Epistemic Limitations and Precise Estimates in Analog Magnitude Representation

This chapter presents a re-understanding of the contents of our analog magnitude representations (e.g., approximate duration, distance, number). The approximate number system (ANS) is considered, which supports numerical representations that are widely described as fuzzy, noisy, and limited in their representational power. The contention is made that these characterizations are largely based on misunderstandings—that what has been called “noise” and “fuzziness” is actually an important epistemic signal of confidence in one’s estimate of the value. Rather than the ANS having noisy or fuzzy numerical content, it is suggested that the ANS has exquisitely precise numerical content that is subject to epistemic limitations. Similar considerations will arise for other analog representations. The chapter discusses how this new understanding of ANS representations recasts the learnability problem for number and the conceptual changes that children must accomplish in the number domain

A Shared Intuitive (Mis)understanding of Psychophysical Law Leads Both Novices and Educated Students to Believe in a Just Noticeable Difference (JND)

Abstract: Humans are both the scientists who discover psychological laws and the thinkers who behave according to those laws. Oftentimes, when our natural behavior is in accord with those laws, this dual role serves us well: our intuitions about our own behavior can serve to inform our discovery of new laws. But, in cases where the laws that we discover through science do not agree with the intuitions and biases we carry into the lab, we may find it harder to believe in and adopt those laws. Here, we explore one such case. Since the founding of psychophysics, the notion of a Just Noticeable Difference (JND) in perceptual discrimination has been ubiquitous in experimental psychology—even in spite of theoretical advances since the 1950’s that argue that there can be no such thing as a threshold in perceiving difference. We find that both novices and psychologically educated students alike misunderstand the JND to mean that, below a certain threshold, humans will be unable to tell which of two quantities is greater (e.g., that humans will be completely at chance when trying to judge which is heavier, a bag with 3000 grains of sand or 3001). This belief in chance performance below a threshold is inconsistent with psychophysical law. We argue that belief in a JND is part of our intuitive theory of psychology and is therefore very difficult to dispel

Approximate Number Sense in Students With Severe Hearing Loss: A Modality-Neutral Cognitive Ability.

The Approximate Number System (ANS) allows humans and non-human animals to estimate large quantities without counting. It is most commonly studied in visual contexts (i.e., with displays containing different numbers of dots), although the ANS may operate on all approximate quantities regardless of modality (e.g., estimating the number of a series of auditory tones). Previous research has shown that there is a link between ANS and mathematics abilities, and that this link is resilient to differences in visual experience (Kanjlia et al., 2018). However, little is known about the function of the ANS and its relationship to mathematics abilities in the absence of other types of sensory input. Here, we investigated the acuity of the ANS and its relationship with mathematics abilities in a group of students from the Sichuan Province in China, half of whom were deaf. We found, consistent with previous research, that ANS acuity improves with age. We found that mathematics ability was predicted by Non-verbal IQ and Inhibitory Control, but not visual working memory capacity or Attention Network efficiencies. Even above and beyond these predictors, ANS ability still accounted for unique variance in mathematics ability. Notably, there was no interaction with hearing, which indicates that the role played by the ANS in explaining mathematics competence is not modulated by hearing capacity. Finally, we found that age, Non-verbal IQ and Visual Working Memory capacity were predictive of ANS performance when controlling for other factors. In fact, although students with hearing loss performed slightly worse than students with normal hearing on the ANS task, hearing was no longer significantly predictive of ANS performance once other factors were taken into account. These results indicate that the ANS is able to develop at a consistent pace with other cognitive abilities in the absence of auditory experience, and that its relationship with mathematics ability is not contingent on sensory input from hearing

Favoreciendo el aprendizaje de la matemática con la tablet: Juguemos con el tiempo, el espacio y las cantidades

Todo el conocimiento que desarrollamos en nuestra etapa escolar está basado en intuiciones básicas. Por ejemplo, en matemática, incluso los niños más pequeños tienen las habilidades básicas necesarias para representar tamaños de objetos, duraciones de tiempo y para estimar la cantidad aproximada de elementos de un conjunto sin contarlos. Esta última habilidad se relaciona con el ‘Sistema Numérico Aproximado’ que se ha propuesto como base para soportar las operaciones simbólicas y, específicamente, el concepto de número. La mayor parte de las investigaciones sobre este sistema muestran que a mayor precisión en la discriminación no simbólica de cantidades, mayor capacidad para la resolución simbólica de problemas matemáticos. Asimismo, estudios previos muestran que el desarrollo de las habilidades básicas de estimación de magnitudes no simbólicas impacta positivamente en el desempeño de las matemáticas simbólicas. A partir de estos estudios previos, el presente proyecto se propone estudiar el efecto de una serie de mini juegos—diseñados específicamente para potenciar el desarrollo de diferentes dimensiones del sistema de magnitudes no simbólicas—sobre las matemáticas simbólicas. Estos juegos podrán ser distribuidos a la población escolar uruguaya a través de las tablets del Plan Ceibal. El entrenamiento de estas habilidades no simbólicas tempranamente puede construir una base sólida para el aprendizaje de las matemáticas en la escuela.Centro de Estudios Fundacion Ceibal, Agencia Nacional de Investigación e Innovació

Determiners are "conservative" because their meanings are not relations: evidence from verification

Quantificational determiners have meanings that are "conservative" in the following sense: in sentences, repeating a determiner's internal argument within its external argument is logically insignificant. Using a verification task to probe which sets (or properties) of entities are represented when participants evaluate sentences, we test the predictions of three potential explanations for the cross-linguistic yet substantive conservativity constraint. According to "lexical restriction" views, words like every express relations that are exhibited by pairs of sets, but only some of these relations can be expressed with determiners. An "interface filtering" view retains the relational conception of determiner meanings, while replacing appeal to lexical filters (on relations of the relevant type) with special rules for interpreting the combination of a quantificational expression (Det NP) with its syntactic context and a ban on meanings that lead to triviality. The contrasting idea of "ordered predication" is that determiners don't express genuine relations. Instead, the second argument provides the scope of a monadic quantifier, while the first argument selects the domain for that quantifier: the sequences with respect to which it is evaluated. On this view, a determiner's two arguments each have a different logical status, suggesting that they might have a different psychological status as well. We find evidence that this is the case: When evaluating sentences like every big circle is blue, participants mentally group the things specified by the determiner's first argument (e.g., the big circles) but not the things specified by the second argument (e.g., the blue things) or the intersection of both (e.g., the big blue circles). These results suggest that the phenomenon of conservativity is due to ordered predication

Is Approximate Number Precision a Stable Predictor

Previous research shows that children's ability to estimate numbers of items using their Approximate Number System (ANS) predicts later math ability. To more closely examine the predictive role of early ANS acuity on later abilities, we assessed the ANS acuity, math ability, and expressive vocabulary of preschoolers twice, six months apart. We also administered attention and memory span tasks to ask whether the previously reported association between ANS acuity and math ability is ANS-specific or attributable to domain-general cognitive skills. We found that early ANS acuity predicted math ability six months later, even when controlling for individual differences in age, expressive vocabulary, and math ability at the initial testing. In addition, ANS acuity was a unique concurrent predictor of math ability above and beyond expressive vocabulary, attention, and memory span. These findings of a predictive relationship between early ANS acuity and later math ability add to the growing evidence for the importance of early numerical estimation skills

Is Approximate Number Precision a Stable Predictor

Previous research shows that children's ability to estimate numbers of items using their Approximate Number System (ANS) predicts later math ability. To more closely examine the predictive role of early ANS acuity on later abilities, we assessed the ANS acuity, math ability, and expressive vocabulary of preschoolers twice, six months apart. We also administered attention and memory span tasks to ask whether the previously reported association between ANS acuity and math ability is ANS-specific or attributable to domain-general cognitive skills. We found that early ANS acuity predicted math ability six months later, even when controlling for individual differences in age, expressive vocabulary, and math ability at the initial testing. In addition, ANS acuity was a unique concurrent predictor of math ability above and beyond expressive vocabulary, attention, and memory span. These findings of a predictive relationship between early ANS acuity and later math ability add to the growing evidence for the importance of early numerical estimation skills

Experimental investigations of ambiguity: the case of most

In the study of natural language quantification, much recent attention has been devoted to the investigation of verification procedures associated with the proportional quantifier most. The aim of these studies is to go beyond the traditional characterization of the semantics of most, which is confined to explicating its truth-functional and presuppositional content as well as its combinatorial properties, as these aspects underdetermine the correct analysis of most. The present paper contributes to this effort by presenting new experimental evidence in support of a decompositional analysis of most according to which it is a superlative construction built from a gradable predicate many or much and the superlative operator -est (Hackl, in Nat Lang Semant 17:63–98, 2009). Our evidence comes in the form of verification profiles for sentences like Most of the dots are blue which, we argue, reflect the existence of a superlative reading of most. This notably contrasts with Lidz et al.’s (Nat Lang Semant 19:227–256, 2011) results. To reconcile the two sets of data, we argue, it is necessary to take important differences in task demands into account, which impose limits on the conclusions that can be drawn from these studies