Top-Down Number Reading: Language Affects the Visual Identification of Digit Strings.

Reading numbers aloud involves visual processes that analyze the digit string and verbal processes that produce the number words. Cognitive models of number reading assume that information flows from the visual input to the verbal production processes-a feed-forward processing mode in which the verbal production depends on the visual input but not vice versa. Here, I show that information flows also in the opposite direction, from verbal production to the visual input processes. Participants read aloud briefly presented multi-digit strings in Hebrew, in which the order of words is congruent with the order of digits (21 = twenty-and-one), and in Arabic, in which the ones word precedes the tens word (one-and-twenty). The error-by-digit-position curve was affected by language: relative to Hebrew, in Arabic the error rate was slightly lower for the unit digit and slightly higher for the decade digit, indicating that in Arabic the unit digit was processed earlier and the decade digit later, in accord with the Arabic word order. This language-dependent processing order originated in the visual level and was not a verbal confound, because it persisted even when I controlled for the serial position of the decade/unit word in the verbal number by using numbers with 0 (two hundred three/two hundred thirty). I conclude that the visual analyzer's digit scanning order, decade-first or unit-first, is not fixed but affected by the language in which the number is produced-a top-down, verbal-to-visual information flow.

Elementary math in elementary school: the effect of interference on learning the multiplication table.

Memorizing the multiplication table is a major challenge for elementary school students: there are many facts to memorize, and they are often similar to each other, which creates interference in memory. Here, we examined whether learning would improve if the degree of interference is reduced, and which memory processes are responsible for this improvement. In a series of 16 short training sessions over 4 weeks, first-grade children learned 16 multiplication facts-4 facts per week. In 2 weeks the facts were dissimilar from each other (low interference), and in 2 control weeks the facts were similar (high interference). Learning in the low-similarity, low-interference weeks was better than in the high-similarity weeks. Critically, this similarity effect originated in the specific learning context, i.e., the grouping of facts to weeks, and could not be explained as an intrinsic advantage of certain facts over others. Moreover, the interference arose from the similarity between facts in a given week, not from the similarity to previously learned facts. Similarity affected long-term memory-its effect persisted 7 weeks after training has ended; and it operated on long-term memory directly, not via the mediation of working memory. Pedagogically, the effectiveness of the low-interference training method, which is dramatically different from currently used pedagogical methods, may pave the way to enhancing how we teach the multiplication table in school.

Brief memory reactivations induce learning in the numeric domain.

Learning of arithmetic facts such as the multiplication table requires time-consuming, repeated practice. In light of evidence indicating that reactivation of encoded memories can modulate learning and memory processes at the synaptic, system and behavioral levels, we asked whether brief memory reactivations can induce human learning in the numeric domain. Adult participants performed a number-fact retrieval task in which they learned arbitrary numeric facts. Following encoding and a baseline test, 3 passive, brief reactivation sessions of only 40 s each were conducted on separate days. Learning was evaluated in a retest session. Results showed reactivations induced learning, with improved performance at retest relative to baseline test. Furthermore, performance was superior compared to a control group performing test-retest sessions without reactivations, who showed significant memory deterioration. A standard practice group completed active-retrieval sessions on 3 separate days, and showed significant learning gains. Interestingly, while these gains were higher than those of the reactivations group, subjects showing reactivation-induced learning were characterized by superior efficiency relative to standard practice subjects, with higher rate of improvement per practice time. A follow-up long-term retention experiment showed that 30 days following initial practice, weekly brief reactivations reduced forgetting, with participants performing superior to controls undergoing the same initial practice without reactivations. Overall, the results demonstrate that brief passive reactivations induce efficient learning and reduce forgetting within a numerical context. Time-efficient practice in the numeric domain carries implications for enhancement of learning strategies in daily-life settings.

Syntactic chunking reveals a core syntactic representation of multi-digit numbers, which is generative and automatic.

Representing the base-10 structure of numbers is a challenging cognitive ability, unique to humans, but it is yet unknown how precisely this is done. Here, we examined whether and how literate adults represent a number's full syntactic structure. In 5 experiments, participants repeated number-word sequences and we systematically varied the order of words within each sequence. Repetition on grammatical sequences (e.g., two hundred ninety-seven) was better than on non-grammatical ones (hundred seven two ninety). We conclude that the participants represented the number's full syntactic structure and used it to merge number words into chunks in short-term memory. Accuracy monotonously improved for sequences with increasingly longer grammatical segments, up to a limit of ~ 4 words per segment, irrespectively of the number of digits, and worsened thereafter. Namely, short chunks improved memorization, whereas oversized chunks disrupted memorization. This chunk size limit suggests that the chunks are not based on predefined structures, whose size limit is not expected to be so low, but are created ad hoc by a generative process, such as the hierarchical syntactic representation hypothesized in Michael McCloskey's number-processing model. Chunking occurred even when it disrupted performance, as in the oversized chunks, and even when external cues for chunking were controlled for or were removed. We conclude that the above generative process operates automatically rather than voluntarily. To date, this is the most detailed account of the core representation of the syntactic structure of numbers-a critical aspect of numerical literacy and of the ability to read and write numbers.

Tracking priors and their replacement: Mental dynamics of decision making in the number-line task.

Several theories of decision making assume that optimal decisions are reached by computing a prior distribution over possible responses, and then updating it according to the evidence received. We show how this prior replacement, with its two processing stages, can be captured with a simple behavioral method: tracking the finger movement as participants point to a response location. On each trial, participants saw a number and pointed to its location on a number line. In two experiments, we manipulated either the prior, via the distribution of target numbers, or the initial finger direction, via explicit instruction. In both experiments, when a trial started the participants pointed towards the instructed direction, and in the last part of the trial they pointed towards the target. Critically, between these two stages there was a third, interim stage in which the participants pointed towards the prior before deviating towards the target. Transient pointing towards the prior was observed even when it induced a brief deviation away from the target. This pattern fits a model wherein decisions are first driven by prior knowledge, followed by the accumulation of trial-specific evidence. We propose that the number-to-position mapping task with finger tracking is a powerful paradigm to investigate fine-grained aspects of priors in a simple decision-making scenario.

Syntactic priming reveals an explicit syntactic representation of multi-digit verbal numbers.

When we say or understand verbal numbers, a major challenge to the cognitive system is the need to process the number's syntactic structure. Several studies showed that number syntax is handled by dedicated processes, however, it is still unclear how precisely these processes operate, whether the number's syntactic structure is represented explicitly, and if it is - what this representation looks like. Here, we used a novel experimental paradigm, syntactic priming of numbers, which can examine in detail the syntactic representation of multi-digit verbal numbers. In each trial, the participants - Arabic-Hebrew bilinguals and Hebrew monolinguals - heard a multi-digit number and responded orally with a random number. The syntactic structure of their responses was similar to that of the targets, showing that they represented the verbal number's syntax. This priming effect was genuinely syntactic, and could not be explained as lexical - repeating words from the target; as phonological - responding with words phonologically-similar to the target; or as a numerical distance effect - producing responses numerically close to the target. The syntactic priming effect was stronger for earlier words in the verbal number and weaker for later words, suggesting that the syntactic representation is capped by working-memory limits. We propose that syntactic priming could become a useful method to examine various aspects of the syntactic representation of numbers.

Serial and syntactic processing in the visual analysis of multi-digit numbers.

The visual analysis of letter strings and digit strings is done by two separate cognitive processes. Recent studies have hypothesized that these processes are not only separate but also qualitatively different, in that they may encode information specific to numbers or to words. To examine this hypothesis and to shed further light on the visual analysis of numbers, we asked adults to read aloud multi-digit strings presented to them for brief durations. Their performance was better in digits on the number's left side than in digits farther to the right, with better performance in the two outer digits than their neighbors. This indicates the digits were processed serially, from left to right. Visual similarity of digits increased the likelihood of errors, and when a digit migrated to an incorrect position, it was most often to an adjacent location. Interestingly, the positions of 0 and 1 were encoded better than the positions of 2-9, and 2-9 were identified better when they were next to 0 or 1. To accommodate these findings, we propose a detailed model for the visual analysis of digit strings. The model assumes imperfect digit detectors in which a digit's visual information leaks to adjacent locations, and a compensation mechanism that inhibits this leakage. Crucially, the compensating inhibition is stronger for 0 and 1 than for the digits 2-9, presumably because of the importance of 0 and 1 in the number system. This sensitivity to 0 and 1 makes the visual analyzer specifically adapted to numbers, not words, and may be one of the brain's reasons to implement the visual analysis of numbers and words in two separate cognitive processes.

Parallel and serial processes in number-to-quantity conversion.

Converting a multi-digit number to quantity requires processing not only the digits but also the number's decimal structure, thus raising several issues. First, are all the digits processed in parallel, or serially from left to right? Second, given that the same digit at different places can represent different quantities (e.g., "2" can mean 2, 20, etc.), how is each digit assigned to its correct decimal role? We presented participants with two-digit numbers and asked them to point at the corresponding locations on a number line, while we recorded their pointing trajectory. Crucially, on some trials, the decade and unit digits did not appear simultaneously. When the decade digit was delayed, the decade effect on finger movement was delayed by the same amount. However, a lag in presenting the unit digit delayed the unit effect by 35 ms less than the lag duration, a pattern reminiscent of the psychological refractory period, indicating an idle time window of 35 ms in the units processing pathway. When a lag transiently caused a display of just one digit on screen, the unit effect increased and the decade effect decreased, suggesting errors in binding digits to decimal roles. We propose that a serial bottleneck is imposed by the creation of a syntactic frame for the multidigit number, a process launched by the leftmost digit. All other stages, including the binding of digits to decimal roles, quantification, and merging them into a whole-number quantity, appear to operate in parallel across digits, suggesting a remarkable degree of parallelism in expert readers.

Track It to Crack It: Dissecting Processing Stages with Finger Tracking.

A central goal in cognitive science is to parse the series of processing stages underlying a cognitive task. A powerful yet simple behavioral method that can resolve this problem is finger trajectory tracking: by continuously tracking the finger position and speed as a participant chooses a response, and by analyzing which stimulus features affect the trajectory at each time point during the trial, we can estimate the absolute timing and order of each processing stage, and detect transient effects, changes of mind, serial versus parallel processing, and real-time fluctuations in subjective confidence. We suggest that trajectory tracking, which provides considerably more information than mere response times, may provide a comprehensive understanding of the fast temporal dynamics of cognitive operations.

Acquisition and processing of an artificial mini-language combining semantic and syntactic elements.

Most artificial grammar tasks require the learning of sequences devoid of meaning. Here, we introduce a learning task that allows studying the acquisition and processing of a mini-language of arithmetic with both syntactic and semantic components. In this language, symbols have values that predict the probability of being rewarded for a right or left response. Novel to our paradigm is the presence of a syntactic operator which changes the sign of the subsequent value. By continuously tracking finger movement as participants decided whether to press left or right, we revealed the successive cognitive stages associated with the sequential processing of the semantic and syntactic elements of this mini-language. All participants were able to understand the semantic component, but only half of them learned the rule associated with the syntactic operator. Our results provide an encouraging first step in elucidating the way in which humans acquire non-verbal syntactic structures and show how the finger tracking methodology can shed light on real-time artificial language processing.

Separate mechanisms for number reading and word reading: Evidence from selective impairments.

Do number reading and word reading use the same cognitive mechanisms? We examined this question through the looking glass of dissociations between impairments in number and word reading. We report two women with selective deficits in number reading, who read words normally. An examination of their impairment pattern indicated that the specific locus of their number reading deficits is in processes that handle the number's structure: both were impaired in parsing a digit string into triplets, and one of them was also impaired in generating the number's verbal structure. In contrast to their structural deficits in number reading, their word reading was completely intact, including the structural processes in word reading (morphological analysis and assembly). We present this dissociation in the framework of a broader effort to examine dissociations between specific components in number and word reading. We went beyond general word-number dissociations: we used detailed cognitive models for word reading and for number reading, and analyzed them in order to identify which components of the number reading process are homologous to which components of words reading. We then show that even these homologous processes are dissociable: an examination of previously-reported dissociations, completed by the case studies presented here, indicated that each of the specific homologous sub-processes of word/number reading can be selectively impaired. We conclude that although the word and number reading pathways show much similarity, they are almost entirely separate.

A cognitive model for multidigit number reading: Inferences from individuals with selective impairments.

We propose a detailed cognitive model of multi-digit number reading. The model postulates separate processes for visual analysis of the digit string and for oral production of the verbal number. Within visual analysis, separate sub-processes encode the digit identities and the digit order, and additional sub-processes encode the number's decimal structure: its length, the positions of 0, and the way it is parsed into triplets (e.g., 314987 → 314,987). Verbal production consists of a process that generates the verbal structure of the number, and another process that retrieves the phonological forms of each number word. The verbal number structure is first encoded in a tree-like structure, similarly to syntactic trees of sentences, and then linearized to a sequence of number-word specifiers. This model is based on an investigation of the number processing abilities of seven individuals with different selective deficits in number reading. We report participants with impairment in specific sub-processes of the visual analysis of digit strings - in encoding the digit order, in encoding the number length, or in parsing the digit string to triplets. Other participants were impaired in verbal production, making errors in the number structure (shifts of digits to another decimal position, e.g., 3,040 â†’ 30,004). Their selective deficits yielded several dissociations: first, we found a double dissociation between visual analysis deficits and verbal production deficits. Second, several dissociations were found within visual analysis: a double dissociation between errors in digit order and errors in the number length; a dissociation between order/length errors and errors in parsing the digit string into triplets; and a dissociation between the processing of different digits - impaired order encoding of the digits 2-9, without errors in the 0 position. Third, within verbal production, a dissociation was found between digit shifts and substitutions of number words. A selective deficit in any of the processes described by the model would cause difficulties in number reading, which we propose to term "dysnumeria".

On-line confidence monitoring during decision making.

Humans can readily assess their degree of confidence in their decisions. Two models of confidence computation have been proposed: post hoc computation using post-decision variables and heuristics, versus online computation using continuous assessment of evidence throughout the decision-making process. Here, we arbitrate between these theories by continuously monitoring finger movements during a manual sequential decision-making task. Analysis of finger kinematics indicated that subjects kept separate online records of evidence and confidence: finger deviation continuously reflected the ongoing accumulation of evidence, whereas finger speed continuously reflected the momentary degree of confidence. Furthermore, end-of-trial finger speed predicted the post-decisional subjective confidence rating. These data indicate that confidence is computed on-line, throughout the decision process. Speed-confidence correlations were previously interpreted as a post-decision heuristics, whereby slow decisions decrease subjective confidence, but our results suggest an adaptive mechanism that involves the opposite causality: by slowing down when unconfident, participants gain time to improve their decisions.

Finger Tracking Reveals the Covert Stages of Mental Arithmetic.

We introduce a novel method capable of dissecting the succession of processing stages underlying mental arithmetic, thus revealing how two numbers are transformed into a third. We asked adults to point to the result of single-digit additions and subtractions on a number line, while their finger trajectory was constantly monitored. We found that the two operands are processed serially: the finger first points toward the larger operand, then slowly veers toward the correct result. This slow deviation unfolds proportionally to the size of the smaller operand, in both additions and subtractions. We also observed a transient operator effect: a plus sign attracted the finger to the right and a minus sign to the left and a transient activation of the absolute value of the subtrahend. These findings support a model whereby addition and subtraction are computed by a stepwise displacement on the mental number line, starting with the larger number and incrementally adding or subtracting the smaller number.

On the origins of logarithmic number-to-position mapping.

The number-to-position task, in which children and adults are asked to place numbers on a spatial number line, has become a classic measure of number comprehension. We present a detailed experimental and theoretical dissection of the processing stages that underlie this task. We used a continuous finger-tracking technique, which provides detailed information about the time course of processing stages. When adults map the position of 2-digit numbers onto a line, their final mapping is essentially linear, but intermediate finger location show a transient logarithmic mapping. We identify the origins of this log effect: Small numbers are processed faster than large numbers, so the finger deviates toward the target position earlier for small numbers than for large numbers. When the trajectories are aligned on the finger deviation onset, the log effect disappears. The small-number advantage and the log effect are enhanced in dual-task setting and are further enhanced when the delay between the 2 tasks is shortened, suggesting that these effects originate from a central stage of quantification and decision making. We also report cases of logarithmic mapping-by children and by a brain-injured individual-which cannot be explained by faster responding to small numbers. We show that these findings are captured by an ideal-observer model of the number-to-position mapping task, comprising 3 distinct stages: a quantification stage, whose duration is influenced by both exact and approximate representations of numerical quantity; a Bayesian accumulation-of-evidence stage, leading to a decision about the target location; and a pointing stage. (PsycINFO Database Record

Steps towards understanding the phonological output buffer and its role in the production of numbers, morphemes, and function words.

The Stimulus Type Effect on Phonological and Semantic errors (STEPS) describes the phenomenon in which a person, following brain damage, produces words with phonological errors (fine â†’ fige), but number words with semantic errors (five â†’ eight). To track the origins of this phenomenon and find out whether it is limited to numbers, we assessed the speech production of six individuals with conduction aphasia following a damage in the left hemisphere, who made phonological errors in words. STEPS was found in all six participants, and was not limited to number words - several other word categories were also produced with semantic rather than phonological errors: function words, English letter names, and morphological affixes were substituted with other words within their category. This supports the building blocks hypothesis: when phonological sequences serve as building blocks in a productive process, they end up having pre-assembled phonological representations, ready for articulation. STEPS reflects a deficit that causes substitutions of one phonological unit with another. In the case of plain content words, this causes substitutions of one phoneme with another, but in the case of pre-assembled phonological units, this causes substitutions of number words with other number words, function words with function words, and morphological affixes with other affixes. An analysis of the participants' functional locus of deficit revealed that they all had a deficit in the phonological output buffer, and this was their only common deficit. We therefore concluded that the pre-assembled phonological units are stored in dedicated mini-stores in the phonological output buffer, which processes not only phonemes but also whole number words, function words, and morphemes. We also found that STEPS depends on the word's role: number words were produced with semantic errors only when they appeared in numeric context, and function words triggered semantic errors only in grammatical context. This suggests that the phonological representation of a word can be obtained either from the phonological output lexicon or from a store of pre-assembled representations in the phonological output buffer, depending on the word's role.

Breaking down number syntax: spared comprehension of multi-digit numbers in a patient with impaired digit-to-word conversion.

Can the meaning of two-digit Arabic numbers be accessed independently of their verbal-phonological representations? To answer this question we explored the number processing of ZN, an aphasic patient with a syntactic deficit in digit-to-verbal transcoding, who could hardly read aloud two-digit numbers, but could read them as single digits ("four, two"). Neuropsychological examination showed that ZN's deficit was neither in the digit input nor in the phonological output processes, as he could copy and repeat two-digit numbers. His deficit thus lied in a central process that converts digits to abstract number words and sends this information to phonological retrieval processes. Crucially, in spite of this deficit in number transcoding, ZN's two-digit comprehension was spared in several ways: (1) he could calculate two-digit additions; (2) he showed good performance in a two-digit comparison task, and a continuous distance effect; and (3) his performance in a task of mapping numbers to positions on an unmarked number line showed a logarithmic (nonlinear) factor, indicating that he represented two-digit Arabic numbers as holistic two-digit quantities. Thus, at least these aspects of number comprehension can be performed without converting the two-digit number from digits to verbal representation.

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.