Figure 5 - uploaded by Sibel Kazak

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# Nelson's branched hair-length factory. We see a male case being produced, which has just received a hair length of 1 from the upper right mixer. Females go to the lower branch and receive a hair length of 1-12 inches.

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In current curriculum materials for middle school students in the US, data and chance are considered as separate topics. They
are then ideally brought together in the minds of high school or university students when they learn about statistical inference.
In recent studies we have been attempting to build connections between data and chance in the...

## Contexts in source publication

**Context 1**

... few months after this first activity, we implemented a branching capability in the software that directs cases with different values on one attribute to different sampling devices (see Figure 5). To introduce the students to it, we revisited the cat factory and pointed out the problem with how the linear factory was often misassigning males and female names. ...

**Context 2**

... is, although students traditionally have difficulty reasoning backwards from data to valid claims about conditional probability, these middle school students seemed quite able to reason forward from conditional situations they knew well to models and resultant data that represented those relations. Figure 5 shows Nelson's model of the hair-length problem, which he described for us: ...

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## Citations

... The preparation process that both students and teachers went through while learning probability and statistics was insufficient (Koparan, 2019;Rodríguez-Alveal, Díaz-Levicoy & Vásquez, 2018). Philosophical debates around the meaning of probability, certain features of probabilistic reasoning, students' misconceptions and difficulties, and the growing diversity of technological resources reveal that teachers need special preparation to teach probability Konold, et al., 2007;Rodríguez-Alveal, Díaz-Levicoy & Vásquez, 2018). While textbooks provide some examples, some texts offer a very narrow view of probability concepts or a single approach to probability. ...

... Simulation is a teaching method in which learners can change their parameters and make experiments one-toone. Simulations provide opportunities to strengthen the understanding of statistical ideas (delMas, Garfield & Chance, 1999;Konold, Harradine & Kazak, 2007;Koparan, 2016b;Koparan, 2022b) and to support learners' learning processes while working on chance experiments Koparan, 2019). Students can build their knowledge through simulation-based activities (Koparan, 2016a;Koparan, 2022b). ...

Solving real-life problems through mathematical modeling is one of the aims of modern mathematics curricula. For this reason, prospective mathematics teachers need to acquire modeling skills and use these skills in learning environments in terms of creating rich learning environments. With this study, it is aimed to examine the reflections of using a simulation on a problem involving uncertainty on the probabilistic thinking of prospective teachers. The activity includes an experimental review of the famous
Hat problem. It was observed that the hat problem, which started as a puzzle, was linked to coding theory and reached the limit of mathematics, statistics, and computer science research. Research findings revealed that the simulation-supported learning environment not only contributes to prospective teachers' probabilistic thinking skills, but also offers the opportunity to experience different methods (such as working with real data, technology assisted learning, modeling) in teaching and learning mathematics. It
has been concluded that simulations have a unique potential that other methods do not have in terms of gaining statistical thinking as well as problem solving and modeling skills

... More recent calls for reform have focused on not only pedagogy and assessment methods, but also course content. Studies have shown that engaging students in modeling and generating distributions (e.g., Doerr & English, 2003;Konold et al., 2007) helps them understand randomness and chance, as well as providing explicit experience with sampling variability. As noted by Lee et al. (2015), prior research (e.g., Garfield et al., 2012;Lane-Getaz, 2007;Saldanha & Thompson, 2002) found that a threetiered approach to constructing an inference problem, problems/models, repeated samples, and sampling distributions, appears to help students and teachers better conceptualize statistical inference. ...

Using simulation-based inference (SBI), such as randomization tests, as the primary vehicle for introducing students to the logic and scope of statistical inference has been advocated with the potential of improving student understanding of statistical inference and the statistical investigative process as a whole. Moving beyond the individual class activity, entirely revised introductory statistics curricula centering on these ideas have been developed and tested. Preliminary assessment data have been largely positive. In this paper, we discuss three years of cross-institutional tertiary-level data from the United States comparing SBI-focused curricula and non-SBI curricula (86 distinct institutions). We examined several pre/post measures of conceptual understanding in the introductory algebra-based course using multi-level modelling to incorporate student-level, instructor-level, and institutional-level covariates. We found that pre-course student characteristics (e.g., prior knowledge) were the strongest predictors of student learning, but also that textbook choice can still have a meaningful impact on student understanding of key statistical concepts. In particular, textbook choice was the strongest “modifiable” predictor of student outcomes of those examined, with simulation-based inference texts yielding the largest changes in student learning outcomes. Further research is needed to elucidate the particular aspects of SBI curricula that contribute to observed student learning gains.

... Simulation is the most appropriate strategy to concentrating on concepts and decrease technical calculations (Borovcnik & Kapadia, 2009). Simulations provide the opportunity to strengthen the perception of statistical ideas (Konold et al., 2007) and support the learning process of students when working on chance experiments (Maxara & Biehler, 2007). Hyman (1970) reviewed the literature in support of games and accordingly presented specific justifications for their use. ...

This study aimed to examine the impact of game and simulation-based learning environments on the success, concept knowledge, and attitude of prospective teachers regarding probability. The sample of the study consisted of 94 prospective mathematics teachers. The study was conducted in accordance with the quasi-experimental study design and the data were collected through achievement tests, concept tests, attitude scale, worksheets, simulations, games and observations made by the researcher. The course was taught with games, worksheets, concrete materials, games and simulations in the experimental group and via theoretical and practical training in the control group. The achievement test, concept test, and Attitude Scale towards Probability and its Teaching were applied to the groups as pre-test and post-test. The data collected were analysed quantitatively using arithmetic average, standard deviation, independent sample t-test, and covariance analysis. According to the findings, it was determined that there was a significant difference between the achievement and concept test scores and the attitude scale towards statistics scores of the prospective teachers in favour of the experimental group. According to the results, it was proposed that teachers and prospective teachers should adopt and utilize games and simulations as part of their teaching methods when teaching probability.

... Simulations are computer applications that allow students the opportunity to observe and interact with real-world experiences [12]. Simulations provide the opportunity to strengthen the understanding of statistical ideas [16] and to support the learning process of students while working on luck experiments [17]. Simulations provide opportunities to strengthen the understanding of statistical ideas [16] and to support learners' learning processes while working on chance experiments [10,17]. ...

... Simulations provide the opportunity to strengthen the understanding of statistical ideas [16] and to support the learning process of students while working on luck experiments [17]. Simulations provide opportunities to strengthen the understanding of statistical ideas [16] and to support learners' learning processes while working on chance experiments [10,17]. Students can build their knowledge using simulation activities [18]. ...

... BBGG, 4 in the form of BGGG, and 1 case in the form of GGGG. There are a total of 16 situations. The state we want is six, with probability 6/16 = 3/8. ...

In this research, the aim was to evaluate a simulation-based learning environment in the context of conditional probability. The study group consisted of 44 prospective mathematics teachers of the Probability and Statistics Teaching course. The data were collected through three probability problems, a survey form for the simulation-based learning environment, and observations. The research was conducted within the scope of the Probability and Statistics Teaching course. In the lessons, conducted in a simulation-based learning environment with distance education, the prospective teachers were asked to solve the questions asked and send the solutions using smartphones. The different ways of thinking that emerged are put forward by the researcher. Then, simulations developed by the researcher were used for the problems, and the prospective teachers were asked to make observations and take notes on important issues. In the last stage, there was a class discussion about the related problems. After the simulation-based learning activities, the prospective teachers were asked to evaluate the learning environment. The data obtained were evaluated qualitatively, and the prospective teachers' ways of thinking about problems, changing thoughts with the use of simulations, and their views on the learning environment are presented as direct quotations. The research findings revealed that the designed learning environment offers unique opportunities for prospective teachers to think about situations with mathematics content and to experience different methods of teaching and learning mathematics. The learning environment provided prospective teachers with the opportunity to understand and evaluate the contribution of simulations to problem-solving and the opportunities they provide for discussion, reflection, and collaboration in a meaningful context. It was concluded that simulations could become a powerful tool and an effective learning environment for learners.

... TinkerPlots (Konold & Miller, 2015) was a contribution from the Digital Technologies curriculum because of its particular features assisting students to display and understand variation in distributions (Konold, 2007;Konold & Lehrer, 2008;Konold et al., 2007). Watson and Fitzallen (2016) examined the software from the perspective of affordances (Chick, 2007;Gibson, 1977), in that it had the capability to foster learning based on its accessible and flexible features, facilitating its incorporation in learning activities by teachers. ...

Integrated STEM activities are espoused as appropriate for enhancing student learning in relation to statistical concepts; however, a greater understanding of the way in which students’ ideas about those concepts develop is needed to maximise the learning potential offered by engagement in STEM activities. For this study, plant growth was chosen as a topic from the Year 6 Australian Science Curriculum as an appropriate context to employ aspects of the four STEM disciplines to explore students’ developing ideas about variation. Sixty-four Year 6 students across three school terms worked in groups of four to trial various treatments and their effects on the growth of radish or wheat seeds. This report considers two aspects of student learning related to this topic based on (i) the formative assessment of features of students’ workbook entries specifically related to variation during the part of the classroom activity based on their TinkerPlots graphs and (ii) the later summative evidence of learning in responses to end-of-year questions on the activity for 56 of the students. The workbook entries are presented via a qualitative analysis to provide evidence of the forming of understanding of variation in a STEM context, with the SOLO Taxonomy being employed to assess the longer-term evidence and developmental nature of that learning. Overall, a broader picture has emerged of the potential for developing appreciation of variation in a STEM context in primary school.

... For example, using dynamic geometry software (DGS), such as Geometry's Sketchpad (Hollebrands, 2007), students can construct geometrical figures, measure lengths or angles of figures, or manipulate elements of a figure with specific properties. In addition to geometry, there are other content areas enhanced by dynamic representation, such as algebra (e.g., SimCalc; Roschelle et al., 2000) and statistics (e.g., TinkerPlots; Konold et al., 2007). However, there are few empirical studies on meaningful use of dynamic technology in elementary mathematics classrooms (Young et al., 2018). ...

In this study, we examine students’ mathematical reasoning within a technological environment designed to support understanding of relationships between quantities with adjustable measuring units. In particular, we provide a cross-sectional snapshot of how 30 elementary students (Grades 3–5) engaged in a series of fraction-as-measurement tasks using a “Dynamic Ruler” that could continuously dilate unit sizes. Screencast recordings were collected from a task-based clinical interview and analyzed to investigate children’s mathematical actions and mathematical ideas. Students’ reasoning patterns were characterized using four distinct types (low attending, holistic estimating, determining, and commeasuring) based on their solution strategies. Findings suggest that the Dynamic Ruler tool can elicit rich conceptions of fractions and even prompt novel approaches such as commeasurement. We conclude by drawing insights into how elementary students might use dynamic technology meaningfully.

... The author suggested that children constructed new ideas based on the interplay of their previous intuitions and their work with the computer resource. Other examples of case studies in a computer simulation experimental setting can be found in [28], in relation to secondary students' ideas of distribution, and of university students understanding of random processes in [29]. ...

Strengthening the teaching of probability requires an adequate training of prospective teachers, which should be based on the prior assessment of their knowledge. Consequently, the aim of this study was to analyse how 139 prospective Spanish mathematics teachers relate the classical and frequentist approaches to probability. To achieve this goal, content analysis was used to categorize the prospective teachers’ answers to a questionnaire with open-ended tasks in which they had to estimate and justify the composition of an urn, basing their answers on the results of 1000 extractions from the urn. Most of the sample proposed an urn model consistent with the data provided; however, the percentage that adequately justified the construction was lower. Although the majority of the sample correctly calculated the probability of an event in a new extraction and chose the urn giving the highest probability, a large proportion of the sample forgot the previously constructed urn model, using only the frequency data. Difficulties, such as equiprobability bias or not perceiving independence of trials in replacement sampling, were also observed for a small part of the sample. These results should be considered in the organisation of probabilistic training for prospective teachers.

... Modeling processes with a digital tool such as TinkerPlots require the development of digital techniques. Digital TinkerPlots techniques for setting up statistical models and simulating data are helpful to introduce key statistical ideas of distribution and probability (Konold, Harradine, & Kazak, 2007). The research by Garfield, delMas, and Zieffler (2012) suggests that students can learn to think and reason from a probabilistic perspective-or, as the authors call it, "really cook" instead of following recipes-by using TinkerPlots techniques to build a model of a real-life situation and to use this model for simulating repeated samples. ...

Digital technology is indispensable for doing and learning statistics. When technology is used in mathematics education, the learning of concepts and the development of techniques for using a digital tool are known to intertwine. So far, this intertwinement of techniques and conceptual understanding, known as instrumental genesis, has received little attention in research on technology-supported statistics education. This study focuses on instrumental genesis for statistical modeling, investigating students’ modeling processes in a digital environment called TinkerPlots. In particular, we analyzed how emerging techniques and conceptual understanding intertwined in the instrumentation schemes that 28 students (aged 14–15) develop. We identified six common instrumentation schemes and observed a two-directional intertwining of emerging techniques and conceptual understanding. Techniques for using TinkerPlots helped students to reveal context-independent patterns that fostered a conceptual shift from a model of to a model for . Vice versa, students’ conceptual understanding led to the exploration of more sophisticated digital techniques. We recommend researchers, educators, designers, and teachers involved in statistics education using digital technology to attentively consider this two-directional intertwined relationship.

... Epistemic forms represent a particular type of convention shared by communities of practice, which has been developed over time to support patterns of activity characteristic of that community (Saxe, 1991). The Tinkerplots Sampler tool as an epistemic form, for example, has highlighted and supported the statistics education communities' interest in emphasizing causes for variability by supporting learners in a "data factory" (Konold, Harradine, & Kazak, 2007) epistemic game. In this game, random devices are configured to approximate real-world mechanisms so that they, in turn, are able to manufacture data that approximates real-world patterns. ...

Data preparation (also called “wrangling” or “cleaning”)—the evaluation and manipulation of data prior to formal analysis—is often dismissed as a precursor to meaningful engagement with a dataset. Here, we re-envision data preparation in light of calls to prepare students for a data-rich world. Traditionally, curricular statistics explorations involve data that are derived from observations that students record themselves or that reflect familiar, relatively closed systems. In contrast, pre-constructed public datasets are much larger in scope and involve temporal, geographic, and other dimensions that complicate inference and blur boundaries between “signal” and “noise.” As a result, students have fewer opportunities to consider sources of variability in such datasets. Due to these constraints, we argue that data preparation becomes an important site for students to reason about variability with public data. Through analyses of repeated task-based interviews with five pairs of adolescent participants, we find that specific actions during data preparation, such as filtering data or calculating new measures, presented opportunities to engage leaners with variability as they prepared and analyzed several public socioscientific datasets. More broadly, our study highlights some changes to theory and curriculum in statistics education that are necessitated by a focus on “big data literacy”.

... The use of models to experience experimental probabilities is not limited to classic cases where the sample space is straightforwardly delineated. Konold, Harradine and Kazak (2007) studied middle school learners as they created a model, described as a "datafactory", to generate data such as the hair length of males and females. The learners were encouraged to compare the data generated to expected frequencies and to edit their model accordingly. ...

We examine the challenges of teaching probability through the use of modelling. We argue how an integrated modelling approach might facilitate a coordinated understanding of distribution by marrying theoretical and data-oriented perspectives and present probability as more connected to the social lives of modern-day students. Research is, however, also showing that learning modelling is non-trivial. For example, students can easily be confused between real-world data and model-generated data. Furthermore, students tend to create models that reflect the structure of the problem context rather than its mathematical structure. The argument we present is that teaching probability through an integrated modelling approach could offer important educational benefits, but this would be an immense challenge to the skills and knowledge of teachers and would demand increased levels of resources for teacher development. The implication is that research needs to identify suitable forms of teacher development and designs for tasks and curricula.