Article

André Andere; Alaina Alves

DOI: 10.22167/2675-6528-202602127

Article derived from a Final Course Work (TCC), with content based on the student’s original work and adapted to the editorial format of the Revista E&S with the support of the ResumeAI tool, an artificial intelligence solution developed by the Pecege Institute for textual synthesis and organization.

Abstract

The relationship between numerical cognition mechanisms and mathematical problem-solving in the final years of Elementary School was investigated, in light of the concept of mathematical literacy. It started from the hypothesis of a bidirectional relationship, in which the development of numerical cognition influences performance in mathematical problems, while certain characteristics of these tasks can favor the strengthening of these mechanisms. The research adopted a qualitative, applied approach, combining theoretical review of the literature in cognitive neuroscience with documentary analysis of textbooks used in public and private networks. An analytical matrix composed of six dimensions was developed, contemplating cognitive, semiotic, and pedagogical aspects of the tasks. The analysis of 123 questions showed a predominance of activities focused on symbolic manipulation and intermediate levels of cognitive demand, with low incidence of tasks that mobilize estimation, approximation, or multiple representations. A significant reduction in the mobilization of numerical cognition aspects in algebraic content was also observed. The results indicated a possible disconnection between the development of numerical cognition and the analyzed didactic practices, suggesting the need for greater integration between meaning construction, diversity of representations, and magnitude exploration. It was concluded that mathematical problems can act as instruments for cognitive development, provided they are structured in a way that mobilizes different dimensions of numerical thinking.

Keywords: Numerical cognition; Mathematical literacy; Textbooks; Mathematics.

1. Introduction

Numerical cognition, fundamental to the human capacity to perceive, represent, and manipulate quantities, forms the basis of formal mathematical thinking. Its development involves a progression from the innate understanding of magnitudes to culturally constructed symbolic representations, as described by Von Aster and Shalev (2007). Non-verbal mechanisms, termed the number sense, are composed of two subsystems: subitizing, which allows for the determination of small quantities without counting, and the approximate number system (ANS), responsible for estimating large numerosities (Feigenson et al., 2004). These primary mechanisms are structured before school experience and serve as a foundation for more complex numerical skills (Dehaene, 1997).

The verbal and symbolic mechanisms, in turn, expand the complexity of numerical processing with the acquisition of Indo-Arabic words and symbols. The verbal/auditory mechanism is structured with language learning, establishing a link between the innate number sense and symbolic representations (Machado, 2011). The symbolic/visual mechanism involves the recognition and use of mathematical symbols, the mastery of systematic counting, and the use of arithmetic operations. These secondary mechanisms, which depend on schooling, enable the transition from intuitive to formal quantitative thinking (Santos et al., 2016). The mental number line, an organized spatial representation, integrates the verbal, symbolic, and quantitative dimensions of numbers, being crucial for mental calculation and estimation (Dehaene, 1997; Von Aster and Shalev, 2007). The development of numerical cognition is central to the improvement of mathematical literacy, which implies the ability to mobilize concepts and procedures to interpret, evaluate, and solve real-world situations (OECD, 2019).

Mathematical literacy goes beyond technical mastery, emphasizing mathematics as a tool for reading and interpreting reality and as support for informed decision-making (OECD, 2023; Machado, 2011). Difficulties in mathematical learning may be associated with deficits in numerical cognition (Piazza, 2010), compromising skills such as estimation and problem-solving. The transition to algebraic language, especially in the final years of Elementary School, seems to imply a reduction in the diversity of experiences related to numerical cognition, with numbers being treated as formal elements in algebraic relationships, to the detriment of exploring broader numerical meanings (Dehaene, 1997).

This suggests a possible disconnection between the development of numerical cognition and the didactic practices analyzed, indicating the need for greater integration between meaning construction, diversity of representations, and magnitude exploration. The literature points to a bidirectional relationship between learning and numerical cognition, in which pedagogical interventions can strengthen fundamental cognitive mechanisms, and the development of these mechanisms favors mathematical learning.

Didactic mathematical problems, when well-structured, can act as instruments for cognitive development, mobilizing different dimensions of numerical thinking. The research objective, therefore, is to analyze the mutual relationship between the mechanisms of numerical cognition, developed in early childhood and in the early school years, and mathematical literacy, with a focus on problem-solving in the final years of Elementary School. It seeks to investigate the extent to which the development of these mechanisms acts as a predictor of school mathematical performance and, conversely, how certain types of mathematical problems, present in textbooks, can favor the consolidation and strengthening of these cognitive structures.

By articulating evidence from cognitive neuroscience with the analysis of pedagogical practices, the study aims to contribute to the advancement of the dialogue between neuroscientific research and the educational field, broadening the understanding of how cognitive processes related to the perception, representation, and manipulation of quantities manifest in real school contexts. Thus, the research seeks to offer subsidies so that the planning of didactic interventions can enhance cognitive processes associated with mathematical learning, promoting an evidence-based pedagogy.

2. Material and Methods

The research adopted a theoretical and qualitative approach, based on documentary analysis, and was organized in two complementary stages. In the first stage, a theoretical review was carried out on the mechanisms of numerical cognition, based on the models proposed by Kucian and Von Aster (2015), by Santos, Silva and Oliveira (2016), and by Dehaene (1997). This review covered non-verbal, verbal/auditory, and symbolic/visual mechanisms, as well as the notion of a mental number line, distinguishing them into primary mechanisms, independent of formal schooling, and secondary mechanisms, dependent on systematized instruction. Studies discussing the relationship between these mechanisms, school mathematical performance, and associated cognitive functions, such as working memory and cognitive flexibility, were analyzed.

In the second stage, a documentary analysis was carried out on two Mathematics textbooks for the Seventh Grade, both in the edition available in schools in 2026. One of the materials was from a collection approved by the National Textbook Program (Ninth Edition, published in 2023), selected for its wide circulation in Brazilian public schools. The other textbook was used in private educational institutions (edition updated in 2025), chosen for its relevance and distinct pedagogical approaches. 123 questions from these materials were analyzed, focusing on the topics of Integers and Equations (65 questions from the public school material, with 34 on integers and 31 on equations; and 58 questions from the private school material, with 33 on integers and 25 on equations). The objective was to identify, categorize, and analyze the types of mathematical problems proposed, classifying them based on previously defined analytical criteria to examine the cognitive demands involved in the tasks. This categorization aimed to distinguish problems with the potential to mobilize and develop the mechanisms of numerical cognition and mathematical literacy from those that are restricted to the mechanical application of procedures and algorithms.

The data analysis was performed qualitatively, seeking to establish correspondences between the cognitive mechanisms described in the literature and the cognitive demands present in the analyzed problems. For this analysis, a matrix composed of six complementary dimensions was developed, which constituted the central instrument of the empirical stage of the research, allowing for a systematic evaluation of the cognitive quality of the analyzed mathematical problems. Dimensions 1 and 6 dialogue directly with numerical cognition, dimensions 3, 4, and 5 examine the semiotic and cognitive structure of the problems, and dimension 2 assesses the degree of symbolic formalization involved.

The dimensions of the analysis matrix were:

• Dimension 1: Type of numerical mobilization. It describes the mathematical treatment of numbers in the task, identifying the numerical cognition mechanisms predominantly mobilized, differentiating primary mechanisms (Magnitude comparison, Subitizing, Approximate estimation, Mental number line mobilization) from secondary ones (Arithmetic fact retrieval, Formal algorithm application, Symbolic manipulation, Procedural exact calculation). The choice was made to assign only the main mobilization in each question.
• Dimension 2: Level of symbolic dependence (syntax). It investigates the degree to which task resolution depends on formal mathematical language, evaluating the weight of symbolic syntax. It considers four levels: Low symbolic dependence, Medium (conceptual-procedural), Medium (guided procedural), and High symbolic dependence.
• Dimension 3: Conversion between record types. Examines the need for conversion between different records (verbal, symbolic, graphic, and tabular). It considers three possibilities: Absence of conversion, Simple conversion, and Multiple conversion.
• Dimension 4: Semantic nature of the problem. Analyzes the meaning structure of the task, distinguishing between problems internal to Mathematics and contextualized problems. It considers four categories: Non-structuring internal, Structuring internal, Pretextual contextual, and Meaningful contextual.
• Dimension 5: Cognitive demand. Assesses the degree of cognitive elaboration required by the task, organized into five categories: Reproduction (mechanical/algorithmic), Direct application, Non-explicit strategy, Modeling, and Argumentation/justification. It allows distinguishing tasks centered on the execution of procedures from those that demand formulation, organization, and validation of reasoning.
• Dimension 6: Development potential of numerical cognition. Analyzes the formative potential of the question in strengthening fundamental mechanisms, considering non-exclusive categories: Estimation, Magnitude comparison, Approximation exploration, Expansion of the mental number line, and Magnitude (understanding the value of a number). The choice was made to consider the main development. This approach starts from the hypothesis that well-structured problems can contribute to the progressive refinement of numerical cognition, establishing a feedback relationship between cognitive mechanisms and formal learning.

To illustrate the application of the analytical matrix, examples of questions are presented along with their respective classifications in the different dimensions:

Example 1: Solve the equation 3x – 5 = 19

Dimension 1: Symbolic manipulation;

Dimension 2: High symbolic dependence;

Dimension 3: Absence of conversion;

Dimension 4: Non-structuring internal;

Dimension 5: Direct application;

Dimension 6: Absence of numerical mobilization.

The task highlights the predominance of symbolic language and algebraic manipulation, without the mobilization of numerical meanings. The number operates as a formal element, not as an interpretable quantity.

Example 2: The temperature in a city was -3°C. After a variation of +5°C, what is the new temperature?

Dimension 1: Procedural exact calculation;

Dimension 2: Medium (guided procedural);

Dimension 3: Simple conversion;

Dimension 4: Contextual pretext;

Dimension 5: Direct application;

Dimension 6: Magnitude.

Despite presenting a daily context, the task is characterized as contextual pretextual, since the situation does not interfere with the structure of the resolution, which is reduced to the direct application of an operation with integers. The context functions as an illustrative element, not requiring the construction of strategies or conceptual analysis.

Example 3: Explain why the sum of two negative integers always results in a negative number.

Dimension 1: Comparison of magnitudes;

Dimension 2: Low symbolic dependence;

Dimension 3: Absence of conversion;

Dimension 4: Structuring internal;

Dimension 5: Argumentation/justification;

Dimension 6: Magnitude.

The issue is internal to Mathematics and has a structuring character, as it promotes the understanding of a fundamental property of integers, not limiting itself to the execution of procedures and requiring the student to build a justification based on the relationships between magnitudes and the interpretation of the sign of the numbers. It is a task that favors generalization and conceptual development.

3. Results and Discussion

The research investigated the relationship between numerical cognition mechanisms and mathematical problem-solving in the final years of Elementary School, from the perspective of mathematical literacy. The analysis of 123 textbook questions revealed a predominance of tasks centered on symbolic manipulation and intermediate levels of cognitive demand. A low incidence of activities that mobilize estimation, approximation, or multiple representations was observed. There was also a significant reduction in the mobilization of numerical cognition aspects in algebraic content, indicating a possible disconnection between the development of numerical cognition and the analyzed didactic practices, suggesting the need for greater integration between meaning construction, diversity of representations, and magnitude exploration. Primary mechanisms

The primary mechanisms of numerical cognition, as discussed in the study, constitute the innate basis of the number sense, characterized by the human capacity to discriminate magnitudes, estimate quantities, and identify elementary numerical relations. These abilities are pre-verbal and biologically rooted, operating independently of language and formal schooling (Dehaene, 1997). They form the foundation upon which more complex mathematical learning is built, being crucial for the subsequent development of mathematical thinking.

This innate numerical sense is organized into two distinct subsystems, which manifest early and remain active throughout life (Feigenson et al., 2004). The first is the exact system, responsible for subitizing, which allows for the rapid apprehension of small numerosities, generally up to three or four elements. Studies indicate that the identification of these sets occurs in milliseconds, without the need for verbal counting, evidencing parallel and pre-attentional processing (Dehaene, 1997). The robustness of this system is observed at various ages and species, suggesting its phylogenetically ancient and ontogenetically early nature.

The structural limitation of the exact system, which is restricted to small quantities, is also historically reflected in symbolic notation systems. Analyses by Dehaene (1997) show that most civilizations represented numbers by repetitions of the symbol for 1 up to the number three, abandoning this practice from four onwards. This occurs due to the difficulty of quickly distinguishing larger sets by visual inspection alone. This finding underscores the necessary transition to more formal mechanisms when dealing with larger quantities.

The second subsystem is the Approximate Number System (ANS), which allows for the comparative estimation of large numerosities, operating in a way that is sensitive to the ratios between sets. Research with infants demonstrates the ability to discriminate numerosities with specific ratios, which progressively refine with development. Neuroimaging studies associate the functioning of the ANS with the bilateral intraparietal sulcus, the lesion of which compromises the ability to estimate and compare quantities (Feigenson et al., 2004). This evidence reinforces the biological basis and the importance of the ANS for numerical cognition.

Anthropological and cultural evidence, such as that observed in Amazonian indigenous groups with restricted linguistic counting systems, corroborates the pre-school nature and lower cultural dependence of these primary mechanisms. These populations preserve subitizing and approximate estimation, despite limitations in the exact enumeration of large quantities (Andrade, 2015). However, research also indicates that infants and young children have difficulty integrating representations of small sets and approximate quantities into a single concept, suggesting the fundamental role of language as a mediator (Marcilese, 2012). Secondary mechanisms

The secondary mechanisms of numerical cognition, in contrast to the primary ones, develop through interaction with culture, language, and formal schooling. This process is not a simple opposition between the innate and the learned, but a progressive reorganization and expansion of initial cognitive dispositions through social and educational experience (Dehaene, 1997). Thus, numerical understanding is a process that involves both early aspects and those that depend on systematic instruction.

The verbal/auditory mechanism emerges with language learning, associating magnitudes with linguistic labels. Von Aster and Shalev (2007) highlight that this stage involves the acquisition of numerical words and the construction of the verbal sequence, essential for oral counting. However, verbal counting is not automatic; young children often use approximate strategies for larger sets, delivering “handfuls” of objects instead of counting systematically (Andrade, 2015). This highlights the cultural nature of counting and the mutual impregnation between language and mathematics (Machado, 2011).

The symbolic/visual mechanism constitutes another central axis of the secondary mechanisms, encompassing the recognition and use of Indo-Arabic symbols, the mastery of systematized counting, and the employment of arithmetic operations. This system depends heavily on schooling, as it requires explicit learning of symbolic conventions and formal procedures. The transition from number as perceived quantity to a manipulable symbolic object is a crucial cognitive transformation for the development of arithmetic and algebra (Dehaene, 1997).

These secondary mechanisms act as mediators between innate intuitive mathematics and formal mathematics. The notion of quantity is fundamental in the human relationship with the world. With the increase in cognitive demands, especially in complex situations, more elaborate operations become necessary, such as systematic counting, estimation, rounding, and the use of orders of magnitude (OECD, 2023). Secondary mechanisms, therefore, integrate initial numerical representations with the symbolic and linguistic practices of school mathematics, facilitating the transition to formal mathematical thinking. Mental number line

The construction of the mental number line represents the final stage in the development of the secondary mechanisms of numerical cognition. This organized spatial representation integrates the verbal, symbolic, and quantitative dimensions of numbers. The formulation of the number line metaphor as the central structure of the mental representation of quantities emerged from the identification of automatic associations between numbers and space, where numbers assume relative positions on a spatial continuum (Dehaene, 1997). Von Aster and Shalev (2007) highlight its importance for mental calculation, estimation, and advanced mathematical reasoning.

The architecture of the mental number line organizes numbers along an oriented spatial axis. In cultures with left-to-right writing, smaller values are represented on the left and larger ones on the right. Experimental evidence shows that the adult brain converts numerals into internal analog magnitudes that preserve proximity relations (Dehaene, 1997). This mechanism explains the numerical distance effect, where adults compare distant numbers more quickly and accurately than close numbers (Andrade, 2015), indicating a continuous and graduated representation.

Beyond directional organization, the mental number line exhibits a non-linear distribution of magnitudes, with over-representation of small numbers. Dehaene (1997) explains that the spatial representation of numbers is compressed, with greater perceptual resolution for small magnitudes and lower relative discrimination for larger values. The spatial position of a number depends on its relative magnitude within the considered interval, highlighting the flexibility and contextual nature of this spatial representation.

The mental number line, by preserving the relationships of order, distance, and magnitude between numbers, is central to estimation, mental calculation, and conceptual understanding of mathematical operations. Von Aster and Shalev (2007) synthesize numerical development into four stages. The first stage, in infancy, involves the central system of magnitude and concrete quantity, with abilities of subitizing, approximation, and comparison, predominating in biparietal regions. The second stage, in preschool, focuses on the verbal number system, with counting strategies and fact retrieval, involving the left prefrontal cortex. In the third stage, in school, the Arabic numeral system and written calculation are developed in occipital regions. Finally, in the fourth stage, also in school, the mental number line and the spatial image of numbers are consolidated, allowing for approximate calculation and arithmetic thinking, with predominance in biparietal regions. The cultural factor in secondary mechanisms

Although the primary mechanisms of numerical cognition have a strong biological basis, secondary mechanisms, especially the construction of the mental number line and the mastery of exact counting, are profoundly influenced by cultural and linguistic factors. Experimental evidence demonstrates that the spatial orientation of the mental number line is not universal, being shaped by cultural practices, such as writing and reading systems. Dehaene’s (1997) studies reveal that Iranian students, literate in right-to-left writing, associate larger numbers with the left side of space, reversing the Western pattern, which suggests the determining role of cultural immersion in the spatial organization of numerical representations.

The notion of numbers greater than three is not natural, but historically and culturally constructed. Etymological analyses by Dehaene (1997) indicate that the Indo-European root of the word ‘three’ refers to “many” or “beyond,” suggesting that, initially, only “one,” “two,” or “one plus another” were clearly differentiated. Anthropological evidence, such as the Warlpiri who use restricted terms for quantity, reinforces this idea. Dehaene (1997) argues that these limitations are lexical, not cognitive, as these groups likely possess non-verbal and approximate quantitative representations for larger numbers, even without linguistic encoding.

Other peoples, such as the Torres Strait Islanders, developed alternative enumeration systems using body parts, highlighting cultural diversity in the construction of counting paradigms (Dehaene, 1997). Studies with Amazonian peoples, such as the Pirahã and Munduruku, reinforce the distinction between a universal system of numerical approximation and a culturally mediated system of exact counting. The Pirahã, with only three terms for numbers, performed close to chance in exact matching tasks (Gordon, 2004). The Munduruku, with numerals up to five, failed at exact arithmetic with larger numbers (Marcilese, 2012). These results support the idea that the ability to estimate quantities is universal, but exact manipulation depends on language and culture.

Spoken language also directly influences the representation, manipulation, and memorization of numbers. Dehaene (1997) shows that Chinese speakers outperform English speakers in numerical sequence memorization tasks due to the phonological brevity of Chinese words. Verbal memory, operating within a limited time window, is impacted by the articulation speed of words. Furthermore, Western languages present greater complexity and irregularity in number nomenclature, resulting in frequent errors in counting learning, such as “twenty and ten,” which do not exist in Asian languages (Dehaene, 1997).

This difference is even more evident in the understanding of the decimal number system. In languages like Chinese, the verbal structure of numerals directly corresponds to the positional logic of Arabic digits, making the concept of base ten transparent. Chinese children, when representing the number 25, tend to select two tens and five units, while American children frequently resort to unit counting (Dehaene, 1997). This data indicates that language not only names numbers but cognitively structures access to numerical relationships, influencing the development of secondary mechanisms of numerical cognition and, consequently, mathematical literacy.

The cultural influence on the secondary mechanisms of numerical cognition is inseparable from the relationship between Mathematics and Mother Tongue. In teaching, this relationship is inevitable, as Mathematics must continually rely on the orality of the mother tongue, which already foreshadows an essential impregnation between the two systems (Machado, 2011). This intertwining is supported by linguistic approaches that attribute a central role to language in the development of higher cognitive skills. From the Vygotskian perspective, natural language acts as a mental tool that mediates the subject’s relationship with the world, developing higher mental functions through language mediation (Marcilese, 2012).

In the field of Mathematics, this linguistic mediation is particularly relevant. Studies indicate that language is involved in the performance of tasks related to multiple mathematical cognitive domains, including problem-solving, calculation, determination of exact numerosities, spatial orientation, and integration of information from different representational systems. This favors inferential and interpretive processes crucial for understanding problem situations (Marcilese, 2012). This interdependence is evident in the acquisition of numerals, which are learned as elements of a hierarchically organized sequence, linked to counting.

The acquisition of the number sequence involves linguistic aspects and the recognition of syntactic and semantic properties, in addition to non-linguistic aspects related to the understanding of the ordinal and cardinal properties of numbers. Although children learn to recite the verbal sequence of numerals early on, understanding the one-to-one correspondence between words and elements of a set is a slow and cognitively demanding process. Language, by providing a specific repertoire for exact quantities, decisively contributes to the consolidation of the concept of a set of individuals and to the emergence of natural numbers, reinforcing the mediating role of the Mother Tongue in the development of the secondary mechanisms of numerical cognition (Marcilese, 2012). Mathematical literacy and the mechanisms of numerical cognition

Mathematical literacy is defined as the ability to reason mathematically, formulate, employ, and interpret mathematics to solve problems in various real-world contexts, mobilizing concepts, procedures, facts, and tools (OECD, 2023). This perspective expands the understanding of mathematics beyond the technical curricular domain, emphasizing its role as an instrument for reading and interpreting reality, as well as support for informed decisions, essential for critical and reflective citizens in contemporary society.

The importance of mathematical literacy in basic education is not restricted to the training of specialists, but to the conscious appropriation of Mathematics as an essential language to represent and understand the world (Machado, 2011). Reducing mathematical learning to the mechanical execution of calculations would be equivalent to reducing the learning of the mother tongue to trivial situations, impoverishing both fields of knowledge. Mathematics, as a privileged field for problem-solving, offers opportunities to build strategies, analyze situations, and validate conclusions based on logical mechanisms, skills that are replicable in other areas (Machado, 2011).

From this perspective, mathematical literacy requires the integrated mobilization of numerical cognition mechanisms, using verbal, visual, and symbolic representations as a basis for its processes. Problems involving the interpretation of graphs, tables, diagrams, or figures demand visuospatial processing. Understanding mathematical statements and terms mobilizes verbal cognitive components. The manipulation of algebraic expressions, algorithms, and formal procedures requires mastery of symbolic mechanisms. This articulation is in line with the nature of mathematical literacy, which involves the ability to move between different representational registers and use them functionally in meaningful contexts, according to PISA (OECD, 2023). Deficits in numerical cognition and learning difficulties in mathematics in the final years of elementary school

The neuroscience literature indicates that formal mathematical learning directly relies on numerical cognition mechanisms, which act as predictors of mathematical performance throughout the school trajectory. Piazza (2010) argues that the notion of number encompasses both the perception of the cardinality of discrete sets, such as estimating candies in a bowl, and abstract entities manipulated symbolically, such as financial market number series. While the former is extracted pre-verbally and shared with other species, the latter is a complex cultural object, whose construction depends on educational and symbolic processes, anchored in the innate and pre-verbal structure.

Studies demonstrate that individual differences in the precision of the Approximate Number System (ANS) and subitizing correlate with school mathematical performance, suggesting that the accuracy of basic numerical representations can predict performance in symbolic arithmetic tasks (Piazza, 2010). These findings indicate that difficulties in mathematical learning may be partially associated with limitations in the refinement of these initial cognitive mechanisms. However, educational challenges become more evident when the mathematics present in these primary and secondary mechanisms serves as a basis for more complex concepts.

Although these systems support magnitude representations and allow for quantitative comparisons, they are insufficient for concepts such as fractions, negative numbers, roots, or irrationals. The construction of this knowledge depends on sophisticated cultural and educational processes, including the mastery of symbolic language and formal instruction. Nevertheless, such concepts remain rooted in basic number sense systems, functioning as cognitive foundations for the development of deeper mathematical thinking (Feigenson et al., 2004). This structural limitation helps explain the recurring difficulties in the final years of Elementary School.

In this stage, the curriculum significantly expands the work with rational numbers, negatives, and decimal representations. Dehaene (1997) argues that the mental number line, which sustains a large part of the rudimentary human quantitative notion, mainly represents positive integers and their proximity relations, offering a limited intuitive basis for understanding other numerical categories. Concepts such as negative, irrational, or complex numbers have historically presented difficulties and continue to be significant obstacles for students, as they do not correspond to pre-existing cognitive categories.

Thus, deficits in numerical cognition can directly impact the development of broader mathematical skills, such as estimation, calculation, and problem-solving. Another explanation for this relationship lies in the role of these mechanisms in promoting cognitive flexibility and optimizing working memory. When basic numerical representations become automated, they free up cognitive resources that can be directed towards the construction and execution of strategies, logical reasoning, and the analysis of more complex problem situations.

A relevant aspect is the cognitive demand associated with the conversion between different forms of mathematical representation. Duval (2003) points out that a large part of the difficulties in learning mathematics stems from the need to move between symbolic, graphical, verbal, and visual registers. Conceptual understanding does not depend solely on the correct manipulation of symbols, but on the ability to translate and articulate meanings between different languages, a process that requires strong integration between verbal, visuospatial, and symbolic cognitive mechanisms, which reinforces the complexity of learning.

In this context, educational literature emphasizes the importance of pedagogical approaches centered on meaning construction. Machado (2013) criticizes the predominance of an excessively syntactic perspective in Mathematics teaching, where concern with formal writing and symbolic manipulation overrides the semantic understanding of concepts. The author argues that by prioritizing rules and notations over meaning, mathematical thinking is subordinated to formal language, compromising learning. The image of Mathematics as arid and inaccessible is linked to this hypertrophy of the syntactic dimension.

Despite the difficulties, the relationship between numerical cognition and mathematical learning is not deterministic. Evidence indicates that numerical abilities are shaped by biological predispositions and cultural and educational processes, with schooling contributing to refine the ANS (Piazza, 2010). This perspective points to a bidirectional relationship, where pedagogical interventions can strengthen fundamental cognitive mechanisms, and the development of these mechanisms favors mathematical learning. Therefore, the teaching of Mathematics in the final years of Elementary School should integrate the construction of meanings, the articulation of languages, and the contextualization of problems.

It is fundamental to prioritize situations that mobilize relationships between Mathematics and the Mother Tongue, promoting interpretation, argumentation, and meaning production, in addition to significant contexts internal and external to Mathematics. This avoids approaches that use contexts merely as illustrative elements (Machado, 2011). This pedagogical orientation can contribute not only to the development of mathematical literacy but also to the strengthening of the mechanisms that support mathematical learning, in a continuous feedback process, ensuring a deeper and more lasting understanding. Mathematical problems, literacy, and the development of numerical cognition

Didactic mathematical problems can play a central role in mobilizing and developing numerical cognition mechanisms. If these mechanisms are cognitive foundations for formal mathematical learning, classroom problems should not be restricted to mechanical exercises or merely illustrative contexts. It is essential that they favor the construction of meanings, the articulation between different languages, and the mobilization of skills such as estimation, modeling, and reflection, promoting deeper engagement with the content.

Mathematical problems that require more than the simple recall of memorized answers, demanding interpretation, strategy selection, comparison of magnitudes, and articulation between representations, tend to mobilize numerical cognition mechanisms more comprehensively. This supports the hypothesis of a bidirectional relationship: problems that develop mathematical literacy strengthen numerical cognition, and the refinement of these mechanisms, in turn, favors performance in more complex problems, creating a virtuous cycle of learning and development.

This perspective is corroborated by Machado’s (2011) critique of the idea that knowledge is built in a unidirectional movement from the concrete to the abstract. The author argues that abstractions are necessary steps in a non-linear chain, where different levels of concreteness can coexist. When the articulation between theory and application is fortuitous or artificial, students feel discomfort and detachment. Thus, mechanical problems with a superficial context do not promote true development to more complex levels of concreteness, limiting the depth of learning.

From the perspective of mathematical literacy, it is crucial to consider the competencies described by PISA, which include the ability to identify, organize, connect information, construct, abstract, justify, explain, interpret, judge, and criticize. Such cognitive actions imply the mobilization of numerical representation mechanisms, the integration between symbolic registers, and cognitive flexibility. Problems that favor these operations not only assess specific content but also strengthen the cognitive foundations of mathematical learning, contributing to a more complete and robust education.

Machado (2011) emphasizes the importance of valuing Mother Tongue orality, the apprehension of significant mathematical objects, the consistent articulation between analysis and synthesis, and the relevance of approximate results and estimations in Mathematics teaching. The author warns against overemphasizing the concrete, reducing it to the manipulation of physical objects. Concreteness, for Machado (2011), lies in the construction of meaning, not in mere visibility or manipulation. A manipulable material can be arbitrary if it lacks meaning, while a symbolic situation can be concrete if it makes cognitive sense.

A classic example by Machado (2011) illustrates this idea with a logical card problem. When presented in an abstract context, the success rate is 19%. With manipulable physical cards, it rises to 49%. However, with a reformulation to a more concrete semantic context, even without physical manipulation, performance reaches 87%. The combination of meaning and material support raises success to 98%. This data reinforces that the concreteness relevant to mathematical learning lies in the construction of meaning, not in the mere manipulation of objects.

Thus, mathematical problems that articulate meaning, multiple representations, estimations, and modeling promote both mathematical literacy and the development and refinement of numerical cognition mechanisms. Persistent difficulties in these tasks may indicate fragilities in the underlying cognitive mechanisms, and not just a specific misunderstanding of content. Therefore, when planning problems for the final years of Elementary School, it is essential to prioritize situations that mobilize estimations, magnitude comparisons, articulations between symbolic and verbal registers, and argumentative reflection, consolidating the feedback between numerical cognition and formal learning. Analysis matrix of the questions

From the theoretical path presented, which articulated contributions from cognitive neuroscience, Mathematics didactics, and discussions on mathematical literacy, it was possible to construct an analytical instrument. This instrument allows for a systematic and qualitative examination of the mathematical problems present in textbooks for Middle School (Ensino Fundamental Anos Finais), evaluating their potential for mobilizing numerical cognition, constructing meanings, and articulating different representational registers. The analysis of problems requires a multidimensional perspective, contemplating cognitive, semiotic, and pedagogical aspects.

The research analyzed 123 questions from two widely used teaching materials: one from the public school system (Ninth Edition, 2023) and another from the private school system (updated edition in 2025), both intended for the Seventh Grade. Questions were selected from the topics Integer Numbers and First-Degree Equations, totaling 65 questions from the public material (34 on integers, 31 on equations) and 58 from the private material (33 on integers, 25 on equations). The inclusion of both materials aimed to broaden the study’s representativeness, encompassing the diversity of pedagogical practices in the Brazilian educational system.

The study of integers mobilizes the construction of meanings related to the expansion of the numerical field and the understanding of negative quantities. Working with equations, in turn, introduces algebraic language more systematically, with an emphasis on symbolic manipulation and generalization of relationships. The analysis of these two themes allowed for the observation of possible continuities and ruptures in the cognitive demands and modes of representation mobilized by the proposed tasks, providing insights into the transition to algebraic thinking.

For this analysis, a matrix composed of six complementary dimensions was developed, which constituted the central instrument of the empirical stage of the research. This matrix allowed for a systematic evaluation of the cognitive quality of the mathematical problems analyzed. It is important to emphasize that the dimensions are not independent, but complementary. Dimensions 1 and 6 dialogue directly with numerical cognition, while dimensions 3, 4, and 5 examine the semiotic and cognitive structure of the problems. Dimension 2, in turn, assesses the degree of symbolic formalization involved in the tasks.

Dimension 1, “Type of numerical mobilization”, describes the mathematical treatment of numbers in the task, identifying which mechanisms of numerical cognition are predominantly mobilized. Primary (I to IV) and secondary (V to VIII) mechanisms are distinguished. The primary ones include Magnitude comparison, Subitizing, Approximate estimation, and Mental number line mobilization. The secondary ones are Retrieval of arithmetic facts, Application of formal algorithms, Symbolic manipulation, and Procedural exact calculation. Although a question may mobilize different mechanisms, only the main mobilization was assigned.

Dimension 2, “Level of symbolic dependence (syntax)”, investigates the degree to which task resolution depends on formal mathematical language, assessing the weight of symbolic syntax. This allows distinguishing high symbolic formalization from high cognitive demand, avoiding confusion between syntactic complexity and conceptual depth. Four levels are considered: Low symbolic dependence (quantitative/relational reasoning, formal calculation dispensable), Medium (conceptual-procedural, semantic understanding precedes algorithm), Medium (guided procedural, direct application of procedures), and High symbolic dependence (intense formal manipulation, focus on rules).

Dimension 3, “Conversion between record types”, aligned with Duval (2003), examines the need for translation between records (verbal, symbolic, graphic, tabular). A greater need for articulation between representations implies a greater cognitive demand. Three possibilities are considered: Absence of conversion (task in the same record), Simple conversion (a transformation between two records), and Multiple conversion (chaining of two or more conversions). This dimension is crucial for understanding how didactic problems promote cognitive flexibility and the integration of different forms of mathematical representation.

Dimension 4, “Semantic Nature of the Problem”, analyzes the meaning structure of the task, differentiating problems internal to Mathematics from contextualized problems, and those that promote conceptual development from those limited to algorithmic application. Four categories are considered: Non-structuring Internal (technical/algorithmic application without relational deepening), Structuring Internal (pure mathematical problem that develops conceptual relations), Pretextual Contextual (merely illustrative context), and Significant Contextual (context whose understanding interferes with resolution). This distinction avoids the false opposition between “real problems” and “mathematical problems”, focusing on the construction of meanings.

Dimension 5, “Cognitive Demand”, assesses the degree of cognitive elaboration required by the task, organized into five categories. Reproduction (mechanical/algorithmic) requires the execution of a known procedure. Direct Application requires recognition of the operation, with an evident resolution path. Non-explicit Strategy demands the selection or construction of a strategy. Modeling requires organizing information from a context to build a mathematical representation. Argumentation/justification requires explanation, justification, or validation of reasoning. This dimension allows distinguishing tasks centered on procedure execution from those that demand formulation, organization, and validation of reasoning.

Dimension 6, “Development potential of numerical cognition”, differs from the first by analyzing what the task can develop, not just what it mobilizes. It assesses the formative potential of the question in strengthening fundamental mechanisms, in non-exclusive categories. The categories include Estimation, Magnitude Comparison, Approximation Exploration, Mental Number Line Expansion, and Magnitude (understanding the value of a number). This approach starts from the hypothesis that well-structured problems can contribute to the progressive refinement of numerical cognition, establishing a feedback relationship between cognitive mechanisms and formal learning. Analysis results

The analysis of the issues, using the proposed matrix, revealed distinct patterns in the mobilization of numerical thinking between the chapters on integers and equations. In the integer chapters, a more balanced distribution was observed among different types of mobilization, with emphasis on exact procedural calculation (29% public, 33% private), comparison of magnitudes (24% public, 27% private), and mobilization of the mental number line (21% public, 18% private). This indicates a significant investment in the construction of numerical meaning, especially in the ordering and location of integers.

In contrast, in the equation chapters, symbolic manipulation (52% public, 32% private) and procedural exact calculation (42% public, 52% private) largely predominated. Categories such as magnitude comparison and mobilization of the mental number line disappeared completely. This pattern suggests a transition towards a more formal and algorithmic treatment of mathematics, where the focus shifts to the symbolic transformation of expressions, at the expense of a more conceptual or magnitude-based approach. The absence of these categories in the algebraic content is a relevant finding.

Another notable aspect is the total absence of categories such as subitizing, approximate estimation, and retrieval of arithmetic facts in all analyzed sets. This reinforces the idea that the proposed tasks prioritize more formal and school-based levels of numerical mobilization, with little emphasis on more intuitive or approximate skills. In general, Dimension 1 reveals a progression, or perhaps a rupture, between the contents: while integers still mobilize different forms of numerical thinking, equations concentrate almost exclusively on symbolic and procedural practices.

Regarding Dimension 2, which assesses the level of symbolic dependency, the results indicated that most questions fall into the medium (conceptual-procedural) and medium (guided procedural) levels. Specifically, the medium (conceptual-procedural) level was predominant, with 53% in public network integers and 61% in private ones, and even more pronounced in equations (81% public, 76% private). Low symbolic dependency was observed in 15% of public and 12% of private integers, but it disappeared completely in equations, indicating that all algebraic tasks require symbolic manipulation.

The lower frequencies for the medium level (guided procedural) and high symbolic dependence suggest that, although the tasks are mostly symbolic, they do not always constitute a direct application of rigid algorithms. This implies that some understanding of the resolution process is often required, even in symbolic tasks. The analysis reinforces the centrality of symbolic language, already evidenced in Dimension 1, and highlights that, in the integer chapters, conceptual understanding still plays a structuring role, while in equations, symbolic manipulation becomes almost universal.

Dimension 3, which analyzes the conversion between record types, revealed a clear predominance of tasks involving some type of conversion. In all chapters, simple conversion was the most frequent, representing 50% of public integers, 52% of private ones, 74% of public equations, and 72% of private ones. This suggests that, although there is articulation between records, such as verbal language, numerical language, and representation on the number line, this articulation tends to occur in a punctual rather than chained manner. Multiple conversion was rare, with 6% in public integers and 12% in private ones, and absent in equations.

This pattern, especially in the equations chapters, where there were no exercises with multiple conversions, indicates that the transformations between verbal statement and algebraic expression tend to be direct and isolated, not demanding more sophisticated coordination between multiple registers. In general, the dimension shows that the conversion between registers is present consistently, but limited in complexity, which may restrict the development of a more integrated understanding of different modes of mathematical representation, according to Duval’s (2003) argument.

Dimension 4, concerning the semantic nature of the problem, showed in the chapters on integers a significant presence of problems internal to Mathematics, with both procedural focus (38% public, 18% private) and potential for conceptual construction (29% public, 30% private). Questions with significant contexts were also relevant (24% public, 33% private). In the chapters on equations, the significant contextual category became more important (39% public, 44% private), suggesting a greater presence of problems that demand interpretation of contexts for the construction of the resolution strategy.

In general, Dimension 4 suggests that, although there is diversity in the semantic nature of problems, the contexts do not always imply greater conceptual complexity or modeling requirements. This reinforces the need to analyze problems in light of other dimensions for a complete understanding of their cognitive demands. The distinction between internal and contextualized problems is important, but the mere presence of a context does not guarantee that it is significant for the structure of the solution.

Dimension 5, which assesses cognitive demand, revealed a predominance of the category “Explicit strategy”, with 44% in public integers and 36% in private ones, and even more pronounced in equations (65% public, 68% private). “Direct application” was also frequent in integers (32% public, 42% private). The higher complexity categories, such as “Modeling” (9% public, 3% private in integers; 3% public, 4% private in equations) and “Argumentation/justification” (0% public, 3% private in integers; 10% public, 0% private in equations), remained absent or underrepresented in all chapters.

Despite the predominance of the non-explicit strategy, the chapters on integers presented greater diversity in cognitive demand levels compared to the chapters on equations, where this category exceeds 60%. In integers, the high frequency of direct application indicates the presence of tasks based on known procedures. In general, Dimension 5 reveals the predominance of tasks that require the organization of problem-solving strategies, but with a low incidence of activities involving more open modeling or explicit construction of arguments, limiting the development of critical and creative thinking skills.

Finally, Dimension 6, which assesses the development potential of numerical cognition, presented revealing results. In the chapters on integers, the categories “Magnitude” (47% public, 52% private) and “Comparison of magnitudes” (32% public, 21% private) predominate, indicating that the tasks contribute to the understanding of numerical value and relationships between numbers. “Expansion of the mental number line” also appears (12% public, 18% private), reinforcing the importance of this representation. However, categories such as “Estimation” and “Exploration of approximation” are absent, suggesting a lack of activities focused on approximate thinking.

In the chapters on equations, the scenario changes significantly. The “Magnitude” category remains predominant (58% public, 60% private), but with a character more associated with the determination of exact values than with the understanding of numerical relationships. The high incidence of the “Absence of numerical mobilization” category (42% public, 40% private) is notable, indicating that many tasks focus on symbolic transformations without directly mobilizing aspects of numerical cognition. There is a complete absence of “Magnitude comparison”, “Mental number line extension”, “Estimation”, and “Approximation exploration”.

This pattern shows a significant reduction in the diversity of cognitive experiences related to numbers in algebraic content. In general, Dimension 6 reveals a possible disconnection between algebraic work and the development of numerical cognition. When moving on to equations, tasks tend to favor symbolic manipulation over the exploration of broader numerical meanings, which can limit the formation of a more robust and flexible mathematical understanding in students.

The intersection between the dimensions allows for the identification of patterns in the organization of didactic problems. One of the most consistent patterns is the association between the level of symbolic dependence (Dimension 2) and cognitive demand (Dimension 5), especially in algorithmic or direct application tasks. Tasks with high symbolic dependence may only require procedural execution, while tasks with moderate symbolic dependence may require strategic elaboration or modeling. There is a strong association between the medium conceptual-procedural level of symbolic dependence and the explicit strategy category in cognitive demand.

This pattern is very evident in all chapters, but even more accentuated in equation problems. This suggests that tasks requiring some non-fully automated symbolic manipulation demand that the student organize a solution path not previously structured in the problem statement. This indicates that, although there is a challenge in organizing the strategy, the nature of the symbolic demand is still a limiting factor for cognitive complexity, especially in contexts where symbol manipulation is central.

Another revealing pattern emerges from the cross-analysis of dimensions 4 (semantic nature) and 5 (cognitive demand). Questions classified as significant contextual (Dimension 4) are generally associated with non-explicit strategies or direct application (Dimension 5). This relationship suggests that not all significant context promotes high cognitive demand, as the association is practically absent when compared to modeling or argumentation/justification. This means that, even with a present and non-pretextual context, it does not necessarily structure reasoning, demanding modeling or validation.

In summary, a dissociation is observed between the presence of contexts and the level of cognitive elaboration required, indicating that, in many cases, the context does not translate into greater intellectual complexity of the task. The comparison between dimensions 1 (numerical mobilization) and 6 (potential for development of numerical cognition) reveals that the chapters on integers mobilize different aspects of numerical cognition, with coherence between what the task mobilizes and what it develops, associating, for example, comparison of magnitudes and number line in both cases.

In the chapters on equations, on the other hand, symbolic manipulation and procedural calculation predominate in Dimension 1, associated, in Dimension 6, with the development of magnitude or the absence of numerical mobilization. This indicates that the transition to algebraic language seems to imply a reduction in the diversity of experiences related to numerical cognition. In these situations, the number ceases to be treated as a quantity to be compared or located and begins to operate as a formal element in algebraic relations, significantly reducing the potential for mobilizing numerical cognition.

The intersection between dimensions 3 (record conversion) and 5 (cognitive demand) aligns with Duval (2003): although most tasks involve some type of conversion, this occurs predominantly in a simple and linear way, associated with intermediate or low levels of cognitive demand. This pattern suggests that conversion, in these cases, is not configured as a coordination process between records, but as a direct and procedural translation. The low incidence of multiple conversions indicates a limitation in exploring situations that require more complex articulation between representations.

This can restrict the development of a more flexible and integrated mathematical understanding. In summary, it is observed, in the four chapters analyzed, that conversion is present as a form, but not with a strong cognitive function. Finally, the comparative analysis of dimensions 4 (semantic nature of the problem) and 2 (level of symbolic dependence) shows that the presence of contexts, even when classified as significant, hardly alters the type of symbolic demand of the tasks.

Most problems, regardless of whether they are internal or contextualized, are concentrated at the medium level of symbolic dependency. Thus, context, in many cases, does not act as a structuring element capable of diversifying strategies or reducing the centrality of symbolic manipulation, functioning predominantly as an external layer to the mathematical organization of the task. This result suggests a relative independence between the semantic dimension and the nature of the symbolic activity required, which may limit the potential of problems to promote more flexible approaches that are less dependent on formal procedures.

In summary, the research identified that, although the didactic materials consider different levels of cognitive demand, there is a predominance of tasks focused on symbolic manipulation and procedural execution, with low occurrence of activities that mobilize estimation, approximation, and comparison of magnitudes. The diversity of cognitive experiences associated with numbers significantly decreases in algebraic content, suggesting a disconnection between the development of numerical cognition and the formal teaching of Mathematics. The results reinforce the need to conceive mathematical problems as devices for cognitive development, which promote the articulation between registers, the construction of meanings, and the mobilization of multiple cognitive mechanisms to strengthen mathematical literacy and numerical cognition.

4. Conclusion

This research analyzed the mutual relationship between numerical cognition mechanisms and mathematical literacy, focusing on problem-solving in the final years of Elementary School. It was found that the mathematical problems present in textbooks, although considering different levels of cognitive demand, prioritize symbolic manipulation and procedural execution. A low incidence of tasks that mobilize estimation, approximation, or multiple representations, crucial elements for the development of a robust number sense, was observed. Furthermore, a significant reduction in the mobilization of numerical cognition aspects in algebraic content was noted, where the number began to be treated predominantly as a formal element in abstract relations. This pattern indicated a possible disconnection between the development of numerical cognition and the analyzed didactic practices, suggesting that the structure of the problems does not always favor the consolidation and strengthening of these cognitive structures.

The study’s main contribution lies in the development of an analytical matrix that allows for a qualitative examination of the cognitive demands of mathematical tasks, fostering dialogue between cognitive neuroscience and the educational field. The results offer subsidies for planning didactic interventions that can enhance cognitive processes associated with mathematical learning, promoting evidence-based pedagogy. However, the analysis was limited to 123 seventh-grade textbook questions, focusing on the topics of integers and first-degree equations, which restricts the generalization of the findings. For future studies, it is suggested to refine the analysis instrument and expand the investigation to other content areas and educational levels, as well as to explore pedagogical practices that more systematically integrate the foundations of numerical cognition into Mathematics teaching.

Bibliographic References

ANDRADE, Paulo Estêvão; PRADO, Paulo Sérgio Teixeira do; CARMO, João dos Santos. Das representações numéricas inatas à matemática culturalmente construída. Temas em Psicologia, v. 23, n. 1, p. 225–242, 2015.

DEHAENE, Stanislas. The number sense: how the mind creates mathematics. New York: Oxford University Press, 1997.

DUVAL, R. Registros de representação semiótica e funcionamento cognitivo da compreensão em matemática. In: MACHADO, S. D. A. (Org.). Aprendizagem em matemática: registros de representação semiótica. Campinas: Papirus, 2003. p. 11-33.

FEIGENSON, Lisa; DEHAENE, Stanislas; SPELKE, Elizabeth. Core systems of number. Trends in Cognitive Sciences, v. 8, n. 7, p. 307–314, 2004.

GORDON, P. Numerical cognition without words: evidence from Amazonia. Science, v. 306, n. 5695, p. 496–499, 2004.

Kucian, K. & von Aster, M. (2015). Developmental dyscalculia. European Journal of Pediatrics, 174(1), 1-13.

MACHADO, Nilson José. Matemática e língua materna: análise de uma impregnação mútua. São Paulo: Cortez, 2011.

MACHADO, Nilson José. Matemática e realidade: das concepções às ações docentes. São Paulo: Cortez, 2013.

MARCILESE, Maria Mercedes. O papel da língua no desenvolvimento da cognição numérica. Alfa – Revista de Linguística, v. 56, n. 4, p. 671-686, 2012. Disponível em: https://www.scielo.br/j/alfa/a/FkxDgpzMMKCMSTXKsXqwnML/?lang=pt. Acesso em: [24 jan 2026].

OCDE. PISA 2022 Assessment and Analytical Framework. Paris: OECD Publishing, 2023. DOI: 10.1787/dfe0bf9c-em.

OECD. PISA 2018 Assessment and Analytical Framework: Mathematics, Reading, Science and Financial Literacy. Paris: OECD Publishing, 2019.

PIAZZA, Manuela. Neural and developmental mechanisms of number representation and processing. Current Opinion in Neurobiology, v. 20, n. 2, p. 177–182, 2010.

Programa Nacional do Livro Didático, 2023 [Referência completa não encontrada no documento original]

SANTOS, F. H.; SILVA, P. A.; OLIVEIRA, R. M. Cognição numérica e educação matemática: interfaces entre neurociência e ensino. Revista Psicologia: Teoria e Prática, v. 18, n. 3, p. 76–89, 2016.

VON ASTER, M.; SHALEV, R. S. Number development and developmental dyscalculia. Developmental Medicine and Child Neurology, London, v. 49, n. 11, p. 868–873, 2007.

Article originating from a Final Course Work on Neuroscience and Learning in Education

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