Relating magnitudes: the brain's code for proportions

Thinking about quantity: the intertwined development of spatial and numerical cognition

There are many continuous quantitative dimensions in the physical world. Philosophical, psychological, and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time and mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, and density) and number can be conceptualized discretely or continuously (i.e., natural vs real numbers). Variation on these quantitative dimensions is typically correlated, e.g., larger objects often weigh more than smaller ones. Number is a distinctive continuous dimension because the natural numbers (i.e., positive integers) are used to quantify collections of discrete objects. This aspect of number is emphasized by teaching of the count word sequence and arithmetic during the early school years. We review research on spatial and numerical estimation, and argue that a generalized magnitude system is the starting point for development in both domains. Development occurs along several lines: (1) changes in capacity, durability, and precision, (2) differentiation of the generalized magnitude system into separable dimensions, (3) formation of a discrete number system, i.e., the positive integers, (4) mapping the positive integers onto the continuous number line, and (5) acquiring abstract knowledge of the relations between pairs of systems. We discuss implications of this approach for teaching various topics in mathematics, including scaling, measurement, proportional reasoning, and fractions.

The Golden Section as Optical Limitation

The golden section, ϕ = (1 + √5)/2 = 1.618... and its companion ϕ = 1/ϕ = ϕ -1 = 0.618..., are irrational numbers which for centuries were believed to confer aesthetic appeal. In line with the presence of golden sectioning in natural growth patterns, recent EEG recordings show an absence of coherence between brain frequencies related by the golden ratio, suggesting the potential relevance of the golden section to brain dynamics. Using Mondrian-type patterns comprising a number of paired sections in a range of five section-section areal ratios (including golden-sectioned pairs), participants were asked to indicate as rapidly and accurately as possible the polarity (light or dark) of the smallest section in the patterns. They were also asked to independently assess the aesthetic appeal of the patterns. No preference was found for golden-sectioned patterns, while reaction times (RTs) tended to decrease overall with increasing ratio independently of each pattern's fractal dimensionality. (Fractal dimensionality was unrelated to ratio and measured in terms of the Minkowski-Bouligand box-counting dimension). The ease of detecting the smallest section also decreased with increasing ratio, although RTs were found to be substantially slower for golden-sectioned patterns under 8-paired sectioned conditions. This was confirmed by a significant linear relationship between RT and ratio (p < .001) only when the golden-sectioned RTs were excluded [the relationship was non-significant for the full complement of ratios (p = .217)]. Image analysis revealed an absence of spatial frequencies between 4 and 8 cycles-per-degree that was exclusive to the 8-paired (golden)-sectioned patterns. The significance of this was demonstrated in a subsequent experiment by addition of uniformly distributed random noise to the patterns. This provided a uniform spatial-frequency profile for all patterns, which did not influence the decrease in RT with increasing ratio but abolished the elevated RTs to golden-sectioned patterns. This suggests that optical limitation in the form of reduced inter-neural synchronization during spatial-frequency coding may be the foundation for the perceptual effects of golden sectioning.

Response trajectories reveal the temporal dynamics of fraction representations

Previous studies on mental arithmetic with fractions have been equivocal with respect to the nature of mental representations that are formed with fractions. It is not clear from present evidence whether fractions form perceptual primitives independent from components or whether component magnitudes must be processed in addition to the holistic magnitude. In the present study, we attempt to resolve this issue by using computer mouse-tracking. We analyzed the dynamics of participants' hand movements as they compared presented fractions to 1/2. We found that before settling to the correct answer, hand trajectories showed competitive influences of component magnitude and overall fraction magnitude, but the influence of components happened much earlier. These data support the idea that in fraction comparison, component magnitudes and holistic magnitude are processed together in a continuous, competitive manner.

Eye movements reflect and shape strategies in fraction comparison

The comparison of fractions is a difficult task that can often be facilitated by separately comparing components (numerators and denominators) of the fractions—that is, by applying so-called component-based strategies. The usefulness of such strategies depends on the type of fraction pair to be compared. We investigated the temporal organization and the flexibility of strategy deployment in fraction comparison by evaluating sequences of eye movements in 20 young adults. We found that component-based strategies could account for the response times and the overall number of fixations observed for the different fraction pairs. The analysis of eye movement sequences showed that the initial eye movements in a trial were characterized by stereotypical scanning patterns indicative of an exploratory phase that served to establish the kind of fraction pair presented. Eye movements that followed this phase adapted to the particular type of fraction pair and indicated the deployment of specific comparison strategies. These results demonstrate that participants employ eye movements systematically to support strategy use in fraction comparison. Participants showed a remarkable flexibility to adapt to the most efficient strategy on a trial-by-trial basis. Our results confirm the value of eye movement measurements in the exploration of strategic adaptation in complex tasks.

Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes

This study presents evidence that humans have intuitive, perceptually based access to the abstract fraction magnitudes instantiated by nonsymbolic ratio stimuli. Moreover, it shows these perceptually accessed magnitudes can be easily compared with symbolically represented fractions. In cross-format comparisons, participants picked the larger of two ratios. Ratios were presented either symbolically as fractions or nonsymbolically as paired dot arrays or as paired circles. Response patterns were consistent with participants comparing specific analog fractional magnitudes independently of the particular formats in which they were presented. These results pose a challenge to accounts that argue human cognitive architecture is ill-suited for processing fractions. Instead, it seems that humans can process nonsymbolic ratio magnitudes via perceptual routes and without recourse to conscious symbolic algorithms, analogous to the processing of whole number magnitudes. These findings have important implications for theories regarding the nature of human number sense - they imply that fractions may in some sense be natural numbers, too.

The role of ANS acuity and numeracy for the calibration and the coherence of subjective probability judgments

The purpose of the study was to investigate how numeracy and acuity of the approximate number system (ANS) relate to the calibration and coherence of probability judgments. Based on the literature on number cognition, a first hypothesis was that those with lower numeracy would maintain a less linear use of the probability scale, contributing to overconfidence and nonlinear calibration curves. A second hypothesis was that also poorer acuity of the ANS would be associated with overconfidence and non-linearity. A third hypothesis, in line with dual-systems theory (e.g., Kahneman and Frederick, 2002) was that people higher in numeracy should have better access to the normative probability rules, allowing them to decrease the rate of conjunction fallacies. Data from 213 participants sampled from the Swedish population showed that: (i) in line with the first hypothesis, overconfidence and the linearity of the calibration curves were related to numeracy, where people higher in numeracy were well calibrated with zero overconfidence. (ii) ANS was not associated with overconfidence and non-linearity, disconfirming the second hypothesis. (iii) The rate of conjunction fallacies was slightly, but to a statistically significant degree decreased by numeracy, but still high at all numeracy levels. An unexpected finding was that participants with better ANS acuity gave more realistic estimates of their performance relative to others.

The Impact of Density and Ratio on Object-Ensemble Representation in Human Anterior-Medial Ventral Visual Cortex

Behavioral research has demonstrated that observers can extract summary statistics from ensembles of multiple objects. We recently showed that a region of anterior-medial ventral visual cortex, overlapping largely with the scene-sensitive parahippocampal place area (PPA), participates in object-ensemble representation. Here we investigated the encoding of ensemble density in this brain region using fMRI-adaptation. In Experiment 1, we varied density by changing the spacing between objects and found no sensitivity in PPA to such density changes. Thus, density may not be encoded in PPA, possibly because object spacing is not perceived as an intrinsic ensemble property. In Experiment 2, we varied relative density by changing the ratio of 2 types of objects comprising an ensemble, and observed significant sensitivity in PPA to such ratio change. Although colorful ensembles were shown in Experiment 2, Experiment 3 demonstrated that sensitivity to object ratio change was not driven mainly by a change in the ratio of colors. Thus, while anterior-medial ventral visual cortex is insensitive to density (object spacing) changes, it does code relative density (object ratio) within an ensemble. Object-ensemble processing in this region may thus depend on high-level visual information, such as object ratio, rather than low-level information, such as spacing/spatial frequency.

Parametric alpha- and beta-band signatures of supramodal numerosity information in human working memory

Numerosity can be assessed by analog estimation, similar to a continuous magnitude, or by discrete quantification of the individual items in a set. While the extent to which these two processes rely on common neural mechanisms remains debated, recent studies of sensory working memory (WM) have identified an oscillatory signature of continuous magnitude information, in terms of quantitative modulations of prefrontal upper beta activity (20-30 Hz). Here, we examined how such parametric oscillatory WM activity may also reflect the abstract assessment of the numerosity of discrete items. We recorded EEG while participants (n = 24) processed the number of stimulus pulses presented in the visual, auditory, or tactile modality, under otherwise identical experimental conditions. Behavioral response profiles showed varying degrees of analog estimation and of discretized quantification in the different modalities. During sustained processing in WM, the amplitude of posterior alpha oscillations (8-13 Hz) reflected the increased memory load associated with maintaining larger sets of discrete items. In contrast, earlier numerosity-dependent modulations of right prefrontal upper beta (20-30 Hz) specifically reflected the extent to which numerosity was assessed by analog estimation, both between and within presentation modalities. Together, the analog approximation-but not the discretized representation-of numerosity information exhibited a parametric oscillatory signature akin to a continuous sensory magnitude. The results suggest dissociable oscillatory mechanisms of abstract numerosity integration, at a supramodal processing stage in human WM.

Psychophysics and the evolution of behavior

Sensory information allows animals to interpret their environment and make decisions. The ways in which animals perceive and measure stimuli from the social and physical environment guide nearly every decision they make. Thus, sensory perception and associated cognitive processing have a strong impact on behavioral evolution. Research in this area often focuses on the unique properties of the sensory system of an individual species, yet certain relevant features of perception and cognition generally hold across taxa. One such general feature is the proportionally based translation of physical stimulus magnitude into perceived stimulus magnitude. This process has been recognized for over a century, but recent studies have begun to consider how a law of proportional psychophysics, Weber's law, exerts selective force in behavioral evolution.

The processing of symbolic and nonsymbolic ratios in school-age children

This study tested the processing of ratios of natural numbers in school-age children. Nine- and eleven-year-olds were presented collections made up of orange and grey dots (i.e., nonsymbolic format) and fractions (i.e., symbolic format). They were asked to estimate ratios between the number of orange dots and the total number of dots and fractions by producing an equivalent ratio of surface areas (filling up a virtual glass). First, we tested whether symbolic notation of ratios affects their processing by directly comparing performance on fractions with that on dot sets. Second, we investigated whether children’s estimates of nonsymbolic ratios of natural numbers relied at least in part on ratios of surface areas by contrasting a condition in which the ratio of surface areas occupied by dots covaried with the ratio of natural numbers and a condition in which this ratio of surface areas was kept constant across ratios of natural numbers. The results showed that symbolic notation did not really have a negative impact on performance among 9-year-olds, while it led to more accurate estimates in 11-year-olds. Furthermore, in dot conditions, children’s estimates increased consistently with ratios between the number of orange dots and the total number of dots even when the ratio of surface areas was kept constant but were less accurate in that condition than when the ratio of surface areas covaried with the ratio of natural numbers. In summary, these results indicate that mental magnitude representation is more accurate when it is activated from symbolic ratios in children as young as 11 years old and that school-age children rely at least in part on ratios of surface areas to process nonsymbolic ratios of natural numbers when given the opportunity to do so.

Non-symbolic halving in an Amazonian indigene group

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.

Fractions: the new frontier for theories of numerical development

Recent research on fractions has broadened and deepened theories of numerical development. Learning about fractions requires children to recognize that many properties of whole numbers are not true of numbers in general and also to recognize that the one property that unites all real numbers is that they possess magnitudes that can be ordered on number lines. The difficulty of attaining this understanding makes the acquisition of knowledge about fractions an important issue educationally, as well as theoretically. This article examines the neural underpinnings of fraction understanding, developmental and individual differences in that understanding, and interventions that improve the understanding. Accurate representation of fraction magnitudes emerges as crucial both to conceptual understanding of fractions and to fraction arithmetic.

Coding of abstract quantity by 'number neurons' of the primate brain

Humans share with nonhuman animals a quantification system for representing the number of items as nonverbal mental magnitudes. Over the past decade, the anatomical substrates and neuronal mechanisms of this quantification system have been unraveled down to the level of single neurons. Work with behaviorally trained nonhuman primates identified a parieto-frontal cortical network with individual neurons selectively tuned to the number of items. Such 'number neurons' can track items across space, time, and modality to encode numerosity in a most abstract, supramodal way. The physiological properties of these neurons can explain fundamental psychophysical phenomena during numerosity judgments. Functionally overlapping groups of parietal neurons represent not only numerable-discrete quantity (numerosity), but also innumerable-continuous quantity (extent) and relations between quantities (proportions), supporting the idea of a generalized magnitude system in the brain. These studies establish putative homologies between the monkey and human brain and demonstrate the suitability of nonhuman primates as model system to explore the neurobiological roots of the brain's nonverbal quantification system, which may constitute the evolutionary foundation of all further, more elaborate numerical skills in humans.

Supramodal numerosity selectivity of neurons in primate prefrontal and posterior parietal cortices

Numerosity, the number of elements in a set, is a most abstract quantitative category. As such, it is independent of the sensory modality of its elements, i.e., supramodal. Because neuronal numerosity selectivity had never been compared directly across different sensory modalities, it remained elusive if and where single neurons encode numerosity irrespective of the items' modality. Here, monkeys were trained to discriminate both the number of auditory sounds and visual items within the same session. While the monkeys performed this task, the activity of neurons was recorded in the lateral prefrontal cortex and ventral intraparietal sulcus, structures critically involved in numerical cognition. Groups of neurons in both areas encoded either the number of auditory pulses, visual items, or both. The finding of neurons responding to numerosity irrespective of the sensory modality supports the idea of a nonverbal, supramodal neuronal code of numerical quantity in the primate brain.