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.

Sensitivity to geometric shape regularity in humans and baboons: A putative signature of human singularity.

Among primates, humans are special in their ability to create and manipulate highly elaborate structures of language, mathematics, and music. Here we show that this sensitivity to abstract structure is already present in a much simpler domain: the visual perception of regular geometric shapes such as squares, rectangles, and parallelograms. We asked human subjects to detect an intruder shape among six quadrilaterals. Although the intruder was always defined by an identical amount of displacement of a single vertex, the results revealed a geometric regularity effect: detection was considerably easier when either the base shape or the intruder was a regular figure comprising right angles, parallelism, or symmetry rather than a more irregular shape. This effect was replicated in several tasks and in all human populations tested, including uneducated Himba adults and French kindergartners. Baboons, however, showed no such geometric regularity effect, even after extensive training. Baboon behavior was captured by convolutional neural networks (CNNs), but neither CNNs nor a variational autoencoder captured the human geometric regularity effect. However, a symbolic model, based on exact properties of Euclidean geometry, closely fitted human behavior. Our results indicate that the human propensity for symbolic abstraction permeates even elementary shape perception. They suggest a putative signature of human singularity and provide a challenge for nonsymbolic models of human shape perception.

Common Neural Functions during Children's Learning from Naturalistic and Controlled Mathematics Paradigms.

Two major goals of human neuroscience are to understand how the brain functions in the real world and to measure neural processes under conditions that are ecologically valid. A critical step toward these goals is understanding how brain activity during naturalistic tasks that mimic the real world relates to brain activity in more traditional laboratory tasks. In this study, we used intersubject correlations to locate reliable stimulus-driven cerebral processes among children and adults in a naturalistic video lesson and a laboratory forced-choice task that shared the same arithmetic concept. We show that relative to a control condition with grammatical content, naturalistic and laboratory arithmetic tasks evoked overlapping activation within brain regions previously associated with math semantics. The regions of specific functional overlap between the naturalistic mathematics lesson and laboratory mathematics task included bilateral intraparietal cortex, which confirms that this region processes mathematical content independently of differences in task mode. These findings suggest that regions of the intraparietal cortex process mathematical content when children are learning about mathematics in a naturalistic setting.

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.

Representation of spatial sequences using nested rules in human prefrontal cortex.

Memory for spatial sequences does not depend solely on the number of locations to be stored, but also on the presence of spatial regularities. Here, we show that the human brain quickly stores spatial sequences by detecting geometrical regularities at multiple time scales and encoding them in a format akin to a programming language. We measured gaze-anticipation behavior while spatial sequences of variable regularity were repeated. Participants' behavior suggested that they quickly discovered the most compact description of each sequence in a language comprising nested rules, and used these rules to compress the sequence in memory and predict the next items. Activity in dorsal inferior prefrontal cortex correlated with the amount of compression, while right dorsolateral prefrontal cortex encoded the presence of embedded structures. Sequence learning was accompanied by a progressive differentiation of multi-voxel activity patterns in these regions. We propose that humans are endowed with a simple "language of geometry" which recruits a dorsal prefrontal circuit for geometrical rules, distinct from but close to areas involved in natural language processing.

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.

The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers.

During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a "geometrical language" with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them.

Entropy, complexity, and maturity in children's neural responses to naturalistic video lessons.

Temporal characteristics of neural signals are often overlooked in traditional fMRI developmental studies but are critical to studying brain functions in ecologically valid settings. In the present study, we explore the temporal properties of children's neural responses during naturalistic mathematics and grammar tasks. To do so, we introduce a novel measure in developmental fMRI: neural entropy, which indicates temporal complexity of BOLD signals. We show that temporal patterns of neural activity have lower complexity and greater variability in children than in adults in the association cortex but not in the sensory-motor cortex. We also show that neural entropy is associated with both child-adult similarity in functional connectivity and neural synchrony, and that neural entropy increases with the size of functionally connected networks in the association cortex. In addition, neural entropy increases with functional maturity (i.e., child-adult neural synchrony) in content-specific regions. These exploratory findings suggest the hypothesis that neural entropy indexes the increasing breadth and diversity of neural processes available to children for analyzing mathematical information over development.

Counterexample Search in Diagram-Based Geometric Reasoning.

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object-wise, could underlie diagram-based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions.

Mental compression of spatial sequences in human working memory using numerical and geometrical primitives.

How does the human brain store sequences of spatial locations? We propose that each sequence is internally compressed using an abstract, language-like code that captures its numerical and geometrical regularities. We exposed participants to spatial sequences of fixed length but variable regularity while their brain activity was recorded using magneto-encephalography. Using multivariate decoders, each successive location could be decoded from brain signals, and upcoming locations were anticipated prior to their actual onset. Crucially, sequences with lower complexity, defined as the minimal description length provided by the formal language, led to lower error rates and to increased anticipations. Furthermore, neural codes specific to the numerical and geometrical primitives of the postulated language could be detected, both in isolation and within the sequences. These results suggest that the human brain detects sequence regularities at multiple nested levels and uses them to compress long sequences in working memory.

Geometric categories in cognition.

At the scale in which we live, space is continuous. Nevertheless, our perception and cognition parse the world into categories, whether physical, like scene or object, or abstract, like infinitesimal point or 7. The present study focuses on 2 categories of special angles in planar geometry, parallels and perpendiculars, and we evaluate how these categories might be reflected in adults' basic angle discrimination. In the first experiment, participants were most precise when detecting 2 parallel or perpendicular lines among other pairs of lines at different relative orientations. Detection was also enhanced for 2 connected lines whose angle approached 90°, with precision peaking at 90°. These patterns emerged despite large variations in the scales and orientations of the angle exemplars. In the second experiment, the enhanced detection of perpendiculars persisted when stimuli were rotated in depth, indicating a capacity to discriminate shapes based on perpendicularity in 3 dimensions despite large variation in angles' 2-dimensional projections. The results suggest that 2 categorical concepts which lie at the foundation of Euclidean geometry, parallelism and perpendicularity, are reflected in our discrimination of simple visual forms, and they pave the way for future studies exploring the developmental and evolutionary origins of these cognitive categories. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

On the role of visual experience in mathematical development: Evidence from blind mathematicians.

Advanced mathematical reasoning, regardless of domain or difficulty, activates a reproducible set of bilateral brain areas including intraparietal, inferior temporal and dorsal prefrontal cortex. The respective roles of genetics, experience and education in the development of this math-responsive network, however, remain unresolved. Here, we investigate the role of visual experience by studying the exceptional case of three professional mathematicians who were blind from birth (n=1) or became blind during childhood (n=2). Subjects were scanned with fMRI while they judged the truth value of spoken mathematical and nonmathematical statements. Blind mathematicians activated the classical network of math-related areas during mathematical reflection, similar to that found in a group of sighted professional mathematicians. Thus, brain networks for advanced mathematical reasoning can develop in the absence of visual experience. Additional activations were found in occipital cortex, even in individuals who became blind during childhood, suggesting that either mental imagery or a more radical repurposing of visual cortex may occur in blind mathematicians.

Shared genetic aetiology between cognitive performance and brain activations in language and math tasks.

Cognitive performance is highly heritable. However, little is known about common genetic influences on cognitive ability and brain activation when engaged in a cognitive task. The Human Connectome Project (HCP) offers a unique opportunity to study this shared genetic etiology with an extended pedigree of 785 individuals. To investigate this common genetic origin, we took advantage of the HCP dataset, which includes both language and mathematics activation tasks. Using the HCP multimodal parcellation, we identified areals in which inter-individual functional MRI (fMRI) activation variance was significantly explained by genetics. Then, we performed bivariate genetic analyses between the neural activations and behavioral scores, corresponding to the fMRI task accuracies, fluid intelligence, working memory and language performance. We observed that several parts of the language network along the superior temporal sulcus, as well as the angular gyrus belonging to the math processing network, are significantly genetically correlated with these indicators of cognitive performance. This shared genetic etiology provides insights into the brain areas where the human-specific genetic repertoire is expressed. Studying the association of polygenic risk scores, using variants associated with human cognitive ability and brain activation, would provide an opportunity to better understand where these variants are influential.

Bayesian validation of grammar productions for the language of thought.

Probabilistic proposals of Language of Thoughts (LoTs) can explain learning across different domains as statistical inference over a compositionally structured hypothesis space. While frameworks may differ on how a LoT may be implemented computationally, they all share the property that they are built from a set of atomic symbols and rules by which these symbols can be combined. In this work we propose an extra validation step for the set of atomic productions defined by the experimenter. It starts by expanding the defined LoT grammar for the cognitive domain with a broader set of arbitrary productions and then uses Bayesian inference to prune the productions from the experimental data. The result allows the researcher to validate that the resulting grammar still matches the intuitive grammar chosen for the domain. We then test this method in the language of geometry, a specific LoT model for geometrical sequence learning. Finally, despite the fact of the geometrical LoT not being a universal (i.e. Turing-complete) language, we show an empirical relation between a sequence's probability and its complexity consistent with the theoretical relationship for universal languages described by Levin's Coding Theorem.

Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain's semantic networks.

Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that 'the sine function is periodical' or that 'London buses are red', activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders.This article is part of a discussion meeting issue 'The origins of numerical abilities'.