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- numerical cognition (25) (entfernen)
Spatial numerical associations (SNAs) are prevalent yet their origin is poorly understood. We first consider the possible prime role of reading habits in shaping SNAs and list three observations that argue against a prominent influence of this role: (1) directional reading habits for numbers may conflict with those for non-numerical symbols, (2) short-term experimental manipulations can overrule the impact of decades of reading experience, (3) SNAs predate the acquisition of reading. As a promising alternative, we discuss behavioral, neuroscientific, and neuropsychological evidence in support of finger counting as the most likely initial determinant of SNAs. Implications of this "manumerical cognition" stance for the distinction between grounded, embodied, and situated cognition are discussed.
The force of numbers
(2021)
The study has two objectives: (1) to introduce grip force recording as a new technique for studying embodied numerical processing; and (2) to demonstrate how three competing accounts of numerical magnitude representation can be tested by using this new technique: the Mental Number Line (MNL), A Theory of Magnitude (ATOM) and Embodied Cognition (finger counting-based) account. While 26 healthy adults processed visually presented single digits in a go/no-go n-back paradigm, their passive holding forces for two small sensors were recorded in both hands. Spontaneous and unconscious grip force changes related to number magnitude occurred in the left hand already 100-140 ms after stimulus presentation and continued systematically. Our results support a two-step model of number processing where an initial stage is related to the automatic activation of all stimulus properties whereas a later stage consists of deeper conscious processing of the stimulus. This interpretation generalizes previous work with linguistic stimuli and elaborates the timeline of embodied cognition. We hope that the use of grip force recording will advance the field of numerical cognition research.
Numbers are omnipresent in daily life. They vary in display format and in their meaning so that it does not seem self-evident that our brains process them more or less easily and flexibly. The present thesis addresses mental number representations in general, and specifically the impact of finger counting on mental number representations. Finger postures that result from finger counting experience are one of many ways to convey numerical information. They are, however, probably the one where the numerical content becomes most tangible. By investigating the role of fingers in adults’ mental number representations the four presented studies also tested the Embodied Cognition hypothesis which predicts that bodily experience (e.g., finger counting) during concept acquisition (e.g., number concepts) stays an immanent part of these concepts. The studies focussed on different aspects of finger counting experience. First, consistency and further details of spontaneously used finger configurations were investigated when participants repeatedly produced finger postures according to specific numbers (Study 1). Furthermore, finger counting postures (Study 2), different finger configurations (Study 2 and 4), finger movements (Study 3), and tactile finger perception (Study 4) were investigated regarding their capability to affect number processing. Results indicated that active production of finger counting postures and single finger movements as well as passive perception of tactile stimulation of specific fingers co-activated associated number knowledge and facilitated responses towards corresponding magnitudes and number symbols. Overall, finger counting experience was reflected in specific effects in mental number processing of adult participants. This indicates that finger counting experience is an immanent part of mental number representations.
Findings are discussed in the light of a novel model. The MASC (Model of Analogue and Symbolic Codes) combines and extends two established models of number and magnitude processing. Especially a symbolic motor code is introduced as an essential part of the model. It comprises canonical finger postures (i.e., postures that are habitually used to represent numbers) and finger-number associations. The present findings indicate that finger counting functions both as a sensorimotor magnitude and as a symbolic representational format and that it thereby directly mediates between physical and symbolic size. The implications are relevant both for basic research regarding mental number representations and for pedagogic practices regarding the effectiveness of finger counting as a means to acquire a fundamental grasp of numbers.
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.
Even simple mental arithmetic is fraught with cognitive biases. For example, adding repeated numbers (so-called tie problems, e.g., 2 + 2) not only has a speed and accuracy advantage over adding different numbers (e.g., 1 + 3) but may also lead to under-representation of the result relative to a standard value (Charras et al., 2012, 2014). Does the tie advantage merely reflect easier encoding or retrieval compared to non-ties, or also a distorted result representation? To answer this question, 47 healthy adults performed two tasks, both of which indicated under-representation of tie results: In a result-to-position pointing task (Experiment 1) we measured the spatial mapping of numbers and found a left-bias for tie compared to non-tie problems. In a result-to-line-length production task (Experiment 2) we measured the underlying magnitude representation directly and obtained shorter lines for tie-compared to non-tie problems. These observations suggest that the processing benefit of tie problems comes at the cost of representational reduction of result meaning. This conclusion is discussed in the context of a recent model of arithmetic heuristics and biases.
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.
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.
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.
Mental arithmetic exhibits various biases. Among those is a tendency to overestimate addition and to underestimate subtraction outcomes. Does such “operational momentum” (OM) also affect multiplication and division? Twenty-six adults produced lines whose lengths corresponded to the correct outcomes of multiplication and division problems shown in symbolic format. We found a reliable tendency to over-estimate division outcomes, i.e., reverse OM. We suggest that anchoring on the first operand (a tendency to use this number as a reference for further quantitative reasoning) contributes to cognitive biases in mental arithmetic.
Mental arithmetic exhibits various biases. Among those is a tendency to overestimate addition and to underestimate subtraction outcomes. Does such “operational momentum” (OM) also affect multiplication and division? Twenty-six adults produced lines whose lengths corresponded to the correct outcomes of multiplication and division problems shown in symbolic format. We found a reliable tendency to over-estimate division outcomes, i.e., reverse OM. We suggest that anchoring on the first operand (a tendency to use this number as a reference for further quantitative reasoning) contributes to cognitive biases in mental arithmetic.
Commentary
(2015)
Finger-based representation of numbers is a high-level cognitive strategy to assist numerical and arithmetic processing in children and adults. It is unclear whether this paradigm builds on simple perceptual features or comprises several attributes through embodiment. Here we describe the development and initial testing of an experimental setup to study embodiment during a finger-based numerical task using Virtual Reality (VR) and a low-cost tactile stimulator that is easy to build. Using VR allows us to create new ways to study finger-based numerical representation using a virtual hand that can be manipulated in ways our hand cannot, such as decoupling tactile and visual stimuli. The goal is to present a new methodology that can allow researchers to study embodiment through this new approach, maybe shedding new light on the cognitive strategy behind the finger-based representation of numbers. In this case, a critical methodological requirement is delivering precisely targeted sensory stimuli to specific effectors while simultaneously recording their behavior and engaging the participant in a simulated experience. We tested the device's capability by stimulating users in different experimental configurations. Results indicate that our device delivers reliable tactile stimulation to all fingers of a participant's hand without losing motion tracking quality during an ongoing task. This is reflected by an accuracy of over 95% in participants detecting stimulation of a single finger or multiple fingers in sequential stimulation as indicated by experiments with sixteen participants. We discuss possible application scenarios, explain how to apply our methodology to study the embodiment of finger-based numerical representations and other high-level cognitive functions, and discuss potential further developments of the device based on the data obtained in our testing.
Recognizing, understanding, and responding to quantities are considerable skills for human beings. We can easily communicate quantities, and we are extremely efficient in adapting our behavior to numerical related tasks. One usual task is to compare quantities. We also use symbols like digits in numerical-related tasks. To solve tasks including digits, we must to rely on our previously learned internal number representations.
This thesis elaborates on the process of number comparison with the use of noisy mental representations of numbers, the interaction of number and size representations and how we use mental number representations strategically. For this, three studies were carried out.
In the first study, participants had to decide which of two presented digits was numerically larger. They had to respond with a saccade in the direction of the anticipated answer. Using only a small set of meaningfully interpretable parameters, a variant of random walk models is described that accounts for response time, error rate, and variance of response time for the full matrix of 72 digit pairs. In addition, the used random walk model predicts a numerical distance effect even for error response times and this effect clearly occurs in the observed data. In relation to corresponding correct answers error responses were systematically faster. However, different from standard assumptions often made in random walk models, this account required that the distributions of step sizes of the induced random walks be asymmetric to account for this asymmetry between correct and incorrect responses.
Furthermore, the presented model provides a well-defined framework to investigate the nature and scale (e.g., linear vs. logarithmic) of the mapping of numerical magnitude onto its internal representation. In comparison of the fits of proposed models with linear and logarithmic mapping, the logarithmic mapping is suggested to be prioritized.
Finally, we discuss how our findings can help interpret complex findings (e.g., conflicting speed vs. accuracy trends) in applied studies that use number comparison as a well-established diagnostic tool. Furthermore, a novel oculomotoric effect is reported, namely the saccadic overschoot effect. The participants responded by saccadic eye movements and the amplitude of these saccadic responses decreases with numerical distance.
For the second study, an experimental design was developed that allows us to apply the signal detection theory to a task where participants had to decide whether a presented digit was physically smaller or larger. A remaining question is, whether the benefit in (numerical magnitude – physical size) congruent conditions is related to a better perception than in incongruent conditions. Alternatively, the number-size congruency effect is mediated by response biases due to numbers magnitude. The signal detection theory is a perfect tool to distinguish between these two alternatives. It describes two parameters, namely sensitivity and response bias. Changes in the sensitivity are related to the actual task performance due to real differences in perception processes whereas changes in the response bias simply reflect strategic implications as a stronger preparation (activation) of an anticipated answer. Our results clearly demonstrate that the number-size congruency effect cannot be reduced to mere response bias effects, and that genuine sensitivity gains for congruent number-size pairings contribute to the number-size congruency effect.
Third, participants had to perform a SNARC task – deciding whether a presented digit was odd or even. Local transition probability of irrelevant attributes (magnitude) was varied while local transition probability of relevant attributes (parity) and global probability occurrence of each stimulus were kept constantly. Participants were quite sensitive in recognizing the underlying local transition probability of irrelevant attributes. A gain in performance was observed for actual repetitions of the irrelevant attribute in relation to changes of the irrelevant attribute in high repetition conditions compared to low repetition conditions. One interpretation of these findings is that information about the irrelevant attribute (magnitude) in the previous trial is used as an informative precue, so that participants can prepare early processing stages in the current trial, with the corresponding benefits and costs typical of standard cueing studies.
Finally, the results reported in this thesis are discussed in relation to recent studies in numerical cognition.
"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.
"BreaThink"
(2021)
Cognition is shaped by signals from outside and within the body. Following recent evidence of interoceptive signals modulating higher-level cognition, we examined whether breathing changes the production and perception of quantities. In Experiment 1, 22 adults verbally produced on average larger random numbers after inhaling than after exhaling. In Experiment 2, 24 further adults estimated the numerosity of dot patterns that were briefly shown after either inhaling or exhaling. Again, we obtained on average larger responses following inhalation than exhalation. These converging results extend models of situated cognition according to which higher-level cognition is sensitive to transient interoceptive states.