Taking the Relational Structure of Fractions Seriously: Relational Reasoning Predicts Fraction Knowledge in Elementary School Children.

Understanding and using symbolic fractions in mathematics is critical for access to advanced STEM concepts. However, children and adults consistently struggle with fractions. Here, we take a novel perspective on symbolic fractions, considering them within the framework of relational structures in cognitive psychology, such as those studied in analogy research. We tested the hypothesis that relational reasoning ability is important for reasoning about fractions by examining the relation between scores on a domain-general test of relational reasoning (TORR Jr.) and a test of fraction knowledge consisting of various types of fraction problems in 194 second grade and 145 fifth grade students. We found that relational reasoning was a significant predictor of fractions knowledge, even when controlling for non-verbal IQ and fractions magnitude processing for both grades. The effects of relational reasoning also remained significant when controlling for overall mathematics knowledge and skill for second graders but was attenuated for fifth graders. These findings suggest that this important subdomain of mathematical cognition is integrally tied to relational reasoning and opens the possibility that instruction targeting relational reasoning may prove to be a viable avenue for improving children's fractions skills.

Symbolic fractions elicit an analog magnitude representation in school-age children.

A fundamental question about fractions is whether they are grounded in an abstract nonsymbolic magnitude code similar to that postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving nonsymbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second- and fifth-grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, nonsymbolic ratios, and mixed symbolic-nonsymbolic pairs. This paradigm allowed us to test three key questions: (a) whether children show an analog magnitude code for rational numbers, (b) whether that code is compatible with mental representations of symbolic fractions, and (c) how formal education with fractions affects the symbolic-nonsymbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth graders' reaction times and error rates showed classic distance and notation effects. Nonsymbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions but had a smaller advantage for nonsymbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance and that fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a nonsymbolic ratio magnitude code and that symbolic fractions can be translated into that magnitude code.

Evidence of stable individual differences in implicit learning.

There is a fundamental psychological and neuropsychological distinction between explicit and implicit memory, and it has been proposed that whereas there are stable trait individual differences in explicit memory ability, there are not such differences across people for implicit learning. There is, however, little evidence about whether or not there are stable trait differences in implicit learning. Here we performed a test-retest reliability study with healthy young adults in which they performed four implicit learning tasks (artificial grammar learning, probabilistic classification, serial response, and implicit category learning) twice, about a week apart. We found medium (by Cohen's guidelines) test-retest reliability for three of the tasks: probabilistic classification, serial response, and implicit category learning, suggesting that differences in implicit learning ability are more stable than originally thought. In addition, implicit learning on all tasks was unrelated to explicit measures: we did not find any correlation between implicit learning measures and independent measures of IQ, working memory, or explicit learning ability. These findings indicate that implicit learning, like explicit learning, varies reliably across individuals.

Developmental dissociation between the maturation of procedural memory and declarative memory.

Declarative memory and procedural memory are known to be two fundamentally different kinds of memory that are dissociable in their psychological characteristics and measurement (explicit vs. implicit) and in the neural systems that subserve each kind of memory. Declarative memory abilities are known to improve from childhood through young adulthood, but the developmental maturation of procedural memory is largely unknown. We compared 10-year-old children and young adults on measures of declarative memory and working memory capacity and on four measures of procedural memory that have been strongly dissociated from declarative memory (mirror tracing, rotary pursuit, probabilistic classification, and artificial grammar). Children had lesser declarative memory ability and lesser working memory capacity than adults, but children exhibited learning equivalent to adults on all four measures of procedural memory. Therefore, declarative memory and procedural memory are developmentally dissociable, with procedural memory being adult-like by age 10years and declarative memory continuing to mature into young adulthood.