Erik Hanke
email: erik.hanke@uni-bremen.de

Angelika Bikner-Ahsbahs

Boundary crossing by design(ing): a design principle for linking mathematics and mathematics education in preservice teacher training

Keywords: Boundary crossing, boundary object, complex analysis, reflection, teacher education

Preservice teachers experience a fragmentation in their study program between two academic areas, university mathematics and mathematics education. To reduce this problem, we conduct a design study linking mathematics and mathematics education on two levels: First, the "dovetailing" of mathematics (here the course on complex analysis) and mathematics education, and second, students' "interlinking" of content and pedagogical content knowledge in the development of learning materials for talented pupils. This paper contributes to the topic on theory and methodology by introducing the design principle "boundary crossing by design(ing)" in "nested design cycles".

Introduction

Many preservice teachers in mathematics experience two types of discontinuities in mathematics teacher training. First, in their views, much of the mathematics they learned in school does not seem very helpful for the rigorous proof-oriented style in university mathematics courses. Second, preservice teachers may not see how the mathematics in their university courses is related to their practice as a teacher [START_REF] Hefendehl-Hebeker | Doppelte Diskontinuität oder die Chance der Brückenschläge[END_REF][START_REF] Prediger | Unterrichtsmomente als explizite Lernanlässe in fachinhaltlichen Veranstaltungen[END_REF]. Several attempts to minimize the perceived gap between university and school mathematics have been developed, for example with special exercises, or through capstone courses (e.g., [START_REF] Bauer | Schnittstellenmodule in der Lehramtsausbildung im Fach Mathematik[END_REF][START_REF] Winsløw | The ecological Klein's double discontinuity revisited: contemporary challenges for universities preparing teachers to teach calculus[END_REF]. Furthermore, empirical studies show that the combination of mathematical knowledge and pedagogical content knowledge for teaching and learning mathematics (i.e. in the sense or similar to that of [START_REF] Ball | Content knowledge for teaching. What makes it special[END_REF]) is important for evaluating situations in the classroom (e.g., [START_REF] Blömeke | Von der Lehrerausbildung in den Beruf -Fachbezogenes Wissen als Voraussetzung für Wahrnehmung, Interpretation und Handeln im Unterricht[END_REF]. [START_REF] Mehlmann | Spotlights Lehre -Ein Ansatz zur Vernetzung von Fachwissenschaft und Fachdidaktik an der Universität Bremen[END_REF] give an overview of recent trends in German mathematics preservice teacher education around the "double discontinuity" and the general criticism of preservice teachers towards their study programs which they experience as fragmented. Because there is barely any explicit research on how preservice teachers may link the scientific disciplines of mathematics and mathematics education 1 concerned with school mathematics, we ask how this fragmentation between the mathematical discipline at university and mathematics education in preservice teacher training can be overcome at least partially by making changes in the curriculum, and how preservice teachers establish interlinkages of the two areas. At the University of Bremen, we conduct the design study "Spotlight-Y" on these issues. Two out of three design cycles have been completed so far, and as a theoretical result from our first design cycles, we will establish the design principle boundary crossing by design(ing) in a nested design cycle to initiate the overcoming of the fragmentation in preservice teacher training. We will show how this design principle is interwoven with the methodology and theory of our study. At the end, examples of empirical findings round off the paper.

Linking mathematics and mathematics education in a two-fold way

Linking mathematics and mathematics education can be viewed as a two-fold phenomenon in higher education: On the institutional level it is a matter of curriculum and we call the linkage of the domains on this level dovetailing (German: Verzahnen). On the students' level we call the linkage of the domains in students' thinking and acting interlinking (German: Vernetzen). This distinction corresponds to a distinction between "curriculum to be taught" and "learned curriculum". Our design study is devoted to an empirically grounded description of these two notions and their connections. Thus, with this terminology our general research questions in the project can be rephrased as: How can dovetailing be realized to initiate interlinking? Which circumstances foster or hinder students' interlinking and which interlinking strategies can be identified? To interlink the two scientific areas, our students have to cross the boundaries between them.

Theoretical background: Boundary crossing

The idea of boundary crossing has origins and applications in the fields of general education and social theory [START_REF] Akkerman | Boundary crossing and boundary objects[END_REF]. Boundaries can be described as "sociocultural differences that give rise to discontinuities in interaction and action" (Akkerman & Bakker, 2011, p. 139) and boundary crossing as the "efforts by individuals or groups at boundaries to establish or restore continuity in action or interaction across practices" (Bakker & Akkerman, 2014, p. 225). Means to achieve this are boundary objects which "are artifacts that articulate meaning and address multiple perspectives" (Akkerman & Bakker, 2011, p. 140), e.g., a concrete task or solution. Aside from this initial understanding of a boundary object, it can also be conceptualized as a "shared problem space" (Akkerman & Bakker, 2011, p. 147) where motives of different domains act together and "materiality derives from action, not from […] 'thing'-ness" (Star, 2010, p. 603).

It is the interpretation of the members of the communities that give objects their meaning as boundary objects in social contexts [START_REF] Star | This is not a boundary object: Reflections on the origin of a concept[END_REF]. Consequently, even if teachers or researchers intend to implement objects as boundary objects in their teaching or research, it may not be the case that the people who work with the objects understand them as such. However, because the aim is "to integrate different types of knowledge typically developed in different practices" (Bakker & Akkerman, 2014, pp. 224-225), boundary objects "may provide learning opportunities" (Akkerman & Bakker, 2011, p. 141), and students working with boundary objects may carry out interlinking actions or even practices, probably implicitly. Interesting for our purpose are three out of four mechanisms for the process of boundary crossing identified by [START_REF] Akkerman | Boundary crossing and boundary objects[END_REF] when considering learning in terms of boundary crossing in the broadest sense. Identification is about finding out particularities of the social practices and seeing the relevance of different practices which meet at a certain border. Coordination means to mediate between social worlds. Reflection differs from identification in the sense that particularities of one community's practice are made explicit for another community (perspective making) or are regarded from the viewpoint of the other community (perspective taking). In Spotlight-Y the notion of reflection is much broader than the boundary crossing mechanism of reflection (see section "Boundary crossing by design(ing) through a nested design approach"), but perspective making and taking can be part of it.

Methodology

Curricular linkage: Dovetailing in the lecture on complex analysis

The project "Spotlight-Y" consists of three cycles of developing the course on complex analysis with the aim to overcome fragmentation (see Hanke & Schäfer 2018a[START_REF] Hanke | Modellprojekt 2: Spotlight-Y (Mathematik)[END_REF]Mehlmann & Bikner-Ahsbahs, 2018, for details). Each cycle takes a year with a lecture in complex analysis being split into a branch for future teachers and future mathematicians after around nine weeks of the lecture. While the future mathematicians continue with more advanced complex analysis, the preservice teachers prepare learning materials (exercise sheets and means to explore and visualize with GeoGebra) for talented secondary school pupils about a mathematical topic they identified in the lecture (e.g., power series expansion, spherical geometry, or differentiation as dilation-rotation) and implement this on a day for experimental mathematics for classes of school students visiting university (XMaSII). In parallel with the complex analysis lecture, the preservice teachers participated in a seminar in mathematics education which also covered the design of tasks. Our assumption is that through boundary crossing in preparing and implementing of this practical teaching, the preservice teachers will overcome fragmentation by showing interlinkage between the knowledge from both areas. So far, 35 students participated in the teacher branch in complex analysis during the last two years.

Approach for designing the lecture

To develop the teaching of complex analysis as described above we conduct a design study [START_REF] Gravemeijer | Design research from a learning design perspective[END_REF]. It aims at achieving a design as well as local theoretical knowledge about how the design works, clarifying circumstances under which the fragmentation in teacher training can be alleviated, and helping to identify obstructions. Each design cycle consists of four elements [START_REF] Prediger | Lehr-Lernprozesse initiieren und erforschen-Fachdidaktische Entwicklungsforschung im Dortmunder Modell[END_REF] see Figure 1, left): First, a learning object or phenomenon has to be restructured. This leads to the development of a (local) design2 based on a specific aim. Then, the design is executed in a design experiment. Insights from these steps are used to theorize the phenomenon, and these insights allow for a better description of the learning object or phenomenon, changes in the curricular implementation, design of the lecture, and execution. The cycle can then start over again. The core idea of our methodological frame is the nature of the design experiment. It contains an embedded cycle of the same structure that initiates "boundary crossing" in students' work (Figure 1,right).

Boundary crossing by design(ing) through nested design cycles

In our design study, two cycles are nested (Figure 1). The outer cycle (grey, left) is the methodical frame for research and teaching: The learning object for us as researchers is dovetailing and interlinking university mathematics and mathematics education in preservice teacher education. The general design of the lecture was described in the section "Curricular linkage: Dovetailing in the lecture on complex analysis" and the design experiment is the course work in the branch for preservice teachers in the lecture on complex analysis. The inner circle (red, right) frames the development and implementation of the learning materials by our students. Here, the learning objects (for students and pupils) are the mathematical phenomena the students choose from the lecture. The process is accompanied with three types of written reflective activities. We understand reflection as anticipatory as well as retrospective (re-)consideration of own work, including experiences of various kinds, thoughts about personal relationships to the two disciplines (mathematics and mathematics education), the own role as a future teacher, and possibilities for the improvement of local designs of learning arrangements. In a preflection, the preservice teachers describe the mathematical phenomenon and anticipate difficulties at the beginning of their planning, in ad-hoc-notes, they reflect on observations right after the execution of XMaSII, and in a final portfolio3 they reflect on the full process and their professional development, each with guiding stimuli (Hanke & Schäfer, 2018a[START_REF] Hanke | Modellprojekt 2: Spotlight-Y (Mathematik)[END_REF]. This is in accordance with the notion of reflection-onaction as proposed by [START_REF] Schön | The reflective practitioner. How professionals think in action[END_REF] and reflection-pre-action by Bikner-Ahsbahs (2017).

Instances of interlinking mathematics and mathematics education can be identified in students' reflections from which we extract interlinking strategies (see results section). Such a strategy may be a certain local mechanism a student implemented, or a personal insight. Since the reflections are written after a teaching episode with classes of school students (the design experiment), we might also reconstruct interlinking strategies which have not appeared in action, but which have been thought about by the students. The reflections have two functions: First, they serve to reconstruct and typify the interlinkages between university mathematics and mathematics education, and second, students experience reflection as a means to learn from own practical experience (planning, executing, assessing) and be aware of interlinking activities [START_REF] Mehlmann | Spotlights Lehre -Ein Ansatz zur Vernetzung von Fachwissenschaft und Fachdidaktik an der Universität Bremen[END_REF].

In the inner cycle, in the branch for the preservice teachers, the boundary of the scientific lecture and mathematics education is crossed by referring to each other. The initial boundary object is the student task to create the XMaSII-day learning arrangement. This task can be interpreted from the mathematical as well as the mathematics education side. The specificities of the implementations and reflections indicate how the students interlink both areas. The work of the students leads to a design process which initiates boundary crossing. All data, in the form of exercise sheets, computer arrangements and different kinds of written reflections shape a dynamic boundary object which is assembled by the students' work at the boundary of mathematics and mathematics education and which, at the same time, is subject to change. We call this design principle (van den Akker, 1999) boundary crossing by design(ing). "By design" refers to the curricular implementation, and "by designing" refers to the processes the students engage in. The boundary crossing or interlinking mechanisms we observe stand for the interlinking practices of students and can be further used for the next cycle of the branch for preservice teachers in the lecture on complex analysis. In that sense, the design principle boundary crossing by design(ing) serves as our epistemic means to describe to which results acting at the boundary of mathematics and mathematics education may lead. In other words, dovetailing is the attempt to initiate boundary crossing and the students' boundary crossing may lead to thinking and acting between mathematics and mathematics education, i.e. interlinking. Thus, boundary crossing is regarded as a mediator between dovetailing and interlinking.

The novelty of boundary crossing by design(ing) is not the fact that mathematics and mathematics education are asked to be combined-this is expected practice of any (future) teacher. It is that the linkage we try to establish is between the two scientific domains, mathematics at university and mathematics education for preservice students' teaching (cf. Introduction).

Exemplary results of the first two cycles

Having developed the methodology and the design principle, we conjecture that implementing the design principle into the third cycle will initiate students' interlinking between complex analysis and mathematics education, and we ask whether we will find the same or new interlinking strategies of the students. But how do these interlinkings show up and what conditions foster or hinder them? Until now, this is answered by analyzing the data of the first two cycles, and here, we give some examples.

Results indicate that interlinking is an additional demand (concerning mathematical expertise and/or time) which is often not accomplished by our preservice teachers, and happens in microscopic ways. Besides others, the students were asked to describe which elements of the course on complex analysis changed their views on the scientific discipline of mathematics and mathematics as a school subject. This question aimed at a retrospection on the course as a whole. One of the students wrote (we note that the learning arrangement of this student did not deal with extension of number sets but with differentiation):

The extension of number sets with the set of complex numbers is very demanding at first. Pupils who encounter negative numbers for the first time might feel similarly. The experiences I made in this course can help me to empathize with pupils and understand why the extension of numbers with the set of irrational numbers can be difficult for them and why they may not consider the number π as a number.

In addition, the experience of the lacking imagination of the graphical representation of complex functions can help to understand the difficulties pupils have with the visualization of graphs of functions. (translated from German) This description expresses the sensitization for pupils' difficulties by analogy. The example in the quote shows an experience-based perspective on problems with extensions of number sets and diagrams which can be regarded as a coordination mechanism with the potential for a transformation mechanism (see: [START_REF] Akkerman | Boundary crossing and boundary objects[END_REF]. For example, a transformation could occur if the student presented a suggestion for how to deal with these issues in class. We may conclude that in this case the student's own experience with subject matter sensitizes for learning processes of pupils. What is not important is that the content in the student's learning process is beyond school but that what is learnt, in each case, is new for the respective learner (student and pupil).

To the stimulus "What did you learn personally? Do you feel assured in your role as a teacher? How and why (not)?" another student expressed (his learning arrangement was about power series), So I find the problem to prepare a very complex and alien (German: fremd) topic for pupils very interesting and instructive because I, myself, had to cope with this topic intensively once again.

Unfortunately, I had to realize that exactly in this transformation very much time and work had to be invested which lacked for example for the precise planning and execution. At any time, the subject specific understanding (German: das fachliche Verständnis) was the most important aspect of the planning. (translated from German)

This utterance shows that the creation of a learning arrangement about a demanding mathematical topic requires resources in order to engage deeply with the subject matter. Here, the need for these resources sidelined the planning actions for the actual planning and implementation.

Discussion and reflection

Reflecting on the two design cycles at the university level we went through, the steps of how the design principle developed becomes clear. Due to the fragmentation in preservice teacher education, our first design principle was to link university mathematics and mathematics education through providing practical experience for our students in one mathematics lecture. The method seemed clear; the students identify a mathematical phenomenon from the lecture, plan and implement learning materials for talented pupils, and thus, overcome fragmentation. But this did not happen with most of the students. Therefore, after the completion of the first design cycle, we clarified the relationship between this method and our research design. That is, our students were actually performing a design cycle similar to ours on the research level. Thus we obtained a more precise methodology, the nested design approach. Since we were not able to rely on previous research, we had to bridge the distinction between dovetailing and interlinking theoretically. The theory of boundary crossing serves this purpose. The concise link between theory and methodology was achieved after the second cycle and on this basis we obtained the current design principle. Boundary crossing by design(ing) condenses the theoretical approach of boundary crossing and links it to the task of creating and implementing learning materials while taking into account two scientific domains. This design principle addresses two different kinds of design: design research at the university level for course development in which the students design learning arrangements. These threads together can lead to insights into the reduction of the fragmentation in teacher training we strive for. Ultimately, our project exemplifies that a suitable design principle does not necessarily stand at the beginning of a design study but can be achieved through development and confluence of methodology and theory.

Our students valued the practical experience in the project. Even when they did not manage to produce explicit interlinkages of the two domains, in some cases of interlinking the two domains we observed a kind of coordinating, a mechanism of boundary crossing. Hence, we should ask what kind of conditions hindered the preservice teachers to experience interlinking mathematics and mathematics education (of school mathematics). We next aim at developing a scaffolding procedure. For example, cognitive overload might be avoided by a concrete example of how a phenomenon of complex analysis could be transposed into a learning arrangement for talented pupils or by a more detailed structure for the task to create a learning arrangement that assists preservice teachers in their boundary crossing.

Last, we highlight an important epistemic obstruction resulting from our methodological approach. The contents and depths in the students' reflections limit the depths of our insights. Also, reflections mainly on a descriptive level are at most an indicator for students' weak interlinking. This means we also have to reconsider the stimuli which we ask our students to consider.

Figure 1 :

 1 Figure 1: Nested design cycles in Spotlight-Y

We understand mathematics education as the scientific discipline about the teaching and learning of mathematics. In German preservice teacher training, mathematics education is mainly concerned with school mathematics.

The word local is used here to distinguish the overall design in the design study "Spotlight-Y" from the designs developed in it: In the outer research cycle it is the course design of complex analysis including teaching strategies, and in the inner cycle it stands for the designs the students develop for their learning arrangement (see Figure2).

A portfolio is a collection of all written work, this includes the exercise sheets, written planning and description as well as answers to final reflective questions (seeHanke & Schäfer, 2018a, 2018b, for details).

Acknowledgment

This project is part of the "Qualitätsoffensive Lehrerbildung", a joint initiative of the Federal Government and the Länder which aims to improve the quality of teacher training. The programme is funded by the Federal Ministry of Education and Research (no. 01JA1612). The authors are responsible for the content of this publication.


Felix Lensing
email: lensinfe@zedat.fu-berlin.de

How concepts turn into objects: an investigation of the process of objectification in early numerical discourse

Keywords: Objectification, reflecting abstraction, Piaget, hypostatic abstraction, Peirce

Mathematical discourse begins where the concrete ends. Despite the intangibility of mathematical objects, we still speak about them in a way that is quite similar to the way in which we speak about material objects. We speak of numbers, classes, and functions as if we were talking about self-sustained entities. But with what right? On the example of the use of number-words in early numerical discourse, I shed some light on the mechanisms by means of which mathematical objects obtain their peculiar ontological status. It turns out that any mathematical object refers back to a concept from which it originates. To better understand the process of objectification, I am thus asking: how do concepts turn into objects? By juxtaposing Piaget's notion of reflecting abstraction with Peirce's notion of hypostatic abstraction, I then propose an explanatory model of the transition from concepts to objects.

Although Patrick (4 years and 1 months) seems to have no problem in recognizing the number-words in the posed questions, he is for some reason unable to cope with them in the first and last question. How can it be that he does not see the pattern common to both questions? What seems unfamiliar to him are not the number-words themselves, but it is the way in which these words are used. In general, we can distinguish between two different uses of number-words (cf. Dummet, 2002, pp. 99-100): Number-words can either occur as adjectives, as in ascriptions of number such as 'There are two giraffes', or as nouns, as in most arithmetic propositions such as 'Three is a prime number'. In their adjectival use, number-words function as predicates. They cannot stand for themselves, but they characterize something else. In that use, a number-word specifies how many there are of a certain kind and thus it must occur as the number of something, or more precisely as the number of a concept ('three lollipops', 'three giraffes', 'three elephants'). In contrast, number-words that occur as nouns stand for something that can be investigated in its own right. In this substantival use, numbers are regarded as self-sustained entities without any intrinsic reference to the concepts which they are possibly ascribed to. Armed with this distinction, we are now able to formulate Patrick's difficulties in a more concise way: While Patrick seems to be familiar with the adjectival use of number-words (see lines 3-9), the substantival one appears foreign to him (see lines 1-2 and 9-10). Consequently, he does not see addition as an operation between numbers as abstract objects, but only as an operation performed on two concepts. To him, adding up means to join two collections of objects. Taking this into consideration, it becomes understandable why he shows difficulties grasping the first question. From an adjectival standpoint, it is of no importance which kinds of objects are conjoined (lollipops, giraffes, elephants, etc.) as long as there are some. An addition cannot be performed in vacuo. The question 'How many is two and one more?' is thus incomplete for Patrick, he feels that there is something missing, namely a pair of concepts with which the operation is executed. These observations gain in importance as soon as it is added that they are by no means a personal exception but rather are typical of many preschool children aged 3 to 4 years (cf. Hughes, 1986, p. 46;Sfard, 2010, p. 47). It is the result of an intricate process of successive steps of objectification that numerals and number-words can be treated as referring to the self--sustained entities we call numbers (cf. Sfard, 2010, p. 136). Within this process, the adjectival use of number-words precedes the substantival use, or bluntly put: numbers are first concepts, and only become objects in a second step.

This developmental sequence, however, is not restricted to the genesis of numbers. Rather, it repeats itself at all levels of mathematical sophistication. For example, [START_REF] Bakker | Diagrammatic reasoning and hypostatic abstraction in statistics education[END_REF] observed in a teaching experiment on the concept of distribution in an 8 th grade in the Netherlands that the students' discourse about the concept of spread took a similar turn:

[M]any used the noun 'spread,' […], whereas students earlier used only predicates such as 'spread out. ' […] Statistically, however, it makes a difference whether we say, 'the dots are spread out' or 'the spread is large.' In the latter case, spread is an object-like entity that can have particular aggregate characteristics that can be measured (for instance by the range or the standard deviation) (Bakker, 2007, p. 24) Or, to provide a historical example: It took several centuries to move from the study of specific examples of functions to the study of functions as such, that is, to the investigation of functions as objects for which certain general characteristics such as continuity or differentiability can be formulated (cf. [START_REF] Kleiner | Evolution of the function concept: A brief survey[END_REF]Sfard, 1991, pp. 14-16). The departure point of this article is the conviction that what we have observed in the case of number can be generalized: At any level of mathematical sophistication, mathematical objects are products, i.e., they are produced in processes of objectification, and at any level of mathematical sophistication, the mathematical objects in question refer to concepts from which they originate.

In this paper, I take early numerical discourse as an example to investigate the transition from concepts to objects as a key element of the process of objectification (in the sense of [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF]. I attempt to provide a model of the mechanisms which underlie this last step in process of objectification. The rationale of the paper is thus a theoretical one and it is precisely here where the novelty of this contribution lies: There are plenty of works in mathematics education (cf. [START_REF] Bakker | Diagrammatic reasoning and hypostatic abstraction in statistics education[END_REF][START_REF] Sfard | On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin[END_REF] in which it is described that this transition happens, but I do not know a single one that provides an explanatory model of how it could actually work. In order to provide such a model, I pursue the following plan: At first, I clarify the use of the words 'object' and 'concept' in order to formulate the theoretical reference-problem as clearly as possible (II.); in the next two sections, I then successively explore the adjectival (III.) and the substantival use of number-words in more detail (IV.); this leads to an explanatory model of the transition from concepts to objects which is developed in the fifth section by juxtaposing Piaget's notion of reflecting abstraction with Peirce's notion of hypostatic abstraction (V.).

II. On the distinction between concept and object

When speaking about material objects, we either use terms that tell us which individual is being talked about, or we use terms that characterize something as an instance of a certain kind. 'Socrates', 'Berlin', and 'Donau' are examples for the former kind of terms, while 'human being', 'city', and 'river' are examples for the latter. To adopt a usual phraseology, we will call these two kinds of terms singular and general terms. A singular term "purports to name one and only one object" (Quine, 1966, p. 205), while a general term does not name the individual object to which it refers at all but, instead, is said to be true of each of many objects of a certain kind (ibid.). The general term 'human being', for example, is true of the object Socrates, but it is not true of the object Berlin. Singular and general terms thus differ in their way of reference: While a singular term refers to one object and one object only, a general term divides its reference. It can always be applied to a multiplicity of objects. What a general term names is thus not any particular object, but at best its own rule of application, i.e., the operation used to decide whether or not the term is true of an object. We want to call these operations concepts: A concept is any operation that is associated with a general term and that allows to decide whether or not the term is true of a particular object.

Therefore, concepts are not to be found in the realm of being but belong to the realm of doing.

We operate upon objects (as they are singled out for our investigation), while we operate through concepts (as it is the conceptual operation by means of which the general term is applied to a particular entity). In contrast to everyday language, objects to which mathematical terms refer are never components of our stream of direct experience. Mathematical discourse begins where the concrete ends. We cannot point at or see mathematical objects such as numbers, classes, or functions. Despite their intangibility, however, we still adopt a jargon that is quite similar to the way in which we speak about material objects. On the one hand, there are singular terms such as ' ', ' ', or 'π', which purport to name one and only one object, on the other hand, there are general terms such as 'prime number', 'continuous', or 'equilateral', which characterize the objects in question as instances or non-instances of a certain concept. In mathematical discourse, the distinction between singular and general terms thus remains intact (see already Peano, 1891Peano, /1973, p. 156), p. 156). The difference between mathematical terms and everyday terms is not the way of reference, but the kinds of objects referred to. 2 In the case of everyday language, the terms are mostly concrete in the sense that the objects referred to can be investigated as parts of our sensory material, while, in the case of mathematical discourse, they are not components of our direct experience and are thus abstract. Besides the division into the singular and the general, there is thus another division that must be taken into consideration, the one into the concrete and the abstract. On the expression layer, we then distinguish between singular and general terms on the one hand, and between concrete and abstract terms on the other hand, while, on the content layer, all singular terms name objects and all general terms stand for concepts.

Note that the two divisions cross each other: There are concrete and abstract singular terms such as 'Socrates' and 'π', and concrete and abstract general terms such as 'human being' and 'prime number' (cf. Quine, 1966, p. 204). An investigation of the relations between these different kinds of terms unveils that we can find correspondences. The abstract singular term 'redness' corresponds to the general term 'red thing'; 'sweetness' is the abstract singular term to the general term 'sweet', and the general term 'round' can be matched to the abstract singular term 'roundness'. This series of correspondences is a systematic feature of our language (cf. Quine, 1966, pp. 204-205): For every general term, be it concrete or abstract, we can come up with a corresponding abstract singular term which names the (abstract) attribute that is 'shared' by all the objects which fall under the concept in question (or, respectively of which the associated general term is true). The term 'roundness', for example, names the attribute that all round things have in common. It names the particular form or gestalt that we recognize in our sensory experience from case to case when encountering a round thing. That also leads us to a hypothesis as to why general terms precede their corresponding abstract singular terms in cognitive development: If the corresponding abstract singular term is what all of the objects that fall under a particular concept have in common, it is hardly possible to recognize this common attribute if the concept in question has not yet been learned.

III. Numbers as concepts

Equipped with this terminology, it is worth returning to the introductory example, which made obvious that number-words can either function as general or singular terms, meaning they can stand for concepts or objects. Let us take a closer look at the adjectival use of number-words first since it precedes the substantival use in the developmental sequence. As a reminder, our introductory thesis was: numbers are first concepts, and only become objects in a second step. Since we have clarified at this point what is meant when speaking of a concept, we can now ask for the operation or rule of application which is associated with a number-word when it functions as a general term. However, before we can do that, there is one more thing that needs to be clarified in advance: the relation of number-words to the sensuous world. It is important not to conceive number-words, not even in their adjectival use, as referring to certain complexes of our direct stream of experience. One and the same sensory impression can always give rise to a majority of equitable ascriptions of number: While looking at one the same external phenomenon, I can say with equal truth both 'It is a group of trees' and 'It is five trees', or both 'Here are four companies' and 'Here are 500 men'. Now what changes here from one judgement to the other is neither any individual object, nor the whole, the agglomeration of them, but rather my terminology. But that is itself only a sign that one concept has been substituted for another. This suggests [...] that the content of a statement of number is an assertation about a concept (Frege, 1970, §46) Whether a certain sensory impression gives rise to one application of a number-word or another, hence solely depends on the way in which we conceptually organize or structure this experience. If a number-word is used as a general term, what falls under the associated concept are again also concepts. What the number-word 'four' stands for is then a second order concept (cf. Frege, 1970, §53), a concept that comprises all of those concepts with exactly four objects. Such a concept does not collect the counted objects or certain aggregates of them, but only the concepts structuring that material. In their adjectival use, number-words are, therefore, abstract general terms. Since a general term cannot stand on its own but is always predicated of something else, in this case of other concepts, it becomes clear why Patrick cannot make sense of the questions without any reference to other concepts. His behavior indicates that he treats the number-words as standing for second order concepts. In this use, there must some concepts upon which the addition is executed. Now we can come back to the question of what the operation or rule of application might look like that is associated with the term 'four'.

In the case of the number-word 'four', this operation might be depicted as following: We coordinate one after the other all the objects that fall under the concept in question with all terms of the number-word sequence starting from 'one' until we come by the word 'four'. If the counting procedure stops at 'four', that is, if all objects were counted up to this point, we say that the given concept falls under the concept four (or, respectively that the term 'four' is true of the particular concept). If the procedure already stops before or if we exceed 'four' without having finished the counting procedure, we will say that the given concept does not fall under the concept four (or, respectively that the term 'four' is not true of the particular concept). Here it is important to emphasize that there is no need at all to suppose an entity such as an attribute that is common to all the concepts comprising exactly four objects. Without a doubt, we can learn and apply the operation described above without supposing the number-words to refer to a separate abstract object of any kind. This observation holds true quite generally: the "use of the general term does not of itself commit us to the admission of a corresponding abstract entity into our ontology" (Quine, 1950, p. 630). Therefore, we should consider the final step of objectification, leading to the treatment of numbers-words as singular terms, as an additional step that requires the use of number-words as general terms already been learned. So, how, we must ask, do numbers become objects?

IV. Numbers as objects

Let us assume we have a box in front of us which contains eleven marbles and we are supposed to determine how many marbles there are in the box. Wanting to communicate our counting result, we have at least two options: (1) 'There are eleven marbles in the box'; (2) 'The number of marbles in the box is eleven'. These two options correspond to our two forms of number-word use: in (1) 'eleven' functions as general term that is predicated of the concept marbles in the box, while in (2) 'eleven' can be said to stand for the same abstract entity as 'the number of marbles in the box', indicating that it functions as an abstract singular term. We argued above that, for a given general term, the corresponding abstract singular term can name the attribute that is shared by all the things which fall under the concept in question.

However, for number-words functioning as general terms the situation is somewhat different. It appears to be quite difficult to come up with a feature or attribute that all of the concepts have in common to which a number word such as 'eleven' can be successfully applied. What is the distance in meters between the penalty spot and the goal? How many corner points does the maple leaf on the Canadian flag have? How many criminals are there in Danny Ocean's gang? How many academy awards did the film Titanic win? And, last but not least, what do these things have to do with our marbles? At the level of content, all these different manifestations of the number eleven obviously have nothing in common. So, what exactly is it that repeats itself from one manifestation to the other?

No matter if we count our marbles, the vertices of the maple leaf of the Canadian flag, or the meters from the penalty spot to the goal, in each of the cases we perform a counting procedure. The sameness does not lie at the level of the concepts upon which the counting procedure is executed, but it lies at the level of the procedure itself, that is, at the level of our operations with these concepts. What repeats itself from one counting act to the other is the fact that all of them end up with the number-word 'eleven'. But this is itself only a sign that one and the same concept is applied: the second order concept associated with the number-word 'eleven'. To go one step further in the process of objectification, a child capable of the adjectival use must thus recognize all the successive counting acts ending up with the same number-word as applications of the same concept. This requires a minimal reflexive loop: In order to sense this kind of 'operational' sameness, the child must make her own operations the object of her own operations. She must relate several successive applications of a number-word to each other and analyze them into sameness and difference of the counting results. However, in order to be able to identify several counting acts that end up with the same number word, a series of factual (the acts might be executed on very different concepts), temporal (the acts are most likely executed one after the other) and social (the acts might be executed by different people) differences between the individual acts must be left out simultaneously.3

V. Piaget versus Peirce: What is abstraction?

What we are facing here is a two-sided process in which something is retained and at the same time something else is left out. Traditionally, this two-sided process is referred to as abstraction (cf. Locke, 1836, book 2, chapter 11, §9). In any process of abstraction there is a certain kind of material or substance upon which the abstraction is carried out. We cannot simply abstract something but rather we always abstract from something. Now, what is unusual in our case is this very substance of abstraction. The substance upon which the abstraction is carried out does not consist of objects of a certain kind but of our own operations. To take account of this reflexivity, Piaget has introduced the notion of reflecting abstraction (cf. Piaget, 2014, pp. 317-323;Glasersfeld, 2003, pp. 103-105). In contrast to what Piaget calls empirical abstraction, a reflecting abstraction is an operation that runs on operations:

[W]hen we are acting upon an object, we can also take into account the action itself, or operation if you will, since the transformation can be carried out mentally. In this hypothesis the abstraction is drawn not from the object that is acted upon, but from the action itself (Piaget, 1971, p. 16) We can now see that the process of relating several counting acts and analyzing them into the sameness and difference of the counting results can be reconstructed as a reflecting abstraction. It runs upon several applications of the second-order concept associated with the number-word 'eleven'. The child must retain what all of the counting acts have in common while simultaneously leaving out all the features in which these successive acts differ from each other. But what is the outcome of this process? Have we already arrived at the substantival use of number-words? To cut it short: yes and no.

In general, we can say that the 'minimal' attribute shared by all things that fall under a certain concept is this purely formal feature, the fact that the things fall under that very concept. What all concepts that fall under the second order concept eleven have in common, is that they bear the same relation to this concept. They fall under it because in each case the counting procedure ends with the word 'eleven'. And to identify this formal sameness, we must perform a reflecting abstraction. In this way, the notion of reflecting abstraction allows extending the notion of a common property or a shared attribute far beyond the immediate, sensuous world. At best, we might then say that in the process of reflecting abstraction the successive mental or communicative acts are 'condensed' into an object-like entity, an attribute, which is the common feature of all these different acts. But, in the end, that is mere speculation. How, one might ask, is this mysterious object-like entity anchored if we obviously cannot perceive it?

In order to provide an answer to this question, we have to move away from the content layer, the series of operations on operations, onto the expression layer, the layer of the use of signs to fix these volatile operations (cf. Sfard, 1991, p. 21). It is precisely here where the notion of reflecting abstraction comes into trouble. It can explain how we get to sense the operational sameness described above, however, what it cannot explain is how this sameness is objectified. We therefore want to complement Piaget's account on abstraction with Peirce's considerations on the very same topic. Peirce has introduced the notion of hypostatic abstraction to describe the exact process we are interested in, the transition from a general term to an abstract singular term or in Peirce's own words:

That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnish us the means of turning predicates from being signs that we think or think through, into being subjects thought of. We thus think of the thought-sign itself, making it the object of another thought-sign. (CP 4.549)

It is noticeable that Peirce gives a beautiful account on the reflexive loop we have already described above: Whenever we apply a general term or a predicate we think through the sign or the concept associated with it in precisely that sense that we always predicate it of something else, while in any attempt to transform it into an object, we must reflect on this very use and make it the subject of another thought. That is to say, we must think of the sign through which we have thought before. So far nothing new, but in the second part of the quotation Peirce goes beyond what we have already described before. We "think", Peirce writes, "of the thought-sign itself, making it the object of another thought-sign" (ibid.). As noticed before, what stays the same from application to application of number words in their adjectival use is nothing but the purely formal feature that the number-words are true of all the concepts to which they are successfully applied. We can now add that, what the corresponding abstract singular term names, is ultimately nothing other than the sign thought through before. Number-words functioning as singular terms name the sign, i.e., the relation between the general term and the particular counting procedure associated with it. This way, the conceptual operation becomes crystallized or frozen. When a number-word is used as a singular term, the operation must not be performed anymore but is rather only implied or pointed at. We say that we 'know' what the numeral '1,098,7877' stands for because it "points to the last element in a familiar counting procedure" (Glasersfeld, 1995, p. 100), a procedure that no longer needs to be executed.

Concluding, I want to generalize this thesis. We do not need to posit some sort of ontological realm to answer the question of how we can speak about numbers, classes, and functions as self-sustained entities. What a singular term in mathematical discourse stands for is ultimately no more than another sign, it is the relation between the general term (from which the singular term in question originates) and the associated conceptual operation.

'MH' stands for Martin Hughes. The transcript is taken fromHughes (1986, pp. 47-48).

For a more detailed discussion of what it is that makes a concept a mathematical concept, see also: De Freitas, Sinclair & Coles (2017).

This distinction between three meaning dimensionsa factual, a temporal, and a social -, in which we can seek for commonalities and differences, stems from the German sociologist NiklasLuhmann (cf. Luhmann, 1995, pp. 59-102).


Andonis Zagorianakos
email: matheart7@math.uoa.gr

A phenomenological methodology based on Husserl's work in the service of mathematics education research

Keywords: Phenomenological methodology, Husserl, intentionality, reduction, bracketing, epochè, phenomenological attitude

The paper introduces a phenomenological methodology based on Husserl's work and on Merleau-Ponty's radical extension of Husserl's work. The phenomenological reduction and bracketing (epochè) are introduced as the critical methodological instruments in exploring the emergence of mathematical understanding, by casting phenomenological light on the learner's intentional cognitive constitutions, from the passive to the active stage of her or his lived experience. An empirical example is used in order to showcase the impact of the researcher's methodological shifting, from the natural to the phenomenological attitude (the reduction and the epochè), and the methods and techniques that are used, in allowing the methodology to produce results. A summarising diagram and some notes for the usefulness of the methodology application for mathematics education research are included.

Introduction

The aim of this paper is to introduce a phenomenological methodology as a valuable frame for mathematics education research (MER). Examples of my use of this methodology have been published in Educational Studies in Mathematics (2015) and other international journals, as well as presented at CERME9, PME (2014PME ( , 2015) ) and BSRLM (2012BSRLM ( , 2013) ) conferences. Critical distinctions emerge, between my use of phenomenology and that of other phenomenological or phenomenologically informed researchers in MER (e.g., Radford, Roth, Nemirovsky). The distinctions are concerned with the principal use of Husserl's work in my research-which is either taken for granted, critically disputed or scarce in the influential work of these scholars. Most importantly, the central difference in my work is the thematisation and application of Husserlian methodological concepts, such as the phenomenological reduction and the epochè. These concepts are used precisely for the purpose introduced by Husserl (i.e., phenomenology as methodology), and they are missing in the aforementioned researchers' phenomenological contributions. In my work, these concepts have been used in order to put in motion Husserl's mature work on the lifeworld (Lebenswelt) and the living present of learners. My conviction is that the same methodology could be applied to research with teachers, mathematical classrooms, curricula etc., each time providing critical distinctions. In this sense, the application of the phenomenological methodology presented here and as applied to learners suggests the viability of its application to other phenomena in mathematics education research.

The key concept of intentionality as a starting point for the application of the methodology

This paper follows Husserl's (1970, p. 168) conceptualisation of the central role of intentionality for the study of mathematical cognitive situations. The intentional aspect of conscious acts is concerned with the fact that every conscious act is about something; I fear something, I hope something, I judge something (as true or false), I love someone, I hate something, and so on. Also, everything perceived is always perceived in a certain manner, with fear, hope, in shaping a judgement, with love, hate, etc. In the same respect the key feature of conscious acts, and in particular the cognitively motivated acts, are about something, present or absent, real or supposed as real, valid or not, but in any case, present for these acts. [START_REF] Husserl | Ideas pertaining to a pure phenomenology and to a phenomenological philosophy: First Book[END_REF] adopted Brentano's conception of intentionality as being the main feature of conscious acts and extended it critically. Mature Husserl (e.g. 1970, p. 168) thematised further the "uncover[ing] and the "clarification of what is accomplished" in the "intentional origins" region of lived experience. What does this mean in terms of the mathematics education research perspective? It means that by focusing on the region of intentional origins we could uncover and clarify what is accomplished in them, since, as he explains "every sort of intentional unity becomes a 'transcendental clue' to guide constitutional 'analyses'…[which are] uncoverings of intentional implications" (Husserl, 1969, p. 245, italics in the original), particularly related to an operative, embodied sense on which abstract knowledge is grounded.

In order to explore the intentional aspect of the mathematical cognitive experience, the phenomenological reduction and the epochè (see the following sections) must be applied, as the two basic methodological instruments. The aim of the methodology is to open up the "currently constituted intentional unity" (1969, p. 245) of the learners' understandings, and thus to understand better how they arrive at their understandings, in their engagement with mathematical tasks and questions.

Epochè (bracketing) and phenomenological reduction-A first introduction

The methodology is supported by a phenomenological perspective that presents an intentional view of the cognitive life-world. This perspective approaches the mathematical lived experience as being intentionally supported; the researcher tries to study it without pre-conceptions on what it ought to be, but merely focuses on the intentionally emerging appearances, as they appear for the learner. The suspension of preconceptions is one of two basic methodological instruments. This suspension amounts to a two-fold task:  To spot and put aside actions that are not relevant to the student's experience;  To suspend the researcher's own intentions, and prevent him or her from interfering with the intentions running through the phenomenon of cognition under study.

In this sense all objectivity is suspended, as was introduced by Husserl (1983, p. 60), who used the term epochè for this process. Husserl employed epochè in order to combine it methodologically with his phenomenological reduction. This is the second basic methodological instrument. It is tantamount to the "leading back" from the natural targets of our concern, to "what seems to be a more restricted viewpoint, one that simply targets the intentionalities themselves" (ibid.).

So a combination of epochè (suspension or bracketing), and phenomenological reduction, shapes the core of the methodology. What are revealed through this combination are not the objects as such, but as they are intended (imaginatively, through remembering, judging, etc.), focusing on the manners or modalities in which such objects are 'given'. In other words, we look at what people normally look through (ibid.).

The application of the phenomenological methodological lens in a designed experiment

Here I present an experiment that is phenomenologically analysed, and organised in accordance with phenomenological investigations. It was an experiment in which the teacher allowed all possible interactions between the students, usually working in groups, but he did not provide any guidance to his students, concerning the open tasks that he introduced. I present the case of Mary (pseudonym), one of 13 students, i.e. prospective teachers of mathematics for secondary education, in a course entitled The nature of mathematics (NoM). As the teacher's part in the collective or individual thought processes of the students' treatments of the tasks was bracketed (epochè, in Husserl's terms) the thinking strategies themselves came to the fore; the research field was enriched by the various approaches to the tasks, by different or by the same students. Reflective and prereflective intentional layers were thus traced as such (see the following section), as they were grounding the students' constitution of the mathematical objects.

As soon as the intentional layers1 of the students' engagement in the task are exclusively targeted the researcher enters the phenomenological attitude, which guides her or him to further analytical reflections, followed by new findings (cf. the unit of phenomenological knowledge production component of the diagram in Figure1). With the use of the phenomenological reduction and the bracketing-considered as a single methodological instrument by the mature Husserl-we are entering the study of the natural attitude, which is "our straightforward involvement with things and the world" (Audi, 1999, p. 405), by describing it with the phenomenological attitude. The phenomenological attitude is "the reflective point of view from which we carry out philosophical analysis of the intentions exercised in the natural attitude and the objective correlates of these intentions". Further, "the process by which we move to the phenomenological attitude is called the phenomenological reduction, a 'leading back' from natural beliefs to the reflective consideration of intentions and their objects" (ibid.).

The application of the phenomenological methodology in this case had the effect that, what were before brilliant ideas that suddenly possessed the students, or desperate efforts to 'survive' by acquiring some sense of the task, now became telling appearances of the operative affordances that each student brought and was able to release actively, either in the team in which the student was working, or individually, during and/or after the sessions.

The transformation of the working field of research was the result of (a) the focusing on the intentional acts that brought the learners' objects to the surface, in a regressive analysis that starts from the 'final' objects and goes back to their genetic appearances, and (b) the neutralising/bracketing (epochè) of factors that played no part in the perception of the objects that were chosen to be activated by the students. From this analysis it occurred that it is the double application of the phenomenological reduction (a) and the epochè (b) that allows the transformation of the cognitive event into a phenomenon-in the Husserlian sense that this methodology suggests.

Methodological affordances and results

The transformation of the data sources' natural appearances into a phenomenon affords us, as researchers, with the following: (a) the ability to discern structures that the students' strategies follow, which could be helpful in teaching design [START_REF] Zagorianakos | The study of the intuition of essence of a prospective teacher of mathematics from a phenomenological perspective[END_REF][START_REF] Zagorianakos | The role of intuition in the process of objectification of mathematical phenomena from a Husserlian perspective: A case study[END_REF]; (b) better understanding of the students' learning horizons, which may also entail further phenomenological analysis, e.g., of a student's change of attitude (Zagorianakos, in press).

Description of the illustrative example-The use of the methodology

The bird's-eye view empirical intuition, as triggering a series of abstract intuitions.

Mary (pseudonym) was a prospective teacher of mathematics in English secondary education.

During the NoM course (see previous section) a task was introduced that was required to be solved in teams of three or four students by involving bodily experience. One of each team actualised a fixed point, 10 meters or steps away from a wall, while the rest of her or his team were trying to embody the track of the sought line, in which each point was supposed to have the same distance from the fixed point and from the wall. The room where the task took place had very wide walls, enough for the four teams of three or four students to explore the task. What will be sampled here from that study is how the methodology allowed the phenomenological detection of the crucial influence of the embodied intuitive stock of Mary. The embodied grounding of Mary's By tracking how Mary intentionally handled her drawing of the wall and the Cartesian plane, I noticed how she transformed her drawing into a different view of the classroom, where she was confusedly trying to embody the solution of the task. Mary's view was an intuition,2 drawing back to the Scholastics' perception of a 'new vision', which allowed her a breakthrough that followed her up to the end of her treatment of the task. It was in an actively embodied rather than in a metaphorical sense that Mary used her perception of a "bird's-eye view". It was this initial embodied intuition that was operating, as a sensory-motor environment to all her understandings, up to the end (see following paragraphs). This phenomenon was lit by full bracketing of nonsignificant issues to her constitutive moments, of the objects that helped her unlock the task. By mentioning non-significant components, I mean all the actions that did not interfere with the student's (constitutive) building process, such as the teacher's non-guidance protocol. Therefore, bracketing is not arbitrary, and it could be responsible for further analyses, as soon as the reduction's furthest application is done (Zagorianakos, in press). Mary's example is not fully analysed here: it has been drawn to showcase the applicability of the methodology, realised by the methods and the techniques, which are introduced in the following section.

Methods and Techniques of the methodology-The student's progress with the task

The methods included participant observation, gathering of documents that included the learners' reflections on their learning processes, and interviews (email or in person). The methods concerned semi-structured live interviews, where the technique was to trigger the flowing of the students' reflections on the actions that brought objects to the surface. Another interview technique was to drive the discussion without force towards the intentional acts that led the learners to their constitutive moments of objects. How did they feel about them? What was special, for them, about these objects? The methods were used so that the intentions that led to the mathematical objects would be revealed; and the techniques were aiming at keeping the researcher in low profile, while bringing the student's narrative vividly to the foreground. Due to such methods and techniques I took advantage of the teacher's self-bracketing in order to study how the students' mathematical experiences were structured.

Let's see how the student's development with the task happened:

After the investigation in the classroom, when the student was in utter confusion about what the sought line was supposed to look like, Mary went home. There she drew on grid paper the wall as a straight line, and a Cartesian plane with the y-axis perpendicular to the 'wall', and the two axes intersecting ten grids away from the line (wall). The student's first intuition appeared, as she grasped her drawing on the grid paper as a bird's-eye-view over her very recent, embodied but unsuccessful classroom activity.

Not metaphorically, but in a very actual sense the student realised due to her bird's-eye-view perception of her diagram how wrong her perception of the task had been in the classroom: she actually observed that the sought line could not be a straight line parallel to the wall (as she had initially supposed), at half way distance from the fixed point and the wall, while being constantly equidistant from the wall and the fixed point. It is evident that her bird's-eye-view perception had equipped her with a movable vision over the classroom, through the diagram, and a trace of this was that her perceptions were time-related: the straight line was equidistant from the wall and the fixed point at some point, but not in the following of the process of a "movable point", in which the student started perceiving the curve. So then she started with three points and she started looking for more points belonging to the sought line, by using a ruler parallel to the line (wall) and between that line and the x-axis, thinking of the sought line as being continuous and therefore aiming at finding the points where her ruler intersected with the sought curve. While being in this operational mode Mary employed the Pythagorean Theorem (PT) as an objectification instrument, to track points belonging to the sought curve and thus unlock its shape. The student's intentional roots were traced; they revealed that the student's recollection of the Pythagorean theorem was in the service of her intention to find more points belonging to the curve, while operatively using her bird's-eye-view perception of her diagram. This insight adds an operatively intentional layer, a clearly embodied layer right at the appearance of the PT as a points' determining instrument.

After finding five points symmetrically located around the y-axis (two of them by using the PT) she had the intuition that the line must have a "bell-shape", from where she thought of the x-square functions. The student's idea of x-square functions was the second intuitive step, described by Husserl as an abstract intuitive step, based on the empirical ("bell-shape") intuition.

Her exploration culminated in a critical abstract intuition which, through the "synthesis of coincidence" (Husserl, 1973, p. 74) brought to the surface the general rule, the general formula for all curves, with any given distance from a 'wall'. And even then, the embodied sense of her understanding of the algebraic solution remained vivid in her recollection: She perceived the curve as an empirically driven "moveable point", since "you could stand on anywhere on he line and you would be equidistant from the wall and a fixed point" (interview extract).

The view that the phenomenological methodological components offered to the study

So, the phenomenological reduction and bracketing (epochè), as distinguished from methodic doubt, and being a suspension compatible with certainty, led me to the phenomenological attitude. This attitude is equivalent to focusing exclusively on the intentions and their correlated objects in the natural attitude, without "excluding of a living conviction, which remains alive" (Husserl, 1983, p. 60). Bracketing enhances focusing and focusing is indicating what needs to be bracketed or seriously studied, phenomenologically or otherwise.

The study of the student's intentional apparatus became my Ariadne's thread in unlocking Mary's lived experience. I finally managed to understand how she imaginatively (with imaginative intention) flew on the wings of her bird's-eye view intuition, over the diagram that she drew at home, on grid paper-i.e., using the mathematical diagram as a re-presentation of the classroom where the embodied treatment of the task took place-and finally, how she developed her bird'seye view intuition in order to accomplish all her subsequent intuitions.

The cultural artefacts (grid paper, Cartesian plane etc.) were imperative for Mary's accomplishments, but it was through (and due to) the researcher's phenomenological reduction and bracketing that the focusing on the intentional apparatus of the learner's lived experience finally disclosed the following aspects: (a) how the first, empirical bird's-eye view intuition (not 'just' as a metaphor but as an actual, embodied perception) was essential in unlocking the task (Zagorianakos & Shvarts, 2015, pp. 152-154); and (b) how her embodied and mental intentionalities were intertwining as she moved further, in activating a general formula for all parabolas (ibid.). The shifting from the natural to the phenomenological attitude (the reduction and the epochè) is a critically innovative methodological move, which can unlock the lived experience of the learner by transforming it into a phenomenon, in its original (Husserlian) sense, for the researcher. So the methodology is concerned with the study of focusing on the learner's object constitution-objects that become mathematical objects as they fulfill the demands of a mathematical task or a question brought in a mathematical context. The focusing is on the objects as they appear for the learner and therefore, as they are intended by the learner. This focusing allows for the tracing of the intentional paths that bring the learner into contact with a particular object, due to the learner's pairing of the object with a certain strategy, while solving a problem:

Through 'something reminds me of something' a pairing of two given things in consciousness first arises. Through reflection we see that association is not a blind mechanism, because I can recreate and understand through which connecting point(s) a pairing originally arose in consciousness. (Held, 2003, p. 43) As an example of the analytical potential that the methodology has made accessible, Mary's intuition on the shape of the curve-as being an x-squared curve, after her empirical intuition of the shape of the 5 points as a bell shape-is phenomenologically perceived as an association generated by a pairing. The empirical image of the bell was paired with the x-squared curves' graph, due to the student's intention to explore a possibility for embodying mathematically the empirical image. This intentional trail from operative (embodied) to active mathematical knowledge production (culminating in Mary's formulas) is a viewpoint on Mary's mathematical lived experience that was developed due to the methodology of intentional focusing (reduction), and teacher bracketing (epochè), in this phenomenological experiment. Practicing this methodology may make the researcher realise the radical view the methodology may offer in the study of cognitive events, particularly when enriched by a phenomenological theoretical viewpoint.

The methods encompass regressive and repetitive reflections on the data, without taking anything for granted, as far as the appearances of the objects for the learners are concerned. The key is to focus exclusively on the acts, and on the intentions that are expressed through them, as they are directed towards objects, as they surface and as they are intended by the learners. Then the focus moves on to objects that triggered the acts and on the acts that brought the latter objects to conscious exploration, within the learner's lived experience. In this sense, this methodology, by using the phenomenological attitude has the effect of a phenomenological mill unit. This is what the unit of phenomenological knowledge production element of the methodology diagram (Figure 1) depicts, as the vertical arrow is moving: The vertical arrow in the diagram is making one full route from the top to the bottom of the rectangle, synchronously to the inner round circle arrow completing one round. Then the whole process is repeated in a more informed manner, phenomenologically, thus producing sounder and more evident 'images' (adumbrations), and according findings of the phenomenon under examination. A top-bottom vertical arrow on the left side of the methodology diagram (Figure 1), which is moving through the unit of phenomenological knowledge production rectangle, is concerned with researcher's own shifting of attitude, from the natural to the phenomenological one. The cognitive phenomenon, as it is unfolding for the researcher, is an orchestration of expressions in the conscious reflective and in the subconscious pre-reflective areas of knowledge production. The methodology shows the switching from a natural attitude to the phenomenological attitude. This is done by specifically thematising the intentional layers that are concealed (taken for granted) in the 'natural' reading of the phenomenon (which is the study field), and can take place by applying the epochè, the neutralising of natural intentions that must occur when we contemplate the intentions that we are studying. The researcher's own change of attitude and the transformative effects on her/his previous ('natural') viewpoint are thus enabled to be at the centre of analytical attention.

Figure 1 :

 1 Figure 1: Methodology diagram

The intentional layers are the layers of intentions of the learner, as he or she is devising an object for a mathematical enquiry.

From late Latin intuitio(n-), from Latin intuit-, intueri 'contemplate', from in-'upon' + tueri 'to look'. [Concise Oxford English Dictionary (Eleventh Edition)]. Intuitive in the sense of a special, immediate regard on what is given, as intended.