Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability

Groups of first-grade (mean age = 82 months), third-grade (mean age = 107 months), and fifth-grade (mean age = 131 months) children with a learning disability in mathematics (MD, n = 58) and their normally achieving peers (n = 91) were administered tasks that assessed their knowledge of counting principles, working memory, and the strategies used to solve simple (4+3) and complex (16+8) addition problems. In all grades, the children with MD showed a working memory deficit, and in first grade, the children with MD used less sophisticated strategies and committed more errors while solving simple and complex addition problems. The group differences in strategy usage and accuracy were related, in part, to the group difference in working memory and to group and individual differences in counting knowledge. Across grade-level and group, the switch from simple to complex addition problems resulted in a shift in the mix of problem-solving strategies. Individual differences in the strategy mix and in the strategy shift were related, in part, to individual differences in working memory capacity and counting knowledge.     

Time and number discrimination in a bisection task with a sequence of stimuli: a developmental approach

Children, aged 5 and 8 years, and adults were tested in a bisection task with a sequence of stimuli in which time and number co-varied. In a counting and a non-counting condition, they were instructed either to process the duration of this sequence while ignoring the number of stimuli (temporal bisection), or to process the number of stimuli while ignoring the duration (numerical bisection). In the temporal bisection task, number interfered with the 5-year-olds' temporal performance, indicating that young children did not process time and number independently in a sequence of stimuli when they had to attend to duration. However, number interference decreased both with age and counting strategy. In contrast, in the numerical bisection task, duration did not interfere with numerical discrimination for any age group.     

Why is number word learning hard? Evidence from bilingual learners

Young children typically take between 18 months and 2 years to learn the meanings of number words. In the present study, we investigated this developmental trajectory in bilingual preschoolers to examine the relative contributions of two factors in number word learning: (1) the construction of numerical concepts, and (2) the mapping of language specific words onto these concepts. We found that children learn the meanings of small number words (i.e., one, two, and three) independently in each language, indicating that observed delays in learning these words are attributable to difficulties in mapping words to concepts. In contrast, children generally learned to accurately count larger sets (i.e., five or greater) simultaneously in their two languages, suggesting that the difficulty in learning to count is not tied to a specific language. We also replicated previous studies that found that children learn the counting procedure before they learn its logic – i.e., that for any natural number, n, the successor of n in the count list denotes the cardinality n + 1. Consistent with past studies, we found that children’s knowledge of successors is first acquired incrementally. In bilinguals, we found that this knowledge exhibits item-specific transfer between languages, suggesting that the logic of the positive integers may not be stored in a language-specific format. We conclude that delays in learning the meanings of small number words are mainly due to language-specific processes of mapping words to concepts, whereas the logic and procedures of counting appear to be learned in a format that is independent of a particular language and thus transfers rapidly from one language to the other in development.     

“Counting” serially presented stimuli by human and nonhuman primates and pigeons

Much of Stewart Hulse's career was spent analyzing how animals can extract patterned information from sequences of stimuli. Yet an additional form of information contained in a sequence may be the number of times different elements occurred. Experiments that required numerical discrimination between different stimulus items presented in sequence are analyzed for primates and pigeons. It is shown that a model based on magnitude discrimination can account well for data from human and nonhuman primate experiments. Similar experiments carried out with pigeons showed a strong recency effect not found with primates. The pigeon data are modeled successfully, however, by assuming that representations of events decay as other events are presented.     

The Idea of an Exact Number: Children's Understanding of Cardinality and Equinumerosity

Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one‐to‐one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood.     

Matching of numerical symbols with number of responses by pigeons

Pigeons were trained to peck a certain number of times on a key that displayed one of several possible numerical symbols. The particular symbol displayed indicated the number of times that the key had to be pecked. The pigeons signalled the completion of the requirement by operating a separate key. They received a food reward for correct response sequences and time-out penalties for incorrect response sequences. In the first experiment nine pigeons learned to allocate 1, 2, 3 or 4 pecks to the corresponding numerosity symbols s ₁, s ₂, s ₃ and s ₄ with levels of accuracy well above chance. The second experiment explored the maximum set of numerosities that the pigeons were capable of handling concurrently. Six of the pigeons coped with an s ₁–s ₅ task and four pigeons even managed an s ₁–s ₆ task with performances that were significantly above chance. Analysis of response times suggested that the pigeons were mainly relying on a number-based rather than on a time-based strategy. Received: 11 October 1999 / Accepted after revision: 27 January 2000     

An artificial intelligent counter

This article, using a counter as an example, explores a novel approach in constructing a cognitive system. The system is a paired memory system of “left and right brain” consisting of a number of memory units. The “left brain” is dedicated to the cognitive process of symbols, and the “right brain” to the representation of the symbols. The left and right subsystems are connected by bundles of internal communication signals. The paired memory system can learn facts and generalize concepts. Its cognitive capability is realized through the communication among these memory units at different cognitive levels within the system. The key claims in this paper are supported by empirical findings and theoretical principles. An AI counter is built and demonstrated in terms of concept learning and responding.     

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.     

Children's mappings of large number words to numerosities

Previous studies have suggested that children's learning of the relation between number words and approximate numerosities depends on their verbal counting ability, and that children exhibit no knowledge of mappings between number words and approximate numerical magnitudes for number words outside their productive verbal counting range. In the present study we used a numerical estimation task to explore children's knowledge of these mappings. We classified children as Level 1 counters (those unable to produce a verbal count list up to 35), Level 2 counters (those who were able to count to 35 but not 60) and Level 3 counters (those who counted to 60 or above) and asked children to estimate the number of items on a card. Although the accuracy of children's estimates depended on counting ability, children at all counting skill levels produced estimates that increased linearly in proportion to the target number, for numerosities both within and beyond their counting range. This result was obtained at the group level (Experiment 1) and at the level of individual children (Experiment 2). These findings provide evidence that even the least skilled counters do exhibit some knowledge of the form of the mapping between large number words and approximate numerosities.     

Exploring the mechanism of dishabituation

In this study we explored elicitation and habituation of the orienting reflex (OR) in the context of indifferent and significant stimuli, particularly aiming to clarify the mechanism driving dishabituation. An in-depth analysis of the mechanisms of electrodermal habituation and dishabituation was conducted, focusing on the role of state measures as determinants of the phasic response profile. Twenty-four young adult participants completed an auditory dishabituation task while electrodermal activity was recorded. Participants listened to a series of 10 innocuous tones of the same frequency (standards), followed by a deviant tone of a different frequency, and succeeded by 2-4 tones of the same frequency as the initial 10 stimuli. All stimuli had a random stimulus onset asynchrony of 5-7 s. Participants completed an indifferent condition in which there was no task in relation to the stimuli, and a significant condition where instruction was given to count the stimuli silently; order was counterbalanced between participants. As predicted, both skin conductance responses (SCRs) and skin conductance levels (SCLs) were larger for the significant than the indifferent condition. The initial phasic ORs were dependent on pre-stimulus arousal level, and there were significant decreases in both SCR and SCL over the first 10 standards in both conditions. Phasic response recovery was apparent to the deviant stimulus, and dishabituation to the following standard stimulus; both effects were enhanced in the significant condition. Sensitisation was apparent in SCL following the initial and deviant stimuli, but the extent of this was confounded with incomplete resolution of the preceding phasic OR in the significant condition. In the indifferent condition, dishabituation was independent of deviant-related sensitisation; this could not be tested in the significant condition. These findings suggest that dishabituation is not a process of sensitisation, but rather, a disruption of the habituation process.