Who can escape the natural number bias in rational number tasks? A study involving students and experts

Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth‐grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions. We conclude that it is possible for experts to be unaffected by the natural number bias on rational number tasks when they use strategies that do not involve natural numbers.     

The Fibonacci Life‐Chart Method (FLCM) as a foundation for Carl Jung's theory of synchronicity

Since the scientific method requires events to be subject to controlled examination it would seem that synchronicities are not scientifically investigable. Jung speculated that because these incredible events are like the random sparks of a firefly they cannot be pinned down. However, doubting Jung's doubts, the author provides a possible method of elucidating these seemingly random and elusive events. The author draws on a new method, designated the Fibonacci Life‐Chart Method (FLCM), which categorizes phase transitions and Phi fractal scaling in human development based on the occurrence of Fibonacci numbers in biological cell division and self‐organizing systems. The FLCM offers an orientation towards psychological experience that may have relevance to Jung's theory of synchronicity in which connections are deemed to be intrinsically meaningful rather than demonstrable consequences of cause and effect. In such a model synchronistic events can be seen to be, as the self‐organizing system enlarges, manifestations of self‐organized critical moments and Phi fractal scaling. Recommendations for future studies include testing the results of the FLCM using case reports of synchronistic and spiritual experiences.     

关于数字卦与六十四卦符号体系之形成问题

将见于商周器物上的数字卦与传逝文献的有关记载结合起来分析,商周时期应已存在用两个符号记写的六十四卦体系;周初陶拍所见易卦与传本《周易》相同的非覆即反的排列方法,也表明当时的筮书应是用两个确定的符号记写的;六十四卦应如文献所记是由八卦重合而成的,而这种八卦形成的前提同样是将其记写符号确定为两个;构成八卦的阴阳爻应是按阴阳观念将偶数记成“一一”,将奇数记戚“—”的产物。     

王弼"得意忘象"解《易》方法辨析

对王弼解《易》的传统理解,学者多认为是"扫象"说。结合王弼《周易略例》和《周易注》进行分析,可以发现,这种看法明显偏颇。诚然,在解《易》的方法上,王弼主张"得意忘象",但他在解《易》过程中意在强调"象"的工具性和"意"的目的性。王弼此种解《易》路数并未"尽黜象数"。实际上,王弼对汉代以来之象数既有所扫,又有所保留。与其说王弼解《易》是"尽黜象数",不如说是"扫象阐理"。王弼"得意忘象"这一解《易》方法,开启了中国传统哲学对经典的解读思路,不但开义理解《易》之先河,也发宋明义理易学之先声。本文通过分析王弼解《易》的这一方法论内容与特点,进一步揭示其在中国哲学史与易学史上的理论意义。     

Beyond quantity: Individual differences in working memory and the ordinal understanding of numerical symbols

In two different contexts, we examined the hypothesis that individual differences in working memory (WM) capacity are related to the tendency to infer complex, ordinal relationships between numerical symbols. In Experiment 1, we assessed whether this tendency arises in a learning context that involves mapping novel symbols to quantities by training adult participants to associate dot-quantities with novel symbols, the overall relative order of which had to be inferred. Performance was best for participants who were higher in WM capacity (HWMs). HWMs also learned ordinal information about the symbols that lower WM individuals (LWMs) did not. In Experiment 2, we examined whether WM relates to performance when participants are explicitly instructed to make numerical order judgments about highly enculturated numerical symbols by having participants indicate whether sets of three Arabic numerals were in increasing order. All participants responded faster when sequential sets (3-4-5) were in order than when they were not. However, only HWMs responded faster when non-sequential, patterned sets (1-3-5) were in order, suggesting they were accessing ordinal associations that LWMs were not. Taken together, these experiments indicate that WM capacity plays a key role in extending symbolic number representations beyond their quantity referents to include symbol-symbol ordinal associations, both in a learning context and in terms of explicitly accessing ordinal relationships in highly enculturated stimuli.     

The representation of negative numbers: Exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.     

Developing number–space associations: SNARC effects using a color discrimination task in 5-year-olds

Human adults’ numerical representation is spatially oriented; consequently, participants are faster to respond to small/large numerals with their left/right hand, respectively, when doing a binary classification judgment on numbers, known as the SNARC (spatial–numerical association of response codes) effect. Studies on the emergence and development of the SNARC effect remain scarce. The current study introduces an innovative new paradigm based on a simple color judgment of Arabic digits. Using this task, we found a SNARC effect in children as young as 5.5 years. In contrast, when preschool children needed to perform a magnitude judgment task necessitating exact number knowledge, the SNARC effect started to emerge only at 5.8 years. Moreover, the emergence of a magnitude SNARC but not a color SNARC was linked to proficiency with Arabic digits. Our results suggest that access to a spatially oriented approximate magnitude representation from symbolic digits emerges early in ontogenetic development. Exact magnitude judgments, on the other hand, rely on experience with Arabic digits and, thus, necessitate formal or informal schooling to give access to a spatially oriented numerical representation.     

In touch with numbers: Embodied and situated effects in number magnitude comparison

The idea of embodied numerosity denotes that seemingly abstract number concepts (e.g., magnitude) are rooted in bodily experiences and situated action. In the present study we evaluated whether there is an embodied representation of the place–value structure of the Arabic number system and if so whether this representation is influenced by situated aspects. In a two-digit number magnitude comparison task participants had to directly touch the larger of two numbers. Importantly, pointing responses were systematically biased toward the decade digit of the target number. Additionally, this leftward bias towards the tens digit was reduced in unit–decade incompatible number pairs. Thereby, we demonstrated an influence of place–value processing on manual pointing movement. Our results therefore corroborate the notion of an embodied representation of the place–value structure of Arabic numbers which is modulated by situated aspects.     

Modelling determinants of walking and cycling adoption: A stage-of-change perspective

Shifting travel away from cars and towards more active modes has proven a formidable policy challenge. This study aims to uncover the determinants of walking and cycling adoption by applying a stage-of-change framework. Drawing on the Transtheoretical Model, this framework models the adoption of active modes as a series of stages from pre-contemplation to maintenance. Ordinal logit models applied to US data (n = 914) illustrate the importance of both observable demographic-personal and perceptual-attitudinal variables for determining stage-of-change membership. Comparing walking and cycling, the model reveals both shared variables (vehicle ownership, self-identity) and differing factors (gender, environmental spatial ability) distinguishing among adoption stages, which has significant implications for transport policy. Results indicate that a model combining both demographic-personal and perceptual-attitudinal factors has the best fit and validity, suggesting that travel behavior interventions would benefit from multivariate segmentation methods that consider an array of individual and group characteristics. This research also gives evidence of different determinants motivating change processes for cycling versus walking. Taken together, results suggest a need for tailored policy interventions to promote behavioural adoption based not merely on the specific mode selection, but also on the usage stage under consideration.     

A working memory account for spatial-numerical associations

Several psychophysical and neuropsychological investigations have suggested that the mental representation of numbers takes the form of a number line along which magnitude is positioned in ascending order according to our reading habits. A longstanding debate is whether this spatial frame is triggered automatically as intrinsic part of the number semantics or whether it constitutes a short-term representation constructed during task execution. Although several observations clearly favor the working memory account, its causal involvement has not yet been demonstrated. In two experiments we show that information stored in working memory get spatially coded in function of its ordinal position in the sequence and that the spatial–numerical associations typically observed in number categorization tasks draw upon this mechanism.