What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through Grade 2

Effects of a 3-factor field intervention on numerical and geometric knowledge in preschool children

The main aim of this study was to develop and test the effects of a field math intervention program on both number and geometry knowledge. The intervention was developed based on three basic skills previously associated with mathematical performance: symbolic number knowledge, mapping processes and spatial reasoning. The participants were 117 preschoolers from six schools in Cali and Bogotá. The children were assigned to an intervention group (N = 55) or a control group (N = 62). The intervention lasted 11 weeks with 3 sessions per week where the children participated in different game-based activities. Tests of numerical and geometric knowledge were administered before and after the intervention. The effects of the intervention were tested twice, immediately after the program ended and six months later. The results show that the children in the intervention group improved more than the control group in both number and geometry. The second posttest revealed a significant intervention effect for geometry, but not for numerical knowledge. The implications of these mixed patterns of results are discussed in the paper.

Arithmetic problem size modulates brain activations in females but not in males

Numerous empirical studies have reported that males and females perform equally well in mathematical achievement. However, still to date, very limited is understood about the brain response profiles that are particularly characteristic of males and females when solving mathematical problems. The present study aimed to tackle this issue by manipulating arithmetic problem size to investigate functional significance using functional magnetic resonance imaging (fMRI) in young adults. Participants were instructed to complete two runs of simple calculation tasks with either large or small problem sizes. Behavioural results suggested that the performance did not differ between females and males. Neuroimaging data revealed that sex/gender-related patterns of problem size effect were found in the brain regions that are conventionally associated with arithmetic, including the left middle frontal gyrus (MFG), left intraparietal sulcus (IPS) and insula. Specifically, females demonstrated substantial brain responses of problem size effect in these regions, whereas males showed marginal effects. Moreover, the machine learning method implemented over the brain signal levels within these regions demonstrated that sex/gender is discriminable. These results showed sex/gender effects in the activating patterns varying as a function of the distinct math problem size, even in a simple calculation task. Accordingly, our findings suggested that females and males use two complementary brain resources to achieve equally successful performance levels and highlight the pivotal role of neuroimaging facilities in uncovering neural mechanisms that may not be behaviourally salient.

What aspects of counting help infants attend to numerosity?

Recent work shows that 18-month old infants understand that counting is numerically relevant-infants who see objects counted are more likely to represent the approximate number of objects in the array than infants who see the objects labeled but not counted. Which aspects of counting signal infants to attend to numerosity in this way? Here we asked whether infants rely on familiarity with the count words in their native language, or on procedures instantiated by the counting routine, independent of specific tokens. In three experiments (N = 48), we found that 18-month old infants from English-speaking households successfully distinguished four hidden objects from two when the objects were counted correctly, regardless of their familiarity with the count words (i.e., when objects were counted in familiar English and in unfamiliar German). However, when the objects were counted using familiar English count words in ways that violated basic counting principles, infants no longer represented the arrays, failing to distinguish four hidden objects from two. Together with previous findings, these results suggest that children may link the procedure of counting with numerosity years before they learn the meanings of the count words.

Links between repeating and growing pattern knowledge and math outcomes in children and adults

This study examined repeating and growing pattern knowledge and their associations with procedural and conceptual arithmetic knowledge in a sample of U.S. children (N = 185; M_age  = 79.5 months; 55% female; 88% White) and adults (N = 93; M_age  = 19.5 years; 62% female; 66% White) from 2019 to 2020. Three key findings emerged: (1) repeating pattern tasks were easier than growing pattern tasks, (2) repeating pattern knowledge robustly predicted procedural calculation skills over and above growing pattern knowledge and covariates, and (3) growing pattern knowledge modestly predicted procedural and conceptual math outcomes over and above repeating pattern knowledge and covariates. We expand existing theoretical models to incorporate these specific links and discuss implications for supporting math knowledge.

Counting many as one: Young children can understand sets as units except when counting

Young children frequently make a peculiar counting mistake. When asked to count units that are sets of multiple items, such as the number of families at a party, they often count discrete items (i.e., individual people) rather than the number of sets (i.e., families). One explanation concerns children's incomplete understanding of what constitutes a unit, resulting in a preference for discrete items. Here we demonstrate that children's incomplete understanding of counting also plays a role. In an experiment with 4- and 5-year-old children (N = 43), we found that even if children are able to name sets, group items into sets, and create one-to-one correspondences with sets, many children are nevertheless unable to count sets as units. We conclude that a nascent understanding of the abstraction principle of counting is also a cause of some children's counting errors.

Acquisition of the counting principles during the subset-knower stages: Insights from children's errors

Studies on children's understanding of counting examine when and how children acquire the cardinal principle: the idea that the last word in a counted set reflects the cardinal value of the set. Using Wynn's (1990) Give-N Task, researchers classify children who can count to generate large sets as having acquired the cardinal principle (cardinal-principle-knowers) and those who cannot as lacking knowledge of it (subset-knowers). However, recent studies have provided a more nuanced view of number word acquisition. Here, we explore this view by examining the developmental progression of the counting principles with an aim to elucidate the gradual elements that lead to children successfully generating sets and being classified as CP-knowers on the Give-N Task. Specifically, we test the claim that subset-knowers lack cardinal principle knowledge by separating children's understanding of the cardinal principle from their ability to apply and implement counting procedures. We also ask when knowledge of Gelman & Gallistel's (1978) other how-to-count principles emerge in development. We analyzed how often children violated the three how-to-count principles in a secondary analysis of Give-N data (N = 86). We found that children already have knowledge of the cardinal principle prior to becoming CP-knowers, and that understanding of the stable-order and word-object correspondence principles likely emerged earlier. These results suggest that gradual development may best characterize children's acquisition of the counting principles and that learning to coordinate all three principles represents an additional step beyond learning them individually.

The Relationship Between Confidence and Conformity in a Non-routine Counting Task With Young Children: Dedicated to the Memory of Purificación Rodríguez

Counting is a complex cognitive process that is paramount to arithmetical development at school. The improvement of counting skills of children depends on their understanding of the logical and conventional rules involved. While the logical rules are mandatory and related to one-to-one correspondence, stable order, and cardinal principles, conventional rules are optional and associated with social customs. This study contributes to unravel the conceptual understanding of counting rules of children. It explores, with a developmental approach, the performance of children on non-routine counting detection tasks, their confidence in their answers (metacognitive monitoring skills), and their ability to change a wrong answer by deferring to the opinion of a unanimous majority who justified or did not justify their claims. Hundred and forty nine children aged from 5 to 8 years were randomized to one of the experimental conditions of the testimony of teachers: with (n = 74) or without justification (n = 75). Participants judged the correctness of different types of counting procedures presented by a computerized detection task, such as (a) pseudoerrors that are correct counts where conventional rules are violated (e.g., first counting six footballs, followed by other six basketballs that were interspersed along the row), and (b) compensation errors that are incorrect counts where logical rules were broken twice (e.g., skipping the third element of the row and then labeling the sixth element with two number words, 5 and 6). Afterwards, children rated their confidence in their detection answer with a 5-point scale. Subsequently, they listened to the testimony of the teachers and showed either conformity or non-conformity. The participants considered both compensation errors and pseudoerrors as incorrect counts in the detection task. The analysis of the confidence of children in their responses suggested that they were not sensitive to their incorrect performance. Finally, children tended to conform more often after hearing a justification of the testimony than after hearing only the testimonies of the teachers. It can be concluded that the age range of the evaluated children failed to recognize the optional nature of conventional counting rules and were unaware of their misconceptions. Nevertheless, the reasoned justifications of the testimony, offered by a unanimous majority, promoted considerable improvement in the tendency of the children to revise those misconceptions.

Children’s learning from others: Conformity to unconventional counting

The current study investigated whether children’s conformity to a majority testimony influenced their willingness to revise their own erroneous counting knowledge. The content of the testimonies focused on conventional rules of counting, by means of pseudoerrors (i.e., unconventional counts) occurring during a detection task. In this work measurements were taken at two different time points. At time 1 children aged 5 to 7 years ( N = 88) first made independent judgments on the correctness of unconventional counting procedures presented by means of a computerized detection task. Subsequently, they watched a video in which four teachers (unanimous majority) or three (non-unanimous majority) made correct claims about the counts and children had to decide whether the informants were right or not, and justify their answers. Our participants conformed significantly more when the correct testimony was provided by a unanimous majority than by a non-unanimous majority. In addition, in two of the three pseudoerrors presented, there was no difference in the children’s tendency to conform to unconventional counts as age increased. At time 2, which was taken to test whether the effect of the testimony was maintained over time, the responses of the 32 children (16 from each age group) who had endorsed the claims of the unanimous majority at time 1 revealed that teachers’ testimonies only had a lasting influence on elementary school children’s understanding of conventional counting rules.

Cognitive Abilities and Mathematical Competencies at School Entry

The aim of this study was to identify mathematical competencies in early childhood and cognitive correlates of those competencies in a prospective longitudinal sample of children (N=1292) in predominantly low-income and nonurban communities in the United States. General mental ability (IQ), processing speed, vocabulary, and the working memory, inhibitory control, and cognitive flexibility components of executive function (EF) were assessed when children were ages 4 and 5. Math ability was assessed prior to school entry using a norm-referenced assessment. Exploratory factor analysis indicated that items from the math assessment loaded onto factors representing conceptual and procedural skill. IQ, processing speed, vocabulary, and a unitary EF composite all related to both conceptual and procedural skill. When EF components were examined separately, however, only the inhibitory control aspect of EF was related to conceptual skill and only the working memory aspect of EF was related to procedural skill.

Understanding arithmetic concepts: The role of domain-specific and domain-general skills

A large body of research has identified cognitive skills associated with overall mathematics achievement, focusing primarily on identifying associates of procedural skills. Conceptual understanding, however, has received less attention, despite its importance for the development of mathematics proficiency. Consequently, we know little about the quantitative and domain-general skills associated with conceptual understanding. Here we investigated 8–10-year-old children’s conceptual understanding of arithmetic, as well as a wide range of basic quantitative skills, numerical representations and domain-general skills. We found that conceptual understanding was most strongly associated with performance on a number line task. This relationship was not explained by the use of particular strategies on the number line task, and may instead reflect children’s knowledge of the structure of the number system. Understanding the skills involved in conceptual learning is important to support efforts by educators to improve children’s conceptual understanding of mathematics.

Longitudinal Predictors of the Overlap between Reading and Math Skills

The predictors of developing reading skill are well known, and there is increasing coherence around predictors of developing math as well. These achievement skills share strong relations. Less knowledge is available regarding the extent to which predictors overlap and predict one another, particularly longitudinally, and across different types of reading and math. We followed kindergarten students (n = 193) for one year, evaluating a range of relevant predictor skills in kindergarten, and a range of relevant achievement outcomes (core, fluency, complex) of reading and math in grade 1. Few predictors differentially predicted math versus reading with some exception (phonological awareness and rapid naming for reading; counting knowledge for math). The pattern was more similar for core and fluency outcomes relative to complex ones. A small set of predictors accounted for much of the overlap among math and reading outcomes, regardless of type (core, fluency, or complex). Results have the potential to inform the development of early screening tools to consider both achievement domains simultaneously, and support the importance of following students identified as at-risk in one domain for their performance in both domains.

Cognitive predictors of children's development in mathematics achievement: A latent growth modeling approach

Research has identified various domain-general and domain-specific cognitive abilities as predictors of children's individual differences in mathematics achievement. However, research into the predictors of children's individual growth rates, namely between-person differences in within-person change in mathematics achievement is scarce. We assessed 334 children's domain-general and mathematics-specific early cognitive abilities and their general mathematics achievement longitudinally across four time-points within the first and second grades of primary school. As expected, a constellation of multiple cognitive abilities contributed to the children's starting level of mathematical success. Specifically, latent growth modeling revealed that WM abilities, IQ, counting skills, nonsymbolic and symbolic approximate arithmetic and comparison skills explained individual differences in the children's initial status on a curriculum-based general mathematics achievement test. Surprisingly, however, only one out of all the assessed cognitive abilities was a unique predictor of the children's individual growth rates in mathematics achievement: their performance in the symbolic approximate addition task. In this task, children were asked to estimate the sum of two large numbers and decide if this estimated sum was smaller or larger compared to a third number. Our findings demonstrate the importance of multiple domain-general and mathematics-specific cognitive skills for identifying children at risk of struggling with mathematics and highlight the significance of early approximate arithmetic skills for the development of one's mathematical success. We argue the need for more research focus on explaining children's individual growth rates in mathematics achievement.

Counting on fine motor skills: links between preschool finger dexterity and numerical skills

Finger counting is widely considered an important step in children's early mathematical development. Presumably, children's ability to move their fingers during early counting experiences to aid number representation depends in part on their early fine motor skills (FMS). Specifically, FMS should link to children's procedural counting skills through consistent repetition of finger-counting procedures. Accordingly, we hypothesized that (a) FMS are linked to early counting skills, and (b) greater FMS relate to conceptual counting knowledge (e.g., cardinality, abstraction, order irrelevance) via procedural counting skills (i.e., one-one correspondence and correctness of verbal counting). Preschool children (N = 177) were administered measures of procedural counting skills, conceptual counting knowledge, FMS, and general cognitive skills along with parent questionnaires on home mathematics and fine motor environment. FMS correlated with procedural counting skills and conceptual counting knowledge after controlling for cognitive skills, chronological age, home mathematics and FMS environments. Moreover, the relationship between FMS and conceptual counting knowledge was mediated by procedural counting skills. Findings suggest that FMS play a role in early counting and therewith conceptual counting knowledge.

The integration of symbolic and non-symbolic representations of exact quantity in preschool children

Preschoolers (n=62) completed tasks that tapped their knowledge of symbolic and non-symbolic exact quantities, their ability to translate among different representations of exact quantity (i.e., digits, number words, and non-symbolic quantities), and their non-symbolic, digit, and spoken number comparison skills (e.g., which is larger, 2 or 4?). As hypothesized, children's knowledge about non-symbolic exact quantities, spoken number words, and digits predicted their ability to map between symbolic and non-symbolic exact quantities. Further, their knowledge of the mappings between digits and non-symbolic quantities predicted symbolic number comparison (i.e., of spoken number words or written digits). Mappings between written digits and non-symbolic exact quantities developed later than the other mappings. These results support a model of early number knowledge in which integration across symbolic and non-symbolic representations of exact quantity underlies the development of children's number comparison skills.

Detection of counting pseudoerrors: What helps children accept them?

This study examines children's comprehension of non-essential counting features (conventional rules). The objective of the study was to determine whether the presence or absence of cardinal values in pseudoerrors and the type of conventional rule violated affects children's performance. A detection task with pseudoerrors was presented through a computer game to 146 primary school children in grades 2 through 4. The same pseudoerrors were presented both with and without cardinal values; the pseudoerrors violated conventional rules of spatial adjacency, temporal adjacency, spatial-temporal adjacency, and left-to-right direction. Half of the participants within each age group were randomly assigned to an experimental condition that included pseudoerrors with a cardinal value, and the other half were assigned to a condition that included pseudoerrors without a cardinal value. The results show that when presented with a cardinal value, children more easily recognize the optional nature of non-essential counting features. Likewise, the type of conventional rule transgressed significantly affected the children's acceptance of pseudoerrors as valid counts. Participants penalized breaches of temporal and spatial-temporal adjacency to a greater degree than breaches of spatial adjacency and left-to-right direction.

Fostering Formal Commutativity Knowledge with Approximate Arithmetic

How can we enhance the understanding of abstract mathematical principles in elementary school? Different studies found out that nonsymbolic estimation could foster subsequent exact number processing and simple arithmetic. Taking the commutativity principle as a test case, we investigated if the approximate calculation of symbolic commutative quantities can also alter the access to procedural and conceptual knowledge of a more abstract arithmetic principle. Experiment 1 tested first graders who had not been instructed about commutativity in school yet. Approximate calculation with symbolic quantities positively influenced the use of commutativity-based shortcuts in formal arithmetic. We replicated this finding with older first graders (Experiment 2) and third graders (Experiment 3). Despite the positive effect of approximation on the spontaneous application of commutativity-based shortcuts in arithmetic problems, we found no comparable impact on the application of conceptual knowledge of the commutativity principle. Overall, our results show that the usage of a specific arithmetic principle can benefit from approximation. However, the findings also suggest that the correct use of certain procedures does not always imply conceptual understanding. Rather, the conceptual understanding of commutativity seems to lag behind procedural proficiency during elementary school.

Preschool predictors of mathematics in first grade children with autism spectrum disorder

Up till now, research evidence on the mathematical abilities of children with autism spectrum disorder (ASD) has been scarce and provided mixed results. The current study examined the predictive value of five early numerical competencies for four domains of mathematics in first grade. Thirty-three high-functioning children with ASD were followed up from preschool to first grade and compared with 54 typically developing children, as well as with normed samples in first grade. Five early numerical competencies were tested in preschool (5-6 years): verbal subitizing, counting, magnitude comparison, estimation, and arithmetic operations. Four domains of mathematics were used as outcome variables in first grade (6-7 years): procedural calculation, number fact retrieval, word/language problems, and time-related competences. Children with ASD showed similar early numerical competencies at preschool age as typically developing children. Moreover, they scored average on number fact retrieval and time-related competences and higher on procedural calculation and word/language problems compared to the normed population in first grade. When predicting first grade mathematics performance in children with ASD, both verbal subitizing and counting seemed to be important to evaluate at preschool age. Verbal subitizing had a higher predictive value in children with ASD than in typically developing children. Whereas verbal subitizing was predictive for procedural calculation, number fact retrieval, and word/language problems, counting was predictive for procedural calculation and, to a lesser extent, number fact retrieval. Implications and directions for future research are discussed.

Development of spatial preferences for counting and picture naming

The direction of object enumeration reflects children's enculturation but previous work on the development of such spatial preferences has been inconsistent. Therefore, we documented directional preferences in finger counting, object counting, and picture naming for children (4 groups from 3 to 6 years, N = 104) and adults (N = 56). We found a right-side preference for finger counting in 3- to 6-year-olds and a left-side preference for counting objects and naming pictures by 6 years of age. Children were consistent in their special preferences when comparing object counting and picture naming, but not in other task pairings. Finally, spatial preferences were not related to cardinality comprehension. These results, together with other recent work, suggest a gradual development of spatial-numerical associations from early non-directional mappings into culturally constrained directional mappings.

Mathematics difficulties in children born very preterm: current research and future directions

Children born very preterm have poorer attainment in all school subjects, and a markedly greater reliance on special educational support than their term-born peers. In particular, difficulties with mathematics are especially common and account for the vast majority of learning difficulties in this population. In this paper, we review research relating to the causes of mathematics learning difficulties in typically developing children, and the impact of very preterm birth on attainment in mathematics. Research is needed to understand the specific nature and origins of mathematics difficulties in very preterm children to target the development of effective intervention strategies.

Prediction and stability of mathematics skill and difficulty

The present study evaluated the stability of math learning difficulties over a 2-year period and investigated several factors that might influence this stability (categorical vs. continuous change, liberal vs. conservative cut point, broad vs. specific math assessment); the prediction of math performance over time and by performance level was also evaluated. Participants were 144 students initially identified as having a math difficulty (MD) or no learning difficulty according to low achievement criteria in the spring of Grade 3 or Grade 4. Students were reassessed 2 years later. For both measure types, a similar proportion of students changed whether assessed categorically or continuously. However, categorical change was heavily dependent on distance from the cut point and so more common for MD, who started closer to the cut point; reliable change index change was more similar across groups. There were few differences with regard to severity level of MD on continuous metrics or in terms of prediction. Final math performance on a broad computation measure was predicted by behavioral inattention and working memory while considering initial performance; for a specific fluency measure, working memory was not uniquely related, and behavioral inattention more variably related to final performance, again while considering initial performance.

A number sense intervention for low-income kindergartners at risk for mathematics difficulties

Early number sense is a strong predictor of later success in school mathematics. A disproportionate number of children from low-income families come to first grade with weak number competencies, leaving them at risk for a cycle of failure. The present study examined the effects of an 8-week number sense intervention to develop number competencies of low-income kindergartners (N = 121). The intervention purposefully targeted whole number concepts related to counting, comparing, and manipulating sets. Children were randomly assigned to either a number sense intervention or a business as usual contrast group. The intervention was carried out in small-group, 30-min sessions, 3 days per week, for a total of 24 sessions. Controlling for number sense at pretest, the intervention group made meaningful gains relative to the control group at immediate as well delayed posttest on a measure of early numeracy. Intervention children also performed better than controls on a standardized test of mathematics calculation at immediate posttest.

Addressing the struggle to link form and understanding in fractions instruction

BACKGROUND

Although making explicit links between procedures and concepts during instruction in mathematics is important, it is still unclear the precise moments during instruction when such links are best made.

AIMS

The objective was to test the effectiveness of a 3-week classroom intervention on the fractions knowledge of grade 5/6 students. The instruction was based on a theory that specifies three sites during the learning process where concepts and symbols can be connected (Hiebert, 1984): symbol interpretation, procedural execution, and solution evaluation. Sample. Seventy students from one grade 5/6 split and two grade 6 classrooms in two public elementary schools participated.

METHOD

The students were randomly assigned to treatment and control. The treatment (Sites group) received instruction that incorporated specific connections between fractions concepts and procedures at each of the three sites specified by the Sites theory. Before and after the intervention, the students' knowledge of concepts and procedures was assessed, and a random subsample of 30 students from both conditions were individually interviewed to measure their ability to make specific connections between concepts and symbols at each of the three sites.

RESULTS

While all students gained procedural skill (p < .001), the students in the Sites condition acquired significantly more knowledge of concepts than the control group (p < .01) and were also better able to connect fractions symbols to conceptual referents (p < .025).

CONCLUSIONS

The current study contributes to the literature because it describes when it might be important to link concepts and procedures during fractions instruction.

Children's understandings of counting: detection of errors and pseudoerrors by kindergarten and primary school children

In this study, the development of comprehension of essential and nonessential aspects of counting is examined in children ranging from 5 to 8 years of age. Essential aspects, such as logical rules, and nonessential aspects, including conventional rules, were studied. To address this, we created a computer program in which children watched counting errors (abstraction and order irrelevance errors) and pseudoerrors (with and without cardinal value errors) occurring during a detection task. The children judged whether the characters had counted the items correctly and were asked to justify their responses. In general, our data show that performance improved substantially with age in terms of both error and pseudoerror detection; furthermore, performance was better with regard to errors than to pseudoerrors as well as on pseudoerror tasks with cardinal values versus those without cardinal values. In addition, the children's justifications, for both the errors and pseudoerrors, made possible the identification of conventional rules underlying the incorrect responses. A particularly relevant trend was that children seem to progressively ignore these rules as they grow older. Nevertheless, this process does not end at 8 years of age given that the conventional rules of temporal and spatial adjacency were present in their judgments and were primarily responsible for the incorrect responses.

One first? Acquisition of the cardinal and ordinal uses of numbers in preschoolers

We studied the acquisition of the ordinal meaning of number words and examined its development relative to the acquisition of the cardinal meaning. Three groups of 3-, 4-, and 5-year-old children were tested in two tasks requiring the use of number words in both cardinal and ordinal contexts. Understanding of the counting principles was also measured by asking the children to assess the correctness of a cartoon character's counting in both contexts. In general, the children performed cardinal tasks significantly better than ordinal ones. Tasks requiring the production of the number for a given quantity or position were solved more accurately than those testing the ability to select a set of n objects or the object in the nth position. Different profiles were obtained for the principles; those principles shared by the two contexts were mastered earlier in the cardinal context. Regarding order (ir)relevance, older children adhered to rigid ways of counting, producing better results in the ordinal context and incorrect rejections in the cardinal trials. Altogether, our data indicate that the acquisitions of cardinal and ordinal meanings of numbers are related, and cardinality precedes the development of ordinality.

Cognitive predictors of achievement growth in mathematics: a 5-year longitudinal study

The study's goal was to identify the beginning of 1st grade quantitative competencies that predict mathematics achievement start point and growth through 5th grade. Measures of number, counting, and arithmetic competencies were administered in early 1st grade and used to predict mathematics achievement through 5th (n = 177), while controlling for intelligence, working memory, and processing speed. Multilevel models revealed intelligence and processing speed, and the central executive component of working memory predicted achievement or achievement growth in mathematics and, as a contrast domain, word reading. The phonological loop was uniquely predictive of word reading and the visuospatial sketch pad of mathematics. Early fluency in processing and manipulating numerical set size and Arabic numerals, accurate use of sophisticated counting procedures for solving addition problems, and accuracy in making placements on a mathematical number line were uniquely predictive of mathematics achievement. Use of memory-based processes to solve addition problems predicted mathematics and reading achievement but in different ways. The results identify the early quantitative competencies that uniquely contribute to mathematics learning.

Mathematical skills in 3- and 5-year-olds with spina bifida and their typically developing peers: a longitudinal approach

Preschoolers with spina bifida (SB) were compared to typically developing (TD) children on tasks tapping mathematical knowledge at 36 months (n = 102) and 60 months of age (n = 98). The group with SB had difficulty compared to TD peers on all mathematical tasks except for transformation on quantities in the subitizable range. At 36 months, vocabulary knowledge, visual-spatial, and fine motor abilities predicted achievement on a measure of informal math knowledge in both groups. At 60 months of age, phonological awareness, visual-spatial ability, and fine motor skill were uniquely and differentially related to counting knowledge, oral counting, object-based arithmetic skills, and quantitative concepts. Importantly, the patterns of association between these predictors and mathematical performance were similar across the groups. A novel finding is that fine motor skill uniquely predicted object-based arithmetic abilities in both groups, suggesting developmental continuity in the neurocognitive correlates of early object-based and later symbolic arithmetic problem solving. Models combining 36-month mathematical ability and these language-based, visual-spatial, and fine motor abilities at 60 months accounted for considerable variance on 60-month informal mathematical outcomes. Results are discussed with reference to models of mathematical development and early identification of risk in preschoolers with neurodevelopmental disorder.

Screening for mathematical disabilities in kindergarten

OBJECTIVE

This article is devoted to the potential early markers for mathematical learning disabilities in kindergarten in order to prevent children from falling further behind and from developing unrecognized mathematical disabilities later on.

METHODS

Performances in preparatory arithmetic tasks were studied in 361 kindergartners focusing on differences between children at risk for mathematical disabilities and children who were at least moderately achieving in numerical arithmetic tasks.

RESULTS

Evidence was found for several markers in kindergarten. Children at risk had lower scores on procedural counting knowledge, conceptual counting knowledge, seriation, classification, conservation and magnitude comparison tasks. Based on these kindergarten abilities, 77% of children who were at risk for mathematical disabilities could be detected.

CONCLUSION

Procedural and conceptual counting knowledge, seriation and classification skills and magnitude comparison abilities could possibly serve as powerful early screeners in the detection of mathematical disabilities.

Detecting children with arithmetic disabilities from kindergarten: evidence from a 3-year longitudinal study on the role of preparatory arithmetic abilities

In a 3-year longitudinal study, 471 children were classified, based on their performances on arithmetic tests in first and second grade, as having persistent arithmetic disabilities (AD), persistent low achieving (LA), persistent typical achieving, inconsistent arithmetic disabilities (DF1), or inconsistent low achieving in arithmetic. Significant differences in the performances on the magnitude comparison in kindergarten (at age 5-6) were found between the AD and LA and between the AD and DF1 groups. Furthermore, the percentage of true-positive AD children (at age 7-8) correctly diagnosed in kindergarten by combination of procedural counting, conceptual counting, and magnitude comparison tasks was 87.50%. When composing clinical samples, researchers should pay attention when stipulating restrictive or lenient cutoffs for arithmetic disabilities and select children based on their scores in 2 consecutive years, because the results of studies on persistent low achievers or children with inconsistent disabilities cannot be generalized to children with persistent arithmetic disabilities.

Knowledge of counting principles: how relevant is order irrelevance?

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality-that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children's conceptual knowledge of counting.