Reading aloud multi-digit numbers is a surprisingly difficult cognitive operation that involves several visual and verbal processes. The main challenge is the need to handle the number’s syntactic structure, however, the extent to which this syntactic processing guides the number-reading process is still unknown. Here, I asked whether syntax-driven processing exists even in the initial visual stage that identifies each digit, via a mechanism that groups the digits according to language-specific conventions (triplets in most western languages: 23,456). Participants read aloud multi-digit numbers presented with a purely visual manipulation: the digits of each number were presented serially, and the inter-digit stimulus-onset-asynchrony (SOA) was constant except one prolonged SOA – either between the thousand and hundred digits, congruent with the standard division of numbers to triplets, or between the hundred and decade digits (incongruent grouping). Accuracy was higher in the congruent condition than in the incongruent one, indicating that the visual analyzer divides the digit string into triplets in the “standard” manner. Critically, this congruency effect was found not only when the participants read the numbers as syntactically-structured number names (“twelve”), but also when they said each number as a syntax-less series of digit names (“one, two”), i.e., triplet-grouping is not a top-down effect arising from the syntactic properties of the current verbal response. I conclude that the visual analyzer consistently divides the digit strings into “standard” triplets, and that this division guides the digit-identification process.

The involvement of Short-Term Memory (STM) in mathematical and numerical processing has been demonstrated repeatedly but its precise role remains unclear. Here, I examined the specific STM mechanisms involved in a fundamental number processing skill: handling symbolic numbers, in particular when writing multi-digit numbers to dictation. Participants heard 3- to 6-digit numbers and wrote down each number on paper with a digitizing pen. As shown previously, the inter-digit temporal gaps follow a consistent hierarchical pattern, reflecting the number’s internal hierarchical representation. Critically, when I imposed a delay between hearing the number and writing it down, thus forcing the participants to maintain the number briefly in an STM buffer, the hierarchical pattern diminished. I conclude that the STM buffer they used was (1) not hierarchical and (2) located in a processing stage that follows the hierarchical representation. When using the buffer (due to the imposed delay), the response was dominated by its flat structure rather by than the hierarchical structure of the preceding processing stage. I propose that this buffer is a Digit Output Buffer that maintains the digit string before writing it down. This means that, similar to the processing of words, number processing is supported by several format-specific STM buffers: some maintain phonological information, as shown in previous studies, and others maintain digits, as shown here.

Reading and writing numbers are handled by highly specialized processes dedicated to this purpose. Are domain-general executive processes involved in this task too, at least when it becomes particularly challenging? To address this, we examined a particularly challenging case in a number dictation task: decade-unit inversion. In Arabic (and several other languages), the unit word is said before the decade word (‘five hundred two and thirty’). In a number dictation task, the decades and units must be mentally inverted in order to write 532 from left to right. To cope with the challenge, people sometimes write the units digit before the decade digit, congruent with word order (5, then 2, then 3 in the middle; ‘UD-writing’). Experiment 1 showed that UD-writing is specifically linked to the inversion effort: it was frequent when the auditory decade-unit pair had two words (as in 532) but rare for single-word decade-unit pairs (530, 503). Experiment 2 used a dual-task method to examine the cognitive process underlying inversion. A phonological dual task increased the UD-writing rate, but adding executive load produced no further increase. We conclude that decade-unit inversion relies on phonological, not executive, resources. Specifically, we propose that it relies on the phonological input buffer, a short-term memory store at the verbal input stage, but not on a domain-general working memory system with executive aspects.

Hierarchical representations of meaningful information are a unique property of human cognition and were studied extensively for language. Here, we show that meaning-carrying hierarchical representations exist in another, non-linguistic domain, namely multi-digit numbers. Participants heard 3- to 6-digit numbers and wrote down each number on paper with a digitizing pen. We recorded the inter-digit temporal gaps, which reflect the post-gap digit processing effort. The gaps followed a systematic hierarchical pattern: short gaps between decade and unit digits (also in the left triplet of 5- and 6-digit numbers), longer gaps between hundred and decade digits, and even longer inter-triplet gaps. We conclude that multi-digit numbers are represented hierarchically, e.g., a 6-digit number (123,456) as [1 & (2 & 3)] & [4 & (5 & 6)]. Critically, this hierarchical pattern was contingent on the stimuli being syntactically structured (‘twenty-three’) and was absent for unstructured stimuli (‘two, three’) – i.e., the hierarchical organization is not a general property of numbers but a unique property of their syntactic structure. We propose that the origin of this hierarchy is in a core syntactic representation of numbers, which can be likened to the internal hierarchical representation of sentences.

The ability to read and write multi-digit numbers is increasingly recognized as a critical component of mathematical literacy. Previous studies showed that this skill takes years to develop, and children are not fluent even by the fourth grade. Here, we examined fifth-grade children as they read aloud briefly-presented digit strings. They were no less accurate than adults. Moreover, several of their number reading patterns resembled those of adults, indicating adult-like functioning of the visual analyzer – the cognitive process that parses digit strings. Specifically, the children scanned digits from left to right, showed an advantage for outer digits over inner digits, and demonstrated a dissociation between digit identification and position encoding. These findings indicate that by the fifth grade, children not only exhibit task proficiency in number reading, but they also rely on cognitive processes that are already mature in key aspects.

Humans exhibit sophisticated mathematical skills, which researchers attribute to our exceptionally large cortex. While most literature has long focused on the involvement of cortical structures in mathematical processes, the potential role of subcortical structures has been largely overlooked. Herein, to bridge this gap, we: 1) Review current evidence potentially indicating the contributions of subcortical structures to mathematical abilities; 2) Propose a potential explanation of how subcortical structures support mathematical abilities; 3) Argue that mathematical skills rely on a dynamic integrative network, involving both cortical and subcortical computations. Finally, given the results of our review, we propose three main future directions for research.

The ability to comprehend oral numbers is central to numerical literacy, yet the mechanisms enabling it are still poorly understood. Here we show that, as several researchers have hypothesized, short-term memory is involved in this process, and we also show how. We report two adults with developmental short-term memory deficit. They performed poorly in writing number to dictation and in other tasks requiring number comprehension, but not in tasks requiring other aspects of number processing. Their performance level was modulated by the memory load imposed by the task, and they made a variety of error types – digit substitutions as well as corruptions of the number’s syntactic structure. We conclude that their deficit was in a short-term memory store which serves the verbal-phonological input of numbers – a phonological input buffer for numbers. A deeper analysis of their error patterns suggests that this buffer serves as a workbench in which a full number is stored prior to parsing its syntactic structure. Based these and previous findings, we propose a detailed cognitive model for the verbal-phonological input of numbers, in which the phonological input buffer has a central role.

Accumulating evidence shows that the phonological production of words does not follow a single pathway, as was assumed in the past. Rather, some words are produced via dedicated production pathways. Previous studies showed such pathway separation for words from different categories (e.g., number versus non-number words) or with different meanings (number words with numerical versus non-numerical meaning, e.g., ‘Nine West’). Here, we show that the separation of phonological production pathways is even greater than that: even the same word with the same meaning can follow different phonological production pathways, depending on the context in which the word is produced. We report HLI, a woman with aphasia. She erred in 5 number-production tasks, all with phonological output but with different input modalities: digits, written number words, dice patterns, encyclopedic questions, and approximate quantities. We conclude she had impaired phonological production of number words. Critically, she performed relatively well in 3 other phonological number production tasks: oral calculation, counting, and repetition. This dissociation indicates multiple phonological production pathways for verbal numbers. We propose that HLI has a deficit in a default phonological production pathway, which serves most number-production tasks. At the same time, other, dedicated pathways of phonological number production, which are recruited uniquely by the calculation, counting, and repetition tasks, are spared. Finally, we propose a concrete model that explains when and how the dedicated, task-specific phonological-access pathways are created, and we show that HLI’s performance satisfies a central prediction of this model.

Number transcoding, the ability to convert digits to words and vice versa, is a critical skill in mathematical literacy and in everyday life. While transcoding is known to be difficult for children, it is unclear whether it challenges adults too, and if so, what the source of that difficulty is. Here, we analyzed the number reading performance of 172 neurotypical adults. Their mean error rate was 6.5%, considerably higher than that typically observed in word-reading tasks, indicating that transcoding is a relatively challenging task even in adulthood. To investigate the cognitive origin of this difficulty, we examined the error types produced by these participants, as well as by a second group of 51 adults with a number-reading deficit (dysnumeria, mean error rate of 28.7%). In both groups, most errors reflected corruptions of the number's syntactic structure. Moreover, among adults with dysnumeria, the most common subtype was syntactic dysnumeria. These findings indicate that, as in children, the primary challenge in adult number reading lies in the processing of the number's syntactic structure.

Reading numbers aloud, a central aspect of numerical literacy, is a challenging skill to acquire, but the origins of this difficulty remain poorly understood. To investigate this matter, we examined the performance of 127 third- and fourth-grade children who read aloud, in Hebrew, numbers with 2-5 digits. We found several key observations. First, we observed a substantial variation among the 3rd graders – 7% and 59% errors in the top and bottom deciles, respectively. Second, the task difficulty stemmed from syntactic processing: most errors were distortions of the number’s syntax, as opposed to digit substitutions or transpositions; and the main factor affecting a specific number’s difficulty was not its magnitude, as is commonly assumed, but rather its syntactic structure. Third, number reading performance was not predicted by a school-like task that assessed syntactic-conceptual knowledge of the decimal system structure, but rather by knowledge of specific syntactic-verbal rules; i.e., the syntactic-verbal knowledge is separate from syntactic-conceptual knowledge. Last, there was a double dissociation between 4-digit numbers and 5-digit numbers, which in Hebrew have completely different syntactic structures: half of the children showed a significant advantage in one number length compared to the other, with equal numbers of children preferring either length. This indicates that the different syntactic-verbal rules are learned relatively independently of each other, with little or no generalization from one rule to another. In light of these findings, we propose that schools should teach number reading explicitly, with explicit instruction of specific syntactic-verbal rules.

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.

The importance of working memory (WM) in executing mental algorithms is well-established in cognitive theories and empirical evidence. However, many of the precise ways in which WM operates during a mental algorithm are still poorly understood. Here, we used the classical example of mental arithmetic to examine how information is managed in WM during the algorithm execution. Participants added, in their head, pairs of two-digit numbers with a decade crossing either at the decades (82+74) or at the units (28+47). They used either of two 3-stage algorithms: [1] add the decades, [2] add the units, [3] merge their sums (20+40; 8+7; 60+15=75); or [1] add-units, [2] add-decades, [3] merge. Addition (stages 1-2) was harder when the sum of the two digits exceeded 10, a situation that increases the WM demands. Importantly, this decade crossing effect was larger in stage 1 than in stage 2. We conclude that the addends of stage-1 were removed from WM once this stage was completed and they were no longer needed; this reduced the WM load in stage 2, leading to better performance. Additionally, the performance in stage 3 (merge) was modulated by the order of the preceding stages (1-2): stage 3 was faster in the decades-then-units case than in the units-then-decades case. We conclude that WM stores data (the decade and unit sums) in an ordered manner by default, even when this is not needed for the task. We discuss the importance of WM management processes, of the kind shown here, for mental algorithms.

Dysnumeria, a learning disorder that disrupts number reading and writing, hinders numerical literacy and thus disturbs everyday life. However, little is known about its prevalence or its origins. Here, we provide data on the number-reading performance of typical readers (n=175) and readers with dysnumeria (n=54), who read aloud 120 multi-digit numbers. The typical readers made 6.5% errors on average, and 5%-9% of them turned out to actually have dysnumeria – similar to the reported prevalence of dyslexia and dyscalculia. We also analyzed the types of reading errors they made, which can be traced back to specific number-reading processes. Both typical readers and those with dysnumeria exhibited a high rate of syntactic errors, indicating that the main challenge in reading numbers is the need to handle the number’s syntactic structure.

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.

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.

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.

A key characteristic of human cognition, hypothesized to underlie the uniqueness of humans, is our ability to represent complex cognitive structures as a set of meaningful syntactic relations. In the domain of numbers, such syntactic representations are the key to the human ability to handle multi-digit numbers, however, we still have little understanding of the precise nature of the core syntactic representation of numbers. Here, I examined whether this core syntactic representation can be formed independently of semantic and lexical representations. Literate adults repeated sequences of number-like nonwords that had the morpho-syntactic structure of numbers but were otherwise unfamiliar and meaningless. Repetition accuracy was higher for grammatical sequences (“palir hundred tugumty-bab”) than for non-grammatical sequences made of the same words (“bab, tugumty, palir-hundred”). This effect, which mirrors similar findings with real numbers, indicates that when a sequence was grammatical, the participants represented its syntactic structure and used this representation to merge the nonwords into chunks in short-term memory, thereby improving memorization. Critically, because the stimuli were nonwords, I could conclude that the syntactic representation can exist even in the absence of semantic and lexical representations; and it can be created even without a number-word lexicon, merely on the basis of morpho-syntactic cues in the verbal stimulus. This fits the view of syntax as a set of separate cognitive processes, relatively independent of semantics.

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 the ability to read and write numbers.

The visual analysis of letter strings is a separate cognitive process from the analysis of digit strings. 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.

A major challenge for elementary school students is memorizing the multiplication table. This is difficult because there are many facts to learn and they are similar to each other, which creates proactive interference in memory. Here, we examined whether reducing interference would improve the memorization of the multiplication table by first graders. In a series of 16 short training sessions over a period of 4 weeks, each child learned 16 multiplication facts – 4 facts per week. Learning was better when the 4 facts in a given week were dissimilar from each other, a situation that reduces the proactive interference among them. 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 some facts over others. The similarity effect persisted 5 weeks after the end of the training period, i.e., proactive interference affected the long-term memory. Furthermore, during training, the similarity effect was not observed immediately but only in later training sessions, and only when examined in the beginning of a session. This indicates that proactive interference affected the long-term memory directly – it did not originate in short-term memory processes and then “leak” to long-term memory. We propose that the effectivity of this low-interference training method, which is dramatically different from currently-used pedagogical methods, calls for a serious reconsideration of the way we teach the multiplication table in school.