How do we convert a number into a finger trajectory?

The left digit effect in an unbounded number line task

The left digit effect in number line estimation refers to the phenomenon where numerals with similar magnitudes but different leftmost digits (e.g., 19 and 22) are estimated to be farther apart on a number line than is warranted. The effect has been studied using a bounded number line task, a task in which a line is bounded by two endpoints (e.g., 0 and 100), and where one must indicate the correct location of a target numeral on the line. The goal of the present work is to investigate the left digit effect in an unbounded number line task, a task that involves using the size of one unit to determine a target numeral's location, and that elicits strategies different from those used in the bounded number line task. In a preregistered study, participants (N = 58 college students) completed four blocks of 38 trials each of an unbounded number line task, with target numerals ranging between 0 and 100. We found a medium and statistically reliable left digit effect (d = 0.70). The study offers further evidence that the effect is not driven by response strategies specific to the bounded number line task. We discuss other possible sources of the effect including conversion of symbols to magnitudes in these and other contexts.

Negation mitigates rather than inverts the neural representations of adjectives

Combinatoric linguistic operations underpin human language processes, but how meaning is composed and refined in the mind of the reader is not well understood. We address this puzzle by exploiting the ubiquitous function of negation. We track the online effects of negation ("not") and intensifiers ("really") on the representation of scalar adjectives (e.g., "good") in parametrically designed behavioral and neurophysiological (MEG) experiments. The behavioral data show that participants first interpret negated adjectives as affirmative and later modify their interpretation towards, but never exactly as, the opposite meaning. Decoding analyses of neural activity further reveal significant above chance decoding accuracy for negated adjectives within 600 ms from adjective onset, suggesting that negation does not invert the representation of adjectives (i.e., "not bad" represented as "good"); furthermore, decoding accuracy for negated adjectives is found to be significantly lower than that for affirmative adjectives. Overall, these results suggest that negation mitigates rather than inverts the neural representations of adjectives. This putative suppression mechanism of negation is supported by increased synchronization of beta-band neural activity in sensorimotor areas. The analysis of negation provides a steppingstone to understand how the human brain represents changes of meaning over time.

Cognitive Foundations of Early Mathematics: Investigating the Unique Contributions of Numerical, Executive Function, and Spatial Skills

There is an emerging consensus that numerical, executive function (EF), and spatial skills are foundational to children's mathematical learning and development. Moreover, each skill has been theorized to relate to mathematics for different reasons. Thus, it is possible that each cognitive construct is related to mathematics through distinct pathways. The present study tests this hypothesis. One-hundred and eighty 4- to 9-year-olds (Mage = 6.21) completed a battery of numerical, EF, spatial, and mathematics measures. Factor analyses revealed strong, but separable, relations between children's numerical, EF, and spatial skills. Moreover, the three-factor model (i.e., modelling numerical, EF, and spatial skills as separate latent variables) fit the data better than a general intelligence (g-factor) model. While EF skills were the only unique predictor of number line performance, spatial skills were the only unique predictor of arithmetic (addition) performance. Additionally, spatial skills were related to the use of more advanced addition strategies (e.g., composition/decomposition and retrieval), which in turn were related to children's overall arithmetic performance. That is, children's strategy use fully mediated the relation between spatial skills and arithmetic performance. Taken together, these findings provide new insights into the cognitive foundations of early mathematics, with implications for assessment and instruction moving forward.

Modeling the left digit effect in adult number line estimation

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over- and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well-captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0-100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

Development of the Mental Number Line Representation of Numbers 0–10 and Its Relationship to Mental Arithmetic

The internal representation of numbers on the mental number line (MNL) was demonstrated by performing the computerized version of the number-to-position (CNP) task on a touchscreen while restricting response time. We found that the estimation pattern is best fit by a sigmoid function, further denoted as the “sigmoidal model”. Two developmental leaps occurring during elementary school were recognized: (1) the division of the number line into two segments and (2) consistent use of different anchor points on the number line—the left endpoint in first grade, the right endpoint in second grade, and finally the midpoint in third grade. Additionally, when examining the differences between the breakpoints, we found that first graders demonstrated a breakpoint close to 6, which linearly decreased over the years until stabilizing close to 5. The relation between the ability to place individual numbers on a number line and performance of mental arithmetic showed that the consistent use of anchor points correlated significantly with faster responses in mental arithmetic.

The influence of locative expressions on context-dependency of endpoint control in aiming

It has been claimed that increased reliance on context, or allocentric information, develops when aiming movements are more consciously monitored and/or controlled. Since verbalizing target features requires strong conscious monitoring, we expected an increased reliance on allocentric information when verbalizing a target label (i.e. target number) during movement execution. We examined swiping actions towards a global array of targets embedded in different local array configurations on a tablet under no-verbalization and verbalization conditions. The global and local array configurations allowed separation of contextual-effects from any possible numerical magnitude biases triggered from calling out specific target numbers.The patterns of constant errors in the target directionwere used to assess differences between conditions. Variation in the target context configuration systematically biased movement endpoints in both the no-verbalization and verbalization conditions. Ultimately, our results do not support the assertion that calling out target numbers during movement execution increases the context-dependency of targeted actions.

Saccadic adaptation shapes perceived size: Common codes for action and perception

Recent findings suggest that perceptual and motor systems share common codes; for instance, perceived object location is known to correlate with motor changes in the oculomotor system. Here, we investigate whether modifying saccade amplitude affects object size perception. Participants saw in peripheral vision a test disk that could vary in size across trials. This disk was then replaced by a small target cross, which was the signal to make a saccade. After the saccade, the target cross was extinguished and replaced by a reference disk (thus seen in foveal vision). Participants had to compare the post- to the pre-saccade disk sizes. Psychometric functions were obtained before and after one session of 142 saccades made toward the cross that either stepped toward the fixation point during the saccade (backward adaptation group) or remained stationary (control group). In the experimental group, stepping the target cross toward fixation during saccades decreased movement amplitude, a phenomenon called saccadic adaptation. We observed a concurrent shift in the psychometric functions reflecting a decrease in perceived object size. Such a perceptual modification did not occur in the control group. Our results reveal that motor changes co-occur with changes in perceived object size. Unlike previous studies evaluating the impact of saccadic adaptation on perceived location, we measured here the perception of another spatial feature (the object size) that is not relevant for the sensorimotor transformation. Theoretical implications of the strong links between oculomotor parameters and object perception are discussed.

A comparison of simple movement behaviors across three different devices

Reaching trajectories have provided a unique tool to observe changes in internal cognitive decisions. Furthermore, technological advances have made devices for measuring reach movements more accessible and researchers have recognized that various populations including children, elderly populations, and non-human primates can easily execute simple movements as responses. As a result, devices such as a three-dimensional (3D) reach tracker, a stylus, or a computer-mouse have been increasingly utilized to study cognitive processes. However, although the specific type of tracking device that a researcher uses may impact behavior due to the constraints it places on movements, most researchers in these fields are unaware of this potential issue. Here, we examined the potential behavioral impact of using each of these three devices. To induce re-directed movements that mimic the movements that often occur following changes in cognitive states, we used a double-step task in which displacement of an initial target location requires participants to quickly re-direct their movement. We found that reach movement parameters were largely comparable across the three devices. However, hand movements measured by a 3D reach tracker showed earlier reach initiation latencies (relative to stylus movements) and more curved movement trajectories (relative to both mouse and stylus movements). Reach movements were also re-directed following target displacement more rapidly. Thus, 3D reach trackers may be ideal for observing fast, subtle changes in internal decision-making processes compared to other devices. Taken together, this study provides a useful reference for comparing and implementing reaching studies to examine human cognition.

How does number magnitude influence temporal and spatial parameters of eye movements?

The influence of numerical processing on individuals' behavior is now well documented. The spatial representation of numbers on a left-to-right mental line (i.e., SNARC effect) has been shown to have sensorimotor consequences, the majority of studies being mainly concerned with its impact on the response times. Its impact on the motor programming stage remains less documented, although swiping movement amplitudes have recently been shown to be modulated by number magnitude. Regarding saccadic eye movements, the few available studies have not provided clear-cut conclusions. They showed that spatial-numerical associations modulated ocular drifts, but not the amplitude of memory-guided saccades. Because these studies held saccadic coordinates constant, which might have masked potential numerical effects, we examined whether spontaneous saccadic eye movements (with no saccadic target) could reflect numerical effects. Participants were asked to look either to the left or to the right side of an empty screen to estimate the magnitude (< or > 5) of a centrally presented digit. Latency data confirmed the presence of the classical SNARC and distance effects. More critically, saccade amplitude reflected a numerical effect: participants' saccades were longer for digits far from the standard (1 and 9) and were shorter for digits close to it (4 and 6). Our results suggest that beyond response times, kinematic parameters also offer valuable information for the understanding of the link between numerical cognition and motor programming.

Decision-making in sensorimotor control

Skilled sensorimotor interactions with the world result from a series of decision-making processes that determine, on the basis of information extracted during the unfolding sequence of events, which movements to make and when and how to make them. Despite this inherent link between decision-making and sensorimotor control, research into each of these two areas has largely evolved in isolation, and it is only fairly recently that researchers have begun investigating how they interact and, together, influence behaviour. Here, we review recent behavioural, neurophysiological and computational research that highlights the role of decision-making processes in the selection, planning and control of goal-directed movements in humans and nonhuman primates.

Repeating Numbers Reduces Results: Violations of the Identity Axiom in Mental Arithmetic

Even simple mental arithmetic is fraught with cognitive biases. For example, adding repeated numbers (so-called tie problems, e.g., 2 + 2) not only has a speed and accuracy advantage over adding different numbers (e.g., 1 + 3) but may also lead to under-representation of the result relative to a standard value (Charras et al., 2012, 2014). Does the tie advantage merely reflect easier encoding or retrieval compared to non-ties, or also a distorted result representation? To answer this question, 47 healthy adults performed two tasks, both of which indicated under-representation of tie results: In a result-to-position pointing task (Experiment 1) we measured the spatial mapping of numbers and found a left-bias for tie compared to non-tie problems. In a result-to-line-length production task (Experiment 2) we measured the underlying magnitude representation directly and obtained shorter lines for tie- compared to non-tie problems. These observations suggest that the processing benefit of tie problems comes at the cost of representational reduction of result meaning. This conclusion is discussed in the context of a recent model of arithmetic heuristics and biases.

Dynamics and development in number-to-space mapping

Young children's estimates of numerical magnitude increase approximately logarithmically with actual magnitude. The conventional interpretation of this finding is that children's estimates reflect an innate logarithmic encoding of number. A recent set of findings, however, suggests that logarithmic number-line estimates emerge via a dynamic encoding mechanism that is sensitive to previously encountered stimuli. Here we examine trial-to-trial changes in logarithmicity of numerosity estimates to test an alternative dynamic model (D-MLLM) with both a strong logarithmic component and a weak response to previous stimuli. In support of D-MLLM, first-trial numerosity estimates in both adults (Study 1, 2, 3, and 4) and children (Study 4) were strongly logarithmic, despite zero previous stimuli. Additionally, although numerosity of a previous trial affected adults' estimates, the influence of previous numbers always accompanied the logarithmic-to-linear shift predicted by D-MLLM. We conclude that a dynamic encoding mechanism is not necessary for compressive mapping, but sequential effects on response scaling are a possible source of linearity in adults' numerosity estimation.

Decoding the processing stages of mental arithmetic with magnetoencephalography

Elementary arithmetic is highly prevalent in our daily lives. However, despite decades of research, we are only beginning to understand how the brain solves simple calculations. Here, we applied machine learning techniques to magnetoencephalography (MEG) signals in an effort to decompose the successive processing stages and mental transformations underlying elementary arithmetic. Adults subjects verified single-digit addition and subtraction problems such as 3 + 2 = 9 in which each successive symbol was presented sequentially. MEG signals revealed a cascade of partially overlapping brain states. While the first operand could be transiently decoded above chance level, primarily based on its visual properties, the decoding of the second operand was more accurate and lasted longer. Representational similarity analyses suggested that this decoding rested on both visual and magnitude codes. We were also able to decode the operation type (additions vs. subtraction) during practically the entire trial after the presentation of the operation sign. At the decision stage, MEG indicated a fast and highly overlapping temporal dynamics for (1) identifying the proposed result, (2) judging whether it was correct or incorrect, and (3) pressing the response button. Surprisingly, however, the internally computed result could not be decoded. Our results provide a first comprehensive picture of the unfolding processing stages underlying arithmetic calculations at a single-trial level, and suggest that externally and internally generated neural codes may have different neural substrates.

Task instructions modulate unit-decade binding in two-digit number representation

Previous studies have found decomposed processes, as well as holistic processes, in the representation of two-digit numbers. The present study investigated the influence of task instruction on such processes. Participants completed both magnitude and parity tasks in one of three instructional conditions, where they were asked to either consider two-digit numbers as a whole or to focus on one specific digit. In two experiments, we found that when participants were asked to consider the two digits as an integrated number, they always exhibited a unit-decade compatibility effect, indicating a failure of selective attention on the digit relevant to the given task. However, the mere presence of the neighboring digit is not a sufficient condition for the compatibility effect: when participants were explicitly asked to process a specific digit, their success/failure to selectively ignore the irrelevant digit depended on task requirements. Further, computer mouse tracking indicated that the locus of the compatibility effect was related to late response-related processing. The results signify the deep involvement of top-down processes in unit-decade binding for two-digit number representation.

Numerical Affordance Influences Action Execution: A Kinematic Study of Finger Movement

Humans represent symbolic numbers as oriented from left to right: the mental number line (MNL). Up to now, scientific studies have mainly investigated the MNL by means of response times. However, the existing knowledge on the MNL can be advantaged by studies on motor patterns while responding to a number. Cognitive representations, in fact, cannot be fully understood without considering their impact on actions. Here we investigated whether a motor response can be influenced by number processing. Participants seated in front of a little soccer goal. On each trial they were visually presented with a numerical (2, 5, 8) or a non-numerical ($) stimulus. They were instructed to kick a small ball with their right index toward a frontal soccer goal as soon as a stimulus appeared on a screen. However, they had to refrain from kicking when number five was presented (no-go signal). Our main finding is that performing a kicking action after observation of the larger digit proved to be more efficient: the trajectory path was shorter and lower on the surface, velocity peak was anticipated. The smaller number, instead, specifically altered the temporal and spatial aspects of trajectories, leading to more prolonged left deviations. This is the first experimental demonstration that the reaching component of a movement is influenced by number magnitude. Since this paradigm does not require any verbal skill and non-symbolic stimuli (array of dots) can be used, it could be fruitfully adopted to evaluate number abilities in children and even preschoolers. Notably, this is a self-motivating and engaging task, which might help children to get involved and to reduce potential arousal connected to institutional paper-and-pencil examinations.

A threshold-free model of numerosity comparisons

A dominant mechanism in the Judgment and Decision Making literature states that information is accumulated about each choice option until a decision threshold is met. Only after that threshold does a subject start to execute a motor response to indicate their choice. However, recent research has revealed spatial gradients in motor responses as a function of comparison difficulty as well as changes-of-mind in the middle of an action, both suggesting continued accumulation and processing of decision-related signals after the decision boundary. Here we present a formal model and supporting data from a number comparison task that a continuous motor planner, combined with a simple statistical inference scheme, can model detailed behavioral effects without assuming a threshold. This threshold-free model reproduces subjects’ sensitivity to numerical distance in reaching, accuracy, reaction time, and changes of mind. We argue that the motor system positions the effectors using an optimal biomechanical feedback controller, and continuous statistical inference on outputs from cognitive processes.

Abandoning and modifying one action plan for alternatives

Visual scenes are often complex and crowded with many different objects. To interact effectively, we must choose one object at a time as a goal for action. Certain external cues can act as a stop signal, quickly cancelling an ongoing action. Less recognized are internal signals. These can come from recent experience, anticipated action outcomes, cognitive states, and when attention is captured by a salient object. These signals elevate one action plan over alternatives and can quickly modify an initial choice. Here, we focus on these internal processes responsible for selecting, abandoning and modifying action plans. We first highlight how the brain resolves competition among multiple action plans. Critical is the existence of parallel motor planning processes, which allow efficient and timely changes. Then, we discuss how the action system interplays with perception, attention and memory processes to bias action selection and suppress or modify erroneous selections. Subsequently, we show how tracking the continuous modification of action trajectories can provide a tool to read out changes in internal cognitive states. Taken together, we shed light on a broader view that sensorimotor networks can continuously modify actions through simultaneous evaluation of alternative activities in concert with widely distributed perceptual and cognitive networks.This article is part of the themed issue 'Movement suppression: brain mechanisms for stopping and stillness'.

Distinct Representations of Magnitude and Spatial Position within Parietal Cortex during Number-Space Mapping

Mapping numbers onto space is foundational to mathematical cognition. These cognitive operations are often conceptualized in the context of a "mental number line" and involve multiple brain regions in or near the intraparietal sulcus (IPS) that have been implicated both in numeral and spatial cognition. Here we examine possible differentiation of function within these brain areas in relating numbers to spatial positions. By isolating the planning phase of a number line task and introducing spatiotopic mapping tools from fMRI into mental number line task research, we are able to focus our analysis on the neural activity of areas in anterior IPS (aIPS) previously associated with number processing and on spatiotopically organized areas in and around posterior IPS (pIPS), while participants prepare to place a number on a number line. Our results support the view that the nonpositional magnitude of a numerical symbol is coded in aIPS, whereas the position of a number in space is coded in posterior areas of IPS. By focusing on the planning phase, we are able to isolate activation related to the cognitive, rather than the sensory-motor, aspects of the task. Also, to allow the separation of spatial position from magnitude, we tested both a standard positive number line (0 to 100) and a zero-centered mixed number line (-100 to 100). We found evidence of a functional dissociation between aIPS and pIPS: Activity in aIPS was associated with a landmark distance effect not modulated by spatial position, whereas activity in pIPS revealed a contralateral preference effect.

Reversing the Manual Digit Bias in Two-Digit Number Comparison

Though recent work in numerical cognition has supported a strong tie between numerical and spatial representations (e.g., a mental number line), less is known about such ties in multi-digit number representations. Along this line, Bloechle, Huber, and Moeller (2015) found that pointing positions in two-digit number comparison were biased leftward toward the decade digit. Moreover, this bias was reduced in unit-decade incompatible pairs. In the present study, we tracked computer mouse movements as participants compared two-digit numbers to a fixed standard (55). Similar to Bloechle et al. (2015) , we found that trajectories exhibited a leftward bias that was reduced for unit-decade incompatible comparisons. However, when positions of response labels were reversed, the biases reversed. That is, we found a rightward bias for compatible pairs that was reduced for incompatible pairs. This result calls into question a purely embodied representation of place value structure and instead supports a competition model of two-digit number representation.

Finger Tracking Reveals the Covert Stages of Mental Arithmetic

We introduce a novel method capable of dissecting the succession of processing stages underlying mental arithmetic, thus revealing how two numbers are transformed into a third. We asked adults to point to the result of single-digit additions and subtractions on a number line, while their finger trajectory was constantly monitored. We found that the two operands are processed serially: the finger first points toward the larger operand, then slowly veers toward the correct result. This slow deviation unfolds proportionally to the size of the smaller operand, in both additions and subtractions. We also observed a transient operator effect: a plus sign attracted the finger to the right and a minus sign to the left and a transient activation of the absolute value of the subtrahend. These findings support a model whereby addition and subtraction are computed by a stepwise displacement on the mental number line, starting with the larger number and incrementally adding or subtracting the smaller number.

The visual number world: A dynamic approach to study the mathematical mind

In the domain of language research, the simultaneous presentation of a visual scene and its auditory description (i.e., the visual world paradigm) has been used to reveal the timing of mental mechanisms. Here we apply this rationale to the domain of numerical cognition in order to explore the differences between fast and slow arithmetic performance, and to further study the role of spatial-numerical associations during mental arithmetic. We presented 30 healthy adults simultaneously with visual displays containing four numbers and with auditory addition and subtraction problems. Analysis of eye movements revealed that participants look spontaneously at the numbers they currently process (operands, solution). Faster performance was characterized by shorter latencies prior to fixating the relevant numbers and fewer revisits to the first operand while computing the solution. These signatures of superior task performance were more pronounced for addition and visual numbers arranged in ascending order, and for subtraction and numbers arranged in descending order (compared to the opposite pairings). Our results show that the "visual number world"-paradigm provides on-line access to the mind during mental arithmetic, is able to capture variability in arithmetic performance, and is sensitive to visual layout manipulations that are otherwise not reflected in response time measurements.

Self-Guided Reading: Touch-Based Measures of Syntactic Processing

A novel online reading methodology termed Self-Guided Reading (SGR) is examined to determine if it can successfully detect well-studied syntactic processing behaviours. In SGR, a participant runs their finger under masked text in order to reveal a sentence. It is therefore similar to self-paced reading in presentation of stimuli, but different in the motion that the participant makes to interact with the stimuli. The phenomena of relative clause, adverb and noun phrase/sentential attachment are utilised to allow comparison to previous research that employed self-paced reading and eye-tracking. SGR was able to detect the predicted processing behaviours in all sentence types. Moreover, once design choices and task effects are accounted for, SGR was the most consistent in triggering a motor movement change at the predicted point in the sentence. Able to provide a semi-continuous reading measure at low cost, SGR should be investigated further to uncover the full potential of the method for psycholinguistic research.

Ocular drift along the mental number line

We examined the spontaneous association between numbers and space by documenting attention deployment and the time course of associated spatial-numerical mapping with and without overt oculomotor responses. In Experiment 1, participants maintained central fixation while listening to number names. In Experiment 2, they made horizontal target-direct saccades following auditory number presentation. In both experiments, we continuously measured spontaneous ocular drift in horizontal space during and after number presentation. Experiment 2 also measured visual-probe-directed saccades following number presentation. Reliable ocular drift congruent with a horizontal mental number line emerged during and after number presentation in both experiments. Our results provide new evidence for the implicit and automatic nature of the oculomotor resonance effect associated with the horizontal spatial-numerical mapping mechanism.

Pushing forward in embodied cognition: may we mouse the mathematical mind?

Freely available software has popularized “mousetracking” to study cognitive processing; this involves the on-line recording of cursor positions while participants move a computer mouse to indicate their choice. Movement trajectories of the cursor can then be reconstructed off-line to assess the efficiency of responding in time and across space. Here we focus on the process of selecting among alternative numerical responses. Several studies have recently measured the mathematical mind with cursor movements while people decided about number magnitude or parity, computed sums or differences, or simply located numbers on a number line. After some general methodological considerations about mouse tracking we discuss several conceptual concerns that become particularly evident when “mousing” the mathematical mind.

Spatial Associations in Numerical Cognition—From Single Digits to Arithmetic

The literature on spatial associations during number processing is dominated by the SNARC (spatial–numerical association of response codes) effect. We describe spatial biases found for single digits and pairs of numbers, first in the “original” speeded parity task and then extending the scope to encompass different tasks, a range of measures, and various populations. Then we review theoretical accounts before surveying the emerging evidence for similar spatial associations during mental arithmetic. We conclude that the mental number line hypothesis and an embodied approach are useful frameworks for further studies.

Doing Arithmetic by Hand: Hand Movements during Exact Arithmetic Reveal Systematic, Dynamic Spatial Processing

Mathematics requires precise inferences about abstract objects inaccessible to perception. How is this possible? One proposal is that mathematical reasoning, while concerned with entirely abstract objects, nevertheless relies on neural resources specialized for interacting with the world—in other words, mathematics may be grounded in spatial or sensorimotor systems. Mental arithmetic, for instance, could involve shifts in spatial attention along a mental “number-line”, the product of cultural artefacts and practices that systematically spatialize number and arithmetic. Here, we investigate this hypothesized spatial processing during exact, symbolic arithmetic (e.g., 4 + 3 = 7). Participants added and subtracted single-digit numbers and selected the exact solution from responses in the top corners of a computer monitor. While they made their selections using a computer mouse, we recorded the movement of their hand as indexed by the streaming x, y coordinates of the computer mouse cursor. As predicted, hand movements during addition and subtraction were systematically deflected toward the right and the left, respectively, as if calculation involved simultaneously simulating motion along a left-to-right mental number-line. This spatial–arithmetical bias, moreover, was distinct from—but correlated with—individuals’ spatial–numerical biases (i.e., spatial–numerical association of response codes, SNARC, effect). These results are the first evidence that exact, symbolic arithmetic prompts systematic spatial processing associated with mental calculation. We discuss the possibility that mathematical calculation relies, in part, on an integrated system of spatial processes.

Compressive mapping of number to space reflects dynamic encoding mechanisms, not static logarithmic transform

The mapping of number onto space is fundamental to measurement and mathematics. However, the mapping of young children, unschooled adults, and adults under attentional load shows strong compressive nonlinearities, thought to reflect intrinsic logarithmic encoding mechanisms, which are later "linearized" by education. Here we advance and test an alternative explanation: that the nonlinearity results from adaptive mechanisms incorporating the statistics of recent stimuli. This theory predicts that the response to the current trial should depend on the magnitude of the previous trial, whereas a static logarithmic nonlinearity predicts trialwise independence. We found a strong and highly significant relationship between numberline mapping of the current trial and the magnitude of the previous trial, in both adults and school children, with the current response influenced by up to 15% of the previous trial value. The dependency is sufficient to account for the shape of the numberline, without requiring logarithmic transform. We show that this dynamic strategy results in a reduction of reproduction error, and hence improvement in accuracy.