In this paper, representational structures of arithmetical thinking, encoded in human minds, are described. On the basis of empirical research, it is possible to distinguish four types of mental number lines: the shortest mental number line, summation mental number lines, point-place mental number lines and mental lines of exact numbers. These structures may be treated as generative mechanisms of forming arithmetical representations underlying our numerical acts of reference towards cardinalities, ordinals and magnitudes. In the paper, the theoretical framework for a (...) formal model of mental arithmetical representations is constructed. Many competitive conceptions of the mental system responsible for our arithmetical thinking may be unified within the presented framework. The paradigm underlying our research may be interpreted philosophically as a neo-Kantian approach to modeling the mind’s representational structures. (shrink)

Recently, an associative learning account of cognitive control has been suggested (Verguts & Notebaert, 2009). In this so-called adaptation by binding theory, Hebbian learning of stimulus–stimulus and stimulus–response associations is assumed to drive the adaptation of human behavior. In this study, we evaluated the validity of the adaptation-by-binding account for the case of implicit learning of regularities within a stimulus set (i.e., the frequency of specific unit digit combinations in a two-digit number magnitude comparison task) and their association with a (...) particular response. Our data indicated that participants indeed learned these regularities and adapted their behavior accordingly. In particular, influences of cognitive control were even able to override the numerical distance effect—one of the most robust effects in numerical cognition research. Thus, the general cognitive processes involved in two-digit number magnitude comparison seem much more complex than previously assumed. Multi-digit number magnitude comparison may not be automatic and inflexible but influenced by processes of cognitive control being highly adaptive to stimulus set properties and task demands on multiple levels. (shrink)

Subjects classified visible 2-digit numbers as larger or smaller than 55. Target numbers were preceded by masked 2-digit primes that were either congruent (same relation to 55) or incongruent. Experiments 1 and 2 showed prime congruency effects for stimuli never included in the set of classified visible targets, indicating subliminal priming based on long-term semantic memory. Experiments 2 and 3 went further to demonstrate paradoxical unconscious priming effects resulting from task context. For example, after repeated practice classifying 73 as larger (...) than 55, the novel masked prime 37 paradoxically facilitated the “larger” response. In these experiments task context could induce subjects to unconsciously process only the leftmost masked prime digit, only the rightmost digit, or both independently. Across 3 experiments, subliminal priming was governed by both task context and long-term semantic memory. (shrink)

Numerical categories such as parity, i.e., being odd or even, have frequently been shown to influence how particular numbers are processed. Mathematically, number parity is defined categorically. So far, cognitive, and psychological accounts have followed the mathematical definition and defined parity as a categorical psychological representation as well. In this manuscript, we wish to test the alternative account that cognitively, parity is represented in a more gradual manner such that some numbers are represented as “more odd” or “more even” than (...) other odd or even numbers, respectively. Specifically, parity processing might be influenced by more specific properties such as whether a number is a prime, a square number, a power of 2, part of a multiplication table, divisible by 4 or by 5, and many others. We suggest that these properties can influence the psychologically represented parity of a number, making it more or less prototypical for odd- or evenness. In the present study, we tested the influence of these numerical properties in a bimanual parity judgment task with auditorily presented two-digit numbers. Additionally, we further investigated the interaction of these numerical properties with linguistic factors in three language groups (English, German, and Polish). Results show significant effects on reaction times of the congruity of parity status between decade and unit digits, even if numerical magnitude and word frequency are controlled. We also observed other effects of the above specific numerical properties, such as multiplication attributes, which facilitated or interfered with the speed of parity judgment. Based on these effects of specific numerical properties we proposed and elaborated a parity continuum account. However, our cross-lingual study also suggests that parity representation and/or access seem to depend on the linguistic properties of the respective language or education and culture. Overall, the results suggest that the “perceived” parity is not the same as objective parity, and some numbers are more prototypical exemplars of their categories. (shrink)

Mental arithmetic is characterised by a tendency to overestimate addition and to underestimate subtraction results: the operational momentum effect. Here, motivated by contentious explanations of this effect, we developed and tested an arithmetic heuristics and biases model that predicts reverse OM due to cognitive anchoring effects. Participants produced bi-directional lines with lengths corresponding to the results of arithmetic problems. In two experiments, we found regular OM with zero problems but reverse OM with non-zero problems. In a third experiment, we tested (...) the prediction of our model. Our results suggest the presence of at least three competing biases in mental arithmetic: a more-or-less heuristic, a sign-space association and an anchoring bias. We conclude that mental arithmetic exhibits shortcuts for decision-making similar to traditional domains of reasoning and problem-solving. (shrink)

The paper presents a formal model of the system of number representations as a multiplicity of mental number axes with a hierarchical structure. The hierarchy is determined by the mind as it acquires successive types of mental number axes generated by virtue of some algebraic mechanisms. Three types of algebraic structures, responsible for functioning these mechanisms, are distinguished: BASAN-structures, CASAN-structures and CAPPAN-structures. A foundational order holds between these structures. CAPPAN-structures are derivative from CASAN-structures which are extensions of BASAN-structures. The constructed (...) formal model unifies two competitive conceptions of cognitive arithmetic: namely, the conception of the mental number line and the conception of parallel individuation. The paper is the continuation of a paper entitled Representational structures of arithmetical thinking, in which rich empirical evidence supporting the model is presented. The main result achieved in the present paper may be philosophically interpreted as an attempt to formalize the Kantian conception of the pure idea of time, understood as the a priori form of human arithmetical thinking. In this way, our theory may be comprehended as a result of applying the hard method of logical reconstruction of fundamental epistemological categories. (shrink)

Recently, priming effects of unconscious stimuli that were never presented as targets have been taken as evidence for the processing of the stimuli’s semantic categories. The present study explored the necessary conditions for a transfer of priming to novel primes. Stimuli were digits and letters which were presented in various viewer-related orientations . The transfer of priming to novel stimulus orientations and identities was remarkably limited: in Experiment 1, in which all conscious targets stood upright, no transfer to unconscious primes (...) in a non-target orientation was found. Experiment 2, in which primes were presented without masks, ruled out the possibility that primes were presented too short to allow congruency effects. In Experiments 3 and 4, in which all targets were presented upside down, priming transferred to upright stimuli with target identities but neither to horizontal stimuli nor to stimuli with novel identities. We suggest that whether a transfer of priming to unpracticed stimuli occurs or not depends on observers’ expectations of specific stimulus exemplars. (shrink)

Are small and large numbers represented similarly or differently on the mental number line? The size effect was used to argue that numbers are represented differently. However, recently it has been argued that the size effect is due to the comparison task and is not derived from the mental number line per se. Namely, it is due to the way that the mental number line is mapped onto the task-relevant output component. Here synesthesia was used to disentangle these two alternatives. (...) In two naming experiments a digit-color synesthete showed that the congruity effect was modulated by number size. These results support the existence of a mental number line with a vaguer numerical representation as numbers increase in size. In addition, the results show that in digit-color synesthesia, colors can evoke numerical representation automatically. (shrink)

Many theories about human number representation stress the importance of a central semantic representation that includes the magnitude information of small integer numbers, and that is conceived as an abstract, compressed number line. However, thus far there has been little or no direct evidence that units and teens are represented on the same number line. In two masked priming experiments, we show that single-digit and two-digit Arabic numerals are equally well primed by an Arabic numeral with the same number of (...) digits as by an equally distant Arabic numeral with a different number of digits (e.g. the priming effect of 7 on the target 9 is the same as the priming effect of 11 on the target 9). The finding was obtained both with a number naming task and with a parity judgement task. This is in line with the hypothesis that units and teens are part of a continuous number line. (shrink)

In 2011, van Dijck and Fias described a positional SNARC effect: the SPoARC. To-be-remembered items presented centrally on a screen seemed to acquire a left-to-right spatial dimension. If confirmed, this spatialization could be crucial for immediate memory theories. However, given the intricate links between visual and spatial dimensions, this effect could be due to the visual presentation, which could have probed the left-to-right direction of reading/writing. To allow a generalization of this effect, we adapted van Dijck and Fias's task using (...) an auditory version of Sternberg's paradigm. Lists of five consonants were auditorily presented at a rate of 3 s/item. A SPoARC effect was observed. The consequences are discussed first from an immediate memory perspective, putting forward the view that order could be coded through spatialization, and then in terms of similarities between SPoARC and SNARC. (shrink)