Reduced choice-confidence in negative numerals.

Negative numbers are central in math. However, they are abstract, hard to learn, and manipulated slower than positive numbers regardless of math ability. It suggests that confidence, namely the post-decision estimate of being correct, should be lower than positives. We asked participants to pick the larger single-digit numeral in a pair and collected their implicit confidence with button pressure (button pressure was validated with three empirical signatures of confidence). We also modeled their choices with a drift-diffusion decision model to compute the post-decision estimate of being correct. We found that participants had relatively low confidence with negative numerals. Given that participants compared with high accuracy the basic base-10 symbols (0-9), reduced confidence may be a general feature of manipulating abstract negative numerals as they produce more uncertainty than positive numerals per unit of time.

The numerator bias exists in millions of real-world comparisons.

Fractions are crucial, from math and science education to daily activities, but they are hard. A puzzling aspect of fractions is that people over-rely on the numerator when comparing a pair of fractions. Previous work has considered this numerator bias mostly as a reasoning mishap. Still, in a vast amount of pairwise comparisons, across many real-world domains, not just education textbooks, we report a high prior probability that the larger fraction has the larger numerator, and, for a relevant case, we provide formal arguments why. The existence of such a regularity suggests that the numerator bias may reflect a rational adaptation that detects and exploits likely events. In a pair of visual-proportion tasks (discrete and continuous fractions), we confirm that the numerator bias in participants adapts to experimented regularities. Even though weak education and math abilities play a role, adaptation to informative priors outside the classroom poses a challenge to educators, learners, and decision-makers.

Intrinsic whole number bias in humans.

Humans have great difficulty comparing quotients including fractions, proportions, and probabilities and often erroneously isolate the whole numbers of the numerators and denominators to compare them. Some have argued that the whole number bias is a compensatory strategy to deal with difficult comparisons. We examined adult humans' preferences for gambles that differed only in numerosity, and not in factors that influence their expected value (probabilities and stakes). Subjects consistently preferred gambles with more winning balls to ones with fewer, even though the probabilities were mathematically identical, replicating prior results. In a second experiment, we found that subjects accurately represented the relative probabilities of the choice options during rapid nonverbal probability judgments but nonetheless showed biases based on whole numbers. We mathematically formalized and quantitatively evaluated cognitive rules based on existing hypotheses that attempt to explain subjects' whole number biases during quotient comparisons. The results show that the whole number bias is intrinsic to the way humans solve quotient comparisons rather than a compensatory strategy. (PsycINFO Database Record

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.

Confidence judgments during ratio comparisons reveal a Bayesian bias.

Rational numbers are essential in mathematics and decision-making but humans often and erroneously rely on the magnitude of the numerator or denominator to determine the relative size of a quotient. The source of this flawed whole number strategy is poorly understood. Here we test the Bayesian hypothesis that the human bias toward large values in the numerator or denominator of a ratio estimate is the result of higher confidence in large samples. Larger values are considered a better (more certain) instance of that ratio than the same ratio composed of smaller values. We collected confidence measures explicitly (Experiment 1) and implicitly (Experiment 2) during subjects' comparisons of non-symbolic proportions (images with arrays of orange and blue dots). We manipulated the discernibility of the fractions to control difficulty and varied the cardinality and congruency of the numerators, denominators, and ratio values (e.g. 8/20 vs. 5/10 and 16/40 vs. 10/20). The results revealed that subjects' confidence during ratio comparisons was modulated by the numerical magnitude of the fraction's components, consistent with a Bayesian perception of relative ratios. The results suggest that the large number bias could arise from greater confidence in large samples.