Caterina Primi

Aisling Leavy

Pedro Arteaga

Daniel Frischemeier

Orlando González

Sibel Kazak

In

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Introduction to the work of TWG5: Probability and Statistics Education

Introduction to the work of TWG5: Probability and Statistics Education

Leaders:

Introduction

The working group gathered 42 participants from 20 countries. 27 papers and 9 posters were accepted. In the first session after the organising team presentation we explained the three Cs of CERME, introduced the aims of the conference and of our TWG. In order to create a collaborative atmosphere, that would support the discussions and feedback over the following days, we started with an ice-breaker activity involving all the participants in a "Bingo" game". Through this activity participants got to know their TG5 colleagues in a low stakes and collaborative context. Following this, the participants were asked to introduce themselves and their expectations for the working group and the conference. Among the expectations communicated were: to receive constructive feedback on their work, to gather new ideas, to get informed and updated on new trends in statistics and probability education, to get better acquainted with emerging research from other countries and develop possible collaborations and networking opportunities.

Participants were introduced to the three subthemes identified as emerging across all submissions. In particular, submissions focused on "Teacher education", "Reasoning about data" and "Statistical and Probabilistic Thinking and Reasoning". However, we remarked that there was some overlap and similarities across subthemes: submissions on Teacher education also focus on subtheme 2 (Reasoning about data) or on subtheme 3 (Statistical and Probabilistic Thinking and Reasoning). Additionally subthemes 2 and 3 are not seen in a disjoint way -which means, in both subthemes 2 and 3 we can of course also find submissions with connections between the "data world" and the "probability world". Following this, participants were split into groups and each group discussed these subthemes. As part of these discussions, the co-leaders introduced and discussed several questions for the discussion in all subthemes during the sessions across the days to follow:

• What experiences can we provide to promote and develop learners understanding when reasoning about data / Statistical and Probabilistic Thinking and Reasoning? • Which (typical) (mis-)conceptions of learners can be identified when reasoning about data / Statistical and Probabilistic Thinking and Reasoning?

• How should tasks and learning environments be designed to enhance reasoning about data / Statistical and Probabilistic Thinking and Reasoning for learners (primary students, high school students, teachers, etc.)? • Which research methods are used / should be used to find about students´ or learners´ reasoning with data / Statistical and Probabilistic Thinking and Reasoning? • How and in which way can we build bridges between data analysis, probability, and inference? Which role does context play? • How and in which ways can the use of digital tools enhance statistical reasoning (reasoning about data and statistical and probability reasoning)? • What about the handling of Big Data in the upcoming Data Science era? How can we (statistics education) profit from the availability of Big and Open Data? Which implications does it have for statistics education?

For all the sessions during the week the participants were split in two groups: TWG5a with Caterina Primi (Leader) and Sibel Kazak and Orlando Rafael Gonzalez (Coleaders) and TWG5b with Aisling Leavy (Leader) and Daniel Frischemeier and Pedro Arteaga (Coleaders).

For each subtheme we organized two sessions managed and chaired by one of the coleaders. We also organized a poster session with all participants (both groups together) where each presenter had 5 minutes to present the poster and the rest of the session had time allocated for the discussion.

The last day we had a culminating session with all participants with the aim being to engage in an in-depth discussion on subthemes and to share the contributions discussed during each session.

Teacher education

From the discussions of various papers addressing pre-and in-service teacher education in our working group, the following ideas and implications emerged, regarding how to prepare teachers to address and foster students' reasoning and thinking about statistics and probability:  to enhance the pedagogical content knowledge facets of statistical knowledge for teaching, and not just focus on the subject matter knowledge facets;  to make teachers familiar with performing statistical tasks and investigations from a procedural, interpretative and contextual perspective, while making connections among fundamental ideas of statistics (e.g., data and randomness), chance, relevant and appealing real-life contexts, software and technology;  to include childhood statistical ideas (both formal and informal) in early years education, and prepare teachers to address and foster them;  to create Professional Learning Communities (i.e., communities of practice within schools and beyond);  to consider the affordances of games, university entrance examination tasks, and real-life problem scenarios as triggers for engaging students in high-quality statistical investigations, grounded on production tasks (as opposed to judgement tasks);

 to develop teachers' statistical reasoning and thinking by engaging them in the interpretation, design, teaching and assessment of stochastical tasks and statistical investigations organized around the three stages of teaching (i.e., lesson design, classroom implementation and reflection);  to develop teachers' assessment skills regarding the most suitable strategies to assess students' statistical reasoning and thinking, in terms of their effectiveness (e.g. peer review among students, projects, written reports);  to raise pre-and in-service teachers' awareness of the importance of providing interdisciplinary experiences to learn statistics (e.g., STEAM);  to raise pre-and in-service teachers' awareness of the importance of providing students with opportunities to work with big data and personalized data; and  to employ qualitative approach (e.g., observations, interviews, analysis of artefacts and final reports), action research, statistical argumentation, and analysis of statistics within social issues as methods to explore teachers statistical and probabilistic knowledge, thinking and reasoning.

Reasoning about data

From the discussions of various papers focusing on the reasoning about data, the following ideas and implications emerged in our working group:

 The Data Science era is emerging: Data are everywhere and the exploration and analysis of Big data and Open data will be a fundamental aspect in reasoning about data. Furthermore, there are other data collection methods which become more relevant (e.g., with Sensors, web scraping, etc.) and which provide new types of data.  The necessity to cooperate with other disciplines like computer science, social science and citizen science is becoming increasingly evident. Especially for Data Science, collaboration with computer science is inevitable.  The use of digital tools when exploring data is fundamental. We need to both consider and make the distinction between educational software tools (like TinkerPlots or Fathom) and professional software tools (like R, Python). Also open source and online tools like Gapminder and CODAP can help develop reasoning about data.  Data competence has to be developed continuously from primary school to adult education. Especially in primary school, data competence and more specifically reasoning about distributions can be enhanced by using pre-formal concepts like modal clumps which offer a pre-stage for the concepts center and spread for example.  The facets data, chance and context are tied strongly together.

Statistical and Probabilistic Thinking and Reasoning

From the discussions of various papers focusing on the statistical and probabilistic thinking and reasoning, the following ideas and implications emerged in our working group:

 The use of context, real problems and visualization tools can support statistical and probabilistic thinking and reasoning in data and chance explorations.

 The focus on interpretation and justification of results is essential to promote statistical and probabilistic thinking and reasoning.  Studies suggest that building on learners' intuitions and emerging conceptions, rather than focusing on their misconceptions, becomes more useful for supporting their thinking and reasoning in data and chance.  The use of real or realistic data can motivate learners to use statistical and probabilistic ideas in their thinking and reasoning.  To move the field forward considering design based research as an additional methodology can inform how to build statistics learning environments that support learners' statistical and probabilistic thinking and reasoning.

Conclusion

The field of statistics and probability education is a fast evolving area and to draw on the words of H.G Wells (cited in Huff, 1954) 'Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write'. As such, we identified several critical areas of focus for research and practice in statistics and probability education. Firstly, gaining access to the 'big ideas' in statistics and probability is important for effective citizenship. Exposing young children, in the early primary years, to informal ideas of statistics and probability is a critical step in order to develop and lay the foundation upon which formal understandings will be built upon. Related to this is the second recommendation that we establish a trajectory of learning experiences to support the continuous development of statistical and probabilistic reasoning and thinking from early years to adulthood. A guiding signpost for both educators and researchers alike, is provided in the third suggestion, and that is to maintain the important connection between data, chance and context as we move forward in the field. Two emerging areas to keep abreast of as we make advancements are the need to be prepared for Data Science (the implementation of Big Data) and its' implications and also to continue to develop and makes use of appropriate digital tools ( such as Gapminder, CODAP, TinkerPlots, Fathom etc.). Finally, we conclude with a reminder to remain mindful of the need to diversify our research design and approaches to inquiry -incorporating attention to fine-grained studies, eyetracking research, larger cross-comparative studies, experimental design, to name but a few -all in an effort to advance our understandings of the complex and evolving areas of statistics and probability education.


Arjen De Vetten
email: a.j.de.vetten@fsw.leidenuniv.nl

Aisling Leavy
email: aisling.leavy@mic.ul.ie

Ronald Keijzer
email: r.keijzer@ipabo.nl

Design principles for short informal statistical inferences activities for primary education

Keywords: Primary education, informal inferential reasoning, informal statistical inference, statistics education

This paper presents the following set of eight design principles for informal statistics inference (ISI) education for upper primary education in settings where limited time is available for ISI: Create awareness of inferential claims before engaging students in ISI via statistical investigations; use a context for which the students have no clear expectations of the outcome or which challenges current beliefs; use a question that highlights the need for an inferential claim; use a large population that makes sampling inevitable; use activities with tangible populations and samples in which generating representative samples and recording data is easy and quick; use data that require little descriptive analyses; and use the principles of growing samples and repeated sampling in order to create sampling variability. Three exemplar ISI activities illustrate how the design principles can be used to design new ISI activities and evaluate existing ISI activities.

Introduction

An introduction to informal statistical inference (ISI) in primary education has been advocated by various researchers [START_REF] Makar | Developing young children's emergent inferential practices in statistics[END_REF][START_REF] Makar | The reasoning behind informal statistical inference[END_REF]. The literature provides examples of activities that introduce ISI in upper primary education (De Vetten, Schoonenboom, Keijzer, & Van Oers, 2018a;[START_REF] Kazak | Sixth grade students' emerging practices of data modelling[END_REF][START_REF] Leavy | Insights into Informal Inferential Reasoning in the Primary Classroom[END_REF]. While design principles for statistics education in general are available [START_REF] Ben-Zvi | Design of statistics learning environments[END_REF], the literature does not report on a comprehensive list of design principles for ISI education for upper primary education. The aim of this paper is to bring together a set of design principles that have been reported in the literature alongside design principles that are distilled from recent research with primary pre-service teachers (De Vetten et al., 2018a;[START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF] with a view to generate design principles that apply to upper primary education. The design principles presented in this paper can be used for the design of ISI activities that require relatively little time (e.g. up to an hour), but still allow for an introduction of all key concepts of ISI.

Theoretical background

ISI can be seen as part of the broader field of statistics. For statistics education in general, it is suggested to focus on developing central statistical ideas, to use real, or realistic, and motivating data, and to integrate the use of technological tools that allow students to explore and analyze data [START_REF] Ben-Zvi | Design of statistics learning environments[END_REF]. Most of these principles are useful also for ISI education in primary education. The design principles we describe below therefore build on these principles. However, for eliciting IIR, these principles are not sufficiently specific.

A number of design principles have been targeted specifically towards ISI. First, activities where students are asked to make predictions about another or a larger sample based on the sample at hand, such as the growing samples heuristic [START_REF] Bakker | Design research in statistics education: On symbolizing and computer tools[END_REF], appear to be an effective principle for activities that aim to engage students in IIR [START_REF] Ben-Zvi | Design of statistics learning environments[END_REF][START_REF] Makar | Developing young children's emergent inferential practices in statistics[END_REF]. Some recent research involving pre-service teachers [START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF] and upper primary school students [START_REF] Kazak | Sixth grade students' emerging practices of data modelling[END_REF] utilizing the growing sample design principle indicates mixed results. The participants used the data as evidence in making predictions of the shape of the distributions of larger samples and thus looked beyond the data. However, pre-service teachers also tended to neglect uncertainty and make deterministic predictions. Moreover, most predictions made of other samples mimic the shape of the original sample, thus suggesting that students did not have a picture in their mind of a sample distribution being the result of an overarching population distribution. Rather, they may have thought that the original sample "produces" the predicted sample. De [START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF] suggest that this could be due to the absence of repeated sampling elements in the activities used. Including such elements confronts students with different sample results, which may make them aware that an inference based on one sample includes some uncertainty.

A second design principle for ISI activities is to have students compare two sample distributions and draw a conclusion about the difference between the means in the population [START_REF] Zieffler | A framework to support research on informal inferential reasoning[END_REF]. However, research evidence shows mixed results in terms of the success of eliciting IIR, due to a number of (contextual) factors. First, without proper teacher intervention, such tasks appear not to sufficiently stimulate students to engage in IIR [START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF]De Vetten, Schoonenboom, Keijzer, & Van Oers, in press;[START_REF] Pfannkuch | Informal inferential reasoning[END_REF], and, in our research, preservice teachers tend to restrict their attention to descriptive analyses. Another factor that may contribute to the absence of IIR in activities where students compare two distributions (De Vetten et al., 2019, in press;[START_REF] Pfannkuch | Informal inferential reasoning[END_REF] was that these activities did not involve repeated sampling, thereby missing the opportunity to view sampling variability. Third, while the sample distributions are visible, the populations remain abstract. Finally, tasks that compare sample distributions require substantial time for the necessary descriptive analyses, which goes at the expense of the available time and energy for inferential reasoning (De Vetten et al., 2018a).

A third design principle is to use a modelling approach in which a population is modelled (often using software, such as Tinkerplots). Samples are drawn from this hypothetical population distribution in order to develop an appreciation for the likelihood of various sample outcomes and an understanding of sampling variability (Manor & Ben-Zvi, in press). This approach requires more time and puts demands on the computer skills of the teacher and the student, and may therefore be less suitable for general upper primary students in settings with limited time and support.

In sum, findings from recent studies points towards a number of conclusions regarding effective design principles for ISI education and requirements for effective design principles: First, the use of growing samples and repeated sampling is recommended in order to create sampling variability and to show that usually larger samples reduce sampling variability and thus reduce the uncertainty of the inference. Second, it appears to be important to make students aware of the distinction between sample and population and when an activity requires a generalization beyond the data. Third, an overemphasis on descriptive analyses, arising from activities involving the comparison of sample distributions, may reduce the opportunity to engage in IIR. Fourth, in many primary school settings with limited available time and support the extensive use of technological tools may not be feasible.

Design principles

Using the above conclusions about (requirements for) effective design principles for statistics education in general and ISI education in particular we formulated eight principles for the design of ISI activities for primary education.

In particular, seven of these design principles are specifications of how statistical investigations activities should look like when the goal is to introduce students to ISI. The first design principle stands apart as it is undertaken prior to students being actively involved in conducting statistical investigations. 2018a) we first engaged pre-service teachers in an activity that was designed to make them aware of the existence and use of inferential claims. This activity appeared to promote an awareness of inference. Throughout the intervention the participants retained this awareness.

2. Use a context for which the students have no clear expectations of the outcome, or for which the outcome challenges students' current beliefs.

When the outcome is unpredictable or challenges students' current beliefs, students are stimulated to engage in inferential investigations and to search for explanations [START_REF] Makar | The reasoning behind informal statistical inference[END_REF]. When the outcome is predictable students may not engage in critical IIR (see for an example De Vetten et al., 2018a).

3. Use a question that clearly highlights that an inferential claim is required and use a population with a sufficiently large number of elements that surveying the entire population is impossible within a limited time frame.

The question posed should lead the student to conclude that merely analyzing the sample data descriptively is insufficient. However, even when considerable care is taken to formulate a good inferential question (e.g. using the guidelines from Pfannkuch, Regan, [START_REF] Pfannkuch | Telling data stories: Essential dialogues for comparative reasoning[END_REF], other factors may cause students to restrict their attention to descriptive analyses [START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF]. Therefore, combining a tangible population (see principle 6) with a population that consists of too many elements to survey in its entirety makes drawing a sample natural and generalization inevitable (De Vetten et al., 2018a).

4. Use activities that support the generation of a representative sample and that stimulate critical reflection on the representativeness of the sample.

The effect of this principle is that students may be willing to accept the sample data as reliable evidence for making inferential claims. In case an ISI activity does not permit students to draw representative samples, but only non-representative ones (such as convenience samples from their own class), students may be reluctant to engage in IIR because generalization based on such nonrepresentative samples is not possible anyway. Ideally, the activity is such that it is possible to draw both representative and non-representative samples, with the effect that discussions are elicited about ways to draw a representative sample.

5. Use activities in which generating samples and recording data is easy and quick and in which students preferably collect data themselves.

The effect of this design principle is that an excessive focus on data collection is avoided. At the same time, the data generation and recording process should allow students to make sense of the data and should facilitate mathematization [START_REF] Treffers | Three dimensions: A model of goal and theory description in mathematics: The Wiskobas project[END_REF] and engagement in IIR. Therefore, it is preferable that students collect their own data.

6. Use activities with a tangible population and sample.

The effect of this design principle is that students remain aware of the distinction between sample and population and that they are required to make a claim that pertains to the entire population.

7. Use data that require little descriptive analyses.

In the PPDAC-cycle, data analyses comes before interpretation of the results, including making inferences [START_REF] Wild | Statistical thinking in empirical enquiry[END_REF]. Spending too much time on descriptive analyses of the sample data leaves little time and energy for inferential reasoning [START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF][START_REF] Leavy | Insights into Informal Inferential Reasoning in the Primary Classroom[END_REF]. The effect of this design principle may be that sufficient time, energy and attention is left for IIR, while at the same time engagement in descriptive analyses helps students to make sense of the results and facilitates further mathematization (De Vetten et al., 2018a).

8. Use the principles of growing samples and repeated sampling in order to create sampling variability.

The effect of using the principle of repeated sampling is that students will consider the uncertainty of their inferences, because sample results will vary [START_REF] Saldanha | Conceptions of sample and their relationship to statistical inference[END_REF]. The effect of using the principle of growing samples is that students will see that larger samples usually reduce sampling variability and thus reduce the uncertainty of the inference [START_REF] Bakker | Design research in statistics education: On symbolizing and computer tools[END_REF].

Applying the ISI design principles -Three example activities

This section describes three ISI activities and how the design principles can be used to design new ISI activities and evaluate existing ISI activities. The first activity was designed with the design principles in mind, while the second and third were not. We evaluate the extent to which they are in line with the design principles.

Activity 1: What is the most frequently used word?

The first activity (De Vetten et al., 2018a;[START_REF] De Vetten | Knowledge in action. The informal statistical inference content knowledge of three pre-service teachers while teaching ISI in upper-primary school[END_REF] was implemented in two settings: in a class of second-year pre-service primary school teachers and in three primary classrooms (grade 3, 5 and 6). The activity centers round a pile of children's novels, that constitutes a population, and the question "Which word is most frequently used in this pile of novels?" First, students make a list of the top 5 words they think are most frequently used in the pile of books. Next, students discuss how to find out which is the most frequently used word. Taking a sample is the expected response. The sampling procedure is discussed. Small groups of students draw small samples and then pool their results in multiple large samples. The inferences for small and large samples are compared. In this way, it could be shown that the proportion that yields the same most frequently used word is greater for large samples than for small samples. If the proportion of samples with the same most frequently used word is high, students could be willing to accept the possibility of making uncertain inferences based on one sample. This activity was designed with the design principles in mind and it aligns well with all but one of the design principles. The activity is clearly inferential and uses a tangible and large population and visible sample (design principles [DP] 3 and 6). It requires that students consider how to take a representative sample and it allows for generating a representative sample (DP 4). Students take a book from the pile and count the occurrence of the top 5 words on a number of lines or a page, making sampling, recording and analyzing simple (DP 5 and 7). Without the need to rely on technological tools growing samples and repeated sampling can be achieved by having individual students draw samples and by combining students' samples (DP 8). The number of students may, however, limit the possibilities for repeated sampling for larger samples. Our research shows that pre-service and primary school students were engaged in inferential reasoning (De Vetten et al., 2018a[START_REF] De Vetten | Knowledge in action. The informal statistical inference content knowledge of three pre-service teachers while teaching ISI in upper-primary school[END_REF]. However, the activity included repeated sampling only for smaller samples. It thus missed the opportunity for students to see the sample results converge for larger samples. Repeated sampling for various sample sizes appears crucial to teach students to balance between sampling representativeness and sampling variability. Finally, the activity might not completely align with DP 2, because students may have beliefs about the outcome, based on information they have about word frequency in books in general (De Vetten et al., 2018a).

Activity 2: How many yellow balls are in the Black Box?

The activity "How many yellow balls are in the Black box?" (Van Dijke, Drijvers, & Bakker, 2018) was implemented in a grade 9 pre-university secondary classroom. The activity has students investigate and estimate the number of yellow balls in a black box, filled with a mix of 1,000 yellow and orange balls, by looking through a small viewing window. It is expected that students will shake the bottle in order to take samples repeatedly and that they will use the average of their sample as an estimate. Findings are recorded and discussed by students as part of a whole class discussion. Following this, a larger viewing window is opened on the black bottle, the experiment repeated, and findings recorded. The contention is that experiences with the physical black box experiment support the development of an informal understanding of samples and sampling variability and promote investigation of the effect of repeated samples and larger sample size. This activity is a standard activity in which students are asked to compare two samples and draw a conclusion whether in the population the groups differ (see for example De [START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF]De Vetten et al., in press). One example of such an activity is shown below, where the activity also involves a growing samples activity. It was implemented and evaluated in three groups of first-year pre-service primary school teachers [START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF]. The driving question was whether in the population there is a difference in math attitude between men and women. During three rounds the students answer this question and make predictions of the shape of the distributions of larger samples.

This activity was not well-aligned with a number of design principles, in particular with the visibility of the population (DP 6), the simplicity of the analyses (DP 7), and the possibilities for repeated sampling (DP 8). Indeed, we showed among the pre-service teachers engaged in this activity, there was a strong tendency to restrict attention to descriptive analyses, rather than to engage in inferential reasoning [START_REF] De Vetten | Pre-service teachers and informal statistical inference: Exploring their reasoning during a growing samples activity[END_REF]. The teacher also had strong ideas of the difference between the groups in the population and, therefore, some had problems treating the data as evidence (DP 2). We also contend that, when the population elements are humans, it may be complex to obtain a representative sample, thus leading students to put less trust in the sample and reducing their willingness to make generalizations (DP 4 and 5). On the positive side, the complexity of the sampling process stimulates critical reflection on sample representativeness (DP 5) and the immensity of the population encourages students to take a sample (DP 3). Although the design principles offer suggestions to improve this activity, in particular using repeated sampling, some elements, such as a visible population from which it is easy to generate a representative sample, might not easily be rectified.

Conclusion

Designing ISI activities for primary education that align with all the design principles is not straightforward, as it demands very good knowledge of ISI and educational design. We have suggested two activities that to a large extent align with the design principles. Teachers can effectively use these activities in their classrooms. The design principles can also be used by educational designers to align their own activities with the design principles and elicit IIR more effectively. For example, the activity in which two samples are compared could incorporate repeated sampling, and the averages and measure(s) of dispersion could be provided, so that more time can be spent on IIR. Some of the design principles for ISI education presented in this paper are well-known and have been reported before, but some are based on recent ISI education research. In this way, we aim to provide a comprehensive set of design principles, based on recent evidence, that can be useful for educational designers and are ready to be corroborated by future research. Moreover, the principles are potentially applicable beyond the primary education (teacher) setting, to other settings where the design for activities is necessary when first introducing students to ISI.

  1. Create awareness of inferential claims and of the distinction between sample and population, before engaging students in ISI via statistical investigations De Vetten et al. (2019) and De Vetten et al. (in press) showed that most pre-service teachers appear to only describe the data, rather than engage in IIR. In De Vetten et al. (

Figure 1 :

 1 Figure 1: Pooled sample results in the activity "What is the most frequently used word?"

  While this activity was not designed based on the list of design principles presented in this paper and its effects on students' reasoning has not yet been reported, it appears to align well with the principles. As the number of yellow balls is determined by the educational designer, students cannot have any expectations of the outcome (DP2). Generalization is natural, as it is obvious that the sample results are only a means to make a claim about the population (DP 3). It has a visible and large population and a visible sample (DP 3 and 6). It is easy to obtain a representative sample as yellow and orange balls are mixed and shaking the bottle results in a random sample. Since obtaining a non-representative sample appears to be hard in this activity, it might not stimulate critical reflection on sample representativeness. However, sampling variation may lead to discussing representativeness (DP 4). The context, question, sampling, recording and analyzing are simple (DP 5 and 7). Repeated sampling is achieved by repeatedly shaking the bottle and counting the yellow balls and by combining small group results in a whole class graph, while growing samples is possible by enlarging the viewing window from 40 to 80 balls (DP 8).

Figure 2 :

 2 Figure 2: Black box, used in the activity "How many yellow balls are in the Black Box?" Activity 3: Comparing math attitude of men and women

Figure 3 :

 3 Figure 3: Overview of the activity "Comparing math attitude of men and women"


Raimundo Elicer
email: raimundo@ruc.dk

The role of decision-making in the legitimation of probability and statistics in Chilean upper secondary school curriculum

Keywords: Critical discourse analysis, curriculum, decision-making, probability, statistics

The role of decision-making in the legitimation of probability and statistics in Chilean upper secondary school curriculum

Raimundo Elicer

Introduction

As a scientific and mathematical field of knowledge, it can be argued that the history of probability and statistics is the history of making decisions when uncertain outcomes are to be evaluated. According to [START_REF] Hacking | The emergence of probability: A philosophical study of early ideas about probability, induction and statistical inference[END_REF] historical and philosophical address, the notion of probability comes to answer whether a statement is an opinion or demonstrable knowledge during the middle ages. Later on the 17th century, probability, induction and statistical inference come to define the fair share when a gamble game is interrupted, whether an accused person is to be condemned, to decide the price of annuities and policy regarding pensions, and to take a stand about scientific hypotheses to be true or not. So risk and decision-making define the "logic" of probability [START_REF] Borovcnik | Risk and Decision Making: The Logic of Probability[END_REF]. Risk, as vague as this concept may be, is the way we evaluate decisions under uncertainty beyond the possible impacts of different choices, but also with some weight given to their likelihood. Probability comes to be an attempt to quantify these levels of likelihood.

As for probability and statistics as teaching-learning subjects, [START_REF] Pfannkuch | Reimagining Curriculum Approaches[END_REF] has identified emerging curricular approaches to be addressed in future research. One proposition is to get "more insight into fostering statistical argumentation including learning how to make evidence-based claims in data-rich environments and critically evaluating data-based arguments in diverse media from a statistical literacy perspective" (p. 407, emphasis in original). This trend is grounded in research experiences focused in the complexity and scaffolding of decision-making tasks under uncertainty. Researching the 'critical lens' of decision processes would require different approaches such as action research, phenomenography, and critical discourse analysis (Petocz, Reid, & Gal, 2018, p. 81).

In this article I attempt to join both propositions by investigating the role of decision-making in current and upcoming probability and statistics official curricula, through a version of Critical Discourse Analysis [CDA]. In order to narrow down and have a sense of what is aim for regarding students' future participation as citizens; I make an exemplary analysis that focuses in the Chilean last two grades of secondary mathematics school curriculum.

The general research questions addressed in this article are: (1) what is the role given to notions of decision-making in the Chilean upper secondary school probability and statistics curriculum? And (2) what has changed from the up-to-date version of the curriculum and the upcoming curricular framework for the same subject and grades?

Interpreting roles and spotting changes are broad intentions, so in the following section I address the conceptual framework and methods concerning a version of CDA, in order to understand these questions in a more precise way.

Methodology

Inspired by [START_REF] Fairclough | A dialectical-relational approach to critical discourse analysis in social research[END_REF], I am describing the methodology as an altogether analytical strategy, since "we cannot so sharply separate theory and method" (p. 234) while constructing the object of research. Starting with theoretical generalities, then I provide the necessary conceptual definitions that methods for selection and analysis require.

Fairclough (2010) defines a general methodology for critical research in an interdisciplinary way. In that line, the present discourse analysis dialogues with statistics education research. Key interrelated concepts are social structures, practices and events: "social structures define what is possible, social events constitute what is actual, and the relationship between potential and actual is mediated by social practices" (Fairclough, 2003, p. 223). In this frame language is a social structure. Among its infinite possibilities, choices are made to produce texts as part of social events, mediated by discursive practices, where discourse is understood as semiosis, i.e. the process of meaning-making.

My analytical strategy consists in selecting texts from the upper secondary school Chilean curriculum which refer to decision-making in probability and statistics, and then extracting elements of textual analysis which illustrate broader discursive practices. Textual analysis describes what is in the texts, and the discursive aspect addresses how these elements give meaning.

Selection of texts

Chilean secondary education is defined from grades 7 to 12. These are called 7th-basic and 8thbasic, and from 1 st -middle (9 th -grade) to 4th-middle (12 th -grade)1 . All of these grades have their own study programs, but more general curricular frameworks are available for 7 th -basic to 2 ndmiddle and the one for 3rd and 4th-middle is under construction. The text selection takes into account 3 rd and 4 th -middle curricula, and the aforementioned upcoming curricular framework, summarized in table 1.

For current versions of the curriculum, I will only take common plan programs, i.e. those directed to every student in the scholar system. Excluded programs are those differentiated mathematics studies defined for scientific education. Therefore, the texts are extracted from grades 3 rd -middle and 4 thmiddle study programs.

Document

Label Study programs are divided into units (numbers, algebra, geometry, and data and chance). Within the units of data and chance, I look for sentences which make references to 'decision' (decisión in Spanish), including conjugations of the verb 'to decide.' These sentences will compose the texts.

The same selection applies the upcoming curricular proposal for these last two grades, within the mathematics subject. It does not still have the form of study programs, but it can be considered to be of the same genre, for it means to become the next version of such. It is a general proposal based on a process that involved national experts, international experiences and consultation to civil society participants. The document is published in MINEDUC's website.

Data analysis

The analysis is performed at two levels: textual and discursive.

Among the several possibilities for identifying and categorising elements in the texts, I will focus in three: modality, collocation, and intertextuality. Modality expresses a commitment the author does with truth and necessity, respectively by epistemic and deontic modalities (Fairclough, 2003, p. 165). It can be evidenced -nonexclusively-by the use of modal verbs, such as 'may' (epistemic) and 'must' (deontic). Collocation is the repeated co-occurrence of certain concepts, such as 'hard working' appears as a common binomial in political speeches. Intertextuality is the more or less explicit presence of elements of other texts and their authors' voices. These elements can be dialogued with, assumed, rejected, and so on (pp. 41-42).

Beyond identification of modality, collocation and intertextuality in texts, I will point out the way they are chosen to be expressed as indicators of discursive practices. In particular, I focus in legitimation strategies. Legitimation is the discursive practice of justifying what is made factual in the texts, through reference to authority, value systems or utility, or conveyed through narrative (Fairclough, 2003, p. 98).

Coming back to the research questions, I define the 'role' of decision-making as choices for the use of legitimation strategies as discursive practices. Additionally, I shall identify changes from current to the upcoming version of the curriculum. Characterising this shift is the final stage of the analysis.

Selected texts

As a way of organisation, Chilean study programs are divided into units. Each unit is described having purpose, previous knowledges, key concepts, contents, abilities, attitudes, expected learning outcomes (and their respective assessment indicators), didactical orientations, and suggested activities for each expected learning outcome. The following are the selected texts, and above them are the contexts within the study programs where they are found. All translations from Spanish are made by me as literal as possible.

In the 3 rd -middle grade current curriculum (3M15), 'decision' appears in the form of 'decision tree representations', as a follow up for the goal of understanding the concept of conditional probability.

I will not take it as part of the analysis, since it actually refers to 'probability tree representations.' Then it forms part of the general didactical orientations for the unit:

3 rd -middle. Unit 4: Data and chance.

Purpose 1 Experimental problems are worked with decision tree representations, which enable a bigger of contents and a tool for probabilistic calculations. (3M15, p. 120, emphasis added)

Didactical orientations

2 In this line, it is fundamental that the teacher promotes the development of random thinking, i.e. that students learn to make decisions with evidence in situations of uncertainty. (3M15, p. 123, emphasis added)

In the last grade current study program (4M15) there are actually two units about data and chance, with no mentions of 'decision' in 'data and chance 2,' which includes graphic notions about binomial and normal distributions. In 'data and chance 1,' 'decision' is part of the didactical orientations as in the 3 rd -middle grade. Later on, there is a mention to 'decision' as a comment for teachers when engaging in activities for the learning goal to critically evaluates information:

4 th -middle. Unit 3: Data and chance 1.

Didactical orientations

3 In this unit, it is expected that students critically evaluate information published on the media and internet, from the analysis, interpretation and synthesis of such information, with which they can obtain results about a population considering its size and the variable's distribution; infer conclusions from the mean, variance and standard deviation; and to make decisions grounded in statistically significant information. (4M15, p. 86, emphasis added)

Suggested activities

4 Furthermore, it is important that the teacher promotes contextualized learning so students develop progressively the statistical literacy, which gives them tools for making grounded decisions. (4M15, p. 88, emphasis added)

As for the public consult document for the upcoming curricular framework (34M17), decision first appears as part of the general purposes of the mathematics subject:

Mathematics: Formative purposes 5 In order to achieve the latter, students will work collaboratively in mathematical modelling of situations, to make grounded decisions disciplinary problems, as well as in the interdisciplinary, social, environmental or economic scope. (34M17, p. 49, emphasis added)

And then, utterances about decisions are part of mathematics learning goals in both grades:

Learning goals for 3rd-middle 6 [It is expected from students to be capable of] 3. Making decisions in situations of uncertainty, with information involving dispersion measures, double entrance tables and conditional probabilities. (34M17, p. 52, emphasis added)

Learning goals for 4th-middle 7 [It is expected from students to be capable of] 3. Solving problems in contexts of uncertainty, through the application of the binomial distribution and calculation of probabilities, for decision-making and critical analysis of statistical information. (34M17, p. 52, emphasis added)

Both learning goals have a parallel in the previous texts. In the current study programs (3M15 and 4M15), topics such as dispersion measures, conditional probabilities and the critical analysis of statistical information are covered.

Textual analysis

I shall first identify and describe elements of textual analysis found in the excerpts as to provide input to the following discussion. I focus in modality, collocation and intertextuality.

Modality

Modal forms expressed as 'it is expected from/that students' (3, 6 and 7) can be identified as epistemic modalities, i.e. as expressions of probability and truth, in this case, the expected and not certain to happen. It can be argued that, given the official character of curricula, these are actual expressions of the necessary, falling into the category of deontic modalities.

Explicit deontic modalities are evidenced as 'it is fundamental that' (2) and 'it is important that' (4), and they express necessity for particular promotions of the teachers in order to provoke students' skills for making decisions.

Collocation

References to decisions do not appear alone. A habitual co-occurrence of the substantive 'decision(s)' comes with 'grounded' as company; both as an adjective as in 'grounded decisionmaking' (4) and 'grounded decisions' (5), and as an adverbial form as in 'decisions grounded in…' (3). A similar adverbial accompanying form is 'decisions with evidence' (2) and 'decisions with information' (6). This collocation suggests a reference to a particular type of decisions or decision-making processes, based on quantitative arguments, distinguishable from a mere act of making a choice.

Intertextuality

Texts 2 and 4 can be read in parallel as having the same structure:

In this line, it is fundamental that the teacher promotes the development of random thinking, i.e. that students learn to make decisions with evidence in situations of uncertainty. (2) Furthermore, it is important that the teacher promotes contextualized learning so students develop progressively the statistical literacy, which gives them tools for making grounded decisions. ( 4)

These texts make references to two different concepts, namely 'random thinking' (2) and 'statistical literacy' (4) in a similar way: as notions to be promoted by the teacher, implying students to make grounded decisions. So intertextuality is found within selected texts. Moreover, these concepts are not defined in the documents, but they are traces of another text published by the Chilean Statistics Society as experts' curricular recommendations [START_REF] Araneda | Recomendaciones para el currículum escolar del eje datos y azar[END_REF]. The term 'random thinking' is not found in the literature, rather I presume the texts intend to refer to 'statistical' [START_REF] Garfield | How students learn statistics revisited: A current review of research on teaching and learning statistics[END_REF] or 'probabilistic thinking' (Chernoff & Sriraman, 2014). Both 'statistical thinking' and 'statistical literacy' are described as core answers to the section 'Why teach statistics?' (pp. 11-17) referencing a paper published in the International Statistical Review [START_REF] Garfield | How students learn statistics revisited: A current review of research on teaching and learning statistics[END_REF]. Overall, texts 2 and 4 are referring to authority, introducing notions without further explanation, which are developed within national and international statisticians associations.

Discussion: Discursive practices

Within Faircoulgh's model, texts are part of social events, and they are signs of broader discursive practices. In the following I point out legitimation strategies evidenced by preceding elements of the textual analysis.

Legitimation

Making decisions appears as a way of justifying the teaching and learning of statistics in the curricula. The texts fulfil the purpose of not only saying what and how to teach and learn, but also why and for what. Fairclough (2003, p. 219) claims that much of the legitimation of a social ordersuch as the inclusion of particular knowledge in the official curriculum-is textual. In particular, the texts show three of the legitimation strategies identified by [START_REF] Van Leeuwen | Legitimation in discourse and communication[END_REF], namely through mythopoesis, rationalisation, and authorisation.

Mythopoesis or legitimation conveyed through narrative is evidence by the collocation of 'decision' with 'grounded' (3, 4 and 5), and similar accompanying adverbial forms such as 'with evidence' (2) and 'with information' (6). This strategy is stable through the texts and makes the case for justifying probability and statistics, since not just any kind of decision is included, but only those rooted in data and mathematical rationality. It allows a steady association between such school subjects, and rational choice.

Merely associating probability and statistics to rational decisions is not enough, it is still necessary to justify the teaching. Aforementioned deontic modalities are evidence of legitimation through rationalisation, which refers to utility. Within the current curriculum (3M15, 4M15) teachers' promotions are 'fundamental' (2) and 'important' (4) for students to make grounded decisions. The upcoming proposal (34M17) provides a similar rationalisation in the form of 'in order to achieve the latter' and 'to make grounded decisions'( 5), but this time it is mediated by the students' action, namely their collaborative work in mathematical modelling situations (5). In a way, this change represents another shift in the agency from teachers to students in the teaching-learning process.

I have already pointed out that traces of an authorisation strategy are found in texts from the current curricula (3M15, 4M15), through the inclusion, without further definition, of 'random thinking' and 'statistical literacy'. What is worth highlighting is the fact that this strategy is not found in the upcoming curricular framework (34M17). This shift resonates with the bottom-up nature of the latter document, where civil society plays a bigger role in the justification of curriculum. This alteration is coherent with another broader mention to 'decision' in the diagnosis chapter of the same document, where students "demand protagonism in decision making and aspire to contribute to solve problems in the world they live in, such as poverty eradication, climate change and sustainable development." (34M17, p. 13)

Concluding remarks

Critical discourse analysis allows us to see texts as part of social structures mediated by discursive practices. In this article I analyse Chilean upper secondary school written curricula as evidence of legitimation strategies, showing that study programs not only describe what is to be taught, but also why. Decision-making appears as a key element of such legitimation, by establishing a narrative of probability and statistics linked to grounded, evidence-based and rational choice.

The analysis illustrates signs of change towards a justification which relies less in professional statisticians and educational researchers as authorities, and more into students' own learning activities as decision makers. This shift is part of the wish for a bigger role in decision processes which go beyond the disciplinary scope.

This article addresses a justification problem in mathematics education research [START_REF] Niss | Goals of mathematics teaching[END_REF]. Future works should address whether it is possible connect decision-based motivations and curricular contents, and how this can be approached in teaching practice. Both possibility and implementation problems are part of the broader PhD project this article is embedded in.

This work was funded by the doctoral scholarship CONICYT BCH/DOCTORADO 72170530.

Sources

Ministerio de Educación (2015a

Table 1 : Sources of selected texts

 1

	Year Curriculum Author

  ). Matemática, Programa de Estudio, Tercero medio [Mathematics, Study Programme, Third middle]. Santiago: Unidad de Currículum y Evaluación, Ministerio de Educación de Chile. Retrieved from: http://www.curriculumenlineamineduc.cl/605/articles-34361_programa.pdf

Not to be confounded with middle school, as in between primary and secondary school. In Chile, enseñanza media (middle education) is the bridge between enseñanza básica (basic education) and educación superior (higher education).

Ministerio de Educación (2015b). Matemática, Programa de Estudio, Cuarto medio [Mathematics, Study Programme, Fourth middle]. Santiago: Unidad de Currículum y Evaluación, Ministerio de Educación de Chile. Retrieved from: http://www.curriculumenlineamineduc.cl/605/articles-34362_programa.pdf Ministerio de Educación (2017). Propuesta curricular para 3º y 4º medio: Documento de consulta pública [Curricular proposal for 3 rd and 4 th -middle: Public consultation document]. Santiago: Unidad de Currículum y Evaluación, Ministerio de Educación de Chile. Retrieved from: http://basesdelfuturo.educarchile.cl/wp-content/uploads/2017/03/Propuesta_Curricular-Consulta-P%C3%BAblica.pdf


Daniel Frischemeier

Aisling M Leavy
email: aisling.leavy@mic.ul.ie

The challenge of constructing statistically worthwhile questions

Keywords: Statistical inquiry, teacher education, primary teaching, posing statistical questions

Recent approaches to statistics education situate the teaching and learning of statistics within cycles of statistical inquiry. This research focuses on the first stepthe preparation of prospective teachers to pose statistical questions. We report on an investigation of prospective elementary teachers in Ireland (n=118) and Germany (n=40) as they design statistical questions. Support was provided through tutorials, think-pair activities, peer-feedback and expert-feedback. We describe the features of statistical questions posed, identify obstacles and difficulties experienced when posing questions and evaluate the effectiveness of both peer and expert feedback.

Introduction

A data analysis cycle such as the PPDAC [START_REF] Wild | Statistical thinking in empirical enquiry[END_REF]) is a structure implemented in statistical inquiries across nearly all age levels. It consists of the phases: problem, plan, data, analysis and conclusion. The first phase, posing a problem (i.e., statistical question), is a critical prerequisite to stimulate a deep and rich analysis of data. It is also critical as the quality of the questions posed informs the types of data collected, determines the representations used and influences the interpretations that can be made. Arnold (2013, p. 19) makes an important distinction between two fundamental types of questions: investigative questions and survey questions. Investigative questions are "the questions to be answered using data" (p. 19), for example 'What patterns are evident in social media use by teenagers?' In contrast, survey questions are "questions which are asked to get the data" (p. 19), for example 'How much time do you spend on social media at the weekend?' One typical misconception is posing survey questions when being asked to pose investigative questions. We concentrate primarily on the generation and development of investigative questions (which we refer to as statistical questions). Our primary research questions are: What is the quality of initial statistical questions posed by prospective teachers and how does the quality develop across several phases (first version of the question, question revised after peer feedback, question revised after expert feedback)?

Literature review

In the literature review we consider different types and categories of statistical questions. [START_REF] Biehler | Statistische Kompetenz von Schülerinnen und Schülern -Konzepte und Ergebnisse empirischer Studien am Beispiel des Vergleichens empirischer Verteilungen[END_REF] distinguishes between the number of variables included in statistical questions. An example of a one-variable question is: "What is the distribution of the variable height?" An example of a two-variable question is: "In which way do boys and girls differ in respect to the variable height?" For two-variable questions, [START_REF] Konold | Students analyzing data: Research of critical barriers[END_REF] distinguish between three different types. The first type, those investigating the relationship of two categorical variables (Are boys or girls more likely to play handball?), are termed categorical ⨉ categorical. Those investigating the relationship of two numerical variables (Is there a relation between hours spent on social media and school grades?) are termed numerical ⨉ numerical. Questions in the numerical ⨉ categorical category are referred to as group comparison questions -for example: "Do those with a curfew tend to study more hours than those without a curfew?" (Konold et al, 1997, p. 7).

These categories index descriptive features of statistical questions. A more evaluative approach is taken by [START_REF] Frischemeier | Stepwise development of statistical literacy and thinking in a statistics course for elementary preservice teachers[END_REF] who have distinguished different qualities and developed a rating system for statistical questions. In this rating system, questions that take into account two variables are rated higher than those that take into account only one variable. In the subset of onevariable questions they distinguish questions leading to a yes/no answer ("Do 60% of the pupils have a mobile phone?", p. 759), questions asking for a specific value ("How many pupils have a personal computer?", p. 759) and more general questions referring to the entire distribution ("How is the variable "mobile phone" distributed?", p. 759). In the subset of two-variable questions a similar categorization is made. Yes/No questions involving two variables are for example "Is there a difference between boys and girls in their time spent on computer use?" (p. 759) and questions aimed at working out differences in group comparison situations are for example "In which regard does computer use differ between boys and girls?" (p. 759). An example of a more sophisticated and complex statistical questions (called open and complex questions) involving two variables is "which differences exist between boys and girls in regard to their leisure time activities?" Makar and Fielding-Wells (2011, pp. 349-350) provide specific characteristics of questions for statistical investigations such as "interesting, challenging, and relevant", "statistical in nature" and "ill-structured and ambiguous" and Arnold (2013, p. 110-111) identified six fundamental criteria for what makes a good investigative question: (1) the variable(s) of interest is/are clear and available, (2) the population of interest is clear, (3) the intent is clear, (4) the question can be answered with the data, (5) the question is one that is worth investigating, that is interesting, that has a purpose, and (6) the question allows for analysis to be made of the whole group. For Criterion 1 this means that the variables are described in a clear way (with regard to the specific situation: summary, comparison, relationship) and have been correctly identified from the actual survey question. Criterion 2 focuses on the population of interest and whether learners focus on individuals, a sample or the population. Another important Criterion 3 is whether the question is clear with regard to whether it is a summary, comparison or relationship question (p. 111). Consideration of whether the question can be answered with the given data is tackled in Criterion 4. Criterion 5 deals with purpose and whether the information arising from answering the question is useful to someone. Finally, Criterion 6 investigates whether the question allows an analysis with regard to a local view (single points, single aspects) or a global view on distributions (characteristics like center, spread, skewness).

Empirical studies examining learners´ reasoning when generating statistical questions reveals that learners across all ages face problems generating adequate statistical questions [START_REF] Arnold | Statistical investigative questions -An enquiry into posing and answering investigative questions from existing data[END_REF][START_REF] Frischemeier | Stepwise development of statistical literacy and thinking in a statistics course for elementary preservice teachers[END_REF][START_REF] Pfannkuch | Developing statistical thinking in a secondary school: A collaborative curriculum development[END_REF]. In their work with secondary schools students, Pfannkuch and Horring (2004, p. 208) observed that students frequently forgot to look back to the original questions and use it to guide their discussion of the data. Switching focus to the teachers, [START_REF] Pfannkuch | Developing statistical thinking in a secondary school: A collaborative curriculum development[END_REF] stated that the "teachers tended to pose narrowly framed statistical questions according to a template" (p. 208). [START_REF] Frischemeier | Stepwise development of statistical literacy and thinking in a statistics course for elementary preservice teachers[END_REF] investigated the development of the quality of statistical questions posed by primary preservice teachers while engaged in a PPDAC-cycle while enrolled in a course on statistical thinking. One fundamental conclusion of [START_REF] Frischemeier | Stepwise development of statistical literacy and thinking in a statistics course for elementary preservice teachers[END_REF] was that the quality of a statistical investigation depends on the statistical question that was posed and subsequently stimulates the investigation. Thus poor statistical questions requiring only a yes/no answer or including only one variable, lead to short and non-sophisticated statistical explorations. Interestingly they concluded that despite the provision of feedback (peer and expert) during two stages of the study, the quality of statistical questions did not improve in a considerable way. Our study reported in this paper investigates how to support prospective primary teachers design statistical questions that motivate a cycle of statistical inquiry suitable for upper grade elementary children.

Methodology

Irish participants (n=118) were 3 rd years in a 4-year undergraduate degree. They were enrolled in a 12-week compulsory module on the teaching of statistics and probability, the first 8 weeks focused on statistics. They were taught in groups of 30-40 students, once a week for 60 minutes (in Autumn 2017). German participants (n=40) were 1 st years in a 3.5-year undergraduate degree. They were enrolled in a 14-week compulsory course on the teaching of geometry and statistics (in winter term 2017/2018), the last three weeks focused on statistics. They met twice a week for a 90-minute lecture and a 60 minute seminar. Courses in both settings were designed around the cycle of statistical investigation which focused attention on the problem, plan, data, analysis and conclusion (PPDAC) components [START_REF] Wild | Statistical thinking in empirical enquiry[END_REF]) and its' implementation in primary classrooms. A core assessment task was the design of a statistical question that would stimulate and drive a cycle of statistical investigation targeted for use with upper elementary grade children (ages 10-12). The content and sequence of the courses was similar across settings and focused broadly on: designing statistical investigations (the PPDAC cycle), types of data, samples and populations, representing and analysing data distributions using graphical visualisations, calculating and interpreting summary statistics (measures of central tendency and variability), and comparing distributions of data.

Study Design

This paper reports on the first part of the investigative cycle: the design of statistical questions. Following instruction on the cycle, participants were placed in pairs and informed that they would be required to design a statistical question, and an ensuing statistical investigation, which they would revise and develop over the course of the semester. The three-phase design consisted of:

Phase 1 (Think): Participants worked in pairs to design a statistical question (SQV1) that promoted comparison of two numerical data sets. Course content aligned with this phase, provided examples of strong and weak research questions and explored statistical components relating to question design. Phase 1 culminated with the design of the first version of the statistical questions (SQV1).

Phase 2 (Peer-feedback): Phase 2 provided peer feedback on initial questions. It was envisioned that providing and receiving feedback would develop skills in identifying desirable features of statistical questions. Each pair received the statistical questions (SQV1) of another pair and used a set of criteria (developed by the authors, Table 1) to guide analysis and feedback. Pairs recorded feedback in the form of questions or statements which were then returned to the original pairs. Phase 2 culminated with participants presenting a revised statistical question (SQV2).

Phase 3 (Expert-feedback): The goal of phase three was the construction of the final research question (SQV3) based on the provision of 'expert' feedback to all participants. Expert feedback was provided during a plenum discussion and was generic in nature in that it identified the general shortcomings, strengths and desirable features that emerged in the revised questions (SQV2). Arising from the plenum, participants revised and improve their statistical questions (SQV3).

Look at the question

Is it meaningful? Will the question sustain interest and curiosity of primary children? Is the intent clear and unambiguous? Think about the variables of interest Is the variable described clearly? Is the variable available/possible to measure? Look at the relationship between the question and the data it will generate Can the question be answered with a simple "yes/no" response [avoid these type of questions]? Will the question generate quantitative data (i.e., numbers)? Will the question motivate a focus on two data sets? Does it promote group comparison of data? Look at (or imagine) the data Can you answer your question with the given data? Is there sufficient data collected to answer the question? Is there sufficient variability in data collected (is there the potential for a wide range of possible data values)?

Data & Data Analysis

The statistical questions at the phases SQV1-SQV3 constituted the data. Hence our analysis focused on these questions. We identified characteristics of questions posed, rated these characteristics and compared their quality across the three phases. A structuring qualitative content analysis method [START_REF] Mayring | Qualitative content analysis: theoretical background and procedures[END_REF] was used for rating statistical questions. We constructed our categories mainly using a deductive approach but also took into account inductive elements from our data as a kind of a "mixed approach" (as it is explained in Kuckartz, 2012, p.69). Due to the fundamental theoretical work (see especially [START_REF] Arnold | Statistical investigative questions -An enquiry into posing and answering investigative questions from existing data[END_REF] carried out on the generation of statistical questions (categories, criteria, etc.) we firstly and mainly used a deductive approach to use and adapt categories and constructs established from existing research studies [START_REF] Arnold | Statistical investigative questions -An enquiry into posing and answering investigative questions from existing data[END_REF][START_REF] Biehler | Statistische Kompetenz von Schülerinnen und Schülern -Konzepte und Ergebnisse empirischer Studien am Beispiel des Vergleichens empirischer Verteilungen[END_REF][START_REF] Frischemeier | Stepwise development of statistical literacy and thinking in a statistics course for elementary preservice teachers[END_REF][START_REF] Konold | Students analyzing data: Research of critical barriers[END_REF]).

The quality of statistical questions was determined using the four question components (Table 1) as an analytical frame. A structured rubric consisting of coding rules and key examples was developed and it carefully guided the assignment of a precise code to each question component. A development process was used to arrive at the components and their sub components (see Column 1, Table 2); however, its description is beyond the scope of this paper. For illustrative purposes, we will focus only on the development process for the first component "Look at the question" since it considers general characteristics of statistical questions like Question_meaningful, Question _interest, Question_clear, Question_variables and Pop_interest.

For instance Question_meaningful considers whether the question is meaningful (connected to a context, e.g., "Do female students spend more time doing homework than male students?") and Question_interest examines whether the question is interesting for primary school students from an affective point of view (e.g., "Do children with pets spend fewer time with their friends than children without pets?"). Both categories are based on Arnold's Criterion 5 (Arnold 2013, p. 111).

The category Question_clear is based on the third criterion of Arnold (2013, p. 111) that "from the question it needs to be clear if it is a summary, comparison or relationship question". Thus we determined whether each statistical question was posed in a clear and unambiguous way. A counterexample for a non-clear statistical question found in our data is for example "What is the probability that a boy in your class has glasses?" This sample question does not refer to a statistical investigation and is neither a summary nor a comparison or relationship question.

The category Question_variables is based on the categorizations and work of [START_REF] Konold | Students analyzing data: Research of critical barriers[END_REF] and [START_REF] Biehler | Statistische Kompetenz von Schülerinnen und Schülern -Konzepte und Ergebnisse empirischer Studien am Beispiel des Vergleichens empirischer Verteilungen[END_REF]. This category checks which kind of variables (categorical, numerical) are embedded in the statistical question and whether they are implemented in a single form (question with regard to a distribution of a single categorical or numerical variable) or as a combination in the statistical question (categorical vs. categorical, numerical vs. categorical, numerical vs. numerical).

The category Pop_interest is based on Arnold´s (2013) second criterion (p. 111). In this category we are interested in the language used in the statistical question and whether the focus is on an individual case, on a sample or on a population. As also mentioned in [START_REF] Arnold | Statistical investigative questions -An enquiry into posing and answering investigative questions from existing data[END_REF], the use of "a" (e.g., a boy) might indicate an individual focus, the use of "the" (e.g., the boys) might refer to cases in the sample and more general the use of "boys" might indicate a focus on a population.

Results

Looking at the question: In Phase 1, the majority of questions posed were meaningful and of interest to children (Rows 2 & 3, Table 2); however, the clarity of these questions posed a greater challenge for Irish participants (see Row 4). Improved clarity was evident across subsequent phases indicating the benefit of peer and expert feedback. By the end of the study, the majority of Irish and Germans (90%, 85%) posed questions involving two variables. A greater number of two-variable questions were posed by Germans in Phase 1 (75%) compared to Irish (44%) (see Row 5, Table 2). Furthermore, the large number of two-variable questions posed in Phase 1 by the Irish (44%) remained stable in Phase 2 (42%) indicating that peer feedback seemed to have little immediate effect. Although there was less room for variation in the German cohort, there was also very little change across the phases. Similarly (see Row 6, Table 2), while the population of interest was clear in 90% of final research questions for both groups, Germans were more successful in identifying the population of interest in Phase 1 (75% versus 58%). The lack of clarity in Irish participants' initial questions may have contributed to the lack of clarity in identifying the population of interest.

Variables in the question: Initially there was a lack of clarity in describing variables with less than 50% being successful in Phase 1 (see Rows 8 & 9, Table 2). For example, examination of the question (Pair 11, Germans) reveals the lack of clarity in the variable 'sporting activity' in terms of what will be measured and the units of measurement. The feedback phases were successful for both groups in improving the clarity of variables (see Pair 11 below). German participants still struggled for clarity in variable description at the end of the study as compared to their Irish counterparts. Are there differences in the sporting activity of boys and girls?

[Pair 11, SQV1] Are there differences in the weekly sporting activities in hours of boys and girls? [Pair 11,SQV3] Relationship between the data and the question: Exploring the relationship between the posed question and the data generated provided insights into how the question opened, or closed, opportunities for exploration of data. Compared with other categories, this category was quite stable over the three phases (see Row 11, Table 2). For both groups, the number of simple survey questions showed a small decrease across the phases. Correspondingly the number of more sophisticated questions showed small increases across phases indicating improvement in the quality of questions posed across the study. Examples of questions categorised as sophisticated in Phase 3 are below.

By how much do the shoes sizes of boys and girls in Class 5 differ? [Pair 3, SQV3, Irish]

Based on your examine of the sugar content of the fruit juices and minerals provided here, explain which drinks are more suitable for consumption. [Pair 1, SQV3, Irish]

Looking at the data: There was no requirement for German participants to submit data sets, hence we report only on Irish data for 3 of the 4 sub-components (see Rows 13-15, Table 2). In Phase 3, 95% were able to answer the question with the given data. Similarly, 95% collected sufficient data to answer the question. When the data sets were examined, the majority (92%) had sufficient structure or variability in the data (that means that a wide range of possible values are given). Both groups demonstrated a steady increase in questions that supported a global view across both phases indicating the benefit of both peer and expert feedback (see Row 16 in Table 2).

Component: Look at the question

Table 1 : Criteria used to provide feedback on statistical questions

 1

Discussion

When provided with appropriate structured support, prospective teachers can develop the skills and understandings to develop rich statistical questions. However, developing good statistical questions is particularly complex and requires considerable support. Multiple considerations are required. Question considerations require attention to the meaningfulness, interest, and clarity of questions in addition to the variables and populations elicited by questions. Consideration of the variable requires attention to the clarity of the variables described and the measurement of those variables. Consideration of the relationship between the question and the type of data generated is also required in an effort to ensure the data support subsequent analysis. Finally, the data themselves require consideration. Attention must be paid to the ability to answer the question from the amount and type of data generated, whether there is sufficient variability in the data and that the question supports a global view of the data. It is not surprising that some of these categories posed lesser or greater difficulty. While participants showed great expertise in addressing the meaningfulness and interest of questions and in identifying the variables available to measure, in contrast, all participants struggled initially to provide clarity when describing the variables. While the purpose of the study was not to compare the Irish and German participants, it is noticeable that German participants performed better across many of the categories in Phase 1. It appears that Irish participants invested their energies in the broader contextual dimensions of questions and in framing questions resulting in the relative neglect of the necessary statistical and content dimensions of the questions posed. The benefits of the peer-feedback process (Phase 2) are not clear from this study. On the one hand by comparing performance across Phases 1 and 2, peer review brought about improvements in some aspects of question design. It appears to have supported the revision and improvement of some question aspects, most notably in clarifying the question and in identifying the population of interest. For other components, peer-feedback did not seem to support any substantial gains in question quality particularly when selecting and identifying the number of variables embedded in the question and in bringing about improvements in the question type. The effect of feedback when generating and developing statistical questions needs further research into the future.


Maynor Jiménez-Castro
email: maynorjc@correo.ugr.es

Pedro Arteaga
email: parteaga@ugr.es

Carmen Batanero
email: batanero@ugr.es

Statistical Graphs Semiotic Complexity, Purpose and Contexts in Costa Rica Primary Education Textbooks

Keywords: Statistical graphs, reading level, semiotic complexity, textbooks, primary education

The aim of this research was analyzing the way in which statistical graphs which are introduced in the new curricular guidelines for Costa Rica primary education are introduced in the textbooks, and compare the results with previous research in other countries. We analyze the distribution of the graph complexity, purpose and context in 167 activities related to statistical graphs selected from the two most widely used textbooks series. Results suggest a non-uniform distribution in both series and suggest ways to enhance the activities proposed in the textbooks.

Introduction

Statistical graphs are a main tool to represent information in the media and professional life and acquiring graphical competence is needed for citizens to manage in the information society. This need led Costa Rica to increase the relevance of statistical graphs in the primary education (M.E.P, 2012), following suggestions such as those of the GAISE project [START_REF] Franklin | Guidelines for assessment and instruction in statistics education (GAISE) report: a pre-k-12 curriculum framework[END_REF]. More specifically, statistical graphs are included in the Costa Rica curriculum in Primary Education as follows:

• In second grade, bar graphs are introduced, with the main objective that the student makes a simple reading of the information and in third grade, the students are expected to construct such graphs and interpret them, using the mode, maximum and minimum. • In the fourth grade, the student must interpret information in dot plots and it is proposed to incorporate the spreadsheet as a support tool in graphic representations. • In the fifth grade, pie charts are included and in the sixth, the line graphs to visualize trends in data series. The graphic comparison of two or more data groups is also suggested.

These goals can only be achieved with a correct teaching of the topic and the use of adequate textbooks. There are however no available empirical studies providing evidence that Costa Rica textbooks follow these guidelines. The aim of this paper is to analyze the graphs included in Costa Rica textbooks and use this information to help teachers to organize the teaching of the topic.

Background Theoretical framework

Statistical graphs are complex semiotic objects, as suggested by [START_REF] Bertin | Semiologie graphique[END_REF], who remarked that interpreting a graph requires first the isolated interpretation of each element, such as the title or scales and finally as a global interpretation of the whole graph. Moreover, [START_REF] Arteaga | Relating graph semiotic complexity to graph comprehension in statistical graphs produced by prospective teachers[END_REF] analyzed the semiotic activity involved in the construction of graphs by prospective teachers, and suggested that different graphs may vary in semiotic complexity, according to the mathematical objects needed in this process. Specifically, the authors described four semiotic levels in statistical graphs that will be used in our analysis: S1. Representing isolated data. Graphs classified in this level only represent some isolated data without considering the whole set from which the data were extracted. S2. Representing a list of data one by one without any attempt to building a distribution. The list of data or a data set is represented in the same order in which the data are located in the list with no use of the numerical order. There is no grouping of similar values of the variable or computation of frequencies. Consequently, although in this graph the idea of variable is used, the distribution is absent. In Figure 1 we present an example. S3. Representing a data distribution. These graphs include the representation of a distribution, with values and frequencies for each value; For quantitative variables, the order of the variable values in the graph axes (if used) is the ordinary numerical order (see figure 2) S4. Representing several distributions on the same graph. At the highest level more than one distribution is represented on the same graph. (an example is given in Figure 3) Following ideas by [START_REF] Kosslyn | Graphics and human information processing[END_REF], we also considered statistical graphs as a resource that can be used with different purposes:

 Analysis; when the main goal is using the graph to identify or discover many features that are hidden in the unorganized data set; for example discovering the mode or the asymmetry of the distribution.  Communication; when the graph is used to transmit to another information about the data and its relationships in an efficient way.  Construction; in traditional teaching, an additional purpose is providing procedures to help students learn how to build a correct graph.

Another aspect analyzed is the context of the graph, using the classification established in the PISA studies, which is described in OECD ( 2016), where it is suggested that the mathematical performance of an individual can be influenced by the context in which he develops his problem situation. In this sense, the context in which the collection, organization and representation of the data are circumscribed may affect the understanding of the graph. This classification contemplates the following contexts:

 Personal context: corresponds to situations in the scope of personal activities of the child, his family or peer group. An example of this context is presented in Figure 3, where the graph shows the activities preferred by a group of boys and girls.  Social context: these contexts correspond to the problems where aspects of the social sphere are addressed, beyond the personal life of the child; for example, the neighborhood, or city. They can correspond to voting systems, public transport, government, public policies, demography, publicity, national statistics or economy. Gambling is also included in this context because it is widely used in society.  Labor or school context: consist of those situations where the problem addresses aspects of the world of work: school, work, employment, production, sale, etc.  Scientific context: this context is related to problems in the field of science and technology, and includes aspects such as: climate, ecology, medicine, space science, genetics, etc.

Previous research

Research analyzing different content in textbooks is increasing today, since these books are a main didactical tool for teachers and students, and constitute an intermediate stage between the official curricular guidelines and the teaching implemented in the classroom [START_REF] Herbel | From intended curriculum to written curriculum: Examining the" voice" of a mathematics textbook[END_REF]. The textbook selected usually provides the main basis why the topic is taught [START_REF] Shield | Assessing the potential of mathematics textbooks to promote deep learning[END_REF].

Our research is based in other studies that analyzed statistical graphs in the textbooks in other countries, in particular, Díaz-Levicoy, Batanero, [START_REF] Díaz-Levicoy | Gráficos estadísticos en libros de texto de Educación Primaria: Un estudio comparativo entre España y Chile. BOLEMA[END_REF], who analyzed the statistical graphs included in three series of Spanish textbooks and three other Chilean series of Primary Education, and compared their results with the curricular orientations. The authors studied the type of graph, level of semiotic complexity, and reading levels implicit in the activity. They conclude the greater presence of the bar chart with little weight of other graphs included in these curricula. Regarding the level of reading, the most frequent were the intermediate ones.

Method

For this study, the curricular orientations referring to the education of statistical graphs in Costa Rica have been analyzed, as well as the two most frequently used series of books in Costa Rica; Asociación Libros para Todos (AL) and Editorial Santillana (S), using content analysis. The sample of publishers was directed or not probabilistic, because we selected the textbooks most spread in the schools of the country (see Appendix). A total of 167 different graphs were analyzed and classified according to the following variables:

 Semiotic complexity level: [START_REF] Arteaga | Relating graph semiotic complexity to graph comprehension in statistical graphs produced by prospective teachers[END_REF] observed in their study as a higher semiotic level in the graphs constructed by a sample of prospective teachers implied a higher level of reading of them by the prospective teachers. Therefore, semiotic levels are related to the reading levels of a graph, that is an important ability related to good levels of graphical competence (Wu, 2004).  Purpose of the graph: [START_REF] Cazorla | [END_REF] shows that statistical graphs are an instrument of both data analysis and information transmission, for all this, considering Kosslyn's ideas we classify the graphs according to their purpose.  Context of the graph: [START_REF] Monteiro | Investigating the interpretation of media graphs among student teachers[END_REF] observed how the interpretation of the graphs mobilizes knowledge and feelings that affect their understanding and how the context of the graph influences the interpretation of it, showing the importance of considering the context of the graphs when interpreting them.

Each graph was analyzed according to these three variables, Maynor, one of the authors of this paper, classified the different graphs as part of his PhD studies, and this process was supervised by the other two authors. The variables analyzed on this paper were previously used in different studies carried out by Díaz-Levicoy and published in relevant national and international journals (Díaz-Levicoy et. al, 2016, Díaz-Levicoy, Giacomone and Arteaga, 2017).

Results

Graph semiotic complexity

Overall the most frequent semiotic complexity level was level 3 (see

Purpose of the graph

The graph is considered according to [START_REF] Kosslyn | Graphics and human information processing[END_REF], as a resource that is used for analysis, communication and in traditional teaching, for purely constructive purposes. This variable has not been taken into account in previous research, so our results represent an original contribution. The activities analyzed in the textbooks tend to use the graphs for analysis purpose with 60.7% of the total graphs used for this purpose, while only 24.4% are used for communication purposes, as shown in Table 3. The constructive purpose does not seem to be so relevant to both publishers and only represents 15% of the total of activities analyzed.

When analyzing the results by grade (Table 4), we observe that that the most common purpose in the first grade is communicating information and little analysis and graph construction are encouraged. In the remaining grades, the purpose of analysis predominates, although there is always an interest in graphic construction, in particular until grade 3. This variable has not been taken into account in previous investigations. The relationship of the context with school grade is shown in Table 6, where the School/work place context dominates along second to the fifth grades, mainly due to situations related to school. In the first year, personal and social contexts are more frequent, in coincidence with Mingorance's research (2014) while social and scientific contexts are more frequent in the sixth grade. Scientific context is scarce in the first school years, but increases after the fourth grade.

Purpose

Conclusions

The two series of books analyzed, concentrate their activities at the semiotic complexity level 3 that is in the representation of a data distribution in coincidence with Díaz-Levicoy et al. ( 2016) results.

The results of the analysis showed that the main goal of graphs in the textbooks is analysis, followed by communication with only 18% graphs oriented only to learn how to build the graph. In our opinion more activities oriented to build a graph should be included because previous literature show the importance of graph construction as part of a good graphical competence (Wu, 2004) and because this ability is recommended in Costa Rica's curricular guidelines (M.E.P., 2012). There were also variation between the different grades with personal contexts more frequent in first grade and educational context more frequent in grades 2 to 5. In grades 1 to 3 scientific contexts hardly appear, while starting from grade 4 they constitute about a quart of all the graphs.

These results are useful for teachers who should take into account the relevance of variables as semiotic complexity, purpose and context in statistics education and select adequate activities when introducing their students to statistical graphs, because previous research show the importance of taking into account these variables in the teaching of statistical graphs [START_REF] Arteaga | Relating graph semiotic complexity to graph comprehension in statistical graphs produced by prospective teachers[END_REF][START_REF] Cazorla | [END_REF][START_REF] Monteiro | Investigating the interpretation of media graphs among student teachers[END_REF][START_REF] Díaz-Levicoy | Gráficos estadísticos en libros de texto de Educación Primaria: Un estudio comparativo entre España y Chile. BOLEMA[END_REF] Textbooks are a teaching and learning resource with an important tradition within the classroom, because they provide support to teachers and students throughout the instructional process. It is therefore necessary to study how the textbooks present statistical topics as a first step to suggest possible improvement in their content and to check that they follow the curricular guidelines.

Project EDU2016-74848-P and group FQM126 (Junta de Andalucía).

Figure 1 :

 1 Figure 1: Example of graph of semiotic complexity S2 (Source: S4, p. 278)

Figure 2 :

 2 Figure 2: Example of graph of semiotic complexity S3 (Source: A5, p. 146)

Figure3:

  Figure3: Example of graph of semiotic complexity S4 (Source: S6, p. 263)

Table 1

 1 78.6% of activities classified at level 3. Similarly, at the S editorial, 62.2% of the activities were classified as level 3 and 30.6% and 7.2% as level 2 and 4, respectively. This general data coincide with the results obtained from the semiotic analysis of the graphs in Díaz-[START_REF] Díaz-Levicoy | Gráficos estadísticos en libros de texto de Educación Primaria: Un estudio comparativo entre España y Chile. BOLEMA[END_REF], where 58.6% of graphs analyzed corresponded to level 3. However, the second most frequent semiotic level proposed in the Spanish study was level 4 with 22.3%, while in Costa Rican textbooks it is level 2 with 24.0%. Contrary to the Spanish study's results, in the Costa Rica textbooks since the highest level of semiotic complexity (S4) represents only 8.4% of graphs.

	Semiotic Complexity AL (n=56) S (n=111)
	S2	10,7	30,6
	S3	78,6	62,2
	S4	10,7	7,2

) since, 67.7% of the graphs represented a distribution; 24.0% of graphs represented a data list (level 2) and only 8.4% corresponded to the highest level of semiotic complexity, representing two or more distributions in the same graph. In the AL editorial, the percentage of activities at level 2 and level 4 was 10.7%, in contrast to

Table 1 : Frequency and percentage of graphs according semiotic complexity and editorial

 1 Regarding the distribution of semiotic complexity by school year, in Table2we show the increment of semiotic complexity with school grade. In the first grade only level 2 activities were found, while in the sixth grade graphs in all the semiotic complexity levels are proposed. Moreover, in grades 2 to 5, complexity level 3 is dominant and only in the 6th grade the most common graphs are those of level 4. In Díaz-Levicoy et al. (2016) research, semiotic complexity level 4 graphs are included from grade 2, and are the most common semiotic level at fifth and sixth grades. This includes a clear difference in the way in which activities are addressed in Spanish and Costa Rican textbooks, where representations of several distributions on the same graph are scarce in Costa Rica.

	School grade

Table 2 : Percentage of graphs by semiotic complexity and school grade

 2

Table 3 . Frequency and percentage of graphs according purpose and editorial

 3

	School grade

Table 4 . Percentage of graphs by purpose and school grade 4.3 Contexts The

 4 contexts of graphs are summarized in Table5, where the most common context used in AL is primarily the school /workplace while in S both this context and personal contexts involve 33, 3%, of the activities each. Both publishers point strongly to the school / work context and to a lesser extent to scientific situations.

	Context	AL (n=56) S (n=111)
	Personal	3,6	33,3
	Social	28,6	18,9
	School /work place 48,2	33,3
	Scientific	19,6	14,4

Table 5 . Frequency and percentage of graphs according context and editorial

 5

Table 6 . Percentage of graphs by context and school grade

 6