Solving olympiad geometry without human demonstrations.

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1-4, owing to their reputed difficulty among the world's best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

Active math and grammar learning engages overlapping brain networks.

We here demonstrate common neurocognitive long-term memory effects of active learning that generalize over course subjects (mathematics and vocabulary) by the use of fMRI. One week after active learning, relative to more passive learning, performance and fronto-parietal brain activity was significantly higher during retesting, possibly related to the formation and reactivation of semantic representations. These observations indicate that active learning conditions stimulate common processes that become part of the representations and can be reactivated during retrieval to support performance. Our findings are of broad interest and educational significance related to the emerging consensus of active learning as critical in promoting good long-term retention.

On the Mathematics of the Fraternal Birth Order Effect and the Genetics of Homosexuality.

Mathematicians have always been attracted to the field of genetics. The mathematical aspects of research on homosexuality are especially interesting. Certain studies show that male homosexuality may have a genetic component that is correlated with female fertility. Other studies show the existence of the fraternal birth order effect, that is, the correlation of homosexuality with the number of older brothers. This article is devoted to the mathematical aspects of how these two phenomena are interconnected. In particular, we show that the fraternal birth order effect implies a correlation between homosexuality and maternal fecundity. Vice versa, we show that the correlation between homosexuality and female fecundity implies the increase in the probability of the younger brothers being homosexual.

The Impairment of Number Transcoding Abilities in Individuals with Amnestic Mild Cognitive Impairment and Alzheimer Disease: Associations With Attentional and Executive Functions.

BACKGROUND: Studies have shown that number transcoding abilities (ie, translating numbers from one numerical code to another) are affected early in the development of Alzheimer disease (AD). However, no study has extensively explored how these abilities are affected in individuals with mild cognitive impairment (MCI). OBJECTIVE: To determine the contribution that number transcoding tasks make to the identification of MCI, and to pinpoint the cognitive correlates of performance in these tasks. METHODS: We compared the performance of 20 individuals with the amnestic subtype of MCI, eight individuals with AD, and 20 healthy controls on three number transcoding tasks. RESULTS: The results confirmed the presence of number transcoding impairment in the individuals with AD. The individuals with MCI were found to be impaired in two of the transcoding tasks; these individuals produced perseverations of the input code-the most noteworthy error type in individuals with AD. In addition, the relationship between impairment in attentional and executive functions and impairment in number transcoding was supported by the correlational analyses. CONCLUSIONS: This study confirmed that number transcoding abilities are impaired in individuals with MCI, although less severely than in individuals with AD. Our results provide evidence for the clinical value of including number transcoding tasks in the assessment of cognitive deficits associated with pathological aging.

Multiscale phenomena and patterns in biological systems: special issue in honour of Hans Othmer.

This special issue on "Multiscale phenomena and patterns in biological systems" is an homage to the seminal contributions of Hans Othmer. He has remained at the forefront of multiscale modelling and pattern formation in biology for over half a century, developing models for molecular signalling networks, the mechanics of cellular movements, the interactions between multiple cells and their contributions to tissue patterning and dynamics. The contributions in this special issue follow Hans' legacy in using advanced mathematics to understand complex biological processes.

Children's discrete proportional reasoning is related to inhibitory control and enhanced by priming continuous representations.

Children can successfully compare continuous proportions as early as 4 years of age, yet they struggle to compare discrete proportions at least to 10 years of age, especially when the discrete information is misleading. This study examined whether inhibitory control contributes to individual differences in discrete proportional reasoning and whether reasoning could be enhanced by priming continuous information. A total of 49 second-graders completed two tasks. In the Hearts and Flowers (H&F) task, a measure of inhibition, children pressed on either the corresponding or opposite side, depending on the identity of the displayed figure. In the Spinners task, a measure of proportional reasoning, children chose the spinner with the proportionally larger red area across continuous and two discrete formats. In the discrete adjacent format, the continuous stimuli were segmented into sections, which could be compatible with the proportional information or misleading; the discrete mixed format interspersed the colored sections from the discrete adjacent conditions. Finally, two priming groups were formed. Children who saw the continuous format immediately before the discrete adjacent format formed the continuous priming group (n = 26). Children who saw the discrete mixed format immediately before the discrete adjacent format formed the discrete priming group (n = 23). Our results showed that children who performed better on the H&F task also had better performance on the discrete counting misleading trials. Furthermore, children in the continuous priming group outperformed children in the discrete priming group, specifically in contexts where discrete information was misleading. These results suggest that children's proportional reasoning may be improved by fostering continuous representations of discrete stimuli and by enhancing inhibitory control skills.

Symbolic fractions elicit an analog magnitude representation in school-age children.

A fundamental question about fractions is whether they are grounded in an abstract nonsymbolic magnitude code similar to that postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving nonsymbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second- and fifth-grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, nonsymbolic ratios, and mixed symbolic-nonsymbolic pairs. This paradigm allowed us to test three key questions: (a) whether children show an analog magnitude code for rational numbers, (b) whether that code is compatible with mental representations of symbolic fractions, and (c) how formal education with fractions affects the symbolic-nonsymbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth graders' reaction times and error rates showed classic distance and notation effects. Nonsymbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions but had a smaller advantage for nonsymbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance and that fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a nonsymbolic ratio magnitude code and that symbolic fractions can be translated into that magnitude code.

Mathematics, executive functioning, and visual-spatial skills in Chinese kindergarten children: Examining the bidirectionality.

This study examined the bidirectional relationships among Chinese children's mathematics, executive functioning, and visual-spatial skills during their transition from kindergarten to primary school. Participants were 172 Cantonese-speaking Hong Kong children (mean age at Time 1 = 62.75 months; 88 male) and their parents. At Time 1 (kindergarten K3), children were administered the measures of mathematics (calculation and applied problems), executive functioning (working memory and inhibitory control), and visual-spatial skills. They were reassessed on these measures at Time 2 (primary 1) 1 year later. Results from the cross-lagged panel model showed that, controlling for child age, gender, and family socioeconomic status, children's visual-spatial skills at Time 1 were significantly predictive of their mathematics at Time 2 and children's executive functioning and visual-spatial skills reciprocally predicted each other across times. However, children's mathematics at Time 1 were not predictive of their executive functioning or visual-spatial skills at Time 2. The findings highlight the desirability of improving children's executive functioning and visual-spatial skills to promote their mathematical performance during the formal school transition.

Influences of basic numerical abilities on graph reading performance.

Understanding graphically presented information is an important aspect of modern mathematical and science literacy. In our study, we investigated the influence of basic numerical abilities on students' ability answer mathematical tasks with information presented in graphs. We analyzed data of 750 German students (grades 9-11) and evaluated the determinants of graph reading performance with multiple regression analysis using predictors of basic numerical abilities (such as number line estimation, basic arithmetic operations, etc.), considering also the influences of general cognitive ability, age, and gender. We found that number line estimation, subtraction, and conceptual knowledge were significant predictors of graph reading performance beyond the influences of general cognitive ability. This indicates that basic numerical abilities are still relevant for real-life problem solving in secondary school. We discuss possible mechanisms which directly (through respective arithmetic procedures) as well as indirectly (through mathematization of the problem) effectuate that basic numerical abilities graph reading performance.

Stimulating numbers: signatures of finger counting in numerosity processing.

Finger counting is one of the first steps in the development of mature number concepts. With a one-to-one correspondence of fingers to numbers in Western finger counting, fingers hold two numerical meanings: one is based on the number of fingers raised and the second is based on their ordinal position within the habitual finger counting sequence. This study investigated how these two numerical meanings of fingers are intertwined with numerical cognition in adults. Participants received tactile stimulation on their fingertips of one hand and named either the number of fingers stimulated (2, 3, or 4 fingers; Experiment 1) or the number of stimulations on one fingertip (2, 3, or 4 stimulations; Experiment 2). Responses were faster and more accurate when the set of stimulated fingers corresponded to finger counting habits (Experiment 1) and when the number of stimulations matched the ordinal position of the stimulated finger (Experiment 2). These results show that tactile numerosity perception is affected by individual finger counting habits and that those habits give numerical meaning to single fingers.

Finger-counting habits, not finger movements, predict simple arithmetic problem solving.

Previous research in embodied mathematical cognition has found differences between those who start counting on their left hand and those who start counting on the right hand. However, if starting hand is a finger-embodied effect, then finger-specific interference may affect these differences between left and right starters. Furthermore, cultures that demonstrate different finger-counting habits may also be differently affected by this interference. In the current study, a total of 66 Canadians and 60 Chinese participants completed a single/dual-task paradigm and were also assessed on their starting hand for counting. The primary task was to verbally answer simple arithmetic problems, while the dual task was to either sequentially tap their fingers or their foot. Contrary to predictions, a specific finger-movement interference pattern that had previously been reported was not evident in this study, despite a much larger sample. Nevertheless, Canadians left starters outperformed right starters for every operation type, which may be further evidence of individual differences in the lateralization of arithmetic processes. Derived from a combination of a replication, a conceptual replication, and a cross-cultural comparison, this investigation suggests that embodied effects in the published literature are in need of both independent replication as well as investigation of individual differences. This study also further validates the differences between left and right starters, and suggests that more research is needed to understand the influence of embodied cognition on mathematical understanding.

The 2019 mathematical oncology roadmap.

Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology-defined here simply as the use of mathematics in cancer research-complements and overlaps with a number of other fields that rely on mathematics as a core methodology. As a result, Mathematical Oncology has a broad scope, ranging from theoretical studies to clinical trials designed with mathematical models. This Roadmap differentiates Mathematical Oncology from related fields and demonstrates specific areas of focus within this unique field of research. The dominant theme of this Roadmap is the personalization of medicine through mathematics, modelling, and simulation. This is achieved through the use of patient-specific clinical data to: develop individualized screening strategies to detect cancer earlier; make predictions of response to therapy; design adaptive, patient-specific treatment plans to overcome therapy resistance; and establish domain-specific standards to share model predictions and to make models and simulations reproducible. The cover art for this Roadmap was chosen as an apt metaphor for the beautiful, strange, and evolving relationship between mathematics and cancer.

Combinatorics of proanthocyanidins in wine.

Condensed tannin are polymers comprised of procyanidin and prodelphinidin units and found in wine, chocolate, apples and many other foods. Current analytical methods to characterize these tannins provide aggregate functional results, such as quantified protein binding, important for its relation to astringency. A more detailed understanding of the constituents has become accessible via mass spectrometry (MS) with high resolution and tandem MS techniques. Analysis of wine tannin by these methods provides thousands of signals, far too many to assess using standard techniques. We propose targeted mass filtering using a table of predicted proanthocyanidin oligomers. Condensed tannin in wine is comprised of 4 constitutionally distinct subunits (catechin, catechin gallate, gallocatechin, and gallocatechin gallate), each of which has 4 stereoisomers. Accounting for all 16 subunits, there are over 5 million possible oligomers from monomer to decamer. Since mass spectroscopy is generally blind to stereoisomeric variations, the number of possible observable MS signals can be reduced to a list of 1000. By applying specialized combinatoric functions, a table for the now manageable compilation of possible proanthocyanidin oligomers has been created containing the compound subunit compositions, molecular formulae, and molecular ion signals for the interpretation of condensed tannin mass spectra. Mathematical formulae for the enumeration of possible compounds in any chemical system composed of polymers with discrete subunits resulted from this endeavor and are presented in general form with specific application to this system. As these condensed tannins react with anthocyanins, forming pigmented tannin, the table created here can also be the foundation of a database for the very complex red wine pigments. The system described here could also be applied to the analysis of proanthocyanidins in chocolate and other foods.

More than number sense: The additional role of executive functions and metacognition in arithmetic.

Arithmetic is a major building block for children's development of more complex mathematical abilities. Knowing which cognitive factors underlie individual differences in arithmetic is key to gaining further insight into children's mathematical development. The current study investigated the role of executive functions and metacognition (domain-general cognitive factors) as well as symbolic numerical magnitude processing (domain-specific cognitive factor) in arithmetic, enabling the investigation of their unique contribution in addition to each other. Participants were 127 typically developing second graders (7- and 8-year-olds). Our within-participant design took into account different components of executive functions (i.e., inhibition, shifting, and updating), different aspects of metacognitive skills (i.e., task-specific and general metacognition), and different levels of experience in arithmetic, namely addition (where second graders had extensive experience) and multiplication (where second graders were still in the learning phase). This study reveals that both updating and metacognitive monitoring are important unique predictors of arithmetic in addition to each other and to symbolic numerical magnitude processing. Our results point to a strong and unique role of task-specific metacognitive monitoring skills. These individual differences in noticing one's own errors might help one to learn from his or her mistakes.

The effect of multimodal information on children's numerical judgments.

Although much research suggests that adults, infants, and nonhuman primates process number (among other properties) across distinct modalities, limited studies have explored children's abilities to integrate multisensory information when making judgments about number. In the current study, 3- to 6-year-old children performed numerical matching or numerical discrimination tasks in which numerical information was presented either unimodally (visual only), cross-modally (comparing audio with visual), or bimodally (simultaneously presenting audio and visual input). In three experiments, we investigated children's multimodal numerical processing across distinct task demands and difficulty levels. In contrast to previous work, results indicate that even the youngest children (3 and 4 years) performed above chance across all three modality presentations. In addition, the current study contributes two other novel findings, namely that (a) children exhibit a cross-modal disadvantage when numerical comparisons are easy and that (b) accuracy on bimodal trial types led to even more accurate numerical judgments under more difficult circumstances, particularly for the youngest participants and when precise numerical matching was required. Importantly, findings from this study extend the literature on children's numerical cross-modal abilities to reveal that, like their adult counterparts, children readily track and compare visual and auditory numerical information, although their abilities to do so are not perfect and are affected by task demands and trial difficulty.

Individual differences in basic numerical skills: The role of executive functions and motor skills.

The aim of the current study was to explore individual differences in basic numerical skills in a normative sample of 151 kindergarteners (mean age = 6.45 years). Whereas previous research claims a substantial link between executive functions and basic numerical skills, motor abilities have been put forward to explain variance in numerical skills. Regarding the current study, these two assumptions have been combined, revealing interesting results. Namely, executive functions (inhibition, switching, and visuospatial working memory) were found to relate to symbolic numerical skills, and motor skills (gross and fine motor skills) showed a significant correlation to nonsymbolic numerical skills. Suggesting that motor skills and executive functions are associated with basic numerical skills could lead to potential avenues for interventions in certain disorders or disabilities such as nonverbal learning disability, developmental dyscalculia, and developmental coordination disorder.

Operational momentum for magnitude ordering in preschool children and adults.

When adding or subtracting quantities, adults tend to overestimate addition outcomes and underestimate subtraction outcomes. They also shift visuospatial attention to the right when adding and to the left when subtracting. These operational momentum phenomena are thought to reflect an underlying representation in which small magnitudes are associated with the left side of space and large magnitudes with the right side of space. Currently, there is limited research on operational momentum in early childhood or for operations other than addition and subtraction. The current study tested whether English-speaking 3- and 4-year-old children and college-aged adults exhibit operational momentum when ordering quantities. Participants were presented with two experimental blocks. In one block of trials, they were tasked with choosing the same quantity they had previously seen three times; in the other block, they were asked to generate the next quantity in a doubling sequence composed of three ascending quantities. A bias to shift attention to the right after an ascending operation was found in both age groups, and a bias to overestimate the next sequential quantity during an ascending ordering operation was found in adults under conditions of uncertainty. These data suggest that, for children, the spatial biases during operating are more pronounced than the mis-estimation biases. These findings highlight the spatial underpinnings of operational momentum and suggest that both very young children and adults conceptualize quantity along a horizontal continuum during ordering operations, even before formal schooling.

On the linear representation of numbers: evidence from a new two-numbers-to-two positions task.

In the number-to-position methodology, a number is presented on each trial and the observer places it on a straight line in a position that corresponds to its felt subjective magnitude. In the novel modification introduced in this study, the two-numbers-to-two-positions method, a pair of numbers rather than a single number is presented on each trial and the observer places them in appropriate positions on the same line. Responses in this method indicate not only the subjective magnitude of each single number but, simultaneously, provide a direct estimation of their subjective numerical distance. The results of four experiments provide strong evidence for a linear representation of numbers and, commensurately, for the linear representation of numerical distances. We attribute earlier results that indicate a logarithmic representation to the ordered nature of numbers and to the task used and not to a truly non-linear underlying representation.

Spatial grounding of symbolic arithmetic: an investigation with optokinetic stimulation.

Growing evidence suggests that mental calculation might involve movements of attention along a spatial representation of numerical magnitude. Addition and subtraction on nonsymbolic numbers (numerosities) seem to induce a "momentum" effect, and have been linked to distinct patterns of neural activity in cortical regions subserving attention and eye movements. We investigated whether mental arithmetic on symbolic numbers, a cornerstone of abstract mathematical reasoning, can be affected by the manipulation of overt spatial attention induced by optokinetic stimulation (OKS). Participants performed additions or subtractions of auditory two-digit numbers during horizontal (experiment 1) or vertical OKS (experiment 2), and eye movements were concurrently recorded. In both experiments, the results of addition problems were underestimated, whereas results of subtractions were overestimated (a pattern that is opposite to the classic Operational Momentum effect). While this tendency was unaffected by OKS, vertical OKS modulated the occurrence of decade errors during subtractions (i.e., fewer during downward OKS and more frequent during upward OKS). Eye movements, on top of the classic effect induced by OKS, were affected by the type of operation during the calculation phase, with subtraction consistently leading to a downward shift of gaze position and addition leading to an upward shift. These results highlight the pervasive nature of spatial processing in mental arithmetic. Furthermore, the preeminent effect of vertical OKS is in line with the hypothesis that the vertical dimension of space-number associations is grounded in universal (physical) constraints and, thereby, more robust than situated and culture-dependent associations with the horizontal dimension.

An outreach program with hands-on, physiology-based exercises generates questions about STEM career expectations.

Scientific advocacy and outreach programs are encouraged to increase public understanding of scientific knowledge and generate interest in science, technology, engineering, and mathematics (STEM) careers. However, evaluation of these events' effectiveness is difficult and somewhat rare. This study's purpose was to better understand how effective an established physiology-based outreach program was in generating interest in STEM careers, while simultaneously providing information that can be used to increase the effectiveness of future events. We partnered with a private school located in Omaha, Nebraska, where 64-80 students participated in 3 h of physiology-based activities presented by volunteers from the University of Nebraska Medical Center. The event included a brief presentation of the eye, sensory, heart, and lung systems, followed by hands-on demonstrations and activities. Each session concluded with 15 min of questions and answers (Q&A), where students were encouraged to engage the volunteers in inquiries about what they just learned, career-related questions, or any topic of their choosing. Each Q&A session was audio recorded and evaluated using thematic analysis to identify patterns in the Q&A data. Two major themes of questions were identified: 1) scientific content (animal circulatory systems and how organs are affected by disease or stimulus); and 2) career-related content, including typical day-to-day activities of a scientist and the volunteers' satisfaction with a scientific career. We conclude that hands-on physiology-based learning opportunities are effective in generating short-term interest in STEM content and careers. The results of this study will also facilitate informed modification of event content to better suit student's interests.