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

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Title: On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts
Language: English
Authors: Lin Li (ORCID 0000-0002-2192-5454), George Zhou
Source: Science & Education. 2025 34(4):2377-2407.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 31
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Descriptors: Scientific Concepts, Taxonomy, Motion, Foreign Students, Physics, Probability, Cognitive Processes, Learning Processes, Reaction Time, Error Patterns, Mathematics
DOI: 10.1007/s11191-024-00545-9
ISSN: 0926-7220
1573-1901
Abstract: Over four decades of conceptual change studies in 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 mixed-method study foregrounds the probabilistic aspect of international students' intuitive-to-counterintuitive conceptions when learning pendulum motion. The probability here is rooted in a moving neural time average in the mind for characterizing these students' cognitive processes (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 and interviewed students to characterize these students' reaction time and error rates in matching the period of a visually presented pendulum, which embodied its mathematical identity: T = 2[pi][square root of]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.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1482208
Database: ERIC
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  Value: <anid>AN0187498037;nmo01aug.25;2025Aug26.02:35;v2.2.500</anid> <title id="AN0187498037-1">On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts </title> <p>Over four decades of conceptual change studies in 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 mixed-method study foregrounds the probabilistic aspect of international students' intuitive-to-counterintuitive conceptions when learning pendulum motion. The probability here is rooted in a moving neural time average in the mind for characterizing these students' cognitive processes (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 and interviewed students to characterize 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.</p> <p>Keywords: Physics education; Pendulum motion; Experimental study; Conceptual change; Psychology and Cognitive Sciences Psychology Education Curriculum and Pedagogy</p> <p>Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</p> <hd id="AN0187498037-2">Introduction</hd> <p>In the context of introducing how hard it was to measure time in history, the science education researcher Matthews ([<reflink idref="bib32" id="ref1">32</reflink>]) notes that "(t)ime measurement, as distinct from timekeeping, could not occur until people thought time was something measurable" (p. 47). A similar lack of awareness of time measurement can also be said about conceptual change studies. As conceptual change research in science education advances from the late 1970s to the present day, researchers have tried various empirical methods for characterizing the dynamics of such a learning process. Most of them have relied on qualitative methods such as surveying, interviews, or a personal reflection of a learning experience (Posner et al., [<reflink idref="bib44" id="ref2">44</reflink>]; Tseitlin & Galili, [<reflink idref="bib55" id="ref3">55</reflink>]; Tamayo Alzate & Sanmartí Puig, [<reflink idref="bib52" id="ref4">52</reflink>]; Zhou et al., [<reflink idref="bib61" id="ref5">61</reflink>]; Moore & Dawson, [<reflink idref="bib37" id="ref6">37</reflink>]). Such methodological choices assume that the temporal aspect of conceptual change processes can be expressed in words or estimated but not measured. However, change occurs in time or implies time. If this is the case, no exception should be made to understanding conceptual change. The question is how time as a dependent variable can be made explicit.</p> <p>Turning to quantitative methods in science education, a few researchers have attempted to introduce precise time measurements for studying conceptual change rather than just coding verbal or nominal data (Babai et al., [<reflink idref="bib2" id="ref7">2</reflink>]; Potvin et al., [<reflink idref="bib46" id="ref8">46</reflink>]). Despite the significance of this type of study, there has been no further extension of such a time-honored research tradition in the field science education. To extend the experimental studies of conceptual change, we conducted two reaction time-based studies highlighting the temporal aspects of learning physics concepts. The studies were part of a mixed methods project that combined the results of both quantitative time measurement and self-reflective verbal reports. The research project was guided by the following question: How is time, an essential aspect of learning a scientific concept such as pendulum motion, operationalized and measured in conceptual change research? One particular aspect of asking such a question deserves to be highlighted: the operationalization, though being "condemned" as a positive dogma in education studies, offers a method of differentiating daily timekeeping or rough time estimation from precise time measurement, a key point needed to understand the brain-enabled learning mechanisms during conceptual change.</p> <p>To clarify an emerging tie between time measurement and conceptual change in science education, we briefly describe how conceptual change studies took their form in a philosophical tradition, which barely noticed time (Hewitt, [<reflink idref="bib18" id="ref9">18</reflink>]; March, [<reflink idref="bib30" id="ref10">30</reflink>]). Next, another time-honoured mental chronometric tradition was traced, highlighting a notion of time as a factor measurable. Then, we critically synthesize a decade of conceptual-change-in-time studies, showcasing their ranges and significant results. What follows are two experimental studies. Finally, we summarize the results and recommend the next move. Overall, we set ourselves in the mood of communicating with science teachers and researchers who might be interested in but unfamiliar with or still skeptical about adopting the experimental paradigm of this type, addressing their concerns and explaining how and why the paradigm has come to be appealing in conceptual change research.</p> <p>In the early 1980s, Posner et al. ([<reflink idref="bib44" id="ref11">44</reflink>]) 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's 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 recognize and explain the time measuring potential of a pendulum or any oscillating object provides a historically meaningful and knowledge-rich experimental platform.</p> <hd id="AN0187498037-3">The Qualitative Aspect of the Origin: Philosophical but Atemporal</hd> <p>The study of conceptual change in science education has a philosophical origin, but time has not been a significant factor in the field. From the late 1950s onwards, the philosophers Thomas Kuhn ([<reflink idref="bib22" id="ref12">22</reflink>]), Stephen Toumin ([<reflink idref="bib54" id="ref13">54</reflink>]), and Imre Lakatos ([<reflink idref="bib23" id="ref14">23</reflink>]) asked how a new theory would replace a previously legitimate one in history. Reflecting on theory change phenomena, Kuhn ([<reflink idref="bib22" id="ref15">22</reflink>]) introduced into the philosophy of science what is now known as a paradigm shift for explaining a scientific revolution, such as replacing the phlogiston theory of fire with its oxidization process conception. According to his paradigm shift perspective, the normal course of scientific development can be dramatically transformed after responding to a crisis or anomaly of an established scientific theory, thus leading to a conceptual change afterwards.</p> <p>In the following years, the influences of such a philosophical origin can be easily spotted in the literature. Joseph Nussbaum and Shimshon Novick ([<reflink idref="bib39" id="ref16">39</reflink>]), for instance, documented another case study of two lessons of teaching particle models, illustrating the effectiveness of accommodating students' alternative frameworks. At the same time, Peter Hewson ([<reflink idref="bib19" id="ref17">19</reflink>]) reported a case study exemplifying his conceptual capture that characterized a reconciliation of a new conception with prior knowledge. Commonly, they suggested a theory of conceptual change for learning an abstract and counterintuitive idea proposed by modern scientists. The influence has also affected science educators such as Posner et al. ([<reflink idref="bib44" id="ref18">44</reflink>]). They drew a parallel between conceptual change in students' learning and Kuhn's paradigm shift ([<reflink idref="bib22" id="ref19">22</reflink>]).</p> <p>Built on this analogy or metaphor, they suggested four time-irrelevant or -insensitive conditions that might help enable a student's conceptual change in learning: (<reflink idref="bib1" id="ref20">1</reflink>) a dissatisfaction with existing conceptions by students; (<reflink idref="bib2" id="ref21">2</reflink>) a new intelligible alternative for them; (<reflink idref="bib3" id="ref22">3</reflink>) the plausibility of the alternative; (<reflink idref="bib4" id="ref23">4</reflink>) its fruitiness. According to these conditions, the students' prior knowledge (conceptual ecology) would determine the direction of a conceptual accommodation if these conditions can be met. The implications of such a conceptual change theory for educational practice were also elaborated elsewhere by Strike and Posner et al. ([<reflink idref="bib44" id="ref24">44</reflink>]).</p> <p>As a result, the discussions have marked a philosophical origin of conceptual change studies, but they are atemporal. Alternatively, it can be said that the temporal aspect of conceptual change, in the eyes of early conceptual change researchers, has been simplified or estimated as a linear and staged one induced by a dissatisfaction: moving from such an uncomfortable stage to another one for accommodating a new counterintuitive concept. However, it is commonly acknowledged that time is an essential aspect of change, whether physical or psychological.</p> <hd id="AN0187498037-4">The Quantitative Aspect of the Origin: Mental Chronometry in History and in Conceptual Change</hd> <p>Although any physical change in the world occurs in time, and scientists are experts in measuring and giving a number to such a duration, it is still a moot point how to conceptualize and measure a learner's conceptual change in time. To some extent, the situation mirrors a significant historical event when the late nineteenth century scientists were finally approaching the threshold of a breakthrough in realizing there is a speed limit to human information processing. It is in that era that time, as a fundamental fabric of human experiences, finally became a then-current frontier for scientific exploration, though it had escaped so many philosophers and scientists' great minds for over a thousand years. Given the realization of the potential for measuring mental activities, researchers of the late nineteenth century sought to obtain and interpret the time lag between the presentation of external events and the onset of a response to the task-relevant stimulus. This duration was named reaction time by Sigmund Exner, an Austrian physiologist (Robert, [<reflink idref="bib48" id="ref25">48</reflink>]), in 1873. Regarding its experimental applications, the Dutch physiologist Frans C. Donders (1818–1889) first demonstrated how to obtain a reaction time difference by subtracting a simpler form of the duration from a more complex one in his subtraction method. Following Donders' demonstration, researchers of the nineteenth and twentieth centuries saw the ups and downs of such an endeavor, stretching the reaction time-based research enterprise from a Golden Age (1850–1900 A.D.) to a Dark Age (1900– 950 A. D.) and its Renaissance (1950 A. D. to present) (Meyer et al., [<reflink idref="bib36" id="ref26">36</reflink>]). When cognitive psychology gradually came to the front stage in the 1970s, mental chronometric measures were recast to tackle human information processing mechanisms, with their impact still felt today.</p> <p>Turning to mental chronometric methods marks another origin of time measurement of conceptual change in science education. The turn helps make explicit the notion of time as a measurable factor, extending the mental chronometric tradition. Babai et al. ([<reflink idref="bib2" id="ref27">2</reflink>]), for example, designed a reaction time study to examine conceptual change in time. In this case, the student's performance was measured by comparing the participants' choice reaction times of judging areas and perimeters, which marked the duration from the onset of presenting a visual pattern to the initiation of a choice. After comparing the averaged durations of students' performances driven by intuitive or counterintuitive rules, they observed a reaction time difference: about 360 ms, roughly one-third of a second. Though brief, this type of evidence significantly differs from verbal responses and personal reflections documented in the literature.</p> <p>First, such a real-time measure is an online index of conceptual change, thus characterizing the change process when its temporal unfolding is still alive in a learner's mind. Moreover, it does not require participants to consciously verbalize their learning experiences, thus digging deeper into cognitive processes. Thirdly, it generates quantitative data at the ratio level contrasting nominal or verbal responses, thus affording uses of advanced statistical analyses. In sum, an interactive dynamical mechanism underlying conceptual change is operationalized and measured through the lens of modern mental chronometry, showing a combined effect of attention and memories. In response to such a methodological turn, other researchers have followed the lead, generating a small but unique body of research literature featuring conceptual change in time. Commonly, they set up at least two experimental conditions for contrasting the students' reaction times. At least, three types of conceptual change in time can be identified considering how the researchers have established their experimental procedures. They are (<reflink idref="bib1" id="ref28">1</reflink>) the simple choice reaction time paradigm (Babai et al., [<reflink idref="bib2" id="ref29">2</reflink>]; Potvin, Sauriol, et al., [<reflink idref="bib46" id="ref30">46</reflink>]; Vosniadou et al., [<reflink idref="bib56" id="ref31">56</reflink>]); (<reflink idref="bib2" id="ref32">2</reflink>) the negative priming procedure (Potvin, Masson, et al., [<reflink idref="bib46" id="ref33">46</reflink>]); and (<reflink idref="bib3" id="ref34">3</reflink>) brain imaging (Zhu et al., [<reflink idref="bib62" id="ref35">62</reflink>]) with reaction time recorded. Commonly, they measured the students' reaction times in rectify a falsely identified cause of a physical phenomenon or in generalizing a mathematical rule. However, none of them has examined the pendulum motion in their experimental studies.</p> <hd id="AN0187498037-5">A False Positive Identification of a "Force" That Drives Pendulum Motion</hd> <p>In the context of surveying students' misconceptions, Clement ([<reflink idref="bib8" id="ref36">8</reflink>]) 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).</p> <p>The problem stated the following:</p> <p></p> <ulist> <item> 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.</item> <p></p> <item> 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).</item> </ulist> <p>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<emph>.</emph> 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.</p> <p>Graph: Fig. 1 The pendulum problem</p> <p>Moreover, Clement ([<reflink idref="bib8" id="ref37">8</reflink>]) summarized the common features of the "motion implies a force" preconception, such as the following:</p> <p></p> <ulist> <item> 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.</item> <p></p> <item> 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.</item> <p></p> <item> The subject may believe that such a force "dies out" or "builds up" to account for changes in an object's speed (p. 69).</item> </ulist> <p>In discussing the implications of such findings, Clement ([<reflink idref="bib8" id="ref38">8</reflink>]) 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).</p> <hd id="AN0187498037-6">The Idealized Simple Pendulum Motion: T=2πl/g</hd> <p>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 ([<reflink idref="bib34" id="ref39">34</reflink>]) documented their teaching and learning experiences of Experimental Control of Simple Pendulum Model. The following aspects of the pendulum were analyzed quantitatively: "vanishing friction, small amplitude, not extensible string, point mass of the body, and vanishing mass of the string" (p. 631). All of these points are the abstraction that are not real, but become helpful when only a limited human information processing resource can be activated at each moment and the distraction from irrelevant factors must be kept at the minimum level. In the laboratory session, their students had to construct a simple pendulum that approaches an ideal one so that the student in a physics laboratory session could analyze the model assumptions which influence its oscillation period. In their own words,</p> <p></p> <ulist> <item> Physical pendulum period</item> <p></p> <item> 1 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi>p</mi><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mrow><mi>I</mi><mo stretchy="false">/</mo><mi>m</mi><mi>g</mi><mi>d</mi></mrow></msqrt></mrow></math> </ephtml></item> </ulist> <p>Graph</p> <p></p> <ulist> <item> where <emph>Tp</emph> represents the period of the physical pendulum, <emph>I</emph> the moment of inertia, <emph>m</emph> the mass of body, <emph>g</emph> the acceleration due to gravity, and <emph>d</emph> the distance between the axis and the center of gravity of the system.</item> <p></p> <item> Ideal simple pendulum period</item> <p></p> <item> 2 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi>s</mi><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mrow><mi>l</mi><mo stretchy="false">/</mo><mi>g</mi></mrow></msqrt></mrow></math> </ephtml></item> </ulist> <p>Graph</p> <p></p> <ulist> <item> where <emph>l</emph> is the length of the string.</item> <p></p> <item> Equation (<reflink idref="bib1" id="ref40">1</reflink>) was deduced assuming:</item> <p></p> <item> A1: negligible friction (the resultant torque on the system about the horizontal axis is solely due to the weight of the body).</item> <p></p> <item> A2: small oscillation amplitudes (in the equation of motion, the sine of the amplitude angle can be replaced by the angle in radians).</item> <p></p> <item> A3: the pendulum is a rigid body (invariable mass distribution, constant moment of inertia).</item> <p></p> <item> A4: the string mass must be negligible.</item> <p></p> <item> A5: the body mass must be concentrated at a point (Medina et al., [<reflink idref="bib34" id="ref41">34</reflink>], p. 632).</item> </ulist> <p>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., [<reflink idref="bib34" id="ref42">34</reflink>], p. 633). In particular, these error terms include the fluctuations "due to friction (ε<emph>f</emph>), initial amplitude (ε<emph>α</emph>), variable length of the string due to a variable tension during oscillation (ε<emph>T</emph>), mass distribution of the body (ε<emph>b</emph>), and mass of the string (εs)" (Medina et al., [<reflink idref="bib34" id="ref43">34</reflink>], 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,</p> <p>Considered separately, within an error of 1%:</p> <p></p> <ulist> <item> an initial amplitude of 23° is "small."</item> <p></p> <item> a sphere, whose diameter is 30% of the length of the string, is "a point mass."</item> <p></p> <item> a mass of the string equal to 10% of the mass of the body is "vanishing."</item> <p></p> <item> any elastic elongation suffered by the string during the static process of loading is negligible, providing the string length is measured after the loading.</item> <p></p> <item> 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., [<reflink idref="bib34" id="ref44">34</reflink>], p. 639).</item> </ulist> <p>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 idealized 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.</p> <p>Table 1 A new taxonomy of physics concepts for learning pendulum motion</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Concept</p></th><th><p>Type 1</p></th><th><p>Type 2</p></th><th><p>Type 3</p></th><th><p>Type 4</p></th></tr></thead><tbody><tr><td><p>Criteria</p></td><td><p>Mathematically defined</p></td><td><p>Half mathematically defined and half verbally defined</p></td><td><p>Verbally defined</p></td><td><p>Non-verbally defined</p></td></tr><tr><td><p>Example</p></td><td><p>An imaginary number, the probability amplitude, simple harmonic motion</p></td><td><p>Force (<italic>F</italic> = <italic>m</italic> × <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac xmlns=""><mtext>dv</mtext><mtext>dt</mtext></mfrac></math><inline-graphic mime-subtype="GIF" href="11191_2024_545_Article_IEq3.gif" />),</p><p>The isochronous pendulum motion (<italic>T</italic><sub>12</sub> = <italic>T</italic><sub>21</sub>)</p></td><td><p>Argumentation, ideology, propositional perception</p></td><td><p>Visual perception, auditory perception, working memory system</p></td></tr></tbody></table> </ephtml> </p> <p>Graph: Fig. 2 Illustration of the new taxonomy of physics concepts for learning pendulum motion</p> <p>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, [<reflink idref="bib7" id="ref45">7</reflink>]; Matthews, [<reflink idref="bib31" id="ref46">31</reflink>]; Rowbottom, [<reflink idref="bib49" id="ref47">49</reflink>]; Worrall, [<reflink idref="bib58" id="ref48">58</reflink>]). 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.</p> <p>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, [<reflink idref="bib31" id="ref49">31</reflink>]). 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: <emph>T</emph>= 2π <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mi>l</mi><mo stretchy="false">/</mo><mi>g</mi></mrow></msqrt></math> </ephtml> 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.</p> <p>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.</p> <hd id="AN0187498037-7">A Mixed-Method Study: Matching the Period of Pendulum Motion</hd> <p>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, [<reflink idref="bib41" id="ref50">41</reflink>]). 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.</p> <hd id="AN0187498037-8">Research Questions and Hypothesis</hd> <p>According to the mathematical relationship describing the pendulum period with a small release angle boundary condition (<emph>T</emph> = 2π <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mi>l</mi><mo stretchy="false">/</mo><mi>g</mi><mspace width="4pt" /></mrow></msqrt></math> </ephtml> ), 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.</p> <hd id="AN0187498037-9">Method</hd> <p>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.</p> <hd id="AN0187498037-10">Participants</hd> <p>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, 20 international students, aged from 18 to 55, were recruited for the first experiment (see Table 2).</p> <p>Table 2 A summary of the twenty participants of experiment 1</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>No</p></th><th><p>Age</p></th><th><p>Gender</p></th><th><p>Edu</p></th><th><p>Major</p></th><th><p>1st language</p></th><th><p>2nd language</p></th></tr></thead><tbody><tr><td><p>01</p></td><td><p>26</p></td><td><p>F</p></td><td><p>BA</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English, Korean</p></td></tr><tr><td><p>02</p></td><td><p>22</p></td><td><p>M</p></td><td><p>BA</p></td><td><p>Physics</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>03</p></td><td><p>32</p></td><td><p>F</p></td><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>04</p></td><td><p>48</p></td><td><p>F</p></td><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>05</p></td><td><p>24</p></td><td><p>M</p></td><td><p>BA</p></td><td><p>Physics</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>06</p></td><td><p>30</p></td><td><p>F</p></td><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>07</p></td><td /><td /><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>08</p></td><td><p>36</p></td><td><p>M</p></td><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>09</p></td><td><p>37</p></td><td><p>F</p></td><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>10</p></td><td><p>18</p></td><td><p>M</p></td><td><p>High School</p></td><td><p>Mathematics</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>11</p></td><td><p>39</p></td><td><p>F</p></td><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>No.</p></td></tr><tr><td><p>12</p></td><td><p>39</p></td><td><p>M</p></td><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>No.</p></td></tr><tr><td><p>13</p></td><td><p>45</p></td><td><p>F</p></td><td><p>Master</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>14</p></td><td><p>55</p></td><td><p>F</p></td><td><p>BA</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>15</p></td><td><p>44</p></td><td><p>F</p></td><td><p>PhD</p></td><td><p>Chemistry</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>16</p></td><td><p>44</p></td><td><p>M</p></td><td><p>BA</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>17</p></td><td><p>32</p></td><td><p>F</p></td><td><p>Master</p></td><td><p>Mathematics</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>18</p></td><td><p>30</p></td><td><p>M</p></td><td><p>PhD</p></td><td><p>Biology</p></td><td><p>Cantonese</p></td><td><p>English</p></td></tr><tr><td><p>19</p></td><td><p>41</p></td><td><p>F</p></td><td><p>PhD</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr><tr><td><p>20</p></td><td><p>19</p></td><td><p>M</p></td><td><p>BA</p></td><td><p>Non-Science</p></td><td><p>Chinese</p></td><td><p>English</p></td></tr></tbody></table> </ephtml> </p> <p>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 summarizes 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."</p> <p>Table 3 The language learning background of the 20 <emph>participants of experiment 1</emph></p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>No</p></th><th><p>Reading</p></th><th><p>Writing</p></th><th><p>Speaking</p></th><th><p>Listening</p></th></tr></thead><tbody><tr><td><p>01</p></td><td><p>5</p></td><td><p>4</p></td><td><p>4</p></td><td><p>4</p></td></tr><tr><td><p>02</p></td><td><p>5</p></td><td><p>5</p></td><td><p>3</p></td><td><p>4</p></td></tr><tr><td><p>03</p></td><td><p>6</p></td><td><p>5</p></td><td><p>5</p></td><td><p>6</p></td></tr><tr><td><p>04</p></td><td><p>5</p></td><td><p>4</p></td><td><p>5</p></td><td><p>4</p></td></tr><tr><td><p>05</p></td><td><p>4</p></td><td><p>4</p></td><td><p>4</p></td><td><p>5</p></td></tr><tr><td><p>06</p></td><td><p>4</p></td><td><p>4</p></td><td><p>5</p></td><td><p>5</p></td></tr><tr><td><p>07</p></td><td><p>4</p></td><td><p>4</p></td><td><p>3</p></td><td><p>4</p></td></tr><tr><td><p>08</p></td><td><p>6</p></td><td><p>5</p></td><td><p>5</p></td><td><p>5</p></td></tr><tr><td><p>09</p></td><td><p>5.5</p></td><td><p>5.5</p></td><td><p>5.5</p></td><td><p>5.5</p></td></tr><tr><td><p>10</p></td><td><p>4.5</p></td><td><p>4</p></td><td><p>5</p></td><td><p>5.5</p></td></tr><tr><td><p>11</p></td><td><p>5.5</p></td><td><p>5</p></td><td><p>5</p></td><td><p>5.5</p></td></tr><tr><td><p>12</p></td><td><p>4</p></td><td><p>3</p></td><td><p>3</p></td><td><p>4</p></td></tr><tr><td><p>13</p></td><td><p>4.5</p></td><td><p>4.5</p></td><td><p>4.5</p></td><td><p>4.5</p></td></tr><tr><td><p>14</p></td><td><p>6</p></td><td><p>5</p></td><td><p>5</p></td><td><p>5</p></td></tr><tr><td><p>15</p></td><td><p>6</p></td><td><p>6</p></td><td><p>5</p></td><td><p>5</p></td></tr><tr><td><p>16</p></td><td><p>5</p></td><td><p>5</p></td><td><p>5</p></td><td><p>5</p></td></tr><tr><td><p>17</p></td><td><p>6</p></td><td><p>6</p></td><td><p>5</p></td><td><p>5</p></td></tr><tr><td><p>18</p></td><td><p>5</p></td><td><p>5</p></td><td><p>5</p></td><td><p>5</p></td></tr><tr><td><p>19</p></td><td><p>6.5</p></td><td><p>5.5</p></td><td><p>6.5</p></td><td><p>6.5</p></td></tr><tr><td><p>20</p></td><td><p>5</p></td><td><p>4</p></td><td><p>5</p></td><td><p>6</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0187498037-11">Apparatus and Stimuli</hd> <p>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, [<reflink idref="bib42" id="ref51">42</reflink>]), an open-source psychophysiological software package with 1-ms 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.</p> <p>Both a standard and the three to-be-matched pendulum stimuli were presented in a rapid serial visual presentation (RSVP) paradigm (see Fig. 3). In each RSVP trial, participants saw a colored standard pendulum shown in the center of the screen on a grey background. They had 5 s 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 500-ms 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 60 cm, participants were instructed to respond to a target-matching pendulum as quickly and accurately as possible.</p> <p>Graph: Fig. 3 Illustration of rapid serial visual presentation experimental procedure</p> <hd id="AN0187498037-12">Design</hd> <p>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 (<reflink idref="bib1" id="ref52">1</reflink>, 2, 3, or 0). The three factors were independently manipulated. The initial release angle was not specifically controlled in this pilot experiment.</p> <hd id="AN0187498037-13">Experimental Procedure</hd> <p>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 ms/item. No feedback was provided in practice, simulating what would occur in the real session.</p> <p>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 5 s. 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 ms 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.</p> <hd id="AN0187498037-14">Data Analyses: ANOVA or Binomial Regression?</hd> <p>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 2000 ms. 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.</p> <p>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.</p> <p>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 <emph>p</emph>-values thresholds of.05 or.01 has been contested as an acceptable good practice (Kuffner & Walker, [<reflink idref="bib21" id="ref53">21</reflink>]; Wasserstein & Lazar, [<reflink idref="bib57" id="ref54">57</reflink>]). 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.</p> <hd id="AN0187498037-15">Results</hd> <p>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 2500 ms. 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.</p> <p>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</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Exp</p></th><th><p>Participant</p></th><th><p>RT0 (ms)</p></th><th><p>RT1 (ms)</p></th><th><p>RT2 (ms)</p></th><th><p>RT3 (ms)</p></th><th><p>Missing</p></th><th><p>Correct%</p></th><th><p>Error%</p></th></tr></thead><tbody><tr><td><p>1</p></td><td><p>1</p></td><td><p>823.6</p></td><td><p>2085</p></td><td><p>1684.5</p></td><td><p>1266.333</p></td><td><p>4</p></td><td><p>32.4</p></td><td><p>67.6</p></td></tr><tr><td><p>1</p></td><td><p>2</p></td><td><p>1759.5</p></td><td><p>1313.429</p></td><td><p>1158.857</p></td><td><p>1438.667</p></td><td><p>2</p></td><td><p>31.4</p></td><td><p>68.6</p></td></tr><tr><td><p>1</p></td><td><p>3</p></td><td><p>1012.059</p></td><td><p>1386</p></td><td><p>1153.909</p></td><td><p>1255.333</p></td><td><p>5</p></td><td><p>65.7</p></td><td><p>34.3</p></td></tr><tr><td><p>1</p></td><td><p>4</p></td><td><p>1789.222</p></td><td><p>2005.5</p></td><td><p>2016</p></td><td><p>1763.6</p></td><td><p>2</p></td><td><p>94.3</p></td><td><p>5.7</p></td></tr><tr><td><p>1</p></td><td><p>5</p></td><td><p>768.0625</p></td><td><p>670.2857</p></td><td><p>718.3529</p></td><td><p>815.6667</p></td><td><p>3</p></td><td><p>89.9</p></td><td><p>10.1</p></td></tr><tr><td><p>1</p></td><td><p>6</p></td><td><p>1131.5</p></td><td><p>2088.5</p></td><td><p>2497.667</p></td><td><p>1358.5</p></td><td><p>13</p></td><td><p>20.3</p></td><td><p>79.7</p></td></tr><tr><td><p>1</p></td><td><p>7</p></td><td><p>973.3889</p></td><td><p>928.0714</p></td><td><p>999.7778</p></td><td><p>1007.063</p></td><td><p>1</p></td><td><p>93</p></td><td><p>7</p></td></tr><tr><td><p>1</p></td><td><p>8</p></td><td><p>664.5211</p></td><td><p>708.2769</p></td><td><p>695.2031</p></td><td><p>711.7578</p></td><td><p>0</p></td><td><p>94.4</p></td><td><p>5.6</p></td></tr><tr><td><p>1</p></td><td><p>9</p></td><td><p>927.9111</p></td><td><p>1010.29</p></td><td><p>713.7378</p></td><td><p>912.9629</p></td><td><p>0</p></td><td><p>91.7</p></td><td><p>8.3</p></td></tr><tr><td><p>1</p></td><td><p>10</p></td><td><p>583.6667</p></td><td /><td><p>855</p></td><td /><td><p>0</p></td><td><p>13.9</p></td><td><p>86.1</p></td></tr><tr><td><p>1</p></td><td><p>11</p></td><td><p>1166.862</p></td><td><p>1233.119</p></td><td><p>1083.355</p></td><td><p>1366.508</p></td><td><p>4</p></td><td><p>82.4</p></td><td><p>17.6</p></td></tr><tr><td><p>1</p></td><td><p>12</p></td><td><p>878.5247</p></td><td><p>977.5772</p></td><td><p>762.1722</p></td><td><p>919.6922</p></td><td><p>1</p></td><td><p>97.2</p></td><td><p>2.8</p></td></tr><tr><td><p>1</p></td><td><p>13</p></td><td><p>929.9567</p></td><td><p>973.7313</p></td><td><p>1029.899</p></td><td><p>893.4729</p></td><td><p>0</p></td><td><p>95.8</p></td><td><p>4.2</p></td></tr><tr><td><p>1</p></td><td><p>14</p></td><td><p>784.2941</p></td><td><p>902.6111</p></td><td><p>784.4706</p></td><td><p>955.8824</p></td><td><p>0</p></td><td><p>95.8</p></td><td><p>4.2</p></td></tr><tr><td><p>1</p></td><td><p>15</p></td><td><p>1014.828</p></td><td><p>1343.771</p></td><td><p>917.91</p></td><td><p>889.02</p></td><td><p>4</p></td><td><p>85.3</p></td><td><p>14.7</p></td></tr><tr><td><p>1</p></td><td><p>16</p></td><td /><td><p>1581.135</p></td><td><p>1374.62</p></td><td><p>3567.84</p></td><td><p>0</p></td><td><p>5.8</p></td><td><p>94.2</p></td></tr><tr><td><p>1</p></td><td><p>17</p></td><td><p>682.9363</p></td><td><p>758.1012</p></td><td><p>612.0661</p></td><td><p>703.2441</p></td><td><p>0</p></td><td><p>94.4</p></td><td><p>5.6</p></td></tr><tr><td><p>1</p></td><td><p>18</p></td><td><p>951.2667</p></td><td><p>1006.278</p></td><td><p>778.1111</p></td><td><p>957.1667</p></td><td><p>2</p></td><td><p>90</p></td><td><p>10</p></td></tr><tr><td><p>1</p></td><td><p>19</p></td><td><p>959.9</p></td><td><p>1079.929</p></td><td><p>964.4615</p></td><td><p>946.2222</p></td><td><p>3</p></td><td><p>66.7</p></td><td><p>33.3</p></td></tr><tr><td><p>1</p></td><td><p>20</p></td><td><p>472.5556</p></td><td><p>416.5</p></td><td><p>376.0556</p></td><td><p>469.2941</p></td><td><p>0</p></td><td><p>98.6</p></td><td><p>1.4</p></td></tr></tbody></table> </ephtml> </p> <p>The mean RT data were submitted to a repeated-measure ANOVA with the temporal position of the candidate pendulum as the four-level factor. The main effect of the temporal position was significant, <emph>F</emph> (<reflink idref="bib3" id="ref55">3</reflink>, 42) = 4.74, <emph>p</emph> <.01, <emph>η</emph><subs>p</subs><sups>2</sups> =.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, <emph>p</emph> <.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, <emph>p</emph> =.08) (see Fig. 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.</p> <p>Graph: Fig. 4 Reaction times as a function of the time positions of the candidate pendulum</p> <p>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 <emph>d</emph> 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.</p> <p>Table 5 Correct response times for the three pendulum length levels in experiment 1</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Exp</p></th><th><p>Participant</p></th><th><p>Longest_Time (ms)</p></th><th><p>Medium_Time (ms)</p></th><th><p>Shortest_Time (ms)</p></th></tr></thead><tbody><tr><td><p>1</p></td><td><p>1</p></td><td><p>1387</p></td><td><p>2085</p></td><td><p>1198.45</p></td></tr><tr><td><p>1</p></td><td><p>2</p></td><td><p>1158.86</p></td><td><p>1412.56</p></td><td><p>1438.67</p></td></tr><tr><td><p>1</p></td><td><p>3</p></td><td><p>1120.41</p></td><td><p>1260.87</p></td><td><p>1110.83</p></td></tr><tr><td><p>1</p></td><td><p>4</p></td><td><p>1921</p></td><td><p>1962.5</p></td><td><p>1793.43</p></td></tr><tr><td><p>1</p></td><td><p>5</p></td><td><p>742</p></td><td><p>708.05</p></td><td><p>781.75</p></td></tr><tr><td><p>1</p></td><td><p>6</p></td><td><p>2333.86</p></td><td><p>2088.5</p></td><td><p>1209.67</p></td></tr><tr><td><p>1</p></td><td><p>7</p></td><td><p>989.96</p></td><td><p>983.15</p></td><td><p>963.68</p></td></tr><tr><td><p>1</p></td><td><p>8</p></td><td><p>720.78</p></td><td><p>685.72</p></td><td><p>678.57</p></td></tr><tr><td><p>1</p></td><td><p>9</p></td><td><p>750.01</p></td><td><p>949.69</p></td><td><p>963.46</p></td></tr><tr><td><p>1</p></td><td><p>10</p></td><td /><td><p>597.5</p></td><td><p>556</p></td></tr><tr><td><p>1</p></td><td><p>11</p></td><td><p>1088.95</p></td><td><p>1231.13</p></td><td><p>1337.99</p></td></tr><tr><td><p>1</p></td><td><p>12</p></td><td><p>796.99</p></td><td><p>938.29</p></td><td><p>925.11</p></td></tr><tr><td><p>1</p></td><td><p>13</p></td><td><p>1013.14</p></td><td><p>933.67</p></td><td><p>921.31</p></td></tr><tr><td><p>1</p></td><td><p>14</p></td><td><p>785.7</p></td><td><p>858.35</p></td><td><p>928.39</p></td></tr><tr><td><p>1</p></td><td><p>15</p></td><td><p>912.03</p></td><td><p>1205.88</p></td><td><p>992.09</p></td></tr><tr><td><p>1</p></td><td><p>16</p></td><td /><td><p>1581.14</p></td><td /></tr><tr><td><p>1</p></td><td><p>17</p></td><td><p>626.55</p></td><td><p>706.26</p></td><td><p>735.58</p></td></tr><tr><td><p>1</p></td><td><p>18</p></td><td><p>803</p></td><td><p>1046.52</p></td><td><p>902.06</p></td></tr><tr><td><p>1</p></td><td><p>19</p></td><td><p>1008.88</p></td><td><p>1044</p></td><td><p>906.08</p></td></tr><tr><td><p>1</p></td><td><p>20</p></td><td><p>427.96</p></td><td><p>421.92</p></td><td><p>450.13</p></td></tr></tbody></table> </ephtml> </p> <p>The mean RT data were submitted to a repeated-measure ANOVA with the length of the candidate pendulum as the three-level factor. The main effect of the length was significant, <emph>F</emph> (<reflink idref="bib2" id="ref56">2</reflink>, 28) = 4.33, <emph>p</emph> <.05, <emph>η</emph><subs>p</subs><sups>2</sups> =.24, indicating the mean RTs differed significantly across the three length levels (see Fig. 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, <emph>p</emph> <.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.</p> <p>Graph: Fig. 5 Reaction times as a function of the pendulum length levels</p> <p>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.</p> <p>Table 6 Correct response times for the three pendulum bob weight levels in experiment 1</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Exp</p></th><th><p>Participant</p></th><th><p>Heavy_Time(ms)</p></th><th><p>Light_Time(ms)</p></th><th><p>Medium_Time(ms)</p></th></tr></thead><tbody><tr><td><p>1</p></td><td><p>1</p></td><td><p>1249.71</p></td><td><p>1493.57</p></td><td><p>1323.63</p></td></tr><tr><td><p>1</p></td><td><p>2</p></td><td><p>952.56</p></td><td><p>1660</p></td><td><p>1544</p></td></tr><tr><td><p>1</p></td><td><p>3</p></td><td><p>1212.93</p></td><td><p>1117.84</p></td><td><p>1188.18</p></td></tr><tr><td><p>1</p></td><td><p>4</p></td><td><p>1820.12</p></td><td><p>1968.91</p></td><td><p>1897.19</p></td></tr><tr><td><p>1</p></td><td><p>5</p></td><td><p>748.19</p></td><td><p>700.68</p></td><td><p>789.11</p></td></tr><tr><td><p>1</p></td><td><p>6</p></td><td><p>1398.25</p></td><td><p>2664.25</p></td><td><p>1973.25</p></td></tr><tr><td><p>1</p></td><td><p>7</p></td><td><p>1015.22</p></td><td><p>931.71</p></td><td><p>981.42</p></td></tr><tr><td><p>1</p></td><td><p>8</p></td><td><p>730.92</p></td><td><p>673.55</p></td><td><p>669.97</p></td></tr><tr><td><p>1</p></td><td><p>9</p></td><td><p>917.41</p></td><td><p>844.75</p></td><td><p>873.48</p></td></tr><tr><td><p>1</p></td><td><p>10</p></td><td><p>564.33</p></td><td><p>593.33</p></td><td /></tr><tr><td><p>1</p></td><td><p>11</p></td><td><p>1254.9</p></td><td><p>1208.41</p></td><td><p>1190</p></td></tr><tr><td><p>1</p></td><td><p>12</p></td><td><p>893.19</p></td><td><p>899.59</p></td><td><p>858.76</p></td></tr><tr><td><p>1</p></td><td><p>13</p></td><td><p>939.02</p></td><td><p>975.78</p></td><td><p>961.27</p></td></tr><tr><td><p>1</p></td><td><p>14</p></td><td><p>850.59</p></td><td><p>877.23</p></td><td><p>845.05</p></td></tr><tr><td><p>1</p></td><td><p>15</p></td><td><p>895.7</p></td><td><p>1082.84</p></td><td><p>1122.33</p></td></tr><tr><td><p>1</p></td><td><p>16</p></td><td /><td /><td><p>1581.14</p></td></tr><tr><td><p>1</p></td><td><p>17</p></td><td><p>702.17</p></td><td><p>694.4</p></td><td><p>664.57</p></td></tr><tr><td><p>1</p></td><td><p>18</p></td><td><p>1036.3</p></td><td><p>807.76</p></td><td><p>898.74</p></td></tr><tr><td><p>1</p></td><td><p>19</p></td><td><p>1030.79</p></td><td><p>1005.71</p></td><td><p>931.31</p></td></tr><tr><td><p>1</p></td><td><p>20</p></td><td><p>408.52</p></td><td><p>468.48</p></td><td><p>425.95</p></td></tr></tbody></table> </ephtml> </p> <p>The mean RT data were also submitted to a repeated-measure ANOVA with the bob weight of the candidate pendulum as the three-level factor. The main effect of the bob weight was insignificant, <emph>F</emph> (<reflink idref="bib2" id="ref57">2</reflink>, 28) =.22, <emph>p</emph> >.05, indicating the mean RTs were not different across the three bob weight levels (see Fig. 6).</p> <p>Graph: Fig. 6 Reaction times as a function of the pendulum bob's weight levels</p> <p>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.</p> <p>As introduced in the data analysis tutorial (Li, [<reflink idref="bib29" id="ref58">29</reflink>]), 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.</p> <p>Table 7 A summary of three binomial logistic models and the statistical indexes</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Dependent variable</p></th><th><p>Predictor</p></th><th><p><italic>df</italic></p></th><th><p><italic>b</italic></p></th><th><p><italic>t</italic></p></th><th><p><italic>p</italic></p></th><th><p><italic>sr</italic><sup>2</sup></p></th><th><p>95% CI</p></th></tr></thead><tbody><tr><td rowspan="12"><p>corrAns</p></td><td><p>position</p></td><td><p>1057</p></td><td><p>0</p></td><td><p>−0.3</p></td><td><p>0.767</p></td><td><p>0</p></td><td><p>[0.00, 0.00]</p></td></tr><tr><td><p><bold>length</bold></p></td><td><p><bold>1057</bold></p></td><td><p><bold>0.04</bold></p></td><td><p><bold>3.12</bold></p></td><td><p><bold>0.002</bold></p></td><td><p><bold>0.01</bold></p></td><td><p><bold>[0.00, 0.02]</bold></p></td></tr><tr><td><p>id</p></td><td><p>1057</p></td><td><p>0</p></td><td><p>1.79</p></td><td><p>0.073</p></td><td><p>0</p></td><td><p>[0.00, 0.01]</p></td></tr><tr><td><p>position</p></td><td><p>1056</p></td><td><p>0</p></td><td><p>−0.31</p></td><td><p>0.757</p></td><td><p>0</p></td><td><p>[0.00, 0.00]</p></td></tr><tr><td><p><bold>length</bold></p></td><td><p><bold>1056</bold></p></td><td><p><bold>0.04</bold></p></td><td><p><bold>3.13</bold></p></td><td><p><bold>0.002</bold></p></td><td><p><bold>0.01</bold></p></td><td><p><bold>[0.00, 0.02]</bold></p></td></tr><tr><td><p>weight</p></td><td><p>1056</p></td><td><p>0</p></td><td><p>−0.26</p></td><td><p>0.797</p></td><td><p>0</p></td><td><p>[0.00, 0.00]</p></td></tr><tr><td><p>id</p></td><td><p>1056</p></td><td><p>0</p></td><td><p>1.79</p></td><td><p>0.073</p></td><td><p>0</p></td><td><p>[0.00, 0.01]</p></td></tr><tr><td><p>position</p></td><td><p>1055</p></td><td><p>0</p></td><td><p>−0.33</p></td><td><p>0.744</p></td><td><p>0</p></td><td><p>[0.00, 0.00]</p></td></tr><tr><td><p><bold>length</bold></p></td><td><p><bold>1055</bold></p></td><td><p><bold>0.04</bold></p></td><td><p><bold>3.15</bold></p></td><td><p><bold>0.002</bold></p></td><td><p><bold>0.01</bold></p></td><td><p><bold>[0.00, 0.02]</bold></p></td></tr><tr><td><p>weight</p></td><td><p>1055</p></td><td><p>0</p></td><td><p>−0.29</p></td><td><p>0.773</p></td><td><p>0</p></td><td><p>[0.00, 0.00]</p></td></tr><tr><td><p><bold>reading</bold></p></td><td><p><bold>1055</bold></p></td><td><p>−<bold>0.07</bold></p></td><td><p>−<bold>6.13</bold></p></td><td><p><bold><.001</bold></p></td><td><p><bold>0.03</bold></p></td><td><p><bold>[0.01, 0.06]</bold></p></td></tr><tr><td><p><bold>id</bold></p></td><td><p><bold>1055</bold></p></td><td><p><bold>0.01</bold></p></td><td><p><bold>3.41</bold></p></td><td><p><bold>0.001</bold></p></td><td><p><bold>0.01</bold></p></td><td><p><bold>[0.00, 0.02]</bold></p></td></tr></tbody></table> </ephtml> </p> <p>Table 8 Results of model comparison results of the three embedded mixed effects models of the experiment 1 data</p> <p> <ephtml> <table frame="hsides" rules="groups"><thead><tr><th><p>Model</p></th><th><p>AIC</p></th><th><p>BIC</p></th><th><p>LogLik</p></th><th><p>DeDeviance</p></th><th><p>χ2</p></th><th><p><italic>df</italic></p></th><th><p><italic>pr</italic> (> <italic>χ</italic><sup>2</sup>)</p></th></tr></thead><tbody><tr><td><p>cc1</p></td><td><p>687.49</p></td><td><p>707.35</p></td><td><p>−339.74</p></td><td><p>679.49</p></td><td /><td /><td /></tr><tr><td><p>cc2</p></td><td><p>689.55</p></td><td><p>714.39</p></td><td><p>−339.78</p></td><td><p>679.55</p></td><td><p>0.0000</p></td><td><p>1</p></td><td><p>1.00000</p></td></tr><tr><td><p>cc3</p></td><td><p>687.06</p></td><td><p>716.86</p></td><td><p>−337.53</p></td><td><p>675.06</p></td><td><p>4.4952</p></td><td><p>1</p></td><td><p>0.03399 *</p></td></tr></tbody></table> </ephtml> </p> <p>*<emph>p</emph> <.05 <emph>cc1</emph> means a model built with R command "corrAns ~ position + length + (1 | id)" whereas <emph>cc2</emph> "corrAns ~ position + length + weight + (1 | id)" and <emph>cc3</emph> "corrAns ~ position + length + weight + reading + (1 | id)."</p> <p>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.</p> <p>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.</p> <p>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.</p> <p>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.</p> <p>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.</p> <hd id="AN0187498037-16">Reflecting on the Pendulum Motion: the Sentence and Beyond</hd> <p>Following the experimental study, I invited five interviewees to share their understanding of the simple pendulum's mathematical identity of the period: <emph>T</emph>= 2π <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mi>l</mi><mo stretchy="false">/</mo><mi>g</mi></mrow></msqrt></math> </ephtml> . In contrast to the salient features of pendulum motion, such as the length of a string or the initial release angles, identifying the period of pendulum motion with a mathematical equation is not so obvious. Despite its abstractness, the equation <emph>T</emph> = 2π <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mi>l</mi><mo stretchy="false">/</mo><mi>g</mi></mrow></msqrt></math> </ephtml> plays a crucial role in the study of pendulum motion and serves as a fundamental mathematical identification for an observer to refine their understanding of oscillatory pendulum motion, though she may find it challenging to grasp this equation's significance due to its mathematical equation's construct. In the interviews, I asked the five interviewees about their experiences with and understanding of the equation. When showing the equation to interviewee no. 1, she responded by commenting on her previous responses during the interview, "Right, it seems like what I was thinking is opposite, haha! And conversely, if its gravitational acceleration is smaller, then its period will be larger." Upon seeing the equation, she immediately checked whether her previous responses were correct. After a brief reflection, she found out the connection of these factors in determining the period by commenting:Interviewee no. 1: No, actually...out of all these variables, it only depends on the length.Interviewer: So you got that just by looking at this formula?Interviewee no. 1: Really?Interviewer: If you were given this formula, what would you think?Interviewee no. 1: I think it's like this, because they are all constants and only L is changing.Interviewer : OK. That's right, you just need to see this formula, and you think the conclusion is only related to L.Interviewee no. 1: Yeah, it's interesting that it's only related to length! Haha!</p> <p>At the end of the first interview, the interviewee added,Interviewee no. 1: "I think physics is pretty amazing. Only this one, for the others with different weights and angles, I thought the period would become shorter, but it's not the case. It turns out that only this one affects it. I think physics is pretty amazing."Interviewer: Yes, it's quite amazing.</p> <p>These final comments displayed the first interviewee's amazement at the potential instructional value of physics knowledge. Given what she said in the interview, I tentatively conclude that as a non-science major student, her experience with the scientific reasoning of using the mathematical equation has been enriching and rewarding, even in this most straightforward case of discussing pendulum motion.</p> <p>Similarly, the interviewer showed the second interviewee the same mathematical equation of the period of pendulum motion and asked for his comments. Without the interviewer finishing the question, the interviewee interrupted and declared, "That is, actually only related to length. Everything else is fixed quantities." After such a realization, the interviewee quickly solved the problem of adjusting the position of the pendulum bob to shorten the period of the pendulum motion of a slowing-down grandfather clock.</p> <p>As a science-stream student in high school, interviewee no. 2's knowledge structure is evidently aligned with the textbook-presented physics knowledge: an energy-based perspective and a force-acceleration-based one. The evidence I observed showed he could switch between these knowledge systems. However, his responses were fragmented when being explored with verbal interview questions. Only after seeing the mathematical equation, he immediately jumped to the correct conclusion without too much reasoning using either the energy-based or the force-acceleration-based conceptual framework and the languages. Compared with the responses of the first interviewee, it can be said that the structure of his physics knowledge was more sophisticated, especially in using mathematical reasoning and the vector-based force language to explain the pendulum motion. However, the knowledge did not guarantee the correct responses, especially when checked from the first responses.</p> <p>As introduced earlier, the third interviewee was the only one who emphasized the mathematical aspect of the equation. Moreover, he described the mathematical relationship in detail. In this aspect, he remarked: "Hmm, gravitational acceleration and pi are constant values. This period only has to do with that L. Hmm...and it's a uh...exponential function relationship, right? A half-power exponential function relationship."</p> <p>Given his responses, I can tell that the third interviewee was the only one who relied on mathematical thinking, which contrasts the way of reasoning employed by the first two interviewees. The sampling and decision-making differences can be attributed to his training in civil engineering and background knowledge in science. He was the only interviewee who continued the mathematical and scientific training after graduating high school. Such a background has enabled his reliance on the mathematical equation and the idealization process to explain pendulum motion.</p> <p>In this aspect, the fourth interviewee was different. She declared that she was not good at physics from the beginning of the interview. Upon seeing the mathematical equation, she noted, "Oh, physics is my weakest subject. Yes, um, I probably haven't seen it before." After being encouraged to express her impression of the equation, she continued:Interviewee no. 4: Um, my first impression would be 2π. Pi is the circumference of a circle. Um, I only remember that r squared is...I probably only remember things about math. Um, I'm not sure what 2π means exactly, like two circumferences...but the square root part feels a bit complicated. But L represents length.Interviewer: Yes, it represents the length of the line.Interviewee no. 4: Length divided by weight.Interviewer: No, that's gravitational acceleration.Interviewee no. 4: Gravitational acceleration! Length divided by gravitational acceleration; I really don't understand that.</p> <p>Her responses sharply contrast the third interviewee's answers to the same interview question. Whereas the latter relied on the symbolic forms of this equation, the fourth interviewee had to figure out the exact meaning of each symbol, let alone the mathematical relationship among these symbols. However, when asked about the implementation question about adjusting the position of the pendulum of a slowing-down pendulum, her answer was the same as the engineering student had given out. Given the evidence, sometimes a correct response to a probing question may not always equal a proper understanding of the same phenomenon.</p> <p>At the end of the last interview, I asked the fifth interviewee the same question about the mathematical equation. She commented on having forgotten it. I also asked her whether she wanted to know more about the pendulum motion. She asked: "Did we learn about pendulum motion in middle school physics?" After hearing as I replied, "Yes!" She further added, "I don't remember anything about pendulum motion." Despite not recollecting learning pendulum motion in high school, she answered the interview questions correctly. It seemed her participation in the experiment improved her learning, though she might need to understand the underlying knowledge centered around pendulum motion.</p> <p>In summary, three of the five interviewees saw the mathematical equation as the key to understanding pendulum motion. Two of them somehow changed their views about the effective factors in determining the swing period of the pendulum motion. The third one even commented on the mathematical aspect of this equation. In contrast, it seems that the other two interviewees have lost meaningful contact with the equation and did not get too much from it. What I have observed in this series of interviews seems to support what Bruce Sherin ([<reflink idref="bib51" id="ref59">51</reflink>]) stressed in discussing reforming introductory physics instruction:I challenge the assumption that in physics or any domain the conceptual and the symbolic elements (the mathematical symbols and their identities or definitions) 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 (introductory) physics cannot be taught without equations. However, it does imply that equation-free courses will result in an understanding of (modern) physics that is fundamentally different from physics as understood by physicists. (Sherin, [<reflink idref="bib51" id="ref60">51</reflink>], p. 524)</p> <p>The interviewees who fused the mathematical understanding with verbal expression showed the advantage in explaining their experiences with the pendulum motion equation, and the equation-enhanced understanding stood the test of time and language change.</p> <hd id="AN0187498037-17">Discussions, Conclusions, and Implications</hd> <p>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 (S-D) 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 <emph>T</emph> = 2π <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mi>l</mi><mo stretchy="false">/</mo><mi>g</mi></mrow></msqrt></math> </ephtml> ) in acting conceptual change learning with the verbal expressions included. Finally, we reflect on the pedagogical implications of these findings.</p> <hd id="AN0187498037-18">A Mathematically Defined Physics Concept: Friend or Foe?</hd> <p>Interestingly, most current conceptual change studies except PER choose to avoid the mathematical contents (Potvin & Cyr, [<reflink idref="bib45" id="ref61">45</reflink>]). For example, diSessa ([<reflink idref="bib11" id="ref62">11</reflink>]) 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, [<reflink idref="bib12" id="ref63">12</reflink>], 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 <emph>T</emph> = 2π <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mi>l</mi><mo stretchy="false">/</mo><mi>g</mi></mrow></msqrt></math> </ephtml> . 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.</p> <p>However, the question remains why the equation-free PER would be different. Without a new theoretical framework, uncertainty remains. As introduced earlier, 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.</p> <p>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. Chalmers ([<reflink idref="bib6" id="ref64">6</reflink>]) 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.</p> <hd id="AN0187498037-19">Unlocking the Learning Brain's Active Sampling and Decision-making</hd> <p>In today's parlance, the brain relies on a network-like structure (Baronchelli et al., [<reflink idref="bib3" id="ref65">3</reflink>]) 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, [<reflink idref="bib10" id="ref66">10</reflink>]; Salmelin & Kujala, [<reflink idref="bib50" id="ref67">50</reflink>]). 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, [<reflink idref="bib17" id="ref68">17</reflink>]), who first identified the independent cellular structure of a neuron. Following his lead, Adrian and Bronk ([<reflink idref="bib1" id="ref69">1</reflink>]) associated neurons' spiking patterns with the axonal and dendritic mechanisms. Later, Hodgkin et al. ([<reflink idref="bib20" id="ref70">20</reflink>]) 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, [<reflink idref="bib35" id="ref71">35</reflink>]; Muller, [<reflink idref="bib38" id="ref72">38</reflink>]; Palva & Palva, [<reflink idref="bib40" id="ref73">40</reflink>]), which implies time is essential for understanding learning.</p> <p>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., [<reflink idref="bib14" id="ref74">14</reflink>]), and (b) the synaptic (neuron-to-neuron) chemical information transmission processes (Bennett, [<reflink idref="bib4" id="ref75">4</reflink>]). First, the glial cells separate the myelinated axonal fibers from the unmyelinated ones. According to Fields et al. ([<reflink idref="bib14" id="ref76">14</reflink>]), 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.</p> <p>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, [<reflink idref="bib9" id="ref77">9</reflink>]; Fields et al., [<reflink idref="bib14" id="ref78">14</reflink>]; Langille & Brown, [<reflink idref="bib24" id="ref79">24</reflink>]), 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, [<reflink idref="bib16" id="ref80">16</reflink>]). Again, the brain processes information in time.</p> <p>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 ([<reflink idref="bib35" id="ref81">35</reflink>]) 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).</p> <p>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.</p> <hd id="AN0187498037-20">The Need for a New Taxonomy of Conceptual Change Studies with Mathematical Symbols</hd> <p>The development of a research project depends on an initial and rudimentary conceptualization that maximally characterizes its unique subject content and on an implicit or explicit taxonomy that categorizes the contents. For example, to most of us, a taxonomy in biology refers to a hierarchical and embedded categorical system for identifying and classifying organisms given their physical and genetic characteristics. With such a categorization, a biologist can categorize the diversity of living organisms into an ordered and accepted theoretical system (Species, Genus, Family, Order, Class, Phylum, Kingdom, Domain) into various sub-categories in a conceptual space. These sub-categories embody a set of premises and organizing principles so that the stability and consistency of these categories and inter- and intra-relationships among living organisms can be represented, with the points representing animals or plants and the areas representing their connections. One of the primary goals of modern ecological science education is to help students understand the taxonomy so that they observe an environment with their mind's eyes and are ready to solve new problems even if they see abnormal data or surprising experimental results. The benefits of using taxonomy in research are also familiar to non-biologists.</p> <p>One widely known example is Bloom's Taxonomy of Educational Objectives ([<reflink idref="bib5" id="ref82">5</reflink>]). After half a century, one of Bloom's students, Lori Anderson and David Krathwohl, who were joined by a group of educational psychologists and educators, published a revised version of Bloom's taxonomy in 2001. In contrast to the biologists' and educators' taxonomies, another famous classification system is the Diagnostic and Statistical Manual of Mental Disorders: DSM-5-TR (First & American Psychiatric Association, [<reflink idref="bib15" id="ref83">15</reflink>]), highlighting its probabilistic nature in categorizing mental health-related conditions. Given the observed mathematical symbols in this study, a new taxonomy of conceptual change research is needed to differentiate their roles in categorizing researchers' conceptions and interpretations.</p> <p>With this taxonomy, it becomes easier to interpret conceptual change studies at different levels: the verbal, scientific, and mathematical descriptions with greater precision. The new taxonomy of concepts based on mathematical symbols reconnects the research with its historical roots. Michael J. Crowe, in his <emph>A History of Vector Analysis: The Evolution of the Idea of a Vectorial System</emph>, depicted a hidden connection between mathematics and academic research. After introducing the history of searching for the concept of numbers, he added,The second tradition, that within the history of physical science, also extends back to ancient times and consists in the search for mathematical entities and operations that represent aspects of physical reality. This tradition played a part in the creation of Greek geometry, and the natural philosophers of the seventeenth century inherited from the Greeks the geometrical approach to physical problems. However in the course of the seventeenth century the physical entities to be represented passed through a transformation. This transformation consisted in the shift in emphasis from such scalar quantities as position and weight to such vectorial quantities as velocity, force, momentum, and acceleration. The transition was neither abrupt nor was it confined to the seventeenth century. Later developments in electricity, magnetism, and optics acted further to transform the space of mathematical physics into a space filled with vectors. (Crowe, 1985, p. 1)</p> <p>Such a historical connection is rarely mentioned in conceptual change studies. However, the physical and cognitive sciences were developed by introducing these mathematical ideas. Avoiding these contents to attract students is not always the best pedagogical strategy. Instead, using the new taxonomy that features the mathematical connection is the first step in the right direction.</p> <p>Next, the new taxonomy helps bridge the gap between educational studies, laboratory teaching, and learning sciences. As demonstrated in the interview data, the interviewees expressed their entangled and constantly changing conceptions about the mathematical identity of the period of simple pendulum motion. Their conceptions were suitable to be described with the notion of probability and the distribution of random errors. Emphasizing the mathematically defined concepts also helps establish the link with these self-reflective routines. With the aid of these interview data, it would be easier to clarify the students' pre-conceptions about scientific concepts presented in textbooks, understand solutions to problems, and demonstrate the underlying idealization principle. In this sense, the new taxonomy provides a probabilistic framework for organizing and categorizing the students' knowledge more psychologically naturally. It recognizes that their pre-conceptions may be in dynamic sampling and decision-making over overlapping knowledge distributions and that the learning is an ongoing process.</p> <p>In brief, the new taxonomy with a probabilistic frame of reference significantly extends Zhou's hybrid learning space (2012) by establishing meaningful connections and offering opportunities to understand a whole range of academic writing activities. It recognizes that their academic written narratives with an emphasis on mathematical symbols, which may be inaccurate or incomplete given a mutually accepted understanding of the writing practice.</p> <hd id="AN0187498037-21">Conclusions and Pedagogical Implications</hd> <p>In this study, 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.</p> <p>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 (Babai et al., [<reflink idref="bib2" id="ref84">2</reflink>]; Posner et al., [<reflink idref="bib44" id="ref85">44</reflink>]; Potvin et al., [<reflink idref="bib47" id="ref86">47</reflink>]; Thagard, [<reflink idref="bib53" id="ref87">53</reflink>]; Worrall, [<reflink idref="bib59" id="ref88">59</reflink>], [<reflink idref="bib60" id="ref89">60</reflink>]), 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.</p> <p>In brief, the new taxonomy with a probabilistic frame of reference significantly contributes to extending the hybrid learning space (Worrall, [<reflink idref="bib60" id="ref90">60</reflink>]) 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.</p> <p>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 (Chalmers, [<reflink idref="bib7" id="ref91">7</reflink>]; Matthews, [<reflink idref="bib31" id="ref92">31</reflink>]; Mayer, [<reflink idref="bib33" id="ref93">33</reflink>]; Rowbottom, [<reflink idref="bib49" id="ref94">49</reflink>]). 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 (Worrall, [<reflink idref="bib60" id="ref95">60</reflink>]). 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 (Worrall, [<reflink idref="bib59" id="ref96">59</reflink>]).</p> <p>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, [<reflink idref="bib13" id="ref97">13</reflink>]; Potvin et al., [<reflink idref="bib47" id="ref98">47</reflink>]; Thagard, [<reflink idref="bib53" id="ref99">53</reflink>]; Worrall, [<reflink idref="bib59" id="ref100">59</reflink>]), the second language learning research (Li, [<reflink idref="bib28" id="ref101">28</reflink>]; Li et al., [<reflink idref="bib25" id="ref102">25</reflink>]; Li & Wong, [<reflink idref="bib27" id="ref103">27</reflink>]; Li & Zhou, [<reflink idref="bib26" id="ref104">26</reflink>]), and domain-specific teaching and learning (Mayer, [<reflink idref="bib33" id="ref105">33</reflink>]).</p> <p>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.</p> <p>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.</p> <hd id="AN0187498037-22">A Caveat on the Limitations of the Current Study and Future Research</hd> <p>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.</p> <p>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., [<reflink idref="bib43" id="ref106">43</reflink>]), 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.</p> <hd id="AN0187498037-23">Declarations</hd> <p></p> <hd id="AN0187498037-24">Ethical Approval</hd> <p>The study was approved by the University's Research Ethics Board.</p> <hd id="AN0187498037-25">Informed Consent</hd> <p>The signed informed consents were obtained from all participants of this study by one of the authors.</p> <hd id="AN0187498037-26">Conflict of Interest</hd> <p>The authors declare that they have no conflict of interest.</p> <hd id="AN0187498037-27">Publisher's Note</hd> <p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p> <ref id="AN0187498037-28"> <title> References </title> <blist> <bibl id="bib1" idref="ref20" type="bt">1</bibl> <bibtext> Adrian ED, Bronk DW. The discharge of impulses in motor nerve fibres. 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Items – Name: Title
  Label: Title
  Group: Ti
  Data: On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Lin+Li%22">Lin Li</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-2192-5454">0000-0002-2192-5454</externalLink>)<br /><searchLink fieldCode="AR" term="%22George+Zhou%22">George Zhou</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Science+%26+Education%22"><i>Science & Education</i></searchLink>. 2025 34(4):2377-2407.
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  Label: Availability
  Group: Avail
  Data: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 31
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Scientific+Concepts%22">Scientific Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Taxonomy%22">Taxonomy</searchLink><br /><searchLink fieldCode="DE" term="%22Motion%22">Motion</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Students%22">Foreign Students</searchLink><br /><searchLink fieldCode="DE" term="%22Physics%22">Physics</searchLink><br /><searchLink fieldCode="DE" term="%22Probability%22">Probability</searchLink><br /><searchLink fieldCode="DE" term="%22Cognitive+Processes%22">Cognitive Processes</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Processes%22">Learning Processes</searchLink><br /><searchLink fieldCode="DE" term="%22Reaction+Time%22">Reaction Time</searchLink><br /><searchLink fieldCode="DE" term="%22Error+Patterns%22">Error Patterns</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics%22">Mathematics</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1007/s11191-024-00545-9
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0926-7220<br />1573-1901
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Over four decades of conceptual change studies in 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 mixed-method study foregrounds the probabilistic aspect of international students' intuitive-to-counterintuitive conceptions when learning pendulum motion. The probability here is rooted in a moving neural time average in the mind for characterizing these students' cognitive processes (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 and interviewed students to characterize these students' reaction time and error rates in matching the period of a visually presented pendulum, which embodied its mathematical identity: T = 2[pi][square root of]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.
– Name: AbstractInfo
  Label: Abstractor
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  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1482208
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1482208
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1007/s11191-024-00545-9
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 31
        StartPage: 2377
    Subjects:
      – SubjectFull: Scientific Concepts
        Type: general
      – SubjectFull: Taxonomy
        Type: general
      – SubjectFull: Motion
        Type: general
      – SubjectFull: Foreign Students
        Type: general
      – SubjectFull: Physics
        Type: general
      – SubjectFull: Probability
        Type: general
      – SubjectFull: Cognitive Processes
        Type: general
      – SubjectFull: Learning Processes
        Type: general
      – SubjectFull: Reaction Time
        Type: general
      – SubjectFull: Error Patterns
        Type: general
      – SubjectFull: Mathematics
        Type: general
    Titles:
      – TitleFull: On a New Taxonomy of Concepts and Conceptual Change: In Search of the Brain's Probabilistic Language of Learning Scientific Concepts
        Type: main
  BibRelationships:
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      – PersonEntity:
          Name:
            NameFull: Lin Li
      – PersonEntity:
          Name:
            NameFull: George Zhou
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          Dates:
            – D: 01
              M: 08
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 0926-7220
            – Type: issn-electronic
              Value: 1573-1901
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              Value: 34
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              Value: 4
          Titles:
            – TitleFull: Science & Education
              Type: main
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