Orsolya Dóra Lócska

Zoltán Kovács
email: kovacs.zoltan@uni-eszterhazy.hu

The "Sense-Making-Algebra" project for Hungarian seventh graders

Keywords: Algebraic reasoning, generalizations, using symbols, algebra in the early grades, lower secondary level education

According to the Hungarian National Curriculum, the systematic study of algebra starts in the seventh grade. In the initial phase, traditional methods focus less on generalization as a cornerstone of algebra and more on algebraic operations. Matched with the traditional syllabus, the authors have incorporated lessons into the teaching process at this early stage of algebra learning that focus on generalization activities and the meaningful use of symbols. The authors investigated how generalization activities manifest themselves during these lessons.

Introduction

The cognitive ability to abstract is given special attention in the Hungarian National Core Curriculum in grades 7-8 (Government Decree, 2012[START_REF]On the Announcement, Introduction, and Application of the National Core Curriculum[END_REF]. The generalization of experiences and the need to justify relationships are also expected. The curriculum also requires the learner to formalize the mathematical content of everyday problems and use letters to denote unknown quantities. To begin teaching using symbols, a procedural approach based on "letter arithmetic," i.e., practicing operations with algebraic expressions based on definitions and rules, is a traditional teaching strategy also presented in Hungarian textbooks. However, less emphasis is on generalization and the meaningful use of symbols in that early stage. Discrepancies between development goals and the practice in teaching early algebra were the impetus of our research. Improving this situation requires recognizing that algebraic thinking, which involves the meaningful use of symbols, is a form of mathematical sense-making related to symbolization [START_REF] Schoenfeld | Early algebra as mathematical sense making[END_REF]. Many authors explain the sense-making process of a mathematical concept by connecting it to prior knowledge [START_REF] Scheiner | New light on old horizon: Constructing mathematical concepts, underlying abstraction processes, and sense making strategies[END_REF]. [START_REF] Palatnik | Four-component decomposition of sense making in algebra[END_REF] found that the criterion for successful implementation of the sense-making process is that students are adequately engaged in the task and that the learning environment, which in their scenario was project work, is supportive. Inspired by these ideas, the authors of this paper added mathematical problem situations to the traditional "letter arithmetic" to support the generalization process and the meaningful use of variables. This paradigm is named the "Sense-Making-Algebra" teaching strategy by the authors of this paper. A rationale behind this teaching strategy is to allow students to see the power of symbolization as early as possible in the algebra learning process. Furthermore, with a problem-oriented teaching approach [START_REF] Kónya | Management of Problem Solving in a Classroom Context[END_REF], the authors aimed to create a supportive learning environment.

The authors' research questions are, (1) how do seventh-grade students' generalization activities manifest themselves during the "Sense-Making-Algebra" intervention, and ( 2) what resources do they employ to create their generalization? We hypothesize that most learners can generalize from experience and argue for generalization using natural language or symbols after the teaching unit. This paper describes a part of the broader experiment: the content and analysis of the first experimental lesson and the related follow-up test. In this lesson, we used the so-called "calendar problem," which has been discussed in various places in the literature, as an effective initiative to encourage generalization [START_REF] Friedlander | Reasoning with Algebra[END_REF][START_REF] Huang | Exploration of Patterns in a Calendar[END_REF].

The first author performed the action research reported in this paper in her three seventh-grade classes. The second author acted as a critical friend during the teaching experiment. The study provides an example of how to put into practice the sense-making use of variables. Although there are some well-prepared collections on teaching strategies and recommendations on algebra learning (e.g., [START_REF] Friedlander | Tasks and Competencies in the Teaching and Learning of Algebra[END_REF], these works are focused on equations. The authors found fewer compendiums of a similar style at the pre-equations stage of algebra learning, with generalization and reasoning in focus, which also motivated this research.

Theoretical underpinning

Several scholars attempted to investigate the nature and content of algebraic thinking. There are several perspectives on what defines algebraic thinking, but many agree that generalization, or the ability to discern the general in the specific, is a crucial element [START_REF] Pittalis | Unpacking 9th grade students' algebraic thinking[END_REF]. The present article focuses mainly on generalization, which may be accomplished using natural language and symbols. While outlining the theoretical basis, we emphasize this perspective, and in this regard, the work of [START_REF] Kaput | What is algebra? What is algebraic reasoning[END_REF] serves as the theoretical framework for our research. [START_REF] Freudenthal | What is algebra and what has it been in history?[END_REF] made an early attempt to conceptualize school algebra and included algebraic thinking in the subject. Freudenthal highlighted that the ability to describe relations and solve procedures in a general way is part of algebra. In this novel perspective, algebra is seen as a human cognitive activity. Thus, those who conceive algebra as reasoning prefer to examine how students think and speak about it. [START_REF] Arcavi | Symbol Sense: Informal Sense-making in Formal Mathematics[END_REF] considers the dichotomy of generalization and symbol usage in school algebra and claims that many students who master algebraic methods often fail to perceive algebra as a tool for comprehending, expressing, conveying generalizations, and constructing mathematical arguments. According to [START_REF] Kieran | Algebraic Thinking in the Early Grades: What Is It[END_REF] model for conceptualizing algebraic activity, algebra is a multidimensional activity that includes numerous ways of thinking. Algebraic thinking approaches quantitative situations that aim to find relationships and structure using not strictly letterssymbolic approaches. [START_REF] Kaput | What is algebra? What is algebraic reasoning[END_REF] emphasizes two core aspects of algebraic reasoning. One of them is generalization, and expressing generalizations in increasingly coherent symbol usage (Core Aspect A). "Increasingly" means that initially, the students use their resources, typically natural language, but later switch to conventional representational forms. Core Aspect B means the syntactically guided action on symbols. Kaput argues that the "Core Aspect B" should come later than the "Core Aspect A" since rule-based actions on symbols depend on knowing the allowed combinations of symbols, particularly which combinations are equivalent to others.

Method

The action research took place from December 2020 till March 2021 in a Hungarian practicing school for teacher training. A total of 68 seventh-grade students aged 13-14 years (hereafter S01-S68) from three classes (hereafter classes b, d, z) participated in the experiment. The groups have three maths lessons a week, each 40 minutes long. The teacher was the same in all three classes. Based on the textbook used in the school, the teacher's syllabus allocated 28 lessons to algebra instruction. The intervention took place in the first teaching unit (Algebraic expressions and operations with algebraic expressions), for which the curriculum allocated 11 lessons. The teacher conducted four experimental lessons in this teaching unit. The authors only report on the first experimental lesson in this paper and the follow-up test relating to it. The authors used the following tools for data collection and analysis:

(1) classroom observation through a visiting teacher's field notes; (2) the teacher's research journal;

(3) copies of students' written outputs.

The first lesson was inspired by [START_REF] Huang | Exploration of Patterns in a Calendar[END_REF], where the authors report a lesson on investigating patterns in calendars, focusing on improving empirical reasoning towards deductive proof. The planning of lessons followed a problem-oriented teaching approach [START_REF] Kónya | Management of Problem Solving in a Classroom Context[END_REF]. It means that students: (1) analyzed mathematical problems; (2) were allowed to reflect on their own and their classmates' thinking critically in classroom discussions; (3) were encouraged to explain and justify their thinking.

The plan of the first lesson is as follows. Part 1: pattern finding. Students looked for patterns in the December 2020 calendar during the preparatory phase of the lesson. First, the teacher chose the diagonal 1-9-17-25 (Figure 1, left). Then, after determining that the numbers in the sequence increased by 8, students worked in pairs to explore similar patterns (Figure 1, right). The purpose of the patternfinding task was to make students aware of the mathematical structure of the calendar, which is already familiar from everyday life. In the follow-up test, two compulsory and one optional assignment were given to the students (Table 1). The first and second problem was close to the calendar task in the lesson. These two tasks were chosen to require generalization and make it easy to argue without using symbols. On the other hand, the third task was assumed to be much more difficult to explain without symbols. To assess the follow-up test, the authors examined the students' solutions, for which they created the following code system (Table 2).

AR Arithmetic level

The student makes relevant calculations only on some numbers.

ALG Algebraic level, without symbol

The student argues generally and correctly, using natural language without using symbols.

ALG+ Algebraic level, using a symbol Three criteria must be fulfilled. (1) A symbol represents a number in the configuration.

(2) Student correctly expresses the relationships between the numbers, and (3) student performs algebraic operations.

Findings and analysis

During the pattern-finding task (part 1 of the lesson), students have skillfully formulated the rules in the natural language during classroom discussions. Moreover, the generalization process emerged naturally:

S45: Is what we are talking about a feature unique to December or not?

S32

[Enthusiastically] There are seven numbers next to each other, that is, they are arranged by week, so for each month, it is [true].

While working on the magician's trick (part 2 of the lesson), the experience was written on the whiteboard (Table 3), and the one who figured out the trick could be the next magician. Finally, students formulated the rule in their own words; they had to divide the sum of three numbers by three to get the number they had chosen.

When questioned why the trick worked, a faulty analogy emerged. Several students suggested that it should be divided by three because three numbers were added. The teacher took advantage of this situation and involved the class in discussing the mistake. The following dialogue is from class b, although almost the same happened in class z. First, S14 tried to explain the rule with his natural resource, which was the basis of the teacher's general example. They concluded with the deductive proof (𝑥 -6) + 𝑥 + (𝑥 + 6) = 3 • 𝑥. It is worth noting that the students explained the rule of merging starting from the problem situation; this is a point where the advantage of the "Sense-Making Algebra" approach is very apparent.

In class d, after the general examples, S46 proposed denoting the numbers with 𝑎, 𝑏, 𝑐, 𝑑, i.e., 𝑎 + 𝑏 + 𝑐 = 𝑑. Although the need to use letters emerged in the student, she did not express the relationships between the symbols. The student's typical failure provided an appropriate opportunity for the teacher to draw attention to the principle that symbols must express the mathematical relations extracted from the situation.

In two classes, the students wanted to create their own magic tricks. The need for generalization was raised naturally in all three classes: do the series apply in other months, does the magic trick work in other directions, or other months? The use of letters and symbols occurs spontaneously in two of the three classes, with minimal teacher guidance. However, more teacher guidance and questions were needed in the third class.

The result of the coding of the follow-up test is shown in Table 4. 39 students gave algebraic level answers (ALG and ALG+) in the first task, i.e., 57% of all students.

The result is similar in the second task, 52%, 35 out of 68 students. 21% of all students used symbols to express their reasoning (ALG+) in the first task, while 18% in the second task. The following examples are representative of each category.

NA (Task 1., S12) "Yes, that is true. Because if a number has a left and a right neighbor, both will be odd or even, and if you add two odd or even numbers together and divide by two, you always get an integer." What student writes is true in itself, but it does not explain the experience.

AR (Task 1, S22) See Figure 2. The student refers to only one particular case and concludes that the rule is always valid. (Translation of the text in the figure: "always true").

ALG (Task 1. S1) "The rule is valid because if you add a number one greater than the number you have chosen and a number one less than the number you have chosen, you will always end up with twice the chosen number." The student correctly expresses the relationship between numbers in natural language; his reasoning is general, but he does not use symbols.

ALG+ (Task 3. S4) See Figure 3. The student represented the central element of the V-shape by 𝑧, correctly determined the other two elements (𝑧 -8, 𝑧 -6), and performed the merge correctly (3𝑧 -14). The number needed is then determined by thinking backward from the sum. Translation of the text: "Add fourteen to the total and divide by three." The follow-up test also showed an example of an unexpected generalization: S15 wrote: "Conclusion: if you add the neighbors of a number (it can be second neighbors, third neighbors, fourth neighbors), you always get twice the number. It is because both neighbors differ by the same difference from the number."

Discussion and pedagogical implications

The authors' first research question was: how do seventh-grade students' generalization activities manifest themselves during the "Sense-Making-Algebra" intervention? Consistent with [START_REF] Friedlander | Reasoning with Algebra[END_REF], our research confirmed that students generalize and reason willingly and satisfactorily at the early algebra learning stage. A possible good choice for this process is the "calendar problem," as presented by [START_REF] Huang | Exploration of Patterns in a Calendar[END_REF]. The mathematical structure is simple, based on addition, so it can be used to pose problems accessible to a wide range of learners. Furthermore, we found that making and justifying conjectures occurs naturally using this type of exploration task. Moreover, this study found that the "problem-oriented" teaching approach supported generalization and reasoning activities, providing a good space for classroom discourse [START_REF] Kónya | Management of Problem Solving in a Classroom Context[END_REF]. Generalization, in many cases, did not occur as an individual activity but as a consequence of group discourse. The starting point for generalization could be the students' questioning in addition to the teacher's initiative.

The second research question was: what resources do students employ to create their generalization? Students expressed their ideas verbally in natural language resources in classroom discourse, which facilitated generalization. The predominance of textual formulation remained in written work until the end of the learning cycle. However, about 20% of the students were already confident using symbols in problem situations in our experiment. Although we did not use project work as a learning environment, we saw an identical process described by [START_REF] Palatnik | Four-component decomposition of sense making in algebra[END_REF], who reported that students developed and justified claims, made generalizations, addressed why-questions, and established coherence among the explored objects.

Figure 1 .

 1 Figure 1. The December 2020 calendar with diagonal patterns

Table 1 :

 1 The follow-up test (for March calendar) Task 1. (Compulsory) Nándi selected a number in the calendar and added its left and right neighbors. When he halved the sum, he got exactly the number he had chosen. Do you think this is always true if you choose a number with a left and a right neighbor? Justify! Task 2. (Compulsory) Csenge chose a number from the calendar with both a lower and an upper neighbor. She added the two neighbors together and then halved the sum. What has she experienced? Will this always be true if you choose a number that has a lower and an upper neighbor? Justify! Task 3. (Optional) Laci had a magician trick. If someone told him the sum of three numbers in the calendar in the shape of a V, he would guess the middle number. (For example, if someone told him that the sum is 22, he would know that the middle number is 12.) However, unfortunately, he forgot how the trick worked. Help him!

  faulty analogy.] Look at the diagonal patterns in the calendar! [She points to the corresponding pattern, see Figure 1 (right) that she deliberately left on the board at the beginning of the lesson.]S03:[Uncertainly] The [number] six is the key.S14:One number is six less and the other six more [than the number one thinks].T:Let us write what he said! Let, for example, the number thought be 11. Thus (

Figure 2 :

 2 Figure 2: AR solution of Task 1 by S22

Figure 3 :

 3 Figure 3: ALG+ solution for Task 3 by S4

Table 2 : The coding system to evaluate students' outcomes

 2

	Code	Description	Definition
	NA	No Answer	One of the following elements: The student has not submitted a solution. No
			explanation. The student did not understand the text. The student tries to give reasons
			but does not give a relevant explanation.

Table 3 : Numerical experience in class d

 3

	The magician	Teacher Teacher Teacher S31	S31
	Sum	33	66	69	73	75
	Number thought 11	22	23	"You made a mistake." 25

Table 4 : Result of the follow-up test

 4

		Task 1 Task 2 Task 3
	NA	23	28	61
	AR	6	5	0
	ALG	25	23	0
	ALG+	14	12	7
	SUM	68	68	68

Acknowledgment

This study was funded by the Scientific Foundations of Education Research Program of the Hungarian Academy of Sciences.


Nelleke Den Braber
email: nelleke.den.braber@nhlstenden.com

Marco Mazereeuw
email: marco.mazereeuw@nhlstenden.com

Jenneke Krüger
email: jenneke.kruger@gmail.com

Wilmad Kuiper
email: w.kuiper@uu.nl

What can be learned from mathematics teachers' experiences who teach an interdisciplinary STEM course

Keywords: mathematics teachers in STEM, secondary education, teacher education, interdisciplinary

In the Netherlands, it is common that teacher training courses for upper secondary education educate for a single school discipline, such as mathematics. At the same time, teachers with a mathematics qualification are also formally competent to teach an interdisciplinary school course called Nature, Life and Technology (NLT), regardless whether it has been addressed in the teacher training programmes. However, teaching NLT is substantially different from teaching mathematics with respect to, for instance, pedagogy and objectives. In this paper we report on a study on experiences of mathematics teachers who, in addition to teaching mathematics, teach NLT. The experiential knowledge is meant to inform mathematics teacher educators who prepare students for interdisciplinary courses such as NLT.

Introduction.

In the Netherlands, most teachers in secondary education are trained to teach a specific school discipline, such as mathematics. Although cooperation between and integrating of school courses are frequently encouraged in the Netherlands (Folmer, Koopmans-van Noorel & Kuiper, 2017), teachers collaborate mainly with colleagues of their own discipline, unless there is a need for cross-curricular collaboration within a school, due to thematic project work or a specific course. One course in which this is different is the interdisciplinary STEM course Nature, Life and Technology (NLT).

Nature, life and Technology is an elective course in upper secondary education. The objectives of the course are to strengthen the cohesion between science and mathematics and to increase the attractiveness of science education [START_REF] Stuurgroep | Contouren van een nieuw bètavak; visie op een interdisciplinair vak: natuur, leven en technologie[END_REF]. In this course, students work on complex real-life problems for which they need knowledge from mathematics, physics, chemistry, biology and physical geography to solve these. The course has a modular structure. With each module, students are working on new problems around a topical issue using teaching materials developed through collaboration between teachers and universities or organizations specialised in the topic. Teachers work within a team of teachers with different competences and backgrounds, mainly mathematics, physics, chemistry, biology and geography. Such team of teachers is called an NLT-team.

Teachers who participate in these kinds of activities are confronted with other learning and teaching situations as in their disciplinary course and as a consequence will have to relate to these experienced differences. For example, NLT requires attention for overarching ideas such as the interplay between the natural sciences and technology, and not merely attention for specific disciplinary concepts, such as finding an exponential formula given two points. The usual pedagogical approaches in NLT are also different from those in mathematics school courses, since a lot of coaching of group work is done in NLT. In addition, NLT often uses different assessment formats than the exercises that are customary in mathematics [START_REF] Braber | Bewust van het perspectief [Aware of the perspective[END_REF]. Knowing that there are such differences between the courses, it can be argued that mathematics teacher training should prepare future teachers for teaching NLT and for dealing with these differences. Especially, when these differences should indeed be dealt with according to mathematics teachers that teach NLT and knowing that it is considered important that mathematic teachers participate [START_REF] Stuurgroep | Contouren van een nieuw bètavak; visie op een interdisciplinair vak: natuur, leven en technologie[END_REF].

However, it has become clear that preparation for NLT is marginal in most mathematics teacher training programmes. Consequently, dealing with the differences will in most cases have to be learned after the teacher training course, during working life, if a teacher chooses to participate in NLT. However, opting for teaching NLT requires some lived experiences in NLT and experiencing the differences with teaching mathematics. However, it is known that mathematics teachers do not seek such experiences in working life easily. Mathematics teachers participate less in NLT than teachers of the other related disciplines such as physics or biology [START_REF] Braber | Bewust van het perspectief [Aware of the perspective[END_REF]. Mathematics teachers indicate that they do not always see a role for themselves in NLT. Teachers who are unfamiliar with NLT may find it difficult to envision what is required to teach NLT and compare teacher requirements to their own knowledge, skills and attitude to determine whether participation is even feasible, let alone appealing. Teachers who have taught NLT will have formed an image of what is required to teach a course such as NLT and will have found ways to act accordingly.

Despite the fact that mathematics teachers currently participate less in NLT than teacher of sciences, there are some mathematics teachers who are successfully part of an NLT-team. The fact that it is possible and can be successful raises the question whether we can learn from the experiences of these mathematics teachers to help prepare future mathematics teachers. It is this question that led to a study with the underlying research question: How do qualified mathematics teachers shape their role within a NLT-team and deal with mathematics in NLT?

In this paper we will describe results regarding to what the experiences teach us about the possibilities for mathematics teacher training programmes to enable lived experiences in teaching NLT for future mathematics teachers.

Conceptual framework

Analysing the experiences of a teacher within NLT requires a framework that describes a learning process of a teacher in his or her work in the broadest sense. It is not only about the knowledge of the teacher but also about the actions, feelings, motives and foresight within the context of a person's working environment. [START_REF] Roth | Vygotsky's Neglected Legacy": Cultural-Historical Activity Theory[END_REF] indicate that cultural-historical activity theory (CHAT), because of its focus on processes and individual and collective agency, can be the analytical lens for learning processes in lifelong learning, both in formal and informal (work) situations as well as in educational settings.

It is for this reason that we make use of a CHAT-based analytical perspective on learning processes while engaged in work processes [START_REF] Mazereeuw | Ik experimenteer dus ik leer [I experiment therefore I learn[END_REF]. This perspective depicts a process in which experiences at the workplace are internalised [START_REF] Zittoun | Internalization: How culture becomes mind[END_REF] and, depending on the interpersonal dialogue are stimuli for envisaging new action possibilities and actions within the work process. Acting on these new perspectives and action possibilities through experimenting can also create new experiences. Hence, experiential learning processes are not linear. Because of the reciprocal nature of the interpersonal process and the social and material consequences in context it is much more dynamic. To emphasize the reciprocal nature of the process we included insights of the Dialogical Self Theory [START_REF] Hermans | The dialogical self: towards a theory of personal and cultural position[END_REF].

The idea of the dialogical self is that a person mirrors experiences with ones knowledge, perspectives, norms and values in an internal dialogue and that these experiences can be re-lived and re-framed beyond the actual experience itself. So, the internal dialogue is an interpersonal process -within the mind-in which a person relates to the lived and re-lived experience. CHAT emphasizes that an internal dialogue can result in the desire to act, especially when frictions are experienced. For instance, a person perceives that contributing to the work calls for knowledge and skills that the person does not feel confident about. When that person wants to continue to participate in the workplace, new action possibilities will be sought out. Therefore, frictions, or the commonly used term contradictions, are often a source of change and development as described by [START_REF] Engeström | From design experiments to formative interventions[END_REF]. We tried to capture the internal dialogical process in four compact questions someone asks aware or unaware. The first question is 'How do I understand what is going on (with me, others in the environment)?'. Understanding includes ones own perceptions of knowledge of the workplace, division of labour and workplace goals, norms, emotions and values. The second question is 'How do I value this?'. Here a person mirrors this conception of the experience with what matters to the person, what the person finds important. The third question is 'Should I do something about this or not?'. This question refers to the motive to become an actor and to contribute to the workplace. The fourth and final question reads 'What can I do to contribute and/or to engage the friction?'. This question refers to the way a person depicts his or her own role in the situation and how that person envisages action possibilities that might help to contribute to a solution or to engage the friction.

The questions above show that this process may well be influenced by previous work and teaching experiences. That is probably also the case for NLT teachers who have experiences in another school course or have worked in different fields before entering the educational system. Figure 1 schematically represents the dialogical processes when interpreted for teaching NLT.

In order to provide insight in the possible frictions that initiate internal dialogue in teachers we conceive teaching of NLT as an activity within an activity system. To portray the activities we make use of the commonly used model of [START_REF] Engeström | Learning by expanding: An activity-theoretical approach to developmental research[END_REF], which builds on the ideas of Vygotsky and Leont'ev. The model shows that human action can be seen as an interaction between a person (subject), a certain goal (outcome) and cultural artefacts that support and shape this action (mediation). Furthermore, the goal-directed actions of an individual cannot be separated from its social and material context. An activity system consists of relevant others (community), with values and norms (rules) and with a division of labour. Teaching NLT in a secondary school can be conceived as a goal-oriented activity in an activity system in secondary education. This is done by using certain teaching materials and supportive (digital) resources (instruments) which are in accordance with a national examination program and a plan for assessment, etc. (rules). Teachers are part of a professional group of teachers.

They have to deal with parents, colleagues within the school who all want NLT to be taught (community). Mathematics teachers can be part of an NLT-team which is another community with another set of (informal) rules, and another division of labour compared to the activity of teaching mathematics.

The outcome of the activity deserves a closer look because the outcome of the activity system NLT is still ambiguous. Even though there is an examination programme, it leaves a lot of room for teachers and NLT-teams to formulate their own objectives, objectives that they feel are fitting for NLT. Moreover, [START_REF] Braber | Mathematics in an interdisciplinary STEM course (NLT) in the Netherlands[END_REF] made clear that the general aims of NLT are still debated on and when it comes to mathematics in NLT are not clear to both mathematics teachers and students. So, looking at the way teachers view NLT we can assume that the outcomes pursued are diverse. It is therefore of interest to find out what mathematics teachers view as the outcome of teaching NLT especially with regards to the role of mathematics. In school mathematics learning mathematical skills is seen as the outcome of the activity system. Within NLT mathematics can be described as an instrument, a means to an end. William and Roth (2019) claim that the more disciplinary courses are a goal by itself, the more difficult it becomes to work interdisciplinary. We started with the idea that through an internal dialogue a person mirrors experiences with ones knowledge, perspectives, norms and values. We portrait the envisaged activity using the model of an activity system where the teacher is the subject.

Data collection and analysis

In order to find out how qualified mathematics teachers shape their role within a NLT-team and deal with mathematics in NLT six male and four female qualified mathematics teachers all teaching NLT were interviewed. The teachers varied in age, work and teaching experience and work in different schools around the country. Following the first interview, an additional interview was conducted with a second member of the same NLT-team to triangulate the data of the first interview and evaluate the interview set-up. This led to small changes in the preferred order of the interview questions.

The interviews were meant to provide insight into how mathematics teachers conceive their role within NLT and what thoughts existed about the role of mathematics within NLT. Questions were asked about frictions that teachers experience when teaching NLT, and whether or not these were due to possible differences with the teaching of mathematics. We also asked how they have dealt with these frictions.

The interviews had the character of a conversation about NLT in which the interviewer also provided information about NLT and gave background information when needed. For example, sometimes the teachers asked questions about the organisation of NLT in other schools. The interview can best be characterized as a semi-structured interview. The interview set-up contained a number of topics and questions related to the conceptual framework. The order in which the questions were asked depended on the course of the interview. However, it always started with a few practical questions about background, teaching experiences, current teaching tasks and past work experiences. This was followed by questions about the experience with NLT, their role in the NLT-team and the possible differences between activities when teaching mathematics and NLT. For example, they were asked whether they experience differences in the use of teaching materials or teaching methods (instruments), collaboration with colleagues (division of labour) and which agreements concerning NLT have been made within the school (rules). When frictions were put forward by the teachers,

questions were asked about the teachers' views of knowledge, skills and attitude towards interdisciplinary teaching and NLT. The role in the NLT-team and the connection between curricula of mathematics and NLT were also addressed. The interview ended with questions that invited the teachers to talk about their thoughts on the role of mathematics in NLT.

The recorded interviews were transcribed. Semantic units (Aviv, 2001) were categorised using elements of the activity system, and the labels (personal) beliefs, knowledge and skills and attitude towards NLT based on a division by [START_REF] Ernest | The knowledge, beliefs and attitudes of the mathematics teacher: A model[END_REF]. For instance, a teacher stating which teaching materials are used in the classroom is labelled under instruments. Frictions experienced by teachers were noted as well as previous experiences when stated by a teacher. For instance, the following semantic unit illustrates an experienced friction concerning the subjects knowledge for teaching NLT compared to teaching mathematics.

Teacher: Mathematics, you can just ask me about that, I know it, but with NLT I sometimes doubt my answers are completely correct.

[…] makes me less comfortable.

Data analysis further focused on the role of each mathematics teacher within NLT. First, we looked at their view on the role within the NLT-team to see if there are similarities or differences. Then we looked at the relationships between how teachers describe their own teaching in NLT and what they see as objectives for NLT. i.e. the subject-instruments-object/outcome relationship. A third analysis focussed on each teacher's orientation towards the objectives of NLT. The emphasis was the teachers' views on intended objectives of mathematics within NLT and views on the overall objectives of teaching NLT. A fourth analysis concerned the frictions each teacher experienced. All frictions mentioned by the teachers were gathered and positioned in an activity system to see if there were similarities or differences in the experienced frictions. Peer-debriefing was used in the process of coding and interpretation.

Results

When we look at division of labour in the NLT-teams, the role of the mathematics teachers vary from teaching a statistical module by themselves, to teaching a science module with a physics teacher. Some are teaching modules with a strong biological component involving the brain or navigation of birds which they had to make their own. With the exception of one, the teachers seem comfortable with what they do because of earlier experiences in which they have acquired knowledge or skills before teaching NLT. The one teacher that was not comfortable assisted teachers from other disciplines only for a few lessons. She was asked to help students with differential equations and modelling with in a spreadsheet. This teacher felt a friction due to her view of her own knowledge of science and the lack of overview of the teaching module.

NLT is seen as a course where mathematics is applied and where mathematics can be useful, especially with regard to mathematical modelling to help solve the problem at hand. So the view of most teachers seems to be consistent with the objectives of NLT. However, often the teachers thoughts of these objectives themselves and their personally desired outcomes do not always correspond with the intended curriculum of NLT. Teachers seem to formulate their own objectives as the formal objectives of NLT are not really known. The latter is specifically the case when it comes to the use of mathematics or the reflection on the role of mathematics in the examination program.

Most teachers are not familiar with objectives in NLT that focus on the importance of mathematics. The ones that do report difficulties translating this for students.

The frictions teachers experience also seem to vary. Much of these frictions have to do with conditionalities. In some schools it is hard to have meetings with the NLT-team because disciplinary faculty meetings are held at the same time and require a teachers attention. Nevertheless, working with other colleagues is viewed as worthwhile and desirable even though it can take some getting used to. Working with and preparing the teaching materials of lessons in NLT generally takes more time than preparing mathematics lessons. On the other hand, teachers also seem to appreciate the freedom they get as they are free to adjust materials, add or skip parts of the content and assignments. Some experience difficulties with the motivation of students because students seem to give priority to the disciplinary courses for which they also take exams. Some teachers indicated that seeing that they only teach one or two modules a year, building a relationship with students can be more difficult than in mathematics lessons.

What stands out in the analyses is that teachers with previous experiences in actual interdisciplinary work environments or with mathematical modelling experience less frictions teaching NLT. In fact, two of those teachers commented that they experience frictions with the way they have to teach mathematics instead of with teaching NLT as it is so different to what they are used to.

Discussion

The freedom to find ways to teach NLT that align with a teacher's personal point of departure has been important for teachers. Especially, since this freedom has been a way for teachers to deal with frictions they experience while teaching NLT. This may indicate that teacher training should focus on the flexibility of working with teaching materials and to encourage experimenting with their pedagogical choices. Even though this is also encouraged in mathematics teacher training, research shows that mathematics teachers are still mainly focused on the textbook [START_REF] Woldhuis | Leermiddelenmonitor 17/18[END_REF]. This is not possible within NLT if a teacher wants to feel comfortable teaching. NLT offers a lot of freedom to make pedagogical choices and many different roles are possible as a teacher, but a teacher has to be aware of the objectives of NLT. Specifically, when it comes to the role of mathematics and the idea of interdisciplinary work. This may have consequences for teacher training programmes, knowing that we saw differences between teachers in dealing with interdisciplinary objectives depending on their background and work experience. This could mean taking into account the different personas in teacher training and take advantage of the lived experience with applied mathematics when present. Teacher training should also support reflection on the role of disciplinary mathematical knowledge when working on an interdisciplinary problem. Seeing mathematics more as an instrument than a 'product'. Something that the some interviewed teachers found difficult to do. Training programs could therefore pay attention to beliefs on mathematics in relation to the sciences. Starting, for instance, with one discipline that is closely connected to mathematics, like physics [START_REF] Nguyen | Interdisciplinarity in school and teacher training programs[END_REF], but perhaps ending with letting students experience mathematics or mathematical modelling in corporate life or interdisciplinary working environments. This could support teachers in letting their students experience how mathematics can help with real-life interdisciplinary questions and discuss the contributions mathematics can make in solving real-life problems.

Figure 1 :

 1 Figure 1: Processes of learning through experiences, based on Mazereeuw (2020) exemplified for teaching NLT Unlike Engeström, we use the model as a means for portraying a teachers' thinking about the activities in which they participate, shown in figure 2. As indicated above, a person ones social and material work environment and places oneself within it. In this study Engeström's model is used to guide the data collection and analysis and find out how teachers think about their own discipline and NLT and the frictions therein.Teaching NLT in a secondary school can be conceived as a goal-oriented activity in an activity system in secondary education. This is done by using certain teaching materials and supportive (digital) resources (instruments) which are in accordance with a national examination program and a plan for assessment, etc. (rules). Teachers are part of a professional group of teachers.They have to deal with parents, colleagues within the school who all want NLT to be taught (community). Mathematics teachers can be part of an NLT-team which is another community with another set of (informal) rules, and another division of labour compared to the activity of teaching mathematics.

Figure 2 :

 2 Figure 2: The envisaged activity system by the NLT teacher of the activity in which the teacher participates.

Acknowledgement

This study was made possible with support from the department of mathematical science at Agder university, Kristiansand, Norway. We especially thank Professor Pauline Vos for her support.


Eszter Kovács-Kószó

Eszter Kónya
email: eszter.konya@science.unideb.hu

Zoltán Kovács
email: kovacs.zoltan@uni-eszterhazy.hu

Planning and implementation: the impact of a professional development program on teachers' and learners' oral manifestations

Keywords: Professional development, problem-oriented instruction, student-centered teaching, class discussion

This paper investigates a professional development program's (PD) effect on classroom discourses.

The PD was based on problem-based curriculum processing and intensive use of classroom discourse. The authors analyze seven pilot lessons of one of the teacher training's participants, providing an example of how a teacher with 20 years of experience in mathematics teaching, but using traditional teacher-centered methods, can apply the new approach in the classroom. Our analysis is based on a combined theoretical approach that starts from students' mathematical thoughts and examines the teacher's responses. We found numerous possibilities for starting classroom discussions building on students' thinking, and the teacher responded to most of them. Furthermore, the teacher's responses also included several invitations for students to classroom discourse, although these elements were almost absent before the PD.

Introduction

Based on the Hungarian tradition and inspired by the ideas of Tamás Varga in mathematics teaching [START_REF] Varga | Mathematics Education in Hungary Today[END_REF], the second and third authors have developed a research-based professional development (PD) program. This program was piloted in 2018-2019, with four voluntary teachers as participants. One teacher's lessons are the basis for this paper, since the authors analyze the PD's effect on classroom discourses during the PD's lessons.

The PD program's two main basis points were problem-oriented curriculum processing and the systematic use of classroom discourses. [START_REF] Kónya | Management of problem solving in a classroom context[END_REF] characterized the problemoriented approach to learning mathematics by three attributes: (1) students analyze mathematical problem situations; (2) students critically adapt their own and their classmates' thinking; (3) students learn to explain and justify their thinking. This approach is very close to the idea of Roh's definition of problem-based learning: it describes a learning environment in mathematics, where problems drive learning [START_REF] Roh | Problem-Based Learning in Mathematics[END_REF]. [START_REF] Cirillo | Teacher Discourse Moves: Supporting Productive and Powerful Discourse[END_REF] believe that mathematics classrooms' discursive nature impacts students' identities as mathematical knowers and doers. [START_REF] Leatham | Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking[END_REF] hypothesize that high cognitive demand tasks support the emergence of mathematically significant moments as possible starting points of mathematical discourses in the classroom. However, teachers' and students' proper reactions are crucial. Whether the active knowledge construction takes place in a discourse is decided by the teacher's reaction. [START_REF] Sfard | Balancing the unbalanceable: The NCTM Standards in light of theories of learning mathematics[END_REF] points out "a productive mathematical discussion (…) turns out to be an extremely demanding and intricate task. The role of discussion coordinator is particularly difficult" (p. 375). Teachers tend to teach and make decisions by routine [START_REF] Shavelson | Research on teachers' pedagogical thoughts, judgments, decisions, and behavior[END_REF], and it is challenging to change this routine. One of the PD's aims was to break the teacher-centered pattern among practiced teachers and create more complex and precious whole-class discussions.

Therefore, this paper concentrates on two aspects: firstly, the possible starting points of classroom discussion generated by students' thoughts identified by [START_REF] Leatham | Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking[END_REF] framework. The second focus is the teacher's reactions to these, analyzed by Sohmer et al.'s (2008) framework. With these tools, our research question is: How does a problem-oriented professional development with additional focus on classroom discourses impact an experienced teacher's lessons with teachercentered instructional habits?

Theoretical frameworks applied for the analysis A student's action is characterized as a Mathematically significant pedagogical Opportunity to build on Students Thinking (MOST) moment when it fulfills six criteria built on each othereach represented by a question (C1…C6) [START_REF] Leatham | Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking[END_REF], see Figure 1. "In their analytic process, the unit of analysis is an instance (…) Typically an idea unit is one conversational turn or physical expression (such as writing a solution on the board)" (Leatham et al., 2015, p. 92). This framework was used to identify MOST moments in the videotaped and transcribed lessons. To find these moments, one should examine each student's utterances with six questions. These questions are built on each other, starting from the very basic point: whether the student's utterance contains understandable mathematical thoughts or not (C1, C2), then examining whether it is accessible for the students (C3) or whether it is the central goal of the lesson (C4). The fifth criteria (C5) can be described by the following question: Does the expression of the student mathematics seem to create an intellectual need (called opening) that, if met, will contribute to understanding the mathematical point of the instance? These expressions can classify into one of the following five groups: (a) a correct answer with novel reasoning, (b) an incorrect answer that involves a common or mathematically rich misconception, (c) a mathematical contradiction, (d) incomplete or incorrect reasoning, (e) why or generalizing questions. The sixth criterium (C6) is about the timing, whether it is worth taking advantage of the student's opening. Later, these criteria will be presented through an example.

Identifying MOST moments provide a quantitative description of a lesson, which can inform about the active and meaningful participation of the students. However, this framework does not provide information about the quality of the teacher's reactions. Therefore, the authors added Sohmer et al.'s (2008) framework to analyze the teacher's reactions to the MOST moments. This framework seems to provide supportive information to assist teachers in making in-the-moment decisions about whether or how to react to those MOST moments. The authors found the combination of these two frameworks highly fruitful.

Researchers try to identify how some types of teacher interaction influence the following student's utterances to support teachers in activating students more effectively [START_REF] Dahl | How to improve teacher students' awareness of critical aspects in a lesson plan[END_REF]. Sohmer et al. (2008) They studied teachers who have been effective in using talk to promote learning. Six moves were identified that can be useful to model and to elicit academically productive talk: (1) revoicing students' utterances, (2) asking to restate someone else's reasoning, (3) asking students to apply their reasoning to someone else's reasoning, (4) prompting students for further participation, (5) asking students to explicate their reasoning and provide evidence, and (6) challenging or providing a counterexample.

The professional development program

The analysis of this research concentrates on a teacher with 20 years of experience in teaching mathematics. She joined the PD program voluntarily, out of an inner urge to renew her practice. The researchers visited her before the PD, observed her class, and discussed the teacher's professional view as a starting point. She used to prefer a teacher-centered way of teaching: explaining the new material, driving the students with direct questions, and rarely initiating open classroom discussions. In an opening workshop, the researchers explained the design of the program and the principles based on work by [START_REF] Varga | Mathematics Education in Hungary Today[END_REF]. The most important of them were the followings:

1. Problem-solving both alone and in pairs or small groups. 2. Improving students' oral and written communication skills encourage independent opinions. 3. Let students learn through experience and using heuristic strategies.

4. The teacher's role should include encouraging group discussions, planning classroom discussions, and implementing students' proposals into the flow of the lesson. 5. Differentiation and individualized treatment for each student.

After selecting the pilot lessons from the teachers' agenda, the researchers developed detailed lesson plans and provided all the teaching materials. These lesson plans explicitly contained reminders and advice on how to realize the principles of the PD. For example, there were time slots devoted to classroom discussions after solving a problem in small groups.

The teachers gave their opinions and suggestions and finalized the lesson plans. The teacher needed to feel that the lesson plan suited her at the end of the collaborative planning. At the end of the lessons, the teacher should reflect on it both alone and accompanied by the researchers.

The researchers organized six teaching cycles (three per semester during one year of the experiment) and concluded the year-long program with one trial lesson (Figure 2). During the planning for the trial lesson, the teacher had to come up with her own lesson plan.

The whole research process and focus are summarized in Figure 3.

The process of analysis

Each 45-minutes lesson was videotaped and transcribed. Two researchers analyzed the transcripts independently and looked for moments where all the six MOST criteria appeared. In a disagreement, the three authors' consensus fixed the MOST moments. The transcripts of each lesson were investigated in the same way:

1. The authors separated those parts of the lesson in which the whole class discussion occurred and identified each observable student utterance according to [START_REF] Leatham | Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking[END_REF]. 2. All of these instances were coded according to the six MOST criteria. 3. Further distinctions between the MOST units were made according to the situation that caused it (see criterion C5) and assigned one of the above codes a, ..., e to each unit. 4. After gathering the MOST moments, the authors examined the teacher's response to these MOST moments in a new analysis process according to [START_REF] Sohmer | Guided construction of knowledge[END_REF] framework. Three categories emerge A) The teacher evaluates the student's action and tells the correct answer if it was incorrect. B) The teacher starts a classroom discussion. C) The teacher does not notice the MOST moment. 5. Case B was further refined, determining the occurred talk move. In line with the work of [START_REF] Sohmer | Guided construction of knowledge[END_REF], the authors used the following codes: the teacher B1) revoicing the student's utterance, or the teacher asks a student for B2) restating someone else's reasoning, B3) commenting on someone else's reasoning, B4) further participating, B5) providing evidence or B6) providing counterexample. The traditional Initiation-Response-Evaluation pattern (IRE) completes the code system as B7.

We illustrate our coding system with an example. How did you know you had to finish here?

Example

This activity is considered as mathematical problem instead of routine task because the student has to define the procedure itself. The teacher does not present the solution as usual, but it appears as the students' activity, as he uses the structure of products to identify all divisors of 54.

The two student manifestations were considered a single action because the teacher's interruption is mathematically insignificant. However, this interruption demonstrates the teacher's accustomed state of controlling everything. The authors classified this student action as a MOST moment because it meets the C1-C6 criteria:

C1. The student is concentrating on mathematical ideas and not offtopic themes. C2. The mathematical point of the instance is to determine all divisors of a number and decompose it into two-factor products in all possible ways. C3. The mathematical point is accessible to all students, but at that point not all students realized it as a helpful tool for the task. C4. A deep understanding of the above procedure is one of the lesson's goals. C5. A deeper analysis of the fifth criterion shows that this is the case (d), i.e., incomplete reasoning, as the student wrote the products in a logical order but did not verbalize the reason behind it. C6. Finally, the timing is considered appropriate as all students were paying attention, and there were still nearly 10 minutes left in the lesson.

The teacher recognized the MOST moment and asked the student to provide evidence, i.e., to explain why the presented procedure is appropriate for finding all the divisors. Therefore, the authors coded the teacher action responding to the MOST moment as B5.

Findings and analysis

The emergence of MOST moments

The analysis found numerous MOST scenes that have emerged applying the problem-oriented lesson plans, which were considered satisfactory from the researchers' point of view (Table 1). Most of the MOST situations in this research emerged from students' incorrect answers (42%), and "MOST" situations based on the student's novel approach or the student's "why" questions were less frequent while mathematical contradictions did not appear.

Teacher's reactions on MOST moments

The teacher identified the MOST scenes effectively (Table 2). The authors attribute this success to the teacher's 20 years of practice, in addition to the influence of the PD. [START_REF] Leatham | Conceptualizing Mathematically Significant Pedagogical Opportunities to Build on Student Thinking[END_REF] also support this view implicitly, as they stated that novice teachers could miss realizing when a MOST scene has developed more often.

Furthermore, 64% of the identified MOST moments do not end with a simple teacher evaluation but lead to a "talk move." The authors consider this to be the result of the PD's approach, as the teacher was aware of the importance of classroom discourse, which she almost neglected in her previous teaching practice.

Therefore, the PD's result is considered to be the realization of the importance of the MOST moments and the use of the appropriate talk moves in the teacher's reactions. In conclusion, the teacher started using the learner's initiative to guide the lesson. The subtle structure of "talk move" reactions

The dominant "talk move" reaction was that the teacher involved others in the conversation (B4, 20 out of 32 talk moves, 62.5%), see Table 3. Thus, B4 has become an almost permanent behavior, especially in the case of novel reasoning by the students. However, the teacher has also used it when a more detailed explanation of the student's own thinking, i.e., elaboration (B5), would have also been adequate. Moreover, since the problem-based approach requires the learner to think critically about his/her thinking, encouraging this elaborative behavior would also have been part of the problembased learning approach.

Conclusions and pedagogical implications

During the PD's lessons, when the teacher got support from training, lesson plans, and collaborative lesson planning, numerous MOST moments mainly emerged from students' incorrect answers or incorrect reasoning. Furthermore, the teacher identified MOST scenes effectively. Most of them do not end with a simple teacher evaluation but lead to a talk move, supporting students' activity. Based on one previously observed lesson and the discussion on her professional view of teaching before the PD, the authors believe that this is the result of the PD's approach. Although further research is needed to prove this finding.

Based on the above result, the authors conclude that the PD is probably suitable to improve the amount and the quality of classroom discourse in an experienced teacher's lessons. It would be worth examining lessons after the PD to explore more about the PD's long-term effect.

It is worth to highlight that identifying mathematically valuable moments is insufficient. Teachers must also be aware of the importance of MOST moments and consciously apply potential talk moves. However, we still know little about how the implementation of talk moves unfold and what is needed to enable teachers to apply them effectively in practice. In connection with it, this paper also argues that the two joined frameworks were beneficial to examine how the demanding mathematics gets leveraged into productive discourse.

Figure 1 :

 1 Figure 1: The MOST Analytic Framework

  identified talk move as a turn at talk that (1) responds to what has gone before; (2) adds to the ongoing discourse; and (3) anticipates or 'sets up' what will come next. A talk move is inextricably tied to the context. It reaches beyond a single turn (p. 107).

Figure 2 :

 2 Figure 2: Scheme of the PD-program

Figure 3 :

 3 Figure 3: Scheme of the research

-Class 9, fourth pilot lesson

	Topic: Divisibility
	Episode: Whole class discussion after finding all divisors of 54 in pair work. (Time: 35:49-36:16)
	Student:	Can I write them as products? (He writes on the blackboard 1•54,2•27)
	Teacher:	[Please use] semicolon…
	Student:	(He corrects and writes 1•54;2•27;3•18; 6•9)
	Teacher:

Table 1 : The emergence of MOST moments

 1

	MOST category	a) Novel reasoning	b) Incorrect answer	c) Mathematical contradiction	d) Incorrect reasoning	e) "Why" question	Total
	Occurrence during the 7 lessons	13	22	0	16	2	53

Table 2 : Teacher's reactions on MOST moments

 2

	MOST category	A (evaluation)	B (talk move)	C (unnoticed)	Total
	Occurrence	18	32	3	53

Table 3 : The subtle structure of "talk move" reactions

 3

	MOST category	B1 revoicing	B4 participate	B5 elaboration	B6 counterexample	B7 IRE	Total
	a) Novel reasoning	0	7	0	0	0	7
	b) Incorrect answer	1	5	3	2	2	13
	d) Incorrect reasoning	0	7	3	1	0	11
	e) "Why" question	0	1	0	0	0	1
	Total	1	20	6	3	2	32

Acknowledgment

This study was funded by the Scientific Foundations of Education Research Program of the Hungarian Academy of Sciences and by the ÚNKP-21-3-SZTE-439 New National excellence program of the Ministry for Innovation and Technology from The Source of the National Research, Development and Innovation Fund.