A Mathematical Model of How People Solve Most Variants of the Number-Line Task.

Current understanding of the development of quantity representations is based primarily on performance in the number-line task. We posit that the data from number-line tasks reflect the observer's underlying representation of quantity, together with the cognitive strategies and skills required to equate line length and quantity. Here, we specify a unified theory linking the underlying psychological representation of quantity and the associated strategies in four variations of the number-line task: the production and estimation variations of the bounded and unbounded number-line tasks. Comparison of performance in the bounded and unbounded number-line tasks provides a unique and direct way to assess the role of strategy in number-line completion. Each task produces a distinct pattern of data, yet each pattern is hypothesized to arise, at least in part, from the same underlying psychological representation of quantity. Our model predicts that the estimated biases from each task should be equivalent if the different completion strategies are modeled appropriately and no other influences are at play. We test this equivalence hypothesis in two experiments. The data reveal all variations of the number-line task produce equivalent biases except for one: the estimation variation of the bounded number-line task. We discuss the important implications of these findings.

Aesthetic Responses to Exact Fractals Driven by Physical Complexity.

Fractals are physically complex due to their repetition of patterns at multiple size scales. Whereas the statistical characteristics of the patterns repeat for fractals found in natural objects, computers can generate patterns that repeat exactly. Are these exact fractals processed differently, visually and aesthetically, than their statistical counterparts? We investigated the human aesthetic response to the complexity of exact fractals by manipulating fractal dimensionality, symmetry, recursion, and the number of segments in the generator. Across two studies, a variety of fractal patterns were visually presented to human participants to determine the typical response to exact fractals. In the first study, we found that preference ratings for exact midpoint displacement fractals can be described by a linear trend with preference increasing as fractal dimension increases. For the majority of individuals, preference increased with dimension. We replicated these results for other exact fractal patterns in a second study. In the second study, we also tested the effects of symmetry and recursion by presenting asymmetric dragon fractals, symmetric dragon fractals, and Sierpinski carpets and Koch snowflakes, which have radial and mirror symmetry. We found a strong interaction among recursion, symmetry and fractal dimension. Specifically, at low levels of recursion, the presence of symmetry was enough to drive high preference ratings for patterns with moderate to high levels of fractal dimension. Most individuals required a much higher level of recursion to recover this level of preference in a pattern that lacked mirror or radial symmetry, while others were less discriminating. This suggests that exact fractals are processed differently than their statistical counterparts. We propose a set of four factors that influence complexity and preference judgments in fractals that may extend to other patterns: fractal dimension, recursion, symmetry and the number of segments in a pattern. Conceptualizations such as Berlyne's and Redies' theories of aesthetics also provide a suitable framework for interpretation of our data with respect to the individual differences that we detect. Future studies that incorporate physiological methods to measure the human aesthetic response to exact fractal patterns would further elucidate our responses to such timeless patterns.

Unlimited capacity parallel quantity comparison of multiple integers.

Research has shown that integer comparison is quick and efficient. This efficiency may be a function of the structure of the integer comparison system. The present study tests whether integers are compared with an unlimited capacity system or a limited capacity system. We tested these models using a visual search task with time delimitation. The data from Experiments 1 and 2 indicate that integers are encoded, identified, and compared within an unlimited capacity system. The data from Experiment 3 indicate that 2nd-order magnitude comparisons are processed with a highly efficient limited capacity system.

Cross-format physical similarity effects and their implications for the numerical cognition architecture.

The sound |faɪv| is visually depicted as a written number word "five" and as an Arabic digit "5." Here, we present four experiments--two quantity same/different experiments and two magnitude comparison experiments--that assess whether auditory number words (|faɪv|), written number words ("five"), and Arabic digits ("5") directly activate one another and/or their associated quantity. The quantity same/different experiments reveal that the auditory number words, written number words, and Arabic digits directly activate one another without activating their associated quantity. That is, there are cross-format physical similarity effects but no numerical distance effects. The cross-format magnitude comparison experiments reveal significant effects of both physical similarity and numerical distance. We discuss these results in relation to the architecture of numerical cognition.

Numerical bias in bounded and unbounded number line tasks.

The number line task is often used to assess children's and adults' underlying representations of integers. Traditional bounded number line tasks, however, have limitations that can lead to misinterpretation. Here we present a new task, an unbounded number line task, that overcomes these limitations. In Experiment 1, we show that adults use a biased proportion estimation strategy to complete the traditional bounded number line task. In Experiment 2, we show that adults use a dead-reckoning integer estimation strategy in our unbounded number line task. Participants revealed a positively accelerating numerical bias in both tasks, but showed scalar variance only in the unbounded number line task. We conclude that the unbounded number line task is a more pure measure of integer representation than the bounded number line task, and using these results, we present a preliminary description of adults' underlying representation of integers.