This paper presents new results for the modulus of families of walks on a graph-a discrete analog of the modulus of curve families due to Beurling and Ahlfors. Particular attention is paid to the dependence of the modulus on its parameters. Modulus is shown to generalize (and interpolate among) three important quantities in graph theory: shortest path, effective resistance, and maxflow or min-cut.
As the volume of studies testing the benefits of active music-making interventions increases exponentially, it is important to document what exactly is happening during music treatment sessions in order to provide evidence for the mechanisms through which music training affects other domains. Thus, to complement systematic and rigorous attention to outcomes of the treatment, we outline four vital components of treatment fidelity and discuss their implementation in nonmusic- and music-based interventions. We then describe the design of Music Impacting Language Expertise (MILEStone), a new intervention that aims to improve grammar skills in children with specific language impairment by increasing sensitivity to rhythmic structure, which may enhance general temporal processing and sensitivity to syntactic structure. We describe the approach to addressing treatment fidelity in MILEStone adapted from intervention research from other fields, including a behavioral coding system to track instructional episodes and child participation, a treatment manual, activity checklists, provider training and monitoring, a home practice log, and teacher ratings of participant engagement. This approach takes an important first step in modeling a formalized procedure for assessing treatment fidelity in active music-making intervention research, as a means of increasing methodological rigor in support of evidence-based practice in clinical and educational settings.
A growing number of studies have shown a connection between rhythmic processing and language skill. It has been proposed that domain-general rhythm abilities might help children to tap into the rhythm of speech (prosody), cueing them to prosodic markers of grammatical (syntactic) information during language acquisition, thus underlying the observed correlations between rhythm and language. Working memory processes common to task demands for musical rhythm discrimination and spoken language paradigms are another possible source of individual variance observed in musical rhythm and language abilities. To investigate the nature of the relationship between musical rhythm and expressive grammar skills, we adopted an individual differences approach in N = 132 elementary school-aged children ages 5–7, with typical language development, and investigated prosodic perception and working memory skills as possible mediators. Aligning with the literature, musical rhythm was correlated with expressive grammar performance (r = 0.41, p < 0.001). Moreover, musical rhythm predicted mastery of complex syntax items (r = 0.26, p = 0.003), suggesting a privileged role of hierarchical processing shared between musical rhythm processing and children’s acquisition of complex syntactic structures. These relationships between rhythm and grammatical skills were not mediated by prosodic perception, working memory, or non-verbal IQ; instead, we uncovered a robust direct effect of musical rhythm perception on grammatical task performance. Future work should focus on possible biological endophenotypes and genetic influences underlying this relationship.
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