Percival G. Matthews

Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance

What basic capacities lay the foundation for advanced numerical cognition? Are there basic nonsymbolic abilities that support the understanding of advanced numerical concepts, such as fractions? To date, most theories have posited that previously identified core numerical systems, such as the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed a ratio-processing system (RPS) that is sensitive to magnitudes of nonsymbolic ratios and may be ideally suited for supporting fraction concepts. We provide evidence for this hypothesis by showing that individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much as the ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.

Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes.

This study presents evidence that humans have intuitive, perceptually based access to the abstract fraction magnitudes instantiated by nonsymbolic ratio stimuli. Moreover, it shows these perceptually accessed magnitudes can be easily compared with symbolically represented fractions. In cross-format comparisons, participants picked the larger of two ratios. Ratios were presented either symbolically as fractions or nonsymbolically as paired dot arrays or as paired circles. Response patterns were consistent with participants comparing specific analog fractional magnitudes independently of the particular formats in which they were presented. These results pose a challenge to accounts that argue human cognitive architecture is ill-suited for processing fractions. Instead, it seems that humans can process nonsymbolic ratio magnitudes via perceptual routes and without recourse to conscious symbolic algorithms, analogous to the processing of whole number magnitudes. These findings have important implications for theories regarding the nature of human number sense - they imply that fractions may in some sense be natural numbers, too.

Fractions We Cannot Ignore: The Nonsymbolic Ratio Congruity Effect

Although many researchers theorize that primitive numerosity processing abilities may lay the foundation for whole number concepts, other classes of numbers, like fractions, are sometimes assumed to be inaccessible to primitive architectures. This research presents evidence that the automatic processing of nonsymbolic magnitudes affects processing of symbolic fractions. Participants completed modified Stroop tasks in which they selected the larger of two symbolic fractions while the ratios of the fonts in which the fractions were printed and the overall sizes of the compared fractions were manipulated as irrelevant dimensions. Participants were slower and less accurate when nonsymbolic dimensions of printed fractions were incongruent with the symbolic comparison decision. Results indicated a robust basic sensitivity to nonsymbolic ratios that exceeds prior conceptions about the accessibility of fraction values. Results also indicated a congruity effect for overall fraction size, contrary to findings of prior research. These findings have implications for extending theory about the nature of human number sense and mathematical cognition more generally.

Assessing Formal Knowledge of Math Equivalence among Algebra and Pre-Algebra Students.

A central understanding in mathematics is knowledge of math equivalence, the relation indicating that 2 quantities are equal and interchangeable. Decades of research have documented elementary-school (ages 7 to 11) children’s (mis)understanding of math equivalence, and recent work has developed a construct map and comprehensive assessments of this understanding. The goal of the current research was to extend this work by assessing whether the construct map of math equivalence knowledge was applicable to middle school students and to document differences in formal math equivalence knowledge between students in pre-algebra and algebra. We also examined whether knowledge of math equivalence was related to students’ reasoning about an algebraic expression. In the study, 229 middle school students (ages 12 to 16) completed 2 forms of the math equivalence assessment. The results suggested that the construct map and associated assessments were appropriate for charting middle school students’ knowledge and provided additional empirical support for the link between understanding of math equivalence and formal algebraic reasoning.

Neurocognitive Architectures and the Nonsymbolic Foundations of Fractions Understanding

Children and adults experience pervasive difficulties understanding symbolic fractions. These difficulties have led some to propose that components of the human neurocognitive architecture, especially systems like the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed neurocognitive architectures—a “ratio processing system” (RPS)—tuned to the holistic magnitudes of nonsymbolic ratios that may be ideally suited for grounding fraction learning. We review this evidence alongside our own recent behavioral and brain imaging work demonstrating that nonsymbolic ratio perception is related to understanding fractions. We argue that this nonsymbolic RPS supports symbolic fraction understanding, similar to how the ANS supports whole-number understanding. We then outline a number of open questions about the RPS and the ways in which it may be used to support fractions learning.

Making Space for Spatial Proportions.

The three target articles presented in this special issue converged on an emerging theme: the importance of spatial proportional reasoning. They suggest that the ability to map between symbolic fractions (like 1/5) and nonsymbolic, spatial representations of their sizes or magnitudes may be especially important for building robust fractions knowledge. In this commentary, we first reflect upon where these findings stand in a larger theoretical context, largely borrowed from mathematics education research. Next, we emphasize parallels between this work and emerging work suggesting that nonsymbolic proportional reasoning may provide an intuitive foundation for understanding fraction magnitudes. Finally, we end by exploring some open questions that suggest specific future directions in this burgeoning area.

From continuous magnitudes to symbolic numbers: The centrality of ratio

Leibovich et al.s theory neither accounts for the deep connections between whole numbers and other classes of number nor provides a potential mechanism for mapping continuous magnitudes to symbolic numbers. We argue that focusing on non-symbolic ratio processing abilities can furnish a more expansive account of numerical cognition that remedies these shortcomings.

Using online compound interest tools to improve financial literacy

ABSTRACT The widespread use of personal computing presents the opportunity to design educational materials that can be delivered online, potentially addressing low financial literacy. The authors developed and evaluated three different educational tools focusing on interest compounding. In the authors’ laboratory experiment, individuals were randomized to one of three display tools: text, linear graph, or volumetric graph. They found that the text and volumetric tools were most effective at improving understanding of interest compounding, whereas individuals using the linear tool made little gains. The superiority of the text over the linear tool runs counter to the prediction of theories that suggest advantages of graphics over text. For researchers, the authors’u2009 findings highlight the importance of pedagogy evaluation. For practitioners, they provide research-validated tools for online dissemination.

Task Constraints Affect Mapping From Approximate Number System Estimates to Symbolic Numbers

The Approximate Number System (ANS) allows individuals to assess nonsymbolic numerical magnitudes (e.g., the number of apples on a tree) without counting. Several prominent theories posit that human understanding of symbolic numbers is based – at least in part – on mapping number symbols (e.g., 14) to their ANS-processed nonsymbolic analogs. Number-line estimation – where participants place numerical values on a bounded number-line – has become a key task used in research on this mapping. However, some research suggests that such number-line estimation tasks are actually proportion judgment tasks, as number-line estimation requires people to estimate the magnitude of the to-be-placed value, relative to set upper and lower endpoints, and thus do not so directly reflect magnitude representations. Here, we extend this work, assessing performance on nonsymbolic tasks that should more directly interface with the ANS. We compared adults’ (n = 31) performance when placing nonsymbolic numerosities (dot arrays) on number-lines to their performance with the same stimuli on two other tasks: Free estimation tasks where participants simply estimate the cardinality of dot arrays, and ratio estimation tasks where participants estimate the ratio instantiated by a pair of arrays. We found that performance on these tasks was quite different, with number-line and ratio estimation tasks failing to the show classic psychophysical error patterns of scalar variability seen in the free estimation task. We conclude the constraints of tasks using stimuli that access the ANS lead to considerably different mapping performance and that these differences must be accounted for when evaluating theories of numerical cognition. Additionally, participants showed typical underestimation patterns in the free estimation task, but were quite accurate on the ratio task. We discuss potential implications of these findings for theories regarding the mapping between ANS magnitudes and symbolic numbers.

Keys to the Gate? Equal Sign Knowledge at Second Grade Predicts Fourth-Grade Algebra Competence

Algebraic competence is a major determinant of college access and career prospects, and equal sign knowledge is taken to be foundational to algebra knowledge. However, few studies have documented a causal effect of early equal sign knowledge on later algebra skill. This study assessed whether second-grade students equal sign knowledge prospectively predicts their fourth-grade algebra knowledge, when controlling for demographic and individual difference factors. Children (Nxa0=xa0177; Magexa0=xa07.61) were assessed on a battery of tests in Grade 2 and on algebraic knowledge in Grade 4. Second-grade equal sign knowledge was a powerful predictor of these algebraic skills. Findings are discussed in terms of the importance of foregrounding equal sign knowledge to promote effective pedagogy and educational equity.