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Understanding Student Errors in Comparing Data Sets with Boxplots

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Title: Understanding Student Errors in Comparing Data Sets with Boxplots
Language: English
Authors: Martin Abt (ORCID 0009-0003-4221-3434), Katharina Loibl (ORCID 0000-0002-1773-1913), Timo Leuders (ORCID 0000-0002-7621-7826), Wim Van Dooren (ORCID 0000-0001-5002-4340), Frank Reinhold (ORCID 0000-0003-4468-024X)
Source: Educational Studies in Mathematics. 2025 120(1):169-193.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 25
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Education Level: Higher Education
Postsecondary Education
High Schools
Secondary Education
Descriptors: College Students, Data Analysis, Graphs, Error Patterns, Difficulty Level, Profiles, High School Graduates, Data Use
DOI: 10.1007/s10649-025-10387-z
ISSN: 0013-1954
1573-0816
Abstract: In the boxplot, the box always represents -- regardless of its area -- the middle half of the data and thus a measure of variability (interquartile range). However, when students first learn about boxplots, they are usual already familiar with other forms of statistical representations (e.g., bar or circle graphs) in which a larger area represents a higher frequency of observations. If students erroneously apply this well-established area-represents-frequency schema to boxplots, it results in a systematic error which we describe as the consequence of an incomplete conceptual change. We empirically validated difficulty-generating characteristics that allow the differentiation between item types with varying complexity (item level) and aimed to identify profiles (person level) that differ depending on which schema was used in which item type. For this purpose, we conducted two cross-sectional studies with N = 100 university students (study 1) and N = 297 participants who finished secondary school or higher (study 2) and used generalized linear mixed models (item level) and k-means clustering with predefined cluster centers (person level) to test our hypotheses. We could replicate the systematic error that was described in previous research and found new difficulty-generating characteristics in boxplot items. Our results support the notion of different profiles potentially emerging based on varying degrees of conceptual change. From an instructional perspective, information about individual progress in conceptual change could be considered for tailoring individualized interventions.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1481701
Database: ERIC
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  Value: <anid>AN0187497173;esm01sep.25;2025Aug26.02:33;v2.2.500</anid> <title id="AN0187497173-1">Understanding student errors in comparing data sets with boxplots </title> <p>In the boxplot, the box always represents – regardless of its area – the middle half of the data and thus a measure of variability (interquartile range). However, when students first learn about boxplots, they are usual already familiar with other forms of statistical representations (e.g., bar or circle graphs) in which a larger area represents a higher frequency of observations. If students erroneously apply this well-established area-represents-frequency schema to boxplots, it results in a systematic error which we describe as the consequence of an incomplete conceptual change. We empirically validated difficulty-generating characteristics that allow the differentiation between item types with varying complexity (item level) and aimed to identify profiles (person level) that differ depending on which schema was used in which item type. For this purpose, we conducted two cross-sectional studies with N = 100 university students (study 1) and N = 297 participants who finished secondary school or higher (study 2) and used generalized linear mixed models (item level) and k-means clustering with predefined cluster centers (person level) to test our hypotheses. We could replicate the systematic error that was described in previous research and found new difficulty-generating characteristics in boxplot items. Our results support the notion of different profiles potentially emerging based on varying degrees of conceptual change. From an instructional perspective, information about individual progress in conceptual change could be considered for tailoring individualized interventions.</p> <p>Keywords: Box plots; Cluster analysis; Conceptual change; Schema-based reasoning; Statistical literacy; Systematic error</p> <p>Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s10649-025-10387-z.</p> <hd id="AN0187497173-2">Introduction</hd> <p>The boxplot (Tukey, [<reflink idref="bib57" id="ref1">57</reflink>]) is a way to display a statistical distribution of one quantitative variable (Fig. 1). A distinctive characteristic of this representation is the juxtaposition of measures of central tendency (median) and variability (range and interquartile range; IQR).</p> <p>Graph: Fig. 1 A boxplot in the manner we use it in the studies presented here: The quartiles are plotted as short vertical lines above an x-axis. The middle half of the data, enclosed by the first (lower) and third (upper) quartile, is represented by a rectangle (box), which gives the representation its name. The median (second quartile) divides the distribution (as well as the box) into two (almost) equal subsets. The upper quarter of the data is represented as a line bounded by the upper quartile and the maximum (upper whisker), and the lower quarter is correspondingly represented as a line bounded by the minimum and the lower quartile (lower whisker) (Dodge, [<reflink idref="bib16" id="ref2">16</reflink>])</p> <p>That is why the boxplot is suitable for a quick and overviewlike comparison of distributions (Kader & Perry, [<reflink idref="bib25" id="ref3">25</reflink>]; Krzywinski & Altman, [<reflink idref="bib28" id="ref4">28</reflink>]) and is frequently found in scientific (Streit & Gehlenborg, [<reflink idref="bib56" id="ref5">56</reflink>]) and curricular (National Council of Teachers of Mathematics [NCTM], [<reflink idref="bib41" id="ref6">41</reflink>]) contexts. At the same time, the boxplot is challenging to learn (Bakker et al., [<reflink idref="bib7" id="ref7">7</reflink>]; Behrens et al., [<reflink idref="bib8" id="ref8">8</reflink>]; Edwards et al., [<reflink idref="bib19" id="ref9">19</reflink>]; Lem et al., [<reflink idref="bib32" id="ref10">32</reflink>]). It is complex because of the condensed representation of multiple descriptive information, and because the meaning of the box area fundamentally conflicts with the idea that a larger area always represents more observations—a usual interpretation in statistical representations (Bakker et al., [<reflink idref="bib7" id="ref11">7</reflink>]). In this paper, we consider learning about the boxplot as a process of conceptual change (Vosniadou & Verschaffel, [<reflink idref="bib67" id="ref12">67</reflink>]) and investigate whether this theory can explain systematic errors made by students when reasoning with boxplots.</p> <hd id="AN0187497173-3">Area-represents-frequency schema</hd> <p>When students first learn about boxplots, they are already familiar with other statistical representations such as circle or bar graphs (NCTM, [<reflink idref="bib41" id="ref13">41</reflink>]). Here, larger areas represent higher frequencies of observations, e.g., if students are asked to decide which party received the most votes in an election based on the two representations (Fig. 2), they can argue by comparing areas. We refer to this well-established type of reasoning as a (problem-solving) schema (Chi et al., [<reflink idref="bib14" id="ref14">14</reflink>]; Mandler, [<reflink idref="bib38" id="ref15">38</reflink>]; Rumelhart et al., [<reflink idref="bib49" id="ref16">49</reflink>]) and call it <emph>Area-Represents-Frequency Schema (ARFS)</emph>; it is part of the acquired domain-specific knowledge about interpreting statistical representations.</p> <p>Graph: Fig. 2 The number of votes received by the three parties A, B and C is displayed in a circle graph (left) and a bar graph (right). In both representations, a larger area represents more observations (here: votes)</p> <hd id="AN0187497173-4">ARFS and systematic errors in boxplot interpretation</hd> <p>If students overgeneralize ARFS with the boxplot, systematic errors can occur since area in the boxplot has a fundamentally different meaning (Abt et al., [<reflink idref="bib3" id="ref17">3</reflink>]; Bakker et al., [<reflink idref="bib7" id="ref18">7</reflink>]; Lem et al., [<reflink idref="bib33" id="ref19">33</reflink>]). The box—regardless of its width—always represents (roughly[<reflink idref="bib1" id="ref20">1</reflink>]) the middle half of the data. The box width—and since the height of the box is arbitrary, the box area as well—indicates how wide the middle half of the data scatters across the x-axis (Fig. 3).</p> <p>Graph: Fig. 3 Both data sets (A and B) include the same number of observations. As the box always represents the middle half of a distribution, both boxes contain the same number of observations as well. Nevertheless, the box widths differ, as the middle half of distribution A scatters over a wider range than the middle half of distribution B</p> <p>This range of the middle half of the data (i.e., interquartile range, IQR) is a robust measure of variability for the entire distribution (Dodge, [<reflink idref="bib16" id="ref21">16</reflink>]). We call tasks in which boxplots are used to decide what percentage of a distribution is above, below or between one or two predefined critical value(s) <emph>Critical-Value-Comparison Task (CVCT)</emph>. Such CVCTs can be considered a typical application scenario for boxplots (Tukey, [<reflink idref="bib57" id="ref22">57</reflink>]) and have been used as diagnostic items in previous studies (Abt et al., [<reflink idref="bib3" id="ref23">3</reflink>]; Lem et al., [<reflink idref="bib33" id="ref24">33</reflink>]): Students are asked to determine which of two schools has more children walking more than 10 min to school in the morning. In the item on the left (Fig. 4a), boxplot A shows a median above the critical value. It can therefore be assumed that more than half of the children have a walk to school of more than 10 min. The median in boxplot B is below the critical value, so that it can be assumed that less than half of the children at school B have a walk to school of more than 10 min; thus, school A is the correct answer. Following the same reasoning, school B is the correct answer in the middle item (Fig. 4b). In the item on the right (Fig. 4c), the question cannot be answered because the number of quartiles above the critical value does not differ in both boxplots. However, if students overgeneralize ARFS, they argue that more area above the critical value also represents a higher frequency of observations and thus select boxplot A in all three items, as boxplot A shows a larger (box) area above the critical value. In the item on the left (Fig. 4a), ARFS consistently yields correct solutions, which is why we refer to such items as <emph>ARFS-congruent</emph>. In the other two item types (Fig. 4b/c), applying ARFS consistently yields incorrect answers, which is why we refer to such items as <emph>ARFS-incongruent</emph>.</p> <p>Graph: Fig. 4 Example task with three different categories of items: (a) is decidable and ARFS-congruent, (b) is decidable and ARFS-incongruent, (c) is undecidable and ARFS-incongruent</p> <p>The occurrence of such systematic errors when using ARFS in CVCTs was first shown by Lem et al. ([<reflink idref="bib33" id="ref25">33</reflink>]). They used congruent and incongruent items and argued that the meaning of area in boxplots—in contrast to the meaning of area in other (statistical) representations—contradicts Tversky's ([<reflink idref="bib58" id="ref26">58</reflink>]) idea of 'the larger the more or the better' as the natural way to interpret area in graphs. On a theoretical level, Lem et al. ([<reflink idref="bib33" id="ref27">33</reflink>]) interpreted the use of ARFS in terms of dual process as a heuristic that prevents analytical reasoning (Evans, [<reflink idref="bib20" id="ref28">20</reflink>]), i.e., although students have acquired the knowledge necessary to interpret boxplots correctly, ARFS—understood as a heuristic that needs to be overcome by analytical thinking—provides a competing, fast, intuitive, and convenient way of interpreting the (box) area. Following these assumptions, the systematic error should have been amplified by increasing time constraints and mitigated by warnings against heuristic reasoning. However, Lem et al. ([<reflink idref="bib33" id="ref29">33</reflink>]) did not find this effect in their study. In a second study, they conducted an intervention which was focusing more on the acquisition of conceptual knowledge; they were able to improve solution rates in incongruent items yet did not entirely eradicate the application of ARFS.</p> <p>The findings that the competing of a heuristic and an analytic thinking process may not be a sufficient explanation for ARFS-related errors and applying ARFS in boxplots has a more fundamental origin serves as starting point for the present studies. We argue that using ARFS in boxplots is a consequence of missing conceptual knowledge about the meaning of the box (area) in boxplots rather than a heuristic. This claim is supported by the positive effect of refutational texts (Lem et al., [<reflink idref="bib30" id="ref30">30</reflink>]) and, conversely, by the finding that traditional instruction on the boxplot is rarely aimed at the acquisition of conceptual knowledge, but rather focuses on procedural skills (Bakker, [<reflink idref="bib6" id="ref31">6</reflink>]; Edwards et al., [<reflink idref="bib19" id="ref32">19</reflink>]; Garfield & Ben-Zvi, [<reflink idref="bib21" id="ref33">21</reflink>]). In the following section, we outline how conceptual change theory can contribute to explaining this systematic error.</p> <hd id="AN0187497173-5">Conceptual change and ARFS-related errors</hd> <p>The term <emph>conceptual change</emph> (Kuhn, [<reflink idref="bib29" id="ref34">29</reflink>]) describes learning "in such domains where the pre-instructional conceptual structures of the students have to be fundamentally restructured in order to allow understanding of the intended knowledge" (Duit & Treagust, [<reflink idref="bib18" id="ref35">18</reflink>], p. 673). Conceptual change is understood as a useful explanatory model for learning processes in mathematics education (Schneider et al., [<reflink idref="bib51" id="ref36">51</reflink>]; Vosniadou, [<reflink idref="bib63" id="ref37">63</reflink>]; Vosniadou & Verschaffel, [<reflink idref="bib67" id="ref38">67</reflink>]), in particular regarding systematic errors (Vamvakoussi et al., [<reflink idref="bib59" id="ref39">59</reflink>]). By now, conceptual change has been used in various areas to describe learning processes in which prior knowledge competes with the new concepts to be acquired. For instance, the understanding of negative numbers (Vlassis, [<reflink idref="bib61" id="ref40">61</reflink>]), the transition from geometric to analytic concepts (Biza & Zachariades, [<reflink idref="bib10" id="ref41">10</reflink>]), as well as the transition from natural to rational numbers (Ni & Zhou, [<reflink idref="bib42" id="ref42">42</reflink>]; Reinhold et al., [<reflink idref="bib48" id="ref43">48</reflink>]; Van Hoof et al., [<reflink idref="bib60" id="ref44">60</reflink>]) can be mentioned. Here, we refer to the <emph>framework theory approach</emph> (Vosniadou, [<reflink idref="bib62" id="ref45">62</reflink>]) and consider conceptual change as a domain-specific integration of newly acquired concepts into existing knowledge structures. Posner et al. ([<reflink idref="bib46" id="ref46">46</reflink>]) assume conceptual change wherever the transfer of existing knowledge to a new task (assimilation) is recognized as inappropriate. This understanding of a radical and conscious change (accommodation) of one's own knowledge structures has been criticized many times (e.g., Caravita & Halldén, [<reflink idref="bib12" id="ref47">12</reflink>]; Vosniadou & Brewer, [<reflink idref="bib65" id="ref48">65</reflink>]). In contrast, Vosniadou ([<reflink idref="bib62" id="ref49">62</reflink>]) describes conceptual change as a long-term and gradual process in which intermediate stages, so-called synthetic (mental) models, can be formed. However, a consequence of this so-called classical approach (Vosniadou, [<reflink idref="bib64" id="ref50">64</reflink>]) is that narrowing the scope of validity of prior knowledge is an essential aspect of conceptual change. We consider the awareness that a mathematical schema often is not inherently correct or incorrect, but that its validity depends on the situation in which it is used to be an essential part of conceptual change (Limón, [<reflink idref="bib37" id="ref51">37</reflink>]; Reinhold, [<reflink idref="bib47" id="ref52">47</reflink>]; Smith et al., [<reflink idref="bib54" id="ref53">54</reflink>]; Spada, [<reflink idref="bib55" id="ref54">55</reflink>]).</p> <p>While ARFS retains its validity in many statistical representations, new schemas have to be integrated into the already established knowledge structures and replace ARFS in boxplots. We describe this process of restructuring the domain-specific knowledge about reasoning in statistical representations in terms of mental models (Johnson-Laird, [<reflink idref="bib24" id="ref55">24</reflink>]). According to Seel et al. ([<reflink idref="bib52" id="ref56">52</reflink>]), a mental model is formed when the assimilation of a new task into an existing schema (Anderson, [<reflink idref="bib4" id="ref57">4</reflink>]) is perceived as unsuccessful and the schema cannot be satisfactorily accommodated by accretion (i.e., adding new components or dimensions to an existing schema) or tuning (i.e., making an existing schema more accurate or fitting to the current task) (Rumelhart & Norman, [<reflink idref="bib50" id="ref58">50</reflink>]). We define the <emph>initial model</emph> (Vosniadou, [<reflink idref="bib62" id="ref59">62</reflink>]) as the state where no narrowing of the scope of validity has taken place. This is where ARFS is used systematically in all statistical representations (including the boxplot) and where no alternative schemas have been acquired (Vosniadou & Brewer, [<reflink idref="bib65" id="ref60">65</reflink>]; Vosniadou & Skopeliti, [<reflink idref="bib66" id="ref61">66</reflink>]). Regarding the use of ARFS in statistical representations, we argue that such intermediate stages (<emph>synthetic models</emph>) emerge when the scope of validity has not been <emph>fully</emph> restricted, implying that ARFS has been replaced by appropriate schemas in some but not all cases.</p> <hd id="AN0187497173-6">The present studies</hd> <p>This study aims to identify the cognitive processes that drive the use of ARFS as a source of errors when comparing distributions with boxplots. From a theoretical perspective, we argue that conceptual change is a necessary part of knowledge acquisition about boxplots. In light of this, it is crucial to identify empirical evidence that supports this explanatory model. We conducted two correlational studies to investigate whether (<reflink idref="bib1" id="ref62">1</reflink>) ARFS holds as an explanation for systematic errors within our data and whether (<reflink idref="bib2" id="ref63">2</reflink>) data can be interpreted as suggesting that different systematic error patterns may be associated with individual synthetic models, which could reflect varying degrees of progression in ongoing conceptual change processes. We expect such evidence on both an item and on a person level. We aimed at empirically validating theory-driven difficulty-generating characteristics in the CVCTs that allow the differentiation between item types with varying complexity. We assumed that items of a given item type are not solved by all students according to a uniform schema; rather, distinct profiles are identifiable according to which schema is employed in specific item types. The existence of these profiles may be interpreted as empirical support for the theoretically hypothesized synthetic model in some students.</p> <hd id="AN0187497173-7">Study 1</hd> <p>Research has focused on items in which a decision in CVCT was possible based on a comparison of medians (<emph>median schema</emph>). We call those items <emph>median items</emph>. In this study, we used an additional <emph>item type</emph> in which the entire box, i.e., the middle half of the data, is above or below the critical value—<emph>box items</emph>; a correct solution can be determined not by comparing the medians but by comparing the position of the boxes (<emph>box schema</emph>). With respect to ARFS, we use both item types (median items and box items) in a congruent and an incongruent configuration, resulting in four <emph>item categories</emph> (Fig. 5).</p> <p>Graph: Fig. 5 Four item categories used in study 1, differing regarding ARFS congruency (ARFS-congruent / ARFS-incongruent) and item type (median item / box item). The correct answer is always A</p> <hd id="AN0187497173-8">Research questions</hd> <p>On an item level, we are interested in whether ARFS is an explanation for errors in solving the above described CVCT, implying that ARFS incongruency is a difficulty-generating characteristic.</p> <hd id="AN0187497173-9">Hypothesis 1a</hd> <p>ARFS-congruent items are more likely to be solved correctly than ARFS-incongruent items.</p> <p>Secondly, research showed that the box is the most salient component of the boxplot and makes other components, such as the whiskers, fade into the background. While the whiskers, the minimum and the maximum are only one-dimensional lines, the box is the only area present in the representation. This may lead students to ignore the fact that the whiskers together also represent 50% of the distribution and instead understand the box as representing the entire distribution (Bakker et al., [<reflink idref="bib7" id="ref64">7</reflink>]; Lem et al., [<reflink idref="bib31" id="ref65">31</reflink>]); implying that the difficulty of median items and box items may differ.</p> <hd id="AN0187497173-10">Hypothesis 1b</hd> <p>Box items are more likely to be solved correctly than median items.</p> <p>A completed conceptual change implies that the scope of validity of ARFS has been fully narrowed by appropriate schemas (Sect. 1.3). We assumed that students who completed the conceptual change will solve median items applying the median schema and box items applying the box schema and will consequently show high solution rates in all item categories, regardless of item type or ARFS congruency (Table 1, <emph>proficient model</emph>). When the scope of ARFS has been narrowed partially but ARFS was not completely replaced by appropriate schemas, we hypothesize that synthetic models may be formed. In median items, a decision can be made based on a central value. If students identify a central value (median) in the boxplot, it is plausible to assume that they use prior knowledge on central values to make a correct decision, whereas (boxplot-specific) knowledge required to correctly solve box items (box schema) must (still) be acquired during the instruction on boxplots. Regardless of ARFS congruency, these students will show high solution rates in median items. However, even box items are not solved randomly but consistently according to ARFS. Therefore, these students will show high solution rates in ARFS-congruent box items and solution rates below guessing probability in ARFS-incongruent box items (Table 1, synthetic model A). Conversely, if students acquired boxplot-specific knowledge that can be used to solve box items while prior knowledge on central values has not (yet) been successfully linked to the boxplot, such students will show high solution rates in box items regardless of ARFS congruency, while they will consistently solve median items according to ARFS and therefore show high solution rates in ARFS-congruent median items and solution rates below guessing probability in ARFS-incongruent median items (Table 1, synthetic model B). If students have not acquired any appropriate schema that is valid in boxplots, we assume that the scope of validity of ARFS has not (yet) been narrowed and that students therefore apply ARFS consistently in all item types and solve ARFS-congruent items with high solution rates and ARFS-incongruent items with a solution rate below guessing probability (Table 1, initial model). This approach, shown in Table 1, is consistent with research on the natural number bias in fraction tasks, in which the systematic variation of task characteristics has been used to infer cognitive processes and reveal specific cognitive biases (Reinhold et al., [<reflink idref="bib48" id="ref66">48</reflink>]).</p> <p>Table 1 Expected pattern in accuracy for the different profiles</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" colspan="3"><p>cognition: applied schemas in specific item types</p></th><th align="left" colspan="4"><p>behavior: expected solution rates in different item categories</p></th></tr><tr><th align="left" /><th align="left" /><th align="left" /><th align="left" colspan="2"><p>median items</p></th><th align="left" colspan="2"><p>box items</p></th></tr><tr><th align="left" /><th align="left" /><th align="left" /><th align="left" colspan="2"><p>ARFS congruency</p></th><th align="left" colspan="2"><p>ARFS congruency</p></th></tr><tr><th align="left"><p>profile</p></th><th align="left"><p>median items</p></th><th align="left"><p>box items</p></th><th align="left"><p>congruent</p></th><th align="left"><p>incongruent</p></th><th align="left"><p>congruent</p></th><th align="left"><p>incongruent</p></th></tr></thead><tbody><tr><td align="left"><p>1 proficient model</p></td><td align="left"><p>median schema</p></td><td align="left"><p>box schema</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td></tr><tr><td align="left"><p>2 synthetic model A</p></td><td align="left"><p>median schema</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td></tr><tr><td align="left"><p>3 synthetic model B</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>box schema</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td></tr><tr><td align="left"><p>4 initial model</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0187497173-11">Hypothesis 1c</hd> <p>We assume that we can identify subgroups in our sample corresponding to the hypothesized profiles (Table 1).</p> <hd id="AN0187497173-12">Method</hd> <p></p> <hd id="AN0187497173-13">Participants</hd> <p> <emph>N</emph> = 100 students participated in Study 1 (48 psychology students, 37 student teachers, 15 computer science students). All students had covered the boxplot in lectures or seminars at the time of the survey. Participants received no gratuity for participation.</p> <hd id="AN0187497173-14">Materials</hd> <p>We designed an item set of four CVCT items per item category (16 items) in the following context: Two boxplots represent the body length of two basketball teams (same number of members) and the participants had to decide in which team more players had a body length of more than 1.75 m. Figure 6a shows all possible arrangements of quartiles around a critical value. In study 1, we only used those items in which a decision is generally possible. Such items are characterized by the fact that the number of quartiles above the critical value is not equal in both boxplots (Fig. 6b). Furthermore, we limited to items in which both boxplots (not necessarily the boxes) overlap the critical value. By definition, median items are items that can be solved with the median schema and box items are items that can be solved with the box schema (indicated in Fig. 6b). We find items that can be solved by applying the median schema <emph>or</emph> by applying the box schema, and can be considered as both, a median and a box item. We excluded such items from study 1, since we were planning to use the accuracy to draw conclusions about the schema that was used. We used the remaining six arrangements presented in Fig. 6c to create the item set. The mirroring on the main diagonal leads to an inversion of boxplot A and B and thus also to an inversion of the correct solution. The arrangements[<reflink idref="bib2" id="ref67">2</reflink>] (<reflink idref="bib1" id="ref68">1</reflink>,<reflink idref="bib2" id="ref69">2</reflink>) and (<reflink idref="bib2" id="ref70">2</reflink>,<reflink idref="bib1" id="ref71">1</reflink>) always lead to ARFS-congruent box items, where exactly one box is completely below the critical value. They cannot be designed ARFS-incongruent, so we created four ARFS-congruent box items from these two arrangements. The arrangements (<reflink idref="bib3" id="ref72">3</reflink>,<reflink idref="bib2" id="ref73">2</reflink>) and (<reflink idref="bib2" id="ref74">2</reflink>,<reflink idref="bib3" id="ref75">3</reflink>) lead to median items and can be designed both, ARFS-congruent and ARFS-incongruent. From these arrangements, we created four ARFS-congruent and four ARFS-incongruent median items. The arrangements (<reflink idref="bib4" id="ref76">4</reflink>,<reflink idref="bib3" id="ref77">3</reflink>) and (<reflink idref="bib3" id="ref78">3</reflink>,<reflink idref="bib4" id="ref79">4</reflink>) lead to box items where one box is completely above the critical value. They can be designed both, ARFS-congruent and ARFS-incongruent. Since the ARFS-congruent items were already created from the arrangements (<reflink idref="bib1" id="ref80">1</reflink>,<reflink idref="bib2" id="ref81">2</reflink>) and (<reflink idref="bib2" id="ref82">2</reflink>,<reflink idref="bib1" id="ref83">1</reflink>), we created four ARFS-incongruent box items from the two arrangements (<reflink idref="bib4" id="ref84">4</reflink>,<reflink idref="bib3" id="ref85">3</reflink>) and (<reflink idref="bib3" id="ref86">3</reflink>,<reflink idref="bib4" id="ref87">4</reflink>). In 10 items, boxplot B, in the remaining 6 items, boxplot A was the correct solution. We presented these 16 items in 4 blocks with each block including all four item categories in a random order which was the same for all participants.</p> <p>Graph: Fig. 6 The matrices show the number of quartiles above the critical value for each of the two boxplots. The characters indicate the schema(s) that can be used to solve the item correctly (M = median schema, B = box schema, M / B = both schemas possible). Arrangements that were not included in study 1 are grayed out</p> <hd id="AN0187497173-15">Procedure</hd> <p>Limesurvey was used for the data collection. Participation was anonymous and voluntary. Participants could give informed consent at the beginning of the survey whether their data may be used for scientific purposes. The items were presented one after the other. Participants had to choose one of both boxplots (A or B) without a time limit. If items were not answered by individual participants, the respective answer was excluded from the analysis without losing the person's remaining answers to other items.</p> <hd id="AN0187497173-16">Data analysis</hd> <p>R (version 2024.04.2 + 764) was utilized for the data analysis. Generalized linear mixed models (GLMMs, lme4, version 1.1–35.5) were used to estimate differences in item difficulty between items with varying difficulty-generating characteristics (fixed effects; Hypotheses 1ab). In addition to these fixed effects, the GLMM can also simultaneously model random effects that can be assumed based on the hierarchical structure of a data set. In this case, <emph>participant</emph> and <emph>item</emph> were included as random effects. To answer Hypothesis 1c of whether the participants showed a solution behavior that corresponded to one of the hypothesized profiles (Table 1), we used a <emph>confirmatory k-means cluster algorithm</emph>, i.e., the initial cluster centers are not chosen randomly, and the number of clusters is not determined by empirical cut-off criteria based on the available data, but predefined based on theoretical hypotheses (four cluster centers, one for each profile, Table 1). The objective of the k-means method is to partition the data points into k groups in a manner that minimizes the sum of squares from points to the assigned cluster centers. In the most optimal scenario, all cluster centers are situated at the mean of their Voronoi sets, which represent the set of data points that are nearest to each cluster center. Each of these four predefined cluster centers consists of the four solution rates (one for each item category, high = 1, low = 0). Additionally, for <emph>non-classifiable</emph> participants, we predefined a fifth cluster center in which we expected the item categories not to be solved by a consistent approach; we assumed solution rates at guessing probability (50%). The algorithm classified each participant into either one of the four hypothesized profiles or into the non-classifiable profile.</p> <hd id="AN0187497173-17">Results</hd> <p>Table 2 shows the estimated marginal means resulting from the GLMM for the four item categories (Fig. 5). On an item level our item variables explained about 88% of the item variance that is considered random in a model without predictors (Nakagawa & Schielzeth, [<reflink idref="bib40" id="ref88">40</reflink>]).</p> <p>Table 2 Estimated marginal mean values for the solution rates in the four item categories in study 1</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>item type</p></th><th align="left"><p>ARFS congruency</p></th><th align="left"><p>estimated M</p></th><th align="left"><p>SE</p></th><th align="left"><p>95%CI</p></th></tr></thead><tbody><tr><td align="left"><p>median item</p></td><td align="left"><p>congruent</p></td><td char="." align="char"><p>0.963</p></td><td char="." align="char"><p>0.013</p></td><td align="left"><p>[0.927, 0.982]</p></td></tr><tr><td align="left" /><td align="left"><p>incongruent</p></td><td char="." align="char"><p>0.690</p></td><td char="." align="char"><p>0.055</p></td><td align="left"><p>[0.574, 0.786]</p></td></tr><tr><td align="left"><p>box item</p></td><td align="left"><p>congruent</p></td><td char="." align="char"><p>0.948</p></td><td char="." align="char"><p>0.018</p></td><td align="left"><p>[0.901, 0.973]</p></td></tr><tr><td align="left" /><td align="left"><p>incongruent</p></td><td char="." align="char"><p>0.776</p></td><td char="." align="char"><p>0.049</p></td><td align="left"><p>[0.666, 0.857]</p></td></tr></tbody></table> </ephtml> </p> <p>In line with Hypothesis 1a, we found that ARFS-congruent items were more likely to be solved correctly than ARFS-incongruent items (<emph>OR</emph> = 7.93, 95%CI [4.39,14.32], <emph>p</emph> < 0.001), whereas we found no effect in line with Hypothesis 1b that box items could be solved more likely than median items in the present sample (Table 3).</p> <p>Table 3 Results of the generalized linear mixed model</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" colspan="3"><p>model 0</p></th><th align="left" colspan="3"><p>model 1</p></th></tr><tr><th align="left" /><th align="left"><p>odds ratios</p></th><th align="left"><p>95% CI</p></th><th align="left"><p>p</p></th><th align="left"><p>odds ratios</p></th><th align="left"><p>95% CI</p></th><th align="left"><p>p</p></th></tr></thead><tbody><tr><td align="left"><p><bold>fixed effects</bold></p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char" /><td char="." align="char" /><td align="left" /></tr><tr><td align="left"><p>ARFS congruency (0 = incongruent, 1 = congruent)</p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char"><p>7.93</p></td><td char="," align="char"><p>[4.39,14.32]</p></td><td align="left"><p> < 0.001</p></td></tr><tr><td align="left"><p>item type (0 = median item, 1 = box item)</p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char"><p>1.19</p></td><td char="," align="char"><p>[0.69,2.08]</p></td><td align="left"><p>0.530</p></td></tr><tr><td align="left"><p><bold>random effects</bold></p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char" /><td char="." align="char" /><td align="left" /></tr><tr><td align="left"><p>student (<italic>N</italic> = <italic>100</italic>)</p></td><td char="." align="char"><p>1.32</p></td><td align="left" /><td align="left" /><td char="." align="char"><p>1.33</p></td><td char="." align="char" /><td align="left" /></tr><tr><td align="left"><p>items (<italic>N</italic> = <italic>16</italic>)</p></td><td char="." align="char"><p>1.28</p></td><td align="left" /><td align="left" /><td char="." align="char"><p>0.16</p></td><td char="." align="char" /><td align="left" /></tr></tbody></table> </ephtml> </p> <p>The confirmatory <emph>k</emph>-means clustering showed results in line with the hypothesized profiles (Fig. 7). For the first profile (proficient model, <emph>n</emph><subs><emph>1</emph></subs> = 56) we assumed high solution rates in all four item categories. For the second profile (synthetic model A, <emph>n</emph><subs>2</subs> = 13) we assumed high solution rates except for ARFS-incongruent box items. For the third profile (synthetic model B, <emph>n</emph><subs>3</subs> = 14) we assumed high solution rates except for ARFS-incongruent median items. For the fourth profile (initial model, <emph>n</emph><subs>4</subs> = 6) we assumed low solution rates for all ARFS-incongruent items and high solution rates for all ARFS-congruent items. Participants who showed a pattern which was not assigned to one of our hypothesized patterns (<emph>n</emph><subs>5</subs> = 11) are found in the fifth profile. However, even these participants did not show a random response pattern (in the sense of guessing), but rather consistent high solution rates in median items and medium solution rates in box items. Since in half of the box items a decision according to the median leads to the correct answer, this pattern could be an indication of a median overgeneralization. However, with the item categories from study 1, we were not able to clearly distinguish such an overgeneralization from guessing—the starting point for study 2.</p> <p>Graph: Fig. 7 Result of the k-means cluster algorithm with predefined cluster centers as described in Table 1. Green graphs represent the hypothesized profiles, orange graphs the profiles actually found in our data. M_AC = ARFS-congruent median items, M_AI = ARFS-incongruent median items, B_AC = ARFS-congruent box items, B_AI = ARFS-incongruent box items</p> <hd id="AN0187497173-18">Study 2</hd> <p>In study 1, we assumed that either the appropriate schema is applied (median schema in median items or box schema in box items), or the decisions are made according to ARFS; we excluded items in which both, the median schema and the box schema would have been appropriate (Sect. 2.2.2). However, in box items a decision can be made by comparing the medians as well, although both medians are above the critical value and therefore the median schema is <emph>never</emph> appropriate. In the items used in study 2, we therefore distinguished a second level of congruency for box items, the <emph>median congruency</emph>. In <emph>median-congruent</emph> box items, the (inappropriate) use of the median schema leads to the same correct answer as the use of the box schema, whereas in <emph>median-incongruent</emph> box items, the use of the median schema leads to the incorrect answer. Thus, in total we distinguish six item categories in study 2 (Fig. 8).</p> <p>Graph: Fig. 8 Six item categories used in study 2, differing regarding ARFS congruency (ARFS-congruent / ARFS-incongruent), median congruency (median-congruent / median-incongruent) and item type (median item / box item)</p> <hd id="AN0187497173-19">Research questions</hd> <p>The rationale of study 2 is identical to study 1.</p> <hd id="AN0187497173-20">Hypothesis 2a</hd> <p>ARFS-congruent items are more likely to be solved correctly than ARFS-incongruent items.</p> <hd id="AN0187497173-21">Hypothesis 2b</hd> <p>Box items are more likely to be solved correctly than median items.</p> <p>Given the extended itemset, we additionally ask whether median incongruency is an additional difficulty-generating characteristic in box items and state the following hypothesis.</p> <hd id="AN0187497173-22">Hypothesis 2c</hd> <p>Box items that are median-congruent are more likely to be solved correctly than box items that are median-incongruent.</p> <p>On a person level, we assumed that in addition to the profiles identified in study 1 two additional profiles (representing additional synthetic models) are plausible within the extended item set. In these two additional profiles, students do not apply ARFS in box items but instead use the median schema. The two profiles differ regarding how median items are solved. If students overgeneralize the median schema and apply it in both, median items and box items, these students show high solution rates in median items regardless of ARFS congruency. In box items, these students systematically use the median schema and therefore show high solution rates in the median-congruent box items and solution rates below guessing probability in median-incongruent box items (Table 4, synthetic model C). If students use the median schema in box items but apply ARFS in median items, these students show high solution rates in ARFS-congruent median items and in median-congruent box items, but solution rates below the guessing probability in the three remaining item categories (Table 4, synthetic model D).</p> <p>Table 4 Expected pattern in accuracy for the different profiles</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" colspan="4"><p>cognition: applied schemas in specific item types</p></th><th align="left" colspan="6"><p>behavior: expected solution rates in different item categories</p></th></tr><tr><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" colspan="2"><p>median items</p></th><th align="left" colspan="4"><p>box items</p></th></tr><tr><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" colspan="2"><p>ARFS congruency</p></th><th align="left" colspan="4"><p>ARFS congruency</p></th></tr><tr><th align="left" /><th align="left"><p>profile</p></th><th align="left"><p>median items</p></th><th align="left"><p>box items</p></th><th align="left"><p>congr</p></th><th align="left"><p>incongr</p></th><th align="left" colspan="2"><p>congr</p></th><th align="left" colspan="2"><p>incongr</p></th></tr></thead><tbody><tr><td align="left"><p>study 1</p></td><td align="left"><p>1 proficient model</p></td><td align="left"><p>median schema</p></td><td align="left"><p>box schema</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left" colspan="2"><p>high</p></td><td align="left" colspan="2"><p>high</p></td></tr><tr><td align="left" /><td align="left"><p>2 synthetic model A</p></td><td align="left"><p>median schema</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left" colspan="2"><p>high</p></td><td align="left" colspan="2"><p>low</p></td></tr><tr><td align="left" /><td align="left"><p>3 synthetic model B</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>box schema</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left" colspan="2"><p>high</p></td><td align="left" colspan="2"><p>high</p></td></tr><tr><td align="left" /><td align="left"><p>4 initial model</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left" colspan="2"><p>high</p></td><td align="left" colspan="2"><p>low</p></td></tr><tr><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" colspan="2"><p>median congruency</p></td><td align="left" colspan="2"><p>median congruency</p></td></tr><tr><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left"><p>congr</p></td><td align="left"><p>incongr</p></td><td align="left"><p>congr</p></td><td align="left"><p>incongr</p></td></tr><tr><td align="left"><p>study 2</p></td><td align="left"><p>1 proficient model</p></td><td align="left"><p>median schema</p></td><td align="left"><p>box schema</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td></tr><tr><td align="left" /><td align="left"><p>2 synthetic model A</p></td><td align="left"><p>median schema</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>low</p></td></tr><tr><td align="left" /><td align="left"><p>3 synthetic model B</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>box schema</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td></tr><tr><td align="left" /><td align="left"><p>4 initial model</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>low</p></td></tr><tr><td align="left" /><td align="left"><p>5 synthetic model C</p></td><td align="left"><p>median schema</p></td><td align="left"><p>median schema</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td></tr><tr><td align="left" /><td align="left"><p>6 synthetic model D</p></td><td align="left"><p>ARFS</p></td><td align="left"><p>median schema</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td><td align="left"><p>high</p></td><td align="left"><p>low</p></td></tr></tbody></table> </ephtml> </p> <p> <bold> <emph>Note:</emph> </bold> congr = congruent, incongr = incongruent</p> <hd id="AN0187497173-23">Hypothesis 2d</hd> <p>We assume that we can identify subgroups in our sample corresponding to the hypothesized and described profiles (Table 4).</p> <hd id="AN0187497173-24">Method</hd> <p></p> <hd id="AN0187497173-25">Participants</hd> <p>For study 2, we used the prolific.com platform to recruit participants (<emph>N</emph> = 297). Prolific (prolific.com) was utilized for participant recruitment due to its reputation for providing a reliable pool of registered users with verified identities. This reduces the potential for issues such as multiple participation and improves the quality of the data collected (Peer et al., [<reflink idref="bib44" id="ref89">44</reflink>], [<reflink idref="bib45" id="ref90">45</reflink>]). Crowdsourced samples, including those from Prolific, have been demonstrated to be reliable alternatives to lab-based samples, yielding consistent results across studies (Buhrmester et al., [<reflink idref="bib11" id="ref91">11</reflink>]; Casler et al., [<reflink idref="bib13" id="ref92">13</reflink>]; Crump et al., [<reflink idref="bib15" id="ref93">15</reflink>]). The survey took place from October 31 to November 8, 2023. Good knowledge of English language was a prerequisite for the survey, so we admitted participants from the following countries: United States of America, Canada, Ireland, United Kingdom, New Zealand, and Australia. We paid an expense allowance of GBP 3 for participation in the survey. The youngest participant was 19 years old, the oldest was 76 years old (<emph>M</emph> = 37.67, <emph>SD</emph> = 12.99), and 50.34% participants were female. The educational attainment at the time of participation ranged from graduating secondary school to holding a doctorate.</p> <hd id="AN0187497173-26">Materials</hd> <p>Participants watched a short introduction video in which the image shown in Fig. 9 was built up step by step. The video had no sound, lasted 1:40 min and described procedurally how the boxplot is created from an ordered list by counting off the quartiles and then drawing them over an x-axis. The introduction video thus simulates the traditional instructional approach to the boxplot as practiced regularly at school and university (Ben-Zvi & Garfield, [<reflink idref="bib21" id="ref94">21</reflink>]; Edwards et al., [<reflink idref="bib19" id="ref95">19</reflink>]). We tested this procedural knowledge in two short tasks within the survey as an implementation check (<emph>M</emph> = 0.71, <emph>SD</emph> = 0.26). These items were based on the prior knowledge test previously used by Lem et al. ([<reflink idref="bib33" id="ref96">33</reflink>]) and can be considered as knowledge about the boxplot representation, available to the students when answering the CVCTs. Since the items were not exclusively requiring dichotomous responses, but rather to read off values or name properties of the boxplot, the result can be understood as a generally robust declarative-procedural prior knowledge of boxplots.</p> <p>Graph: Fig. 9 Introduction video on boxplots</p> <p>In addition to the restrictions mentioned in Sect. 2.2 which also apply to the item creation process of study 2 (Fig. 10a), we excluded another two quartile arrangements and used only the four arrangements (Fig. 10b). In this way, we used the arrangements (<reflink idref="bib2" id="ref97">2</reflink>,<reflink idref="bib3" id="ref98">3</reflink>) and (<reflink idref="bib3" id="ref99">3</reflink>,<reflink idref="bib2" id="ref100">2</reflink>) for median items and (<reflink idref="bib3" id="ref101">3</reflink>,<reflink idref="bib4" id="ref102">4</reflink>) and (<reflink idref="bib4" id="ref103">4</reflink>,<reflink idref="bib3" id="ref104">3</reflink>) for box items. The arrangement (<reflink idref="bib1" id="ref105">1</reflink>,<reflink idref="bib2" id="ref106">2</reflink>), which was used in study 1, always results in ARFS-congruent items; with the restriction to the arrangements (<reflink idref="bib2" id="ref107">2</reflink>,<reflink idref="bib3" id="ref108">3</reflink>) and (<reflink idref="bib3" id="ref109">3</reflink>,<reflink idref="bib4" id="ref110">4</reflink>), each arrangement could be designed both, ARFS-congruent and ARFS-incongruent. In total we created 36 items, six for each of the six item categories (Online Resource 1).</p> <p>Graph: Fig. 10 The matrices show the number of quartiles above the critical value for each of the two boxplots. The characters indicate the schema(s) that can be used to solve the item correctly (M = median schema, B = box schema). Arrangements that were not included in study 2 are grayed out</p> <hd id="AN0187497173-27">Procedure</hd> <p>Registered prolific.com users were able to take part in the study voluntarily. Participants gave informed consent at the beginning of the survey whether their data may be used for scientific purposes. They were then redirected from the prolific.com website to our Limesurvey site. After watching the introduction video and completing the implementation check, participants answered the 36 items in succession. We presented these 36 items in 6 blocks with each block including all six item categories in a random order which was the same for all participants. Each item was visible for a maximum of 10 s, but the answer could still be given afterwards. Each item had to be answered, and no item could be skipped. The median total completion time for the study was 12.32 min (<emph>IQR</emph> = 4.58).</p> <hd id="AN0187497173-28">Data analysis</hd> <p>The analytic procedure was identical in study 1 and study 2. In study 2, profiles in which the median schema is used in box items could now be distinguished.</p> <hd id="AN0187497173-29">Results</hd> <p>Table 5 shows the estimated marginal means resulting from the GLMM for the six item categories (Fig. 8). On an item level our item variables explained about 84% of the item variance that is considered random in a model without predictors (Nakagawa & Schielzeth, [<reflink idref="bib40" id="ref111">40</reflink>]).</p> <p>Table 5 Estimated marginal mean values for the solution rates in the six item categories in study 2</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left"><p>item type</p></th><th align="left"><p>ARFS congruency</p></th><th align="left"><p>median congruency</p></th><th align="left"><p>estimated M</p></th><th align="left"><p>SE</p></th><th align="left"><p>95%CI</p></th></tr></thead><tbody><tr><td align="left"><p>median item</p></td><td align="left"><p>congruent</p></td><td align="left" /><td char="." align="char"><p>0.941</p></td><td char="." align="char"><p>0.012</p></td><td align="left"><p>[0.911, 0.961]</p></td></tr><tr><td align="left" /><td align="left"><p>incongruent</p></td><td align="left" /><td char="." align="char"><p>0.475</p></td><td char="." align="char"><p>0.053</p></td><td align="left"><p>[0.373, 0.579]</p></td></tr><tr><td align="left"><p>box item</p></td><td align="left"><p>congruent</p></td><td align="left"><p>congruent</p></td><td char="." align="char"><p>0.898</p></td><td char="." align="char"><p>0.020</p></td><td align="left"><p>[0.851, 0.931]</p></td></tr><tr><td align="left" /><td align="left" /><td align="left"><p>incongruent</p></td><td char="." align="char"><p>0.738</p></td><td char="." align="char"><p>0.041</p></td><td align="left"><p>[0.650, 0.811]</p></td></tr><tr><td align="left" /><td align="left"><p>incongruent</p></td><td align="left"><p>congruent</p></td><td char="." align="char"><p>0.551</p></td><td char="." align="char"><p>0.053</p></td><td align="left"><p>[0.447, 0.651]</p></td></tr><tr><td align="left" /><td align="left" /><td align="left"><p>incongruent</p></td><td char="." align="char"><p>0.329</p></td><td char="." align="char"><p>0.047</p></td><td align="left"><p>[0.244, 0.428]</p></td></tr></tbody></table> </ephtml> </p> <p>In line with Hypothesis 2a, we found that ARFS-congruent items were more likely to be solved correctly than ARFS-incongruent items (<emph>OR</emph> = 8.95, 95%CI [6.23,12.85], <emph>p</emph> < 0.001). There was an opposite effect to Hypothesis 2b. Median items were more likely to be solved correctly than box items (<emph>OR</emph> = 0.53, 95%CI [0.36,0.77], <emph>p</emph> < 0.01). Hypothesis 2c was supported by our data; we found that median-congruent box items were more likely to be solved correctly than median-incongruent box items (<emph>OR</emph> = 2.51, 95%CI [1.35,4.66], <emph>p</emph> < 0.01) (Table 6).</p> <p>Table 6 Results of the generalized linear mixed model</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" colspan="3"><p>model 0</p></th><th align="left" colspan="3"><p>model 1</p></th></tr><tr><th align="left" /><th align="left"><p>odds ratios</p></th><th align="left"><p>95% CI</p></th><th align="left"><p>p</p></th><th align="left"><p>odds ratios</p></th><th align="left"><p>95% CI</p></th><th align="left"><p>p</p></th></tr></thead><tbody><tr><td align="left"><p><bold>fixed effects</bold></p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char" /><td char="." align="char" /><td align="left" /></tr><tr><td align="left"><p>ARFS congruency (0 = incongruent, 1 = congruent)</p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char"><p>8.95</p></td><td char="," align="char"><p>[6.23,12.85]</p></td><td align="left"><p> < 0.001</p></td></tr><tr><td align="left"><p>item type (0 = median item, 1 = box item)</p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char"><p>0.53</p></td><td char="," align="char"><p>[0.36,0.77]</p></td><td align="left"><p> < 0.010</p></td></tr><tr><td align="left"><p>median congruency (−0.5 = incongruent, 0 = median item, 0.5 = congruent)</p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char"><p>2.51</p></td><td char="," align="char"><p>[1.35,4.66]</p></td><td align="left"><p> < 0.010</p></td></tr><tr><td align="left"><p>ARFS congruency x median congruency</p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char"><p>1.25</p></td><td char="," align="char"><p>[0.52,3.02]</p></td><td align="left"><p>0.503</p></td></tr><tr><td align="left"><p><bold>random effects</bold></p></td><td char="." align="char" /><td align="left" /><td align="left" /><td char="." align="char" /><td char="." align="char" /><td align="left" /></tr><tr><td align="left"><p>student (<italic>N</italic> = <italic>297</italic>)</p></td><td char="." align="char"><p>1.60</p></td><td align="left" /><td align="left" /><td char="." align="char"><p>1.59</p></td><td char="." align="char" /><td align="left" /></tr><tr><td align="left"><p>items (<italic>N</italic> = <italic>36</italic>)</p></td><td char="." align="char"><p>1.76</p></td><td align="left" /><td align="left" /><td char="." align="char"><p>0.28</p></td><td char="." align="char" /><td align="left" /></tr></tbody></table> </ephtml> </p> <p>The results of the confirmatory <emph>k</emph>-means clustering were in line with Hypothesis 2d (Fig. 11). For the first profile (proficient model, <emph>n</emph><subs>1</subs> = <emph>55</emph>) we assumed high solution rates in all six item categories. For the second profile (synthetic model A, <emph>n</emph><subs><emph>2</emph></subs> = <emph>36</emph>) we assumed high solution rates except for ARFS-incongruent box items. For this cluster we found a less appropriate fit compared to the other clusters. For the third profile (synthetic model B, <emph>n</emph><subs>3</subs> = <emph>47</emph>) we assumed high solution rates except for ARFS-incongruent median items. For the fourth profile (initial model, <emph>n</emph><subs>4</subs> = <emph>73</emph>) we assumed low solution rates for all ARFS-incongruent items and high solution rates for all ARFS-congruent items. For the fifth profile (synthetic model C, <emph>n</emph><subs><emph>5</emph></subs> = <emph>62</emph>) we assumed high solution rates in median items, in box items only if the box item is median-congruent. The sixth profile (synthetic model D) with high solution rates in ARFS-congruent median items and median-congruent box items and low solution rates in ARFS-incongruent median items and median-incongruent box items could not be identified. Students who showed a pattern in solution rate which did not match with one of our hypothesized patterns (<emph>n</emph><subs>7</subs> = <emph>23</emph>) can be found in the seventh profile. Except for the sixth profile, we were able to identify all the hypothesized profiles.</p> <p>Graph: Fig. 11 Result of the k-means cluster algorithm with predefined cluster centers as described in Table 4. Green graphs represent the hypothesized profiles, orange graphs the profiles actually found in our data. M_AC = ARFS-congruent median items, M_AI = ARFS-incongruent median items, B_AC_MC = ARFS- and median-congruent box items, B_AC_MI = ARFS-congruent and median-incongruent box items, B_AI_MC = ARFS-incongruent and median-congruent box items, and B_AI_MI = ARFS- and median-incongruent box items</p> <hd id="AN0187497173-30">Discussion</hd> <p>When students learn about the boxplot, they are already familiar with the idea that in statistical graphs larger areas typically represent higher frequency (ARFS). If ARFS is applied in boxplots, this can lead to errors. Using ARFS-congruent and ARFS-incongruent items, both studies provide evidence that ARFS-related reasoning frequently lead to a systematic error, not only in median items (replication of Lem et al., [<reflink idref="bib33" id="ref112">33</reflink>]) but also in those items where the median schema could not be applied (box items). We showed that not only ARFS congruency but also item type (median items & box items) and median congruency are difficulty-generating characteristics. This underpins the theoretical rationale for differentiating the items into six categories according to these characteristics empirically.</p> <p>Previous research has shown that students tend to ignore whiskers and see the box as representative of the whole distribution (Bakker et al., [<reflink idref="bib7" id="ref113">7</reflink>]). We thus hypothesized that items in which exactly one box is completely above the critical value are more likely to be solved correctly than items in which both boxes covered the critical value. However, we found a contrary effect in study 2. This may indicate that the error (students ignore the whiskers) described by Bakker et al. ([<reflink idref="bib7" id="ref114">7</reflink>]) may not have been a relevant issue in both our samples.</p> <hd id="AN0187497173-31">Person-oriented analysis</hd> <p>Our person-oriented correlational findings offer robust empirical support for the assumption that the area bias may be associated with an incomplete conceptual change, where ARFS has not been fully replaced by alternative schemas adequately applicable in boxplots—with our correlational findings laying the groundwork for exploring the causal aspect of our hypothesis in future research. In both studies, an initial profile was identified in which participants consistently used ARFS to solve CVCTs, showing the area bias in both item types. In contrast, in both studies a proficient profile could be identified in which learners for both item types used adequate problem-solving schemas, instead of ARFS. Participants of the two other profiles used ARFS in one item type, but not the other. The different profiles suggest that the participants differ in how far they succeeded or still need conceptual change.</p> <p>In this work, conceptual change relates to the scope of validity of a problem-solving schema (here: ARFS). This perspective could also be referred to as schema-related conceptual change. If there are individuals in the sample who have completed this conceptual change process, it is plausible to assume that these participants can be found in this proficient profile. Contrary schema-related conceptual change has not yet taken place in the initial profile. If participants show the area bias in only one item type, then the scope of validity of the area schema in boxplots has not yet been fully restricted by adequate schemas. In terms of a schema-related conceptual change, these profiles can be described as synthetic (Vosniadou, [<reflink idref="bib62" id="ref115">62</reflink>]).</p> <p>More precisely, it is plausible to assume that some participants already have conceptualized the box as representing the middle half of the data (or in some cases, incorrectly, as the entire distribution) or at least perceived it more informally (through the area and shading) as the salient main part of the distribution and therefore compared the position of the boxes in box items relative to the critical value instead of using ARFS. At the same time, it is possible that these participants have not acquired any conceptual knowledge about the meaning of quartiles as cut-off points in general and therefore substitute a missing median schema with ARFS. In the synthetic profile B, we see a solution pattern that fits these assumptions: Participants exhibit the area bias predominantly in median items, while box items were mostly solved correctly.</p> <p>Central values are introduced early in the curriculum and are preferred by students to make data-driven decisions (Abt, Leuders, Loibl, & Reinhold, [<reflink idref="bib1" id="ref116">1</reflink>]; Biehler, [<reflink idref="bib9" id="ref117">9</reflink>]; Kramer et al., [<reflink idref="bib27" id="ref118">27</reflink>]; Masnick & Morris, [<reflink idref="bib39" id="ref119">39</reflink>]; Obrecht et al., [<reflink idref="bib43" id="ref120">43</reflink>]). The median is a very salient feature in the boxplot. Therefore, it is plausible to assume, that there are some participants recognizing the median in the boxplot and linking it to prior knowledge of central values. These participants would solve median items consequently correctly, even if they did not acquire any boxplot-specific knowledge. Remarkably, the corresponding profile in which learners showed the area bias predominantly in box items, while median items were solved correctly, could only be definitively identified in the first study, while the identified group in the second study was a less good fit. This indicates that—at least in the second study—participants who acquired the median schema did not use the ARFS in box items, even if they did not apply the box schema.</p> <p>Instead, the synthetic profile C, which was additionally identified by the inclusion of new item categories in study 2, indicates that individuals who used the median schema in median items but may not have acquired an adequate box schema, did not generally substitute the missing problem-solving schema with ARFS, but rather overgeneralized the median schema and showed a <emph>median bias</emph> in box items. In other words, the question of whether ARFS was applied in CVCTs in study 2 was essentially determined by whether participants had acquired conceptual knowledge of the median or not.</p> <p>From the systematic use of ARFS, it follows that participants incorrectly assume the proportional relationship between area and represented frequency known from other statistical representations in the boxplot representation as well. Consequently, for these participants there is a need for both a schema-related conceptual change and conceptual change regarding the meaning of the box area.</p> <p>In addition to the differences that emerged between studies 1 and 2 due to the different research designs (additional item categories), there are two other substantial differences: Only study 2 found a difference in the difficulty of the item types. This is probably because the short introduction video (shown to the participants before answering the items in study 2) was not able to compensate for the prior knowledge advantage that participants in study 1 had due to their academic discipline (mathematics, computer science, and psychology). At the same time, however, this video may have activated prior knowledge of the median as the mean value. This prior knowledge can also be assumed in people who are not professionally engaged with statistical issues, and it is plausible to assume, that it therefore was overgeneralized by some participants from study 2 in all items (median bias). This may also be an explanation for the second substantial difference, which is that only in study 1 we identified a group of people who exhibit area bias in box items despite having acquired the median schema.</p> <hd id="AN0187497173-32">Limitations and future studies</hd> <p>We assume that people who solve CVCT without knowledge on boxplots will systematically use ARFS and can be found in the initial profile. Following this assumption, theoretically, participants in the proficient profile should have completely replaced ARFS in CVCTs with adequate problem-solving schemas, which we interpret as a complete schema-related conceptual change (i.e., narrowing the scope of validity of ARFS). The individual process of conceptual change itself, however, cannot be made visible without longitudinal data. In addition, it is unclear whether the participants in the proficient profile (as well as in the synthetic profile C) have completed a representation-related conceptual change, i.e., interpreting the box area as a measure of variability, for example, and not assuming a proportional relation between area and frequency. It may be that these participants have narrowed the scope of validity of ARFS because they started interpreting the box area as a measure of variability (IQR) or that problem-solving schemata have replaced the ARFS despite that no such representation-related conceptual change has taken place. The research design of the studies does not allow for any differentiation in this regard.</p> <p>If the objective of research is to detect a completed representation-related conceptual change, this requires tasks with a functional context in which the consideration of variability is relevant to the solution (Abt, Leuders, Loibl, & Reinhold, [<reflink idref="bib1" id="ref121">1</reflink>]). Even if the synthetic profiles and the results of the GLMM indicate that different schemata are used in the two item types and, in particular, that there are participants who have acquired isolated knowledge about single quartiles such as the median, it is unclear whether there are also participants who use a single schema for both item types that does not include any conceptual knowledge about the boxplot components. Such a mere procedural schema could be a <emph>counting schema</emph>, where participants count the quartiles above the critical value. Participants systematically applying a counting schema in all items would be assigned to the proficient profile in our studies—and the presented research design is not able to distinguish them from participants using the median schema and the box schema.</p> <p>While it is highly plausible that the median schema is based on a comparison of the medians, this is not just as straightforward for ARFS and the box schema. We have used the umbrella term <emph>box schema</emph>, which opens up questions regarding the specific cognitive processes participants undergo when utilizing a box schema. One conceivable possibility is that the box schema involves a pairwise comparison of the first quartiles—similar to the assumed pairwise comparison of the medians in the median schema. In such a case, however, it is hardly plausible why participants from profile 3 would not have successfully applied the pairwise comparison of the medians in median items as well. We therefore assume that a more plausible explanation is that participants perceived the box 'as a whole'.</p> <p>In just the same way, it is not clear whether ARFS is mainly activated by the third quartile, or whether it also depends on the ratio of the box area above and below the critical value. Further research is needed in this area, in which eye-tracking methodology has proven to be a promising approach (Abt, Leuders, Loibl, Strohmaier, et al., [<reflink idref="bib2" id="ref122">2</reflink>]). The question about why participants do not succeed in solving a task, even though they are assumed to have already acquired the necessary knowledge, is the starting point for the use of dual process theories (Evans, [<reflink idref="bib20" id="ref123">20</reflink>]; James, [<reflink idref="bib23" id="ref124">23</reflink>]; Kahneman, [<reflink idref="bib26" id="ref125">26</reflink>]; Sloman, [<reflink idref="bib53" id="ref126">53</reflink>]; Wason & Evans, [<reflink idref="bib68" id="ref127">68</reflink>]) also in the field of mathematics education (Gillard et al., [<reflink idref="bib22" id="ref128">22</reflink>]). In earlier studies, the dual process mechanism was used as an explanatory model for the occurrence of ARFS (Lem et al., [<reflink idref="bib32" id="ref129">32</reflink>], [<reflink idref="bib33" id="ref130">33</reflink>], [<reflink idref="bib34" id="ref131">34</reflink>]; Lem, Onghena, et al., [<reflink idref="bib35" id="ref132">35</reflink>]; Lem, Vandebeek, et al., [<reflink idref="bib36" id="ref133">36</reflink>]). Even if participants completed conceptual change, participants may fall back to ARFS in the sense of a fast heuristic when making decisions and comparisons. If we look at the proficient model in our studies, we see descriptively lower solution rates in the ARFS-incongruent items. This can be interpreted as an indication of such a heuristic reasoning. Combining dual process and conceptual change as explanatory theories may be the subject of further research, e.g., assigning learners from the proficient profile to conditions in which heuristic reasoning is enforced or analytic reasoning is hindered (e.g., by time pressure, Lem et al., [<reflink idref="bib33" id="ref134">33</reflink>]).</p> <hd id="AN0187497173-33">Instructional implications</hd> <p>Our results indicate that incomplete conceptual change should be considered an explanation for ARFS-related errors in the interpretation of boxplots. We therefore suggest that the typically limited time available for teaching the boxplot should be spent less on students practicing how to create the boxplot but more on addressing the particular challenges of interpretation (e.g., different meaning of area). Specifically, instruction should focus more on promoting and initiating conceptual change regarding the box area; refutational texts (Asterhan & Resnick, [<reflink idref="bib5" id="ref135">5</reflink>]; Lem et al., [<reflink idref="bib30" id="ref136">30</reflink>]; Schneider et al., [<reflink idref="bib51" id="ref137">51</reflink>]) and inducing cognitive conflict (Dreyfus et al., [<reflink idref="bib17" id="ref138">17</reflink>]) seem suitable for such conceptual change interventions. A cognitive conflict can be created when participants switch frequently between the boxplot and the dotplot representation (Bakker et al., [<reflink idref="bib7" id="ref139">7</reflink>]) or if they use the boxplot in the context of tasks that require a comparison of variability (Abt, Leuders, Loibl, & Reinhold, [<reflink idref="bib1" id="ref140">1</reflink>]).</p> <p>The profiles described could be the basis for differentiated instruction. Students in a particular profile may need specific kinds of refutation or cognitive conflict, different from students in other profiles.</p> <p>The findings of this work can be understood as further empirical evidence that the procedural focus repeatedly described for statistics teaching is not a sufficient prerequisite for conceptual knowledge acquisition.</p> <hd id="AN0187497173-34">Authors contribution</hd> <p>This research is part of the cumulative doctoral thesis of the first author, Martin Abt. Contributor roles for the present research (not the entire doctoral thesis) are as follows. <uline>Conceptualization</uline>: Martin Abt, Katharina Loibl, Timo Leuders, Wim Van Dooren, Frank Reinhold; <uline>Data Curation</uline>: Martin Abt, Katharina Loibl; <uline>Funding Acquisition</uline>: Katharina Loibl, Timo Leuders, Frank Reinhold; <uline>Formal Analysis</uline>: Martin Abt, Frank Reinhold; <uline>Investigation</uline>: Martin Abt; <uline>Methodology</uline>: Martin Abt, Katharina Loibl, Timo Leuders, Wim Van Dooren, Frank Reinhold; <uline>Supervision</uline>: Katharina Loibl, Timo Leuders, Frank Reinhold; <uline>Project Administration</uline>: Martin Abt; <uline>Resources</uline>: Martin Abt; <uline>Validation</uline>: Martin Abt, Katharina Loibl, Timo Leuders, Wim Van Dooren, Frank Reinhold; <uline>Visualization</uline>: Martin Abt; <uline>Writing – Original Draft</uline>: Martin Abt; <uline>Writing – Review & Editing</uline>: Martin Abt, Katharina Loibl, Timo Leuders, Wim Van Dooren, Frank Reinhold.</p> <hd id="AN0187497173-35">Funding</hd> <p>Open Access funding enabled and organized by Projekt DEAL. This work was supported by the Baden-Wuerttemberg Ministry of Science, Research and Arts [grant number: 43-7742.35/24/1]; and the University of Education Freiburg [grant number: 20204023].</p> <hd id="AN0187497173-36">Data availability</hd> <p>The data that support the findings of this study are available from the authors upon reasonable request.</p> <hd id="AN0187497173-37">Declarations</hd> <p></p> <hd id="AN0187497173-38">Competing interests</hd> <p>The authors have no competing interests to declare that are relevant to the content of this article. The participants whose data were evaluated in this study provided their informed consent.</p> <hd id="AN0187497173-39">Supplementary Information</hd> <p>Below is the link to the electronic supplementary material.</p> <p>Graph: Supplementary file1 (PDF 3441 KB)</p> <hd id="AN0187497173-40">Publisher's Note</hd> <p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p> <ref id="AN0187497173-41"> <title> References </title> <blist> <bibl id="bib1" idref="ref20" type="bt">1</bibl> <bibtext> Abt, M, Leuders, T, Loibl, K, & Reinhold, F. (2024). Developing initial notions of variability when learning about box plots. Mathematical Thinking and Learning, 1–24. https://doi.org/10.1080/10986065.2024.2421412</bibtext> </blist> <blist> <bibl id="bib2" idref="ref63" type="bt">2</bibl> <bibtext> Abt, M, Leuders, T, Loibl, K, Strohmaier, A. R, Van Dooren, W, & Reinhold, F. (2024). 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  Data: In the boxplot, the box always represents -- regardless of its area -- the middle half of the data and thus a measure of variability (interquartile range). However, when students first learn about boxplots, they are usual already familiar with other forms of statistical representations (e.g., bar or circle graphs) in which a larger area represents a higher frequency of observations. If students erroneously apply this well-established area-represents-frequency schema to boxplots, it results in a systematic error which we describe as the consequence of an incomplete conceptual change. We empirically validated difficulty-generating characteristics that allow the differentiation between item types with varying complexity (item level) and aimed to identify profiles (person level) that differ depending on which schema was used in which item type. For this purpose, we conducted two cross-sectional studies with N = 100 university students (study 1) and N = 297 participants who finished secondary school or higher (study 2) and used generalized linear mixed models (item level) and k-means clustering with predefined cluster centers (person level) to test our hypotheses. We could replicate the systematic error that was described in previous research and found new difficulty-generating characteristics in boxplot items. Our results support the notion of different profiles potentially emerging based on varying degrees of conceptual change. From an instructional perspective, information about individual progress in conceptual change could be considered for tailoring individualized interventions.
– Name: AbstractInfo
  Label: Abstractor
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  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1481701
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1481701
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1007/s10649-025-10387-z
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 25
        StartPage: 169
    Subjects:
      – SubjectFull: College Students
        Type: general
      – SubjectFull: Data Analysis
        Type: general
      – SubjectFull: Graphs
        Type: general
      – SubjectFull: Error Patterns
        Type: general
      – SubjectFull: Difficulty Level
        Type: general
      – SubjectFull: Profiles
        Type: general
      – SubjectFull: High School Graduates
        Type: general
      – SubjectFull: Data Use
        Type: general
    Titles:
      – TitleFull: Understanding Student Errors in Comparing Data Sets with Boxplots
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Martin Abt
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            NameFull: Katharina Loibl
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            NameFull: Timo Leuders
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            NameFull: Wim Van Dooren
      – PersonEntity:
          Name:
            NameFull: Frank Reinhold
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          Dates:
            – D: 01
              M: 09
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 0013-1954
            – Type: issn-electronic
              Value: 1573-0816
          Numbering:
            – Type: volume
              Value: 120
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: Educational Studies in Mathematics
              Type: main
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