On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts

On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts Lin Li, George (Guoqiang) Zhou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4485936/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Over four decades of conceptual change studies in science education have been based on the assumption that learners come to science classrooms with functionally fixated intuitive ideas. However, it is largely ignored that such pre-instructional conceptions are probabilistic, reflecting some aspects of an idiosyncratic sampling of their experiences and intuitive decision-making. This study foregrounds the probabilistic aspect of international students' intuitive and counterintuitive conceptions when learning pendulum motion. The probability here is rooted in a moving neural time average in the mind for characterizing these students' cognition (sampling and decision-making) and learning processes (resampling and making a new decision). To sharpen the said focus, we would argue that a new taxonomy of physics concepts is needed to save the mathematical identification of the isochrony of pendulum motion. To connect the mathematical core-based taxonomy with reality, we conducted an experimental study to characterising these students' reaction time and error rates in matching the period of a visually presented pendulum, which embodied its mathematical identity: T = 2π√l/g. The reaction times and error rates data have converged on the probabilistic aspects of the students' active learning mechanisms in their mind. The pedagogical implications of such a probabilistic cognitive mechanism have also been discussed. Physics education pendulum motion experimental study conceptual change Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Introduction Imagine a jigsaw puzzle - a pile of variously shaped pieces spread on the table. In expectation, the curly and wiggly-edged cardboard fragments would fit together to form a complete visual representation of something familiar: a flower, a watch, or even a pendulum clock. Regardless of age, culture, or language background, the joy of seeing a complete picture of some sort emerging out of seemingly chaotic bits is a memorable experience which may have a lasting effect on their problem-solving in future. However, occasionally, the joyful puzzle solver may come across a new piece shaped rather strangely. At first glance, it is a bit like part of a flower petal that produces nectar and attracts pollinators. However, after a careful re-examination, it also has the features of a pendulum bob. Bearing these alternative classificatory possibilities in mind, the solver sees the edge of the unidentified puzzle piece as having information so bizarre that it cannot fit into the already-in-place boundary conditions of a half-way-completed picture of the flower or the clock. Even worse is such a scenario where she suddenly realizes there might be unknown missing ones waiting to be filled up. The revelation is alarming because solving the puzzle with previously unknown pieces signals that the current solution is not tenable, and the semi-completed "flower" may be something else. To proceed, she may have to think out of the "box" and reconceptualize what the whole jigsaw puzzle is really about: a flower, a pendulum clock or something else. When the unknown is assumed to represent a counterintuitive scientific concept to be learned, the characterization of playing such a hypothetical game is close to what initially caught the attention of a group of Cornell and Witwatersrand scholars forty years ago: conceptual change. Both the opening example and conceptual change research focus on the unique uncertain aspect of working with a new concept or a conceptual system in problem-solving. Commonly, they assume a conceptual space for representing a problem's essential information, sampling thus-defined problem-solving sub-spaces, and deciding on possible solutions in time. For some researchers, the space is filled with emotional experiences or features of socio-cultural processes. We, in this study, would assume a cognitive science perspective and take a closer look at how a new anchoring concept can be constructed, developed, upgraded, or even repurposed for solving a puzzling problem. In the early 1980s, Posner, Strike, Hewson, and Gertzog ( 1982 ) published the article Accommodation of a scientific conception: Toward a theory of conceptual change in Science Education. Since then, thousands of theoretical and empirical studies have been documented in the literature using the same classifying term conceptual change, covering at least a narrow and broad sense of this classifier. Narrowly speaking, it refers to students' conceptual physics- or classical mechanisms-related learning phenomena. In this sense, the research focuses on transforming pre-college or college science students' misconceptions of some aspects of nature so that they can align with and appreciate a qualified theoretical or experimental physicist' view of the same phenomenon, such as the Brownian motion or the isochrony of pendulum motion. Broadly, the same classifier also refers to changing cultural, social, and philosophical attitudes or belief systems about science and science education systems over time, with a learner in such a social group changing her conceptions of a natural phenomenon laterally and concurrently. The subject matter of this study is the first, narrower, and domain-specific sense of conceptual change. In this sense, how students recognise and explain the time measuring potential of a pendulum or any oscillating object provides a historically meaningful and knowledge rich experimental platform. A False Positive Identification of a "Force" That Drives Pendulum Motion In the context of surveying students' misconceptions, John Clement ( 1982 ) documented how the student conceptual primitive of the relationship between force and acceleration was misunderstood at the qualitative level in the context of a pendulum problem. The pendulum problem was designed to elicit the "motion implies a force" preconception (p. 67). The problem stated: (a) A pendulum is swinging from left to right as shown below (above). Draw arrows showing the direction of each force acting on the pendulum bob at point A. Do not show the total net force and do not include frictional forces. Label each arrow with a name that says what kind of force it is. (b) In a similar way, draw and label arrows showing the direction of each force acting on the pendulum bob when it reaches point B. (p. 67) Facing such a pendulum motion problem, college students often drew the dashed line representing the driving force of the pendulum (See Fig. 1 ). In contrast, the physicist only labelled two types of forces as shown in the two solid lines: gravity and the tension force. For an illustration of the difference, the dashed line was added as evidence to show these students' preconception of the implied force. The students believed that there must be an independent force acting on the bob in the direction of its movement. In terms of human information processing, the identification of something non-existent as real is a false positive. Particularly in this case, they tended to add an extraneous "force" to the visual representation of the pendulum, in addition to the necessary tension and gravity. This non-existent intuitive force was seen as an "essential" component that would drive the swinging pendulum. Moreover, Clement ( 1982 ) summarized the common features of the "motion implies a force" preconception, such as the following: (1) Continuing motion, even at a constant velocity, can trigger an assumption of the presence of a force in the direction of motion that acts on the object to cause the motion. (2) Such invented forces are especially common in explanations of motion that continues in the face of an obvious opposing force. In this case the object is assumed to continue to move because the invented force is greater than the opposing force. (3) The subject may believe that such a force "dies out" or "builds up" to account for changes in an object's speed. (p. 69) In discussing the implications of such findings, Clement ( 1982 ) noted that the preconceptions are "not likely to disappear simply because students have been exposed to the standard view in their physics courses. More likely, Newtonian ideas are simply misperceived or distorted by students to fit their existing preconceptions" (p. 70). He further suggested Galileo might be aware of the teaching challenges faced by a modern physics instructor for "his dialogs represent a marvelous attempt to deal directly with the common preconceptions and prevailing theories of his time at a qualitative level… One might do worse than to take these aspects of Galileo's teaching technique as a model for pedagogy today" (p. 70). The idealised Simple Pendulum Motion: T = 2π \(\sqrt{\mathbf{l}/\mathbf{g}}\) To contrast the students' false positive identification and explanation of pendulum motion, we briefly summarize a pendulum motion lab exercise that features the quantitative aspect of pendulum motion. César Medina, Sandra Velazco, and Julia Salinas of Argentina (2004) Physical pendulum period Tp = 2π \(\sqrt{I /mgd}\) (1) where T p represents the period of the physical pendulum, I the moment of inertia, m the mass of body, g the acceleration due to gravity and d the distance between the axis and the center of gravity of the system. b) Ideal simple pendulum period Ts = 2π \(\sqrt{l/g}\) (2) where l is the length of the string. Equation (1) was deduced assuming: A1: negligible friction (the resultant torque on the system about the horizontal axis is solely due to the weight of the body). A2: small oscillation amplitudes (in the equation of motion, the sine of the amplitude angle can be replaced by the angle in radians). A3: the pendulum is a rigid body (invariable mass distribution, constant moment of inertia). … A4: the string mass must be negligible. A5: the body mass must be concentrated at a point. (Medina et al., 2004 , p. 632) After showing how a physical pendulum can be mathematically associated with an ideal one, they used a section focusing on error analysis. From such a viewpoint, there are random errors in addition to a systematic error due to "the fact that assumptions A1 to A5 are not fulfilled" (Medina et al., 2004 , p. 633). In particular, these error terms include the fluctuations "due to friction (ε f ), initial amplitude (ε α ), variable length of the string due to a variable tension during oscillation (ε T ), mass distribution of the body (ε b ), and mass of the string (εs)" (Medina et al., 2004 , p. 633). With the analysis of error considered, they showed that the model assumptions could be accomplished in laboratory exercises within a reasonably small range of experimental errors. They concluded that, Considered separately, within an error of 1%: – an initial amplitude of 23 ◦ is "small". – a sphere, whose diameter is 30% of the length of the string, is "a point mass". – a mass of the string equal to 10% of the mass of the body is "vanishing". – any elastic elongation suffered by the string during the static process of loading is negligible, providing the string length is measured after the loading. – without loosing its property of 'not extensible', the string may vary its length during oscillation (due to a variable tension), providing this variation is less than the measurement error of the string length. (Medina et al., 2004, p. 639) In their view, such a quantitative laboratory experimental demonstration advances a better understanding of scientific practices, promoting a deeper comprehension of the pendulum motion-related physics concepts. Moreover, the epistemological implications of the analysis of errors are also rich. Taken together the qualitative and quantitative aspects of learning pendulum motion, it reasonable to expect an idealised conceptual change process as the one that overcomes the false positive identification and approaches a scientifically identification of the period of pendulum motion. To provide a new conceptual basis to accommodate such a view of conceptual change, I put forward a new taxonomy of of concepts, which is depicted in Table 1 . With this taxonomy, it becomes possible to categorize and analyze the processes of conceptual change at different levels of verbal, scientific, and mathematical descriptions with greater precision. Figure 2 is an example of applying such a taxonomy to reorganize the pendulum knowledge systems. For dramatizing the effects of using the new taxonomy, a pictorial and symbolic mixed artwork is created to foreground the conception of mathematically defined physics concepts, especially in the case of learning pendulum motion. The keywords of the new taxonomy in Fig. 2 characterize a knowledge system of pendulum motion phenomena, statements, data, and a structural realist' theory about them (A. F. Chalmers, 1999 ; Matthews, 2015 ; Rowbottom, 2019 ; Worrall, 2007 ). At the bottom of the upward swing, the physical objects, natural processes, and simple events of our world occurred naturally, without the involvement of any form of symbolic processing in any language. Moving up a bit, it is human psychological aspects of observing what has happened in the world and the symbolization in English. Next, human perception-driven statements or scientific narratives about the experience of understanding pendulum motion are located at the verbally defined level or propositional perception (Matthews, 2015 ). Following the level, error-term-characterized empirical observations are represented as half mathematically and half verbally defined as raw scientific data. At the top of the upward swing sits the structural realist's mathematical core: T = 2π \(\sqrt{l/g}\) for small swing amplitudes and the real natural phenomenon hidden behind the veil of so-called "reality": an isochronous simple harmonic oscillator. Together, the upward movement of a pendulum acts as a conceptual linchpin for characterizing the scope of knowledge involved in learning pendulum motion, which spans from an event, propositional perception, and the underlying continuous mathematical identification of this phenomenon. The downward swing by the side of the upward one shows a recurring information integration episode in a learner's mind. The dashed lines indicate the probabilistic nature of human information processing. Most importantly, we contend that all these episodes of information processing take time, regardless of which aspect is of interest. In this sense, students' conceptional change toward understanding the notion of a mathematical idealized simple pendulum can be tested empirically in a series of experiments. Thus, the overall effect of students' intuitive pre-instructional conceptions on their real-time responses can be measured. The last two levels of such a knowledge system must be learnt with effort over time. In light of this, an active learning mechanism is also needed to explain the conceptual change effect during such effortful science learning. An Experimental Pilot Study: Matching the Period of Pendulum Motion The purpose of the first experimental study was to prototype the quantitative conceptual change studies and to run the new online experimental platform: Pavlovia (Pavlovia, n.d.). The difference between this experiment and the others was its use of the Rapid Serial Presentation technique (a visual stimuli train featuring the sub-second or millisecond presentation) to highlight the temporal aspect of visual experiences. Research Questions and Hypothesis According to the mathematical relationship describing the pendulum period with a small release angle boundary condition (T = 2π \(\sqrt{l/g }\) ), the length is the only factor that would affect the oscillation period of the pendulum motion. In contrast, other factors of the visual pendulum stimuli should not determine the time. If a learner understands the underlying reasoning, she ought to ignore the other factors when seeing them in this experiment. Experiment 1 was designed to test such a possibility while test-running the online open science platform Pavlovia with Chinese-English international students. The research question asked whether visual changes in a candidate pendulum's length, bob weight, or temporal position would affect these bilinguals' matching choices. The null hypothesis of this experiment was that there would be no reaction time or accuracy differences existed among the bilinguals' responses among the experimental conditions. In contrast, the alternative hypothesis was that these participants' period-matching reaction times on these conditions would differ, reflecting their various levels of understanding of the mathematically defined pendulum motion and their sampling and decision-making processes on the fly. Method The first experiment was designed to measure the bilingual participants' reaction times with a within-participants design. To address the research question, I varied three visual perceptual levels of a computer-controlled display of a pendulum (length, weight, and the temporal position of a candidate pendulum). Each participant was presented with a set of such four pendulums as an experimental unit through their home computers. Upon seeing the standard and matching pendulums unit of such visual inputs, the participant indicated her responses on each experimental trial with the mouse of the visual stimuli-presenting computer. All other visual features of the stimuli were irrelevant to the purpose of this experiment. In other words, the two levels of the visual features of pendulum motion (its length and bob weight) and one level of temporal position initial release angle) were manipulated in a within-participants design, with the participants' reaction times and accuracies recorded as the dependent variables. Participants A power analysis was conducted to find an adequate sample size for the within-participants experimental design. A variance and an effect size were estimated, given the currently reported similar experimental results in the literature. Then the Hotelling-Lawley Trace Statistical test with a Type 1 error of .05 was calculated. The analysis result showed that a total sample size of about 35 would yield a power level of .8. Due to the onset of the COVID-19 pandemic, however, twenty international students, aged from 18 to 55, were recruited for the first experiment (See Table 2). They were contacted through the community of a southeastern Canadian university through contacting its Chinese Scholars and Students Association and recruitment postings. Most of them were native Chinese speakers who could speak English, and one of them also reported Korean as one of her known foreign languages. Their participation was compensated with a $10.00 e-gift card. All the participants were naïve to the purpose of this experiment, and they reported having normal or corrected to normal vision. Table. 3 summarises the self-reports of their English language skills in reading, writing, speaking, and listening. The scale used was a 7-point one, with 1 representing "very poor" and 7 "native-like." Apparatus and Stimuli As the experiment was administered by an open science online experimental site Pavlova, we did not know specific details of the apparatus the participants used. However, they were set up to present all the stimuli at the refresh rate of 60 Hz. Psychopy (Peirce, 2007), an open-source psychophysiological software package with 1 msec precision, was used to create the experiment and upload it to the online platform Pavlovia, which synchronized the stimuli generation and data collection. Students' responses were registered through the mouse of their computers. Both a standard and the three to-be-matched pendulum stimuli were presented in a rapid serial visual presentation (RSVP) paradigm (See Figure. 3). In each RSVP trial, participants saw a coloured standard pendulum shown in the centre of the screen on a grey background. They had five seconds to closely examine this pendulum, estimating its period and other features of their interest. After clicking a CONTINUE button, a series of three candidate pendulums would be rapidly presented, with a 500ms blank duration separating them. Each of the three pendulums was equally likely to be presented at the 1st, 2nd or 3rd position. Moreover, they were equally likely to be the target pendulum or one of the two distractor pendulums. Viewing from a distance of approximately 60cm, participants were instructed to respond to a target-matching pendulum as quickly and accurately as possible. Design The experiment used a within-participants design. This minimized inter-participant variability's impact across the different experimental conditions. There were three factors: (a) the length of the pendulum (long, middle, and short); (b) the weight of the pendulum bob (heavy, medium, light), and (c) the position of the target pendulum (1, 2, 3 or 0). The three factors were independently manipulated. The initial release angle was not specifically controlled in this pilot experiment. Experimental Procedure After signing off the consent form, the participants would receive an experimental link, their unique participant numbers, and a session number. The experiment would automatically start after typing in the numbers through the link. The written instructions were presented in the middle of the screen. After reading the introduction, they would complete a language history questionnaire. Next, the practice session of this experiment was presented at the same rate as that of the real one. Only their responses were not recorded. For the practice session, the item presentation rate started from 500 msec/item. No feedback was provided in practice, simulating what would occur in the real session. Before the actual experimental session started, the participants had a chance to take a break. When they were ready again, each participant could start the real experiment by clicking the CONTINUE button on the screen. Each trial would begin with presenting a standard pendulum for viewing as long as they liked within a limit of five seconds. After they chose to click the CONTINUE button, a series of two distractor pendulums and a target one was to be presented one at a time for 500 msec at the same location in the centre of the screen. Each stream ended with a choice display showing "1", "2", "3", and "0." The participants were instructed to click on the number indicating the temporal position of the period-matching the target pendulum, with "0" indicating "no matching", while ignoring all the other visual stimuli. Because the candidate pendulums' visual features differed from the standard pendulum, the participants could not use these visual features as clues to match the standard pendulum at the sensory or perceptual level. Instead, they would have to use their understanding of the fundamental scientific laws to choose the target correctly (also known as a hit). They were also instructed to respond as quickly and as accurately as they could by clicking on the indicating number of the temporal position of the candidate pendulum. Data Analyses: ANOVA or Binomial Regression? Given that the experimental condition groups' mean reaction times are the reaction times of the correct probe responses, the reaction time analyses must be limited to those probe trials in which the participants correctly identified a matching target pendulum. Moreover, the correct reaction times were constrained by a latency range between 200 and 2,000 msec. We expected most results observed in Experiment 1 to be within this range. Further analyses did not include the participants' data with over 30% error rates, which may indicate a lapse of attention in a simple selection task. The alpha level was set at .05. Given the within-participants design, the mean reaction time data were submitted to a repeated-measure ANOVA with pendulum motion features as the within-participant factors. I expected to find the main effect of the pendulum feature, showing their understanding of the fundamental scientific laws. Quantitative data can take many forms in education research, such as time or accuracy. In contrast to measuring learners' performance in milliseconds, it is common for education researchers to collect students' responses and mark them as right or wrong given a pre-defined theoretical position. Both teachers and researchers in education tend to draw definitive conclusions from analyzing such data, particularly after seeing a statistically significant result, yet reporting significance or not based on p-values thresholds of .05 or .01 has been contested as an acceptable good practice (Kuffner & Walker, 2019; Wasserstein & Lazar, 2016). The unsuitability is of great concern when the data modelling method may not be appropriate. Although researchers in education and psychology may be more familiar with the former, its underlying assumptions are not satisfied given the underlying nature of the error rates data. For them, adopting a logistic regression approach is more appropriate. Results Table 4 shows the mean correct RTs for the candidate pendulum presented at the first, the second, and the third temporal position, with 0 representing no match or a correct rejection of Experiment 1. In the same table, the error rates are also displayed. The mean RTs were the response latencies of correctly matching a candidate pendulum with the standard one. Thus, the RT analyses were limited to those probe trials in which the participants correctly identified the pendulum oscillation period. Moreover, the correct RTs were constrained by a latency range between 200 and 2,500 msec. In this experiment, 98.9% of the correct RTs were within this range. The data of five participants with over 60% error rates were not included in further analyses. The alpha level was set at .05. The mean RT data were submitted to a repeated-measure ANOVA with the temporal position of the candidate pendulum as the 4-level factor. The main effect of the temporal position was significant, F (3, 42) = 4.74, p < .01, η p 2 = .25, indicating mean RTs differed significantly across the three time points and a non-matching level. A post hoc pairwise comparison using the Bonferroni correction showed that it took an increased response time to decide the candidate pendulum presented at the first temporal position than at the second one (1026.67 vs 907.03, p < .05). Also, making a no-matching decision (correct rejection) was approaching statistically significant level as compared with doing that at the first temporal position (931.75 vs 1026.67, p = .08) (See Figure 4). No other pairwise comparisons had reached the statistically significant level. Therefore, we can conclude that the results of the repeated ANOVA have indicated a significant time effect for matching the pendulum oscillation period as measured in time by RTs. In addition to the time position of a candidate pendulum, similar quantitative data analyzing procedures were also implemented to analyze the effects of the other two independent variables: the length of the pendulum d and the bob's weight. Table 5 shows the mean correct RTs for the candidate pendulum presented with the longest, medium, and shortest length in Experiment 1. The mean RTs were the response latencies of correctly matching a candidate pendulum with the standard one, given the length of the pendulum. The data screening considerations were the same as the analysis of RTs for the temporal positions. The alpha level was also set at .05. The mean RT data were submitted to a repeated-measure ANOVA with the length of the candidate pendulum as the 3-level factor. The main effect of the length was significant, F (2, 28) = 4.33, p < .05, η p 2 = .24, indicating the mean RTs differed significantly across the three length levels (See Figure 5). A post hoc pairwise comparison using the Bonferroni correction showed that it took an increased response time to make a decision about the candidate pendulum with the medium-length rod than that of the longest one (995.73 vs 913.82, p < .05). No other pairwise comparisons had reached the statistically significant level. Therefore, I can conclude that the repeated ANOVA results have indicated a significant length effect for matching the pendulum oscillation period as measured in time by RTs. Table 6 shows the mean correct RTs for the candidate pendulum presented with the heavy, medium, and light bob in Experiment 1. The mean RTs were the response latencies of correctly matching a candidate pendulum with the standard one, given the weight of the pendulum bob. The data screening considerations were the same as the analysis of RTs for the temporal positions. The alpha level was also set at .05. The mean RT data were also submitted to a repeated-measure ANOVA with the bob weight of the candidate pendulum as the 3-level factor. The main effect of the bob weight was insignificant, F (2, 28) = .22, p > .05, indicating the mean RTs were not different across the three bob weight levels (See Figure 6). In brief, two statistically significant results have been identified in Experiment 1 following the repeated ANOVA methods. First, the time position of a candidate pendulum did affect the participants' decision-making, increasing their matching response times when the first candidate pendulum had to be selected from the other alternatives. Second, the oscillation period of the mid-length pendulum took more time to be judged as the same as that of the standard pendulum. All other experimental manipulations of Experiment 1 did not have the same response time-extending effects, such as the weight of pendulum bobs. As introduced in the data analysis tutorial (L. Li, 2023), the ANOVA methods may not be suitable for analyzing error rates. The logistic regression-based modelling of accuracy data was adopted for the error rates observed in this experiment (See Table 7). The analyses of the error rates have added some informative data to the response time results (See Table 8). Commonly, the length of the pendulum has been singled out as a significant predictor of making a correct decision about its oscillation period. However, the time position of a candidate pendulum has not been shown as another significant one. Most importantly, when the participants' reading levels were added to the logistic regression equation, it was identified as a significant predictor of making a correct decision in a pendulum period-matching trial. All other predictors have not been shown by both the response time and error rate analyses as significant. Several aspects of the observed patterns of the data are worth noting. First, there was a temporal position effect in identifying and matching the period of a pendulum. As this is one of the first studies using a series of candidate pendulums to measure students' responses, the outcome needs particular consideration. The time difference between making a decision about the first and the second candidate pendulum reveals a serial effect of human memory retrieval mechanisms. It indicates that regaining the pendulum matching information from the represented distribution of the first candidate pendulum was more challenging than another overlapping distribution of the second time position. The structure of the candidate pendulum representation distributions determined the signature of such a time course in matching the period of the pendulum motion. Second, the evidence suggests that deciding on a correct rejection was faster than hitting the first candidate pendulum, though only a marginally significant time difference had been observed in Experiment 1. This result was likely caused by the extent of memory loading involved in making such a negative response. In other words, less memory loading was needed for "saying-no" to a series of candidate pendulums. With the no-matching visual stimulus sampled after seeing a trial, it seems that the participants did not need to keep activated any knowledge distributions activated for completing the task, thus only using less time to click the "no-matching" selection. Given the marginally significant result, there are other possibilities worth further exploration. Besides the role of the time position of a candidate pendulum in determining the time course of matching a pendulum period, the third observed aspect was a significant effect of the length of the pendulum in this experiment. The time difference observed between deciding the candidate pendulum with the medium-length one, and that of the longest one seems to reveal a differentiating effect of human memory retrieval mechanisms. It indicates that regaining the pendulum matching information from the represented distribution of a mid-length pendulum was more challenging than from another overlapping distribution of the longest one. Again, the structure of the candidate pendulum representation distributions determined the characteristic feature of such a time course in matching the period of the pendulum motion. No similar results have been observed for varying the weight of pendulum bobs over the experimental conditions. The results indicate that the participants paid attention to the key determining factor of the pendulum, showing the effect of students' knowledge of the pendulum motion. Fourth, it is worth noting that no interaction effects were observed between the time position and the length of the pendulum. This result indicates that the candidate pendulums presented in another time position may be processed similarly. This is likely because only one candidate pendulum needed to be selected out of the series once the distributed information of the first two-time positions had been processed. There was no further need to process the last temporal position in an experimental trial, or it would be easier to process the last. At last, the analyses of the error rates have confirmed the response time results. By a binomial logistic regression-based technique, the length of the pendulum has also been singled out as a significant predictor of making a decision correctly. However, the time position of a candidate pendulum has not been shown as another significant predictor. Interestingly, when the participants' reading levels were added to the logistic regression equation, it was identified as a significant predictor of correctly matching the pendulum period. All other predictors have not been shown by both the response time and error rate analyses as significant. Discussions, Conclusions, and Implications The results have documented at least two types of evidence to highlight the organizing role of the mathematical identity expressed in students' sampling and decision-making in the pendulum period-matching and explaining tasks. The time has come to restate the thesis that conceptual change can be viewed as an active S-D process over an overlapping knowledge distribution in students' conceptual spaces. Whether in the intuitive or counterintuitive information processing or in the overlapping middle area of the represented knowledge distributions, the probability is the key to unlocking what has changed or not. By embracing a probabilistic frame of reference, we have proposed in this study to advance conceptual change in science education by holding tight to the mathematical definition of a physics concept and embodying the caveat "Don't throw the baby out with the bathwater!" In the two experiments and the interviews, the role of the mathematical definition of a physics concept in organizing these participants' conceptual spaces has been revealed through a pendulum period-matching task, which is complemented by the interviewee's verbal expression of understanding such a mathematical identity. In this chapter, we take a closer look at how to re-integrate a mathematically defined physics concept (such as T = 2π \(\sqrt{l/g}\) ) in acting conceptual change learning with the verbal expressions included. Finally, we reflect on the pedagogical implications of these findings. A Mathematically Defined Physics Concept: Friend or Foe? Interestingly, most current conceptual change studies except PER choose to avoid the mathematical contents (Potvin & Cyr, 2017 ). For example, Andrea A. diSessa ( 2014 ) noted "(t)he conceptual change paradigm is less often applied to other areas of science, and much less in mathematics" (p. 88). More directly, he later argued that "(u)nderstanding mathematics and its use in science is a worthy topic, but I believe it is secondary to deep qualitative, conceptual understanding" (diSessa, 2017 , p. 26). However, the data collected in this study have suggested otherwise, even in the simplest case of matching a simple single pendulum motion task. The most relevant significant factors are those already included in the mathematical equation T = 2π \(\sqrt{l/g}\) . The mathematical expression is, in effect, the mathematical definition of the concept: the oscillation period. Given what I observed in the experiments and the interviews, the participants struggled to understand such a mathematically defined physics concept with their ordinary senses. When the boundary condition of a small initial release angle has not been met, only mathematical or experimental knowledge, rather than other types of verbal expressions, can provide a satisfactory explanation. In this aspect, I concur with Bruce L. Sherin when he commented on the conceptual physics program (Hewitt, 1971 ) or the similar Physics for Poets (March, 2003 ). He contested, I challenge the assumption that in physics or any domain the conceptual and the symbolic elements (the mathematical identity or definition) of a practice can be separated for the purposes of instruction. Removing equations from the mix changes the nature of understanding. This does not imply that physics cannot be taught without equations. However, it does imply that equation-free courses will result in an understanding of physics that is fundamentally different from physics as understood by physicists. (Sherin, 2001, p. 524) However, the question remains why the equation-free PER would be different. Without a new theoretical framework, uncertainty remains. As introduced in Chap. 4, I have attempted to draw a wider picture of conceptual learning that centers on a psychologically plausible and probabilistic mechanism: the sampling and decision-making over an overlapped knowledge distribution. In this S-D framework, the sampling process provides a front end for the mind to take in new information, whereas the decision-making drives the learning outcomes. If this assumption is reasonable and correct, it implies that the equation-free physics learning programs or the conceptual change at the verbal level only promote a biased sampling strategy while leaving relevant mathematical contents out of the equation. we argue that mathematical elements are inevitable to understand learner sampling and decision-making fully. In general, using the mathematical form departs from the established conceptual change research traditions rooted in the philosophy of science, which may result in interpreting the history of science in a new light, especially when the philosophical tradition may sometimes become misleading. Alan Chalmers ( 2009 ) has reminded science education researchers that losing the experimental contact with reality had failed the philosophical atomism as a general heuristic conceptual structure to maintain a productive role in guiding modern atomic physics research. If the fate of philosophical atomism has revealed something soberingly informative, it also reminds conceptual change researchers not to lose mathematical and experimental contact with reality. Unlocking the Learning Brain's Active Sampling and Decision-making In today's parlance, the brain relies on a network-like structure (Baronchelli et al., 2013 ) to enable us to sample and make a decision, thus changing conceptions. Such a neural network has often been approximately characterized by its components and connections: neurons and synapses (Dehaene & Naccache, 2001 ; Salmelin & Kujala, 2006 ). As we know, a single neuron affords the basic cell-level information processing unit. Its conception was conceived more than 100 years ago by Santiago Ramóny Cajal (Haines, 2007 ), who first identified the independent cellular structure of a neuron. Following his lead, Adrian and Bronk ( 1929 ) associated neurons' spiking patterns with the axonal and dendritic mechanisms. Later, Hodgkin, Huxley, and Eccles ( 1963 ) demonstrated the ionic mechanisms inside and outside neurons' membranes. Together, these single neuron-based mechanisms demystified the brain's neural impulse trains - the information-carrying mechanism of human cognition. Regardless of describing or explaining axonal or ionic neuronal mechanisms, time is a fundamental aspect of them (Mesulam, 1998 ; Muller, 2000 ; Palva & Palva, 2012 ), which implies time is essential for understanding learning. The importance of time in understanding the brain's language of information processing can be found beyond the single neurons. In effect, it also has implications for other neuronal structures, such as (a) the supportive glial cells-based mechanisms (Fields et al., 2013 ), and (b) the synaptic (neuron-to-neuron) chemical information transmission processes (Bennett, 2000 ). First, the glial cells separate the myelinated axonal fibres from the unmyelinated ones. According to Fields et al. ( 2013 ), the glial cell-based mechanisms are still largely absent from thinking about representing and processing information because the glial cells do not generate electrical impulses. However, they form crucial cell-cell interaction that shapes the cellular mechanisms of learning and cognition. More importantly, they couple neurons into functional units for short-term and long-term information storage and transformation, thus enabling learning and cognition. In other words, learning is in time. Furthermore, time is also involved in inter-neuronal connections. As for the chemical agent-based neuron-to-neuron communication, the specialized neuronal structure at the axon terminal is called a synapse (Debanne, 2004 ; Fields et al., 2013 ; Langille & Brown, 2018 ), the gap for diffusing and relaying neurotransmitters from one sending neuron to a receiving one. The diffusion process starts by releasing the functional molecules from pre-synaptic neurons' membranes into the synaptic gap. Over the gap, the ionic channels of post-synaptic neurons would enable a membrane-fusing process, binding these molecules in a lock-and-key manner. The binding thus opens and closes the membrane ionic channels. The exchange would permit some kind of neuron-to-neuron information transmission, spreading neuronal information forward in a neuronal network. The three components (single neurons, glial cells, and synapses) help form a neuron-based information processing network in the central and peripheral neural systems. As for learning and cognition, small neocortical networks form large-scale neural networks to support reshaping the dynamics and structures of such a network (Gastner & Ódor, 2016 ). Again, the brain processes information in time Such a vast time-based information processing network is necessary for researchers to conceptualize problem-solving, conceptual change, and metacognitive processes. One powerful way to characterize the synaptic connection-based structure is to use a hierarchical structure for a functional approximation. For example, Mesulam ( 1998 ) introduced six degrees of synaptic separations to capture the essence of such a network (i.e., its primary sensory-motor function, unimodal associative representing function, and hetero-modal associative and the paralimbic and limbic representing function). More specifically, the primary sensory-motor function of such an information processing network interfaces the initial processing of "raw" sensory inputs and the generation of behaviourally significant responses. Close to it, the unimodal associative function of the network maintains the fidelity of the "raw" sensory inputs. In contrast, the hetero-modal associative function of the network serves to provide a cross-sensory-modality representation of the input data. At last, the paralimbic and limbic functions provide reciprocal access to the hypothalamus. Collectively, such a characterization offers a framework to ground cognition. In other words, cognitive processes are defined as the neural information processes between the obligatory processing of "raw" sensory inputs and the generation of behaviourally significant responses in such a network. Meanwhile, cognitive processes manifest contextual effects, memory guidance, and other task-bound constraints realized in the network. Again, the brain's information processing over six degrees of separation can be seen as a conceptional change process that occurred in time. These considerations contribute to building a solid scientific foundation for reconceptualizing conceptual change through active sampling and decision-making. Conclusions and Pedagogical Implications In this experiment, participants' pendulum period-matching was measured in the rapid serial presentation format by varying a range of factors. To our knowledge, this is the first study that has demonstrated how to measure it and the first study that has given an initial estimate of its magnitude. The results pointed out a unique structure of intuitive and nonintuitive in their mind: an overlapping binomial distribution-like conceptual structure. The binomial distribution-like knowledge structure has unique characteristics that distinguish it from those verbal definitions of a conceptual change space. Specifically, it exhibits an overlapping middle area encompassing intuitive and non-intuitive knowledge. It can explain conceptual change as a sample and decision-making process within this conceptual space. Given such a theoretical construct, the conceptual change process can be viewed as a time-based procedure with a different sampling tendency over the knowledge distribution. While a complete understanding of conceptual change remains elusive (G. J. Posner et al., 1982 ; Thagard, 1990 b; Babai, Levyadun, et al., 2006 b; Zhou, 2010a, 2012a; Potvin et al., 2020 b), this study has provided a unique and informative reference point for future research into the active sampling and decision-making mechanisms involved in conceptual change. In brief, the new taxonomy with a probabilistic frame of reference significantly contributes to extending the hybrid learning space (Authors, 2012) by establishing meaningful connections and offering opportunities to understand international students' science learning experiences in verbal expressions and their reaction times. It recognizes that these international students come to the science classroom with intuitive pre-instructional ideas, which may be inaccurate or incomplete given mutually accepted scientific understanding and practice. By seeing these misconceptions as sampling and decision making, science educators can have a conceptual handle to help students resample and make a new decision, thus promoting a more accurate and comprehensive understanding of scientific concepts. Moreover, foregrounding the role of a matrix of event, propositional perception, and mathematical functions in influencing students' real-time responses help to reconnect sociocultural conceptual change studies with the structural realist's outlook of the fundamental science and science education (A. F. Chalmers, 1999 ; Matthews, 2015 ; Mayer, 2004 ; Rowbottom, 2019 ). Furthermore, the results of this study help promote a new line of discussion of measurement theories in conceptual change studies. Most importantly, the evidence has confirmed the co-existing view of students' intuitive ideas and the scientific notion of pendulum motion, as advocated in the conceptual advancement view of conceptual change (Zhou, 2012). Such evidence will further support the student-centred approaches in PER, which let students speak out their intuitive ideas first; and then offer culturally and linguistically appropriate feedback to improve their science learning (Zhou, 2010). The conceptual change view of science learning is not new, but a new taxonomy based on concepts and conceptual change is, especially when considering international students' learning experiences. Although the probabilistic taxonomy has not been explicitly discussed before, similar ideas have been explored in the intersections of conceptual change studies (Duit & Treagust, 2003; Potvin et al., 2020 ; Thagard, 1990 ; Author, 2010 ), the second language learning research (J. Li et al., 2021 ; J. Y. Li & Zhou, 2007 ; K. C. Li & Wong, 2021 ; L. Li, 2016), and domain-specific teaching and learning (Mayer, 2004 ). The classificatory scheme helps researchers refocus on what culturally and linguistically diverse international students really bring to Canadian science classrooms. More importantly, the taxonomy promotes more generalized thinking in science education, seeing previous separate conceptual change studies as special cases of reweighting personal knowledge distributions. Although it is always challenging to characterize a still-evolving research field, reviewing some fundamental aspects of conceptual change studies still helps us consolidate what we have learned so far. The pedagogical implications of such a probabilistic cognitive "revolution" are manifold: (a) the probabilistic re-orientation can enhance science students', domestic or international, understanding of scientific concepts and scientists' conceptions. By using mathematically defined conceptual tools, the students can gain a deeper understanding of scientific concepts that may have previously appeared difficult to comprehend. The new taxonomy allows students to view scientific concepts and conceptions through a physics-compatible lens, which can significantly help clarify the underlying theoretical principles and the organizing key notions; (b) the new taxonomy helps guide new curriculum design endeavours to bridge the gap between abstract mathematical concepts and their physical interpretations. In the tradition of conceptual change studies, physics education research has relied heavily on qualitative approaches to understanding physical phenomena, often overlooking the importance of idealized quantitative reasoning and its error terms. By incorporating sampling and decision-making theory into physics education research, researchers and students alike are more likely to appreciate a deeper understanding of the underlying mathematical structures that govern physical phenomena, which entails conceptional change; (c) the taxonomy and its experimental manifestations deliberately promote a positive attitude toward the interdisciplinary learning since the experimentation and statistical modelling are not limited to physics research alone, and many other fields such as mathematical psychology, artificial intelligence, and educational assessment depend heavily on mathematical reasoning. By incorporating the new taxonomy into the researchers' teaching practices, they, in effect, help develop the student's skills necessary to apply scientific and mathematical reasoning across a wide range of disciplines. ­A Caveat on the Limitations of the Current Study and Future Research This study was designed when the COVID-19 pandemic was still affecting every aspect of students' lives. One limitation of this study is its lack of an interactive component in the pendulum period-matching task. Therefore, the experiment has not fully explored the participants' active learning. Further studies should consider the possibility of adding such a component as a participant self-controlled matching procedure. It will be informative to find out whether their active exploration would increase their response times. The absence of an emotional component is the second limitation of the current study. Since the publication of Beyond Cold Conceptual Change: The Role of Motivational Beliefs and Classroom Contextual Factors in the Process of Conceptual Change (Pintrich et al., 1993) , the emotional aspect of conceptual change processes has attracted scholars' attention. For some non-math students, their motivational and emotional experiences may significantly influence their conceptual change learning. Future experiments should explore this possibility. Declarations Author contributions: G. Zhou contributed to the study’s conceptualization, resource management, supervision, revising the manuscript, and approving it. L. Li contributed to the literature search, manuscript’s writing, validation, analysis of the results, editing, and finalization. Conflict of Interest: There are no conflicts of interest between the authors. Ethical Approval: The study was approved by the University's Research Ethics Board. 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A Cultural Perspective of Conceptual Change: Re-examining the Goal of Science Education. McGill Journal of Education / Revue Des Sciences de l’éducation de McGill , 47 (1), 109–129. https://doi.org/10.7202/1011669ar Tables Table 1. A New Taxonomy of Physics Concepts for Learning Pendulum Motion. Concept Type 1 Type 2 Type 3 Type 4 Criteria Mathematically Defined Half mathematically defined and half verbally defined Verbally defined Non-verbally defined Example An imaginary number, the probability amplitude, Simple harmonic motion Force (F = m × ), The isochronous pendulum motion (T 12 = T 21 ) Argumentation, Ideology, Propositional perception Visual perception, Auditory perception, Working memory system Table 2. A Summary of the Twenty Participants of Experiment 1. No Age Gender Edu Major 1st language 2nd language 01 26 F BA Non-Science Chinese English, Korean 02 22 M BA Physics Chinese English 03 32 F Master Non-Science Chinese English 04 48 F Master Non-Science Chinese English 05 24 M BA Physics Chinese English 06 30 F Master Non-Science Chinese English 07 Master Non-Science Chinese English 08 36 M Master Non-Science Chinese English 09 37 F Master Non-Science Chinese English 10 18 M High School Mathematics Chinese English 11 39 F Master Non-Science Chinese No. 12 39 M Master Non-Science Chinese No. 13 45 F Master Non-Science Chinese English 14 55 F BA Non-Science Chinese English 15 44 F PhD Chemistry Chinese English 16 44 M BA Non-Science Chinese English 17 32 F Master Mathematics Chinese English 18 30 M PhD Biology Cantonese English 19 41 F PhD Non-Science Chinese English 20 19 M BA Non-Science Chinese English Table 3. The Language Learning Background of the Twenty Participants of Experiment 1. No Reading Writing Speaking Listening 01 5 4 4 4 02 5 5 3 4 03 6 5 5 6 04 5 4 5 4 05 4 4 4 5 06 4 4 5 5 07 4 4 3 4 08 6 5 5 5 09 5.5 5.5 5.5 5.5 10 4.5 4 5 5.5 11 5.5 5 5 5.5 12 4 3 3 4 13 4.5 4.5 4.5 4.5 14 6 5 5 5 15 6 6 5 5 16 5 5 5 5 17 6 6 5 5 18 5 5 5 5 19 6.5 5.5 6.5 6.5 20 5 4 5 6 Table 4. Correct Response Times and Mean error rates (% error) for the Three Time Positions of the Candidate Pendulum in Experiment 1, with No-Matching as the Zero Position. Exp Participant RT0(ms) RT1(ms) RT2(ms) RT3 (ms) Missing Correct% Error% 1 1 823.6 2085 1684.5 1266.333 4 32.4 67.6 1 2 1759.5 1313.429 1158.857 1438.667 2 31.4 68.6 1 3 1012.059 1386 1153.909 1255.333 5 65.7 34.3 1 4 1789.222 2005.5 2016 1763.6 2 94.3 5.7 1 5 768.0625 670.2857 718.3529 815.6667 3 89.9 10.1 1 6 1131.5 2088.5 2497.667 1358.5 13 20.3 79.7 1 7 973.3889 928.0714 999.7778 1007.063 1 93 7 1 8 664.5211 708.2769 695.2031 711.7578 0 94.4 5.6 1 9 927.9111 1010.29 713.7378 912.9629 0 91.7 8.3 1 10 583.6667 855 0 13.9 86.1 1 11 1166.862 1233.119 1083.355 1366.508 4 82.4 17.6 1 12 878.5247 977.5772 762.1722 919.6922 1 97.2 2.8 1 13 929.9567 973.7313 1029.899 893.4729 0 95.8 4.2 1 14 784.2941 902.6111 784.4706 955.8824 0 95.8 4.2 1 15 1014.828 1343.771 917.91 889.02 4 85.3 14.7 1 16 1581.135 1374.62 3567.84 0 5.8 94.2 1 17 682.9363 758.1012 612.0661 703.2441 0 94.4 5.6 1 18 951.2667 1006.278 778.1111 957.1667 2 90 10 1 19 959.9 1079.929 964.4615 946.2222 3 66.7 33.3 1 20 472.5556 416.5 376.0556 469.2941 0 98.6 1.4 Table 5. Correct Response Times for the Three Pendulum Length Levels in Experiment 1. Exp Participant Longest_Time (ms) Medium_Time (ms) Shortest_Time (ms) 1 1 1387 2085 1198.45 1 2 1158.86 1412.56 1438.67 1 3 1120.41 1260.87 1110.83 1 4 1921 1962.5 1793.43 1 5 742 708.05 781.75 1 6 2333.86 2088.5 1209.67 1 7 989.96 983.15 963.68 1 8 720.78 685.72 678.57 1 9 750.01 949.69 963.46 1 10 597.5 556 1 11 1088.95 1231.13 1337.99 1 12 796.99 938.29 925.11 1 13 1013.14 933.67 921.31 1 14 785.7 858.35 928.39 1 15 912.03 1205.88 992.09 1 16 1581.14 1 17 626.55 706.26 735.58 1 18 803 1046.52 902.06 1 19 1008.88 1044 906.08 1 20 427.96 421.92 450.13 Table 6. Correct Response Times for the Three Pendulum Bob Weight Levels in Experiment 1. Exp Participant Heavy_Time(ms) Light_Time(ms) Medium_Time(ms) 1 1 1249.71 1493.57 1323.63 1 2 952.56 1660 1544 1 3 1212.93 1117.84 1188.18 1 4 1820.12 1968.91 1897.19 1 5 748.19 700.68 789.11 1 6 1398.25 2664.25 1973.25 1 7 1015.22 931.71 981.42 1 8 730.92 673.55 669.97 1 9 917.41 844.75 873.48 1 10 564.33 593.33 1 11 1254.9 1208.41 1190 1 12 893.19 899.59 858.76 1 13 939.02 975.78 961.27 1 14 850.59 877.23 845.05 1 15 895.7 1082.84 1122.33 1 16 1581.14 1 17 702.17 694.4 664.57 1 18 1036.3 807.76 898.74 1 19 1030.79 1005.71 931.31 1 20 408.52 468.48 425.95 Table 7. A Summary of Three Binomial Logistic Models and the Statistical Indexes. Dependent Variable Predictor df b t p sr 2 95% CI corrAns position 1,057 0 -0.3 0.767 0 [0.00, 0.00] length 1,057 0.04 3.12 0.002 0.01 [0.00, 0.02] id 1,057 0 1.79 0.073 0 [0.00, 0.01] position 1,056 0 -0.31 0.757 0 [0.00, 0.00] length 1,056 0.04 3.13 0.002 0.01 [0.00, 0.02] weight 1,056 0 -0.26 0.797 0 [0.00, 0.00] id 1,056 0 1.79 0.073 0 [0.00, 0.01] position 1,055 0 -0.33 0.744 0 [0.00, 0.00] length 1,055 0.04 3.15 0.002 0.01 [0.00, 0.02] weight 1,055 0 -0.29 0.773 0 [0.00, 0.00] reading 1,055 -0.07 -6.13 < .001 0.03 [0.01, 0.06] id 1,055 0.01 3.41 0.001 0.01 [0.00, 0.02] Table 8. Results of Model Comparison Results of the Three Embedded Mixed Effects Models of the Experiment 1 Data. Model AIC BIC LogLik DeDeviance χ2 df pr (> χ2) cc1 687.49 707.35 -339.74 679.49 cc2 689.55 714.39 -339.78 679.55 0.0000 1 1.00000 cc3 687.06 716.86 -337.53 675.06 4.4952 1 0.03399 * * p < .05 Note. cc1 means a model built with R command “corrAns ~ position + length + (1 | id)” whereas cc2 “corrAns ~ position + length + weight + (1 | id)” and cc3 “corrAns ~ position + length + weight + reading + (1 | id).” Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4485936","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":321023277,"identity":"4f5384e4-1197-4253-a081-3241d8c3d175","order_by":0,"name":"Lin Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtklEQVRIiWNgGAWjYBACPmbmAwc+MDDwMDAkEKmFjZkt8eAM0rQw8Bgf5gEzidbCzpZw2LbtsIw5e/rDxxUMdvJEOIz5wOHctsM8lj1vjA3PMCQbNhDWArQlt+02j8GNHDbJBoYDjERo4TE4bAnWkv4MpMWeOC2MYC0JZiAtiUQ57GDPuf88BmeAfmkwSE4mqIWf//DhDz/K0uwNjqc/fNhQYWdLUAsaMCBR/SgYBaNgFIwC7AAACyk5gOr2n+0AAAAASUVORK5CYII=","orcid":"","institution":"University of Windsor","correspondingAuthor":true,"prefix":"","firstName":"Lin","middleName":"","lastName":"Li","suffix":""},{"id":321023280,"identity":"8f49b2e8-b065-4c56-84b2-2a4f16ed7df6","order_by":1,"name":"George (Guoqiang) Zhou","email":"","orcid":"","institution":"University of Windsor","correspondingAuthor":false,"prefix":"","firstName":"George","middleName":"(Guoqiang)","lastName":"Zhou","suffix":""}],"badges":[],"createdAt":"2024-05-27 15:10:46","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4485936/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4485936/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":59468907,"identity":"3f5842d4-9964-4ffc-b282-9ed88e24c202","added_by":"auto","created_at":"2024-07-02 07:12:38","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":5482,"visible":true,"origin":"","legend":"\u003cp\u003eThe Pendulum Problem.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4485936/v1/4f161a1055ae2053971ab2bb.png"},{"id":59468908,"identity":"3ed9c698-5a71-4c41-bbf6-394324de1911","added_by":"auto","created_at":"2024-07-02 07:12:38","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":58767,"visible":true,"origin":"","legend":"\u003cp\u003eIllustration of the New Taxonomy of Physics Concepts for Learning Pendulum Motion.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4485936/v1/df59023af95b2520ca39a2d1.png"},{"id":59468906,"identity":"1a27f8bd-1013-4c43-9a80-1f126a0132f6","added_by":"auto","created_at":"2024-07-02 07:12:38","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":29688,"visible":true,"origin":"","legend":"\u003cp\u003eIllustration of Rapid Serial Visual Presentation Experimental Procedure.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4485936/v1/f7a4637723b4452c2080d0c8.png"},{"id":59468331,"identity":"fa634f18-05ab-4395-ac5c-d3653a782336","added_by":"auto","created_at":"2024-07-02 07:04:38","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":27913,"visible":true,"origin":"","legend":"\u003cp\u003eReaction Times as a Function of the Time Positions of the Candidate Pendulum.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4485936/v1/7988f89e9ecb8c16d52638e4.png"},{"id":59468327,"identity":"d2e6ea82-031e-4139-a3b7-0a5bb97cf3c2","added_by":"auto","created_at":"2024-07-02 07:04:38","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":28305,"visible":true,"origin":"","legend":"\u003cp\u003eReaction Times as a Function of the Pendulum Length Levels.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4485936/v1/7aff55066bd5d0de8c82b9b4.png"},{"id":59468328,"identity":"1e5a7da9-de50-46d3-8c67-e3a13b5f0c08","added_by":"auto","created_at":"2024-07-02 07:04:38","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":27919,"visible":true,"origin":"","legend":"\u003cp\u003eReaction Times as a Function of the Pendulum Bob's Weight Levels.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4485936/v1/10e3b3129ac033fc3c27635a.png"},{"id":62277106,"identity":"6c3f0890-5f7f-4355-b72e-4a9470ac6fba","added_by":"auto","created_at":"2024-08-12 11:36:07","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1230638,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4485936/v1/f380bb7f-9640-451b-a772-6a228731907f.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts","fulltext":[{"header":"Introduction","content":"\u003cp\u003eImagine a jigsaw puzzle - a pile of variously shaped pieces spread on the table. In expectation, the curly and wiggly-edged cardboard fragments would fit together to form a complete visual representation of something familiar: a flower, a watch, or even a pendulum clock. Regardless of age, culture, or language background, the joy of seeing a complete picture of some sort emerging out of seemingly chaotic bits is a memorable experience which may have a lasting effect on their problem-solving in future. However, occasionally, the joyful puzzle solver may come across a new piece shaped rather strangely. At first glance, it is a bit like part of a flower petal that produces nectar and attracts pollinators. However, after a careful re-examination, it also has the features of a pendulum bob.\u003c/p\u003e\n\u003cp\u003eBearing these alternative classificatory possibilities in mind, the solver sees the edge of the unidentified puzzle piece as having information so bizarre that it cannot fit into the already-in-place boundary conditions of a half-way-completed picture of the flower or the clock. Even worse is such a scenario where she suddenly realizes there might be unknown missing ones waiting to be filled up. The revelation is alarming because solving the puzzle with previously unknown pieces signals that the current solution is not tenable, and the semi-completed \u0026quot;flower\u0026quot; may be something else. To proceed, she may have to think out of the \u0026quot;box\u0026quot; and reconceptualize what the whole jigsaw puzzle is really about: a flower, a pendulum clock or something else. When the unknown is assumed to represent a counterintuitive scientific concept to be learned, the characterization of playing such a hypothetical game is close to what initially caught the attention of a group of Cornell and Witwatersrand scholars forty years ago: conceptual change.\u003c/p\u003e\n\u003cp\u003eBoth the opening example and conceptual change research focus on the unique uncertain aspect of working with a new concept or a conceptual system in problem-solving. Commonly, they assume a conceptual space for representing a problem\u0026apos;s essential information, sampling thus-defined problem-solving sub-spaces, and deciding on possible solutions in time. For some researchers, the space is filled with emotional experiences or features of socio-cultural processes. We, in this study, would assume a cognitive science perspective and take a closer look at how a new anchoring concept can be constructed, developed, upgraded, or even repurposed for solving a puzzling problem.\u003c/p\u003e\n\u003cp\u003eIn the early 1980s, Posner, Strike, Hewson, and Gertzog (\u003cspan class=\"CitationRef\"\u003e1982\u003c/span\u003e) published the article Accommodation of a scientific conception: Toward a theory of conceptual change in Science Education. Since then, thousands of theoretical and empirical studies have been documented in the literature using the same classifying term conceptual change, covering at least a narrow and broad sense of this classifier. Narrowly speaking, it refers to students\u0026apos; conceptual physics- or classical mechanisms-related learning phenomena. In this sense, the research focuses on transforming pre-college or college science students\u0026apos; misconceptions of some aspects of nature so that they can align with and appreciate a qualified theoretical or experimental physicist\u0026apos; view of the same phenomenon, such as the Brownian motion or the isochrony of pendulum motion. Broadly, the same classifier also refers to changing cultural, social, and philosophical attitudes or belief systems about science and science education systems over time, with a learner in such a social group changing her conceptions of a natural phenomenon laterally and concurrently. The subject matter of this study is the first, narrower, and domain-specific sense of conceptual change. In this sense, how students recognise and explain the time measuring potential of a pendulum or any oscillating object provides a historically meaningful and knowledge rich experimental platform.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eA False Positive Identification of a \u0026quot;Force\u0026quot; That Drives Pendulum Motion\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIn the context of surveying students\u0026apos; misconceptions, John Clement (\u003cspan class=\"CitationRef\"\u003e1982\u003c/span\u003e) documented how the student conceptual primitive of the relationship between force and acceleration was misunderstood at the qualitative level in the context of a pendulum problem. The pendulum problem was designed to elicit the \u0026quot;motion implies a force\u0026quot; preconception (p. 67).\u003c/p\u003e\n\u003cp\u003eThe problem stated:\u003c/p\u003e\n\u003cp\u003e(a) A pendulum is swinging from left to right as shown below (above). Draw arrows showing the direction of each force acting on the pendulum bob at point A. Do not show the total net force and do not include frictional forces. Label each arrow with a name that says what kind of force it is.\u003c/p\u003e\n\u003cp\u003e(b) In a similar way, draw and label arrows showing the direction of each force acting on the pendulum bob when it reaches point B. (p. 67)\u003c/p\u003e\n\u003cp\u003eFacing such a pendulum motion problem, college students often drew the dashed line representing the driving force of the pendulum (See Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e). In contrast, the physicist only labelled two types of forces as shown in the two solid lines: gravity and the tension force. For an illustration of the difference, the dashed line was added as evidence to show these students\u0026apos; preconception of the implied force. The students believed that there must be an independent force acting on the bob in the direction of its movement. In terms of human information processing, the identification of something non-existent as real is a false positive. Particularly in this case, they tended to add an extraneous \u0026quot;force\u0026quot; to the visual representation of the pendulum, in addition to the necessary tension and gravity. This non-existent intuitive force was seen as an \u0026quot;essential\u0026quot; component that would drive the swinging pendulum.\u003c/p\u003e\n\u003cp\u003eMoreover, Clement (\u003cspan class=\"CitationRef\"\u003e1982\u003c/span\u003e) summarized the common features of the \u0026quot;motion implies a force\u0026quot; preconception, such as the following:\u003c/p\u003e\n\u003cp\u003e(1) Continuing motion, even at a constant velocity, can trigger an assumption of the presence of a force in the direction of motion that acts on the object to cause the motion.\u003c/p\u003e\n\u003cp\u003e(2) Such invented forces are especially common in explanations of motion that continues in the face of an obvious opposing force. In this case the object is assumed to continue to move because the invented force is greater than the opposing force.\u003c/p\u003e\n\u003cp\u003e(3) The subject may believe that such a force \u0026quot;dies out\u0026quot; or \u0026quot;builds up\u0026quot; to account for changes in an object\u0026apos;s speed. (p. 69)\u003c/p\u003e\n\u003cp\u003eIn discussing the implications of such findings, Clement (\u003cspan class=\"CitationRef\"\u003e1982\u003c/span\u003e) noted that the preconceptions are \u0026quot;not likely to disappear simply because students have been exposed to the standard view in their physics courses. More likely, Newtonian ideas are simply misperceived or distorted by students to fit their existing preconceptions\u0026quot; (p. 70). He further suggested Galileo might be aware of the teaching challenges faced by a modern physics instructor for \u0026quot;his dialogs represent a marvelous attempt to deal directly with the common preconceptions and prevailing theories of his time at a qualitative level\u0026hellip; One might do worse than to take these aspects of Galileo\u0026apos;s teaching technique as a model for pedagogy today\u0026quot; (p. 70).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eThe idealised Simple Pendulum Motion: T\u0026thinsp;=\u0026thinsp;2\u0026pi;\u003c/strong\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sqrt{\\mathbf{l}/\\mathbf{g}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eTo contrast the students\u0026apos; false positive identification and explanation of pendulum motion, we briefly summarize a pendulum motion lab exercise that features the quantitative aspect of pendulum motion. C\u0026eacute;sar Medina, Sandra Velazco, and Julia Salinas of Argentina (2004)\u003c/p\u003e\n\u003cp\u003ePhysical pendulum period\u003c/p\u003e\n\u003cp\u003eTp\u0026thinsp;=\u0026thinsp;2\u0026pi; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sqrt{I /mgd}\\)\u003c/span\u003e\u003c/span\u003e (1)\u003c/p\u003e\n\u003cp\u003ewhere T\u003cem\u003ep\u003c/em\u003e represents the period of the physical pendulum, \u003cem\u003eI\u003c/em\u003e the moment of inertia, \u003cem\u003em\u003c/em\u003e the mass of body, \u003cem\u003eg\u003c/em\u003e the acceleration due to gravity and \u003cem\u003ed\u003c/em\u003e the distance between the axis and the center of gravity of the system.\u003c/p\u003e\n\u003cp\u003eb) Ideal simple pendulum period\u003c/p\u003e\n\u003cp\u003eTs\u0026thinsp;=\u0026thinsp;2\u0026pi; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sqrt{l/g}\\)\u003c/span\u003e\u003c/span\u003e (2)\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003el\u003c/em\u003e is the length of the string.\u003c/p\u003e\n\u003cp\u003eEquation (1) was deduced assuming:\u003c/p\u003e\n\u003cp\u003eA1: negligible friction (the resultant torque on the system about the horizontal axis is solely due to the weight of the body).\u003c/p\u003e\n\u003cp\u003eA2: small oscillation amplitudes (in the equation of motion, the sine of the amplitude angle can be replaced by the angle in radians).\u003c/p\u003e\n\u003cp\u003eA3: the pendulum is a rigid body (invariable mass distribution, constant moment of inertia).\u003c/p\u003e\n\u003cp\u003e\u0026hellip;\u003c/p\u003e\n\u003cp\u003eA4: the string mass must be negligible.\u003c/p\u003e\n\u003cp\u003eA5: the body mass must be concentrated at a point. (Medina et al., \u003cspan class=\"CitationRef\"\u003e2004\u003c/span\u003e, p. 632)\u003c/p\u003e\n\u003cp\u003eAfter showing how a physical pendulum can be mathematically associated with an ideal one, they used a section focusing on error analysis. From such a viewpoint, there are random errors in addition to a systematic error due to \u0026quot;the fact that assumptions A1 to A5 are not fulfilled\u0026quot; (Medina et al., \u003cspan class=\"CitationRef\"\u003e2004\u003c/span\u003e, p. 633). In particular, these error terms include the fluctuations \u0026quot;due to friction (\u0026epsilon;\u003cem\u003ef\u003c/em\u003e ), initial amplitude (\u0026epsilon;\u003cem\u003e\u0026alpha;\u003c/em\u003e), variable length of the string due to a variable tension during oscillation (\u0026epsilon;\u003cem\u003eT\u003c/em\u003e ), mass distribution of the body (\u0026epsilon;\u003cem\u003eb\u003c/em\u003e), and mass of the string (\u0026epsilon;s)\u0026quot; (Medina et al., \u003cspan class=\"CitationRef\"\u003e2004\u003c/span\u003e, p. 633). With the analysis of error considered, they showed that the model assumptions could be accomplished in laboratory exercises within a reasonably small range of experimental errors. They concluded that,\u003c/p\u003e\n\u003cp\u003eConsidered separately, within an error of 1%:\u003c/p\u003e\n\u003cp\u003e\u0026ndash; an initial amplitude of 23\u003csup\u003e◦\u003c/sup\u003e is \u0026quot;small\u0026quot;.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026ndash; a sphere, whose diameter is 30% of the length of the string, is \u0026quot;a point mass\u0026quot;.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026ndash; a mass of the string equal to 10% of the mass of the body is \u0026quot;vanishing\u0026quot;.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026ndash; any elastic elongation suffered by the string during the static process of loading is negligible, providing the string length is measured after the loading.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u0026ndash; without loosing its property of \u0026apos;not extensible\u0026apos;, the string may vary its length during oscillation (due to a variable tension), providing this variation is less than the measurement error of the string length. (Medina et al., 2004, p. 639)\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn their view, such a quantitative laboratory experimental demonstration advances a better understanding of scientific practices, promoting a deeper comprehension of the pendulum motion-related physics concepts. Moreover, the epistemological implications of the analysis of errors are also rich. Taken together the qualitative and quantitative aspects of learning pendulum motion, it reasonable to expect an idealised conceptual change process as the one that overcomes the false positive identification and approaches a scientifically identification of the period of pendulum motion. To provide a new conceptual basis to accommodate such a view of conceptual change, I put forward a new taxonomy of of concepts, which is depicted in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. With this taxonomy, it becomes possible to categorize and analyze the processes of conceptual change at different levels of verbal, scientific, and mathematical descriptions with greater precision. Figure \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e is an example of applying such a taxonomy to reorganize the pendulum knowledge systems.\u003c/p\u003e\n\u003cp\u003eFor dramatizing the effects of using the new taxonomy, a pictorial and symbolic mixed artwork is created to foreground the conception of mathematically defined physics concepts, especially in the case of learning pendulum motion. The keywords of the new taxonomy in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e characterize a knowledge system of pendulum motion phenomena, statements, data, and a structural realist\u0026apos; theory about them (A. F. Chalmers, \u003cspan class=\"CitationRef\"\u003e1999\u003c/span\u003e; Matthews, \u003cspan class=\"CitationRef\"\u003e2015\u003c/span\u003e; Rowbottom, \u003cspan class=\"CitationRef\"\u003e2019\u003c/span\u003e; Worrall, \u003cspan class=\"CitationRef\"\u003e2007\u003c/span\u003e). At the bottom of the upward swing, the physical objects, natural processes, and simple events of our world occurred naturally, without the involvement of any form of symbolic processing in any language. Moving up a bit, it is human psychological aspects of observing what has happened in the world and the symbolization in English.\u003c/p\u003e\n\u003cp\u003eNext, human perception-driven statements or scientific narratives about the experience of understanding pendulum motion are located at the verbally defined level or propositional perception (Matthews, \u003cspan class=\"CitationRef\"\u003e2015\u003c/span\u003e). Following the level, error-term-characterized empirical observations are represented as half mathematically and half verbally defined as raw scientific data. At the top of the upward swing sits the structural realist\u0026apos;s mathematical core: T\u0026thinsp;=\u0026thinsp;2\u0026pi; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sqrt{l/g}\\)\u003c/span\u003e\u003c/span\u003efor small swing amplitudes and the real natural phenomenon hidden behind the veil of so-called \u0026quot;reality\u0026quot;: an isochronous simple harmonic oscillator. Together, the upward movement of a pendulum acts as a conceptual linchpin for characterizing the scope of knowledge involved in learning pendulum motion, which spans from an event, propositional perception, and the underlying continuous mathematical identification of this phenomenon.\u003c/p\u003e\n\u003cp\u003eThe downward swing by the side of the upward one shows a recurring information integration episode in a learner\u0026apos;s mind. The dashed lines indicate the probabilistic nature of human information processing. Most importantly, we contend that all these episodes of information processing take time, regardless of which aspect is of interest. In this sense, students\u0026apos; conceptional change toward understanding the notion of a mathematical idealized simple pendulum can be tested empirically in a series of experiments. Thus, the overall effect of students\u0026apos; intuitive pre-instructional conceptions on their real-time responses can be measured. The last two levels of such a knowledge system must be learnt with effort over time. In light of this, an active learning mechanism is also needed to explain the conceptual change effect during such effortful science learning.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAn Experimental Pilot Study: Matching the Period of Pendulum Motion\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe purpose of the first experimental study was to prototype the quantitative conceptual change studies and to run the new online experimental platform: Pavlovia (Pavlovia, n.d.). The difference between this experiment and the others was its use of the Rapid Serial Presentation technique (a visual stimuli train featuring the sub-second or millisecond presentation) to highlight the temporal aspect of visual experiences.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResearch Questions and Hypothesis\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAccording to the mathematical relationship describing the pendulum period with a small release angle boundary condition (T\u0026thinsp;=\u0026thinsp;2\u0026pi; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sqrt{l/g }\\)\u003c/span\u003e\u003c/span\u003e ), the length is the only factor that would affect the oscillation period of the pendulum motion. In contrast, other factors of the visual pendulum stimuli should not determine the time. If a learner understands the underlying reasoning, she ought to ignore the other factors when seeing them in this experiment. Experiment 1 was designed to test such a possibility while test-running the online open science platform Pavlovia with Chinese-English international students. The research question asked whether visual changes in a candidate pendulum\u0026apos;s length, bob weight, or temporal position would affect these bilinguals\u0026apos; matching choices. The null hypothesis of this experiment was that there would be no reaction time or accuracy differences existed among the bilinguals\u0026apos; responses among the experimental conditions. In contrast, the alternative hypothesis was that these participants\u0026apos; period-matching reaction times on these conditions would differ, reflecting their various levels of understanding of the mathematically defined pendulum motion and their sampling and decision-making processes on the fly.\u003c/p\u003e"},{"header":"Method ","content":"\u003cp\u003eThe first experiment was designed to measure the bilingual participants\u0026apos; reaction times with a within-participants design. To address the research question, I varied three visual perceptual levels of a computer-controlled display of a pendulum (length, weight, and the temporal position of a candidate pendulum). \u0026nbsp;Each participant was presented with a set of such four pendulums as an experimental unit through their home computers. Upon seeing the standard and matching pendulums unit of such visual inputs, the participant indicated her responses on each experimental trial with the mouse of the visual stimuli-presenting computer. All other visual features of the stimuli were irrelevant to the purpose of this experiment. \u0026nbsp;In other words, the two levels of the visual features of pendulum motion (its length and bob weight) and one level of temporal position initial release angle) were manipulated in a within-participants design, with the participants\u0026apos; reaction times and accuracies recorded as the dependent variables.\u003c/p\u003e\n\u003cp\u003eParticipants\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eA power analysis was conducted to find an adequate sample size for the within-participants experimental design. A variance and an effect size were estimated, given the currently reported similar experimental results in the literature. Then the Hotelling-Lawley Trace Statistical test with a Type 1 error of .05 was calculated. The analysis result showed that a total sample size of about 35 would yield a power level of .8. Due to the onset of the COVID-19 pandemic, however, twenty international students, aged from 18 to 55, were recruited for the first experiment (See Table 2).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThey were contacted through\u0026nbsp;the community of a southeastern Canadian university through contacting its Chinese Scholars and Students Association and recruitment postings. Most of them were native Chinese speakers who could speak English, and one of them also reported Korean as one of her known foreign languages. Their participation was compensated with a $10.00 e-gift card. \u0026nbsp;All the participants were na\u0026iuml;ve to the purpose of this experiment, and they reported having normal or corrected to normal vision. Table. 3 summarises the self-reports of their English language skills in reading, writing, speaking, and listening. The scale used was a 7-point one, with 1 representing \u0026quot;very poor\u0026quot; and 7 \u0026quot;native-like.\u0026quot;\u003c/p\u003e\n\u003cp\u003eApparatus and Stimuli\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAs the experiment was administered by an open science online experimental site Pavlova, we did not know specific details of the apparatus the participants used. However, they were set up to present all the stimuli at the refresh rate of 60 Hz. \u0026nbsp;Psychopy\u0026nbsp;(Peirce, 2007), an open-source psychophysiological software package with 1 msec precision, was used to create the experiment and upload it to the online platform Pavlovia, which synchronized the stimuli generation and data collection. Students\u0026apos; responses were registered through the mouse of their computers. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eBoth a standard and the three to-be-matched\u0026nbsp;pendulum stimuli were presented in a rapid serial visual presentation (RSVP) paradigm (See Figure. 3). \u0026nbsp;In each RSVP trial, participants saw a coloured standard pendulum shown in the centre of the screen on a grey background. They had five seconds to closely examine this pendulum, estimating its period and other features of their interest. After clicking a CONTINUE button, a series of three candidate pendulums would be rapidly presented, with a 500ms blank duration separating them. Each of the three pendulums was equally likely to be presented at the 1st, 2nd or 3rd position. Moreover, they were equally likely to be the target pendulum or one of the two distractor pendulums. Viewing from a distance of approximately 60cm, participants were instructed to respond to a target-matching pendulum as quickly and accurately as possible.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eDesign\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe experiment used a within-participants design. This minimized inter-participant variability\u0026apos;s impact across the different experimental conditions. \u0026nbsp;There were three factors: (a) the length of the pendulum (long, middle, and short); (b) the weight of the pendulum bob (heavy, medium, light), and (c) the position of the target pendulum (1, 2, 3 or 0). \u0026nbsp; The three factors were independently manipulated. \u0026nbsp;The initial release angle was not specifically controlled in this pilot experiment. \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eExperimental Procedure\u003c/p\u003e\n\u003cp\u003eAfter signing off the consent form, the participants would receive an experimental link, their unique participant numbers, and a session number. The experiment would automatically start after typing in the numbers through the link. The written instructions were presented in the middle of the screen. After reading the introduction, they would complete a language history questionnaire. Next, the practice session of this\u0026nbsp;experiment was presented at the same rate as that of the real one. Only their responses were not recorded. For the practice session, the item presentation rate started from 500 msec/item. No feedback was provided in practice, simulating what would occur in the real session.\u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eBefore\u0026nbsp;the actual experimental session started, the participants had a chance to take a break. When they were ready again, each participant could start the real experiment by clicking the CONTINUE button on the screen. \u0026nbsp;Each trial would begin with presenting a standard pendulum for viewing as long as they liked within a limit of five seconds. \u0026nbsp;After they chose to click the CONTINUE button, a series of two distractor pendulums and a target one was to be presented one at a time for 500 msec at the same location in the centre of the screen. \u0026nbsp;Each stream ended with a choice display showing \u0026quot;1\u0026quot;, \u0026quot;2\u0026quot;, \u0026quot;3\u0026quot;, and \u0026quot;0.\u0026quot; The\u0026nbsp;participants\u0026nbsp;were instructed to click on the number indicating the temporal position of the period-matching the target pendulum, with \u0026quot;0\u0026quot; indicating \u0026quot;no matching\u0026quot;, while ignoring all the other visual stimuli. \u0026nbsp; Because the candidate pendulums\u0026apos; visual features differed from the standard pendulum, the participants could not use these visual features as clues to match the standard pendulum at the sensory or perceptual level. Instead, they would have to use their understanding of the fundamental scientific laws to choose the target correctly (also known as a hit).\u0026nbsp;They were also instructed to respond as quickly and as accurately as they could by clicking on the indicating number of the temporal position of the candidate pendulum.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eData Analyses: ANOVA or Binomial Regression?\u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eGiven that the experimental condition groups\u0026apos; mean reaction times are the reaction times of the correct probe responses, the reaction time analyses must be limited to those probe trials in which the participants correctly identified a matching target pendulum. \u0026nbsp;Moreover, the correct reaction times were constrained by a latency range between 200 and 2,000 msec. \u0026nbsp;We expected most results observed in Experiment 1 to be within this range. \u0026nbsp;Further analyses did not include the participants\u0026apos; data with over 30% error rates, which may indicate a lapse of attention in a simple selection task. \u0026nbsp; The alpha level was set at .05.\u003c/p\u003e\n\u003cp\u003eGiven the within-participants design, the\u0026nbsp;mean\u0026nbsp;reaction time data were submitted to a repeated-measure ANOVA with pendulum motion features as the within-participant factors. \u0026nbsp;I expected to find the main effect of the pendulum feature, showing their understanding of\u0026nbsp;the fundamental scientific laws.\u0026nbsp; \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eQuantitative data can take many forms in education research, such as time or accuracy. In contrast to measuring learners\u0026apos; performance in milliseconds, it is common for education researchers to collect students\u0026apos; responses and mark them as right or wrong given a pre-defined theoretical position. Both teachers and researchers in education tend to draw definitive conclusions from analyzing such data, particularly after seeing a statistically significant result, yet reporting significance or not based on p-values thresholds of .05 or .01 has been contested as an acceptable good practice (Kuffner \u0026amp; Walker, 2019; Wasserstein \u0026amp; Lazar, 2016). The unsuitability is of great concern when the data modelling method may not be appropriate. Although researchers in education and psychology may be more familiar with the former, its underlying assumptions are not satisfied given the underlying nature of the error rates data. For them, adopting a logistic regression approach is more appropriate.\u0026nbsp;\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003eTable 4 shows the mean correct RTs for the candidate pendulum presented at the first, the second, and the third temporal position, with 0 representing no match or a correct rejection of Experiment 1. In the same table, the error rates are also displayed. The mean RTs were the response latencies of correctly matching a candidate pendulum with the standard one. \u0026nbsp; Thus, the RT analyses were limited to those probe trials in which the participants correctly identified the pendulum oscillation period. Moreover, the correct RTs were constrained by a latency range between 200 and 2,500 msec. \u0026nbsp;In this experiment, 98.9% of the correct RTs were within this range. \u0026nbsp;The data of five participants with over 60% error rates were not included in further analyses. \u0026nbsp;The alpha level was set at .05.\u003c/p\u003e\n\u003cp\u003eThe\u0026nbsp;mean\u0026nbsp;RT data were submitted to a repeated-measure ANOVA with the temporal position of the candidate pendulum as the 4-level factor. \u0026nbsp;The main effect of the temporal position was significant, \u003cem\u003eF\u003c/em\u003e (3, 42) = 4.74, \u003cem\u003ep\u003c/em\u003e \u0026lt; .01, \u003cem\u003eη\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e = .25, indicating mean RTs differed significantly across the three time points and a non-matching level. A post hoc pairwise comparison using the Bonferroni correction showed that it took an increased response time to decide the candidate pendulum presented at the first temporal position than at the second one (1026.67 vs 907.03, p \u0026lt; .05). Also, making a no-matching decision (correct rejection) was approaching statistically significant level as compared with doing that at the first temporal position (931.75 vs 1026.67, p = .08) (See Figure 4). No other pairwise comparisons had reached the statistically significant level. Therefore, we can conclude that the results of the repeated ANOVA have indicated a significant time effect for matching the pendulum oscillation period as measured in time by RTs.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn addition to the time position of a candidate pendulum, similar quantitative data analyzing procedures were also implemented to analyze the effects of the other two independent variables: the length of the pendulum d and the bob's weight. Table 5 shows the mean correct RTs for the candidate pendulum presented with the longest, medium, and shortest length in Experiment 1. The mean RTs were the response latencies of correctly matching a candidate pendulum with the standard one, given the length of the pendulum. \u0026nbsp; The data screening considerations were the same as the analysis of RTs for the temporal positions. The alpha level was also set at .05.\u003c/p\u003e\n\u003cp\u003eThe\u0026nbsp;mean\u0026nbsp;RT data were submitted to a repeated-measure ANOVA with the length of the candidate pendulum as the 3-level factor. \u0026nbsp;The main effect of the length was significant, \u003cem\u003eF\u003c/em\u003e (2, 28) = 4.33, \u003cem\u003ep\u003c/em\u003e \u0026lt; .05, \u003cem\u003eη\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e = .24, indicating the mean RTs differed significantly across the three length levels (See Figure 5). A post hoc pairwise comparison using the Bonferroni correction showed that it took an increased response time to make a decision about the candidate pendulum with the medium-length rod than that of the longest one (995.73 vs 913.82, p \u0026lt; .05). \u0026nbsp;No other pairwise comparisons had reached the statistically significant level. Therefore, I can conclude that the repeated ANOVA results have indicated a significant length effect for matching the pendulum oscillation period as measured in time by RTs.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTable 6 shows the mean correct RTs for the candidate pendulum presented with the heavy, medium, and light bob in Experiment 1. The mean RTs were the response latencies of correctly matching a candidate pendulum with the standard one, given the weight of the pendulum bob. The data screening considerations were the same as the analysis of RTs for the temporal positions. \u0026nbsp; The alpha level was also set at .05.\u003c/p\u003e\n\u003cp\u003eThe\u0026nbsp;mean\u0026nbsp;RT data were also submitted to a repeated-measure ANOVA with the bob weight of the candidate pendulum as the 3-level factor. The main effect of the bob weight was insignificant, F (2, 28) = .22, p \u0026gt; .05, indicating the mean RTs were not different across the three bob weight levels (See Figure 6). \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn brief, two statistically significant results have been identified in Experiment 1 following the repeated ANOVA methods. First, the time position of a candidate pendulum did affect the participants' decision-making, increasing their matching response times when the first candidate pendulum had to be selected from the other alternatives. Second, the oscillation period of the mid-length pendulum took more time to be judged as the same as that of the standard pendulum. All other experimental manipulations of Experiment 1 did not have the same response time-extending effects, such as the weight of pendulum bobs.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAs introduced in the data analysis tutorial\u0026nbsp;(L. Li, 2023), the ANOVA methods may not be suitable for analyzing error rates. The logistic regression-based modelling of accuracy data was adopted for the error rates observed in this experiment (See Table 7).\u0026nbsp;The analyses of the error rates have added some informative data to the response time results (See Table 8). Commonly, the length of the pendulum has been singled out as a significant predictor of making a correct decision about its oscillation period. However, the time position of a candidate pendulum has not been shown as another significant one. Most importantly, when the participants' reading levels were added to the logistic regression equation, it was identified as a significant predictor of making a correct decision in a pendulum period-matching trial. All other predictors have not been shown by both the response time and error rate analyses as significant.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eSeveral\u0026nbsp;aspects\u0026nbsp;of the observed patterns of the data are worth noting. First, there was a temporal position effect in identifying and matching the period of a pendulum. As this is one of the first studies using a series of candidate pendulums to measure students' responses, the outcome needs particular consideration. \u0026nbsp;The time difference between making a decision about the first and the second candidate pendulum reveals a serial effect of human memory retrieval mechanisms. It indicates that regaining the pendulum matching information from the represented distribution of the first candidate pendulum was more challenging than another overlapping distribution of the second time position. The structure of the candidate pendulum representation distributions determined the signature of such a time course in matching the period of the pendulum motion. \u0026nbsp; \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eSecond, the evidence suggests that deciding on a correct rejection was faster than hitting the first candidate pendulum, though only a marginally significant time difference had been observed in Experiment 1. This result was likely caused by the extent of memory loading involved in making such a negative response. In other words, less memory loading was needed for \"saying-no\" to a series of candidate pendulums. With the no-matching visual stimulus sampled after seeing a trial, it seems that the participants did not need to keep activated any knowledge distributions activated for completing the task, thus only using less time to click the \"no-matching\" selection. Given the marginally significant result, there are other possibilities worth further exploration. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eBesides the role of the time position of a candidate pendulum in determining the time course of matching a pendulum period, the third observed aspect was a significant effect of the length of the pendulum in this experiment. The time difference observed between deciding the candidate pendulum with the medium-length one, and that of the longest one seems to reveal a differentiating effect of human memory retrieval mechanisms. It indicates that regaining the pendulum matching information from the represented distribution of a mid-length pendulum was more challenging than from another overlapping distribution of the longest one. Again, the structure of the candidate pendulum representation distributions determined the characteristic feature of such a time course in matching the period of the pendulum motion. No similar results have been observed for varying the weight of pendulum bobs over the experimental conditions. The results indicate that the participants paid attention to the key determining factor of the pendulum, showing the effect of students' knowledge of the pendulum motion. \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFourth, it is worth noting that no interaction effects were observed between the time position and the length of the pendulum. This result indicates that the candidate pendulums presented in another time position may be processed similarly. This is likely because only one candidate pendulum needed to be selected out of the series once the distributed information of the first two-time positions had been processed. There was no further need to process the last temporal position in an experimental trial, or it would be easier to process the last.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAt last, the analyses of the error rates have confirmed the response time results. By a binomial logistic regression-based technique, the length of the pendulum has also been singled out as a significant predictor of making a decision correctly. However, the time position of a candidate pendulum has not been shown as another significant predictor. Interestingly, when the participants' reading levels were added to the logistic regression equation, it was identified as a significant predictor of correctly matching the pendulum period. All other predictors have not been shown by both the response time and error rate analyses as significant.\u003c/p\u003e"},{"header":"Discussions, Conclusions, and Implications","content":"\u003cp\u003eThe results have documented at least two types of evidence to highlight the organizing role of the mathematical identity expressed in students' sampling and decision-making in the pendulum period-matching and explaining tasks. The time has come to restate the thesis that conceptual change can be viewed as an active S-D process over an overlapping knowledge distribution in students' conceptual spaces. Whether in the intuitive or counterintuitive information processing or in the overlapping middle area of the represented knowledge distributions, the probability is the key to unlocking what has changed or not. By embracing a probabilistic frame of reference, we have proposed in this study to advance conceptual change in science education by holding tight to the mathematical definition of a physics concept and embodying the caveat \"Don't throw the baby out with the bathwater!\" In the two experiments and the interviews, the role of the mathematical definition of a physics concept in organizing these participants' conceptual spaces has been revealed through a pendulum period-matching task, which is complemented by the interviewee's verbal expression of understanding such a mathematical identity. In this chapter, we take a closer look at how to re-integrate a mathematically defined physics concept (such as T\u0026thinsp;=\u0026thinsp;2π\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sqrt{l/g}\\)\u003c/span\u003e\u003c/span\u003e ) in acting conceptual change learning with the verbal expressions included. Finally, we reflect on the pedagogical implications of these findings.\u003c/p\u003e \u003cp\u003eA Mathematically Defined Physics Concept: Friend or Foe?\u003c/p\u003e \u003cp\u003eInterestingly, most current conceptual change studies except PER choose to avoid the mathematical contents (Potvin \u0026amp; Cyr, \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). For example, Andrea A. diSessa (\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) noted \"(t)he conceptual change paradigm is less often applied to other areas of science, and much less in mathematics\" (p. 88). More directly, he later argued that \"(u)nderstanding mathematics and its use in science is a worthy topic, but I believe it is secondary to deep qualitative, conceptual understanding\" (diSessa, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2017\u003c/span\u003e, p. 26). However, the data collected in this study have suggested otherwise, even in the simplest case of matching a simple single pendulum motion task. The most relevant significant factors are those already included in the mathematical equation T\u0026thinsp;=\u0026thinsp;2π \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sqrt{l/g}\\)\u003c/span\u003e\u003c/span\u003e. The mathematical expression is, in effect, the mathematical definition of the concept: the oscillation period. Given what I observed in the experiments and the interviews, the participants struggled to understand such a mathematically defined physics concept with their ordinary senses.\u003c/p\u003e \u003cp\u003e When the boundary condition of a small initial release angle has not been met, only mathematical or experimental knowledge, rather than other types of verbal expressions, can provide a satisfactory explanation. In this aspect, I concur with Bruce L. Sherin when he commented on the conceptual physics program (Hewitt, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1971\u003c/span\u003e) or the similar Physics for Poets (March, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2003\u003c/span\u003e). He contested,\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eI challenge the assumption that in physics or any domain the conceptual and the symbolic elements (the mathematical identity or definition) of a practice can be separated for the purposes of instruction. Removing equations from the mix changes the nature of understanding. This does not imply that physics cannot be taught without equations. However, it does imply that equation-free courses will result in an understanding of physics that is fundamentally different from physics as understood by physicists. (Sherin, 2001, p. 524)\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHowever, the question remains why the equation-free PER would be different. Without a new theoretical framework, uncertainty remains. As introduced in Chap.\u0026nbsp;4, I have attempted to draw a wider picture of conceptual learning that centers on a psychologically plausible and probabilistic mechanism: the sampling and decision-making over an overlapped knowledge distribution. In this S-D framework, the sampling process provides a front end for the mind to take in new information, whereas the decision-making drives the learning outcomes. If this assumption is reasonable and correct, it implies that the equation-free physics learning programs or the conceptual change at the verbal level only promote a biased sampling strategy while leaving relevant mathematical contents out of the equation. we argue that mathematical elements are inevitable to understand learner sampling and decision-making fully.\u003c/p\u003e \u003cp\u003eIn general, using the mathematical form departs from the established conceptual change research traditions rooted in the philosophy of science, which may result in interpreting the history of science in a new light, especially when the philosophical tradition may sometimes become misleading. Alan Chalmers (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2009\u003c/span\u003e) has reminded science education researchers that losing the experimental contact with reality had failed the philosophical atomism as a general heuristic conceptual structure to maintain a productive role in guiding modern atomic physics research. If the fate of philosophical atomism has revealed something soberingly informative, it also reminds conceptual change researchers not to lose mathematical and experimental contact with reality.\u003c/p\u003e \u003cp\u003eUnlocking the Learning Brain's Active Sampling and Decision-making\u003c/p\u003e \u003cp\u003eIn today's parlance, the brain relies on a network-like structure (Baronchelli et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2013\u003c/span\u003e) to enable us to sample and make a decision, thus changing conceptions. Such a neural network has often been approximately characterized by its components and connections: neurons and synapses (Dehaene \u0026amp; Naccache, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Salmelin \u0026amp; Kujala, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2006\u003c/span\u003e). As we know, a single neuron affords the basic cell-level information processing unit. Its conception was conceived more than 100 years ago by Santiago Ram\u0026oacute;ny Cajal (Haines, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2007\u003c/span\u003e), who first identified the independent cellular structure of a neuron. Following his lead, Adrian and Bronk (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1929\u003c/span\u003e) associated neurons' spiking patterns with the axonal and dendritic mechanisms. Later, Hodgkin, Huxley, and Eccles (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e1963\u003c/span\u003e) demonstrated the ionic mechanisms inside and outside neurons' membranes. Together, these single neuron-based mechanisms demystified the brain's neural impulse trains - the information-carrying mechanism of human cognition. Regardless of describing or explaining axonal or ionic neuronal mechanisms, time is a fundamental aspect of them (Mesulam, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e1998\u003c/span\u003e; Muller, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Palva \u0026amp; Palva, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2012\u003c/span\u003e), which implies time is essential for understanding learning.\u003c/p\u003e \u003cp\u003eThe importance of time in understanding the brain's language of information processing can be found beyond the single neurons. In effect, it also has implications for other neuronal structures, such as (a) the supportive glial cells-based mechanisms (Fields et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), and (b) the synaptic (neuron-to-neuron) chemical information transmission processes (Bennett, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2000\u003c/span\u003e). First, the glial cells separate the myelinated axonal fibres from the unmyelinated ones. According to Fields et al. (\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), the glial cell-based mechanisms are still largely absent from thinking about representing and processing information because the glial cells do not generate electrical impulses. However, they form crucial cell-cell interaction that shapes the cellular mechanisms of learning and cognition. More importantly, they couple neurons into functional units for short-term and long-term information storage and transformation, thus enabling learning and cognition. In other words, learning is in time.\u003c/p\u003e \u003cp\u003eFurthermore, time is also involved in inter-neuronal connections. As for the chemical agent-based neuron-to-neuron communication, the specialized neuronal structure at the axon terminal is called a synapse (Debanne, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Fields et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Langille \u0026amp; Brown, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), the gap for diffusing and relaying neurotransmitters from one sending neuron to a receiving one. The diffusion process starts by releasing the functional molecules from pre-synaptic neurons' membranes into the synaptic gap. Over the gap, the ionic channels of post-synaptic neurons would enable a membrane-fusing process, binding these molecules in a lock-and-key manner. The binding thus opens and closes the membrane ionic channels. The exchange would permit some kind of neuron-to-neuron information transmission, spreading neuronal information forward in a neuronal network. The three components (single neurons, glial cells, and synapses) help form a neuron-based information processing network in the central and peripheral neural systems. As for learning and cognition, small neocortical networks form large-scale neural networks to support reshaping the dynamics and structures of such a network (Gastner \u0026amp; \u0026Oacute;dor, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). Again, the brain processes information in time\u003c/p\u003e \u003cp\u003eSuch a vast time-based information processing network is necessary for researchers to conceptualize problem-solving, conceptual change, and metacognitive processes. One powerful way to characterize the synaptic connection-based structure is to use a hierarchical structure for a functional approximation. For example, Mesulam (\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e1998\u003c/span\u003e) introduced six degrees of synaptic separations to capture the essence of such a network (i.e., its primary sensory-motor function, unimodal associative representing function, and hetero-modal associative and the paralimbic and limbic representing function).\u003c/p\u003e \u003cp\u003eMore specifically, the primary sensory-motor function of such an information processing network interfaces the initial processing of \"raw\" sensory inputs and the generation of behaviourally significant responses. Close to it, the unimodal associative function of the network maintains the fidelity of the \"raw\" sensory inputs. In contrast, the hetero-modal associative function of the network serves to provide a cross-sensory-modality representation of the input data. At last, the paralimbic and limbic functions provide reciprocal access to the hypothalamus. Collectively, such a characterization offers a framework to ground cognition. In other words, cognitive processes are defined as the neural information processes between the obligatory processing of \"raw\" sensory inputs and the generation of behaviourally significant responses in such a network. Meanwhile, cognitive processes manifest contextual effects, memory guidance, and other task-bound constraints realized in the network. Again, the brain's information processing over six degrees of separation can be seen as a conceptional change process that occurred in time. These considerations contribute to building a solid scientific foundation for reconceptualizing conceptual change through active sampling and decision-making.\u003c/p\u003e \u003cp\u003eConclusions and Pedagogical Implications\u003c/p\u003e \u003cp\u003eIn this experiment, participants' pendulum period-matching was measured in the rapid serial presentation format by varying a range of factors. To our knowledge, this is the first study that has demonstrated how to measure it and the first study that has given an initial estimate of its magnitude. The results pointed out a unique structure of intuitive and nonintuitive in their mind: an overlapping binomial distribution-like conceptual structure.\u003c/p\u003e \u003cp\u003eThe binomial distribution-like knowledge structure has unique characteristics that distinguish it from those verbal definitions of a conceptual change space. Specifically, it exhibits an overlapping middle area encompassing intuitive and non-intuitive knowledge. It can explain conceptual change as a sample and decision-making process within this conceptual space. Given such a theoretical construct, the conceptual change process can be viewed as a time-based procedure with a different sampling tendency over the knowledge distribution. While a complete understanding of conceptual change remains elusive (G. J. Posner et al., \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e1982\u003c/span\u003e; Thagard, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e1990\u003c/span\u003eb; Babai, Levyadun, et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2006\u003c/span\u003eb; Zhou, 2010a, 2012a; Potvin et al., \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003eb), this study has provided a unique and informative reference point for future research into the active sampling and decision-making mechanisms involved in conceptual change.\u003c/p\u003e \u003cp\u003eIn brief, the new taxonomy with a probabilistic frame of reference significantly contributes to extending the hybrid learning space (Authors, 2012) by establishing meaningful connections and offering opportunities to understand international students' science learning experiences in verbal expressions and their reaction times. It recognizes that these international students come to the science classroom with intuitive pre-instructional ideas, which may be inaccurate or incomplete given mutually accepted scientific understanding and practice. By seeing these misconceptions as sampling and decision making, science educators can have a conceptual handle to help students resample and make a new decision, thus promoting a more accurate and comprehensive understanding of scientific concepts.\u003c/p\u003e \u003cp\u003eMoreover, foregrounding the role of a matrix of event, propositional perception, and mathematical functions in influencing students' real-time responses help to reconnect sociocultural conceptual change studies with the structural realist's outlook of the fundamental science and science education (A. F. Chalmers, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Matthews, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Mayer, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Rowbottom, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Furthermore, the results of this study help promote a new line of discussion of measurement theories in conceptual change studies. Most importantly, the evidence has confirmed the co-existing view of students' intuitive ideas and the scientific notion of pendulum motion, as advocated in the conceptual advancement view of conceptual change (Zhou, 2012). Such evidence will further support the student-centred approaches in PER, which let students speak out their intuitive ideas first; and then offer culturally and linguistically appropriate feedback to improve their science learning (Zhou, 2010).\u003c/p\u003e \u003cp\u003eThe conceptual change view of science learning is not new, but a new taxonomy based on concepts and conceptual change is, especially when considering international students' learning experiences. Although the probabilistic taxonomy has not been explicitly discussed before, similar ideas have been explored in the intersections of conceptual change studies (Duit \u0026amp; Treagust, 2003; Potvin et al., \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Thagard, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e1990\u003c/span\u003e; Author, \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2010\u003c/span\u003e), the second language learning research (J. Li et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; J. Y. Li \u0026amp; Zhou, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; K. C. Li \u0026amp; Wong, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; L. Li, 2016), and domain-specific teaching and learning (Mayer, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2004\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe classificatory scheme helps researchers refocus on what culturally and linguistically diverse international students really bring to Canadian science classrooms. More importantly, the taxonomy promotes more generalized thinking in science education, seeing previous separate conceptual change studies as special cases of reweighting personal knowledge distributions. Although it is always challenging to characterize a still-evolving research field, reviewing some fundamental aspects of conceptual change studies still helps us consolidate what we have learned so far.\u003c/p\u003e \u003cp\u003eThe pedagogical implications of such a probabilistic cognitive \"revolution\" are manifold: (a) the probabilistic re-orientation can enhance science students', domestic or international, understanding of scientific concepts and scientists' conceptions. By using mathematically defined conceptual tools, the students can gain a deeper understanding of scientific concepts that may have previously appeared difficult to comprehend. The new taxonomy allows students to view scientific concepts and conceptions through a physics-compatible lens, which can significantly help clarify the underlying theoretical principles and the organizing key notions; (b) the new taxonomy helps guide new curriculum design endeavours to bridge the gap between abstract mathematical concepts and their physical interpretations. In the tradition of conceptual change studies, physics education research has relied heavily on qualitative approaches to understanding physical phenomena, often overlooking the importance of idealized quantitative reasoning and its error terms. By incorporating sampling and decision-making theory into physics education research, researchers and students alike are more likely to appreciate a deeper understanding of the underlying mathematical structures that govern physical phenomena, which entails conceptional change; (c) the taxonomy and its experimental manifestations deliberately promote a positive attitude toward the interdisciplinary learning since the experimentation and statistical modelling are not limited to physics research alone, and many other fields such as mathematical psychology, artificial intelligence, and educational assessment depend heavily on mathematical reasoning. By incorporating the new taxonomy into the researchers' teaching practices, they, in effect, help develop the student's skills necessary to apply scientific and mathematical reasoning across a wide range of disciplines.\u003c/p\u003e\n\u003cp\u003e\u0026shy;A Caveat on the Limitations of the Current Study and Future Research\u003c/p\u003e\n\u003cp\u003eThis study was designed when the COVID-19 pandemic was still affecting every aspect of students\u0026apos; lives. One limitation of this study is its lack of an interactive component in the pendulum period-matching task. Therefore, the experiment has not fully explored the participants\u0026apos; active learning. Further studies should consider the possibility of adding such a component as a participant self-controlled matching procedure. It will be informative to find out whether their active exploration would increase their response times. \u0026nbsp; \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe absence of an emotional component is the second limitation of the current study. Since the publication of Beyond Cold Conceptual Change: The Role of Motivational Beliefs and Classroom Contextual Factors in the Process of Conceptual Change\u0026nbsp;(Pintrich et al., 1993)\u003cem\u003e,\u0026nbsp;\u003c/em\u003ethe emotional aspect of conceptual change processes has attracted scholars\u0026apos; attention. For some non-math students, their motivational and emotional experiences may significantly influence their conceptual change learning. Future experiments should explore this possibility. \u0026nbsp;\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAuthor contributions:\u003c/strong\u003e G. Zhou contributed to the study\u0026rsquo;s conceptualization, resource management, supervision, revising the manuscript, and approving it. L. Li contributed to the literature search, manuscript\u0026rsquo;s writing, validation, analysis of the results, editing, and finalization.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of Interest:\u0026nbsp;\u003c/strong\u003eThere are no conflicts of interest between the authors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthical Approval:\u0026nbsp;\u003c/strong\u003eThe study was approved by the University\u0026apos;s Research Ethics Board.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eInformed Consent:\u0026nbsp;\u003c/strong\u003eThis The signed informed consents were obtained from all participants of this study by one of the authors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability:\u003c/strong\u003e The data supporting our findings are available upon request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAdrian, E. D., \u0026amp; Bronk, D. W. (1929). 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Concepts and conceptual change. \u003cem\u003eSynthese\u003c/em\u003e, \u003cem\u003e82\u003c/em\u003e(2), 255\u0026ndash;274. https://doi.org/10.1007/BF00413664\u003c/li\u003e\n\u003cli\u003eWorrall, J. (2007). Miracles and Models: Why reports of the death of Structural Realism may be exaggerated. \u003cem\u003eRoyal Institute of Philosophy Supplements\u003c/em\u003e, \u003cem\u003e61\u003c/em\u003e, 125\u0026ndash;154. https://doi.org/10.1017/S1358246100009772\u003c/li\u003e\n\u003cli\u003eAuthor. (2010). Conceptual Change in Science: A Process of Argumentation. \u003cem\u003eEurasia Journal of Mathematics, Science \u0026amp; Technology Education\u003c/em\u003e, \u003cem\u003e6\u003c/em\u003e(2).\u003c/li\u003e\n\u003cli\u003eAuthor. (2012). A Cultural Perspective of Conceptual Change: Re-examining the Goal of Science Education. \u003cem\u003eMcGill Journal of Education / Revue Des Sciences de l\u0026rsquo;\u0026eacute;ducation de McGill\u003c/em\u003e, \u003cem\u003e47\u003c/em\u003e(1), 109\u0026ndash;129. https://doi.org/10.7202/1011669ar\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Tables","content":"\u003cp\u003eTable 1. A New Taxonomy of Physics Concepts for Learning Pendulum Motion.\u0026nbsp;\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.746753246753247%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eConcept\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.051948051948052%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eType 1\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.88961038961039%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eType 2\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.051948051948052%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eType 3\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.25974025974026%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eType 4\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.746753246753247%\" valign=\"top\"\u003e\n \u003cp\u003eCriteria\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.051948051948052%\" valign=\"top\"\u003e\n \u003cp\u003eMathematically Defined\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.88961038961039%\" valign=\"top\"\u003e\n \u003cp\u003eHalf mathematically defined and half verbally defined\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.051948051948052%\" valign=\"top\"\u003e\n \u003cp\u003eVerbally defined\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.25974025974026%\" valign=\"top\"\u003e\n \u003cp\u003eNon-verbally defined\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.746753246753247%\" valign=\"top\"\u003e\n \u003cp\u003eExample\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.051948051948052%\" valign=\"top\"\u003e\n \u003cp\u003eAn imaginary number, the probability amplitude,\u003c/p\u003e\n \u003cp\u003eSimple harmonic motion\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.88961038961039%\" valign=\"top\"\u003e\n \u003cp\u003eForce (F = m \u0026times;\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAgCAIAAACO148VAAABMElEQVRIDe2Vba3FIAyGawENtVAPlXA0YAEHdYADFKAAAxjAwTz0JmtCll124Hz9W38sGZSH9y2sA/1BwA+YugqFPRYVrEJzzgCryat5pZSvQWutRAQAzjmDikgvBTMDgIicyjJRiogxRlU92iciGyylMPOJqDo7qG75aD+lhIiqysyttZehzjkTlVICgG3bDIGIj8fjv3GbndhPKbk9YowAEEKwZSa273ESO4Geshdfb+hloeySv/Q01l3Ty5q+PbFUU/vwh71juPESVFVDCN+HikiHElEpZajRBidKrXFYkzao977f3Cv0M2hrDQBqrSf7zHyFmys9Nvaj/Y+gtdau1HtPRCaEmXPO2x7Dyj6zb64BABFFxDmXUlJV+/cx892k36npcM10cHJQ0/XDhD+RpYkj4E9MnAAAAABJRU5ErkJggg==\" width=\"28\" height=\"32\"\u003e\u0026nbsp;),\u003c/p\u003e\n \u003cp\u003eThe isochronous pendulum motion (T\u003csub\u003e12\u003c/sub\u003e = T\u003csub\u003e21\u003c/sub\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.051948051948052%\" valign=\"top\"\u003e\n \u003cp\u003eArgumentation,\u003c/p\u003e\n \u003cp\u003eIdeology,\u003c/p\u003e\n \u003cp\u003ePropositional perception\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.25974025974026%\" valign=\"top\"\u003e\n \u003cp\u003eVisual perception,\u003c/p\u003e\n \u003cp\u003eAuditory perception,\u003c/p\u003e\n \u003cp\u003eWorking memory system\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eTable 2. A Summary of the Twenty Participants of Experiment 1.\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eNo\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eAge\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eGender\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eEdu\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eMajor\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e1st language\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e2nd language\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eBA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish, Korean\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eBA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003ePhysics\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e04\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eBA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003ePhysics\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e08\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eHigh School\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eMathematics\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eNo.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eNo.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eBA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003ePhD\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eChemistry\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eBA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eMaster\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eMathematics\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003ePhD\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eBiology\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eCantonese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eF\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003ePhD\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"6.652806652806653%\" valign=\"top\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.108108108108109%\" valign=\"top\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.305613305613306%\" valign=\"top\"\u003e\n \u003cp\u003eM\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.8004158004158%\" valign=\"top\"\u003e\n \u003cp\u003eBA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.126819126819125%\" valign=\"top\"\u003e\n \u003cp\u003eNon-Science\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.671517671517673%\" valign=\"top\"\u003e\n \u003cp\u003eChinese\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.334719334719335%\" valign=\"top\"\u003e\n \u003cp\u003eEnglish\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eTable 3. The Language Learning Background of the Twenty Participants of Experiment 1.\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eNo\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eReading\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eWriting\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eSpeaking\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eListening\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e04\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e08\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e5.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5.5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e4.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5.5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e5.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5.5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e4.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e4.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e4.5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e6.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e5.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e6.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e6.5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.725274725274724%\" valign=\"top\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.87912087912088%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.439560439560438%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"21.978021978021978%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eTable 4. Correct Response Times and Mean error rates (% error) for the Three Time Positions of the Candidate Pendulum in Experiment 1, with No-Matching as the Zero Position.\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eExp\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eParticipant\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eRT0(ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eRT1(ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eRT2(ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eRT3 (ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eMissing\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eCorrect%\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eError%\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e823.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2085\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1684.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1266.333\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e32.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e67.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1759.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1313.429\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1158.857\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1438.667\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e31.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e68.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1012.059\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1386\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1153.909\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1255.333\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e65.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e34.3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1789.222\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2005.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2016\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1763.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e94.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e5.7\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e768.0625\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e670.2857\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e718.3529\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e815.6667\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e89.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e10.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1131.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2088.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2497.667\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1358.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e20.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e79.7\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e973.3889\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e928.0714\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e999.7778\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1007.063\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e664.5211\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e708.2769\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e695.2031\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e711.7578\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e94.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e5.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e927.9111\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1010.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e713.7378\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e912.9629\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e91.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e8.3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e583.6667\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n 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\u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1083.355\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1366.508\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e82.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e17.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e878.5247\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e977.5772\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e762.1722\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e919.6922\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e97.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e929.9567\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e973.7313\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1029.899\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e893.4729\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e95.8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e4.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n 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\u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1014.828\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1343.771\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e917.91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e889.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e85.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e14.7\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1581.135\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1374.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e3567.84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e5.8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e94.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e682.9363\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e758.1012\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e612.0661\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e703.2441\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e94.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e5.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e951.2667\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1006.278\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e778.1111\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e957.1667\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e90\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e959.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1079.929\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e964.4615\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e946.2222\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e66.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e33.3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.286689419795222%\" valign=\"top\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e472.5556\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e416.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e376.0556\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e469.2941\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.262798634812286%\" valign=\"top\"\u003e\n \u003cp\u003e98.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.921501706484642%\" valign=\"top\"\u003e\n \u003cp\u003e1.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTable 5. Correct Response Times for the Three Pendulum Length Levels in Experiment 1.\u0026nbsp;\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"501\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eExp\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eParticipant\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eLongest_Time (ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eMedium_Time (ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eShortest_Time (ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e1387\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e2085\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1198.45\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e1158.86\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1412.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1438.67\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e1120.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1260.87\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1110.83\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e1921\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1962.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1793.43\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e742\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e708.05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e781.75\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e2333.86\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e2088.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1209.67\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e989.96\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e983.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e963.68\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e720.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e685.72\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e678.57\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e750.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e949.69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e963.46\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e597.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e556\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e1088.95\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1231.13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1337.99\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e796.99\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e938.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e925.11\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e1013.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e933.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e921.31\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e785.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e858.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e928.39\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e912.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1205.88\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e992.09\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1581.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e626.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e706.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e735.58\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e803\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1046.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e902.06\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e1008.88\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e1044\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e906.08\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"9.3812375249501%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.962075848303392%\" valign=\"top\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.55489021956088%\" valign=\"top\"\u003e\n \u003cp\u003e427.96\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e421.92\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"24.550898203592816%\" valign=\"top\"\u003e\n \u003cp\u003e450.13\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eTable 6. Correct Response Times for the Three Pendulum Bob Weight Levels in Experiment 1.\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"558\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eExp\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eParticipant\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eHeavy_Time(ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eLight_Time(ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003eMedium_Time(ms)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e1249.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e1493.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e1323.63\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e952.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e1660\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e1544\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e1212.93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e1117.84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e1188.18\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e1820.12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e1968.91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e1897.19\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e748.19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e700.68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e789.11\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e1398.25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e2664.25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e1973.25\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e1015.22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e931.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e981.42\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e730.92\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e673.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e669.97\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e917.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e844.75\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e873.48\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e564.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e593.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e1254.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e1208.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e1190\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e893.19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e899.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e858.76\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e939.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e975.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e961.27\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e850.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e877.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e845.05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e895.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e1082.84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e1122.33\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e1581.14\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e702.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e694.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e664.57\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e1036.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e807.76\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e898.74\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e1030.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e1005.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e931.31\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.196779964221825%\" valign=\"top\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.99463327370304%\" valign=\"top\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"23.613595706618963%\" valign=\"top\"\u003e\n \u003cp\u003e408.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"22.54025044722719%\" valign=\"top\"\u003e\n \u003cp\u003e468.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.65474060822898%\" valign=\"top\"\u003e\n \u003cp\u003e425.95\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eTable 7. A Summary of Three Binomial Logistic Models and the Statistical Indexes.\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"580\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.655172413793103%\"\u003e\n \u003cp\u003eDependent Variable\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.10344827586207%\"\u003e\n \u003cp\u003ePredictor\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e\u003cem\u003edf\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e\u003cem\u003eb\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e\u003cem\u003et\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e\u003cem\u003ep\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e\u003cem\u003esr\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.06896551724138%\"\u003e\n \u003cp\u003e95% CI\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.655172413793103%\" rowspan=\"12\"\u003e\n \u003cp\u003ecorrAns\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.10344827586207%\"\u003e\n \u003cp\u003eposition\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e1,057\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e-0.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e0.767\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.03448275862069%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.06896551724138%\"\u003e\n \u003cp\u003e[0.00, 0.00]\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003e\u003cstrong\u003elength\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e1,057\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.04\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e3.12\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.002\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.01\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e\u003cstrong\u003e[0.00, 0.02]\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003eid\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e1,057\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e1.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0.073\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e[0.00, 0.01]\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003eposition\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e1,056\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e-0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0.757\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e[0.00, 0.00]\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003e\u003cstrong\u003elength\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e1,056\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.04\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e3.13\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.002\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.01\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e\u003cstrong\u003e[0.00, 0.02]\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003eweight\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e1,056\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e-0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0.797\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e[0.00, 0.00]\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003eid\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e1,056\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e1.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0.073\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e[0.00, 0.01]\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003eposition\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e1,055\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e-0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0.744\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e[0.00, 0.00]\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003e\u003cstrong\u003elength\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e1,055\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.04\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e3.15\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.002\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.01\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e\u003cstrong\u003e[0.00, 0.02]\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003eweight\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e1,055\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e-0.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0.773\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e[0.00, 0.00]\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003e\u003cstrong\u003ereading\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e1,055\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e-0.07\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e-6.13\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026lt; .001\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.03\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e\u003cstrong\u003e[0.01, 0.06]\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.353535353535353%\"\u003e\n \u003cp\u003e\u003cstrong\u003eid\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e1,055\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.01\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e3.41\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.001\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.929292929292929%\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.01\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20%\"\u003e\n \u003cp\u003e\u003cstrong\u003e[0.00, 0.02]\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTable 8. Results of Model Comparison Results of the Three Embedded Mixed Effects Models of the Experiment 1 Data.\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"557\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.053859964093357%\" valign=\"top\" style=\"width: 8.706%;\"\u003e\n \u003cp\u003eModel\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003eAIC\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003eBIC\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.721723518850988%\" valign=\"top\" style=\"width: 12.8215%;\"\u003e\n \u003cp\u003eLogLik\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.183123877917415%\" valign=\"top\" style=\"width: 12.6632%;\"\u003e\n \u003cp\u003eDeDeviance\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e\u0026chi;2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.258527827648114%\" valign=\"top\" style=\"width: 7.1231%;\"\u003e\n \u003cp\u003e\u003cem\u003edf\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.0556552962298%\" colspan=\"2\" valign=\"top\" style=\"width: 14.721%;\"\u003e\n \u003cp\u003e\u003cem\u003epr\u0026nbsp;\u003c/em\u003e(\u0026gt;\u0026nbsp;\u0026chi;2)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.053859964093357%\" valign=\"top\" style=\"width: 8.706%;\"\u003e\n \u003cp\u003ecc1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e687.49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e707.35\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.721723518850988%\" valign=\"top\" style=\"width: 12.8215%;\"\u003e\n \u003cp\u003e-339.74 \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.183123877917415%\" valign=\"top\" style=\"width: 12.6632%;\"\u003e\n \u003cp\u003e679.49 \u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.258527827648114%\" valign=\"top\" style=\"width: 7.1231%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.0556552962298%\" colspan=\"2\" valign=\"top\" style=\"width: 14.721%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.053859964093357%\" valign=\"top\" style=\"width: 8.706%;\"\u003e\n \u003cp\u003ecc2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e689.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e714.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.721723518850988%\" valign=\"top\" style=\"width: 12.8215%;\"\u003e\n \u003cp\u003e-339.78 \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.183123877917415%\" valign=\"top\" style=\"width: 12.6632%;\"\u003e\n \u003cp\u003e679.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e0.0000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.258527827648114%\" valign=\"top\" style=\"width: 7.1231%;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.0556552962298%\" colspan=\"2\" valign=\"top\" style=\"width: 14.721%;\"\u003e\n \u003cp\u003e1.00000 \u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"10.053859964093357%\" valign=\"top\" style=\"width: 8.706%;\"\u003e\n \u003cp\u003ecc3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e687.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e716.86\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.721723518850988%\" valign=\"top\" style=\"width: 12.8215%;\"\u003e\n \u003cp\u003e-337.53 \u0026nbsp;\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.183123877917415%\" valign=\"top\" style=\"width: 12.6632%;\"\u003e\n \u003cp\u003e675.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.8491921005386%\" valign=\"top\" style=\"width: 10.2889%;\"\u003e\n \u003cp\u003e4.4952 \u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.43806104129264%\" colspan=\"2\" valign=\"top\" style=\"width: 7.2814%;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"17.0556552962298%\" valign=\"top\" style=\"width: 15.5125%;\"\u003e\n \u003cp\u003e0.03399 *\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e* \u003cem\u003ep\u003c/em\u003e \u0026lt; .05\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eNote. \u003cem\u003ecc1\u003c/em\u003e means a model built with R command \u0026ldquo;corrAns ~ position + length + (1 | id)\u0026rdquo; whereas \u003cem\u003ecc2\u003c/em\u003e \u0026ldquo;corrAns ~ position + length + weight + (1 | id)\u0026rdquo; and \u003cem\u003ecc3\u003c/em\u003e \u0026ldquo;corrAns ~ position + length + weight + reading + (1 | id).\u0026rdquo;\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Physics education, pendulum motion, experimental study, conceptual change","lastPublishedDoi":"10.21203/rs.3.rs-4485936/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4485936/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eOver four decades of conceptual change studies in science education have been based on the assumption that learners come to science classrooms with functionally fixated intuitive ideas. However, it is largely ignored that such pre-instructional conceptions are probabilistic, reflecting some aspects of an idiosyncratic sampling of their experiences and intuitive decision-making. This study foregrounds the probabilistic aspect of international students' intuitive and counterintuitive conceptions when learning pendulum motion. The probability here is rooted in a moving neural time average in the mind for characterizing these students' cognition (sampling and decision-making) and learning processes (resampling and making a new decision). To sharpen the said focus, we would argue that a new taxonomy of physics concepts is needed to save the mathematical identification of the isochrony of pendulum motion. To connect the mathematical core-based taxonomy with reality, we conducted an experimental study to characterising these students' reaction time and error rates in matching the period of a visually presented pendulum, which embodied its mathematical identity: T = 2π√l/g. The reaction times and error rates data have converged on the probabilistic aspects of the students' active learning mechanisms in their mind. The pedagogical implications of such a probabilistic cognitive mechanism have also been discussed.\u003c/p\u003e","manuscriptTitle":"On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-07-02 07:04:33","doi":"10.21203/rs.3.rs-4485936/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b02614ad-52c5-401d-ac47-8b9c29b739ad","owner":[],"postedDate":"July 2nd, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-08-12T11:27:59+00:00","versionOfRecord":[],"versionCreatedAt":"2024-07-02 07:04:33","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4485936","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4485936","identity":"rs-4485936","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}