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.     

Predictors of early numeracy: Is there a place for mistakes when learning about number?

It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy‐two 5‐year‐olds were given a battery of nine tasks on each of three sessions (at 3‐monthly intervals). Tasks measured procedural proficiency, conceptual understanding (using an error‐detection paradigm) and the ability to compare sets using number knowledge. Procedural skills remained fairly stable over the 6‐month period, and preceded children's ability to detect another's violations to those procedures. Regression analysis revealed that children who are sensitive to procedural errors in another's counting and sharing are more likely to recognize the significance of cardinal numbers for set comparisons. We suggest that although children's conceptual understanding of well‐rehearsed routines is often limited, conceptual insight might be achieved by setting tasks that require reflection rather than practice.     

Conceptual Competence as a Component of Second Language Fluency

In this article we argue that conceptual competence should be seen as a component of second language (L2) communicative competence. Abstract concepts are highly expressed by means of metaphors, metonymies, idioms and other types of figurative language. In literature it is suggested that knowledge and appropriate use of these lexical segments are closely related to L2 mastery and therefore conceptual instruction is expected to facilitate L2 learning. To test the relationship between conceptual and L2 competence we conducted an experiment in which Modern Greek learners were encouraged to express their views on the concept of happiness. The results showed their weak performance in conveying their ideas in a coherent and acceptable manner.     

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

The development of conceptual and procedural knowledge about counting was explored for children in kindergarten, Grade 1, and Grade 2 (N = 255). Conceptual knowledge was assessed by asking children to make judgments about three types of counts modeled by an animated frog: standard (correct) left-to-right counts, incorrect counts, and unusual counts. On incorrect counts, the frog violated the word-object correspondence principle. On unusual counts, the frog violated a conventional but inessential feature of counting, for example, starting in the middle of the array of objects. Procedural knowledge was assessed using speed and accuracy in counting objects. The patterns of change for procedural knowledge and conceptual knowledge were different. Counting speed and accuracy (procedural knowledge) improved with grade. In contrast, there was a curvilinear relation between conceptual knowledge and grade that was further moderated by children's numeration skills (as measured by a standardized test); the most skilled children gradually increased their acceptance of unusual counts over grade, whereas the least skilled children decreased their acceptance of these counts. These results have implications for studying conceptual and procedural knowledge about mathematics.     

Defining and measuring conceptual knowledge in mathematics

A long tradition of research on mathematical thinking has focused on procedural knowledge, or knowledge of how to solve problems and enact procedures. In recent years, however, there has been a shift toward focusing, not only on solving problems, but also on conceptual knowledge. In the current work, we reviewed (1) how conceptual knowledge is defined in the mathematical thinking literature, and (2) how conceptual knowledge is defined, operationalized, and measured in three mathematical domains: equivalence, cardinality, and inversion. We uncovered three general issues. First, few investigators provide explicit definitions of conceptual knowledge. Second, the definitions that are provided are often vague or poorly operationalized. Finally, the tasks used to measure conceptual knowledge do not always align with theoretical claims about mathematical understanding. Together, these three issues make it challenging to understand the development of conceptual knowledge, its relationship to procedural knowledge, and how it can best be taught to students. In light of these issues, we propose a general framework that divides conceptual knowledge into two facets: knowledge of general principles and knowledge of the principles underlying procedures.     

A cognitive analysis of number-series problems: Sources of individual differences in performance

Two experiments were conducted to identify the roles of three hypothesized procedures in the solution of simple number-series problems and to determine whether individual differences in these solution procedures are related to performance on a number-series subtest from a standardized test of intelligence. The three procedures are recognition of memorized series, calculation, and checking. Subjects verified whether number sequences formed rule-based series. True series included both memorized counting series (e.g., “5 10 1520”) and unfamiliar noncounting series (e.g., “14 710”). False series could not be described by simple rules. The results of Experiment 1 indicated that (1) counting series were verified more quickly than were noncounting series, and (2) partial counting information in false series facilitated rejection. In Experiment 2, reliable differences in recognition of memorized sequences and calculational efficiency were found between individuals who scored well on a standardized test of number-series completion and those who scored poorly. The results provide a basis for understanding how individual differences in knowledge influence performance on problems often used to assess inductive reasoning skill.     

Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.     

Evidentiality in language and cognition

What is the relation between language and thought? Specifically, how do linguistic and conceptual representations make contact during language learning? This paper addresses these questions by investigating the acquisition of evidentiality (the linguistic encoding of information source) and its relation to children's evidential reasoning. Previous studies have hypothesized that the acquisition of evidentiality is complicated by the subtleness and abstractness of the underlying concepts; other studies have suggested that learning a language which systematically (e.g. grammatically) marks evidential categories might serve as a pacesetter for early reasoning about sources of information. We conducted experimental studies with children learning Korean (a language with evidential morphology) and English (a language without grammaticalized evidentiality) in order to test these hypotheses. Our experiments compared 3- and 4-year-old Korean children's knowledge of the semantics and discourse functions of evidential morphemes to their (non-linguistic) ability to recognize and report different types of evidential sources. They also compared Korean children's source monitoring abilities to the source monitoring abilities of English-speaking children of the same age. We found that Korean-speaking children have considerable success in producing evidential morphology but their comprehension of such morphology is very fragile. Nevertheless, young Korean speakers are able to reason successfully about sources of information in non-linguistic tasks; furthermore, their performance in these tasks is similar to that of English-speaking peers. These results support the conclusion that the acquisition of evidential expressions poses considerable problems for learners; however, these problems are not (necessarily) conceptual in nature. Our data also suggest that, contrary to relativistic expectations, children's ability to reason about sources of information proceeds along similar lines in diverse language-learning populations and is not tied to the acquisition of the linguistic markers of evidentiality in the exposure language. We discuss implications of our findings for the relationship between linguistic and conceptual representations during development.     

Curricular Treatments of Length Measurement in the United States: Do They Address Known Learning Challenges?

Extensive research has shown that elementary students struggle to learn the basic principles of length measurement. However, where patterns of errors have been documented, the origins of students’ difficulties have not been identified. This study investigated the hypothesis that written elementary mathematics curricula contribute to the problem of learning length measurement. We analyzed all instances of length measurement in three mathematics curricula (grades K–3) and found a shared focus on procedures. Attention to conceptual principles was limited overall and particularly for central ideas; conceptual principles were often presented after students were asked to use procedures that depended on them; and students often did not have direct access to conceptual principles. We also report five groupings of procedures that appeared sequentially in all three curricula, the conceptual principles that underlie those procedures, and the conventional knowledge that receives substantial attention by grade 3.     

Patterns of strengths and weaknesses in children's knowledge about fractions

The purpose of this study was to explore individual patterns of strengths and weaknesses in children's mathematical knowledge about common fractions. Tasks that primarily measure either conceptual or procedural aspects of mathematical knowledge were assessed with the same children in their fourth- and fifth-grade years (N=181, 56% female and 44% male). Procedural knowledge was regressed on levels of conceptual knowledge, and vice versa, to obtain residual scores. Residual scores capture variability in each kind of math knowledge that is not shared with the other type of knowledge. Cluster analysis using residuals indicated four distinct knowledge profiles in fourth graders: (a) higher than expected conceptual knowledge and relatively lower procedural knowledge, (b) relatively lower conceptual knowledge and higher procedural knowledge, (c) lower concepts but expected levels of procedural knowledge, and (d) relatively higher than expected levels of both procedural and conceptual knowledge. In fifth grade, another cluster emerged that showed lower procedures but expected levels of conceptual knowledge. In general, students with relatively lower than expected conceptual knowledge showed poorer accuracy on measures used to form the clusters and also word problem setups and estimation of sums. Implications for explaining seemingly conflicting results from prior work across studies are discussed.     

Do children understand fraction addition?

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, 2017) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments examining fourth to eighth graders' estimates of fraction sums. We found that roughly half of estimates of sums were smaller than the same child's estimate of one of the two addends in the problem. Moreover, children's estimates of fraction sums were no more accurate than if they had estimated each sum as the average of the smallest and largest possible response. This weak performance could not be attributed to poor mastery of arithmetic procedures, poor knowledge of individual fraction magnitudes, or general inability to estimate sums. These results suggest that a major source of difficulty in this domain is that many children's learning of fraction arithmetic procedures develops unconstrained by conceptual understanding of the procedures. Implications for education are discussed.     

Effects of training with added difficulties on RADAR detection

Three experiments simulating military RADAR detection addressed a training difficulty hypothesis (training with difficulty promotes superior later testing performance) and a procedural reinstatement hypothesis (test performance improves when training conditions match test conditions). Training and testing were separated by 1 week. Participants detected targets (either alphanumeric characters or vehicle pictures) occurring among distractors. Two secondary tasks were used to increase difficulty (a concurrent, irrelevant tone‐counting task and a sequential, relevant action‐firing response). In Experiment 1 , involving alphanumeric targets with rapid displays, tone counting during training degraded test performance. In Experiment 2, involving vehicle targets with both sources of difficulty and slower presentation times, training under relevant difficulty aided test accuracy. In Experiment 3, involving vehicle targets and action firing with slow presentation times, test accuracy tended to be worst when neither training nor testing involved difficult conditions. These results show boundary conditions for the training difficulty and procedural reinstatement hypotheses. Copyright © 2010 John Wiley & Sons, Ltd.     

Conceptual and motor learning in music performance

Are the mental plans for action abstract or specified in terms of the movements with which they are produced? We report motor independence for expert music performance but not for novice performance in a transfer-of-learning task. Skilled adult pianists practiced musical pieces and transferred to new pieces with the same or different motor (hand and finger) requirements and conceptual (melodic) relations. Greatest transfer in sequence duration was observed when the same conceptual relations were retained from training to transfer, regardless of motor movements. In a second experiment, novice child pianists performed the same task. More experienced child pianists showed transfer on both the motor and the conceptual dimensions; the least experienced child pianists demonstrated transfer only to sequences with identical motor and conceptual dimensions. These findings suggest that mental plans for action become independent of the required movements only at advanced skill levels.     

Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge

Competence in many domains rests on children developing conceptual and procedural knowledge, as well as procedural flexibility. However, research on the developmental relations between these different types of knowledge has yielded unclear results, in part because little attention has been paid to the validity of the measures or to the effects of prior knowledge on the relations. To overcome these problems, we modeled the three constructs in the domain of equation solving as latent factors and tested (a) whether the predictive relations between conceptual and procedural knowledge were bidirectional, (b) whether these interrelations were moderated by prior knowledge, and (c) how both constructs contributed to procedural flexibility. We analyzed data from 2 measurement points each from two samples (Ns = 228 and 304) of middle school students who differed in prior knowledge. Conceptual and procedural knowledge had stable bidirectional relations that were not moderated by prior knowledge. Both kinds of knowledge contributed independently to procedural flexibility. The results demonstrate how changes in complex knowledge structures contribute to competence development.     

The social-cognitive model of achievement motivation and the 2 x 2 achievement goal framework

Two studies examined hypotheses drawn from a proposed modification of the social-cognitive model of achievement motivation that centered on the 2 x 2 achievement goal framework. Implicit theories of ability were shown to be direct predictors of performance attainment and intrinsic motivation, and the goals of the 2 x 2 framework were shown to account for these direct relations. Perceived competence was shown to be a direct predictor of achievement goals, not a moderator of relations implicit theory or achievement goal effects. The results highlight the utility of attending to the approach-avoidance distinction in conceptual models of achievement motivation and are fully in line with the hierarchical model of achievement motivation.     

Conceptual models and task representation in using a command language

In most cases, there is no direct correspondence between the objectives that users of an interactive device may be led to formulate and the commands that enable them to attain those objectives. In order to set up what Young (1981) calls a “mapping” between the “task and action arenas”, users must (1) reorganize their objectives in order to adapt them to the capabilities of the device and (2) if necessary, combine several commands to reach a given objective. In doing so, they are assumed to rely on a “conceptual model” of the device, a notion whose cognitive status remains to be clarified. In this article, the editing keys of the MS/DOS operating system are taken as an example and described according to two different conceptual models. The links between the task and action arenas, in these two conceptual models and in a third one called the “mixed model”, are analysed in detail. Finally, the performance of three groups of subjects, while editing a line of text, are compared. All three of the dependent variables analysed are shown to vary according to how the objective is assumed to be formulated in the three conceptual models. Some hypotheses are presented concerning the cognitive mechanisms involved in such a situation.     

Categorical flexibility in preschoolers: contributions of conceptual knowledge and executive control

The current study evaluated the relative roles of conceptual knowledge and executive control on the development of categorical flexibility, the ability to switch between simultaneously available but conflicting categorical representations of an object. Experiment 1 assessed conceptual knowledge and executive control together; Experiment 2 differentiated conceptual knowledge from costly executive processes. In Experiment 1, 3‐ to 5‐year‐olds were given a three‐choice (taxonomic, thematic, and nonassociate) match‐to‐sample task and asked to match two associates. In Experiment 2, same‐aged children were assessed on another match‐to‐sample task that reduced executive costs by presenting thematic and taxonomic associates on separate trials. By comparing performance across tasks, age‐related changes resulting from conceptual knowledge and executive control indicated that conceptual knowledge of superordinate relations showed gains between 3 and 4 years, whereas gains in executive control were seen between 4 and 5 years, suggesting a décalage in the development of conceptual and executive processes underlying categorical flexibility.     

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.     

Motivational climate, achievement goals, perceived sport competence, and involvement in physical activity: structural and mediator models

Students (N=231) were tested on involvement in physical activity, motivational climate, perceived sport competence, and goal orientations. Multiple regression, partial correlation, and LISREL analyses indicated that mastery goal adoption is positively correlated with a mastery climate. Performance-approach goal adoption is positively correlated with a performance climate. Mastery climate, mastery goal, and perceived sport competence are all positively correlated with involvement in physical activity. LISREL analyses supported three mediational hypotheses: (I) the positive correlation between the performance-approach goal and involvement in physical activity is mediated by (high) perceived sport competence, (II) the negative correlation between the performance-avoidance goal and involvement in physical activity is mediated by (low) perceived sport competence, (III) the positive correlation between mastery climate and involvement in physical activity is mediated by (high) mastery goal orientation. An alternative structural model with perceived competence as the last latent construct in the path was also tested.