Do children learn the integers by induction?

According to one theory about how children learn the meaning of the words for the positive integers, they first learn that "one," "two," and "three" stand for appropriately sized sets. They then conclude by inductive inference that the next numeral in the count sequence denotes the size of sets containing one more object than the size denoted by the preceding numeral. We have previously argued, however, that the conclusion of this Induction does not distinguish the standard meaning of the integers from nonstandard meanings in which, for example, "ten" could mean set sizes of 10, 20, 30,... elements. Margolis and Laurence [Margolis, E., & Laurence, S. (2008). How to learn the natural numbers: Inductive inference and the acquisition of number concepts. Cognition, 106, 924-939] believe that our argument depends on attributing to children "radically indeterminate" concepts. We show, first, that our conclusion is compatible with perfectly determinate meanings for "one" through "three." Second, although the inductive inference is indeed indeterminate - which is why it is consistent with nonstandard meanings - making it determinate presupposes the constraints that the inference is supposed to produce.     

Children's understanding of counting

This study examines the abstractness of children's mental representation of counting, and their understanding that the last number word used in a count tells how many items there are (the cardinal word principle). In the first experiment, twenty-four 2- and 3-year-olds counted objects, actions, and sounds. Children counted objects best, but most showed some ability to generalize their counting to actions and sounds, suggesting that at a very young age, children begin to develop an abstract, generalizable mental representation of the counting routine. However, when asked "how many" following counting, only older children (mean age 3.6) gave the last number word used in the count a majority of the time, suggesting that the younger children did not understand the cardinal word principle. In the second experiment (the "give-a-number" task), the same children were asked to give a puppet one, two, three, five, and six items from a pile. The older children counted the items, showing a clear understanding of the cardinal word principle. The younger children succeeded only at giving one and sometimes two items, and never used counting to solve the task. A comparison of individual children's performance across the "how-many" and "give-a-number" tasks shows strong within-child consistency, indicating that children learn the cardinal word principle at roughly 3 1/2 years of age. In the third experiment, 18 2- and 3-year-olds were asked several times for one, two, three, five, and six items, to determine the largest numerosity at which each child could succeed consistently. Results indicate that children learn the meanings of smaller number words before larger ones within their counting range, up to the number three or four. They then learn the cardinal word principle at roughly 3 1/2 years of age, and perform a general induction over this knowledge to acquire the meanings of all the number words within their counting range.     

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.     

Contextual control in teaching numbers to a child with intellectual disability

The aim of this work was to teach the discrimination of "equal" and "different" in numbers. The experiment was carried out a seven-year-old child with intellectual disability. The problem was analysed from the contextual control perspective. The learning procedure consisted of explicitly teaching a second-order conditional discrimination, and transfer to a novel second-order conditional discrimination was tested. In this study, the boy learned that, in presence of X1 (equal), he had to select the comparison B1 (the number one), given the sample A1 (the word one) and B2 (the number two), given A2 (the word two). He also he learned that, in the presence of X2 (different), he had to select the comparison B2 (the number two), given the sample A1 (the word one) and B1 (the number one), given A2 (the word two). We subsequently presented the contextual stimuli with two new numbers: three and four. The results showed the occurrence of learning transfer without explicit training in the new numbers.     

材料类型和颜色典型性影响颜色-物体Stroop效应

通过两个实验考查材料类型和颜色典型性对颜色-物体Stroop效应的影响。实验1,考查颜色-物体（图片）Stroop效应。结果颜色典型性差异显著,命名图片的颜色和图片的名称都产生显著的颜色-物体Stroop效应。实验2,考查颜色-物体（词语）Stroop效应。结果颜色典型性差异显著,命名词语的颜色产生颜色-物体Stroop效应,命名词语的名称未产生颜色-物体Stroop效应。结论,材料类型和颜色典型性影响颜色-物体Stroop效应。     

Characteristics of spatial memory span: Is there an analogy to the word length effect, based on movement time?

In studies of verbal short-term memory it has been shown that the length of words to be remembered affects the size of memory span. This word-length effect is attributed to relationships between the rate of rehearsal of verbal material and the time it takes to speak the words being rehearsed. For spatial memory span there may also be an internal rehearsal system linked to overt responding, and if there is a strong analogy to be drawn between the verbal and spatial domains then movement time between spatial targets should predict the number of spatial locations that can be recalled. In the experiments reported here the time taken to move between spatial targets is varied by altering the size of targets and the distance between them. No difference between span performance on a nine-block spatial span task were found, either on immediate recall or on recall after an interval. When recall is of items from an array of 27, grouped in nine sets of three, with only one location in any set being presented on any trial, there is an effect of display size. This effect is consonant with the argument that movement time is related to spatial rehearsal, but other explanations are also possible. However, if recall in this task is scored over the nine sets rather than over the 27 items, then there is no difference between the displays. The results indicate that performance on the normal nine-block spatial-span task cannot be predicted by movement time.     

Grey parrot number acquisition: The inference of cardinal value from ordinal position on the numeral list

A Grey parrot (Psittacus erithacus) had previously been taught to use English count words ("one" through "sih" [six]) to label sets of one to six individual items (Pepperberg, 1994). He had also been taught to use the same count words to label the Arabic numerals 1 through 6. Without training, he inferred the relationship between the Arabic numerals and the sets of objects (Pepperberg, 2006b). In the present study, he was then trained to label vocally the Arabic numerals 7 and 8 ("sih-none", "eight", respectively) and to order these Arabic numerals with respect to the numeral 6. He subsequently inferred the ordinality of 7 and 8 with respect to the smaller numerals and he inferred use of the appropriate label for the cardinal values of seven and eight items. These data suggest that he constructed the cardinal meanings of "seven" ("sih-none") and "eight" from his knowledge of the cardinal meanings of one through six, together with the place of "seven" ("sih-none") and "eight" in the ordered count list.     

Number gestures predict learning of number words

When asked to explain their solutions to a problem, children often gesture and, at times, these gestures convey information that is different from the information conveyed in speech. Children who produce these gesture‐speech “mismatches” on a particular task have been found to profit from instruction on that task. We have recently found that some children produce gesture‐speech mismatches when identifying numbers at the cusp of their knowledge, for example, a child incorrectly labels a set of two objects with the word “three” and simultaneously holds up two fingers. These mismatches differ from previously studied mismatches (where the information conveyed in gesture has the potential to be integrated with the information conveyed in speech) in that the gestured response contradicts the spoken response. Here, we ask whether these contradictory number mismatches predict which learners will profit from number‐word instruction. We used the Give‐a‐Number task to measure number knowledge in 47 children (M_age = 4.1 years, SD = 0.58), and used the What's on this Card task to assess whether children produced gesture‐speech mismatches above their knower level. Children who were early in their number learning trajectories (“one‐knowers” and “two‐knowers”) were then randomly assigned, within knower level, to one of two training conditions: a Counting condition in which children practiced counting objects; or an Enriched Number Talk condition containing counting, labeling set sizes, spatial alignment of neighboring sets, and comparison of these sets. Controlling for counting ability, we found that children were more likely to learn the meaning of new number words in the Enriched Number Talk condition than in the Counting condition, but only if they had produced gesture‐speech mismatches at pretest. The findings suggest that numerical gesture‐speech mismatches are a reliable signal that a child is ready to profit from rich number instruction and provide evidence, for the first time, that cardinal number gestures have a role to play in number‐learning.     

A Nonparametric,Multiple Imputation-Based Method for the Retrospective Integration of Data Sets

Complex research questions often cannot be addressed adequately with a single data set. One sensible alternative to the high cost and effort associated with the creation of large new data sets is to combine existing data sets containing variables related to the constructs of interest. The goal of the present research was to develop a flexible, broadly applicable approach to the integration of disparate data sets that is based on nonparametric multiple imputation and the collection of data from a convenient, de novo calibration sample. We demonstrate proof of concept for the approach by integrating three existing data sets containing items related to the extent of problematic alcohol use and associations with deviant peers. We discuss both necessary conditions for the approach to work well and potential strengths and weaknesses of the method compared to other data set integration approaches.     

Evolutionary history versus current causal role in the definition of disorder: reply to McNally

The harmful dysfunction (HD) analysis (Wakefield, American Psychologist 47 (1992a) 373) asserts that "disorder" means "harmful dysfunction", where "harm" is a value concept anchored in social values and "dysfunction" is a factual concept referring to failure of a mechanism to perform a natural function. Additionally, the HD analysis claims that a mechanism's natural functions are its naturally selected effects. McNally (Behaviour Research and Therapy (2000) pp. 309-314) argues to the contrary that "dysfunction" is a value concept referring to negative failures of function, that "function" refers to current causal roles and not evolutionarily designed causal roles, and that "disorder" consequently means "harmful failure of a mechanism to perform a valued current causal role." I reply by showing that McNally's proposals lack the HD analysis's power to explain common judgments about function, dysfunction, and disorder. "Dysfunction" cannot be a negative value concept because many dysfunctions are positive or neutral; "function" cannot refer to current causal roles because many current causal roles are not functions and some functions are not current causal roles; and "disorder" cannot refer to harmful failures of current causal roles because that definition allows almost any negative condition whatever to be a disorder and thus fails to explain the distinctions we make between disorder and non-disorder.     

The time-based word length effect and stimulus set specificity

The word length effect is the finding that short items are remembered better than long items on immediate serial recall tests. The time-based word length effect refers to this finding when the lists comprise items that vary only in pronunciation time. Three experiments compared recall of three different sets of disyllabic words that differed systematically only in spoken duration. One set showed a word length effect, one set showed no effect of word length, and the third showed a reverse word length effect, with long words recalled better than short. A new fourth set of words was created, and it also failed to yield a time-based word length effect. Because all four experiments used the same methodology and varied only the stimulus sets, it is argued that the time-based word length effect is not robust and as such poses problems for models based on the phonological loop.     

An associative account of the development of word learning

Word learning is a notoriously difficult induction problem because meaning is underdetermined by positive examples. How do children solve this problem? Some have argued that word learning is achieved by means of inference: young word learners rely on a number of assumptions that reduce the overall hypothesis space by favoring some meanings over others. However, these approaches have difficulty explaining how words are learned from conversations or text, without pointing or explicit instruction. In this research, we propose an associative mechanism that can account for such learning. In a series of experiments, 4-year-olds and adults were presented with sets of words that included a single nonsense word (e.g. dax). Some lists were taxonomic (i.,e., all items were members of a given category), some were associative (i.e., all items were associates of a given category, but not members), and some were mixed. Participants were asked to indicate whether the nonsense word was an animal or an artifact. Adults exhibited evidence of learning when lists consisted of either associatively or taxonomically related items. In contrast, children exhibited evidence of word learning only when lists consisted of associatively related items. These results present challenges to several extant models of word learning, and a new model based on the distinction between syntagmatic and paradigmatic associations is proposed.     

认知诊断属性分类一致性信度区间估计三种方法

摘要：引入了三种可以估计认知诊断属性分类一致性信度置信区间的方法：Bootstrap法、平行测验法和平行测验配对法。用模拟研究验证和比较了这三种方法的表现,结果发现,平行测验法和Bootstrap法在被试量比较少、题目数量比较少的情况下,估计的标准误和置信区间较接近,但是随着被试量的增加,Bootstrap法的估计精度提高较快,在被试量大和题目数量较多时基本接近平行测验配对法的结果。Bootstrap法的所需时间最少,平行测验配对法计算过程复杂且用时较长,推荐用Bootstrap法估计认知诊断属性分类一致性信度的置信区间。     

Young children use their hands to tell their mothers what to say

Children produce their first gestures before their first words, and their first gesture+word sentences before their first word+word sentences. These gestural accomplishments have been found not only to predate linguistic milestones, but also to predict them. Findings of this sort suggest that gesture itself might be playing a role in the language‐learning process. But what role does it play? Children's gestures could elicit from their mothers the kinds of words and sentences that the children need to hear in order to take their next linguistic step. We examined maternal responses to the gestures and speech that 10 children produced during the one‐word period. We found that all 10 mothers ‘translated’ their children's gestures into words, providing timely models for how one‐ and two‐word ideas can be expressed in English. Gesture thus offers a mechanism by which children can point out their thoughts to mothers, who then calibrate their speech to those thoughts, and potentially facilitate language‐learning.     

Does learning to count involve a semantic induction?

We tested the hypothesis that, when children learn to correctly count sets, they make a semantic induction about the meanings of their number words. We tested the logical understanding of number words in 84 children that were classified as "cardinal-principle knowers" by the criteria set forth by Wynn (1992). Results show that these children often do not know (1) which of two numbers in their count list denotes a greater quantity, and (2) that the difference between successive numbers in their count list is 1. Among counters, these abilities are predicted by the highest number to which they can count and their ability to estimate set sizes. Also, children's knowledge of the principles appears to be initially item-specific rather than general to all number words, and is most robust for very small numbers (e.g., 5) compared to larger numbers (e.g., 25), even among children who can count much higher (e.g., above 30). In light of these findings, we conclude that there is little evidence to support the hypothesis that becoming a cardinal-principle knower involves a semantic induction over all items in a child's count list.     

The Frequency with which a Group of Unselected College Students Experience Colored Dreaming and Colored Hearing

The principal aim of this report has been to clear the air with regard to number development theory and research. We began with a brief overview of the status of the number concept in pure mathematics and psychology. It was argued that although there are clear methological differences between pure mathematical and psychological approaches to the study of number, the two approaches are substantively complementary. Both are attempts to found the number concept on underlying psychological processes.

Three influential mathematical theories of number and three metamathematical criteria by which such theories are judged (consistency, completeness, categoricalness) then were summarized. The first theory, originally proposed by Peano (30) and von Neumann (49), postulates that the number concept may be reduced to a wholistic property of the natural numbers: viz., their inherent ordering. It was found that Peano-von Neumann's ordinal theory allows one to construct natural numbers without contradiction (consistency) and to construct all the natural numbers (completeness). However, the ordinal theory is not only a theory of number; it is a theory of all ordered progressions. The second theory, originally proposed by Russell (39, 40, 41, 42, 51, 52) and Frege (15), postulates that the number concept may be reduced to an atomistic property of the natural numbers: viz., the fact that each natural number is a class which includes all classes containing a particular number of elements. It was found that Russell-Frege's class theory does not allow one either to construct natural numbers without contradiction (inconsistency) or to construct all the natural numbers (incompleteness). The third theory, originally proposed by Piaget (31, 32), is a combination of the Peano-von Neumann and Russell-Frege theories; i.e., it founds number simultaneously on the wholistic property of order and the atomistic property of class. Piaget's theory was found to be contradictory (inconsistent) and to contain a superfluous undefined concept. A comparative analysis of the three theories revealed that although none of them is universally accepted as an ultimate theory of number, Peano-von Neumann's ordinal theory is by far the most satisfactory of the three by metamathematical criteria.

Each of the mathematical theories of number was translated into a cognitive developmental theory by substituting “developmental priority” of “mathematical priority.” The first theory postulated that children's number concepts derive from a prior understanding of the quantification of ordinal relations (ordination). The second theory postulated that children's number concepts derive from a prior understanding of the quantification of classes (cardination). The third theory postulated that children's number concepts derive from a prior understanding of both ordination and cardination. Published research which pertains to the postulates of each of these theories was reviewed (2, 8, 9, 10, 23, 33, 45, 46). Because of incomplete data analyses and poor operational definitions, it was concluded that the existing evidence does not clearly support any of the three theories.

Behavioral methods of assessing ordination, cardination, and natural number competence in children's thinking then were constructed. Care was taken to insure that each behavioral test was the most obvious counterpart of the mathematical definition of the notion that the test was designed to measure. It was argued in this regard that scrupulously precise operational definitions—as opposed to the loose operational definitions that characterize Piaget's (33) original studies of number development—are essential if we hope to decide which of the competing theories of number development is the most tenable. Two developmental studies were reported. In the first study, there were three principal findings: (a) Ordination was found to emerge in three stages (no ordering, spatial ordering, ordination). (b) Cardination was found to emerge in three stages (no correspondence, one-to-many and many-to-one correspondence, one-to-one correspondence). (c) Ordination was found to emerge long before cardination. In the second study, there were three principal findings: (a) The major findings of the first study were replicated. (b) Ordination was found to emerge prior to natural number competence. (c) Cardination was not found to emerge prior to natural number competence. It was concluded that the theory of number development which corresponds to the most satisfactory of the three mathematical theories—namely, the ordinal theory—is consistently supported by the developmental findings.

Possible explanations of the cognitive-developmental sequences reported in the two studies were considered. It was suggested that Flavell's (14) notion of item structure probably accounts for all the sequences. Flavell's notion of item structure specifies that a developmental sequence of the form X₁ → X₂ obtains between two cognitive items when the use of X₂ logically presupposes the use of X₁. It was argued in this regard that the sequences reported in the two studies each reflect logical relationships mentioned in the earlier review of mathematical number theories. Finally, the epistemological question of why such a close relationship exists between logical priorities and cognitive-developmental sequences was considered. It was concluded that there must be at least some structural isomorphism between the domains of cognition and pure mathematics.     

Match: A program to assist in matching the conditions of factorial experiments

In most experiments that involve between-subjects or between-items factorial designs, the items and/or the participants in the various experimental groups differ on one or more variables, but need to be matched on all other factors that can affect the outcome measure. Matching large groups of items or participants on multiple dimensions is a difficult and time-consuming task, yet failure to match conditions will lead to suboptimal experiments. We describe a computer program, "Match", that automates this process by selecting the best-matching items from larger sets of candidate items. In most cases, the program produces near-optimal solutions in amatter of minutes and selects matches that are typically superior to those obtained using hand matching or other semiautomated processes. We report the results of a case study in which Match was used to generate matched sets of experimental items (words varying in length and frequency) for a published study on language processing. The program was able to come up with better-matching item sets than those hand-selected by the authors of the original study, and in a fraction of the time originally taken up with stimulus matching.     

Six does not just mean a lot: preschoolers see number words as specific

This paper examines what children believe about unmapped number words - those number words whose exact meanings children have not yet learned. In Study 1, 31 children (ages 2-10 to 4-2) judged that the application of five and six changes when numerosity changes, although they did not know that equal sets must have the same number word. In Study 2, 15 children (ages 2-5 to 3-6) judged that six plus more is no longer six, but that a lot plus more is still a lot. Findings support the hypothesis that children treat number words as referring to specific, unique numerosities even before they know exactly which numerosity each word refers to.     

Event-related potentials and the interaction between orthographic and phonological information in a rhyme-judgment task

Event-related potentials (ERPs) were recorded from one midline and three pairs of lateral electrodes in three experiments involving a rhyme-judgment task. Experiment 1 employed sequentially presented word pairs consisting of orthographically similar and dissimilar rhyming and nonrhyming items (RUNG-SUNG, MAKE-ACHE, BEAD-DEAD, GIFT-ROAD). Comparison of the ERPs elicited by the dissimilar pairs revealed a rhyme/nonrhyme difference in the form of an increase in the amplitude of a late negative component (N450) for nonrhyming pairs; this effect was confined almost entirely to right-hemisphere electrodes. By contrast, rhyme/nonrhyme differences in the ERPs to orthographically similar word pairs were smaller in magnitude, later in onset, and bilaterally distributed. Experiment 2 showed that this pattern of ERP effects with orthographically similar items depended upon orthographic and not visual similarity. Experiment 3 tested the hypothesis that the lack of a right-hemisphere based N450 effect with orthographically similar items resulted from the operation of an orthographic priming mechanism. ERPs to nonrhyming pairs containing a word with an inconsistent segment (COAST-FROST) were compared with visually matched controls (SPARSE-CREASE). The rhyme/nonrhyme differences in the N450 components from these two conditions were indistinguishable, although subjects found it as difficult to make nonrhyme responses to "COAST-FROST" pairs as to the orthographically similar nonrhyming items in Experiment 1. It was concluded that while "orthographic priming" accounted for the behavioral data from these experiments, it could not explain the interaction between phonology and orthography observed in the concurrently recorded ERP data.     

Effects of types of featural information on interpretation by young children of novel words at the superordinate level

In this experiment with a Novel Label Task, 48 children ages 5 to 6 years were given a novel word for a target item, e.g., a dog. They were also given one of two types of featural information for the target item, a feature naturally common to animals, i.e., "This has a heart inside," or an accidental feature uncommon to animals, i.e., "This gets a splinter." As a result, the number of children who interpreted the novel word at the superordinate level (animal) increased significantly when they were given the feature naturally common to animals. On the other hand, there was no significant increase for an accidental feature. Further, the children were given the instruction that all animal items in this task had the same features as the target item. As a result, although the number of children who interpreted the novel word at the superordinate level (animal) increased significantly when they were given both the feature naturally common to animals and also the accidental feature, there were more when the instruction was with the feature naturally common to animals than with the accidental feature. The findings were discussed in relation to the factors corresponding to young children's interpretation of a novel word at the superordinate level.