have to vary the input quantity, discretize the situation,  organise the work: work backwards and forwards, keep your approachchange your approachboth at the right moment.

Fig. 1 :

 1 Fig. 1: Mathematical Content and Methods

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ii There is a broad discussion whether mathematics is produced by humans or discovered and for both standpoints and for the idea, that mathematics is a mixture of both there are good arguments. For this paper it makes no difference if one changes the verb "producing" by "discovering".

iii The original is in German language, this is an own translation. iv Dörner uses the word "method" not in the same way "scientific method" is used in this paper. The word method has more the meaning of an algorithm here.

v [START_REF] Pólya | How to solve it. A new aspect of mathematical methods[END_REF] gave examples from mathematics for the heuristic strategies he mentioned but also examples from outside mathematics for most of the heuristic strategies shown.