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Students persistently misinterpret histograms. Based on a literature review we conjectured that the confusion of histograms with case-value plots was leading to inappropriate interpretation strategies. Hence, the question for this research is: what are the most common strategies for secondary school students when estimating the mean in histograms and case-value plots? In a task with twelve graphs (histograms or case-value plots) we measured students’ eye movements (N=10, grade 10–11. The most common strategies students use are: a case-value plot interpretation strategy and a computational strategy both applied to histograms. Furthermore, some students reported strategies that are not in line with their gaze data nor their estimated mean.

Content uploaded by Lonneke Boels

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All content in this area was uploaded by Lonneke Boels on Jul 16, 2019

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... Despite many carefully designed interventions to tackle misinterpretations (e.g., Kaplan et al., 2014), students' difficulties with histograms remain (e.g., Cooper, 2018). We, therefore, decided to use eye tracking to study in depth how students interpret histograms (Boels et al., 2018(Boels et al., , 2019a(Boels et al., , 2022a. ...

... Three common strategies were identified (Table 1): a histogram strategy ( Fig. 5-a correct strategy that reads off the estimation on the horizontal weight axis), a case-value plot strategy ( Fig. 2-a strategy that would be correct for a case-value plot but is incorrect for finding the mean from a histogram as it returns the mean frequency, read on the vertical frequency axis), and a count-and-compute strategy (an incorrect 6 strategy that, for example, adds the height of the bars, hence the frequencies, and divides by the number of bars-resulting in a kind of zig-zag pattern of horizontal and vertical gazes, see Boels et al., 2022a for more details). Both the case-value plot strategy and count-and-compute strategy relate to the same misinterpretation: interpreting the histogram as a case-value plot (Boels et al., 2019a;Cooper, 2018); the difference is whether students estimated (case-value plot strategy) or calculated (count-and-compute strategy) the mean. Hence, almost all strategies can be attributed to one of two classes: one in which students correctly interpreted the graph as a histogram and one in which students incorrectly interpreted the graph as a case-value plot. ...

... In contrast to most eye-tracking studies, in the qualitative study, we looked at the perceptual form of the scanpath, for example, the vertical gaze pattern (Fig. 5), and refer to this as a stable scanpath if it includes multiple aligned fixations and saccades along this scanpath and was explicitly mentioned by at least some students as being relevant for their strategy (e.g., Boels et al., 2018Boels et al., , 2019aBoels et al., , 2022a. This vertical line is formed by looking back and forth between the balance point of the graph on the horizontal axis and the height of the bars as a weighting factor. ...

As a first step toward automatic feedback based on students’ strategies for solving histogram tasks we investigated how strategy recognition can be automated based on students’ gazes. A previous study showed how students’ task-specific strategies can be inferred from their gazes. The research question addressed in the present article is how data science tools (interpretable mathematical models and machine learning analyses) can be used to automatically identify students’ task-specific strategies from students’ gazes on single histograms. We report on a study of cognitive behavior that uses data science methods to analyze its data. The study consisted of three phases: (1) using a supervised machine learning algorithm (MLA) that provided a baseline for the next step, (2) designing an interpretable mathematical model (IMM), and (3) comparing the results. For the first phase, we used random forest as a classification method implemented in a software package (Wolfram Research Mathematica, ‘Classify Function’) that automates many aspects of the data handling, including creating features and initially choosing the MLA for this classification. The results of the random forests (1) provided a baseline to which we compared the results of our IMM (2). The previous study revealed that students’ horizontal or vertical gaze patterns on the graph area were indicative of most students’ strategies on single histograms. The IMM captures these in a model. The MLA (1) performed well but is a black box. The IMM (2) is transparent, performed well, and is theoretically meaningful. The comparison (3) showed that the MLA and IMM identified the same task-solving strategies. The results allow for the future design of teacher dashboards that report which students use what strategy, or for immediate, personalized feedback during online learning, homework, or massive open online courses (MOOCs) through measuring eye movements, for example, with a webcam.

... The dotplot was part of Item16. The histogram was part of Item05, not further discussed here (for more details, see Boels, Bakker, et al., 2022). ...

... Another scanpath pattern was a horizontal gaze pattern indicating that this student (incorrectly) tried to make all bars equally high which resulted in the mean of the frequencies instead of the mean weight. In total, five different scanpath patterns were found for students estimating and comparing means of histograms, each related to a specific strategy (Boels, Bakker, et al., 2022). Other AOIs did not emerge as relevant to these students' task-specific strategies. ...

... Details on participants, the eye-tracking method, and two items (Item02 and Item11) were reported previously in a qualitative study (Boels, Bakker, et al., 2022). Two items were used previously in a machine learning analysis (Item02 and Item20; Boels, Garcia Moreno-Esteva, et al., accepted) but with a different aim, namely, to examine how a machine learning algorithm (MLA) could identify students that used a correct or incorrect strategy-for solving the item-purely based on their gaze data on the graph area of this item. ...

Many students persistently misinterpret histograms. Literature suggests that having students solve dotplot items may prepare for interpreting histograms,as interpreting dotplots can help students realize that the statistical variable is presented on the horizontal axis. In this study, we explore a special case of this suggestion, namely, how students’ histogram interpretations alter during an assessment. The research question is: In what way do secondary school students’ histogram interpretations change after solving dotplot items? Two histogram items were solved before solving dotploti tems and two after. Students were asked to estimate or compare arithmetic means. Students’ gaze data, answers, and cued retrospective verbal reports were collected. We used students’ gaze data on four histogram items as inputs for a machine learning algorithm (MLA; random forest). Results show that the MLA can quite accurately classify whether students’ gaze data belonged to an item solved before or after the dotplot items. Moreover, the direction (e.g., almost vertical) and length of students’ saccades were different on the before and after items. These changes can indicate a change in strategies. A plausible explanation is that solving dotplot items creates readiness for learning and that reflecting on the solution strategy during recall then brings new insights. This study has implications for assessments and homework. Novel in the study is its use of spatial gaze data and its use of anMLA for finding differences in gazes that are relevant for changes in students’ task-specific strategies. This article is published open access on the website of the journal. Please note that the original article contains a minor error that does not influence the conclusions. We notified the editor and hope this will be corrected soon.

... Despite being promising, gaze data are still uncommon in statistics education [1,2]. In a review, Strohmaier et al. [3] found only four studies in statistics education using eye-tracking [4][5][6][7]. Strohmaier et al. described the added value of gaze data so that students' solution processes become visible-including approaches that students never articulated-and that these processes are neither disturbed nor interrupted, and that gaze data can make some complex cognitive processes visible (eg, for statistical thinking). Gaze data provide teachers and researchers with a new window into these processes. ...

... In addition, data triangulation, for example through cued recall (retrospective reporting with students' own gazes as a cue), will be needed until clear patterns have been found for specific tasks and topics and in different communities. We found such patterns for estimating means from histograms and case-value plots for university students [5], teachers [22], and high school students [4,8] in the Netherlands. Future research is needed to find out if gaze patterns on those tasks are similar in different cultural settings and educational F I G U R E 4 Gaze pattern of a student correctly interpreting a messy (top) and stacked (bottom) dotplot. ...

... Although eye-tracking has been around for some time, its use is still in its infancy in statistics education. Besides our conference papers [4,5,22], only two other studies were found in a literature review [3]. One study was on Bayesian reasoning strategies [6]. ...

Gaze data are still uncommon in statistics education despite their promise. Gaze data provide teachers and researchers with a new window into complex cognitive processes. This article discusses how gaze data can inform and be used by teachers both for their own teaching practice and with students. With our own eye-tracking research as an example, background information on eye-tracking and possible applications of eye-tracking in statistics education is provided. Teachers indicated that our eye-tracking research created awareness of the difficulties students have when interpreting histograms. Gaze data showed details of students' strategies that neither teachers nor students were aware of. With this discussion paper, we hope to contribute to the future usage and implementation of gaze data in statistics education by teachers, researchers, educational and textbook designers, and students.

... We found four different strategies for single dotplots. The most common strategy is a strategy that we previously called a histogram (interpretation) strategy (Boels et al., 2019b): Students estimate the mean by finding the point of the graph, or a of dots. When students apply this strategy, a vertical scanpath pattern is visible in their gaze data. ...

... The height of the resulting stack is then estimated. We previously (Boels et al., 2019b) called this a case-value plot (interpretation) strategy. The difference between this strategy ( Figure 3, left) and a histogram strategy (Figure 3, middle) is clearly visible in the heatmaps by the difference in horizontal spread-outness of gazes. ...

Dotplots can increase students’ reasoning about variability and distribution in statistics education but literature shows mixed results. To better understand students’ strategies when interpreting non-stacked dotplots, we examine how and how well upper secondary school students estimate and compare means of dotplots. We used two item types: single dotplots requiring estimation of the mean and double ones requiring comparison of means. Gaze data of students solving six items were triangulated with data from stimulated recall. Most students correctly estimated means from single dotplots; results for comparison were mixed. A possible implication is that single, non-stacked dotplots can be seen as a step towards teaching students to interpret univariate graphs but further research is needed for comparing graphs.

... As empirical analysis with eye-tracking shows, cultural perception of histograms requires specific sensory-motor strategies (Boels et al., 2019). We created the fields of promoted actions that would solicit the actions that match cultural sensory-motor strategies for building a histogram. ...

Different approaches to embodied learning-conceptual learning of curricular content grounded in a new capacity for enacting forms of purposeful physical movement in interaction with the environment-have become increasingly central to mathematics-education research. This research forum provides participants with an up-to-date overview of diverse and complementary theoretical perspectives on embodied learning, principles derived from these perspectives governing the design of environments for learning various mathematical content, and demonstrations thereof. We speculate on promising directions for future embodied design research.

... global view of data Numerous studies in mathematics education research have shown that eye-tracking has a high potential to provide new insights into students' mathematical thinking and learning (for an overview, see Strohmaier et al., 2020). Recent studies in statistics education illustrated that eye-tracking is an effective method to study students' strategies and difficulties when interpreting and comparing statistical graphs such as histograms (e.g., Boels et al., 2019) In a first step of our ongoing research project, hypotheses for differences in certain eyetracking measures (fixation count, saccade amplitude, saccade direction) between students with a local and global view of data were theoretically derived and empirically investigated with a sample of 25 6 th grade students (Schreiter & Vogel, in press). The results confirmed our hypotheses by showing that students with a global compared to a local view of data had on average significantly fewer fixations, longer saccade amplitudes and a higher relative number of horizontal saccades. ...

Many students tend to perceive a data distribution as a collection of individual values rather than as a conceptual entity (local vs. global view of data). These difficulties seem to persist even after instruction in statistics. This study uses a methodological triangulation of eye-tracking and stimulated recall interviews to examine and contrast 6th and 8th grade students' (N = 49) viewing patterns and statistical thinking when comparing data distributions. Results showed no significant differences between 6th and 8th graders. Regardless of students' grade level, the empirical data confirmed our theoretically derived hypotheses for differences in certain eye-tracking measures (fixation count, saccade amplitude, saccade direction) between students with a local and global view of data.

... This interaction can be traced as a system of sensorimotor processes and perceptual action loops, such as noticing a bump on a ski slope and going around it. Another example is projecting a height of a bar in a graph onto the vertical axis to assess its height (Boels et al. 2022a). For this reason, we provided students with an environment that allowed for sensorimotor processes and direct mathematical actions with mathematical objects, rather than having students manipulate sliders or enter numbers and let the digital environment perform the mathematical actions. ...

Density histograms can bridge the gap between histograms and continuous probability distributions, but research on how to learn and teach them is scarce. In this paper, we explore the learning of density histograms with the research question: How can a sequence of tasks designed from an embodied instrumentation perspective support students’ understanding of density histograms? Through a sequence of tasks based on students’ notions of area, students reinvented unequal bin widths and density in histograms. The results indicated that students had no difficulty choosing bin widths or using area in a histogram. Nevertheless, reinvention of the vertical density scale required intense teacher intervention suggesting that in future designs, this scale should be modified to align with students’ informal notions of area. This study contributes to a new genre of tasks in statistics education based on the design heuristics of embodied instrumentation. Keywords: Density histograms, Design-based research; Embodied design; Statistics education. Preprint available on request.

... The field of statistics, for example, was identified as a domain with rarely any eye-tracking studies (Strohmaier et al., 2020), although aspects of visualization and mental representations play a vital role in this domain. There are just a few very recent studies that used eye-tracking technology to study students' strategies and difficulties when interpreting and comparing statistical graphs such as histograms (e.g., Boels et al., 2019;Lyford and Boels, 2022). Eye-tracking is a suitable method to obtain information about visual attention and cognitive processing while students are solving problems, especially when visual strategies are involved (e.g., Andrà et al., 2009;Klein et al., 2018;Malone et al., 2020). ...

Comparing data distributions is a fundamental activity in statistics and a motivating learning opportunity in schools to initiate statistical thinking. Research has shown that many students tend to perceive a data distribution as a collection of individual values rather than as a conceptual entity that has certain features such as center, spread, and shape. These difficulties are reflected in students' tendency to focus on local details of the distribution (so-called local view of data) instead of referring to differences between the distributions as a whole (so-called global view of data). While many authors refer to school students' conceptions and difficulties related to their view of data, there is, to the best of our knowledge, no empirical study that investigated their actual viewing behavior (local vs. global) when comparing distributions. The central assumption of this study is that specific eye-tracking measures constitute indicators for the perceiving and processing of local vs. global distributional features. For this purpose, hypotheses for differences in certain eye-tracking measures (fixation count, saccade amplitude, and saccade direction) between students with a local and global view of data were theoretically derived and empirically investigated using a methodological combination of eye-tracking and stimulated recall interviews. We analyzed data of 25 sixth-grade students who each completed four items on distributional comparisons. The results showed strong positive inter-item correlations for all eye-tracking measures, indicating high internal consistency in participants' gaze behavior across all items. Based on the analysis of the eye-tracking stimulated recall interviews, we split our sample into those students who perceived and processed global features in half or more of the items (global view) and those below that threshold (local view). In line with our theoretically derived hypotheses, students with a global compared to a local view of data had on average significantly fewer fixations, longer saccade amplitudes, and a higher relative number of horizontal saccades. These results suggest that eye-tracking can assist in identifying students' conceptions and difficulties related to a local vs. global view of data. Implications for school practice and further research are discussed.

An approach to interdisciplinary data workshops has been developed that brings together mathematical science students and practitioners to work on problems with real data in the practitioners’ area of expertise. It combines ideas from training statisticians for statistical consultancy and developing data literacy skills in non-specialists. The approach was conceived through collaborative research projects aimed at tackling development challenges in Africa using data-based approaches to corruption in public procurement and farmer experimentation in agriculture. Implementation of the approach in four workshops in three African countries is presented. The workshops provided important learning outcomes both for students, experiencing a problem-solving approach to working with real data in a genuine context, and for practitioners to gain data awareness, literacy, and skills.KeywordsData skillsStatistical consultancy trainingStatistics training for non-specialistsData problem solvingCooperative learning

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