Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task

Whether two-digit numbers are represented holistically (each digit pair processed as one number) or compositionally (each digit pair processed separately as a decade digit and a unit digit) remains unresolved. Two experiments were conducted to examine the distance, magnitude, and SNARC effects in a number-matching task involving two-digit numbers. Forty undergraduates were asked to judge whether two two-digit numbers (presented serially in Experiment 1 and simultaneously in Experiment 2) were the same or not. Results showed that, when numbers were presented serially, unit digits did not make unique contributions to the magnitude and distance effects, supporting the holistic model. When numbers were presented simultaneously, unit digits made unique contributions, supporting the compositional model. The SNARC (Spatial-Numerical Association of Response Codes) effect was evident for the whole numbers and the decade digits, but not for the unit digits in both experiments, which indicates that two-digit numbers are represented on one mental number line. Taken together, these results suggested that the representation of two-digit numbers is on a single mental number line, but it depends on the stage of processing whether they are processed holistically or compositionally.     

Flexible and unique representations of two-digit decimals

We examined the representation of two-digit decimals through studying distance and compatibility effects in magnitude comparison tasks in four experiments. Using number pairs with different leftmost digits, we found both the second digit distance effect and compatibility effect with two-digit integers but only the second digit distance effect with two-digit pure decimals. This suggests that both integers and pure decimals are processed in a compositional manner. In contrast, neither the second digit distance effect nor the compatibility effect was observed in two-digit mixed decimals, thereby showing no evidence for compositional processing of two-digit mixed decimals. However, when the relevance of the rightmost digit processing was increased by adding some decimals pairs with the same leftmost digits, both pure and mixed decimals produced the compatibility effect. Overall, results suggest that the processing of decimals is flexible and depends on the relevance of unique digit positions. This processing mode is different from integer analysis in that two-digit mixed decimals demonstrate parallel compositional processing only when the rightmost digit is relevant. Findings suggest that people probably do not represent decimals by simply ignoring the decimal point and converting them to natural numbers.     

Exploring the developmental changes in automatic two-digit number processing

Even when two-digit numbers are irrelevant to the task at hand, adults process them. Do children process numbers automatically, and if so, what kind of information is activated? In a novel dot-number Stroop task, children (Grades 1-5) and adults were shown two different two-digit numbers made up of dots. Participants were asked to select the number that contained the larger dots. If numbers are processed automatically, reaction time for dot size judgment should be affected by numerical characteristics. The results suggest that, like adults, children process two-digit numbers automatically. Based on the current findings, we propose a developmental trend for automatic two-digit number processing that goes from decomposed sequential (activation of decade digit followed by that of unit digit) to decomposed parallel processing (simultaneous activation of decade and unit digits).     

Multi-digit number processing beyond the two-digit number range: a combination of sequential and parallel processes

Investigations of multi-digit number processing typically focus on two-digit numbers. Here, we aim to investigate the generality of results from two-digit numbers for four- and six-digit numbers. Previous studies on two-digit numbers mostly suggested a parallel processing of tens and units. In contrast, the few studies examining the processing of larger numbers suggest sequential processing of the individual constituting digits. In this study, we combined the methodological approaches of studies implying either parallel or sequential processing. Participants completed a number magnitude comparison task on two-, four-, and six-digit numbers including unit-decade compatible and incompatible differing digit pairs (e.g., 32_47, 3<4 and 2<7 vs. 37_52, 3<5 but 7>2, respectively) at all possible digit positions. Response latencies and fixation behavior indicated that sequential and parallel decomposition is not exclusive in multi-digit number processing. Instead, our results clearly suggested that sequential and parallel processing strategies seem to be combined when processing multi-digit numbers beyond the two-digit number range. To account for the results, we propose a chunking hypothesis claiming that multi-digit numbers are separated into chunks of shorter digit strings. While the different chunks are processed sequentially digits within these chunks are processed in parallel.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

Working memory and two-digit number processing

The processing of two-digit numbers in comparison tasks involves the activation and manipulation of magnitude information to decide which number is larger. The present study explored the role of different working memory (WM) components and skills in the processing of two-digit numbers by examining the unit-decade compatibility effect with Arabic digits and number words. In the study, the unit-decade compatibility effect and different WM components were evaluated. The results indicated that the unit-decade compatibility effect was associated to specific WM skills depending on the number format (Arabic digits and number words). We discussed the implications of these results for the decomposed view of two-digit numbers.     

Working memory and two-digit number processing

The processing of two-digit numbers in comparison tasks involves the activation and manipulation of magnitude information to decide which number is larger. The present study explored the role of different working memory (WM) components and skills in the processing of two-digit numbers by examining the unit–decade compatibility effect with Arabic digits and number words. In the study, the unit–decade compatibility effect and different WM components were evaluated. The results indicated that the unit–decade compatibility effect was associated to specific WM skills depending on the number format (Arabic digits and number words). We discussed the implications of these results for the decomposed view of two-digit numbers.     

Memory comparison time for complex digit stimuli: The effect of number length

Woodworth (1938) reported that naming latency increased linearly with the number of digits per number (number length). In the present study, the Sternberg memory scanning paradigm was utilized to investigate this effect. It was found that the slope of the memory scanning function increased linearly with number length: memory scanning time was 40 msec for one-digit numbers, 70 msec for two-digit numbers and 101 msec for 3 digit numbers. The intercepts of the memory scanning functions did not differ for the three types of numbers. Thus the increase in latency may be due to the memory comparison stage of processing. The data suggest that a memory comparison operation occurs for each digit position of the complex memory items composed of more than one digit.     

Numerical comparison of two-digit numbers: How differences at encoding can involve differences in processing

The study of two-digit numbers processing has recently gathered a growing interest. Here, we examine whether differences at encoding of two-digit oral verbal numerals induce differences in the type of processing involved. Twenty-four participants were submitted to a comparison task to 55. Differences at encoding were introduced by the use of dichotic listening and synchronous (synchronous condition) or asynchronous presentation (tens-first and units-first conditions) of the two-digit numerals' components. Our results showed that differences at the encoding stage of two-digit numerals involve: (1) different comparison processes (tens-first and units-first conditions: parallel comparison; synchronous condition: parallel and holistic comparison); and (2) differences in the weight of the tens- and units-effects. Therefore, attentional mechanisms determining at the encoding stage how much attention is paid to the two-digit numerals' components might account for the different types of processing found with two-digit numbers.     

Inversion Errors in Arabic Number Reading: Is There a Nonsemantic Route?

This article reports on a mildly aphasic patient with major disorders in reading, writing, and number processing. His predominant error type in reading aloud Arabic numbers and in matching heard numerals with Arabic numbers was the violation of the inversion rule of the German Arabic number reading system. According to this rule most of the two-digit numbers or numbers in the final and prefinal position of longer digit strings have to be read beginning with the final digit (e.g., 26 → sechsundzwanzig (literally translated: six-and-twenty)). It is argued that AT's inversion errors (e.g., 26 → zweiundsechzig (literally translated: two-and-sixty)) are not consistent with the predictions of single route models of Arabic number reading but are in agreement with proposals of a visually based asemantic reading routine in addition to a semantically mediated reading routine.     

Memory impairment following ruptured aneurysm of the anterior communicating artery

This study reports the case of a 42-year-old man who suffered a ruptured aneurysm of the anterior communicating artery. His memory capabilities were assessed after a considerable recovery period during which many of his memory deficits ameliorated. His scan revealed a left frontal lesion and many of his deficits were characteristic of frontal impairment. He was impaired on temporal discrimination, and he showed marked source forgetting. He also performed badly on the Brown-Peterson task, and we suggest that this is another task that may be characteristic of frontal impairment. In contrast, the patient showed normal or near normal performance on some memory tasks but not on others. It is concluded that the patient's frontal signs are similar to those found in Korsakoff's Syndrome, but that his memory impairment is qualitatively different from that encountered in patients with the amnesic syndrome.     

Automaticity for numerical magnitude of two-digit Arabic numbers in children

This paper examines the automatic processing of the numerical magnitude of two-digit Arabic numbers using a Stroop-like task in school-aged children. Second, third, and fourth graders performed physical size judgments on pairs of two-digit numbers varying on both physical and numerical dimensions. To investigate the importance of synchrony between the speed of processing of the numerical magnitude and the physical dimensions on the size congruity effect (SCE), we used masked priming: numerical magnitude was subliminally primed in half of the trials, while neutral priming was used in the other half. The results indicate a SCE in physical judgments, providing the evidence of automatic access to the magnitude of two-digit numbers in children. This effect was modulated by the priming type, as a SCE only appeared when the numerical magnitude was primed. This suggests that young children needed a relative synchronization of numerical and physical dimensions to access the magnitude of two-digit numbers automatically.     

Memory while shadowing

Subjects were asked to repeat verbally (shadow) English words which were presented to one ear. They were then tested for their memory of two-digit numbers which were presented to their other ear. It is demonstrated that subjects have no memory for these digits if they must continue to shadow for 20 sec. before being tested on their memory for the digits. However, if tested immediately after the digit presentation, they do remember some digits. Hence, verbal material presented on non-attended channels gets into a short-term memory, but is not transferred into long-term memory.     

In touch with numbers: Embodied and situated effects in number magnitude comparison

The idea of embodied numerosity denotes that seemingly abstract number concepts (e.g., magnitude) are rooted in bodily experiences and situated action. In the present study we evaluated whether there is an embodied representation of the place–value structure of the Arabic number system and if so whether this representation is influenced by situated aspects. In a two-digit number magnitude comparison task participants had to directly touch the larger of two numbers. Importantly, pointing responses were systematically biased toward the decade digit of the target number. Additionally, this leftward bias towards the tens digit was reduced in unit–decade incompatible number pairs. Thereby, we demonstrated an influence of place–value processing on manual pointing movement. Our results therefore corroborate the notion of an embodied representation of the place–value structure of Arabic numbers which is modulated by situated aspects.     

两位数的整体与局部加工

关于两位数的加工方式有整体加工说和局部加工说,实验证据主要来自数字数量控制/主动加工任务。本研究主要考察在数字数量自动加工任务中两位数的加工方式。实验一要求被试完成数量大小比较和物理大小比较两个任务,实验二只要求被试完成物理大小比较任务。结果是在数量比较任务和物理比较任务中都存在显著的个位十位一致性效应和数量物理一致性效应,这表明在两位数的数量主动和自动加工任务中均存在整体加工和局部加工两种方式。     

When visual and verbal memories compete: Evidence of cross-domain limits in working memory

Recently, investigators have suggested that visual working memory operates in a manner unaffected by the retention of verbal material. We question that conclusion on the basis of a simple dual-task experiment designed to rule out phonological memory and to identify a more central faculty as the source of a shared limitation. With a visual working memory task in which two arrays of color squares were to be compared, performance was unaffected by concurrent recitation of a two-digit list or a known seven-digit sequence. However, visual working memory performance decreased markedly when paired with a load of seven random digits. This was not a simple tradeoff, inasmuch as errors on the visual array and high digit load tasks tended to co-occur. Working memory for digits and visual information thus are both subject to at least one type of shared limit, not just domain-specific limitations. The nature of the shared limit is discussed.     

Remembering a list of two-digit numbers

The way subjects remember a list of two-digit numbers has been examined in some detail. It is found that intrusions in free recall are not random. They resemble omissions in having the same first digit but not in other ways. This non-randomness of recall errors has been used to construct recognition tests of varying difficulty. Numbers which occurred commonly as intrusions were difficult to distinguish from the correct items when used as distractors in recognition tests. The experiments suggest that the previously observed relationship between recognition efficiency and number of alternatives (Davis, Sutherland and Judd, 1961) can be attributed to the increased probability that such intrusions will be included when the total number of distractors is increased.     

On-line syntactic processing under concurrent memory load

Thirty-six university students were tested in a plausibility judgment task using a self-paced listening paradigm under no-interference and two-digit load conditions. Listening times were longer at syntactically more complex portions of syntactically more complex sentences, and greater loads led to increased listening times. However, listening times at syntactically more complex positions in syntactically more complex sentences did not increase more than listening times at comparable positions in syntactically simple sentences under digit load conditions. The results indicate that a concurrent memory load does not reduce the availability of working memory resources used for on-line syntactic processing and, thus, provide evidence that the working memory system used for assigning syntactic structure is separate from that measured by standard working memory tasks.     

The effects of familiarity in a perceptual matching task

Latencies of same-different judgments to pairs of two-digit numerals were recorded for stimuli presented in familiar or unfamiliar (inverted) orientation. Familiar stimuli were responded to more quickly than unfamiliar. For both stimulus types, latencies were correlated with the syllable length of the verbal representation of the numerals, allowing the interpretation that the effect of stimulus orientation is on encoding processes. In two other experiments, it was found that familiarity had no effect on different judgments when the stimuli were relatively simple (e.g., a single digit), but did affect different judgments with more complex stimuli. These results were related to the hypothesis that the complexity of verbal material determines whether different judgments are instigated by visual or by verbal representations of the stimuli.