Bridging cognitive neuroscience and education: Insights from EEG recording during mathematical proof evaluation

B S


Rigor and intuition in mathematics education
Established in France in 1935, Nicolas Bourbaki-the collective pseudonym of an influential group of mathematicians-made a significant impact on mathematics and mathematics education.Central to the Bourbaki movement was its emphasis on symbolic formalisms rather than visuals and intuitions.This approach to doing mathematics spilled over to teaching mathematics; for example, Bourbaki directly influenced both the Moderne Mathématique movement that modernized mathematics education in France and the New Math movement in the United States [1].
Historically speaking, mathematics is thousands of years old; the modern emphasis on symbolic formalisms is a relatively recent phenomenon.Indeed, debates regarding formalisms versus intuition continue to the present day in the context of symbolic (i.e., formal) versus non-symbolic (intuitive/visual) arguments in mathematics education (e.g., [2,3]).For example, the mathematical notion of continuity-taught in analysis and calculus courses-can be intuitively visualized as a curve made by freely leading the hand, which is how the celebrated Swiss mathematician Euler described it in the 18th century.At present, high school students are instead taught a formal definition of continuity reliant on a formal definition of a limit. 2 There are valid reasons for this definition, but they are beyond the scope of this paper, couched in nineteenth-century analytic methods developed to avoid certain problems arising in more advanced mathematical settings.To clarify, this is not a criticism of symbolic formalisms, which are a necessary component of modern mathematics.Instead, a relevant pedagogical question is whether a highly technical, formal, and symbolic definition is the appropriate starting point for an introduction to the idea of continuity.After all, students might benefit more from starting with more intuitive visual representations of continuous functions and smooth change (see [4]).In the context of more elementary mathematics, we might wonder whether students benefit more from first learning fractions as mathematical abstractions or first learning fractions as representations of concrete objects (e.g., a quarter of a pie).More generally, the central pedagogical question is whether learning could be improved by prioritizing more abstract and symbolic or more intuitive and non-symbolic mathematical representations [5,6].While a good deal of empirical work has already investigated this question, there is, as of yet, no clear answer or consensus [3,7,8].
When a line of empirical research points in multiple directions, it may be useful to consider an alternative approach to the question.In this paper, we approach this problem through neuroimaging, that is, by asking what goes on in the brains of individuals who engage in either symbolic or non-symbolic mathematical reasoning.We can pursue this question because many mathematical ideas can be presented in both symbolic and non-symbolic formats, such as the parabola f(x)=x 2 , which allows us to present mathematically equivalent ideas as phenomenologically distinct stimuli.Specifically, we use electroencephalogram (EEG) recordings to explore the neural activity underlying mathematical reasoning involving symbolic and non-symbolic mathematical proofs-making use of algebraic proofs for symbolic reasoning and related geometric proofs for non-symbolic reasoning.A distinct advantage of using EEG recordings is that we can build on related, more basic previous research that posits neurally distinct systems for processing different formats in mathematics [9].
In contrast to previous work in this area, we investigate more complex mathematical reasoning in naturalistic situations that take longer (minutes rather than milliseconds) and involve more complex stimuli (mathematical proofs rather than arithmetic).Yet we are building on more basic research because it is reasonable to assume that complex mathematical reasoning also utilizes and even builds on more basic mathematical processes identified in earlier work.Thus, we additionally investigated whether the neural correlates of more basic symbolic and non-symbolic processing also appear when engaged in more advanced and authentic forms of mathematical reasoning.

Mathematical cognition
It is argued that we are born with an intuitive sense of quantity [10] that develops in the first two years of life [11,12].This non-symbolic type of reasoning helps learners explore the concept of numbers by estimating, comparing, and combining sets of visual stimuli such as dots or geometric forms [13].In contrast, symbolic skills are culturally acquired [13], developing approximately from the age of two and a half years onward [14].These acquired skills include the ability to represent numbers verbally (strings of words) and visually (strings of Arabic number symbols) but do not contain any semantic information about the meaning of the number words and symbols (see, e.g., the triple-code model [15,16]).
In educational settings, this cognitive distinction becomes important when students struggle with mathematics.The "access deficit hypothesis" assumes that having difficulties learning math can be attributed to inadequately developed symbolic skills [17], even alongside normally developed non-symbolic skills.One source of evidence supporting different mathematical development processes comes from longitudinal studies examining the relations between non-symbolic and symbolic skills and their relative effects on math achievement.Those studies show that progress towards mastery of both skills directly predicts math achievement [18,19].In addition, the effect of non-symbolic skills can be mediated by symbolic skills [20], and some researchers have argued that symbolic skills enhance non-symbolic skills during development, and vice versa [21].In short, mathematical cognition relies upon both symbolism and intuitive processes, which mutually influence each other.

Mathematics and brain activity
Mathematical processing activates a widespread neural network, including parietal regions [22].Parietal areas specifically recruited by diverse mathematical tasks are found within and around the horizontal intraparietal sulcus [23,24].Neuropsychological models posit that numerical quantity is expressed in a symbolic format bilaterally in cortical regions surrounding the intraparietal sulci [25].While the left parietal areas are involved in quantity estimations independent of format, some right parietal areas are more associated with non-symbolic than symbolic processing, suggesting that there are two different but overlapping networks for the two processing categories [9].Even though the parietal area findings are widely reported, the frontal cortex is also consistently activated in mathematical processing, even during simple tasks [9].Therefore, it has been suggested that "mathematics" is represented in a frontoparietal network.For example, although algebraic tasks and arithmetic tasks involve bilateral parietal brain regions, algebra relies more on the semantic network, while arithmetic relies more on the phonological and visuospatial networks [26].
Other cortical areas crucial for mathematical tasks are related to more generic cognitive processes like intrinsic motivation for learning and training [27], task difficulty [28], response execution [29], error processing [30], task switching [31], and emotional processing [32].In short, many different processes are involved when we engage with mathematics.
Moreover, different EEG oscillations are associated with various cognitive processes, and specific frequency bands play distinct roles in processing different mathematical formats.Beta-band activity (12-30 Hz) in parietal regions is known to be linked and associated with visuospatial processing and the integration of visual features [33,34].This suggests a potential role for beta oscillations in mathematical formats relying more on visuospatial skills, such as non-symbolic processing [35,36].Gamma oscillations (30-40 Hz) have been implicated in semantic and linguistic operations [37].The unique role of gamma oscillations in semantic and linguistic functions positions them as promising candidates for further exploration in the context of symbolic mathematics.Mathematical symbols and notations have been said to be a form of "language" created to represent mathematical concepts (e.g., algebraic expressions, equations, and mathematical symbols convey complex ideas in a concise and standardized way [38]).Gamma oscillations have also been associated with more complex algebraic problem solving [39] and symbolic representations [36].
In summary, previous research posits distinct systems for processing different formats in mathematics.The evidence suggests that these systems are behaviorally and neurologically distinct.Thus, it is reasonable to assume the existence of some neuronal dynamics that map to the behavioral distinction between symbolic (algebraic) and nonsymbolic (geometric) processing.A limitation of prior work is that it is generally done with simple visual stimuli involving millisecond presentation times.More complex mathematical reasoning takes longer and involves more complex stimuli.Yet, it is reasonable to assume that complex mathematical reasoning also utilizes, and possibly even builds on, more basic mathematical processes identified in the previously mentioned work.Thus, we also investigated whether the neural correlates of more basic symbolic and non-symbolic processing appear when reasoning with more advanced and authentic mathematical content.

Present study
The present study underscores the value of incorporating naturalistic data alongside experimental data, recognizing their complementary roles in advancing our understanding of complex cognitive processes of real-world phenomena [6,[40][41][42].In our study, we explore neural dynamics in the context of naturalistic and, thus, more complex stimuli: algebraic (which we use as a stand-in for symbolic) and geometric (i.e., non-symbolic) proofs.Specifically, building on previous research, we asked: Are mathematically complex tasks (algebraic and geometric proof solving) segregated through functional networks (activity in parietal and frontal regions) in different frequency ranges (lower gamma 30-40 Hz, and lower beta 12-17 Hz)?
The decision to incorporate electroencephalogram (EEG) recording in our repeated measures study stems from our interest in uncovering the neural processes involved in more naturalistic symbolic and nonsymbolic mathematical reasoning.Utilizing EEG allows us to capture real-time neural activity, providing a novel perspective on the mechanisms associated with these different activities.While our main goal is neuroscientific exploration, our study also has the potential for educational implications.Examining the neural signatures tied to symbolic and non-symbolic proofs offers insights into how the brain neurally processes these forms of reasoning.This information could offer insights into potential cognitive preferences, assisting in decisions about the sequential introduction of mathematical concepts.For example, if the neural data indicates a more efficient or distinct processing pattern for one format over the other, we may be justified in structuring learning sequences in a way that aligns effectively with such cognitive processes.Repeated measures studies utilizing advanced mathematical symbolic versus non-symbolic stimuli are scarce in research on mathematical cognition, particularly in EEG studies.
Drawing from historical perspectives [1] and contemporary discussions [2,3], the first hypothesis is (1) Students will find non-symbolic (geometric) proofs more accessible compared to symbolic (algebraic) proofs, leading to higher ratings on metrics such as "sufficient time," "better understanding," and "more engagement."This aligns with the broader notion that intuitive visual representations may serve as a more natural entry point into conceptual understanding [43].Conversely, given the prevalence of symbolic mathematics in traditional educational settings [5], the second hypothesis is (2) Students will perceive symbolic (algebraic) proofs as "more familiar" than non-symbolic (geometric) proofs.
In light of the existing literature on neuro-spatial distinctions in mathematical processing [9,13,15], we hypothesize that there will be distinctive patterns in neural oscillations, particularly in the gamma and beta bands, during the evaluation of algebraic and geometric proofs.Given the involvement of gamma oscillations in complex mathematical tasks [39] and the association of symbolic numerical processing with gamma frequencies over the frontocentral region [36], we hypothesized that (3) While solving symbolic (algebraic) proofs, students will show higher gamma oscillations in frontal electrodes than when solving related non-symbolic (geometric) proofs.Due to the potential sensitivity of beta oscillations to visual processing of stimuli, as well as enhanced involvement of visual processing in non-symbolic mathematics [34,35], and the association of non-symbolic numerical processing with beta frequencies around the parietal lobe [36], we hypothesize the following: (4) While solving symbolic (algebraic) proofs, students will show higher gamma oscillations in frontal electrodes than when solving related non-symbolic (geometric) proofs.

Participants
At a research university in the German-speaking part of Switzerland, 46 students were recruited from a range of majors, including mathematics, art, economics, and literature students.The experimental sample consisted of 42 participants (15 female, 27 male), a sample size similar to previous work [36,44].
The study was conducted under the Declaration of Helsinki and approved by the local Ethics Commission.All participants were righthanded, reported no hearing loss or history of neurological illnesses, and provided written informed consent.Table 1 shows the descriptive data of the sample.

Procedure and materials
To ensure comparability between the algebraic (symbolic) and geometric (non-symbolic) proofs, we identified mathematical proofs that could be matched as closely as possible with respect to length, complexity, and familiarity.Following an online pilot study with mathematics experts and novices, eight proofs were identified as acceptable for use (see supplementary materials for all proofs).The eight proofs varied from each other in number of slides (mean 7, with a range of 4-12 slides) and duration (mean 33 s, with a range of 13-68 s).In the first slide, the mathematical problem was stated.In the second slide the proof of the problem started, meaning that the symbolic or non-symbolic aspect of the proof started on the second slide.The 16 tasks (8 proofs x 2 formats) were programmed in MATLAB using the Psychophysics Toolbox extensions [45,46].Fig. 1 shows a schematic example of the task, while all tasks are in supplementary materials.
The participants were tested in a tutor-student situation in an educational setting.First, they completed a short version of the Berlin Intelligence Scale (to estimate numerical IQ -BIS [47]), provided the number of hours a week spent on math, and recorded an EEG baseline condition (eyes open) for 90 s.Next, they watched sixteen proofs: eight in algebraic and eight in geometric format.To counterbalance for order effects, half of the participants started with algebraic proofs, followed by their geometric counterparts; the other half started with geometric proofs, and then saw their algebraic counterparts.The students were asked to watch and make sense of the proofs.After each proof, they were asked about their agreement with the following statements on a 4-point Likert scale: A. I had enough time to follow the proof.B. I am familiar with the proof.C. I understood the proof.D. I engaged with the mathematical proof.
Participants answered using a four-button response box.The total length of the entire mathematical proof task was approximately 15 min.Similar to a tutor-student situation, the students were asked to explain the proofs in their own words to the tutor (experimenter) after they watched and rated all proofs.However, subsequent tutoring discussion is out of the scope of the current study and, therefore, not reported here.

Measures
Data were recorded using a mobile scalp EEG system to keep the experimental context as naturalistic as possible and not restrict students' movements (Wave-Guard™ EEG cap and eego™ mylab amplifier; ANT Neuro, b.v., Hengelo, the Netherlands).The EEG electrodes were placed according to the extended international 10-20 system [48].Eye movements were recorded via electrooculography (EOG), using four external electrodes placed below, above, and on the left side of the left eye as well as on the right side of the right eye.The ground electrode was placed behind the right ear.Electrode impedance was kept below 30 kΩ, and data were digitized at a rate of 2048 Hz.The experimental setup included triggers that marked the beginning of each proof and format.During the proofs, additional triggers were utilized to indicate changes in slides.All triggers were transmitted wirelessly to the computer using the Lab Streaming Layer (LSL).We focused on electrode clusters relevant to our hypotheses (see Fig. 2).

Analysis
Behavioral data.For the self-reflection questions, sum scores were formed for each scale (time, familiarity, understanding, and engagement).The raw score was analyzed for all other measures (reported weekly math hours and numerical IQ).Furthermore, paired t-tests were used to compare self-reflections between the symbolic proofs and their non-symbolic counterparts.
EEG Analysis.Neurophysiological data underwent preprocessing using the open-source toolbox MNE-Python [49].The recorder signals were filtered at 1 Hz for DC components (high-pass) and at 50 Hz for power-line contamination (notch) -both were finite impulse response, zero-phase filters with 0.0194 passband ripple and 53 dB stopband attenuation.Signals were downsampled to 512 Hz and visually inspected.A total of 20 channels with poor signal quality or missing signals across 6 participants were identified and interpolated using the spherical spline method [50].The method consists of projecting the recordings to a unit sphere and computing a mapping matrix from the projections of good channels to poor channels (in both space and time).The average of all electrodes served as the reference (common-average reference).Artifact Subspace Reconstruction (ASR) was employed as an automated and online technique in our process for handling artifacts in our multi-channel EEG recordings.ASR [51] is a component-based method designed to remove transient or large-amplitude artifacts effectively (e.g., eye movement artifacts).During our analysis, we utilized a Python implementation of the standard ASR algorithm (cutoff=10 Hz).Furthermore, we used independent component analysis (ICA) decomposition using the "fastica" method to remove physiological artifacts and project the data back without the rejected artifactstypically 2 to 12 components related to heartbeat were removed and projected back to the original channels.Drawing on previous research [36,52,53] and an fMRI meta-analysis [9], we selected a frontocentral cluster of electrodes, namely FC1, FC2, FC3, FC4, and FCz and a parietal cluster of electrodes, namely P1, P2, Pz, and POz electrodes, again based on previous research [36,54,55].
To measure the changes in EEG oscillation, a fast Fourier transform Fig. 2. Schematic example of the selected electrodes.
(FFT [56]) was used to estimate the absolute power spectrum density (μV2) in Beta (12)(13)(14)(15)(16)(17) and Gamma (30-40 Hz) bands.Each epoch reflected a proof comprising a presentation of 4-12 slides.The duration of slides varied between 13 and 68 s (M = 33 s).We observed that most existing EEG studies on math cognition have primarily concentrated on short epochs.To evaluate and potentially replicate these findings [36], we similarly explored shorter epochs during our math demonstrations.This approach allows us to assess whether findings obtained from short epochs milliseconds after stimulus onset (specifically 100-200 ms after stimulus onset [57]) differ from findings considering the entire duration of the proof.To maintain consistency and capture comparable effects, we specifically examined shorter epochs, precisely 100 ms after stimulus onset, within a defined interval of 100 ms.Thus, for these epochssimilar to the full proof -"time zero" was the onset of the second slide of each proof.
We used repeated measures multilevel models to predict power by Electrode clusters (frontal and parietal), Band (Gamma and Beta), Proof type (algebra and geometric), and Math Hours as predictors.These data analyses were performed using R 4.3.2statistics software [58].Standardized parameters and 95 % Confidence Intervals (CIs) and p-values were computed using a Wald t-distribution approximation.Further analyses can be found in supplementary materials.

Results
The answers to the self-reflection questions were compared in a within-subject manner using paired t-tests (see Fig. The correlation matrix shows (see Table 2) a weak correlation of numerical IQ with self-reflection ratings.Engagement with math hours was positively related to having enough time to follow the non-symbolic and symbolic proofs, and engagement with math hours was positively associated with understanding the symbolic proofs.
We employed a repeated measures multilevel model, estimated using REML and the Nelder-Mead optimizer, to predict EEG Power from predictors Electrode Location, Frequency Band, Proof Type

Discussion
This study explores the neural correlates of different proof formats (symbolic and non-symbolic) within a naturalistic classroom simulation.Our investigation encompasses students' self-reflections and examines potential differences in oscillatory dynamics [36] during the evaluation   of advanced mathematical proofs, extending prior research on behavioral and neuro-spatial distinctions in mathematical processing [9,13,15].Contrary to Hypothesis 1, students rated geometric proofs as more challenging to understand than their algebraic counterparts.This surprising finding seems to contradict our assumptions about the intuitive appeal of more visual, non-symbolic reasoning.However, in support of Hypothesis 2, algebraic proofs were rated as more familiar by students.
One explanation for why students rated algebraic (symbolic) versions as easier to understand and more familiar than geometric (nonsymbolic) proofs involves the different conceptual developmental paths toward understanding mathematics [43].At present, the Bourbaki model-symbolic rigor over visual intuition-is the standard approach to formal mathematics instruction.Thus, students might be more familiar and comfortable with the symbolic (algebraic) format because this is what they experienced throughout their education.There is some evidence for this explanation in the correlation matrix.Specifically, although experience increased the understanding of both proof formats, students with more mathematical experience were more likely to express a preference for algebraic (symbolic) proofs but not geometric (non-symbolic) proofs.However, it is important to note that our study does not directly assess students' prior exposure to different mathematical formats, and therefore, this assertion remains speculative.
The exploration of neural oscillations in processing geometric and algebraic proofs revealed distinctive patterns.Parietal electrodes exhibited greater activation than frontal ones in both the more naturalistic long presentation of proofs and the first 200 ms, supporting the claims regarding the frontoparietal network's involvement in mathematical processing [9,26].Notably, gamma oscillations, linked to complex mathematical tasks [39], displayed an interaction with Math Hours, indicating greater parietal activation in individuals who spend more time with mathematics.This result aligns with previous studies indicating that the manipulation of symbolic numbers involves Gamma-band neuronal activation patterns, possibly influenced by individuals' familiarity and fluency with learned and linguistically based numerical concepts [36,59,60].Contrary to expectations and previous research [36], beta band activation did not differ based on the proof format in the full proof.Analyzing the early stages of proof evaluation (100-200 ms), we observed a parietal activation interaction with non-symbolic (geometric) format, hinting at specific time windows for beta oscillation differences.This indicates that the analysis of shorter intervals can replicate previous lab findings, yet it also underscores the complexity of beta oscillations, potentially influenced by stimulus onset and specific non-symbolic stimuli.The differences in results could have been caused by the dissimilarities between our stimuli and those used previously [36].In contrast to our more naturalistic symbolic (algebraic) and non-symbolic (geometric) proofs, previous stimuli comprised basic Note.This participant has shown a large difference in the ratings between symbolic and non-symbolic proofs.Thus, we chose this participant as an example for visualization.Fig. 6.Topography showing spatial distribution of average absolute spectral power for all conditions averaged across all participants.
V. Gashaj et al. magnitudes presented either as Arabic numerals or as dot clusters.Another explanation might be the visual complexity of the proofs we presented in both formats.Although it did not rely on symbols, the presentation of the geometric proof was still geometrically complex.The parietal cortex is associated not only with numerical processing but also with visual attention and visual processing in general (see [61] for a review).Previous neurophysiological studies have shown that global continuous perception (Gestalt perception) is associated with stronger beta oscillations in parietal electrodes [62].Thus, our results could reflect Gestalt perception of the visually presented proofs.Furthermore, it is possible that the effect may not have been replicated in the main analysis due to potential limitations in sample size.The null result obtained could simply indicate that our study lacked sufficient power for a repeated measures multilevel model to detect this effect, assuming it exists within the population being studied.In supplementary materials, however, we have added a repeated measures ANOVA, which successfully replicated prior findings in the short time window, but not the full proof, and demonstrated sufficient statistical power.
Naturalistic designs in research offer distinct advantages and challenges compared to more controlled settings of experiments.One major benefit is enhanced ecological validity, as the data emerges from behaviors and responses in a more authentic scenario.This authenticity is crucial for understanding complex phenomena, such as mathematical reasoning in educational settings, where interactions are influenced by various factors not easily replicated in a lab.Naturalistic data can provide a more holistic view, capturing the complexity and dynamics that might be overlooked or intentionally controlled for in lab experiments.By employing more authentic tasks, the study aims to provide insights into the cognitive processes students employ when authentically engaging in symbolic (algebraic) and non-symbolic (geometric) reasoning.However, this approach comes with challenges, including reduced control over variables and difficulty in isolating specific factors of interest.The uncontrolled nature of the environment may introduce noise or confounding variables, impacting the precision and replicability of results.The richness and applicability of insights gained from more naturalistic data make it a valuable complement to more controlled experimental data.

Limitations
While the present study offers insights into the neural correlates of naturalistic mathematical reasoning, several limitations should be considered.First, we did not administer pretests specifically designed to assess participants' knowledge about symbolic and non-symbolic proofs nor did we assess prior knowledge of the specific domain in mathematics, relying instead on numerical IQ and self-reported weekly engagement with mathematics (in hours) as proxy measures.Second, the study utilized complex mathematical tasks, reflecting a spectrum of cognitive skills, potentially influencing participants' task processing and proof comprehension.Future investigations could benefit from incorporating baseline comparison conditions, involving stimuli with complexities similar to proofs but lacking mathematical meaning.Third, basing our sample size on previous EEG studies with a within-subjects design could have led to the study being underpowered, especially regarding the more complex analysis method (repeated measures multilevel model).However, repeated measures ANOVA (in supplementary materials) shows similar results patterns and demonstrated sufficient statistical power, as the sample size exceeded the threshold for detecting a large effect size (f = 0.4), estimated at N = 23 (using G*Power 3.1.9.2).In future studies, researchers should include a power analysis to estimate the proper sample size while they are designing the study.Last, our EEG setup focused on capturing lower-frequency gamma power (30-40 Hz) due to experimental constraints, limiting the exploration of high-frequency gamma power (above 50 Hz) associated with more complex mathematical processing.These limitations should be considered when interpreting the findings and designing future studies in the realm of naturalistic mathematical cognition.

Conclusions
Historically, mathematicians developed knowledge in different ways: some by relying on visualizations and imagery, while others relied more on a consistent buildup of symbolic, more formal mathematics from the beginning [63].We are not concerned with prioritizing either symbols or imagery-both are invaluable-but with characterizing the differences between these types of reasoning and, ultimately, whether mathematical ideas can be "presented in forms that are potentially meaningful to [students]" [64] (p.18).Understanding the neural processing of different mathematics formats could be valuable to both basic cognitive research and the practice of mathematics education.In mathematics education, the success of the formalization of mathematics became the rationalization for the formalization of mathematics education itself.However, in terms of cognition and learning, this may be the equivalent of putting the cart before the horse.
From a neuroimaging perspective, our knowledge of brain activity during engagement with various mathematical formats is very limited.Our study makes initial strides in this direction and, notably, finds no difference in mathematical reasoning considered over more naturalistic time scales (the mean presentation time of an argument was about 13 s).Thus, there is a need for more research with naturalistic stimuli if neuroscience is to inform education effectively.Basic research alone may not fully reflect the complexities of naturalistic research.
Behaviorally, our study yielded unexpected results: students consistently rated symbolic math proofs as more familiar and understandable than non-symbolic proofs.One plausible explanation is that the widespread use of symbolic examples in teaching has cultivated greater familiarity and understanding among students, thereby influencing their perceptions in favor of symbolic proofs.In other words, while nonsymbolic presentations, such as the geometric ones used in our study, may be more familiar to a theoretical student encountering mathematics for the first time, the reality is different.By the time they graduate from secondary school, virtually all students are exposed to more symbolic representations than non-symbolic ones.This interpretation underscores the complex interplay between pedagogical approaches and student comprehension in mathematics education, prompting further investigation into the effects of instructional methods on learning outcomes.
Our study also highlights some challenges of using EEG methods to investigate mathematical reasoning in ecologically valid settings.Although laboratory studies have provided insights into the neural mechanisms underlying mathematical cognition, their use of controlled and simplified stimuli may not adequately capture the complexity and variability of real-world mathematical problem-solving.The combination of naturalistic approaches with controlled experiments holds the potential to develop a more nuanced and thorough understanding of the neural basis of authentic mathematical reasoning.

Ethical statement
We assure that the following is fulfilled: 1) This material is the authors' own original work, which has not been previously published elsewhere.
2) The paper is not currently being considered for publication elsewhere.
3) The paper reflects the authors' own research and analysis in a truthful and complete manner.

Fig. 1 .
Fig. 1.Schematic example of the procedure Note.All proofs can be found in supplementary materials.
Note. * p < .05;** p < .01;Math in hours = self-reported weekly engagement with math, estimated in hours; NS = non-symbolic; S = symbolic; the p-values are not corrected for multiple testing.

Fig. 4 .
Fig. 4. Math Hours Interaction with proof type and frequency band for whole proof and 100-200 ms interval after stimulus onset.

Fig. 5 .
Fig. 5. Topography of one participant showing spatial distribution of average absolute spectral power for all conditions.Note.This participant has shown a large difference in the ratings between symbolic and non-symbolic proofs.Thus, we chose this participant as an example for visualization.
Financial disclosureThis work was supported by the UKRI Economic and Social Research Council [grant number ES/W002914/1].
Note:Numerical IQ = raw score on Berlin Intelligence Scale (BIS); Weekly Math Hours = self-reported weekly hours in which the participant is engaged in mathematical activities, such as math lessons or work-related mathematical tasks.