Electrophysiological Correlates of Cognitive Flexibility in Fraction Representation

Electrophysiological Correlates of Cognitive Flexibility in Fraction Representation | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 11 July 2025 V1 Latest version Share on Electrophysiological Correlates of Cognitive Flexibility in Fraction Representation Authors : Chen-Yu Yao 0009-0008-8818-935X , Hui-Yu Hsu [email protected] , Zai-Fu Yao 0000-0002-9823-9110 , and Tsu-Jen Ding Authors Info & Affiliations https://doi.org/10.22541/au.175224251.10789993/v1 234 views 120 downloads Contents Abstract Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This study investigates cognitive flexibility in fraction representation by examining behavioral performance, event-related potentials (ERPs), and neural complexity measured via multiscale entropy (MSE). Thirty college students completed a fraction comparison task involving figure and symbol fraction representations under switch (FS, SF) and non-switch conditions (FF, SS). Behavioral analyses unexpectedly revealed higher accuracy, faster reaction times, and greater efficiency in switch trials compared to non-switch trials, contrasting with traditional task-switching paradigms. ERP analyses showed that the purely graphical (FF) condition elicited significantly larger N2 amplitudes (indicating higher conflict monitoring) and larger P3 amplitudes (reflecting greater working-memory updating) compared to other conditions, particularly at frontal and central electrode sites. Additionally, MSE results demonstrated sustained higher neural complexity for the FF condition within the central region during late processing stages, indicating increased cognitive load from continuous visuospatial integration. These findings highlight the predominance of representation transformation complexity over traditional switch costs. By integrating behavioral data, ERP components, and MSE analyses, this study provides novel insights into cognitive flexibility in mathematical cognition, suggesting that graphical fraction representations may inherently demand greater cognitive resources than symbolic representations. Electrophysiological Correlates of Cognitive Flexibility in Fraction Representation Chen-Yu Yao 1 , Hui-Yu Hsu 23 *, Zai-Fu Yao 1,3,4,5 , Tsu-Jen Ding 6 1Department of Educational Psychology and Counseling, National Tsing Hua University, Hsinchu City, 30013, Taiwan 2Graduate Institute of Mathematics and Science Education, National Tsing Hua University, Hsinchu City, 30013, Taiwan 3Research Center for Education and Mind Sciences, National Tsing Hua University, Hsinchu City, 30013, Taiwan 4Department of Kinesiology, National Tsing Hua University, Hsinchu City, 30013, Taiwan 5College of Education, National Tsing Hua University, Hsinchu City, 30013, Taiwan 6Department of Environmental and Cultural Resources, National Tsing Hua University, Hsinchu City, 30013, Taiwan *Correspondences: 300 No.521, Nanda Rd., Hsinchu City 30014, Taiwan, +886-3-571-5131 ext.78603, [email protected] Abstract This study investigates cognitive flexibility in fraction representation by examining behavioral performance, event-related potentials (ERPs), and neural complexity measured via multiscale entropy (MSE). Thirty college students completed a fraction comparison task involving figure and symbol fraction representations under switch (FS, SF) and non-switch conditions (FF, SS). Behavioral analyses unexpectedly revealed higher accuracy, faster reaction times, and greater efficiency in switch trials compared to non-switch trials, contrasting with traditional task-switching paradigms. ERP analyses showed that the purely graphical (FF) condition elicited significantly larger N2 amplitudes (indicating higher conflict monitoring) and larger P3 amplitudes (reflecting greater working-memory updating) compared to other conditions, particularly at frontal and central electrode sites. Additionally, MSE results demonstrated sustained higher neural complexity for the FF condition within the central region during late processing stages, indicating increased cognitive load from continuous visuospatial integration. These findings highlight the predominance of representation transformation complexity over traditional switch costs. By integrating behavioral data, ERP components, and MSE analyses, this study provides novel insights into cognitive flexibility in mathematical cognition, suggesting that graphical fraction representations may inherently demand greater cognitive resources than symbolic representations. Introduction Executive Functions (EF) refer to high-level cognitive processes that regulate, monitor, and adjust thought and behavior to achieve goals (Diamond, 2013; Miyake & Friedman, 2012); cognitive flexibility, as a core component of EF, involves the ability to rapidly shift between different task rules or representations while effectively managing attentional resources (Hölig & Berti, 2010; Norman & Shallice, 2000). However, extant literature has predominantly examined cognitive flexibility using verbal or purely symbolic tasks (e.g., Monsell, 2003; Rogers & Monsell, 1995), with limited investigation into the dynamic switching between graphical fraction representations (fraction figures) and symbolic fraction representations (fraction symbols). To address this gap, the present study comprehensively examines both behavioral effects and neural correlates of fraction representation transformation. To investigate the cognitive flexibility underlying fraction representation transformation, this study employs the well‐established task‐switching paradigm (Koch et al., 2018; Monsell, 2003; Ravizza & Carter, 2008). In this paradigm, participants alternate rapidly between different representations within a task or among tasks, which allows quantification of the “switch cost”. Switch cost is defined as the performance decrement (e.g., increased reaction times or decreased accuracy) observed when switching among different tasks compared to repeating the same task (Wylie & Allport, 2000). This cost is attributed to the cognitive demands involved in reconfiguring one’s mental set, which requires adjustments in task parameters such as goals, rules (Logan & Gordon, 2001; Norman & Shallice, 2000), and attentional focus (Hölig & Berti, 2010), thereby placing increased demands on prefrontal executive control mechanisms (Sauseng et al., 2006). Beyond the switching mechanism itself, the characteristics of representational systems may modulate switch cost. Dual‐Coding Theory, proposed by Paivio (1990), posits that cognition utilizes both verbal and nonverbal representational systems. Symbolic representation (S) is processed by a verbal system, whereas figural representation (F) is handled by a nonverbal system. Cognitive Load Theory further suggests that figural representations, by relying on intuitive visuospatial mechanisms, impose lower working-memory demands than abstract symbolic formats (Chandler & Sweller, 1991; Mayer, 2005; Sweller, 1988). In the context of fraction instruction, Ni and Zhou (2005) found that graphical fraction representations reduce whole‐number bias, and Arcavi (2003) demonstrated that visualization tools enhance understanding of mathematical concepts. However, these studies have largely focused on single representations, leaving the cognitive load and neural mechanisms of switching between figure and symbol representations underexplored. Thus, the present study specifically examines how fraction representation transformation affect switch cost. We used event-related potentials (ERPs) to examine the timing of neural processes underlying representation transformation, focusing on the N2 and P3 components. The N2 component emerges roughly 220–300 ms after stimulus onset and reflects early conflict detection and inhibitory control when resolving conflicts between different representations, with amplitude increases indicating greater engagement of monitoring mechanisms (Enriquez-Geppert et al., 2010; Folstein & Van Petten, 2008). The P3 component, appearing around 300–500 ms, reflects the allocation of attention and updating of working memory representations necessary for reconfiguring mental sets; larger P3 amplitudes signal more efficient integration of new information into existing cognitive frameworks (Friedman et al., 2001; Kok, 2001; Meinhardt & Pekrun, 2003; Polich, 2007). By examining both N2 and P3, we can distinguish the initial conflict-monitoring phase from the subsequent representational-updating phase. Previous studies confirm that representational format affects these components: Groom and Cragg (2015) observed enhanced N2 amplitudes in high-conflict tasks, while Nguyen et al. (2016) reported increased P3 amplitudes during successful inhibitory control. In terms of neural indices, event‐related potentials (ERP) reveal time‐locked EEG changes associated with stimuli or cognitive processes (Luck, 2014), but their temporal components may not fully capture the dynamic complexity of brain activity. Multiscale Entropy (MSE), introduced by Costa et al. (2005), quantifies signal complexity across multiple time scales. Recent studies have demonstrated that MSE can complement ERP analyses by uncovering brain dynamic patterns under various cognitive states (Heisz & McIntosh, 2013). Therefore, applying MSE to ERP data in fraction representation transformation offers a more comprehensive understanding of neural flexibility and allows us to assess how different switching conditions affect brain signal complexity. Building on the above literature, this study aims to integrate behavioral, ERP, and MSE analyses to examine fraction representation switching along two dimensions: the task switching factor and the representation transformation factor. Hypothesis 1 (H1): We will compare switch versus non-switch conditions in terms of reaction time, accuracy, ERP components (N2 at 220–300 ms, P3 at 300–500 ms), and MSE values. We predict that switch conditions, relative to non-switch conditions, will produce significantly longer reaction times, lower accuracy, larger N2 and P3 amplitudes, and increased MSE, reflecting increased demands for mental set reconfiguration and inhibition of previous task rules (Rogers & Monsell, 1995; Sauseng et al., 2006). Hypothesis 2 (H2): We will compare the four representation transformation (FF, SS, FS, SF) on reaction time, accuracy, ERP components (N2, P3), and MSE values. We predict that the FF condition, compared to SS, FS, and SF, will yield the shortest reaction times, highest accuracy, smallest N2 and P3 amplitudes, and smallest MSE values, reflecting the lowest cognitive load among the four formats (Ni & Zhou, 2005; Paivio, 1990; Sweller, 1988). 2. Method 2.1 Participant Thirty right-handed college students (15 males, age: M=20.5, SD=1.89) without known dyscalculia were recruited for this study. All subjects were asked to read and sign an informed consent form before the experiment. The study was approved by the Research Ethics Committee of National Tsing Hua University 2.2 Experimental Stimuli and Procedure The experimental stimuli comprised fraction representations in two formats: symbol and figure. The symbolic stimuli consisted of three target fractions (1/2, 1/3, 1/4) presented in black Arial font (72-point) on a gray background. The probe stimuli were generated by multiplying both the numerator and the denominator of each target by the integers 2 through 6. Thus for the target 1/2 the probes were 2/4, 3/6, 4/8, 5/10 and 6/12; for the target 1/3 they were 2/6, 3/9, 4/12, 5/15 and 6/18; and for the target 1/4 they were 2/8, 3/12, 4/16, 5/20 and 6/24. Participants judged whether each probe-target pair represented the same numerical value. Half of the trials were congruent, involving numerically equivalent pairs, and half were incongruent, with each probe paired with a different target. The figural stimuli consisted of blue pie charts displayed on the same gray background. Each chart measured 200 pixels in diameter and had the corresponding fraction of its area shaded in blue; for example, half of the circle was shaded for 1/2 and one third was shaded for 1/3. To examine the impact of representation transformation and switching demands on cognitive load, four trial types were defined by pairing these formats across presentations (Table 1): Figure to Figure (FF), Symbol to Symbol (SS), Figure to Symbol (FS), and Symbol to Figure (SF). In FF trials, two figure stimuli were presented consecutively, requiring no shift in format; in SS trials, two symbol stimuli appeared in succession, also constituting a non-switch condition. Switch trials comprised FS, in which an initial figure stimulus was followed by a symbol stimulus, and SF, where a symbol stimulus preceded a figure stimulus. Table 1. Table 1. Experimental Conditions: Representation transformation and Task Switching Structure Participants were seated approximately 70 cm from a 24-inch monitor in a sound‑attenuated, dimly lit room and instructed to minimize movement and blinking during trials. Each experimental session began with a brief instruction screen explaining the match/non-match task and button mappings (left key = “same,” right key = “different”), followed by 16 practice trials (four per condition) during which participants received corrective feedback. The main experiment consisted of four blocks corresponding to the four representation transformation (FF, SS, FS, SF), presented in a counterbalanced Latin square order. Each block contained 40 trials, with an equal number of congruent and incongruent probe–target pairs randomized within the block. At the start of each block, participants were prompted to press any key to begin and reminded of the current condition (e.g., “Figure→Symbol block”). Short rest intervals (approximately 30 seconds) were provided between blocks, during which participants were encouraged to relax and blink as needed. On each trial, a centrally located fixation cross appeared for 500 ms, followed by the first stimulus (S1) displayed for 1000 ms. After a 500 ms blank interval, the second stimulus (S2) was presented for 1500 ms. Participants were instructed to respond during the S2 interval by pressing the appropriate key. If no response was made within 1500 ms, the trial was recorded as a missed response and the next trial commenced. A blank screen of 500 ms then separated consecutive trials. Reaction times were measured from S2 onset, and accuracy (correct/incorrect) was recorded (Figure 1). Figure 1 Trial Sequence and Experimental Conditions for Fraction Comparison Task : This figure illustrates the trial sequence for the fraction comparison task, with S1 and S2 corresponding to the four experimental conditions outlined in Table 1. 2.3. EEG Measures EEG recordings were conducted in a sound-attenuated, electrically shielded laboratory. Stimuli were presented using E-Prime 3.0, and participants responded using their right and left index fingers. EEG data were acquired using a SynAmps RT 64-channel amplifier (COMPUMEDICS Neuroscan) with a Quick-Cap 64 electrode cap based on the international 10–20 system. The signal was referenced to the average of the left (M1) and right (M2) mastoids, with electrode impedances maintained below 5 kΩ. Data were recorded at 1000 Hz and downsampled offline to 250 Hz. Preprocessing was conducted in EEGLAB (sccn.ucsd.edu/eeglab) and ERPLAB (erpinfo.org/erplab/), with band-pass filtering (0.1–30 Hz, IIR Butterworth, 24 dB/octave), epoch segmentation (−200 to 800 ms relative to stimulus onset), and baseline correction using the −200 to 0 ms interval. Trials contaminated by artifacts were excluded using a moving window peak-to-peak threshold method (threshold: 50 μV; window size: 200 ms; step: 100 ms). Approximately 16.2% of trials were rejected. Only correct-response trials were retained for ERP averaging. 2.4. Data Analysis Behavioral measures—reaction time (RT), accuracy, and inverse efficiency score (IES; calculated as mean RT divided by proportion correct to integrate speed and accuracy)—and ERP indices were analyzed using JASP Version 0.19.3. ERP analyses centered on the N2 (220–300 ms) and P3 (300–500 ms) components, with mean amplitudes extracted from nine electrodes grouped into three scalp regions: frontal (Fz, F3, F4), central (Cz, C3, C4), and parietal (Pz, P3, P4). Multiscale entropy (MSE) was computed from ERP data collected during the fraction comparison task. MSE analyses were applied to signals from three anatomically defined ROIs: Frontal (Fz, F3, F4), Central (Cz, C3, C4), and Parietal (Pz, P3, P4). Sample entropy was calculated across 100 temporal scales, linearly corresponding to the ERP window of interest, by down sampling and coarse-graining the time series at each scale. For Hypothesis 1 (task switching factor), separate one-way repeated-measures ANOVAs were performed on reaction time (RT), accuracy, and inverse efficiency score (IES) with task switching (switch: FS, SF vs. non-switch: FF, SS) as the within-subject factor. Corresponding two-way repeated-measures ANOVAs were conducted on N2 (220–300 ms) and P3 (300–500 ms) mean amplitudes with task switching (switch vs. non-switch) and scalp region (frontal, central, parietal) as within-subject factors to assess topographical variations in Task effects on neural indices. One-way ANOVAs compared sample entropy values across task switching (switch: FS, SF vs. non-switch: FF, SS) at each scale for each ROI. For Hypothesis 2 (representation transformation factor), one-way repeated-measures ANOVAs were conducted on behavioral measures (RT, accuracy, IES) with representation transformation (FF, SS, FS, SF) as the within-subject factor. ERP data were subjected to two-way repeated-measures ANOVAs with representation transformation (FF, SS, FS, SF) and scalp region (frontal, central, parietal) as within-subject factors to examine topographical variations in N2 and P3 amplitudes across conditions. One-way ANOVAs compared sample entropy values across four representation transformation (FF, SS, FS, SF) at each scale for each ROI. Significant main effects and interactions were followed by Bonferroni-adjusted post hoc comparisons. Partial eta squared (\(\eta_{p}^{2}\)) is reported as the effect size for all ANOVA results, and statistical significance was set at α = .05. 3. Results 3.1. Behavioral Results Regarding the task switching effect, we predicted longer reaction times (RT), lower accuracy (Acc), and higher inverse efficiency scores (IES) during switch conditions (FS and SF) compared to non-switch conditions (FF and SS). However, our results contrasted with prior literature (Figure 2). The ANOVA revealed significantly higher accuracy in the switch conditions (M = 96.944%) compared to non-switching conditions (M = 95.778%), F (1, 29) = 15.458, p < .001,\(\eta_{p}^{2}\) = 0.348. Additionally, reaction times were significantly faster in the switch conditions (M = 587.707 ms) compared to the non-switching conditions (M = 613.499 ms), F (1, 29) = 11.283, p < .01, \(\eta_{p}^{2}\) = 0.280. Furthermore, the inverse efficiency scores were significantly lower under switch conditions (M = 6.064 ms) than under non-switch conditions (M = 6.422 ms), F (1, 29) = 15.381, p < .001,\(\eta_{p}^{2}\) = 0.347, indicating higher processing efficiency during task switching. Figure 2. Behavioral performance (Accuracy, Reaction Time, Inverse Efficiency Score) across switch and non-switch conditions. Error bars indicate standard errors. Regarding the representation transformation effect, we proposed that cognitive load, as indexed by performance measures (RT, Acc, IES), would vary across the four representation transformation (FF, SS, FS, SF), with the lowest cognitive load predicted in the FF condition (shortest RT, highest Acc), the highest load in the SS condition, and intermediate loads for FS and SF conditions. ANOVA results confirmed significant differences across representation transformation. Accuracy significantly differed among the representation transformation, F (3, 87) = 8.829, p < .001, \(\eta_{p}^{2}\) = 0.233. Reaction times also showed significant differences across conditions, F (3, 87) = 40.681, p < .001, \(\eta_{p}^{2}\) = 0.584, and inverse efficiency scores likewise revealed significant variation, F (3, 87) = 39.869, p < .001, \(\eta_{p}^{2}\) = 0.579. Post-hoc comparisons revealed significant differences in accuracy between FF and SS (\(p_{\text{bonf}}\) < .05), FF and FS (\(p_{\text{bonf}}\) < .01), and FF and SF (\(p_{\text{bonf}}\) < .01). Reaction times significantly differed between FF and SS (\(p_{\text{bonf}}\) < .001), FF and FS (\(p_{\text{bonf}}\) < .001), FF and SF (\(p_{\text{bonf}}\) < .01), SS and SF (\(p_{\text{bonf}}\)< .001), and FS and SF (\(p_{\text{bonf}}\) < .001). For inverse efficiency scores, significant differences were observed between FF and SS (\(p_{\text{bonf}}\) < .001), FF and FS (\(p_{\text{bonf}}\) < .001), FF and SF (\(p_{\text{bonf}}\)< .001), SS and SF (\(p_{\text{bonf}}\) < .001), and FS and SF (\(p_{\text{bonf}}\) < .001). Notably, contrary to expectations, the FF condition demonstrated the highest inverse efficiency scores, indicating the lowest processing efficiency. Figure 3. Behavioral performance (Accuracy, Reaction Time, Inverse Efficiency Score) across the four representation transformation (FF, SS, FS, SF). Error bars indicate standard errors. As can be seen in the Figure 3, FF condition had the much lower accuracy when compared to the other three conditions, required much longer time to respond the items and caused the highest IES. These behavioral findings suggest a need to reconsider the traditional notion of switch costs, because in the fraction comparison task, remaining in a purely graphical format (FF) appears to impose a greater cognitive load than switching between formats (FS and SF). This pattern indicates that representational complexity itself may dominate cognitive demands, highlighting the primacy of representation transformation over switching per se. 3.2 ERP Analysis 3.2.1. N200 Component Mean N200 amplitudes were analyzed using a 2 (task switching: switch vs. non-switch) × 3 (scalp region: frontal, central, parietal) repeated-measures ANOVA. The task switching and scalp region interaction was not significant, F (2, 58) = 0.515, p = .6,\(\eta_{p}^{2}\) = 0.017, nor was the main effect of task switching, F (1, 29) = 0.938, p = .341, \(\eta_{p}^{2}\) = 0.031, indicating that N200 responses did not differ between switch and non-switch trials. In contrast, the main effect of scalp region was significant, F (2, 58) = 39.50, p < .001,\(\eta_{p}^{2}\) = 0.577. Bonferroni-adjusted comparisons demonstrated that frontal electrodes evoked larger N200 amplitudes than central (\(p_{\text{bonf}}\) < .01) and parietal sites (\(p_{\text{bonf}}\) < .001), and that central sites produced greater amplitudes than parietal sites (\(p_{\text{bonf}}\) < .001). To assess the influence of fraction representation transformation on N200 mean amplitudes, a separate 4 (representation transformation: FF, SS, FS, SF) × 3 (scalp region) repeated-measures ANOVA was conducted. This analysis revealed a significant of representation transformation and scalp region interaction, F (6, 174) = 23.92, p < .001, \(\eta_{p}^{2}\) = 0.452, confirming that the effect of format depended on electrode location. Simple main effects analyses were conducted to further characterize the interaction. A one-way ANOVA of representation transformation at each scalp region revealed significant effects: at frontal sites, F (3, 87) = 15.894, p < .001, \(\eta_{p}^{2}\) = 0.354; at central sites, F (3, 87) = 13.374, p < .001, \(\eta_{p}^{2}\) = 0.316; at parietal sites, F (3, 87) = 10.994, p < .001, \(\eta_{p}^{2}\) = 0.275. Conversely, one-way ANOVAs of scalp region within each format condition showed significant regional differences for all formats: FF: F (2, 58) = 39.821, p < .001, \(\eta_{p}^{2}\) = 0.578; SS: F (2, 58) = 36.951, p < .001, \(\eta_{p}^{2}\) = 0.560; FS: F (2, 58) = 63.400, p < .001, \(\eta_{p}^{2}\) = 0.686; SF: F (2, 58) = 12.579, p < .001,\(\eta_{p}^{2}\) = 0.302. These simple main effects confirm that both representation transformation and scalp region systematically influence N200 amplitude across conditions. Follow-up Bonferroni‐adjusted pairwise comparisons within each scalp region. At frontal sites, FF produced larger N2 amplitudes than SS (\(p_{\text{bonf}}\) < .05) and FS (\(p_{\text{bonf}}\) = .001), and SF exceeded both SS (\(p_{\text{bonf}}\) < .001) and FS (\(p_{\text{bonf}}\) < .001). At central sites, FF again exceeded SS (\(p_{\text{bonf}}\) < .001) and FS (\(p_{\text{bonf}}\) < .01), and SF was larger than SS (\(p_{\text{bonf}}\) < .001) and FS (\(p_{\text{bonf}}\)< . 05). At parietal sites, FF elicited greater amplitudes than SS (\(p_{\text{bonf}}\) = .001) and SF (\(p_{\text{bonf}}\)< .05), and FS exceeded SS (\(p_{\text{bonf}}\) < .001) and SF (\(p_{\text{bonf}}\) < .05). These region‐specific contrasts demonstrate that the FF sequence consistently elicited the largest N2 amplitudes across all scalp regions. Moreover, SF produced similarly elevated N2 responses at frontal and central sites, whereas FS yielded the second‐highest amplitudes at parietal sites. In contrast, SS evoked the smallest N2 across regions. Figure 4. Grand-average N200 amplitudes (µV) at frontal, central, and parietal electrode clusters. Panel A: Grand-average N200 waveforms for the non-switch (black) and switch (red) conditions, recorded at three electrode clusters; Panel B: Grand-average N200 waveforms for each of the four representation transformation: FF (black), SS (green), SF (blue) and FS (red), plotted at the same Frontal, Central and Parietal clusters. The same analysis window is highlighted. Panels C quantify these effects in bar‐graph. 3.2.2. P300 Component Mean P300 amplitudes were analyzed using a two-way repeated-measures ANOVA. The task switching and scalp region interaction was significant, F (2, 58) = 7.37, p = .001, \(\eta_{p}^{2}\) = 0.203, indicating that switching effects varied across electrode sites. To clarify the interaction, simple main effects of task switching were examined at each scalp region: switching did not significantly modulate P300 amplitude at frontal sites, F (1, 29) = 2.34, p = .137, \(\eta_{p}^{2}\) = 0.075, central sites, F (1, 29) =0.61, p = .442, \(\eta_{p}^{2}\) = 0.021; or parietal sites, F (1, 29) =2.10, p = .158, \(\eta_{p}^{2}\) = 0.068. Conversely, simple main effects of scalp region were significant within both task switching: in non-switch trials, F (2, 58) =37.54, p < .001, \(\eta_{p}^{2}\) = 0.564; and in switch trials, F (2, 58) =45.99, p < .001,\(\eta_{p}^{2}\) = 0.613. These results indicate that P300 amplitude is primarily determined by scalp region, with limited modulation by task switching. A two-way repeated-measures ANOVA on mean P300 amplitudes revealed a significant representation transformation and scalp region interaction, F (6, 174) = 9.82, p < .001, \(\eta_{p}^{2}\) = 0.253, indicating that the influence of representation transformation on P300 amplitude varied by cortical site. Simple main effects analyses further characterized this interaction. At frontal sites, representation transformation differed significantly, F (3, 87) = 3.60, p < .05, \(\eta_{p}^{2}\) = 0.111; at central sites, F (3, 87) = 4.97, p < .01, \(\eta_{p}^{2}\) = 0.147; and at parietal sites, F (3, 87) = 12.28, p < .01,\(\eta_{p}^{2}\) = 0.298. Conversely, when examining scalp region within each representation transformation, regional differences were significant for all formats: FF: F (2, 58) = 21.566, p < .001, \(\eta_{p}^{2}\) = 0.426; SS: F (2, 58) = 37.252, p < .001, \(\eta_{p}^{2}\) = 0.561; FS: F (2, 58) = 51.032, p < .001, \(\eta_{p}^{2}\) = 0.638; SF: F (2, 58) = 33.358, p < .001,\(\eta_{p}^{2}\) = 0. 535. These simple main effects confirm that P300 amplitudes differed significantly among the four representation transformation across scalp regions. Follow-up Bonferroni‐adjusted pairwise comparisons within each region demonstrated that, at frontal electrodes, P3 amplitudes were significantly greater for FF than for SF (\(p_{\text{bonf}}\)< .05). In the central region, FF again elicited larger P3 responses than SF (\(p_{\text{bonf}}\) < .05), and SS exceeded FS (\(p_{\text{bonf}}\) < .05). At parietal sites, FS produced significantly higher P3 amplitudes than FF (\(p_{\text{bonf}}\)< .01), SS (\(p_{\text{bonf}}\) < .001), and SF (\(p_{\text{bonf}}\) < .001). These region‐specific contrasts indicate that FF place greater demands on frontal and central updating processes, whereas FS engage parietal updating mechanisms most strongly. Figure 5 Grand-average P300 amplitudes (µV) at frontal, central, and parietal electrode clusters. Panel A: Grand-average P300 waveforms for the non-switch (black) and switch (red) conditions, recorded at three electrode clusters; Panel B: Grand-average P300 waveforms for each of the four representation transformation: FF (black), SS (green), SF (blue) and FS (red), plotted at the same Frontal, Central and Parietal clusters. The same analysis window is highlighted. Panels C quantify these effects in bar‐graph. 3.3. Multiscale entropy MSE curves for switch (FS, SF) and non‑switch (FF, SS) conditions were compared across all three ROIs (Frontal, Central, Parietal; Figure 6). One‑way repeated‑measures ANOVAs conducted separately at each scale (1–100) revealed a significant main effect of Task‑Switching. One‑way repeated‑measures ANOVAs at each scale (1–100) revealed a significant task switching effect only in the central ROI at scales 60–80 (p < .05). In this range (~500–600 ms), switch trials showed lower sample entropy than non‑switch trials, indicating a brief reduction in neural complexity when formats were switched. Across all regions, sample entropy peaked around scales 30–40 (~100–200 ms), reflecting maximal neural variability early in processing; however, only the central ROI exhibited a distinct dip for switch trials between scales 60–80, whereas the frontal and parietal curves overlapped almost entirely, suggesting minimal switch‑related modulation outside the central region. Figure 6. Group‑averaged sample entropy curves are shown for non‑switch (blue) and switch (red) trials across Frontal, Central, and Parietal ROIs. Representation transformation were compared by conducting one-way repeated-measures ANOVAs on sample entropy across three ROIs (Frontal, Central, Parietal) and 100 time scales (1–100), with Bonferroni-corrected post-hoc tests identifying specific differences (Figure 7). In the interval following stimulus onset (0–800 ms), one-way ANOVAs conducted separately at each scale revealed no significant condition differences in sample entropy for the Frontal and Parietal ROIs (all scales 1–100: p > .05). In contrast, within the Central ROI, one-way ANOVAs at each scale showed a significant main effect of condition for scales 80–100 (corresponding to ~600–800 ms). Post hoc comparisons demonstrated that FF exhibited higher sample entropy than SS ( p < .001). These findings indicate that, during the late processing window, only the Central region exhibits increased complexity, whereas the Frontal and Parietal regions remain unaffected by representation transformation. Figure 7. MSE Curves for Representation transformation Conditions. Group‑averaged sample entropy curves for FF (blue), SS (red), FS (green), and SF (orange) across ROIs. 4. Discussion In the present study, we investigated how task switching between figure and symbol fraction representations influences behavioral performance, ERP components (N2, P3), and neural complexity as indexed by multiscale entropy (MSE). Contrary to classical switch-cost models, participants were both faster and more accurate in switch trials (FS, SF) than in non-switch trials (FF, SS). Moreover, ERP and MSE measures revealed that the figure to figure (FF) condition, rather than the mixed-format switch conditions, elicited the largest N2 and P3 amplitudes and sustained elevated entropy in the central ROI during late processing (scales 60–80). Together, these findings suggest that representation transformation may outweigh the costs of switching per se. One key factor accounting for the discrepancy with classical switch‐cost findings is the FF condition. Traditional task-switching frameworks predict slower responses, lower accuracy, and greater neural engagement in switch relative to repeat conditions (Alport et al., 1994; Arbuthnott & Frank, 2000; Hsieh & Allport, 1994; Monsell, 2003; Wylie & Allport, 2000). In our study, it was the FF blocks rather than format switching that drove both behavioral and neural differences. During FF conditions, participants performed continuous visuospatial translation of pie-chart representations, a process that recruits parietal mapping of analog magnitudes and sustained working-memory updating. As a result, FF conditions elicited the largest N2 amplitudes (indicating heightened conflict monitoring) and the largest P3 amplitudes (indicating intensified context updating) of all conditions. Behaviorally, FF conditions also produced slower reaction times and higher inverse-efficiency scores compared with mixed-format blocks, underscoring the added cognitive load of repeated graphical encoding. Because FF conditions require uninterrupted visuospatial integration and continuous part–whole translation, they may place greater demands on cognitive resources than either symbolic repetition or switching conditions. One plausible explanation for these results is that handling two figure representations requires more mental work. Understanding a fraction shown as a figure involves not only recognizing shapes but also grasping the embedded mathematical meaning (Arcavi, 2003; Duval, 1995). People first notice how much of the shape is shaded versus unshaded, then translate that visual ratio into a mental sense of quantity. This step recruits number processing regions in the parietal cortex, consistent with the triple-code model of numerical cognition (Dehaene & Cohen, 1997). By contrast, symbol fractions (e.g., “3/4”) allow direct propositional encoding in the phonological loop (Paivio, 1991), facilitating automatic retrieval once conventions are well learned. Accordingly, FF conditions elicited larger N2 amplitudes, reflecting greater conflict monitoring and larger P3 amplitudes, reflecting intensified working-memory updating than SS conditions. Multiscale entropy further showed that FF trials maintained higher neural complexity in the central ROI between ~500 and 600 ms, suggesting sustained integrative processing when two figure representations are presented in succession. These findings extend Dual-Coding Theory (Paivio, 1990) by highlighting how individual differences in imagery versus verbal cognitive style (Richardson, 1977) may interact with educational history. University students, having years of practice with symbolic fractions in formal mathematics instruction, may automatize symbolic representations to the point that consecutive symbol conditions impose minimal cognitive load. In contrast, processing two figure trials consecutively prevents benefit from verbal automatization, forcing sustained visuospatial integration. Accordingly, SS conditions imposed the least processing demands, switch conditions (FS, SF) required a moderate level of effort, and FF conditions remained the most taxing. The findings require further examination to examination of the four conditions involving fraction figure and fraction symbol with different age participants in order to understand the dominance of representation system in relation to task switching and representation preference. One major type of task switching experimental design is to arrange representations into different stages in order to purely examine the effect of switch cost. The current study follows this experimental design specific to task switching paradigm. However, as the current study requires participants to grasp the meaning both fraction figure and fraction symbols into quantity sense. In this regard, translating the fraction figure or fraction symbols into quantity and using the quality to evaluate the fraction representation in the second stage might involve different cognitive efforts in remembering the fraction representation and translating the representation into quantity. In this regard, designing different experiments involving both fraction figure and fraction symbol (e.g., showing both presentations in the same stage) may provide further information to understand task switching paradigm with mathematics representation. In the present study, we deliberately adopted a block design for three main reasons. First, by presenting the same format repeatedly within each block, participants can quickly establish a stable task set, minimizing the momentary confusion that frequent format changes might induce (Alport et al., 1994); this stability, in turn, enhances the signal-to-noise ratio of the averaged ERPs (e.g., N2, P3), yielding more reliable component waveforms (Handy, 2005; Teplan, 2002). Second, blockwise presentation ensures that participants engage in the same cognitive strategy throughout, so that the neural dynamics we observe reflect the intrinsic processing demands of each format rather than the extra load imposed by random switching. Finally, for complex mathematical representation tasks, a predictable block sequence reduces anxiety or arousal fluctuations associated with uncertainty, allowing us to focus squarely on the core comparison of graphical versus symbolic processing. Although future work could introduce intermixed or cued-trial designs to probe instantaneous switch mechanisms, the block design used here is optimal for highlighting sustained processing effects across different fraction representation formats. It should also vary stimulus complexity and graphical formats and include participants from different age groups and levels of mathematical proficiency to determine whether the cross-representation switching advantage generalizes. 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Information & Authors Information Version history V1 Version 1 11 July 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Authors Affiliations Chen-Yu Yao 0009-0008-8818-935X National Tsing Hua University Department of Educational Psychology and Counseling View all articles by this author Hui-Yu Hsu [email protected] National Tsing Hua University Graduate Institute of Mathematics and Science Education View all articles by this author Zai-Fu Yao 0000-0002-9823-9110 National Tsing Hua University Department of Educational Psychology and Counseling View all articles by this author Tsu-Jen Ding National Tsing Hua University Department of Environmental and Cultural Resources View all articles by this author Metrics & Citations Metrics Article Usage 234 views 120 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Chen-Yu Yao, Hui-Yu Hsu, Zai-Fu Yao, et al. Electrophysiological Correlates of Cognitive Flexibility in Fraction Representation. 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