Introduction

Numbers serve as essential instruments for humans to grasp and interpret external phenomena, with their processing influenced by both their representational forms and intrinsic semantics. Studies on numerical cognition have demonstrated that numbers significantly influence human thought and behavior. Various forms of numerical representation—be it non-symbolic, symbolic, or culturally specific symbolic forms—profoundly affect individuals’ cognitive processing of numerical information1,2. Non-symbolic numbers are typically represented through dot arrays, sequential stimuli (such as sounds), or even body parts (such as using two fingers to signify '2')2. Non-symbolic numerical perception refer to the ability to intuitively comprehend and process numerical information without depending on specific symbols or textual representations 3. Symbolic numbers assume diverse representational forms across different cultural contexts. Common examples include Arabic numerals like '2' and textual representations such as '二' or ‘two’ in various languages. These symbolic forms encompass dozens of variations4. Symbolic numerical representation entails grasping the relationship between numbers and their corresponding symbols, manifesting as the mapping between numerical figures and the quantities they signify5. Across different forms of numerical representation, individuals demonstrate the capacity to abstract specific quantities, evidenced in the processes of quantity estimation and quantity comparison. Quantity estimation involves determining the number of discrete objects, such as assessing the number of dots in an array or the number of distinct tones heard simultaneously6. The process of quantity estimation is relatively swift, enabling individuals to rapidly assess potential threats or items within complex environments. Quantity comparison, an extension of quantity estimation, refers to the ability to compare the number of objects in different sets without explicit counting. For example, when observing two clusters of fruit, individuals can swiftly determine which cluster contains more and opt to pick from it first.

Besides the spatial representations, world semantics plays a pivotal role in numerical cognition, influencing how numerical attributes (e.g., cardinality and ordinality) are processed across different representational formats. According to the Semantic Consistency Model7, world semantics refers to the integration of real-world knowledge with numerical reasoning, shaping how individuals interpret numerical relationships and execute mathematical operations8,9. World semantics is characterized by the solver’s non-mathematical, daily-life knowledge about the elements of the problem statement as well as the relations between them9. For example, world semantics may include knowledge that flowers can be put into vases, that there is a co-hyponym relation between oranges and apples, or that to go from the first to the third floor of a building one must pass by the second floor first. This particular understanding of social and cultural contexts, as well as environmental situations, not only shapes individuals’ cognitive approaches and choices of problem-solving strategies but also plays a crucial role in effectively tackling real-world issues. For example, when solving a problem such as "There are 5 birds and 3 worms, how many more birds are there than worms?", the way the question is framed affects how solvers approach the solution. By contrast, a numerically equivalent problem, "How many birds won’t get a worm?", is generally easier for children to solve, as it is more intuitively linked to real-world experiences10. In Chinese, such calculation problems are considered word problems. The core of these problems lies in creating everyday life scenarios that require computation. By incorporating these scenarios, numerical calculations are imbued with world semantics. Moreover, scenarios such as balloon counting or elevator weight limits may more effectively engage fundamental arithmetic skills than contexts involving building heights or study durations, which require different cognitive processing. These distinct contexts evoke two fundamentally different concepts of numerical situations11. In the process of solving mathematical word problems, the scenarios are derived from everyday life, reflecting the world semantics of numbers.

The specific text of word problems triggers different world semantics due to the variation in contexts, leading to two fundamentally different numerical attributes: cardinality and ordinality. The relationship between cardinal and ordinal numbers pertains to the core concept of numeracy. Cardinality, by definition, describes the numerical property of a set, answering the question 'how many?' and is determined through counting. Ordinality, in contrast, defines an object’s relative position within a set, answering questions such as 'which one?' or 'in what order?'8. According to the concept of world semantics proposed by Gros in 2017, problems involving collection, price, and weight can be encoded as cardinal number issues, primarily reflecting set operations. Conversely, problems involving duration, distance, and floors are conceptualized as ordinal number tasks12. This classification of numerical attributes primarily depends on the context in which the numbers are used13. Research into numerical cognition cannot be divorced from the understanding and application of cardinal and ordinal issues. Thus, the world semantics embedded in applied problem contexts shape how cardinality and ordinality are attributed to numbers. These semantic influences interact with numerical representation methods, affecting individuals’ cognitive strategies and decision-making in mathematical problem-solving. Real-life mathematical scenarios can be constructed through verbal expressions to explore how world semantics of numbers affect numerical cognitive processing8. This enhances our understanding of individuals’ numerical sense and the essence of numerical cognition in real-world contexts.

Building on this foundation, an important question is whether numbers, in both non-symbolic and symbolic forms, share similar world semantics. Firstly, non-symbolic numerical representations, such as dot arrays or sequences, provide an intuitive perception of quantity2. This form of representation transcends cultural limitations and emphasizes the intrinsic value of numbers, allowing individuals to directly perceive quantity. Secondly, symbolic representation of numbers is an abstract cognitive process that has been enriched over time through human language and cultural contexts. In symbolic representation, numbers can be categorized as either cardinal or ordinal, as seen in English with terms like “two” and "second." In the Chinese context, the distinction is more explicit, using "个" (ge) for cardinal and "第" (di) for ordinal representations, such as "三个" (three) and "第三" (third). In symbolic representation, world semantics are encoded through specific morphemes, such as "第" (ordinal) and "个" (cardinal) in Chinese. For instance, the morpheme "个" explicitly conveys cardinality. Finally, word problems, as a unique form, require solvers to read and comprehend problem statements, process numerical values, and execute computational algorithms to arrive at correct answers. This understanding necessitates reliance on world semantics to ensure the correct integration of numerical attributes with problem contexts. In word problem contexts, world semantics more significantly influence the distinction of numerical attributes than mere reliance on morphemes, requiring solvers to discern cardinal or ordinal properties. Therefore, the role of world semantics becomes progressively more complex across non-symbolic, symbolic, and word problem contexts.

The cognition of quantity in individuals is closely associated with spatial information, a phenomenon known as the spatial-numerical associations of response codes (SNARC) effect. The SNARC effect is characterized by the tendency for smaller numbers (such as 1 or 2) to be more easily associated with responses on the left side of the body, while larger numbers (such as 8 or 9) are more readily associated with responses on the right. Two predominant theories explain the origin of the SNARC effect: the Mental Number Line theory and the working memory account 14. The Mental Number Line theory posits that numbers possess spatial properties within the human brain, with smaller and larger numbers arranged sequentially from left to right in a mental spatial representation15. Additionally, research in "near-far" dimension experiments has found that smaller numbers are associated with the “near” dimension, while larger numbers are linked to the “far” dimension, regardless of movement direction 16. The working memory theory suggests that numerical position within a working memory sequence influences response speed, as earlier-positioned items are associated with left-space responses, while later-positioned items are linked to right-space responses. Based on the research by Van Dijck and Fias, the SNARC effect may not originate from inherent spatial representations of numbers in long-term memory, but rather is closely related to the ordinal encoding of information in working memory14. Items in working memory are spatially encoded based on their position within a sequence, with items at the beginning associated with the left space and those at the end associated with the right space14. This dynamic encoding mechanism can explain the flexibility and task dependency of the SNARC effect14. Additionally, Van Dijck and Fias further noted that the flexibility of the SNARC effect—such as range dependency and task dependency—and its attenuation under high working memory load support the critical role of working memory in spatial-numerical associations, rather than solely relying on the mental number line representation in long-term memory14. However, in their experiment on the "near-far" dimension, Santens and Gevers discovered that small numbers are consistently associated with the “near” dimension and large numbers with the “far” dimension, irrespective of movement direction16.

Empirical studies has demonstrated that the SNARC effect is prevalent across various forms of numbers and emerges as early as childhood 17,18,19. Adolescents’ mathematical learning process is inextricably linked to acquiring fundamental numerical skills. From the perspective of the Mental Number Line theory, these effects associated with numerical magnitude may relate to the spatial representation of numbers. However, working memory theory tends to explain them in terms of working memory resources and spatial positioning. Beyond theoretical considerations, task demands and numerical attributes can influence the SNARC effect. According to Prpic, task requirements significantly affect the role of ordinal and magnitude information in the SNARC effect20. In direct tasks, ordinal information predominates, whereas in indirect tasks, magnitude information plays a more significant role. This suggests that the SNARC effect may result from the combined influence of two independent mechanisms: the Ordinal Representation Mechanism (ORM) and the Magnitude Representation Mechanism (MRM). These mechanisms exhibit varying dominance depending on the task requirements20. The quantitative characteristics of numbers are first mapped to different positions on the mental number line based on their magnitude. The interaction between numerical directionality and response directionality ultimately leads to the SNARC effect21. However, no experiment has entirely ruled out the possibility of the SNARC effect being influenced by numerical order properties. Previous studies using English letters, Greek letters, or exam grades, representing only ordinal properties, have also elicited the SNARC effect 22,23. Additionally, research by Gevers demonstrated that ordinal information, such as months and letters, is spatially encoded in a manner similar to numbers, exhibiting the SNARC effect24. Items at the beginning of a sequence tend to be associated with the left space, while those at the end are linked to the right space. Notably, this spatial encoding effect persists even when it is irrelevant to the task, indicating that the spatial representation of ordinal information can be automatically activated24. Abrahamse and van Dijck proposed that the understanding of cardinality, ordinality, and magnitude is constructed within a low-dimensional ranking space based on a spatial framework25. In this model, ordinal and magnitude information derive their meaning through relative position encoding within goal-directed cognitive processes, while cardinal information relies more on direct labeling mechanisms such as counting or estimation25. Both ordinality and cardinality may contribute to the SNARC effect, but their mechanisms and influence can vary depending on task demands. Ordinal relationships can automatically activate spatial representations in both task-relevant and task-irrelevant contexts, whereas cardinal relationships significantly influence the SNARC effect primarily when the task requires direct processing of numerical magnitude14,24,26,27,28.

In the context of world semantics, the question arises as to how different forms of numerical representation affect the SNARC effect in numerical cognitive processing. Studies by Lengyel and Krajcsi (2024) and Prpic et al. (2023) compared the SNARC effect between symbolic and non-symbolic numbers29,30. Their findings revealed that symbolic numbers exhibited a significant SNARC effect, whereas non-symbolic numbers did not29,30. These findings suggest that the SNARC effect is predominantly linked to the spatial representation of symbolic numbers. In contrast, the spatial representation of non-symbolic numbers may follow a more complex process or involve distinct cognitive mechanisms. Similarly, the axial representation of ordinal numbers, akin to the mental number line, may lead to a more pronounced SNARC effect. Within the framework of the semantic consistency model, this study explores how the quantitative information of two numerical attributes impacts the SNARC effect under varying intensities of world semantic conditions in different forms of representation. Drawing on world semantics theory and the Mental Number Line theory, we propose the following hypotheses: (1) When world semantics are absent, the SNARC effect should be observable for both numerical attributes; (2) When world semantics are present, ordinal numbers, due to their axial representation consistent with the mental number line, should elicit a stronger SNARC effect than cardinal numbers.

Experiment 1: the impact of numerical attributes on the SNARC effect in non-symbolic representations

This experiment employs a numerical magnitude judgment task to observe whether the intensity of the SNARC effect differs between cardinal and ordinal numbers in a non-symbolic format.

Participants

Using G*Power 3.1.9.7, we calculated the required sample size under the conditions of Power (1-β) = 0.80 and f = 0.25. At a significance level α = 0.05, the total sample size needed is 48. We recruited 66 university students (14 males, 52 females, aged 18–22), all of whom were right-handed and had normal or corrected-to-normal vision. In this study, all participants provided written informed consent. Before the commencement of the research, participants were thoroughly informed about the study’s purpose, procedures, potential risks and benefits, and their right to withdraw at any time. The research team ensured that all participants fully understood this information and voluntarily agreed to participate. This study adhered to relevant ethical standards and guidelines. And this study received ethical approval from the Ethics Committee of Scientific Research Projects at the School of Psychology, Northwest Normal University, under approval number 2022128.

Materials

This experiment adopts a numerical magnitude judgment task31, dot arrays are a commonly used experimental material to investigate how humans associate numerical information with spatial positions32. Cardinal numbers are represented by dot-arrays, while ordinal numbers are indicated by dot-arrows, which points to a dot to represent its order33. To control for the influence of the approximate number system and the exact number system, we used numbers from the approximate number system (11–14, 16–19). Adriano suggests that when using dot arrays as experimental materials, it is essential to control irrelevant variables such as convex hull, total surface area, density, item size, and total circumference32. Accordingly, we regulated the diameter of each dot, the length of the array, and the circumference of the dots in the images.

To rule out the potential influence of participants’ judgment strategies on the SNARC effect, we conducted a supplementary experiment in which participants were asked to report the strategy they used for numerical judgment after completing the experiment 1. The results indicated that the three primary response strategies were the quantity-based strategy (30.07%), the length-based strategy (28.67%), the density-based strategy (32.17%) and 13 participants (9.09%) reported using other strategies. An ANOVA was conducted with the three primary judgment strategies as the independent variable and reaction time as the dependent variable, the results showed that judgment strategies had no significant effect on reaction time (F (2,128) = 0.687, p = 0.505). For more details, please refer to the supplementary materials.

Design

A 2 (Numerical Attribute: Cardinal, Ordinal) × 2 (Response Type: Congruent, Incongruent) within-subjects design was used. Based on previous research, congruent responses were defined as small numbers responded to with the left hand and large numbers with the right hand; incongruent responses were defined as small numbers with the right hand and large numbers with the left hand. The main effect of response type was used as an indicator to test for the presence of the SNARC effect34,35. The dependent variable was the reaction time for correct trials.

Procedure

The experimental tasks were presented using the Credamo online platform. To ensure participants understood the experimental procedure, the experimenter explained the required responses before the formal experiment. Each trial began with a fixation cross "+" randomly displayed at the center of the screen for 300 to 1000 ms, followed by a non-symbolic representation of numbers 11, 12, 13, 14, 16, 17, 18, or 19 displayed at the center for 3000 ms. The specific experimental flow is shown in Fig. 1. Participants were asked to judge whether the presented non-symbolic number was greater than or less than 15. Before the formal experiment, participants were informed of the definition of small and large numbers in this experiment and underwent a practice session with 8 trials, following the same procedure as the formal experiment. The formal experiment consisted of two programs, A and B, each with 32 trials. In Program A, participants responded to numbers 11–14 with their left hand pressing 'F' and numbers 16–19 with their right hand pressing 'J'. In Program B, participants responded to numbers 11–14 with their right hand pressing 'J' and numbers 16–19 with their left hand pressing 'F'. Participants with odd student ID numbers performed Program A first, followed by Program B; those with even student ID numbers performed Program B first, followed by Program A, to balance the sequence of hand responses and potential practice effects. We included a post-experiment question asking participants to report their method for evaluating numerical magnitude. The SNARC effect was assessed using the methodology of Shi et al. and Han Meng et al., where the SNARC effect is indicated by shorter reaction times for congruent responses compared to incongruent responses34,35.

Fig. 1
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Flowchart of experiment 1.

Data analysis and results

Data preprocessing was conducted using RStudio software under the R version 4.3.1 environment (R Core Team, 2023), while statistical analyses were performed with SPSS 29.0.1 software. Following data exclusion methods from previous studies 31,36, we excluded data that fell beyond ± 2 standard deviations for each participant and each level (239 trials) and trials with incorrect responses (826 trials), accounting for 25.21% of the total data. Additionally, five participants with an accuracy rate below 40% were removed, leaving 61 participants for analysis.

Using repeated measures ANOVA revealed a significant main effect of Numerical Attribute, F(1, 60) = 10.80, p < 0.01, ηp2 = 0.23, with reaction times for cardinal numbers (841.39 ± 220.87 ms) significantly shorter than those for ordinal numbers (911.19 ± 224.68 ms). The main effect of Response Type was also significant, F(1, 60) = 5.32, p < 0.05, ηp2 = 0.08, with reaction times for congruent trials (852.77 ± 216.19 ms) significantly shorter than those for incongruent trials (899.81 ± 232.10 ms), indicating the presence of the SNARC effect in Experiment 1. The interaction between Numerical Attribute and Response Type was not significant, F(1, 60) = 0.00, p = 0.97.

Given the significant main effect of Response Type but the non-significant interaction with Numerical Attribute, paired-sample t-tests were conducted to examine differences in Response Type under the conditions of Cardinal and Ordinal attributes. For Cardinal numbers, a paired-sample t-test showed that reaction times for congruent responses (818.13 ± 207.47 ms) were significantly shorter than for incongruent responses (864.64 ± 232.89 ms), t(60) = -1.95, p < 0.05, d = -0.25, 95% CI [-0.50, 0.01]. For Ordinal numbers, reaction times for congruent responses (887.41 ± 220.85 ms) were significantly shorter than for incongruent responses (934.98 ± 227.78 ms), t(60) = -1.66, p = 0.05, d = -0.21, 95% CI [-0.47, -0.04] (see Fig. 2).

Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
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Reaction time differences in different contextual semantic responses.

Experiment 2: the impact of numerical attributes on the SNARC effect in symbolic representations

This experiment employs a numerical magnitude judgment task to observe whether the SNARC effect differs between cardinal and ordinal numbers when presented in a symbolic format.

Participants

The sample size was determined using G*Power 3.1.9.7, set at Power (1-β) = 0.80 and effect size f = 0.25. At a significance level of α = 0.05, the required sample size was 48. Sixty university students were recruited (10 males, 50 females, aged 18–22), all right-handed with normal or corrected vision. In this study, all participants provided written informed consent. Before the commencement of the research, participants were thoroughly informed about the study’s purpose, procedures, potential risks and benefits, and their right to withdraw at any time. The research team ensured that all participants fully understood this information and voluntarily agreed to participate. This study adhered to relevant ethical standards and guidelines. And this study received ethical approval from the Ethics Committee of Scientific Research Projects at the School of Psychology, Northwest Normal University, under approval number 2022128.

Materials

This experiment adopted a numerical magnitude judgment task31 and utilized Chinese words that encode both quantity and order as experimental materials. In the Chinese linguistic context, the representation of numbers typically combines “numeral classifiers” with “morphemes” to express cardinal and ordinal numbers. For instance, cardinal numbers are often represented using structures like '三' (three) combined with the classifier '个' (ge), whereas ordinal numbers incorporate the combined '第' (di) followed by a numeral, such as '第三' (third). Accordingly, the stimuli in this experiment were presented in a format that integrates numeral classifiers and morphemes, aligning with the natural linguistic representation of numbers in Chinese.

Design

A 2 (Numerical Attribute: Cardinal, Ordinal) × 2 (Response Type: Congruent, Incongruent) within-subjects design was employed. Following prior research, congruent responses were defined as small numbers responded to with the left hand and large numbers with the right hand; incongruent responses were the reverse. The main effect of response type served as the indicator for the SNARC effect34,35. The dependent variable was the reaction time for correct trials.

Procedure

Tasks were presented using the Credamo platform. To ensure comprehension, the experimenter explained the required responses before the formal session. Each trial began with a fixation cross " + " randomly displayed at the screen’s center for 300 to 1000 ms, followed by a symbolic representation of numbers 1, 2, 3, 4, 6, 7, 8, or 9 for 3000 ms. The specific experimental flow is shown in Fig. 3. Participants judged if the number was greater or less than 5. Before the formal experiment, participants were informed about the definition of small and large numbers and completed a practice session with 8 trials, mirroring the main experiment’s flow. The main experiment consisted of two programs, A and B, each with 32 trials. Program A required left-hand responses with 'F' for numbers 1–4 and right-hand responses with 'J' for numbers 6–9. Program B reversed this order. Participants with odd student ID numbers started with Program A, then Program B; even-numbered IDs started with Program B, then A, to balance the sequence of hand responses and potential practice effects.

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
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Flowchart of experiment 2.

Data analysis and results

Data preprocessing was carried out using RStudio within the R version 4.3.1 environment37, with statistical analyses performed using SPSS 29.0.1 software. Employing data exclusion methods from prior studies31,36, we removed data exceeding ± 2 standard deviations for each participant at each level (231 trials) and trials with incorrect responses (203 trials), accounting for 11.30% of the total. Additionally, one participant with an accuracy rate below 20% and two participants with reaction times beyond three standard deviations were excluded, leaving 57 participants for analysis.

Using repeated measures ANOVA revealed no significant main effect of Numerical Attribute, F(1, 56) = 0.68, p = 0.42. However, the main effect of Response Type was significant, F(1, 56) = 5.06, p < 0.05, ηp2 = 0.08, with reaction times for congruent trials (646.50 ± 113.90 ms) significantly shorter than for incongruent trials (676.14 ± 117.23 ms), indicating the presence of the SNARC effect in Experiment 2. The interaction between Numerical Attribute and Response Type was significant, F(1, 56) = 6.48, p < 0.05, ηp2 = 0.10. Further simple effects analysis showed that for incongruent responses, reaction times for cardinal numbers (664.20 ± 96.64 ms) were significantly shorter than for ordinal numbers (688.08 ± 134.54 ms), F(1, 56) = 5.78, p < 0.05, ηp2 = 0.09. For congruent responses, the simple effect of Numerical Attribute was not significant. At the cardinal level, the simple effect of Response Type was not significant; however, at the ordinal level, reaction times for congruent responses (641.14 ± 115.32 ms) were significantly shorter than for incongruent responses (688.08 ± 134.54 ms), F(1, 56) = 8.1, p < 0.01, ηp2 = 0.13 (see Figs. 4 and 5).

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
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Reaction time differences in different contextual semantic responses.

Fig. 5
Fig. 5The alternative text for this image may have been generated using AI.
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Reaction time differences in various response modes under different contextual semantics.

Experiment 3: the impact of numerical attributes on the SNARC effect in symbolic representations within applied problem contexts

This experiment employs a numerical magnitude judgment task to observe whether the SNARC effect differs between cardinal and ordinal numbers when presented in the context of applied problem scenarios.

Participants

Using G*Power 3.1.9.7, the required sample size was calculated under the conditions of Power (1-β) = 0.80 and effect size f = 0.25. At a significance level of α = 0.05, the total sample size needed was 34. A total of 56 university students (13 males, 43 females, aged 18–27) were recruited. All participants were right-handed with normal or corrected vision. In this study, all participants provided written informed consent. Before the commencement of the research, participants were thoroughly informed about the study’s purpose, procedures, potential risks and benefits, and their right to withdraw at any time. The research team ensured that all participants fully understood this information and voluntarily agreed to participate. This study adhered to relevant ethical standards and guidelines. And this study received ethical approval from the Ethics Committee of Scientific Research Projects at the School of Psychology, Northwest Normal University, under approval number 2022128.

Materials

The experiment adopted a numerical magnitude judgment task31. Stimuli were presented within applied problem scenarios, requiring participants to judge the results of simple arithmetic operations. The problems were designed to yield results between 1–4 and 6–9, utilizing balanced addition and subtraction tasks. Based on the research by Gros (2017), We considered duration, height and number of floors as ordinal values because their ordinal component is salient in daily life, putting emphasis on successorship relation and on comparison12. Similarly, number of elements, price and weight were used as cardinal values because the world semantics attached to such quantities evoke the unordered grouping of elements assigned to values and the partition of a whole into its component parts12. The word problems used in the study were controlled for length at 27.25 ± 2.05 characters. A length difference t test was conducted for problems involving different numerical attributes, showing no significant differences (t(30) = 1.037, p = 0.308). Similarly, a length difference test was performed for texts with different calculation methods, and the difference was also not significant (t(30) = 1.359, p = 0.184). Additionally, 70 participants were recruited to evaluate the reading and response difficulty of the problems. The results indicated that 0% of participants found the problems difficult to understand, and 0% found them difficult to calculate.

Design

A 2 (Numerical Attribute: Cardinal, Ordinal) × 2 (Response Type: Congruent, Incongruent) within-subjects design was employed. Following prior research, congruent responses were defined as small numbers responded to with the left hand and large numbers with the right hand; incongruent responses were the reverse. The main effect of response type served as the indicator for the SNARC effect34,35. The dependent variable was the reaction time for correct trials.

Procedure

Tasks were presented using the Credamo platform. To ensure participants understood the procedure, the experimenter explained the required responses before the formal session. Each trial began with a fixation cross "+" displayed at the screen’s center for 300 to 1000 ms, followed by the presentation of applied problem materials. The specific experimental flow is shown in Fig. 6. Participants judged whether the non-symbolic numerical result was greater or less than 5. No maximum duration was set for the presentation of materials in Experiment 3. Before the formal experiment, participants were informed of the defined ranges for large and small numbers in this study and participated in a practice session. The procedure was similar to that of the formal experiment. After participants became familiar with the experimental process, they proceeded to the formal testing phase, which consisted of two parts: Part A and Part B. Participants were required to complete both parts. Those with student ID numbers ending in an odd digit first completed Part A followed by Part B, while those with student ID numbers ending in an even digit completed Part B first followed by Part A. This counterbalancing was designed to control for potential hand order effects and practice effects.

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
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Flowchart of experiment 3.

Each part (A and B) consisted of 32 trials. In Part A, participants responded to smaller numbers (1–4) by pressing the "F" key with their left hand and to larger numbers (6–9) by pressing the "J" key with their right hand, representing a response-congruent condition. In Part B, participants responded to smaller numbers (1–4) by pressing the "J" key with their right hand and to larger numbers (6–9) by pressing the "F" key with their left hand, representing a response-incongruent condition.

Data analysis and results

Data preprocessing was conducted using RStudio within the R version 4.3.1 environment37, and statistical analyses were performed using SPSS 29.0.1 software. Following data exclusion methods from prior studies31,36, we removed data exceeding ± 2 standard deviations for each participant at each level (187 trials) and trials with incorrect responses (250 trials), accounting for 12.19% of the total. Additionally, two participants with an accuracy rate below 25% and three participants with reaction times beyond three standard deviations were excluded, resulting in a final sample of 51 participants.

Using repeated measures ANOVA revealed a significant main effect of Numerical Attribute, F(1, 50) = 91.37, p < 0.001, ηp2 = 0.65, with reaction times for cardinal numbers (2038.83 ± 575.78 ms) significantly shorter than for ordinal numbers (2658.23 ± 989.82 ms). The main effect of Response Type was not significant, F(1, 50) = 0.72, p = 0.40, ηp2 = 0.11, indicating that the SNARC effect did not appear in Experiment 3. The interaction between Numerical Attribute and Response Type was also not significant, F(1, 50) = 0.93, p = 0.34 (see Fig. 7).

Fig. 7
Fig. 7The alternative text for this image may have been generated using AI.
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Reaction time differences in various response modes under different contextual semantics.

Given the significant main effect of Numerical Attribute but the non-significant main effect of Response Type, we conducted paired-sample t-tests under both cardinal and ordinal conditions to ensure that no level-specific SNARC effect was overlooked. In the cardinal condition, a paired-sample t-test revealed no significant difference in reaction times between the congruent (2013.03 ± 494.18 ms) and incongruent (2064.64 ± 651.22 ms) conditions, t(50) = − 0.54, p = 0.30. Similarly, in the ordinal condition, no significant difference was found in reaction times between the congruent (2580.47 ± 767.35 ms) and incongruent (2735.99 ± 1173.85 ms) conditions, t(50) = − 0.95, p = 0.17. These results indicate that the SNARC effect was absent under both cardinal and ordinal conditions in Experiment 3.

To mitigate the potential influence of “Operational Momentum” and "Spatial-Numerical Association of Response Codes" as discussed by McCrink and Wynn (2009), we conducted a 2 (calculation method: addition, subtraction) × 2 (numerical attribute: cardinal, ordinal) × 2 (response mode: congruent, incongruent) repeated-measures ANOVA. The results revealed a significant main effect of numerical attribute (F(1, 50) = 103.546, p < 0.001, ηp2 = 0.657), with reaction times for cardinal numbers (2112.805 ± 75.383 ms) being significantly shorter than those for ordinal numbers (2748.760 ± 117.510 ms). However, the main effect of calculation method was not significant (F = 1.513, p = 0.224), and no significant interactions were observed. These findings suggest that the balanced design of addition and subtraction tasks in Experiment 3 effectively eliminated their potential influence on the results.

Discussion

Influence of numerical representation forms on numerical attributes

The representation of numbers is regarded as a mathematical context, with different representations eliciting distinct cognitive pathways38,39,40. Concrete fading is a commonly used method in mathematics education that helps students transition from concrete to abstract understanding of mathematical concepts and problem-solving. This method gradually reduces concrete information, ultimately guiding students to independently grasp abstract mathematical concepts and problem-solving skills41,42. This study, incorporating the concept of concrete fading, designed three experiments to investigate the influence of different numerical representations on the SNARC effect across varying levels of representation. The experimental materials ranged from the most abstract “dot arrays” to “verbal numerals” with semantic information, to “word problems” encompassing concrete contexts, with increasing world semantic information and varying degrees of concreteness. In Experiment 1, “dot arrays” were used as the stimulus material. The quantity of dots was represented as a set to indicate cardinality, while arranging the dots in a sequence with an arrow pointing to a specific dot represented ordinality. This method disregards cultural symbols, allowing participants to intuitively perceive numerical magnitude without incorporating world semantics. Additionally, because numbers less than four are not only small but also fall within the subitizing system, which allows for immediate and precise quantity recognition, we refined previous studies by setting “15” as the threshold between large and small numbers, thereby eliminating the influence of differences between the approximate and exact number systems on experimental results.

In Experiment 2, using Chinese character symbols, we employed morphemes in conjunction with Chinese numeral characters as the stimulus material. Morphemes, as the basic units of language symbols, carry rich semantic information. They are not merely parts of grammar but also reflect cultural and social backgrounds. The use of morphemes in language mirrors human knowledge, experience, and cultural heritage. The morphemes "个" (ge) and "第" (di) adequately represent cardinal and ordinal quantities, respectively.

In Experiment 3, using simple word problems that could be solved in one step as the stimulus material, requiring participants to respond to the calculated answers via keystrokes. Unlike abstract morphemes, word problem contexts are richer and more concrete in world semantics, effectively ensuring that participants process numerical semantics while also engaging with practical contexts. In solving these word problems, participants abstract algorithms from real-world problem contexts, further ensuring the role of world semantics. This setup better simulates actual mathematical problem-solving processes, thus exploring whether the SNARC effect manifests differently in real-life scenarios. This study increased the role of world semantics across the three experiments.

The results showed that in both Experiments 1 and 2, reaction times for cardinal numbers were significantly shorter than for ordinal numbers, while the differences in Experiment 2 were not significant. This aligns with our pre-experiment expectations. The insignificant reaction time differences in Experiment 2 likely stem from incomplete processing of numerical attributes by participants. Morphemes, as abstract concepts controlling numerical attributes, allow participants to compare numerical magnitudes even without full processing. In Experiments 1 and 3, numerical attributes could not be overlooked.

The reaction time differences brought by numerical attributes can be interpreted from developmental and familiarity principles. From a developmental perspective, Gentner and Medina’s pioneering research on counting principles proposes that children’s understanding of cardinal and ordinal properties of numbers develops separately43. They posited that to develop proficient counting abilities, children must grasp three counting principles: the one-to-one correspondence principle, the stable order principle (critical for ordinal development), and the cardinal principle (indicating the total number of items in a set). Collectively, these principles constitute the "how to count" principles, essential for understanding the set nature of cardinal numbers11. Miller et al. noted that ordinal acquisition occurs slightly later than cardinal acquisition44. Cardinal meanings are more common in everyday scenarios like weight, crowds, or item counting during shopping, aligning with the familiarity principle. Early acquisition and frequent exposure can jointly explain why reaction times for cardinal attributes are shorter, indicating that cardinal acquisition might be more intuitive and natural in mathematical learning due to its alignment with daily experiences.

Influence of numerical attributes on the SNARC effect

The SNARC effect, representing the influence of spatial position on numerical magnitude responses, has been extensively studied. This research’s novelty lies in introducing numerical attributes as a new variable affecting numerical cognition and examining its relationship with the SNARC effect under different representation forms. In traditional numerical magnitude judgment tasks, consistent trials involve pressing a left-hand key for small numbers and a right-hand key for large numbers, while inconsistent trials involve pressing a left-hand key for large numbers and a right-hand key for small numbers. The SNARC effect is considered present if reaction times differ between consistent and inconsistent trials.

In Experiment 1, using non-symbolic dot arrays to represent numbers, the SNARC effect was observed. Simple effect analysis indicated that the SNARC effect was more pronounced for cardinal levels than ordinal levels, contrary to our hypothesis that ordinal numbers, similar to the mental number line representation, would show a stronger SNARC effect. Post-experiment discussions with participants revealed that some participants, despite instructions to judge sequence from the left, used the right side of the sequence as the origin for magnitude estimation, potentially diluting the non-symbolic ordinal SNARC effect.

Experiment 2 revealed a significant SNARC effect and interaction, with further analysis showing the SNARC effect only at the ordinal level, supporting our hypothesis. Both ordinal numbers and the mental number line are represented axially, corroborating the mental number line theory as an explanation for the SNARC effect. Experiment 3, however, did not find a SNARC effect or interaction, though the main effect size of numerical attributes in word problem contexts was significantly larger than in non-symbolic and symbolic contexts. This suggests that the extensive processing of world knowledge in numerical attributes might influence the SNARC effect.

Limitations and future directions

This study advances the exploration of the spatial-numerical relationship, enhancing the understanding of the formation process of numerical spatial representation at the spatial representation stage of quantitative information, and holds significant implications for comprehensively understanding the SNARC effect. However, some limitations remain.

In numerosity research, dot arrays are a more common choice as stimulus material. Dot arrays differ from dot sequences by being non-linearly arranged, precluding quantity judgment based on length. While cardinal quantities expressed through dot arrays can be represented, ordinal sequences cannot. Due to potential differences in processing dot arrays and dot sequences, Experiment 1 used dot sequences to represent cardinal quantities, potentially confounding or diluting the SNARC effect. Additionally, the fixed total number of dots in ordinal sequences and the use of the right side as an origin for magnitude estimation by some participants might have further diluted the non-symbolic ordinal SNARC effect.

We also compared participants’ reaction times across the three experiments and found significant differences. Reaction times were longest in Experiment 3 (word problem scenarios), followed by Experiment 1 (non-symbolic representations), and shortest in Experiment 2 (symbolic representations) (F = 344.788, p < 0.001, ηp2 = 0.806). This result does not align with the influence of semantic magnitude on reaction time but is better explained by Bruner’s “concreteness fading” theory45. Concreteness fading refers to the process in which the physical, concrete representation of a concept gradually becomes more abstract over time41. This process involves three stages: concrete representations, iconic representations, and abstract representations. Based on the level of semantic magnitude, the tasks across the three experiments can be ranked as follows: Experiment 3 (word problem scenarios) involved the greatest level of semantic magnitude, followed by Experiment 2 (symbolic representations), and Experiment 1 (non-symbolic representations) had the lowest. According to the concreteness fading theory, symbolic representations are the product of complete abstraction, retaining only numerical information, while non-symbolic representations involve partial abstraction, conveying visual information such as images. Due to the varying levels of concreteness fading, non-symbolic representations require more time to extract numerical information, which explains the observed differences in reaction times.

Conclusion

This study investigated how different numerical attributes shape the SNARC effect across dot sequences, Chinese character symbols, and word problem contexts. The results of three experiments suggest that extensive processing of world semantics in numerical attributes may attenuate the SNARC effect. Furthermore, ordinal representation, which aligns with the mental number line, elicits a stronger SNARC effect than cardinal representation. This research provides further evidence for the existence of the SNARC effect in both numerical quantity and ordinal processing. It also advances research in mathematical cognition by examining how world semantic numeral information, across different representational formats, influences the SNARC effect.