Diverse mathematical knowledge among indigenous Amazonians.

We investigate number and arithmetic learning among a Bolivian indigenous people, the Tsimane', for whom formal schooling is comparatively recent in history and variable in both extent and consistency. We first present a large-scale meta-analysis on child number development involving over 800 Tsimane' children. The results emphasize the impact of formal schooling: Children are only found to be full counters when they have attended school, suggesting the importance of cultural support for early mathematics. We then test especially remote Tsimane' communities and document the development of specialized arithmetical knowledge in the absence of direct formal education. Specifically, we describe individuals who succeed on arithmetic problems involving the number five-which has a distinct role in the local economy-even though they do not succeed on some lower numbers. Some of these participants can perform multiplication with fives at greater accuracy than addition by one. These results highlight the importance of cultural factors in early mathematics and suggest that psychological theories of number where quantities are derived from lower numbers via repeated addition (e.g., a successor function) are unlikely to explain the diversity of human mathematical ability.

Evaluating the impact of short educational videos on the cortical networks for mathematics.

Many teaching websites, such as the Khan Academy, propose vivid videos illustrating a mathematical concept. Using functional magnetic resonance imaging, we asked whether watching such a video suffices to rapidly change the brain networks for mathematical knowledge. We capitalized on the finding that, when judging the truth of short spoken statements, distinct semantic regions activate depending on whether the statements bear on mathematical knowledge or on other domains of semantic knowledge. Here, participants answered such questions before and after watching a lively 5-min video, which taught them the rudiments of a new domain. During the video, a distinct math-responsive network, comprising anterior intraparietal and inferior temporal nodes, showed intersubject synchrony when viewing mathematics course rather than control courses in biology or law. However, this experience led to minimal subsequent changes in the activity of those domain-specific areas when answering questions on the same topics a few minutes later. All taught facts, whether mathematical or not, led to domain-general repetition enhancement, particularly prominent in the cuneus, posterior cingulate, and posterior parietal cortices. We conclude that short videos do not suffice to induce a meaningful lasting change in the brain's math-responsive network, but merely engage domain-general regions possibly involved in episodic short-term memory.

Spatiotemporal Dynamics of Successive Activations across the Human Brain during Simple Arithmetic Processing.

Previous neuroimaging studies have offered unique insights about the spatial organization of activations and deactivations across the brain; however, these were not powered to explore the exact timing of events at the subsecond scale combined with a precise anatomical source of information at the level of individual brains. As a result, we know little about the order of engagement across different brain regions during a given cognitive task. Using experimental arithmetic tasks as a prototype for human-unique symbolic processing, we recorded directly across 10,076 brain sites in 85 human subjects (52% female) using the intracranial electroencephalography. Our data revealed a remarkably distributed change of activity in almost half of the sampled sites. In each activated brain region, we found juxtaposed neuronal populations preferentially responsive to either the target or control conditions, arranged in an anatomically orderly manner. Notably, an orderly successive activation of a set of brain regions-anatomically consistent across subjects-was observed in individual brains. The temporal order of activations across these sites was replicable across subjects and trials. Moreover, the degree of functional connectivity between the sites decreased as a function of temporal distance between regions, suggesting that the information is partially leaked or transformed along the processing chain. Our study complements prior imaging studies by providing hitherto unknown information about the timing of events in the brain during arithmetic processing. Such findings can be a basis for developing mechanistic computational models of human-specific cognitive symbolic systems.

A bird's-eye view of research practices in mathematical cognition, learning, and instruction: Reimagining the status quo.

Research on mathematical cognition, learning, and instruction (MCLI) often takes cognition as its point of departure and considers instruction at a later point in the research cycle. In this article, we call for psychologists who study MCLI to reflect on the "status quo" of their research practices and to consider making instruction an earlier and more central aspect of their work. We encourage scholars of MCLI (a) to consider the needs of educators and schools when selecting research questions and developing interventions; (b) to compose research teams that are diverse in the personal, disciplinary, and occupational backgrounds of team members; (c) to make efforts to broaden participation in research and to conduct research in authentic settings; and (d) to communicate research in ways that are accessible to practitioners and to the general public. We argue that a more central consideration of instruction will lead to shifts that make research on MCLI more theoretically valuable, more actionable for educators, and more relevant to pressing societal challenges.

Uncovering the interplay between drawings, mental representations, and arithmetic problem-solving strategies in children and adults.

There is an ongoing debate in the scientific community regarding the nature and role of the mental representations involved in solving arithmetic word problems. In this study, we took a closer look at the interplay between mental representations, drawing production, and strategy choice. We used dual-strategy isomorphic word problems sharing the same mathematical structure, but differing in the entities they mentioned in their problem statement. Due to the non-mathematical knowledge attached to these entities, some problems were believed to lead to a specific (cardinal) encoding compatible with one solving strategy, whereas other problems were thought to foster a different (ordinal) encoding compatible with the other solving strategy. We asked 59 children and 52 adults to solve 12 of those arithmetic word problems and to make a diagram of each problem. We hypothesized that the diagrams of both groups would display prototypical features indicating either a cardinal representation or an ordinal representation, depending on the entities mentioned in the problem statement. Joint analysis of the drawing task and the problem-solving task showed that the cardinal and ordinal features of the diagrams are linked with the hypothesized semantic properties of the problems and, crucially, with the choice of one solving strategy over another. We showed that regardless of their experience, participants' strategy use depends on their problem representation, which is influenced by the non-mathematical information in the problem statement, as revealed in their diagrams. We discuss the relevance of drawing tasks for investigating mental representations and fostering mathematical development in school.

Visual foundations of Euclidean geometry.

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as "natural". We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry - i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3-34 years), and 25 participants from the Amazon (age 5-67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in "sense"). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans' knowledge in Euclidean geometry could possibly be grounded.

Do children with mathematical learning disabilities use the inversion principle to solve three-term arithmetic problems?: The impact of presentation mode.

Numerical inversion is the ability to understand that addition is the opposite of subtraction and vice versa. Three-term arithmetic problems can be solved without calculation using this conceptual shortcut. To verify that this principle is used, inverse problems (a + b - b) can be compared with standard problems (a + b - c). If this principle is used, performance on inverse problems will be higher than performance on standard problems because no calculation is required. To our knowledge, this principle has not been previously studied in children with mathematical learning disabilities (MLD). Our objectives were (a) to study whether 10-year-olds with MLD are able to use this conceptual principle in three-term arithmetic problems and (b) to evaluate the impact of the presentation mode. A total of 64 children with or without MLD solved three-term arithmetic problems (inverse and standard) in two presentation modes (symbolic and picture). The results showed that even though children with MLD have difficulties in performing arithmetic problems, they can do so when the inverse problem is presented with pictures. The picture presentation mode allowed children with MLD to efficiently identify and use the conceptual inversion shortcut and thus to achieve a similar performance to that of typically developing children. These results provide interesting perspectives for the care of children with MLD.

A primarily serial, foveal accumulator underlies approximate numerical estimation.

The approximate number system (ANS) has attracted broad interest due to its potential importance in early mathematical development and the fact that it is conserved across species. Models of the ANS and behavioral measures of ANS acuity both assume that quantity estimation is computed rapidly and in parallel across an entire view of the visual scene. We present evidence instead that ANS estimates are largely the product of a serial accumulation mechanism operating across visual fixations. We used an eye-tracker to collect data on participants' visual fixations while they performed quantity-estimation and -discrimination tasks. We were able to predict participants' numerical estimates using their visual fixation data: As the number of dots fixated increased, mean estimates also increased, and estimation error decreased. A detailed model-based analysis shows that fixated dots contribute twice as much as peripheral dots to estimated quantities; people do not "double count" multiply fixated dots; and they do not adjust for the proportion of area in the scene that they have fixated. The accumulation mechanism we propose explains reported effects of display time on estimation and earlier findings of a bias to underestimate quantities.

Number line development of Chilean children from preschool to the end of kindergarten.

Children's performance on number line tasks reflects their developing number system knowledge. Before 5 years of age, most children perform poorly on even the simplest number lines (i.e., 0-10). Our goal was to understand how number line skills develop before formal schooling. Chilean preschoolers attempted a 0-10 number line task three times over 2 years: at the beginning of pre-kindergarten (M = 4:7 [years;months]; Time 1), at the end of pre-kindergarten (M = 5:0; Time 2), and at the end of kindergarten (M = 5:10; Time 3). We used latent class analysis to group children according to their patterns of performance across number targets. At Time 1, 86% of children had error patterns indicating that they randomly placed estimates on the line. At Time 2, 56% of children continued to respond randomly. At Time 3, 56% of children showed competent performance across the number line, 23% were accurate only near the endpoints, and 21% were accurate only for low target numbers near the origin. Latent transition analyses showed that the transition from less to more proficient estimation classes was predicted by children's number identification skills. Thus, number line performance changed dramatically from 4 to 6 years of age as children began to develop the cognitive and numerical skills necessary to accurately estimate numbers on a number line.

Non-symbolic and symbolic number lines are dissociated.

People use mental number lines for both symbolic numerals and numerosity, but little is known about how these two mental number lines are related. The current study investigated the association in effect size, directionality of the mental number line, and development between symbolic and non-symbolic mental number lines to determine if they were related to or independent from each other. We collected data from numerosity- and digit-matching tasks that used the following numbers: 11, 14, 17, 20, 23, 26, and 29. Tasks were performed by college undergraduates and the fifth-grade primary school students. The results showed that none of the effects for non-symbolic numbers was related to any of the effects for symbolic numbers, and vice versa, in both adults and children. Another notable finding was that the correlation between the SNARC (spatial-numerical association of response code) effect size and mathematical ability was negative in the adult group. These results are consistent with the dissociated processes hypothesis and suggest that mental number lines are notation-dependent. As shown by the SNARC effect, the mental number line might result in interference in the current task by an automatically activated spatial notation-dependent representation of numbers.

Mapping subcomponents of numerical cognition in relation to functional and anatomical landmarks of human parietal cortex.

Human functional imaging has identified the middle part of the intraparietal sulcus (IPS) as an important brain substrate for different types of numerical tasks. This area is often equated with the macaque ventral intraparietal area (VIP) where neuronal selectivity for non-symbolic numerical stimuli (sets of items) is found. However, the low spatial resolution and whole-brain averaging analysis performed in most fMRI studies limit the extent to which an exact correspondence of activations in different numerical tasks with specific sub-regions of the IPS can be established. Here we acquired high-resolution 7T fMRI data in a group of human adults and related the activations in several numerical contrasts (implying different numerical stimuli and tasks) to anatomical and functional landmarks on the cortical surface. Our results reveal a functional heterogeneity within human intraparietal cortex where the retinotopic visual field maps in superior/medial parts of the IPS and superior parietal gyrus respond preferentially to the visual processing of concrete sets of items (over single Arabic numerals), whereas lateral/inferior parts of the IPS are predominantly recruited during numerical operations such as calculation and quantitative comparison. Since calculation and comparison-related activity fell mainly outside the retinotopic visual field maps considered the human functional equivalent of the monkey VIP/LIP complex, the areas most activated during such numerical operations in humans are likely different from VIP.

Further insights into the operation of the Chinese number system: Competing effects of Arabic and Mandarin number formats.

Here we report the results of a speeded relative quantity task with Chinese participants. On each trial a single numeral (the probe) was presented and the instructions were to respond as to whether it signified a quantity less than or greater than five (the standard). In separate blocks of trials, the numerals were presented either in Mandarin or in Arabic number formats. In addition to the standard influence of numerical distance, a significant predictor of performance was the degree of physical similarity between the probe and the standard as depicted in Mandarin. Additionally, competing effects of physical similarity, defined in terms of the Arabic number format, were also found. Critically the size of these different effects of physical similarity varied systematically across individuals such that larger effects of one compensated for smaller effects of the other. It is argued that the data favor accounts of processing that assume that different number formats access different format-specific representations of quantities. Moreover, for Chinese participants the default is to translate numerals into a Mandarin format prior to accessing quantity information. The efficacy of this translation process is itself influenced by a competing tendency to carry out a translation into Arabic format.

Intuitive errors in learners' fraction understanding: A dual-process perspective on the natural number bias.

Although a good rational number understanding is very important, many learners struggle to understand fractions. Recent research attributes many of these difficulties to the natural number bias - the tendency to apply natural number features in rational number tasks where this is inappropriate. Previous correlational dual process studies found evidence for the intuitive nature of the natural number bias in learners' response latencies. However, the reported correlations do not ascertain the causality that is assumed in this ascription. In the present study we therefore experimentally elicited intuitive responses in a fraction comparison task in educated adults by restricting reaction time. Results show that the natural number bias has an intuitive character. Findings also indicate that the detection of conflict between the natural number-based answer and the correct answer seems to work at an intuitive level.

Semantic associations between arithmetic and space: Evidence from temporal order judgements.

Spatial biases associated with subtraction or addition problem solving are generally considered as reflecting leftward or rightward attention shifts along a mental numerical continuum, but an alternative hypothesis not implying spatial attention proposes that the operator (plus or minus sign) may favour a response to one side of space (left or right) because of semantic associations. We tested these two accounts in a series of temporal order judgement experiments that consisted in the auditory presentation of addition or subtraction problems followed 200 ms (Experiments 1-2) or 800 ms (Experiment 3) later by the display of two lateralized targets in close temporal succession. To dissociate the side where the operation first brought their attention from the side they had to respond to, we asked participants to report which of the left or right target appeared first or last on screen. Under the attention-orienting account, addition should elicit more rightward responses than subtraction when participants have to focus on the first target, but more leftward responses when they have to focus on the last target, because the latter is opposite to the side where the operation first brought their attention. Under the semantic account, addition should elicit more rightward responses than subtraction, no matter the focus is on the first or last target, because participants should systematically favour the side conceptually linked to the operator. The results of the three experiments converge to indicate that, in lateralized target detection tasks, the spatial biases induced by arithmetic operations stem from semantic associations.

Visual form perception predicts 3-year longitudinal development of mathematical achievement.

Numerous studies have demonstrated an association between approximate number system (ANS) acuity and mathematical performance. Studies have also shown that ANS acuity can predict the longitudinal development of mathematical achievement. Visual form perception in the current investigation was proposed to account for the predictive role of ANS acuity in the development of mathematical achievement. One hundred and eighty-eight school children (100 males, 88 females; mean age = 12.2 ± 0.3 years) participated in the study by completing five tests: numerosity comparison, figure matching, mental rotation, nonverbal matrix reasoning, and choice reaction time. Three years later, they took a mathematical achievement test. We assessed whether the early tests predicted mathematical achievement at the later date. Analysis showed that the ANS acuity measured via numerosity comparison significantly predicted mathematical achievement 3 years later, even when controlling for individual differences in mental rotation, nonverbal matrix reasoning, and choice reaction time, as well as age and gender differences. Hierarchical regression and mediation analyses further showed that the longitudinal predictive role of ANS acuity in mathematical achievement was interpreted by visual form perception measured with a figure-matching test. Together, these results indicate that visual form perception may be the underlying cognitive mechanism that links ANS acuity to mathematical achievement in terms of longitudinal development.

Extended mathematical cognition: external representations with non-derived content.

Vehicle externalism maintains that the vehicles of our mental representations can be located outside of the head, that is, they need not be instantiated by neurons located inside the brain of the cogniser. But some disagree, insisting that 'non-derived', or 'original', content is the mark of the cognitive and that only biologically instantiated representational vehicles can have non-derived content, while the contents of all extra-neural representational vehicles are derived and thus lie outside the scope of the cognitive. In this paper we develop one aspect of Menary's vehicle externalist theory of cognitive integration-the process of enculturation-to respond to this longstanding objection. We offer examples of how expert mathematicians introduce new symbols to represent new mathematical possibilities that are not yet understood, and we argue that these new symbols have genuine non-derived content, that is, content that is not dependent on an act of interpretation by a cognitive agent and that does not derive from conventional associations, as many linguistic representations do.

A distinct cortical network for mathematical knowledge in the human brain.

How does the brain represent and manipulate abstract mathematical concepts? Recent evidence suggests that mathematical processing relies on specific brain areas and dissociates from language. Here, we investigate this dissociation in two fMRI experiments in which professional mathematicians had to judge the truth value of mathematical and nonmathematical spoken statements. Sentences with mathematical content systematically activated bilateral intraparietal sulci and inferior temporal regions, regardless of math domain, problem difficulty, and strategy for judging truth value (memory retrieval, calculation or mental imagery). Second, classical language areas were only involved in the parsing of both nonmathematical and mathematical statements, and their activation correlated with syntactic complexity, not mathematical content. Third, the mere presence, within a sentence, of elementary logical operators such as quantifiers or negation did not suffice to activate math-responsive areas. Instead, quantifiers and negation impacted on activity in right angular gyrus and left inferior frontal gyrus, respectively. Overall, these results support the existence of a distinct, non-linguistic cortical network for mathematical knowledge in the human brain.

Shared and distinct neural circuitry for nonsymbolic and symbolic double-digit addition.

Symbolic arithmetic is a complex, uniquely human ability that is acquired through direct instruction. In contrast, the capacity to mentally add and subtract nonsymbolic quantities such as dot arrays emerges without instruction and can be seen in human infants and nonhuman animals. One possibility is that the mental manipulation of nonsymbolic arrays provides a critical scaffold for developing symbolic arithmetic abilities. To explore this hypothesis, we examined whether there is a shared neural basis for nonsymbolic and symbolic double-digit addition. In parallel, we asked whether there are brain regions that are associated with nonsymbolic and symbolic addition independently. First, relative to visually matched control tasks, we found that both nonsymbolic and symbolic addition elicited greater neural signal in the bilateral intraparietal sulcus (IPS), bilateral inferior temporal gyrus, and the right superior parietal lobule. Subsequent representational similarity analyses revealed that the neural similarity between nonsymbolic and symbolic addition was stronger relative to the similarity between each addition condition and its visually matched control task, but only in the bilateral IPS. These findings suggest that the IPS is involved in arithmetic calculation independent of stimulus format.

Metric error monitoring in the numerical estimates.

Recent studies have shown that participants can keep track of the magnitude and direction of their errors while reproducing target intervals (Akdogan & Balci, 2017) and producing numerosities with sequentially presented auditory stimuli (Duyan & Balci, 2018). Although the latter work demonstrated that error judgments were driven by the number rather than the total duration of sequential stimulus presentations, the number and duration of stimuli are inevitably correlated in sequential presentations. This correlation empirically limits the purity of the characterization of "numerical error monitoring". The current work expanded the scope of numerical error monitoring as a form of "metric error monitoring" to numerical estimation based on simultaneously presented array of stimuli to control for temporal correlates. Our results show that numerical error monitoring ability applies to magnitude estimation in these more controlled experimental scenarios underlining its ubiquitous nature.

Origins of the brain networks for advanced mathematics in expert mathematicians.

The origins of human abilities for mathematics are debated: Some theories suggest that they are founded upon evolutionarily ancient brain circuits for number and space and others that they are grounded in language competence. To evaluate what brain systems underlie higher mathematics, we scanned professional mathematicians and mathematically naive subjects of equal academic standing as they evaluated the truth of advanced mathematical and nonmathematical statements. In professional mathematicians only, mathematical statements, whether in algebra, analysis, topology or geometry, activated a reproducible set of bilateral frontal, Intraparietal, and ventrolateral temporal regions. Crucially, these activations spared areas related to language and to general-knowledge semantics. Rather, mathematical judgments were related to an amplification of brain activity at sites that are activated by numbers and formulas in nonmathematicians, with a corresponding reduction in nearby face responses. The evidence suggests that high-level mathematical expertise and basic number sense share common roots in a nonlinguistic brain circuit.