Simone Jablonski
email: jablonski@math.uni-frankfurt.de

Matthias Ludwig
email: ludwig@math.uni-frankfurt.de

Mathematical Arguments in the Context of Mathematical Giftedness -Analysis of Oral Argumentations with Toulmin

Keywords: argumentation, giftedness, Toulmin

Being defined as a main learning goal of mathematical education, arguments and argumentative skills are of high relevance. The development of these skills in the context of a deeper understanding is an important aspect for mathematical teaching and the support of mathematically gifted students, whose interest in mathematics should be increased. To focus more in detail on arguments in the context of mathematical giftedness, the paper poses the question whether it exists a relation between mathematical giftedness on the one hand, and mathematical argumentations on the other hand. For this purpose, the arguments of primary students from an enrichment program for mathematically gifted and interested students are gathered in an interview setting and analyzed by means of the Toulmin scheme. The results show that giftedness might influence the content and quality of arguments, but not the need for argumentations.

Introduction

For curricular, societal and mathematical reasons, argumentative skills are claimed as central learning goals in mathematical education. Especially the view on mathematics as a deductive organized system with theorems and proofs forces fundamental skills in argumentation (e.g. [START_REF] Hanna | Proof, explanation and exploration: an overview[END_REF]. Nevertheless, former research shows that (German) students have deficits in formulating arguments in written and oral form [START_REF] Cramer | Everyday argumentation and knowledge construction in mathematical tasks[END_REF]. It is therefore an important aim to support the development of argumentative competences, not solely, but also with regards to mathematical giftedness. The idea that gifted students perform better in the formulation of mathematical arguments seems legitimate, even though the relation is not clear [START_REF] Fritzlar | Zum Beweisbedürfnis im jungen Schulalter [On the need for proof in young school ages[END_REF]. In this context, the study focuses on this relation through analyzing gifted students' oral arguments.

Theoretical Framework

Arguments in Mathematical Education

Not only in every day's communication, but also in mathematical education, arguments are of high relevance. The aim of an argumentation is to convince the communication partner with the help of shared respected statements [START_REF] Cramer | Everyday argumentation and knowledge construction in mathematical tasks[END_REF]. Nevertheless, real disputes in mathematical education are rare, so that argumentations are best described in the context of a problem solving process as "a type of dialogical or dialectical game […] that is associated with collaborative meaning-making." (Baker, 2003, p. 48). In mathematical education, the process of an argumentation involves activities such as to formulate assumptions on mathematical characteristics, to reason on relations, as well as to question assumptions [START_REF] Bezold | Förderung von Argumentationskompetenzen durch selbstdifferenzierende Lernangebote [Support of argumentative competences through self-differentiating learning opportunities[END_REF]. The paper defines the term 'argument' as a product which results from the processes of former described activities. With this product-oriented focus, it is possible to analyze arguments among different categories, e.g. structure and content. Apart from a structural analysis, it is furthermore necessary to analyze arguments according to their content as the analysis by means of the Toulmin scheme does not state anything on the argument's quality, e.g. whether a warrant is based on authority or a mathematical rule [START_REF] Koleza | Primary and secondary students' argumentation competence: a case study[END_REF]. Following Toulmin's definition, an argument is called analytic if W includes all relevant information for the step from D to C [START_REF] Toulmin | The Uses of Argument[END_REF], e.g. mathematical rules or laws that are adequate in the specific context. Arguments that do not fulfil this condition are claimed as substantial [START_REF] Toulmin | The Uses of Argument[END_REF]. These arguments leave open or fail to answer (further critical) questions [START_REF] Koleza | Primary and secondary students' argumentation competence: a case study[END_REF]), e.g. "My calculation is correct, because Anna has the same result" could be rebutted by the question "But what if both of you did a wrong calculation?" [START_REF] Fetzer | Wie argumentieren Grundschulkinder im Mathematikunterricht?[END_REF].

Mathematical Giftedness

Mathematical giftedness is a complex construct that lacks a standardized definition and diagnosis as it is not directly observable [START_REF] Bardy | Mathematisch begabte Grundschulkinder [Mathematically gifted primary students[END_REF]. It is more a potential that might develop into an outstanding mathematical performance by means of an advantageous interplay of genetic and environmental factors [START_REF] Käpnick | Mathematisch begabte Kinder [Mathematically gifted children[END_REF]. Therefore, the construct is approximated by means of lists of characteristics, including special skills such as the memorizing and transferring of mathematical structures, creativity and problem solving competences [START_REF] Käpnick | Mathematisch begabte Kinder [Mathematically gifted children[END_REF].

State of the Art

A main interest of studies on argumentations in the educational context is on the analysis of arguments with help of frameworks and models in order to make characteristics observable (e.g. [START_REF] Nussbaum | Argumentation, Dialogue Theory, and Probability Modeling. Alternative Frameworks for Argumentative Research in Education[END_REF]. Mostly, studies observe arguments in classroom interactions and collaborative argumentations (e.g. [START_REF] Forman | Learning what counts as a mathematical explanation[END_REF]. For this purpose, the Toulmin scheme, and especially its core, is often used (e.g. [START_REF] Koleza | Primary and secondary students' argumentation competence: a case study[END_REF]. Further studies focus on the evaluation of argumentative skills by means of models and methods to improve them (e.g. [START_REF] Bezold | Förderung von Argumentationskompetenzen durch selbstdifferenzierende Lernangebote [Support of argumentative competences through self-differentiating learning opportunities[END_REF]. The studies show quite consistently that arguments in primary school have a low significance and real disputes are rare, so that the teacher has to initiate arguments and show the need for arguments [START_REF] Schwarzkopf | Argumentation Processes in Mathematics Classrooms -Social Regularities in Argumentation Processes[END_REF]. With regards to Toulmin, arguments by primary school children are typically characterized by the missing of a warrant and by substantial arguments [START_REF] Fetzer | Wie argumentieren Grundschulkinder im Mathematikunterricht?[END_REF]. [START_REF] Koleza | Primary and secondary students' argumentation competence: a case study[END_REF] assume that primary students are able to formulate arguments which can be analyzed using the Toulmin scheme, even though basic elements are not mentioned through a lack of need for argumentation. Focusing on argumentative skills of mathematically gifted students, the relation is not explicitly clarified [START_REF] Fritzlar | Zum Beweisbedürfnis im jungen Schulalter [On the need for proof in young school ages[END_REF]. On the one hand, [START_REF] Fritzlar | Zum Beweisbedürfnis im jungen Schulalter [On the need for proof in young school ages[END_REF] assumes that mainly other factors apart from giftedness influence argumentative skills and the need for argumentations. Other models see the potential of gifted students (inter alia through supportive characteristics, such as creativity) to develop outstanding argumentative skills, which can be forced through training [START_REF] Bardy | Mathematisch begabte Grundschulkinder [Mathematically gifted primary students[END_REF].

Research Questions and Methodology

Based on this theoretical outline and former research findings, the research questions of this paper is formulated as follows: What are the characteristics of mathematically gifted and interested primary students' arguments? In how far do they differ from general research findings on the arguments of primary students with special regards to the Toulmin scheme and the need for arguments?

The research question is answered in a qualitative study. The sample is taken from the participants of the extra-curricular enrichment program "Young Math Eagles Frankfurt". The aim of the program is to support mathematical interested and gifted students and to increase their mathematical interest and joy on a regular and long-term level. In the school term 2017/2018, about 50 students between 8 and 10 years from 14 schools participate in the program after nomination of their teachers.

To focus on the characteristics of the oral arguments of the participants, task-based and problemorientated interviews are created [START_REF] Goldin | A scientific perspective on structured, task-based interviews in mathematical education research[END_REF]. This interview form is characterized by the openness of answers while focusing on a special problem. The interview material includes tasks that emphasize arguments and ask primary students to formulate warrants through irritation or the task itself [START_REF] Bezold | Förderung von Argumentationskompetenzen durch selbstdifferenzierende Lernangebote [Support of argumentative competences through self-differentiating learning opportunities[END_REF]. The task formats focus on number pyramids and numerical lattices with regard to special number relations. The formats can be expected to be known by the primary students and do not ask for difficult calculations. Therefore, the task formats' focus is basically on argumentation and it can be expected that the students are able to use their findings on number relations as basis for warrants that can be analyzed by means of the Toulmin scheme. In each interview, different components of the formulation of arguments are included. In particular, the students are asked to comment on a wrong assumption e.g. through giving a counter example, to argue on different mathematical relations and to generalize their detections. One task is to answer the question how the basic stones of the number pyramid/the arrow numbers of the numerical lattices have to be arranged in order to maximize the top of the pyramid/the result of the numerical lattices (see Figure 2). Every interview is done individually and includes four tasks which are done within 15 minutes. To make the interviews comparable, they are directed with help of a guideline. The interviews are transcribed on the basis of an audio recording and an observational protocol.

The task formats, as well as the guideline were tested during the pilot phase regarding their appropriateness in terms of the students' age, needed time and mathematical content. Further, they were analyzed according to the need for argumentations. The results show that the formats are adequate in the named categories. Most students were able to identify relevant mathematical characteristics and some were even able to generalize them to some extent. In all interviews, the students formulated arguments which could be reconstructed with the Toulmin scheme. Therefore, the piloting could confirm that the tasks emphasize arguments, which shows the suitability of the tasks for the research question. Nevertheless, many arguments were initiated by the interviewer. Through these initiation processes in the interview, it is therefore possible to analyze the independence of and need for arguments.

On the basis of the results in the piloting and the theoretical framework, a scheme for the analysis of the arguments within the interviews is created. The analyzed categories within this paper are (1) Structure of Arguments: Within this part, it is analyzed which of the elements Data, Conclusion and Warrant could be observed during the focus on the specific task.

(2) Independency of Arguments: Here the focus is led to whether a warrant is initiated by the interviewer through questions/given after mediation, or given on an independent level.

(3) Content of Arguments: This part involves a categorization of the warrant in terms of its mathematical content (inacceptable/wrong, substantial, analytic).

The use of the Toulmin scheme as a basis for the analysis can be legitimated as the results are to be compared to former research findings on arguments in the primary age and the Toulmin layout is frequently used in these studies. Through adapting the independency and the content, its limitations apart from a structural analysis for this setting are recognized.

During May and June 2018, 32 participants (all students with parental consent form and presence during the interview dates) were interviewed. Within the interviews, 128 argumentation tasks are analyzed. Beforehand, no special training in the formulation and awareness of arguments or an education on the formulation of analytic warrants took place. Argumentation and reasoning tasks were part of some lessons, but not to a greater extent than the focus on other competences, such as problem solving. First observations within the enrichment program did not seem to show obvious differences in the need for argumentations. Nevertheless, the detection of mathematical findings, might be an important influence on the arguments of gifted children. The following results from the analysis should specify the relation of argument and giftedness on a systematic level.

Results

Table 1 shows an example of an interview extract with a functional analysis according to the elements of the core of the Toulmin scheme. At the date of the interview, the student (S) was in ten years old. His interview bases on the task format numerical lattices. The sample analysis focuses on the task within the interview, in which the participant has to argue in which case the result is the highest. In advance, he does calculations with changed orders of the arrow numbers and concludes that the results change. Through initiation of the interviewer, he gives a substantial warrant for the relation of the basic elements and result. His argumentation bases on the order of the boxes of the numerical lattices and the switch of numbers comparing both tasks.

On the basis of this observation (which is coded as Data for the following task), he is able to identify the case in which the result is the highest, namely when the arrow with the higher number shows downwards (see Figure 2). Without initiation, he uses a substantial warrant and through initiation he supports a conclusion through an analytic warrant by transforming it to a general case. I: […] In which case do you receive the highest result? You can use your former calculation.

Introduction of Task

S: When the higher number is taken plus here (shows on arrow downwards of numerical lattices).

Conclusion

Because here, there are three and here there are solely two (shows on boxes of numerical lattices). and therefore I think, this is always like that.

Conclusion

With the former task as data, a conclusion is drawn and supported through a warrant. The task is therefore coded as Data -Conclusion -Warrant. For the warrant, the highest observable category before initiation is substantial, and after initiation analytic. The development of the warrant after initiation can be observed frequently, so correspondingly, two phases within each interview task are distinguished for the following analysis: Phase 1 describing the argument respectively the warrant on an independent level, and Phase 2 after an initiation of a warrant.

Table 2 shows the results from the structural analysis of all interview tasks through categorization according to the included elements on an independent level, namely without a question as initiation of reasoning (Phase 1). Without initiation processes, only 24.2% of all analyzed tasks include a warrant on an independent level. In the majority of the tasks, a conclusion is drawn, but ungrounded. Taking those tasks that are coded within the category Conclusion or Data -Conclusion on an independent level and in which a warrant was initiated afterwards, in 87%, a warrant could be formulated after being initiated. By combining the independently formulated and initiated arguments, within 98 of all 128 tasks the complete core according to the Toulmin scheme (Data -Conclusion -Warrant) could be reconstructed. Nevertheless, within 67 of these 98 tasks, the warrant was only formulated after being initiated. To focus more in detail on the mathematical content of the arguments, the tasks including a warrant are categorized as not acceptable/wrong, substantial and analytic. Again, the highest observable category is coded, e.g. a task including a substantial and an analytic warrant before initiation is coded as analytic. In Table 3, the independently given warrants from Table 2 are categorized (N=31). In addition to the former finding that in about 75% of all tasks no warrant is coded on an independent level, one can observe that two thirds of the independently given warrants are substantial. In Table 4, a further categorization is made before and after initiation (Phase 1 and 2) for those tasks in which an initiation of a warrant is coded (N=95). Even though in 10.5% of initiated tasks still no warrant is coded, this number decreases significantly after initiation. In addition, it becomes obvious that the proportion of analytic warrants increases after a question on (further) reasoning is posed.

Discussion and Conclusion

A comparison of the results with former research findings gives different insights into the arguments of mathematically gifted and interested primary students. On the one hand, the results can confirm the general conclusion by [START_REF] Schwarzkopf | Argumentation Processes in Mathematics Classrooms -Social Regularities in Argumentation Processes[END_REF] that arguments have to be initiated at least partly. In two thirds of all tasks including a warrant, the warrant was only mentioned after initiation of the interviewer. Therefore, one can assume that gifted primary students do not seem to have an exceptionally sensible need for reasoning and arguing. Without initiation processes, the research findings by [START_REF] Fetzer | Wie argumentieren Grundschulkinder im Mathematikunterricht?[END_REF] on the lack of a warrant in primary students' arguments seem to be accurate for this sample. Also the independently formulated warrants are mostly substantial what fits the findings by [START_REF] Fetzer | Wie argumentieren Grundschulkinder im Mathematikunterricht?[END_REF]. On the other hand, it is nevertheless noticeable that nearly 40% of the analyzed tasks with initiation involve an analytic warrant after an initiation process took place. Further, only in 10.5%, still no warrant is mentioned after initiation. The focus on initiation emphasizes the hypothesis that a large group of the students is able to formulate (analytic) warrants after being forced to. Nevertheless, only the minority of the students formulates (analytic) warrants on an independent level.

With special regard to the research questions, one can formulate the hypothesis that mathematical giftedness might not influence the need for argumentations. Nevertheless, most of the gifted students are able to formulate warrants after initiation, some of them even analytic. Mathematical giftedness therefore might have an impact on the mathematical basis for the formulation of analytic warrants in order to support (creative) detections and findings. So, incomplete arguments without initiation do not seem to be caused by a lack of ability, but by the low significance of arguments' needs or the missing of structural knowledge of arguments. This finding is further supported by the fact that the students were not taught on arguments in advance. Hereby, former studies of mathematically gifted students' intuition (e.g. [START_REF] Käpnick | Intuitionen. Ein häufiges Phänomen beim Problemlösen mathematisch begabter Grundschulkinder [Intuitions. A common phenomenon in problem solving of mathematically gifted primary children[END_REF] might be relevant in order to explain the low need of giving warrants. In the interpretation of these findings, one has to take some limitations of the study into consideration. The interviews were analyzed on a qualitative level without a control group as it was not possible to exclude many different influencing factors (e.g. differences in school, teacher, language skills and mother tongue). Therefore, the comparison was built on former research findings and studies by different researchers, which mostly show consent in the analyzed categories and therefore allow a certain generalization of the findings for different settings. The comparison nevertheless has to be seen under varying task formats and different settings.

We take up the findings as a basis for a longitudinal study on the changes of the arguments of the students in terms of the structure and content, as well as the need for arguments. During this, the focus will be laid on the individual changes in arguments of the children with special regards to their environmental background in terms of language and family. Moreover, the study focuses on the question whether mathematical giftedness as a potential for outstanding mathematical performance can further be a potential for arguments.

Literature

Figure 1 :

 1 Figure 1: Core of the Toulmin Scheme (Toulmin, 1958/2003)

Figure 2 :

 2 Figure 2: Task formats number pyramid (left) and numerical lattices (right)

  you think this is always the case or solely in our example? Initiation S: Well, I think this is in every example like this. Conclusion I: Can you tell me why this is the case or/? Initiation S: (shows on arrows of numerical lattices) Because when you calculate here the higher number plus at the arrow downwards, then there is one box more, where you can calculate plus and here on top, there is no more Warrant (analytic)

Table 1 : Extract from the transcript (translated from German by the author)

 1

Table 2 : Structural categorization according to the Toulmin scheme on independent level

 2

	N=128	No Conclusion	Conclusion or	Data -Conclusion -
			Data -Conclusion	Warrant
	Argument on an	6 (4.7%)	91 (71.1%)	31 (24.2%)
	independent level

Table 3 : Content-related analysis of the tasks with an independently formulated warrant

 3

	N=31	Not acceptable	Substantial	Analytic
	Independent Warrants 3 (9.7%)	21 (67.7%)	7 (22.6%)

Table 4 : Content-related analysis of the tasks with initiation

 4

	N=95	No warrant	Not acceptable	Substantial	Analytic
	Before initiation 77 (81.0%)	2 (2.1%)	15 (15.8%)	1 (1.1%)
	After initiation	10 (10.5%)	7 (7.4%)	42 (44.2%)	36 (37.9%)


Leander Kempen
email: kempen@khdm.de

Petra Carina Tebaartz
email: petra.c.tebaartz@math.uni-giessen.de

Miriam Krieger
email: miriamkrieger@ish.de

On the effect of using different phrasings in proving tasks

Keywords: Proof, argumentation, socio-mathematical norms, semiotic norms

The research presented in this paper is about the question, if and how the phrasing of a proving task influences students' proof productions. In our study, 381 first-year preservice teachers were asked to work on a proof questionnaire involving two proving tasks, where the phrasings "prove that", "show that", "reason", and "explain" were used alternatively. Students' proof productions were analyzed concerning the kind of reasoning, the use of algebraic variables and the number of words used. While we found statistically significant differences in the proof productions for the easier first task, there were no remarkable differences in the case of the harder second task.

Introduction

The learning about proof and proving is considered to be a main hurdle for beginning university students. Several studies have been conducted to describe and to analyze the low proving competences of freshmen (see [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF] for a good overview). Despite these results, only little effort has been made to investigate the relationship between research results, the teaching of proving and the proving tasks used in research. [START_REF] Dreyfus | Why Johnny can't prove[END_REF] gives a description of several aspects that have to be considered in the teaching of proving. Dreyfus concludes (p. 103):

The examples in Section 2 provide ample room for questioning what is expected by the different formulations used, including 'explain' [..], 'justify' [..], 'prove' [..], and 'show that ' [..]. Does 'show that' mean 'formally prove' or 'use an example to demonstrate that' (or something intermediate between these two)? Does 'explain' mean explain to a fellow student or explain in such a way as to convince the teacher that you understand the reasoning behind the claim?

It was this idea of Dreyfus that made us conduct a study on how students' proof constructions vary due to the phrasing of the proving tasks. We chose the four phrasings "prove that", "show that", "reason" and "explain" to investigate possible systematic differences concerning students proof productions. While "prove that" and "show that" are genuine phrasings in proving tasks, "reason" and "explain" are also phrasings that are used in studies to investigate students' proof competencies. In this paper, we will describe our research project and outline the main results of this study.

Theoretical background

There are various phrasings that can be used to formulate a mathematical proving task. In the German school system for example, the concrete phrasings are meant as follows (KMK, 2012; our translation; the added words in quotation marks are the German translation of the former phrasing):

 to prove ["beweisen"]: to verify statements mathematically by using known facts and deduction, starting from the assumptions given  to show ["zeigen"]: to verify statements by using valid forms of reasoning, calculations, derivations or logical connections  to reason ["begründen"]: to trace data back to principles or to causal connections by using rules and mathematical relationships  to explain ["erklären"]: to clarify and to make comprehensible data by using personal knowledge and to arrange it reasonably into mathematical relationships So, there are some definitions or specific requirements combined with these phrasings. Following this differentiation, the mathematical solutions could vary in some detail due to the phrasing used to formulate a mathematical task.

In the mathematical classroom, the meaning of the different phrasing of a mathematical proving task can be considered to be an aspect of sociomathematical norms in the sense of [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF]. In the concrete learning contexts, the teacher and the students negotiate what is accepted, when a reasoning, an explanation, or a proof is asked for. Accordingly, the students learn in their daily mathematical class, what they have to do, when dealing with a task starting with "prove", "show", "reason", and "explain". In this sense, the way students are responding to a specific mathematical (proving) task is a result of a socialization process. This perspective offers the very possibility of existing effects concerning students' proof productions when using different phrasings in proving tasks and also makes it possible to give an explanation for them.

In line with the theory of sociomathematical norms, semiotic norms have to be considered. The concept of semiotic norms covers the idea that people may develop preferences concerning communicating ideas (e.g. with mathematical symbols, using representations or giving concrete examples) when being confronted with a given and known formulation of a task. [START_REF] Kempen | Pre-service teachers' perceptions of generic proofs in elementary number theory[END_REF] transferred the idea of [START_REF] Dimmel | What details do geometry teachers expect in students' proofs? A method for experimentally testing possible classroom norms[END_REF] to explain students' preference of using algebraic variables when being asked "to prove" a mathematical statement. There are some suggestions in the literature that the phrasing of a mathematical task influences students' solutions. [START_REF] Schupp | Zur didaktischen Analyse des Teilungsproblems[END_REF] discusses the 'problem of points' and gives several examples how students' solutions may vary in reference to a specific phrasing of the task. Knipping et al. (2015) adopt this idea and compare several phrasings of the same task to discuss different solutions obtained in different studies.

The authors also consider the influence of sociological factors: students' solutions in different school contexts (grammar school class and comprehensive or mixed school ["Oberschule"]) differed clearly from each other. Finally, it seems reasonable to link the phrasing of a mathematical task with emotions on students' side. Hemmi (2006, p. 145) showed that while most of the students felt positive when being asked to solve a task starting with "Show that…", about 40% of them stated a negative feeling. One might conclude that a negative feeling concerning a given task may lead to the result that a student might not seriously try to answer the task.

The theoretical considerations above led us to consider several hypotheses. Students may link different phrasings of proving tasks with different mathematical activities. According to the meaning of the phrasing "show that" in common speech (in German it is "Zeigen Sie"), this phrasing may lead to an answer of a proving task, where a student is justifying a given claim by only testing one or more examples. Whereas this phrasing might lead to an empirical-inductive answer, a phrasing like "reason" is considered to evoke the formulation of arguments (compare research question 1). In line with the idea of semiotic norms, the different phrasings of proving tasks may lead to the use of different notational systems. One might assume that "prove that" may lead to the use of algebraic variables, whereas "explain" might be combined with the use of (verbal) language (compare research questions 2 and 3). For a deeper analysis of the data we also considered social factors (like gender, age, former math courses at school, …) and also investigated the data concerning the use of word variables, the use of representations, the use of concrete examples, quality of deductive approach and the structure of the proof. But due to the length of this paper, we will only report on the aspects "kind of reasoning", "use of algebraic variables" and "number of words used".

We finally came up with the following research questions:

1. In how far do students' proof attempts vary significantly concerning the kind of reasoning used (empirical-inductive or deductive) with respect to the phrasing of the task? 2. In how far does the occurrence of (algebraic) variables vary significantly with respect to the phrasing of the task? 3. In how far does the number of words students used vary significantly with respect to the phrasing of the task?

Methodology

Following the winter term in October 2015 in Germany, 381 mathematical freshmen, who took part in a degree program leading to different teacher accreditations at the Universities of Gießen, Münster and Paderborn, participated in this study. These students were asked to work voluntarily on an anonymous entrance test. All students were told, that their results would not affect their marks in any ways.

The mathematical statements used in this survey arise from the field of elementary number theory. Both tasks were chosen due to their manageable amount of mathematic operations as well as their contiguousness to topics dealt with in German grammar schools and are as follows:

(1) The sum of an odd natural number and its double is always odd and (2) The product of three consecutive natural numbers is always divisible by 6. Each proving task allows for different approaches, e.g. generic or formal solutions. The first statement can easily be proven by making use of generic examples, by using figurate numbers, or by using algebraic variables (see for example [START_REF] Kempen | Pre-service teachers' perceptions of generic proofs in elementary number theory[END_REF]). We felt the need to include a task that is as easy to understand as the first one, but harder to prove, because in the case of the second task, the mere use of algebraic variables and respective computations does not automatically lead to a proof of the statement ( ). Here, some additional arguments are necessary to prove the given claim. In this case, a narrative justification seems to be the simplest way of proof. One possible narrative proof for this task might be: If you have three consecutive numbers, one will be a multiple of 3 as every third number is in the three times table. Furthermore, at least one number will be even and all even numbers are multiples of 2. If you multiply the three consecutive numbers together, the answer must have at least one factor of 3 and one factor of 2. Accordingly, the result will always be divisible be 6 (compare with [START_REF] Healy | A study of proof conceptions in algebra[END_REF]. Each statement was introduced with one of the phrasings "Proof", "Show", "Justify", and "Explain". This allowed for twelve different questionnaire versions in total, as identical operators were excluded. Furthermore, in order to prevent cheating, different colored sheets of paper were assigned to the twelve versions of the questionnaire. Having developed an initial questionnaire, this prototype was piloted in mathematical courses for teacher education at the Justus Liebig University Giessen and the Westphalian Wilhelms-University Münster in May 2015 (N=48). We used the following set of categories to identify the kind of reasoning (see research question 1). Here, the category "empirical-inductive" comprises two aspects. Any mere testing of one or more concrete examples without further arguments or ideas is located in this category. Furthermore, inductive approaches, where the truth of the given statement is asserted on the basis of purely empirical considerations, belong to this category. We combined these two aspects to one category to stress the overall difference to deductive approaches. It has to be mentioned that this distinction of empirical-inductive and deductive reasoning is independent from the correctness respectively the completeness of the argument given; an incorrect deductive argument still counts as deductive. ) deductive The student mentions ideas or further arguments that could be used to prove the statement.

Moreover, the mere use of algebraic symbols is considered to be a kind of deductive reasoning.

(two times odd equals even odd + even equals odd accordingly, the claim is proved) Concerning the use of algebraic variables, we applied this code if any algebraic variable had been presented in the answer to a proving task or not. While the use of algebraic variables is measured as a category to code its appearance, we also counted the number of words used in students' proving attempts. In this case, we were not only interested in the appearance of words, because more or less any proof attempt will make use of some words somehow. Accordingly, we had to count the words to investigate a special kind of shift in students' proof attempts leading to more narrative approach-es. When counting the number of words used in an answer of the proving tasks, we did not count the words for structuring the proof production (like "example:", "proof:" or "q.e.d") or symbols (like "+" or "…"). Following this categorization, the first example given above in table 1 is considered to be an empirical-deductive answer, with no use of variables and zero words. The second example is a deductive answer without variables containing 14 words.

Results

The following results are based on the answers of 381 first-year preservice teachers at the University of Gießen, Münster and Paderborn. The results concerning the kind of reasoning in accordance to the given phrasings are shown in Figure 1. Concerning proving task (1), the percentage of empirical-inductive answers is the highest for the phrasing "show that" (26%) and the lowest for "explain" (17%). This difference is statistically significant with small effect size [Chi²-test, p=.021 with Cramer's V=.167]. In the second task, the results are quite different. Around half of the students give empirical-inductive answers. In this case, there is no statistically significant difference between the kind of reasoning in accordance to the different phrasings of the task. This might be due to the fact that the second task is harder to prove (see above). Accordingly, students might have only given examples because they did not know how to proceed otherwise.

When analyzing the data concerning the use of variables and the number of words used, we only referred to the answers making use of a deductive approach, as the 'empirical-inductive' proof attempts will obviously include no variables and contain fewer words. The results concerning the use of algebraic variables with regard to the deductive attempts in accordance to the given phrasings are shown in figure 2. Having a look at the answers to the first proving task, the phrasing "prove that" leads to the highest percentage of the use of algebraic variables (76%), followed by "show that" with 62%. Concerning students' proof productions for task (1), there are several statistically significant differences considering the use of algebraic variables with small and medium effect size (see table 2). One might assume that something like an implicit semiotic norm would lead to these differences. In task (2), the minor differences concerning the use of algebraic variables in accordance to the given phrasings are not statistically significant (Chi²-test). Here again, the students' problem in dealing with the second proving task can be considered as an explanation why no differences concerning the use of algebraic variables in accordance to the given phrasings could be observed. However, the results concerning task (1) hint that a construct like semiotic norms should be considered, because there, the use of algebraic variables differs in accordance to the given phrasings. P value (Chi²-test) effect size (Cramer's V) "prove that" vs. "show that"

.021 .158 "prove that" vs. "reason" <.001 .391 "prove that" vs. "explain" <.001 .376 "show that" vs. "reason" <.001 .241 "show that" vs. "explain"

.

.225

Table 2: Statistical data concerning the differences about the "use of algebraic variables" in accordance to the given phrasings. P value (Chi²-test) and effect size (Cramer's V)

Students used 25.68 words on average to answer task (1) and 18.83 to answer task (2) (see table 3).

Having a look at the arithmetic means of number of words used in accordance to the given phrasings in task (1), there are remarkable and statistically significant differences (see table 4). Whereas the phrasings "reason" and "explain" lead to an increased use of words (27.35 respectively 27.52 on average), "prove that" and "show that" lead to a minor use of words (14.83 respectively 13.11 on average). In this case, we consider the phenomenon of an increased use of words due to the phrasing of a proving task as a matter of semiotic norms.

Summary and final remarks

The focus of this paper is on if and how the phrasing of a proving task might influence students' proof productions. In the case of the easier proving task (1) about the claim that the sum of an odd number and its double is always odd, remarkable differences could be observed. Here, the phrasing "show that" led to statistically significant more answers consisting only of empirical evidence compared to the phrasing "explain". While "show that" and "prove that" evoked the use of algebraic variables, "reason" and "explain" led to an increased use of words. While these differences could be observed in the case of the first task, there were no such results in the case of the second task. One explanation might be that the second task (about the divisibility by six of the product of three consecutive natural numbers) was too hard for the students. If the students do not know how to solve a problem in any way, they cannot vary concerning the way they prove the claim. This possible explanation is supported by the fact that much more students only gave empirical-inductive arguments to answer the second proving task (44% vs. 18% in the case of the first task).

Following our theoretical considerations, the differences in students' proof attempts in the context of task (1) can be explained by the framework of sociomathematical norms (in the sense of [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF]. Students had acquired throughout their daily (school) life and their mathematics classes what they are expected to do when being asked "to prove" a statement, "to show" that something is true, "to reason" or "to explain" a given fact. The emergence of respective sociomathematical norms evolves from the (implicit and explicit) discourse taking part in the classroom, where students and teachers negotiate their expectations and requirements. In the case of task (1), the phrasing "show that" led to statistically significant more empirical-inductive approaches than the phrasing "explain". As a result of sociomathematical norms, semiotic norms can be developed. Students might link a task or the phrasing of a task with the use of certain semiotic resources. In this study, the phrasings "prove that" and "show that" led to an increased use of algebraic variables, whereas "explain that" led to an increased use of words. While we claim, that certain sociomathematical norms have been developed concerning the area of proof and reasoning, we can only guess, which experiences have led to respective phenomena. Here, classmates and (mathematics) teachers will have played a decisive role when negotiating norms. However, other influences from real life might also have affected students' attitude and behavior.

To sum up, the phrasing of a proving task can influence students' proof productions. In this study we could identify differences concerning the kind of reasoning, the use of algebraic variables and the number of words used. This result should be considered in the teaching of proof and in the research in this domain. The teachers at school and at university should be aware of the fact that students seem to combine different (implicit) norms for phrasings of a proving task when designing tasks for their students. Researcher should consider the fact that the phrasing of a proving task might affect their research results when analyzing the data. But more research has to be done to get deeper into the effects different phrasings may evoke. Beside the formulation of task, also other aspects have to be considered, like social and sociological factors, individual preferences or mathematical thinking styles. These considerations have to be investigated by future research. Finally, as was shown above, the findings of this study seem to be closely related to the two tasks used in this study. More research is needed to confirm or to specify the results obtained in the context of different proving tasks in different domains.

Figure 1 :

 1 Figure 1: Results concerning the "kind of reasoning"

Figure 2 :

 2 Figure 2: Results concerning the "use of algebraic variables"

Table 1 : Set of categories to investigate the "kind of reasoning"

 1

Table 3 : Arithmetic means concerning the number of words used in students proof productions in accordance to the given phrasings

 3

	P value (t-test)	effect size (Cohen's d)

Table 4 : Statistical data concerning the differences about the "number of words used" in proving task (1) in accordance to the given phrasings. P value (t-test) and effect size (Cohen's d)

 4


Kotaro Komatsu
email: kkomatsu@shinshu-u.ac.jp

Gabriel J Stylianides
email: gabriel.stylianides@education.ox.ac.uk

Andreas J Stylianides

Task design for developing students' recognition of the roles of assumptions in mathematical activity

Keywords: Assumptions, Task design principles, Classroom intervention, Teacher's role

Mathematics education researchers have highlighted the importance of assumptions in school mathematics given their vital roles in mathematical practice. However, there is scarcity of research aiming at enhancing students' recognition of different roles that assumptions play in mathematical activity. In this paper, we begin to address this issue by formulating two task design principles and reporting on the implementation of a classroom intervention where lower secondary school students in two classes worked, with the same teacher, on a task designed following the proposed principles. Our analysis shows how the task, together with the purposeful teacher's actions in implementing it, led to students' developing appreciation of two roles that assumptions play in mathematical activity.

Introduction

Assumptions refer to statements that doers of mathematics use or accept (often implicitly) and on which their claims are based [START_REF] Stylianides | Promoting prospective elementary teachers' knowledge about the role of assumptions in mathematical activity[END_REF]. Several researchers in mathematics education have highlighted the importance of helping students be more aware of the roles that assumptions play in mathematics, especially in relation to the notion of proof. For example, [START_REF] Fawcett | The nature of proof[END_REF], who treated proof as a key notion in school mathematics, listed four criteria for checking students' understanding of the nature of proof, among which two are closely related to assumptions: students understand "[t]he necessity for assumptions or unproved propositions" and " [t]hat no demonstration proves anything that is not implied by assumptions" (p. 10).

Despite the importance of assumptions in school mathematics, assumptions have received little research and pedagogic attention in school mathematics. Furthermore, existing studies (e.g., [START_REF] Fawcett | The nature of proof[END_REF][START_REF] Jahnke | Understanding what a proof is: A classroom-based approach[END_REF][START_REF] Komatsu | Fostering empirical examination after proof construction in secondary school geometry[END_REF] have mainly focused on geometry at the secondary school level. Research in other mathematical domains is needed that will build on and widen the scope of application of prior research in geometry, given also the recent calls for curriculum reform (see, e.g., NGA & CCSSO, 2010, in the United States) to introduce proof-related mathematical activity across content areas as well as in primary school (e.g., [START_REF] Stylianides | Proving in the elementary mathematics classroom[END_REF].

To develop students' recognition of the roles of assumptions in mathematical activity, we focus on the design of mathematical tasks, which have potential to affect significantly students' experiences with and understanding of the subject. Specifically, in this paper we propose two principles that can be used to support task design in the area of assumptions, we report on an intervention implemented in two classes where lower secondary school students worked on a task designed following the proposed principles, and we discuss the influence on students' understanding of the roles of assumptions in mathematical activity.

Task design principles for the roles of assumptions

Types of assumptions

Assumptions often refer to statements that are accepted as true without proofs, and their typical examples are axioms like the fifth postulate in Euclidean geometry whose modification led to the formation of non-Euclidean geometries. To expand the opportunity for students to experience the relativity of truth in mathematics, we extend the mainstream notion of assumptions to include two additional types of assumptions, both related to mathematical tasks and their formulation: (1) conditions of tasks (including premises of statements mentioned in tasks), and (2) definitions of terms mentioned in tasks. This paper reports on an intervention concerning the former type in the context of a function task. We are in the process of designing another intervention addressing the latter type.

Roles of assumptions

Assumptions play multiple roles in the discipline of mathematics. Their primary role has been to introduce the relativity of truth, i.e., whether a proposition is true or false cannot be absolutely determined, but hinges on assumptions [START_REF] Fawcett | The nature of proof[END_REF]. For example, Euclidean geometry was originally accepted as the standard form of geometry that was deemed to be true in an absolute sense. However, following mathematicians' unsuccessful attempts to deduce the fifth postulate from other postulates, mathematicians created new geometries, the non-Euclidean geometries, by adopting axioms different from the fifth postulate. It was thus recognised that some propositions in Euclidean geometry are true only under the set of Euclidean axioms, and that the conclusions are different under different sets of axioms.

A second role is to mediate disputes that occur concerning the truth of conjectures; mathematicians have managed to resolve such disputes and reach consensus by delving into the assumptions underlying the conjectures. Consider, for example, the conjecture that the limit of any convergent series of continuous functions is itself continuous. According to [START_REF] Lakatos | Proofs and refutations: The logic of mathematical discovery[END_REF], while Cauchy provided a proof for this conjecture in the 19th century, a counterexample to the conjecture was found in Fourier's work. Mathematicians attempted various explanations for this puzzling situation where the proof for and counterexamples against the conjecture coexisted. Eventually, Seidel analysed Cauchy's proof and discovered a hidden lemma, which once incorporated into the conjecture as a condition (related to the concept of uniform convergence), validated the conjecture.

Learning goals

Considering the importance of promoting authentic mathematical practice in school mathematics [START_REF] Lampert | Practices and problems in teaching authentic mathematics[END_REF], our study aims to help cultivate students' sense of the two roles of assumptions we discussed earlier. Specifically, we set up the following two learning goals for the intervention in our study (similar to the goals discussed in [START_REF] Stylianides | Promoting prospective elementary teachers' knowledge about the role of assumptions in mathematical activity[END_REF], for prospective teachers):

Learning Goal 1: To recognise that a conclusion is dependent on the assumptions on which the argument that led to it was based.

Learning Goal 2: To recognise that making the underlying assumptions explicit is crucial for reaching a consensus on the conclusion.

Task design principles

Based on our previous studies [START_REF] Komatsu | Fostering empirical examination after proof construction in secondary school geometry[END_REF][START_REF] Komatsu | Task design principles for heuristic refutation in dynamic geometry environments[END_REF]Stylianides, 2016, chapter 4;[START_REF] Stylianides | Promoting prospective elementary teachers' knowledge about the role of assumptions in mathematical activity[END_REF], we formulated the task design principles below in order to address the above learning goals:

Principle 1: Designing tasks that are subject to different legitimate assumptions by leaving purposefully some of the assumptions of tasks implicit or unspecified.

Principle 2: Allowing tasks to have different legitimate answers based on different legitimate assumptions about the tasks.

At the centre of these principles is the deliberate act of keeping some of the assumptions of the tasks implicit or unspecified. The typical way of showing a task to students and promptly clarifying its assumptions eliminates the possibility of multiple interpretations of these assumptions and reinforces the intuitive connection between the clarified assumptions and the expected answer. In our on going research (part of which we report here), we intend to break this norm by creating a situation where an assumption of a task is purposefully unstated by the teacher so that the students can work on the task under their own (implicit and potentially varied) assumptions. We anticipate that the disagreement in the proposed answers likely to emerge in the whole-class discussion will prompt the students to explore the reasons for the inconsistency. This discussion, purposefully orchestrated by the classroom teacher, is expected to help students realise that their answers depend on their respective assumptions and that an explicit assumption is needed to justify each answer.

Method Task design

Based on the principles stated above, we designed a task shown in Figure 1, which we adapted from a task implemented in the 2016 National Assessment of Academic Ability in Japan (NIEPR, 2016). With respect to Principle 1, the functional relationship between x and y is not specified in the task so that students can work on the task under different assumptions about functional relationships. With respect to Principle 2, the answer to the task is, for example, y = 6 if y is inversely proportional to x, and y = -6 if y is a linear function of x. Despite the divergence between these answers (as well as an infinite number of other legitimate answers not discussed here), both of them are, or can be, correct as long as the respective functional relationship is assumed.

The table below represents y-values corresponding to the given x-values. Find the y-value when x = 6.

x … 2 3 … 6 … y … 18 12 … …

Participants

We conducted an intervention study using the task in Figure 1 in two ninth-grade classes (approximately 40 students in each class, aged 14-15 years) in a Japanese lower secondary school affiliated with a national university. The intervention consisted of one lesson (50 minutes) and was implemented in the two classes by the same teacher who had five years of teaching experience.

We worked together with the teacher in planning the intervention. Specifically, the first author showed to the teacher the task in its original form from the national assessment, and also discussed the intended learning goals and task design principles we presented in the previous section. The teacher then created a lesson plan and discussed it with the first author. Based on the discussion, we decided to use the task as in Figure 1 that was modified from the original form. The teacher then implemented the task following closely the lesson plan as outlined below. The participating students were not introduced to the notion of assumptions before, but they were familiar with the definition of functions and several kinds of functions: proportion (y = ax), inverse proportion (y = a/x), linear function (y = ax + b), a special case of quadratic function (y = ax 2 ), and step function.

Data collection and analysis

The data include the transcripts of the videotaped lessons, the students' worksheets, and the field notes taken by the first author during the lessons. The data analysis aimed to determine whether the learning goals described earlier were achieved in each class. We focused our analysis on the whole-class discussions to examine the learning trajectory of each class as a whole, rather than attempting to trace individual students' thinking. Due to space limitations, results from analysis of students' worksheets are referred to only for triangulation, complementing the results pertaining to the whole-class activity. Because the two classes followed nearly identical paths, we report only the results from one class (38 students) in this paper. All student names are pseudonyms.

Classroom intervention

The relative correctness of answers

The teacher began the lesson by presenting the task in Figure 1; the students individually worked on it for about five minutes. During this work, some students provided the answer y = 6, while others the answer y = -6. This disagreement surfaced during the subsequent whole-class discussion. Misaki first said, "I think y is 6", but a student objected to her answer, "That's wrong". Aoi then said, "[y =] -6", but Shun questioned her answer saying "What?" After that, the teacher posed to the students the pre-planned question of which answer, y = 6 or y = -6, was actually correct:

31 Teacher: Well, now there are two answers. Which is a correct answer? 32 Students:

Both are correct answers. 33 Teacher:

Really? Are both correct answers? Please think everyone.

[…] [The students start to discuss with their neighbours.] 34 Nanami:

What? What is the conclusion? 35 Takumi: I don't know.

As seen in this interaction, an anticipated sense of confusion emerged among students. While some of them considered both answers to be correct (line 32), others had no idea about which the correct answer was (lines 34, 35). To address this confusion, the teacher then asked students to think more about whether both answers were correct and to write their thoughts on their worksheets. The students then shared their thoughts in the whole-class discussion: Um, I also think both are correct. If we change how we read the table, it becomes an inverse proportion and also a linear function.

Ren and Riko's responses (lines 39, 41) are representative of students' responses and are relevant to Learning Goal 1: recognising that a conclusion is dependent on the assumptions on which the respective argument that led to it was based. That is, these students understood that the correctness of their answers was relative to their assumptions, implying that y is 6 if y is inversely proportional to x, and y is -6 if y is a linear function of x. All students in the class agreed that both answers could be correct and most students shared or wrote similar thoughts. For example, Mizuki wrote on her worksheet, "the answer is 6 if we interpret it as an inverse proportion, but -6 if it is a linear function". Similarly, Nanami wrote, "The graph becomes a curve where the y-value becomes 6, and also becomes a line where the y-value becomes -6, so I think both values are correct depending on how we think about it". These students' responses show that the task has afforded them the opportunity to recognise the connection between different answers and respective assumptions.

Two ways for pinning down the answer

While the students recognised the relativity of their answers, they started to feel dissatisfied with the task they worked on: Nanami said, "That's not good. I don't know what I am solving". The teacher responded to the students' feeling of dissatisfaction by posing two pre-planned questions (written on the blackboard): "(1) What are the reasons for the answer to be ambiguous? ( 2) What can we do to address the ambiguity?" The students were given approximately ten minutes to tackle these questions, before sharing their thoughts in the whole-class discussion. The students' responses can be divided into two groups, of which Kenta's and Shun's contributions are representative:

75 Kenta: Um, um, [regarding question] number 1, [this] is because there is no explanation about whether y is proportional to x or inversely proportional. Um, if there are only points (2, 18) and (3, 12), equations are possible both for y = 6 and -6. Um, if we interpret it as the graph of the inverse proportion of y = 36/x, um, we have 6. If we interpret it as the graph of the proportion of -6x + 30, we have -6, I think. Well, [regarding question] number 2, I think, we should add an explanation, like when y is proportional to x or when y is inversely proportional to x. [Kenta misspoke here. He meant linear functions when he talked about proportions.] 88 Shun:

Um, because there are only two given points, the answer changes depending on whether we connect two points with a line or connect [them] with a curve.

[Regarding question] Number 2, I think, if we determine the values of more than two points, the answer will be [uniquely] determined.

According to first group of students (illustrated by Kenta's comment), the reason for the answer to the task to be ambiguous was the absence from the task description a specification of the functional relationship between x and y; the way this group suggested to address the ambiguity was to clarify the relationship in the task. According to the second group of students (illustrated by Shun's comment), the ambiguity was because only two pairs of values were given in the task; the way this group suggested to address the ambiguity was to provide in the task more points with given values.

Recognition of the necessity of assumptions

The teacher had anticipated these two kinds of student responses and made a planned move to address them, beginning with the second group's suggestion to increase the number of points with given xand y-values. He plotted the points (2, 18) and (3, 12) on the coordinate plane on the blackboard and said: "Suppose that in the case of […] 4, we have 6 [meaning y = 6 for x = 4]. Is this a linear function?

We know three points." Here the teacher selected (4, 6), because the y-value is 6 when x = 4, if the function is assumed to be linear.

The alternative possibility of step functions was then proposed by Shota. Shota's suggestion greatly surprised other students, who later began to agree with him and acknowledged that, even if three pairs of x-and y-values were given, the function in the task could not be uniquely determined. Below is the whole-class discussion after Shota drew the graph shown in Figure 2. In the above interaction, the students stated that even if another point (4, 6) was added to the table (Figure 1), the function could be a step function rather than linear, and that in this case, the y-value when x = 6 could not be uniquely determined (lines 143, 145-147). The class thus realised that increasing the number of points was not a viable option to address the ambiguity in the task. The consideration of step functions motivated the class to seek another way to pin down the answer, returning to the way that was represented earlier by Kenta (line 75), i.e., specifying the functional relationship between x and y in the task. At the end, the teacher summarised the lesson as follows:

194 Teacher:

Making this clear [meaning to specify the functional relationship] is good. I introduce this [pointing "in the case of a linear function" and "in the case of an inverse proportion" written on the blackboard], this, the one shown at the beginning is, […] this is called an assumption, assumption. So, I believe you see that the answer becomes definite by this assumption.

In this comment, the teacher introduced the term assumption and clarified that the conditions, such as "y is a linear function of x", were assumptions one could make based on the task's phrasing. He also mentioned that making the assumptions explicit was crucial for reaching a consensus answer. This is relevant to Learning Goal 2, which was reached by the class as illustrated by students' comments at the end of the lesson when they summarised their learning. For example, Takumi said, "If there is no assumption [made explicit], even if we know many points, the graph cannot be fixed into one function's graph". Similarly, Yuka said, "If there is no assumption, different functions can be considered. So, when we want to make the answer definite, we write the assumption".

Discussion

In this paper, we described a classroom intervention involving ninth-grade students with a particular mathematical task (Figure 1), designed based on two principles (Principles 1 and 2). Our analysis showed that the task and its purposeful implementation were useful for stimulating the students to recognise that the correctness of their answers was relative to their respective assumptions (Learning Goal 1), and that it was necessary to make the assumption of the task explicit to reach a consensus answer (Learning Goal 2). Hence, the principles can be regarded as helpful for designing tasks that aim to help students appreciate the roles of assumptions in mathematical activity.

Although in this paper we focused on the role of task design in promoting particular learning goals, the results of our intervention highlight also the important role that the teacher played for achieving these goals. Prior to the intervention, we held meetings with the teacher to discuss the task, the two design principles, and to explain the task's intended purpose. The teacher well appreciated the learning goals of the task and was able to devise a detailed lesson plan where he anticipated students' responses to the task and planned some questions strategically, preparing himself to capitalise on these responses during the lesson so as to effectively manage students' contributions and steer class progression towards the learning goals. The intervention played out in both classrooms as expected, thereby limiting the need for teacher to improvise and make in-the-moment decisions; the lesson naturally progressed towards the learning goals, while the students' contributions were respected and were integrated into the discussions [START_REF] Sherin | A balancing act: Developing a discourse community in a mathematics classroom[END_REF].

A significant aspect of the teacher's role was asking probing questions. For example, in response to students' different answers (y = 6 and -6), the teacher asked, "What are the reasons for the answer to be ambiguous? What can we do to address the ambiguity?" These questions triggered students' interest in exploring ways to obtain a consensus answer, and created an "intellectual need" [START_REF] Harel | Two dual assertions: The first on learning and the second on teaching (or vice versa)[END_REF] for the class to be introduced to the notion of assumptions. Another important aspect of the teacher's role was orchestrating whole-class discussions [START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF]. When planning the intervention, the teacher had predicted that some students would come up with responses similar to Kenta's (line 75) and Shun's (line 88). When implementing the intervention, the teacher 'filtered' and 'sequenced' the students' responses [START_REF] Sherin | A balancing act: Developing a discourse community in a mathematics classroom[END_REF][START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF] so that Shun's and Kenta's ideas were examined, one at a time, in the whole-class discussion. When the students understood that Shun's idea was not viable, he helped students recognize the advantage of Kenta's idea (line 75) to make explicit the functional relationship in the task.

Figure 1 :

 1 Figure 1: The designed task, which we adapted from a task derived from NIEPR (2016)

Figure 2 :

 2 Figure 2: Step function drew by Shota (the dotted vertical line was added later by the teacher) 138 Teacher: Well, in this case, what happens? This, this, well, what did you draw? 139 Shota: A step function. 140 Teacher: Oh, yes, thank you. Is this a linear function? 141 Students: No. 142 Teacher: Yeah. […] In the case of 6, this 5, 6, the point in this case… [The teacher added the dotted vertical line in Figure 2, and suggested a question asking the y-value when x = 6.] 143 Nanami: Anything [would be] okay [meaning that the y-value can be any values]. 144 Teacher: Can you predict [the value of y]? 145 Nanami: Anything [would be] okay. 146 Moe: Like quite suddenly [it is] changing. 147 Nanami: It could change suddenly, it is changing.

  Um, I think both [answers] are correct. These two. Since we don't know what the function of this graph is, since we can interpret it as both an inverse proportion and a linear function, I think both are correct.

	40 Teacher:	You've described your opinion that both are correct. Well, let's listen to
		another student. Riko, can you share [your opinion]? What do you think?
	41 Riko:


Dorte Moeskaer Larsen

Camilla Hellsten Østergaard

Questions and answers … but no reasoning!

Keywords: Mathematical reasoning, statistical reasoning, study and research path

Statistical reasoning has often been used synonymously with statistical thinking. In this paper, we focus on the reasoning part and we analyse mathematic lessons about statistics in a primary school class, using a construct from the Anthropological Theory of Didactics approach, called Study and Research Paths. By comparing the a priori tree-diagram of the course with the a posteriori treediagram of the observed teaching, it becomes clear that the teacher's questions never make the students engage in statistical reasoning and the students' questions are more concerned with practical and organizational issues than with obtaining a greater understanding of statistical reasoning.

Introduction

Teaching reasoning and justification requires more than just asking students to explain their answers or to pose open problems [START_REF] Ball | Making mathematics reasonable in school[END_REF]. Teachers' and students' questions and answers are crucial. This paper contributes further insight into teaching reasoning in elementary mathematics classrooms by analysing a teacher and students' questions and answers during a statistics course. We aim to contribute to a more systematic methodology to study what actually takes place in the classroom and to determine if it is possible to do this by looking at the questions and answers of students and teachers during their lessons.

To analyse the students' reasoning processes in the classroom, we will use a tool from the Anthropological Theory of Didactics (ATD), namely, Study and Research Paths (SRP). SRP provides to model mathematical knowledge from a didactical perspective [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF]. The analysis needs detailed information about context, contents, order of questions and answers. In the last decade several studies has focused on the potentials of SRP; [START_REF] Winsløw | Study and research courses as an epistemological model for didactics[END_REF] examine how SRP and a new diagrammatic representations can be used to analyse didactic processes; [START_REF] Barquero | A study and research path on mathematical modelling for teacher education[END_REF] illustrate how SRP can be used in professional programs for teachers; and [START_REF] Jessen | Study and Research Paths at Upper Secondary Mathematics Education: a Praxeological and Explorative study[END_REF] studies how SRP can support the development of knowledge in bidisciplinary settings.

We investigate whether the teacher and the students ask questions and answer questions in a way that allows the students to engage in mathematical reasoning related to statistics; we also reflect on the potentials and limitations we found when using SRP as an analytical tool. More precisely, we ask the following research question: What is the content of the students' and teachers' questions and answers in the "Youngsters and ICTs" intervention, and does it support the students' opportunities to produce statistical reasoning?

Theoretical considerations: mathematical and statistical reasoning

Researchers and educators around the world have advocated that a primary goal of mathematics education for all grades should be the development of mathematical reasoning [START_REF] Ball | Making mathematics reasonable in school[END_REF]. Nevertheless, there is no consensus about the definition of mathematical reasoning in the research literature [START_REF] Jeannotte | A conceptual model of mathematical reasoning for school mathematics[END_REF][START_REF] Mariotti | Argumentation and proof[END_REF]. Reasoning in statistics can be seen as a particular form of reasoning in mathematics. del [START_REF] Del Mas | A comparison of mathematical and statistical reasoning[END_REF] argues that mathematical and statistical reasoning should place similar demands on a student and should display similar characteristics when the students are asked to reason with highly abstract concepts. At first, mathematical and statistical reasoning appear to be similar, but the nature of the tasks in statistics and mathematics are somewhat different. In mathematical reasoning, context may not play a large role, but in the practice of statistics, the inquiry will always be dependent on data and typically grounded within a context [START_REF] Del Mas | A comparison of mathematical and statistical reasoning[END_REF]. Statistical reasoning is often used to define the same capabilities as statistical thinking, but [START_REF] Ben-Zvi | The challenge of developing statistical literacy, reasoning and thinking[END_REF] try to separate the two concepts. They define statistical reasoning as "understanding and being able to explain statistical processes and being able to fully interpret statistical results" (p. 7). Statistical thinking, on the other hand, involves an understanding of why and how statistical investigations are conducted, and also when to use appropriate methods of data and analysis. Both statistical thinking and reasoning can be involved when working on the same task, so the two types of activities cannot necessarily be separated. del [START_REF] Del Mas | A comparison of mathematical and statistical reasoning[END_REF], however, writes that it is possible to distinguish them through the nature of the task: "For example, a person who knows when and how to apply statistical knowledge and procedures demonstrates statistical thinking. By contrast, a person who can explain, why a conclusion is justified demonstrates statistical reasoning." (p. 85). [START_REF] Brousseau | Didactical Handling of Students' Reasoning Processes in Problem Solving Situations[END_REF] argue that the teaching of reasoning used to be conceived as a presentation of model proofs, which then had to be faithfully reproduced by the students. Teachers today see reasoning as an activity, which cannot be learned as a simple recitation of a memorized proof; instead, it is necessary to confront students with problems, where they naturally engage in reasoning. If students are presented with model-proofs today, they are meant to serve as a model of others' reasoning, which students then can use to produce their own original or creative forms of reasoning. However, as [START_REF] Brousseau | Didactical Handling of Students' Reasoning Processes in Problem Solving Situations[END_REF] 14) noted, "There is always the risk of reducing problem solving to an application of recipes and algorithms, which eliminates the possibility of actual reasoning". When it comes to statistical reasoning, most teachers tend to teach concepts and procedures and hope that reasoning will develop as a result [START_REF] Cobb | Principles of instructional design for supporting the development of students' statistical reasoning[END_REF]). [START_REF] Cobb | Principles of instructional design for supporting the development of students' statistical reasoning[END_REF] argue that, in statistical reasoning, students must reason about data rather than attempt to recall procedures for manipulating numerical values.

Study and Research Paths

Study and Research Paths (SRP) is a recent construct in the Anthropological Theory of Didactics (ATD) [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF]. Within ATD SRP were introduced as a design tool for teaching within the paradigm of "Questioning the world" [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF]. The aim is to focus on important and meaningful "big" questions and not just "visit monuments", meaning a set of rules prescribing, what is to be studied with no place to raise "What for?" or "So What?" questions [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF]. The fundamental dialectics between questions and answers are at the root of the idea of SRP [START_REF] Winsløw | Anthropological theory of didactic phenomena: some examples and principles of its use in the study of mathematics education[END_REF]. A group or an individual develops knowledge as a result of working with an overall question, Q. Students identify the "official" knowledge that can help them answer Q; the students use this "studying" to justify their answers to Q by engaging in reasoning, which is the "research" about Q. Elaborating on Q generates the "path" [START_REF] Winsløw | Study and research courses as an epistemological model for didactics[END_REF]. SRP is normally used to design lesson plans but [START_REF] Winsløw | Study and research courses as an epistemological model for didactics[END_REF] introduce SRP as a modelling tool to analyse didactic processes. [START_REF] Jessen | How can study and research paths contribute to the teaching of mathematics in an interdisciplinary settings[END_REF] argues that a tree-diagram of the SRP is a strong tool for analysing didactical processes. We will use a tree-diagram (an example can be seen in figure 1) to make a SRP analysis in an elementary mathematics classroom. Conducting a SRP a priori analysis entails exploring what questions and answers could occur from one particular overall question (the generating question, called Q0). The tree-diagram refers to the possible path the students could follow after generating Q0. The Q0 must be so strong that students can derive new questions, Qi, from it. The answers to the derived questions add up to an answer to the original question, Q0. The Q0 must be of real interest to the students. This continues with more questions and more answers and could leads to a tree-diagram of pairs of questions and answers [START_REF] Jessen | How can study and research paths contribute to the teaching of mathematics in an interdisciplinary settings[END_REF]. The questions in black are ask by the teachers, and the questions in white are made by the students. The grey-coloured questions are those that are created in collaboration between the teacher and the students. The numbers next to the questions and answers indicate the order of the questions. In the SRP-process the media-milieus dialectics must be taken seriously: the information the teacher brings into the class; the answers available through different books, articles, videos or online resources; and the classroom milieu where the teacher and students manage to establish meaningful actions.

"Youngsters and ICTs"

"Youngsters and ITCs" is a 15-lesson statistics course, which is taught over 3 weeks and intended for grade 6 students; it was designed by C. K. Skott and the second author of this paper, [START_REF] Skott | How does an ICT-competent mathematics teacher benefit from an ICT-integrative project[END_REF] 1 . Before teaching the course, the teacher participated in a professional development workshop (6 lessons) that focused on statistical reasoning and digital technologies. The purpose of the course is to improve students' reasoning in statistics. The course frames and proposes ways that teachers can engage students in statistical investigations; formulate statistical problems; generate, analyse, and reason about data; interpret results; and disseminate them both inside and outside of a school context. The emphasis of the course is to create new habits of classrooms interactions, in which the students raise questions and explore the context with their teacher, who challenges them to come up with new questions and reflect further on possible answers. This approach breaks away from teaching a succession of more or less independent "chapters" where only "small" questions are raised.

Methodological approach

We employ a micro ethnographic design; this kind of approach is well-suited for describing, analysing, and interpreting a specific aspect of a group's shared behaviour [START_REF] Garcez | Microethnography[END_REF]. We use a case study research design to obtain a thick description of the observed teaching and to understand, how all the questions and answers in the classroom operate together in this context, which is part of a complex system [START_REF] Stake | The art of case study research[END_REF].

Within a two-year period, we observed a teacher in 31 classroom lessons; 16 observations from the course and 15 from before or after the course. All the observed lessons were video-recorded. Four audio-recorded semi-structured interviews with the teacher were conducted. The choice of semistructured interviews was chosen to get a deep understanding of the background of the classroom context. All the interviews were transcribed, and 15 lessons selected from the observations were transcribed.

"Children and young people spend too much time on media ... or?" "Youngsters and ITCs" was not originally designed as an SRP project. The course was designed by the five design principles by [START_REF] Cobb | Principles of instructional design for supporting the development of students' statistical reasoning[END_REF] including using technology to support students development of statistical reasoning, establishing norms for students statistical argumentation and the importance of making students explore realistic data -in this case questioning the students media habits, which can be compared with "Questioning the World". The overall question that "Youngsters and ITCs" asks is: "Children and young people spend too much time on media ... or?" (Q0). To answer the overall question, the students must explore the TV patterns and the media habits of each student in their class through a circular working process, which entails collecting data, analysing data, and making conjectures about habits to discuss whether or not the students spend too much time on media. A really important part of the work is to justify how and why the students' conjectures are accurate or true.

The institutional framework for the lessons analysed and presented in this paper is a small elementary school in Denmark. The teacher, Ea, is an experienced mathematics teacher. Her normal teaching is dominated by working with skills framed by a textbook or by iPad exercises. Ea characterizes herself as "a bit old-fashioned. I think it is most important they have skills". When describing dialogs in the classroom, Ea explains, "I fail to tell them anything in twenty minutes ... I do not think they listen when I stand at the blackboard".

In the present article we analyse two lessons (2x45 minutes) of the course (Q3). In this section, the students investigate a set of data that includes 49 students' TV-consumption. They analyse and interpret the data, and make a conclusion, based on that background, about whether the students watch too much TV. The idea is, that the students can use their experiences from the smaller dataset in (Q3), when they work with the authentic and more motivational question Q0. The students are explicitly asked to explain and justify their conclusions.

Presentation of the a priori and a posteriori diagrams

A comparison between the a priori and a posteriori diagrams illustrates (respectively, Figure 2 and Figure 3) how the intended teaching goals and the actual enacted lessons are different: In Figure 2, it is evident that we expect the students to consider and ask many questions in collaboration with the teacher. All the questions from Q3,1 to Q3,9 have extensions like the one shown for Q3,8 to the right. The expectation is that the students' answers will include an explanation of their results and of their choices of models/representations, and that students will use statistical arguments to justify their claims when they generalise about conjectures. The posteriori tree-diagram of the observed classroom is constructed by analysing the collected observations from the classroom. Q3 is a question from the teacher Ea, which give rise to three quick answers from the students (A3,1, A3,2, & A3,3) and one new question from Ea (Q3,1). Q3,1 produces six answers (A3,1,1-A3,1,6), and six new questions (Q3,1,1 -Q3,1,8), all with no connection to mathematics. We only see a few questions from the students that focus on specific solutions and procedures (e.g. Q3 ,1,3,3,1,3,3,2).

Comparing the a priori tree-diagram with the posteriori tree-diagram

In comparing the two tree-diagrams we do not see any of the expected questions (Q3,1-3,9) from the a priori tree-diagram. Ea does not ask many questions in this lesson, although she does ask a interpretative questions such as Q3,1.1: "Would the students' parents agree with your conclusions?" The questions Ea asks, do not enable the students to elaborate, explain, or justify their choices.

Instead the students answer Ea's questions with examples from their everyday life and their beliefs about what parents think. In the posteriori tree-diagrams, it also becomes clear that the students ask many questions that are organizational and procedural, including Q3,1,8 ("How do we upload in Showbie?") and Q3,1,7 ("What do you want me to say [in the presentation]?"). Studying the students' answers in the posteriori tree-diagram, it is possible to see that there are not any answers that include statistical argumentation like the questions in the a priori tree-diagram Q3,8. Mostly, the students use their own rationale as argumentation: "19 hours a week is ok, but 20 hours is way too much" (A3,1,1,1). However, some answers include calculations of different descriptors, like A3,1,3: "The greatest value is 30, the minimum value is 0, the average is 9,3 and the most common number is 6."

Discussion

The tree-diagrams reveal a lot of answers that indicate that the students may be unaware of the purpose of the generating question, how to ask questions, and how to answer those questions. The students answer the question, Q3, immediately without any inquiry and without any mathematical argumentations. For example, they say, "yes, [children watch too much TV] if the weather is good" (A3,1), or "yes, [children watch too much TV] if they watch a series" (A3,2). These quick answers could indicate that the students have not understood the premise about "inquiry" that is crucial to this type of investigative work, and the SRP does not contribute to establish a milieu with new norms for working with statistical argumentation. The tree-diagram indicates that the students do not discuss appropriate statistical descriptors. The students simply repeat an approach to analyse data that they have made in earlier lessons by calculating descriptors. Most of the students found the average number of hours spent on media without using any argumentation whatsoever for why it makes sense to calculate the average number of hours or what the average actually says about the dataset. The interpretation of what actually happens in this class may be characterized as 'model reasoning' [START_REF] Brousseau | Didactical Handling of Students' Reasoning Processes in Problem Solving Situations[END_REF]. Overall, the a posteriori tree-diagram does not indicate any joint dialogue in which the students raise questions and explore the context in collaboration with their teacher, who challenges them to create and reflect further on their statistical reasoning. However, we do see that the students apply statistical knowledge and procedures and demonstrate statistical thinking, e.g. when the students draw up charts (A3,1,6), fill tables (A3,1,4), and calculate descriptors (A3,1,3,2). Therefore, this approach can be seen more as "visiting monument" than "questioning the world" [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF]. To understand the students' development of statistical reasoning, it was rewarding to use SRP as an analytical tool. It helped us to distinguish different strategies for raising and answering questions and to interpret the content in the questions and answers. However, the SRP model has not helped us to clarify the amount of time spent on the different questions and answers and to know if certain areas received a greater focus. The teacher's treatment of students' answers and the possible feedback of students' answers are not visible. Finally, in the model we do not explicit focus on medias and the lack of integrating new medias, and we do not include the teacher's explanations and introductions; e.g. Ea explicitly explained how to make a frequency-table: "So you start counting how many one hours there are in the survey, and then you type the number here…[points at the Excel sheet]." Ea also explicitly explains how to calculate the average: "… to figure out how much the average is in this week per day -you just sum it together and divide it by seven." To make the diagrams, the teacher explains: "Just get Excel to make the diagrams for you by plotting the numbers in the sheet." These instructions can possibly be seen as a basis for 'model reasoning' and are therefore very important for the interpretation of the students' reasoning processes.

The gap between the a priori and the a posteriori tree-diagrams is in many ways not surprising. Ea participated in the 6-lesson professional development workshop and she worked collaboratively with her colleague during this program, but the intended statistical course is very different compared to Ea's normal teaching method and the milieu established in the classroom, which mostly focuses on mathematical skills. It is also important to notice, that the students also experience a very different way of learning mathematics; an approach, which no matter what, stresses some adaptation. The statistical course challenged Ea's view of classroom dialogs and it was difficult for her to establish new practices: "We'll take the arguments afterwards, because we'll first see all the presentations." However, in the end of the lesson Ea did not prioritize to hear the argumentation from the students.

Conclusion

The comparison of the two SRP tree-diagrams displaying the teacher and the students' questions and answers show that the content was focused more on 'model reasoning' and statistical thinking and that the course in many ways did not support the students' opportunities for production of statistical reasoning. The SRP tree-diagram has been found to be a convincing tool for analysing classroom practice. A comparison of the two diagrams provides a rich view of the questions and answers in the classroom, which is closely connected to the students' development of statistical reasoning; however, the use of the model had limitations, in the sense that we did not directly include media aspects as feedback, explanations, and introductions. It could nevertheless be interesting to design a "tree diagram" including not only question and answers, but which also include feedback, explanations and introductions and study if this model in any way gives a deeper understanding. Furthermore, it could be interesting also to focus on the processes and relations between the questions and the answersto explore why the students respond as they do?

Figure 1 :

 1 Figure 1: An example of a SRP tree-diagram

Figure 2 :

 2 Figure 2: The a priori tree-diagram of the course, Youngsters and ICTs, for Q3

Figure 3 :

 3 Figure 3: The a posteriori tree-diagram for Q3 in the observed classroom

The lesson plan (in Danish) is available at: http://auuc.demonstrationsskoler.dk/materialer/innovation/forloeb3/faser/hvordan-bruges-statistik-i-hverdagen.


Serpa Ary

Sabrine Neto

Carmen Hemmes

Ana Barbas

Ana Beiderlinden

Emmanuel Fernandez-Bustamante

Ognjen Futier

Mohamed Gajic

Abdulmohsin El-Tahan

Ersin Ghamdi

Günay

Association between driving pressure and development of postoperative pulmonary complications in patients undergoing mechanical ventilation for general anaesthesia: a meta-analysis of individual patient data

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Edmund M Clarke

Thomas A Henzinger

Helmut Veith

Roderick Bloem

Igor Konnov

E Allen Emerson

Joseph Sifakis

Under consideration for publication in Formal Aspects of Computing Handbook of Model Checking

Keywords:

This book is a comprehensive guide on model checking

the approaches for dealing with procedures and concurrency in software model checking. These techniques mainly built upon the framework of push-down automata.

Chapters 16, 19, 20 build the bridges between model checking and other approaches to ensuring system correctness: data-flow analysis and abstract interpretation, automated theorem proving, and automated testing. Although these approaches were initially thought of as orthogonal, they are much closer now, due to cross fertilization.

Chapters 21, 22, 28, 30, 31 introduce the techniques that bring model checking beyond the standard application domains. Whereas the classical model checking deals with finite-state Kripke structures, parameterized model checking answers the question, whether a temporal property holds for the systems comprising an arbitrary number of components. Chapter 21 presents several prominent techniques that answer this question for special kinds of parameterized protocols. A recent survey on this topic has also appeared in [BJK + 15]. Chapter 22 introduces model checking techniques for security protocols. These protocols are modeled as term transformers in the Dolev-Yao model. Chapter 28 discusses model checking of probabilistic systems. In these systems, events happen with certain probabilities. Instead of checking, whether such a system satisfies a temporal property, one is interested in answering the question of whether a temporal property holds with a given probability. Chapter 30 presents techniques for hybrid systems, in which some variables may change continuously, and their evolution is constrained with differential equations. Chapter 31 deals with model checking of infinite-state systems by fix-point computations over regions in the framework of µ-calculus.

Chapters 23-25 introduce applications of model checking in hardware industry. Chapter 23 summarizes the success of model checking in hardware verification and pitfalls that one meets when transferring theoretical results in practice. Chapter 24 discusses practical specification languages that are used in hardware industry for expressing the expected behavior of a hardware circuit. Related to that, Chapter 25 presents the technique of symbolic trajectory evaluation, which is used in industry for circuit verification.

Chapters 26, 29, 32 discuss the well-studied extensions of model checking techniques µ-calculus, timed automata, and process algebras. Finally, Chapter 27 presents the techniques for synthesis of reactive systems, as opposite to verification.

Obviously, it is hard to imagine that one reads the Handbook from cover to cover. When this book was lying on my desk for several months, I would browse through chapters on this or that topic. What I like the most about the book is that many chapters contain contributions by authors from different teams. This gives the reader a balanced view on every approach.

As the large part of the coordination and copy-editing was shared among three teams in Austria, including Helmut Veith's group, I was lucky to witness the work on the Handbook. This was a long-term project that could be only finished thanks to the energy of the editors and contributions by the authors, reviewers, and the editing team. To a shock to all of us, Helmut Veith tragically passed away, when the Handbook was in the final stage. We are missing his brilliance, energy, and unordinary thinking.

Finally, I have to give a few comments about the format of the Handbook. Owing to the high-quality paper and its 1234 pages, the hardcover is truly heavy, it weighs over two kilograms. Although it looks amazing on a bookshelf, it is rather an exercise to hold the Handbook in hands for longer than five minutes. I would thus recommend buying the eBook and printing the chapters of interest, when needed.