A memory-based account of automatic numerosity processing

We investigated the mechanisms responsible for the automatic processing of the numerosities represented by digits in the size congruity effect (Henik & Tzelgov, 1982). The algorithmic model assumes that relational comparisons of digit magnitudes (e.g., larger than {8,2}) create this effect. If so, congruity effects ought to require two digits. Memory-based models assume that associations between individual digits and the attributes "small" and "large" create this effect. If so, congruity effects ought only to require one digit. Contrary to the algorithmic model and consistent with memory-based models, congruity effects were just as large when subjects judged the relative physical sizes of small digits paired with letters as when they judged the relative physical sizes of two digits. This finding suggests that size congruity effects can be produced without comparison algorithms.     

Immediate judgments of learning are insensitive to implicit interference effects at retrieval

We conducted three experiments to determine whether metamemory predictions at encoding, immediate judgments of learning (IJOLs) are sensitive to implicit interference effects that will occur at retrieval. Implicit interference was manipulated by varying the association set size of the cue (Experiments 1 and 2) or the target (Experiment 3). The typical finding is that memory is worse for large-set-size cues and targets, but only when the target is studied alone and later prompted with a related cue (extralist). When the pairs are studied together (intralist), recall is the same regardless of set size; set size effects are eliminated. Metamemory predictions at retrieval, such as delayed JOLs (DJOLs) and feeling-of-knowing (FOK) judgments accurately reflect implicit interference effects (e.g., Eakin & Hertzog, 2006. In all three experiments, we found that DJOLs and FOKs accurately predicted set size effects on retrieval but that IJOLs did not. The findings provide further evidence that metamemory predictions are inferred from information other than direct access to the state of the memory trace, as well as indicate that inferences are based on different sources depending on when in the memory process predictions are made.     

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.     

Finding similarity in a model of relational reasoning

Similarity plays a central role in the study of perception and cognition. Previous attempts to model similarity have captured effects of either featural or structural similarity, but typically not both. We simulated both by fitting similarity data with the LISA model of relational reasoning [Hummel, J. E., & Holyoak, K. J. (1997). Distributed representations of structure: A theory of analogical access and mapping. Psychological Review, 104, 427–466, Hummel, J. E., & Holyoak, K. J. (2003a). A symbolic-connectionist theory of relational inference and generalization. Psychological Review, 110, 220–264]. The same mechanisms LISA uses to simulate analogy also provide a natural account of feature-based similarity effects (e.g., violations of symmetry), structural effects (e.g., the advantage of alignable over non-alignable differences), and the combined effects of featural and structured information (i.e., MIPs and MOPs; “Matches In/Out of Place”) on similarity judgments. Our approach differs from most models of similarity in that LISA was not originally designed to simulate similarity judgments, but rather analogical reasoning. LISA’s incidental ability to simulate diverse similarity effects speaks to the plausibility of the model’s account of human knowledge representation.     

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.     

The Wisdom of Individuals: Exploring People's Knowledge About Everyday Events Using Iterated Learning

Determining the knowledge that guides human judgments is fundamental to understanding how people reason, make decisions, and form predictions. We use an experimental procedure called 'iterated learning,' in which the responses that people give on one trial are used to generate the data they see on the next, to pinpoint the knowledge that informs people's predictions about everyday events (e.g., predicting the total box office gross of a movie from its current take). In particular, we use this method to discriminate between two models of human judgments: a simple Bayesian model ( Griffiths & Tenenbaum, 2006 ) and a recently proposed alternative model that assumes people store only a few instances of each type of event in memory (Min K ; Mozer, Pashler, & Homaei, 2008 ). Although testing these models using standard experimental procedures is difficult due to differences in the number of free parameters and the need to make assumptions about the knowledge of individual learners, we show that the two models make very different predictions about the outcome of iterated learning. The results of an experiment using this methodology provide a rich picture of how much people know about the distributions of everyday quantities, and they are inconsistent with the predictions of the Min K model. The results suggest that accurate predictions about everyday events reflect relatively sophisticated knowledge on the part of individuals.     

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.     

Is three greater than five: The relation between physical and semantic size in comparison tasks

In this study, subjects were asked to judge which of two digits (e.g., 3 5) was larger either in physical or in numerical size. Reaction times were facilitated when the irrelevant dimension was congruent with the relevant dimension and were inhibited when the two were incongruent (size congruity effect). Although judgments based on physical size were faster, their speed was affected by the numerical distance between the members of the digit pair, indicating that numerical distance is automatically computed even when it is irrelevant to the comparative judgment being required by the task. This finding argues for parallel processing of physical and semantic information in this task.     

Comparative longitudinal structural analyses of the growth and decline of multiple intellectual abilities over the life span.

Latent growth curve techniques and longitudinal data are used to examine predictions from the theory of fluid and crystallized intelligence (Gf-Gc theory; J. L. Horn & R. B. Cattell, 1966, 1967). The data examined are from a sample (N approximately 1,200) measured on the Woodcock-Johnson Psycho-Educational Battery-Revised (WJ-R). The longitudinal structural equation models used are based on latent growth models of age using two-occasion "accelerated" data (e.g., J. J. McArdle & R. Q. Bell, 2000; J. J. McArdle & R. W. Woodcock, 1997). Nonlinear mixed-effects growth models based on a dual exponential rate yield a reasonable fit to all life span cognitive data. These results suggest that most broad cognitive functions fit a generalized curve that rises and falls. Novel multilevel models directly comparing growth curves show that broad fluid reasoning (Gf) and acculturated crystallized knowledge (Gc) have different growth patterns. In all comparisons, any model of cognitive age changes with only a single g factor yields an overly simplistic view of growth and change over age.     

Working memory load differentially affects tip-of-the-tongue states and feeling-of-knowing judgments

Tip-of-the-tongue states (TOTs) are judgments of the likelihood of imminent retrieval for items currently not recalled, whereas feeling-of-knowing judgments (FOKs) are predictions of successful recognition for items not recalled. The assumption has been that similar metacognitive processes dictate these similar judgments. In Experiment 1, TOTs and FOKs were compared for general information questions. Participants remembered four digits (working memory load) during target retrieval for half of the questions, and there was no memory load for the other questions. Working memory did not affect recall but decreased the number of TOTs and increased FOKs. In Experiment 2, participants maintained six digits during retrieval. TOTs decreased in the working memory condition, but FOKs remained constant. Experiment 3 replicated the results of Experiment 2 while asking for FOKs for recall. In each of the first three experiments, positive metacognitive judgments also affected working memory performance, supporting the idea that working memory and metamemory use similar monitoring processes. In Experiment 4, visual working memory did not affect TOTs or FOKs. The data support a view that TOTs and FOKs are separable metacognitive entities.     

Why is word recognition impaired by disorientation while the identification of single letters is not?

Past research has shown that speed of identifying single letters or digits is largely indifferent to orientation, whereas the recognition of single words or connected text is markedly disrupted by disorientation. In a series of four experiments, we attempted to reconcile these findings. The results suggest that disorientation does not impair the identification of the characters but disrupts the perception of their spatial arrangement. When spatial order information is critical for distinguishing between different stimuli, disorientation is disruptive because some rectification process is required to restore order information. Utilizing the similarity between the letter B and the number 13, we found strong effects of orientation when a stimulus was interpreted as the two-digit number 13 but not when interpreted as the single letter B. This, however, occurred only when the set of numbers to be classified included permutations of the same digits (Experiments 1 and 2). Odd-even decisions on single-digit and two-digit numbers (Experiment 3) yielded strong effects of stimulus orientation for order-dependent numbers (e.g., 32), weaker effects for order-independent numbers (e.g., 24), and none for repeated-digit (e.g., 22) or single-digit numbers. Classification time for two-letter Hebrew words evidenced strong effects of orientation for words that differed only in letter order but much weaker effects for words that had no letters in common, even when these were embedded within some words that did (Experiment 4).     

Invisible Stimuli,Implicit Thresholds: Why Invisibility Judgments Cannot be Interpreted in Isolation

Some studies of unconscious cognition rely on judgments of participants stating that they have “not seen” the critical stimulus (e.g., in a masked-priming experiment). Trials in which participants gave invisibility judgments are then treated as those where the critical stimulus was “subliminal” or “unconscious,” as opposed to trials with higher visibility ratings. Sometimes, only these trials are further analyzed, for instance, for unconscious priming effects. Here I argue that this practice requires implicit assumptions about subjective measures of awareness incompatible with basic models of categorization under uncertainty (e.g., modern signal-detection and threshold theories). Most importantly, it ignores the potential effects of response bias. Instead of taking invisibility judgments literally, they would better be employed in parametric experiments where stimulus visibility is manipulated systematically, not accidentally. This would allow studying qualitative and double dissociations between measures of awareness and of stimulus processing per se.     

Conditioned inhibition produced by extinction-mediated recovery from the relative stimulus validity effect: a test of acquisition and performance models of empirical retrospective revaluation

Empirical retrospective revaluation is a phenomenon of Pavlovian conditioning and human causal judgment in which posttraining changes in the conditioned response (Pavlovian task) or causal rating (causal judgment task) of a cue occurs in the absence of further training with that cue. Two experiments tested the contrasting predictions made by 2 families of models concerning retrospective revaluation effects. In a conditioned lick-suppression task, rats were given relative stimulus validity training, consisting of reinforcing a compound of conditioned stimuli (CSs) A and X and nonreinforcement of a compound of CSs B and X, which resulted in low conditioned responding to CS X. Massive posttraining extinction of CS A not only enhanced excitatory responding to CS X, but caused CS B to pass both summation (Experiment 1) and retardation (Experiment 2) tests for conditioned inhibition. The inhibitory status of CS B is predicted by the performance-focused extended comparator hypothesis (J. C. Denniston, H. I. Savastano, & R. R. Miller, 2001), but not by acquisition-focused models of empirical retrospective revaluation (e.g., A. Dickinson & J. Burke, 1996; L. J. Van Hamme & E. A. Wasserman, 1994).     

Asymmetries in the processing of Arabic digits and number words

Numbers can be represented as Arabic digits ("6") or as number words ("six"). The present study investigated potential processing differences between the two notational formats. In view of the previous finding (e.g., Potter & Faulconer, 1975) that objects are named slower, but semantically categorized faster, than corresponding words, it was investigated whether a similar interaction between stimulus format and task could be obtained with numbers. Experiment 1 established that number words were named faster than corresponding digits, but only if the two notation formats were presented in separate experimental blocks. Experiment 2 contrasted naming with a numerical magnitude judgment task and demonstrated an interaction between notation and task, with slower naming but faster magnitude judgment latencies for digits than for number words. These findings suggest that processing of the two notation formats is asymmetric, with digits gaining rapid access to numerical magnitude representations, but slower access to lexical codes, and the reverse for number words.     

A model of exact small-number representation

To account for the size effect in numerical comparison, three assumptions about the internal structure of the mental number line (e.g., Dehaene, 1992) have been proposed. These are magnitude coding (e.g., Zorzi & Butterworth, 1999), compressed scaling (e.g., Dehaene, 1992), and increasing variability (e.g., Gallistel & Gelman, 1992). However, there are other tasks besides numerical comparison for which there is clear evidence that the mental number line is accessed, and no size effect has been observed in these tasks. This is contrary to the predictions of these three assumptions. Moreover, all three assumptions have difficulties explaining certain symmetries in priming studies of number naming and parity judgment. We propose a neural network model that avoids these three assumptions but, instead, uses place coding, linear scaling, and constant variability on the mental number line. We train the model on naming, parity judgment, and comparison and show that the size effect appears in comparison, but not in naming or parity judgment. Moreover, no asymmetries appear in primed naming or primed parity judgment with this model, in line with empirical data. Implications of our findings are discussed. This work was supported by Grant P5/04 from the Interuniversity Attraction Poles Program—Belgian Science Policy and by a GOA grant from the Ghent University Research Council to W.F.     

Masked priming of number judgments depends on prime validity and task

The influence of brief masked primes (42 or 50 msec) on number target judgments is shown to be highly sensitive to the list-wide validity of the primes for performing a particular target task. Odd/even judgments were facilitated on parity-valid trials (e.g., 1-7) relative to parity-invalid trials (e.g., 6-7), especially when .8 rather than .2 of the trials were parity valid. The opposite pattern was observed with magnitude judgments relative to 5: Responses were facilitated on magnitude- valid trials (e.g., 6-7) relative to magnitude-invalid trials (e.g., 1-7), especially when .8 of the trials were magnitude valid. These results are consistent with Bodner and Masson's (2001) claim that a processing episode constructed during a masked prime event is more likely to be recruited when there is a high probability that it will facilitate responding to the target.     

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.     

On the form of ROCs constructed from confidence ratings

A classical question for memory researchers is whether memories vary in an all-or-nothing, discrete manner (e.g., stored vs. not stored, recalled vs. not recalled), or whether they vary along a continuous dimension (e.g., strength, similarity, or familiarity). For yes-no classification tasks, continuous- and discrete-state models predict nonlinear and linear receiver operating characteristics (ROCs), respectively (D. M. Green & J. A. Swets, 1966; N. A. Macmillan & C. D. Creelman, 1991). Recently, several authors have assumed that these predictions are generalizable to confidence ratings tasks (J. Qin, C. L. Raye, M. K. Johnson, & K. J. Mitchell, 2001; S. D. Slotnick, S. A. Klein, C. S. Dodson, & A. P. Shimamura, 2000, and A. P. Yonelinas, 1999). This assumption is shown to be unwarranted by showing that discrete-state ratings models predict both linear and nonlinear ROCs. The critical factor determining the form of the discrete-state ROC is the response strategy adopted by the classifier.     

Adults' strategies for simple addition and multiplication: verbal self-reports and the operand recognition paradigm

Accurate measurement of cognitive strategies is important in diverse areas of psychological research. Strategy self-reports are a common measure, but C. Thevenot, M. Fanget, and M. Fayol (2007) proposed a more objective method to distinguish different strategies in the context of mental arithmetic. In their operand recognition paradigm, speed of recognition memory for problem operands after solving a problem indexes strategy (e.g., direct memory retrieval vs. a procedural strategy). Here, in 2 experiments, operand recognition time was the same following simple addition or multiplication, but, consistent with a wide variety of previous research, strategy reports indicated much greater use of procedures (e.g., counting) for addition than multiplication. Operation, problem size (e.g., 2 + 3 vs. 8 + 9), and operand format (digits vs. words) had interactive effects on reported procedure use that were not reflected in recognition performance. Regression analyses suggested that recognition time was influenced at least as much by the relative difficulty of the preceding problem as by the strategy used. The findings indicate that the operand recognition paradigm is not a reliable substitute for strategy reports and highlight the potential impact of difficulty-related carryover effects in sequential cognitive tasks.