Chrysanthi Skoumpourdi
email: kara@aegean.gr

Inquiry-based implementation of a mathematical activity in a kindergarten classroom

Keywords: Inquiry-based education, early childhood education, mathematical activity, plane geometrical shapes

Although inquiry-based education is an approach that supports student-centered ways of teaching involving students in the development of their own learning, it seems that inquiry-based instruction has not been employed broadly in the subject area of mathematics and the few studies that have been conducted are not focused on very young children. In this paper an inquiry-based implementation of a mathematical activity in a kindergarten classroom is presented investigating the way it was applied by the teacher and the way students participated into it. The activity was related with plane geometrical shapes as a mathematical construct that cause difficulties to young students. From the results it emerged that even though the inquiry-based implementation is a demanding approach it had many benefits for both teacher and students.

Introduction

A mathematical construct that cause difficulties to young students is the plane geometrical shapes. It is a concept that needs to be communicated multimodally [START_REF] Clements | Early childhood mathematics learning[END_REF], because their iconic representations need 'observation', 'reading' and parallel 'description of this reading' in order to be understood [START_REF] Skoumpourdi | Different modes of communicating geometric shapes, through a game, in kindergarten[END_REF]. The 'observation' factor has to do with shape's visual perception. The 'reading' factor is focused on shape's figural properties. The 'description of this reading' factor requires the description/communication of the above. The above three factors can help students face their difficulties, so as, to develop gradually the shape's concept, but through the appropriate teaching method, since, geometrical thinking is main influenced by instruction.

Inquiry-based teaching is a theoretical approach that supports more student-centered ways of teaching. It is an approach that actively involves students in the development of their own learning supporting their active participation to the teaching/learning process. But, the effects of inquirybased teaching remain limited and the inquiry-based education has been slow in mathematics due to several reasons such as the very low attention that has been paid to that approach, the imprecise definition of the approach, as well as, the peculiarities and the demands that are required for its adoption and promotion.

Taking the above into consideration an educational design research study is conducted, aiming to investigate how teachers and students adopt and apply the inquiry-based approach in the mathematics classrooms. In this paper an inquiry-based implementation of a mathematical activity related with plane geometrical shapes in a kindergarten classroom is presented investigating the benefits and the demands of such an implementation in order to design the gradual adoption of the approach in the subsequent educational levels. The research questions were the following: 1. How the teacher implements a mathematical activity related to plane geometrical shapes, in a kindergarten classroom, grounded on the inquiry-based approach? 2. How the students participate in the inquiry-based implementation of a mathematical activity related to plane geometrical shapes?

Geometrical Shapes in the Kindergarten

Kindergarten children identify and describe shapes through levels. The first one is the prerecognition level [START_REF] Clements | Early childhood mathematics learning[END_REF], in which children may recognize only a subset of a shape's visual characteristics. In this level, children cannot distinguish the basic shapes from their counterexamples. The other levels are the first two of Van Hiele's levels of thought [START_REF] Yin | Young children's concept of shape: van Hiele visualization level of geometric thinking[END_REF]. At the first one, 'visualization', children identify shapes according to appearance and cannot form mental images of them. At the second level, 'analysis', children start analyzing and naming properties of geometric figures. They use shapes' properties to recognize and characterize shapes.

Circle, square, triangle and rectangle, when represented in a prototype way can be recognized and named by kindergarten children. Shape's features and whether they can be defined (closeness, straight sides and exact number of sides) or not (orientation, position, sides' length and angles' size) influence the accuracy of children recognitions [START_REF] Satlow | When is a triangle not a triangle? Young children's developing concepts of geometric shape[END_REF]. Young children identify circles accurately, they are less accurate in identifying triangles and their average accuracy for rectangles is low [START_REF] Clements | Early childhood mathematics learning[END_REF].

Children in the age of five-years-old have a difficulty in describing shapes. The language that children use to describe shapes is gradually developed [START_REF] Clements | Early childhood mathematics learning[END_REF]. Initially they answer the question "what shape?" with the use of just a word, the shape's name. Then they match this word with specific shape examples. After that, they combine the correct shapes' names with the prototypical examples of the shapes. When children are asked to explain their decision, they are induced to describe shapes' features, using, initially, a subset of the shape's visual characteristics, such as the quantity and the size of sides [START_REF] Yin | Young children's concept of shape: van Hiele visualization level of geometric thinking[END_REF]. Although children's primary descriptions may include various terms and attributes they usually base these descriptions in the comparison with the specific shape's representation. Later, children become more capable of distinguishing attributes and adding to their descriptions other visual-spatial elements like right angles.

Children's written schematisation for a circle is something closed and 'rounded', for a square something closed with almost equal sides and nearly right angles, for triangle something closed and 'pointy' and for a rectangle something closed with 'long' like-parallel sides [START_REF] Clements | Early childhood mathematics learning[END_REF].

Instruction influences the development of geometric thinking more than age does (Van Hiele, 1986). But in traditional geometry programs, during formal and informal instruction, little emphasis is given on understanding plane shapes, incorrect statements and terminology are used, no new content is added [START_REF] Clements | Early childhood mathematics learning[END_REF] and inappropriate examples of geometric shapes are used. On the other hand, in contexts such as team games, that teacher's guidance is limited, it seemed that kindergartners recognized shapes in different types, sizes and orientations and could describe some of the shapes' basic attributes such as sides and angles [START_REF] Skoumpourdi | Different modes of communicating geometric shapes, through a game, in kindergarten[END_REF]. But children did not use more sophisticated descriptions related to shapes' specific characteristics like the equality of shapes' sides, and the schematisations they used in the game, to describe an atypical shape, were not accurate. Thus, it was hypothesized that inquiry-based approach could be an effective method of teaching plane geometrical shapes in the kindergarten.

Inquiry-Based Mathematics Education

Inquiry-based mathematics education, is one of the main aspirations of innovation in the field of mathematics education focusing more on cooperative problem solving, on understanding, on learning processes and strategies, on posing and answering questions and on cultivating exploratory practices, than learning mathematical skills only [START_REF] Calder | Student wonderings: scaffolding student understanding within student-centred inquiry learning[END_REF]. Students in that approach are in the center of the learning process. They are actively involved in explorations and discussions, in posing questions, in making assumptions, in expressing, explaining and communicating their opinions, presenting their solutions to a problem, justifying them in a way that is understood both by the other students and the teacher [START_REF] Eckhoff | Partners in inquiry: A collaborative life science investigation with preservice teachers and kindergarten students[END_REF]. Teachers create opportunities to students and work cooperatively with them to facilitate and support their mathematical constructions. They transfer the problem-solving responsibility to the students by posing questions linked with the inquiry-based activity, giving meaning to student's actions and developing their understanding without, at least initially, intervening and suggesting ways of thinking.

Research results highlight the benefits and demands of inquiry-based method on teaching practices. When analyzing a teacher's practice following an inquiry-based program, it emerged that although the teacher at the beginning of the program had very limited knowledge of the subject, after the program he was able to apply inquiry-based practices [START_REF] Towers | Learning to teach mathematics through inquiry: a focus on the relationship between describing and enacting inquiry-oriented teaching[END_REF]. Also, in a research program, it came out that although the process brought uncertainties and tensions within the teachers' practice and between the practices of teachers and didacticians, the value of the approach for the learning emerged by the knowledge gained by participants [START_REF] Goodchild | Critical alignment in inquiry-based practice in developing mathematics teaching[END_REF]. In a study on how prospective teachers adopt inquiry-based approach in their professional development it seemed that they: 1. resisted to the adoption of the inquiry approach, 2. partially adopted the inquiry approach, or 3. fully adopted the inquiry approach. At the end of the course all students stated that they built a theoretical understanding of the inquiry teaching and they felt a bit more able to design and implement inquiry-based activities (Σκουμπουρδή, 2017). Teachers' difficulties to adopt inquiry-based pedagogies, to develop and manage collaborative cultures within the classroom and plan lessons that adapt to the emerging needs of learners [START_REF] Swan | Designing and using professional development resources for inquiry-based learning[END_REF] are mentioned in research. Other obstacles and difficulties that the teachers face in adopting the inquiry-based method for teaching are the lack of experience and of complete understanding of the method, the lack of enough allocated teaching hours, as well as the lack of believe that students can be prepared for succeeding in tests with that method [START_REF] Chin | A survey of the practice of a large-scale implementation of inquirybased Mathematics Teaching -from Taiwan's perspective[END_REF]. The lack of teachers' familiarity with the method and the need of advanced professional development, which will equip them with multiple teaching strategies to create an adaptive and enriched learning environment is also indicated in research [START_REF] Wu | Inquiry-based mathematics curriculum design for young childrenteaching experiment and reflection[END_REF]. In sum, active teachers lack knowledge of the inquirybased approach, they have difficulty in adopting and applying this approach in practice, as well as they have difficulty in designing inquiry-based activities. These obstacles might cause the lack of inquiry-based instruction being employed broadly in the subject area of mathematics and the lack of studies being conducted with a focus on very young children.

Taking into consideration the above results an 'inquiry-based implementation' of an activity related to plane geometrical shapes, was conducted in a kindergarten, investigating the way it was realized by the teacher as well as the way that kindergartners participated in such an intervention.

Method

For the main research study, which aims to investigate how teachers and students adopt and apply the inquiry-based approach in the mathematics classrooms, the design-based research is used, as the methodology to develop learning theories and tools designed to support learning [START_REF] Cobb | Design experiments in educational research[END_REF]. Through this method, which is intrusive in nature, changes and revisions are taken place, in every stage of the main research which enables the simultaneous development and testing of the inquiry-based approach as an innovative learning environment, giving an insight into whether, how and under which circumstances this approach can work. In this paper the benefits and the demands of the approach in a kindergarten classroom is presented.

The kindergarten classroom and the mathematical activity

The kindergarten classroom, where the intervention took place, had 25 students divided by the teacher in four teams and was chosen because the teacher uses student-centered methods, cultivating the conditions for math discussion. She is also positive to innovations.

The author explained to the teacher the basic characteristics of the inquiry-based approach and encouraged her to use it through the Framework for Inquiry-Based implementation of Activities (FIBA) [START_REF] Skoumpourdi | A framework for designing inquiry-based activities (FIBA) for early childhood mathematics[END_REF]. The FIBA consists of seven stages: 1. Τask: A problem-based task is invented and presented. 2. Exploration: Children use their own problem-solving strategies to explore the problem. 3. Presentation: Children share their explorations with the whole class by presenting their solutions. 4. Connection: Teacher, in cooperation with students, connect the presented solutions with each other, with the task that was explored and with the mathematical aims. 5. Generalization: Teacher is generalizing students' solutions. 6. Translation: Students translate their solutions through different modes-verbalizing, gesturally and schematizing. 7. Expansion: Expanded problems are posed. For the implementation of the mathematical activity by the teacher the FIBA was used.

The mathematical activity, designed by the author, was developed in such a way as to problematize and incite children, engaging them in a problem solving and posing process. The problem to be solved was a non-standard, 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 and discussion. The scenario of the task was the following: "Once upon a time, a strange magician, who envied the happiness of the Shape-Town, captured with his spells its inhabitants (shapes) on the wings of butterflies flying around his enchanted forest. This strange magician would be good enough to solve the spells only if the children would help the inhabitants-shapes to liberate. The spells would be solved only if the children managed to recognize the shapes hidden in the butterflies' wings". The butterfly had in its wings the four-basic plane geometrical shapes (circle, square, rectangle and triangle) in various types and orientations, as well as counterexamples of them (Figure ,1).

Results

In the beginning, the problem-based task was presented by the teacher through the context of the 'Enchanted Butterfly' (FIBA's 1 st stage: Task). The teacher in this stage asked each team of children to mark one kind of shape. The 1 st team had to mark the shapes they thought to be circles, the 2 nd team the squares, the 3 rd team the rectangles and the 4 th team the triangles.

Each team independently explored the shapes on the Butterfly, posing questions to each other and to the teacher in order to identify and mark the correct shapes giving a right answer to the problem (FIBA's 2 nd stage: Exploration). Although, in general, each child thought about the problem on their own and marked a shape, each in turn, there were instances in which wrong shapes were marked and thus interesting discussions between children of each team and with the teacher took place regarding the identity of the shapes. Debates of whether a shape is a triangle or not, is a square or not, is a rectangle or not, is a circle or not, took place. An example is described in the following dialogue [The letter T refers to the teacher and S1, S2 etc. refer to the young students. The symbol ( ) is used when many children speak simultaneously and our notes are in brackets]: S1: Is this a circle? T:

You will discuss with your team and you will find out.

():

It is circle, it is not, it is circle, it is not, it is circle but is a small circle [a girl tried to mark it but the other girl pushes her hand away] …., it is not a circle, is an oval. Similar debates were also carried out for the other shapes and especially for the non-prototypical examples and the counterexamples. The identity of the triangles, the squares and the rectangles were justified, by the children, by counting shapes' angles. But this strategy was not always effective. For example, when a girl showed a counterexample as a triangle, another child told her: "No, this is a 'square' because it has four angles" and the other team members disagreed by saying "No, it is not, it is a shuttle". The orientation of the shapes did not seem to confuse the children. This might have occurred due to the way the team members sat around the table, allowing them to have different perspectives of the shapes. Once they explored the shapes and discussed about them each team marked 'their shapes'.

After that, each team shared their explorations with the whole class by presenting/describing their shapes (FIBA's 3 rd stage: Presentation). The teacher, in that stage, observed and organized each team's presentation, posing questions to help children describe, explain and open out their explorations, as well as to ensure effective and understandable presentations. For example, during the 'circle team' presentation the following discussion took place:

T: How did you understand that all these shapes are circles?

S2:

Because it is round, and this is round, and this is round … T: I think that this is also round [showed an oval, placed horizontally].

S3:

No, this is like an egg.

T: So, is not a circle? (): No.

T: Is that shape circle [showed an oval placed vertically]?

():
No, this is not a circle.

T:

But in my opinion is round, why it is not a circle? S2:

Because it is oblong [showed with her hands] and the circle is not oblong.

T:

Why are these shapes circles? S3: Because they do not have angles.

S2:

And because circles are thicker than ovals

For the other shapes, children's answers in the question 'why these shapes are triangles (squares/rectangles)?' were focused on the amount of their angles as well as the amount of their sides, with the usual answer being that: "It is triangle because it has 3 angles and 3 lines/sides".

The teacher posed advanced questions to the members of each team. For instance, when a boy presented his teams' shapes and said that "this shape is rectangle because it has four sides", the teacher continued, by showing a square near it, asking "So I can assume that this shape is also a rectangle because of its four sides?" and vice versa. Or, she was addressing to the children supposing that "this shape (showing counterexamples) is e.g. triangle (rectangle)" calling them to reflect on that statement. Children, in general, had answers to those questions, justifying them by relying on the specific iconic representation. Their answers were as follows: "It is not a rectangle because it has different height", "It is not a square because it is thin", "It is not a triangle because this shape has seven lines". When children had difficulty in their justifications they used gestures in order to present their solution: "It is not a rectangle because it is like that … (trying to show the curvature sides by gestures)".

After the presentations, teacher, in cooperation with all the students, orchestrated a mathematical discussion trying to connect presentations with each other, with the 'Enchanted Butterfly' task, as well as with plane geometrical shapes concept so that a shared meaning to emerge (FIBA's 4th stage: Connection). She also tried to generalize plane geometrical shapes concept connecting it with students' previous knowledge (FIBA's 5 th stage: Generalization). She posed combining questions to the students, bringing back, for negotiation, issues that were discussed such as: "Does the circle looks like the triangle?" or "Does the square looks like the rectangle?". In the first question, children' one-word answers, through the re-questions of the teacher, were transformed into 'documentations', such as "Circle does not look like triangle because circle does not have angles", "the triangle is pointed … the circle is not". Teacher's second question seemed to confuse children and their answers differed, claiming that they look like, but the rectangle is bigger or that they do not resemble because they are different shapes with different sizes. These issues, such as the similarities and differences of shapes (circle and oval, square and rectangle), expanded the activity and could be the context of new activity in order to be discussed more thoroughly (FIBA's 7 th stage: Expansion). At the end of the activity, students were asked to make a painting in order to explain to their parents what they had done in that activity (FIBA's 6 th stage: Translation). Children's schematizations looked like the visual prototypes except the rectangle which was placed vertically (Figure 2).

Conclusions-Discussion

An inquiry-based implementation of a mathematical activity is a complex situation, that requires from the teacher time, effort and deep knowledge of both the approach and the subject matter and from the students to develop various skills, abilities and capabilities to be actively involved in their own learning, constructing mathematics concepts and ideas. From the results of the implementation it seemed that the FIBA, as a readymade framework, helped the teacher to adopt and realize an inquiry-based teaching process encouraging exploration, presentation, connection, generalization, translation and expansion, cultivating cooperation, communication and justification, through a child-centered approach. In all the FIBA's stages, teacher activated students' questions, ideas, actions and comments and tried to turn them into learning opportunities. She also successfully scaffolded the mathematical discussion between the students and herself, guided team work and the whole classroom as a community of learning. Inquiry-based approach helped students to develop their cooperation, communication and justification capabilities. It also helped them to construct the idea of plane geometrical shapes.

The research results about young children' abilities for plane geometrical shapes were confirmed in this paper. But it also became obvious that although there were children in the pre-recognition level, recognizing only some of shape's visual characteristics, they moved on the visualization and the analysis level using shapes' properties to recognize and to characterize typical and atypical shapes through the cooperative exploration, discussion and justification process. Children, initially, described or compared the shapes using specific shapes' representation features. All shape's features either defined or undefined were recognized by children because of the way the activity presented and implemented. Students had identified and named plane geometrical shapes in various forms and orientations, recognized them among counterexamples, communicating their descriptions, discussing about their characteristics and particularities. They had also translated their solutions and constructions to schematic representations that looked like the visual prototype but there were also variations.

Figure

  Figure 1: The 'Enchanted butterfly'


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

Dina Tirosh

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

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

Shedding light on preschool teachers' self-efficacy for teaching patterning

Keywords: Repeating patterns, self-efficacy, preschool teachers, professional development

As teacher educators, we recognize the importance of considering teachers' self-efficacy for teaching mathematics. In this study, we investigate preschool teachers' self-efficacy for teaching repeating patterns, both before and after participating in a professional development program. Findings from questionnaires indicated that self-efficacy related to subject-matter knowledge changed little, while self-efficacy related to pedagogical-content knowledge, increased. Interviews with teachers shed light on these findings.

Introduction

Recently, several countries have introduced specific guidelines for introducing mathematical concepts during the early years. Along with new guidelines comes a need for enhancing preschool teachers' knowledge for teaching mathematics. Knowledge, however, may not be sufficient. Teacher self-efficacy is another factor which may impact on teachers' classrooms interactions and student achievement (e.g., [START_REF] Guo | Relations among preschool teachers' self-efficacy, classroom quality, and children's language and literacy gains[END_REF]. Teacher self-efficacy may be conceptualized as "a teacher's individual beliefs in their capabilities to perform specific teaching tasks at a specified level of quality in a specified situation" (Dellinger et al. 2008, p. 752). With regard to mathematics, [START_REF] Hackett | An exploration of the mathematics self-efficacy/mathematics performance correspondence[END_REF] defined mathematics self-efficacy as, "a situational or problem-specific assessment of an individual's confidence in her or his ability to successfully perform or accomplish a particular [mathematics] task or problem" (p.262). This paper focuses on preschool teachers' self-efficacy for engaging young children with repeating pattern activities. Repeating patterns are patterns with a cyclical repetition of an identifiable 'unit of repeat' [START_REF] Zazkis | Generalization of patterns: The tension between algebraic thinking and algebraic notation[END_REF]. For example, the pattern ABBABBABB… has a minimal unit of repeat of length three. According to the Israel National Mathematics Preschool Curriculum (INMPC) (2008), "patterning activities provide the basis for high-order thinking, requiring the child to generalize, to proceed from a given unit, to a pattern in which the unit is repeated in a precise way " (p. 23). In this paper we take a close look at a group of preschool teachers who attended the Repeating Patterns professional development Program (RPP) aimed at enhancing their knowledge and self-efficacy for teaching repeating patterns. We examine their self-efficacy before and after the program, and through interviews, we attempt to untangle variations in their self-efficacy beliefs.

Background

For the past several years, we have been employing the Cognitive Affective Mathematics Teacher Education (CAMTE) framework for investigating preschool teachers' knowledge and self-efficacy for teaching mathematics, as well as planning for professional development with preschool teachers [START_REF] Tirosh | Using children's patterning tasks for professional development for preschool teachers[END_REF]. The framework is based on [START_REF] Shulman | Those who understand: knowledge growth in teaching[END_REF] recognition of teachers' subject-matter knowledge (SMK) and pedagogical-content knowledge (PCK), as well as [START_REF] Ball | Content knowledge for teaching what makes it special?[END_REF] Regarding teachers' knowledge of patterning, several studies pointed out the need to promote teachers' knowledge of the language of patterns. For example, [START_REF] Waters | Mathematical patterning in early childhood settings[END_REF], described how one preschool teacher showed the children dress materials with, what she called, patterns. Some of the materials contained a mixture of shapes, colors, hearts, and stars, and demonstrated random designs without any regularities. The teacher did mention the need for repetitions, but did not focus on what exactly is repeated. Similarly, in our study of preschool teachers' patterning knowledge (Tirosh, Tsamir, Levenson, Barkai, & Tabach, 2018), we found that teachers have difficulties verbalizing what exactly is repeated, although most referred in some way to the unit of repeat and to the notion of repetition. We also found that when asked to draw or continue repeating patterns, teachers mostly drew repeating patterns that have three cycles of the unit of repeat and a unit length of three. In addition, teachers' continuations of given patterns indicated a strong tendency to end patterns with a complete unit of repeat.

For each knowledge cell in the framework, there is a related self-efficacy cell, emphasizing teachers' mathematics self-efficacy as well as their pedagogical-mathematics self-efficacy, i.e. their self-efficacy related to the pedagogy of teaching mathematics (see Table 1). This differentiation was also pointed out by [START_REF] Bates | Linking preservice teachers' mathematics self-efficacy and mathematics teaching efficacy to their mathematical performance[END_REF], who investigated the relationship between early childhood (pre-K to third grade) preservice teachers' mathematics self-efficacy and their mathematics teaching self-efficacy. Results of that study showed that teachers who reported higher mathematics selfefficacy were more confident in their ability to teach mathematics than teachers with a lower mathematics self-efficacy. However, participants with a high mathematics teaching self-efficacy did not necessarily perform well on the mathematics skills test. Some teachers who scored low on the skills test still felt confident to teach mathematics. We also note that self-efficacy beliefs are not only domain-specific (e.g. mathematics, history, science) and content-specific (e.g., within the domain of mathematics there is numeracy, patterns, geometry, etc.), but also task-specific (e.g., extending patterns, duplicating patterns, etc.) [START_REF] Zimmerman | Attainment of self-regulation: a social cognitive perspective[END_REF].

The aim of this paper is to report on preschool teachers' mathematics self-efficacy as well as their pedagogical-mathematics self-efficacy, before and after the Repeating Patterns Program (RPP). The teachers in this study met seven times (21 hours in total) over a period of about four months. All lessons were planned by the four authors of this paper. The fourth author did the actual teaching.

The program revolved around patterning tasks that we designed, which teachers could implement with children, but could also be used to engage teachers with the mathematics involved in patterning, and promote their knowledge of patterning tasks and children's ways of solving those tasks. Towards the end of the program, teachers implemented patterning tasks with children in their classes, video-taped their implementations, and then brought those recordings to the program for discussion. (See [START_REF] Tirosh | Using children's patterning tasks for professional development for preschool teachers[END_REF] for additional information regarding this program.) Specifically, we ask: (1) Was there a change in teachers' self-efficacy from before to after the program, and if so, what was the nature of this change? (2) When reflecting on self-efficacy, how do teachers explain their self-efficacy beliefs?

Methodology

Participants in this study were 18 preschool teachers enrolled in the RPP. All had a first degree in education, between 1 and 38 years of teaching experience in preschools, and were currently teaching children ages 4-6 years in municipal kindergartens. All sessions were videotaped and transcribed.

Before the program began, and again during the last session, teachers were asked to fill out a questionnaire which began with the following self-efficacy statements: I am able to say what a repeating pattern is; if shown a repeating pattern along with several suggestions for continuing the pattern, I am able to choose appropriate continuations; I am able to point out repeating patterns that most preschool children are able to continue appropriately; I am able to choose tasks for investigating children's patterning knowledge. A four-point Likert scale was used to rate participants' agreements with self-efficacy statements: 1 -I do not agree that I am capable; 2 -I somewhat agree that I am capable; 3 -I agree that I am capable; 4 -I strongly agree that I am capable.

Approximately one month after the program was over, seven teachers, chosen to reflect a variety of responses (e.g., no change in self-efficacy, increased self-efficacy, decreased self-efficacy) were interviewed individually. The aim of the interview was to further investigate teachers' self-efficacy beliefs for teaching patterning by evoking their reflections on these beliefs, and focusing on changes that may have occurred with regard to self-efficacy. In general, the interview questions were of the form: "Before the program began, you wrote that you strongly agree (somewhat agree/agree/do not agree) that you are capable of ______. At the end of the program you wrote ________. Can you tell me more about this?" The blanks were filled in with the different tasks taken from the questionnaire and teachers' self-efficacy assessments in the pre-and posttests. In addition, teachers were encouraged to freely reflect on their self-efficacy beliefs. Interviews were recorded and transcribed.

Findings Results from the questionnaires

Table 2 shows the mean values for teachers' self-efficacy scores before and after the program (on a scale from 1-4). As can be seen, teachers had a rather high regard, both before and after the program, in their ability to define a pattern and to choose appropriate continuations for a given pattern, with their self-efficacy rising slightly after the program. Teachers also had a high selfefficacy for identifying patterns that children could continue, with seemingly no change between before and after the program. The greatest change was noted in the last question. Before the program, teachers were not so confident in their ability to choose appropriate pattern tasks for investigating children's pattern knowledge, whereas after the program, their self-efficacy was noticeably higher.

Self Taking a closer look at the distribution of self-efficacy scores, Table 3 presents the frequencies of teachers whose self-efficacy scores increased, stayed the same, or decreased. In general, approximately two-thirds of the teachers did not feel a change in their self-efficacy related to SMK, while the rest of the teachers' self-efficacy increased. However, when it came to PCK-related self-efficacy beliefs, findings were more complex. With regard to knowledge of students, approximately half of the teachers did not feel any change, but 25% of the teachers felt a rise in their self-efficacy and 25% reported a decrease. With regard to knowledge of tasks, approximately 75% of the teachers' self-efficacy scores increased, with the rest showing no change.

Self-efficacy

Statement

Interviews: Reflecting on self-efficacy

As seen in the previous section, most teachers' SMK-related self-efficacy beliefs were high and did not seem to change. When asked to comment on her SMK-related self-efficacy, Michelle (this and all other names are pseudonyms), who rated her self-efficacy both before and after the course at level 4, said, "Before the course, I don't think I knew how to define a pattern. I'm trying to remember when I learned this topic. But after we studied this in depth during the program…I know a lot more now." While Michelle does not explain why in the beginning of the course she had such a high self-efficacy, thinking back, she realizes that her SMK regarding defining a pattern actually increased.

With regard to being able to choose appropriate continuations to a pattern, Anne explains why she rated her self-efficacy as very high before the course, "Before the course, I knew that a pattern could be continued in several ways." However, she adds, "But now, I understand the reasoning behind these options, and I can explain it better to the children." In other words, although her selfefficacy rating has not changed, she feels that her knowledge has increased. Rina, who also had rated her self-efficacy as 4 both before and after the course, had a different reason for why no change was reported:

In the beginning of the course, when they asked me these questions In other words, Rina reflects back on before the course, and can now say that before the program, she did not really know what was meant by a repeating pattern, and was not aware that patterns do not have to end with a complete unit of repeat. This is similar to Michelle. Both teachers seem to acknowledge that their self-efficacy ratings in the beginning of the course were incorrectly high. In fact, when Rina was asked what score she should have given herself in the beginning of the course, she answered, "Now, I'm a 3-4. Then, maybe a 2, because I did know a little."

When it came to reflecting on their PCK self-efficacy beliefs related to knowing children's patterning abilities, teachers again had varied responses. Prior to the program, Anne had evaluated her self-efficacy at level 3, and at the end she reported level 4. She told the interviewer that her self-efficacy increased and explained, "Today, I can define it [a repeating pattern] better. I know that there is a minimal unit of repeat, and that it is made up of elements. This definition organizes my thoughts and adds to my self-efficacy." Although Anne was asked to relate to her ability to point out patterns that most preschool children can continue, her response focused on her ability to define a repeating pattern. It could be that for Anne, her knowledge of the centrality of the minimal unit of repeat, affected her belief in her ability to assess children's knowledge.

Rina, who consistently rated her self-efficacy to assess children's knowledge at 4, said, "If, in the beginning of the course, I didn't really understand about repeating patterns, then of course, I couldn't really know about the children." Rina is acknowledging her over-confidence in the beginning of the course. Moreover, she connects her SMK with her PCK, by connecting her knowledge of patterns to her knowledge of students. Sharon, who consistently rated her selfefficacy at level 3 said, "I'm not so sure of myself here. I think I need more experience working with children to feel more sure of myself in this area." This explains why currently, her self-efficacy is a 3 and not a 4. When asked to reflect back, she adds, "Before (the course), I would give myself a 2. I didn't engage with patterns as much as I do now, and so I'm more self-confident now that I can do things better." She now acknowledges that her self-efficacy in the beginning of the course was too high.

Lottie's self-efficacy for assessing children's patterning knowledge decreased from a 4 to a 3. During her interview, she stated the following:

For me, before the course, I would have children make patterns by giving them a red sticker, and then a yellow sticker, and so on. This course was a wake-up call. Patterns can be made with different materials… I was stuck on one type. There is also the pattern ABB. Before, I engaged the children with patterns, but without reflecting on what I did.

Lottie states that before the program, she engaged children with patterning tasks. This could be the reason that in the beginning, she felt very strongly that she could say which patterns children would be able to complete. After the course, Lottie acknowledges that her previous patterning knowledge was limited. Perhaps, this causes her now to be unsure regarding her knowledge of children.

Finally, we review teachers' reflections regarding their self-efficacy for choosing appropriate tasks.

Recall that most teachers' self-efficacy for choosing tasks increased. Rina, who increased her selfefficacy from a 3 to a 4, commented, "I did work with repeating patterns with children, but they were not so varied, they did not end in the middle (with a partial unit of repeat)." Sharon, who went from level 2 to 4, said, "The program gave me lots of tools, and also self-confidence." However, she qualifies her response by adding, "I think I still need more experience." Similarly, Anne, who also went from a self-efficacy of 2 to 4, said, "The patterns that Ruthi (the program instructor) gave us in the course, really helped me to organize my thoughts and implement them in the kindergarten. I know that another teacher also uses them, and even when in the yard she uses sound patterns, because it's very nice, and we really liked it."

Summary and discussion

This study examined preschool teachers' mathematics self-efficacy as well as their pedagogicalmathematics self-efficacy, related to teaching repeating patterns, as well as teachers' reflections on these beliefs. Regarding teachers' SMK-related self-efficacy beliefs, results indicated that no significant change was felt. That teachers reported high SMK-related self-efficacy beliefs at the end of the program, was satisfying. However, we are left with two questions: How is it that teachers had such a high self-efficacy before the program began? In addition, even though the reported selfefficacy beliefs did not change, did teachers feel some change?

Regarding the first question, it could simply be that teachers indeed knew how to complete these tasks in the beginning, although to a lesser extent than they did in the end. In a previous study (Tirosh et al., 2018), we found that most preschool teachers did recognize that a repeating pattern must have unit of repeat, but missed stating that the unit must be structured. On the other hand, they could all construct and extend given repeating patterns. Regarding the task of choosing an appropriate continuation for a given repeating pattern, in general, teachers were able to choose appropriate continuations, although they had greater success in choosing appropriate continuations for the patterns which ended with a complete unit, than for the pattern which ended in a partial unit.

Teachers' interviews shed additional light on the issue of their self-efficacy. During the interviews, some teachers acknowledged that their first self-efficacy reports were higher than they should have been. They explained these initial high self-efficacy reports by saying that the topic of repeating patterns was not new to them. Thus, it might be that teachers did feel a change in their knowledge, but that this change was not reflected in the questionnaires. This finding raises the complexity of investigating self-efficacy beliefs. First, it could be that a scale of 1-4 was not sensitive enough to capture changes. In addition, this study, like most self-efficacy studies (e.g., [START_REF] Bates | Linking preservice teachers' mathematics self-efficacy and mathematics teaching efficacy to their mathematical performance[END_REF], relied on Likert-scales. It was the addition of interviews that revealed teachers' changes in their beliefs. Thus, we encourage the use of interviews when investigating self-efficacy.

Regarding teachers' PCK-related self-efficacy beliefs, from the questionnaires, it seemed that teachers did not feel a change in their self-efficacy related to knowing children's patterning abilities. Taking a closer look, we found that some teachers' self-efficacy increased, some stayed the same, and some teachers' self-efficacy decreased. As one aim of the course was to support teachers' self-efficacy beliefs, this result is surprising. One explanation is that after the program, teachers realized how much more there is to learn. They also mentioned feeling the need for more experience engaging children with patterning activities. Thus, this may actually be positive outcome of the program.

Regarding teachers' self-efficacy for choosing patterning tasks that can investigate children's knowledge, we ask, why is it that before the course this particular self-efficacy belief was relatively low? One possibility is that we asked teachers if they could choose tasks for the purpose of investigating children's pattern knowledge. It could be that if they were asked to choose tasks that had the potential to promote, rather than assess knowledge, their self-efficacy might have been greater. In a previous study of preschool teachers' knowledge and self-efficacy for teaching threedimensional figures [START_REF] Tsamir | Preschool teachers' knowledge and self-efficacy needed for teaching geometry: Are they related[END_REF], we found that teachers' self-efficacy related to designing tasks for promoting knowledge was greater than their self-efficacy related to designing evaluation tasks.

Finally, this study highlighted the connection between teachers' SMK-related self-efficacy and their PCK-related self-efficacy. When asked about her self-efficacy for knowing children's pattern abilities, Anne referred to her improved ability to define a repeating pattern. Rina and Lottie also connected their knowledge of repeating patterns, including that a repeating pattern does not have to end a complete unit of repeat, to their knowledge of children and patterns. While previous educators (e.g., [START_REF] Ball | Content knowledge for teaching what makes it special?[END_REF][START_REF] Shulman | Those who understand: knowledge growth in teaching[END_REF] connected teachers' SMK to PCK, this finding connects teachers' SMK-related self-efficacy to their PCK-related self-efficacy, and suggests that if we wish to promote positive self-efficacy for teaching mathematics, we should also consider teachers' self-efficacy for solving mathematical tasks.

Table 1 : The CAMTE framework illustrated within the context of repeating patterns

 1 refinement of these elements.

		Subject-matter	Pedagogical-content
		Solving	Evaluating	Students	Tasks
	Knowledge	Cell 1: Defining, drawing, extending repeating patterns	Cell 2: Identifying examples of repeating	Cell 3: Knowledge of students' ways for drawing, extending,	Cell 4: Designing and choosing tasks for promoting and evaluating children's
			patterns	and comparing	knowledge of repeating
				repeating patterns	patterns
	Self-efficacy	Cell 5: Mathematics self-efficacy related to Defining, drawing,	Cell 6: Mathematics self-efficacy related to	Cell 7: Pedagogical-mathematics self-efficacy related to	Cell 8: Pedagogical-mathematics self-efficacy related to
		extending	evaluating	children's	designing and evaluating
		repeating patterns	solutions	conceptions	tasks

Table 2 : Means and SD per self-efficacy statement before and after the program (N=18)

 2

		-efficacy statement	Pre	Post
			Mean SD Mean SD
	SMK I can say what a pattern is.	3.3	.59 3.8	.44
		I can choose appropriate continuations for a repeating	3.6	.62 3.8	.38
		pattern.
	PCK	I can point out patterns that most preschool children can	3.2	.56 3.3	.69
		continue.
		I can choose appropriate tasks for investigating children's	2.2	.79 3.4	.72
		patterning knowledge.

Table 3 : Changes in self-efficacy beliefs between pre and posttests (N=18)

 3

  , I think I didn't really know what they [repeating patterns] meant. At the end of the course, now, I understand what is meant by a repeating pattern, that it can end in the middle [with a partial unit of repeat]. Now, the question is clear. I don't remember what I wrote in the beginning.

Acknowledgment

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


Marianna Tzekaki
email: tzekaki@auth.gr

Evaggelia Papadopoulou

Generalization in early arithmetic

Keywords: arithmetic learning, number sense, generalization

Research in the field of early arithmetic and relevant international educational programs present a variety of approaches (activities, tasks and material) that aim at teaching children from early age to count verbally or count quantities or to recognize and compare without counting, etc. However, a deeper insight into underlying arithmetic thinking, reasoning and early generalization always remains required. In our study, 23 preschoolers (of 5 to 6 years) participated in a program focusing on generalizing in early arithmetic. Task-based interviews before and after this program were conducted to examine children's improvement in conceptual development and generalization related to numbers. Findings indicate that appropriated teaching approaches with relevant tasks and discussion supported children to extract more general ideas about numbers, their properties and interrelations.

INTRODUCTION

Arithmetic learning is one of the most well-known and recognized mathematical field of teaching and learning in early years, thereby, there is an extensive related literature with different approaches and views. Programs around the world suggest a wide variety of projects, tasks and materials aimed at effective numerical learning. Thus, children from early age learn to recognize and compare quantities without counting, recite the arithmetic sequence, count objects, etc. What is though genuine arithmetic learning and how can we ensure it? Researchers have long ago pointed out that daily numerical activities or involvement with arithmetical elements with numbers and operations could not assure the development of numerical meaning. First references regarding this issue started with [START_REF] Brownell | The place of meaning in the teaching of arithmetic[END_REF] who attempted to distinguish 'meaning of' from 'meaning for' that essentially differentiates the use of numbers in real life from generalizations that lead to mathematical conceptualization.

Long ago also, researchers highlighted the gap between everyday use of numbers and development of mathematical number concepts in education [START_REF] Carraher | Mathematics in the streets and in schools[END_REF]. The value of spontaneous or informal use of mathematics by students or practitioners is generally recognized and studied, but this use does not automatically link mathematical elements to relevant mathematical concepts. It could be supported that counting quantities or realizing other numerical tasks in everyday situations keep a 'concrete' aspect that is not simply transferred out of its context. Thus, the development of number sense towards the correspondent mathematical meaning demands a shift from the specific situation and content in which it functions to a broaden level through systematic generalization processes. As it was recently underlined by many researchers related to the learning and teaching of arithmetic in early years "…our knowledge of how children develop accurate and general understanding of basic arithmetic principles and make use of them as strategies for arithmetic problem solving remains incomplete…" (Björklund et al., 2018, p. 126)".

In this paper, we will attempt to highlight different levels of generalization that pre-schoolers can achieve in arithmetic learning when they are systematically encouraged by relevant teaching approaches.

Arithmetic Learning

Research in the field of early arithmetic and consequent educational programs presents a variety of supplementary or complementary approaches, activities, tasks and material, emphasizing their importance for later mathematical proficiency. However, while all these advances support the need of meaningful arithmetic knowledge and the development of number sense, cognitive expectations and educational outcomes remain to be clarified by teachers and educators. Moving away from the (initial) Piagetian logical operations related to arithmetic learning and from other relevant psychological research, newer approaches such as subitizing and counting-based arithmetic appear to focus more on quantity and number sense (e.g. [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF]. Current suggestions orient also their interest to number structure and connections [START_REF] Venkat | Mediating primary mathematics: theory, concepts, and a framework for studying practice[END_REF].

Researchers who emphasized the importance of counting as a base upon which number concepts could be developed, supported that counting connects knowledge coming from outside with inschool understanding [START_REF] Fuson | Children's counting and concepts of number[END_REF]. However, when children count quantities by reciting a sequence of number names and pointing out one by one different objects (even respecting the relevant principles: ordered names, one-to-one and cardinal principle, [START_REF] Carpenter | Young children's mathematics: Cognitively guided instruction in early childhood education[END_REF] follow a rather procedural than conceptual understanding (LeFevre et al, 2005). Studies have revealed that while young students, knowing in general object counting, attempted to complete a simple task of change (e.g. 4 to 8 or 8 to 5) stayed more depended on counting or perceptual information (use of fingers or blocks) than on numerical facts. Subsequently later, research in early arithmetic learning (e.g. [START_REF] Baroody | The developmental bases for early childhood number and operations standards[END_REF]Clements, 1999, etc.) studied the representational flexibility of subitizing that reinforced children to interconnect and generalize numerical concepts, maintaining however the importance of counting and recognition of number interrelations for arithmetic learning.

In this brief presentation, we attempted to justify why, despite the volume of research related to early arithmetic, a deeper insight into underlying arithmetic thinking, reasoning and early generalization remains always required.

Generalizing in early arithmetic

Generalization is an important prerequisite for arithmetic learning, similarly to every other mathematical learning. As presented in an earlier paper, generalization [START_REF] Tzekaki | Teaching intervention for developing generalization in early childhood: the case of measurement[END_REF], p. 1926) "…could be firstly identified as the level at which students, starting from specific situations, proceed to more general ideas and conclusions, identifying patterns, structures, relationships etc." (see, e.g. Döfler, 1991;[START_REF] Kaput | Teaching and learning a new algebra[END_REF], with fundamental elements the "reflection over actions" and the "expression of more generalized ideas or rules". Despite its importance in the mathematical development, research in early childhood generalization is mainly limited to patterns.

On the other hand, research and proposals in early arithmetic place its focal points on counting, subitizing, ordering and connecting quantities and numbers. Relevant proposals underline the need of pupil's shift from 'quantity sense' to 'number sense', by first perceiving the magnitude of numbers, then understanding their multiple interrelations and finally operating on them [START_REF] Wagner | Feeling number: grounding number sense in a sense of quantity[END_REF], thus, moving beyond perceptual or procedural practices, both in counting and subitizing or in operating with other means such as fingers or relevant material [START_REF] Dehaene | The number sense: How the mind creates mathematics[END_REF].

Combining views on generalization with proposals for early arithmetic we attempted to examine whether young children were able to overcome perceptual subitizing or verbally object counting in a concrete content and apprehend numbers in a more generalized level. Thus, in our study we examined whether preschoolers managed to approach and verbally express more global ideas about quantities, numbers and their interrelations, beyond perceptual or procedural practices in the four strands of early arithmetic (1) verbal counting up to 10, (2) subitizing, (3) analyzing and combining quantities and numbers, (4) relating and ordering quantities and numbers [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF]. Moreover, we investigated whether these ideas could be evolved when working in teaching approaches that systematically encouraged generalizations.

Methodology

23 preschoolers aged 5 to 6 years (selected by classroom teachers on the basis of their common age and attendance in the same school and school area) participated in an 8-month teaching program focusing on generalization related to all fields of early mathematics education (shapes, patterns, measurement and numbers, [START_REF] Tzekaki | Teaching intervention for developing generalization in early childhood: the case of measurement[END_REF], with the number module to be the last, form late April to late May. Task-based interviews were conducted to examine pre-schoolers' arithmetic achievement and levels of generalization before and after the teaching program (following a design study). The content of the task-based interviews followed the strands mentioned earlier: counting quantities up to 10 (4 cases), subitizing quantities up to 10 (3 cards), combining numbers (1 task with quantities in dot cards) and transforming numbers (2 tasks of change, see tasks in Table 1). The interviews' questions aimed at examining statements related to children's conceptualization of numbers concerning their way of acting (e.g. what did you do to …, how did you do it…), children's thinking (e.g. what did you think to …, what did you notice to...), children's justification (e.g. why did you do it…) and finally children's more general ideas (e.g. Is it always the same or it may be different…, how can we find that…) [START_REF] Dougherty | Developing concepts and generalization to build algebraic thinking: The reversibility, flexibility and generalization approach[END_REF]. The interviews were conducted individually, recorded and transcribed.

To approach children's ideas about numbers, we analyzed their utterances in the transcribed interviews and categorized them in a three-stage generalization model, following current literature. Preschoolers' explanations at a first stage do not present any characteristics or properties or relationships related to quantities or numbers, as children remain verbally reciting the number sequence (see, e.g. [START_REF] Baroody | The developmental bases for early childhood number and operations standards[END_REF]). Children's explanations at a second stage show a perceptual or procedural approach of characteristics, properties or relationships related to quantities, as children order or combine quantities mainly by counting or by subitizing small quantities and recognize only a part of interrelations (see, e.g. [START_REF] Clements | Subitizing: What is it? Teaching children mathematics[END_REF]). Children's expressions at a third stage start presenting a certain awareness of characteristics, properties or interrelations of numbers, as children conceptually subitize and stabilize number facts (see, e.g. [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF]; due to space restrictions examples of each stage will be presented along with the results). After their initial examination and categorization, these 23 pre-schoolers, individually or in teams, dealt in their classroom with rather usual arithmetic tasks, materials and games for four weeks (e.g. games for verbal counting quantities forward and backward, games for subitizing dot cards with different configurations, tasks for comparing, ordering or analyzing quantities, recognizing numbers in numbers cards, etc., see [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF]. During this program, they were systematically encouraged by their teacher to discuss about their actions, strategies, methods, decisions, reasoning and to proceed to verbal formulations regarding more general ideas related to their outcomes of their activity. Thus, they were reinforced to move from a local level with conclusions made out of a specific task or activity (e.g. way of doing it) to a broader level with assumptions about mathematical ideas (e.g. number 4) and finally formulation of rules or propositions (e.g. numbers' relations) [START_REF] Dougherty | Developing concepts and generalization to build algebraic thinking: The reversibility, flexibility and generalization approach[END_REF] and they were reexamined in similar tasks. After working for several months on other mathematical fields, these children improved, in general, their flexibility to find common elements in different contents, to reflect on their actions and produce more general conclusions deriving from specific situations.

Results

The table below (Table 1) shows differences between the initial and final examination of the children as they appear in the transcripts of their interviews. These results provide an overview about pre-schoolers' achievement and progress in relevant tasks after a rather short teaching period.

Items

Quantities Correct answers in this table concern: for verbal counting, correctly counting in all four cases (one by one or by twos, forward and backward), for subitizing, correctly recognizing all dot cards with 5, 7 and 8 dots without counting, for combining, putting together dot cards in all possible combinations (5+1, 3+3, 4+2 and 2+2+1) and for transforming from 7 to 4 and 2 to 6, correctly completing both changes.

Seeking for a deeper insight into preschoolers' number conceptualization and, thus, their underlying more generalized ideas about numbers and interrelations, we analyzed their explanations in the interviews according to the three-stage generalization model, presented earlier (in brief: reciting the number sequence, perceptually subitizing or procedurally approaching relationships, conceptually subitizing and stabilizing number facts). In general, there were no children at the first stage.

-In the initial examination of verbal counting [START_REF] Baroody | The developmental bases for early childhood number and operations standards[END_REF]), all 23 children were able to recite the arithmetic sequence one by one forward, but did not show the same stability in all counting tasks (by twos or backward, by ones or twos). Seven children who were identified as being at the second stage (mostly procedural) were able to count one by one, but could not do it backward, hesitating especially for numbers after 7, missing one or two numbers in the sequence or starting counting with their fingers. They had greater difficulties in counting forward or backward by twos, not being, thus, able to realize that number sequence was stable (e.g. 5,7… so backwards 7, 5…) or to perceive relationships between numbers. For example, a child said, "It may be something else..." and another one "10, 8, 6, 2, 0, … there must be some numbers missing". Eleven children, who were considered as being at the third stage (more advanced procedural), recited the arithmetic sequence fluently, but did not have the same fluency in counting backward or counting by twos, while the rest ( 5) with the same procedural approach were able to count by using the one by one correspondence and to explain based on it, e.g. "0, 2, 4 (plus 1), 5, (plus 1), 6, …", showing, thus, an awareness of stability and simple relationships between numbers (see Table 2).

In the final examination of verbal counting, 22 children verbally counted correctly in all cases and, furthermore, declared that the order of number sequence was stable, being identified thus at the third stage. However, their explanations regarding this order might be indicative of different levels of generalization, as 6 of them who recognized certain relationships (e.g. 8 and 1 is 9 or 3 and 2 is 5) could not generalize them throughout the whole number succession, or other 5 children who had stabilized knowledge of succession one by one, were not in the same state when counting by twos. Finally, with more or less complete understanding, 11 children seemed to be aware of numbers' interrelations that generate the number sequence, appearing to be at a more advanced level (see Table 2).

-In subitizing dot cards [START_REF] Clements | Subitizing: What is it? Teaching children mathematics[END_REF], initially most of the children (15) could not recognize quantities or explain how to do it by perceiving subgroups of quantities and interrelations between them, being thus at the second stage. Very few children (3) could recognize quantities with more than 3 dots and, except one, they said that they counted silently. Some other also (5) used the combinations they knew, e.g. "I saw 3+3 that is 6 and then 7…" (partly recognizing and partly counting). These 8 children were identified as being at the third stage. In the final examination, after working with relevant tasks and discussing on number characteristics and relationships, 15 children declared that they recognized subgroups and did not use counting, e.g. a child said "No, I didn't count, I did 3+3 and plus 2…". Some children (6) made use of patterns of previous cards and said "this was 7, then this is 8". Some other still (2) combined subitizing small quantities and counting. These different approaches (some procedural and some conceptually subitizing) show also different kinds of generalization at the third stage (see Table 2).

-While combining dot cards to form the number 6 [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF], 15 children initially presented only the combination 5+1 of one 5-dot card and one 1-dot card. The children (8 in total) who provided all the right combinations were divided into two groups: in the first group 3 preschoolers mainly used a 'trial and error' method and counted the dots one by one, without knowing if always 4 plus 2 was equal to 6. For example, a child started with a 5-dot card and put it together with a 2-dot card and counted all the dots, as the result was 7 s/he changed with a smaller card. In the second group, 5 children put cards together using the relationships they already knew, but they checked the result by counting. This group of children remained mainly at a procedural and, thus, second stage.

In the final examination, 15 children were able to present all the combinations for 6 (third stage) while 8 remained in 1 or 2 -3 of them (second stage). Eleven of the 15 pupils did not use the method of "trial and error" or of counting, but, in general, they were able to put together groups of quantities, showing, thus, an awareness of relationships of quantities and numbers. They could be considered being in a course of a conceptual understanding, as they were competent to explain with numbers e.g. "I said 3+2 makes 5, plus 1, 6…" and "…then 3+3 makes 6". For the other four pupils of this stage, the interrelations between quantities and numbers were generally less developed, as they verified every suggestion by counting, showing that they remained at a procedural stage.

-Finally, in the task of transforming 7 to 4 or 2 to 6, initially all the children were at the second stage. Twelve of them counted in different ways, e.g. a child said "I counted with my fingers and (showing) if I put out 3 remains 4 fingers", or, more generally, they did it with number sequence, "I put out 1, 6 and then 5, so we get 4", while 11 children were not able to find a way of doing it (see Table 2).

But in the final examination, 12 children were identified as being at the third stage, but in different levels of conceptualization. Thereby, a group of 5 children explained the changes with numbers in a more general way e.g. "3 and 3 and 1 equals 7, if you keep 4 you have 3..." showing a conceptual understanding, while 7 others did the same without being sure about the results (between conceptual and procedural). E.g. for 7 to 4, a child said "I can see a group with 3 and a group with 4… so (hesitates)… then 3…" or they put groups together and counted (in a procedural stage). The rest 11 children were still not able to do it correctly. This analysis is summarized in the Table 2 in which the initial three-stage model was analyzed in more details, as there was no child in stage 1 but, as it was previously presented, differences were revealed in other stages.

Pre-schoolers' explanations of stage 2 are divided in three sub-stages: 2a, in which their utterances revealed a procedural recognition of quantities and basic relationships, with no specific awareness; 2b, in which they presented a wider procedural recognition of quantities and relationships and 2c, in which they showed children mainly able to manage procedurally quantities and numbers. For example, in the task of combining dot cards to form a quantity of 6 dots, children of 2a put together cards with 5 and 1 (basic) dots, without being able to present other combinations, while children of 2b started putting together more cards by counting them and children in 2c were able to find cards and in case of errors (e.g. 4-dot card and 3-dot card) to directly change with other dot cards but also count for checking.

Pre-schoolers' explanations of stage 3 were also divided in three sub stages: 3a, in which their utterances revealed recognition of a part of groups/subgroups of quantities and numbers or their interrelations, 3b, in which they presented a wider recognition of interrelations of quantities or numbers but without consolidation and (a final) 3c, in which children recognized numbers, their order and their interrelations. For example, in the task of subitizing, children of 3a started by recognizing small quantities and continuing by counting, while children of 3b recognized some of the proposed cards and explained e.g. for a 5-dot card "I saw these 4 and this 1", but still misjudged a 7-dot card, finally 3c children recognized all cards and explained e.g. "2 plus 2 and 1 equals 5" or "3 plus 3 plus 2 equals 8" showing the relevant groups in the cards.

Discussion

The results presented indicate in brief that the participating children were initially able to count verbally up to 10, but they did not recognize more generalized interrelations between quantities or numbers, remaining mainly in this procedural counting. Conversely, after working in the teaching program that systematically encouraged them to reflect over this counting and express more generalized ideas related to quantities and numbers, most of these children (almost 19 out of 23) started developing flexibility in combining quantities and numbers, in realizing numbers' relationships and consolidating several numbers' facts. Despite the small sample that does not allow wider conclusions, these outcomes suggest that guiding young children to focus on dynamic and relational aspects of numbers and express more general ideas related to their numerical tasks could support decisive numerical understanding and more generalized arithmetic knowledge.

Arithmetical learning has many approaches, but the dominance of counting tasks does not often make apparent whether children develop number sense, structural numerical elements or relationships, as this study attempted to show. Many deficiencies in the first arithmetic learning prove to cause difficulties to students to conceptualize the multiplicative, relational and repetitive structure of numerical system. Teaching approaches that encourage children to generalize seem to be required for numbers as they are for any other mathematical concept, since learning mathematics is not only to practice in situations with material, but also to reflect and draw more general ideas out of this practice

Table 1 . Number of correct answers before and after teaching period in numbers.

 1

		or Numbers	Initial	Final
	Verbal counting	Up to 10, one by one or by twos, forward and	6	18
		backward
	Subitizing	Dot-cards with 5, 7, 8 dots	3	15
	Combining	Quantities with dot-cards to form 6	8	17
	Transforming	7 to 4 and 2 to 6 (in numbers)	3	15

Table 2 . Number of children in different generalization stages in the initial and final examination

 2 This table provides an overview of these children's improvement in approaching and verbally expressing more global ideas about quantities, numbers and interrelations indicating, thus, a more substantial conceptual understanding.

			Stage 2: perceptual and	Stage 3: procedural and
			procedural approach		conceptual approach
			Stage 2a Stage 2b Stage 2c Stage 3a Stage 3b Stage 3c
	Reciting	Initial	4	3	0	11	5	0
		Final	0	0	1	6	5	11
	Subitizing	Initial	8	5	2	5	2	1
		Final	0	0	0	2	6	15
	Combining	Initial	15	3	5	0	0	0
		Final	4	4	0	3	1	11
	Transforming	Initial	11	9	3	0	0	0
		Final	4	4	3	7	3	2


Yuly Vanegas
email: ymvanegas@ub.edu

Montserrat Prat
email: montserrat.prat@uab.cat

Alba Rubio
email: arubio@escolesgarbi.cat

Characterisation of the learning trajectory of children aged six to eight years old when acquiring the notion of length measurement

Keywords: Learning trayectories, length measurement, early years

HAL is

Introduction

From the standpoint of the pre-and primary school education mathematics curriculums, measurement is a fundamental area of content and strongly related to the mathematics used in everyday life [START_REF] Battista | Understanding the development of students' thinking about length[END_REF][START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF][START_REF] Chamorro | Didáctica de las matemáticas para primaria[END_REF]Smith, van der Heuvel-Panhuizen, & Teppo, 2011;[START_REF] Antonopoulos | Measurement activities and teaching interaction in early childhood education[END_REF]. Nonetheless, sometimes not enough effort is invested in teaching length measurement formally because it is considered social knowledge that will be acquired by students outside school [START_REF] Chamorro | Didáctica de las matemáticas para primaria[END_REF].

Length measurement has the potential to connect different mathematical domains such as numeracy and geometry [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF]Smith, van der Heuvel-Panhuizen, & Teppo, 2011). It is also a tool without which many descriptions of a spatial type would be vague and imprecise [START_REF] Battista | Understanding the development of students' thinking about length[END_REF] and it provides a practical skill that is important in everyday life [START_REF] Buys | Domain description Measurement[END_REF].

On the other hand, like Sarama, Clements, Barret, Van Dine and McDonel (2011) we believe that an understanding of the learning trajectory involved in length measurement gives us, as teachers, resources to improve teaching and learning strategies that support children's acquisition of this competence. The goal of this study is to identify the learning trajectories followed by pupils aged six to eight years old in order to construct the notion of length measurement.

Learning length measurement

Research on measurement learning at an early age shows that educators place the focus on arithmetic (counting, comparing quantities) and geometry (constructions, shapes and patterns) rather than on measurement [START_REF] Benz | Attitudes of Kindergarten Educators about Math[END_REF], despite the huge presence of measurement in children's day-to-day lives. [START_REF] Copley | The young child and mathematics[END_REF] expresses himself in similar terms when noting that measurement is an often-neglected mathematics topic in the education of young children; and also, Barret et al. (2012), for whom measurement learning is little connected to its mathematical foundations, often meaning that educators miss opportunities to strengthen measurement concepts such as the idea of the unit. And as pointed out by [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF] mechanical learning of the notion of measurement should be avoided. [START_REF] Piaget | The Child's Conception of Geometry[END_REF] define the notion of length measurement as the synthesis of subdivision and change of position, which includes extracting a part of the total and iterating this unit throughout the entire object. Measurement for [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF] "can be defined as the process of assigning a number to a magnitude of some attribute of an object, such as its length, relative to a unit" (p.163). These attributes are measured in continuous quantities so that they can always be divided into smaller quantities [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF]. For these authors "length is a characteristic of an object found by quantifying how far it is between the end points of the object" (p. 164). Thus, to determine the length it is necessary to first subdivide the object into equal units of measurement and iterate them along the object [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF].

According to [START_REF] Clements | Early Childhood Mathematics learning[END_REF]2009), learning of measurement, and in particular of length measurement, involves the comprehension of eight concepts: (i) understanding of the attribute (of length); (ii) conservation (of length); (iii) transitivity; (iv) equal partitioning of the object to be measured; (v) units and unit iteration; (vi) accumulation of distance and additivity (the number of iterations indicates the space covered by the units up to a specific point); (vii) origin; and, (viii) relation between number and measurement.

Hypothetical Learning Trajectories

The construct of the Hypothetical Learning Trajectory (here in after HLT) was developed by [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] as part of his teaching cycle model. According to this author, an HLT is built around three components: a) the learning goal, b) the mathematical tasks that will be used to promote learning, and c) the hypotheses about the learning process. The aim is to make a prediction of how students can learn a certain mathematical content based on their previous knowledge and experience. [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF] state that children follow natural processes of development when learning mathematics. These developmental paths form the basis of learning trajectories. Thus, in a similar way to what was proposed by [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF], these authors consider that learning trajectories involve three essential components: a) a mathematical goal; b) a developmental path; and c) a set of instructive activities or tasks typical of the levels of thinking of the path. They also suggest that the use of learning trajectories can help answer key questions concerning teaching and learning processes: the goals to be set; where to start; how to decide the direction of the next step; and how to achieve that next step. Specifically, in the case of length measurement, [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF] propose an HLT organized at different levels of development, with some associated instructional tasks. It is assumed that these levels indicate the path taken by children while constructing meanings for length measurement. Some learning goals defined by key aspects of the curriculums of each pre and primary school class level are considered in order to specify these levels. Importance is given to the fact that children initially have to identify the attributes of measurement and compare objects using these attributes. Later they must be able to order objects according to their length. As the children advance in their understanding of numbers, their learning should be geared towards the recognition of connections between the metric and numeral systems, making measurements that imply the use of units and counting those units. Finally, children must be able to recognize the need to have units of equal length, through the use of standard units, and be able to identify and use measurement tools.

The levels proposed by [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF] for the learning trajectory involved in length measurement are as follows: Pre-Length Quantity Recognizer (PR); Length Quantity Recognizer (R); Length Direct Comparer (DC); Indirect Length Comparer (IC); Serial Orderer to 6+; End-to-End Length Measurer (EE); Length Unit Relater and Repeater (RR).

To improve children's understanding of length measurement, it is necessary to identify how they act in the various situations that involve this notion. We see the HLTs as a construct that serves to verify this understanding and thereby provide key elements for the definition of specific actions related to the teaching and learning processes.

In this study, we use the HTL construct in order to design our instrument and also as an analytical framework. Particularly we compare the current performance of each child with the hypothetical learning trajectory about length measurement.

Methodology

Our research uses qualitative methodology. An ad-hoc data collection instrument featuring open and manipulative activities was designed to respond to the research goal. The instrument was used in interview format with 20 students aged six to eight years old. The data was recorded audiovisually. The following factors were considered when defining the sample: a) Gender, there were eight boys and twelve girls. b) The time of the year when they were born, with six participants born in the first four months of the year, six in the second four months and eight born during the last four, the children were distributed by age, which resulted in a more homogeneous sample. c) Diversity in mathematical performance, we find children who are able to solve different mathematical activities and others who usually have some difficulty in carrying them out.

The data collection instrument considered the components of the HLT. It consisted of seven activities related to the key concepts underlying measurement: recognition, comparison, ordering and iteration. By way of an example, Table 1 presents instrument activities 1 and 3, consisting of the goal, the task, the description and a series of questions that guide the dialogue during its implementation. An object is placed on the table (notebook or box) that prevents the child from seeing both pencils at the same time. A pencil is placed on one side of the notebook and the child is asked to place his/her pencil on the other side. The child can look at and touch the pencils without moving them from their place. They can use any object they have available.

Goal

Can you use the piece of string to discover which pencil is longer? Can you use another object to measure them?

Table 1: Data collection instrument activities 1 and 3

The data collected came from 20 videos, one per student, lasting between 9' 21'' and 26' 28''. Each student was tagged with a number in chronological order. The analysis focused on the actions and processes performed by the children in order to situate their level (L) on the trajectory for each activity (Act.) carried out. Using the levels proposed by [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF], some indicators associated with each of the activities are described (Table 2).

Act.

L Indicators

R

The student is capable of identifying the attribute of length 2 DC The student is capable of making a direct comparison 3 IC The student is either capable of making an indirect comparison or does not do so properly, but he/she certainly understands the notion that two objects can be measured using a third one in order to know which is longer 4 O The student is capable of ordering the five strips using length as a criterion.

EE

The student takes into account the origin and purpose of the objects when measuring them but does not show any signs of understanding that the accumulation of units of measurement serves to measure the total length. He/she overlaps the units of measurement, leaves gaps between them, or does not show an understanding of the possible subdivision of units of measurement 6 EE/RR The student takes into account the origin and purpose of the objects when measuring them and shows some signs of understanding that the accumulation of units of measurement serves to measure the total length. He/she does not perform the procedures needed to carry out the measurement in the right way (i.e.by avoiding overlaps, gaps and unequal positioning of units of measurement) 7 RR The student takes into account the origin and purpose of the objects when measuring them, and shows that he/she understandsthat the accumulation of units of measurement serves to measure a total length Once the videos were transcribed, the dialogues generated by each activity were established as units of analysis. The information about each student was then organized according to the following factors: the activity, the unit of analysis, the graphic evidence and the researchers' observations. At this point the level evidenced by the children in each unit of analysis was identified (Table 3).

Results

The results of this research visualise the individual trajectory of each student and these trajectories provide a characterization that encompasses the whole sample. Thus, it can be seen in Table 3 that student A10 is capable of recognizing the attribute of length, of making direct and indirect comparisons, and ordering. To contrast the hypothetical learning trajectory with the real trajectory of each student, we drew up a two-dimensional adhoc graph that relates the activities to the levels. According to Figure 1, student A10 performs the first activities (Act 1 -Act4) in consonance with the HLT. However, in the activities geared towards measurement levels (Act 5 -Act 7) there is no consonance. As shown by the data collected in Table 3 and Figure 1, we can position student A10 on the IC level. At this level children are able to use rulers but often show a lack of understanding or ability. We only find student A10 performing actions on the objects on the EE level during the last activity.

It is important to clarify that the purpose of the graph contrasting the HLT with the real trajectory (Figure 1) is to provide a visual overview that shows where each child is situated during the performance of each activity. For this reason, although the transition from one level to another is not always the same, when drawing up the graph the levels have been positioned equidistantly. This makes the HLT appears to rise on a steady gradient, which is not really the case.

In Figure 2, the results obtained by the sample are presented visually, indicating in what levels the students are for the set of activities. Each point on the graph represents one student's level in each of the respective tasks of the instrument. Once the individual trajectories of the children had been established, it was observed that 70% of the group were either moving between the EE and RR levels or on the RR level. Only 20% showed no signs of the RR level, with 15% located at the EE level and 5% at the CI level. On the other hand, we can see that in activities 5, 6 and 7 related to the levels of measurement, the students were moving between the EE, EE/RR and RR levels but they did not necessarily stay at the same level during the performance of the three activities. In activities 1 and 4 all students were situated at the corresponding HLT levels. In Activity 2, all students except one were on the DC level.

Discussion

This study provides an overview of the group on the basis of the children's levels and trajectories. All the children, except one, were situated on levels EE and RR or were moving between them. According to [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF] and [START_REF] Barrett | Evaluating and improving a learning trajectory for linear measurement in elementary grades 2 and 3: A longitudinal study[END_REF] children reach the EE level of measurement when around six years old and the RR level when around seven. This affirmation concurs with the data generated by this study. The transition between these two levels is very subtle and for this reason we decided to divide it into two sub-levels so that we could see more clearly where the children were situated.

During the analysis of the data, it was observed that children, although they are shown to be working mostly on one of the HLT levels, apply concepts and perform actions on the objects that correspond to levels considered higher. And, of course, they also apply strategies and perform actions on objects that belong to lower HLT levels [START_REF] Sarama | Evaluation of a learning trajectory for length in the early years[END_REF].

Once the data was analysed, we noted that despite the fact that all the children except one were above the EE level, not all of them were able to cope with the challenge of transitive inference. Although we are aware that the study is limited by the performance of only one activity linked to the IC during data collection, and the fact of working with only 20 children, in our case we suggest the following explanation: the EE level is either usually reached a little before the possibility of transitive inference, or work on specific mathematical experiences related to this inference is needed prior to its development. In this respect [START_REF] Sarama | Evaluation of a learning trajectory for length in the early years[END_REF] conclude in their study: "(...) children achieved the Indirect Length Comparer level of thinking just prior to the End to End measurer level" (p. 678).

In this study we found students who, despite being mostly on the RR level, were not able to make explicit transitive inferences. This finding also supports the idea that children do not need to acquire conservation, nor the concept of transitivity, in order to acquire certain notions about measurement [START_REF] Hiebert | Cognitive development and learning linear measurement[END_REF]Clements, 1999, cited by Sarama et al., 2011).

It is noteworthy that in this type of study is relevant to consider the influence of the interaction between the child and the tutor, as it could make a difference in the evaluation of actions in the various activities. In some cases, this could be considered a limitation of the study. In our case, the tutor who participated is a teacher who knew the children and was involved in the research.

A knowledge of the Hypothetic Learning Trajectory and its levels can enhance the teaching of length measurement [START_REF] Barrett | Evaluating and improving a learning trajectory for linear measurement in elementary grades 2 and 3: A longitudinal study[END_REF]. This study showed us where the six-to-eight-year-olds in our sample were situated and which mathematics topics demand greater attention to enable them to move on to the next level of the trajectory and thereby consolidate their learning.

  length. The idea is children talk about the attributes of the pieces of string during the activity and that they clearly refer to the notion of length.What differences do you observe between these pieces of string? Are the four the same? How are they different? Can you see any other features that are different? How do you say it when they are different in their "bigness?

  's it. Tutor: That's it? How many paperclips long is it? Student: (counts) 6. Tutor: 6? Ok Student: 1,2,3,4,5,6. Tutor: And why do you do it like that? Can you tell me? Student: Because I put them in 3s. 1,2,3, 1,2,3 and that's how I know. Tutor: Ok. And why have you put them on both sides. Wasn't one side enough? Student: No... Tutor: Did you have to put them like that? Student: Yes, I had to put them like that. Tutor: (...) Student: I think it's 6 metres. Tutor: 6 metres? Or 6 paperclips? Student Well, 6 metres I think 7a 7. Do you know what number that is? A 1 and a 7? Student: Twenty.... seven? Isn't it? Tutor: I don't know, I'm asking you.... Student: 17. Tutor: 17 what? Studen: 17 metres.

Figure 1 :

 1 Figure 1: Real learning trajectory of student A10 in relation to the HLT

Figure 2 :

 2 Figure 2: Learning trajectories in the sample

Table 2 : Correspondence between activities and levels

 2

Table 3 : Example from analysis instrument corresponding to student A10

 3

Acknowledgement

This work is part of projects: EDU2015-65378-P and EDU2013-46083-R, MINECO/Spain