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Understanding the concept of equivalent fractions is not only about memory of a fact or application of a procedure, but more importantly about making connections between different representations (Lesh et al., 1983). This study aims to investigate how students detect equivalence in fractions shown in two representation systems: figure and symbol. We designed a test consisting of 180 problems, which requires students to verify whether two given fractions shown by either graph or symbol representations were equivalent. The test was implemented using the E-prime software, and students' response accuracy (Acc) and reaction time to the correct response (RTc) were adopted as indicators for problems' cognitive complexity. Brainwave activities during problem-solving processes were recorded and analyzed using event-related potential (ERP) techniques. The participants were 30 university students. Analysis of behavioral data indicated that students performed more accurately and had a shorter response time on inequivalent fractions than on equivalent ones. This finding suggests lower cognitive complexity when verifying inequivalence in fractions than equivalnece ones. Fractions shown in symbol representation format had higher accuracy and shorter reaction time than those shown in figure representation. The result indicates that verfication by symbol representations is easier than by figural representations. For brain activity, we particulary focused on how students detected the inequivalence by examining the N2 and N400 components. The N2 component refers to error awareness, while the N400 component is involved in error recognition after semantic processing. The study found both N2 and N400 components in students' brainwaves. However, significant differences was found in the N400 but not the N2 when comparing between inequivalent fractions and equivalent ones . The finding indicates that no matter inequilavence or equivalence require students similar mental effort in being aware of the error. However, verification on inequivalent fractions ask for higher mental effort in the semantic processing for error recognition than that on equivalent fractions. Additionally, symbol representations elicited a larger N400 than figure representations, indicating greater semantic processing when verifying by symbols than by figures. [ABSTRACT FROM AUTHOR]

Spatial ability is essential to development of different subject domain areas including geometry and cartography. Dissociating spatial ability into distinct sub-constructs is one of the goals in psychological research. In this study, we hypothesize that geometry and cartography, especially related to self-localization by map, require different types of spatialability. That is, geometry problem-solving often has students to perform mental-rotation actions, while self-localization necessitates the use of perspective-taking ability. To examine the hypothesis, the ERP (Event-Related Potential) techniques were adopted. High school students were randomly assigned to geometry group (37 students) and cartography group (39 students). We designed two tests. One is geometry test that asks to perform mental rotation based on mental images of triangles created by students. Self-localization test needs one to perform perspective-taking ability to identify the relationship between map symbols and real-world landmarks. The ERP brainwaves were analysed based on Gunia et al. (2021) as they indicated that spatial ability often occurs in parietal-occipital brain areas. The analysis showed that both geometry group and cartography group all had P300 component in the POz channel located between parietal and occipital brain areas. This is aligned with what Gunia et al. have found in their study. In particular, we found that RRN (rotation-related negativity) component occurred in geometry group but not in cartography group, which confirmed geometry requires the performance of mental rotation actions. We further checked Cz, C3, and C4 channels in central brain area as literature indicated that the three channels are crucial to check if one performs perspective-taking ability. In addition, the negativity peaks in C3 and C4 allow to understand ones' motor intension of body-movements oriented to either right side (C3) or left side (C4). Our analysis demonstrated that cartography group had significant negativity peak in Cz but not geometry group. C3 and C4 also only occurred in cartography group but not geometry group. The ERP analysis results confirmed that geometry and cartography require different types of spatial ability. Moreover, the ERP techniques can be a useful tool in distinguishing the two types of spatial ability. [ABSTRACT FROM AUTHOR]