Attila Krajcsi

Numerical abilities in Williams syndrome: Dissociating the analogue magnitude system and verbal retrieval

Two numerical systems—the analogue magnitude system and verbal retrieval—were investigated in Williams syndrome (WS) with three numerical tasks: simple addition, simple multiplication, and number comparison. A new matching technique was introduced in selecting the proper control groups. The WS group was relatively fast in the addition and multiplication tasks, but was slow in number comparison. No reverse numerical effect was observed in the comparison task, and the distance effect was stronger than that in the control groups. The findings indicate a profile with an impaired analogue magnitude system and less impaired verbal retrieval in Williams syndrome.

Inhibition and interference in the think/no-think task

Five experiments using the think/no-think (TNT) procedure investigated the effect of the no-think and substitute instructions on cued recall. In Experiment 1, when unrelated A–B paired associates were studied and cued for recall with A items, recall rates were reliably enhanced in the think condition and reliably impaired below baseline in the no-think condition. In Experiments 2 and 5, final recall was cued with B items, leading to reliably higher recall rates, as compared with baseline, in both the think and no-think conditions. This pattern indicates backward priming of no-think items. In Experiments 3 and 4, the no-think instruction was replaced with a thought substitution instruction, and participants were asked to think of another word instead of the studied one when they saw the no-think cued items. As in Experiments 1 and 2, the same amount of forgetting of B items was observed when A items were the cues, but in contrast to Experiment 2, there was no increase in the recall performance of A items when B items were the cues. These results suggest that not thinking of studied items or, alternatively, thinking of a substitute item to avoid a target item may involve different processes: the former featuring inhibition and the latter interference.

Processing negative numbers by transforming negatives to positive range and by sign shortcut

Numerals are processed by a phylogenetically old analogue magnitude system. Can culturally new negative numerals be processed using this same representation? To find out whether magnitude representation could be used, we contrasted three possible processing mechanisms: an extended magnitude system for both positive and negative numbers, a mirroring mechanism that could transform negative values to the positive range to be processed on the positive magnitude system, and a sign shortcut strategy that can process the signs of numbers independently of the absolute values of numerals. To test these three hypotheses, a comparison task was used and the reaction time pattern, numerical distance, and Spatial-Numerical Association of Response Codes (SNARC) effect was analysed. The results revealed a mirroring process along with a sign shortcut mechanism. The SNARC effect was observed only when positive numbers were compared.

The Source of the Symbolic Numerical Distance and Size Effects

Human number understanding is thought to rely on the analog number system (ANS), working according to Weber’s law. We propose an alternative account, suggesting that symbolic mathematical knowledge is based on a discrete semantic system (DSS), a representation that stores values in a semantic network, similar to the mental lexicon or to a conceptual network. Here, focusing on the phenomena of numerical distance and size effects in comparison tasks, first we discuss how a DSS model could explain these numerical effects. Second, we demonstrate that the DSS model can give quantitatively as appropriate a description of the effects as the ANS model. Finally, we show that symbolic numerical size effect is mainly influenced by the frequency of the symbols, and not by the ratios of their values. This last result suggests that numerical distance and size effects cannot be caused by the ANS, while the DSS model might be the alternative approach that can explain the frequency-based size effect.

Numerical distance and size effects dissociate in Indo-Arabic number comparison

Numerical distance and size effects (easier number comparisons with large distance or small size) are mostly supposed to reflect a single effect, the ratio effect, which is a consequence of activation of the analog number system (ANS), working according to Weber’s law. In an alternative model, symbolic numbers can be processed by a discrete semantic system (DSS), in which the distance and size effects could originate in two independent factors: the distance effect depending on the semantic distance of the units, and the size effect depending on the frequency of the symbols. Whereas in the classic view both symbolic and nonsymbolic numbers are processed by the ANS, in the alternative view only nonsymbolic numbers are processed by the ANS, but symbolic numbers are handled by the DSS. The present work contrasts the two views, investigating whether the sizes of the distance and size effects correlate in nonsymbolic dot comparison and in symbolic Indo-Arabic comparison tasks. If a comparison is backed by the ANS, the distance and size effects should correlate, because the two effects are merely two ways to measure the same ratio effect. However, if a comparison is supported by another system—for example, the DSS—the two effects might dissociate. In the present measurements, the distance and size effects correlated very strongly in the dot comparison task, but they did not correlate in the Indo-Arabic comparison task. Additionally, the effects did not correlate between the Indo-Arabic and dot comparison tasks. These results suggest that symbolic number comparison is not handled by the ANS, but by an alternative representation, such as the DSS.

The Role of Number Notation: Sign-Value Notation Number Processing is Easier than Place-Value

Number notations can influence the way numbers are handled in computations; however, the role of notation itself in mental processing has not been examined directly. From a mathematical point of view, it is believed that place-value number notation systems, such as the Indo-Arabic numbers, are superior to sign-value systems, such as the Roman numbers. However, sign-value notation might have sufficient efficiency; for example, sign-value notations were common in flourishing cultures, such as in ancient Egypt. Herein we compared artificial sign-value and place-value notations in simple numerical tasks. We found that, contrary to the dominant view, sign-value notation can be applied more easily than place-value notation for multi-power comparison and addition tasks. Our results are consistent with the popularity of sign-value notations that prevailed for centuries. To explain the notation effect, we propose a natural multi-power number representation based on the numerical representation of objects.

The interplay of holistic shape, local feature and color information in object categorization

Although it is widely accepted that colors facilitate object and scene recognition under various circumstances, several studies found no effects of color removal in tasks requiring categorization of briefly presented animals in natural scenes. In this study, three experiments were performed to test the assumption that the discrepancy between empirical data is related to variations of the available meaningful global information such as object shapes and contextual cues. Sixty-one individuals categorized chromatic and achromatic versions of intact and scrambled images containing either cars or birds. While color removal did not affect the classification of intact stimuli, the recognition of moderately scrambled achromatic images was more difficult. This effect was accompanied by amplitude modulations of occipital event-related potentials emerging from approximately 150ms post-stimulus. Our results indicate that colors facilitate stimulus classification, but this effect becomes prominent only in cases when holistic processing is not sufficient for stimulus recognition.

Symbolic Number Comparison Is Not Processed by the Analog Number System: Different Symbolic and Non-symbolic Numerical Distance and Size Effects

HIGHLIGHTS We test whether symbolic number comparison is handled by an analog noisy system. Analog system model has systematic biases in describing symbolic number comparison. This suggests that symbolic and non-symbolic numbers are processed by different systems. Dominant numerical cognition models suppose that both symbolic and non-symbolic numbers are processed by the Analog Number System (ANS) working according to Webers law. It was proposed that in a number comparison task the numerical distance and size effects reflect a ratio-based performance which is the sign of the ANS activation. However, increasing number of findings and alternative models propose that symbolic and non-symbolic numbers might be processed by different representations. Importantly, alternative explanations may offer similar predictions to the ANS prediction, therefore, former evidence usually utilizing only the goodness of fit of the ANS prediction is not sufficient to support the ANS account. To test the ANS model more rigorously, a more extensive test is offered here. Several properties of the ANS predictions for the error rates, reaction times, and diffusion model drift rates were systematically analyzed in both non-symbolic dot comparison and symbolic Indo-Arabic comparison tasks. It was consistently found that while the ANS models prediction is relatively good for the non-symbolic dot comparison, its prediction is poorer and systematically biased for the symbolic Indo-Arabic comparison. We conclude that only non-symbolic comparison is supported by the ANS, and symbolic number comparisons are processed by other representation.

Interference Between Number Magnitude and Parity

Interference between number magnitude and other properties can be explained by either an analogue magnitude system interfering with a continuous representation of the other properties or by discrete, categorical representations in which the corresponding number and property categories interfere. In this study, we investigated whether parity, a discrete property which supposedly cannot be stored on an analogue representation, could interfere with number magnitude. We found that in a parity decision task the magnitude interfered with the parity, highlighting the role of discrete representations in numerical interference. Additionally, some participants associated evenness with large values, while others associated evenness with small values, therefore, a new interference index, the dual index was introduced to detect this heterogeneous interference. The dual index can be used to reveal any heterogeneous interference that were missed in previous studies. Finally, the magnitude-parity interference did not correlate with the magnitude-response side interference (Spatial-Numerical Association of Response Codes [SNARC] effect) or with the parity-response side interference (Markedness Association of Response Codes [MARC] effect), suggesting that at least some of the interference effects are not the result of the stimulus property markedness.

Development of Understanding Zero

While the knowledge about the development of understanding positive integers is rapidly growing, the development of understanding zero is not well-known. Here we tested several components of preschoolers’ understanding zero: whether they can use empty sets in numerical tasks, whether they can use empty sets as soon as they understand the cardinality principle, whether they know what the word “zero” refers to, and whether they categorize zero as a number. The results show that preschoolers can handle empty sets in numerical tasks as soon as they understand the cardinality principle or even earlier, and some of them know that these sets are labeled as “zero.” However, they are unsure whether zero is a number. We argue that preschoolers might understand numbers as the properties of items or objects in a set. In this view, zero cannot be a number, because an empty set does not include any items, and the missing items cannot have any property, excluding also the number property. This model might explain why zero is handled correctly in numerical tasks, while it is not regarded to be a number.