text:
Jani Hannula
email: jani.hannula@helsinki.fi

Subject matter knowledge and pedagogical content knowledge in the learning diaries of prospective mathematics teachers

Keywords: Teacher knowledge, mathematics teacher education, learning diaries, subject matter knowledge, pedagogical content knowledge

In this study, 26 learning diaries by prospective mathematics teachers were analysed in order to describe the characteristics of mathematical and pedagogical knowledge discussed therein and to evaluate the potential and limitations of the learning diary in mathematics teacher education. Conceptualisations of teacher knowledge are typically discussed in terms of subject matter knowledge, pedagogical knowledge and pedagogical content knowledge. A central goal of mathematics teacher education is to strengthen all of these areas of competency. The results of this study indicate that, although the learning diary is a potential learning tool, prospective mathematics teachers tend to emphasise pedagogical content knowledge, placing less stress on subject matter knowledge. Consequently, more structured learning diary tasks could be used to support all the components of mathematical knowledge for teaching.

Introduction

Finnish mathematics teacher education consists of three somewhat distinct parts: subject matter studies, educational studies and practical teacher training at schools. Subject matter studies at mathematics departments form a major part of the mathematics teacher studies. Finnish mathematics teachers, however, report that university-level subject studies in mathematics lack a clear connection to the mathematics taught at school [START_REF] Koponen | Teachers and their educators' views on contents and their development needs in mathematics teacher education[END_REF]. The transition to university-level mathematics requires a major change in mathematical thinking (e.g., [START_REF] Tall | The transition to advanced mathematical thinking: Functions, limits, infinity and proof[END_REF]. At university, mathematics courses typically emphasise formal reasoning, meaning reasoning based on axioms, definitions and proven theorems [START_REF] Viholainen | Prospective mathematics teachers' informal and formal reasoning about the concepts of derivative and differentiability[END_REF]. Informal reasoning is based on visual or physical interpretations of mathematical concepts [START_REF] Viholainen | Prospective mathematics teachers' informal and formal reasoning about the concepts of derivative and differentiability[END_REF]. Some empirical studies (e.g., [START_REF] Chin | Making sense of mathematics: Supportive and problematic conceptions with special reference to trigonometry[END_REF][START_REF] Viholainen | Prospective mathematics teachers' informal and formal reasoning about the concepts of derivative and differentiability[END_REF] have shown that prospective teachers may have difficulties connecting formal and informal reasoning.

On the other hand, prospective mathematics teachers may emphasise the importance of a teacher's personal characteristics and pedagogical knowledge, while diminishing the importance of subject matter knowledge (e.g., [START_REF] Hoffkamp | Dimensions of mathematics teaching and their implications for mathematics teacher education[END_REF]. Subject matter knowledge nevertheless plays a significant role in a teacher's professional knowledge. Firstly, subject matter knowledge is typically seen as theoretically necessary for developing pedagogical content knowledge [START_REF] Baumert | Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress[END_REF]. The quality of prospective teachers' subject matter knowledge also affects their pedagogical choices when participating in practical training [START_REF] Even | Teacher subject matter knowledge and pedagogical content knowledge[END_REF]. In addition, subject matter knowledge along with pedagogical content knowledge can be seen as a foundation for effective teaching, as a teacher's professional knowledge affects student achievement (e.g., [START_REF] Baumert | Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress[END_REF].

This study is a part of a design-based research and development project that has been carried out at the University of Helsinki. The aim of the research is to develop instructional practices in order for Thematic Working Group 20

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prospective teachers to both strengthen their subject matter knowledge and build up their pedagogical content knowledge. The research also aims to give insight into prospective mathematics teachers' conceptions of the relationship between school and university mathematics.

In this study, the specific focus is on learning diaries written by prospective mathematics during a six-week seminar. The seminar focused on finding connections between the mathematics studied both at university and at school and on discussions of mathematical content from the teacher's point of view. That is, the aim of the seminar was for prospective teachers to, first, strengthen their (structural) knowledge of mathematical topics (such as derivative) and, second, enhance their pedagogical content knowledge with relation to these topics. The aim of this study was to examine the potential and limitations of learning diaries as a learning tool in this context and to conceptualise the kinds of knowledge these prospective mathematics teachers discussed in their diaries.

Theoretical background

Theories used in design-based research can be divided into grand theories, orienting frameworks, frameworks for action and domain-specific instructional theories [START_REF] Disessa | Ontological innovation and the role of theory in design experiments[END_REF]. In this study, the idea of constructive alignment [START_REF] Biggs | Teaching for quality learning at university[END_REF], which provided the instructional design of the research setting, is used as a framework for action. The data analysis for this study is based on domain-specific conceptualisation of teacher knowledge [START_REF] Ball | Content knowledge for teaching what makes it special[END_REF]. In the next two subsections, these frameworks will be discussed in more detail.

Teacher Knowledge

The distinctions between content knowledge (or subject matter knowledge), pedagogical knowledge and pedagogical content knowledge [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF] are an established starting point for conceptualisations of teacher knowledge [START_REF] Scheiner | Lessons we have (not) learned from past and current conceptualizations of mathematics teachers' knowledge[END_REF]. Especially the distinction between subject matter knowledge and pedagogical content knowledge has gained significant attention and generated a great amount of research and further development of the conceptualisations of teacher knowledge. According to the Mathematical Knowledge for Teaching (MKT) model [START_REF] Ball | Content knowledge for teaching what makes it special[END_REF], a teacher's content knowledge consists of common content knowledge (CCK), specialised content knowledge (SCK) and horizon content knowledge (HCK). In the MKT model, pedagogical content knowledge, in contrast, is divided into knowledge of content and students (KCS), knowledge of content and teaching (KCT) and knowledge of content and curriculum.

The components of MKT model have been shown as important for effective teaching [START_REF] Jakobsen | Delineating issues related to horizon content knowledge for mathematics teaching[END_REF]. Hence, the model is valuable for this study, which aims to offer insight into prospective mathematics teachers' discussions of their learning diaries and to use this information for further development of instructional practices in teacher education.

Constructive alignment and learning diaries

Present research and development of instructional practices in higher education is typically based on the constructivist view of learning and concepts, such as learner approaches to learning, selfregulation and reflection. Constructive alignment is based on the constructivist view of learning and suggests that the intended learning outcomes, implementation of teaching and assessment should be carefully aligned and support active learning. [START_REF] Biggs | Teaching for quality learning at university[END_REF] suggest that by using more active ways of learning (such as problem-based learning) even 'less academic' students can achieve more advanced levels of learning, such as applying and theorising.

In this study, the seminar was designed in the spirit of constructive alignment (e.g., the students worked in groups and specified their own study/discussion topics). The learning diary task was one of the ways to promote active learning and reflection among students. Typically, learning diaries are seen as texts that include both the central arguments of a course or a seminar and the writer's own interpretation of and reflection on these themes. That is, learning diaries are not supposed to promote knowledge telling writing [START_REF] Bereiter | The psychology of written composition[END_REF], which is understood as writing based on memorised facts. Instead, learning diaries promote knowledge transforming writing [START_REF] Bereiter | The psychology of written composition[END_REF], which is based on problem analysis and reflection.

Research questions

In Finnish higher education, learning diaries have been used successfully in subjects such as research methodology [START_REF] Kyttälä | Learning diary as a learning tool for prospective special education teachers in a quantitative research methods course[END_REF]. Journaling has also been found to be useful in studying university-level mathematics [START_REF] Meel | Journal writing: Enlivening elementary linear algebra[END_REF]. There is, however, a lack of research evaluating the use of learning diaries in mathematics teacher education from the point of view of the teacher's knowledge. Additionally, more insight into prospective mathematics teachers' mathematical and pedagogical thinking is needed for further development of instructional practices in mathematics teacher education. Thus, the following research questions were formed.

1. Can learning diaries be used to promote knowledge transforming writing in mathematics teacher education?

2. What kinds of professional knowledge do the prospective mathematics teachers discuss in their learning diaries?

The first research question was posed in order to evaluate whether learning diaries have potential as a reflective learning method in mathematics teacher education. The second research question was posed in order to characterise the prospective mathematics teachers' discussions on teacher knowledge. The question of whether some/certain aspects of teacher knowledge would be emphasised in the diaries was also considered, as prior research has shown that prospective teachers may emphasise pedagogical knowledge and diminish the importance of subject matter knowledge (e.g., [START_REF] Hoffkamp | Dimensions of mathematics teaching and their implications for mathematics teacher education[END_REF].

Method

The data was collected during a seminar held in autumn 2014. The students (prospective mathematics teachers) attending the seminar formed small groups of 4-5 members. All groups prepared an introduction to a topic (such as dot product), so that both mathematical and pedagogical ideas were covered. These introductions led to group discussion and, as homework, the students reflected on their ideas by writing a learning diary. In their diaries, the students were asked to discuss 1) What was discussed and how do the topics of discussion relate to other contexts?; 2) What did I learn and what was its meaning for me?; 3) Was something missing or unclear?

Participants were mainly mathematics students at the end of their studies. Three students were studying another subject (such as physics) with minor studies in mathematics. Also, six students were second-or third-year students and, thus

, not yet at the end of their five-year studies. The participants were studying in a subject teacher programme that qualifies them to work as a teacher in the last years of comprehensive school (with students aged 13 to 16 years) and upper secondary school (with students aged 16 to 19 years).

Student learning diaries (N=26) were analysed using content analysis [START_REF] Elo | The qualitative content analysis process[END_REF]. The units of observation were first placed in categories from the MKT model using deductive content analysis. Subcategories were then formed using inductive content analysis. In addition, the individual diary entries were classified either as knowledge telling writing or knowledge transforming writing in order to classify the entire diary either as knowledge telling or knowledge transforming.

The author of the present article created the coding. As it was not possible to use two independent coders, during the process, the author reread the diaries and the coding to ensure that his thinking remained constant during the coding process. The components of the MKT model may be difficult to distinguish from one another and this boundary problem has been highlighted in the research literature. This poses a challenge for coding, as two researchers may create different categorisations. The most problematic category seems to be HCK. In this study, HCK was understood, as defined by Jacobsen et al. ( 2013), as 'an orientation to and familiarity with the discipline (or disciplines) that contribute to the teaching of the school subject at hand, providing teachers with a sense for how the content being taught is situated in and connected to the broader disciplinary territory'.

The coding of knowledge telling writing and knowledge transforming writing was based on a prior study by [START_REF] Kyttälä | Learning diary as a learning tool for prospective special education teachers in a quantitative research methods course[END_REF]. When coding each diary entry as either knowledge telling or knowledge transforming, the former was used if the entry included only repetition of the information discussed in the seminar and the latter code was used if the entry included personal reflection. Knowledge telling writing included excerpts such as 'This week we discussed linear algebra. Firstly, we discussed vectors in R 2 ', whereas knowledge transforming writing included personal reflection such as 'I soon realised that I didn't remember much about dot product. I remembered that it had something to do with lengths and the perpendicularity of vectors.'

If at least half of the entries were labelled as knowledge transforming the entire diary was labelled accordingly. This methodology was chosen to ease the comparison of the results of this study to prior studies in the Finnish higher education context.

Results

The results of the study are presented in three parts. First, writing strategies (knowledge telling vs. knowledge transforming) are discussed. Then, in the following two subsections, the subject matter knowledge and pedagogical content knowledge observed in the diaries are discussed.

Knowledge telling writing vs. knowledge transforming writing

23 of the 26 diaries featured knowledge transforming writing. This seems to indicate that learning diaries can be used to promote reflective learning in mathematics teacher education as they have in other educational contexts, as [START_REF] Kyttälä | Learning diary as a learning tool for prospective special education teachers in a quantitative research methods course[END_REF] has suggested. However, while most of the diaries were categorised as knowledge transforming, the content discussed in the diaries varied significantly. In some of the diaries, both mathematical and pedagogical topics/issues were discussed comprehensively, whereas in others, the mathematical content was discussed only cursorily and the pedagogical issues were discussed in depth. The coding of subject matter knowledge and pedagogical content knowledge aimed to highlight this variation in greater detail.

Subject matter knowledge

The distinguished subcategories of subject matter knowledge are presented in Table 1. The frequency of each category is indicated in brackets. In the main categories, percentages are also given. Discussing representations of mathematical content was labelled as SCK, as according to [START_REF] Ball | Content knowledge for teaching what makes it special[END_REF], knowledge of 'how to choose, make, and use mathematical representations' belongs to SCK. The category 'nature of mathematics' included utterances such as 'In mathematics you don't prove absolute truths. Instead, the proofs are based on chains where assumptions lead to something.' These can be also seen in connection to HCK, but as these comments were general, they were coded as CCK. Additionally, some discussion of the curriculum, such as 'Does knowing probability require knowing set theory? I suppose so. It would be good if there would be more of that in secondary school', were categorised as SCK instead of KCC. These comments also seemed connected to HCK, but were more focused on rethinking school mathematics and were consequently categorised as SCK. Common content knowledge was mainly discussed in terms of giving a list of concepts (related to the subject), giving a definition of a concept (such as limit) or giving a theorem (such as 'If function f is derivative, then function f is continuous'). This discussion was typically limited to telling the facts and no explanations or proofs were given. The specialised content knowledge typically focused on discussing the representations of mathematical content. Only one student adapted an example so that different versions of a problem were considered. The least discussed aspect of subject matter knowledge was horizon content knowledge; only 46 units of observation included discussion of the hierarchical relationship of mathematical concepts or an application of a mathematical entity or method. Overall, discussion of subject matter knowledge was somewhat focused on SCK. More specifically, discussing the different representations of mathematical content was common in many diaries.

Pedagogical content knowledge

The distinguished subcategories of pedagogical content knowledge are given in Table 2. In many diaries, the secondary school and university curricula were compared. Students discussed, for instance, the content of secondary school calculus courses and university analysis courses. The knowledge of content and students sections mainly focused on difficulties or misconceptions that school students may have. For example, affect and learning (e.g., emotions) were little discussed and the cognitive studies in mathematics education were mainly used as references. The knowledge of content and teaching focused on discussing different means of approaching mathematical content in teaching and different teaching methods. For example, no imaginary learning situations were introduced and only one student pondered the answering of school students' questions.

Overall, PCK was discussed more than subject matter knowledge. However, the PCK typically discussed in the diaries can be described as content-driven, as it was mainly placed in subcategories such as 'Ways to approach the mathematical content in teaching' or 'Difficult content for students'.

Discussion and conclusion

It is worth noticing that the results of this study cannot be generalised to whole student populations or other contexts. This study contributes only case-specific information, which can, however, be used in further development of the specific learning environment. In addition, the reliability of this study could be enhanced by using two independent researchers in the data analysis phase. Nevertheless, this study found that in this specific context, many prospective mathematics teachers adopted a knowledge transforming writing strategy in their learning diaries. The knowledge discussed in the diaries was somewhat focused on SCK and PCK. More specifically, the most discussed topics were representations of mathematical content, curricula, student knowledge and teaching methods. Some of the learning diaries discussed both subject matter knowledge and pedagogical content knowledge. Some of the diaries, however, were more focused on pedagogical content knowledge. This seems to indicate that some of the prospective teachers emphasised pedagogical topics, while other prospective teachers discussed teacher knowledge more comprehensively. Further research would be needed to discuss this variation in detail and, especially, to compare students who are at different stages of their studies. In addition, it is notable that horizon content knowledge was rarely discussed in the diaries. This was somewhat surprising as the aim of the seminar was to connect the content of university-level mathematics and school mathematics. If HCK is understood as [START_REF] Jakobsen | Delineating issues related to horizon content knowledge for mathematics teaching[END_REF] have presented it, connecting mathematics as a discipline to school mathematics means discussing horizon content knowledge. In addition, some aspects of SMK and PCK (such as modifying tasks) also received little attention. This implies that learning diaries may lead to reflections that are not fully aligned with the intended learning outcomes. Further research is needed to determine whether more structured learning diary tasks would help students to better discuss desired sides of mathematical knowledge for teaching.

Table 1 : Subcategories of subject matter knowledge distinguished in the diaries

 1

	Common content knowledge	Specialised content knowledge	Horizon content knowledge
	(66; 25 %)	(158; 59 %)	(46; 17 %)
	 Giving a list of concepts (20)	 Discussing representations of	 Discussing the hierarchical
	 Giving a definition (11)	mathematical content given in	relationship of
	 Giving a theorem (9)	textbooks (72)	mathematical concepts
	 Explaining a property of a	 Discussing alternative	(28)
	mathematical entity (8)	representations of mathematical	 Giving an application of a
	 Giving a solution strategy (8)	content (65)	mathematical entity or
	 Giving alternative definitions	 Discussing relationship	method (18)
	(7)	between mathematical
	 Discussing the nature of	knowledge and curriculum (11)
	mathematical knowledge (6)	 Going through some history of
	 Giving a mathematical	mathematics (6)
	example (2)	 Giving and discussing
		matriculation examination tasks
		(5)
		 Reflecting on a mathematical
		example (2)
		 Modifying an example (1)

Table 2 : Subcategories of pedagogical content knowledge distinguished in the diaries

 2

Proceedings of CERME10

Acknowledgment

This research was supported by the Jenny and Antti Wihuri Foundation.


domain: math shs

text:
Jennie Golding
email: j.golding@ucl.ac.uk

Subject professional association activity: What can it offer teachers of mathematics and their students?

Keywords: Professional association, professional development, mathematics, affect. Thematic Working Group 20

Professional association activity is commonly regarded as a professional 'good', yet there remains little systematic evidence of its impact. This paper reports on a small study that asked English teachers of mathematics participating in such activity what contribution they believe it makes to the development of their knowledge, skills and affect, and how that then impacts on their students. Participants claimed a range of significant and pervasive benefits, many of which are distinctive to this form of professional development. These include a renewed commitment to their role as teachers of mathematics, refreshment and inspiration, and a deep and lasting impact on both their own learning and that of their students.

Background

Within mathematics education in the UK, there are four national 'classroom-facing' PAs, funded entirely from membership and each attracting up to several thousand members: The Mathematical Association (MA), the Association of Teachers of Mathematics (ATM), the National Association for Numeracy and Mathematics in Colleges (NANAMIC), and NAMA, the National Association for Mathematics Advisers. All have a core purpose of supporting the teaching and learning of preuniversity mathematics through working with teachers and others.

Their annual conferences, residential for most, offer sessions that might focus on mathematics pedagogic knowledge or skills, learner enrichment or teachers' own mathematics enhancement and/or enjoyment, or mathematics curriculum, leadership or assessment issues. There is time for networking and also for social activities. Conferences are usually held in teachers' holidays and teachers who attend often fund themselves, so they clearly value what such activity offers. Between them the PAs also offer a range of day conferences, bespoke courses, professional periodicals, local groups, policy debate, and a variety of social media opportunities, so building professional communities of up to national scale. Additionally, the larger ones operate working days and weekends when resources are developed for publication, and [START_REF] Hodgen | Reflection, identity and belief change in primary mathematics[END_REF] suggests this in itself can develop teacher reflection, knowledge and hence practice. Teachers can therefore personalize the extent and type of their involvement and the professional development (PD) targeted. I adopt [START_REF] Cobb | Cognitive and situative learning perspectives in theory and practice[END_REF] conceptualization of teacher PD as any planned experience intended to develop teachers' professional functioning (for the ultimate benefit of their students' learning)and that process as both enculturation and construction. We know something of what makes PD effective, for example that it has a content focus, is coherent with teacher's prior learning and needs, active, sustained, features collective participation (e.g. [START_REF] Desimone | Improving impact studies of teachers' professional development: towards better conceptualizations and measures[END_REF]. [START_REF] Golding | Mathematics teacher capacity for change[END_REF] suggests that for sustained development it should develop positive work-related identity and affect, as in [START_REF] Hodgen | Reflection, identity and belief change in primary mathematics[END_REF] and [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF] respectively, including self-efficacy, resilience, enthusiasm and feeling valued. The importance of effective PD, job satisfaction and recognition to teacher retention is also well evidenced [START_REF] Lynch | Engaging Teachers: NFER Analysis of Teacher Retention Slough: NFER[END_REF]: critical when, as in England, there is a shortage of effective teachers of mathematics at all levels (Ofsted, 2012).

There are, though, some sizeable gaps in the PD literature, e.g. there is limited evidence of the impact on students' learning [START_REF] Joubert | Researching CPD for teachers of mathematics: a review of the literature[END_REF], and the affordances of online activity continue to change. Further, I can find no systematic study of the contribution professional association (PA) activity can make to teachers ' development, though Chetwin (2010) suggests there are likely to be gains from networking, growth of knowledge and/or skills, and taking responsibility for one's development. This small study therefore investigated the contribution of PA activity as perceived by participant English teachers of mathematics (n=185). It asked  How is PA activity aligned with what is known about effective PD?  What contribution can PA activity make to teachers' professional development, and to the development of their students?  Is any of that distinctive?

The study I conducted semi-structured individual interviews with a purposive sample of four anonymized participants from each of the four 2016 PA annual conferences (Table 1), drawing on a variety of phase/experience/PA background from those currently, or recently, active in the profession. Questions (Figure 1) offered opportunity for development of a grounded account [START_REF] Charmaz | Constructing grounded theory: a practical guide through qualitative analysis[END_REF].

All 2016 conference participants were invited to complete a (usually online) questionnaire designed in part to validate interview responses with a wider sample. The interview questions were complemented by Likert-style questions (Table 2) designed to elicit the value attributed to aspects of conference activity known to be particular to this form of PD. Additionally, I scrutinised documents and publications available on PA websites or at conferences. Data consisted of questionnaire responses (n=185), transcriptions of recorded interviews (n=16), and my notes from documents scrutinized. There is no claim to generalisation from this highly selective sample, though the study shows clearly the breadth and depth of perceived impact on some teachers.

Grounded analysis of all qualitative data was by open, axial and selective coding [START_REF] Charmaz | Constructing grounded theory: a practical guide through qualitative analysis[END_REF]. Documentary evidence was used to validate participant claims, and interviewees validated all written interpretations of their talk; additionally, a colleague acted as a 'critical friend' -particularly important given I have a history of involvement in PA activity myself.

Throughout, though, the study is framed by its reliance on teacher accounts. The status of such claims has been contested: do teacher narratives represent warranted true belief, and if not, to what extent can they be represented as 'truth'? This issue is addressed in the literature, though [START_REF] Desimone | Improving impact studies of teachers' professional development: towards better conceptualizations and measures[END_REF] argues concerns can be over-stated in relation to accounts of PD; I adopt here [START_REF] Doyle | Heard any good stories lately? A critique of the critics of narrative in educational research[END_REF] position applied to teacher development, that the study aims to develop understanding of a highly contextualised and personal phenomenon, through access to participants' stories of intentions, motives, purposes and perceptions of effectiveness, rather than a universally knowable phenomenon susceptible to legislation through policy.

Findings

Coding exposed themes around identity and values, specific gains for teachers and/or students, strengths and limitations of professional association activity, including some apparently distinctive benefits, and threats or disincentives to that. I consider each of those in turn. Whilst it was not possible to enact either questionnaires or interviews in precisely comparable ways across the four associations, I argue that differences did not fundamentally influence the nature of responses.

Identity and values

Interviewees were keen to talk about impact on their professional identity or values, often in terms of affirmation, empowerment and meeting with like-minded people who support and challenge them professionally. For 11 of 16 this was their first focus in response, often centred around face to face participationa sense of community, sometimes built up over years, and talk about refreshment and renewal, often contrasting that with the draining nature of teaching. Claims were often extravagant: "It's been a life-changer, it builds me up as a maths teacher so I can do a better job in the classroom" (Kim), and for four teachers this was specifically linked with retention:

PA activity helps me analyse and then be proactive about developing what I value. Hugely empowering, and … that keeps me in the profession despite the grinding demands. (Lara)

Here is about personal PD, affirmation, valueschallenge too, but support for…your long-term growth and enablement, that enables you to go back refreshed and keeps you committed to what can be a …very draining profession -I just couldn't stay in my job long-term without that injection of positivity and recharging. (Billy)

For interviewees with a background in Further or Adult Education, this identity work was talked about in even more fundamental terms:

In FE, very often you don't even see other teachers of maths... So NANAMIC gives you that identitythere are other people out there struggling with you, valuing some of the same things as you do -otherwise you're just functioning in isolation, far too often. (Sally)

Specific gains for teachers and/or students

Interviewees commonly (10 of 16) talked about the high quality of PA publications, sessions and resources in terms of direct benefit to themselves and to their students:

Supportinspiration -resources: I return brimming over with ideas and enthusiasm, with knowledge about innovations across the country, catholic ideas and approaches that have worked in different circumstances. The resources are creative and engaging, they really probe deep understanding and the students love them. (Terry)

Often the benefit was claimed to spread beyond the interviewee concerned:

I've worked with teachers using these materials and boy are they effective. If they can make the right selection and the right tweaks, and we work on that, then they see real and immediate impact on learning. (Graham)

Many respondents (25 open questionnaire responses and 7 interviewees) greatly valued informal networking opportunities, claiming explicit benefits also to their colleagues and studentsboth immediate and also for sustained learning and positive disposition towards mathematics:

The specific numeracy ideas, I took them straight back to my classroom and my students are already showing the benefits, in a couple of weeksto confidence as well as skill. There are also 'seed' ideas, things that ...will come to fruition over a longer timescale. (Susan)

For some (seven interviewees and over 25 open questionnaire responses), the opportunity to be better informed about, and contribute to, national policy debate is valued; for others (in 6 interviews and some 15 questionnaires), the chance to engage with cutting-edge research relevant to their practice and reflect on its application is important. Teachers who engage in local branch meetings claim similar, but less extravagant, benefits. Questionnaire open responses were generally consistent with these interview response strands.

Strengths and limitations of PA activity

As well as the specific benefits to professional skills and knowledge, and to professional affect and identity, teachers identified the eclectic nature of professional association activity, and the fact that they can easily personalise it to their own professional needs, as underpinning its effectiveness. Many described it as 'uniformly high qualitythe best professional development I get' or similar. It was often reported as having long-term benefits for both teachers and their students, sometimes in contrast to other courses which "focus on short-term skill or particular knowledge." (Janet).

Five interviewees talked about the benefits of being physically removed from their work environment and the luxury of sustained unhindered time committed to their professional growth. Several teachers described the desirability of also participating in institution-based development alongside colleagues, with access to familiar resources, and with whom they could contextualise new ideas. In three interviews they extolled the particular advantages of engaging in branch or conference activity with at least one colleague. On the other hand, teachers said they found distance learning can be effective and efficient for pure dissemination of information. Four identified PA activity as often limited by a 'light touch' for more substantial knowledge or skill development, particularly where there was a need for substantial subject or subject pedagogical knowledge, perhaps better provided in a series of inputs interspersed by classroom embedding. This was true in particular for two of the one-day NANAMIC conference participants. One commented:

And of course then you go back into college and there's no-one to share it with…so unless you're really committed, those interesting ideas and good intentions might well get lost. (Angela)

However, all of those interviewed and virtually all of those completing questionnaires identified face to face PA activity as a central and rich component of their effective impact on students. These comments were echoed in questionnaire responses, though typically in less depth. Questionnaire responses added no significantly different responses.

Likert scale items in questionnaires largely concentrated on features of conferences. Table 2 shows mean response, on a scale of 1 (of little importance to me) to 5 (very important to me), together with standard deviations s. As quantitative data it is of limited robustness but gives some indication of the ranking of different aspects, similar but not identical for the different conferences. For the range of participants, working with others from a variety of roles and experiences is highly valued, as are opportunities to engage with new ideas or mathematics. These teachers also value opportunities to construct a programme that meets their individual needs.

Threats or disincentives to such activity

There are, though, some clear threats to participation, of which funding was mentioned by nearly all interviewees. Although "cheaper for several days of exceptionally high quality development than many mediocre commercial courses" (Rachel), teachers routinely talked of schools and colleges prioritising performance-framed one-off courses for funding, and leadership teams not valuing the "deeper, wider learning that is supported by face to face PA opportunities" (Janet). Others said that colleagues "thought they were mad to spend holiday time at a …conference when there are so many pressures during term time you just want to curl up and die when you get to a holiday" (Kathy). Four interviewees claimed their schools/colleges would not pay for conference attendance because a betterinformed teacher was more likely to move. Funding was a particular issue for those working in FE, with a majority of those respondents reporting little or no employer support for subject-specific development, so no choice but to fund such development themselves.

Four interviewees suggested that Primary or FE teachers without a strong mathematical background or a specific mathematics responsibility were unlikely to prioritise, or be confident to participate in, subject specific and self-funded professional development. They suggested incentives for Primary teachers to 'bring a local friend' might increase both confidence and impact, and identified day conferences as a good first step "where it's often desperately needed" (Janet).

Discussion

It is important to note that there is no claim to generalisability here: these are teachers in an English education culture who choose to attend these conferences in their own time, and often self-funded. They are therefore highly committed to their own development as teachers, but also claim that they gain motivation and energy for their work from PA activity, often in contrast to other opportunities available to their context. It is striking that almost all interviewees privilege talk about values, affirmation of professional identity, and improved self-efficacy in their accounts, together with deep, wide and reflective mathematical (subject and subject pedagogical) learning for the long-term effective exercise of their professional role. They commonly contrast that with much external PD and often generic local provision. With both preservice and inservice education in England increasingly adopting generic rather than subject-specific approaches, [START_REF] Joubert | Researching CPD for teachers of mathematics: a review of the literature[END_REF] show such subject-specific opportunities are central to the development of a deeply effective teaching profession.

The benefits described align well with [START_REF] Desimone | Improving impact studies of teachers' professional development: towards better conceptualizations and measures[END_REF] and [START_REF] Golding | Mathematics teacher capacity for change[END_REF] criteria for effective PD: showing a clear content focus, active and coherent with teacher's prior learning needs, featuring collective participation, and contributing very positively to teachers' affective and identity needs. PA activity can be sustained (sometimes over years) in the sense of offering longitudinal stimulus interspersed with everyday teaching, but not usually in the sense of a critical mass of hours focused on particular knowledge or skills, for which other avenues would appear to be more effective. The benefits claimed for teachers, and for their students, are significant, deep, wide-reaching and longlasting, including a renewed commitment to retention in the profession. There is no a priori reason why such benefits should not be experienced by far greater numbers, and it is important that perceived threats to participation are addressed. These are not just about funding, but, as in [START_REF] Lynch | Engaging Teachers: NFER Analysis of Teacher Retention Slough: NFER[END_REF], about the value teachers perceive management to give to PA activity and to teachers' PD beyond the short term specific needs of their school/college.

All teachers in this study were teachers of mathematics, but in the English context there are comparable professional associations in other curriculum areas, and an obvious question is whether the benefits cited here, particularly in relation to subject-related identity, apply also to them.

Benefits distinctive to professional association activity

Some distinctive benefits of PA activity appear to emerge. First, there is the opportunity to mould a PD programme to one's own professional needs, whether in terms of reading, resources or face to face development. Many teachers also referred explicitly to working with mathematicsfor its own sake as well as for possible classroom benefits. In England, in contrast to many other jurisdictions, this is unusual after initial qualification, yet this aspect of PA activity was commonly highly valued.

There appears to be a great deal of professional affirmation, networking and identity work taking place. Some teachers particularly value the access to recent research offered by the PAs, and/or the informed policy work facilitated by PAs and based on professional discussion, and I would argue it is healthy for policy systems to be informed by such knowledge and discussion. Finally, through the PAs it is relatively easy for teachers to be able to offer something back to the PD of others, whether through writing about their professional work, giving a session at a conference, disseminating, receiving critique and discussing in a varied but informed peer group, producing resources or developing courses, or engaging in writing work that responds to deep desires to improve mathematics learning. This is itself is recognized to be developmental:

Writing for journals has been very developmental, and the support you get for doing that. Very high quality writing sessions: a richness of ideas from people whose ideas have since shaped my practice, that I admire and aspire to. (Charles)

There are, though, other providers of different subsets of the cited activities, and it is an open question how the affordances and constraints of such provisions compare with those of PA activity.

Conclusions

These English mathematics teachers claim a wide range of benefits from PA activity (especially face to face events), some of which are perceived as either exclusive to such activity or most effectively provided by it. They say it gives them deep and wide professional learning which impacts on their students both through specific pedagogical tools and approaches, and through teacher refreshment and re-commitment. They claim an affirmation of their professional identity through sharing goals and values with others, and increased self-efficacy through peer validation and personalisation of development. Teachers value the subject-specific nature of PA activity, that in the English context contrasts with much school-based, typically generic, development. They appreciate the range of ideas and ways of thinking that far exceed what is available within one school/college or group of institutions. For many, these claimed benefits have been developed and sustained over years.

The study offers evidence to management and policy makers about the value these teachers of mathematics place on subject-specific development that affirms their professional identity and their values, recharges and renews their commitment and enthusiasm, and engages them actively in deep and reflective subject and subject pedagogic learning. With current performance pressures and limited budgets, it is not surprising English schools and colleges often privilege perceived immediate curriculum needs, but the cost of PA activity outside teaching time is small when compared with the costs to students of a stale and drained teacheror no teacher at all. There is a need to identify politically-acceptable ways to invest in teachers' longer term subject-specific development, so that more teachers are encouraged to participate in such activity.

Table 1 : Interviewees

 1

	Source	Pseudonym and teaching context	PA activity(years)
	ATM	Alice (11-18, special education), Lara (11-18), Terry (11-18), Billy	6/8/ 2/'many'
		(11-18, Higher Education, Adviser)
	MA	Jackie (5-11, 11-18, Higher Education), Janet (5-11, Adviser), Kim	41/35/3/10
		(11-18), Rachel (5-11)
	NAMA	Charles (11-18, Higher Education), Gail (5-11, Higher Education),	16/20/16/'many'
		Graham (11-18, Adviser), Kathy (11-18)
	NANAMIC Sally (11-18, 16+), David (11-18, adults), Angela (16+), Susan (11-18,	'Many'/0/'12/10
		16+)
		Figure 1: Interview structure

Table 2 : How important are the following aspects of the conference to you?

 2

	Association (responses from active teachers or	NANAMIC	NAMA	MA	ATM	Overall (185)
	those actively working with teachers)	(n=10)	(n=29)	(n=56)	(n=90)
	Meeting people in comparable roles	4.2	3.7	4.3	4.2	4.2 (s=0.7)
	Face to face rather than at a distance	4.4	4.5	4.3	4.3	4.4 (s=0.7)
	Meeting people from other phases in education
	or with different roles or from different areas of	4.2	3.9	4.7	4.7	4.6 (s=0.6)
	the country
	A mix of beginners and experienced colleagues	4.4	4.0	4.5	4.6	4.5 (s=0.7)
	Sessions that are grounded in the classroom	4.1	3.4	4.4	4.2	4.1 (s=0.5)
	Social activities	-	2.3	3.3	3.9	3.5 (s=1.1)
	Immersion -it's residential	-	2.3	4.5	4.3	4.0 (s=0.7)

Proceedings of CERME10


text:
Gonzalo Espinoza-Vásquez
email: gonzalo.espinoza.v@gmail.com

Diana Zakaryan
email: diana.zakaryan@pucv.cl

José Carrillo Yañez

Use of analogies in teaching the concept of function: Relation between knowledge of topics and knowledge of mathematics teaching

Keywords: Function concept, mathematics teachers' specialised knowledge, analogies. Thematic Working Group 20

This paper addresses the knowledge about the function concept and the knowledge on how to teach this concept using an analogy. The analysis of an episode of one lesson in which the analogy between a washing machine and the concept of a function is shown allows identifying specialised knowledge about the function concept and teaching strategies. The study findings reveal links between knowledge of the topic and knowledge of mathematics teaching, permitting identification of potentialities and limitations of the analogy used.

Introduction

Function is one of the most important concepts in mathematics, fundamental for development of mathematical analysis and mathematics in general [START_REF] Ponte | The history of the concept of function and some educational implications[END_REF]. This concept is not only present in many areas of mathematics, but is prevalent in the school curriculum and is, of course, studied as a part of the mathematics teacher training programs. Given its importance and great complexity [START_REF] Dubinsky | The Nature of the Process Conception of Function[END_REF], it is essential that the mathematics teacher's knowledge considers both the discipline-specific knowledge of the function concept, as well as knowledge about how it is taught and learned.

Teachers' knowledge has been widely studied from different perspectives, using a wide range of theoretical models (e.g., [START_REF] Shulman | Those who understand: knowledge growth in teaching[END_REF][START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF][START_REF] Carrillo | Un marco teórico para el conocimiento especializado del profesor de matemáticas[END_REF]. The Mathematics Teacher's Specialised Knowledge (MTSK) model is presented both as a conceptualisation for mathematics teachers' knowledge and as an analytical tool for acquiring this knowledge [START_REF] Flores | Oportunidades que brindan algunos escenarios para mostrar evidencias del MTSK[END_REF]. Studying the subdomains and categories proposed by the MTSK model, and their relationships, allows us to advance the understanding and analysis of teachers' knowledge [START_REF] Sosa | Conocimiento del profesor acerca de las características de aprendizaje del álgebra en bachillerato[END_REF]. Part of this knowledge is related to the depth of the teacher's understanding of a concept (in this case the function) as a mathematical concept and as an object of teaching. Several studies conducted to date focused on the understanding of the function concept on students, pre-service teachers, and practicing teachers (e.g., [START_REF] Even | Subject matter knowledge for teaching and the case of functions[END_REF][START_REF] Breidenbach | Development of the process conception of function[END_REF]. Other studies highlight the importance of fully understanding the concept (e.g., [START_REF] Sierpinska | On understanding the notion of the concept of function[END_REF], identification of its representations (e.g., [START_REF] Even | Subject matter knowledge for teaching and the case of functions[END_REF], as well as difficulties in learning (e.g., [START_REF] Dubinsky | The Nature of the Process Conception of Function[END_REF] and relating their different representations (e.g. [START_REF] Ponte | The history of the concept of function and some educational implications[END_REF][START_REF] Figueiredo | Concepción de función: definición, presentación y ejemplificación en la enseñanza y aprendizaje[END_REF]. The focus of the present investigation is on the relationship between teachers' knowledge about the function concept and the knowledge about its teaching from the perspective of the MTSK, particularly when a teacher uses an analogy to make the function concept understandable. In this study we ask what knowledge about functions and their teaching can be inferred from the use of an analogy? How are these knowledge types related?

Analogies

According to [START_REF] Treagust | Science teachers' use of analogies: observations from classroom practice[END_REF], analogy is achieved through a comparison of structures in two distinct domains, one of which is familiar (source or analogue), while the other is unfamiliar (target).

An analogy refers to comparisons of structures between domains. An analogy is a relation between parts of the structures of two conceptual domains and may be viewed as a comparison statement on the grounds that these structures bear some resemblance to one another. (Treagust et al., 1992, p. 413) The use of analogies in teaching, particularly as a didactic strategy in the teaching-learning process, has been extensively studied (e.g., [START_REF] Duit | On the role of analogies and metaphors in learning science[END_REF][START_REF] Treagust | Science teachers' use of analogies: observations from classroom practice[END_REF]. For example, [START_REF] Curtis | The use of analogies in written text[END_REF] analysed instructional text and provided a classification of the ways in which the relationship between the source and the target domain is established. According to the authors, the relationship can be (1) structural, referring to physical similarity or similar construction; (2) functional, referring to the way of functioning of both structures; and (3) structural-functional, formed by combining the previous two. They add that analogies occur in two forms: verbal and pictorial-verbal, whereby the former is achieved solely via the use of words, while in the latter words are complemented by an image.

Teachers tend to produce analogies automatically when answering questions or explaining the concepts they are teaching (Ünver, 2009). According to [START_REF] Figueiredo | Concepción de función: definición, presentación y ejemplificación en la enseñanza y aprendizaje[END_REF], function as a machine is an example of such analogies and this comparison will show only some aspect of the concept. Function can be descripting operationally as a computational process or structurally as a set of ordered pairs [START_REF] Sfard | On the dual nature of mathematical conceptions: reflections on processes and object as different sides of the same coin[END_REF]. To present the function trough this analogy conducts to understand the concept as an input-output process in an operational way.

Mathematics Teacher's Specialised Knowledge

The MTSK model (Figure 1), in the spirit of contributions by [START_REF] Shulman | Those who understand: knowledge growth in teaching[END_REF] and [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF], proposes, within the teacher's knowledge, a discipline-specific component (MK, mathematical knowledge) and a didactic component (PCK, pedagogical content knowledge). A further component related to beliefs about mathematics and about teaching and learning it is introduced in the middle of the model.

According to [START_REF] Carrillo | Un marco teórico para el conocimiento especializado del profesor de matemáticas[END_REF], MK corresponds to knowledge specific to the discipline being taught, and comprises of three subdomains: Knowledge of Topics (KoT), Knowledge of the Structure of Mathematics (KSM), and Knowledge of the Practice of Mathematics (KPM). KoT considers the phenomenology, definitions, properties, procedures, and foundations of the topic, as well as the ways of recording and representing it. On the other hand, KSM pertains to the conceptual connections among mathematical concepts, relating a concept to prior contents (simplification), later contents (adding complexity), or to contents with a common property (transverse connections), and the auxiliary connections among objects. Finally, KPM is related to knowledge about the characteristics of mathematical work, namely how to proceed and create knowledge in mathematics, practices linked to mathematics in general, and practices linked to a specific topic. PCK corresponds to didactic knowledge specific to teaching work in the process of teaching and learning mathematics. Once again, it comprises of three subdomains, namely Knowledge of Features of Learning Mathematics (KFLM), Knowledge of Mathematics Teaching (KMT) and Knowledge of Mathematics Learning Standards (KMLS). KFLM considers teachers' knowledge of their students' learning styles, strengths and difficulties associated with learning, way of interacting with mathematical content, students' conceptions of mathematics, and personal or institutional theories of mathematics learning. KMT is knowledge about mathematical content conditioned by its teaching, including knowledge about personal or institutional teaching theories, physical and virtual resources, and strategies, activities, examples, and help. Finally, KMLS pertains to knowledge about required mathematical concepts to be taught, knowledge about the level of conceptual and procedural development expected, and the sequencing of the various topics.

The aforementioned division into subdomains allows us to deepen our understanding of the elements of knowledge that are utilised in an integrated and interconnected manner. The MTSK model is a suitable analytical tool for meeting the objective of the present study because, in addition to highlighting mathematics, its categories, and subdomains, it allows focusing on teachers' knowledge about the function concept and its teaching (for example, the definition, its properties, representation, and the strategies used when teaching).

Methods

This research is grounded in an interpretative paradigm and is based on the instrumental case study design [START_REF] Stake | Investigación con estudio de casos[END_REF]. The aim is to investigate from the perspective of MTSK the knowledge manifested by a high-school teacher when teaching the concept of function. The teacher that is in the focus of the study, henceforth referred to as Arturo, has ten years of teaching experience, teaching classes from fifth to twelfth grade. He is also a university teacher in first-year classes for engineering students and for pre-service teachers. He has also worked as a teacher in continuing education courses for primary-school teachers and has taken courses connected to university teaching, curricular updates in geometry, and curricular reform. At the time of this study, Arturo was teaching ninth-grade classes, where he had planned to introduce the function concept. According to the information provided by Arturo, his group of students is familiar with the use of algebraic language, equation solving techniques, and Cartesian plane, among others.

To collect the data, classes in which Arturo planned to introduce the function concept were observed and video-recorded. The videos were transcribed and the transcripts served as the principal source of information. The resulting data was subjected to content analysis [START_REF] Bardin | El análisis de contenido[END_REF], whereby class episodes were determined according to the tacit or explicit goals of the teacher. The units of analysis correspond to the teacher's interventions and responses provided by his students. In addition, only those that present evidence of teacher's specialised knowledge have been considered [START_REF] Flores | Oportunidades que brindan algunos escenarios para mostrar evidencias del MTSK[END_REF]. In the first class, Arturo introduces the concept of function and its definition, and provides some examples of functions. An episode was selected from this class, in which the teacher uses an analogy to promote students' understanding of the definition of function. The episode was analysed in relation to the KoT and KMT sub-categories of the MTSK.

Results and discussion

Arturo defined function as a correspondence between elements of two sets in which each element of the input set corresponds to a single element in the output set. Knowledge of this definition is part of his KoT. To make this definition understandable, Arturo introduces a washing machine as an analogy for a function, alluding to a family context for his students. To know this context is not part of the MTSK, but it allows us to reflect on the scope and applicability of the analogy according to the type of students to whom they are presented. In the following extract, Arturo presents the analogy between function and washing machine: Knowledge of analogies, as elements that enhance the teaching of a concept, is part of the teacher's specialised knowledge [START_REF] Carrillo | Un marco teórico para el conocimiento especializado del profesor de matemáticas[END_REF]. The use of analogy shows teacher's knowledge about when to give any specific help to his students (KMT). The use of analogy favours the understanding and visualisation of abstract objects in students, besides being a motivation for a new theme. This analogy presents the function as a process and allows the students to better understand this concept [START_REF] Sfard | On the dual nature of mathematical conceptions: reflections on processes and object as different sides of the same coin[END_REF][START_REF] Figueiredo | Concepción de función: definición, presentación y ejemplificación en la enseñanza y aprendizaje[END_REF]. Moreover, different components can be identified, namely domain, co-domain, pre-image, and image explaining the connection between the source and the target domain. The analogy is presented in two formats: in the intervention described (verbal) and when the teacher draws a washing machine on the whiteboard (pictorial). This illustration (Figure 2a) shows the function as an input-output process, in which the object that enters is modified (in this case, a dirty article of clothing comes out clean). In this example, the objects are the same at entry and exit (Tshirts in both cases). Other analogies for this concept can relate objects of different nature. For example, function can be represented as a coffee dispensing machine into which money is entered in order to obtain a cup of coffee, highlighting the arbitrariness of related sets [START_REF] Even | Subject matter knowledge for teaching and the case of functions[END_REF]. In this sense, the analogy of the washing machine impedes the association of arbitrary sets, which may result in students gaining a partial understanding of the concept. Likewise, the washing machine does not show other conceptions for the function as, for example, the co-variation of magnitudes. Similarly, it does not facilitate representation of more complex functions or the complexity of the concept itself (function algebra, composition).

Arturo takes advantage of this relation between input and output to clarify the definition of the concepts of image and pre-image, which are parts of his KoT. Similarly, evidence of his knowledge about the domain and co-domain of the function appears in the analogy as "dirty clothes" and "clean clothes." This knowledge and the exposed relations between the source and target domains account for the use of analogy as a strategy for teaching the function, evidencing a relationship between its KoT and its KMT. Figure 2a shows Arturo's knowledge of the notation f(x) = y (knowledge about representations as a part of his KoT) that allows him to show the relationship between two domains of the analogy (as knowledge of strategies -KMT) and to introduce new ways to represent the function.

In the following excerpt, Arturo explains the relationship between the linked domains, source and target, using the analogy, in which we interpret the structural and functional character of the analogy presented [START_REF] Curtis | The use of analogies in written text[END_REF].

Arturo:

In our context, our function was the washing machine, washing. Set A would be dirty clothes and Set B clean clothes. If this is our washing machine, and it carries out its function; dirty clothes go in, and how do they come out?

Student:

Clean! Arturo:

The same as what we did here. The function was applied to this kind of T-shirt that I drew that was dirty. What will it equal?

Student: Clean, clean clothes.

Arturo:

The same T-shirt, but clean. These two elements also have names. This element here is called the "image" of what I sent in. [...] And these elements here are called "pre image." What is the clean T-shirt?

Student:

Image.

Arturo:

The image of what?

Student:

Of the dirty T-shirt.

The structural characteristic is shown in establishing the correspondences of the Set A with the dirty clothes, the Set B with the clean clothes, and the washing machine with the arithmetic process carried out by the function. That is, the structure of the laundry process is analogous to the evaluation process in the function. When Arturo presents the analogy "function as a washing machine," he also refers to its functional character, as he establishes a comparison between the operation of the machine and the function. After this intervention, Arturo represented verbally and as an algebraic expression (representations in his KoT) an example of function (Figure 2b), thus deepening the analogy between machine and function.

Arturo: With numbers, the function isn't going to do the washing. It's going to add two to whatever comes in [he writes "f(x) = x+2"]. Whatever comes into the function, to the machine, I add two to it. If this is my machine that adds two to whatever comes in, if a one enters, how does it come out? [Student: Three.]

It should be noted that, when Arturo teaches "with numbers," he aims to work in the target domain of functions. Consequently, the situation created becomes an example (knowledge of examples -KMT) that will allow him to present the functional characteristic of the analogy and propose a two-way process for understanding the relationship between a function and a washing machine (Figure 2b).

The relationship between an algebraic expression (function) and a washing machine presents functions as an input-output process, meaning of the concept that we consider part of his KoT. In the same way, the articulation and selection of representations for the function (the analogy, f(x) = y, algebraic expression and natural language) accounts for Arturo's KoT, relating to his KMT.

In this last intervention, by associating a function with a machine, Arturo highlights the process role of a function (part of his KoT). We do not have evidence of Arturo highlighting the object role of a function supported by the analogy studied here, although the structural character of the analogy may be the first approach to this conception.

Conclusion

According to the results yielded by the analyses presented above, the relations between the teacher's KoT and KMT are demonstrated in the articulation between knowledge of representations of the function and the choice and use of these representations as examples and analogies for teaching the Thematic Working Group 20 concept. In addition, the choice of analogies and examples given by Arturo reflect his KMT, which is nurtured and influenced by his KoT. The analogy shows the function as a correspondence in coherence with the definition given, while also permitting articulation of different representations: sagittal and algebraic diagram (Figure 2b). They also allow the teacher to produce other representations as a Cartesian graph or a table of values, thereby expanding the concept image for the students. Likewise, the analogy allows the students to appreciate the univalence character of the function [START_REF] Even | Subject matter knowledge for teaching and the case of functions[END_REF] and to extend the range of situations in which the concept of function is present (phenomenology -KoT).

The analogy utilised in this case is suitable for teaching and learning given that it is connected to the students' prior everyday experience. Moreover, its functional and structural character allows them to understand the concept from different perspectives and using different representations. Establishing a bidirectional relationship between the function and the washing machine can allow students to utilise the concept of a function they have learned to identify the concept in other areas of mathematics and other areas of knowledge. It should be noted that, in spite of the benefits presented by this analogy, students may have a partial notion of function if the conception of function as a machine is maintained. Thus, it is essential that other aspects of the concept be highlighted, such as the arbitrariness of the sets involved [START_REF] Even | Subject matter knowledge for teaching and the case of functions[END_REF]. It is also important to relate it to other representations [START_REF] Figueiredo | Concepción de función: definición, presentación y ejemplificación en la enseñanza y aprendizaje[END_REF]. Lack of articulation between representations can cause a limitation in the development of the conception of function from the structural perspective [START_REF] Sfard | On the dual nature of mathematical conceptions: reflections on processes and object as different sides of the same coin[END_REF][START_REF] Breidenbach | Development of the process conception of function[END_REF]. On the other hand, this presentation of the concept may constitute an obstacle to learning, for example, the algebra of functions or its composition (How do I sum two machines?). Likewise, students' direct experience with the laundry process can be detrimental in understanding the function: "the function did not fulfil its function." However, it is not the purpose of the study to evaluate the methodological proposal or Arturo's knowledge, but rather to approach it with the intention of improving our understanding of such knowledge, that of subdomain relationships in particular. In that sense, the analysis using the MTSK model is useful for advancing this understanding.

Figure 1 :

 1 Figure 1: Sub-domains of the MTSK model (Carrillo et al., 2014)

  more names, the function works like a kind of machine. An example could be a washing machine. A washing machine carries out a function. What is its function? do? You take an article of clothing. It's dirty. You put it in the washing machine. How does it come out? machine fulfil its function? Yes. The dirty article of clothing would be a member of the input set, and the clean article of clothing would be a member of the output set. This is what the function does. Here [he points to a diagram] we would have the dirty article of clothing. The function does what it does, depending on the machine, and arrives at the other side. In the case of a washing machine, it arrives clean.

Figure 2a :

 2a Figure 2a: Presentation of the analogy. 2b: Relationship between the source and the target domain.

Proceedings of CERME10

f: washing machine A: dirty clothes B: clean clothes

Acknowledgments 2015 Conicyt Doctorate Grant (21151243); 2016 Conicyt Complementary Grant (R.E.: 5152/2016); Proyect EDU2013-44047-P, Ministry of Economy and Competitiveness (Spain)


text:
Adriana Breda
email: adriana.brea@ulagos.cl

Luis R Pino-Fan

Vicenç Font
email: vfont@ub.edu

Valderez Marina
email: valderez.lima@pucrs.br

Rosário Lima

Didactic assessment over a final work of a master for mathematics teachers in service

Keywords: Didactic assessment, suitability criteria, masters thesis

. As a result of the analysis it was possible to notice that the teacher, in addition to using all didactical suitability criteria proposed by the OSA implicitly, highlights the importance of finding a balance among the suitability criteria to achieve the learning of the didactic proposal carried out by him.

Introduction

The tendency to achieve an international convergence in the planning of College programs and, particularly, those related to the Professional Master education centered on the education of teachers, has fostered a series of reforms in different countries, so that there is a model organized by a sort of refinement and evolution around professional skills. In the Brazilian context, in an attempt to train Mathematics teachers who were currently working in the area, the Professional Master Program in Mathematics in the National Network (PROFMAT) was launched in 2010 by recommendation of the Conselho Técnico-Científico da Educação Superior da Capes. The program is an on-site and long-distance program, offered throughout the Brazilian territory, coordinated by the Sociedade Brasileira de Matemática (SBM), which has a main objective to support Mathematics teachers who work in the primary education level, especially in public schools. It is important to highlight that, although PROFMAT's main objective is to foster teaching of mathematics at all levels [START_REF] Brasil | Uma análise quali-quantitativa de perfis de candidatos ao Mestrado Profissional em Matemática em Rede Nacional (PROFMAT)[END_REF] it is configured as a program composed almost entirely of mathematical disciplines. In addition, at the end of the course, the students must present a final work (ETM) consisting of design a sequence of tasks, not assigning the mandatory implementation.

The work presented in this document is part of a larger research (Breda & Lima, 2016;[START_REF] Breda | Análise das Propostas de Inovação nos Trabalhos de Conclusão de Curso de um Programa de Mestrado Profissional em Matemática[END_REF][START_REF] Breda | Establishing criteria for teachers' reflection on their own practices[END_REF]Breda, Pino-Fan & Font, in press), in which, through the analysis of 29 Master's Thesis Works (EMTs) of PROFMAT [START_REF] Breda | Melhorias no ensino de matemática na concepção de professores que realizam o mestrado PROFMAT no Rio Grande do Sul: uma análise dos trabalhos de conclusão de curso[END_REF] and was concluded that the teachers who implemented the design of their sequence of tasks performed a much more refined and balanced didactic analysis compared to the teachers who did not implement the proposal. In addition, it was evidenced that when the teachers' opinions were clearly evaluative, they were organized implicitly using some characteristics of the components of the didactical suitability criteria proposed by the Onto-Semiotic Approach (OSA) to mathematical knowledge and instruction [START_REF] Godino | The onto-semiotic approach to research in mathematics education[END_REF].

So, the objective of this paper is to present a case study that analyses what are the criteria used by a teacher in his reflection process (explained in his EMT of PROFMAT), who will be addressed as Mr. Lopes, in order to improve the design and implementation of new contents related to the Riemman integral in Elementary School.

Theoretical framework

In the field of Mathematics Education there is no consensus on the notion of "quality" and, in particular, there is no consensus on the "methods for assessing and improving the teaching and learning of mathematics". There are basically two ways to address this problem, from a positivist perspective or from a consensual perspective [START_REF] Font | Inicio a la investigación en la enseñanza de las matemáticas en secundaria y bachillerato[END_REF]. From the first, the scientific research in the area of Mathematics Education tell us what are the causes to be modified to achieve the effects as objectives to be achieved, or at least tell us what are the conditions and restrictions that must be taken into account to achieve them. From the consensual perspective, that tells us how to guide the improvement process of mathematics instruction, which must come from the argumentative discourse of the scientific community, when it is aimed at achieving a consensus on "what can be considered as the best".

The notion of didactical suitability criteria proposed by the Onto-Semiotic Approach (OSA) to mathematical knowledge and instruction (OSA, from now onwards) [START_REF] Godino | The onto-semiotic approach to research in mathematics education[END_REF] is positioned in the consensual perspective. Such notion is a partial answer to the following problematic: What criteria should be used to design a sequence of tasks to assess and develop mathematical competence of students and what changes should be made in its redesign to improve the development of this competition? Suitability criteria can first serve to guide the teaching and learning of mathematics and, second, to assess their implementation. Suitability criteria are rules of useful correction in two stages of the processes of mathematical study. A priori, the suitability criteria are the principles that guide "how things should be done". In hindsight, the criteria used to assess the study process effectively implemented. According to these authors, the didactical suitability criteria are: 1) Epistemic suitability, to evaluate if the Mathematics being taught are "good Mathematics"; 2) Cognitive suitability, to evaluate, prior to the beginning of the instruction process, if what is intended to be taught is at a reasonable distance from what the students already know, and after the process, if the knowledge acquired is any close to what was intended to teach; 3) Interactional suitability, to evaluate if the interactions contribute to clear doubts and difficulties students encounter; 4) Mediational suitability, to evaluate the adaptation of material and timerelated resources used in the process of instruction; 5) Emotional suitability, to evaluate the implications (interests, motivations,…) of students during the process of instruction; 6) Ecological suitability, to evaluate the adaptation of the instruction process to the educational project of the school, the curricular guidelines, the social and professional environmental conditions (Font, Planas & Godino, 2010, p. 101).

Methodological aspects

We chose to conduct a case study [START_REF] Ponte | O estudo de caso na investigação em educação matemática[END_REF] where the didactic analysis performed by a mathematics teacher in service, as part of his master degree, is investigated. To analyse our case, we used the indicators of didactical suitability proposed by the OSA [START_REF] Godino | The onto-semiotic approach to research in mathematics education[END_REF][START_REF] Godino | Indicadores de la idoneidad didáctica de procesos de enseñanza y aprendizaje de las matemáticas. XIII Conferência Interamericana de Educação Matemática (CIAEM-IACME)[END_REF][START_REF] Breda | A noção de idoneidade didática e seu uso na formação de professores de matemática[END_REF], as theoretical model to analyse the reflections performed by the teachers regarding ways to improve their teaching practices, related to the implementation of the didactical activities proposed as part of their EMT.

Research context

According to the guiding document of the PROFMAT, the EMT should, preferably, consist of a project with direct application to the mathematics classroom in Basic Education, thus contributing to the enrichment of the teaching of said discipline.

In this work we proceed on the assumption that the End of Master's Thesis (EMT) is a task that involves, implicitly, a didactical analysis exercise, since in their EMTs teachers must explain a didactical proposal and justify why it represents an improvement in teaching. In this sense, the reason for choosing the case of Professor Lopes is that, in addition to having applied the sequence of tasks with the students, he presents in his reflection aspects that "did not work" or that should be improved in future implementations.

Professor Lopes' didactical proposal

Professor Lopes' EMT (2014), entitled "A review of the introduction of Riemann's sums into High School Education", presents the design and the implementation of a didactic proposal for a group of third-year high school students (students aged 16 to 17) in order to intuitively introduce the integral calculus through the study of the areas of 2D geometric forms. [START_REF] Lopes | Um relato sobre à introdução às somas de Riemann na Educação Básica[END_REF] explains that it is possible to introduce methods and notions of the integral calculus in High School Education intuitively, starting with area-calculation problems for curvilinear shapes. That is, the aim is to broaden the calculation of areas habitually studied in Elementary Education through the study of area-calculation of curvilinear shapes, using both Archimedes' and Riemann's methods.

To be specific, Lopes' EMT ( 2014) is organized into four chapters; in the first, the professor presents, using literature reviews, the argument: "Should integral calculus be used in Elementary School?" On the basis of this question, the professor seeks to justify-through the study of literaturethe use of two methods: Archimedes' method (used to calculate the area of a circle) and Riemann's method (used to calculate the area of three curvilinear shapes: circles, eclipses and polynomial shapes with an x axis). In the second chapter, [START_REF] Lopes | Um relato sobre à introdução às somas de Riemann na Educação Básica[END_REF] explains the didactic unit which was implemented with a group of third-year students from a state secondary/high school in Brazil. The group was formed of 41 students but at the beginning of the year, only 36 students attended the classes and participated willingly in the project. In this second chapter, the professor also explains in detail the initial self-evaluation he performed with the students and, in particular, he explains the method for evaluation previous knowledge on certain geometry topics, on mathematical software knowledge and also on the expectations of the project.

In the third chapter, professor Lopes describe the implementation he carried out. This section is a sequential report in which the author explains what happened during the implementation of the didactic sequence, placing emphasis on the set tasks, what the students learnt and the interactions made during the implementation. We are looking at a review written from the perspective of the professor but, in his very review, the professor ensures he presents evidence of the statements he makes. In the fourth and last chapter, the professor presents his reflections and conclusions on the implementation he carried out. In this way, it can be said that Lopes' proposal (2014) covers the four phases of didactic design (preliminary study, design, implementation and evaluation), which other models of mathematics teachers' knowledge also cover, in order to answer the most fundamental question: "What knowledge should a mathematics teacher have to be able to appropriately manage their students' learning?" [START_REF] Pino-Fan | Towards a methodology for the characterization of teachers' didactic-mathematical knowledge[END_REF].

Professor Lopes' analysis on his own implementation project

When teachers have to reflect on a didactic proposal that implies a change to or an innovation in their own practices, they implicitly employ some of the didactical suitability criteria. Lopes' EMT (2014) has also allowed us to deduce the use of some of these criterions in the justification and reflection on the suggested proposal. In the following subsections, we show the extent to which the author considered-implicitly and explicitly-the suitability criteria put forward by OSA in attempt to defend his didactic proposal as improvement for mathematics teaching.

Epistemical suitability

Lopes (2014) justifies the 'innovative and creative' nature of his proposal by pointing out that it encourages students to perform relevant mathematical processes, in particular that of mathematical modelling. In his own words, he explains: "In this way, the application process, divided in three stages, aims to build knowledge through the use of mathematical models. Starting with the first construction, on the basis that the topic is studied in depth and new elements arise, other models are built based on the previous ones […]". (Lopes, 2014, p. 22) The professor also considered that his innovative proposal allows students to perform other relevant mathematical processes such as connections, meaningful constructions, problem-solving, etc. "In this sense, the aim is to (…) awaken the student's creativity and enthusiasm to learn geometry, to create geometrical models with the students, making connections with reality, and to provide situational problems with a geometric focus…" (Lopes, 2014, p. 21).

It is evident in his review that some of the processes mentioned were in fact developed during the implementation of his proposal. In his thesis, the professor generally presents explicit reflections on the fact that his didactic proposal for teaching area-calculation is more representative (since it thoroughly explores the area-calculation of curvilinear figures) than the proposals that are commonly implemented at high school level.

Cognitive suitability

In Lopes' work (2014) there are comments, reflections, etc., that allow concluding that the author takes into account, in an implicit way most of the times, the indicator of cognitive suitability.

Background knowledge. The teacher carries out an initial evaluation in order to find out if the students had the necessary background knowledge for the study of the intended content. Furthermore, he makes sure that the students have such background knowledge, and specifically, he dedicates part of the time intended for the implementation, to revise the calculation of the area of triangles and quadrilaterals, and the study of trigonometric ratios. On the other hand, the learning objectives were attained by the students, "and there is confirmation that the Archimedes and Riemann methods are in the students' zone of proximal development" (Lopes, 2014, p. 19).

Curricular adaptation to individual differences. With the narration of the teacher it is not possible to conclude if he considers at some point complementary or reinforcement activities. However, when he assesses the learning related to the Riemann method, he concludes that many students will not achieve such learning and adds: "…it would be necessary to have a more extensive study period, to be able to ask the students (…) to interpret results more thoroughly, considering that each student is unique and as such, needs a shorter or longer time to learn" (Lopes, 2014, p. 92).

Regarding the learning intended, the teacher states in a very clear way that he has to carry out evaluations to verify that his innovative proposal helps the students to achieve the learning objectives. Therefore, apart from the initial evaluation, the teacher carries out three formative evaluations that show the acquisition of the competences/learning implemented. With these evaluations, the teacher concludes that the learning related to the calculation of areas of quadrilaterals and triangles, and the Archimedes method was acquired, but the same cannot be said about the learning of the Riemann method, which he justifies with lack of time.

High cognitive demand. The author considers that his proposal requires a high cognitive demand from the students, since it activates relevant cognitive processes.

Interactional suitability

Teacher-Student interaction. The teacher describes a "teacher-large group interaction", through a dynamic of questions asked by the teacher and answers given by the students, which, according to him "facilitates comprehension among students" (Lopes, 2014, p. 32). He also presents some examples of how this type of interaction helps to clarify doubts that the students might have.

Interaction among students.

In his narrative, the teacher also mentions that the students worked in small groups and although he did not comment if such dynamic has solved the student's semiotics conflicts, he concludes that this organization allowed some students that hardly participated in the classroom to express themselves in a larger group.

Autonomy. It is possible to conclude that there were moments in which the autonomy of students was encouraged. On the one hand, "the students had to do homework" (Lopes, 2014, p. 67); on the other hand, there were some moments in which it was possible to observe that the responsibility to study (exploration, formulation and validation) was assumed by the students.

Formative evaluation. As mentioned in the cognitive suitability section, the teacher carried out a formative evaluation that allowed a systematic observation of the cognitive process of the students.

Mediational suitability

It was possible to observe the use of material resources such as calculators and computers. The teacher explains that he used the GeoGebra software and the calculator during the teaching process. Regarding GeoGebra, he presents some implicit evaluative comments about the advantages of including this software of dynamic geometry in the teaching process.

Number of students, Schedule and classroom conditions. Regarding this aspect, the teacher makes several comments. In a relevant way, he explains that the number of students and the conditions of the classroom (both the physical space as well as the computer laboratory) somehow determined the use of GeoGebra. Thus, the software was mainly used by the teacher to illustrate and show mathematical practices (e.g., the calculation of the areas of quadrilaterals and triangles).

Regarding the timeof group teaching and learning -, the teacher makes comments and assessments about three indicators of this component: the adaptation of intended meanings in the available time, the time spent in the most important and relevant contents, and the time spent in the contents that were more difficult for the students. In connection to the first indicator, the teacher states very clearly that he could not adapt the intended meanings in the time that was available. Particularly, he states that he did not have enough time to finish explaining all he had planned regarding the Riemann method. For the second indicator, the teacher states that it took him a lot of time to ensure the required background knowledge, and that, on the other hand, he did not have time to solve the initial problem that was contextualized in order to later introduce the Archimedes and Riemann methods. Finally, regarding the third indicator, it is possible to infer from the teacher's comments that it was impossible to carry out the whole study due to lack of time (e.g., there was not enough time to explain the Riemann method in depth).

Emotional suitability

In connection to this suitability indicator, no comments regarding the interests and needs of the students were found in Lopes' EMT (2014). No comments about the attitudes of the students were found neither. Regarding emotions, the teacher states that the implementation he carried out promotes the students' self-esteem.

Ecological suitability

According to the criteria and objectives that the teachers had to consider for the elaboration of their projects, professor Lopes adds that his proposal is a didactical innovation that adapts to the Elementary school curriculum and, according to his students, contributes to social and professional integration (social and labour utility) and that presents an intra-mathematical connection to higher level Mathematics (intra and interdisciplinary connections).

Final reflections

The analysis of the EMT of Professor Lopes shows how the indicators of didactical suitability proposed by OSA are -implicitly-present in his reflection processes on their own practice. An important aspect to highlight is that this EMT clearly demonstrates the issue of finding a balance between each of the suitability criteria. On one hand, the author plans an innovation with high epistemic suitability and he demonstrates in his review that he also made a substantial effort to achieve high cognitive suitability. On the other hand, however, he also demonstrates that he was obliged to neglect part of the content he had planned; in particular he could not solve the initial problem which was the very motive of his didactic proposal and the learning was not complete (in particular, the Riemann method) due to the fact that the suitability of means was not adequate; to be precise, there was not enough time.

We could say that, in terms of the suitability criteria, Lopes concludes that if in future implementations, cognitive and epistemic suitability are not to be neglected, and then it would be necessary to allow more time. One aspect, which is difficult to explain, is the reason why the criteria of didactic suitability function as implicit patterns in the Lopes' discourse, when he has to evaluate instruction processes without specific training on the use of this analysis tool. One possible explanation is that the training that he has received in the PROFMAT allows him to do, implicitly, this type of analysis. However, Caldatto's research (2015) and Caldatto [START_REF] Caldatto | O PROFMAT e a Formação do Professor de Matemática: uma análise curricular a partir de uma perspectiva processual e descentralizadora[END_REF], leads us to believe that the characteristics of the PROFMAT do not encourage this kind of reflection. Now, other answer to this very question is related to the origins of this construct. In the OSA, the didactic suitability criteria, its components and characteristics were constructed on the basis that they should be constructs which rely on a certain amount of consensus within the Mathematics Education community, albeit the local one. Therefore, one of the plausible explanations that the suitability criteria can be considered as teachers' reflections patterns is related to the extensive consensus that they themselves generate amongst persons involved in Mathematics Education. Therefore, another possible explanation, for Lopes case, could be based on his previous training and his experience, which would lead him to participate in this consensus.

The analysis of the EMT of Professor Lopes shows that, when the teachers' opinions were clearly evaluative, they were organized implicitly using some characteristics of the components of the didactical suitability criteria proposed by the OSA. This result has been evidenced in other investigations [START_REF] Breda | Melhorias no ensino de matemática na concepção de professores que realizam o mestrado PROFMAT no Rio Grande do Sul: uma análise dos trabalhos de conclusão de curso[END_REF][START_REF] Seckel | Competencia en análisis didáctico en la formación inicial de profesores de educación general básica con mención en matemática[END_REF]Breda, Pino-Fan, Font, in press) in which it is also suggested that the suitability indicators can be taught as powerful methodological tools to organize the teacher reflection -as it has already been done in different processes of teacher training in Spain, Ecuador, Chile and Argentina [START_REF] Giménez | El papel del trabajo final de Máster en la formación del profesorado de Matemáticas[END_REF]Valls & Vanegas, 2015;Posadas, 2015;Pochulu, Font & Rodriguez, 2016;[START_REF] Seckel | Competencia en análisis didáctico en la formación inicial de profesores de educación general básica con mención en matemática[END_REF] -, that aim at the fostering of the "meta" dimension of didactical-mathematical knowledge (DMK) of Mathematics teachers [START_REF] Pino-Fan | Towards a methodology for the characterization of teachers' didactic-mathematical knowledge[END_REF][START_REF] Pino-Fan | Assessing key epistemic features of didacticmathematical knowledge of prospective teachers: the case of the derivative[END_REF].

Proceedings of CERME10

Acknowledgements

This work has been developed in the framework of Research Projects: FONDECYT Nº11150014 (CONICYT-Chile), EDU2015-64646-P (MINECO/FEDER, UE), REDICE16-1520 (ICE-UB) and the PDSE program CAPES/Brazil process number 99999.004658/2014-00.