skip to main content

This content will become publicly available on July 12, 2023

Title: Large numbers cause magnitude neglect: The case of government expenditures
Four studies demonstrate that the public’s understanding of government budgetary expenditures is hampered by difficulty in representing large numerical magnitudes. Despite orders of magnitude difference between millions and billions, study participants struggle with the budgetary magnitudes of government programs. When numerical values are rescaled as smaller magnitudes (in the thousands or lower), lay understanding improves, as indicated by greater sensitivity to numerical ratios and more accurate rank ordering of expenses. A robust benefit of numerical rescaling is demonstrated across a variety of experimental designs, including policy relevant choices and incentive-compatible accuracy measures. This improved sensitivity ultimately impacts funding choices and public perception of respective budgets, indicating the importance of numerical cognition for good citizenship.
Authors:
; ; ;
Award ID(s):
2017651 1851702
Publication Date:
NSF-PAR ID:
10371701
Journal Name:
Proceedings of the National Academy of Sciences
Volume:
119
Issue:
28
ISSN:
0027-8424
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract Whether and how the brain encodes discrete numerical magnitude differently from continuous nonnumerical magnitude is hotly debated. In a previous set of studies, we orthogonally varied numerical (numerosity) and nonnumerical (size and spacing) dimensions of dot arrays and demonstrated a strong modulation of early visual evoked potentials (VEPs) by numerosity and not by nonnumerical dimensions. Although very little is known about the brain's response to systematic changes in continuous dimensions of a dot array, some authors intuit that the visual processing stream must be more sensitive to continuous magnitude information than to numerosity. To address this possibility, we measured VEPs of participants viewing dot arrays that changed exclusively in one nonnumerical magnitude dimension at a time (size or spacing) while holding numerosity constant and compared this to a condition where numerosity was changed while holding size and spacing constant. We found reliable but small neural sensitivity to exclusive changes in size and spacing; however, exclusively changing numerosity elicited a much more robust modulation of the VEPs. Together with previous work, these findings suggest that sensitivity to magnitude dimensions in early visual cortex is context dependent: The brain is moderately sensitive to changes in size and spacing when numerosity ismore »held constant, but sensitivity to these continuous variables diminishes to a negligible level when numerosity is allowed to vary at the same time. Neurophysiological explanations for the encoding and context dependency of numerical and nonnumerical magnitudes are proposed within the framework of neuronal normalization.« less
  2. Abstract This paper studies the political determinants of inequalities in government interventions under majoritarian (MAJ) and proportional representation (PR) systems. We propose a probabilistic voting model of electoral competition with highly targetable government interventions and heterogeneous localities. We uncover a novel relative electoral sensitivity effect that affects government interventions only under MAJ systems. This effect tends to reduce inequality in government interventions under MAJ systems when districts are composed of sufficiently homogeneous localities. This effect goes against the conventional wisdom that MAJ systems are necessarily more conducive to inequality than PR systems. We illustrate the empirical relevance of our results with numerical simulations on possible reforms of the US Electoral College.
  3. Abstract

    Idealized numerical simulations of weak tropical cyclones (e.g., tropical depressions and tropical storms) in sheared environments indicate that vortex tilt reduction and convective symmetrization are key structural changes that can precede intensification. Through a series of ensembles of idealized numerical simulations, this study demonstrates that including radiation in the simulations affects the timing and variability of those structural changes. The underlying reason for those effects is a background thermodynamic profile with reduced energy available to fuel strong downdrafts; such a profile leads to weaker lower-tropospheric ventilation, greater azimuthal coverage of clouds and precipitation, and smaller vortex tilt with radiation. Consequently, the simulations with radiation allow for earlier intensification at stronger shear magnitudes than without radiation. An unexpected finding from this work is a reduction of both vortex tilt and intensity variability with radiation in environments with 5 m s−1 deep-layer shear. This reduction stems from reduced variability in nonlinear feedbacks between lower-tropospheric ventilation, cold pools, convection, and vortex tilt. Sensitivity experiments confirm the relationship between those processes and suggest that microphysical processes (e.g., rain evaporation) are major sources of uncertainty in the representation of weak, sheared tropical cyclones in numerical weather prediction models.

  4. Rational numbers (i.e., fractions, percentages, decimals, and whole-number frequencies) are notoriously difficult mathematical constructs. Yet correctly interpreting rational numbers is imperative for understanding health statistics, such as gauging the likelihood of side effects from a medication. Several pernicious biases affect health decision-making involving rational numbers. In our novel developmental framework, the natural-number bias—a tendency to misapply knowledge about natural numbers to all numbers—is the mechanism underlying other biases that shape health decision-making. Natural-number bias occurs when people automatically process natural-number magnitudes and disregard ratio magnitudes. Math-cognition researchers have identified individual differences and environmental factors underlying natural-number bias and devised ways to teach people how to avoid these biases. Although effective interventions from other areas of research can help adults evaluate numerical health information, they circumvent the core issue: people’s penchant to automatically process natural-number magnitudes and disregard ratio magnitudes. We describe the origins of natural-number bias and how researchers may harness the bias to improve rational-number understanding and ameliorate innumeracy in real-world contexts, including health. We recommend modifications to formal math education to help children learn the connections among natural and rational numbers. We also call on researchers to consider individual differences people bring to health decision-making contexts and howmore »measures from math cognition might identify those who would benefit most from support when interpreting health statistics. Investigating innumeracy with an interdisciplinary lens could advance understanding of innumeracy in theoretically meaningful and practical ways.« less
  5. Manufacturing-induced residual stresses in carbon/epoxy 3D woven composites arise during cooling after curing due to a large difference in the coefficients of thermal expansion between the carbon fibers and the epoxy matrix. The magnitudes of these stresses appear to be higher in composites with high throughthickness reinforcement and in some cases are sufficient to lead to matrix cracking. This paper presents a numerical approach to simulation of development of manufacturing-induced residual stresses in an orthogonal 3D woven composite unit cell using finite element analysis. The proposed mesoscale modeling combines viscoelastic stress relaxation of the epoxy matrix and realistic reinforcement geometry (based on microtomography and fabric mechanics simulations) and includes imaginginformed interfacial (tow/matrix) cracks. Sensitivity of the numerical predictions to reinforcement geometry and presence of defects is discussed. To validate the predictions, blind hole drilling is simulated, and the predicted resulting surface displacements are compared to the experimentally measured values. The validated model provides an insight into the volumetric distribution of residual stresses in 3D woven composites. The presented approach can be used for studies of residual stress effects on mechanical performance of composites and strategies directed at their mitigation.