Children’s mathematical development involves acquiring a system of knowledge, such as basic addition and multiplication facts, as well as conceptual understandings of mathematics, such as the structure of the base-ten system or equality (Geary, 1994). From this perspective, knowledge is information that can be applied in problem-solving or related contexts, such as knowing how to trade from one column to the next when solving complex arithmetic problems (e.g., 38 + 45), whereas conceptual understanding involves more abstract features of mathematics that are generalizable across contexts and accurately representable across modalities. An example is the equality represented by the ‘=’ means the same thing across mathematical contexts, even if different patterns of knowledge are needed to solve problems in each of these contexts.

Cognitive, instructional, and brain-imaging studies have revealed much about students’ understanding and misunderstanding of core mathematics concepts (Alibali et al., 2007; Ansari, 2008; Fuson et al., 1997; Siegler & Braithwaite, 2017; Ünal et al., 2024), but these leave unaddressed the issue of how humans are able to form abstract mathematical concepts, such as the meaning of the ‘=’ sign, at all when other species cannot (Beran et al., 2015). It is possible that the system that supports sensitivity to approximate magnitudes – approximate number system (ANS) (Feigenson et al., 2004) – is recycled or remodeled to support abstract mathematics (Dehaene & Cohen, 2007), but this does not appear to be sufficient. This is because it does not explain many conceptual insights (e.g., that each successive number in the count string is exactly one more than the number before it) that emerge during mathematical development (Carey et al., 2017; Cheung et al., 2017; Chu et al., 2015; Geary & vanMarle, 2018; Shusterman et al., 2016), although it may contribute to the emergence of children’s understanding of the quantities represented by small numbers (vanMarle et al., 2018).

The assumption here is that abstract, formal mathematics is an evolutionarily novel, cultural invention (Geary, 1995, 2024) and, with the exception of the ANS, does not have an evolved system to support its learning. For evolved domains (e.g., involving social dynamics), the formation of explicitly accessible (e.g., can be verbally described) and generalizable concepts appears to depend on the controlled semantic cognition system that has been elaborated during hominid evolution. An appreciation of how this system builds generalizable understandings provides a unique and overlooked perspective on mathematical development. I overview the basic features of this system in the first section and in the second integrate these with brain imaging and cognitive research on mathematical development. More practically, features of the controlled semantic cognition system include statistical learning, which has implications for structuring mathematics instruction and identifying the source of common conceptual understandings and misunderstandings.

“Semantic cognition refers to our ability to use, manipulate and generalize knowledge that is acquired over the lifespan to support innumerable verbal and non-verbal behaviors” (Lambon Ralph et al., 2017, p. 42). The system supports the ability to categorize different features of the world, such as dogs and cats, and form generalizations based on related experiences. These experiences extract statistical regularities – statistical learning – across episodes, contexts, and exemplars that eventually form cognitive and neural representations that capture common features of the category or concept (Raviv et al., 2022). The development of a conceptual understanding of the core features of dogs, based on frequent and variable interactions with them, allows these features to be integrated cross modalities such that this conceptual understanding can be accessed through auditory (e.g., bark), visual, and action (e.g., jumping on someone while standing on hind legs) cues. The automatic identification of commonalities (e.g., dogs and cats have a similar four-legged body) and differences (e.g., dogs are more sociable) across similar entities supports higher-order abstractions (e.g., animals) and a fine-grained understanding of specific concepts.

There is still debate regarding how these categorical and conceptual understandings are represented in the brain (Humphreys et al., 2021; Lambon Ralph et al., 2017; Mahon & Caramazza, 2011; Martin, 2016; Pobric et al., 2007), but Lambon Ralph and colleagues’ hub-spoke model captures the basic idea. Following Warrington and Shallice (1984), the transmodal or amodal representations that capture the core conceptual features of the category are represented in the bilateral anterior temporal lobe (ATL), and for some concepts more posterior temporal regions, and is part of a wider controlled semantic cognition system (Humphreys et al., 2021). The center of Figure 1 shows the transmodal hub that is integrated with domain-specific systems. The latter support modality-specific representations of specific instances, as in naming a picture of a dog or identifying the animal by sound (e.g., bark). The hub can also act to integrate the expression of concepts across modalities, such that verbal descriptions of the category might be accompanied by actions (e.g., gestures) that highlight the core categorical features.

Pre-existing white matter tracks integrate the ventrolateral (bottom, outer) ATL with areas that support these domain-specific competencies (e.g., speech) as well as with prefrontal and parietal lobes associated with cognitive and attentional control. The control component involves the use of categorical distinctions and conceptual understandings in here-and-now contexts. Accessing this semantic information engages several areas of the frontal cortex, including the bilateral (but left dominant) inferior frontal gyrus and the dorsomedial prefrontal cortex, as well as portions of the temporal and parietal cortices (Jackson, 2021; Noonan et al., 2013). The control component partially overlaps the domain-general fronto-parietal cognitive control system (Fedorenko et al., 2013; Menon & D’Esposito, 2022). The extent to which this cognitive-control network is engaged, however, depends on the complexity and familiarity of the context-specific demands. A master mechanic will readily (without engagement of the cognitive control networks) identify the tools and actions needed to fix an engine, but his novice apprentice will need to explicitly engage relevant knowledge to estimate (and test) the best course of action. Moreover, the semantic system overlaps the default mode network, and this enables use of conceptual and other internally represented information to be used for planning ahead when not focused on specific external tasks (Binder et al., 2009).

The representations in the transmodal hub are graded but are systematically organized (Figure 2). The organization here is simplified but captures general patterns (Binder et al., 2009; Braunsdorf et al., 2021; Hung et al., 2020). More ventral (bottom) areas are integrated with higher-order visual (posterior temporal and occipital) regions and engaged with processing visual information such as faces and objects. More dorsal (top) areas are integrated with auditory (e.g., music) and language regions and engaged by this type of information, although the language bias is more in the left hemisphere (see Blazquez Freches et al., 2020). The posterior regions are engaged with more concrete information, such as the features of an individual face or the words of a simple utterance, as well as aspects (along with parietal areas) of action knowledge (e.g., how to use a tool), whereas the anterior lateral (outer) portions of the ATL involve increasingly complex concepts, especially as related to living things (Campanella et al., 2010; Humphreys & Riddoch, 2003). These would range from understanding the gist of a verbal argument to mentalizing or making inferences about the emotional state or intentions of someone based on more concrete cues (e.g., facial expression, body posture).

The controlled semantic cognition system has been largely overlooked in studies of mathematical development and has not been consistently identified in brain imaging studies of number and arithmetic processing (first section below), but it is nonetheless a candidate for contributing to children’s emerging conceptual understanding of mathematics. This is because the system is involved in conceptual learning across many domains and this learning emerges automatically from statistical regularities across related experiences. The latter means the system will automatically extract generalizable regularities across these experiences, and it is unlikely that the system is not engaged during the more than a decade of exposure to mathematics.

If so, then children’s conceptual understandings and misunderstandings of mathematics should be strongly influenced by how mathematics problems are presented during instruction and in textbooks (second section below), given the statistical learning property of the controlled semantic cognition system. The hub-and-spoke organization of the system not only explains but anticipates that mathematical concepts can be expressed verbally, visually, and with gesture. Finally, the evolutionary integration of this system with the cognitive control systems means that people are able to access conceptual understandings and represent them multimodally (e.g., explain them to others) when asked to do so, that is, outside of the contexts in which the conceptual understandings emerged. More generally, we would expect that individual differences in the emergence of mathematical knowledge and conceptual understandings will be strongly influenced by executive functions and working memory, and this is indeed the case (Bull & Lee, 2014; Geary et al., 2023; Geary et al., 2017; Lee & Bull, 2016).

At the end of secondary schooling, the mathematical competencies that students take with them to the workplace or to higher education have long-term influences on their occupational, financial, and personal well-being (Reyna et al., 2009; Ritchie & Bates, 2013). Given this, it is not surprising that modern societies devote considerable resources to teaching mathematics, developing associated curricula, and studying how children learn and understand mathematics (National Mathematics Advisory Panel, 2008). The bulk of these efforts has been guided by pragmatics, such as determining the mathematics children need to know to prepare them for adulthood and presenting this material to children to study how they come to learn it. These are useful and productive approaches, but can be augmented by stepping back and, in this case, considering the broader – including evolutionary – nature of learning (Geary, 1995, 2024). In this case, distinguishing between knowledge – facts and procedures that are used piecemeal in problem-solving contexts – and conceptual understanding – abstract representations of features that are common across related experiences or problems that can be correctly generalized and can be expressed through language, visually, or in gesture. Issues related to knowledge and conceptual understanding have been the focus of much, often vigorous, debate within and across the mathematics education and the mathematics learning and cognition communities, which is sometimes dubbed the math wars (Loveless, 2001; National Mathematics Advisory Panel, 2008).

My strategy is to approach the question by first examining the brain and cognitive systems that support conceptual learning and understanding across domains and species and thus to focus, in part, on the controlled semantic cognition system (Lambon Ralph et al., 2017). If this approach is viable, then the general properties of this system should align with the results from empirical studies that were conducted without explicit consideration of this system. Brain imaging findings that implicate areas of the intraparietal sulcus as critical to number processing (e.g., determining that 7 > 4) and simple arithmetic (4 + 2 = 6, yes or no) appear to undermine the viability of my approach (Ansari, 2008; Dehaene et al., 2003). As noted, however, performance on these types of tasks do not seem to require much conceptual understanding of mathematics, at least for adults (Cappelletti et al., 2001), and studies that involve access to such conceptual knowledge do appear to engage the temporal regions that are part of the controlled semantic cognition system (Amalric & Dehaene, 2019; Liu et al., 2019).

The results of behavioral studies, in contrast, are more consistent with engagement of the semantic cognition system, as these have revealed that the frequency and ways in which mathematical problems are presented strongly influence students’ either implicit (e.g., expressed through gesture) or explicit (e.g., verbally stated) understanding (correct or not) of mathematical concepts (McNeil et al., 2006; Siegler, 2024). These findings are in keeping with the statistical learning property of the system and speak to a core issue in the math wars; whether learning should focus on concepts and discovery learning or involve solving lots of problems (sometimes referred to as drill and kill). Although the interrelation between procedural skills and concept learning has been recognized for some time (Rittle-Johnson et al., 2001), the statistical learning approach adds to this: Generalizable concept learning requires repeated and varied exposure to problems that are exemplars of the same and similar mathematical concepts (to set boundary conditions). Students can certainly memorize verbal definitions of concepts, but this is not the same as a deep conceptual understanding of them. If the controlled semantic cognition system is engaged during mathematics learning, then conceptual understanding should be expressible verbally, visually, and through gesture; this seems to be the case (Alibali & Goldin-Meadow, 1993).

The evolutionary framing is also important because abstract mathematical concepts, such as equality or variables, are evolutionarily novel. Thus, there are not built in perceptual, cognitive, or motivational biases to ensure that students seek out and engage in the range of experiences needed to acquire these concepts through statistical learning (Geary, 2024). Unguided discovery learning will almost certainly result in a biased and incomplete sampling of the types of problems and contexts needed to fully develop mathematical concepts. Exposure to these appropriate problems and contexts must come from instructional materials (e.g., classroom experiences, textbooks) and related experiences outside of formal materials (e.g., at home, with tutors). As reviewed by Siegler (2024), the current patterns of problem exposure typically are not sufficient to ensure a full conceptual understanding of the associated concept, such as equality (McNeil et al., 2006) and often results in an inaccurate or incomplete (i.e., not fully generalizable) understanding. Finally, the hub-and-spoke organization of the controlled semantic cognition system suggests that assessments of students’ conceptual understandings should be multi-modal, with the goal of achieving converging mathematically correct expressions of the concept across these modalities.