text:
Renaud Chorlay
email: renaud.chorlay@espe-paris.fr

Mathematical analysis of informal arguments: A case-study in teacher-training context

Keywords: History of mathematics, fractions, teacher-training, argumentation, generic examples

We will endeavour to show that some historical documents which are probably too difficult to be used in the classroom can nevertheless be fruitfully used in teacher-training, in order to provide teachers with tools for the analysis of informal or semi-formal justifications. We will analyze an extract from the Nine Chapters, so as to spell out tool for the analysis of three school documents bearing on the multiplication of decimal or general fractions.

Introduction

We will endeavour to show that some historical documents which are probably too difficult to be used in the classroom can nevertheless be fruitfully used in teacher-training, in order to provide teachers with tools for the analysis of informal or semi-formal justifications. We will analyze an extract from the Nine Chapters, so as to spell out tool for the analysis of three school documents bearing on the multiplication of decimal or general fractions.

A historical example: Multiplying fractions in the Nine Chapters

The Nine Chapters on the Mathematical Art (九章算術) is a Classic that was compiled during the Han Dynasty (206 BC -220 AD). It contains an organized list of problems with general procedures to solve them, yet with no attempts at justification. In 263 AD, scholar Liu Hui wrote an extensive commentary in which endeavoured to justify all the procedures, and identify key subprocedures of general scope. We will comment on the passage in chapter one which deals with the multiplication of fractions, on the basis of the critical edition and French translation [START_REF] Chemla | Les neuf chapitres : Le classique mathématique de la Chine ancienne et ses commentaires[END_REF] 1 . Problem 1.19. Now given a [rectangular] field, 4/7 bu in breadth and 3/5 bu in length. Tell: what is the area? Answer: 12/35 bu.

(…)2 Procedure for the multiplication of parts: multiply the denominators to make up the divisor; multiply the numerators to make up the dividend. Divide the dividend by the divisor.

(…) [Liu Hui]: In all cases when a dividend does not fill a divisor, then they are called denominator and numerator. If there are parts, expand the corresponding dividend by multiplication, then, when the divisor is filled, the division yields an integer. If, moreover, one multiplies something by the numerator, the denominator must therefore divide in return [baochu]. To divide in return is to "divide the dividend by the divisor". Now, "the numerators are multiplied one by the other", hence both denominators have to divide in return. Whence the multiplication of the two denominators, and the division at one go [by their product].

The mere length and width of a rectangular field leave no room for a general explanation. Let someone ask: "20 horses are valued at 12 jin of gold. Now 20 horses are sold and the proceeds are shared by 35 persons. How much does each one get?" The answer is 12/35 jin. To solve this by the procedure for dividing into parts, take 12 jin of gold as dividend, and 35 persons as divisor. Now, change [the problem] to: "5 horses are valued at 3 jin of gold; now 4 horses are sold, and [the proceeds] are shared by 7 persons. How much does each one get?" Answer: each one gets 12/35 of jin. To solve this, you need to homogenize these quantities (shu) of people and of gold; it is then completely similar to the first problem, that of dividing into parts. If so, "multiply the numerators to make up the dividend" does homogenize this quantity of gold; "multiply the numerators to make up the dividend" does homogenize this quantity of people. Equalizing the denominators yields 20, but that plays no part: we just need the homogenized. Moreover, when 5 horses are valued at 3 jin of gold, these are the lü in whole numbers. If we express them with parts, 1 horse is valued at 3/5 jin of gold. Let 7 people sell 4 horses, 1 person does sell 4/7 of a horse. (…) As far as expression is concerned, that's different; yet as far as quantities are concerned, the three procedures boil down to the same. Even if the procedure in quite easy to understandand familiar to the reader -Liu Hui's commentary is probably hard to decipher. Even without a clear understanding of the details, one can probably see at least two things: the goal of Liu is to justify the general procedure of the classic; he gives two different justifications, one which does not rely on a semantic context nor on specific numbers, and one which does.

Let us first focus on the first justification. Liu first distinguishes between two cases: a basic case in which the fractions represent whole numbers; in this case, the procedure for multiplication is already known. Of course, the goal is here to justify the new procedurein the case where the divisions have non-zero remainders -on the basis of the basic procedure. The word "procedure" is important, since what is to be justified is the validity of an algorithm. More precisely, two sequences of calculations are to be compared: If the two sequences relied on operations defined previously and independently, the goal would have been to prove the universal equivalence of these two algorithms (that is, same entries yield the same output). However, if one is not to beg the question, the first sequence actually contains the undefined operation multiply, since the final multiplication may involve a multiplication of fractions 3 . So the goal here is rather to justify that sequence 2 (which involves only well defined operations on integers) should be universally equivalent to sequence 1, thereby implicitly defining "multiply" in terms of "multiply."

To perform this, Liu uses terms which are specific to the commentary: whereas the Classic uses "divide" and "multiply", Liu uses "expand by multiplication", and "divide in return" (baochu); these terms point to the role of these operations. However, it is important to know that throughout the Nine Chapters, "baochu" denotes a division whose role is to compensate for a multiplication by the same number, so that these two steps of the procedure do not affect the final output of the sequence of operations. On this basis, one can see that Liu's justification involves a third sequence of operations: Sequence 3: 1. multiply a by b, so that the first factor becomes ab / b, which is equal to a 2. multiply c by d, so that the second factor becomes cd / d, which is equal to c 3. multiply the two factors, which yields ac 4. divide by bd, which is the product of the two denominators This sequence is not to be actually performed, it is mentioned for justificatory purposes only. Since step 4 is a baochu-division, it compensates steps 1 and 2. The line of argumentation can thus be reconstructed as: sequence 1 (which is yet undefined) should be universally equivalent to sequence 3, and sequence 3 boils down to sequence 2.

This line of argumentation may seem far-fetched. However, it is quite close to the following contemporary proof of the following theorem: the only map f from ℚ × ℚ * which is ℤ-bilinear and which is a prolongation of integer multiplication (i. Diving both sides by (bd) completes the proof. One could argue that, although Liu Hui states the prolongation requirement quite explicitly, never does he say anything resembling the bilinearity requirement. Even if this requirement is not explicit, it should be stressed that the bilinearity of the operation being defined (i.e. multiply) can be rephrased as follows: multiplying either of the two factors by and integer n should multiply the product by n; which is exactly the property which justifies the role of the baochu division.

The second part of Liu Hui's commentary clearly follows another line of argumentation. The first part does not rely on a semantic context of interpretation, nor does it use specific numbers. The second part, however, depends on a semantic context which Liu Hui introduces out of the blue after dismissing the context provided by the Classic (rectangular areas), and uses the numbers from Problem 19.

In the second part, several in-context problems are mentioned, and their relationships discussed. Although Pb2 and 2' differ from a semantic viewpoint, they are equivalent in the following sense: the numerical answers to Pb 2 and 2' are equal, since the second parts are the same, and the answers depends only on the ratio between horses and jin of gold. The argument would hold for any factor, it so happens that "4" is more relevant than for what follows 5 . This is quite explicit in the text, since throughout the Nine Chapters the technical term lü denotes either a ratio, or numbers which are to be considered up to multiplication by a common factor. The relationship between Pb2 and Pb 2' is the same as that between Pb2' and Pb2'':

Pb. 2'': 5x4 horses are valued at 3x4 jin of gold. Now 4x5 horses are sold and [the proceeds] are shared by 7x5 persons. How much does each one get?

Now, Pb 2'' is the same as Pb 1, so their numerical answers are the same: 12/35.

But there is another way to solve Pb 2. The numerical answer to Pb 2 is the same as that of Pb 2''': 1 horse is valued at 3/5 jin of gold. Now 4/7 horses are sold and [the proceeds] are shared by 1 person. How much does this person one get?

We want to define multiplication of fractions so that the following property of integer multiplication remains valid: total value = unit value times quantity. As a consequence, the product A more formal summary would go as follows (we use ⊗ to denote to multiplication of fractions, to be defined here in terms of ):

3 5 ⊗ 4 7 = 3 × 4 5 × 4 ⊗ 4 × 5 7 × 5 = 3 × 4 5 × 4 ⊗ 4 × 5 7 × 5 = 3 × 4 7 × 5
As for the extra-steps mentioned in the first argument, none of these intermediary operations are to be performed when actually multiplying fractions; they are here for justificatory purposes only. This formal summary does not do justice to the nature of the argumentation, since it is the context only which justifies the equalities: dependence on ratios only (first equality); then reduction to a problem involving only two integer data instead of four, and which can be solved by ordinary division. Just as well, the fact that any given fraction can be freely replaced by any equivalent fraction can be justified 5 We will not comment on the use of technical terms such as "equalizing" and "homogenizing", which Liu Hui introduced in the context of the addition of fractions: by the context; for instance, the unit price depends only on the ratio between the number of horses and their total value.

Tools for the analysis of arguments in today's textbooks

Reflecting on the Nine Chapters from a teacher-training perspective

The obscurity of the Chinese text probably makes it impossible to use in the classroom. In this paper, we would like to show how the mathematical analysis of this text that we carried out in the first part can help raise the awareness of prospective teachers as to several general features of argumentation in rather informal contexts (this is our Hypothesis #1); and even supply them with tools which are useful in everyday teaching, when they are to decide how to (somehow) justify some mathematical definition or property (this is our Hypothesis #2) 6 .

The analysis of the Nine Chapters drew our attention to several facts, some of which bear specifically on fraction multiplication, and some of which are of a more general scope:

 There are several ways to justify the rule for the multiplication of fractions.  The use of letters is not the only way to express a general line of argumentation, as Liu Hui's first argument illustrates.  The rule for fraction multiplication is not only an equality between formulae; it can also be seen as the equivalence of two different algorithms (one with two divisions followed by a multiplication, and one with two multiplications followed by a division). This equivalence of algorithm can be established using general arguments within the algorithmic framework (i.e. arguments about algorithms), as opposed to the rewriting of formulae within an algebraic framework.  Argumentation is not limited to properties or theorems: the choice of a definition (here: for the multiplication fractions) can also be justified. With fraction multiplication we are dealing with a case domain extension: some notion was already defined for a given class of objects; the to-be-defined notion has to apply to a class of objects that encompasses the first class, and to coincide with the former notion on the first class. In our case, a very weak form of justification would be: this definition of fraction multiplication boils down to ordinary multiplication (of integers) should the fractions denote integers (this could be called the control case). The Nine Chapters suggests a stronger kind of justification: the extended definition of "multiply" should preserve some structural properties of the former definition, properties that we value. For instance: bilinearity; or the validity of the formula "total value = unit value times quantity". On the other hand, we are ready to give up on some properties, such as: the product of two numbers is greater than or equal to either of them. To some extent, when justify the choice of a definition, these required properties play the part that hypotheses play in the proof of a theorem.  In some cases, from a mathematical viewpoint, these requirements completely characterize the new notion. In these cases, whether or not this uniqueness property should be proved in 6 The decisions as to how to justify a given definition or property is closely related to the decision as to whether or not it should be, and to the knowledge of ways to justify it (can it be defined, or not, in a given mathematical context?). We will not have time to discuss the interplay of these three aspects in this short paper.

Thematic Working Group 12

the classroom calls for another decision. Even if the teacher thinks the uniqueness aspect cannot / need not be made explicit, the Nine Chapters suggest another lead. Making explicit the properties that we want to extend also has a heuristic value, suggesting the path to a definition of the extended notion.  Liu Hui's second line of argumentation also suggests many fruitful leads, which we can only mention in passing for lack of space: o A rather unusual form of argument: proving that two formulae or two algorithms are universally equal / equivalent by exhibiting a class of in-context problems for which the answer is uniquefor any given set of initial data -, and for whichfor reasons that can be spelled outboth formulae or algorithms work out the solution. Of course, this form of argumentationthat can be called semantic immersionraises questions as to the context-dependence of the argument. For instance, multiplication plays a part in the context of commercial transactions and in the context of area calculation, and nothing proves that the fact that the relevant operations coincide in both contexts when the data are integers will still hold with all rational data. o Liu Hui's second argument is not only in-context (a horse-deal), it is also an arguments which clearly claims full generality while dwelling on a single specific case, that of 3/5 times 4/7. This raises the well-kown but deep questions pertaining to the notion of generic example (Balachef, 1987, 157). Let us mention two such questions: (1) from a mathematical viewpoint, which conditions guarantee the minimal level of nonspecificity that an example should enjoy to be potentially seen as generic (for instance: dealing with 1/5, 7/5, or 15/5 instead of 3/5 would probably make this case less generic)? (2) The genericity of a case is not a mathematical property but is a multilayered property of the relationship between it and the students; it involves pragmatic aspects (students should be able to adapt the reasoning to other cases), and epistemological aspects (students should regard this case as a mere representative of all cases, thus providing some form a general argument); what are the general conditions for a didactical genesis of this relationship?

Field-work

We shall end with some applied work. The following three documents are extracts from contemporary French textbooks or standard teacher-training documents, for students in the second year of middleschool (Figures 2 and3) or in the last year of primary school (Figure 4). In cases 2 and 3 they are meant for students who are somewhat familiar with fractions and their addition, but with no knowledge of multiplication. As to Figure 4, students are familiar with integer multiplication, and with multiplication and division of decimals by powers of 10. These extracts are what most French textbooks call "introductory activities", in which students are to experience a new notion from a hands-on, guided but non-dogmatic approach. We suggest these documents be analyzed in TWG12 in order to test Hypotheses #1 and #2.

Conclusion

The goal of this short paper was to contribute to the ongoing reflection on the use of original historical sources in the classroom or in teacher-training; a reflection which bears both on the goals of this use, and on the relevant ways to use such documents.

Within this general field of research, the main features of this specific contribution are: we dealt with teacher-training only, since we do not think this excerpt from the Nine Chapters can be used with students who learn to multiply fractions; we endeavoured to show that even for such documents, they can be used in teacher-training for other purposes than simply teaching some history of mathematics, or enriching teacher's image of mathematics. As teachers, it is or will be part of their everyday work to analyze/assess teaching documents and make decisions on the basis of this analysis/assessment. Among these documents, some will be of an argumentative nature: sometimes in the form of pretty formal and academic proofs; more often than not, these documents will explicitly display or rather implicitly point to other forms of argumentation. We think the conceptual analysisan analysis which has a mathematical component and an epistemological component, and which is to be distinguished from both the historical analysis and the didactical analysisof some historical documents can give teacher-trainers opportunities to make explicit some tools which can be of constant use in the analysis and assessment of argumentative documents.

In this paper, we dealt with pretty informal arguments and elementary mathematics. However, we do not think this general scheme is specific to such contexts, since this work is a continuation of our CERME8 paper [START_REF] Chorlay | The making of a proof-chain: Epistemological and didactical perspectives[END_REF], in which a similar approach was used in a formal and rather advanced mathematical context (proof of the relationship between the variations of a function and the sign of its derivative).

Figure 1 :

 1 Figure 1: (Chemla & Shuchun, 2004, 169-171). Free translation: R. Chorlay

  Sequence 1: divide a by b; divide c by d; multiply the two quotients.Sequence 2: multiply a by c; multiply b by d; divide the first product by the second product.

=

  e. if m and n are integers, then 𝑓(𝑚, 𝑛) = 𝑚 × 𝑛), a, b, c, d denote integers, with bd  0). The proof goes as follows: 𝑓(𝑎, 𝑐) by a Lemma 4 which relies only on addition in ℚ = (𝑎 × 𝑐) (prolongation requirement)

  terms) = 𝑏𝑎 𝑏 = 𝑎 (assuming b ∈ ℕ * ). Thematic Working Group 12 Proceedings of CERME10Pb. 1: 20 horses are valued at 12 jin of gold [for all 20 horses]. Now 20 horses are sold and the proceeds are shared by 35 persons. How much does each one get? Pb. 2: 5 horses are valued at 3 jin of gold [for all 5]. Now 4 horses are sold and [the proceeds] are shared by 7 persons. How much does each one get? Problems 1 and 2 can be connected by a series of problems. Starting from Problem 2, on can consider: Pb. 2': 5x4 horses are valued at 3x4 jin of gold. Now 4 horses are sold and [the proceeds] are shared by 7 persons. How much does each one get?

  to Pb 2''', hence to Pb 2 just as well, hence equal to 12/35.

Figure 2 :Figure 4 :

 24 Figure 2: Two ways to work out a rectangular area (Malaval, 2014, 69)

An independent English translation is also available[START_REF] Crossley | The nine chapters on the mathematical art: Companion and commentary[END_REF]. As far as this passage is concerned, the two translations differ significantly; our interpretation is based on Chemla's translation and interpretation.

I edited out two similar examples (with proper fractions), as well as Li Chunfeng's commentary.Thematic Working Group 12Proceedings of CERME10

When the remainder in the Euclidean division is non-zero, the quotient, the remainder and the dividend were read as a mixed number. For instance, the division of 7 by 3 has quotient 2 and remainder 1, so its result is 2 and 1/3.Thematic Working Group 12Proceedings of CERME10

Proceedings of CERME10

Document retrieved from a teacher-training slideshow: http://docslide.fr/documents/techniques-operatoires-cycles-2-et-3-multiplication-jean-luc-despretz-cpc-landivisiau-avril-2010-lacquisition-des-mecanismes-en-mathematiques-esttoujours.html (slide 22). Accessed September 14, 2016.


domain: math shs

text:
Introduction to the papers of TWG12: History in mathematics education

Renaud Chorlay 1 ; Kathy Clark 2 ; Katalin Gosztonyi 3 ; Snezana Lawrence 4 1 Uni. Paris Diderot & ESPE de l'Académie de Paris, France; renaud.chorlay@espe-paris.fr 2 Florida State University, USA 3 Eötvös Loránd University, Budapest, Hungary 4 Anglia Ruskin University, UK History of mathematics in mathematics education continues to receive much attention. However, empirical research and coherent theoretical/conceptual frameworks within this area have emerged relatively recently. The purpose of this TWG is to provide a forum to approach mathematics education in connection with history and epistemology dedicated primarily to theory and research on all aspects of the role, effect, and efficacy of history and epistemology as elements in mathematics education.

TWG12 welcomes both empirical and theoretical research papers, and poster proposals related to one or more of the following issues:

1. Design and/or assessment of teaching/learning materials using the history of mathematics, preferably with conclusions based on empirical data; all levels can be considered, from earlyage mathematics to tertiary education and teacher training.

2. Surveys on the existing uses of history or epistemology in curricula, textbooks, and/or classrooms in primary, secondary, and tertiary levels;

3. History of mathematics education;

4. Relationships between, on the one hand frameworks for and empirical studies on history in mathematics education and, on the other hand, theories, frameworks and studies in other parts of mathematics education research.

Even though the creation of this TWG is fairly recentit started in CERME6 ( 2009)it has deeper institutional roots within the maths education research community. Indeed, the HPM study-group (History and Pedagogy of Mathematics) was created at the 1972 ICME conference; it has been organizing satellite conferences to the ICME meetings since 1984, and has several active regional branches (HPM-Americas, European Summer Universities). In CERME10, 16 papers and 2 posters were presented in TWG12, for a total of 23 participants affiliated to this group, covering a large range of European countries (from Ireland to Russia) and beyond (Brazil, Mexico, the U.S.). A short survey showed this TWG attracts newcomers to the CERME community from the HPM community, since 9 participants were CERME first-timers, yet only two had never attended any HPM-related event.

Before going into any details, it should be stressed that this TWG has four general but distinctive features which give these meetings their specific flavour. Firstly, its topic lies at the intersection of different fields of researchmaths education research and history of mathematicswhich requires versatility and methodological vigilance [START_REF] Fried | Can mathematics education and history of mathematics coexist[END_REF][START_REF] Chorlay | History of science, epistemology, and mathematics education research[END_REF]. Secondly, the strength of the historical and the HPM community varies greatly among countries, and these meetings play a crucial role for researchers working in relative isolation, and with difficult access to resources in the field. Thirdly, the scope of TWG covers both history in mathematics education and history of mathematics education, which are two significantly different research topics (TSG 24 and 25 in ICME13); connecting the two lines of investigations is a constant challenge. Fourthly, since the topic of TWG12 is neither specific to one level of the educational system (from primary education to teacher-training) nor to any single mathematical topic (be it fraction concepts, algebra, proof, etc.), the work in TWG12 intersects that of most other TWGs. This time, intersection with TWG1 (Proof and argumentation), TWG18 (Teacher education), and TWG22 (Resources and task design) was significant. It should be noted that, for this edition, there was little intersection with what was covered in TWG8 (Affects and mathematical thinking), TWG10 (Diversity and maths education), in spite of the fact that it is not uncommon for outsiders of the HPM research communityamong which most policy-makers and curriculum-designersto ascribe such goals to the historical perspective in teaching.

These four features made this meeting not only useful but also challenging and exciting. As the final discussion made clear, the general feeling among the participants was that one of the main outcomes of this meeting is that we actually learned a lot from the one another, both from their papers and from the lively discussions. Let us now highlight some of the significant feature of the 2017 conference.

For quite some time it has been stressed that more attention should be paid to the actual effects of the use of historical sources, either in the classroom or in teacher training [START_REF] Chorlay | Historical sources in the classroom and their educational effects[END_REF][START_REF] Jankvist | A categorization of the "whys" and "hows" of using history in mathematics education[END_REF]. This year, at least two papers contributed to this line of research. For example, the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project is a five-year project funded by the National Science Foundation in the United States, which will create and test 25 full-length Primary Source Projects (PSPs) and 30 oneday "mini-PSPs." Each PSP is designed to cover its topic in about the same number of course days as mathematics classes would otherwise. With PSPs, rather than learning a set of ideas, definitions, and theorems from a modern textbook, students learn directly from the original work of mathematicians such as Leonhard Euler, Augustin-Louis Cauchy, or Georg Cantor. The project includes an extensive "research with evaluation" study, which will seek to address several evaluation and research questions and enable both formative and summative evaluation of the project activities. Data sources to inform the research are pre-and post-course surveys (of both students and instructors), post-PSP surveys, student interviews, student PSP work samples, video captures of selected classroom instruction and audio captures of selected small group student work, and instructor post-implementation reports. By the end of the project it is expected that some 50 instructors and over 1000 students will participate in undergraduate mathematics classrooms where PSPs are used.

On a smaller scale, Areti Panaoura studies the manifold difficulties faced by an "ordinary" teacher attempting to use a textbook activity on Egyptian multiplication. It raises many questions for our research community to investigate further: as to the level and naturemathematical, didactical, historicalof expertise required from the teacher; as to our (as researchers and teacher-trainers) criteria for assessing such teaching sessions; as to the relevant theoretical frameworks for the description and analysis of teacher-practice (in particular the use of pedagogical documents). Along with these questions, it shows the importance of leaving our comfort-zone, a zone in which the teaching sessions are implemented by the researcher who designed them or by teachers with a significant experience in the field.

As is customary in HPM-related meetings, a large number of papers carry out detailed content analysis. Let us restrict ourselves to those dealing with numbers and early-algebra: Antonio Oller-Marcén and Vicente Meavilla describe forms of argumentation about equations of the type in a 16 th century Spanish treatise, and endeavour to make sense of what we would consider to be errors or flaws; Chorlay studies arguments justifying the rule for fraction multiplication in a Chinese treatise from the Han dynasty and compares them with arguments found in today's textbooks; Maria Sanz and Bernardo Gómez devise a structural classification of sharing problems on the basis of a large historical sample, and complement this classification by showing the variety of methodsboth arithmetic and algebraicfor solving them. Coming from a perspective of history of education, Rui Candeias discusses in details a pedagogical approach to operation on decimals, in a context which combines proportionality and magnitudes. Although this line of investigation may seem to be very content-oriented, its connections to didactical questionsbe they theoretical or more appliedare manifold. First, it is hardly necessary to say that content-analysis is a central part of a priori analysis, and thaton a par with a purely mathematical analysisinvestigations into the history of mathematical knowledge and mathematical practices provide key background data. Second, the work presented in some of papers is explicitly described as a first phase in a larger research project focusing either on learners (Sanz-Gómez) or teachers (Chorlay). Third, the content-analyses presented in these papers contribute to the general theoretical discussion on some important didactical concepts, such as "epistemological obstacle" (Oller-Marcèn) or "generic example" (Chorlay).

As far as history of education is concerned, let us highlight the contribution of Katalin Gostonyi. Her comparative study of the works of mathematics educators T. Varga (in Hungary) and G. Brousseau (in France) shed light on the origins of theoretical frameworks which are still very much alive in mathematics education research.

Finally, the paper of Liz de Freitas contributes to the ongoing work in the HPM community from a new perspective. Being a philosopher of mathematics, she draws on both her personal researchin the continental tradition of the philosophy of mathematics and mathematical practiceand her experience in the training of maths teachers to suggest a large number of research questions which are relevant for the historian and the maths education researcher alike. Here, we briefly mention two such issues which we feel would be worth investigating further. A first series of questions bears on diagrams: the way they are drawn and read; their cognitive impact and their epistemological significance; the historical evolution of the meta-rules governing the use of diagrams, in themselves and in their relations to other elements of mathematical texts. A second series of questions bears on the image of mathematics maths teachers have, its impact on their teaching, and the way teachertraining modules may impact this image. Investigating this second series of questions could bridge the gap between the maths education community and the science education community, a community in which research on the Nature of Science (NOS) is a central research topic (see for instance Abd-El-Khalick, 2013).

Proceedings of CERME10


text:
Jorryt Van Bommel
email: jorryt.vanbommel@kau.se

Hanna Palmér
email: hanna.palmer@lnu.se

Slow down, you move too fast!

Keywords: Abstraction, combinatorics, representations, pre-school

Short description of the research topic and theoretical framework

The focus of this poster is on the links between representations and solutions when 6-year-olds work on combinatorial tasks. The task can be explained as follows: In how many ways can three bears sit on a sofa? Studies of young children's representations often focus on informal and formal representations. Abstraction is often perceived as a goal and teachers often value when young children use abstract representations. [START_REF] Heddens | Bridging the gap between the concrete and the abstract[END_REF] introduced semi-concrete and semi-abstract as connections between concrete (objects) and abstract (signs) representations. He referred to representations of real situations, for example, pictures of real items, as semi-concrete, whereas he referred to symbolic representations of concrete items, where the symbols or pictures do not look like the objects they represent, as semi-abstract. Connected to the task presented above, concrete implies real bears and a sofa, semi-concrete implies images of (resembling) bears, semi-abstract implies a symbolic, not resembling, representation of the bears and abstract implies the use of formal signs. As for the concrete level, the 'real' bears can be replaced by three play-bears, or by engaging in a role-play, enacting the situation. In both semi-concrete and semi-abstract representations, the colours of the bears and their symbols are often kept identical.

When conducting the combinatorial task in 13 pre-school classes in Sweden (125 children, age 6), children showed different levels of abstraction in their solutions of the problem. In a previous paper we problematized the apparent relation between the level of abstraction and the number of duplicate solutions. Surprisingly we found that the children who used semi-concrete representations in their documentations were more systematic in the process of finding solutions with less duplication than the children who used semi-abstract representations. We argued that a more abstract level seemed to reduce the problem from bears on a sofa to putting three coloured dots on a paper. It seemed that children moved too fast to the next level of abstraction. Internalisation of the problem had not occurred yet which led to a reduction of the problem as described above. Simultaneously, we also concluded that documentation within the semi-concrete level was very time-consuming for children, as it takes a lot of time for the children to draw bears [START_REF] Palmér | Exploring the role of representations when young children solve a combinatorial task[END_REF].

Methodology

How could we slow down these children without slowing down the process of documentation? An application was especially designed to let children work on the semi-concrete level, and simultaneously, the time-consuming issue of drawing the bears was taken into account. The aim of the application was to provide an opportunity to develop an understanding of combinatorics and its Thematic Working Group 13 Proceedings of CERME10 systematisation by letting children work within the semi-concrete level. The design-principles for the application concerned a possibility to adjust the level of the problem, resulting in a choice of the number of bears and the size of sofa. It also resulted in the possibility to save solutions as well as to offer an insight to all outcomes. By doing so, we wanted to explore the semi-concrete level more explicitly and focus on the learning opportunities this level of abstraction can offer the children.

During the autumn of 2016, 6-year-olds (about 60) from different preschool-classes have tested the application. Our reflections concern the learning opportunities created by using the digital form of documentation (in addition to the paper and pencil forms). After using the application, we let the children work on similar combinatorial tasks but only by using paper and pencil.

Preliminary results

Our preliminary results show that the children who have worked with the application do not document as many duplicates when using semi-abstract representations as the children in our previous study (Palmér & van Bommel, in press). Additionally, we see an increase in the systematic way they organise and search for solutions. These first results of the study also indicate that the children indeed develop significant skills when working within the semi-concrete level, for instance they seem to have obtained a good understanding of what the concept different combinations entails. A special note has to be made towards the possibility to adjust the level of the task in the application. For example, some children explored the situations of two bears on a three-seat sofa and four-seat sofas with different number of bears. Such explorations led to discussions of similarities between two and three bears on a three-seat sofa and mathematical aspects of the tasks were discussed at another level compared to paper and pencil lessons.

Proceedings of CERME10


text:
Yuly Vanegas
email: yulymarsela.vanegas@uab.cat

Joaquín Giménez

Marjorie Samuel
email: msamuel@ucm.cl

Discussing school mathematical narratives in early childhood future teacher education

Keywords: Early childhood, future teachers, mathematical reasoning, problem solving. Thematic Working Group 13

In this work we recognise how a group of future teacher of Early Childhood Education, analyse narratives about rich school experiences. We presented a professional task, in which we wanted to see how future teachers recognize initially the potential of the mentioned experiences to promote mathematical processes. We recognise that future teachers give a limited value to the problemsolving process and have difficulties in recognizing the processes of reasoning and proof. We found it is not because of a mathematical previous weak formation, but rather it points to the need of analysing school practices and narratives as good examples of action.

Introduction

 NCTM (2000NCTM ( , 2013)), as other new national curricula, suggested that teachers could design better rich school activities, if they can identify the power to develop mathematical processes in their classrooms. In Spain, authors like [START_REF] Alsina | Procesos matemáticos en Educación infantil: 50 ideas clave[END_REF], pointed out that to achieve a quality of mathematics education for early years, it is important to implement curricula focusing on mathematical processes in a systematic way. It is important for the teaching and learning of mathematics to use mathematical models in relevant everyday contexts. Preservice teacher's mathematical knowledge plays an important role when teaching mathematics. It is clear that for early years, almost no one remembers about his own experience. Therefore, their knowledge for teaching at early years is limited, and based upon personal theories and preconceptions [START_REF] Jaworski | Educating new mathematics teachers[END_REF]. "Little improvement is possible without direct attention to the practice of teaching ... [h]ow well teachers know mathematics is central" (Ball, Hill & Bass, 2005:14). In this paper, we assume that narratives are a wonderful way to allow students to personalize mathematics [START_REF] Kurz | Conceptualizing mathematics using narratives and art[END_REF] and develop mathematical knowledge. In such a framework, the use of narratives is a powerful tool for teacher professional development and an useful research methodology for those interested in the study of teachers, teaching and teacher development [START_REF] Ponte | Professional narratives in mathematics teacher education[END_REF]. [START_REF] Llinares | Changes in how prospective teachers anticipate secondary students' answers[END_REF] pointed out that for teacher education purposes, it is important to promote that future teachers (FT) grow on mathematical understanding by noticing mathematical aspects when future teachers analyse school experiences. It is also important for FT to know about designing rich practices. What about Early Childhood Education? In some countries as Spain, there is a global curriculum for Kindergarten, without any explicit mathematical goals for children's mathematical knowledge. Therefore, FT must have a preparation to understand the emergence of mathematical objects and processes from good early school practices. In the case of Spain, it is difficult to involve FT in implementing designed tasks, as it is for continuous training. There is a danger when considering the practice of teachers, and therefore presumably the experience of learners, of focusing exclusively on pedagogic practices, without reflective processes [START_REF] Shulman | Those who understand knowledge growth in teaching[END_REF]. It seems to be the main reason for the need of mathematics reflective activities for prospective kindergarten teachers. In this presentation, we want to analyse the initial position of future teachers when they analyse narratives about school experiences, in order to find how they relate the richness of school actions and the emergence of mathematical objects and processes.

Theoretical framework

As [START_REF] Ponte | Professional narratives in mathematics teacher education[END_REF] wrote, we use teacher narratives as a way to represent a school experience for oneself or for others. A narrative involves three basic elements: i) a situation involving some conflict or difficulty, ii) one or more agents that act on that situation with their own intentions, and iii) a temporal sequence of related events in which the conflict is resolved in a certain way. It involves people, settings, and events that take place in a given time. The acceptance of a story is oriented by convention and by "narrative necessity" [START_REF] Bruner | The narrative construction of reality[END_REF]. In pre-service teacher education, narratives such as these provide good starting points to discuss issues faced by a teacher in making curriculum decisions and conducting classroom instruction. Some authors, like [START_REF] Chapman | Narratives in mathematics teacher education[END_REF] and [START_REF] Ponte | Professional narratives in mathematics teacher education[END_REF], mainly use narratives to reflect on future mathematics teachers' thinking and actions in relation to mathematics and mathematics teaching and learning, aiming to broaden their understanding of new curriculum orientations.

In this paper, we use narratives to study teacher's knowledge when analysing innovative teaching practices [START_REF] Ponte | Histórias de investigações matemáticas[END_REF]. In the current study, we focus on identifying, planning and enriching mathematics practices, made by others. We consider such activity as a fruitful starting point for inquiring into how FT should anticipate the enactment that will occur. Some future teachers of early childhood want to know and copy nice school experiences. Instead of this, this research leads to understand that richness of a school activity relates to the possibilities for emergence of mathematics objects and processes. Our aim is to recognise the initial reflection of future teachers about the emergence of mathematical objects and processes, as a noticing professional competency (Jacobs, Lamb & Philips, 2010). In this presentation, we focus on the analysis of problem solving processes, and analysis of the reasoning and proof processes. We assume that the practices in reading, analysing, and discussing narratives generate a number of insights that provoked to modify future teachers' planning for instruction. Many studies have reported that narrative data helps to validate the learning results as a basis for understanding of human actions [START_REF] Polkinghorne | Validity issues in narrative research[END_REF], and to understand the role of intentionally drawn school practices [START_REF] Font | The emergence of objects from mathematical practices[END_REF].

Methodological issues

We use a qualitative, naturalistic research perspective [START_REF] Creswell | Qualitative inquiry and research design[END_REF] focusing on capturing and interpreting the participants' thinking about narratives as a case study. To achieve our aim, we designed a professional task. The professional task is structured in two parts. In the first part, FT are asked to read two articles: "The map of a treasure" (De [START_REF] Castro | El mapa del tesoro: Un proyecto sobre la representación del espacio en la educación infantil[END_REF] and "Where is Paula?" [START_REF] Feixes | ¿Dónde está Paula? Aula de Infantil[END_REF].These articles describe two school experiences about geometrical itinerary aspects and spatial references, made with children of 5-6 years old. We select these two narratives because they describe rich (Woodham & Pennat, 2014) and high quality (NAEYC & NCTM, 2002) of school experiences for children from 3 to 6 years of age. Both school experiences use a continuous dialogue in the classroom. They also explain didactical orientations to planning and managing activities, connecting geometrical objects and processes. During the first experience, the idea is to promote a problem solving approach to "know about" the space of the school, as a provocation to find the place where a treasure is. Children spontaneously use paper and pencil itineraries, written codifications, and gestures to discuss in groups how to go from one point to another among other decisions. During the second experience, the teacher proposed to talk with Paula (a child that left the school to come back to her country, Uruguay). The five years-old children immediately ask where Paula is. A nice discussion about at what time do we call her, helps the children to discuss how the difference of time relates to the difference places in Earth, and different hemispheres. Our interest in choosing these two narratives is to look at two sides of the problem of situating points in the real world space: the local or short distance problem (narrative 1) and global world problem (narrative 2).

In the second part of the professional task, four questions are posed to guide the analysis of the two narratives read in the first part. These questions are: 1) Talk about what raise your attention after first reading. 2) If you should recommend to your teacher friends these experiences, what do you explain for them? 3) What is the role of the teacher in both experiences? 4) Why do you think there are rich practices that develop the emergence of children's mathematical thinking?

The participants in our study are 33 FT of Early Childhood Education at the Barcelona University. To describe how they explain the emergence of mathematical objects and processes in the two narratives, we collect all the individual responses to the questions proposed in the second part of professional task. To analyse the data, we used a tool raised by [START_REF] Coronata | Evaluation of the mathematical processes in the practices of teaching and learning in childhood education[END_REF]. This tool includes five categories that correspond to the five mathematical processes proposed by the NCTM, (2000). For each of these categories, 6-7 indicators are provided for evaluation. In our study we only consider the indicators of problem solving and indicators of reasoning and proof processes.

The analysis takes three moments: a) The research team answer as experts giving a set of the processes observed; b) the future teachers' answers are analysed by using the methodological tool cited above; c) the research team explain some hypothesis about why the results appear. We assume that some text is related to one or more indicators, if there is a sentence evocating such principle by means of discursive argumentation [START_REF] Gee | An introduction to discourse analysis[END_REF]. General sentences are not considered. For instance, the sentence "The teacher develop the capacity of creating arguments to explain children's curiosities in reference to mathematical concepts as distance, space and time" is assumed as relating to an indicator of reasoning and proof. But a sentence like "It is considered the interest of children" is not assigned to the indicator, because it is a fuzzy simple comment without any explanation given to what is the text or mathematical idea in the narrative.

Results

Many mathematical objects are easily identified by almost all the FT, but it is not enough explained how these objects emerge from the examples given in the narratives. In general comments, many FT talk about the differences among space by using time, and the idea of having different periods in a year. They talk about the meridian as a reference for timing. They assume the need of codes in order to represent itineraries among other geometrical objects.

Many FT also consider that problem solving is a common framework in both school experiences, but they explained some of the mathematical processes superficially (See Table 1). We find that future teachers identify issues related to problem solving more easily than other processes, and have difficulties in explaining aspects related to the processes of "reasoning and proof".

In Table 1, we associate examples of the responses made by FT to each indicator and we include an expert comment when FT identify problem solving process in the narrative 1, as an example of the use of indicators for problem solving. Some FT tell us that children's participation in narrative 2 stimulate imagination and creativity. Nevertheless, we only observed two out of 33 FT on where is it possible to see such promotion of creativity as an inquiry problem solving process.

The main indicator found is related to the assumption that contextualisation plays a role in problem solving activity. Nevertheless almost a half of the students write sentences in which FT talk about problem solving without any explicit indicator In some cases, as FT14 and FT 24, they do not express any sentence about contextualisation and interest. We also see that 27% of the FT talking about mediators and interest are the same as those who talk about contextualisation. It is possible that the lack of processes recognised relate to the mathematics background of the FT. Children construct and read maps in big spaces with the Earth globe as a powerful semiotic mediator. It emerges the idea of meridian line as a reference.

Indicators

The experience itself has an emotional background. Reinforce the process using different support 6% "…using trial and error, children structure math knowledge…"(FT14)

The teacher promotes the use of the starting situation to promote the use of spatial relations and references

Table 1: Responses associated to the indicators of problem solving process (narrative 1)

We can see that the FT do not relate some verbal aspects as discussing as a part of the problem solving process for narrative 2 (See Table 2). They realise that contextualisation is the main aspect behind problem solving activity for having good answers. It is expected that the FT talk about the role of mediators in this initial moment of analysis, but the justification they give is very limited focusing on having a "meaningful task" without any relation to a specific mathematical knowledge. They talk about "learn from the surrounding environment" without explaining that the need of a reference line (Greenwich Meridian) appears when we have numbers to indicate points in the space (initial idea of geographical coordinates). This could indicate a rather weak mathematical background of the FT, and the need for having a professional reflection about what mathematical knowledge emerge from a mathematically interesting activity like this. Precisely, this is the role of training process and professional activity never done before. The results show that narratives help to focus the reader's attention for recognising more mathematical connections (distance/speed; codification/itineraries; real world/representation) than expected. The maps, are used to identify itineraries, distances, directionality It is difficult to find explicit children's arguments and reasoning in the narrative of "Where is Paula?" In fact, the teachers tell us many sentences (as "we always ask why") about the use of argumentation and reasoning, without explaining all the details. Nevertheless, in the children's pictures we can observe that they talk and argue when they observe the Earth globe, or when they talk about "Uruguay is far away". We only find general statements about reasoning and proof as we can see in Table 3 and Table 4.

Indicators % FT n=33 Examples of FT's responses Responses from experts

Helping to develop student's thinking 12 % "Children structure their mathematical knowledge" (FT 13).

The Teacher promotes reflections and arguments about the invariance of day/month/year but different time and season, by seeing to the Earth globe. Inviting to explain conjectures 6 % "The need to have good questioning"(FT4) "revealing initial ideas and preconceptions" (FT 16) The practice promotes to use arguments relating conjecturing about the need for having time references Promoting to control conjectures 3% "to establish hypothesis to understand zones having the same time…to understand why it happens in the world… and which are the lines that make the difference, when situating places in the map" (FT 16) The teacher promotes some deductive reasoning. To argue, Bernat uses if…then as a deductive reasoning. Describes the relation between numbers (+1) and going to the right. Also (-1) means going to the left. [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF], tell us about the need for promoting reasoning since early years, however, FT in the research do not identify the amount of possible quotations in the narratives relating reasoning and proof. We observe that, in general, many of the future teachers do not offer specific mathematical examples from the narratives to illustrate the reasoning and proof processes.

Questioning

It is surprising that inquiry attitude not seems to be considered as part of a problem solving process, perhaps because FT have the belief that the most important for a problem solving is to have a right solution. In fact, many of the future teachers' comments do not pay attention to the role of the teacher giving opportunities for continuous problem posing moments, promoting hypothesis and conjectures. The FT explain that the teacher in both narratives promote the use of arguments, but none of the FT mentions the importance of feedback. After this professional task, we devote some time for collective reflection not detailed in this paper. Some new processes appear as: "The teacher drives arguments, and promotes different possible contents and meanings" (FT 19) or "by dialoguing, the teacher gives immediate answers to students, reinforcing children's knowledge about coordinates" (FT 16), or "Children almost prove their conjectures, in a way that surprises us" (FT 9).

Conclusion

Future teachers are able to identify many mathematical objects and some processes implicit in the narratives analyzed. We see less process than it was expected. The main one is problem solving. The analysis promoted by the two professional tasks has allowed us not only to characterize some aspects of professional noticing of the future teachers, but also to establish a basis for recognizing the role of mathematical discourse [START_REF] Adler | A framework for describing mathematics discourse in instruction and interpreting differences in teaching[END_REF]. In fact, after the implementation of the professional task, we find a more structured discourse of future teachers, more connections, didactic arguments, recognition of a greater number of processes and more justifications. With this research, we enlarge the conjecture done by [START_REF] Llinares | Changes in how prospective teachers anticipate secondary students' answers[END_REF] that noticing also promote deeper subject-matter understanding of pre-school's future teachers. Therefore, we consider that the implementation of this type of tasks is relevant in the training of future teachers of early childhood education.

Finally, it stands out that discussing and reflecting on school narratives such as those presented here has allowed future teachers to contrast school practices, different from those they have traditionally observed giving mathematics knowledge improvement.

Table 2 : Responses associated to the indicators of problem solving process (narrative 2)

 2

Table 3 : Responses associated to the indicators of reasoning and proof process (narrative 1)

 3

Table 4 : Responses associated to indicators of reasoning and proof process (narrative 2)

 4

	Indicators	%	Examples of FT's responses	Responses given from experts
		FT
		n= 33
	Helping to develop student's thinking	15 %	"In this practice… the teacher ask questions to influence 19) children's reasoning, to improve their thinking" (FT	The teacher tell us explicitly the use of the students' natural environment to promote spatial thinking
	Inviting to explain conjectures	3 %	"The teacher tries to improve autonomy to reflect, to produce hypothesis about the ways of coding"(FT28)	The teacher promotes the emergence of Students' conjectures about maps and big distances o represent travelling by using descriptions.
	Promoting to control conjectures	6 %	"The teacher permits that children explore information given by the maps, and select which information is relevant and which one is not"(FT 19)	The teacher tells about reflections and argumentation, but it is not explicit how the teacher controls the conjectures, because the focus is the codification process.
	Questioning to		"Sharing and contrasting (FT	They discuss to have a common result, but
	evaluate	6 %	17)"Discussing about his or	different representations and evaluate the
	arguments		her discoveries"(FT 23)	representation used
	Promote reasoning by giving feedback	--	--	The teacher find that David, use a comparison (small globe vs big Earth) to reason that from Madrid to Lanzarote you must use a flight.
	Promoting divergent thinking when arguing	--	--	The teacher promote that Children adjust their arguments about which objects must be in the map, where to place and how to represent them.
	Promoting, analysis and arguments	6 %	"..revealing initial ideas and preconceptions" (FT 16)	The teacher promote the emergence of Students' ideas (Luke : Itinerary as a set of steps; David what we can learn from Earth globe

Proceedings of CERME10

Acknowledgement

This research is part of the Project Construction of school mathematical knowledge: Discourse of teachers and teaching activity. MINECO/Spain (EDU2015-65378-P).


text:
Marianna Tzekaki
email: tzekaki@auth.gr

Evangelia Papadopoulou

Teaching intervention for developing generalization in early childhood: The case of measurement

Keywords: Generalization, abstraction, measurement, early childhood, teaching intervention

Abstraction and generalization, lying at the heart of mathematical activity, attract the interest of many researchers who mostly examine generalizing processes in patterns and algebra. Given that earlier approaches questioned the possibility of developing generalizing capabilities in early ages, in our research we attempt to examine if an appropriate teaching intervention could change this initial assumption. For the needs of this work, 23 preschoolers participated in a seven months teaching intervention with relevant tasks in various topics: figures, patterns, measures and numbers. In this paper only for length measurement is presented. The children were pre and post examined in tests designed to examine their knowledge and initial generalizing abilities in this topic. The results indicated that their long-term involvement helped these young children to improve their abilities to (a) reflect on their own activity, (b) express their ideas and (c) reach to some concluding remarks, first related to their own personal experiences and later to more general thoughts.

Introduction

A 6-year-old child saying, "When two distances begin and end together, this does not mean they are equal, we need to measure them...." shows a preschooler able to express in a more detached from his/her experience and general way a relation concerning length comparison. This reveals a generalizing ability that, lying in the heart of mathematical activity, attracts the interest of research in mathematics education.

Current studies related to generalization and generalizing processes concern mainly patterns and approaches in algebra for older students [START_REF] Hitt | Generalization, covariation, functions and calculus[END_REF][START_REF] Warren | Research on the teaching and learning of algebra[END_REF][START_REF] Zazkis | The role of examples in forming and refuting generalizations[END_REF][START_REF] Lannin | Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities[END_REF], while relevant investigations of younger children's generalizing abilities also are connected to patterns and structures [START_REF] Mulligan | Early awareness of mathematical pattern and atructure[END_REF].

In our study we attempted to explore generalizing skills in early ages. More specifically, our research question focused on whether an appropriate teaching intervention could support young students' disposition related to noticing properties, relations and structure and developing a first level of more general conclusions. Our research supports the efforts for a meaningful and effective early mathematics education, by accepting generalizing abilities as an important component of it.

Abstraction and generalization

Humanity starts dealing with ideas related to abstraction and generalization quite early, pursuing answers related to the development of concepts in human's mind. Similarly, psychologists systematically investigated abstraction and generalization, as integral parts of humans' conceptual formation, supporting, thus, many explorations in mathematics education. For mathematics, abstraction is considered as a process during which students directly reorganize former structured mathematics in a new mathematical structure [START_REF] Hershkowitz | Abstraction in context: Epistemic actions[END_REF]. This process (as part of the generalization process) has been studied by many researchers. We could underline the advances of [START_REF] Dubinsky | Reflective abstraction in advanced mathematical thinking[END_REF] and APOS, [START_REF] Sfard | Operational origins of mathematical objects and the quandary of reification-The case of function[END_REF] and the theory of reification, Hershkowith, [START_REF] Hershkowitz | Abstraction in context: Epistemic actions[END_REF] and epistemic actions, as well as a variety of combinations of these approaches [START_REF] Tzur | Distinguishing two stages of mathematics conceptual learning[END_REF].

Generalization could be identified as the level at which students, starting from specific situations, proceed to more general ideas and conclusions identifying patterns, structures, relationships, rules etc. [START_REF] Kaput | Teaching and learning a new algebra[END_REF]. Similarly the idea of generalization, as an integral part of mathematical development, holds an important place in mathematics education. Starting from [START_REF] Harel | The general, the abstract, and the generic in advanced mathematics[END_REF], [START_REF] Tall | Concept image and concept definition in mathematics, with special reference to limits and continuity[END_REF], [START_REF] Fischbein | The theory of figural concepts[END_REF], [START_REF] Bills | Examples, generalisation and proof[END_REF], [START_REF] Radford | Factual, contextual and symbolic generalizations in algebra[END_REF], Becker and Rivera (2003) and [START_REF] Sriraman | Reflective abstraction, uniframes and the formulation of generalizations[END_REF], many researchers have proposed models regarding theoretical approaches for this conceptual elaboration. These models refer to concept formation (as important stage of generalization), reflection and communication (as important reason for generalization), different kinds of generalization (extension, reconstruction, disjunction/ results and process of generalization), elements of generalization (grasping and expressing) and generalization as objectification (factual, contextual, symbolic).

As mentioned earlier, most of this research is related to generalizing processes in patterns and algebraic elements, and certainly examines these processes in older students. Their findings can be organized in the following strands: generalizing abilities, generalizing procedures and strategies, teaching intervention aiming at improving generalizing skills. Related to generalizing abilities, [START_REF] Yeap | Elementary Singapore classrooom school students engaging in making generalisation: A glimpse from a Signapore classroom[END_REF] studied structure and pattern recognition with the use of heuristic tools and technology as well as the development of metacognition and critical thought. Related to teaching intervention, [START_REF] Zazkis | The role of examples in forming and refuting generalizations[END_REF] investigated the use of appropriate examples and tasks.

The elements or actions and elaborations that consist the integral parts of generalization cannot be defined so easily. However, elements such as reflection of action and expression of more generalized properties, rules, relationships and structures deriving from specific tasks, problems and practical experience could be identified as being essential in this conceptual elaboration and are placed in the center of students' mathematical development. Therefore, mathematical education needs to cultivate these skills from early ages and progressively improve them in the course of mathematics education.

Starting from this basic question, our research aimed at examining (a) the possibility of developing generalizing capabilities in early childhood, and (b) teaching interventions that could improve these skills in young children. Our wider research concerns teaching and findings related to four topics of mathematics curriculum: geometric figures, patterns, geometric measurement and numbers. In previous communication [START_REF] Tzekaki | Generalization in early childhood[END_REF] we presented results concerning geometric figures and numbers, while in this paper we are limited to data related to geometric length measurement. We believe that the most interesting part of this presentation is not exactly the specific topic but the development of generalization capabilities in 5 years old children.

Measurement and generalization

Measurement is an important every day activity, that mathematically connects geometry and numbers, but its approach is not so evident in early years or even later [START_REF] Bragg | A measure of rulers -the importance of units in a measure[END_REF].

Length measurement can be identified as a procedure that divides a continuous magnitude of an object to a number of specific parts (units) and connects it to the number of iteration of these parts. Thus, the conceptualization of length measurement could be analyzed in several steps that demand understanding and generalization [START_REF] Battista | Levels of sophistication in elementary students'. reasoning about length[END_REF]:

1. Identification of length as a special (unchangeable) attribute of an object.

2. Transfer of this attribute to another object (intermediate).

Comparison of lengths by means of intermediate (arbitrary or conventional).

4. Covering with different equal units (arbitrary and conventional) and quantify this covering.

5. Connecting unit iteration (and thus measurement) to a number [START_REF] Sarama | Early childhood mathematics education research. learning trajectories for young children[END_REF].

Our research related to length measurements examines whether young children succeed to understand and express different generalized ideas for three of these steps: transfer, comparison and covering. More specifically, generalizing levels of each of these measurement steps was expressed and examined, both as procedure and relation, as following:

-Generalization related to the transfer of a length to an intermediate (2) means that children could identify the equality of the lengths of objects and intermediates.

-Generalization related to the comparison of lengths (or distances) by means of intermediate (3) means that children could identify the starting and ending points of objects and intermediates and the equality of their lengths.

-Generalization related to covering with different units for the needs of a measurement means that children could understand the role of equal units, the different quantification when the units are different and the role of numbers in the procedure of measurement.

Children's explanations both in tests and in classroom interchanges during the teaching intervention were classified according to this analysis.

Methodology

For the needs of this research related to length measurement, 23 preschoolers participated in a three weeks teaching intervention with relevant measurement tasks. There were also pro-and post examined in tests specially designed to examine their knowledge about measurement and their generalization level, before and after the intervention.

Pre and Post tests

The tests items in length measurement included four (4) items in accordance to the aforementioned analysis:

1. For the transfer of a length, a real object (for example a frame) hung on a specific height on the wall had to be transposed to another place.

2. For the comparison of lengths, cards of different width and height had to be ordered (their dimension were changing inversely, the thinner was the taller, and the thicker was the shorter).

3. For covering with different units and comparison there were two items: the children had to compare unequal distances in zigzags with common starting and ending points drawn on worksheets and covered with equal units, as well as equal distances in zigzags with common starting and ending points, but covered with different (non equal) units.

Two of the items concerned lengths, while the other two distances. Also, two of them were real situations with material while the other two were representations of comparison situations. The children were individually interviewed on these tasks and were questioned about their reflections and conclusions. Their answers were organized in the following stages.

1 st Stage: The child does not express any procedure or relation related to measurement.

2 nd Stage: The child starts with a holistic measurement approach of procedure or relations that can explain his/her doing, e.g. transferring an object a child says "I thought to put it in the same height.." showing an imaginary line with his hand.

3 The children explain all the relations that are needed for the measurement, e.g. comparing also equal distances in zigzags by means of matches, a child explains "they start together and end together and have the same number of matches".

Teaching intervention

After the pre-test the children were involved in "generalizing experiences" concerning length measurement. They first worked with relevant measurement tasks in groups and then presented their results in the whole class. The teaching intervention consisted of five (5) lessons with eight (8) tasks.

The tasks concerned all five steps of length measurement (length/distance -widthheight, presented earlier) with -Length transfer, direct and indirect comparisons and estimations.

-Indirect comparisons by means of arbitrary of conventional units (unit covering and number assignment).

-Indirect comparison by means of conventional units with iteration and number assignment.

-Length and height measurement by means of meter.

Tasks with length transfer proposed the use of batons or strings to measure, for example, the height of the position of a frame that has to be hung on a different wall. Indirect comparisons suggested covering with meters or other convectional units to measure and compare, for example, itineraries in stories or dimensions of furniture. Similarly, other comparisons presupposed the iteration of units for these measurements or comparisons.

The most important part of this intervention was the closure of activities, when children were systematically encouraged (a) to identify common characteristics, relationships and properties in the different encountered situations, (b) to express more general ideas and (c) to formulate conclusions or other overall rules about length measurement.

The discussion and the generalizing questions aimed at the understanding of measurement procedure and its principles and, more specifically, of the equal partitioning of the continuous attribute of objects, the unit iteration and the equality relations (with starting and ending points).

Results

The results coming from the comparison of the pre and post tests indicated that, after a series of tasks and systematic discussions, young children were able to improve their abilities to reflect on their own activity, to express their ideas and reach to some kind of conclusions, initially 'locally' related to their own personal experiences or the specific task (e.g. way of doing it) and later to a more general level regarding mathematical ideas (e.g. whether are equal or nor) or even formulating a rule or proposition (e.g. to measure two distances we have to…).

Findings related to measurement: Pre and post tests

The pre-test findings indicated that children initially didn't recognize the procedure of measurement, but approached comparisons globally and estimated lengths and distances based on their own experiences. For example, comparing distances covered with batons they didn't count the number of batons but based their responses on personal judgments of closeness. Their justifications were idiosyncratic, practical or kinesthetic, e.g. "because I think so…", or "This line is bigger because in this place there is only one animal, so the thunderbolt killed more…". They ordered objects based on their figure, while they covered them with units that were overlapping or had gaps. They understood that there was an equality or inequality of dimensions, but explained it based on morphological elements, e.g. "it is more pointy…"..

The analysis of children's responses after the intervention indicated a significant improvement regarding the identification of the procedure of measurement, as they did successful measurements and explained their actions based on measurement principles. They identified the iteration of units (with no overlapping and gaps) explain e.g. "we don't put the sticks as we like, but one after the other, watching that they are not overlapping…". They were able to order objects based on metrical characteristics (length and height) and explained the measurement they have done. The children's improvement is illustrated in the following table (Tab. 1).

Issues of Tests

Students' conclusions related to measurement

Examining the children's answers to the interviews and the records from their work in the classroom, we can support that the preschoolers are able to conceptualize length measurement and its principals and moreover describe it, as procedure and relations at a more general level. The following examples present the ideas developed by the young children and the level of generalization achieved after involving in generalizing exchanges. At the end, the teacher gathered the children's conclusion about measurement in drawings and utterances.

The children's utterances expressing their concluding remarks about the transfer of heights:

-We must measure one distance before doing the other -The two distances must start form the same point

The children's utterances expressing their concluding remarks about covering with units:

-It shouldn't be one over the other and crooked -We don't leave gaps, we put start to start and back to back -We put them in a line

The children's utterances expressing their concluding remarks about the measurement of distances:

-When two distances are straight and start together and end together, then they are equal -When two distances are zigzag and start and end together, it doesn't mean that they are equal, we have to measure to see which one is longer and which one is shorter -When two distances are zigzag and don't start and end together, it doesn't mean that they are not equal, we have to measure them.

Discussion

In general, this study shows that the development of generalizing capabilities in early childhood is possible on the basis of appropriate teaching approaches that encourage reflection, activity justification and concluding communication. The development of these abilities depends generally on a modification of young children's focus from personal to 'impersonal' ideas, and, thus, from 'local' to more general as a way of functioning in the mathematical class. This development was observed and recorded in all topics: figures, patterns, measurement and numbers [START_REF] Tzekaki | Generalization in early childhood[END_REF].

The development of children's ideas and generalizing remarks reaffirmed the importance of teaching approaches that orient the class to generalizing experiences [START_REF] Sriraman | Reflective abstraction, uniframes and the formulation of generalizations[END_REF]. More specifically in the identification of the measurement procedure and relations by the children, their utterances showed that they succeeded to overcame the specific content of the tasks and proceeded to the conceptualization of measurement principles and invariants [START_REF] Bills | Examples, generalisation and proof[END_REF]. The children passed gradually from local generalizations related to every one of their actions and tasks to more general related to all their activity and finally to general principles of the measurement that recorded in the specific for their age way. The choice and the sequence of tasks, the group work in the classroom and the exchange between groups aiming at arriving systematically to more general remarks are the main factors that led to this result.

The findings gathered from this topic, as well as from all other topics, justify why it is possible but also of imperative to exercise children from early age, with appropriate activities and elaborations with respect to their age and their way of thinking, to generalizing skills as significant part of their mathematical development [START_REF] Tzekaki | Mathematical activity in early childhood: is it so simple?[END_REF].

  rd Stage: The child starts conceptualizing length measurement and can explain the procedure presenting some relations, e.g. comparing distances in zigzags by means of batons, a child

says "for this side (showing the starting point) they are the same, but from the other side (showing the end) this one (showing) looks longer…" 4 th Stage: The child starts conceptualizing the measurement as iteration of units and can explain almost the whole procedure and relations, e.g. comparing distances in zigzags by means of matches, a child explains "this one is longer because it has 1,2,3,…,6 matches and the other has 1,2,3,4 matches…" 5 th Stage:

Table 1 : Results before and after the teaching intervention

 1

	Success pre, % Success post, %

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