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.     

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).     

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.     

The place-value of a digit in multi-digit numbers is processed automatically

The automatic processing of the place-value of digits in a multi-digit number was investigated in 4 experiments. Experiment 1 and two control experiments employed a numerical comparison task in which the place-value of a non-zero digit was varied in a string composed of zeros. Experiment 2 employed a physical comparison task in which strings of digits varied in their physical sizes. In both types of tasks, the place-value of the non-zero digit in the string was irrelevant to the task performed. Interference of the place-value information was found in both tasks. When the non-zero digit occupied a lower place-value, it was recognized slower as a larger digit or as written in a larger font size. We concluded that place-value in a multi-digit number is processed automatically. These results support the notion of a decomposed representation of multi-digit numbers in memory. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

How do we convert a number into a finger trajectory?

How do we understand two-digit numbers such as 42? Models of multi-digit number comprehension differ widely. Some postulate that the decades and units digits are processed separately and possibly serially. Others hypothesize a holistic process which maps the entire 2-digit string onto a magnitude, represented as a position on a number line. In educated adults, the number line is thought to be linear, but the “number sense” hypothesis proposes that a logarithmic scale underlies our intuitions of number size, and that this compressive representation may still be dormant in the adult brain. We investigated these issues by asking adults to point to the location of two-digit numbers on a number line while their finger location was continuously monitored. Finger trajectories revealed a linear scale, yet with a transient logarithmic effect suggesting the activation of a compressive and holistic quantity representation. Units and decades digits were processed in parallel, without any difference in left-to-right vs. right-to-left readers. The late part of the trajectory was influenced by spatial reference points placed at the left end, middle, and right end of the line. Altogether, finger trajectory analysis provides a precise cognitive decomposition of the sequence of stages used in converting a number to a quantity and then a position.     

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.     

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.     

The processing of two-digit numbers in bilinguals

We explored possible between-language influences when bilinguals processed two-digit numbers. Spanish/English bilinguals and German/English bilinguals performed a number comparison task with Arabic digits and verbal numbers in their first language (L1) and second language (L2) while the unit-decade compatibility was manipulated. The two bilingual groups showed regular compatibility effect with Arabic digits. In L1, Spanish/English bilinguals showed reverse compatibility effect, while German/English bilinguals showed regular compatibility effect. However, both groups of bilinguals presented reverse compatibility effect in English (L2), which suggested that the bilinguals' L1 did not determine the processing of number words in their L2. The results indicated that bilinguals processed two-digit number words selectively in their L1 and L2 and that they did not transcode L2 numbers into Arabic format.     

Is place-value processing in four-digit numbers fully automatic? Yes,but not always

Knowing the place-value of digits in multi-digit numbers allows us to identify, understand and distinguish between numbers with the same digits (e.g., 1492 vs. 1942). Research using the size congruency task has shown that the place-value in a string of three zeros and a non-zero digit (e.g., 0090) is processed automatically. In the present study, we explored whether place-value is also automatically activated when more complex numbers (e.g., 2795) are presented. Twenty-five participants were exposed to pairs of four-digit numbers that differed regarding the position of some digits and their physical size. Participants had to decide which of the two numbers was presented in a larger font size. In the congruent condition, the number shown in a bigger font size was numerically larger. In the incongruent condition, the number shown in a smaller font size was numerically larger. Two types of numbers were employed: numbers composed of three zeros and one non-zero digit (e.g., 0040–0400) and numbers composed of four non-zero digits (e.g., 2795–2759). Results showed larger congruency effects in more distant pairs in both type of numbers. Interestingly, this effect was considerably stronger in the strings composed of zeros. These results indicate that place-value coding is partially automatic, as it depends on the perceptual and numerical properties of the numbers to be processed.     

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。     

Testing the limits of the phonetic mnemonic system

Three experiments tested the efficacy of the phonetic mnemonic system under varying conditions of application. The first study attempted unsuccessfully to replicate and extend the work of Morris and Greer (1984), who had shown training in the phonetic mnemonic method to facilitate memorization of a serial list of two-digit numbers. In the present study, subjects trained in the phonetic mnemonic method failed to learn lists of two-, four-, and six-digit numbers better than control subjects. The second experiment partially replicated the first, the differences being that training in the phonetic mnemonic method was strengthened, and time allotted for number recall was extended. Under these circumstances the phonetic mnemonic group recalled the two-, four-, and six-digit numbers significantly better than the control groups, a finding conforming with Morris and Greer's (1984) results. The third experiment partially replicated the second, everything being the same except that, in this case, subjects constructed their own key words representing each number, instead of these words being supplied by the experimenter. Under these conditions, subjects trained in the phonetic mnemonic method recalled significantly fewer numbers than control subjects.     

Comparing two-digit numbers: the importance of being presented together

The effect of presentation mode on magnitude comparisons of two-digit (2D) numbers was examined using the stimuli set developed by Nuerk, Weger, and Willmes (2001). In Experiment 1, only number pairs from difference decades were presented either simultaneously or sequentially. In the former case there was evidence for the parallel processing of both the units and decades digits and for a components representation, consistent with previous findings. In contrast, in the latter case there was evidence for the processing of mainly the decades digits. In Experiment 2, within-decade number pairs were added to make both digits task relevant. The results from the simultaneous condition were again consistent with a components representation, while results from the sequential presentation were in line with a holistic representation, in line with Zhang and Wang's (2005) research. Results therefore suggest that the processing of 2D numbers depends on the way they are presented.     

On the perceptual generality of the unit-decade compatibility effect

Number magnitude is assumed to be holistically represented along a single mental number line. Recently, we have observed a unit-decade-compatibility effect which is inconsistent with that assumption (Nuerk, Weger, & Willmes, 2001). In two-digit Arabic number comparison, we have demonstrated that compatible comparisons in which separate decade and unit comparisons lead to the same decision (32_47, 3 < 4 and 2 < 7) were faster than incompatible trials (37_52, 3 < 5, but 7 > 2). Because overall distance was matched, a holistic model could not account for the compatibility effect. However, one could argue that the compatibility effect was due to the specific vertical perceptual arrangement of the two-digit numbers in Nuerk et al.'s (2001) experiment where the decade digits and unit digits were presented column-wise above each other. To examine this objection, we studied the perceptual generality of the compatibility effect with diagonal presentation. We replicated the compatibility effect with diagonal presentation. It is concluded that the compatibility effect is not due to encoding characteristics imposed by the perceptual setting of the original experiment. In particular, the assumption of an overall analog magnitude representation for two-digit numbers is not consistent with these data.     

Magnitude representation in sequential comparison of two-digit numbers is not holistic either

There is accumulating evidence suggesting that two-digit number magnitude is represented in a decomposed fashion into tens and units rather than holistically as one integrated entity. However, recently, it has been claimed that this property does not hold for the case when two to-be-compared numbers are presented sequentially. In the present study, we pursued this issue in two experiments by evaluating perceptual as well as strategic aspects arising for sequential stimulus presentation in a magnitude comparison task. We observed reliable unit-decade compatibility effects indicating decomposed processing of tens and units in a magnitude comparison task with sequential presentation of the to-be-compared numbers. In particular, we found that both confounding low-level perceptual features and stimulus set characteristics determining cue validity of the units influenced the compatibility effect. Taken together, our results clearly indicate that decomposed representations of tens and units seem to be a general characteristic of multi-digit number magnitude processing, rather than an exception occurring under very specific conditions only. Implications of these results for the understanding of number magnitude representations are discussed.     

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.     

The differential influence of decades and units on multidigit number comparison

What role do the magnitudes of the constituent digits play in three-digit number comparison (e.g., choosing the larger one of two numbers)? The present study addressed this question by examining compatibility effects between hundred and decade digits and between hundred and unit digits. For example, the number pair 372-845 is hundred-decade incompatible because the larger number contains the smaller decade digit, but hundred-unit compatible because the larger number contains the larger unit digit. We obtained significant effects ofhundred-decade and hundred-unit compatibility on number comparison times. However, the effect of hundred-unit compatibility was largely restricted to the hundred-decade-compatible condition. These results suggest that place-value information, through decomposition, is automatically taken into account when multidigit numbers have to be compared. Implications of our findings for models of number processing are discussed.     

The differential influence of decades and units on multidigit number comparison

What role do the magnitudes of the constituent digits play in three-digit number comparison (e.g., choosing the larger one of two numbers)? The present study addressed this question by examining compatibility effects between hundred and decade digits and between hundred and unit digits. For example, the number pair 372–845 is hundred–decade incompatible because the larger number contains the smaller decade digit, but hundred–unit compatible because the larger number contains the larger unit digit. We obtained significant effects of hundred–decade and hundred–unit compatibility on number comparison times. However, the effect of hundred–unit compatibility was largely restricted to the hundred–decade-compatible condition. These results suggest that place-value information, through decomposition, is automatically taken into account when multidigit numbers have to be compared. Implications of our findings for models of number processing are discussed.