The number line task is often used to assess children and adults' underlying representation 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 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 and using these results, we present a preliminary description of adults' underlying representation of integers.The number line task is often used to assess children and adults' underlying representation of integers (e.g., Berteletti, Lucaneli, Piazza, Dehaene & Zorzi, 2010;Geary, Hoard, Nugent & Byrd-Craven, 2008;Opfer & DeVries, 2007;Siegler & Booth, 2004;Siegler & Opfer, 2003). The traditional task consists of a bounded number line with labeled endpoints (e.g. 0-100) and a target number. On each trial, the participant is asked to indicate the position on the number line that the target number would occupy. Bounded number line tasks, however, have limitations (e.g., Barth & Paladino, In Press). Here, we (a) demonstrate the bounded number line task is an invalid measure of integer representation, (b) present a new task, an unbounded number line task, that overcomes the limitations of the bounded number line task and produces data consistent with integer representation, and (c) present a preliminary description of adults' underlying representation of integers, using the unbounded number line data.One's intuitive understanding of an integer (also referred to as an analogue quantity representation or analogue magnitude) can be described as a distribution of quantities. For example, each time we see 15, we understand its quantity to be slightly different (sometimes greater than 15, sometimes less). The distribution of quantities one associates with an integer describes one's psychological understanding of that integer. There is much debate in the literature concerning the placement and variance of these psychological distributions in relation to each other. The debate concerns whether integers are best described by a psychological representation in which (a) the mean distances between successive integers are logarithmically spaced and the perceptual errors associated with successive integers has a fixed variance (logarithmic models; e.g

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.

The sound |faiv| 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 (|faiv|), 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.

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 modelled 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.

The sound |faIv| 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 (|faIv|), 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.