Non-numerical Magnitudes Research Articles

When numbers get heavy: is the mental number line exclusively numerical?

The mental number line, with its left-to-right orientation of increasing numerical values, is often regarded as evidence for a unique connection between space and number. Yet left-to-right orientation has been shown to extend to other dimensions, consistent with a general magnitude system wherein different magnitudes share neural and conceptual resources. Such observations raise a fundamental, yet relatively unexplored, question about spatial-numerical associations: What is the nature of the information represented along the mental number line? Here we show that this information is not exclusive to number, simultaneously accommodating numerical and non-numerical magnitudes. Participants completed the classic SNARC (Spatial-Numerical Association of Response Codes) task while sometimes wearing wrist weights. Weighting the left wrist–thereby linking less and more weight to right and left, respectively–worked against left-to-right orientation of number, leaving no behavioral trace of the mental number line. Our findings point to the dynamic integration of magnitude dimensions, with spatial organization instantiating representational currency (i.e., more/less relations) shared across magnitudes.

Overlapping and distinct brain regions involved in estimating the spatial position of numerical and non-numerical magnitudes: An fMRI study

How are numerical and non-numerical magnitudes processed in the brain? Brain imaging research, primarily using comparison paradigms (i.e. judging which of two magnitudes is larger), has provided strong evidence demonstrating that the intraparietal sulcus (IPS) is a key region for processing both numerical (e.g. Arabic numerals, arrays of dots) and non-numerical magnitudes (e.g. height, brightness). These studies have suggested that there is both activation overlap and segregation in the brain regions involved in processing different dimensions of magnitude. In the present functional Magnetic Resonance Imaging (fMRI) study, we extended this line of investigation by probing the brain mechanisms underlying the mapping of numerical (Arabic numerals) and non-numerical magnitudes (brightness levels) onto a number line. Consistent with previous studies the present results revealed that number and brightness estimation was associated with overlapping activation within right lateralized areas of the posterior IPS. In addition, the contrast between number and brightness estimation revealed that bilateral anterior regions of the IPS are specifically involved in the process of estimating the position of symbolic numbers onto a number line. Furthermore, we found a significant influence of landmark reference points (0, 50 and 100) on brain activation in the right IPS for number estimation only. No regions were found to be specifically associated with brightness estimation. The results of this study reveal that the estimation of both numerical and non-numerical magnitude are associated with the engagement of a right lateralized magnitude system, but that symbolic number estimation is associated with additional engagement of bilateral regions of the anterior IPS.

The Logic of Proportionality: Reasoning with Non-Numerical Magnitudes

This paper aims at explaining the basic logical structure of proportionality assessments, under the assumption that such assessments are based on quantitative reasoning, even when no numbers are explicitly given.First an analysis of practical rationality is proposed, including the endorsement of values, the adoption of goals to be pursued, the selection of plans of actions to be executed, In particular it is argued that possible plans for action are to be assessed with regard to their impacts on all relevant values.Then the way in which norms have an impact on practical rationality is considered, distinguishing two kinds of norms: value-norms and action-norms. Value norms determine what values an agent should consider, in what scale of importance, to assess the merits of its goal-directed choice; action norm determine what actions the agent should execute. The reasoning involved in assessing the merits of an action with regard to a set of values is then considered, and a novel approach is proposed. The idea is that this assessment is performed by processing mental magnitudes, concerning the impact of the action on the realization of values, the proportional utilities so delivered, and the weights of the values. Even though this processing does not use symbolically expressed numbers, it still deals with quantities, and has to comply with the common laws of arithmetic. Relying on some work on cognitive and evolutionary psychology it is observed that processing non-symbolic approximate continuous magnitudes is a fundamental cognitive capacity. We share it with animals and it supports our understanding of numerical mathematics, and our ability to learn it.Moving more specifically into the analysis of proportionality in constitutional adjudication, it is argued that it consists in examining the extent to which value-norms directed at the legislator have been complied with. The traditional standards of suitability, necessity and proportionality are examined and reformulated under the perspective of reasoning with non-numerical magnitudes. Finally some specific issues are considered, such as how action-norms can be obtained on the basis of value assessments, and how proportionality assessments are constrained by the requirement of consistency with precedents.

Size Matters: Non-Numerical Magnitude Affects the Spatial Coding of Response

It is known that small and large numbers facilitate left/right respectively (the SNARC effect). Recently, it has been proposed that numerical magnitude is just one example of a range of quantities, which have a common cognitive/neural representation. To investigate this proposition, response congruency effects were explored for stimuli which differed according to their: (a) numerical size, (b) physical size, (c) luminance, (d) conceptual size and (e) auditory intensity. In a series of experiments, groups of undergraduate participants made two-alternative forced choice discriminations with their left or right hands. There were clear interactions between magnitude and responding hand whereby right hand responses were faster for stimuli with (a) large numbers, (b) large physical size, (c) low luminance, and (d) a reference to large objects. There was no congruency effect for the auditory stimuli. The data demonstrate that the response congruency effect observed for numbers also occurs for a variety of other non-numerical visual quantities. These results support models of general magnitude representation and suggest that the association between magnitude and the left/right sides of space may not be related to culture and/or directional reading habits.

Domain‐specific and domain‐general changes in children's development of number comparison

The numerical distance effect (inverse relationship between numerical distance and reaction time in relative number comparison tasks) has frequently been used to characterize the mental representation of number. The size of the distance effect decreases over developmental time. However, it is unclear whether this reduction simply reflects developmental changes in domain-general speed of processing and whether it is specific to numerical compared with non-numerical magnitude. To examine these open questions, we conducted a cross-sectional study with 6-, 7-, and 8-year-old children as well as adult college students. Participants performed comparisons on Arabic numerals, arrays of squares, squares of varying luminance and bars of varying height. To control for general age-related changes in reaction time, a measure of speed of processing was used as a covariate in the analysis. A significant developmental decrease in the distance effect was found across numerical and non-numerical comparison tasks over and above general changes in processing speed. However, this change was not found to differ as a function of format. These data suggest that developmental changes in the distance effect are reflective of changes in a domain-general comparison process, rather than domain-specific developmental changes in number representations. However, analysis of overall reaction times revealed significantly greater developmental changes for numerical relative to non-numerical comparison tasks. These findings highlight the importance of taking multiple measures into account when characterizing developmental changes in numerical magnitude processing. Implications for theories of numerical cognition and its development are discussed.

Dyscalculia

Dyscalculia

Does the Parietal Cortex Distinguish between “10,” “Ten,” and Ten Dots?

It is well established that the intraparietal sulcus (IPS) plays an important role in the processing and representation of numerical magnitude. Two recent studies by Piazza et al. and Cohen Kadosh et al. published in this issue of Neuron used fMRI adaptation to explore the extent to which parietal number processing is dependent upon or independent of a numbers' surface format. Their results, while slightly different, converge to suggest that the answer may be neither, but rather that it depends on the hemisphere.