Complexity in Science Learning: Measuring the Underlying Dynamics of Persistent Mistakes
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| Title: | Complexity in Science Learning: Measuring the Underlying Dynamics of Persistent Mistakes |
|---|---|
| Language: | English |
| Authors: | Fleuchaus, Ethan, Kloos, Heidi, Kiefer, Adam W., Silva, Paula L. |
| Source: | Journal of Experimental Education. 2020 88(3):448-469. |
| Availability: | Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals |
| Peer Reviewed: | Y |
| Page Count: | 22 |
| Publication Date: | 2020 |
| Document Type: | Journal Articles Reports - Research |
| Descriptors: | Science Instruction, Misconceptions, Preschool Children, Human Body, Individual Differences, Statistical Analysis, Motion, Motor Reactions |
| DOI: | 10.1080/00220973.2019.1660603 |
| ISSN: | 0022-0973 |
| Abstract: | Mistaken beliefs pose a barrier to science learning. For this reason, it is important to understand the circumstances in which they emerge and change. In the current paper, we apply complexity theory to shed light on the nature of mistaken beliefs. The strength of this approach lies in conceptualizing beliefs as dynamic stabilities, a well-defined construct that can be indexed precisely. For example, Recurrence Quantification Analysis (RQA) can determine the presence of dynamic stabilities by analyzing variability in time-series data. We applied this analytical tool to probe for mistaken beliefs in a beam-balancing task, a task that is known to elicit mistaken beliefs in preschoolers. Using a case-study design with four preschoolers, we tracked children's hand position with motion sensors as they balanced various beams. The resulting time series of hand position was submitted to RQA, yielding two important results: First, we found that consistent mistakes in trying to balance the beams were not always accompanied by dynamic stability. This undermines the common assumption that overt consistency in task performance is sufficient to conclude the presence of beliefs. Second, we found strong individual differences over time, as children explored the balance beams. Applications to science education are discussed. |
| Abstractor: | As Provided |
| Entry Date: | 2020 |
| Accession Number: | EJ1254571 |
| Database: | ERIC |
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| FullText | Links: – Type: pdflink Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwGERqK1eljFaQZvnDD_7FElAAAA4zCB4AYJKoZIhvcNAQcGoIHSMIHPAgEAMIHJBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDKvdx_ALxORjoCLh1wIBEICBm_ZSND57RLikvy0hWSvk6S8F1Ffp2YnOXNV1LtZCEbJvLRqRa4yGvbMc-pPl7pPNrUFTUNm1M9CxJRTINutHLDjMUxJ4RAY5tUwnsG5maTu6huXrAumEMBeycXRZ1pi_IXfCgooANqWPtyqkceIORDVHgs4ar5xvjGegxO5neG-mJ265BeShExk5QszgdCu9AOSPJxIV4kpfSKGA Text: Availability: 1 Value: <anid>AN0143251723;jxe01jul.20;2020May19.04:20;v2.2.500</anid> <title id="AN0143251723-1">Complexity in Science Learning: Measuring the Underlying Dynamics of Persistent Mistakes </title> <p>Mistaken beliefs pose a barrier to science learning. For this reason, it is important to understand the circumstances in which they emerge and change. In the current paper, we apply complexity theory to shed light on the nature of mistaken beliefs. The strength of this approach lies in conceptualizing beliefs as dynamic stabilities, a well-defined construct that can be indexed precisely. For example, Recurrence Quantification Analysis (RQA) can determine the presence of dynamic stabilities by analyzing variability in time-series data. We applied this analytical tool to probe for mistaken beliefs in a beam-balancing task, a task that is known to elicit mistaken beliefs in preschoolers. Using a case-study design with four preschoolers, we tracked children's hand position with motion sensors as they balanced various beams. The resulting time series of hand position was submitted to RQA, yielding two important results: First, we found that consistent mistakes in trying to balance the beams were not always accompanied by dynamic stability. This undermines the common assumption that overt consistency in task performance is sufficient to conclude the presence of beliefs. Second, we found strong individual differences over time, as children explored the balance beams. Applications to science education are discussed. A classical task of beam balancing was used to explore the underlying dynamics of children's mistaken beliefs. Moment-to-moment hand movements were tracked and subjected to a multi-dimensional recurrence quantification analysis (RQA). Dynamic stability was captured through percent laminarity (%LAM), a measure of rigidity in children's explorations. The RQA measure of %LAM shed light on patterns of stability that were not available from the analysis of overt behavior. In line with complexity theory, a model of persistent mistakes is offered that has important implications for science education.</p> <p>Keywords: Balance beams; center of mass; misconceptions; conceptual change; recurrence quantification analysis; laminarity</p> <hd id="AN0143251723-2">Introduction</hd> <p>A CENTRAL ASPECT OF science learning is the presence of mistaken beliefs, sometimes referred to as naïve beliefs, naïve representations, misconceptions, preconceptions, intuitive conceptions, alternative conceptions, or children's science (for a review see Gilbert &amp; Watts, [<reflink idref="bib19" id="ref1">19</reflink>]).[<reflink idref="bib1" id="ref2">1</reflink>] The problem with mistaken beliefs is that they persist despite obvious shortcomings (Mazens &amp; Lautrey, [<reflink idref="bib45" id="ref3">45</reflink>]; Murphy &amp; Alexander, [<reflink idref="bib47" id="ref4">47</reflink>]; Pfundt &amp; Duit, [<reflink idref="bib53" id="ref5">53</reflink>]; Vosniadou, [<reflink idref="bib82" id="ref6">82</reflink>]; Vosniadou &amp; Brewer, [<reflink idref="bib83" id="ref7">83</reflink>]). Consider, for example, a balancing task in which children have to balance a beam on a fulcrum (Karmiloff-Smith &amp; Inhelder, [<reflink idref="bib28" id="ref8">28</reflink>]; Cheung &amp; Wong, [<reflink idref="bib7" id="ref9">7</reflink>]). Preschoolers will sometimes ignore proprioceptive information about the beam and insist instead that the beam should balance in the middle. This perseverance makes mistaken beliefs a common topic in science education. In the current paper, we use complexity theory to offer a path forward.</p> <p>In order to motivate our study, we first review fundamental principles of complex systems and how these principles map onto science beliefs. We then describe an analytical tool that can identify mistaken beliefs in time-series data: Recurrence Quantification Analysis (RQA; Webber &amp; Zbilut, [<reflink idref="bib85" id="ref10">85</reflink>]). This tool examines the moment-to-moment variability in behavior in order to characterize the underlying organization of the system. Finally, we present the findings of a case study with four preschoolers who completed a beam-balancing task.</p> <hd id="AN0143251723-3">A complexity view of mistaken beliefs</hd> <p>The unique strength of complexity theory is that it can explain persistent behavior without needing to reduce it to a fixed entity (Bak, [<reflink idref="bib1" id="ref11">1</reflink>]; Deacon, [<reflink idref="bib14" id="ref12">14</reflink>]; Kugler &amp; Turvey, [<reflink idref="bib41" id="ref13">41</reflink>]; Mitchell, [<reflink idref="bib46" id="ref14">46</reflink>]; Ulanowicz, [<reflink idref="bib78" id="ref15">78</reflink>]). The secret lies in two characteristics of complex systems: that elements of the system are affected by the outside and each other. To explain, consider two iconic complex systems: the weather and ecosystems. In the case of the weather, elements are air molecules. They are affected by the outside temperature (via the transfer of heat) and each other (as they collide). Under the right circumstances, these two forces give rise to persistent behavior, such as a tornado. In the case of ecosystems, the elements are the individual species (birds, worms, bacteria). They are affected by outside features (e.g., light, sound) and each other (via the transfer of calories). Here too we see persistent behavior, such as the survival of an ecosystem through an unusually dry summer.</p> <p>Persistence emergences when three aspects align: (<reflink idref="bib1" id="ref16">1</reflink>) the outside "energy cluster" that elements are sensitive to, (<reflink idref="bib2" id="ref17">2</reflink>) the cohesion by which elements affect each other, and (<reflink idref="bib3" id="ref18">3</reflink>) the pressure on the system to dissipate the outside energy cluster (Kondepudi, Kay, &amp; Dixon, [<reflink idref="bib37" id="ref19">37</reflink>], [<reflink idref="bib38" id="ref20">38</reflink>]; Swenson &amp; Turvey, [<reflink idref="bib73" id="ref21">73</reflink>]). Applied to tornadoes, the three aspects translate into: the amount of heat concentrated in one area, the viscosity between molecules, and the difference in temperature between hot and cold areas. If these aspects align just right, tornados emerge to dissipate the cluster of heat. Applied to ecosystems, the three aspects translate into: the amount of local resources, the speed by which the resources are consumed, and the pressure that comes from fluctuations in resources. If these aspects align just right, persistent food webs emerge to bridge the fluctuation in resources.</p> <p>The principle that gives rise to persistent structures is known as <emph>dynamic stability</emph>, a kind of self-sustaining organization that originates in elements changing each other in a circular fashion (Kelso, [<reflink idref="bib29" id="ref22">29</reflink>]; Prigogine &amp; Nicolis, [<reflink idref="bib58" id="ref23">58</reflink>]; Pecora &amp; Carroll, [<reflink idref="bib51" id="ref24">51</reflink>]). In the case of the tornado, the hot air molecules that bump into cool molecules leave behind a vacuum that pulls in the very same molecules that were pushed aside by the hot molecules. This, in turn, leaves behind a vacuum that pulls in the hot air molecules that started the movement. The same causal circularity applies to ecosystems: As a predator catches prey, the predator becomes food for another predator, which then becomes food for another predator, until the circle closes. The circularity protects the emergent structure from outside perturbations: The air molecules caught inside a tornado contribute to trapping heat inside the tornado, to the expense of changing to fluctuations in temperature independently. And the species of an ecosystem contribute to the accumulation of calories, to the expense of responding to outside resources independently.</p> <p>In the language of complexity, a belief is a persistent structure that exhibits dynamic stability: a kind of tornado or ecosystem of the mind (cf. Colunga &amp; Smith, [<reflink idref="bib11" id="ref25">11</reflink>]; Gabora &amp; Steel, [<reflink idref="bib18" id="ref26">18</reflink>]; Kloos, Baker, &amp; Waltzer, [<reflink idref="bib35" id="ref27">35</reflink>]). Even though there are numerous differences between the mind and other complex systems, the same general principles apply: Elements are affected by the outside and each other, and their coordination yields persistent structures when three aspects align: the outside energy cluster, the cohesion between elements, and the pressure on the system. The elements of the mind are <emph>impressions</emph>, more commonly referred to as representations or schemas (e.g., Piaget, [<reflink idref="bib54" id="ref28">54</reflink>]). Impressions are mental snapshots of what is being experienced in a given moment. When a child balances a beam, for example, an impression forms of the visual, haptic, and acoustic layout of the event. Impressions are affected by the outside in that they represent an outside event (cf. perception, sensation, attention). In contrast, impressions are affected by each other in that they coordinate on the basis of compatible content (cf. constructivism, connectionism, representational re-description; Elman, Bates, &amp; Johnson, [<reflink idref="bib16" id="ref29">16</reflink>]; Karmiloff-Smith, [<reflink idref="bib27" id="ref30">27</reflink>]; Thagard, [<reflink idref="bib75" id="ref31">75</reflink>]). For example, an impression of a red object coordinates with an impression of another red object on the basis of the match in the color red.</p> <p>The outside "energy cluster" that impressions are sensitive to is the information that is locally available. Some children might be able to perceive minuscule aspects of an information cluster, such as the weight distribution of beams. Other children might be sensitive to only obvious information clusters, such as the visual appearance of a beam. The cohesion among impressions pertains to the depth of coordination among impressions. Some children might be able to "see" multiple points of connection between old and new impressions, while other children might only connect impressions locally, seeing only minimal correspondences across impressions. Finally, the pressure on the system pertains to interruptions of information. More generally, it can refer to uncertainty about how to perform in a task. Pressure is minimal if a task is simple, such as balancing a beam on a wide fulcrum. Pressure increases when the fulcrum is narrow or there is a time limit.</p> <p>Beliefs form when there is an alignment between (<reflink idref="bib1" id="ref32">1</reflink>) available information, (<reflink idref="bib2" id="ref33">2</reflink>) cohesion among impressions, and (<reflink idref="bib3" id="ref34">3</reflink>) task pressure. Consider, for example, a sequence of experiences with three balancing trials. Say a child successfully balances a symmetrical beam on the first trial, but then fails to balance the next two beams (one being symmetrical and one being asymmetrical). If cohesion is sufficiently strong to allow these impressions to coordinate with each other, the balancing difficulty on the two latter trials will increases the pressure on the system (i.e., the uncertainty about how to balance beams). This will strengthen the already existing coordination among impressions, in an attempt to dissipate the mounting pressure. Whether the emerging coordination can successfully dissipate the pressure depends on the outside information available to the child. In the current example, the outside information is that beams either balanced at the geometric middle or not at all. This compatibility of experiences can fuel the circular coordination among impressions to give rise to a belief (e.g., that all beams either balance at their middle or not at all). This belief reduces the uncertainty about how to balance beams and thus persists.</p> <hd id="AN0143251723-4">Capturing dynamic stability</hd> <p>Unlike tornados and ecosystems, beliefs cannot be seen directly: The coordination between impressions remains hidden behind overt behavior. It is for this reason that the presence of beliefs is commonly inferred from patterns of overt behavior (e.g., Kloos &amp; Van Orden, [<reflink idref="bib33" id="ref35">33</reflink>]; Kohn, [<reflink idref="bib36" id="ref36">36</reflink>]; Penner &amp; Klahr, [<reflink idref="bib52" id="ref37">52</reflink>]). For example, the mistaken belief that beams either balance at their geometric middle or not at all is commonly inferred from consistent attempts of a child to balance the beams at their geometric middle (Pine &amp; Messer, [<reflink idref="bib57" id="ref38">57</reflink>], [<reflink idref="bib55" id="ref39">55</reflink>], [<reflink idref="bib56" id="ref40">56</reflink>]; Krist, Horz, &amp; Schönfeld, [<reflink idref="bib39" id="ref41">39</reflink>]; Cheung &amp; Wong, [<reflink idref="bib7" id="ref42">7</reflink>]). In this section, we describe an alternative for determining the presence of dynamic stability, using advances made in the area of motor behavior.</p> <p>Advances in motor research offer unique opportunities to capture beliefs. Note that the motor system meets the requirements of complex systems: It is composed of elements (e.g., neurons, muscles, joints, limbs) that are affected by the outside (via perceptual systems) and each other (via their coordination). Importantly, like mental activity, motor behavior displays systematic patterns that are difficult to change at will. For example, beginner drummers will often demonstrate a spontaneous persistence to move the drumming sticks in a one-to-one pattern (Hodges, Chua, &amp; Franks, [<reflink idref="bib24" id="ref43">24</reflink>]; Hodges &amp; Franks, [<reflink idref="bib23" id="ref44">23</reflink>]; Zanone &amp; Kelso, [<reflink idref="bib88" id="ref45">88</reflink>],[<reflink idref="bib89" id="ref46">89</reflink>]). Such persistence is pervasive across motor tasks, including walking, running, pointing, etc. (e.g., Buchanan, Kelso, &amp; de Guzman, [<reflink idref="bib6" id="ref47">6</reflink>]; van Emmerik &amp; Wagenaar, [<reflink idref="bib79" id="ref48">79</reflink>]).</p> <p>Traditionally, persistent motor behavior has been linked to fixed entities such as an internal optimization rule to minimize jerk, effort, or discomfort (Cruse &amp; Brüwer [<reflink idref="bib12" id="ref49">12</reflink>]; Flash &amp; Hogan, [<reflink idref="bib17" id="ref50">17</reflink>]; Hasan [<reflink idref="bib22" id="ref51">22</reflink>]; Todorov &amp; Jordan, [<reflink idref="bib74" id="ref52">74</reflink>]). However, fixed-entity accounts fail to explain the full pattern of results, such as the variability in behavior (e.g., Bickhard &amp; Terveen, [<reflink idref="bib4" id="ref53">4</reflink>]; Newell &amp; Corcos, [<reflink idref="bib50" id="ref54">50</reflink>]; Searle, [<reflink idref="bib63" id="ref55">63</reflink>]; Shaw, [<reflink idref="bib64" id="ref56">64</reflink>]). Variability is ubiquitous in motor behavior. A blacksmith, for example, despite being highly skilled at hitting the chisel every time, will nevertheless display variability in the trajectories of individual body segments (Bernstein, [<reflink idref="bib3" id="ref57">3</reflink>]). Fixed-entity approaches cannot account for such variability without augmenting their theory (Jaric &amp; Latash, [<reflink idref="bib26" id="ref58">26</reflink>]; Yang &amp; Scholz, [<reflink idref="bib87" id="ref59">87</reflink>]).</p> <p>The link between persistent motor behavior and dynamic stability was formalized in the well-known HKB model (Haken, Kelso, &amp; Bunz, [<reflink idref="bib20" id="ref60">20</reflink>]; Kelso, [<reflink idref="bib29" id="ref61">29</reflink>]; Kugler, Kelso, &amp; Turvey, [<reflink idref="bib40" id="ref62">40</reflink>]). The model illustrates the process by which increased pressure on the system narrows down the ways in which elements coordinate with each other. Building upon this model, extensive research established that dynamic stability can be uncovered in the structure of variability in behavior (Richardson, Schmidt &amp; Kay, [<reflink idref="bib60" id="ref63">60</reflink>]; Riley, Balasubramaniam, &amp; Turvey, [<reflink idref="bib61" id="ref64">61</reflink>]; Balasubramaniam, Riley, &amp; Turvey, [<reflink idref="bib2" id="ref65">2</reflink>]). Specifically, the more the underlying variability of behavior deviates from randomness, the more the behavior is dynamically stable.</p> <p>Recurrence quantification analysis (RQA) is an analysis that can assess the underlying structure of behavioral variability and identify the presence of dynamic stabilities. It is a non-linear technique designed to uncover subtle time correlations in a time series that has irregular, noisy, and non-stationary patterns (Webber &amp; Zbilut, [<reflink idref="bib85" id="ref66">85</reflink>]). Specifically, this analysis is designed to detect a "recurrent event" in a time series: the repetition of a behavioral state. It returns metrics such as <emph>percent laminarity</emph> (<emph>%LAM</emph>) to quantify the pattern of underlying repetition over time. Recurrence patterns captured by <emph>%LAM</emph> (among other measures) vary from purely random (i.e., no stability in the dynamic coupling) to completely predictable (i.e., maximal stability in the dynamic coupling). Thus, this measure is a quantitative reflection of persistence that stems from dynamic stability.</p> <p>To apply RQA, the first step is to obtain a behavioral time series from a system of interest (e.g., the position of a body segment at various points in time). This can be done with the use of motion-capture technology (e.g., video recordings, motion sensor, computer vision). The resulting data stream is used (a) to reconstruct the movement trajectories of a given body segment during task performance, and (b) to quantify the fluctuations in trajectories over task repetitions. There is extensive evidence that RQA measures can capture the underlying dynamics of a complex system. For example, high <emph>%LAM</emph> revealed sustained production of a required level of force, while low <emph>%LAM</emph> revealed a trial-and-error approach that was responsive to moment-to-moment variation in environmental inputs (Kuznetsov &amp; Riley, [<reflink idref="bib42" id="ref67">42</reflink>]).</p> <p>RQA measures have also proved relevant to study cognitive behavior (Leonardi, [<reflink idref="bib43" id="ref68">43</reflink>]; Richardson, Dale, &amp; Kirkham, [<reflink idref="bib59" id="ref69">59</reflink>]; Shockley, Santana, &amp; Fowler, [<reflink idref="bib66" id="ref70">66</reflink>], Stephen &amp; Dixon, [<reflink idref="bib71" id="ref71">71</reflink>]). A representative example is the investigation of children's insights during a gear task (Stephen &amp; Dixon, [<reflink idref="bib71" id="ref72">71</reflink>]). Children were asked to decide the turning direction of a gear in the chain of interlocked gears, using information about the turning direction of the first gear in the chain. The task set up the pressure that needed to be dissipated, and the stable arrangement of interlocked gears provided the cluster of information. Results show that dynamic stability decreased prior to the transition between stable strategies, signaling a change in sensitivity to new clusters of information. Finding that body movements offered a window into cognition motivates our choice of looking at children's hand movements to investigate their beliefs.</p> <hd id="AN0143251723-5">Overview of the current study</hd> <p>In the current study, we seek to apply ideas from complexity to a science learning task that involves balancing various beams on a fulcrum. To perform correctly in this task, the child has to find the beam's center of mass, which is determined by the weight distribution of the beam. The visual shape of the beam can provide an approximation of a beam's center of mass, namely when the weight is distributed uniformly. However, when the weight is distributed unevenly (e.g., when additional mass is hidden inside the beam), the visual shape of the beam is no longer a reliable guide to the beam's center of mass. The crucial finding with this task is that young children sometimes make persistent mistakes: They insist that the beams either balance at their geometric center or not at all (Karmiloff-Smith &amp; Inhelder, [<reflink idref="bib28" id="ref73">28</reflink>]; Pine &amp; Messer, [<reflink idref="bib57" id="ref74">57</reflink>]).</p> <p>Our procedure followed the procedure in Karmiloff-Smith and Inhelder's ([<reflink idref="bib28" id="ref75">28</reflink>]):[<reflink idref="bib2" id="ref76">2</reflink>] Preschoolers (<emph>N</emph> = 4) were presented with one beam at a time and asked to balance it on a fulcrum, either with their eyes open or with their eyes closed. Some beams balanced at their geometric center; other beams balanced off their geometric center, and one beam did not balance at all. During the first two sessions, children were asked to balance the beams with their eyes open. In the third session, children were asked to close their eyes for the duration of a trial. This eyes-closed manipulation was expected to perturb children's balancing performance and thus add to the richness of findings.</p> <p>Children's hand movements were tracked as they balanced the beams. The time series obtained for each trial was then subjected to RQA. Of interest was the <emph>%LAM</emph> measure to capture the dynamical organization underlying overt performance. Details about this measure are presented in <emph>Data Preparation</emph>. We also analyzed each child's overt balancing behavior to capture the level of balancing success. Of interest was whether children could balance the beams successfully. The case-study format allowed us to explore the details of how the complexity of the underlying dynamics interfaced with overt balancing performance in each session and over time.</p> <hd id="AN0143251723-6">Method</hd> <p></p> <hd id="AN0143251723-7">Participants</hd> <p>Four preschool children participated: one 4.3-year-old boy (referred to as B1), one 4.7-year-old boy (referred to as B2), one 4.7-year-old girl (referred to as G1), and one 5.8-year-old girl (referred to as G2). Children were recruited through flyers advertising the study. Parents reported that their child had no known disabilities, and they confirmed that their child attended regular preschool environments.</p> <hd id="AN0143251723-8">Stimuli</hd> <p>Stimuli are shown in Figure 1. Experimental beams were made out of 1.3 cm thick fiberboard, cut into rectangular shapes (each 15 × 5 cm) and painted red. <emph>Center beams</emph> (Beams A-D in Figure 1) balanced at their geometric middle: Beam A was a single fiberboard shape; Beam B had two fiberboard shapes stacked on top of each other; Beam C had two fiberboard shapes off-set by 7.5 cm; and Beam D had four fiberboard shapes, paired to stack on top of each other and off-set by 7.5 cm. The <emph>no-balance beam</emph> (Beam E in Figure 1) could not be balanced. It was composed of three fiberboard shapes, off-set in such a way that the beam's center of mass fell outside of the area defined by the bottom shape.</p> <p>Graph: Figure 1. Picture of the stimuli used during the balance-beam task. Beams A-D were "center beams," Beams F-I were "off-center beams," and Beam E was the "no-balance beam."</p> <p> <emph>Off-center beams</emph> (Beams F-I in Figure 1) balanced off their geometric middle. Beams F and G had a 5-cm cube glued to one side of the fiberboard (Beam F: one fiberboard shape; Beam G: two fiberboard shapes). Thus, these two beams were visually asymmetrical and balanced at about 2.4 cm off the geometric middle. In contrast, Beams H and I had a hidden cavity at one side of the beam that was filed with metal weights. (Beam H: two fiberboard shapes; Beam I: four fiberboard shapes). The balancing point of these two beams was 2 cm and 2.7 cm off the center, respectively. The fulcrum was 1 cm wide, 2.5 cm tall, and 25 cm long, mounted on a wooden base.</p> <p>In addition to the nine experimental beams, there was also a familiarization beam. It was made out of the same fiberboard, but cut into rectangles that were a bit narrower than the experimental beam (15 × 3.7 cm). Three shapes were stacked on top of each other and glued together, yielding a beam that was 15 cm long, 3.7 cm wide, and 3.9 cm tall. A cavity was carved out at one side of the beam and filled with metal weights. Adding this weight made it so that the beam's center of mass was 2 cm off the geometric middle. Thus, despite looking symmetrical, the familiarization beam did not balance at its geometric center. It was painted purple.</p> <hd id="AN0143251723-9">Apparatus</hd> <p>A Samsung Galaxy S7 phone was used for video recording. It was mounted on a tripod in a position suitable for capturing the child's hand movements, as he or she attempted to balance the beams. A portable Polhemus Patriot Motion tracker, connected to a laptop, was used to track the child's hand movements. A wired sensor (6 degrees of freedom, 60 Hz capture rate) was placed on the top of each hand and held down with soft Velcro straps. To secure the wire of a sensor, an additional strap was wrapped around each wrist, and the wire was routed up over the child's shoulder and taped down on the back of the shirt. This configuration did not hinder the child's natural movements.</p> <hd id="AN0143251723-10">Design</hd> <p>The experiment consisted of three sessions, administered back-to-back in one visit. Each experimental beam appeared once during a session, presented in random order. The difference among sessions was only whether the child completed the task with eyes open (Sessions 1 and 2) or with eyes closed (Session 3).</p> <hd id="AN0143251723-11">Procedure</hd> <p>The procedure was approved by the university's Institutional Review Board. Parent permission and child assent was obtained prior to the testing event. Children were tested in a quiet room located in the lab. Parents were present as well. They were instructed to remain quiet and refrain from interacting with the child for the duration of the task. Two experimenters were present for data collection (E1 and E2). No other individuals interacted with the children during testing.</p> <p>Figure 2 shows the set-up. The child sat next to E1 and across from E2. The beams were arranged inside an opaque box placed next to E1, away from the child (i.e., the child could not see the beams in the box). The first step was to familiarize the child with the motion-capture equipment. The child was told that the sensors placed on the child's hands were part of a special camera that records movements. The child was allowed to refuse to wear them, but no child did. Once the sensors were affixed, the fulcrum was placed in front of the child, and E1 introduced the task. Specifically, E1 said: "Today we are going to play a game with balancing beams. Let me show you a balance beam, and I'll show you what I mean by balance."</p> <p>Graph: Figure 2. Schematic of the experimental set-up.</p> <p>E1 retrieved the familiarization beam and placed it on the fulcrum so that the long axis of the beam was perpendicular to the fulcrum. Upon releasing it, the beam tipped over. E1 said "Oh look, see how it fell over? That means it's not balanced." E1 picked up the familiarization beam and placed it on the fulcrum again, adjusting it so that it balanced on its own. "See how the beam stays up like that? Now the beam is balanced." Next it was the child's turn to balance the beams. The instructions were: "I will hand you different beams to explore. See if you can make them balance. Some beams won't balance at all. Give it a try."</p> <p>E2 started the video camera and motion-tracking unit at the onset of the experiment proper. The child was asked to raise both hands overhead at the beginning of a session. This movement was distinct and easily recognizable in the movement data, allowing for precise matching of the time series with the appropriate video frame. After the child placed the hands back on the table, E1 handed the child the first beam. The child was not given a time limit during which to balance the beam. However, after 30 seconds, E1 asked if the child would like to try the next one. If the beam was balanced successfully within the allocated time, E1 praised the child (e.g., "Good job. Now let's try this one"). If the child gave up or failed to balance the beam within the allocated time, the feedback given was: "Remember, some beams won't balance at all. Let's try a different beam."</p> <p>E1 and E2 remained quiet while the child attempted to balance a beam (e.g., no encouragement, no questions or hints, and no obvious reactions). The only exception was when the child attempted to balance a beam in ways that were not in line with the instruction. This included task-irrelevant exploratory behavior, such as using the beam to tell a story (e.g., "This one looks like Oklahoma"). It also included balancing the beam parallel to the fulcrum (rather than perpendicular). In these cases, E1 redirected the child by saying "What I really want to see is if you can make this beam balance the long way, like I showed you before." If the child did not want to follow instruction (or could not), E1 moved on to the next beam.</p> <p>At the end of each session, E2 stopped the recording and saved the data. For the third session, E1 said: "You've done a great job so far, but I want to see if you can do it one more time a little differently. This time I want to see if you can balance beams with your eyes closed. Do you want to give it a try?" At this point, the child was given the option to stop the experiment, should they rather not want to close their eyes. When the child decided to continue, E1 handed the child a beam and remind them to keep their eyes closed and not peek. All other aspects of the task remain the same.</p> <hd id="AN0143251723-12">Data preparation</hd> <p>Two data sets were prepared for each trial, synched using a custom MATLAB script: the video recording of overt balancing behavior and the time series of angular hand positions obtained from the motion tracking. Upon inspecting the video clips, we learned that children often placed the beam parallel to the fulcrum, rather than perpendicular. To capture this pattern, we devised a four-level scoring system: Level 0 means the child could not balance the beam within the allocated time. Level 1 means that the child only balanced the beam parallel to the fulcrum (against task instructions), ignoring the experimenter's prompt. Level 2 means the child balanced the beam parallel to the fulcrum at first but then corrected the orientation of the beam after the experimenter's prompt and balanced it successfully. Finally, Level 3 means the child could balance the beam as instructed (without needing reminders).</p> <p>Children's hand movements between trials differed substantially: Sometimes children rested their hands on the table, and other times they stretched their arms out. Thus, we deemed it necessary to exclude these movements from the time series and identify a strict start and end time of a trial: The start of a trial was the moment the experimenter released the beam and the child took full control of it. The end of a trial was the moment the child released a balanced beam. If the child was unable to balance the beam and gave up, the end time was the moment the experimenter touched the beam. The resulting time series were standardized and detrended (see Figure 3 for an example trial).</p> <p>Graph: Figure 3. Example time series for standardized hand exploratory movements during one session, with an example segment taken from the original series.</p> <p>In order to probe for dynamic stability, we used multi-dimensional RQA (Wallot, Roepstorff, &amp; Mønster, [<reflink idref="bib84" id="ref77">84</reflink>]). The difference between multi-dimensional RQA and standard RQA lies in how the number of dimensions is determined that are necessary to represent the data. In the classical RQA, the dimensions are reconstructed using time-delayed copies of a single time series. In multi-dimensional RQA, in contrast, the dimensions are instead a set of measured variables. In the current case, we used the three-dimensional angular positions of a child's hands as the representative dimensions. This returns six dimensions: dimensions x, y, and z for each hand. Our reasoning was that this dimensionality can reflect the underlying dynamics of the system best.</p> <p>After reconstructing the time series in the identified 6-dimensional phase space, the next step is to create a recurrence plot—a square matrix that marks each time a state recurs (i.e., when a particular movement in the six-dimensional space is repeated). We first computed the distance of each point in the phase space trajectory (<emph>x<subs>i</subs></emph>) to every other point (<emph>x<subs>j</subs></emph>). If the distance between any pair (<emph>xi, xj</emph>) was smaller than a given radius (<emph>r</emph>), the two points count as recurrent. A darkened pixel appears in the recurrent plot for each pair that is recurrent. Crucial is the appropriate selection of the radius, which effectively functions as an inclusion criterion for recurrence. The radius has to be large enough to reveal the recurrence structure of the system, but not so large that it includes spurious recurrences. Ideally, the radius is such that recurrence rate stays between 2 and 5% of all possible recurrent points (cf. Shockley, [<reflink idref="bib65" id="ref78">65</reflink>]; Zbilut, Thomasson, &amp; Webber, [<reflink idref="bib91" id="ref79">91</reflink>]).</p> <p>Because the explorations varied dramatically from child to child (and even from trial to trial), setting a fixed radius for the whole sample would violate the criterion to limit recurrence to 5%. To circumvent this problem, we utilized a variable radius, one that resulted in a pre-selected recurrence rate of 4.5% (<emph>M</emph><subs>Obtained Recurrence</subs> = 4.58%; <emph>SD</emph> = 0.33%). The advantage of this method is that the density of recurrence points was preserved, allowing for comparisons of recurrence patterns across trials of different lengths without the need to normalize the data (Marwan, Carmen Romano, Thiel, &amp; Kurths, [<reflink idref="bib44" id="ref80">44</reflink>]). The radius was correlated with trial length (<emph>r</emph> = 0.38; longer trials required a larger radius to capture 5% recurrence). It was uncorrelated with the standard deviation of hand displacements (<emph>p</emph> &lt; 0.20).</p> <p>Inspections of recurrence plots revealed a prevalence of vertical lines in our data (see Figure 4). These vertical lines mark the time intervals in which the state of the system changes very slowly or not at all (cf. Marwan et al., [<reflink idref="bib44" id="ref81">44</reflink>]). They represent periods in which the system remains "trapped" in a particular exploratory state, reflective of a certain degree of stickiness of the ongoing coupling among the system's elements (Davis, Pinto, &amp; Kiefer, [<reflink idref="bib13" id="ref82">13</reflink>]). The measure of <emph>%LAM</emph> quantifies the recurrence patterns clustered around vertical lines. It is computed as the percentage of total recurrence points that fall on vertical lines (defined as 3 consecutive points).</p> <p>Graph: Figure 4. Example recurrence plots for three trials.</p> <p>Figure 4 illustrates how recurrence plots differ with various levels of <emph>%LAM</emph>. The recurrence plot on the left has the highest <emph>%LAM</emph> (99%), characterized by solid squares indicative of stickiness to a few preferred patterns of behavior. The checkbox character of the squares in the middle (92%) and right panels (60%) indicates that the child moved more freely in and out of preferred states, a pattern that is aligned with a greater impermanency in the child's behavior. Note that the recurrence structure illustrated in the plots and quantified by <emph>%LAM</emph> captures differences in the dynamical stability of hand movements that is not readily apparent in the angular position time series. <emph>%LAM</emph> was unaffected by trial length or standard deviation of hand displacements, <emph>p</emph>s &gt;.20.</p> <hd id="AN0143251723-13">Results</hd> <p>Results were analyzed in three steps: First, we looked at overt behavior, in line with what is traditionally done to establish the presence of a belief. Second, we looked at <emph>%LAM</emph> to identify the presence of dynamic stability. Finally, we looked at how those two measures combine. Appendix A shows detailed trial-by-trial information on children's balancing performance, including children's interaction with the beams, their balancing success, fluctuations in the radius obtained from RQA, and fluctuations in <emph>%LAM</emph>.</p> <hd id="AN0143251723-14">Overt balancing behavior</hd> <p>There were 108 trials in total. For 84% of these trials, children interacted with the beams in a task-relevant way (79% of center beams, 90% of off-center beams, 83% of no-balance beams, 83% of eyes-open trials, 89% of eyes-closed trials). While every child showed at least some task interactions that were irrelevant, the proportion of this type of interaction was small (maximally 18%). Whether children interacted with the beam in relevant ways affected balancing success, but not in the expected direction. Specifically, the average level of balancing success (treating balancing success as an interval scale) was lower when the interactions were relevant (<emph>M</emph> = 1.88, <emph>SD</emph> = 1.10) than irrelevant (<emph>M</emph> = 2.47, <emph>SD</emph> = 1.18), <emph>t</emph>(<reflink idref="bib106" id="ref83">106</reflink>) = 2.0, <emph>p</emph> &lt; 0.05. This means that irrelevant task interactions nevertheless allowed children to carry out the task as instructed. Given these findings, type of interaction was excluded from subsequent analyses.</p> <p>Figure 5 presents children's balancing performance as a function of beam type and condition. As expected, the no-balance beam (Beam E) was most difficult to balance, with only one successful trial.[<reflink idref="bib3" id="ref84">3</reflink>] For center beams, on the other hand, balancing was largely successful: Level 3 performance was found in 67% of the trials (i.e., correct balancing without prompt), and Level 2 performance was found in 12% of the trials (i.e., correct balancing after prompt). Finally, balancing success of off-center beams was mixed. While Level 3 performance occurred in 42% of the trials, Level 1 performance occurred in 40% of the trials (i.e., incorrect balancing parallel to the fulcrum). The association between balancing success (Level 3 vs. Level 1) and beam type (center, off-center) was statistically significant (Fisher's exact test <emph>p</emph> &lt; 0.01).</p> <p>Graph: Figure 5. Balancing performance across children, separated by beam type and condition. The number of trials is shown in parentheses. Balancing performance was captured on a 4-level scale (0 = beam could not be balanced; 1 = beam was balanced parallel to the fulcrum; 2 = beam could be balanced correctly after prompt; 3 = beam could be balanced correctly without prompt). The y-axis shows the proportion of trials per level (bars). It also shows average balance success (open circles; treating the levels as interval scale divided by number of trials).</p> <p>In order to define overt balancing success, we combined Levels 0 and 1 (i.e., no balancing; incorrect balancing), and Levels 2 and 3 (i.e., correct balancing with prompt; correct balancing without prompt). We also omitted the no-balance beam. U<emph>nsuccessful balancing</emph> across a session is characterized by failing to correctly balance at least three off-center beams. It was observed in five sessions: four eyes-open sessions (B2, G2) and one eyes-closed sessions (B2). In contrast, <emph>successful balancing</emph> across a session is characterized by correctly balancing at least three off-center beams. It was observed in seven sessions: four eyes-open sessions (B1, G1) and three eyes-closed sessions (B1, G1, G2). These findings are consistent with previous research with this task: approximately half of the trials are typically characterized by mistakes in balancing (Krist, Horz, &amp; Schönfeld, [<reflink idref="bib39" id="ref85">39</reflink>]; Pine &amp; Messer, [<reflink idref="bib55" id="ref86">55</reflink>], [<reflink idref="bib56" id="ref87">56</reflink>]).</p> <hd id="AN0143251723-15">Findings from RQA's %LAM</hd> <p>Looking across sessions and children, a 2 by 2 ANOVA (condition by beam type) revealed no effect of condition, beam type, or condition-by-beam interaction on <emph>%LAM</emph>, <emph>p</emph>s &gt;.10. Two distributional properties of <emph>%LAM</emph> were of interest within a session: the average <emph>%LAM</emph> (low vs. high), and the standard deviation of <emph>%LAM</emph> (stable vs. fluctuating). To define these properties mathematically, we first calculated the average <emph>%LAM</emph> and standard deviation for each session. This yielded three averages and three standard deviations per child (one for each session). We then transformed the averages and standard deviations into z-scores, using a child's overall average as point of comparison. A z-sore with a magnitude of 1 or higher was considered different from the other z-scores. For example, an average <emph>%LAM</emph> with a z-score of +1 was considered "high" for this child, while all other averages were considered "low." And a standard deviation of +1 was considered "fluctuating," while all other standard deviations were considered "stable."</p> <p>Using these criteria, we found that high averages were always paired with low standard deviations and, vice versa, low averages were always paired with high standard deviations. A low-and-fluctuating <emph>%LAM</emph> is indicative of reduced dynamic stability. It was observed in six sessions: three eyes-open sessions (B2, G1) and three eyes-closed session (B1, B2, G2). A high-and-stable <emph>%LAM</emph> is indicative of dynamic stability. It was observed in six sessions: five eyes-open sessions (B1, B2, G2) and one eyes-closed session (G1).</p> <hd id="AN0143251723-16">Combining overt balancing consistency with patterns of %LAM</hd> <p>Recall that children performed either consistently correct (successful balancing), or they sought to balance the beams at their geometric middle (unsuccessful balancing). Such consistency is intuitively taken as indication for children's beliefs about the domain (children either having a valid or a mistaken belief about the domain). The pattern of <emph>%LAM</emph> fluctuation provides yet another tool to detect the presence of children's beliefs (assuming that beliefs are dynamic stabilities of a complex system). Findings show that these two methods did not yield the same outcome (see Figure 6). Unsuccessful balancing was paired with high-and-stable <emph>%LAM</emph> in only some sessions (three eyes-open session). It was paired with low-and-fluctuating <emph>%LAM</emph> in the other sessions (one eyes-open session; one eyes-closed session). Similarly, successful balancing was paired with high-and-stable <emph>%LAM</emph> in only some sessions (two eyes-open sessions; two eyes-closed sessions). It was paired with low-and-fluctuating <emph>%LAM</emph> in other sessions (two eyes-open sessions; one eyes-closed session).</p> <p>Regarding the transition from Session 1 to 2 (both eyes-open sessions), note that the pattern of performance did not change for three of the four children. The fourth child, B2, changed from unsuccessful balancing with reduced dynamic stability to unsuccessful balancing with dynamic stability. Regarding the transition from Session 2 to 3 (from eyes-open to. eyes-closed), all children changed their pattern of performance. B1 balanced the beams successfully and changed from high to low dynamic stability; B2 balanced the beams unsuccessfully and changed from high to low dynamic stability; G1 balanced the beams successfully and changed from low to high dynamic stability; andG2 went from balancing the beams unsuccessfully with dynamic stability to balancing the beams successfully with reduced dynamic stability.</p> <hd id="AN0143251723-17">Discussion</hd> <p>Our goal was to apply the lens of complexity to identify the presence of mistaken beliefs in the classic task of beam balancing. In addition to looking at overt balancing behavior, we also applied the analytical tool of RQA to capture persistence. RQA was applied to the time series obtained from children's hand movements, as they tried to balance the beams. Results show (<reflink idref="bib1" id="ref88">1</reflink>) a dissociation between overt performance consistency and underlying dynamic stability, and (<reflink idref="bib2" id="ref89">2</reflink>) strong individual differences across the learning trajectory. These findings have theoretical and practical implications for science education, as discussed next.</p> <hd id="AN0143251723-18">Overt consistency versus dynamic stability</hd> <p>A common recommendation in science education is to probe for mistaken beliefs prior to deploying a science pedagogy. Our findings challenge this line of reasoning. When using a more stringent way of indexing the presence of beliefs, we found that consistency in overt behavior is an unreliable marker for stable knowledge structures. Only half of the sessions characterized by overt consistency were also characterized by dynamic stability. All other sessions had reduced dynamic stability, despite overt consistency. This discrepancy between consistency in overt task performance and underlying stability mimics what was found with motor behavior (Kiefer, Riley, Shockley, Villard, &amp; Van Orden, [<reflink idref="bib32" id="ref90">32</reflink>]; Vaz, Kay, &amp; Turvey, [<reflink idref="bib81" id="ref91">81</reflink>]). It calls for a more differentiated understanding of how to interpret consistent task performance.</p> <p>Complexity theory provides initial intuitions about the source of consistent behavior. Relevant is the distinction between whether elements of a complex system are primarily affected by the outside or each other. At one extreme, elements are affected primarily by each other, to the expense of outside influences. Tornados and ecosystems are examples of this extreme: Overt consistency in these examples is the result of elements affecting each other in a push-pull configuration, to the expense of sensitivity to outside influences. Beliefs too are examples of this extreme: Individual experiences coordinate with each other in a way that amplifies the "glue" that keeps them together, to the expense of novel information.</p> <p>At the other extreme, elements are affected primarily by the outside, to the expense of their effect on each other. An example of this kind of consistent behavior is air rising from a hot cup of coffee. The heat trapped in the coffee cup will heat up the surrounding air molecules, which then will move on an approximately linear trajectory, without forming turbulences. Another example is the trajectory of animals escaping a forest fire. The heat of the forest fire will override established predator-prey relations between species and push for an approximately linear trajectory. Thus, consistency without dynamic stability is the result of clusters of information that are available to the system continuously.[<reflink idref="bib4" id="ref92">4</reflink>] Children's consistent task performance—when paired with reduced dynamic stability—is likely to follow the same principle: Rather than reflecting a stable mental structure, it stems from outside regularities that are available continuously during the task.</p> <p>Note that all four participants performed consistently with reduced dynamic stability during at least one session. In some cases, this reduced dynamic stability was paired with consistently correct performance. G2, for example, was able to balance the beams successfully (Session 3), likely by attending to the weight distribution of the beams trial after trial. In other cases, reduced dynamic stability was paired with consistently incorrect performance. B2, for example, failed to balance many of the beams (e.g., Session 1). He might have taken advantage of the specific layout of task space that made it easiest to balance the beams at their middle. Thus, even though there was nothing explicit in the task that encouraged children to try to balance a beam at its geometric center, the task might have inadvertently encouraged them to do so.</p> <p>These considerations highlight an important shortcoming of science tasks employed to assess children's a-priori beliefs. It stems from the fact that science truisms are embedded in a rich surrounding of relevant and irrelevant features. The movement of objects, for example, is not merely a mathematical statement. Moving objects differ richly in their size, shape, color, heaviness, function, etc.—features that vary independently of the relevant features of force, weight distribution, density, or torque. Given such rich surrounding, science tasks inevitably harbor local correlations between relevant and irrelevant features. Children might tune into this local regularities. Indeed, merely changing the task context can change children's performance drastically (Kloos, Fisher, &amp; Van Orden, [<reflink idref="bib34" id="ref93">34</reflink>]). Thus, what appears like a mistaken belief, when judged by overt performance consistency, could be an indication of irrelevant feature correlations available in the task.</p> <p>There is yet another problem with using science tasks to probe for children's mistaken beliefs: In addition to mis-identifying the presence of mistaken beliefs, tasks that harbors regularities irrelevant to science truisms can induce a mistaken belief. We found this to be the case for B2. This child started out exploring the beams with a trial-and-error approach in Session 1, but then displayed a mistaken belief in Session 2. It is unlikely that the mistaken belief was "dormant" during Session 1. Instead, it is likely that B2 developed the mistaken belief during the course of the task, after coordinating with an irrelevant regularity first (see Figure A2 in Appendix A for his trial-by-trial performance). Thus, a science task designed to merely probe for the presence of beliefs runs to risk of contributing to the emergence of mistaken beliefs.</p> <hd id="AN0143251723-19">Individual differences despite equivalent surroundings</hd> <p>Despite the small sample size, our findings show strong individual differences in how children learn. Consider, for example, the performance of B2 and G1 during Session 1. Both of these children used a trial-and-error approach to explore the beams. Yet, additional exposure to the beams did not have the same effect on them: B2 formed a mistaken belief during Session 2, while G1 continued to use a trial-and-error approach. Or consider the performance of B2 and G2 during Session 2. Even though both children showed a mistaken belief, their performance in Session 3 was no longer equivalent: B2 reverted to mistaken trial-and-error, while G2 managed to explore the beams successfully. Finally, even though B1 and G2 had different patterns of performance during the two eyes-open sessions, these children ended up with the same patterns of performance when they closed their eyes.</p> <p>The prevalent differences in learning trajectories are a product of non-linear relationships that characterize all complex systems: Small changes in initial conditions can cause large, unexpected effects (Holland, [<reflink idref="bib25" id="ref94">25</reflink>]; Prigogine &amp; Nicolis, [<reflink idref="bib58" id="ref95">58</reflink>]). For example, the exact timing and path of a tornado can be influenced by minor perturbations, even by the flapping of butterfly wings. Ecosystems too are characterized by a certain degree of unpredictability. For example, while a forest remains intact after a wild fire, sea life will change dramatically with a mere 1° increase in ocean temperature. And while a pine forest can recover from expansive deforestation, a rain forest is often lost. Even the killing of predators can have unpredictable effects on the population of prey: While killing foxes can increase the population of its prey, killing seals was found to have the opposite effect on its prey (Hansen &amp; Harding, [<reflink idref="bib21" id="ref96">21</reflink>]).</p> <p>Unpredictability in a child's learning trajectory calls for reframing the goal of science education: Rather than seeking to change a mistaken belief with a fixed pedagogy, a more appropriate goal is to shape a learning system continuously (Chiu, Chou, &amp; Liu, [<reflink idref="bib9" id="ref97">9</reflink>]). Complexity theory offers insights about how this can be accomplished: It requires an ongoing calibration of two opposing aspects: (<reflink idref="bib1" id="ref98">1</reflink>) the outside regularity that allows impressions to coordinate with each other, and (<reflink idref="bib2" id="ref99">2</reflink>) the task pressure that pushes for the emergence of circular coordination (cf. Kloos et al., [<reflink idref="bib35" id="ref100">35</reflink>]). If outside regularity is continuously available, but without sufficient task pressure, there is no impetus for elements to coordinate with each other into self-sustaining networks. This is analogous to having training wheels permanently attached to a bike. Vice versa, if task pressure is too high, given the available regularities, the coordination between impressions is not sufficiently strong to yield self-sustaining networks. This is analogous to trying to bike while blindfolded.</p> <p>Research from motor rehabilitation provides a roadmap for how to calibrate the availability of outside regularities with task pressure (Bonnette et al., [<reflink idref="bib5" id="ref101">5</reflink>]; Kiefer &amp; Myer, [<reflink idref="bib31" id="ref102">31</reflink>]): Sensors can be attached to patients strategically so that effective motor configurations can be distinguished from less effective configurations. The output of the sensors can then be transformed to produce visual shapes (e.g., a square for an effective motor configuration; Kiefer et al., [<reflink idref="bib30" id="ref103">30</reflink>]). The mapping of motor configurations to unique shapes constitutes the outside regularities available to patients. And the task to produce certain shapes via movement patterns constitutes the task pressure, adjusted to fit the unique and changing skills of the patient. This method was found to be more effective than merely instructing patients about how to move (e.g., Silva, Kiefer, Riley, &amp; Chemero, [<reflink idref="bib68" id="ref104">68</reflink>]).</p> <p>Translated into science pedagogy, the continuous calibration pertains to the salience of science-relevant information and the pressure to apply this information in a task. Too little task pressure occurs when children experience their environment without the pressure to organize experiences coherently. This is perhaps what happened with G1: The relevant information of a beam's weight distribution was readily available for her through trial-and-error explorations. This experience made successful performance possible, but without leading to a stable belief about beam balancing. The solution here would be to strategically increase the task pressure. Indeed, when G1 was asked to close her eyes, a valid belief emerged (cf. Stephen, Boncoddo, Magnuson, &amp; Dixon, [<reflink idref="bib70" id="ref105">70</reflink>]; Zanone &amp; Kostrubiec, [<reflink idref="bib90" id="ref106">90</reflink>]).</p> <p>Alternatively, task pressure is elevated when the setting does not make obvious the information cluster that is relevant to the science concept in question. This is perhaps what happened with B2: He could not detect the relevant weight distribution through initial trial and error. Instead, the information cluster that became most obvious was irrelevant to science of balancing. It is possible that science domains prone to mistaken beliefs might lack obvious indicators of science truisms. For example, children might have difficulty predicting an object's movement because the relevant weight distribution of an object is less salient than visual features of symmetry. The solution here would be to strategically increase the salience of science-relevant information, for example with conceptual maps (cf. Wiser &amp; Smith, [<reflink idref="bib86" id="ref107">86</reflink>]).</p> <hd id="AN0143251723-20">Limitations of the current study</hd> <p>Several aspects of our study potentially limit the degree to which our findings can be generalized. A first limitation is our sample size. A case-study design was necessary to obtain a full picture of the methods we employed here, especially given that RQA methods are relatively new in the domain of children's learning. At the same time, there are shortcomings of a case-study design. For example, we cannot answer questions about the prevalence of response patterns. A larger sample would be necessary to calculate the likelihood of different types of behaviors and how they change during children's exploration.</p> <p>A second limitation is the specific task we chose: Even though beam balancing involves prevalent science concepts (center of mass, mass distribution, torque), the task is not anchored in the science-education literature. It is instead part of the cognitive-development literature (Siegler, [<reflink idref="bib67" id="ref108">67</reflink>]). Thus, while this task allowed us to build upon a rich literature on children's beliefs, it is limited in the degree to which we can derive explicit recommendations for science education. To strengthen our claims about science education, it would be necessary to adapt science-learning tasks to fit the requirements of complexity methods.</p> <p>Finally, our study did not test the complexity angle directly. The justification for applying complexity theory to mistaken beliefs was based on findings in cognition and child development more broadly (e.g., Clancy, [<reflink idref="bib10" id="ref109">10</reflink>]; Spivey, Dale, &amp; Ross, [<reflink idref="bib69" id="ref110">69</reflink>]; Stephen, Dixon, &amp; Isenhower, [<reflink idref="bib72" id="ref111">72</reflink>]; Thelen, Corbetta, Kamm, Spencer, Schneider, &amp; Zernicke, [<reflink idref="bib76" id="ref112">76</reflink>]; Thelen, Schöner, Scheier, &amp; Smith, [<reflink idref="bib77" id="ref113">77</reflink>]; Van Geert, [<reflink idref="bib80" id="ref114">80</reflink>]). In that sense, our argument is a logical one: that children's beliefs follow the principles of complexity because the mind follows the principles of complexity. The empirical test of this argument is still missing.</p> <hd id="AN0143251723-21">Conclusion</hd> <p>The purpose of this paper was to contribute to the topic of mistaken beliefs using insights from complexity theory. Specifically, we defined beliefs in complexity terms and then applied a complexity-based analysis to index mistaken beliefs. Our findings raise questions about how to assess mistaken beliefs. We found that merely analyzing overt performance consistency is not a reliable gauge of children's knowledge structures. Thus, there is a fallacy in equating consistent mistaken performance in children's behavior with the presence of persistent beliefs. Mistaken performance can stem from irrelevant information available inadvertently in the task context. Or it can stem from a mistaken belief that developed after children explored the task setting that was supposed to detect the mistaken belief.</p> <p>Our findings also cast doubt on the effectiveness of a fixed pedagogy. We found evidence for strong individual differences across the learning trajectory, even when children started out with the same knowledge structures. Such prevalence of individual differences calls for an adaptive pedagogy, one that considers a child's changing knowledge structures throughout the entire phase of learning. Complexity theory offers insights about the details of such a dynamic pedagogy: Adaptive pedagogy needs to calibrate the amount of science-relevant information with the pressure to apply the science-relevant information in a task. Further work is needed to test these conclusions and further sharpen the link between complexity theory and science education.</p> <hd id="AN0143251723-22">Appendix A: Trial-by-trial performance of individual children</hd> <p> <emph>Performance of 4.3-year-old B1 </emph>(Figure A1). In Session 1, B1 correctly balanced all center and off-center beams without needing a prompt. He only failed to balance the no-balance beam. Average <emph>%LAM</emph> was high and <emph>%LAM</emph> fluctuations were relatively small. The same type of performance was observed in Session 2: high balancing success (other than with the no-balance beam), high average <emph>%LAM</emph>, and low <emph>%LAM</emph> fluctuation. Performance patterns changed only in Session 3: While overt balancing performance remained successful across beam types (other than with the no-balance beam), the average <emph>%LAM</emph> decreased, and the <emph>%LAM</emph> fluctuations increased.</p> <p> <emph>Performance of 4.7-year-old B2</emph> (Figure A2). In Session 1, the most prevalent pattern was to balance the beams parallel to the fulcrum (three off-center beams) or fail to balance the beams altogether (two center beams; one off-center beam; no-balance beam). B2 correctly balanced two center beams. The average <emph>%LAM</emph> was low, paired with high <emph>%LAM</emph> fluctuations. In Sessions 2, balancing was again largely unsuccessful: B2 balanced eight beams parallel to the fulcrum. He balanced one center beam correctly. Average <emph>%LAM</emph> increased and <emph>%LAM</emph> fluctuations decreased. Balancing remained unsuccessful in Session 3: B2 balanced the beams either parallel to the fulcrum (one center beam, four off-center beams; no-balance beam) or failed to balance them altogether (one center beam). He correctly balanced two center beams. Average <emph>%LAM</emph> was low, paired with high <emph>%LAM</emph> fluctuation.</p> <p>Graph: Figure 6. Combination of Overt Balancing Success and %LAM.</p> <p>Graph: Figure A1. Performance profile of B1 across three sessions. Dashed lines represent averages of a given session. Letters indicate the type of beams. Children were asked to close their eyes during the last session.</p> <p>Graph: Figure A2. Performance profile of B2 across three sessions. Dashed lines represent averages of a given session. Letters indicate the type of beams. Children were asked to close their eyes on the last session.</p> <p>Graph: Figure A3. Performance profile of G1, across three sessions. Dashed lines represent averages of a given session. Letters indicate the type of beams. Children were asked to close their eyes on the last session.</p> <p>Graph: Figure A4. Performance profile of G2 across three sessions. Dashed lines represent averages of a given session. Letters indicate the type of beams. Children were asked to close their eyes on the last session.</p> <p> <emph>Performance of 4.7-year-old G1</emph> (Figure A3). In Session 1, the most prevalent pattern was to balance the beams correctly with prompt (two off-center beams) or without prompt (all center beams; two off-center beams). G1 failed to balance the no-balance beam. Average <emph>%LAM</emph> was low and <emph>%LAM</emph> fluctuation was high. A similar pattern appeared in Session 2: G1 correctly balanced two beams with prompt (off-center beams) and five beams without prompt (four center beams; one off-center beam). She failed to balance the no-balance beam, and she placed one off-center beam parallel to the fulcrum. Average <emph>%LAM</emph> was again low, paired with high <emph>%LAM</emph> fluctuation. In Session 3, overt balancing performance was again highly successful, with prompt (one center beam; one off-center beam) or without prompt (two center beams; three off-center beams). She failed to balance the no-balance beam and a center beam. Average <emph>%LAM</emph> increased, and %LAM fluctuation decreased.</p> <p> <emph>Performance of 5.8-year-old G2 </emph>(Figure A4). In Session 1, the most prevalent pattern was to balance the beams parallel to the fulcrum (one center beam; three off-center beams) or to fail to balance the beams altogether (one off-center beam; no-balance beam). She correctly balanced the remaining three beams without prompt (center beams). Average <emph>%LAM</emph> was high, with virtually no <emph>%LAM</emph> fluctuation across trials. A similar pattern appeared in Session 2: G2 balanced four beams parallel to the fulcrum (one center beam; three off-center beams). She succeeded in balancing the remaining beams with prompt (two center beams) or without prompt (one center beam; one off-center beam; no-balance beam). Average <emph>%LAM</emph> was again very high, with virtually no <emph>%LAM</emph> fluctuation. Finally, in Session 3, the most prevalent pattern was to balance the beams correctly, either with prompt (three center beams; two off-center beams) or without prompt (one center beam; one off-center beam). She balanced the remaining beams parallel to the fulcrum (off-center beam; no-balance beam). Average <emph>%LAM</emph> decreased and <emph>%LAM</emph> fluctuations increased.</p> <ref id="AN0143251723-23"> <title> Notes </title> <blist> <bibl id="bib1" idref="ref2" type="bt">1</bibl> <bibtext> The literature is inconsistent about what to name the mental structure that gives rise to persistent mistakes. For example, while science education typically uses the term "misconception" (e.g., Sawyer, [62]), the term "belief" often appears in the literature on belief change (e.g., Chi, Slotta, &amp; De Leeuw, [8]; Murphy &amp; Mason, [48]; National Research Council, [49]). 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| Items | – Name: Title Label: Title Group: Ti Data: Complexity in Science Learning: Measuring the Underlying Dynamics of Persistent Mistakes – Name: Language Label: Language Group: Lang Data: English – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Fleuchaus%2C+Ethan%22">Fleuchaus, Ethan</searchLink><br /><searchLink fieldCode="AR" term="%22Kloos%2C+Heidi%22">Kloos, Heidi</searchLink><br /><searchLink fieldCode="AR" term="%22Kiefer%2C+Adam+W%2E%22">Kiefer, Adam W.</searchLink><br /><searchLink fieldCode="AR" term="%22Silva%2C+Paula+L%2E%22">Silva, Paula L.</searchLink> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="SO" term="%22Journal+of+Experimental+Education%22"><i>Journal of Experimental Education</i></searchLink>. 2020 88(3):448-469. – Name: Avail Label: Availability Group: Avail Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals – Name: PeerReviewed Label: Peer Reviewed Group: SrcInfo Data: Y – Name: Pages Label: Page Count Group: Src Data: 22 – Name: DatePubCY Label: Publication Date Group: Date Data: 2020 – Name: TypeDocument Label: Document Type Group: TypDoc Data: Journal Articles<br />Reports - Research – Name: Subject Label: Descriptors Group: Su Data: <searchLink fieldCode="DE" term="%22Science+Instruction%22">Science Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Misconceptions%22">Misconceptions</searchLink><br /><searchLink fieldCode="DE" term="%22Preschool+Children%22">Preschool Children</searchLink><br /><searchLink fieldCode="DE" term="%22Human+Body%22">Human Body</searchLink><br /><searchLink fieldCode="DE" term="%22Individual+Differences%22">Individual Differences</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Analysis%22">Statistical Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Motion%22">Motion</searchLink><br /><searchLink fieldCode="DE" term="%22Motor+Reactions%22">Motor Reactions</searchLink> – Name: DOI Label: DOI Group: ID Data: 10.1080/00220973.2019.1660603 – Name: ISSN Label: ISSN Group: ISSN Data: 0022-0973 – Name: Abstract Label: Abstract Group: Ab Data: Mistaken beliefs pose a barrier to science learning. For this reason, it is important to understand the circumstances in which they emerge and change. In the current paper, we apply complexity theory to shed light on the nature of mistaken beliefs. The strength of this approach lies in conceptualizing beliefs as dynamic stabilities, a well-defined construct that can be indexed precisely. For example, Recurrence Quantification Analysis (RQA) can determine the presence of dynamic stabilities by analyzing variability in time-series data. We applied this analytical tool to probe for mistaken beliefs in a beam-balancing task, a task that is known to elicit mistaken beliefs in preschoolers. Using a case-study design with four preschoolers, we tracked children's hand position with motion sensors as they balanced various beams. The resulting time series of hand position was submitted to RQA, yielding two important results: First, we found that consistent mistakes in trying to balance the beams were not always accompanied by dynamic stability. This undermines the common assumption that overt consistency in task performance is sufficient to conclude the presence of beliefs. Second, we found strong individual differences over time, as children explored the balance beams. Applications to science education are discussed. – Name: AbstractInfo Label: Abstractor Group: Ab Data: As Provided – Name: DateEntry Label: Entry Date Group: Date Data: 2020 – Name: AN Label: Accession Number Group: ID Data: EJ1254571 |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1080/00220973.2019.1660603 Languages: – Text: English PhysicalDescription: Pagination: PageCount: 22 StartPage: 448 Subjects: – SubjectFull: Science Instruction Type: general – SubjectFull: Misconceptions Type: general – SubjectFull: Preschool Children Type: general – SubjectFull: Human Body Type: general – SubjectFull: Individual Differences Type: general – SubjectFull: Statistical Analysis Type: general – SubjectFull: Motion Type: general – SubjectFull: Motor Reactions Type: general Titles: – TitleFull: Complexity in Science Learning: Measuring the Underlying Dynamics of Persistent Mistakes Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Fleuchaus, Ethan – PersonEntity: Name: NameFull: Kloos, Heidi – PersonEntity: Name: NameFull: Kiefer, Adam W. – PersonEntity: Name: NameFull: Silva, Paula L. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Type: published Y: 2020 Identifiers: – Type: issn-print Value: 0022-0973 Numbering: – Type: volume Value: 88 – Type: issue Value: 3 Titles: – TitleFull: Journal of Experimental Education Type: main |
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