Marlova Estela Caldatto
email: maracaldatto@yahoo.com

Miguel Ribeiro

An analysis of the nature of the knowledge disseminated by a mathematics teacher training policy: The PROFMAT case

Keywords: Mathematics teacher's specialized knowledge, PROFMAT, public policy

This paper discusses the nature of the knowledge disseminated by the Brazilian public policy PROFMAT for training teachers on natural numbers, based on the analysis of one of the textbooks that outline the formative activities developed in this program. This discussion was structured and developed from a conceptual and analytical model of the mathematical knowledge of teachers-Mathematics Teacher's Specialized Knowledge (MTSK). The analyses and discussions developed suggest that this public policy favors only part of the knowledge taken by the literature as essential for the practice of the mathematics teacher in the school.

Introduction

Educational policies are configured as sources pertinent to the investigative processes that seek to identify the specificities of the teacher's knowledge [START_REF] Stylianides | Studying the mathematical knowledge needed for teaching: The case of teachers' knowledge of reasoning and proof[END_REF], since they demonstrate the response produced by the government bodies to an issue that permeates the educational discussions related with what teachers need to know in order to teach mathematics adequately [START_REF] Ball | Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge[END_REF].

Thus, the object of discussion of this article is a Master's Program in Mathematics (PROFMAT), aimed at the continuous training of mathematics teachers (lower and upper secondary), developed in all Brazilian states and financed by the country's governmentassuming the character of Public Policy. According to the official documents, such a Program aims to provide in-depth and relevant mathematical training for teaching in Basic Education, seeking to give teachers a certified qualification for the mathematics teacher profession [START_REF] Bsm | PROFMAT: a reflection and some results[END_REF].

The role played by the mathematical knowledge of teachers to improve the quality of learning opportunities justifies, at least in part, the growing focus of research on teachers' mathematical knowledge [START_REF] Stylianides | Viewing "mathematics for teaching" as a form of applied mathematics: Implications for the mathematical preparation of teachers[END_REF]. This knowledge has been widely studied and, from different points of view, diverse conceptualizations of teacher knowledge have emerged (e.g., Knowledge Quartet - [START_REF] Rowland | Developing and Using the 'Knowledge Quartet': A Framework for the Observation of Mathematics Teaching[END_REF], MKT -Ball, Thames & Phelps, 2008;COACTIV -Kleickmann et al., 2015;MfT-Stylianides & Stylianides, 2014;MTSK-Carrillo et al., 2018). In this text, we focus on the Mathematics Teacher's Specialized Knowledge -MTSK conceptualization [START_REF] Carrillo-Yañez | The Mathematics Teacher's Specialised Knowledge (MTSK) Model[END_REF] as the analytical model of this type of knowledge. Thus, we will support the MTSK to discuss teacher training developed within the scope of PROFMAT, seeking to answer the following question: what knowledge about Natural Numbers and their teaching are presented (and not presented) in the textbook that guides a part of the training developed in PROFMAT?

Theoretical framework

The MTSK is an analytical model that assumes that the professional practice of the math teacher requires a set of specific knowledge strictly related to the teaching of mathematics. This includes the meanings, properties and definitions of particular topics, means that favor the construction of the understanding of the subject, connections between mathematical items, knowledge of mathematics teaching and characteristics associated with learning mathematics, among others [START_REF] Carrillo-Yañez | The Mathematics Teacher's Specialised Knowledge (MTSK) Model[END_REF]. In this sense, the MTSK model (Figure 1 The KoT includes phenomenological aspects, meanings of specific concepts and examples that characterize specific aspects of the topic addressed, aside from contemplating the disciplinary content of the mathematics addressed by textbooks and other materials of a pedagogical nature. KSM refers to the understanding that the knowledge of teachers, besides including concepts as isolated elements, must integrate them into a system of connections, which allows the teacher to understand certain advanced concepts from an elementary perspective and develop certain concepts from an advanced perspective.

KPM covers aspects related to mathematical thinking, such as knowledge of the different ways of defining, arguing or proving in mathematics, as well as knowledge of mathematical syntax. The teaching action involves the knowledge of how this teaching can and should be developed, so the KMT contemplates knowledge as knowing different teaching strategies that allow the teacher to foster the development of procedural and conceptual mathematical capacities. In the same way, this subdomain predicts that the teacher needs to know examples that awaken the intuition in the student about some concepts, as well as resources that allow the teacher to induce his students to learn, through manipulation, certain mathematical concepts. Knowing how students learn mathematical content is knowledge that every teacher should possess. Thus, KFLM encompasses the knowledge of the characteristics of the process of understanding the different contents by the students, the errors, difficulties and obstacles associated with each concept and the language used by students in relation to the concept worked in the classroom. The KMLS refers especially to the teacher's knowledge of the curriculum adopted by the institution in all stages / levels of education. This knowledge can be complemented with information present in the productions originating from research in the area of mathematics education, with information provided by experienced teachers about the expected learning in each stage (For further deepening and detailing the subdomains see [START_REF] Carrillo | Determining specialised knowledge for mathematics teaching[END_REF][START_REF] Flores-Medrano | El Papel del MTSK como Modelo de Conocimiento del Profesor en las Interrelaciones entre los Espacios de Trabajo Matemático[END_REF], Carrillo et al. 2017).

The context of the study: The PROFMAT

PROFMAT is a teacher training program promoted by the Brazilian Federal Government and developed in partnership with 96 higher education institutions (universities and colleges) that train teachers of mathematics (lower and upper secondary), organized in the format of professional masters. This program has a length of 2 years and is basically composed of a list of 9 subjects, each with a 120-hour workload. Of these, 7 of them are considered as manditory and titled "Real Numbers and Functions", "Discrete Mathematics", "Arithmetic", "Geometry", "Problem Solving", "Calculus Fundamentals" and "Analytical Geometry". The "elective" disciplines can be chosen by students from the following list: Mathematics and Actuality II. The final course work refers to the production, by the academic under the guidance of a university professor, of a product/work that can be presented in different formats (dissertation, literature review, article, patent, applications, teaching and instructional materials , media programs) and that addresses topics relevant to the Mathematics curriculum of Basic Education and promotes the impact on didactic practices in the classroom.

These disciplines are based on the collection of textbooks entitled "PROFMAT Collection", whose elaboration is the responsibility of mathematicians, and each discipline is based on a textbook from the collection, which is adopted by all 96 institutions where PROFMAT is developed.

Here we focus on analyzing and discussing one of the elements that make up the teacher training process developed in the scope of PROFMAT from the analytical model MTSK. However, the analyzes presented here are part of a project that has analyzed a multiplicity of elements of the PROFMAT: policy texts, textbooks that guide the training activities; class episodes; and PROFMAT academic productions (answers to questionnaires, works, etc.).

In this analysis, we classify all the contents that compose the textbooks (definitions, axioms, theorems and demonstrations, properties, activities, exercises, explanations, etc.), from the subdomains of the MTSK framework, with the objective of identifying the presence and absence of knowledge related to the teacher's practice in mathematics teaching, as proposed by Carrillo and contributors [START_REF] Carrillo | Determining specialised knowledge for mathematics teaching[END_REF][START_REF] Flores-Medrano | El Papel del MTSK como Modelo de Conocimiento del Profesor en las Interrelaciones entre los Espacios de Trabajo Matemático[END_REF], Carrillo et al., 2017[START_REF] Carrillo-Yañez | The Mathematics Teacher's Specialised Knowledge (MTSK) Model[END_REF].

It is highlighted in this scenario that, through the following discussion presented, we will seek to note indications of the nature of the knowledge disseminated by PROFMAT, since the knowledge produced from a textbook depends on the manipulation that is made of this didactic material, particularly, the manipulation developed by the teacher who uses it.

Due to limitations of space, we focus here on two chapters of one of the textbooks used in PROFMAT called "real numbers and functions" (which describes the work developed in the mandatory subject that has the same name). We chose to analyze this textbook because it guides the teaching activities of one of the four compulsory subjects developed during the course of the PROFMAT, besides addressing mathematical themes that permeate themes also common to the school curriculum, such as numerical sets and functions (NCTM, 2000). In this analysis, we will focus on "Natural Numbers" and for the analysis, we use MTSK as a theoretical lens as a way of discussing the nature and focus of such a course. With this, in the sequence, we will present some representative examples of the analyzes that we developed.

Analysis and discussion

The textbook "Real Numbers and Functions" [START_REF] Lima | Real Numbers and Functions (PROFMAT Collection)[END_REF] is structured from the following sequence of themes (chapters): Sets; Natural Numbers; Cardinal Numbers; Real Numbers; Related Functions; Quadratic Functions; Polynomial Functions; Exponential and Logarithmic Functions; Trigonometric Functions. This organization is justified by the author, who states "Mathematics deals primarily with numbers and space. Therefore, the sets most frequently found in Mathematics are numerical sets, geometric figures (which are sets of points) and the sets that derive from them, such as the function sets, those of matrices" (Lima, 2013, pp. 02). This structure presents opportunities for the development of MK, in particular KSM, by teachers in training, since it favors the (potential) development of connections between the numerical sets and functions that establish relations between subsets of R.

The chapter "Natural Numbers" is initiated by the author with a description of the meaning of the term "numbers" as being "abstract entities, developed by man as models that allow to count and measure, therefore to evaluate the different quantities of a magnitude" (Lima, 2013, pp. 22). "Mathematical definition" is described as "a convention consisting of using a name, or a brief sentence, to designate an object or property, the description of which would normally require the use of a longer sentence" (Lima, 2013, pp. 23), axiom and primitive concept. These descriptions can be associated with MTSK by means of KPM, as it returns to the fundamentals associated with mathematical proceedings.

The author also presents a description of the axiomatic method and in relating it to the school argues that "[...] it is not appropriate to present Mathematics in an axiomatic form. But it is necessary for the teacher to know it can be organized in the manner outlined above " (Lima, 2013, pp. 23). As this fragment shows, the author argues that it is necessary for the teacher to know that mathematics can be organized in the axiomatic form (KPM), but it does not elucidate, for example, why it is necessary for the teacher to know this form of mathematical exposition (KoT), nor "how" (KMT) and "when" (KMLS) this method would aid in the teaching of mathematics. Lima (2013, pp. 26) links natural numbers to historical aspects (potential indications of KPM) by saying that "Slowly, to the measure that it has become civilized, mankind has taken over this abstract model of counting (one, two, three , four, ...) which are the natural numbers ". The author also argues that, currently, the set of natural numbers can be described as "concise" and "precise" by Peano's theory, which states that "N is a set, whose elements are called natural numbers. The essence of the characterization of N lies in the word "successor" (if n, n`∈ N, then n` is the successor of n means that n` comes soon after n, there being no other natural numbers between n and n`), which is considered a primitive concept because it is not explicitly defined. The author then mentions that "everything" about natural numbers "can be demonstrated" from Peano Axioms1 .

According to Lima (2013, pp. 27), "an ingenious process, called a decimal number system, allows us to represent all natural numbers with the help of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. In addition, the first natural numbers have names: the successor of number one is called two, the successor of two is called three, etc. " This presentation of decimal numbering systems can be linked to KoT (knowing the set) on the one hand, and on the other, to KPM (knowing the role of the successor and the associated mathematical structuree.g., continuity, density). Meanwhile, the author avoided discussing, for example, the different aspects of the mathematical knowledge underlying the construction and use of the decimal system (KoT), such as the notion of grouping, the language involved in the reading of numbers and the idea of positional value (KoT; KPM). In this case, it is important to highlight the lost opportunity for elaborating some discussions on, for example, the resources and examples in and for teaching such topic (KMT), concerning the students' difficulties (KMLM) and the content of the official documents (KMLS). Lima (2013, pp. 27) then argues that the set N = {1, 2, 3, ...} is a "sequence of abstract objects that are, in principle, empty of meaning" and that "each of these objects has only one place determined in this sequence ". Since "Every number has a (single) successor and, with the exception of 1, it also has a single predecessor. Viewed in this way, we can say that natural numbers are ordinal numbers: 1 is the first, 2 is the second, etc. ". In this fragment, Lima presents a powerful scenario for an exploratory work of KoT (when discussing the definition of natural numbers); however, although the author recognizes that the objects of the sequence of natural numbers are abstract and are in principle empty of meaning, he does not discuss, for example, what the teacher's work could be in addressing this content in the school environment (PCK), in addition to the symbol of "..." being placed in the representation of the set of natural numbers, but without making any comment about the meaning of this symbol (KoT and KPM). Lima also does not present indicatives of "how" (KMT) and "when" (KMLS) Peano axioms could be addressed in math classes at school.

The author also defines the "fundamental" operations among natural numbers, and the addition of numbers n, p ∈ N corresponds with the sum "n + p" and multiplication associates the product "np" [START_REF] Lima | Real Numbers and Functions (PROFMAT Collection)[END_REF]. It also re-elaborates them in such a way as to present the sum "n + p" as "the natural number that is obtained from applying n p times following the operation of taking the successor", so that "n + 1 is the successor of n, n + 2 is the successor of the successor of n, etc. " (Lima, 2013, pp. 29) and presents, as an example, the addition of 2 + 2 = 4, where " 4 is the successor of the successor of 2". For the product and defined as n.1 = n and, when p ≠ 1, "np is the sum of p plots equal to n". In this context, it is argued that "until we know how to use natural numbers to make counts, there is no point in speaking in 'p times' and 'p plots'. Therefore, fundamental operations must be defined by induction" (Lima, 2013, pp. 29). In the following, Lima defines Addition and Multiplication by induction (KPM -from a level of knowing how to perform). However, the operations are not linked to the practices of mathematics teachers, which could have been done by discussing how these operations are approached in the school curriculum (KMLS), what kind of resources -and their intentionality -could be used to explore such operations with students (KMT) in order to understand what they do and why; and the students learning processes linked to understanding multiplication mainly as a solely repeated additions (KFLM).

In addition, the author constructs the set of natural numbers and only two of their operations (addition and multiplication), omitting the definition of the operations of subtraction and division between natural numbers (KoT), which are also used by teachers in school. The author also does not link the definitions of addition and multiplication (and consequently of subtraction and division) to the algorithms commonly used by school materials (KMLS) and used in calculating these operations (KoT). Another relevant observation refers to the non-linking of this work with one of the main practices of the Mathematics teacher, to give meaning to the operations among the natural ones through problem situations and problem history (KMT).

In addressing the Natural Numbers, [START_REF] Lima | Real Numbers and Functions (PROFMAT Collection)[END_REF] presents the order relationship between natural numbers in terms of addition ("Given m, n ∈ N, it is said that m is less than n, and we write m < n to mean that there is some p ∈ N in which n = m + p ") and presents the properties of Transitivity, Trichotomy, Monotonicity of Addition and Multiplication, and Good Ordination. This fragment highlights the potential of the work in the development of KoT and KPM. However, the author does not present any indication of discussions regarding PCK subdomains.

In the thread of the work the author presents and demonstrates properties of N: Associative property of Addition, Commutative property of Addition, Distributive property, Commutative property of Multiplication, Cancellation Law for Addition, Transitive Order Relationship, Trichotomy, Monotonicity, Cancellation Law for Inequalities and the Principle of Good Ordination. Lima argues that the adoption of this approach was because it expresses "[...] some basic facts about natural numbers, which are used very often, most of the time without us stopping to ask how to prove them." He argues further that the "[...] objective is to show how such facts result from Peano axioms. There is no creative reasoning or elaborate methods to prove them. In all demonstrations, the central role is played by the Induction Axiom" (Lima, 2013, pp. 31).

This approach of N properties again highlights the potential of the material in relation to the development of KoT and KPM. However, at no point in the work does the author present any discussion that may be associated with the development of PCK. For example the author does not discuss, whether these demonstrations are accessible to students of basic education, whether they need to be adapted (KMT) to become accessible to school students (KFLM) and in which years it is possible to use them (KMLS).

Final comments

The developed analyses of a textbook that guides the formative practice in a discipline of PROFMAT show indications of the nature of the knowledge disseminated through this public policy of continuous training of teachers that aims to impact the quality of the learning of Brazilian students. Thus, based on the MTSK, it was found that elements of MK are considerably favored, particularly those associated with properties and definitions (KoT) and mathematical practice (KPM), such as the forms of validation and demonstration and the use of formal language. On the other hand, this material presents very few indications of the presence of knowledge related to the PCK, such as those related to the characteristics of the teaching (KMT) and learning (KFLM) of Natural Numbers and how this content is included in the school curriculum (KMLS).

The results presented provide us with indications that this public policy favors only part of the knowledge taken by the literature [START_REF] Carrillo | Determining specialised knowledge for mathematics teaching[END_REF][START_REF] Flores-Medrano | El Papel del MTSK como Modelo de Conocimiento del Profesor en las Interrelaciones entre los Espacios de Trabajo Matemático[END_REF][START_REF] Carrillo | Mathematics Teacher's Specialized Knowledge (MTSK) in the "dissecting an equilateral triangle" problem[END_REF][START_REF] Scheiner | What Makes Mathematics Teacher Knowledge Specialized? Offering Alternative Views[END_REF][START_REF] Carrillo-Yañez | The Mathematics Teacher's Specialised Knowledge (MTSK) Model[END_REF] as essential for the practice of the mathematics teacher in school. With this, it is possible to affirm that PROFMAT will not be able to fully achieve the objectives it proposes, which is to change the mathematical education of the country qualitatively and quantitatively, especially if we consider that the work developed in this program follows rigid evaluation standards that are guided by the textbooks (see [START_REF] Caldatto | The PROFMAT Program and Mathematics Teacher Education: a curricular analysis from a procedural and decentralizing perspective[END_REF][START_REF] Caldatto | The PROFMAT Program and Mathematics Teacher Education: a curricular analysis from a procedural and decentralizing perspective[END_REF].

  ) considers three subdomains in Mathematical Knowledge (MK) and three in Pedagogical Content Knowledge (PCK). In this way, MK is subdivided into: Knowledge of Topics (KoT), Knowledge of the Structure of Mathematics (KSM) and Knowledge of the Practice of Mathematics (KPM), while PCK is subdivided into: Knowledge of Mathematics Teaching ), Knowledge of Features of Learning Mathematics (KFLM) and Knowledge of Mathematics Learning Standards (KMLS).

Figure 1 :

 1 Figure 1: Sub-domains of the MTSK (Carrillo et al., 2018)

I) Every natural number has a single successor; II) Different natural numbers have different successors; III) There is a single natural number, called one and represented by the symbol 1, which is not a successor to any other; IV) X is a set of natural numbers (that is, X Ϲ N). If 1 ∈ X and if, in addition, the successor of every element of X still belongs to X, then X = N.


Cosette Crisan
email: c.crisan@ucl.ac.uk

Empowering teachers conceptually and pedagogically through supporting them in seeing connections between school mathematics and relevant advanced mathematics knowledge

Keywords: Advanced mathematics knowledge, mathematics knowledge for teaching, specialized knowledge, continuous professional development (CPD)

Empowering teachers conceptually and pedagogically through supporting them in seeing connections between school mathematics and relevant advanced mathematics knowledge

Cosette Crisan

Introduction

There is agreement amongst researchers all over the world that teachers need to have both subject knowledge of mathematics per se, and mathematical knowledge for teaching in order to teach effectively. There is also agreement that teachers must know in detail the school mathematics they are expected to teach and a bit more, beyond the level they are assigned to teach. But 'how much more? ' and 'More of what'? This body of more mathematics knowledge acquired through studying mathematics beyond school level is referred to in literature as 'academic mathematics' or 'advanced mathematics'. [START_REF] Moreira | Academic mathematics and mathematical knowledge needed in school teaching practice: Some conflicting elements[END_REF] refer to academic mathematics as that large part of the mathematics that a 'major' of mathematics is required to learn and which consists of that "scientific body of knowledge as produced and organized by the professional mathematicians" (p. 24). Similarly (and somehow a more influential terminology) is Advanced Mathematical Knowledge (AMK) put forward in 2009 by Zazkis and Leikin and defined as "systematic formal mathematical knowledge beyond secondary mathematics curriculum, likely acquired during undergraduate studies" (p.2368).

The Research Question

This paper thus reports on a study which was aimed at investigated if and in what ways (re)engagement with relevant AMK related to a school mathematics topic, functions in particular, empowers teachers conceptually and pedagogically.

Theoretical Influencesbrief overviews Teachers' knowledge for teaching

The study of teachers' knowledge of subject matter and its relationship to the quality of classroom instruction has grown substantially since Lee Shulman launched a call for researching teachers' different components of a professional knowledge base for teaching (Shulman, 1986). While there is no agreement amongst the mathematics education community about the relationship between these components, research flourished in an effort to conceptualize mathematics teachers' professional knowledge base for teaching.

One of such effort which builds on and refines Shulman's (1986) initial categorization of types of knowledge of a teacher of any subject, namely subject matter knowledge and pedagogical content knowledge, and which has proven to be very to be influential is the mathematics specific framework advanced by [START_REF] Ball | Content knowledge for teaching: What makes it special?[END_REF]. Their Mathematical Knowledge for Teaching (MKT) framework lays the foundation for a practice-based theory for mathematical knowledge for teaching. The authors divided Shulman's second category of Pedagogical Content Knowledge (PCK) into two other sub-domains, Knowledge of Content and Students (KCS) and Knowledge of Content and Teaching (KCT), while Shulman's third category of Curricular Knowledge (CK) was also relocated under PCK as Knowledge of Content and Curriculum.

Similarly, Shulman's category of Subject Matter Knowledge was divided into three sub-domains: Common Content Knowledge (CCK), Specialized Content Knowledge (SCK), and Horizon Content Knowledge (HCK). Recently, a few other researchers (e.g., Zaskis & Leikin 2010[START_REF] Even | The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: practitioners' views[END_REF] proposed positioning advanced mathematical knowledge (AMK) as an important aspect of MKT and in the following we will briefly review these categories and how they complement each other.

Common Content Knowledge (CCK) and Specialized Content Knowledge (SCK)

CCK is knowledge used in the work of teaching, but also used in ways that correspond with how it is used in settings other than teaching. SCK encompasses knowledge of mathematics needed by teachers, but not necessarily used by others, such as knowledge of a particular mathematical model or representation useful for teaching a certain concept.

Development of such SCK usually starts in teacher education programmes. Indeed, teachers acquire SCK through learning about how to use number lines, hot-air balloons or any other representations and metaphors that would enable them to teach about operations with negative numbers. For instance, while an engineer knows that the product of two negative numbers is a positive number, s/he does not need to know or give a mathematical reason for why this rule works or be able to provide a conceptually-sound explanation for the 'minus and minus make plus' metaphor. This kind of knowledge and reasoning should be an intrinsic part of a teacher's everyday classroom teaching, knowledge of the mathematics underlying rules, approaches, representations. However, far too often the implicit assumption is that prospective teachers already know the mathematics, to include the what and the why. But this is not the case. Many studies (e.g., [START_REF] Tirosh | Prospective elementary teachers' conceptions of rational numbers[END_REF] have revealed that school teachers possess a limited knowledge of mathematics, including the mathematics they teach. The mathematical education they received, both as pupils in school education and in teacher preparation, more often than not did not provide them with appropriate or sufficient opportunities to learn mathematics relationally (Skemp, 1976) and as a result, teachers themselves may know the facts and procedures that they teach but often have little or weak understanding of the conceptual basis underlying those rules and procedures [START_REF] Ball | The mathematical understandings that prospective teachers bring to teacher education[END_REF].

Horizon Content Knowledge

HCK (horizon content knowledge), the third sub-domain of subject matter knowledge in the MKT framework was tentatively defined by Ball and colleagues as 'an awareness of how mathematical topics are related over the span of mathematics included in the curriculum' (Ball et al., 2008, p. 403). Defined, interpreted and re-interpreted, HCK means different things to different researchers (e.g., Ball & Bass, 2009, Jakobsen et al., 2013); what is common ground amongst these interpretations is that it is knowledge that goes beyond that included in school mathematics curriculum, that influences teaching! [START_REF] Ball | Content knowledge for teaching: What makes it special?[END_REF] themselves describe HCK domain of knowledge as "a kind of mathematical 'peripheral vision' needed in teaching, a view of the larger mathematical landscape that teaching requires" (p.1), including "the vision useful in seeing connections to much later mathematical ideas" (p.403). The authors acknowledged that "we do not know how horizon knowledge can be helpfully acquired and developed" (ibid, 2009, p.11)

Advanced Mathematics Knowledge

While engaging with the MKT framework, Zaskis and Mamolo (2011) proposed to view HCK through the notion of viewing elementary (school) mathematics from an advanced standpoint, thus positioning advanced mathematical knowledge (AMK) as an important aspect of MKT. The notion of horizon content knowledge is given by Zazkis and Mamolo in terms of the application of the notion of 'advanced mathematical knowledge', which they define as the "knowledge of the subject matter acquired during undergraduate studies at colleges or universities" (Zazkis & Leikin, 2010 p. 264). [START_REF] Wasserman | Abstract algebra for algebra teaching: Influencing school mathematics instruction[END_REF], and later Stockon and Wasserman (2017) narrowed down the description of AMK to knowledge outside the typical scope of what a school mathematics teacher would likely teach, in that AMK is relevant, the advanced mathematical ideas are connected to the content of school mathematics, but also that these forms of knowledge of advanced mathematics are in some way productive for the teaching of school mathematics content.

Since Zaskis and Leikin's (2009) first conceptualisation of AMK, the authors also launched a call for further research to determine whether teachers' ability to identify explicit connections between AMK and the mathematics taught in school is a rare gift of only a few teachers or whether specific prompting is needed to bring this ability to surface.

The study: the England, UK context

Rather than relying on individual teachers' gift to identify explicit connections between AMK and the mathematics they are expected to teach, in this paper we propose that teachers of mathematics should be explicitly supported in becoming aware of these connections.

However, unlike other countries where teacher training is undertaken alongside undergraduate mathematics studies, and were such opportunities could be offered to prospective teachers as part of their undergraduate studies (e.g., [START_REF] Stockton | Forms of Knowledge of Advanced Mathematics for Teaching[END_REF], the UK context is different. The teachers in the UK complete their training in a one-year postgraduate course, meaning that they would have studied advanced mathematics as part of their undergraduate studies not related with teacher education. Thus, for them the study of advanced mathematics had no explicit relation to school mathematics content.

Hence the study presented in this paper is a step forward in response to Zaskis and Leikin's (2009) call, but suited to the UK approach to teacher training, and where such interventions are offered to teachers after they completed their undergraduate studies, either as part of their initial teacher training postgraduate course or as a professional development opportunity after they qualify as teachers.

Methodology

In this paper, I will be reporting on a two-hour CPD workshop designed to support the teaching of specific areas of the school curriculum, namely functions, and aimed at increasing teachers' familiarity with a variety of representations of functions in the school mathematics and their awareness of how these representations interconnect. The CPD workshop was designed and taught by the author of this paper, and the tasks attempted were of both mathematical and pedagogical nature, aimed at developing a deeper conceptual and pedagogical understanding of this topic, in ways suggested by research findings on ways of teaching about functions at various levels of students' education (e.g., [START_REF] Ayalon | Development of students' understanding of functions throughout school years[END_REF][START_REF] Nardi | Domain, co-domain and relationship: Three equally important aspects of a concept image of function[END_REF].

Each workshop was designed to start by posing a school mathematics question or a problem situation that teachers could do but where they may encounter some difficulties in answering it correctly and completely; in order to overcome the difficulties, the teachers will be guided towards recalling/reengaging with some relevant AMK; this will then be followed by a classroom-inspired scenario, of a pedagogical nature, where the teachers will be applying their new learning and become explicitly aware how their new learning supports their teaching.

Data sources

Textual data was collected through field notes that detailed some of the group interactions. Postsession written reflections were solicited and collected at the end of the session. The teachers were asked to comment on the activities in relation to their own learning, their pupils' mathematical learning, and their preparedness of teaching this topic. Written notes about the teachers' comments, questions, written work, were also made down by the myself throughout the workshop.

The participants

The participants were eight early career teachers of mathematics attending the two-hour CPD workshop. They were practicing mathematics teachers who wanted to refresh their knowledge about functions, given the high profile of this topic in the new re-vamped mathematics curriculum in England, UK Features of and operations with functions such as: domain, range, inverse and composite functions, which were traditionally studied by older learners (16 to 18 year old learners or even undergraduates) are now to be studied by 14 to 16 year old learners. Hence, all secondary mathematics teachers need to be able to teach this topic. The eight teachers gained their qualified teacher status as a result studying on a one-year teacher training course. As expected, they studied some mathematics at undergraduate level: six teachers studied for mathematics degrees, one had an engineering background, while one other teacher had an economics background and introduced himself as a non-specialist mathematics teacher. Consent to collect any observational and written notes throughout and after the workshop was sought from all the eight teachers.

Data Analysis and discussion

The workshop started by posing a school mathematics question or a problem situation that teachers could do but where it was envisaged they may encounter some difficulties in answering it correctly and completely, as suggested by previous research (e.g., [START_REF] Nardi | Domain, co-domain and relationship: Three equally important aspects of a concept image of function[END_REF]. The teachers were provided with an activity which required sketching the graph of functions that shared the same rule (one such example being f(x) = x 2 ,) but which had different domains of definition The teachers worked in pairs, and each pair was provided with a different domain for the function. The domains were: the whole set of real numbers, open and closed intervals, and discrete sets of real numbers. At the start of the activity, the teachers were not aware of the functions allocated to the other pairs.

Teachers' engagement with (partial) representations of a function:

The graphs produced by each pair looked more or less the same; a smooth curve in the shape of a parabola, carefully drawn to look symmetric about the y-axis. When the graphs were shared with the whole group, the teachers became aware of the similarities, but also the differences in their tasks; despite all sharing functions described by the same rule, the domains for each function were different, and so the discussion led naturally to a discussion about what a function was.

Teachers' re-engagement with the formal definition of a function: In the discussion, some teachers seemed to recall having studied about the formal definition of a function in their undergraduate studies, while others did not seem to have such a recollection or even an awareness of ever encountering such a definition. In what followed, disparate suggestions from teachers were recalled and put forward such as domains, co-domains, ranges, one-to-one, correspondence, notation conventions, and with some guidance from me, they reached the formal definition of a function of one real variable, which some recognised as having encountered them in their undergraduate mathematics course. This is evidence of teachers reaching for more advanced mathematics knowledge (in this case, the formal definition) related to functions, in the need to complete the task successfully. The 'starter' activity provided them with the impetus to reach for more than the school mathematics they were all too familiar with. Tapping into that knowledge, and once recalled (or newly learned, in the case of two teachers) there was evidence that it supported teachers completing the activity successfully. The teachers revisited the graphs they initially produced and each pair produced different graphs: either a smooth continuous parabola, or a pointwise graph, or a piece-wise graph, depending of the given domain of definition. Even though a few of the teachers had an awareness of the formal definition of functions and were able to recall some 'bits' of it, they commented that "it did not occur to me to relate this activity with the formal definition", and "that was high level mathematics not much used after the [undergraduate] course". The teachers seemed to be much more influenced by the current limited description of functions in the school curriculum, where domains and ranges of functions are not explicitly considered until the more advanced years of school level education.

Pedagogical implications: developing an awareness of building up to the formal definition of a function:

The 'starter' activity was then followed with a discussion of how the concept of function develops in the school mathematics curriculum. Representations of functions as they chronologically appear in the school curriculum were discussed: One-to-one or many-to-one mappings, Input/output machines, Relations between particular x-values and y-values; Expressions to calculate the y-values from given x-values, and Graphs. Each time, the teachers were encouraged to relate these representations with their recalled or newly learned AMK about functions. In doing so, the teachers came to realise that each of these representations explains particular aspects/features of the concept without being able to describe it completely! And a realization that overreliance on one representations or lack of connections between such representations gives way to misconceptions when working with functions, just as pointed out by [START_REF] Ayalon | Development of students' understanding of functions throughout school years[END_REF].

Indeed, the teachers themselves became explicitly "aware of stages of building up to the definition of a function". A teacher in particular was able to illustrate this new learning eloquently. She stated that she learned about: "Different representations of functions -I've always seen them as disconnected representations, but they complement each other nicely towards understanding functions" and exemplified with how in the lower secondary school curriculum, functions are portrayed as a computational process and are seen as an input-output machine that processes input values into output values. Such representation emphasizes the rule aspect in the definition of functions, seeing thus functions as an instruction to calculate one numerical set from another. This view leads to a perception of graphical representation of functions amongst pupils as points (usually with integer coordinates) plotted on the set of axes, which are then joined up with segments, with no explicit awareness that any other point lying on those segments could be just as good a candidate in the table of values. And in fact, the teacher herself realized that she never discussed or pointed this out in her teaching when plotting graphs.

Pedagogical empowerment -addressing ambiguities in the treatment of school maths topics: For the final task of this workshop, the teachers were asked to think about how they introduce and teach pupils of different ages about square roots. Sharing of own experiences immediately led into disputes over the numerical value of the square root of a number: a positive, a negative or a  value.

Most teachers defended the  value, and attempted to justify their answers in a variety of ways: "this is how I was taught myself", "this is how it is presented in textbooks", "this is how it appears in marking schemes of examination board", "This has always been the case." I then prompted the teachers to think about how this school maths concepts related to their learning about functions gained in the session up to that point and I suggested that perhaps it could help them settle the inconsistency in their answers.

The teachers were indeed able to call upon their recollection of the more advanced knowledge about functions (extended at this point to inverse functions and relationships between domains and ranges) and agreed on the positive answer only, despite feeling uncomfortable about dismantling a long-held belief about a piece of knowledge about square roots, they themselves inherited from their secondary school maths education and unfortunately still perpetuated by current mathematics school textbooks [START_REF] Crisan | The case of the square root: ambiguous treatment and pedagogical implications for prospective mathematics teachers[END_REF].

Concluding remarks

The teachers on this CPD course had revisited some of their AMK of functions, which provided them with a better understanding and an awareness of the developmental trajectory in learning about functions. They became aware of how representations in the school curriculum are particular instances of the concept itself, and how teaching towards a complete/full understanding of functions requires teachers to be aware of what each of these representations contribute to the complete understanding of the concept of functions: "Today's session helped me understand how I could have addressed the [pupils'] errors and how I can clarify things in the future.", while another teacher shared his learning in the session: "What I have learnt today? About advanced mathematics knowledge and its place in classroom and planning.". On a pedagogical level, AMK of functions empowered the teachers to justify why pupils make mistakes, and thus increased their knowledge for teaching this topic in the future.

The task in which the participants were involved in this workshop provided a context in which they recalled AMK related to functions, while in the case of some other teachers, they acquired new AMK (the teachers with an engineering and economics background). The teachers gained conceptually, as the mathematics tasks created some instability in what they knew about functions and their graphical representations, and in order to address the differences in their tasks, the teachers need to engage with more advanced mathematics knowledge of functions was brought out into the open.

However, this study has shown that even when teachers posses the AMK, they are not necessarily aware of manifestations of AMK in the school mathematics curriculum, thus they need to be supported in develop such awareness and make it explicit. One cannot simply assume that teachers will make connections without some intervention. This paper proposes that, in the UK context this should be the remit of courses preparing teachers of mathematics (either initial teacher training and/or CPD). All teachers, irrespective of their academic background, should be supported to look at school mathematics from an advanced standpoint and to examine school mathematics topics by engaging with advanced mathematics knowledge, while guidance is provided in terms of what AMK is relevant and how it informs school mathematics.


Rosa Delgado-Rebolledo
email: rosamdelgadorebolledo@gmail.com

Diana Zakaryan
email: diana.zakaryan@pucv.cl

Exemplifying mathematics teacher's specialised knowledge in university teaching practices

Keywords: lecturer knowledge, specialised knowledge, university level, real analysis, real numbers

Extant research on the teachers' knowledge includes limited studies focusing on teachers at university level. In this work, based on the Mathematics Teacher's Specialised Knowledge (MTSK) model and through an instrumental case study, knowledge of a lecturer in a real analysis course for prospective mathematics

Introduction

The mathematics education research focusing on the teachers' knowledge has traditionally been conducted at the elementary and secondary level. However, the study of mathematics lecturer's knowledge has recently emerged as a line of research that seeks to understand this knowledge, its development, and how it is reflected in university teaching practices [START_REF] Biza | Research on Teaching and Learning Mathematics at the Tertiary Level: State-of-the-art and Looking Ahead[END_REF]. Among the scarce account of research about mathematics lecturer's knowledge, the work of [START_REF] Breen | An analysis of university mathematics teaching using the Knowledge Quartet[END_REF] is a first approximation of teaching at the university level using Knowledge Quartet model [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF]. In addition, the studies conducted by [START_REF] Vasco | The characterisation of the specialised knowledge of a university lecturer in linear algebra[END_REF] and [START_REF] Vasco | Relationships between the knowledge and beliefs about mathematics teaching and learning of two university lectures in linear algebra[END_REF] highlight the utility of the Mathematics Teacher's Specialised Knowledge model [START_REF] Carrillo | Determining specialised knowledge for mathematics teaching[END_REF] and its corresponding analytical categories for understanding mathematics lecturers' knowledge. Other research in the same line, indicate that it is important to obtain empirical evidence pertaining to the components of specialised mathematics lecturer's knowledge [START_REF] Delgado-Rebolledo | Knowledge of the practice in mathematics in university teachers[END_REF] and deepening the understanding of these components in situations other than lecturers' classroom practices (Vasco & Climent, 2018).

Taking into account this background, we propose the following research question: Which knowledge, according to the Mathematics Teacher's Specialised Knowledge model, arises in the teaching practice of a mathematics lecturer? Teaching practice is considered in a broad sense, including lesson planning, liaising with colleagues, giving lessons, and taking time to reflect on them afterwards [START_REF] Carrillo-Yañez | The mathematics teacher's specialized knowledge (MTSK) model[END_REF]. In this sense, with the aim of answering the research question, we have studied one mathematics lecturer teaching a real analysis course in a mathematics teachers' training program, and reflecting on his performance. In this paper, we present examples of the different subdomains of this lecturer's specialised knowledge. We do not consider his characteristics as a mathematics teacher educator. A discussion about mathematics teacher educator's knowledge is exposed in the paper of Almeida, Ribeiro and Fiorentini (in this volume).

Theoretical Framework

In the Mathematics Teacher's Specialised Knowledge (MTSK) model, two domains of teacher knowledge are distinguished, Mathematical Knowledge (MK) and Pedagogical Content Knowledge (PCK). The model also considers teacher's beliefs about mathematics and about mathematics teaching and learning [START_REF] Carrillo-Yañez | The mathematics teacher's specialized knowledge (MTSK) model[END_REF].

MK refers to teacher knowledge of mathematics within the educational context, considering some of its characteristics as a scientific discipline. MK includes the Knowledge of Topics (KoT), Knowledge of the Structure of Mathematics (KSM) and Knowledge of Practices in Mathematics (KPM) subdomains. KoT contains knowledge about definitions, properties and their foundations, procedures, registers of representation, phenomenology, and applications (e.g., knowledge of the field properties of rational numbers; knowledge of definitions of real numbers). KSM encompasses knowledge of connections among mathematical items: connections associated with an increase in complexity or with simplification, and inter-conceptual connections (e.g., knowledge of relationships between infinity and the Archimedean property of the real numbers). KPM comprises knowledge about demonstrating, justifying, defining, making deductions and inductions, giving examples, and understanding the role of counterexamples (e.g., knowledge of how to prove the density property of the rational numbers in real numbers).

On the other hand, PCK is a specific type of knowledge of pedagogy in which the mathematical content determines the teaching and learning that takes place. PCK includes the Knowledge of Mathematics Teaching (KMT), Knowledge of Features of Learning Mathematics (KFLM), and Knowledge of Mathematics Learning Standards (KMLS) subdomains. KMT includes knowledge of theories of mathematics teaching, teaching resources, and strategies, techniques, tasks, and examples (e.g., knowledge of particularities of a real analysis textbook that makes it more convenient than others for use in a course, or use of analogy to illustrate the features of the existential quantifier). KFLM comprises the knowledge of theories of mathematical learning, strengths and weaknesses associated with learning, ways in which students interact with mathematical content, and emotional aspects of learning mathematics (e.g., awareness of analysis being more difficult for students than calculus; knowledge of difficulties students encounter when working with real numbers). Finally, KMLS contains the knowledge of sequencing of topics, expected learning outcomes, and the expected level of conceptual or procedural development (e.g., the sequencing of the completeness theorem topics, characterization of the greatest element, and the Archimedean property of the real numbers).

Methodological aspects

In this research, based on an interpretive paradigm and a qualitative methodology, an instrumental case study [START_REF] Stake | The art of case study research[END_REF] was conducted. The case pertains to a lecturer and mathematics researcher that was developing a real analysis course. Real analysis is a second-year course in a mathematics teachers' training program in a Chilean university. In the first two years, the students take calculus, algebra, and geometry courses, whereas, from the third year, the courses focus on teaching practices, general pedagogy, and didactics of mathematics (numerical systems, functions, geometry, and statistics).

The lecturer, who will be called Diego, has more than 20 years of teaching experience at the university level and this is the sixth time in recent years that he has developed the real analysis course. Diego's classes are of 90-minute duration, and each was videotaped, transcribed, and organized in class episodes according to Diego's tacit or explicit goals. For example, we consider a class episode the period since Diego begins until he finishes presenting a definition. In each episode, Diego's interventions that show knowledge according to the MTSK model (a knowledge that could be classified in an analytical category of some subdomain of the model) are chosen as analysis units. The analysis units that allow us to affirm the presence of a teacher knowledge were named evidence. Others, where we suspected the existence of teacher knowledge, but additional information was needed in order to confirm or refute this suspicion, were denoted as indication. An indication provided a reason to investigate in more detail the lecturer's knowledge. Hence, the differentiation between evidence and indication (Moriel-Junior & Carrillo, 2014) is considered with the aim to refine our interpretations and deepen the understanding of the subdomains of lecturer's knowledge.

The data obtained through video recordings was complemented with a semi-structured interview that was divided in two sessions for a total duration of three hours. The interview was audiorecorded and subsequently transcribed, reproducing Diego's speech with the highest fidelity possible. A template of questions was constructed to prompt Diego to think about some of his expressions and performances during the classes. For example, in one of Diego's classes, he referred to a YouTube video as a complementary material. Thus, in the interview, we asked Diego why he used that video and what was its intended objective. In addition, some questions were formulated with the intention to examine indications of knowledge and to explore the subdomains of specialised knowledge that has not been present in class episodes. Video clips of the classes were shown to stimulate Diego's recollection, because the interview was conducted once the course was completed. Transcriptions of the interview were analyzed in a similar manner to classes, considering each response to a particular question as an analysis unit.

Results

In this section, we exemplify with evidence the subdomains of Diego's specialised knowledge during a class session on real numbers. Diego starts this class by enunciating some concepts and properties studied in the previous class, such as the least-upper bound property and the Archimedean property of real numbers. Next, he uses these elements with the aim to construct, together with the students, the density of the rational and irrational numbers in the real numbers proof.

Diego's mathematical knowledge

As a part of his KoT, Diego knows axioms for the real numbers, constructions of this numerical system (e.g., Cauchy sequences or Dedekind cuts), definition of real numbers as a complete ordered field, and properties of real numbers. For example, Diego knows the Archimedean property of the real numbers (KoT) because he enunciates the property ( such that ) and comments on that in the following way: Diego: And, this property, which was not specified by Archimedes, but Euclid, is equivalent to saying that, for all small natural number , there exists a natural number such that .

Later, Diego writes on the blackboard the following proposition: Let , and , if is a set of closed and bounded intervals, there exists , such that , . Next, he makes a comment regarding this statement.

Diego:

This proposition indicates that, every time that I intercept closed and bounded intervals, that are nested… I forgot to say that they have to be nested, if not, it is not true. If the intervals are not nested, then the intersection is empty. For example, if you take the [0,1] interval and later the [10,12] interval, in the intersection you will have nothing because they are not nested. Then, to say that they are nested, I add to the proposition [writes on the blackboard, ]

Diego knows the nested intervals property (KoT), and deepening his discussion on the property, he emphasizes that the intervals must be nested in order to have an intersection that is not empty. Also, he understands how this sufficient condition gives sense to the implication expressed in the property. This knowledge of use of formal language as a way of communicating the mathematical idea expressed in the nested intervals property, belongs to the KPM subdomain.

On the other hand, starting from an indication of Diego's knowledge about the importance of the density of in property, in the interview, when we asked Diego about the meaning of this property, he responded as follows:

Diego:

The density is everywhere, because you cannot do anything if you do not have a dense and numerable set. The fact that the rational numbers are dense in the real numbers is very important, because you can take any real number and approximate it by a rational number. The real numbers have a cardinality greater than aleph 0, so you cannot count using them…Then, in this process of approximation, the rational numbers are important because, in practice, our calculations are limited to rational numbers.

In this excerpt, Diego demonstrates his knowledge of a connection to items within the same topic (KoT) because he points out that properties of such as its density in and its numerability are essential to work approximation and calculation processes with real numbers. When the lecturer said, "in practice, our calculations are limited to rational numbers", he referred to the processes mentioned above, which are also important in applications of mathematics, such as in modelling or numerically solving differential equations. In this sense, Diego's knowledge of uses and applications of properties of is identified as a part of his KoT subdomain.

Likewise, Diego refers to the density property of the rational and irrational numbers in real numbers. Given an interval , it holds and . Diego particularizes the proposition for the case of an interval with and . Using the Archimedean property, he establishes the following lemma: Given , and , the interval ) contains a rational number and an irrational number . Hence, Diego shows his knowledge of establishing preliminary results to facilitate the development of the density of rational numbers in real numbers proof. This knowledge of a way of proceeding in mathematics is a part of Diego's KPM. Moreover, Diego considers the case of a positive rational number , extending his previous arguments, and demonstrates his knowledge of process of particularization and generalization of a proposition about real numbers (KPM) as a way of proceeding in mathematics.

Continuing with the development of the density of the irrational numbers in real numbers proof, the lecturer expresses:

Diego: Using the previous lemma, there exists a number , real and not rational number, such that is between and . Then, if I add to this inequality, I have [writes on the blackboard is a rational number and is an irrational number, then, where is ? . . . If I can argument that is not in , then I can find an irrational number in . Why is not in ? Because, if was a rational number and I add another rational number, I will have to obtain a rational, because the rational numbers are a field. So, when I add two rational numbers, the answer is a rational number. Then, what can I add to , conveniently, to get a contradiction? Student: Diego:

Ok, if I add to the inequality, the addition is equal to That should be a rational, but I know that this is not true, so cannot be a rational number. Ok, I win.

In this episode, lecturer's knowledge of ways of validating in mathematics is identified. Diego demonstrates his knowledge of how proofs by contradiction method are done (KPM). The lecturer understands the logic underpinning this method of proof because he exposes what should be assumed ( is in ), how a contradiction is constructed (adding -conveniently considering properties of the rational numbers) and what must be concluded ( is not in ).

Diego's pedagogical content knowledge

Regarding Diego's PCK, in the class, Diego referred to the textbooks that he uses to develop the course. Later, in the interview, Diego elaborated on the reasons behind this literature selection.

Diego:

In the real analysis course, I use the Spanish edition of a famous textbook; the original edition is in Portuguese, but in Spanish, it has two editions, detailed and summarized. I do not use the detailed version because that textbook has a lot of information. Instead, I use the summarized edition that provides the most essential parts and, if I am lacking something, then I complement it with other textbooks.

In the exposed fragment, we observe that Diego not only knows both editions of the analysis textbook (detailed and summarized), he also knows the specific characteristics of each one, which allows him to select the textbook that makes it more convenient to develop the real analysis course. This knowledge of the teaching resource belongs to Diego's KMT subdomain.

Likewise, Diego exposes the reasons behind the use of YouTube videos as a complementary material to the classes.

Diego:

I like these videos because they are produced by the author of the textbook that we use in the course. The teacher worked in a prestigious university in Brazil and these videos are from a course intended for postgraduate students. The course duration is two months. I present the course [the same content] in a semester, with more time and I share the videos with the students, although the videos are in Portuguese and the language could be a problem.

Diego highlights the advantages of YouTube videos (they are created by the textbooks' author and he can develop the content in more detail). However, Diego knows that this digital resource has the limitation of language that is unfamiliar for some students. In this sense, Diego shows his knowledge of the advantages and limitations associated with the videos as a digital teaching resource. This knowledge is included in Diego's KMT subdomain.

On the other hand, an indication of Diego's knowledge was confirmed in the interview when Diego talked about students' understanding of the real numbers.

Diego:

The trouble with real numbers is the least-upper bound property, a historical difficulty that comes from Greeks. To humanity, it took more than 2000 years to comprehend the continuum . . . and maybe more, because I say until Newton, but actually, the formulation comes from Weierstrass. Then, this difficulty, this epistemological obstacle, inevitably emerges when you talk about real numbers.

Diego knows students' weaknesses pertaining to the understanding of real numbers (KFLM). He explains this issue considering historically intrinsic difficulties in conceptualizing real numbers due to the presence of the continuum as an epistemological obstacle.

Linked with the above, Diego comments:

Diego:

The other issue is that, in high school, real numbers are reduced to an algorithmic point of view only. I do not claim that this is wrong, but it should not be restricted to only that.

In the previous statement, Diego shows his knowledge of how are taught real numbers in high school. He refers to "an algorithmic point of view" highlighting deep level regarding the approach given to real numbers in learning standards in some Chilean high schools. This knowledge of the expected level of procedural development belongs to Diego's KMLS subdomain.

Final remarks

In this research, we have shown some examples of how the mathematics teacher's specialised knowledge could be identified and characterized in the case of a mathematics lecturer. In the pedagogical content knowledge, we exemplified the three subdomains (KMT, KFLM, and KMLS) and in mathematical knowledge we exemplified the KoT and KPM subdomains. Evidence of KSM was not supported by our findings in line with the results reported by [START_REF] Vasco | The characterisation of the specialised knowledge of a university lecturer in linear algebra[END_REF]. However, we obtained some indications of knowledge which supply several ideas to deepen in the KSM in the case of mathematics lecturers. Furthermore, descriptors of KPM regarding ways of communicating and ways of validating in mathematics are exposed in this work. Descriptors about ways of proceeding and ways of validating were reported by Delgado-Rebolledo and Zakaryan (2018), although the descriptors' grouping into categories is still under study. Establishing categories of KPM is an important topic of research in the MTSK model, and descriptors obtained in this study focusing on one mathematics lecturer are a good starting point for investigating components of this subdomain in teachers at other educational levels, given that scarce evidence of KPM has been reported in the research literature [START_REF] Zakaryan | ¿Cómo los profesores hacen prácticas matemáticas en sus aulas? [How teachers do practices in mathematics in class?[END_REF].

Additionally, in some examples, we observed possible relationships among the subdomains of knowledge. For instance, when the lecturer exposes the nested intervals property, his KPM regarding use of formal language allows him to understand this property in his KoT. Also, in the lemma used to develop the density of the irrational numbers in the real numbers proof (KPM), the lecturer relies on his knowledge of the Archimedean property (KoT). In a similar manner, relationships within pedagogical content knowledge domain could be established. For example, the lecturer ascribes students' weaknesses to the epistemological obstacle associated to real numbers, which allows him to understand the procedural approach to this notion in high school. Then, when the lecturer refers to the approach to real numbers adopted in high school (KMLS), he also demonstrates his awareness of students' weaknesses when working with real numbers (KFLM).

Thus far, relationships between disciplinary knowledge and pedagogical content knowledge have been reported in studies of secondary teachers (e.g., [START_REF] Sherin | When teaching becomes learning[END_REF]. We propose that this type of relationships could also be established in the knowledge of mathematics lecturer. These relationships are an opportunity to investigate the development of lecturer's mathematical and pedagogical knowledge.

In line with the above, the identification of the mathematics lecturer's specialised knowledge contributes to understand mathematics lecturers' knowledge and how this knowledge is reflected in their teaching practice [START_REF] Biza | Research on Teaching and Learning Mathematics at the Tertiary Level: State-of-the-art and Looking Ahead[END_REF]. However, more research is necessary to deepen the understanding of the nature and the components of mathematics lecturer's knowledge. In this sense, we propose to explore the relationships between the different subdomains of mathematics lecturer's knowledge as a topic for further research.

Acknowledgment

Beca de Doctorado año 2017 CONICYT, folio 21170442.


Lena Karlsson
email: lena.karlsson@lnu.se

Applying the Knowledge Quartet to student teachers' lesson plans: An intervention

The Knowledge Quartet

The Knowledge Quartet is a theoretical framework for analysing and developing mathematics teaching developed by [START_REF] Rowland | Developing primary mathematics teaching: Reflecting on practice with the Knowledge Quartet[END_REF] and elaborated on by [START_REF] Rowland | The Knowledge Quartet: The genesis and application of a framework for analysing mathematics teaching and deepening teachers' mathematics knowledge[END_REF]. The framework comprises four categories -Foundation, Transformation, Contingency, and Connectionthat together constitute the mathematical knowledge needed by teachers. Within each of these four categories, several codes have been developed with which to identify and evaluate the knowledge of student teachers based on observations of classroom teaching. This poster presents and discusses an intervention addressing whether and how the Knowledge Quartet can also be used to evaluate the knowledge of student teachers based on their lesson plans. In teacher education, there are often few occasions to observe student teachers' teaching, which is why it is of interest to explore whether some of the knowledge to be revealed (or revealed to be missing) by using the Knowledge Quartet can also be revealed in reflections of lesson plans.

The intervention

The intervention was conducted within a mathematics education course in lower primary school teacher education. In this course, the student teachers were to produce lesson plans for a series of lessons, implement one to two of these lessons, and then reflect on the series of lessons and on their implementation. The lesson implementation, however, was not observed by the teacher educators. The task was conducted in two groups of student teachers. In one group, the students were to write their reflections, and in the other group, the students were to record their reflections orally in video. For this poster, five reflections were randomly selected from each group.

In the written reflections, Foundation and Connection were the two dominant categories that were evident. In Foundation, a huge variety of codes was evident, the two most common ones being reflections on student beliefs (code A1) and reflections on the course (code A6). Connection was the second-largest category, and here the most common code was the student teacher's expectations about classroom complexity (code C1). In Transformation, the most common code was representations used in lessons (code B2). Contingency was only evident in one written reflection, in which one student teacher wrote about how she responded to and acted on her pupils' answers in the lesson (code D2).

In the oral reflections, Foundation and Connection were also the two dominant categories. In Foundation, several reflections addressed how assessment was to be carried out in the classroom (code A8), and another common code was the importance of using correct mathematical terminology and mathematical language even in lower primary school (code A7). Connection was the second largest category, and here the most common code was student teacher expectations about complexity in the classroom (code C1). The classroom setting was also common in the oral reflections code (code C6). In Transformation, the recordings of oral reflections featured how the student teachers chose examples and questions to be able to reveal student knowledge and understanding (code B1). Contingency was the least common category in the oral reflections, though it was more common here than in the written reflections. The most common codes were how students could deviate from the lesson plan (code D3) as well as how the student teachers responded to their pupils' questions and answers (code D2). Even code D1, which deals with how the student teachers handle and act with pupils during instructions occurred.

New codes and sub-codes detected during analysis of the students' lesson plans

In Foundation, a new code was added concerning students' thoughts about assessment. Among other things, the students reflected on complexity in assessment and teaching, and on the difficulty of finding the right teaching level and of fairly judging pupil knowledge. In Connection, two new sub-codes of code C1 (expectations of classroom complexity) were defined. One sub-code focused on pupils needing special support, in which the student teachers reflected on and discussed how to teach mathematics so as to include all pupils. This was obvious in both the written and oral reflections. The second sub-code connected to C1 was self-reflections on teaching, in which the student teachers reflected on the lesson structure and outcome, though this sub-code was only evident in the oral reflections. Another common sub-code, of code C6, in both the written and oral reflections was classroom framing. Socio-mathematical norms as well as ability grouping versus inclusion were important elements of this code; how pupils' language abilities influence their mathematics learning was also included as a sub-heading.

Reflection

The Knowledge Quartet is a theoretical framework designed to be applied to observations of student teachers in the classroom. In this study the framework has been applied to two different forms of examination in a teacher program which generated new codes and sub codes not visible in the original framework. These codes and sub-codes might be specific to observe students' reflections on teaching. Through the new codes and sub-codes, it is visible that there is a tendency that the student teachers´ reflections become clearer and broader examined through the film than through the text assignment. Hence, based on the intervention, I would argue that the refined framework of the Knowledge Quartet can be used to analyse students´ reflections in lesson plans. However, the analysis of lesson plans differs from that of teaching as performed, as it indicates only whether the student teachers have the theoretical knowledge needed to teach a lesson, not whether they have the practical skills. Though, there are often few occasions to observe student teachers' teaching in teacher education, which is why it is of interest to explore knowledge to be revealed in reflections of lesson plans.


Hulya Kilic
email: hulya.kilic@yeditepe.edu.tr

Oguzhan Dogan

Nil Arabaci

Sena Simay Tun

Preservice teachers' noticing of mathematical opportunities

Keywords: Noticing, preservice teachers, mathematical opportunities, faculty-school collaboration

The aim of this study was to investigate the effects of a faculty-school collaboration program on preservice mathematics teachers' professional knowledge and skills. Seven preservice teachers worked with a group of students from the collaboration school such that they attempted to understand and support students' mathematics through the implementation of mathematical tasks. Video tapes and written documents were used to detect any changes in preservice teachers' noticing of mathematical opportunities emerged from students' mathematics. A coding scheme was developed to understand the pattern in their noticing skills. The results revealed that preservice teachers were able to attend to mathematical opportunities during the implementations as well as in oral and written reflections took place after implementations. Furthermore, they began to enrich their interpretations of students' mathematics by giving examples from students' work and their interactions with students.

Introduction

For effective teaching, teachers should support students' mathematical understanding by organizing appropriate learning environment for students and giving various learning opportunities for students as well as providing necessary scaffolding for their learning [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. Teachers' such ability of eliciting and interpreting students' mathematics and scaffolding their understanding is a sign of both their Mathematical Knowledge for Teaching (MKT) [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF][START_REF] Stockero | Learning to notice important student mathematical thinking in complex classroom interactions[END_REF] and also their noticing skills (Jacobs, Lamp, & Philipp, 2010;[START_REF] Schoenfeld | Noticing matters. A lot. Now what[END_REF][START_REF] Van Zoest | Attributes of instances of student mathematical thinking that are worth building on in whole-class discussion[END_REF]. Because MKT involves in knowledge of how students learn and understand mathematics while noticing entails attending to learning opportunities occurred during instruction, interpreting students' thinking and taking an action for fostering students' understanding [START_REF] Jacobs | Professional noticing of children's mathematical thinking[END_REF]. In other words, MKT and noticing skills of teachers are interrelated [START_REF] Stockero | Learning to notice important student mathematical thinking in complex classroom interactions[END_REF][START_REF] Van Zoest | Attributes of instances of student mathematical thinking that are worth building on in whole-class discussion[END_REF]. Indeed, in-service teachers have opportunity to improve such knowledge and skills in schools throughout years such that they could enrich their repertoire of students' thinking styles, capabilities and needs in mathematics as well as teaching strategies that are appropriate for their students. However, recent studies on preservice teachers' training (PSTs) revealed that intervention studies conducted under teacher education programs were likely to support development of PSTs' MKT and noticing skills to some extent as well (e.g., [START_REF] Barnhart | Studying teacher noticing: Examining the relationship among preservice science teachers' ability to attend, analyze and respond to student thinking[END_REF]. In the light of such findings, we developed a faculty-school collaboration program for PSTs to investigate the nature and the development of their professional knowledge and noticing skills. As a requirement of the program, PSTs worked with a group of students on the mathematical tasks in a school setting throughout a year such that they videotaped all implementation sessions and then reflected on the implementations both orally and in written.

Theoretical framework

Noticing entails paying attention to important instances occurred in the classroom however it is difficult to decide what is noteworthy or not. In this study, we decided to focus on students' mathematical thinking and understanding and we used Leatham and his colleagues' [START_REF] Leatham | Conceptualizing mathematically significant pedagogical opportunities to build on student thinking[END_REF] definition of Mathematically Significant Pedagogical Opportunity to Build on Student Thinking (MOST) as a unit of analysis to investigate PSTs' noticing skills. They described MOST as a composition of three sequential components such that it should be emerged from student's mathematical thinking, be mathematically significant and be a pedagogical opportunity (see for details [START_REF] Leatham | Conceptualizing mathematically significant pedagogical opportunities to build on student thinking[END_REF]. For instance, finding the answer of -8+5 as -13 might be counted as a MOST if it occurs while teaching operations with integers. In this case, the student thinks that addition of a negative and a positive integer is utilized as addition of natural numbers such that the sign of the first number is omitted first but then it is attached to the result. Because such an answer is observed while teaching integers, it is both mathematically significant and also a pedagogical opportunity to be discussed at that time in the class. Leatham and his colleagues also noted that not only students' misconceptions or incomplete reasoning but also their correct answers based on use of different strategies or approaches, mathematical contradictions or "why" questions might be coded as a MOST instance. We adopted Jacobs and her colleagues' [START_REF] Jacobs | Professional noticing of children's mathematical thinking[END_REF] definition of professional noticing of students' thinking to analyze PSTs' noticing skills. They defined noticing as having three interrelated components as attending to students' strategies, interpreting their understanding and deciding how to respond to students' mathematics. In this study, we analyzed PSTs' noticing in terms of whether or not they attended to MOST instance, how explicitly and accurately they interpreted students' thinking or student mathematics and how they responded to the MOST instance. We attempted to investigate PSTs' noticing during implementations (in-the-moment) and after implementations (in oral and written reflections). The implementation videos provided data about attending and responding while written and oral reflections used for attending and interpreting components of noticing. Although we asked PSTs to comment on their responding actions in oral and written reflections we did not count them as responding component but as a sign of interpretation of student mathematics as well as an indicator of their MKT.

Methodology Research setting and participants

This study was conducted under a university-school collaboration program between a large university in Turkey and a local middle school in the neighborhood of the university. In the line of the collaboration, we took the responsibility of administration of an elective mathematics course offered for the seventh grade students in the school such that we grouped students and assigned a volunteer PST to work with these students on the mathematical tasks.

Seven preservice teachers attended to the program both in fall and spring semesters. Four of them were sophomore (Asya, Aydan, Ayla, Aysun) and three of them were junior (Bahar, Berna, Beste) undergraduate students of mathematics education program.

At the beginning of fall semester, a couple of weeks, we discussed about design and implementation of mathematical tasks as well as students' common misconceptions, and effective ways of understanding and supporting students' mathematical thinking. We discussed these issues via some sample videos and student work that we had in our repertoire from our earlier studies. Then we assigned a group of four students for each PST that they would work with in the school for a year. Each group consisted of mixed ability students in terms of mathematics achievement.

We followed a 4-step intervention process for PSTs: pre-implementation discussion, task implementation, post-implementation discussion and written reflection. During pre-discussions we talked about the students' possible performance on the tasks as well as any modifications in the tasks before the implementation. Then, PSTs implemented the tasks in the school such that they allowed students worked on the tasks individually at first and then students discussed their answers as a group and finally PSTs began to interact with students to address mathematical opportunities occurred during the implementations. During post-discussions, each PST talked about how students performed on the tasks and how they addressed to students' mistakes or misconceptions. Each of these discussion sessions and implementations were videotaped. Then we asked PSTs to write reflection reports based on their videos and students' worksheets.

As the research team, we prepared 20 tasks such that 5 of them were about numbers, 7 of them were algebra, 5 of them were geometry and 3 of them were data and statistics. We also asked PSTs to prepare at least one task for each content area for their own groups and implement them. We used 2 lesson hours (80 min.) for each task implementation and we spent a total of 25 weeks in the school.

Data collection and analysis

We used discussion and task implementation videos, and PSTs' written reflections as the main source for PSTs' noticing skills. Furthermore, we used task implementation videos and students' worksheets to identify MOST instances. We determined MOST instances according to Leatham and his colleagues' (2015) framework. Based on Jacobs and her colleagues' (2010) definition of noticing we developed a coding scheme to identify PSTs' noticing skills in terms of attending to MOST, interpreting student mathematics and responding to students. While developing the coding framework we used similar frameworks in the literature (e.g., [START_REF] Van Es | A framework for learning to notice student thinking[END_REF] and also made a workshop with math educators to get their suggestions. The scheme that we used for coding is presented in Table 1. We used different identifiers and levels for interpreting component of written and oral reflections because during oral reflections we gave a limited time for PSTs to talk however, they had enough time to write about implementation and students' performances. Furthermore, as the research team we discussed all MOST instances and codes for PSTs' noticing skills together. Thus, we have achieved a consistency in coding of each PSTs' noticing and MOST instances, in other words, we fully achieved interrater reliability.

Findings

In this paper, we present the data collected from seven PSTs who attended to the program in both semesters. A total of 354 MOST instances (approximately two MOST instances per week for each PST) were detected throughout the year. Although [START_REF] Leatham | Conceptualizing mathematically significant pedagogical opportunities to build on student thinking[END_REF] discussed various sources of MOST instances, including students' alternative solutions or their answers for "why" questions, in this study majority of MOSTs emerged from students' misconceptions. Furthermore, because we mentioned about students' possible misconceptions during pre-implementation discussions, PSTs were somewhat familiar to such instances before the implementations. Indeed, 45% of MOST instances were already mentioned in the pre-implementation discussions while 55% of them were new for the PSTs. The frequency distribution of how PSTs attended to those MOST instances is presented in Table 2. When we analyzed data about noticing during implementations, we observed that the PSTs attended to majority of MOST instances in both fall and spring semesters (88% and 89%, respectively). In terms of responding to MOST instances, it was observed that PSTs attempted to guide students in some ways (totally 91% in Fall and 98% in Spring) but they mostly preferred answered-focused actions in both semesters (totally 88% for each semester). However, it was seen that in the spring semester they attempted to elicit students' mathematics in the form of exploration and elaboration more in comparison to fall semester (totally 3% in Fall, 11% in Spring).

Interaction

In terms of written and oral reflections after implementations it was also observed that PSTs attended to MOST instances occurred during implementations. During post-implementation discussions the instructors from the research team asked about how implementation went and what PSTs observed about students' mathematical performances. When discussion videos were analyzed it was observed that PSTs mentioned about the MOST instances occurred during the implementations (68% in Fall and 81% in Spring). Furthermore, they attempted to give justifications for students' performances such that the percentage of such specific comments about students' mathematics increased in spring semester (50% in Fall, 69% in Spring).

The PSTs attended to some of the MOST instances that they missed during the implementations in their written reports such that in total they attend to 10 of the 18 missed MOSTs in fall (approx. 56%) and 13 of the 21 missed MOSTs in spring (approx. 62%). However, there were cases that PSTs did not write about the MOST instances even though they attended to them during the interactions. When we compared the semesters, the number of such cases decreased in spring semester from 17% to 9% as seen in Table 2. That is, PSTs attended to MOST instances in both during interaction with students and in their written reports (71% in Fall and 80% in Spring). We also analyzed how PSTs interpreted student mathematics in their written reports. It was observed that all PSTs attempted to interpret students' mathematical thinking behind the MOST instances.

Although in fall semester the percentage of the cases that PSTs only wrote about what students did (Level 1 in coding scheme) was high, it decreased in the spring semester (43% in Fall, 25% in Spring). That is, they began to comment on students' mathematics by providing justifications from their interactions with students and students' written work (totally 56% in Fall and 75% in Spring).

We also analyzed how each PST's noticing varied in fall and spring semesters. As shown in Table 3, we analyzed their responding actions during interactions in terms of answer-focused vs mathematical-understanding focused. We also categorized their interpretation of students' mathematics in terms of providing justifications or not. As seen Table 3, Aydan showed a progress in terms of attending to MOST instances in the moment of implementations. In terms of responding actions, except Aysun, PSTs attempted to elicit or elaborate students' mathematical understanding during the interactions such that there were more instances coded as "3" or "4" in spring semester. The PSTs also attended to MOST instances and specifically mentioned about students' mathematics during oral reflections. This fact was more evident in Asya's, Aydan's and Ayla's cases as shown in Table 3. Although all PSTs attempted to provide justifications for their interpretations of students' thinking in their reports, the change in Aysun's and Bahar's written reflections during spring semester was noteworthy.

Discussion

In this study, we attempted to create an opportunity for PSTs to investigate nature and development of their in-the-moment noticing skills as well as noticing based on analysis of their own videos. As aligned with the findings of intervention studies mentioned in the literature [START_REF] Barnhart | Studying teacher noticing: Examining the relationship among preservice science teachers' ability to attend, analyze and respond to student thinking[END_REF][START_REF] Stockero | Learning to notice important student mathematical thinking in complex classroom interactions[END_REF] this faculty-school collaboration contributed to PSTs' noticing. Firstly, PSTs were able to attended majority of MOST instances occurred during implementations. Pre and post implementation discussions might help PSTs to recognize the MOST instances because we talked about students' possible misconceptions and difficulties as well as how to eliminate such misconceptions during the discussions. Furthermore, as they got to know their students they got better in predicting their students' performances so that they had prepared for MOST instances which were likely to occur during implementations. This fact could be thought as an improvement in their MKT specifically in their specialized content knowledge and knowledge of students and content. Secondly, we observed changes in PSTs' responding actions throughout the year such that almost all PSTs achieved a transition from "No attempt" to "Elaboration" type of responding. However, such transition was not in progressive fashion [START_REF] Barnhart | Studying teacher noticing: Examining the relationship among preservice science teachers' ability to attend, analyze and respond to student thinking[END_REF] most probably because variations in students' prior knowledge and the context of the tasks [START_REF] Kilic | Supporting preservice teachers' in-themoment noticing[END_REF]. For further studies, PSTs might be trained more for eliciting and elaborating type of responding actions before implementations of the tasks with students. Thirdly, analysis of own videos supported PSTs' noticing skills [START_REF] Barnhart | Studying teacher noticing: Examining the relationship among preservice science teachers' ability to attend, analyze and respond to student thinking[END_REF][START_REF] Stockero | Learning to notice important student mathematical thinking in complex classroom interactions[END_REF] such that they were able to recognize and wrote about the MOST instances that they missed during interactions in their reflections. Furthermore, written reflections provided an evidence for how PSTs interpreted students' thinking which was implicit in implementation videos.

Briefly, in addition to their theoretical courses, teacher educators should provide hands-on experiences for PSTs to promote their professional knowledge and skills. Although analysis of own teaching videos contributed to PSTs' noticing skills, for in-the-moment noticing, they should be given opportunities to work with students or teach in a classroom for a long period of time.

Table 1 : Framework for teachers' noticing skills

 1

	the moment (Interaction)	Attending Responding	0 1 Answer-focused Missed the MOST Attended to the MOST 0 (No attempt) Only tells to the students that their answers/solutions are wrong; no guidance for students 1 (Explanation) S/he or other students tells/explains the procedure or solution 2
	In

(Orientation)

Attempts to make students find out the correct answer through short-answer, Yes/No type, prompting (directs students to correct answer like "…,isn't it?), no-follow up, non-specific type of questions or b) asking them to re-read, re-do, re-think

Table 2 : Coding of preservice teachers' noticing skills

 2

  For instance, in fall semester Bahar was just writing about what students did or she blamed students having lack of knowledge however in spring semester she began to explain the reasoning behind students' mathematics by providing examples from student work or sample vignettes of group discussions.

		Interaction	Oral reflection	Written reflection
		Att.	Res.	Att.	Int.	Att.	Int.
		1	0--2 3--4	1	0--1 2	IR	0--2 3--4
	Asya	Fall Spring 83% 86%	95% 5% 83% 17%	71% 79%	33% 67% 21% 79%	62% 76%	60% 40% 76% 24%
	Aydan	Fall Spring 89% 58%	100% 0% 92% 8%	68% 89%	68% 32% 22% 78%	32% 78%	50% 50% 50% 50%
	Ayla	Fall Spring 89% 90%	100% 0% 76% 24%	71% 86%	48% 52% 21% 79%	81% 89%	53% 47% 43% 57%
	Aysun	Fall Spring 85% 86%	100% 0% 100% 0%	76% 69%	57% 43% 38% 62%	81% 85%	74% 26% 53% 47%
	Bahar	Fall Spring 90% 100%	92% 8% 85% 15%	72% 83%	40% 60% 31% 69%	80% 83%	95% 5% 71% 29%
	Berna	Fall Spring 93% 91%	100% 0% 96% 4%	65% 77%	48% 52% 58% 42%	74% 81%	76% 24% 73% 27%
	Beste	Fall Spring 97% 100%	96% 4% 94% 6%	50% 82%	64% 36% 27% 73%	80% 73%	80% 20% 83% 17%

Note: Att.: Attending, Res.: Responding, Int.: Interpreting

Table 3 : Variations in preservice teachers' noticing skills

 3

Acknowledgment

This research was supported by The Scientific and Technological Research Council of Turkey (TUBITAK, Grant no: 215K049)