How to learn the natural numbers: inductive inference and the acquisition of number concepts

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A. (2006). Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51-B60.] argue that such an inductive inference is consistent with a representational system that clearly does not express the natural numbers and that possession of the natural numbers requires further principles that make the inductive inference superfluous. We argue that their critique is unsuccessful. Provided that children have access to a suitable initial system of representation, the sort of inductive inference that Rips et al. call into question can in fact facilitate the acquisition of larger integer concepts without the addition of any further principles.     

How counting represents number: what children must learn and when they learn it

This study compared 2- to 4-year-olds who understand how counting works (cardinal-principle-knowers) to those who do not (subset-knowers), in order to better characterize the knowledge itself. New results are that (1) Many children answer the question "how many" with the last word used in counting, despite not understanding how counting works; (2) Only children who have mastered the cardinal principle, or are just short of doing so, understand that adding objects to a set means moving forward in the numeral list whereas subtracting objects mean going backward; and finally (3) Only cardinal-principle-knowers understand that adding exactly 1 object to a set means moving forward exactly 1 word in the list, whereas subset-knowers do not understand the unit of change.     

Working memory and arithmetic calculation in children: the contributory roles of processing speed, short-term memory, and reading

The cognitive underpinnings of arithmetic calculation in children are noted to involve working memory; however, cognitive processes related to arithmetic calculation and working memory suggest that this relationship is more complex than stated previously. The purpose of this investigation was to examine the relative contributions of processing speed, short-term memory, working memory, and reading to arithmetic calculation in children. Results suggested four important findings. First, processing speed emerged as a significant contributor of arithmetic calculation only in relation to age-related differences in the general sample. Second, processing speed and short-term memory did not eliminate the contribution of working memory to arithmetic calculation. Third, individual working memory components--verbal working memory and visual-spatial working memory--each contributed unique variance to arithmetic calculation in the presence of all other variables. Fourth, a full model indicated that chronological age remained a significant contributor to arithmetic calculation in the presence of significant contributions from all other variables. Results are discussed in terms of directions for future research on working memory in arithmetic calculation.     

When the mental number line involves a delay: the writing of numbers by children of different arithmetical abilities

Our study focused on number transcoding in children. It investigated how 9-year-olds with and without arithmetical disabilities wrote Arabic digits after they had heard them as number words. Planning time before writing each digit was registered. Analyses revealed that the two groups differed not only in arithmetical abilities but also in verbal and reading abilities. Children with arithmetical disabilities were overall slower in planning Arabic digits than were control children with normal arithmetical abilities. In addition, they showed a number size effect for numbers smaller than 10, suggesting a semantically mediated route in number processing. Control children did not need more planning time for large numbers (e.g., 8) than for small numbers (e.g., 3), suggesting a direct nonsemantic route. For both two- and three-digit numbers, both groups of children showed a number size effect, although the effect was smaller each time for control children. The presence of the stronger number size effect for children with arithmetical disabilities was seen as a delay in the development of quick and direct transcoding. The relation between transcoding problems and arithmetical disabilities is discussed. A defect in the linking of numerical symbols to analog numerical representations is proposed as an explanation for the transcoding problems found in some children.     

Learning your way around town: how virtual taxicab drivers learn to use both layout and landmark information

By having subjects drive a virtual taxicab through a computer-rendered town, we examined how landmark and layout information interact during spatial navigation. Subject-drivers searched for passengers, and then attempted to take the most efficient route to the requested destinations (one of several target stores). Experiment 1 demonstrated that subjects rapidly learn to find direct paths from random pickup locations to target stores. Experiment 2 varied the degree to which landmark and layout cues were preserved across two successively learned towns. When spatial layout was preserved, transfer was low if only target stores were altered, and high if both target stores and surrounding buildings were altered, even though in the latter case all local views were changed. This suggests that subjects can rapidly acquire a survey representation based on the spatial layout of the town and independent of local views, but that subjects will rely on local views when present, and are harmed when associations between previously learned landmarks are disrupted. We propose that spatial navigation reflects a hierarchical system in which either layout or landmark information is sufficient for orienting and wayfinding; however, when these types of cues conflict, landmarks are preferentially used.     

Some strangeness in the proportion,or how to stop worrying and learn to love the mechanistic forces of darkness

Understanding humans requires viewing them as mechanisms of some sort, since understanding anything requires seeing it as a mechanism. It is science’s job to reveal mechanisms. But science reveals much more than that: it also reveals enduring mystery—strangeness in the proportion. Concentrating just on the scientific side of Selinger’s and Engström’s call for a moratorium on cyborg discourse, I argue that this strangeness prevents cyborg discourse from diminishing us.     

The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: evidence for a segmented linear model

This experiment aimed to expand previous findings on the development of mental number representation. We tested the hypothesis that children's familiarity with numbers is directly reflected by the shape of their mental number line. This mental number line was expected to be linear as long as numbers lay within the range of numbers children were familiar with. Five- to 9-year-olds (N=78) estimated the positions of numbers on an external number line and additionally completed a counting assessment mirroring their familiarity with numbers. A segmented regression model consisting of two linear segments described number line estimations significantly better than a logarithmic or a simple linear model. Moreover, the change point between the two linear segments, indicating a change of discriminability between numbers, was significantly correlated with children's familiar number range. Findings are discussed in terms of the accumulator model, assuming a linear mental representation with scalar variability.     

From domain-generality to domain-sensitivity: 4-Month-olds learn an abstract repetition rule in music that 7-month-olds do not

Learning must be constrained for it to lead to productive generalizations. Although biology is undoubtedly an important source of constraints, prior experience may be another, leading learners to represent input in ways that are more conducive to some generalizations than others, and/or to up- and down-weight features when entertaining generalizations. In two experiments, 4-month-old and 7-month-old infants were familiarized with sequences of musical chords or tones adhering either to an AAB pattern or an ABA pattern. In both cases, the 4-month-olds learned the generalization, but the 7-month-olds did not. The success of the 4-month-olds appears to contradict an account that this type of pattern learning is the provenance of a language-specific rule-learning module. It is not yet clear what drives the age-related change, but plausible candidates include differential experience with language and music, as well as interactions between general cognitive development and stimulus complexity.     

From grammatical number to exact numbers: early meanings of 'one', 'two', and 'three' in English, Russian, and Japanese

This study examined whether singular/plural marking in a language helps children learn the meanings of the words 'one,' 'two,' and 'three.' First, CHILDES data in English, Russian (which marks singular/plural), and Japanese (which does not) were compared for frequency, variability, and contexts of number-word use. Then young children in the USA, Russia, and Japan were tested on Counting and Give-N tasks. More English and Russian learners knew the meaning of each number word than Japanese learners, regardless of whether singular/plural cues appeared in the task itself (e.g., "Give two apples" vs. "Give two"). These results suggest that the learning of "one," "two" and "three" is supported by the conceptual framework of grammatical number, rather than that of integers.     

Wyckoff's observing response: Pigeons learn to observe stimuli for free food but not stimuli for extinction

Four pigeons received periods of free food delivery alternating with periods of extinction. The experimental chamber was divided in half. Initially the subjects could produce stimuli selectively associated with these schedules by standing on the right side of the chamber and later by standing on the left side. In both phases, subjects produced the free food stimulus most of the time it was available but did not increase above baseline the time spent producing the extinction stimulus. Thus, when alternative stimuli are available, the pigeon prefers the stimulus associated with the greater frequency of reinforcement although the choice results in no biological advantage.     

THE BACON not the bacon: how children and adults understand accented and unaccented noun phrases

Two eye-tracking experiments examine whether adults and 4- and 5-year-old children use the presence or absence of accenting to guide their interpretation of noun phrases (e.g., the bacon) with respect to the discourse context. Unaccented nouns tend to refer to contextually accessible referents, while accented variants tend to be used for less accessible entities. Experiment 1 confirms that accenting is informative for adults, who show a bias toward previously-mentioned objects beginning 300 ms after the onset of unaccented nouns and pronouns. But contrary to findings in the literature, accented words produced no observable bias. In Experiment 2, 4 and 5 year olds were also biased toward previously-mentioned objects with unaccented nouns and pronouns. This builds on findings of limits on children's on-line reference comprehension [Arnold, J. E., Brown-Schmidt, S., & Trueswell, J. C. (2007). Children's use of gender and order-of-mention during pronoun comprehension. Language and Cognitive Processes], showing that children's interpretation of unaccented nouns and pronouns is constrained in contexts with one single highly accessible object.     

Feeling the need for (personalized) speed: how natural controls and customization contribute to enjoyment of a racing game through enhanced immersion

Prior research suggests that video game features that appear natural or that otherwise allow players to identify with their in-game experience will promote enjoyment. Using a 2 × 2 experiment, this study demonstrates the positive effects of a steering-wheel controller and the opportunity to customize the driven vehicle on enjoyment of a console driving game, as mediated by transportation and challenge-skill balance. The role of presence is also probed, with results suggesting no direct link to enjoyment.     

Beyond objectivity: The performance impact of the perceived ability to learn and solve problems

The purpose of this research was to develop and provide initial validation evidence for the performance impact of a measure of an individual's perceived ability to learn and solve problems (PALS). Building on the self-efficacy literature and the importance of learning and problem solving, the fundamental premise of this research was that PALS would significantly explain employee performance. In addition to demonstrating that PALS represented a distinct construct, PALS was a significant predictor of performance for managerial and entry-level employees in two different organizational contexts. Moreover, PALS explained additional variance in performance beyond general mental ability, personality, and similar constructs related to learning and problem solving.     

Experience matters: 11-month-old infants can learn to use material information to predict the weight of novel objects

In contrast to previous findings, this study demonstrates that 11-month-old infants are able to learn the relationship between object material and object weight when exploring different objects that provided a systematic covariation of both object features. This guides their action in a subsequent preferential-reaching task.     

Left to right: representational biases for numbers and the effect of visuomotor adaptation

Adaptation to right-shifting prisms improves left neglect for mental number line bisection. This study examined whether adaptation affects the mental number line in normal participants. Thirty-six participants completed a mental number line task before and after adaptation to either: left-shifting prisms, right-shifting prisms or control spectacles that did not shift the visual scene. Participants viewed number triplets (e.g. 16, 36, 55) and determined whether the numerical distance was greater on the left or right side of the inner number. Participants demonstrated a leftward bias (i.e. overestimated the length occupied by numbers located on the left side of the number line) that was consistent with the effect of pseudoneglect. The leftward bias was corrected by a short period of visuomotor adaptation to left-shifting prisms, but remained unaffected by adaptation to right-shifting prisms and control spectacles. The findings demonstrate that a simple visuomotor task alters the representation of space on the mental number line in normal participants.     

To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count

Recent accounts of number word learning posit that when children learn to accurately count sets (i.e., become “cardinal principle” or “CP” knowers), they have a conceptual insight about how the count list implements the successor function – i.e., that every natural number n has a successor defined as n + 1 (Carey, 2004, 2009; Sarnecka & Carey, 2008). However, recent studies suggest that knowledge of the successor function emerges sometime after children learn to accurately count, though it remains unknown when this occurs, and what causes this developmental transition. We tested knowledge of the successor function in 100 children aged 4 through 7 and asked how age and counting ability are related to: (1) children’s ability to infer the successors of all numbers in their count list and (2) knowledge that all numbers have a successor. We found that children do not acquire these two facets of the successor function until they are about 5½ or 6 years of age – roughly 2 years after they learn to accurately count sets and become CP-knowers. These findings show that acquisition of the successor function is highly protracted, providing the strongest evidence yet that it cannot drive the cardinal principle induction. We suggest that counting experience, as well as knowledge of recursive counting structures, may instead drive the learning of the successor function.     

邵雍先天易“天地自然之数”一解——对冯友兰先生《邵雍的“先天学”》的两点补充

作者通过对冯友兰先生《邵雍的“先天学”》的两点补充,认为．邵雍“先天易”图的六十四卦卦序按二进制六位数的大小依次排列而来．此即是宋代道学家所谓的“天地自然之数”。作者由此进一步揭示了先天卦序与阴阳生息的卦气思想之同．在一定程度上的某种暗合与实质上的相互矛盾。     

Improving adolescents' standardized test performance: An intervention to reduce the effects of stereotype threat

Standardized tests continue to generate gender and race gaps in achievement despite decades of national attention. Research on “stereotype threat” (Steele & Aronson, 1995) suggests that these gaps may be partly due to stereotypes that impugn the math abilities of females and the intellectual abilities of Black, Hispanic, and low-income students. A field experiment was performed to test methods of helping female, minority, and low-income adolescents overcome the anxiety-inducing effects of stereotype threat and, consequently, improve their standardized test scores. Specifically, seventh-grade students in the experimental conditions were mentored by college students who encouraged them either to view intelligence as malleable or to attribute academic difficulties in the seventh grade to the novelty of the educational setting. Results showed that females in both experimental conditions earned significantly higher math standardized test scores than females in the control condition. Similarly, the students—who were largely minority and low-income adolescents—in the experimental conditions earned significantly higher reading standardized test scores than students in the control condition.