Monkeys display classic signatures of human symbolic arithmetic

Spatial coding of arithmetic operations in early learning: an eye tracking study in first-grade elementary school children

A growing body of evidence indicates that mental calculation in adults is accompanied by horizontal attention shifts along a mental continuum representing the range of plausible answers. The fast deployment of spatial attention suggests a predictive role in guiding the search for the answer. The link between arithmetic and spatial functions is theoretically justified by the need to alleviate the cognitive load of mental calculation, but the question of how this link establishes during development gives rise to opposing views emphasizing either biological or cultural factors. The role of education, in particular, remains debated in the absence of data covering the period when children learn arithmetic. In this study, we measured gaze movements, as a proxy for attentional shifts, while first-grade elementary school children solved single-digit additions and subtractions. The investigation was scheduled only a few weeks after the formal teaching of symbolic subtraction to assess the role of spatial attention in early learning. Gaze patterns revealed horizontal- but not vertical- attentional shifts, with addition shifting the gaze more rightward than subtraction. The shift was observed as soon as the first operand and the operator were presented, corroborating the view that attention is used to predictively identify the portion of the numerical continuum where the answer is likely to be located, as adult studies suggested. The finding of a similar gaze pattern in adults and six-year-old children who have just learned how to subtract single digits challenges the idea that arithmetic problem solving requires intensive practice to be linked to spatial attention.

Unraveling the small tie problem mystery: Size effects from finger counting to mental strategies in addition

Determining how children solve arithmetic problems when they stop using their fingers is a real challenge. To take it up, the evolution of problem-size effects for tie and non-tie problems was observed when 6-year-olds (N = 65) shift from finger counting to mental strategies. These observations revealed that the problem-size effect remained the same for non-tie problems, whereas it drastically decreased for tie problems. Moreover, the solving strategy for tie problems switched directly from the representation of both operands on fingers to retrieval without transition through the representation of only one operand on fingers. This direct switch could be made possible by the relative ease to commit symmetrical representations to memory (in the case of tie problems) rather than non-symmetrical ones (in the case of non-tie problems). This would explain why, early during development, small tie problems are solved quickly and present null or negligible size effects. All in all, our results and interpretations provide an answer to the long-standing question as to why tie problems have a special cognitive status. Our results also nuance the classical description of the developmental pattern reported in all textbook chapters devoted to numerical cognition according to which a finger strategy where only one operand is represented on fingers constitutes a developmental stage between the representation of two operands on fingers and retrieval. We demonstrate here that it is true only for non-tie problems.

The calculating brain

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

Passive Grouping Enhances Proto-Arithmetic Calculation for Leftward Correct Responses

Baby chicks and other animals including human infants master simple arithmetic. They discriminate 2 vs. 3 (1 + 1 vs. 1 + 1 + 1) but fail with 3 vs. 4 (1 + 1 + 1 vs. 1 + 1 + 1 + 1). Performance is restored when elements are grouped as 2 + 1 vs. 2 + 2. Here, we address whether grouping could lead to asymmetric response bias. We recoded behavioural data from a previous study, in which separate groups of four-day-old domestic chicks underwent an arithmetic task: when the objects were presented one-by-one (1 + 1 + 1 vs. 1 + 1 + 1 + 1), chicks failed in locating the larger group irrespective of its position and did not show any side bias; Experiment 1. When the objects were presented as grouped (2 + 1 vs. 2 + 2), chicks succeeded, performing better when the larger set was on their left; Experiment 2. A similar leftward bias was also observed with harder discriminations (4 vs. 5: 3  +  1 vs. 3  +  2), with baby chicks succeeding in the task only when the larger set was on the left (Experiments 3 and 4). A previous study showed a rightward bias, with tasks enhancing individual processing. Despite a similar effect in boosting proto-arithmetic calculations, individual processing (eliciting a right bias) and grouping (eliciting a left bias) seem to depend on distinct cognitive mechanisms.

Young Children Intuitively Divide Before They Recognize the Division Symbol

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.

Rhesus Monkeys Have a Counting Ability and Can Count from One to Six

Counting ability is one of the many aspects of animal cognition and has enjoyed great interest over the last couple of decades. The impetus for studying counting ability in nonhuman animals has likely come from more than a general interest in animal cognition, as the analysis of animal abilities amplifies our understanding of human cognition. In addition, a model animal with the ability to count could be used to replace human subjects in related studies. Here we designed a behavioral paradigm to train rhesus monkeys to count 1-to-6 visual patterns presented sequentially with long and irregular interpattern intervals on a touch screen. The monkeys were required to make a response to the sixth pattern exclusively, inhibiting response to any patterns appearing at other ordinal positions. All stimulus patterns were of the same size, color, location, and shape to prevent monkeys making the right choice due to non-number physical cues. In the long delay period, the monkey had to enumerate how many patterns had been presented sequentially and had to remember in which ordinal position the current pattern was located. Otherwise, it was impossible for them to know which pattern was the target one. The results show that all three monkeys learned to correctly choose the sixth pattern within 3 months. This study provides convincing behavioral evidence that rhesus monkeys may have the capacity to count.

Approximate arithmetic training does not improve symbolic math in third and fourth grade children

BACKGROUND

Prior studies reported that practice playing an approximate arithmetic game improved symbolic math performance relative to active control groups in adults and preschool children (e.g. Park & Brannon, 2013, 2014; Park et al., 2016; Szkudlarek & Brannon, 2018). However, Szkudlarek, Park and Brannon (2021) recently failed to replicate those findings in adults. Here we test whether approximate arithmetic training yields benefits in elementary school children who have intermediate knowledge of arithmetic.

METHOD

We conducted a randomized controlled trial with a pre and post-test design to compare the effects of approximate arithmetic training and visuo-spatial working memory training on standardized math performance in third and fourth grade children.

RESULTS

We found that approximate arithmetic training did not yield any significant gains on standardized measures of symbolic math performance.

CONCLUSION

A Bayesian analysis supports the conclusion that approximate arithmetic provides no benefits for symbolic math performance.

Absolute Numerosity Discrimination as a Case Study in Comparative Vertebrate Intelligence

The question of whether some non-human animal species are more intelligent than others is a reoccurring theme in comparative psychology. To convincingly address this question, exact comparability of behavioral methodology and data across species is required. The current article explores one of the rare cases in which three vertebrate species (humans, macaques, and crows) experienced identical experimental conditions during the investigation of a core cognitive capability – the abstract categorization of absolute numerical quantity. We found that not every vertebrate species studied in numerical cognition were able to flexibly discriminate absolute numerosity, which suggests qualitative differences in numerical intelligence are present between vertebrates. Additionally, systematic differences in numerosity judgment accuracy exist among those species that could master abstract and flexible judgments of absolute numerosity, thus arguing for quantitative differences between vertebrates. These results demonstrate that Macphail’s Null Hypotheses – which suggests that all non-human vertebrates are qualitatively and quantitatively of equal intelligence – is untenable.

Spatial biases in mental arithmetic are independent of reading/writing habits: Evidence from French and Arabic speakers

The representation of numbers in human adults is linked to space. In Western cultures, small and large numbers are associated respectively with the left and right sides of space. An influential framework attributes the emergence of these spatial-numerical associations (SNAs) to cultural factors such as the direction of reading and writing, because SNAs were found to be reduced or inverted in right-to-left readers/writers (e.g., Arabic, Farsi, or Hebrew speakers). However, recent cross-cultural and animal studies cast doubt on the determining role of reading and writing directions on SNAs. In this study, we assessed this role in mental arithmetic, which requires explicit number manipulations and has revealed robust leftward or rightward biases in Western participants. We used a temporal order judgement task in French and Arabic speakers, two languages that have opposite reading/writing directions. Participants had to solve subtraction and addition problems presented auditorily while at the same time determining which of a left or right visual target appeared first on a screen. The results showed that the right target was favoured more often when solving additions than when solving subtractions both in the French- (n = 31) and Arabic-speaking (n = 25) groups. This was true even in Arabic-speaking participants whose preference for ordering of various series of numerical and non-numerical stimuli went from right to left (n = 10). These results indicate that SNAs in mental arithmetic cannot be explained by the direction of reading/writing habits and call for a reconsideration of current models to acknowledge the pervasive role of biological factors in SNAs in adults.

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.

From number sense to number symbols. An archaeological perspective

How and when did hominins move from the numerical cognition that we share with the rest of the animal world to number symbols? Objects with sequential markings have been used to store and retrieve numerical information since the beginning of the European Upper Palaeolithic (42 ka). An increase in the number of markings and complexity of coding is observed towards the end of this period. The application of new analytical techniques to a 44-42 ka old notched baboon fibula from Border Cave, South Africa, shows that notches were added to this bone at different times, suggesting that devices to store numerical information were in use before the Upper Palaeolithic. Analysis of a set of incisions on a 72-60 ka old hyena femur from the Les Pradelles Mousterian site, France, indicates, by comparison with markings produced by modern subjects under similar constraints, that the incisions on the Les Pradelles bone may have been produced to record, in a single session, homologous units of numerical information. This finding supports the view that numerical notations were in use among archaic hominins. Based on these findings, a testable five-stage scenario is proposed to establish how prehistoric cultures have moved from number sense to the use of number symbols.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Finding numbers in the brain

After listing functional constraints on what numbers in the brain must do, I sketch the two's complement fixed-point representation of numbers because it has stood the test of time and because it illustrates the non-obvious ways in which an effective coding scheme may operate. I briefly consider its neurobiological implementation. It is easier to imagine its implementation at the cell-intrinsic molecular level, with thermodynamically stable, volumetrically minimal polynucleotides encoding the remembered numbers, than at the circuit level, with plastic synapses encoding them.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

The problem with percentages

A great many students at a major research university make basic conceptual mistakes in responding to simple questions about two successive percentage changes. The mistakes they make follow a pattern already familiar from research on the difficulties that elementary school students have in coming to terms with fractions and decimals. The intuitive core knowledge of arithmetic with the natural numbers makes learning to count and do simple arithmetic relatively easy. Those same principles become obstacles to understanding how to operate with rational numbers.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

The Art of Pharmacotherapy: Reflections on Pharmacophobia

PURPOSE/BACKGROUND

This commentary deals with the neglected issue of the art of psychopharmacology by recounting the authors' journeys.

METHODS/PROCEDURES

First, a model of medical science situated within the history of medicine is described including (1) a limitation of the mathematical model of science, (2) the distinction between mechanistic science and mathematical science, (3) how this distinction is applied to medicine, and (4) how this distinction is applied to explain pharmacology to psychiatrists. Second, the neglected art of psychopharmacology is addressed by explaining (1) where the art of psychopharmacotherapy was hiding in the first author's psychopharmacology research, (2) how the Health Belief Model was applied to the art of medicine, (3) how the second author became interested in the Health Belief Model, and (4) his studies introducing the Health Belief Model in psychopharmacology. The authors' collaboration led to: (1) study of the effect of pharmacophobia on poor adherence and (2) reflection on the limits of the art of psychopharmacology.

FINDINGS/RESULTS

Low adherence was found in 45% (116/258) of psychiatric patients with pharmacophobia versus 22% (149/682) in those with no pharmacophobia, providing an odds ratio of 2.9 (95% confidence interval, 2.2-4.0) and an adjusted odds ratio of 2.5 (95% confidence interval, 1.8-3.5) after adjusting for other variables contributing to poor adherence.

IMPLICATIONS/CONCLUSIONS

Different cognitive patterns in different patients may contribute to poor adherence. Specific interventions targeting these varying cognitive styles may be needed in different patients to improve drug adherence.

Theory of Animal Mind: Human Nature or Experimental Artefact?

Are animals capable of empathy, problem-solving, or even self-recognition? Much research is dedicated to answering these questions and yet few studies have considered how humans form beliefs about animal minds. Evidence suggests that our mentalising of animals is a natural consequence of Theory of Mind (ToM) capabilities. However, where beliefs regarding animal mind have been investigated, there has been slow progress in establishing the mechanism underpinning how this is achieved. Here, we consider what conclusions can be drawn regarding how people theorise about animal minds and the different conceptual and methodological issues that might limit the accuracy of conclusions currently drawn from this work. We suggest a new empirical framework for better capturing the human theory of animal mind, which in turn has significant political and social implications.

Ratio abstraction over discrete magnitudes by newly hatched domestic chicks (Gallus gallus)

A large body of literature shows that non-human animals master a variety of numerical tasks, but studies involving proportional discrimination are sparse and primarily done with mature animals. Here we trained 4-day-old domestic chicks (Gallus gallus) to respond to stimuli depicting multiple examples of the proportion 4:1 when compared with the proportion 2:1. Stimuli were composed of green and red dot arrays; for the rewarded 4:1 proportion, 4 green dots for every red dot (e.g. ratios: 32:8, 12:3, and 44:11). The birds continued to discriminate when presented with new ratios at test (such as 20:5), characterized by new numbers of dots and new spatial configurations (Experiment 1). This indicates that chicks can extract the common proportional value shared by different ratios and apply it to new ones. In Experiment 2, chicks identified a specific proportion (2:1) from either a smaller (4:1) or a larger one (1:1), demonstrating an ability to represent the specific, and not relative, value of a particular proportion. Again, at test, chicks selectively responded to the previously reinforced proportion from new ratios. These findings provide strong evidence for very young animals’ ability to extract, identify, and productively use proportion information across a range of different amounts.