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The present thesis deals with the mental representation of numbers in space. Generally it is assumed that numbers are mentally represented on a mental number line along which they ordered in a continuous and analogical manner. Dehaene, Bossini and Giraux (1993) found that the mental number line is spatially oriented from left-to-right. Using a parity-judgment task they observed faster left-hand responses for smaller numbers and faster right-hand responses for larger numbers. This effect has been labelled as Spatial Numerical Association of Response Codes (SNARC) effect. The first study of the present thesis deals with the question whether the spatial orientation of the mental number line derives from the writing system participants are adapted to. According to a strong ontogenetic interpretation the SNARC effect should only obtain for effectors closely related to the comprehension and production of written language (hands and eyes). We asked participants to indicate the parity status of digits by pressing a pedal with their left or right foot. In contrast to the strong ontogenetic view we observed a pedal SNARC effect which did not differ from the manual SNARC effect. In the second study we evaluated whether the SNARC effect reflects an association of numbers and extracorporal space or an association of numbers and hands. To do so we varied the spatial arrangement of the response buttons (vertical vs. horizontal) and the instruction (handrelated vs. button-related). For vertically arranged buttons and a buttonrelated instruction we found a button-related SNARC effect. In contrast, for a hand-related instruction we obtained a hand-related SNARC effect. For horizontally arranged buttons and a handrelated instruction, however, we found a buttonrelated SNARC effect. The results of the first to studies were interpreted in terms of weak ontogenetic view. In the third study we aimed to examine the functional locus of the SNARC effect. We used the psychological refractory period paradigm. In the first experiment participants first indicated the pitch of a tone and then the parity status of a digit (locus-of-slack paradigma). In a second experiment the order of stimulus presentation and thus tasks changed (effect-propagation paradigm). The results led us conclude that the SNARC effect arises while the response is centrally selected. In our fourth study we test for an association of numbers and time. We asked participants to compare two serially presented digits. Participants were faster to compare ascending digit pairs (e.g., 2-3) than descending pairs (e.g., 3-2). The pattern of our results was interpreted in terms of forwardassociations (“1-2-3”) as formed by our ubiquitous cognitive routines to count of objects or events.
The embodied cognition framework suggests that neural systems for perception and action are engaged during higher cognitive processes. In an event-related fMRI study, we tested this claim for the abstract domain of numerical symbol processing: is the human cortical motor system part of the representation of numbers, and is organization of numerical knowledge influenced by individual finger counting habits? Developmental studies suggest a link between numerals and finger counting habits due to the acquisition of numerical skills through finger counting in childhood. In the present study, digits 1 to 9 and the corresponding number words were presented visually to adults with different finger counting habits, i.e. left- and right-starters who reported that they usually start counting small numbers with their left and right hand, respectively. Despite the absence of overt hand movements, the hemisphere contralateral to the hand used for counting small numbers was activated when small numbers were presented. The correspondence between finger counting habits and hemispheric motor activation is consistent with an intrinsic functional link between finger counting and number processing.
There is much recent interest in the idea that we represent our knowledge together with the sensory and motor features that were activated during its acquisition. This paper reviews the evidence for such "embodiment" in the domain of numerical cognition, a traditional stronghold of abstract theories of knowledge representation. The focus is on spatial-numerical associations, such as the SNARC effect (small numbers are associated with left space, larger numbers with right space). Using empirical evidence from behavioral research, I first describe sensory and motor biases induced by SNARC, thus identifying numbers as embodied concepts. Next, I propose a hierarchical relationship between grounded, embodied, and situated aspects of number knowledge. This hierarchical conceptualization helps to understand the variety of SNARC-related findings and yields testable predictions about numerical cognition. I report several such tests, ranging from cross-cultural comparisons of horizontal and vertical SNARC effects (Shaki and Fischer in J Exp Psychol Hum Percept Perform 38(3): 804-809, 2012) to motor cortical activation studies in adults with left- and right-hand counting preferences (Tschentscher et al. in NeuroImage 59: 3139-3148, 2012). It is concluded that the diagnostic features for each level of the proposed hierarchical knowledge representation, together with the spatial associations of numbers, make the domain of numerical knowledge an ideal testing ground for embodied cognition research.
Freely available software has popularized "mousetracking" to study cognitive processing; this involves the on-line recording of cursor positions while participants move a computer mouse to indicate their choice. Movement trajectories of the cursor can then be reconstructed off-line to assess the efficiency of responding in time and across space. Here we focus on the process of selecting among alternative numerical responses. Several studies have recently measured the mathematical mind with cursor movements while people decided about number magnitude or parity, computed sums or differences, or simply located numbers on a number line. After some general methodological considerations about mouse tracking we discuss several conceptual concerns that become particularly evident when "mousing" the mathematical mind.
We examined the spontaneous association between numbers and space by documenting attention deployment and the time course of associated spatial-numerical mapping with and without overt oculomotor responses. In Experiment 1, participants maintained central fixation while listening to number names. In Experiment 2, they made horizontal target-direct saccades following auditory number presentation. In both experiments, we continuously measured spontaneous ocular drift in horizontal space during and after number presentation. Experiment 2 also measured visual-probe-directed saccades following number presentation. Reliable ocular drift congruent with a horizontal mental number line emerged during and after number presentation in both experiments. Our results provide new evidence for the implicit and automatic nature of the oculomotor resonance effect associated with the horizontal spatial-numerical mapping mechanism.
There is accumulating evidence suggesting an association of numbers with physical space. However, the origin of such spatial-numerical associations (SNAs) is still debated. In the present study we investigated the development of two SNAs in a cross-sectional study involving children, young and middle-aged adults as well as the elderly: (1) the SNARC (spatial-numerical association of response codes) effect, reflecting a directional SNA; and (2) the numerical bisection bias in a line bisection task with numerical flankers. Results revealed a consistent SNARC effect in all age groups that continuously increased with age. In contrast, a numerical bisection bias was only observed for children and elderly participants, implying an U-shaped distribution of this bias across age groups. Additionally, individual SNARC effects and numerical bisection biases did not correlate significantly. We argue that the SNARC effect seems to be influenced by longer-lasting experiences of cultural constraints such as reading and writing direction and may thus reflect embodied representations. Contrarily, the numerical bisection bias may originate from insufficient inhibition of the semantic influence of irrelevant numerical flankers, which should be more pronounced in children and elderly people due to development and decline of cognitive control, respectively. As there is an ongoing debate on the origins of SNAs in general and the SNARC effect in particular, the present results are discussed in light of these differing accounts in an integrative approach. However, taken together, the present pattern of results suggests that different cognitive mechanisms underlie the SNARC effect and the numerical bisection bias.
Is more always up?
(2017)
It has been argued that the association of numbers and vertical space plays a fundamental role for the understanding of numerical concepts. However, convincing evidence for an association of numbers and vertical bimanual responses is still lacking. The present study tests the vertical Spatio-Numerical-Association-of-Response-Codes (SNARC) effect in a number classification task by comparing anatomical hand-based and spatial associations. A mixed effects model of linear spatial-numerical associations revealed no evidence for a vertical but clear support for an anatomical SNARC effect. Only if the task requirements prevented participants from using a number-hand association due to frequently alternating hand-to-button assignments, numbers were associated with the vertical dimension. Taken together, the present findings question the importance of vertical associations for the conceptual understanding of numerical magnitude as hypothesised by some embodied approaches to number cognition and suggest a preference for ego-over geocentric reference frames for the mapping of numbers onto space.
It has been debated whether negative number concepts are cognitively represented on the same mental number line as positive number concepts. The present study reviews this debate and identifies limitations of previous studies. A method with nonspatial stimuli and responses is applied to overcome these limitations and to document a systematic implicit association of negative numbers with left space, thus indicating a leftward extension of the mental number line. Importantly, this result only held for left-to-right counting adults. Implications for the experiential basis of abstract conceptual knowledge are discussed.
Spatial-numerical associations (SNAs) have been studied extensively in the past two decades, always requiring either explicit magnitude processing or explicit spatial-directional processing. This means that the typical finding of an association of small numbers with left or bottom space and of larger numbers with right or top space could be due to these requirements and not the conceptual representation of numbers. The present study compares explicit and implicit magnitude processing in an implicit spatial-directional task and identifies SNAs as artefacts of either explicit magnitude processing or explicit spatial-directional processing; they do not reveal spatial conceptual links. This finding requires revision of current accounts of the relationship between numbers and space.
Numerical knowledge, including number concepts and arithmetic procedures, seems to be a clear-cut case for abstract symbol manipulation. Yet, evidence from perceptual and motor behaviour reveals that natural number knowledge and simple arithmetic also remain closely associated with modal experiences. Following a review of behavioural, animal and neuroscience studies of number processing, we propose a revised understanding of psychological number concepts as grounded in physical constraints, embodied in experience and situated through task-specific intentions. The idea that number concepts occupy a range of positions on the continuum between abstract and modal conceptual knowledge also accounts for systematic heuristics and biases in mental arithmetic, thus inviting psycho-logical approaches to the study of the mathematical mind.
"Left" and "right" coordinates control our spatial behavior and even influence abstract thoughts. For number concepts, horizontal spatial-numerical associations (SNAs) have been widely documented: we associate few with left and many with right. Importantly, increments are universally coded on the right side even in preverbal humans and nonhuman animals, thus questioning the fundamental role of directional cultural habits, such as reading or finger counting. Here, we propose a biological, nonnumerical mechanism for the origin of SNAs on the basis of asymmetric tuning of animal brains for different spatial frequencies (SFs). The resulting selective visual processing predicts both universal SNAs and their context-dependence. We support our proposal by analyzing the stimuli used to document SNAs in newborns for their SF content. As predicted, the SFs contained in visual patterns with few versus many elements preferentially engage right versus left brain hemispheres, respectively, thus predicting left-versus rightward behavioral biases. Our "brain's asymmetric frequency tuning" hypothesis explains the perceptual origin of horizontal SNAs for nonsymbolic visual numerosities and might be extensible to the auditory domain.
Number to me, space to you
(2022)
Recent work has shown that number concepts activate both spatial and magnitude representations. According to the social co-representation literature which has shown that participants typically represent task components assigned to others together with their own, we asked whether explicit magnitude meaning and explicit spatial coding must be present in a single mind, or can be distributed across two minds, to generate a spatial-numerical congruency effect. In a shared go/no-go task that eliminated peripheral spatial codes, we assigned explicit magnitude processing to participants and spatial processing to either human or non-human co-agents. The spatial-numerical congruency effect emerged only with human co-agents. We demonstrate an inter-personal level of conceptual congruency between space and number that arises from a shared conceptual representation not contaminated by peripheral spatial codes. Theoretical implications of this finding for numerical cognition are discussed.
Among the different meanings carried by numerical information, cardinality is fundamental for survival and for the development of basic as well as of higher numerical skills. Importantly, the human brain inherits from evolution a predisposition to map cardinality onto space, as revealed by the presence of spatial-numerical associations (SNAs) in humans and animals. Here, the mapping of cardinal information onto physical space is addressed as a hallmark signature characterizing numerical cognition.
According to traditional approaches, cognition is defined as complex forms of internal information processing, taking place in the brain (cognitive processor). On the contrary, embodied cognition approaches define cognition as functionally linked to perception and action, in the continuous interaction between a biological body and its physical and sociocultural environment.
Embracing the principles of the embodied cognition perspective, I conducted four novel studies designed to unveil how SNAs originate, develop, and adapt, depending on characteristics of the organism, the context, and their interaction. I structured my doctoral thesis in three levels. At the grounded level (Study 1), I unfold the biological foundations underlying the tendency to map cardinal information across space; at the embodied level (Study 2), I reveal the impact of atypical motor development on the construction of SNAs; at the situated level (Study 3), I document the joint influence of visuospatial attention and task properties on SNAs. Furthermore, I experimentally investigate the presence of associations between physical and numerical distance, another numerical property fundamental for the development of efficient mathematical minds (Study 4).
In Study 1, I present the Brain’s Asymmetric Frequency Tuning hypothesis that relies on hemispheric asymmetries for processing spatial frequencies, a low-level visual feature that the (in)vertebrate brain extracts from any visual scene to create a coherent percept of the world. Computational analyses of the power spectra of the original stimuli used to document the presence of SNAs in human newborns and animals, support the brain’s asymmetric frequency tuning as a theoretical account and as an evolutionarily inherited mechanism scaffolding the universal and innate tendency to represent cardinality across horizontal space.
In Study 2, I explore SNAs in children with rare genetic neuromuscular diseases: spinal muscular atrophy (SMA) and Duchenne muscular dystrophy (DMD). SMA children never accomplish independent motoric exploration of their environment; in contrast, DMD children do explore but later lose this ability. The different SNAs reported by the two groups support the critical role of early sensorimotor experiences in the spatial representation of cardinality.
In Study 3, I directly compare the effects of overt attentional orientation during explicit and implicit processing of numerical magnitude. First, the different effects of attentional orienting based on the type of assessment support different mechanisms underlying SNAs during explicit and implicit assessment of numerical magnitude. Secondly, the impact of vertical shifts of attention on the processing of numerical distance sheds light on the correspondence between numerical distance and peri-personal distance.
In Study 4, I document the presence of different SNAs, driven by numerical magnitude and numerical distance, by employing different response mappings (left vs. right and near vs. distant).
In the field of numerical cognition, the four studies included in the present thesis contribute to unveiling how the characteristics of the organism and the environment influence the emergence, the development, and the flexibility of our attitude to represent cardinal information across space, thus supporting the predictions of the embodied cognition approach. Furthermore, they inform a taxonomy of body-centred factors (biological properties of the brain and sensorimotor system) modulating the spatial representation of cardinality throughout the course of life, at the grounded, embodied, and situated levels.
If the awareness for different variables influencing SNAs over the course of life is important, it is equally important to consider the organism as a whole in its sensorimotor interaction with the world. Inspired by my doctoral research, here I propose a holistic perspective that considers the role of evolution, embodiment, and environment in the association of cardinal information with directional space. The new perspective advances the current approaches to SNAs, both at the conceptual and at the methodological levels.
Unveiling how the mental representation of cardinality emerges, develops, and adapts is necessary to shape efficient mathematical minds and achieve economic productivity, technological progress, and a higher quality of life.