text:
Dina Tirosh

Pessia Tsamir
email: pessia@post.tau.ac.il

Ruthi Barkai
email: ruthibar@post.tau.ac.il

Esther Levenson
email: levenso@post.tau.ac.il

Preschool teachers' variations when implementing a patterning task

Keywords: Repeating patterns, tasks, preschool teachers

It is often recommended to engage young children with patterning activities. As part of a professional development program, nine preschool teachers were introduced to repeating patterns and were given the materials and instructions with which to implement an extension task with children. This study presents the various ways teachers implemented this task and investigates the impact of the various implementations on children's success in extending the repeating patterns.

Introduction and background

For several years, we have been providing professional development for preschool teachers guided by the Cognitive Affective Mathematics Teacher Education (CAMTE) framework (e.g. [START_REF] Tsamir | Developing preschool teachers' knowledge of students' number conceptions[END_REF]. Our aims are to promote teachers' knowledge and selfefficacy for teaching mathematics to young children. An essential element of pedagogical-content knowledge is knowledge of tasks [START_REF] Sullivan | Converting mathematics tasks to learning opportunities: An important aspect of knowledge for mathematics teaching[END_REF]. In Israel, where this study took place, there is a mandatory mathematics preschool curriculum, but few curricular materials are available. Thus, introducing preschool teachers to appropriate mathematical tasks is essential. However, studies have shown that even when providing teachers with a task, and with explicit instructions for carrying out that task, teachers may implement the task in different ways [START_REF] Bieda | Enacting proof-related tasks in middle school mathematics: Challenges and opportunities[END_REF]. In turn, different implementations may affect the cognitive load of a task or a student's conceptualization of key mathematical ideas [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF]. This paper investigates the various ways preschool teachers implement a given repeating pattern task.

Repeating patterns are patterns with a cyclical repetition of an identifiable 'unit of repeat'. For example, a pattern of the form ABBABBABB… has a (minimal) unit of repeat of length three. The importance of engaging young children with pattern activities is supported by mathematicians, mathematics education researchers, and curriculum developers [START_REF] Sarama | Early childhood mathematics education research: Learning trajectories for young children[END_REF]. Pattern exploration and recognition may support children as they learn a variety of mathematical skills developed at this age. For example, recognizing repeating patterns may help children develop skip counting, such as 5, 10, 15, 20, 25, 30 ... where the ones digit forms the pattern 5, 0, 5, 0, … Recognition and analysis of patterns can also provide a foundation for the development of algebraic thinking and provide children with the opportunity to observe and verbalize generalizations as well as to record them symbolically [START_REF] Threlfall | Repeating patterns in the primary years[END_REF].

Several studies have investigated ways in which young children engage with repeating patterns. For example, [START_REF] Seo | What is developmentally appropriate in early childhood mathematics education? Lessons from new research[END_REF] found that young children build block towers with an ABAB pattern. [START_REF] Fox | Child-initiated mathematical patterning in the pre-compulsory years[END_REF] observed young children painting stripes in ABAB patterns as well as one child who painted four sets of an ABC pattern and then said, "Look at my pattern" (p. 317). [START_REF] Waters | Mathematical patterning in early childhood settings[END_REF] observed a young girl who created a necklace out of game materials and described her necklace as "diamond, funny shape, diamond, funny shape" (p. 326). Papic,Mulligan,& Thematic Working Group 13 Proceedings of CERME10 [START_REF] Papic | Assessing the development of preschoolers' mathematical patterning[END_REF] found that some preschool children may be able to draw an ABABAB pattern from memory by recalling the pattern as single alternating colors of red, blue, red, blue, basically recalling that after red came blue and after blue came red. However, when shown a more complicated pattern such as ABBC, they could not replicate the pattern.

This study focuses specifically on the task of extending a repeating pattern. Pattern extension tasks mostly include showing the child a pattern and requesting the child to continue it. [START_REF] Papic | Assessing the development of preschoolers' mathematical patterning[END_REF] reported that many children succeed at these tasks without necessarily recognizing the unit of repeat. Instead, they use the "matching one item at a time" strategy, also known as the "alternation strategy" which is especially successful with simple ABAB patterns. [START_REF] Rittle-Johnson | Emerging Understanding of Patterning in 4-Year-Olds[END_REF] found that some children reverted to producing an ABAB pattern while others could not produce more than one unit of repeat correctly when extending an ABB pattern. Similarly, Swoboda (2010) found that for some four-year old children, continuing a pattern means duplicating the unit of repeat once, and no more. In other words, both the complexity of the unit of repeat, as well as the number of times the unit is to be repeated, seemed to contribute to the difficulty of the task.

Another factor which may impact on the difficulty of extending repeating patterns is whether (or not) the given pattern ends with a complete unit of repeat. In one study, children were shown given repeated patterns and asked to consider extending the patterns by choosing between different possible continuations, some appropriate and some not appropriate. Children had greater success extending patterns which ended with a complete unit, than extending patterns which ended in a partial unit [START_REF] Tsamir | Which continuation is appropriate? Kindergarten children's knowledge of repeating patterns[END_REF]. Furthermore, several of the appropriate continuations would have extended the pattern in such a way as to end the pattern with an incomplete unit of repeat. Fewer children chose those possibilities as appropriate, even though they were correct extensions.

From the above studies, we see that there are several variables that may be taken into consideration when engaging with pattern extension activities: the structure of the pattern, the length of the unit of repeat, the number of times a unit appears in a pattern, and if the presented pattern ends in a complete unit of repeat or not. However, these variable all have to do with the repeating pattern. Might there be other variables that need to be considered when requesting children to extend a repeating pattern? Our first question is: Given an extension task and a set of repeating patterns, what are the various ways preschool teachers implement the task? Considering variation theory of learning and that learners may experience objects in various ways (Ling & Marton, 2012), our second question is: What can we learn about children's patterning abilities from the different implementations?

Methodology

This study took place within the context of a professional development course for preschool teachers, focusing on patterning for young children. Twenty-three preschool teachers participated in the program. All had a first degree in education and between 1 and 38 years of teaching experience in preschools. The entire program was planned for 21 hours. The teachers met seven times over a period of about four months in the local professional development center in their area. Approximately five of the seven sessions were devoted to repeating patterns with the other two focusing on number concepts. All lessons and tasks were planned by the four authors of this paper.

During the program, teachers were introduced to different patterning tasks as a tool for promoting their mathematical and pedagogical knowledge for teaching patterns in preschool. For the final project of the program, teachers were instructed to choose two of the tasks that were presented and analyzed during the course, and implement and video those chosen activities with one child. Those videos were then analyzed and discussed together in terms of children's solutions. In this paper we investigate teachers' implementations of one task (see Figure 1). Nine teachers (T1-T9) implemented this task, each with one child from their preschool (C1-C9). It is important to note that the task, along with the explicit instructions, was presented to the teachers by the teacher educator, who demonstrated how the task should be implemented. Furthermore, this task was not meant to be an instructive task, but instead an evaluation task in the sense that it was meant to assess children's ability to extend various repeating patterns.

Present the child with one pattern at a time. For each pattern prepare two or three separate containers, each container containing cutouts of triangles, squares, or circles. For example, when presenting the first pattern, place before the child two containers, one with blue squares and one with red triangles. For each pattern ask: What comes next? This question is repeated three times so that in the end, the child will have added three elements to the pattern. Note that there are basically three different structures, from the simpler AB, to the more complex ABC, and the even more complex ABB (e.g., Rittle-Johnson, el al., 2013). In addition, the first three patterns end in a complete unit of repeat and the last three do not. In other words, the sequencing of patterns goes from the simple to the more complex.

Various ways of implementing the task

We first note that none of the teachers changed the given patterns. Some variations came about from not implementing the tasks according to the instructions. For example, although teachers were told to prepare five separate containers for each possible cutout, according to shape and color, and only present to the child those containers which contained cutouts for that pattern, only one teacher actually followed this instruction. Four teachers did separate the elements into five containers according to shape and color, but then kept all of the containers on the table, no matter which pattern was being extended. Four other teachers separated the elements into only three containers according to shape (e.g., putting blue and red squares in one container), and then placed all three containers on the table, no matter which pattern was being extended. Another explicit instruction which was not followed was the sequencing of the patterns. Two teachers did not present the patterns in the order given above. One teacher showed the patterns in the following order: P4, P6, P3, P4, P1, and P2. The second teacher used the following order: P4, P3, P2, P1, P6, and P5. Some variations in implementation seemed to come about because no explicit instructions were given as to what to say to the child before beginning the task. That is, teachers were instructed to ask for each pattern "What comes next?", but were not told what to say when sitting down with the child and introducing the task. Six teachers stated at the beginning of the task, as they placed the pattern down on the table, "Here is a pattern." Five teachers asked the children to say out loud each element from the beginning of the pattern. It might be that the teachers thought that saying out loud the elements would allow children to hear the repetition of the unit of repeat and thus enable the children to pick out the correct next element. For example, at first, T5 did not ask C5 to read out loud each element of the pattern. She told her that there was a pattern and then asked her to pick out the element that should come next. After waiting a bit and seeing that the child sat still and did nothing, she then requested the child to say out loud each element of the pattern from the beginning. After that, C5 continued with the task and picked out the next element (correctly). T5 then continued with this instruction for each additional pattern (and answered each one correctly). Other teachers did not wait to see what would happen, but from the beginning requested that the child say out loud each element. T7 began her interview with C7 by saying, "Let's read the pattern together, let's read." At that point, C7 did not read the pattern but stretched out her arm to take a blue square (the correct element for extending the pattern) from one of the containers and place it at the end of the given pattern. T7 stopped her, despite that C7 chose the correct way of extending the pattern, and said, "No, sweetie. Wait a minute. First, let's read it." Three of the five teachers who requested children to read out loud the patterns, and an additional two teachers, asked the child they interviewed to say what was in the containers.

Variations in task implementation also occurred while the child was actively engaged with the task. Some of those variations were queries into why the child chose one or another element. For example, when engaged in the second pattern (P2), one child mistakenly took a blue square, but then immediately switched it with a correct red square. The teacher then inquired, "Why didn't you put down the blue square and why did you put down the red one?" This type of intervention did not interfere with the child's performance, but was instead a way for the teacher to listen to the child's way of thinking. In this case, the child answered, "because here (pointing to the pattern), the square is red." In other instances, the teacher's intervention came about even before the child took action. For example, when placing on the table P4 (the first pattern that did not end with a complete unit of repeat) T4 said to the child, "Now pay attention." After placing on the table the last pattern, T4 said, "Now look closely at the pattern, and also look carefully where it ends." This type of intervention has the potential to alternate a child's performance. In this case, despite all these warnings, C4 incorrectly extended all of the last three patterns (those that did not end with a complete unit). Some teachers intervened when the child chose an incorrect way of extending the pattern. For example, when T6 asked C6 to extend P3 (the pattern with an ABB structure), C6 incorrectly added ABA. The teacher then pointed to each element in the unit of repeat and said, "Look closely. Square, triangle, and …" C6 then responded correctly, "triangle." Interestingly, C6 had previously extended P2 in an incorrect manner. Although placing the correct shapes to extend the pattern, the child chose incorrect colors. In that case, T6 did not intervene, and instead said, "Good." Perhaps the teacher was satisfied that at least the child had chosen the correct shapes. However, when it came to placing incorrect shapes for P3, the teacher (T6) intervened.

Children's performances

Results of children's performances on the task, for each pattern, are shown in Table 1 according to structure and if the pattern ended in a complete unit (Comp.) or an incomplete (Inc.) unit. An extension of the pattern was only considered correct if the child successfully extended the pattern by three elements. As can be seen, children performed better extending a pattern that ended in a complete unit of repeat than a pattern which did not end in a complete unit. We now address the question of whether different implementations affected the children's success in extending the pattern. Comparing results of children who read out loud the elements of the pattern before extending the pattern, and those who did not, the relative frequency of success was slightly greater for those children who did not first read out loud the pattern (see Table 2).

Structure

Reads out loud the pattern (N=5) Does not read out loud the pattern (N=4) P1 (AB Comp.) 4( 80 Regarding the placing of elements in containers, results (see Table 3) indicated that in general, the way the elements were presented made little difference to the children's ability to extend the pattern. Taking a closer look, for the first three patterns that ended in a complete unit of repeat, there was a higher success rate when the elements were separated by shape and color. However, when the patterns did not end with a complete unit of repeat, there was either no difference or there was a higher success rate when the elements were separated only by shape. Regarding other differences in implementations, few affects were noticed. For example, among the six children who were told explicitly before beginning the task that there was a pattern which needed to be extended, three children extended correctly only two of the six patterns; the other three correctly extended three, five, and six of the patterns. Among those who were not explicitly told that there was a pattern (three children), a similar variance in success rates was found. The same variance in success was noted regarding children who were requested to say which elements were in each of the containers.

When analyzing the errors made by children, we found that the most prevalent mistake when attempting to extend a pattern that did not end with a complete unit of repeat was to continue the pattern as if it had ended in a complete unit, i.e., adding the first three elements from the beginning of the pattern. For example, C2 continued P4 by adding a square, triangle, and then a square. Another type of mistake which occurred for patterns both that ended and did not end with a complete unit of repeat, was to continue the pattern with ABAB despite there being a different structure to the given pattern. This occurred for C6 who added BA to an ABC structure, and also an ABB structure. Likewise, C9 continued P6 by adding BABABABA. Another type of mistake was taking the correct shape, but with the wrong color, as was demonstrated above by C2. This last type of mistake was directly related to the way the task was implemented. Obviously, if only the correct colors of shapes would have been on the table, this type of mistake could not occur.

Summary and discussion

As part of the professional development program, teachers were supplied with the materials for implementing the repeating pattern task. They were given laminated strips of paper with the patterns printed on them in color. They were given the matching pictures of colored squares, triangles, and circles to be cut out and placed in containers. They were even told what to ask each child. Yet, many variations occurred when implementing the task. Some of the variations occurred in the setup of the task, specifically with placing the elements in containers. Some of the variations occurred in the midst of implementing the task. When reflecting with the teachers on their implementations, it became apparent that these variations occurred spontaneously, without planning for them ahead of time. And yet, several of the teachers had the same ideas, such as having the children read out loud the pattern. Knowledge of tasks includes knowing the affordances and constraints of that task [START_REF] Watson | Seeing an exercise as a single mathematical object: Using variation to structure sense-making[END_REF]. It could be that the teachers saw this task as affording the opportunity to review with the children names of two-dimensional figures. According to [START_REF] Zaslavsky | Meeting the challenges of mathematics teacher education through design and use of tasks that facilitate teacher learning[END_REF], teaching tools include not only materials, but other kinds of resources, such as language and time. It could be that teachers were incorporating the tool of language into this given task, having children say the names of the shapes in the pattern. It could also be, similar to other studies [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF] that teachers were attempting to lighten the cognitive load of the students by telling them that there was a pattern and having them say out loud the names of the shapes which repeated themselves. In any case, as teacher educators it is important to be aware that teachers may implement a given task in various ways. In fact, in our program, the teachers brought the videos of their implementations back to the program, and as a group, we viewed them together. This, in turn, enhanced the teachers' knowledge of tasks, including their knowledge of the way children engage with repeating patterns tasks.

Despite the variations in implementations, most of the outcomes were consistent with previous research. For example, children had greater success when extending patterns which ended in a complete unit of repeat than those which did not [START_REF] Tsamir | Which continuation is appropriate? Kindergarten children's knowledge of repeating patterns[END_REF]. Children in this study made similar errors as children in other studies, such as extending an ABB pattern with ABA (Rittle- [START_REF] Rittle-Johnson | Emerging Understanding of Patterning in 4-Year-Olds[END_REF]. Can we conclude then that variations seen in this study had no impact? Certainly, a study with nine children is not enough to make such a conclusion, but it does leave us with an interesting question. How come the variations seen in this study (e.g., telling the children that there was a pattern, reading out loud the elements of a pattern, placing the elements in various containers) did not seem to impact on children's performance? The answer to this question perhaps lies in acknowledging the essence of repeating patterns, which is the unit of repeat and its structure. None of the variations in implementations focused the child on the unit of repeat. What seemed to impact on results was the complexity of the structure as well as if the pattern ended in a complete unit of repeat. By reviewing these results with preschool teachers, noticing the variations as well as the little affect they had on children's performances, we strengthen teachers' appreciation for the structure of a pattern, and promote their knowledge for teaching repeating patterns in preschool.

Figure 1 :

 1 Figure 1: What comes next?

Table 1 : Frequency (%) of successfully extending each pattern (N=9)

 1

		P1 (AB	P4 (AB	P2 (ABC	P5 (ABC	P3 (ABB	P6 (ABB
		Comp.)	Inc.)	Comp.)	Inc.)	Comp.)	Inc.)
	Frequency	8 (89)

Table 2 : Frequency (%) of success per variations in reading out loud the pattern

 2

Table 3 : Frequency (%) of success per variations in containers.

 3

Proceedings of CERME10

(67) 6 (67) 6 (67) 2 (22) 4 (44)

Acknowledgment

This research was supported by The Israel Science Foundation (grant No. 1270/14).


domain: math shs

text:
Chrysanthi Skoumpourdi
email: kara@aegean.gr

A framework for designing inquiry-based activities (FIBA) for early childhood mathematics

Keywords: Inquiry-based activities, early childhood mathematics, framework

Nowadays it is widely agreed that students should be involved in critical and creative thinking about mathematics concepts and ideas in order to organize and consolidate their thinking, as well as to reflect and construct their own learning. Inquiry-based mathematics education, as a teaching and learning method among many others, seems to be a fundamental way to initiate and support students' effective involvement to the teaching/learning process. In this method, in which the role of students, teachers and activities is changing, the teachers must be also able to design and implement mathematics activities that are open to an inquiry-based process. In this paper, a framework for designing inquiry-based activities (FIBA) for early childhood mathematics is proposed, accompanied with an example, aiming to assist teachers in their instructional design.

Introduction -Theoretical background

In the last decades, it is argued that instructional design should support a critical [START_REF] Skovsmose | Towards a philosophy of critical mathematics education[END_REF] and a creative mathematics education [START_REF] Leikin | Bridging research and theory in mathematics education with research and theory in creativity and giftedness[END_REF]. The teaching that was based on the simple reproduction of given conceptual contents and the transmission of knowledge has yielded space to the teaching based on student's own activities, that contributes to a meaningful learning and stimulation of their own identity as an emerged process (Van Oers, 2010).

In this new context, in which mathematics education is based on inquiry1 , teaching includes exploration, practical action, developmental thinking, connection of mathematical ideas and contents, problem solving and posing, collaborative learning, as well as autonomy to the development of ideas and methods [START_REF] Artigue | Inquiry in mathematics education[END_REF][START_REF] Ulm | IBME in schools: Overview and examples in international contexts[END_REF]. [START_REF] Clarke | Challenging and effective teaching in junior primary mathematics: What does it look like[END_REF] describing the effective mathematics teaching, set as a starting point the use of unusual problems without suggestions for their solution and the use of materials and other resources relevant to the content of the problems but also to the students' interests and needs. Moreover, they propose to address the classroom as a 'community of learning', that develops mathematical discourse and encourages the expression of students' ideas and strategies, focuses on the big ideas of mathematics, uses informal assessment methods to support instructional decisions, as well as facilitates students to act and think. In a "community of inquiry" teachers learn from students and students learn from teachers solving and posing mathematical problems to each other [START_REF] Goodchild | Critical alignment in inquiry-based practice in developing mathematics teaching[END_REF]. In this perspective, qualitative teaching, takes place by encouraging the creative and critical participation of students [START_REF] Radford | Signifying and meaning-making in mathematical thinking, teaching, and learning[END_REF] and adoption of exploratory and expressive ways of learning. As Sierpinska argues, "it makes sense to assume that learningmathematics or mathematics teachingis either inquiry-based learning or it is not learning at all" (2016, p. 55).

In the inquiry-based mathematics education, the roles of students, teachers and activities, as factors that jointly shape the instructional design, are changing. Students are no longer considered as the passive recipients of knowledge but as co-constructors and co-researchers of their knowledge assuming greater responsibility for their actions [START_REF] Lau | Constructing mathematics in an interactive classroom context[END_REF]. They are at the center of the learning process; they actively participate in it and are strongly engaged with mathematics problems. They are actively involved in explorations and discussions, and they pose questions, that were usually posed by teachers, in order to express an opinion, to share and explain their ideas, to make assumptions and generalizations, to present their solutions to a problem, to justify their creations and to support or prevent the ideas of their peers. When students are forced to describe their strategies in detail and justify them, they promote their understanding. Students' creative participation in the teaching and learning process of mathematics is related with producing new ideas, giving meaning to symbols, materials and tools, facts and procedures, understanding math problems, making plans and devising ways to solve them, as well as finding ways to assess the logic of their solution [START_REF] Haylock | A framework for assessing mathematical creativity in schoolchildren[END_REF].

The role of the teachers, since they are not the key persons of the teaching/learning process, is also changing and instead of being the sole sources of mathematical knowledge, who only focused on practicing the mathematical operations and procedures, they become supporters of children' mathematics constructions [START_REF] Lau | Constructing mathematics in an interactive classroom context[END_REF]. They work cooperatively with students to create the appropriate conditions and organize the process and the ongoing opportunities for them, to explore, to make connections, to build mental representations and to develop mathematics concepts based on their prior or/and informal knowledge through the interactions with others. They cultivate to students the need to communicate their actions, their creations, their solutions with the use of materials or other auxiliary means, to discuss their mathematical ideas, to share their thoughts, to cooperate [START_REF] Shriki | Working like real mathematicians: Developing prospective teacher's awareness of mathematical creativity through generating new concepts[END_REF]. They help students to look for multiple solution methods and multiple outcomes for a situation [START_REF] Tsamir | Multiple solution methods and multiple outcomes -is it a task for kindergarten children?[END_REF], to identify relationships and common characteristics in different cases, to reflect and justify their thoughts and actions. Teachers transfer the problem-solving responsibility to students without, at least initially, intervening and suggesting ways of thinking. They encourage children to think and present their solutions to the rest of the class. The discourse, which includes the presentation of students' results, solutions, strategies and methods, highlights students' own considerations and encourages the exchange of their ideas [START_REF] Ryan | Children's mathematics 4-15. Learning from errors and misconceptions[END_REF]. Teachers pose questions related to students' ideas in such a way, as to explain their thinking and imagining, as well as giving meaning to their actions and developing their own understanding. It is important that the posed questions are linked with the task and its solving. The questions are not supposed to fulfill teachers' desire to teach mathematics, but have to encourage the emergence of multiple strategies, to make clear the relationships between these strategies, as well as to give opportunities to students to integrate the pieces of their knowledge.

Activities' type, in the new cooperative, critical and creative classroom context, is changing. Activities arise from tasks that are problem-based with sufficient openness for inquiry-based learning [START_REF] Ulm | IBME in schools: Overview and examples in international contexts[END_REF]. The context of these tasks can be authentic or not, known or unknown to students and can be associated with everyday activities, stories, games, workshops etc. Complex and creative situations allow various solutions and reasoning. They enable the use of students' informal knowledge and empirical reality. They are 'realistic' experiences, supported by a variety of materials and other means (Varol & Farran, 2006). Materials that are related to the task are used and aim to help students to investigate the situation, to find answers to the questions posed, as well as to pose their own questions. The main aim of the activities is not just to be carried out by the students, because it is the lesson of mathematics and they should do so, but the challenge to deal with the situation presented to them. They challenge students to connect mathematics with their daily life (Van den [START_REF] Van Den Heuvel-Panhuizen | The role of contexts in assessment problems in mathematics[END_REF] and encourage them to describe the different ways in which they perceive things. Activities that are realized in many ways require students to conjecture, to interpret, as well as to justify their thoughts (NCTM, 2007). They encourage reflection and communication for students, to construct mathematical meaning using skills and knowledge they already hold [START_REF] Varol | Early mathematical growth: How to support young children's mathematical development[END_REF]. Moreover, they have to be interesting, effective and developmentally appropriate inducing the active participation of students [START_REF] Tzur | Fine grain assessment of students' mathematical understanding: participatory and anticipatory stages in learning a new mathematical conception[END_REF] to mathematics discourse. The solution of the problem is not readily apparent. To solve the problem, students have to use something more than a routine solution or an algorithmic process. The solution of the problem may be proposed by a student or a group of students. It may occur during the discourse or be suggested by the teacher. The mathematical discussion is completed when the whole class agrees to a mutually acceptable solution/s for the task, which can consist of many answers. In each case, children "earn" if the discussion, that took place, has set the basis for them to understand that the proposed solution is reasonable for this reflection. Then, when possible, the solution is modified and generalized. The generalization begins when reasoning gets independent, when it does not refer to the specific context in which it was created, and continues when the common characteristics of the different ideas are combined. Tasks can be either structured or unstructured but are open to an inquiry-based implementation.

The design of creative and critical activities that can be realized within an inquiry-based process that encourages students to decide for themselves when and how to use a method, a process or a strategy, can lead to meaningful learning. Such activities allow the creation and communication of students' reasoning cultivating skills such as prediction, fast perception of information, systematic reasoning, critical and creative thinking, problem solving and posing, assessment, comparison and correlation, as well as generalization, skills that are necessary for the future citizens [START_REF] Sarama | Early childhood mathematics education research: Learning trajectories for young children[END_REF]. This type of activities is not usual in the mathematics classrooms because its design and implementation is difficult [START_REF] Sierpinska | Inquiry-based learning approaches and the development of theoretical thinking in the mathematics education of future elementary school teachers[END_REF]. It is easier for teachers to teach facts, procedures and problem solving processes through structured situations even if these situations are often not understood by the students and do not influence their thinking. Children, often, because they have to deal with routine and structured mathematical tasks, that require the implementation of specific strategies, do not sense when and how to use a mathematical procedure to solve a problem. Research results pinpoint that it is essential for the mathematics education to include also inquirybased activities [START_REF] Artigue | Inquiry in mathematics education[END_REF]. Thus, a framework for designing inquiry-based activities for early childhood mathematics (FIBA) is proposed, accompanied with an example, to assist teachers in their instructional design.

Framework for designing and implementing inquiry-based activities (FIBA)

Taking into consideration the above main points of inquiry-based mathematics education, a framework for designing and implementing activities is proposed. The framework consists of seven stages (figure 1): through a context, resulting from children's interests, experiences, knowledge, queries (emergent from a previous free talk and sharing experiences with the students in the class which is based on the curriculum and is in accordance with a mathematical purpose). The task, which may have one or more solutions, is designed in such a way that it problematizes and incites children, in order to engage them in a problem solving and posing process. The problem to be solved could be a nonstandard, unfamiliar, a bit complex and novel situation in order not to be solved just by applying existing knowledge and already-known strategies, but through exploration.

2. Exploration (key person: students): Children (individually or in groups) use their own (informal) problem-solving strategies to explore the problem introduced by the scenario, to choose/use materials and other auxiliary means, to make conjectures, to pose questions to each other and to the teacher for understanding/grasping the situation and to suggest solutions, in order to 'solve' the problem. In that stage, students have the opportunity to reflect and think about the problem on their own, before sharing their thoughts with their peers. They are also free to discuss their ideas about the problem with their peers before presenting them to the whole class.

3. Presentation (key person: students): Children share their explorations with the whole class by presenting/describing their ideas, constructions, solutions, experiences etc. Teacher, in that stage, is an observer and organizer of each team's presentation, orchestrating students' contributions, posing questions to help children describe, explain and open out their explorations. He/She also encourages students to pose questions to their peers, from other groups, in order to ensure that they understand all the presentations.

4. Connection (key person: teacher): Teacher, in cooperation with students, summarizes the results, poses questions and encourages students to ask questions that connect the presented ideas with each other, with the task that was explored and with the mathematical aims, in order to construct the common meaning that the classroom would share. Teacher's questions have to encourage mathematical thinking and reasoning and can be of several types [START_REF] Carlsen | Orchestration of mathematical activities in the kindergarten: The role of questions[END_REF]. At that stage, it will become apparent if cooperative grouping strategies are effective in promoting classroom discourse.

5. Generalization (key person: teacher): Teacher is generalizing (and mathematizing when and if possible) students' actions, shaping the mathematical concept, connecting it with students' previous knowledge and giving feedback to them.

6. Translation (key person: students): At that stage, students are asked to communicate to others (students from another class, family etc.), their solutions, creations, understandings etc. through different modes-verbalizing, gesturally and schematizing (Skoumpourdi, 2016a).

7. Expansion: Students are asked to pose/solve related/changed/expanded problems.

In all the above stages, teachers are also responsible to take students' questions and comments into consideration, turn them into learning opportunities incorporating them to their instructional design, creating a new problem-based task.

The Pattern King: An example of an inquiry-based activity

Given that, the importance of patterning is increasingly highlighted in recent years and that, to further improve the performance of children on patterning, teaching interventions are necessary from the early years of schooling [START_REF] Skoumpourdi | Patterns for kindergartners: A developmental framework[END_REF], in this chapter, an example of a pattern task 2 is presented, through the seven stages of the FIBA, supporting an inquiry-based teaching/learning process of mathematics.

Task: "The Pattern King forgot the password of his secret room and thus he cannot enter.

Fortunately, he has photographs of the passwords he used to use and he asks your help to construct them". The students, (individually or) in groups, pick a drawing with chain schematization/password 3 .

Exploration: Each (student or) group explores its schematization. They pose questions to the teacher to understand the situation. They observe how it is made and suggest ways for creating it. After the discussion, they choose the associated materials (connected shapes 4 ) in order to construct the pattern/password and they create a/the chain.

Presentation: Each (student or) group presents the structure of its creation to the whole class, 2 This is just an example of an inquiry based pattern activity. Taking into consideration the pattern's four functional characteristics [START_REF] Skoumpourdi | Patterns for kindergartners: A developmental framework[END_REF], as well as students' pattern abilities and their type of performance [START_REF] Skoumpourdi | Kindergartners' performance levels on patterning[END_REF] a wide range of critical and creative pattern tasks can be designed which can engage children in challenging patterning experiences. A different combination of these characteristics leads to different pattern tasks with a varying degree of difficulty.

3 Examples of chain schematizations/passwords bringing side to side the construction with the schematic representation. The teacher is posing questions to ensure effective and understandable presentations. Also, students are encouraged to pose questions to their peers. Generalization: Teacher is shaping the pattern concept, connecting it with students' previous knowledge to generalize it. Questions are more general and are related to the pattern construct. For example: "Are these patterns the same"? "What is repeated in this pattern?" etc.

Translation: "Can you draw your creation to show it to the students of the other class?" In this stage students 'translate' their constructs and understandings for patterns and shapes to the paper.

Expansion: "A magician invited the King to play a game. He was sure that he would win him, to get his palace. The magician has two dice, one with numbers and one with shapes. Each of them-the king and the magician-would throw the dice and would take as many shapes as the dice indicate.

The one that would create the longest chain with patterns will be the winner. Come to play the game to help the King make a very long pattern chain." In that stage students expand their understanding by creating their own patterns.

At the end of the inquiry process, students would have used the connected shapes to copy the patterns or to construct their own ones. They would have translated their constructions to schematic representations, using paper and pencil, as well as verbalized them describing the pattern construct. They would also have discussed about the shapes' names (circle, triangle and rectangle) and their orientations, as well as about length comparison.

Conclusions

Teachers, in their educational practice, are asked to invent mathematical tasks for their students, to design activities relied on them and implement them, as well as to reflect on the outcomes of the implementation. Yet it is not easy for teachers to design inquiry-based activities and implement them in classroom. They prefer to stay within the comfort of the usual school tasks which are routine and structured and they design teacher-centered activities that reproduce the given conceptual contents and are solved by memorizing facts and processes.

But if we take into consideration the argumentations of the last decades for the inquiry-based mathematics education, the design and implementation of inquiry-based mathematics activities, which are beyond the usual school tasks, will help students to develop their curiosity and creativity, their ability for critical exploration, reflection and reasoning and their autonomy as learners leading them to mathematical understanding [START_REF] Artigue | Inquiry in mathematics education[END_REF].

Using the proposed framework one can design and implement activities, for early childhood, supporting an inquiry-based teaching/learning process of mathematics. Through the seven stages of the framework which include: 1. the formulation of the task, 2. the exploration of the problem posed by the task, 3. the presentation of the explorations, 4. the connection of the presentations with mathematical aims, 5. the generalization of the connections and mathematical concepts and ideas, 6. the translation of the understandings on other modes and 7. the expansion of the initial task to a modified one, students' and teachers' roles alternate. During their cooperation they are both the key persons, the co-constructors and co-researchers of the teaching/learning process.

We do not support that all of mathematics activities should be designed and implemented in an inquiry-based process but we do believe that such activities should be adopted to develop different skills in students and to actively involve them in their own learning.

Figure 1 :

 1 Figure 1: The seven stages of the FIBA 1. Τask (key person: teacher): A problem-based task is invented and presented, by the teacher,through a context, resulting from children's interests, experiences, knowledge, queries (emergent from a previous free talk and sharing experiences with the students in the class which is based on the curriculum and is in accordance with a mathematical purpose). The task, which may have one or more solutions, is designed in such a way that it problematizes and incites children, in order to engage them in a problem solving and posing process. The problem to be solved could be a nonstandard, unfamiliar, a bit complex and novel situation in order not to be solved just by applying existing knowledge and already-known strategies, but through exploration.

  The questions can be: "Can you describe us what did you construct? Which shapes did you use? Which shapes are similar? How many shapes did you use? Why did you put the triangle in this place? Which shapes are repeated? Do you observe any pattern?" etc. Connection: Teacher, in cooperation with students, orchestrates a mathematical discourse trying to connect presentations/creations with each other, with the Pattern King's problem, as well as with the pattern concept in order for a shared meaning to emerge. Questions can be: "What do you observe to this construct? In what, these constructs differ? To what are these constructs similar?" etc.

The IBME (Inquiry Based Mathematics Education) perspectives are based on approaches that have been developed in the field of mathematics education, such as the problem-solving tradition, among others, which tend to shape it in a particular way. But, in the problem-solving tradition the focus was on 'teaching problem solving' by developing problem solving skills and associated metacognitive competences, whereas in the IBME the focus is on 'teaching via problem solving'.Thematic Working Group 13Proceedings of CERME10

Proceedings of CERME10


text:
Esther Levenson
email: levenso@post.tau.ac.il

Manya Raman-Sundström
email: manya.sundstrom@umu.se

Young children's aesthetic development in the context of mathematical explanation

Keywords: Aesthetics, explanation, even numbers, triangular numbers, affect

Mathematicians routinely report that beauty is both a reward and a motivation for the work they do. However, how and to what extent children can appreciate mathematical beauty is an open question. This exploratory study looks at young children (ages 6-12, with a focus on the younger years) as they evaluate different explanations of claims about even numbers and triangular numbers. While our results are fairly speculative, we provide case studies which illustrate possible kinds of aesthetic reactions, and some of the factors which might impact on those reactions.

Introduction

There is little doubt that mathematicians have rich, aesthetic lives [START_REF] Sinclair | The roles of the aesthetic in mathematical inquiry[END_REF]. While recent research has attempted to characterize what exactly this aesthetic life consists of [START_REF] Raman-Sundström | Mathematical fit: A case study[END_REF] or what is meant by aesthetics in mathematics in the first place [START_REF] Rota | The phenomenology of mathematical beauty[END_REF], evidence suggests that aesthetic reactions are common in, and perhaps even central to, the working lives of mathematicians. According to some, aesthetic reactions include pleasure, tension, surprise, and a sense of being compelled (or repelled) [START_REF] Marmur | Surprise and the aesthetic experience of university students: A design experiment[END_REF]. Interestingly, neuroscientists suggest that the same region of the brain is involved for judging mathematical equations and works of art [START_REF] Zeki | The experience of mathematical beauty and its neural correlates[END_REF].

Recently, there has been interest in studying aesthetics in mathematics education. While some of those studies have investigated school children's aesthetic reactions while working on mathematical problems (e.g., [START_REF] Sinclair | Mathematics and beauty: Aesthetic approaches to teaching children[END_REF], few have focused on young children. Yet, research has shown that young children are capable of sophisticated proof-like reasoning [START_REF] Maher | The development of the idea of mathematical proof: A 5year case study[END_REF], justification, and argumentation [START_REF] Tatsis | Kindergarten children discussing the fairness of probabilistic games: The creation of a primary discursive community[END_REF]). Thus, it seems reasonable to ask whether young children could be capable of aesthetic experiences, and if so, at what age. Are young children able to experience the satisfaction of finding a good explanation? Do they find pleasure in coming to understand an explanation?

This paper presents an explorative study of the possible aesthetic experiences of young children. Mathematics, like other aesthetic subjects, provides experiences that have the potential to draw people in. It also provides a reward or a sense of satisfaction [START_REF] Sinclair | The roles of the aesthetic in mathematical inquiry[END_REF]. What makes the experience "complete" is the presence of both a startup phase and a reward phase. We study the presence or absence of aesthetic reactions of young children (age 6-7) by comparing their behavior to an older cohort (ages 9-10). We find that the younger children, while lacking some of the insight of the older children, have what we might call "aesthetic dispositions" that allow them to enjoy and to be curious about fairly complex mathematical tasks.

Theoretical background

One of the central questions of aesthetics is whether beauty is objective or subjective. According to [START_REF] Marmur | Surprise and the aesthetic experience of university students: A design experiment[END_REF], those that consider mathematical beauty as objective (such as [START_REF] Dreyfus | On the aesthetics of mathematical thought[END_REF]) list characteristics such as clarity, simplicity, brevity, and conciseness when judging theorems and proofs. In other words, beauty is an intrinsic property of the mathematical object. Those that consider beauty to be subjective, claim that mathematical beauty is in the eye of the beholder and that experience, age, knowledge, and culture contribute to aesthetical views. [START_REF] Marmur | Surprise and the aesthetic experience of university students: A design experiment[END_REF] integrate both views, concluding that when discussing school mathematics, we may hypothesize that a mathematical problem might elicit an aesthetic experience among students because of its simplicity or surprising result. Ultimately, however, students may or may not have an aesthetic experience depending on, among other things, the pedagogical setup of the problem.

In their study of university students, Marmur & Koichu (2009) found that surprise was integral to experiencing mathematical beauty. They found that students who referred to a solution as beautiful had first struggled with the problem, had put significant effort into finding the solution, and ultimately were surprised at the simple and unexpected solution. Struggle was also a factor in [START_REF] Brinkmann | Mathematical beauty and its characteristics: A study on the student's point of view[END_REF] study of middle and upper school students' appreciation of mathematical beauty. A problem was considered to be beautiful if it had a certain degree of complexity, yet felt solvable. [START_REF] Eberle | The role of children's mathematical aesthetics: The case of tessellations[END_REF] investigated students' (ages 8-10) aesthetic attractions when evaluating geometric tessellations. Students referred to several characteristics of the geometric objects which contributed to their appreciation, such as real world connections, color, complexity, and uniqueness. [START_REF] Eberle | The role of children's mathematical aesthetics: The case of tessellations[END_REF], as well as Sinclair (2001) also noted the generative role of aesthetics when students were involved in inquiry-based tasks. In both studies, aesthetics led students to engage and play with the mathematics, guiding them when deciding which direction to pursue.

The above studies related to aesthetics with regard to problems, solutions, and geometric objects. In our study, we focus on mathematical explanations and young children's appreciation of those explanations. Previously, [START_REF] Levenson | Fifth-grade students' use and preferences for mathematically and practically based explanations[END_REF] found that fifth-grade students have preferences regarding different types of explanations. Students' preferences were based on clarity, brevity, relatedness, and because the explanation was perceived as fun. Although some of the reasons students mentioned for their preferences are reminiscent of aesthetic evaluations given by older students, and even mathematicians, the focus of that study was not specifically on aesthetic appreciation or satisfaction from an explanation. In this study, we draw on a theory of explanation developed by [START_REF] Gopnik | Explanation as orgasm and the drive for causal knowledge: The function, evolution, and phenomenology of the theory formation system[END_REF], which helps explain what makes certain explanations satisfying. Gopnik suggests that an explanation consists of two parts, the why? (the 'hmmm….' phase) and the because! (the aha! or wow! phase depending on how surprising the result is for the individual). An exploration can have a why? without a because! and vice versa. Both phases are needed for an explanation to be found satisfying. Moreover, Gopnik suggests that both the seeking and the satisfaction from finding a good explanation are part of our human nature.

The aim of this study is to begin an exploration of young children's appreciation of mathematical beauty. Specifically, we ask: Do young students' aesthetic reactions (or non-reactions) to mathematical inquiry and explanation differ from that of older students? Are younger and older students capable of an experience that contains both the hmmm… and aha! phase of a mathematical explanation?

Methodology

Data was collected using semi-structured interviews with two cohorts of children, one aged 6-7 years old, and one aged 9-10 years old. The older children worked with explanations about triangular numbers, namely that the number of dots in the n th triangular number is n (n+1)/2. This cohort included four fifth grade girls, all from the same class, who sat together in a group with the interviewer in one of the girl's houses. The discussion began with introducing the girls to triangular numbers, discussing the number of dots in the first five triangles, and then asking them to come up with the number of dots in the 100 th triangle. After giving them time to work on the problem, the girls were shown two solutions and asked to evaluate each solution.

The younger cohort of three children worked individually with the interviewer with explanations of the claim: the sum of two even numbers is always even. Two interviews were conducted in the house of the child and one was conducted in the house of the child's grandmother. Each interview began in the same way, asking the child to say if he or she could give examples of even numbers and to say why those numbers were even. After confirming that the children were familiar with even numbers they were given the following question: what would happen if you add two even numbers, would the answer be even or odd? Children were given time to think and reply. Although the interviewer had several explanations on hand for the children to evaluate, as will be shown in the next section, only one child was asked to evaluate explanations.

One of the difficulties of studying aesthetics in children, or even with mathematicians, is how to detect an aesthetic experience. While there may be bodily clues, such as changes in eye-dilation or neural correlates (e.g., [START_REF] Zeki | The experience of mathematical beauty and its neural correlates[END_REF], a natural place to start is by simply listening to what people say (see [START_REF] Wickman | Aesthetic experience in science education: Learning and meaning-making as situated talk and action[END_REF] and watching for engagement (or disengagement) during the experience. In this study, we take this approach as a first approximation, using key words (taken from the background studies) such as "Wow!" and "Funny!" as markers for a general aesthetic experience.

Findings

Below we present four episodes, one with older girls and three with younger children. What is striking about the young children is that there is no sense of surprise. The children seem to lack the 'hmm….' needed to build the wow! or to even warrant an explanation.

Fifth-grade girls, age 11

Trying to figure out how many dots will be in the 100 th triangle proves challenging to the girls. When they realize that they would have to sum all of the numbers from 1 to 100, the interviewer gives them some time to work this out and then explains to the girls the Gaussian method for summing an arithmetic sequence. She lines up the numbers from 1 to 100 in one row and on top of that row, lined up the numbers from 100 to 1 (see Figure 1), explaining that this shows how many dots are in each row of the 100 th triangle. Girls: Ahh!

Esther:

1 and 100, 2 and 99, 3 and 98.

Trina: Wow! It's great. It's the same thing.

The girls are surprised that the sums all add to the same number, 101. Their remarks of "Ahh" and "Wow" indicate their pleasure in this simple conclusion. After showing the girls this explanation, the interviewer shows them a second method, that of drawing two congruent triangles, inverting one and placing it next to the first, thus creating a rectangle. The number of dots in the triangle is then equal to the area of the rectangle divided by two (see Figure 2). After establishing that the girls understand both explanations, the girls are asked to compare the two methods.

Esther: Which explanation of the method gives you more satisfaction?

(All of the girls point to the second explanation with the dots.) Amanda: You simply see that you do this times this, and then divide by 2 because you have 2 triangles.

Hailey:

Because instead of computing it all, this is easier and simpler and in front of your eyes.

Amanda: Also, it draws attention more. It's more fun, but not just more fun, it's like, it goes more into your head.

In the second segment, the girls claim to like the second explanation better because it is simpler and because they can "see it." This hints at their appreciation for the aesthetic value of efficiency, which might have been enhanced by their struggle to find the solution.

Zev, age seven

Zev is seven years old, attends first grade in Israel, where learning about even and odd numbers is part of the curriculum in school. Zev is able to list several even numbers as well as several odd numbers. When asked why eight is an even number he says, "because four is even and … because four is even and it's… and also… the second four is even." Note that he does not stress that 8 could be written as the sum of two equal whole numbers, but rather that both of the addends, in this case fours, were both even numbers. When asked why 10 is an even number, the following discussion ensues:

Esther: OK. Is ten an even number?

Zev:

Yes.

Esther: How do you know that ten is even?

Zev:

Because odd plus odd is even.

Here Zev was probably thinking that 10 results from 5 + 5, both odd numbers. Realizing that Zev seems already familiar with summing odd numbers, the interviewer asks about the sum of two even numbers:

Esther: And what about an even number plus an even number?

Zev: Even.

Esther: Always?

Zev: Yes.

Esther: Can you tell me how you know that even plus even is always even?

Zev: Four plus four, eight. Eight pus eight, sixteen.

Esther: How do you know that sixteen is even?

Zev: Because ten is even, and six is even.

For Zev, it seems that his working definition of an even number is of a number that can be written as the sum of two other even numbers. Although this is a recursive definition, it does not seem to bother him and the outcome of this conception is that Zev does not even recognize the question of what might be the sum of two even numbers. Zev has no sense of hmm…. When Zev is shown another explanation for why the sum of two even numbers is always even (that every even number can be written as the sum of twos and thus the sum of two even numbers can also be written as the sum of twos), Zev says that the explanation is boring because he already knew that.

Anna, age six

Anna is six years old and attends kindergarten in Israel. Although even and odd numbers are not part of the kindergarten curriculum, Anna was able to list several even and odd numbers and claimed to have learned about them from her teacher. When asked why eight is an even number she responded, "Because each one has a partner." When she was asked to explain what an even number is, she said, "That the two of them have a partner. That each of them has a partner." What Anna is alluding to in her own language is that an even number may be written as the sum of twos. After talking about even numbers for a few minutes, we discuss the sum of two even numbers:

Esther: What would happen if I added an even number with another even number?

Anna: It would be even.

Esther: How do you know?

Anna:

Because both of them are not separate, it never separates from the second.

Esther: What do you say! Are you sure that this is always this way?

Anna:

Yes. It will never be that, uhmm, a partner doesn't run away from the pair.

Anna has a very specific conceptualization of even numbers, that an even number represents an inseparable pair. She draws on this conceptualization to explain why the sum of two even numbers must always be even. Engaging further with the problem, Anna takes out a bunch of wooden blocks from a basket (without counting) and proceeds to pair up blocks. When the interviewer asks if she can say if she took out an even or odd number of blocks she readily says that she took out an even number of blocks because on the table, every block is paired off. To summarize, Anna is drawn in to the problem, has an explanation, but she does not struggle with the claim.

Leila, age seven

Leila is from Sweden, has just turned 7, and attends kindergarten. Although learning about even and odd numbers is not part of the curriculum, she is able to list several even numbers (perhaps hearing about them from her parents or friends). Although prompted to think about the possible sum of any two even numbers, Leila shows no interest and instead, with the use of colorful cubes, begins to explore a self-generated conjecture involving pairs of twos. She puts together six pairs of cubes to represent an even number and then wonders if seven pairs will still be even.

Leila: I have a question. You can't have an odd number of twos.

Manya: How are you thinking?

Leila: Like, if I have these (pointing to the six pairs of cubes), you can never have an odd number. Look here. I have 1, 2, 3, 4, 5, 6. If I have this many (she takes another pair), now I have enough…. odd numbers of twos… because there's seven… does it get an odd number or an even? I think it is actually funny. If you have an odd number of twos, it even gets an even number. That I think is really funny.

Manya: I see. Why is that?

Leila: I don't know, because it feels like it's actually pretty funny.

Manya: Do you think it's funnier to have an even number made of an odd number of twos than an even?

Leila: What do you mean -an even number of odd number …? Yeah. It gets an even number if you have an odd number of twos. Because I figured it out of these.

This discussion suggests that autonomy might play a role in having an aesthetic experience. Leila generates her own question (What is the sum of two odd numbers?) and the resolution of the question (that an odd number of twos can be even). She thinks that this conclusion is funny. While she might have had a similar reaction to a given statement, the affect seems to be closely correlated to ownership of ideas. In the last two lines, the researcher asks her to compare two statements. Leila replies "what do you mean"? The statement is funny to her because she "figured it out of these."

Comparing the older cohort to the younger cohort

The following table summarizes the observations from the data presented above and the children's aesthetic experiences (AE). A full circle represents that an AE took place, a dotted circle reflects a partial experience, and an empty circle that no AE was detected. In all of these cases, we can see that the path to having an aesthetic experience consists of several distinct phases. One needs to be engaged; there should be some build-up, some crucial moment, and then some release. Only the 10 year olds seem to have a complete experience. Build-up takes place when they explore the question, and their interest increases as the interviewer shows the explanation involving pairs of numbers. After having time to digest this information, the interviewer shows another explanation which further increases their interest. Each time a new explanation is understood, the girls say words like "wow!" and "aha!". The fact that they have this kind of reaction, we claim, is because they had time to explore and to start generating their own explanations. We also cannot rule out that there might be some developmental issues, such as the children being old enough to abstract and/or take in the explanations given.

In contrast to the fifth-graders, the younger students had limited or no aesthetic experience. Zev is strikingly uninterested in any explanation at all. We suspect that his disinterest came from the fact that he had been told in school that even + even is even, so there was no tension left to resolve. Leila has some interest in explanation, but not for the question given to her. Rather, she generates her own question about whether an odd number of pairs can result in an even number. She finds this result funny, indicating some level of surprise, which she quickly believed despite her initial expectation. Anna is drawn in to the explanation activity, but did not seem to have a full aesthetic experience. Unlike the fifth graders who could "see" why the explanations held, Anna simply states her conception of even numbers in terms of pairs or partners, and claims that any sum of pairs will still be even. She does not give an actual reason, which might be because she did not experience the hmmm… phase of explanation. She is not bothered by any alternative, so no relief or satisfaction is expressed.

Conclusion

One of the challenges of this study was to find tasks that might elicit an aesthetic reaction. We attempted to find tasks that would be suitably challenging, yet accessible to each of the age groups.

In the end, the older children worked on a new task, presented not only in general manner, but with an iconic illustration, while the younger children worked on familiar (at least for two children) general characteristics of numbers. Thus, it might be that the different conditions affected the aesthetic experiences. Taking these limitations into consideration, there is still the possibility of developmental differences in aesthetic experiences. In the naïve view, children have and rely on concepts, but are not yet puzzled. Because of this lack of puzzlement (an essential ingredient according to several researchers (e.g., [START_REF] Gopnik | Explanation as orgasm and the drive for causal knowledge: The function, evolution, and phenomenology of the theory formation system[END_REF]Marmur & Koichu, 2009)), there is no tension in their mathematical exploration, nothing to be resolved, and so no aesthetic experience is possible. In contrast, in the mature view, children are engaged and puzzled. They are more open to explanations because they themselves have struggled with the questions. This kind of behavior is possible among quite young children (Leila, at age 7, has a very small amplitude aesthetic experience when she generated her own conjecture), but might be more likely to occur the more autonomy is given to the students, the more challenging the task, and the more supported the students are to not to give up when they think they already have the answer. As an exploratory study, this paper has begun a discussion regarding young children's possible aesthetic experiences when working on mathematics. Additional research is needed to continue this discussion.

Figure 1 :Figure 2 :

 12 Figure 1: Summing the numbers from 1 to 100

Table 1 . Types of aesthetic experiences for the triangular number and even number tasks

 1

	Children Age Task	AE	Behavior
	Trina,	10	Triangular		Have an aesthetic experience, marked by both surprise
	Hailey,		numbers		and satisfaction. They appreciate a solution which is
	Amanda				simple and "goes into their head".
	Zev	7	Even		Has a conception of even numbers, but is uninterested in
			numbers		explaining why the sum of evens is even. He "already
					knows that".
	Anna	6	Even		Has a conception of even numbers which goes along with
			numbers		the claim about the sum of even numbers. Is engaged and
					involved, but is lacking a "hmmm…."
	Leila	7	Even		Takes control. Explores her own hypothesis. Tests if an
			numbers		odd number of twos can be even. Finds the result "funny".

Thematic Working Group 13Proceedings of CERME10

Proceedings of CERME10


text:
Sigurd Per

Ingvald Hundeland
email: per.s.hundeland@uia.no

Martin Erfjord

Carlsen

A kindergarten teacher's revealed knowledge in orchestration of mathematical activities

Keywords: Design research, kindergarten, Knowledge Quartet, mathematics, orchestration

The aim of this study is to employ the Knowledge Quartet, proposed by Rowland, Huckstep, and Thwaites (2005), in order to characterise the kindergarten teacher's competence when orchestrating researcher designed mathematical activities for 5-year-olds. We are drawing on design research as a methodology in which design principles such as playful learning and inquiry are implemented in the activities. Our analyses show that knowledge-in-action and knowledge-ininteraction are revealed in the orchestration. Hence, the Knowledge Quartet is applicable in the kindergarten contexthowever, with some modifications due to kindergarten particularities.

Introduction

For about a decade, mathematics has been part of the Norwegian curriculum for kindergarten, a curriculum situated within a social pedagogy tradition. In a kindergarten setting in Norway, a kindergarten teacher (KT) is supposed to empower the children's mathematical explorations. However, the curriculum is not explicit as regards how KTs are to facilitate the children's mathematical explorations. In this study we characterise one KT's orchestration of mathematical activities for 5-year-olds in terms of knowledge employed by the KT. The following research questions are formulated for the study:

In what ways may the Knowledge Quartet be employed in order to characterise the kindergarten teacher's competence? What knowledge-in-action and knowledge-in-interaction does a kindergarten teacher reveal in her orchestration of a mathematical activity on geometrical shapes?

The activities were designed by researchers in mathematics education (the authors of this paper among others), as part of an ongoing research and development project called "The Agder project" i , in order for the participating children to engage with mathematical concepts and ideas.

In this study we use the metaphor of orchestration. By this metaphor we mean that it is the KT who is in charge of and leads the mathematical activity. She has to plan, think forward, act in the moment, follow up children's questions and comments, adapt questions for each of the children, etc.

an important and by no means easy role to fulfil. Our admiration of this complexity is thus considerable when we set out to design the mathematical activities and study how the KTs orchestrate the activities.

Theoretical lens: The Knowledge Quartet

In order to analyse in depth how the Knowledge Quartet may be employed in order to characterise the kindergarten teacher's competence and the revealed knowledge-in-action and knowledge-ininteraction on behalf of the KT, we draw heavily on the Knowledge Quartet coined by Rowland, Thematic Working Group 13 Proceedings of CERME10 [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF]. The Knowledge Quartet was developed by [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] as a theoretical construct drawing on the profound and substantial work of [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF]. Rowland and colleagues drew on videotapes from mathematics classroom lessons where pre-service teachers utilised their mathematical and pedagogical knowledge in their teaching. This quartet is used as a theoretical lens through which we analyse video data of one KT´s orchestration of one of the developed activities. We limit ourselves to one KT taking into account that "the quartet is comprehensive as a tool for thinking about the ways that subject knowledge comes into play in the classroom" (Rowland, Huckstep, & Thwaites, 2003, p. 97).

The Knowledge Quartet differs from the framework of [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF], mathematical knowledge for teaching (MKT), in that the former focuses on situations in the mathematics classroom through which the teacher's mathematics-related knowledge may be observed. Ball et al.'s (2008) framework describes different kinds of mathematics teachers' knowledge. Mosvold, Bjuland, Fauskanger, and Jacobsen (2011) used the framework of [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF] to analyse mathematics teaching at the kindergarten level. [START_REF] Mosvold | Similar but differentinvestigating the use of MKT in a Norwegian kindergarten setting[END_REF] found that the MKT framework needs to be adjusted to the kindergarten context in order to be appropriately used. This is because (1) the work situation of orchestrating mathematical activities for a Norwegian kindergarten teacher is very different from the teaching situation of a U.S. mathematics school teacher; and (2) the tasks of teaching in the kindergarten setting are significantly different from that of the school setting. It is pedagogical activities which is the basis for learning activities in Norwegian kindergartens, not mathematical activities as such. These differences have been emphasised by [START_REF] Erfjord | Kindergarten teachers' accounts of their developing mathematical practice[END_REF], through the lens of the didactic triangle.

In adopting a grounded approach to the analyses of data, [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] identified what they labelled the Knowledge Quartet, four dimensions along which mathematics teachers' "mathematicsrelated knowledge" (p. 255) may be analysed. These four dimensions are termed foundation, transformation, connection and contingency. Foundation encompasses the knowledge background of the KT; transformation and connection encompass how and to what extent knowledge is revealed in action as the KT implements and orchestrates the activity, hence dimensions described as knowledge-in-action. Contingency encompasses the KT's knowledge as this unfolds in interaction with the children, hence knowledge-in-interaction.

Foundation is a dimension of propositional knowledge used, adapted to our case, to address the KT's mathematical knowledge, the KT's knowledge of mathematics education, and the KT's view upon the purpose of mathematics education and how children learn mathematics. Analytically, we use this dimension to characterise both the mathematics and the didactical insights revealed by the KT in her orchestration of the activity.

Transformation is a dimension of knowledge-in-action which addresses the KT's choices of representations, demonstrations, and use of examples in her orchestration of the activity. The dimension focuses on the KT's ability to transform the mathematics "in ways designed to enable students to learn it" (Rowland et al., 2005, p. 265). Analytically, we use this dimension to describe the KT's orchestration of the activity on characterising two-dimensional geometrical shapes.

Connection is a dimension of knowledge-in-action as well, addressing to what extent the KT draws connections between various mathematics concepts and connections between various mathematics procedures, alternative meanings for these concepts and different ways of carrying out procedures. Analytically, we use this dimension to characterise how the KT makes connections between the geometrical shapes and their features, as well as various ways of deciding what shape is what.

Contingency is a dimension of knowledge-in-interaction, a dimension that addresses how the KT interacts with the children through appropriately responding to children's contributions, to what extent she takes advantage of learning opportunities that emerge, and to what extent she makes the activity her own and deviates from the goals and foci of the activity. Analytically, we use this dimension to characterise how the KT responds to the children's ideas, the questions and comments she uses, whether she draws the children's attention towards particular mathematical ideas, and whether she makes the geometry activity her own and are unbounded by its original foci.

Design principles and context

In our design of the mathematical activities we drew on two main principles, playful learning and inquiry approach to the teaching and learning of mathematics. The design principle of playful learning emphasises that for children play and learning is one and the same thing. Playful learning encompasses both free play (child-initiated and child-directed play) and guided play (adult-initiated and child-directed play), where guided play is the principle used here. In guided play the KT orchestrates and literally guides the play in an adequate direction in order to reach pre-formulated aims for the children's play and to nurture the children's interest, curiosity, engagement, and (mathematical) sense-making [START_REF] Weisberg | Making play work for education[END_REF]. The design principle of adopting an inquiry approach to the teaching and learning of mathematics stems from [START_REF] Jaworski | Learning communities in mathematics: Creating an inquiry community between teachers and didacticians[END_REF] inquiry as "a way of being in practice" (p. 103). That is, the KT and the children collaborate in order to achieve meaningful answers to prompts and questions, the children's curiosity is taken advantage of, and the children are guided into mathematical inquiries through the use of questions, being curious and excited about mathematical issues.

The KT that takes part in our study is a participant in the Agder Project, and by that also a participant in a professional development program. The professional development program within this project was based on four workshops of two days duration, focusing on Number, Measurement, Geometry and Statistics, combinatory and probability. In the third workshop, the KTs participated in lectures and group discussions on Geometry. Our data collection took place during the period when the KTs tried out two geometry activities. The observed KT is in her forties, educated for three years (180 ECTS) at university level, and is a well experienced kindergarten teacher.

Context

We as researchers are interested in scrutinising the KTs' processes of orchestrating our designed mathematical activities. Moreover, the KTs have been participants in a professional development program where we have contributed with lectures and feedback on previous orchestrations of mathematical activities. We study the case of Wilma and videotaped her orchestration of a geometry activity. Wilma had received a written instruction note for the geometry activity a couple of weeks in advance of our data collection.

Analysis and results

In order to employ the Knowledge Quartet (KQ) on empirical data, one has to go into the code level from which the four dimensions are extracted (Rowland, 2016, personal communication). Our analytical process started with a first phase of collectively reflecting on our collected data in total, consisting of two days with data collection in each of four different kindergartens. In doing that, we had the KQ in mind, seeking to use KQ terminology and codes to analyse our data. The second phase consisted of us collectively looking at video excerpts from the different kindergartens, resulting in our choice of the case of Wilma. The reason for choosing one of the sessions of Wilma was that this session, from our collective looking at the video, turned out to be the most promising in addressing our research questions. The KT observably revealed her knowledge in action and interaction, the children contributed with oral statements and questions, and interaction amongst the children was taking place. The third analytical phase consisted of transcribing the video from Wilma's session. The fourth phase consisted of us conducting collective in-depth analyses of the natural talk-in-interaction adopting the codes developed in the Knowledge Quartet.

The activities were presented in written form by us with the three headings Equipment, Intention, and Implementation. The particular activity we consider here, is the first part of an activity focusing on two-dimensional geometrical shapes. Under Equipment we wrote: Geometrical shapes: triangles, squares, rectangles, circles, trapezium, rhombus, while for Intention we wrote: The children are supposed to get experience in recognizing properties of different two-dimensional shapes. Furthermore, the children are supposed to practice mathematical argumentation with respect to features of the various shapes. As regards Implementation, we wrote: Let the children investigate the shapes and their characteristics. Let the children discover the shapes' differences.

Excerpt 1: Foundation

The following excerpts are taken from the initial phase of Wilma's orchestration of the geometry activity. Wilma shows and shakes a box containing two-dimensional paper shapes. One of the children responds that "It's shapes". Upon this, Wilma continues their conversation. The analytical contributory codes are: "awareness of purpose; identifying errors; overt subject knowledge; theoretical underpinning of pedagogy; use of terminology; use of textbook; reliance on procedures" (Rowland et al., 2005, p. 265).

Wilma:

Yes, that's correct. And with mathematical terminology we call them geometrical shapes. Are you able to pronounce that?

Sam: I think it is cookies (Smiles as he says it).

Wilma: John, are you able to pronounce that? Geometrical shapes?

John: Geometrical shapes. (Several children simultaneously say "Geometrical shapes") Wilma: Yes, that is what they are called with mathematical terminology. Inside this box there are several of such shapes (She opens the box and shows it to all the children so that they may look inside the box).

John: It looks like a puzzle.

Wilma: Yes, it looks like a puzzle. That's true.

Sam:

Yes. Are we going to puzzle with them?

Wilma: At least we are going to work with them, yes we are.

Ken:

Can you pour them out?

Wilma: I was thinking of pouring them out. Then I want you to take a look at them. Currently, there are quite a few shapes and some of them are almost identical. Now you may take a look at them. (She pours the shapes out on the table; the children take some shapes each and say "that is small" and "a triangle".)

In this excerpt we argue that Wilma's orchestration is characterised by her foundation, both with respect to mathematics insights and didactical insights. The mathematical insights are revealed through Wilma's emphasis on the mathematical terminology through the twice expressed term "geometrical shapes". Furthermore, her mathematical insights are revealed through her choice of shapes that are congruent, shapes that are similar, and the variety of shapes included (various triangles, various quadrilaterals, circles of various sizes, ellipses, hexagons and octagons). The written material made by us suggested "triangles, squares, rectangles, circles, trapezium, rhombus" as shapes to work on in the activity while Wilma introduced many more shapes and terminology. Thus, we argue that the foundation here is Wilma's and not just her use of external provided foundation from us in the written form.

The didactical insights of Wilma are revealed through her way of establishing interest and curiosity among the children by shaking the box. The children's interest and curiosity about the shapes are nurtured further by her showing of the various shapes in the box. Wilma confirms that the shapes look like pieces of a puzzle and by that she makes a link between these shapes and the apparently well-known activity of puzzling. By establishing that link, Wilma also communicates a playful way of engaging with the shapes. Finally, in this excerpt, Wilma's didactical insights are revealed when she asks the children to inquire into the various shapes. By orchestrating the activity as playful, Wilma signals that she appreciates inquiry as a tool in order to learn mathematics. Additionally, the children's interest, curiosity, engagement, and mathematical sense-making are nurtured.

Excerpt 2: Transformation and Connection

Approximately ten minutes later, the following dialogue occurred, exemplifying the dimensions of transformation and connection. Relative to transformation, the analytical codes are: "choice of representation; teacher demonstration; choice of examples" (Rowland et al., 2005, p. 265). Relative to connection, the analytical codes are: "making connections between procedures; making connections between concepts; anticipation of complexity" (Rowland et al., 2005, p. 265). Some of the children had picked up quadrilaterals that they found interesting since they did not know their names and they were different from rectangles and squares. Then the KT said:

Wilma: Do you know what? These two quadrilaterals actually have other names with mathematical terminology. They have four edges (She counts "one, two, three, four" aloud while simultaneously pointing at the edges).

Susie:

But what are they called then?

Wilma: That one is called a rhombus (she points at the rhombus while speaking).

Susie:

Rhombus.

Wilma:

Rhombus. And that one, do you notice that two and two edges are equal (she points at the parallelogram she shows). That edge and that edge (slides her finger along the two opposite, parallel edges), are equal, and that edge and that edge are equal (slides her finger along the two other, opposite, parallel edges). Its name is actually a parallelogram.

Sam:

A paragram?

Wilma: Yes, a parallel o gram.

In this excerpt we see that Wilma carefully introduces two new shapes for the children. She gets everyone's attention by holding up and showing one shape at a time. She emphasises that both shapes are conceptually associated with quadrilaterals, by overtly counting the edges. At the same time she makes it obvious that the two shapes are particular kinds of quadrilaterals. This is pinpointed by saying that they have "other names", implicitly distinguishing these shapes from the familiar quadrilaterals square and rectangle.

Wilma's orchestration in this excerpt exemplifies transformation in that she uses one example of each of the two new shapes. She points at each of them in accordance with what she is saying, and she demonstrates how to classify geometrical shapes by counting their edges. Her orchestration also exemplifies contingency due to the deviations from the agenda set by the researchers in the designed activity.

The dimension of connection is also exemplified in this excerpt as Wilma draws the children's attention towards the particular features of one of the shapes. Wilma carefully focuses at the two pairs of edges that are parallel, one pair at a time, by sliding her index finger along the edges. She neither uses the mathematical concept of parallel nor equal length at this occasion. She only uses the feature "equal". The feature of parallelism is thus only implicitly focused. However, by sliding her index finger along the two edges, her gesture signals that they are of equal length.

In the end of the dialogue we also notice that Wilma is particularly focused at offering opportunities for the children to learn the name of the new shape. On two occasions, Wilma uses the name "parallelogram" for the shape. Both times she puts emphasis on the expression "parallelogram". We see that Sam tries to pronounce the name, but only partly succeeds. Wilma then slows down the pace in her pronunciation of the word, in order for Sam, and the other children, to pay attention to the new word. Hence, we observe that Wilma is eager to name new objects mathematically correct.

Naming is an important element in these children's mathematical learning process. However, both in this example and in other examples the naming of shapes comes at a late stage, after the children have presented their sorting of shapes and the KT has orchestrated a discussion of the properties of the shapes. We consider this as evidence of her taking an inquiry approach in her orchestration.

Excerpt 3: Contingency

To exemplify this dimension of the KQ in Wilma's orchestration, we include examples of single moves which illustrate how she responded to the children's ideas, the questions and comments she used to make the children pay attention to various mathematical ideas. Even though she occasionally addressed her questions to one particular child, the other children still paid attention.

The analytical codes used were: "responding to children's ideas; use of opportunities; deviation from agenda" (Rowland et al., 2005, p. 266).

Wilma drew the children's attention towards the mathematical concept of sorting. Sam said: "Can we sort them?" (19). Then, some moves, but only a few seconds later, Wilma said: "Sam, what does it mean to sort?" (27). Wilma also drew the children's attention towards mathematical features of the various shapes. One example was found when Sam said: "Yes, but these are small (points at the short edges of the rectangle). These two are equally long" (144). A few seconds later, Wilma responded to this utterance and asked all of the five children: "Does anybody know what the shape is called when two edges are quite long and two edges are shorter?" (147). With this question, she seeks to establish interest and curiosity, and thus to nurture playfulness in her orchestration.

Wilma likewise focused their communication around the similarities between the shapes. Susie talks about the fact that two shapes, two congruent triangles, may be joined in order to make a rectangle. She says: "Jack's shapes have such…, but Ken's do not have such when he puts them together" (196). Upon this statement of observation, Wilma responds immediately: "What happens when you put them together?" (197). Furthermore, Wilma addressed the children's mathematical reasoning through questions like: "How did you figure out that one (points at one of the hexagons)?" (72); and "Do you want to tell the other children?" (227). These examples show that Wilma adopts inquiry as a way of being (cf. [START_REF] Jaworski | Learning communities in mathematics: Creating an inquiry community between teachers and didacticians[END_REF] and responds to the children's ideas and uses the opportunities that unveil in their interaction.

Discussion

In this study we have considered the following questions: In what ways may the Knowledge Quartet be employed in order to characterise the kindergarten teacher's competence? and What knowledgein-action and knowledge-in-interaction does a kindergarten teacher reveal in her orchestration of a mathematical activity on geometrical shapes? From what is possible to include in this relatively short paper, we are able to discern instances of all four knowledge dimensions through which her orchestration of the mathematical activity is informed. As such, the Knowledge Quartet [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] has proven to be analytically useful when seeking to wrap up how a KT is able to orchestrate researcher-designed mathematical activities. At the same time, we observe that the KQ unfolds slightly different in the kindergarten context than in the school context, the context from which the KQ was originally elaborated and developed. In the Norwegian kindergarten context, it is highly rare, even inappropriate to orchestrate mathematical activities through long introductions and demonstrations. Concerning the issue of time, we see that the periods where the KT has the word are quite short, often only 5-10 seconds. Furthermore, in the kindergarten context it is unusual to give children extensive time to inquire into the mathematics without KT interference. This is however a usual case in the school context. KQ contributory codes, such as use of textbook, reliance on procedures, teacher demonstration, and making connections between procedures, are partially inappropriate and inapplicable in the kindergarten context. Thus, our modification of the KQ materialised as not taking these codes into consideration.

It is challenging to separate the dimensions from each other in the dialogues. In the dialogues the dimensions are intertwined, where one move may simultaneously exemplify several dimensions. The excerpts above are thus not mutually exclusive when it comes to the four dimensions of the

Proceedings of CERME10

i The Agder project is funded by the Research Council of Norway (NFR no. 237973), The Sørlandet Knowledge Foundation, The Development and Competence Fund of Aust Agder, Vest Agder County, Aust Agder County, University of Agder and University of Stavanger.

KQ. For the purpose of this paper, however, the excerpts are deliberately chosen to illustrate how the KT reveals her knowledge-in-action and knowledge-in-interaction with respect to this particular mathematical activity.