The first component postulated in Krutetskii’s model is obtaining mathematical information, which involves the ability to formally perceive mathematical material. This component most closely aligns with the notion that mathematically gifted children ‘see the world through mathematical eyes’. This component appears similar to two concepts studied in the domain of mathematical cognition: spontaneous focusing on numerosity (SFON) and attention to numbers (ATN). SFON is characterized by an individual’s tendency to recognize and focus on numerical features without explicit instruction to do so (McMullen et al., 2019). ATN refers to an automatic attentional process that supports the recognition and discrimination of numbers (Chan & Mazzocco, 2017). Although the definitions of both concepts are similar, the tasks typically used to measure each differ. SFON is primarily assessed in children using behavioral (e.g., imitating an action demonstrated by an experimenter like feeding a parrot a certain number of berries and observing whether the child feeds the same number of berries without being explicitly told to focus on numbers) and verbal tasks (e.g., children describe a picture and it is observed whether the child gives numerical responses such as there are four ducks in the pond; Li et al., 2025). The main task to assess ATN is a picture matching task in which one must select the picture that best matches a target based on visual features such as color, shape, location, and number. A match solely based on number is considered an indication of ATN. Therefore, ATN is sometimes argued to be a ‘pure’ laboratory-based indicator of SFON tendencies (Mazzocco et al., 2020). However, although differences in children’s SFON have been reliably associated with mathematics learning and performance (for a review, see McMullen et al., 2019), the relationship between ATN and mathematical achievement is less clear: some studies report a positive relationship while others do not (Mazzocco et al., 2020). In the present study, we aim to assess Krutetskii’s MCOM component of obtaining mathematical information by tasks rooted in both SFON and ATN.

The second proposed component, processing of mathematical information, comprises six subfacets (Krutetskii, 1976, p. 350): a) Logic: Engaging in logical thought regarding quantitative and spatial relationships, as well as number and letter symbols; this includes the ability to think in mathematical symbols. b) Generalization: Rapidly and broadly generalizing mathematical objects, relations, and operations. c) Curtailment: Shortening the process of mathematical reasoning and the corresponding system of operations. d) Flexibility: Switching between different mental operations during mathematical activities. e) Elegance of solutions: Striving for clarity, simplicity, economy, and rationality in solutions. f) Reversibility: Rapidly and freely reconstructing the direction of a mental process, switching from a direct to a reverse train of thought in mathematical reasoning.

Based on recent literature in mathematical cognition we consider flexibility to be the most important subfacet of this component. Krutetskii (1976) defined flexibility of thought as the ability to switch between mental operations, approach problem-solving from multiple perspectives, and break free from conventional thinking, allowing for adaptive problem-solving. He assessed it using tasks that allowed for multiple solutions, whereby a problem could be solved in different ways. This is consistent with current definitions of mathematical creativity and corresponding assessment tasks. Mathematical creativity has been defined as the process that results in unusual or novel as well as useful solutions to a given mathematical problem (Sipahi & Bahar, 2025; Sriraman et al., 2013) and is commonly measured using divergent thinking tasks that require individuals to generate multiple potential solutions to mathematical problems (Sipahi & Bahar, 2025; Suherman & Vidákovich, 2022). Similar to creativity in other domains, mathematical creativity is typically characterized by three core components: fluency, flexibility, and originality. Fluency refers to the ability to produce a large number of ideas, flexibility to the ability to generate ideas across different categories or solution strategies, and originality to the ability to generate unusual or unique ideas. Performance in mathematical creativity tasks has been consistently linked to mathematical achievement both in children and adolescents (e.g., Bicer et al., 2021) as well as in adults (e.g., Meier et al., 2021). Additionally, mathematical creativity has been considered the essence of mathematics (Elgrably & Leikin, 2021; Kozlowski et al., 2019) and is inherently connected with mathematical giftedness (Leikin, 2021; Singer, 2018; Sriraman et al., 2013). Against this background, in this study we assess processing of mathematical information with flexibility, fluency, and originality derived from two mathematical divergent thinking tasks.

While not explicitly included in Krutetskii’s model, motivational aspects are frequently acknowledged in his book as important for mathematical abilities and giftedness. By interviewing professional mathematicians, Krutetskii identified general personality traits that contribute to success in mathematics. These traits include a focused interest in mathematics, persistence, a drive for mathematical knowledge, and a need to engage in mathematics. Based on these interviews, Krutetskii concluded that a strong level of interest in mathematics is particularly important for high mathematical achievement. In a more recent model of mathematical giftedness, it was reported that professional mathematicians emphasized the link between MCOM and the intrinsic motivation to learn (Subotnik et al., 2009). These factors can drive students to attempt more challenging forms of mathematics. A related concept in current personality research is the need for cognition (NFC), defined as ’the intrinsic motivation to engage in and enjoy effortful analytic activity‘ (Cacioppo & Petty, 1982, p. 116). This concept has significant implications for academic performance, as a recent meta-analysis demonstrated a positive relationship between NFC and academic achievement (r = .20; Liu & Nesbit, 2024). Although the authors defined academic achievement very broadly (e.g., course grades, grade point averages) and did not examine specific subjects, such as mathematics, separately, they reported a significant correlation of r = .17 between NFC and academic achievement in the natural sciences. Given the significant role that motivational aspects play in achieving high levels in mathematics (e.g., Breit et al., 2025; Myers et al., 2017), we aim to capture motivational tendencies as part of MCOM by assessing interest, self-concept, and NFC.

To our knowledge, there is only one study that aimed to measure MCOM using tasks from current mathematical cognition research. Bakker et al. (2022) examined the characteristics of high achievers in mathematics by comparing 81 high achievers and 81 average achievers among 10- to 12-year-old children. The ATN task mentioned above was used as a measure of MCOM. Additionally, the researchers administered the adaptive number task. In this task, children had to reach a target number using four to five given numbers along with the four arithmetic operations (McMullen et al., 2016). They also performed the variability on demand task. In this task, children were required to solve complex multidigit arithmetic operations. They were then asked to explain their solution strategy and whether they knew other solution strategies (Torbeyns et al., 2009). Both of those tasks were intended to measure flexibility, with the latter also assessing the striving for elegance. Furthermore, NFC, spatial visualization ability, and working memory were tested. The results of this study demonstrated that high achievers outperformed the average achievers on all tasks except for the ATN task. The authors argued that the ATN task may not have been sensitive enough to detect subtle differences in children's MCOM. However, two important points should be noted here. First, the study viewed MCOM and flexibility as separate constructs. In our study, we will examine the structure of the MCOM and hypothesize that flexibility is nested within MCOM. Second, the authors did not include verbal and figural intelligence as control variables.

This is particularly crucial because a recent meta-analysis of the role of learner characteristics in mathematical achievement (Breit et al., 2025) showed that particularly the verbal and figural components of intelligence are highly important predictors of mathematical achievement. Verbal intelligence is important because language skills facilitate learning in mathematics, and mathematics requires receptive and productive linguistic comprehension. Figural intelligence, incorporating spatial abilities, is important because spatial representations are used as a cognitive strategy to understand and solve many mathematical problems, and mathematics comprises several spatial areas, such as geometry.

The main goal of the present study was to examine the structure of MCOM in a sample of adults with varying levels of mathematical achievement. The first research objective was to test whether individuals’ performance on selected tasks are positively correlated with one another and form an overarching factor that represents MCOM (see Figure 1). We operationalized obtaining mathematical information using a SFON task and an ATN task. Processing of mathematical information was assessed via flexibility, fluency, and originality in two well-established mathematical creativity tasks drawing on divergent thinking. Motivational tendencies were measured using questionnaires on interest, self-concept, and NFC in mathematics. The second research objective was to determine whether MCOM, as an overarching factor or as separate components, is significantly associated with mathematical achievement beyond verbal and figural intelligence. Mathematical achievement was assessed at both a more fundamental arithmetic and a more knowledge-based level.

The sample comprised 82 young adults (52 female, 30 male), aged between 18-37 years (M = 23.23, SD = 3.41). Of these, 63 individuals (77%) were university students. 50 (60%) participants studied psychology and the remainder were enrolled in various other subjects (e.g., mathematics, law, German philology). The remaining 19 participants were employed across diverse industries, and roughly half had completed tertiary education. All participants were native German speakers and reported no learning disabilities such as dyslexia or dyscalculia. Two participants reported a diagnosis of Attention-Deficit-Hyperactivity-Disorder; however, as their performance on most tasks fell within the normal range, they were not excluded. For an exploratory factor analysis with eight variables, a ratio of cases per variable between 4 to 1 and 20 to 1 is recommended, resulting in 32 to 160 participants for our study (Lorenzo-Seva & Ferrando, 2024). A post-hoc sensitivity analysis using G-power (Faul et al., 2007) indicated that for single regression coefficients in a linear multiple regression with six predictors, effect sizes of f² = .10 and larger can be detected with a power of .80 using the present sample size. Recruitment involved sending an e-mail to all students at the local university and personal outreach to friends and acquaintances by the researchers. Written informed consent was obtained from all participants, and the local ethics committee approved the study.

There are several potential reasons why, contrary to our initial predictions, we did not find one general factor of MCOM. One possible explanation relates to the way we interpreted the theoretical model. We chose to see MCOM as a general, overarching factor. But there are also interpretations of Krutetskii's work that could lead to different conclusions. Krutetskii outlined the structure of mathematical abilities by identifying four main aspects: obtaining mathematical information’ as the first aspect, ‘processing mathematical information’ as the second, ‘retaining mathematical information’ as the third, and a ‘general synthetic component: mathematical cast of mind’. This framework allows for the interpretation of MCOM as an independent factor rather than a general overarching factor including the three other components. This alternative interpretation has been adopted by Bakker et al. (2022), who operationalized MCOM solely through the ATN task. Krutetskii (1976, p. 351), however, also emphasized that ‘these components are closely interrelated, influencing one another and forming in their aggregate a single integral system, a distinctive syndrome of mathematical giftedness, the mathematical cast of mind’. This assertion supports our decision to consider MCOM as a general factor, as it highlights the interconnectedness of the components within the structure of giftedness.

Another reason for the lack of an overarching factor of MCOM could be that we did not assess all components of Krutetskii's model. As noted above, another component in Krutetskii’s model is ‘retention of mathematical information‘, which occurs after the information has been obtained and processed. However, retaining mathematical information is not merely a matter of memorizing symbols and numbers, rather it involves a generalized memory for mathematical relations, types of properties, argument schemes, proofs, and methods of problem solving. For now, this aspect has not been included because there is very little empirical evidence that the concept of retaining mathematical information, as defined by Krutetskii, is important for achievement in mathematics. Most studies on memory in mathematics focus on the conceptually different working memory or short-term memory (see Breit et al., 2025, for a review). Furthermore, it is debatable whether how mathematical information is stored can be considered a predictor of achievement in mathematics or whether it should be considered a criterion of mathematical knowledge. Nevertheless, future studies should attempt to operationalize this component and include it in the full MCOM model.

Our operationalization of the first component of MCOM may provide another explanation for our findings. Compared to the other two factors, the factor obtaining mathematical information performed rather badly. First, SFON and ATN had low sampling adequacy (MSA) values (.36 and .39, respectively), whereas all other variables had MSA values above .50. Typically, variables with MSA values lower than .50 are excluded from factor analysis, as low values may indicate that the variables do not represent the same construct (Lorenzo-Seva & Ferrando, 2021). However, because one of our primary aims was to examine which variables in our model capture shared variance, we retained SFON and ATN in the analysis (results excluding SFON and ATN can be found in the OSF project). Second, this concern is further reflected in the non-significant correlation between SFON and ATN (r = .20) and their relatively low factor loadings. Third, neither SFON nor ATN correlated significantly with any of the other variables (see Table A1 in the Appendix). The factor representing the ability to obtain mathematical information correlated negatively, albeit non-significant, with motivational aspects (r = -.06) and processing of mathematical information (r = -.13). These points should be considered when interpreting findings related to this factor. Additionally, our measurements of obtaining mathematical information may have captured only a limited aspect of a potentially broader spectrum. While the literature on spontaneous focusing tendencies originally centered on numerosity, more recent research has extended this to include a focus on numerical order and quantitative relationships (Harju et al., 2024; McMullen et al., 2019). Furthermore, the literature on mathematical experts highlights that recognizing patterns in various contexts, not just in terms of numerosity, is an essential component of high achievement (Mehta et al., 2016; Meier et al., 2023). Therefore, future studies should consider expanding the concept of obtaining mathematical information beyond numerosity to include other aspects of mathematics, such as the recognition of patterns, which are often describe as fundamental for mathematical thinking and learning (Fyfe et al., 2017; Vogel, 2005).

Furthermore, future studies should aim to select tasks that theoretically cover all subcomponents and that use different assessment modes within each subcomponent, as this is a limitation of the current study. SFON and ATN are both forced-choice tasks, while flexibility, fluency, and originality are based on open-ended tasks, and motivational tendencies are based on Likert rating scales. Therefore, tasks within each factor share a common method, while the methods vary systematically across factors. Consequently, the factors may be construed not only because they reflect a shared theoretical or cognitive construct, but also because they share common assessment methods. However, this common method bias can lead to biased estimates of the reliability and validity of latent variables. Unfortunately, post hoc statistical methods to control for common method bias are not yet promising (Podsakoff et al., 2024). This highlights the importance of taking common method bias into account when selecting tasks for future studies.

Finally, as this study is the first attempt to measure MCOM in adults, we focused on a general student population. It should be noted, however, that while a few participants were from STEM fields, the majority were Psychology students. Additionally, relative to the overall adult population, our sample was highly educated and likely underrepresented the lowest-performing and the highest-performing individuals of mathematical ability, given that participants did not have a math-focused background. Future studies should aim to include larger and more diverse samples. Additionally, since the concept of MCOM was designed to explain high achievement in mathematics, our originally proposed model may hold true if restricted to gifted individuals. Thus, future research should investigate MCOM using different populations. This could include individuals from a broader range of backgrounds, from those with dyscalculia to gifted students. Furthermore, many Fields Medal laureates, who are considered the mathematical equivalent of Nobel Prize winners, describe mathematics as omnipresent and a constant, intuitive facet of their lives (Mehta et al., 2016). This could be interpreted as seeing the world through mathematical eyes and having a MCOM. Therefore, an empirical comparison between mathematics experts and experts in non-mathematical subjects along dimensions of MCOM would be desirable.

Based on the results from the EFA we examined how the three factors, obtaining mathematical information, processing of mathematical information, and motivational tendencies, predicted arithmetic fluency and mathematical achievement, with verbal and figural intelligence included as control variable. Our analysis revealed a differentiated picture for arithmetic fluency and mathematical achievement. Overall, the model explained 20% of the variance in arithmetic fluency and 57% of the variance in mathematical achievement. This is consistent with the MCOM model, which aims to explain mathematical giftedness—a construct characterized by high mathematical achievement but not necessarily high arithmetic fluency (Sipahi & Bahar, 2024). Additionally, intelligence is known to be more strongly related to more complex mathematical tasks (Peng et al., 2019), which aligns with the observed pattern of explained variance, as the items used in our mathematical achievement questionnaire are more complex compared to the arithmetic fluency task. The MCOM factors accounted for an additional 20% of the variance in mathematical achievement, but only 9% in arithmetic fluency, further supporting these theoretical considerations. Notably, only motivational tendencies significantly predicted both arithmetic fluency and mathematical achievement, whereas the factors representing the obtaining and processing of mathematical information did not.

Contrary to our expectations, the first factor, obtaining mathematical information could not predict either arithmetic fluency or mathematical achievement. This is regardless whether one uses the latent factor, or the two manifest variables SFON and ATN (this can be also found in the OSF project). This contrasts with findings of both Ben-Shachar et al. (2020) and Mazzocco et al. (2020). Ben-Shachar and colleagues (2020) examined data from 77 undergraduate students and assessed participants' mathematical performance by having them report their quantitative reasoning scores from their college entrance exam. This exam measures an individual's ability to use numbers and mathematical knowledge to solve quantitative problems presented in both verbal and visual (e.g., tables, graphs) formats. A logistic regression analysis demonstrated that individuals with higher mathematical ability were more likely to focus on the numerosity of the stimuli in the SFON task. Mazzocco et al. (2020) examined the relationship between the ATN task and measures of math performance in a sample of 90 undergraduate students. Participants completed an arithmetic fluency task, and the authors collected data from a standardized college entrance exam measuring mathematical problem-solving. While they found a positive association with mathematical achievement (r = .25), no significant correlation emerged for arithmetic skills (r = -.03). In contrast, Bakker et al. (2022) compared high-achieving children in mathematics with average achievers in the ATN task and found no difference in performance between both groups. Additionally, the ATN was not correlated with performance in two other mathematics-specific task. However, it is important to consider that when comparing our findings to those from children and adolescents, SFON and ATN may be subject to developmental changes. While a meta-analysis examining the relationship between SFON and mathematical performance (Li et al., 2025) found this association to be stronger in older children and adolescents, no studies involving adults were included. Thus, it is challenging to draw firm connections to our work.

Surprisingly, we did not find a significant positive relationship between the second factor, processing of mathematical information, where we used divergent thinking tasks as a proxy, and mathematical achievement. A substantial body of evidence demonstrates that mathematical creativity is linked to mathematical achievement in both children and adults. A meta-analysis by Bicer et al. (2021) reported a positive correlation between mathematical creativity and mathematical achievement in children and adolescents, studies involving adults have shown similar results (e.g., Meier et al., 2021). Furthermore, creativity is often intrinsically linked to giftedness in both domain-general and domain-specific models of giftedness (Kaufman & Sternberg, 2019; Sriraman et al., 2013). Empirical evidence has also shown that mathematically gifted children often outperform their typically developing peers in mathematical creativity tasks (Kroesbergen & Schoevers, 2017). Despite finding small effects (β = .19 and .14), they were not significant in our multiple regressions. While our regression analysis included only the combined creativity score (flexibility, fluency, and originality), flexibility had the strongest factor loading. This pattern, in which flexibility shows stronger loadings on an overarching factor of mathematical creativity compared to fluency and originality, has also been reported by others (e.g., Kattou et al., 2013). It aligns with the research of Bakker et al. (2022), which demonstrated that high-achieving children scored higher on tasks assessing flexibility in mathematical processing compared to average-achieving children. Furthermore, this finding aligns with our expectations regarding the significance of flexibility, as supported by literature emphasizing the importance of mathematical flexibility for mathematics (Hickendorff et al., 2022; Newton et al., 2020). Additionally, when examining the full correlation matrix (see Table A1 in the Appendix), flexibility correlated more strongly with mathematical achievement (r = .42) compared to fluency (r = .27) and originality (r = .21).

While we have chosen to consider flexibility, fluency and originality, assessed with divergent thinking tasks as one aspect of how individuals process mathematical information, Krutetskii (1976) proposed several other aspects. Among these, he discusses the reversibility of mental processes in mathematical reasoning, which allows individuals to switch from a direct to a reverse train of thought, an ability that, in our opinion, also requires flexibility. Another aspect worth investigating in future studies is the elegance of solutions, characterized by the pursuit of clarity, simplicity, economy, and rationality of solutions. This quality has not only been identified as a characteristic of high-achieving children (Bakker et al., 2022), but professional mathematicians also cite aesthetic aspirations and striving for mathematical beauty as driving force for their work (Mehta et al., 2016; Sa et al., 2024). Thus, future studies on MCOM should broaden the ways in which the processing of mathematical information is assessed.

The third factor, motivational tendencies, was the strongest predictor of arithmetic fluency and mathematical achievement, even when verbal and figural intelligence and the two other factors of the MCOM were included. Krutetskii (1976) emphasized the importance of personality factors, such as motivation, attitudes, and interest, for mathematical performance. These aspects were addressed in our questionnaires on interest in mathematics, mathematical self-concept, and domain-specific need for cognition. Our findings are consistent with empirical studies that link motivational aspects and mathematical achievement (Schukajlow et al., 2023; Strobel et al., 2024). Breit et al. (2025) conducted an extensive review of meta-analyses that summarized the extent to which various learner characteristics influence mathematical achievement. In addition to knowledge, skills, and aptitude, the authors examined personality traits, such as non-cognitive personality traits, attitudes, and emotional and motivational variables. Mathematical self-concept had the strongest effect size of all personality variables (r = .43), even having a larger effect on mathematical achievement than many cognitive abilities. This strong influence is assumed to be caused by the reciprocal relationship between achievement and self-concept. High levels of achievement foster a more positive self-concept, which encourages greater engagement with the subject and results in higher achievement. But also enjoyment of mathematics (r = .31) and attitudes toward mathematics (r = .12) were positively associated with mathematical achievement (Breit et al., 2025). As shown in Table A1 in the Appendix, in our study self-concept, interest, and NFC all showed strong relationships with arithmetic fluency (r = .31 – .38) and mathematical achievement (r = .52 – .57).

It is worth mentioning that among the control variables, figural intelligence was a significant predictor of both arithmetic fluency and mathematical achievement. This link has been consistently reported throughout the literature (e.g., Breit et al., 2025), probably reflecting the role of visual-spatial skills in mathematical problem-solving and the ability to manipulate abstract concepts (Atit et al., 2022). Verbal intelligence also contributed to mathematical achievement, likely due to the task demands. Unlike the arithmetic fluency task, the mathematical achievement task involved a variety of mathematical problems, many of which were presented in a verbal format. Previous studies have also demonstrated the role of verbal intelligence in higher mathematical achievement (e.g., Berkowitz & Stern, 2018). However, it should be noted that, due to time constraints, we only used one task to assess verbal and figural intelligence instead of the three tasks stated in the manual. Therefore, our scores only provide a rough estimate of verbal and figural intelligence, and future studies should aim to assess verbal and figural intelligence more comprehensively.

In conclusion, while this study does not provide evidence for a general MCOM factor in our adult sample, it underscores the significance of factors such as motivation and cognitive flexibility, in contributing to mathematical achievement beyond verbal and figural intelligence. Further research is needed to explore whether these findings hold true in samples of high-achievers and gifted individuals, as proposed by Krutetskii. Nevertheless, the current work provides a solid starting point to empirically test the model of MCOM and to critically evaluate whether it is a reliable construct that can predict high mathematical achievement or merely a term used to describe mathematically gifted individuals.