### Refine

#### Has Fulltext

- yes (8) (remove)

#### Document Type

- Postprint (6)
- Doctoral Thesis (2)

#### Keywords

- numerical cognition (8) (remove)

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.

Commentary
(2015)

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.

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.