Kyeong-Hwa Lee

Yeong Ok

Chong
email: yochong@gin.ac.kr

What is taught and learnt on confidence interval? A case study

Keywords: Anthropological theory of the didactic, didactic transposition, confidence interval, sample size

This study investigated what is taught and learnt with regard to relations of confidence interval and sample size when students were facilitated to reason about the graph of sampling distribution of sample means. For this aim, a teaching experiment was designed and implemented by researchers and teachers in the model of co-learning inquiry. Based on anthropological theory of the didactic, the praxeological analysis was conducted. Observed similarities and differences between the taught and learnt praxeology were discussed.

Introduction

The concept of confidence interval is essential in inferential statistics. However, many students are in difficulty with understanding confidence interval [START_REF] Hagtvedt | Teaching confidence intervals using simulation[END_REF]. In particular, some research reported students' confusing on the relationship between confidence interval and other related concepts such as sample size. For example, [START_REF] Fidler | Teaching confidence intervals: Problems and potential solutions[END_REF] disclosed students' misconceptions that the width of confidence interval increases as sample size increases or is not affected by sample size. However, little research was carried on the teaching and learning of confidence interval (Pfannkuch et al., 2012).

To understand confidence interval, sample size, and their relationships, opportunities should be given for students to engage in mathematical and statistical reasoning [START_REF] Delmas | A comparison of mathematical and statistical reasoning[END_REF]. This study investigated what is taught and learnt regarding relations between confidence interval and sample size when students were facilitated to reason about the sampling distribution of sample means [START_REF] Chance | Reasoning about sampling distributions[END_REF]. For this aim, a teaching experiment was conducted in an upper secondary classroom. The framework of the Anthropological Theory of the Didactic (ATD) was used in the analysis [START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF]. This paper reports on some preliminary results of the analysis.

Theoretical Background Anthropological Theory of the Didactic

ATD is a theoretical approach characterized by its institutional perspective on mathematical knowledge. This view builds on the notion of didactic transposition [START_REF] Chevallard | Didactic transposition in mathematics education[END_REF], which assumes that knowledge in classroom setting originates in the scholarly institution and is transformed as it adapts to different institutions. Bosch and Gascón (2014, p. 70) proposed four main institutions related to school didactic system (Figure 1). Our study focuses on the transposition between taught knowledge of teachers to learnt knowledge of students. According to [START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF], researchers should take their own epistemological stance on the knowledge they address to avoid bias in the analysis. The stance is formulated as a Reference Epistemological Model (REM) and is expressed by a praxeology. We briefly introduce our REM on the concept of confidence interval below.

REM on confidence interval

The idea of confidence interval is founded on Frequentism, as opposed to Bayesianism [START_REF] Vanderplas | Frequentism and bayesianism: a python-driven primer[END_REF]. To the frequentist, a probability is defined as the relative frequency of occurrence of a repeatable event in the long run. In this sense, confidence intervals with % confidence level mean that we would expect % of the confidence intervals under repeated sampling to contain the fixed value of the parameter.

To illustrate the relationship between confidence interval and sample size, we further need another probability theory in our REM. The central limit theorem (Dekking et al., 2005, p. 197) implies that holds for the sample mean , the population mean , the (known) standard deviation of population , and sufficiently large sample size . From this property, we obtain a formula, where is the quantile of the standard normal distribution. [START_REF] Behar | Experts and students' conceptions regarding confidence intervals[END_REF] introduced two logically equivalent versions of this formula:

where refers to the margin of error. Formula (1) can be interpreted as the probability of the parameter being in the interval , called confidence interval, equals to . Contrasting to formula (1), formula (2) represents that the probability of the sample mean being in the interval equals to . The probability theory shows that the probability of lying in the interval is equal to the integral of probability density function of over the interval (Dekking et al., 2005, pp. 57-58). The aforementioned meanings of confidence interval, confidence level, and probability equations (1), (2) constitute the theory in our REM.

Building on the theory , we can consider two techniques and their technology about the type of tasks of finding the relationship between the width of confidence interval and sample size. Based on the formula (1), the first technique refers to applying the equation , which is often found in curriculum materials (e.g. Lee et al., 2011, p. 181). By proportional reasoning [START_REF] Post | Proportionality and the development of prealgebra understandings[END_REF], which includes interpreting of the algebraic representation of proportionality ( ), it is explained that the increasing sample size decreases width of confidence interval. Another technique can be found on the formula (2). refers to comparing the area under graphs of sampling distribution within for different sample sizes. This technique requires justification by reasoning about statistical or probabilistic idea including "describing what a sampling distribution would look like for different sample sizes based on shape" ( ) and "interpreting areas under the sampling distribution curve as probability statements about sample means". (

) (Chance et al., 2004, p. 301). Also, the explanation about relations between and confidence interval is included in the technology [START_REF] Behar | Experts and students' conceptions regarding confidence intervals[END_REF]

(

). These kinds of technology should be integrated to justify as follows. Since the spread of sampling distribution decreases as sample size increases, the margin of error satisfying the area under the sampling distribution over equals to become smaller as sample size increases and, accordingly, confidence interval become narrower as sample size increases.

Method

This study is qualitative research performed through a case study of a designed teaching experiment [START_REF] Yin | Case study research: Design and methods[END_REF]. We choose our design-based research methodology based on the model of co-learning inquiry [START_REF] Jaworski | Insiders and outsiders in mathematics teaching development: The design and study of classroom Activity[END_REF]. In this model, teachers are treated as partners of researchers rather than passive participants of research, and teachers and researchers collaboratively design, implement, and evaluate lessons in the iterative process. The main power of co-learning inquiry model is that the result of the research can contribute to "sustainable new practices" of teachers, since "teachers engage in their own purposeful activity" in this model (ibid., p. 19).

Participants in this study were a high school teacher from a public upper secondary school in Korea and his 35 students. Around the topic of statistical inference, five lessons were designed and implemented. Audio and video recordings of the lessons and students' worksheets were collected.

In this study, we analyzed the last lesson of which learning objective was to understand the relations among confidence interval, confidence level, and sample size. We emphasize that students were encouraged to use graphs of sampling distribution to explain these relations. The collected data were analyzed by praxeological analysis [START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF]. The focus of the analysis was on the didactic transposition between taught knowledge to learnt knowledge about relations between confidence interval and sample size. Based on our elaborated REM on confidence interval (Table 1), we analyzed the four elements of the taught and learnt praxeology and revealed their similarities and differences. The designed activities were interpreted as type of tasks in ATD sense. The teacher's utterance and the teaching materials were mainly analyzed to figure out the other components of the taught praxeology. On the other hand, worksheets of students and their audio recordings were analyzed to find out the learnt praxeology. In particular, students' answers on the following task were analyzed to describe technique, technology, and theory parts of the learnt praxeology: Consider three confidence intervals with 90% of confidence level. Size of each sample is 4, 25, and 100. Array these intervals according to their width. Explain your answer using sampling distributions.

Type of tasks

: To find the relations between the width of confidence interval and sample size Technique : To apply (Lee et al., 2011, p. 181) : To compare the area under graphs of sampling distribution function within for different sample sizes Technology (for ): Proportional reasoning [START_REF] Post | Proportionality and the development of prealgebra understandings[END_REF] (for ): Reasoning about the impact of sample size on graphs of sampling distribution [START_REF] Chance | Reasoning about sampling distributions[END_REF] (for ): Graphical interpretation of probability statements about sample means [START_REF] Chance | Reasoning about sampling distributions[END_REF] (for ): Explanation about relations between and confidence interval [START_REF] Behar | Experts and students' conceptions regarding confidence intervals[END_REF] Theory : The meaning of confidence interval and confidence level (VanderPlas, 2014); the meaning of and [START_REF] Behar | Experts and students' conceptions regarding confidence intervals[END_REF]

Theory

The meaning of confidence interval with confidence level; The meaning of and ( ) The meaning of each statement together with confidence interval and confidence level was explained by the teacher in the sense of Frequentism.

Teacher:

As you can see, among 100 confidence intervals, only 5 do not contain and the other 95 contain (the population mean). … It means that the level of confidence is 95%.

Grounded on the meaning of probability, the relations between and confidence interval were explained. Graphic representation of the probability statements and was used with the explanation.

Teacher:

Here's the summary. The probability of being contained between and is equivalent to the probability of population mean contained in this one (

). [PowerPoint slides that teacher used with this explanation. Numbers in slides were chosen from an exercise without contextual meanings.]

Although the technology and theory part of ground the robust foundation for its potential practice, this praxeology was 'in a vacuum' in that "what kind of problems it can help to solve" is not known (Barbé et al., 2005, p. 237).

Table 3 describes components of another taught praxeology .

Type of tasks To compare the probability of sample mean being in a given range when graphs of sampling distribution with different (hidden) sample sizes are given Technique To compare the shape of graphs and area under graphs of sampling distribution within a given range ( )

Technology

Reasoning about the impact of sample size on graphs of sampling distribution

Graphical interpretation of probability statements about sample means ( ) Theory Empty

Table 3: Components of the taught praxeology was organized around a type of tasks, to compare the probability of sample mean being in a given range when graphs of sampling distribution with hidden sample size (4, 25, 100) are given. The technique of consisted of two subsequent sub-techniques. The first sub-technique was to compare the shape of sampling distribution function to figure out hidden sample sizes. This sub-technique was justified by the technology , which was the discourse about the impact of sample size on graphs of sampling distribution. The following sub-technique was to compare the area under the graphs of sampling distribution. This sub-technique required interpreting areas under the sampling distribution curve as probability statements about sample means ( ).

One may note that the technique is slightly different from in that did not require the explanation about confidence interval. Hence, the technology and theory related to the concept of confidence interval were missing in . The theory about probability sentences was also void in , which made the theory part of empty.

What is learnt?

The praxeology of the taught and learnt praxeology are summarized in Table 4. Although the taught praxeology and did not contain as a technique, some students (8%) used to explain the relationship between the width of confidence interval and sample size. For example, a student explained that his technique was justified "since the width of confidence interval equals to , the width becomes smaller as increases". This kind of justification corresponds to proportional reasoning .

Excluding non-replied answers, remaining 30 students (85%) applied However, their technology differed in degree (Figure 2). 13 students ( only relied on the reasoning about the impact of sample size on the shape of sampling distribution (

). The other 17 students (48%) could integrate with when they justified their technique. These two technological elements were also found in the taught praxeology . However, , which was another technological element of the taught praxeology , was missing in the learnt praxeology even though was essential to fully justify technique of the learnt praxeology. The meaning of confidence interval and confidence level (in the sense of Frequentism) was implicit in students' worksheets or in their dialogues. Moreover, the expression or was hardly dealt with in students works, which leads to the theory part of learnt praxeology lacking the meaning of those expressions.

Discussion

In this study, we analyzed what is taught and learnt in a didactic setting which encouraged reasoning about sampling distribution. Some notable similarities and differences between them were observed.

Firstly, technology and were both taught and learnt in the classroom setting. Agreeing with Chance et al. (2004, p. 300) that and are kinds of reasoning "students should be able to do with their knowledge of sampling distributions", we argue that essential reasoning about sampling distribution could be integrated in the teaching of relationships between confidence interval and sample size. Furthermore, this kind of reasoning would contribute to the understanding of the relationship between confidence interval and sample size since the notion of sampling distribution and its representation are closely related with each of those concepts.

Secondly,

as well as theory underneath it were found in the taught praxeology but were entirely implicit in the learnt praxeology. These technology and theory were only contained in the local know-how . Similar phenomenon was also appeared at [START_REF] Barbé | Didactic Restrictions on the Teacher's Practice: The Case of Limits of Functions in Spanish High Schools[END_REF] in the case of teaching of limits of functions. We infer that the reason of the incongruity between the taught and learnt praxeology was partly because the related practice of the taught praxeology was unknown. Hence, we expect that another taught praxeology with a concrete task whose technique is justified by and its theory would derive more strict discourse in learnt praxeology.

Finally, with was not taught but learnt in this lesson. This result shows the potential that diverse techniques, including and , could be employed in the activity of searching relations between confidence interval and sample size. Discussing different techniques and their technology, different local praxeology of students could be connected and integrated into regional praxeology [START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF].

Figure 1 :

 1 Figure 1: The didactic transposition process (Bosch & Gascón, 2014, p. 70) In ATD, mathematical activities within each institution are modeled by the notion of praxeology. A praxeology consists of four intertwined elements, type of tasks ( ), technique ( , technology ( , and theory ( . Type of tasks denotes similar problems which are solved with certain technique. The technique is produced, justified, explained, or reasoned by the discourse named technology, and the technology, in turn, is justified by theory.

-

  To compare the shape of graphs and area under graphs of sampling distribution within a given range ( -To compare the area under graphs of sampling distribution within for different sample sizes ,

[Figure

  Figure 2: Examples of with (left) and with and (right)

  The research was funded by the Korean Ministry of Education and the National Research Foundation of Korea (NRF-2016S1A5A2A01024273)

Table 1 : REM on confidence interval Result What is taught?

 1

	Two taught praxeology ,	were revealed in the lesson. The components of	are summarized
	in Table 2.
	Type of tasks Empty
	Technique	Empty
	Technology	Graphical interpretation of probability statements about sample means (	)
		Explanation about relations between	and confidence interval
		(	)

Table 2 : Components of the taught praxeology was

 2 a local know-how which consisted of only technology and theory parts. The theory part of included the meaning of probability statements of sample means, and

Table 4 : Components of the taught and learnt praxeology

 4


Susanne Podworny
email: podworny@math.upb.de

Connecting context, statistics and software for understanding a randomization test: a case study

Keywords: Statistical inference, randomization test, simulation

Drawing statistical inference with the help of simulations has gained prominence in statistics education as well as introducing statistical inference with randomization tests. This paper describes some selected results of a case study of preservice primary teachers who attended a short learning trajectory on statistical inference with randomization tests. It will be shown how the participants address the context, statistics and software level when conducting a randomization test with software and how the conscious linking of the three levels can support the learning process and help to understand certain elements of a randomization test.

Introduction

Statistical reasoning is a cornerstone on which statistical practice is based. In almost all areas of daily life, data and thus also conclusions drawn from data play an important role. It is impossible to imagine statistical practice without computer-supported evaluations and methods. In many fields, like in industry, medicine, or politics, decisions are increasingly being made on the basis of data. When looking at a newspaper, a television report or an entry on the World Wide Web, interested citizens increasingly come across the keywords "A study has shown ..." or "The effect of X is Y". However, it is often suggested that the results and interpretations delivered in this way are by no means certain, as is often suggested in the media.

There are two big areas in inferential statistics: parameter estimation and hypothesis testing and there are two kinds of inferences that may be drawn: generalizations beyond a given sample and causation for a given treatment for a given sample. The latter should be an integral part of stochastic education, as called for in the basic article by [START_REF] Wild | Statistical Thinking in Empirical Enquiry[END_REF].

Statistics education should really be telling students something every scientist knows, 'The quest for causes is the most important game in town.' It should be saying: 'Here is how statistics helps you in that quest'. (Wild & Pfannkuch, 1999, p. 238) One statistical method for drawing causal inferences is the randomization method. About ten years ago, [START_REF] Cobb | The Introductory Statistics Course: A Ptolemaic Curriculum?[END_REF] strongly advocated introducing the logic of inference via randomization tests. A randomization test is a non-parametric method that allows easy access to inferential reasoning via computer-based simulations. Through simulations, nearly no formulas or calculations are needed, and this is one of the main reasons for the easiness of this method. The "core logic of inference" [START_REF] Cobb | The Introductory Statistics Course: A Ptolemaic Curriculum?[END_REF] can be in the center and conclusions are possible even for data from small or nonrandom samples.

From this perspective, a learning trajectory on inferential reasoning with randomization tests was developed by the author to be implemented in an existing course on statistics and probability for preservice primary school teachers to complete the general statistics education of these preservice teachers. Eight weeks after the learning trajectory a case study was conducted with participants who conducted a randomization test for a given problem with computer-based simulations. The objective of this paper is to better understand the reasoning process of these learners. Selected findings of this study will be presented in this paper. [START_REF] Cobb | The Introductory Statistics Course: A Ptolemaic Curriculum?[END_REF] gave an impetus to rethink the introduction to inference statistics, especially at college level, and to get a new introduction with randomization tests. Some curricula for introductory statistics courses emerged since then (e.g. [START_REF] Rossman | Concepts of statistical inference: Approach, scope, sequence and format for an elementary permutation-based first course[END_REF][START_REF] Tintle | An active approach to statistical inference[END_REF]Zieffler & Catalysts for Change, 2013) and some shorter learning trajectories for introducing inferential reasoning were created (e.g. [START_REF] Budgett | Dynamic visualizations for inference[END_REF][START_REF] Frischemeier | Design and exploratory evaluation of a learning trajectory leading to do randomization tests facilitated by TinkerPlots[END_REF]. All these teaching proposals are based on the use of computer-based simulations and focus on the logic of inferential reasoning rather than on calculations. In addition, there are some few empirical studies focusing on special aspects of the process when learners conduct a randomization test.

Literature review

A common factor of all these learning units is that they use a plan or a scheme to structure the reasoning process. A compilation of elements of these schemes by the author has resulted in nine elements that can be considered central when conducting a randomization test. The first element is the random allocation of experimental units to groups. Explaining this is a core component in understanding the underlying design [START_REF] Pfannkuch | Experiment-to-causation inference: Understanding causality in a probabilistic setting[END_REF]. To find possible explanations for observed differences between two groups of an experiment [START_REF] Pfannkuch | Experiment-to-causation inference: Understanding causality in a probabilistic setting[END_REF] is the second element. One explanation can be that the treatment is effective, another explanation can be that the observed differences are due to the random allocation of units to the groups. The third element is to pose or reconstruct the research question for the experiment [START_REF] Wild | Statistical Thinking in Empirical Enquiry[END_REF]. Analyzing the observed data and identifying a difference between the two groups is the fourth element [START_REF] Biehler | Preservice teachers' reasoning about uncertainty in the context of randomization tests[END_REF][START_REF] Rossman | Concepts of statistical inference: Approach, scope, sequence and format for an elementary permutation-based first course[END_REF]. The fifth element is setting up the null model with the null and alternative hypotheses [START_REF] Biehler | Preservice teachers' reasoning about uncertainty in the context of randomization tests[END_REF]. Transferring the null model to a simulation model is the sixth element [START_REF] Biehler | Preservice teachers' reasoning about uncertainty in the context of randomization tests[END_REF][START_REF] Lee | Simulation approaches for informal inference: Models to develop understanding[END_REF]. The seventh element is the production of test statistics and the sampling distribution [START_REF] Lee | Simulation approaches for informal inference: Models to develop understanding[END_REF]. The eighth element is to identify the p-value [START_REF] Biehler | Preservice teachers' reasoning about uncertainty in the context of randomization tests[END_REF][START_REF] Rossman | Concepts of statistical inference: Approach, scope, sequence and format for an elementary permutation-based first course[END_REF]. And the last element is drawing possible conclusions [START_REF] Cobb | The Introductory Statistics Course: A Ptolemaic Curriculum?[END_REF]. Each of these elements has its own difficulties, but one difficulty across all elements is to combine the levels of context, statistics and software.

In our own research [START_REF] Biehler | Preservice teachers' reasoning about uncertainty in the context of randomization tests[END_REF], we identified three levels when working on a randomization test: the context level, the statistics level, and the software level. Following up on this, [START_REF] Noll | TinkerPlots model construction approaches for comparing two groups: Student perspectives[END_REF] conducted a case study with eight learners who created different models in TinkerPlots for the "dolphin therapy problem" (Noll & Kirin, 2017, p. 219) and analyzed what influenced the learners' reasoning about models. One of their results was, that "Students did not spend much time discussing attribute labels or what type of devices they wanted to use" (Noll & Kirin, 2017, p. 232). This may be an explanation for the reported result that "the concept of no difference between two groups is difficult to operationalize into a TinkerPlots model" (Noll & Kirin, 2017, p. 235). Naming the (simulated) attributes was identified by Noll and Kirin as an important aspect of the modeling process, because it can act as a bridge between the context and the tool (Noll & Kirin, 2017, p. 236). But they also state, that they "see this work as in its infancy in that we need more research focused on why students create the models" (Noll & Kirin, 2017, p. 240). They also noted that it is not reasonable to separate the three worlds like they interpreted the work of [START_REF] Biehler | Preservice teachers' reasoning about uncertainty in the context of randomization tests[END_REF]. At this connection between context, statistics and software this paper ties in.

The learning trajectory "Inferential reasoning with randomization tests"

A learning trajectory "Inferential reasoning with randomization tests" for preservice primary school teachers was created in the sense of the design based research paradigm [START_REF] Cobb | Design Experiments in Educational Research[END_REF]. The aim of the trajectory was to introduce preservice teachers at university to the logic of inference like in the sense of [START_REF] Cobb | The Introductory Statistics Course: A Ptolemaic Curriculum?[END_REF] in a short period of time. Some design ideas were adapted from [START_REF] Pfannkuch | Experiment-to-causation inference: Understanding causality in a probabilistic setting[END_REF], who successfully introduced inferential reasoning in a short learning sequence to a similar target group. The learning trajectory was designed for the end of an existing course on elementary statistics at Paderborn University. The existing course consisted of three modules data analysis, combinatorics, and probability and used the software TinkerPlots from the beginning. The new learning trajectory was designed for three 90 minutes sessions.

In the first session the students should get in touch with a first example of a randomization test with a significant p-value [START_REF] Rossman | Concepts of statistical inference: Approach, scope, sequence and format for an elementary permutation-based first course[END_REF] and with the logic of inferential reasoning as a continuation of group comparisons [START_REF] Makar | Comparing Two Distributions: Investigating Secondary Teachers' Statistical Thinking[END_REF], which were already discussed in the data analysis module. The second session focused on the performance of a randomization test by the students. First, a hands on simulation with pen and paper was to be carried out and then transferred to a computer-based simulation, like proposed for example by [START_REF] Gould | Enhancing conceptual understanding with data driven labs[END_REF]. In the third session, the nine elements (see literature review) were to be discussed in detail and possible difficulties addressed.

To get insights into the cognitive processes of students conducting a randomization test with TinkerPlots, an interview study was designed as part of the Ph.D. study of the author.

Methodology

The research question for the case studythat is focused on in this paperis How do the participants relate the three levels context, statistics and software to each other?

The problem given to students was an adaption of the "Fish oil and blood pressure task" of [START_REF] Pfannkuch | Experiment-to-causation inference: Understanding causality in a probabilistic setting[END_REF]. This task contains real data from a medical experiment with 14 volunteers on the question of whether fish oil supplements have a blood pressure-lowering effect compared to normal oil supplements. The blood pressure of the participants was measured at the beginning and after four weeks and the blood pressure reductions for the two groups were recorded like in The observed data was visualized on the worksheet for the students like in Figure 1 and accompanied by the statement and the question "The observed data are shown in Figure 1 and show that the blood pressure reduction in the fish oil group tends to be greater than those in the 'normal oil' group". What can be concluded here?" The interview study took place as a semi structured interview [START_REF] Mayring | Einführung in die qualitative Sozialforschung: eine Anleitung zu qualitativem Denken[END_REF] with a large part of the participants working independently on the task. Questions of the participants were allowed in this phase, and some help could be given by the interviewer. This part was followed by interview questions relating to the individual steps and arguments of the working phase.

A total of six participants working in three pairs took part in the interview study. All of them attended the whole course and the learning trajectory, so this was their educational background. Participation in the study was voluntary, so participants cannot be considered representative of the 236 participants in the whole course. The participants' age ranged between 21 and 25 years, being in the third/fourth semester of University. The interviews took place eight weeks after the course and none of the participants used TinkerPlots in between. The conversations were recorded together with the screen activities and then transferred into a transcript. This transcript included the conversations as well as the other activities, with and without software.

The analysis of the transcripts was carried out by means of interaction analysis [START_REF] Krummheuer | Grundlagen und Beispiele Interpretativer Unterrichtsforschung[END_REF]. First, the transcripts were divided into 15 interaction units using methods of linguistic conversation analysis [START_REF] Egbert | Introduction to conversation analysis with examples from audiology[END_REF]. The next step was to reconstruct the solution process interpretatively with interaction analysis. A detailed turn-by-turn analysis took place here and was discussed with other interpreters. For each of the nine elements (see literature review), the level (context, statistics, software) at which the participants communicated was examined. As a third step, the use of the software in the solution process was examined in detail. Summary and comparative analyses between the pairs formed the fourth step.

Results

The levels at which the participants communicate regarding the nine elements are described below.

Table 2 shows which level the participants address linguistically when they talk about and work on the various elements. Table 2 shows a clear pattern. The first two elements are addressed by all three pairs both at the context level and at the statistics level. These are related to the observed data and no reference to the software is necessary. The research question is formulated by the participants at the contextual level. The corresponding interaction units are very short, because this element is obviously clear for all participants. Like the first two elements, the observed data are discussed at the contextual and at the statistical level. As the data had already been evaluated on the worksheet, there was no need to work with the software and therefore no need to talk about it.

The null model element includes the formulation of the null hypothesis and the alternative hypothesis. Communication takes place both at the context level and at the statistics level. However, the null hypothesis formulated by Rebecca and Selina reveals only a small contextual reference. These two formulate "Random group allocation is the cause of observed differences". Such a formulation is almost arbitrarily applicable to different situations, since a reference to the direct context is not recognizable. The formulation of the other pairs for the null hypothesis are "There is no difference in blood pressure reduction in the effect between both supplements" (Fabia and Laura) and "It does not matter which oil is taken to lower the blood pressure but the results are due to the random allocation" (Mandy and Alisa). Both show a clear connection to the context. Perhaps this explains why Rebecca and Selina do not end up drawing conclusions in context. However, Fabia and Laura do not draw these either, although their null hypothesis is clearly related to the context.

The only element that is addressed by all three pairs on all three levels is the development of the simulation model in TinkerPlots. For this element, all three pairs needed additional help from the interviewer on a technical level, for example how to copy all the observed values into a box of the sampler. The same need of help could be observed for the next two elements for which TinkerPlots had to be used. But, unlike in our earlier research [START_REF] Biehler | Preservice teachers' reasoning about uncertainty in the context of randomization tests[END_REF], renaming the attributes of the sampler and of the new groups for the new allocation was a topic for all three pairs. An interesting dialogue for this came about between Rebecca and Selina. This shows that the concept of random allocation developed linguistically by linking the three levels (that was identified as a difficulty by [START_REF] Noll | TinkerPlots model construction approaches for comparing two groups: Student perspectives[END_REF]). The sampler built by Rebecca (R) and Selina (S) so far is shown in Figure 2. This sampler is built correctly with all observed blood pressure values in the first box and independently these will be drawn in one of two new groups. TinkerPlots labelled the two stacks in the second device in "a" and "b" automatically. Selina first renames the first stack in "fish" like in Figure 2, but then retains the letters a and b and the following dialogue arises.

R: Do we have to call them that, because we actually divide them up in such a way that we can do it later, for example/. So I'm not quite sure, but it could be that we/.

S: /That's right, we don't really need it. R:

/put a fish oil person with a normal oil person/. S:

/You're right/. R: /Together. I think a and b is actually quite neutral. S:

You can actually take a and b, because then that's more. We want to prove now that it has nothing to do with the oil anymore. R:

Yes, exactly we will say it/. S:

/So you have to/ R:

We are representatives of the null hypothesis. S:

Exactly, because then in the one group possibly always both come, a person with fish oil and one with normal oil. It is now only about lowering blood pressure. R:

Yes, that's exactly what it can be/. S: /Yes. R:

That you have maybe five of them (points to the fish oil group in the plot in TinkerPlots) and then (.) the rest of them (points to the "normal oil" group in the plot). Well, it can be that way/. S: /You're right. (laughs) Well, I would spontaneously agree with you. (laughs) In the end, Rebecca and Selina decide not to rename the new groups.

For the next two elements, sampling distribution and p-value, none of the three pairs have a conversation at context level. They operate only on the software level and on the statistics level.

Conclusions are correctly formulated at the statistical level by all three pairs, which is a good result for the whole process. The written formulations are as follows:

Rebecca & Selina: Research hypothesis can be accepted and null hypothesis can be rejected, but with slight uncertainty Fabia & Laura:

We reject the null hypothesis and accept the research hypothesis at a P-value of 1%. Mandy & Alisa: The null hypothesis can be rejected. Blood pressure reduction depends on the type of oil.

However, only Mandy and Alisa apply their conclusion to the treatment carried out. A deeper look at the transcripts shows that the greatest difficulties are with the last element formulating contextual conclusions, and that a statistical formulation seems to be the easiest for the participants.

Discussion

Like [START_REF] Noll | TinkerPlots model construction approaches for comparing two groups: Student perspectives[END_REF], who did not find a separation of the three levels context, statistics, and software to be beneficial, connecting the three levels seems to be extremely useful in the present work. Only when the levels are differentiated a conscious connection between them can take place and thus promote the learning process. An example for a helpful connection can be seen in the selected dialogue between Rebecca and Selina, in which they clarify the meaning of random assignment and its implementation in the software.

For the process of understanding it can be concluded that the dialogue between Rebecca and Selina is one of the most important parts of the conversation, because here the two make it clear that during the simulation (level of software) a random reallocation (level of statistics) of the persons with the blood pressure values (level of context) happens independently (level of statistics) of the oil consumed (level of context). Such discussion would have been desirable for further settings at the software level, as it would enhance understanding by linking the three levels. Unfortunately, very few such dialogues have taken place throughout the case studies. Even though the data are taken from a real scientific experiment, the participants seem to have been given too little opportunity to deal with the context in detail. This would be a better prerequisite for a deeper interpretation of the context.

However, the context caused a confrontation with the data, as reported by [START_REF] Gonzáles | Mathematics teachers' conceptions of how to promote decision-making while teaching statistics: The case of Japanese secondary school teachers[END_REF], in order to promote understanding of the need of decision-making in situations with uncertainty. The demands of [START_REF] Cobb | The Introductory Statistics Course: A Ptolemaic Curriculum?[END_REF] were implemented here in a learning trajectory and the learners were successfully introduced to statistical inference.

Figure 1 :

 1 Figure 1: Visualized data on the participants' worksheet

Figure 2 :

 2 Figure 2: Sampler created by Rebecca and Selina

Table 1 .

 1

	Fish oil group	8	12	10	14	2	0	0
	"normal oil" group	-6	0	1	2	-3	-4	2

Table 1 : Data on blood pressure reduction after four weeks

 1

Table 2 : Overview of the levels at which the pairs communicate during the randomization test process

 2

	Element	Context	Statistics	Software
	Random alloca-	Rebecca & Selina	Rebecca & Selina	-
	tion	Fabia & Laura	Fabia & Laura	-
		Mandy & Alisa	Mandy & Alisa	-
	Possible explana-	Rebecca & Selina	Rebecca & Selina	-
	tion	Fabia & Laura	Fabia & Laura	-


Caterina Primi
email: primi@unifi.it

Maria Anna Donati

Sara Massino

Elisa Borace

Edoardo Franchi

Kinga Morsanyi

Measuring probabilistic reasoning: the development of a brief version of the Probabilistic Reasoning Scale (PRS-B)

Keywords: Assessment, item response theory, short form, probabilistic reasoning, validity

The assessment of probabilistic reasoning skills is important in various context. The aim of this study was to develop an abbreviated, open-ended version of the Probabilistic Reasoning Scale (PRS-B), applying Item Response Theory (IRT). The analyses based on the open-ended version of

the 16-item scale suggested the exclusion of seven items that did not perform well in measuring the latent trait. The resulting 9-item scale (PRS-B), which included highly discriminative items, covered a wider range of the measured trait than the original scale and it showed high measurement precision. Concerning validity, the results showed the expected correlations with numerical skills, math anxiety, and statistics achievement. In conclusion, the PRS-B can be is a shorter, open-ended version of the PRS that maintains excellent reliability and validity.

Introduction

The ability to think about uncertain outcomes and to make decisions on the basis of probabilistic information is relevant in many fields (e.g., business, medicine, politics, law, and psychology). However, people often struggle with interpreting probability information (see [START_REF] Chernoff | Commentary on Probabilistic Thinking: Presenting Plural Perspectives[END_REF]. In particular, when they solve probability problems, people tend to make some typical mistakes instead of applying formal analytic procedures (see e.g., Gilovich, Griffin, & Kahneman, 2002). These mistakes, including heuristics, biases and misconceptions, make it difficult for people to interpret and critically evaluate probabilistic information, understand data-related arguments, and make reasoned judgments and decisions (e.g. Garfield & del Mas, 2010[START_REF] Morsanyi | The effects and side-effects of statistics education: Psychology students'(mis-) conceptions of probability[END_REF]Pratt & Kazak, 2017).

Regarding education in probabilistic reasoning, previous studies demonstrated the difficulty of improving probabilistic reasoning ability or, rather, a resistance to eliminate probabilistic reasoning biases once these had consolidated (for a summary of the literature, see e.g., Gilovich et al., 2002; for adolescents, see [START_REF] Klaczynski | A dual-process model of adolescent development: Implications for decision making, reasoning, and identity[END_REF]. Nevertheless, training activities that target specific difficulties can reduce some misconceptions (e.g. [START_REF] Fong | Immediate and delayed transfer of training effects in statistical reasoning[END_REF]. Additionally, in statistic courses, which have been incorporated into a wide range of school and university programs in many countries, students often encounter difficulties and, eventually, many of them fail to pass the exams. Some authors [START_REF] Konold | Reconnecting data and chance[END_REF] pointed out that one of the reasons students have difficulties in learning basic data analysis stem from a lack of basic understanding of probability and in grasping the fundamental ideas of probability.

Given the important role of probabilistic reasoning skills in various contexts, it is important to accurately measure probabilistic reasoning skills, in order to improve these skills. This, in turn, can lead to other beneficial outcomes, such as better academic achievement. For this reason, recently we developed a new scale, the Probabilistic Reasoning Scale -PRS [START_REF] Primi | Measuring probabilistic reasoning: The construction of a new scale applying item response theory[END_REF] to measure probabilistic reasoning ability, which can be used to identify people with difficulties in this domain. The PRS was developed with a focus on some typical biases and fallacies that are known to lead to incorrect responses. Indeed, the PRS is a useful tool in educational contexts to identify individuals who could be targeted by specific interventions.

However, the scale has the limitation that the items measure probabilistic reasoning skills most precisely in the lower ability ranges and it is not ideal for the measurement of ability levels around the mean. Additionally, the scale is composed of 16 items and it does not seem appropriate for large, multivariate studies in which many tests and scales need to be administered at the same time. Indeed, with research questions becoming increasingly complex, and involving a growing number of constructs, such as when investigating the role of probabilistic reasoning in decision making (e.g., [START_REF] Schiebener | Decision making under objective risk conditions-a review of cognitive and emotional correlates, strategies, feedback processing, and external influences[END_REF] or in risk taking [START_REF] Donati | Prevention of problematic gambling behavior among adolescents: Testing the efficacy of an integrative intervention[END_REF], shorter scales potentially offer added value.

In sum, the aim of this study was to develop an abbreviated, open-ended version of the PRS. To achieve this goal, we used Item Response Theory (IRT) analyses, which make it possible to select items that offer the most information in measuring the targeted underlying trait, that is, probabilistic reasoning. Specifically, IRT has potential benefits in shortening a scale because it makes it possible to evaluate the amount of information provided by each item of the scale for each trait level on the trait dimension through the Item Information Function (IIF). In other words, if the amount of information is large, the trait level can be estimated with high precision, if the amount of information is small, the trait cannot be accurately estimated. Thus, on the basis of item information, it is possible to select items that convey the higher amount of information along the entire range of the measured trait. Through the selection of items that perform better and assure adequate information along the different levels of the trait, a well-performing shortened scale can be obtained.

Additionally, IRT provides the Test Information Function (TIF), which evaluates the precision of the test at different levels of the measured construct instead of providing a single value (e.g., Cronbach's α) for reliability (Embretson & Reise, 2000). More precisely, the TIF provides information on how accurate the test is at estimating a trait along the whole range of trait scores. The more information the test provides at a particular trait level, the smaller the error associated with ability estimation, and the higher the local reliability. Since the TIF is generated by aggregating the IIFs, in general, longer tests will measure an examinee's attribute with greater precision than shorter tests. Nonetheless, in the IRT framework, item selection can be done ensuring that the TIF of the shortened scale maintains an adequate amount of information along the trait continuum, which is similar to the original scale.

Finally, given that a short form of a test should meet the same standards of validity as the full form [START_REF] Smith | On the sins of short-form development[END_REF], validity measures were administered to provide evidence that the abbreviated scale still measures probabilistic reasoning adequately. Thus, we expected to replicate the pattern of relationships established for the construct as measured by the long form of the test. In particular, we investigated the relationship between the PRS-B, numerical skills, anxiety and self-confidence related to numbers, and statistics achievement.

Methods

Participants

The participants in this study were 316 psychology students (mean age = 20.53; SD = 2.9; 76% female) enrolled in an undergraduate introductory statistics course at the University of Florence in Italy. All students participated on a voluntary basis.

Materials

The original Probabilistic Reasoning Scale (PRS, [START_REF] Primi | Measuring probabilistic reasoning: The construction of a new scale applying item response theory[END_REF] consists of 16 multiple-choice questions. In this study, the scale was administered in an open-ended format, which required some minor changes in the wording of the items. The items include questions about simple, conditional and conjunct probabilities, and the numerical data are presented in frequencies or percentages (e.g., "A ball was drawn from a bag containing 10 red, 30 white, 20 blue, and 15 yellow balls. What is the probability that it is neither red nor blue?").

The Mathematics Prerequisites for Psychometrics (MPP; [START_REF] Galli | Measuring mathematical ability needed for "nonmathematical" majors: the construction of a scale applying IRT and differential item functioning across educational contexts[END_REF] was developed with the aim of measuring the mathematics skills needed by students enrolling in introductory statistics courses. The test consists of 30 problems, and it has a multiple choice format (one correct out of four alternatives). A single composite score, based on the sum of correct responses, was calculated. In the present sample, Cronbach's α was .74. We used this measure as an estimate of students' math knowledge.

The Abbreviated Math Anxiety Scale (AMAS; [START_REF] Hopko | The abbreviated math anxiety scale (AMAS) construction, validity, and reliability[END_REF]Italian version: Primi, Busdraghi, Tomasetto, Morsanyi, & Chiesi, 2014) measures math anxiety experienced by students in learning and test situations. Participants have to respond on the basis of how anxious they would feel during the events specified (for example, "Listening to another student explain a math formula"). High scores on the scale indicate high math anxiety. A single composite score was obtained, based on participants' ratings of each statement. In the present sample, Cronbach's α was .84.

The Subjective Numeracy Scale [START_REF] Fagerlin | Measuring numeracy without a math test: development of the Subjective Numeracy Scale[END_REF]) is a subjective measure (i.e., self-assessment) of quantitative ability. An example item is "How good are you at working with fractions?" The items have to be rated on a 6-point Likert scale. A single composite score was computed based on participants' ratings of each item. Coefficient α in the current sample was .78.

Measure of statistics achievement. As a measure of achievement, we used the final examination grade. The exam consisted of a written task that included three problems to be solved by a paperand-pencil procedure without the support of a statistics computer package, 5 multiple-choice and 2 open-ended questions (e.g., describe the properties of a normal distribution) and 1 output of data analyses conducted with R-Commander to interpret. For the problems, students were given a data matrix (3-4 variables, 10-12 cases) and they had to compute descriptive indices, draw graphs, and choose and apply appropriate statistical tests (identifying the null and the alternative hypotheses, finding the critical value, calculating the value of the test, and making a decision regarding statistical significance). Grades range from 0-30. From 0 to 17 the grade is considered insufficient based on the Italian University Grading System. Thus, only students who obtain 18 or higher grades pass the examination.

Procedure

Participants completed the measures individually in a self-administered format in the classroom. Each task was briefly introduced, and instructions for completion were given. The answers were collected in a paper-and-pencil format. All participants completed the scale during the first week of an introductory statistics course. Achievement in statistics was measured at the end of the course during the exam session.

Data Analysis

Preliminarily, we tested the unidimensionality of the PRS evaluating local dependence (LD). LD is an excess of covariation among item responses that is not accounted for by a unidimensional IRT model and it was assessed using the χ 2 LD statistic [START_REF] Chen | Local dependence indices for items pairs using item response theory[END_REF], computed by comparing the observed and the expected frequencies in each of the two-way cross tabulations between responses to each pair of items. This diagnostic statistic is approximately distributed as standardized χ 2 . Given this approximation, as a rule of thumb, values of 10 or greater indicate the presence of LD.

After having verified this assumption, unidimensional IRT analyses were performed. IRT models use the original response data for estimating probabilities of responses as a function of the latent trait θ (i.e., in the current study, probabilistic reasoning), which is defined as a continuous variable that conventionally has a mean of zero and SD of 1.0. This function describes the relation between the probability of endorsing a response given not only the respondent's level of θ but also the item's characteristics. A model with two parameters (2PL) was tested in order to estimate the item difficulty and discrimination parameters. The parameters were estimated by employing the marginal maximum likelihood estimation method with the Expectation-Maximization (EM) algorithm implemented in the IRTPRO software [START_REF] Cai | IRTPRO: Flexible, multidimensional, multiple categorical IRT modeling[END_REF]. In the 2PL model, the two item parameters are item difficulty and item discrimination. The item difficulty parameter (β) or "location" represents the latent trait level corresponding to a .50 probability of correctly endorsing the item. The item discrimination parameter (a) or "slope" represents the item's ability to differentiate between people at contiguous levels of the latent trait. This parameter describes how rapidly the probabilities change with trait levels. In order to test the adequacy of the model, the fit of each item under the 2PL model was tested computing the S χ 2 statistics. Additionally, for each item was calculated the IIF, graphically represented by the Item Information Curve (IIC), that describes the amount of information that a particular item provides across the entire continuum of the latent construct, and it depends on both the discrimination and location parameters. Thus, we used IIFs to select the items that conveyed the higher amount of information along the range of the trait measured by the PRS, looking at the area above the IICs, which equals both the size of the a parameters and the spread of the b parameters. Once the shortened scale was defined, all the above described analyses were repeated for the brief scale in order to confirm the item and test psychometric properties. In particular, we investigated the reliability of the shortened scale. IRT makes it possible to assess the measurement precision of the test through the TIF.TIF is generated by aggregating the IIFs of items in a single measure and it allows to compute the information (I), that is, the expected value of the inverse of the standard error (SE), provided by the test at each level of the trait. Thus, the more information the test provides at a particular trait level, the smaller the error associated with trait estimation and the higher the test's reliability. Graphically, the TIF shows how well the construct is measured at different levels of the underlying construct continuum, and the peak of the TIF is where measurement precision is greatest. Regarding validity, Pearson product-moment correlations were computed.

Results

Preliminarily, we tested the unidimensionality of the probabilistic reasoning construct, a fundamental criterion underlying IRT models, evaluating the presence of LD. The results confirmed that a single factor model adequately represented the structure of the scale, as none of the LD statistics were greater than 10. Additionally, the factor loadings ranged from .54 to .80. Each item had a non-significant S-χ 2 value, indicating that all items fit under the 2PL model. Concerning the difficulty parameters (b), the results showed that the parameters ranged across the continuum of the latent trait from -2.68.53 to .49.12 logits (i.e., the logarithm of the odd, that is, the ratio of the probability of producing a correct response and the probability of responding incorrectly). Compared to the difficulty of the original scale (-2.97.5 to -.07.08 logits), results showed that the parameters spanned a wider range of the latent trait. With regard to the discrimination parameters (a), following Baker's ( 2001) criteria, all items showed adequate discrimination levels (a values over .60). Having verified the preliminary assumptions for IRT modelling, we looked at the IICs to select the items that conveyed the higher amount of information along the range of the trait measured by the open-ended PRS (see Figure 1). The figure clearly shows that items 1, 2, 3, 5, 13,15 and 16 provide lower amount of information.

Consequently, we retained nine items and we repeated the analyses for this shortened version of the PRS scale with an open-ended format. None of the LD statistics were greater than 10, indicating the absence of LD. All factor loadings were significant, ranging from .49 to .83. Each item had a nonsignificant S-χ 2 value, indicating that all items fit under the 2PL model. Concerning the difficulty parameters (b), the results showed that the parameters ranged from -1.57 (.29) to .46 (.11) logits across the continuum of the latent trait. With regard to the discrimination parameters (a), all items were above .60, indicating adequate discriminative power. Thus, still concerning item difficulty, the shortened version showed that the items had a low-medium level of difficulty. Nonetheless, the discriminative measures showed that the items could discriminate individuals with different trait levels. Next, we investigated the TIF that provides test reliability estimations indicating the precision of the whole test for each level of the latent trait. As shown in Figure 2, from 2 standard deviations below the mean to 0.5 above the mean (fixed at 0), the amount of test information was equal to or greater than 4 indicating that the instrument was sufficiently informative for this range of the trait. Thus, compared to the original scale that adequately measured low levels of probabilistic reasoning ability, the PRS-B accurately measures a wider range of the underlying trait. Finally, we looked at the validity of the scale. Table 1 presents descriptive statistics for all measures in the study, and the relations between the PRS-B, and all other variables. As expected, in line with the original scale, the PRS-B was significantly and positively correlated with mathematics skills and subjective numeracy, and negatively with mathematics anxiety. Concerning the relationship between PRS-B and statistics achievement, we found a significant positive correlation. To ascertain the predictive value of probabilistic reasoning as measured by the PRS-B, we tested a regression model in which the PRS-B was entered as a predictor of statistics achievement along with math competence (MPP), math anxiety (AMAS), and subjective numeracy (SN). The results showed that when all predictors were entered in the regression analysis together (F(4,133)=3.29, p=.013; R=.30; R 2 =.09), the PRS-B was a significant predictor of statistics exam performance (β =.26; p=.009), whereas the MPP, SN, and AMAS did not significantly predict statistics achievement.

Conclusion

In this study, we presented the PRS-B scale to assess probabilistic reasoning skills. Applying IRT, we obtained a shorter version of the original PRS, where all remaining items had good discriminative power and they measured a large spectrum of the trait which covered a wider range than the original scale. Additionally, the PRS-B accurately measures average levels of probabilistic reasoning ability, and it is helpful in identifying individuals who have difficulties in this domain.

Finally, the validity results confirm that the shortened form replicates the pattern of relationships established for the construct as measured by the long form, with positive relations with mathematics and statistics skills and subjective confidence, and a negative relationship with mathematics anxiety.

In conclusion, the PRS-B could be a useful tool in educational contexts, as well as in research.

Figure 1 :

 1 Figure 1: Item information curve (IIC) of the original 16-item PRS open format. Latent trait (theta) is shown on the horizontal axis and the amount of information and the SE yielded by the test at each trait level is shown on the vertical axis

Figure 2 :

 2 Figure 2: Test information function of the Probabilistic Reasoning Scale-Brief

Table 1 : Descriptive statistics for the measures, and correlations between the PRQ-B and MPP, AMAS, SN, and statistics achievement

 1

	1	2	3	4	5


Susanne Schnell
email: susanne.schnell@math.upb.de

Daniel Frischemeier

Primary school students reasoning about and with the median when comparing distributions

Keywords: statistics education, early statistics, primary school, comparing groups, median

Statistical reasoning can be enhanced at an early stage by facilitating informal statistical reasoning and developing conventional measures based on individual conceptions. This paper aims at identifying young students´ conceptions of the median and their relations to its definition. For this, we present a design research study with grade 3 students (age 9) focusing on the comparison of two groups. The qualitative analysis of complementary interviews shows that young students already develop an initial understanding of the median but struggle to find the conventional reference for the middle. Consequences will be drawn for further design cycles.

Introduction: Early statistics in German curriculum

Statistical literacy is a fundamental component for active citizenship [START_REF] Engel | Statistical Literacy for active Citizenship: A Call for Date Science Education[END_REF]. Some facets of statistical literacy can be approached as early as primary schoolan overview is presented in [START_REF] Leavy | Statistics in Early Childhood and Primary Education: Supporting Early Statistical and Probabilistic Thinking[END_REF]. In Germany, nation-wide standards for primary education in mathematics require students to gain competences in dealing with "data, frequency, and chance" [START_REF] Hasemann | Daten, Häufigkeit und Wahrscheinlichkeit[END_REF]. The core idea is to develop conceptual fundaments, e.g. by posing statistical questions, collecting data, and creating and interpreting representations of data [START_REF] Hasemann | Daten, Häufigkeit und Wahrscheinlichkeit[END_REF]. German textbooks and teaching materials for primary school, however, often emphasize singular aspects such as reading a diagram rather than introducing context-rich and complex data analysis projects as sustainable teaching-learning arrangement [START_REF] Bakker | Reasoning about shape as a pattern in variability[END_REF]. Such an arrangement aims at developing students' competences in drawing and dealing with informal statistical inferences [START_REF] Makar | The reasoning behind informal statistical inference[END_REF] and building on existing informal conceptions in order to develop conventional ideas [START_REF] Bakker | Reasoning about shape as a pattern in variability[END_REF]. Hence, researching initiated learning processes is an important task [START_REF] Makar | Young children's explorations of average through informal inferential reasoning[END_REF]. This paper presents a design-based research project [START_REF] Prediger | Lehr-Lernprozesse initiieren und erforschen-fachdidaktische Entwicklungsforschung im Dortmunder Modell[END_REF] which was conducted over four cycles so far in order to develop a statistics teaching-learning arrangement for classrooms in German primary schools. After summarizing theoretical considerations, the main design principles and framework of the teaching-learning setting are discussed. Lastly, we will present empirical insights of students' statistical reasoning about and with the median from interviews with students grade 3 (age 9) conducted after the whole-classroom activities.

Statistical projects in primary school

Going beyond calculation and algorithms in statistics education means enabling students to deal with statistical inquiries and applying the phases problem, plan, data, analysis and conclusions of the PPDAC-cycle [START_REF] Wild | Statistical thinking in empirical enquiry[END_REF]. Especially at the starting point of a statistical inquiry e.g., when posing statistical questionsthe desire arises to not only consider the distribution of one variable, but to look for a relationship between two or more variables. So, for example, the question whether third graders tend to have more games on their smartphones than fourth graders leads to the comparison of two groups. While easily motivated, group comparisons are also sophisticated statistical activities taking into account many of the fundamental statistical ideas like data, variation, representation, etc. [START_REF] Burrill | Fundamental statistical ideas in the school curriculum and in training teachers[END_REF]. [START_REF] Frischemeier | Statistisch denken und forschen lernen mit der Software TinkerPlots[END_REF] identified several important and sustainable group comparison elements like center, spread, skewness, shift, p-based and q-based comparisons. While these concepts can be quite complex, the fundamental ideas behind them can be addressed already in primary school. Informal comparisons are a fruitful task to provide in-depth insights into students' conceptual development and informal statistical reasoning (e.g. [START_REF] Schnell | Individual concepts of students comparing distributions[END_REF][START_REF] Makar | Young children's explorations of average through informal inferential reasoning[END_REF]. In this paper, we focus on using (informal) conceptions of the median as well as visual features of distributions for comparing groups in early statistics.

To describe the center of a distribution is a key idea of statistics (e.g. [START_REF] Bakker | Reasoning about shape as a pattern in variability[END_REF][START_REF] Makar | Young children's explorations of average through informal inferential reasoning[END_REF]. The concept of the median can be introduced on an enactive level via "animated statistics" in the sense of embodied cognition by Lakoff & Núnez (2000). In this case the students are statistical units themselves, line up in an ascending order in regard to a certain attribute (e. g. their height) and identify the middle. Even the more formal procedure of determining the median (sorting values and finding the middle value) is easily and non-formally presentable, but the interpretation and a deeper conceptual understanding can be challenging (Mokros & Russel, 1995) as it is interwoven with ideas like representativeness and typicality. However, these ideas can be pre-formally existent from even young students' everyday experiences (Mokros & Russel, 1995). Mokros and Russel (1995) identified approaches to the average, which include among others the average as an algorithm without predominant focus on contextual interpretation or the average as midpoint which is chosen as representative of the data and draws on the 'middle' which alternately is the median, the middle of the X-axis or the middle of the range. To be able to interpret the median correctly and in a meaningful way, it has to be understood as a value in reference to the rest of the distribution (e.g. cutting the data set in two halves). Overall, even students who are familiar with formal ways to determine the median sometimes do not apply it when comparing distributions (e.g. [START_REF] Watson | The beginning of statistical inference: Comparing two data sets[END_REF][START_REF] Konold | Students analyzing data: Research of critical barriers[END_REF]. This might be due to a lack of understanding that measures such as the median are representative of a distribution [START_REF] Konold | Students analyzing data: Research of critical barriers[END_REF]. However, [START_REF] Makar | Young children's explorations of average through informal inferential reasoning[END_REF] shows how even young students of age 8 can be supported in developing a fundamental understanding of the median and using it for informal inferential reasoning when presented with an adequate teaching-learning arrangement. The key is to give students room to develop informal ideas which serve as important base for building foundational knowledge of statistical concepts [START_REF] Makar | Young children's explorations of average through informal inferential reasoning[END_REF][START_REF] Bakker | Reasoning about shape as a pattern in variability[END_REF][START_REF] Smith | Misconceptions Reconceived -A Constructivist Analysis of Knowledge in Transition[END_REF].

Especially in terms of visual aspects, young students draw on a variety of different strategies to describe and compare distributions (e. g. [START_REF] Watson | The beginning of statistical inference: Comparing two data sets[END_REF]. For instance, they focus on 'hills' to describe the shape of the data [START_REF] Bakker | Reasoning about shape as a pattern in variability[END_REF]. To facilitate a proto-concept of center, [START_REF] Konold | Students' use of modal clumps to summarize data[END_REF] introduced so-called modal clumps as "a range of data in the heart of a distribution of values" (p. 1) in order to find ways to identify the center in stacked dot plots and "allow students to express simultaneously what is average and how variable the data are" (p.1). [START_REF] Bakker | Learning to reason about distributions[END_REF] build on these concepts to lead students from a local perspective (focusing on singular dates such as the minimum) to a global perspective on distributions (with a certain shape, spread etc.) and see modal clumps as proto-concepts not only for center but also for spread and therefore provide students with the opportunity to compare two distributions.

Overall, our aim is to design and realize a teaching-learning arrangement to enhance the statistical reasoning of primary school students and to enable them to compare groups with proto-concepts like modal clumps and concepts like medians when working on statistical projects. This paper focusses on exploring which concepts of the median were developed by the children during the teaching-learning unit and applied in the interview. Therefore, the research question is: Which conceptions of the median do students verbalize in the interviews and what is their (visual) point of reference for the median?

Research Design Methodological framework: Design Research

This study is in the methodological framework of topic-specific didactical design research [START_REF] Prediger | Lehr-Lernprozesse initiieren und erforschen-fachdidaktische Entwicklungsforschung im Dortmunder Modell[END_REF]. Thus, it aims at designing a specific teaching-learning arrangement for data competence in primary school on the one hand, and the development of local theories of teaching and learning conceptual basics at an early age on the other hand. These intentions are addressed in iterative research cycles of (re-)designing, implementing, and analyzing the materials [START_REF] Prediger | Lehr-Lernprozesse initiieren und erforschen-fachdidaktische Entwicklungsforschung im Dortmunder Modell[END_REF]. So far, four cycles were conducted, exploring different settings with whole classes and special interest groups in grade 3 and 4. After the classroom-activities, additional interviews for indepth analysis of students' reasoning were conducted.

Design principles and teaching-learning arrangement

For the design of the teaching-learning unit we implemented several recommendations for the design of Statistical Reasoning Learning Environments [START_REF] Garfield | Developing students' statistical reasoning[END_REF]) such as using real data sets and using digital tools like TinkerPlots [START_REF] Konold | TinkerPlots 2.0[END_REF], which can help young students to manipulate and modify the data in regard to a specific question. Furthermore, we facilitated collaborative work on statistical projects (first with small datasets, later with larger datasets), introduced the students to data analysis concepts on different representation levels (enactive, iconic, symbolic) and promoted the use of informal concepts such as modal clumps [START_REF] Konold | Students' use of modal clumps to summarize data[END_REF].

In the following we refer mainly to the version of the teaching-learning arrangement by [START_REF] Breker | Verteilungen vergleichen mit der Software TinkerPlots -Eine Interviewstudie mit Schülerinnen und Schülern der Klasse 3[END_REF] from the last design research cycle in which 22 third grade students participated. All of the students have no specific statistical pre-knowledgethey have only been introduced to the reading and interpreting of pie graphs and tallies. The teaching-learning arrangement consisted of seven modules spread over 13 lessons (45 minute each): The median was introduced in module 4. As mentioned above, one fundamental idea was to present data analysis concepts on different representation levels. Hence the median was introduced on an enactive level with animated statistics: The students were asked to line up in a row ascending by their height and determined the child in the middle. After completing a fill-in-the-blank text with the characteristics of the median, the teacher demonstrated how to use TinkerPlots to identify the median in a distribution. The median was then used by the students to compare different distributions.

Our aim was to investigate the ways in which the students use statistical concepts such as the median to compare groups in data sets in TinkerPlots after completing the teaching-learning arrangement in class. Hence we conducted subsequent interviews with a convenience sample of participants. Here, we provided them with a real data set including variables about leisure time activities and media use of about 680 primary school students (grade 3 and 4). The participants' task was to investigate whether the third or the fourth graders have more games on their smartphones. The interview consisted of two main activities with regard to the task: First the students were asked to conjecture two dot plots (grade 3 vs. grade 4) based on their expectation. Second the students were asked to use TinkerPlots to compare the groups of third and fourth graders in the real data set.

Data collection and analysis

Ten students participated in the interview study of cycle 4 after they completed the teachinglearning arrangement with their class. In the interviews, the students worked in pairs of two. All interviews, including the activities in TinkerPlots, were recorded and fully transcribed. The analysis was conducted under the interpretative paradigm, scrutinizing the transcripts for sequences in which the students use the median and then reconstructing their cognitive activities and conceptions.

Findings The middle of what? -The case of Paula and Linda

Paula's and Linda's initial hypothesis is that fourth graders have more games on their smartphone. After creating two conjectured dot plots, the researcher asks the students to explain the meaning of the median. The center point of the/ well it's the center of the dot plot.

Paula calls it the 'center point' and seems to be aware that she has to specify the point of reference of the center. Her offered reference 'the center of the dot plot' (line 81) is ambiguous as it could refer to the X-axis, the range of data dots or the distribution of data dots (similar to average as midpoint, Mokros & Russel, 1995). For determining the median, she counts only the columns of dots (with the same value) rather than every single dot, which equals determining the center of the range. Both girls interpret this median correctly in terms of the higher median indicating more games for fourth graders.

When introduced to the real data set, the girls first determine and interpret the hills. Asked to summarize the findings in regard to the overall question (line 218), Paula argues with the median which they have not determined for this distribution, yet. Yes, definitely. But really, if you just take half somehow, then it really looks like the 3 rd graders were furth/ had more (incomprehensible)

Describing how the distribution of fourth graders "moved" more to the right in comparison to the third graders, Paula describes the center as a point of reference (line 223). It is likely that Paula is not thinking of the correctly determined median (which -in contrast to her expectation -is smaller for the fourth graders than for the third graders), but rather her individually determined center of the range. However, her remark about the X-axis shows again awareness of the reference for this value.

While both children are at first confident in their answer of the 4 th graders having more games, Linda mentions doubts in line 224. By "taking half", she could either refer to the left half of dots (e.g. the data between 0 and 16 games) or possibly to the lower "half" (e.g. the range of 0 to 36 games but cutting of each column above a certain frequency). This idea is not followed up upon, though.

When the girls determine the median in TinkerPlots, it seems to differ from their expectation:

238 Paula: They just took it of the hills. And in the hills, 3 rd graders have more. 239 Linda:

Right. Yes, in this case, where they used only the hills, it's the 3 rd graders. But if you had done it without the hills, it would be the 4 th graders.

Paula explains the unexpected median by the way she thinks it was determined in the program, i.e. by determining the center of the hill's width (cf. figure 1). Thus, she still interprets the TinkerPlots median as referential but differing from the learned definition. In line 247, Paula suspects the true median of the 4 th graders to be farther in the middle (i.e. right). This indicates that she stays with her initial interpretation of the median as center of the range. In line 250, Linda suggests a manipulation of the data by "cutting of the hill". As she says the median would then be between 12 and 16, she might refer to cutting of the dot plot horizontally, thus evening out the number of dots per number of games and making Paula's procedure of determining the median correct. Thus, Linda twice offers ways to manipulate the data set and thinks about the effects on the median. Overall, the girls use the idea of two different medians in order to support her initial expectation that fourth graders have more games on the phone (line 272).

In regard to the research question, the girls developed conceptions of the median as the middle of the range and median as the middle of the modal hill. They are very consistent with their way of determining it, which might have its roots in the animated statistics activity. It is remarkable that they consider different options of manipulating the data set and their effects on the median, which hints that they perceive it as a value that is closely connected to the distribution and that has representative value. By assuming that TinkerPlots uses the hills' boundaries rather than the whole distribution for determining the median, the girls establish a pre-formal connection between the visual approach and the calculated measure.

Additional interpretations of the median

The students in the interview study showed a variety of different conceptions of the median:

 Layla and Sandra as well as Burak and Tarik call the median "the middle (of the dots)" (LS_line 483, BT_line 103) but do not specify it further other than using the learned procedure to determine it  median as unspecified middle  Johannes and Nils interpret it as the center with "roughly the same amount [of dots]" in each half (JN_line 53)  median as the middle of two halves  Lara and Lia seem to make a connection between the modal value and median "A median is the middle of the biggest. Of the biggest amount." (LL_line 171)  median as the middle of modal clump

Conclusion

This paper aimed at identifying young students´ conceptions of the median and their relations to the conventional measure [START_REF] Bakker | Reasoning about shape as a pattern in variability[END_REF]. The analysis shows how the students are integrating visual and calculated aspects of the distribution in order to compare groups. Building on [START_REF] Mokros | Children's concepts of average and representativeness[END_REF], we could identify several conceptions of the median which draw on different references in the data set: median as middle of the range, median as middle of the modal hill, median as the middle of two halves, median as the middle of modal clump or column and median as unspecified middle. While all pairs were able to interpret the shift of the medians correctly, the indepth analysis shows their struggle with the referential concept. Even though these conceptions

were not yet in-line with the conventional conception of the median, they serve as important starting points for the development of more formal conceptions (Smith, diSessa, & Rochelle, 1991).

Overall, the teaching-learning arrangement clearly supports young students in informal reasoning about statistical concepts while comparing groups. For instance, Linda's considerations of manipulating the data set in order to change the median are promising starting points for further design cycles.

Another implication from the analysis is to be careful when introducing the median via animated statistics. When choosing this way, a scale should be prepared so that the distances between the children (i.e. data dots) become apparent and stacked distributions should be discussed. For a hypothetical learning pathway, we suggest to introduce first the modal clump as main area in the heart of the data distribution and then the median as a special value in relation to the modal clump, indicating the center of the distribution. Next discuss different distributions in which the position of the clump and the median differ. These ideas will be realized in the next design cycle of the teaching-learning arrangement.

  data analysis and getting to know the questionnaire 2 Statistical representations on different representational levels (enactive, iconic, symbolic) 3 Introduction to data analysis with TinkerPlots and creating stacked dot plots in TinkerPlots Getting to know modal clumps, medians and hatplots to compare

  explain it again: what is the median, Paula? 81 Paula:

Figure 1 :

 1 Figure 1: Paula and Linda's TinkerPlots screen with drawn-in hills and later added medians (median of 3 rd graders: 8, median of 4 th graders: 4) 218 R: Okay. Good. Now, what did you find out for your question, when you look at the hills? 3 rd or 4 th graderswho has more games on the smartphone? (…) 223 Paula: 4th graders. (…) Because 4 th graders moved farther to the right with their dots and if you take the exact center, now, only of the dots, not of the stripes [i. e. X-axes]. Because otherwise the median would be the same. Just from the dots, the median of the 4 th graders is farther in the middle than of the 3 rd graders. 224 Linda:Yes, definitely. But really, if you just take half somehow, then it really looks like the 3 rd graders were furth/ had more (incomprehensible)


Balgeum Song
email: bgsong@snu.ac.kr

Kyeong-Hwa Lee

Yeong Ok

Chong
email: yochong@ginue.ac.kr

A case study on design and application of creative tasks for teaching percentage bars and pie charts

Keywords: Mathematical creativity, percentage bar, pie chart, statistical thinking

Introduction

There is a need to discuss on teaching and learning situations which include graph comprehension and construction to promote students' conceptual understanding of graphs, however, little attention was given to this until the present day [START_REF] Friel | Making sense of graphs: Critical factors influencing comprehension and instructional implications[END_REF]. Since several research have indicated that creative tasks and teaching strategies effectively enhance students' conceptual understanding and mathematical thinking [START_REF] Luria | Enhancing equity in the classroom by teaching for mathematical creativity[END_REF], in this study, we investigate how creative tasks and instructional strategies for lessons on percentage bars and pie charts are used to elicit students' growth in conceptual understanding and statistical thinking.

Theoretical framework

Percentage bars, Pie charts and Statistical thinking

Percentage bars and pie charts show how sectors constitute a whole (Reys et al., 2012). Learning these graphs relates to percentage, proportional relationship between the area of each sectors and their percentages, and interpreting graphs connecting data and the context (Reys et al., 2012;[START_REF] Curcio | Comprehension of mathematical relationships expressed in graphs[END_REF]. In terms of statistical thinking, it is necessary to guide students to develop global view from local view as we can discern the general feature of the data set with global view [START_REF] Ben-Zvi | Junior high school students' construction of global views of data and data representations[END_REF].

Mathematical creativity

Four main components of mathematical creativity are fluency, flexibility, originality and elaboration [START_REF] Luria | Enhancing equity in the classroom by teaching for mathematical creativity[END_REF]. Here we define creative task as open-ended and multiple solution task which can be performed in various ways, inclusive of informal method and representations, especially, applying one's experience and prior knowledge using fluency, flexibility, originality and elaboration.

Methodology

In this study, case study methodology was used and a class of 28 sixth-grade students participated. These pupils had not learnt about percentage bars and pie charts before. The first task asked students to discuss the claim which includes certain percentage value and to visualize given percentage value.

The second was about construction of percentage model from given data table. In the last task, each group of students was given a percentage bar or a pie chart without label. They were asked to infer the percentage and the item of each sector and then interpret the given graph. Discussions among students were conducted for social interaction. All teaching-experiment sessions were videotaped and recorded. These data were transcribed later for analysis. The students' individual worksheets were collected. Researchers attended all lessons and recorded field notes. We established four phases of learning percentage bars and pie charts based on Curcio(1987)'s three levels of graph comprehension and Ben-Zvi and Arcavi( 2001)'s research about the process of the construction of global view from local view. Students' responses and models were classified according to these four phases in order to verify if students' conceptual understanding and statistical thinking were improved.

Findings and Conclusion

The students constructed and developed percentage models, and interpreted percentage bars and pie charts using their fluency, flexibility, originality and elaboration in this study. Meanwhile, they showed increases in the phases of learning percentage bars and pie charts. If the tasks did not allow diverse approaches and representations using students' own method to perform the tasks, we believe that it would be hard to expect these increases. The findings showed that creative tasks with proper instructional strategies promote conceptual understanding and statistical thinking by stimulating students' creativity. Furthermore, the analysis indicated that the creative tasks provide various learning opportunities for students who differ in their levels. This study provides empirical evidence to support previous claims that the use of creative tasks and instructional strategies in the mathematics classroom enhance students' conceptual understanding and mathematical thinking [START_REF] Luria | Enhancing equity in the classroom by teaching for mathematical creativity[END_REF].

This study is part of "Siheung City -Seoul National University Partnership for Primary/Secondary School Education Program" funded by Siheung City, Gyeonggi-do, Republic of Korea.

Table 1 : Phases of learning percentage bars and pie charts

 1

	Phase	Construction	Data	Interpretation	Context
	0	Focused on context		Considering personal experience
	1	Focused on pointwise information		Considering appropriate context situation
	2	Focused on the difference between individual data Comparison of the meaning of each part
	3	Focused on the proportional relationship among data	Suggestion of alternative, prediction, implication


Ödön Vancsó

Peter Fejes-Tóth

Zsuzsanna Jánvári

Introduction of inferential statistics in high school in Hungary

Introduction of inferential statistics in high school in Hungary

Ödön Vancsó 1 , Peter Fejes-Tóth 2 , and Zsuzsanna Jánvári 3 1 Eötvös Loránd University, Budapest, Hungary; vancso@caesar.elte.hu 2 Szent István University, Budapest, Hungary; fejestothpeter@yahoo.com 3 Szerb Antal Gimnázium, Budapest, Hungary; zsjanvari@gmail.com Keywords: Statistical inference, probability distribution, interdisciplinary approach.

Introduction, theoretical background

Inferential statistics in not part of the secondary school curriculum in Hungary. The current plans to reform the national curriculum include the introduction of basic inferential statistics. A working group of the Hungarian Academy of Sciences, of which these authors are members, is exploring possibilities and designing tasks and feasible learning paths. Descriptive statisticsthough mainly from a didactical rather than a mathematical point of view -is taught in teachers' education since the early 2000s, while teachers learn basic probability at the university since 1959. Because of this lack in teachers' background, the aim of the project is to design an acceptable way of introducing inferential statistics in high-school curricula both for students and their teachers. This paper is linked to the project "Complex Mathematics Education". Our work is based on the ideas about teaching probability and statistics of T. Varga and T. Nemetz. We have designed several experiments and tested them by one of the authors (Jánvári) in Szerb Antal Secondary School (grade 11, 17 year-old students). The experiments are determined by their fit into our theoretical views on modelling probability and to the approach of teaching classical and Bayesian statistics together developing the concepts in parallel [START_REF] Vancsó | Parallel discussion of classical and Bayesian ways as an introduction to statistical inference[END_REF][START_REF] Vancsó | How visualisation using software helps understanding classical and Bayesian statistics[END_REF]. In these studies, teacher's and students' beliefs and skills were developed by a "parallel course", which helped to understand classical and Bayesian ideas. While the informal approaches to teaching inference in [START_REF] Garfield | Learning to reason about statistical inference[END_REF] focus on simplifying statistical inference by re-randomisation and Bootstrap, we try to make inference more understandable by integrating the Bayesian view. This is in line with [START_REF] Borovcnik | Informal inference -Some thoughts to reconsider[END_REF], who describes shortcomings of informal inference. It also fits to an analysis of statistical inference from the statistician's perspective by Vic Barnett, which is summarised in a comparative educational study of statistical inference by [START_REF] Borovcnik | A comparative educational study of statistical inference[END_REF].

Conclusion and results

Our first experiment is rolling dice judging whether the die used is well-balanced or loaded by a probability statement. Though this method was introduced in earlier papers (e.g., [START_REF] Lawton | An exercise for illustrating the logic of hypothesis testing[END_REF][START_REF] Dambolena | An intuitive introduction to hypothesis testing[END_REF], our approach places greater emphasis on the hands-on inclass experiment. It is used to introduce hypothesis testing. In this case, the Bayesian way is technically too complicated but becomes manageable through the use of software. The second example is a reaction test using a stopwatch. This interactive game among students unfolds connections between different school subjects such as physics, biology, and mathematics. Furthermore, it is also suitable to discuss about discrete and continuous quantities. It is our goal to use more games in teaching, which is motivating for students and provides an excellent opportunity to discuss basic notions and misconceptions in probability and statistics. Also, application of modern ICT tools in itself can be interesting and challenging for secondary-school students. Moreover, the gamification element (see [START_REF] Papp | Gamification effects on motivation and learning: application to primary and college students[END_REF] can increase their involvement and consequently their motivation and understanding of the scientific material. This paper was supported by the Hungarian Academy of Sciences' "The Complex Mathematics Education Project".