Transcoding abilities in typical and atypical mathematics achievers: The role of working memory and procedural and lexical competencies

Transcoding between numerical systems is one of the most basic abilities acquired by children during their early school years. One important topic that requires further exploration is how mathematics proficiency can affect number transcoding. The aim of the current study was to investigate transcoding abilities (i.e., reading Arabic numerals and writing dictation) in Brazilian children with and without mathematics difficulties, focusing on different school grades. We observed that children with learning difficulties in mathematics demonstrated lower achievement in number transcoding in both early and middle elementary school. In early elementary school, difficulties were observed in both the basic numerical lexicon and the management of numerical syntax. In middle elementary school, difficulties appeared mainly in the transcoding of more complex numbers. An error analysis revealed that the children with mathematics difficulties struggled mainly with the acquisition of transcoding rules. Although we confirmed the previous evidence on the impact of working memory capacity on number transcoding, we found that it did not fully account for the observed group differences. The results are discussed in the context of a maturational lag in number transcoding ability in children with mathematics difficulties.     

Basic number processing deficits in ADHD: a broad examination of elementary and complex number processing skills in 9- to 12-year-old children with ADHD-C

ADHD (attention-deficit hyperactivity disorder) and academic difficulties are frequently associated, but to date this link is poorly understood. In order to explore which components of number processing and calculation skills may be disturbed in children with ADHD we presented a series of respective tasks to 9- to 12-year-old children with ADHD-combined type and matched children without ADHD (of any type) without concomitant dyscalculia and/or dyslexia. Overall, group differences were not significant regarding overlearned and explicitly trained simple and complex calculation skills. More basic number processing skills are - for instance - the ability to compare one-digit numbers according to their magnitude (so-called magnitude comparison), to count or to transcode numbers, i.e. to write down an Arabic number '21' in verbal dictation. Significant differences favouring control children in basic number processing skills were obtained in a number comparison task and in a dot enumeration task. Importantly, our results cannot be explained by group differences regarding specific working memory and executive function components. Thus, number processing skills and in particular the processing of numerical magnitude should be investigated in children diagnosed with ADHD even when no comorbid learning disabilities are reported.     

Possible number systems

Number systems—such as the natural numbers, integers, rationals, reals, or complex numbers—play a foundational role in mathematics, but these systems can present difficulties for students. In the studies reported here, we probed the boundaries of people’s concept of a number system by asking them whether “number lines” of varying shapes qualify as possible number systems. In Experiment 1, participants rated each of a set of number lines as a possible number system, where the number lines differed in their structures (a single straight line, a step-shaped line, a double line, or two branching structures) and in their boundedness (unbounded, bounded below, bounded above, bounded above and below, or circular). Participants also rated each of a group of mathematical properties (e.g., associativity) for its importance to number systems. Relational properties, such as associativity, predicted whether participants believed that particular forms were number systems, as did the forms’ ability to support arithmetic operations, such as addition. In Experiment 2, we asked participants to produce properties that were important for number systems. Relational, operation, and use-based properties from this set again predicted ratings of whether the number lines were possible number systems. In Experiment 3, we found similar results when the number lines indicated the positions of the individual numbers. The results suggest that people believe that number systems should be well-behaved with respect to basic arithmetic operations, and that they reject systems for which these operations produce ambiguous answers. People care much less about whether the systems have particular numbers (e.g., 0) or sets of numbers (e.g., the positives).     

Low working memory capacity impedes both efficiency and learning of number transcoding in children

This study aimed to evaluate the impact of individual differences in working memory capacity on number transcoding. A recently proposed model, ADAPT (a developmental asemantic procedural transcoding model), accounts for the development of number transcoding from verbal form to Arabic form by two mechanisms: the learning of new production rules that enlarge the range of numbers a child can transcode and the increase of the mental lexicon. The working memory capacity of 7-year-olds was evaluated along with their ability to transcode one- to four-digit numbers. As ADAPT predicts, the rate of transcoding errors increased when more production rules were required and when children had low working memory capacity, with these two factors interacting. Moreover, qualitative analysis of the errors produced by high- and low-span children showed that the latter have a developmental delay in the acquisition of the production rules.     

《河图》五行数与《周易》四象数之间的关系

<河图>的五行成数不能与<周易>四象数混同."六、七、八、九"四数在<周易>与<河图>中的含义并不一致;五行成数实际存在两种说法,先秦文献中使用的土的成数是"五",而不是"十",因而五行数到"九"为止;土有成数"十"最初见于汉代,实际是五行学说在<易>理指导下发展的产物.     

Extracting parity and magnitude from Arabic numerals: developmental changes in number processing and mental representation.

Children from Grades 2, 3, 4, 6, and 8 (7.8, 9.2, 9.8, 11.7, and 13. 6 years old, respectively) made speeded, bimanual parity (odd/even) judgments of the Arabic numerals 0-9. Analysis of response times indicated that from fourth grade on, parity information is retrieved directly from memory rather than being extracted by means of a mental calculation strategy. As early as Grade 3, children exhibited the SNARC (Spatial-Numerical Association of Response Codes) effect, where small numbers were responded to faster with the left hand than with the right hand, while the converse held true for large numbers. This finding, previously demonstrated only in adults, confirms that (a) children represent magnitude information in the form of a left-to-right oriented mental number line, and (b) this information is accessed obligatorily even when irrelevant. Finally, although the SNARC effect remained strong at Grade 4, it was attenuated at Grades 6 and 8 by a linguistic effect based on associations between the unmarked adjectives "even" and "right" and between the marked adjectives "odd" and "left." Copyright 1999 Academic Press.     

The interrelatedness between infants’ communicative gestures and lexicon size: A longitudinal study

Research has shown a close relationship between gestures and language development. In this study, we investigate the cross-lagged relationships between different types of gestures and two lexicon dimensions: number of words produced and comprehended. Information about gestures and lexical development was collected from 48 typically developing infants when these were aged 0;9, 1;0 and 1;3. The European Portuguese version of the MacArthur–Bates Communicative Development Inventory: Words and Gestures (PT CDI:WG) was used. The results indicated that the total number of actions and gestures and the number of early gestures produced at 0;9 and at 1;0 year predicted the number of words comprehended three months later. Actions and gestures’ predictive power of the number of words produced was limited to the 0;9–1;0 year interval. The opposite relationship was not found: word comprehension and production did not predict action and gestures three months later. These results highlight the importance of non-verbal communicative behavior in language development.     

Gendered Patterns of Experience in Social and Cultural Transition: The Case of English-Speaking Immigrants in Israel

This study aimed to connect between work on the impact on the self as a consequence of the immigration process an theories of gender. It was based on semistructured interviews with 12 English-speaking immigrants to Israel. The data were analyzed using grounded theory (A. Strauss & J. Corbin, 1990, 1994) to examine the difficulties that the immigrants felt when leaving and since being in Israel, their methods of coping and their views of what constitutes a successful immigration. Two major patterns were identified, each of which rested on a different self need that appeared to have been affected by the immigration process: (1) the need to feel competent and (2) the need to feel a sense of belonging. Although most of the interviewees talked of both needs being damaged or affected to some extent, there was a strong tendency for the women respondents to place belonging needs in the foreground of their accounts and for men to place competence needs in theirs. Such results can be understood in the context of feminist theories of self that distinguish between women's sense of self, built in relation through connection to others, and men's sense of self, built around the concept of separation.     

Apparative therapy of enuresis nocturna, experience and problems

Eleven patients aged 5 7/12 to 15 9/12 years were treated with the "pillow wakener" (a device developed by the author and his associates) within the framework of an exploratory study of apparatus-assisted treatment of bed-wetting. The critical analysis of this device and form of therapy has as its objective to encourage investigators to make further studies in the field of treatment of bed-wetting. Particular difficulties such as technical deficiencies, the need for the interruption stimulus immediately following the activity of the bladder the problem of deep sleep, etc. are pointed out.     

Lexical search speed in children and adults

The retrieval of words from lexical memory was investigated in kindergartners (Age 6), first-graders (Age 7), third-graders (Age 9), and adults. A lexical decision task with auditorily presented words and nonwords was used, with decision latency as the dependent variable. Multiple regression analyses were used to examine the effects of word frequency, age of word acquisition, and number of word meanings. Results showed that word frequency contributed to decision latency for all age groups, age of word acquisition contributed to all groups but the third-grade subjects, and number of word meanings did not contribute for any age group. Results are discussed in relation to a model of the lexicon in children and adults. It is suggested that factors from the external linguistic community are internalized in the lexicon at an early age.     

大小判断任务中正负号及其异同对SNARC效应的影响

SNARC效应(Spatial-Numerical Association of Response Codes)是指被试对数字做按键反应时,对于较小的数字,按左键的速度快于按右键;对于较大的数字,按右键的速度快于按左键。本研究以ERP作为测量手段,采用修正的大小判断任务,旨在探究数字正负号及其异同对SNARC效应的影响。行为结果发现,在反应时上,当目标数字与基线数字正负号相同且基线数字为+5时,一致条件显著快于不一致条件。ERP结果发现,当目标数字与基线数字正负号相同时,无论基线数字为+5还是–5,在反应选择阶段,不一致都比一致条件更负且均诱发了P3。当目标数字与基线数字正负号相异时,若基线数字为+5,一致比不一致条件在刺激呈现阶段诱发了波幅显著更小的N300;若基线数字为–5,一致比不一致条件在反应执行阶段诱发了更正的LPP。无论目标数字与基线数字正负号相同还是相异,在反应选择阶段,不一致都比一致条件更负且均诱发了P3,表明出现了SNARC效应。同时,SNARC效应的出现激活了额叶头皮位置,负数加工伴随左额叶的激活,而正数加工伴随右额叶的激活,溯源分析结果进一步表明SNARC效应定位于额叶与顶叶。这些结果说明负数按实际大小表征在心理数字线上,支持了负数空间表征的个体发展论假说;表明符号捷径机制会改变SNARC效应的发生时间;同时证明了负数与正数的空间表征具有不同的优势半球。     

The effect on language development of the special characteristics of speech addressed to children

Studies of adults' speech to children indicate that adults make a number of modifications in their speech when interacting with a young child and that some of these modifications may facilitate the child's language acquisition. Modified repetitions of the child's utterances, along with certain syntactic, semantic, and cognitive simplifications, appear to be potentially effective language-teaching devices. However, the special lexicon, phonological simplification, and higher pitch that also characterize this baby-talk style are much less likely to play important roles in the child's language learning. The child's role in affecting these modifications in adult speech is briefly discussed, and an optimal pattern of adult-child language interaction is suggested.     

Images of numbers, or "When 98 is upper left and 6 sky blue".

Some people declare that they possess a personal visual representation of numbers: some automatically "see" the numbers they are confronted with in a precise location in a structured mental space, others "associate" specific colours with given numbers. Such visuo-spatial representations of numbers were first described by Galton in 1880 but have since received little attention from psychologists. It is the aim of this article to describe these mental representations and discuss their role in number processing. The authors first review Galton's observations, and then present their own. Finally, they discuss the relevance of these visuo-spatial representations in relation to contemporary debates on number representation and calculation.     

Repetition priming across distinct contexts: Effects of lexical status,word frequency,and retrieval test

Previous studies have shown an interference of task-irrelevant numerical information with the spatial parameters of visuomotor behaviour. These findings lend support to the notion that number and space share a common metric with respect to action. Here I argue that the demonstration of the structural similarity between scales for number and space would be a more stringent test for the shared metrics than a mere fact of interference. The present study investigated the scale of number mapping onto space in a manual estimation task. The physical size of target stimuli and the magnitudes of task-irrelevant numbers were parametrically manipulated in the context of the Titchener illusion. The results revealed different scaling schemas for number and space. Whereas estimates in response to changes in stimulus physical size showed a gradual increase, the effect of number was categorical with the largest number (9) showing greater manual estimate than the other numbers (1, 3, and 7). Possible interpretations that are not necessarily incompatible with the hypothesis of shared metrics with respect to action are proposed. However, the present findings suggest that a meticulous scale analysis is required in order to determine the nature of number–space interaction.     

The cognitive profile of Chinese children with mathematics difficulties

This study examined how four domain-specific skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed) may account for Chinese children’s mathematics learning difficulties. Children with mathematics difficulties (MD) of two age groups (7-8 and 9-11 years) were compared with age-matched typically achieving children. For both age groups, children with MD performed significantly worse than their age-matched controls on all of the domain-specific and domain-general measures. Further analyses revealed that the MD children with literacy difficulties (MD/RD group) performed the worst on all of the measures, whereas the MD-only group was significantly outperformed by the controls on the four domain-specific measures and verbal working memory. Stepwise discriminant analyses showed that both number fact retrieval and place value concept were significant factors differentiating the MD and non-MD children. To conclude, deficits in domain-specific skills, especially those of number fact retrieval and place value understanding, characterize the profile of Chinese children with MD.     

Mechanisms for accessing lexical representations for output: evidence from a category-specific semantic deficit

We report the performance of a neurologically impaired patient, JJ, whose oral reading of words exceeded his naming and comprehension performance for the same words--a pattern of performance that has been previously presented as evidence for "direct, nonsemantic, lexical" routes to output in reading. However, detailed analyses of JJ's reading and comprehension revealed two results that do not follow directly from the "direct route" hypothesis: (1) He accurately read aloud all orthophonologically regular words and just those irregular words for which he demonstrated some comprehension (as indicated by correct responses or within-category semantic errors in naming and comprehension tasks); and (2) his reading errors on words that were not comprehended at all (but were recognized as words) were phonologically plausible (e.g., soot read as "suit"). We account for these results by proposing that preserved sublexical mechanisms for converting print to sound, together with partially preserved semantic information, serve to mediate the activation of representations in the phonological output lexicon in the task of reading aloud. We present similar arguments for postulating an interaction between sublexical mechanisms and lexical output components of the spelling process.     

词汇表征和加工理论及其认知神经心理学证据

认知神经心理学为探讨语言心理表征和加工理论提供了新途径 ,使得人们对心理词典结构的认识不断深入细化。研究表明词汇不同性质的信息在表征上相互独立 ,词典系统至少包括语义系统、语音输入词典与语音输出词典、字形输入词典与字形输出词典等模块。本文着重介绍了这一词典系统及其认知神经心理学证据     

Number line estimation in highly math‐anxious individuals

In this study, we aimed to investigate the difficulties highly math‐anxious individuals (HMA) may face when having to estimate a number's position in a number line task. Twenty‐four HMA and 24 low math‐anxiety (LMA) individuals were presented with four lines with endpoints 0–100, 0–1,000, 0–100,000, and 267–367 on a computer monitor on which they had to mark the correct position of target numbers using the mouse. Although no differences were found between groups in the frequency of their best‐fit model, which was linear for all lines, the analysis of slopes and intercepts for the linear model showed that the two groups differed in performance on the less familiar lines (267–367 and 0–100,000). Lower values for the slope and higher values for the intercept were found in the HMA group, suggesting that they tended to overestimate small numbers and underestimate large numbers on these non‐familiar lines. Percentage absolute error analyses confirmed that HMA individuals were less accurate than their LMA counterparts on these lines, although no group differences were found in response time. These results indicate that math anxiety is related to worse performance only in the less familiar and more difficult number line tasks. Therefore, our data challenge the idea that HMA individuals might have less precise numerical representations and support the anxiety–complexity effect posited by Ashcraft and colleagues.     

The Declarative/Procedural Model of Lexicon and Grammar

Our use of language depends upon two capacities: a mental lexicon of memorized words and a mental grammar of rules that underlie the sequential and hierarchical composition of lexical forms into predictably structured larger words, phrases, and sentences. The declarative/procedural model posits that the lexicon/grammar distinction in language is tied to the distinction between two well-studied brain memory systems. On this view, the memorization and use of at least simple words (those with noncompositional, that is, arbitrary form-meaning pairings) depends upon an associative memory of distributed representations that is subserved by temporal-lobe circuits previously implicated in the learning and use of fact and event knowledge. This declarative memory system appears to be specialized for learning arbitrarily related information (i.e., for associative binding). In contrast, the acquisition and use of grammatical rules that underlie symbol manipulation is subserved by frontal/basal-ganglia circuits previously implicated in the implicit (nonconscious) learning and expression of motor and cognitive skills and habits (e.g., from simple motor acts to skilled game playing). This procedural system may be specialized for computing sequences. This novel view of lexicon and grammar offers an alternative to the two main competing theoretical frameworks. It shares the perspective of traditional dual-mechanism theories in positing that the mental lexicon and a symbol-manipulating mental grammar are subserved by distinct computational components that may be linked to distinct brain structures. However, it diverges from these theories where they assume components dedicated to each of the two language capacities (that is, domain-specific) and in their common assumption that lexical memory is a rote list of items. Conversely, while it shares with single-mechanism theories the perspective that the two capacities are subserved by domain-independent computational mechanisms, it diverges from them where they link both capacities to a single associative memory system with broad anatomic distribution. The declarative/procedural model, but neither traditional dual- nor single-mechanism models, predicts double dissociations between lexicon and grammar, with associations among associative memory properties, memorized words and facts, and temporal-lobe structures, and among symbol-manipulation properties, grammatical rule products, motor skills, and frontal/basal-ganglia structures. In order to contrast lexicon and grammar while holding other factors constant, we have focused our investigations of the declarative/procedural model on morphologically complex word forms. Morphological transformations that are (largely) unproductive (e.g., in go—went, solemn—solemnity) are hypothesized to depend upon declarative memory. These have been contrasted with morphological transformations that are fully productive (e.g., in walk—walked, happy—happiness), whose computation is posited to be solely dependent upon grammatical rules subserved by the procedural system. Here evidence is presented from studies that use a range of psycholinguistic and neurolinguistic approaches with children and adults. It is argued that converging evidence from these studies supports the declarative/procedural model of lexicon and grammar.