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Learning a Novel Number System: The Role of Compositional Rules and Counting Procedures

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Title: Learning a Novel Number System: The Role of Compositional Rules and Counting Procedures
Language: English
Authors: Sebastian Holt, David Barner
Source: Cognitive Science. 2025 49(6).
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 31
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Education Level: Adult Education
Descriptors: Computation, Numbers, Adult Students, Number Concepts, Multiplication, Memory, Rote Learning, Numeracy, Word Lists
DOI: 10.1111/cogs.70071
ISSN: 0364-0213
1551-6709
Abstract: Humans count to indefinitely large numbers by recycling words from a finite list, and combining them using rules--for example, combining sixty with unit labels to generate sixty-one, sixty-two, and so on. Past experimental research has focused on children learning base-10 systems, and has reported that this rule learning process is highly protracted. This raises the possibility that rules are slow to emerge because they are not needed in order to represent smaller numbers (e.g., up to 20). Here, we investigated this possibility in adult learners by training them on a series of artificial number "languages" that manipulated the availability of rules, by varying the numerical base in each language. We found (1) that the size of a base--for example, base-2 versus base-5--had little effect on learning, (2) that learners struggled to acquire multiplicative rules while they learned additive rules more easily, (3) that memory for number words was greater when they were taught as part of a sequential count list, but (4) that learning numbers as part of a rote list may impair the ability to map them to magnitudes.
Abstractor: As Provided
Notes: https://osf.io/rwqk7
Entry Date: 2025
Accession Number: EJ1475024
Database: ERIC
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  Value: <anid>AN0186163275;cgn01jun.25;2025Jun27.04:02;v2.2.500</anid> <title id="AN0186163275-1">Learning a Novel Number System: The Role of Compositional Rules and Counting Procedures </title> <p>Humans count to indefinitely large numbers by recycling words from a finite list, and combining them using rules—for example, combining sixty with unit labels to generate sixty‐one, sixty‐two, and so on. Past experimental research has focused on children learning base‐10 systems, and has reported that this rule learning process is highly protracted. This raises the possibility that rules are slow to emerge because they are not needed in order to represent smaller numbers (e.g., up to 20). Here, we investigated this possibility in adult learners by training them on a series of artificial number "languages" that manipulated the availability of rules, by varying the numerical base in each language. We found (<reflink idref="bib1" id="ref1">1</reflink>) that the size of a base—for example, base‐2 versus base‐5—had little effect on learning, (<reflink idref="bib2" id="ref2">2</reflink>) that learners struggled to acquire multiplicative rules while they learned additive rules more easily, (<reflink idref="bib3" id="ref3">3</reflink>) that memory for number words was greater when they were taught as part of a sequential count list, but (<reflink idref="bib4" id="ref4">4</reflink>) that learning numbers as part of a rote list may impair the ability to map them to magnitudes.</p> <p>Keywords: Number; Artificial languages; Counting; Syntax; Base systems</p> <hd id="AN0186163275-2">Introduction</hd> <p>Humans represent large, precise magnitudes, such as <emph>two‐thousand and five</emph>, using culturally transmitted number words that are characterized by both memorized symbols and rules for combining them. Most number systems in use today feature a base‐10 structure, in which words for 1–10 are combined to generate large numbers—for example, by composing the unit labels for 1–9 with decade labels to generate number words up to 99, such as <emph>sixty‐six</emph> (6*10+6). However, a much wider range of systems is attested on the historical record, including systems with bases as small as 2 and as large as 20 (Hammarström, [<reflink idref="bib35" id="ref5">35</reflink>]; Comrie, [<reflink idref="bib17" id="ref6">17</reflink>]; Table 1). For example, the Western Tribe of Torres Straits had a number system featuring only two lexical items, <emph>urapun</emph> (one) and <emph>okosa</emph> (two). These could be combined to create <emph>okosa urapun</emph> (three), <emph>okosa okosa</emph> (four), <emph>okosa okosa urapun</emph> (five), and <emph>okosa okosa okosa</emph> (six) (Haddon, [<reflink idref="bib34" id="ref7">34</reflink>]). These differences between historically attested systems are important because they may impact how number words are used, understood, and learned. Systems with small bases tend to differ in many ways from larger bases. For example, small bases may require less rote memorization because they feature only a few lexical roots, but may also require more frequent use of syntactic rules to construct larger numbers.[<reflink idref="bib1" id="ref8">1</reflink>] By contrast, systems with large bases may require more memorization of lexical roots, while depending less on rules to represent the same numbers. Although historically it was primarily adults who created and transmitted number systems, our current understanding of how they are learned has come overwhelmingly from studies of young children learning base‐10 number systems. Consequently, it remains unclear whether our current understanding of how humans learn number words generalizes beyond base‐10, where compositional rules are only relevant to learning relatively large numbers, which for children are only encountered relatively late in development. In the present study, we examined how the structure of number systems might impact learning by investigating adult learners, who unlike children are capable of quickly learning new symbolic systems, and by training these learners on diverse number systems, with different bases and systems of rules.</p> <p>1 Table Several base systems: Western Torres (Haddon, 1890), Ventureño (Beeler, 1964 ; 1986), Kómnzo (Döhler, 2018), and English</p> <p> <ephtml> <table><thead><tr><th align="left">Language</th><th align="center">Base</th><th align="center">Example roots</th><th align="center">Example composed numbers</th></tr></thead><tbody><tr><td>Torres</td><td>2</td><td>1, urapun; 2, okosa</td><td>3, okosa‐urapun; 4, okosa‐okosa</td></tr><tr><td>Ventureño</td><td>4</td><td>1, pakeʼet, 2, ʼiškom̓, 3, masǝx, 4, tskumu</td><td>6, yǝti'iškom̓; 12, masǝx tskumu</td></tr><tr><td>Kómnzo</td><td>6</td><td>1, näbi, 2, eda; 3, etha; 4, asar; 5, tabuthui; 6: nibo<xref ref-type="fn" rid="fn13" /></td><td>7, nibo a näbi, 8, nibo a eda ...</td></tr><tr><td>English</td><td>10</td><td>1, one; 2, two; ... 10, ten<xref ref-type="fn" rid="fn14" /></td><td>16, sixteen; 60, sixty</td></tr></tbody></table> </ephtml> </p> <hd id="AN0186163275-3">How are number words usually learned?</hd> <p>Studies of number word learning in base‐10 systems ranging from English, to Cantonese, to Hindi find that children acquire number words in a protracted process that takes 3–4 years, first learning meanings for small words like "one," "two," and "three," then learning to accurately count large sets. While children initially appear to treat the count routine as a meaningless, rote procedure, they ultimately learn rules for combining number words by around 5‐and‐a‐half to 6 years of age (Almoammer et al., [<reflink idref="bib2" id="ref9">2</reflink>]; Barner, Chow, & Yang, [<reflink idref="bib4" id="ref10">4</reflink>]; Carey & Barner, [<reflink idref="bib12" id="ref11">12</reflink>]; Ceylan & Aslan, [<reflink idref="bib14" id="ref12">14</reflink>]; Condry & Spelke, [<reflink idref="bib18" id="ref13">18</reflink>]; Davidson, Eng, & Barner, [<reflink idref="bib21" id="ref14">21</reflink>]; Jara‐Ettinger, Piantadosi, Spelke, Levy, & Gibson, [<reflink idref="bib39" id="ref15">39</reflink>]; Le Corre, Li, Huang, Jia, & Carey, [<reflink idref="bib46" id="ref16">46</reflink>]; Le Corre & Carey, [<reflink idref="bib45" id="ref17">45</reflink>]; Marchand, Lovelett, Kendro, & Barner, [<reflink idref="bib48" id="ref18">48</reflink>]; Meyer, Barbiers, & Weerman, [<reflink idref="bib53" id="ref19">53</reflink>]; Negen & Sarnecka, [<reflink idref="bib55" id="ref20">55</reflink>]; Nikoloska, [<reflink idref="bib56" id="ref21">56</reflink>]; Piantadosi, Jara‐Ettinger, & Gibson, [<reflink idref="bib59" id="ref22">59</reflink>]; Sarnecka, Kamenskaya, Yamana, Ogura, & Yudovina, [<reflink idref="bib64" id="ref23">64</reflink>]; Sarnecka & Carey, [<reflink idref="bib63" id="ref24">63</reflink>]; Spaepen, Gunderson, Gibson, Goldin‐Meadow, & Levine, [<reflink idref="bib70" id="ref25">70</reflink>]; Wagner, Kimura, Cheung, & Barner, [[<reflink idref="bib73" id="ref26">73</reflink>], [<reflink idref="bib72" id="ref27">72</reflink>]]; Wege, Bourque, Merkley, & Cheung, [<reflink idref="bib74" id="ref28">74</reflink>]; Wynn, [[<reflink idref="bib77" id="ref29">77</reflink>]]). This change occurs at around the same time that children begin to show an appreciation for how counting relates to number—for example, that moving one place in the count list entails increasing the cardinality of the counted set by +1 (Davidson et al., [<reflink idref="bib21" id="ref30">21</reflink>]; Spaepen et al., [<reflink idref="bib70" id="ref31">70</reflink>]). Evidence that these older children use rules to produce larger number words comes from studies that ask them to count as high as they can, noting where errors occur. If children have memorized their count list without the use of rules, then errors should occur at random points—for example, at 34, 47, and so on. However, if they use a rule to compose the words for "one" through "nine" with familiar decade labels, errors should be clustered at decade transitions where children must either memorize the decade label (e.g., "sixty") or must have knowledge of a relatively sophisticated multiplicative rule (e.g., six‐ty equals six‐tens). Using this metric, previous studies report that while some children count up to random numbers, the most frequent error of 4‐ and 5‐year‐olds is to count up to decade transitions like 29, 39, or 49 (Fuson, Richards, & Briars, [<reflink idref="bib27" id="ref32">27</reflink>]; Gould, [<reflink idref="bib33" id="ref33">33</reflink>]; Schneider et al., [<reflink idref="bib66" id="ref34">66</reflink>], [<reflink idref="bib67" id="ref35">67</reflink>], [<reflink idref="bib68" id="ref36">68</reflink>]; Siegler & Robinson, [<reflink idref="bib69" id="ref37">69</reflink>]; Wright, [<reflink idref="bib76" id="ref38">76</reflink>]). Importantly, these children, unlike those who count to random numbers, are often able to continue counting upwards when prompted with the next number—for example, counting up to 49 when told that the number after 39 is "forty"—consistent with the use of an additive rule.</p> <p>Because previous work has focused exclusively on children learning base‐10 counting systems, it remains unclear whether this protracted learning sequence generalizes to other possible number systems, or whether rules might emerge earlier given other systems, such as those with smaller bases. One possibility is that children exposed to a base‐10 system struggle to learn rules because of conceptual limitations, or because these rules depend on knowledge acquired at earlier stages of number word learning. However, it is also possible that rule learning emerges late not because it is especially difficult, but because rule structures are not frequent within the small range of numbers that children hear in their language input and actually use. In English, the basis for many conclusions in the child literature, numbers up to 20 must be memorized, and rules are only salient in numbers greater than 30 (the first point at which the 1–9 rule is repeated). This is important because corpus studies find that although children are likely to hear small number words with high frequency in caregiver speech, larger words that exhibit rules are much less frequent (Willits, Jones, & Landy, [<reflink idref="bib75" id="ref39">75</reflink>]). Relatedly, number words are often taught to children as part of a memorized counting routine, which may encourage rote memorization over rule learning (Fuson, [<reflink idref="bib26" id="ref40">26</reflink>]). Meanwhile, historically humans have used a much more diverse set of number systems, with both smaller and larger bases, and often without the use of memorized counting systems (Hammarström, [<reflink idref="bib35" id="ref41">35</reflink>]; Comrie, [<reflink idref="bib17" id="ref42">17</reflink>]). This raises the question of whether different number systems, used in alternative ways, might lead to easier learning of numerical rule structures.</p> <p>Some previous studies have begun to address how learners represent the syntactic rules of base systems using experimental methods (Barrouillet, Thevenot, & Fayol, [<reflink idref="bib5" id="ref43">5</reflink>]; Cheung, Dale, & Le Corre, [<reflink idref="bib15" id="ref44">15</reflink>]). However, these studies have not directly tested how variability in the structure of number systems—like that found on the historical record—might impact learning. For example, Cheung et al. ([<reflink idref="bib15" id="ref45">15</reflink>]) taught English‐ and Cantonese‐speaking 4‐ to 6‐year‐olds a novel number term, <emph>gobi</emph>, referring to groups of three. They found that children speaking both languages could generalize a digit‐multiplier construction (e.g., "one <emph>gobi</emph> houses" = 3 houses) to identify sets of six ("two <emph>gobi</emph> houses"), while only Cantonese‐speaking children frequently succeeded in generalizing a bare form (e.g., "<emph>gobi</emph> houses" to "two <emph>gobi</emph> houses"). While there may be many linguistic or cultural reasons for this different performance (Cantonese's more transparent compositional syntax, its use of classifiers, etc.), this study demonstrated that children speaking both languages can learn a compositional rule that is central to a base‐3 system. Barrouillet et al. ([<reflink idref="bib5" id="ref46">5</reflink>]) used a serial recall task to test whether children have implicit representations of number syntax, even if they are not yet capable of consciously using those representations. Reasoning that implicit representations of syntactic structure in long‐term memory should facilitate short‐term memory storage and recall (e.g., Hulme, Maughan, & Brown, [<reflink idref="bib37" id="ref47">37</reflink>]; Gathercole, [<reflink idref="bib28" id="ref48">28</reflink>]), they asked 5‐year‐olds to repeat sequences of number words that were either grammatically well‐formed (e.g., "three hundred forty five") or ill‐formed ("five three forty hundred"). They found that children were more accurate when repeating well‐formed number words, suggesting that they already had an implicit representation of the valid rules of composition. Although these studies suggest that children are sensitive to existing rules and can learn new ones, neither addressed how learning might differ across different systems, and whether it is impacted by factors such as the size of the memorized base, or the types of compositional rules present in a system (e.g., additive, multiplicative, exponential).</p> <hd id="AN0186163275-4">Does counting facilitate rule learning?</hd> <p>Aside from leaving open how diversity in number systems impacts learning, previous studies also leave open how rule learning is impacted by the way in which we teach number words. As already noted, when children learn number words, it is typical for caregivers and early educators to explicitly train them on both the count list and related counting procedures. Such training typically includes learning to recite a subset of the count list in order (e.g., labels for 1–10 or ultimately 1–100), and the practice of placing these labels in one‐to‐one correspondence with objects in an array (e.g., pointing at each object while reciting the number words; Blevins‐Knabe & Musun‐Miller, [<reflink idref="bib8" id="ref49">8</reflink>]; Gibson, Gunderson, & Levine, [<reflink idref="bib30" id="ref50">30</reflink>]; LeFevre et al., [<reflink idref="bib47" id="ref51">47</reflink>]; Turan & De Smedt, [<reflink idref="bib71" id="ref52">71</reflink>]). Notably, this is not how children learn other words in language. For example, although adult learners of a second language may memorize morphological paradigms, children learning their first language generally acquire rules by hearing different forms used in conversation—for example, abstracting the past tense or pluralization rules from hearing many instances of regular forms in caregiver speech, not by rote memorizing a verbal paradigm within an ordered sequence (e.g., <emph>help</emph>, <emph>helped</emph>, <emph>helping</emph>, etc.). Also notable is that not all attested number systems in the historical record were accompanied by counting practices. For example, in the number system of the Western Tribe of Torres Straits, described above, expressions <emph>urapun</emph> (one), <emph>okosa</emph> (two), and <emph>okosa urapun</emph> (three) were not memorized in a list and taught in scripted routines and procedures, but instead were learned like other morphological forms of the language, akin to English quantity expressions like "a couple," "several," and "a few" (Haddon, [<reflink idref="bib34" id="ref53">34</reflink>]). Similarly, although children learning languages like Slovenian and Arabic are taught counting routines and procedures, they are not taught routines for learning singular and dual morphology in their languages, although these also encode number (Almoammer et al., [<reflink idref="bib2" id="ref54">2</reflink>]; Marušič et al., [[<reflink idref="bib52" id="ref55">52</reflink>], [<reflink idref="bib51" id="ref56">51</reflink>]]). Instead, these forms and the sometimes complex rules they entail are learned from exposure in conversation.</p> <p>While there is evidence that early mastery of counting may have long‐term educational benefits (Koponen, Aunola, & Nurmi, [<reflink idref="bib41" id="ref57">41</reflink>]; Martin, Cirino, Sharp, & Barnes, [<reflink idref="bib50" id="ref58">50</reflink>]), we know of no study that tests whether training on rote counting routines facilitates learning above and beyond frequent exposure to number words in conversational speech, nor how different types of training impact rule learning. One possibility is that rote training enhances a learner's ability to discover the rules that govern counting, by providing them with a robustly represented sequence that they can then analyze and decompose. That is, memorizing a count sequence may provide a dataset that supports further learning. Alternatively, it is possible that training with counting routines has a negligible or even negative effect on rule learning. For example, if the main impetus for constructing rules is to generate novel forms, memorization may supplant this need, by providing an alternative representational format. If learners receive sufficient rote training on numbers 1–100, for example, they may not need to construct rules that generate these forms. On this possibility, learners who are exposed to numbers in both everyday language use and a memorized routine may actually be less able to generate novel forms versus learners who are exposed to the forms only in language use.</p> <hd id="AN0186163275-5">Why study adults?</hd> <p>Although almost all previous work on number word learning has been conducted in child learners, in the present study, we began the investigation of these questions by testing fully numerate adults. We did so as a first step in this research program for two main reasons. First, whereas we know that children take years to learn even labels for 1, 2, and 3 and are difficult to train on even a single new number word (Carey, Shusterman, Haward, & Distefano, [<reflink idref="bib13" id="ref59">13</reflink>]; Huang, Spelke, & Snedeker, [<reflink idref="bib36" id="ref60">36</reflink>]; Spaepen et al., [<reflink idref="bib70" id="ref61">70</reflink>]), adults can more easily learn a large number of novel words in a single session. Also, testing adults allows us to dissociate the problem of learning a combinatorial rule system from the various other factors that may limit learning in children, including both domain general limits, like working memory and attention, but also factors specific to number, such as the fact that number words are almost always trained as part of a counting routine. By testing adults, we were able to train learners on complete number systems featuring multiple rules and lexical items, while also manipulating whether words were presented as part of a sequential counting routine. Finally, testing adults allowed us to administer multiple tasks, including tests of recall, generalization to untrained numbers, and semantic understanding of novel combinations.</p> <p>Second, we know little about how adult learners learn novel number systems despite the fact that, historically, it was primarily adults who created and used number systems. This is important both because adult learners have greater cognitive capacities that may lead to the creation of different rule structures, but also because adult learners bring additional conceptual understanding to the problem of number word learning. In fact, there is evidence that some attested number systems emerged historically in communities that used other, pre‐existing number systems. For example, the Torres Straits system existed in parallel with a separate, noncompositional counting routine (Haddon, [<reflink idref="bib34" id="ref62">34</reflink>]). A similar pattern is found in the current day Nadahup languages, where some dialects are restricted to small numbers, some have small numbers and a rote body counting system, and other dialects extend this body counting practice into a system of large, compositional, verbal numerals (Epps, [<reflink idref="bib23" id="ref63">23</reflink>]). In multilingual contexts, existing number systems are often supplemented by imported systems, resulting in compound number systems with distinct linguistic sources (e.g., Garifuna, Mundurukú, Tsimané), or in which multiple number systems are used side‐by‐side, often for different purposes (e.g., Korean, Kómnzo). Finally, early written numerals (e.g., in Mesopotamia) were slow to develop, but almost certainly did so in the context of existing verbal number systems (Overmann, [<reflink idref="bib57" id="ref64">57</reflink>]). These historical observations suggest that the evolution of number systems often occurs in contexts of adult "multi‐numeracy" (Marques, [<reflink idref="bib49" id="ref65">49</reflink>]), through the interaction of already‐familiar number symbols with new symbols that are being constructed.</p> <p>Testing adult learners allowed us to ask a series of novel questions. For example, we reasoned that if systems with smaller bases are harder to learn than systems with larger bases, then this difficulty should be reflected in the learning behaviors of adult participants. Similarly, if additive rules (e.g., adding unit labels for 1–9 to decades) are easier to learn than multiplicative rules (e.g., multiplying units by a base), then we expect that this should be reflected in the data of adults. Although it is possible that prior exposure to a base‐10 system might make it easier for adult learners to acquire a new base‐10 system, we did not expect such experience to transfer to learning systems with smaller (or larger) bases, since the additive and multiplicative rules in different base systems result in dramatically different structures for the same finite range of numbers. To make this concrete, consider some examples of base‐2, 4, or 5 number systems that participants might encounter in the studies we describe here (see examples 1–4). Like English, these languages use rules to combine numbers additively or multiplicatively, expressing exponential powers of the base using a new basic word (e.g., English 10<sups>2</sups> is <emph>hundred</emph>, 10<sups>3</sups> is <emph>thousand</emph>). A base‐2 system, such as in (<reflink idref="bib1" id="ref66">1</reflink>), expresses "three" compositionally as <emph>2+1</emph>. By contrast, in a base‐4 or base‐5 system, as in (<reflink idref="bib2" id="ref67">2</reflink>) or (<reflink idref="bib3" id="ref68">3</reflink>), it has a dedicated syllable.</p> <p></p> <p> <ephtml> <table><tbody><tr><td align="left">(1) Base‐2</td><td align="left" /></tr><tr><td align="left">  Root Lexemes:</td><td align="left">(1) ka; (2) si; (4) tu<xref ref-type="fn" rid="fn2" />; (8) he</td></tr><tr><td align="left">  Numbers 1–10:</td><td align="left">ka, si, sika, tu, tuka, tusi, tusika, he, heka, hesi</td></tr><tr><td align="left">(2) Base‐4</td><td align="left" /></tr><tr><td align="left">  Root Lexemes:</td><td align="left">(1) ho; (2) na; (3) se; (4) ki</td></tr><tr><td align="left">  Numbers 1–10:</td><td align="left">ho, na, se, ki, kiho, kina, kise, naki, nakiho, nakina</td></tr><tr><td align="left">(3) Base‐5</td><td align="left" /></tr><tr><td align="left">  Root Lexemes:</td><td align="left">(1) ti; (2) ku; (3) so; (4) ne; (5) ha</td></tr><tr><td align="left">  Numbers 1–10:</td><td align="left">ti, ku, so, ne, ha, hati, haku, haso, hane, kuha</td></tr></tbody></table> </ephtml> </p> <p>Notably, systems with different bases vary with respect to the number of root lexemes that a learner must memorize, the number of words that are composite and involve rules, and also the frequency of more complex rules. For example, in a base‐10 system, the numbers 11–19 can be generated by a purely additive rule, by composing the lexical label for 10 (e.g., <emph>zu</emph>) with the labels for 1–9 (e.g., <emph>ka</emph>, <emph>do</emph>, <emph>go</emph>, etc.) to derive a composite numeral (e.g., <emph>zu‐ka</emph>, <emph>zu‐do</emph>, <emph>zu‐go</emph>, etc.). By contrast, a base‐3 system requires learners only to memorize three basic unit labels and one exponential label (for 9), but to compose numbers between 11 and 19 using both multiplicative and additive rules. For example, 16 might be expressed as 9+2*3+1. Nonetheless, learners of either system might be inclined to rote memorize numbers in their training set—compositional or not—and might remain unaware of the generative rule structure governing each system. An important question is whether learners in fact notice and exploit the rule‐governed structure of smaller base systems, and if so, which properties they learn and how they make use of these rules.</p> <hd id="AN0186163275-6">The present study</hd> <p>In Experiment 1, we investigated a small sample of possible number systems that varied in numerical base, which, therefore, also varied in many of the properties described above, such as the prevalence of additive rules or size of lexicon. In the task, participants learned a number system exhibiting one of six different numerical bases (<reflink idref="bib2" id="ref69">2</reflink>, 3, 4, 5, 8, 10). We began by pairing the Arabic numeral "1" with a randomly generated mono‐syllabic novel number word, for example, "ka," and then immediately tested the participant by asking them to type the name for the number 1 in this new language. Next, they saw the labels for 1 and 2 presented in sequence with their Arabic numeral translations, and then were tested on both. This progressive introduction of numbers continued up to the number 10 akin to a counting routine, with the result that smaller number words were trained more frequently than larger ones, resulting in frequencies loosely corresponding to those in natural language (e.g., BNC Consortium, [<reflink idref="bib9" id="ref70">9</reflink>]). This Training Phase exposed participants to a memorized list, but also allowed us to test memory for novel items, and whether memory differed across the different base sizes, and rule structures. However, the Training Phase did not necessarily require learners to notice or acquire the rules that characterized the different systems, since they could choose to simply memorize the sequences, without further analysis. For this reason, we also administered a Generalization Phase, in which we asked participants to generate numbers beyond their training set (e.g., 11, 12). Here, we expected that the relative difficulty of noticing—and then deploying—different rule types might be more apparent in participants' responses, even if it is not obvious from errors in the Training Phase data.[<reflink idref="bib3" id="ref71">3</reflink>]</p> <p>In Experiment 2, we explored two additional questions. First, whereas learners in Experiment 1 were always exposed to numbers in an ordered sequence, analogous to how children are often taught to count, in Experiment 2, we sought to isolate the effect of counting on learning from the effect of being exposed to a compositional rule structure. To do so, we trained participants in three distinct conditions, in which they learned number systems with (<reflink idref="bib1" id="ref72">1</reflink>) compositional syntax and a consistently ordered count list as in Experiment 1, (<reflink idref="bib2" id="ref73">2</reflink>) compositional rules but no consistent order of exposure, or (<reflink idref="bib3" id="ref74">3</reflink>) <emph>no</emph> compositional rules, but a consistently ordered count list. This allowed us to ask whether learning a system in the context of an ordered count routine impacted memory for the training set, but also whether it affected the ability to generate numbers beyond the training set. One possibility, for example, is that adding a count sequence to training improves memory for items and consequently improves the ability to extract rules and generate larger numbers. Alternatively, it may be that adding a count sequence to training supplants the need to extract rules. If so, learners might have good memory of items in the training set, but an impaired ability to generate larger numbers. Finally, in addition to testing these questions, we added a semantic task to Experiment 2, and asked participants to make quantity comparison judgments for trained numbers (i.e., judging which of two numerals represents a larger cardinality). This allowed us to probe whether the properties of different number systems impacted only the learning of grammatical forms, or also impacted the ability to reason about the semantic representations of these forms.</p> <hd id="AN0186163275-7">Experiment 1: Base systems</hd> <p>In Experiment 1, participants were assigned to learn one of six novel number systems that differed with respect to the size of their numerical base, and, therefore, their rule structures. In particular, in Experiment 1a, we tested bases 2, 3, 4, and 5, and in Experiment 1b, we tested bases 8 and 10 in order to explore a more diverse range of memorized versus rule‐based number words. Participants first received a Training Phase in which we taught them labels for numbers in sequence and tested their memory for these labels, and then a Generalization Phase, in which we tested their ability to generate novel labels for numbers up to 15 in Experiment 1a, and 20 in Experiment 1b. Using these methods, we asked (<reflink idref="bib1" id="ref75">1</reflink>) whether some of these numerical bases are more or less learnable than others, given the relative difficulty of learning memorized words and rules, and (<reflink idref="bib2" id="ref76">2</reflink>) how participants generate number words, and whether some rules are easier to extract and deploy than others. For example, if additive rules are easier to learn and use than multiplicative rules, we may expect that learners of a base‐8 system will find it easier to generate numbers up to 20 than learners of a base‐4 system, which requires more frequent use of multiplicative rules.</p> <hd id="AN0186163275-8">Methods</hd> <p>Hypotheses and analyses for Experiments 1 and 2 were preregistered on the Open Science Framework at https://osf.io/rwqk7/. Materials and code for both experiments are available at: https://github.com/SebastianHolt/counting_syntax.</p> <hd id="AN0186163275-9">Participants</hd> <p>Based on a preregistered power analysis, 235 adult participants were recruited online via Prolific, an online crowdsourcing platform commonly used for scientific research. Of these, 53 were excluded due to our preregistered criteria, for example, less than 50% accuracy during the Training Phase, accuracy 3 standard deviations below the mean, or navigating away from the experiment screen on more than five trials. In Experiment 1a, <emph>n</emph> = 31, 30, 32, and 30 participants were recruited for base‐2, 3, 4, and 5, respectively. In Experiment 1b, 30 were recruited for base‐8, and 29 for base‐10. Compensation was targeted at $14/h, though usually exceeded this with a performance bonus (<emph>mean</emph> = $1.90). The most commonly reported first languages were English (<emph>n</emph> = 87), Polish (<emph>n</emph> = 23), Portuguese (<emph>n</emph> = 21), and Spanish (<emph>n</emph> = 13).</p> <hd id="AN0186163275-10">Stimuli</hd> <p>Number systems for each game combined lexical roots via a compositional syntax. Mono‐syllabic roots were randomly generated from consonant‐vowel bigrams (e.g., <emph>ka</emph>) such that no consonant or vowel was used in more than one root per number system. Rules for the different bases were constructed according to the following logic (see Table 2). For each base, <emph>b</emph>, (<reflink idref="bib1" id="ref77">1</reflink>) lexical roots denoted quantities up to <emph>b</emph>, (<reflink idref="bib2" id="ref78">2</reflink>) quantities larger than <emph>b</emph> but not divisible by it were represented by additive rules, where roots were added to base, (<reflink idref="bib3" id="ref79">3</reflink>) quantities of 2 × <emph>b</emph> or greater used multiplication, where the largest compatible unit was placed before the <emph>b</emph> (like digit‐multiplier constructions, such as <emph>six‐dozen</emph>, 6*12, in English), and (<reflink idref="bib4" id="ref80">4</reflink>) quantities of <emph>b</emph> × <emph>b</emph> or greater used an exponential root (like <emph>hundred</emph>, 10<sups>2</sups>), composed like the unit terms. Across the two experiments, there were six bases: 2, 3, 4, 5, 8, and a base‐10 system with 10 roots and no rules.</p> <p>2 Table Sample base‐3 number system</p> <p> <ephtml> <table><thead><tr><th align="left" /><th align="center">Base‐3 example</th><th align="center">English example</th></tr></thead><tbody><tr><td>Additive</td><td>4 is hi‐no (3+1)</td><td>Fifteen = 5 + 10</td></tr><tr><td>Multiplicative</td><td>6 is ka‐hi (2*3)</td><td>Twenty = 2 * 10</td></tr><tr><td>Exponential</td><td>9 (3<sup>2</sup>), se, is a lexical root</td><td>Thousand = 10<sup>3</sup></td></tr><tr><td>Roots (Base‐3)</td><td>1 = no; 2 = ka; 3 = hi; 9 = se</td></tr><tr><td>1−10 (Base‐3)</td><td>no, ka, hi, hino, hika, kahi, kahino, kahika, se, seno</td></tr><tr><td>Roots (Base‐4)</td><td>1 = no; 2 = ka; 3 = hi; 4 = se</td></tr><tr><td>1−10 (Base‐4)</td><td>no, ka, hi, se, seno, seka, sehi, kase, kaseno, kaseka</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>. The compositional rules of an example artificial base‐3 number system, showing how four lexical roots are combined to label quantities up to 10, with reference to analogous rules of composition in English.</p> <hd id="AN0186163275-11">Training phase</hd> <p>After reading an instructions page and passing a quiz that probed their comprehension of the instructions (see Supplementary Materials), participants entered the Training Phase, during which they learned a sequence of 10 novel number words (Fig. 1). Each participant learned a sequence that was generated by one of the six base systems, depending on condition. The first time they encountered a number word, they were explicitly taught its meaning (e.g., the screen read "1 = <emph>ka</emph>"), and they were asked to type the number word in an input field below. Subsequently, every time they were cued with that cardinality, they had to recall the number word from memory (e.g., the screen read "1 = ?"). Before each new numeral was introduced, the participant was tested on all previously trained numbers: First, they were trained and tested on the words for 1 and 2, then they were trained on 3 and tested on 1, 2, and 3, and so on until they reached 10. In this way, training resulted in the memorization of an ordered sequence, in which the frequency of exposure to each number corresponded roughly to their frequency in natural language. At the end of this process, participants rehearsed the sequence one last time, from 1 to 10. Responses for these 64 recall trials (all responses in the Training Phase except the first exposure) were used to assess the relative learnability of the different base systems, in addition to data from the Generalization Phase. To motivate accurate responding, participants received a bonus for accuracy: $0.03 per correct guess on the first try,$0.02 if correct on the second try, and $0.01 correct on the third try. Accuracy and response times were calculated only for the first try, and participants were shown the correct answer after responding incorrectly on all three tries. After submitting a correct response, or after their third try, participants were advanced to the next word in the sequence. They were instructed to complete trials "as quickly and accurately as possible," although they were not given a time limit.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/CGN/01jun25/cogs70071-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="cogs70071-fig-0001.jpg" title="1 Experiment phases. (a) In the Training Phase, participants repeatedly typed a counting sequence, learning one additional number word with each repetition, until they had counted up to 10. (b) In the Generalization Phase, participants typed the entire counting sequence, and then also labeled quantities from 11 to 15 (in Experiment 1a) or from 11 to 20 (Experiment 1b). The sequence of both old and new numbers appeared together on the same screen. (c) Experiment 2 also contained a Magnitude Comparison task, in which participants clicked on the larger of every pair of number words they had trained on." /> </p> <p></p> <hd id="AN0186163275-13">Generalization phase</hd> <p>After completing the Training Phase of the experiment, participants completed the Generalization Phase, wherein they were shown a table with Arabic numerals from 1 to 15 (or 1–20, in Experiment 1b) in the left‐hand column, and empty text‐entry fields on the right. They were then asked to enter number words from the language they had learned in the text entry fields, one at a time. Answers could not be edited after being entered. The last five numbers (i.e., 11–15 in Experiment 1a) or 10 numbers (11–20 in Experiment 1b) required participants to generalize the number system to novel cardinalities that they had not yet encountered.</p> <hd id="AN0186163275-14">Results</hd> <p></p> <hd id="AN0186163275-15">Training phase</hd> <p>To test whether base size impacted the learnability of a counting system, we assessed both accuracy and response time. To measure response time, we recorded the time that elapsed between receiving a prompt and the participant's first keystroke, and added this to the time they spent typing the response, divided by its character length.[<reflink idref="bib4" id="ref81">4</reflink>] Overall, accuracy during the Training Phase was near ceiling across conditions (.957, 95%‐CI: [.953, .96]; see Fig. 2), and the mean response time across conditions was 2.034 s (95%‐CI: [2.002, 2.066]). Accordingly, mean accuracy did not vary much across base conditions (base‐2: m = .96, sd = .04; base‐3: m = .93, sd = .05; base‐4: m = .98, sd = .05; base‐5: m = .94, sd = .07; base‐8: m = .97, sd = .03; base‐10: m = .94, sd = .07), while response time varied more while showing similar qualitative trends (base‐2: m = 1.74, sd = .96; base‐3: m = 2.13, sd = 1.08; base‐4: m = 2.12, sd = .96; base‐5: m = 2.12, sd = .87; base‐8: m = 1.75, sd = .72; base‐10: m = 2.04, sd = .84). Thus, overall participants performed with very high accuracy, resulting in few qualitative differences across conditions.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/CGN/01jun25/cogs70071-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="cogs70071-fig-0002.jpg" title="2 Counting in recall and generalization. Left Panel: Training Phase accuracy for recalling number words (light gray), compared to Generalization Phase accuracy permitting alternative rules or "counting up" strategy (medium gray), and Generalization Phase accuracy permitting alternative rules but not "counting up" (dark gray). Red dots are participant means. Error bars are 95% confidence intervals. Right Panel: Same data, collapsed across bases." /> </p> <p></p> <p>To assess these data statistically, we constructed separate linear mixed effects regressions predicting accuracy and response time from base size, with random intercepts for each participant. We found no effect of base size on response time during the Training Phase in either Experiment 1a (bases 2–5: <emph>b</emph> = 22.29 ms, <emph>t</emph> = .307, <emph>p</emph> = .759) or Experiment 1b (bases 8 and 10: <emph>b</emph> = −26.61, <emph>t</emph> = −.741, <emph>p</emph> = .460). Meanwhile, we found a negative effect of base size on accuracy in Experiment 1b, such that accuracy was poorer for the base‐10 system relative to the base‐8 system (<emph>b</emph> = −.223, <emph>z</emph> = −3.103, <emph>p</emph> = .002), though there was no such effect on accuracy in Experiment 1a (<emph>b</emph> = −.005, <emph>z</emph> = −.046, <emph>p</emph> = .963), or Experiment 1 as a whole (<emph>b</emph> = −.012, <emph>z</emph> = −.315, <emph>p</emph> = .753). At first pass, these results appear to suggest that these different systems are similarly easy to learn. However, as we note below, each system may be similarly easy if learners approach the task purely as one of rote memorization, since the same number of words was learned in each base system. If participants instead extract rules from training, then two measures may distinguish base systems from each other. First, the specific rules of composition within each base system—such as additive or multiplicative rules—may be more or less easy for participants to learn. Second, participants learning different base systems may differ in their ability to generalize to new exemplars, where knowledge of rule structures is more important.</p> <p>We, therefore, first asked whether different syntactic rules were easier or harder for participants in the Learning Phase. To do so, we conducted a post hoc analysis of data from the Training Phase to test whether participants' responses were faster or more accurate when the number words they learned contained more (or less) compositional rules. We coded each number word for the presence of an additive rule, a multiplicative rule, or a lexical root that represented an exponential meaning (i.e., 4 in the base‐2 condition and 9 in base‐3). These codes were then entered as predictors in a regression that also included each word's length in syllables, as well as which base system it belonged to. This analysis found that the presence of multiplicative rules, but not additive rules, was significantly related to both increased response time (<emph>b</emph> = 580.21, <emph>t</emph> = 3.89, <emph>p</emph><.001), and lower accuracy (<emph>b</emph> = −.929, <emph>t</emph> = −1.966, <emph>p</emph> = .049). The greater difficulty of expressions containing multiplicative rules suggests that those words were harder to learn than words composed from additive rules (Fig. 3), and, therefore, that at least also some learners were impacted by the presence of rules (whether they were aware of them or not).[<reflink idref="bib5" id="ref82">5</reflink>] At the suggestion of an anonymous reviewer, we conducted an additional post hoc analysis in which we restricted data to only those pairs of expressions in which one member of the pair was composed of two numbers added together (e.g., in base‐3, the number 5 was composed as 3+2), and the other member was composed of those same numbers multiplied together (e.g., 6 is 2*3). This allowed for a more tightly controlled test of additive versus multiplicative rules, as the only difference between each minimal pair was the additive or multiplicative rule itself. One such pair of numbers existed in each of the base‐3 (3+2 and 2*3), base‐4 (4+2 and 2*4), and base‐5 (5+2 and 2*5) conditions, but not the other bases. We removed irrelevant predictors from the above linear models, leaving only base size (<reflink idref="bib3" id="ref83">3</reflink>, 4, or 5), and additive versus multiplicative rule type. Responses to the multiplicative forms were slower (<emph>b</emph> = 652.599, <emph>t</emph> = 4.462, <emph>p</emph><.001) though not significantly less accurate (<emph>b</emph> = −0.437, <emph>z</emph> = −1.636, <emph>p</emph> = .102) than additive forms. This result is consistent with our primary analyses, which suggest that expressions containing multiplicative rules were more difficult to learn than those that contained only additive rules.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/CGN/01jun25/cogs70071-fig-0003.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="cogs70071-fig-0003.jpg" title="3 Use of different rule‐types. (a) Word recall accuracy (left) and response time measures (right) in the Training Phase of Experiment 1, for number words featuring no compositional rules (i.e., lexical roots), additive rules, and multiplicative rules. Red dots are participant means. (b) Strategy use in the Generalization Phase. Light gray indicates the proportion of number words that would be expected to show additive and multiplicative rules of composition for the extended range of numbers (11–15 for Experiment 1a, or 11–20 for 1b) under the rules used to generate each number system. Dark gray indicates the proportion of actual participant responses that could be parsed as using additive or multiplicative rules in that range." /> </p> <p></p> <hd id="AN0186163275-18">Generalization phase</hd> <p>Data from the Training Phase assessed participants' ability to recall words they had just learned, and provided only a preliminary test of how they mentally represented the rules of each system. Although rules were available to be learned in most systems, learners could nevertheless ignore these rules and learn the number words by rote. To explore this question, we, therefore, analyzed data from the Generalization Phase.</p> <p>Participants' Generalization Phase accuracy for Old Numbers (learned in Training Phase) was perfect (100%), confirming that they had learned the number system up to 10. Our first question was how many participants produced an "Exact Match" to target generalizations (i.e., forms that would be expected if participants had learned the rules that were used to generate their training set). For example, if a participant in the base‐4 condition had learned <emph>ho</emph>, <emph>se</emph>, <emph>na</emph>, and <emph>ki</emph> to represent the numbers 1–4, then an exact match for the number 13 during generalization would be <emph>na‐ki‐ho</emph> (3*4+1). Of 182 participants across the different base conditions, 17 (9%) generalized the number system exactly in accordance with those rules through the entire generalization range (up to 15 in Experiment 1a, or up to 20 in 1b). Across participants, 37.7% of responses (454 out of 1205) were an exact match to the expected forms (CI: [0.349, .404]). This suggests that, despite being fully numerate users of a base‐10 system featuring additive and multiplicative rules, most adult participants in our study did not learn the full set of additive, multiplicative, and exponential rules that were used to generate the number systems, even when trained in a base‐10 system. A post hoc binomial linear regression with base size as a predictor, random intercepts for participants, and a random slope for number revealed a significant effect of base size on the probability that participants would provide an exact match on each trial (<emph>b</emph> = 0.435, <emph>z</emph> = 2.057, <emph>p</emph> = .04). This effect was driven by the base‐8 condition and disappeared when base‐8 data were removed from our analysis (<emph>b</emph> = 0.162, <emph>z</emph> = 0.848, <emph>p</emph> = .396). This advantage for base‐8 was likely due to the fact that, in this case, an alternative "counting up" strategy could generate the same responses as the target rules for all numbers from 11 to 15. Having been explicitly taught 8+1 and 8+2, base‐8 participants could continue this pattern until 8+7.[<reflink idref="bib6" id="ref84">6</reflink>]</p> <p>Note that although most participants did not make Exact Match responses, this does not mean that most failed to use rules. Instead, participants may have constructed rules that deviated from those in their training set. We, therefore, asked whether participants may have constructed alternative rule systems. For example, if trained in the <emph>ho</emph>, <emph>se</emph>, <emph>na</emph>, <emph>ki</emph> base‐4 system, participants may have created a purely additive expression for 13, such as <emph>ki‐ki‐ki‐ho</emph> (4+4+4+1) or <emph>na‐na‐se‐ki‐ho</emph> (3+3+2+4+1), different from the target label but still mathematically coherent. They also may have created multiplicative expressions that differed from target responses, featuring both additive and multiplicative rules of composition, such as <emph>se‐ki‐ki‐ho</emph> (2*4+4+1). We, therefore, coded participants' responses by first identifying substrings that were part of the number system they had learned (e.g., <emph>ki</emph> or <emph>ki‐se</emph>) and then inferring rules of composition based on which parse produced the number closest to the target quantity. For example, while the response <emph>se‐ki‐ki‐ho</emph> is ambiguous between a multiplicative (2*4+4+1 = 13) or purely additive parse (2+4+4+1 = 11), if it was produced as a label for the number 13, then we identified it as multiplicative because only this matched the target number. Because in natural language (and in the artificial language stimuli of this experiment), multiplicative rules of composition typically place the smaller number expression before the larger number (e.g., Mandarin <emph>èrshí</emph>, "two‐ten," is 2*10) rather than vice‐versa (e.g., <emph>shíèr</emph>, "ten‐two," is 10+2), we only entertained multiplicative compositions for bigrams in which the first element denoted a smaller number than the second element.</p> <p>Using this framework, we found that under a more Expanded Additive coding scheme, which included the Exact Match scheme as well as the above set of additive rules, participants produced responses that could be considered to be a correct product of rules on 53% of trials (CI: [.506, .562]), which was significantly greater than the 37.7% for the Exact Match scheme (χ<sups>2</sups>(<reflink idref="bib2" id="ref85">2</reflink>, _I_N_i_ = 1205 trials) = 120.405, <emph>p</emph> < .001). Meanwhile, we found that under the more permissive Expanded Multiplicative + Additive scheme, participants produced correct responses 58.9% of the time (CI: [.561, .617]), significantly greater than under the Expanded Additive scheme (χ<sups>2</sups>(<reflink idref="bib2" id="ref86">2</reflink>, 1205 trials) = 14.529, <emph>p</emph><.001), and, therefore, also greater than the Exact Match scheme.[<reflink idref="bib7" id="ref87">7</reflink>] To better understand this result, we considered how often we might expect participants' responses to fall into these categories by chance. For every label generated by a participant, we asked how likely it was that a random string of the same length would have been marked as correct under the same coding scheme. For example, if a base‐4 participant's label for 11 was formed by combining "2*4+3," this was coded as consistent with the Expanded Multiplicative + Additive scheme. To assess chance, we asked how many randomly generated trigrams from the base‐4 lexicon could also be interpreted as 11, using additive or multiplicative rules. For responses marked under Expanded Additive, we computed chance to be 6.9% on average (highest in base‐3 at 11.1%, lowest in base‐10 at 5.2%), and 8.4% for Expanded Additive+Multiplicative (highest in base‐3 at 10.7%, lowest in base‐10 at 6.3%). These rates thus tended to be much lower than the number of responses assigned to each strategy within our coding scheme.</p> <p>Although base‐10 knowledge appeared to be of little help in learning the target rules of novel base systems, it may nevertheless have influenced how learners generalized when using these alternative rule systems. For example, in the "Expanded Additive" and "Counting Up" strategies, it is possible that some learners may have directly calqued Arabic numerals by associating each digit with a number word. In a base‐2 system, for example, a learner might express 13 using the words for "one" (a lexical root) and "three" (a composition of 2‐1), resulting in an expression glossed as "1‐2‐1." This occurred quite rarely in base‐2 (3.2%, 95%‐CI: [0.4, 6.0]), base‐3 (5.3%, CI: [1.7, 8.9]), base‐4 (3.1%, CI: [0.4, 5.8]), base‐5 (4.0%, CI: [0.9, 7.1]), and base‐8 (0.0%, CI: [0.0, 0.0]), though was somewhat common in base‐10 (29.7%, CI: [24.4, 34.9]). Another influence from base‐10 may have been responses that combined the expression for 10 (e.g., "8‐2" in the base‐2 system) with unit words, for example, expressing 13 in the base‐2 system as "(8‐2)‐(2‐1)," that is, "10‐3." This occurred somewhat more often in base‐2 (31.6%, CI: [24.3, 39.0]), but less in base‐3 (16.7%, CI: [10.7, 22.7]), and less still in base‐4 (10.0%, CI: [5.3, 14.7]), base‐5 (0.0%, CI: [0.0, 0.0]), base‐8 (4.0%, CI: [1.8, 6.2]), and base‐10 (0.0%, CI: [0.0, 0.0]). Thus, we found that learners sometimes adopted base‐10 strategies, and that when they did, both the prevalence and type of base‐10 strategy differed across systems, and occurred most frequently but in different ways for the smallest and largest bases.</p> <p>Our next question was whether participants who employed alternative strategies were more or less likely to over‐rely on additive rules (as opposed to multiplicative rules) relative to learners who exactly matched the target rules. To examine this, we compared the frequency of these rule types under the two expanded coding schemes just described to their prescribed frequency in the Exact Match scheme. For example, according to the Exact Match scheme, in the base‐8 system, nine numbers between 11 and 20 feature an additive rule (i.e., 11–15 and 17–20), while five numbers feature a multiplicative rule (i.e., 16–20). Relative to an Exact Match scenario, we found that participants showed a stronger bias to use more additive rules than multiplicative rules for creating new number words (χ<sups>2</sups>(<reflink idref="bib3" id="ref88">3</reflink>, _I_N_i_ = 1205 trials) = 143.625, <emph>p</emph> < .001). This result is compatible with the earlier conclusion that additive rules may be easier to learn than multiplicative ones, and also that they may be easier to deploy in generalization.</p> <p>In addition to considering these alternative additive and multiplicative systems, we also asked whether participants might have used a simpler alternative that involved counting up. For example, consider a base‐4 participant who generated the label <emph>se‐ki‐ho</emph> (2*4+1) to represent 9. Remembering that in the counting sequence, <emph>ho</emph> is followed by <emph>se</emph>, this participant might represent 10 by "counting up" and replacing the syllable <emph>ho</emph> with <emph>se</emph>—for example, <emph>se‐ki‐se</emph> (2*4+2). Although such a heuristic would produce responses consistent with the Exact Match scheme in most cases, it may also sometimes generate incorrect answers. For example, instead of correctly counting up from <emph>ho</emph> in the previous example, a participant might instead count up from <emph>se</emph>, resulting in the expression <emph>na‐ki‐ho</emph> to represent 10 (which in fact adds to 8). In order to explore alternative rule uses, we coded cases like this—that is, counting up from any component of a complex expression—as correct responses in the Counting Up coding scheme. Also, although in the example just provided <emph>na‐ki‐ho</emph> was produced by counting up from a correct response of <emph>se‐ki‐ho</emph>, we also considered cases where participants counted up from incorrect responses. So, if they replied <emph>ho‐ho</emph> (1+1) when prompted with 4 and then provided <emph>ho‐se</emph> for 5, we also coded this as correct. Under this scheme, accuracy was 79.3% (CI: [.77, .815]), which was higher than in the Expanded Multiplicative + Additive scheme (χ<sups>2</sups>(<reflink idref="bib2" id="ref89">2</reflink>, _I_N_i_ = 1205 trials) = 302.953, <emph>p</emph> < .001) and, therefore, also higher than in the Additive and Exact Match schemes. This finding suggests that at least some participants created novel numbers using a simple counting heuristic, rather than additive or multiplicative rules of composition.</p> <hd id="AN0186163275-19">Discussion</hd> <p>In Experiment 1, we found five main results. First, we found that participants readily recalled the number words that they were trained on, with no important differences between small and large bases. Second, we found that participants were both faster and more accurate at recalling number words in their training set when they were composed of additive rules versus multiplicative rules. Third, we found that when participants were asked to produce numbers beyond their training set, they produced forms that exactly matched the target rules of their training set on just over 1/3 of trials, though only 9% of participants did so consistently. While we found a post hoc effect of base size in the generalization phase, this appeared to be driven by participants in the base‐8 condition, where ad hoc rules, like counting up, mirrored target rules. Fourth, compatible with this last finding, we found that participants often created their own, alternative, rules to generalize beyond their training set. They created labels consistent with the use of an expanded set of additive rules on just over 50% of trials, and consistent with a combination of multiplicative and additive rules on nearly 60% of trials. When we also coded "counting up" responses as correct, nearly 80% of all participant responses reflected some kind of rule structure. Finally, consistent with the finding that expressions with additive rules may have been easier for participants to learn, when participants created novel rules, these were more likely than expected to be additive than multiplicative.</p> <p>Overall, these results suggest that participants were very good at rote learning of diverse counting systems, that many learned target rules, and that most participants imposed some kind of rule‐like structure on their input when asked to generalize. Many used some form of compositional system, while others used a simpler counting‐up strategy. Finally, despite being fully numerate, most participants learned systems that did not directly reflect existing base‐10 knowledge. When they did so, the kinds of base‐10 knowledge they applied to this problem also differed between bases.</p> <p>In Experiment 1, counting systems were always presented in a sequential order, akin to a counting routine, together with a compositional rule structure. In the Introduction, we noted that the presence of a counting routine might enhance the ability to rote memorize a number sequence, potentially reducing the need to derive rule structures. Given this, and given the finding that many participants sometimes used counting‐up in lieu of compositional rules, a natural question is whether there might be competition between these learning strategies, and whether learning a number system that is embedded in a sequential counting procedure might inhibit the acquisition of compositional rules. To explore this question, in Experiment 2, we asked how learning number words as part of an ordered counting sequence impacts learning. In particular, we asked whether counting routines facilitate learning or if they may instead inhibit the ability to extract compositional rules.</p> <hd id="AN0186163275-20">Experiment 2</hd> <p>In Experiment 1, we investigated the learnability of different base systems, all of which were trained as part of a sequence that was ordered in terms of magnitude. On one hand, the presentation of number words in an ordered counting sequence may be important to learning the meanings of number words, since their meanings are defined by their ordinal relations to one another. On the other hand, it is possible that although children in the West are typically taught number words as part of an ordered procedure, this ordering is not actually essential to learning their meanings, or to extracting rules for creating new numbers. Instead, learners may be able to learn these rules just as they acquire many other grammatical rules, via cross‐situational learning that includes many distinct tokens of words across different utterances (Akhtar & Montague, [<reflink idref="bib1" id="ref90">1</reflink>]; Fisher, Hall, Rakowitz, & Gleitman, [<reflink idref="bib24" id="ref91">24</reflink>]; Gleitman, [<reflink idref="bib31" id="ref92">31</reflink>]; Pinker, [<reflink idref="bib61" id="ref93">61</reflink>]; Aslin & Newport, [<reflink idref="bib3" id="ref94">3</reflink>]). Also, as noted in the Introduction—and suggested by the results of Experiment 1—it is possible that the opportunity to memorize numbers as part of a sequence may even supplant the need to learn rules, therefore, interfering with rule learning.</p> <p>In Experiment 2, we explored this question by comparing three groups of participants presented with number systems featuring (<reflink idref="bib1" id="ref95">1</reflink>) compositional rules and a consistently ordered count list, (<reflink idref="bib2" id="ref96">2</reflink>) compositional rules but no consistent order of exposure, and (<reflink idref="bib3" id="ref97">3</reflink>) no compositional rules, but a consistently ordered count list. As part of testing this question, Experiment 2 also included a magnitude comparison task in which participants were presented two words from the novel system and were asked to judge which word represented a larger amount. This allowed us to not only probe participants' acquisition of combinatorial rules, but also their acquisition of number word meanings, and whether this learning was impacted by exposure to compositional rules or a sequential counting procedure.</p> <hd id="AN0186163275-21">Methods</hd> <p>Experiment 2 was preregistered on the Open Science Framework (for link, see Methods, Experiment 1).</p> <hd id="AN0186163275-22">Participants</hd> <p>We recruited 119 adult participants via Prolific, of whom 28 were excluded due to our preregistered criteria, for example, less than 50% accuracy during the Training Phase, or accuracy 3 standard deviations below the mean. Samples were <emph>N</emph> = 30, 31, and 30 in the Order+Rules, Order‐Only, and Rules‐Only conditions, respectively. Compensation was targeted at $14/h + performance bonus (<emph>mean</emph> = $2.36). The most commonly reported first languages were English (<emph>n</emph> = 38), Polish (<emph>n</emph> = 16), Portuguese (<emph>n</emph> = 10), and Spanish (<emph>n</emph> = 7).</p> <hd id="AN0186163275-23">Training phase</hd> <p>Because Experiment 1 found no consistent effects of base size, all participants in Experiment 2 were taught a base‐5 system, selected because it exhibits combinatorial structure over half the count list (from 1 to 10), and requires only additive and multiplicative rules (no exponents) to label any number up to 20 (the highest cardinality appearing in the experiment). In order to independently manipulate the presence of combinatorial rules and an ordered counting procedure, we divided participants into three conditions. In Condition 1 (Order+Rules), number words featured compositional rules, and were trained as a consistent, ordered list, as in Experiment 1. In Condition 2 (Order‐Only), words were trained in an ordered list, but the words generated via compositional rules were remapped to an arbitrary, different, cardinality within the count list. In other words, these combinations of syllables did not correspond to meaningful units (morphemes) and, therefore, were arbitrary, and could not be analyzed via rules. For example, if the lexicon learned in the Order+Rules condition was {1 = <emph>ni</emph>, 2 = <emph>se</emph>, 3 = <emph>ta</emph>, 4 = <emph>ho</emph>, 5 = <emph>ku</emph>, 6 = <emph>ku‐ni</emph>, etc.}, the lexicon learned in the Order‐Only condition might be {1 = <emph>ku</emph>, 2 = <emph>ta</emph>, 3 = <emph>ku‐ni</emph>, 4 = <emph>ni</emph>, 5 = <emph>ho</emph>, 6 = <emph>se</emph>, etc.}. In Condition 3 (Rules‐Only), the lexicon was identical to the Order+Rules condition, but the words were trained in a random order (i.e., not in ascending order of magnitude). Critically, the frequency of each number word in this condition was matched with the synonymous word in the Order+Rules condition, such that the word for 1 was most frequent, followed by the word meaning 2, then 3, and so on.</p> <hd id="AN0186163275-24">Generalization phase</hd> <p>The Generalization Phase was identical to Experiment 1b, and tested numbers from 11 to 20.</p> <hd id="AN0186163275-25">Magnitude comparison</hd> <p>After the Generalization Phase, participants completed the Magnitude Comparison task, in which they compared the magnitudes of number words they had learned. In each trial, they were presented with a pair of unequal number words from the numbers they had just learned, and asked to click on the one that represented the "bigger" value (Fig. 1c). Each trial had a time limit of 4 s, during which a progress bar at the top of the screen shrank to indicate remaining time. Over the task, all 45 combinations of number words were queried.</p> <hd id="AN0186163275-26">Results</hd> <p></p> <hd id="AN0186163275-27">Training phase</hd> <p>Accuracy during the Training Phase was comparable to Experiment 1 (.939, 95%‐CI: [.932, .946]), as were mean response times 2.230 s (95%‐CI: [2.167, 2.293]). Accuracy was generally high across all three conditions, though important differences existed, as discussed below (Rules+Order: m = .96 sd = .02; Rules‐Only: m = .91, sd = .05; Order‐Only: m = .95, sd = .02). Response times also exhibited some variability (Rules+Order: m = 1.73, sd = .79; Rules‐Only: m = 2.76, sd = .98; Order‐Only: m = 2.20, sd = .99). To test the relative contributions of rules and order on recall of learned words, we constructed mixed effects linear regressions predicting accuracy and response time from task condition, target number, cumulative frequency of that number, an interaction between condition and frequency, and random intercepts per participant. Participants in the Rules‐Only condition were significantly slower (<emph>b</emph> = 1404.78, <emph>t</emph> = 5.493, <emph>p</emph><.001) and less accurate (<emph>b</emph> = −1.57, <emph>z</emph> = −3.238, <emph>p</emph> = .001) in recalling trained number words than in the Order+Rules condition, suggesting that learning an ordered count sequence helped participants remember words that they were explicitly trained on. Compatible with this, participants in the Order‐Only condition were just as accurate as those in the Order+Rules condition (<emph>b</emph> = −.469, <emph>z</emph> = −.945, <emph>p</emph> = .345; Fig. 4a), suggesting that the addition of compositional rules did not improve recall of trained words. Nonetheless, participants in the Order+Rules condition were slightly faster to respond than those in the Order‐Only condition, suggesting that compositional rules may slightly facilitate recall (<emph>b</emph> = 688.78, <emph>t</emph> = 2.716, <emph>p</emph> = .008). This model also showed that participants were less accurate when responding to larger numbers (<emph>b</emph> = −0.141, <emph>z</emph> = −4.84, <emph>p</emph><.001), although they also made their responses more quickly (<emph>b</emph> = −44.16, <emph>t</emph> = −3.282, <emph>p</emph> = .001). While participants did not generally become more accurate with increased exposure (<emph>b</emph> = −0.011, <emph>z</emph> = −0.191, <emph>p</emph> = .849), they did become faster to respond (<emph>b</emph> = −50.61, <emph>t</emph> = −2.257, <emph>p</emph> = .024). With increased exposure, participants in the Order‐Only condition were especially faster than in the Order+Rules condition (<emph>b</emph> = −78.74, <emph>t</emph> = −2.615, <emph>p</emph> = .009) but not more accurate (<emph>b</emph> = 0.114, <emph>z</emph> = 1.414, <emph>p</emph> = .158), whereas those in the Rules‐Only condition were both faster (<emph>b</emph> = −130.13, <emph>t</emph> = −4.284, <emph>p</emph><.001) and more accurate (<emph>b</emph> = 0.44, <emph>z</emph> = 5.282, <emph>p</emph><.001). Together, these results suggest that—at least for numbers within a training set—learners benefited from learning number words in an ordered counting sequence and that the presence of transparent rules may have helped learners to recall those words faster, but not necessarily more accurately. While increased exposure (i.e., cumulative frequency) did not always predict higher accuracy, this may be due to the structure of trials in both the Order‐Only and the Order+Rules conditions. Because the counting procedure increased in length each time it was repeated (i.e., 12, 123, 1234, etc.), successive trials of each number were also interrupted by a longer interval of trials for different numbers.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/CGN/01jun25/cogs70071-fig-0004.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="cogs70071-fig-0004.jpg" title="4 Counting and rules in number‐word learning. (a) When participants learn numbers in a counting procedure (O+R and Order), they recall words better (black) but make less accurate judgments of magnitude (red). (b) Participants learning numbers in a stable counting order also recall number words faster (black), but do not judge the magnitude of numbers faster (red). Dotted line in each panel represents average of the base‐5 condition from Experiment 1. Dots in both panels show mean Magnitude Comparison performance of individual participants." /> </p> <p></p> <hd id="AN0186163275-29">Generalization phase</hd> <p>Although data from the Training Phase address the impact of rules and order on learning trained items, they do not address how these factors impact the acquisition of rules that can be used to generate and comprehend larger numbers. Therefore, as in Experiment 1, we asked whether participants generalized the number system to the range 11–20 using expressions exactly matching our algorithm for producing artificial number words, whether they used other additive or multiplicative rules, and whether they showed signs of a counting up strategy. Because the system learned by participants in the Order‐Only condition featured no compositional rules, these analyses could only be meaningfully applied to the Rules‐Only and Order+Rules conditions. In these two conditions, exact matches to the target rules occurred on 35.8% of 600 trials (CI: [.32, .397]), comparable to Experiment 1. Participants' rate of success did not increase under an Expanded Additive coding scheme (.362, CI: [.323, .4]; χ<sups>2</sups>(<reflink idref="bib2" id="ref98">2</reflink>, _I_N_i_ = 600 trials) = 0.029, <emph>p</emph> = .865), but reached 53.3% under the Expanded Multiplicative + Additive scheme, which was significantly greater than under the Expanded Additive scheme (.533, CI: [.493, .573]; χ<sups>2</sups>(<reflink idref="bib2" id="ref99">2</reflink>, _I_N_i_ = 600 trials) = 76.589, <emph>p</emph> < .001). Accuracy was higher still, at 74.3%, when also allowing a counting up strategy (CI: [.708, .778]; χ<sups>2</sups>(<reflink idref="bib2" id="ref100">2</reflink>, _I_N_i_ = 600 trials) = 138.687, <emph>p</emph> < .001). Across these analyses, we found no effect of condition, suggesting that learning number words in a sequential order neither helped nor hindered participants' acquisition of compositional rules.[<reflink idref="bib8" id="ref101">8</reflink>]</p> <hd id="AN0186163275-30">Magnitude comparison task</hd> <p>The results presented thus far suggest that exposure to an ordered counting sequence helps learners to recall number words, but not to extract rules that generalize to novel forms. Left open by these analyses of how participants learn compositional rules is whether differences in training conditions impacted how the <emph>meanings</emph> of number words are acquired. To explore this question, we constructed mixed effects regressions to predict accuracy and response time on the Magnitude Comparison task, which required participants to reason about the magnitudes that novel number words represented. This included terms for condition (Rules‐Only, Order+Rules, Order‐Only), the sum of the compared numbers (a+b), the interaction between condition and sum, as well as intercepts for each participant.[<reflink idref="bib9" id="ref102">9</reflink>]</p> <p>These models revealed that participants in the Order+Rules condition were faster (<emph>b</emph> = −1218.046, <emph>t</emph> = −6.481, <emph>p</emph><.001) and more accurate (<emph>b</emph> = 2.019, <emph>z</emph> = 4.171, <emph>p</emph><.001) than those in the Order‐Only condition. Participants in the Rules‐Only condition were even more accurate than those in the Order+Rules condition (<emph>b</emph> = 2.586, <emph>z</emph> = 4.299, <emph>p</emph><.001), though not significantly faster (<emph>b</emph> = −132.906, <emph>t</emph> = −0.692, <emph>p</emph> = .49). Thus, these models showed a marked difference in performance between conditions, and that participants were generally less accurate when trained with an ordered counting sequence, while providing responses with equal speed. They also revealed that the magnitude of compared numbers impacted performance. In the Order+Rules condition, larger summed cardinality of the compared numbers predicted lower accuracy (<emph>b</emph> = −0.107, <emph>z</emph> = −6.203, <emph>p</emph><.001) and longer response times (<emph>b</emph> = 13.292, <emph>t</emph> = 2.048, <emph>p</emph> = .041). This was also observed in the Rules‐Only condition: lower accuracy (<emph>b</emph> = −0.233, <emph>z</emph> = −8.862, <emph>p</emph><.001) and longer response times (<emph>b</emph> = 19.722, <emph>t</emph> = 2.889, <emph>p</emph> = .004). It was only in the Order‐Only condition that larger summed cardinality predicted faster response times (<emph>b</emph> = −25.504, <emph>t</emph> = −4.004, <emph>p</emph><.001), though it did not predict significant differences in accuracy (<emph>b</emph> = −0.016, <emph>z</emph> = −1.133, <emph>p</emph> = .257). Thus, overall it seems that participants showed lower accuracy for larger numbers. However, our interpretation of the summed cardinality effect is limited by a nonlinear relationship between cardinalities and word lengths that existed only in the Order+Rules and Rules‐Only conditions: pairs of small number words had the same word length (i.e., one syllable), as did pairs of large number words (i.e., two syllables), but not pairs containing both a small and a large number, which also summed to cardinalities in between small‐small and large‐large pairs. We return to the question of word length below.</p> <p>The previous analyses indicated that participants made Magnitude Comparison judgments differently across conditions. We next asked whether these judgments were compatible with a numerical distance effect, in which magnitude judgments are faster and more accurate for numbers that are farther from each other on a number line than for numbers that are close to each other (e.g., 1 and 9 are easier to distinguish than 8 and 9). To do this, we constructed a set of linear mixed effects models predicting accuracy and response time from condition, the sum of numbers in each pair (<emph>a+b</emph>), and the numerical distance between them (<emph>a</emph>−<emph>b</emph>), with a random effect of participant. This model showed qualitatively similar estimates for the main effect of each condition, and a main effect of summed cardinality consistent with the average of the effects reported above—that is, no overall effect on reaction time (<emph>b</emph> = 0.173, <emph>t</emph> = 0.047, <emph>p</emph> = .963), and a negative effect of summed cardinality on accuracy (<emph>b</emph> = −0.077, <emph>z</emph> = −7.88, <emph>p</emph><.001). It also revealed a main effect of numerical distance, in which larger differences between each pair of numbers resulted in faster responses (<emph>b</emph> = −106.55, <emph>t</emph> = −15.85, <emph>p</emph><.001) that were also more accurate (<emph>b</emph> = 0.507, <emph>z</emph> = 19.095, <emph>p</emph><.001).</p> <p>Observing this, we asked whether this numerical distance effect differed between conditions, and if so, why. Previous studies (e.g., Krajcsi, Lengyel, & Kojouharova, [<reflink idref="bib42" id="ref103">42</reflink>]) suggest that the sources of the numerical distance effect may differ between judgments of perceptual stimuli (e.g., dot arrays) and symbolic representations (e.g., Arabic numeral digits). The methods used here differ from previous studies in two important ways, which we address in turn. First, the number words in two of the three conditions—Order+Rules and Order‐Only—were learned in a counting sequence. We, therefore, asked whether the distance effect was impacted by the presence of this counting procedure. Second, the symbolic numbers compared in this task varied in length between one and two syllables, and when pairs of words differed in length, choosing the longer word was a reliable heuristic for choosing the larger number only in the Order+Rules and the Rules‐Only condition. We, therefore, asked whether participants were more likely to show a numerical distance effect in the rules‐based conditions, where the length of each number word being compared could also serve as a heuristic for magnitude. Finally, we asked whether the use of such a heuristic would fully account for any distance effect in these groups, or whether other characteristics of our artificial stimuli (e.g., being symbols, being counted) might also mediate the effect.</p> <p>To explore these questions, we added an interaction term between numerical distance and task condition to the above model. This revealed that numerical distance predicted both higher accuracy and faster reaction time across conditions. We found greater accuracy for greater numerical distances for Order+Rules (<emph>b</emph> = 0.825, <emph>z</emph> = 12.42, <emph>p</emph><.001), Rules‐Only (<emph>b</emph> = 0.742, <emph>z</emph> = 8.707, <emph>p</emph><.001), and Order‐Only (<emph>b</emph> = 0.353, <emph>z</emph> = 11.293, <emph>p</emph><.001). And we found faster reaction times for greater numerical distances for Order+Rules (<emph>b</emph> = −149.11, <emph>t</emph> = −12.796, <emph>p</emph><.001), Rules‐Only (<emph>b</emph> = −132.35, <emph>t</emph> = −10.813, <emph>p</emph><.001), and Order‐Only (<emph>b</emph> = −47.925, <emph>t</emph> = −4.36, <emph>p</emph><.001). These results suggest that participants can learn to associate novel words with numerical magnitudes, both in the presence of syntactic rules or a counting procedure. Further, the numerical distance effect was strongest in the Order+Rules condition and not significantly weaker in the Rules‐Only condition for accuracy (<emph>b</emph> = −0.083, <emph>z</emph> = −0.771, <emph>p</emph> = .441) or response time (<emph>b</emph> = 16.76, <emph>t</emph> = 0.992, <emph>p</emph> = .321), suggesting that a counting sequence did not contribute much to the numerical distance effect over and above syntax. The numerical distance effect was also significantly weaker in the Order‐Only condition compared to the Rules‐Only condition for both accuracy (<emph>b</emph> = −0.388, <emph>z</emph> = −4.28, <emph>p</emph><.001) and response time (<emph>b</emph> = 84.425, <emph>t</emph> = 5.131, <emph>p</emph><.001), and, therefore, also weaker than the Order+Rules condition. This suggests that rules were more helpful than a counting procedure for learning the numerical content of novel number words.</p> <p>We next asked what accounted for the numerical distance effect in the Order+Rules and Rules‐Only conditions. In both conditions, many pairs of numbers could be successfully distinguished by comparing the length of words—any two‐syllable number word always encoded a larger value than a one‐syllable number word. To assess this, we added one further predictor to the original model (condition + cardinality + distance), which was whether the pair of number words on each trial differed in length by 1 syllable, or contained the same number of syllables. Overall, we found that although word length was a significant predictor of performance, it did not fully explain the numerical distance effect. Adding word length improved model fit for both accuracy (χ<sups>2</sups> = 102.35, <emph>p</emph><.001) and reaction time (χ<sups>2</sups> = 143.53, <emph>p</emph><.001), and predicted higher accuracy for both Order+Rules (<emph>b</emph> = 2.412, <emph>z</emph> = 7.142, <emph>p</emph><.001) and Rules‐Only (<emph>b</emph> = 3.716, <emph>z</emph> = 5.027, <emph>p</emph><.001) conditions, with a smaller positive effect in the Order‐Only condition (<emph>b</emph> = 0.365, <emph>z</emph> = 3.243, <emph>p</emph> = .001).[<reflink idref="bib10" id="ref104">10</reflink>] Word length also predicted faster reaction time in Order+Rules (<emph>b</emph> = −781.54, <emph>t</emph> = −10.893, <emph>p</emph><.001) and Rules‐Only conditions (<emph>b</emph> = −811.69, <emph>t</emph> = −11.126, <emph>p</emph><.001), but not in the Order‐Only condition (<emph>b</emph> = −4.81, <emph>t</emph> = −0.105, <emph>p</emph> = .917). However, numerical distance by itself continued to predict higher accuracy in the Order‐Only condition (<emph>b</emph> = 0.348, <emph>z</emph> = 11.151, <emph>p</emph><.001) as well as faster response times (<emph>b</emph> = −47.92, <emph>t</emph> = −4.456, <emph>p</emph><.001). Numerical distance also continued to predict higher accuracy in the Order+Rules (<emph>b</emph> = 0.502, <emph>z</emph> = 6.219, <emph>p</emph><.001) and Rules‐Only (<emph>b</emph> = 0.338, <emph>z</emph> = 3.076, <emph>p</emph> = .002) conditions, but it no longer significantly predicted faster reaction times in the Order+Rules (<emph>b</emph> = −24.75, <emph>t</emph> = −1.534, <emph>p</emph> = .125) or Rules‐Only (<emph>b</emph> = −4.87, <emph>t</emph> = −0.294, <emph>p</emph> = .769) conditions.</p> <hd id="AN0186163275-31">General discussion</hd> <p>In two experiments, we trained numerate adults on a series of novel number systems and manipulated the availability of rule structures and ordered count sequences. This approach produced several important results. In Experiment 1, we found that base size had little effect on participants' ability to memorize number words in their training set. Also, generalization data from Experiment 1 showed that less than 10% of participants extracted rules matching those in their training, with little impact of base size on performance. While we found no overall effect of base size, we did find that associated features (like the presence of rules) did affect the learnability of each system. For example, generalization data also revealed that additive rules were easier to learn than multiplicative rules, and that participants often imposed novel structures onto their input, including additive, multiplicative, and counting‐up strategies, in ways that varied with the base system they had learned. Thus, despite being numerate adults with prior exposure to both multiplicative and additive rules, most participants in our study struggled to learn the precise systems they were trained on, and instead hypothesized rules that were not contained within the training data. In Experiment 2, we found similar results, and also that learning number words in a consistently ordered count list helped participants to recall those words, but not to generalize the number system to describe novel quantities. Also, magnitude comparison data from Experiment 2 suggest that participants were more likely to learn the meanings of number words when trained on a system that featured only rules, compared to one that featured only a sequential counting structure, or counting and rule structures combined. This suggests that presenting number words in a count routine may help learners memorize the words in their training set, but may also impair the ability to access quantity information.[<reflink idref="bib11" id="ref105">11</reflink>]</p> <hd id="AN0186163275-32">Effects of base size and counting on number word learning</hd> <p>In the Introduction, we noted that most previous studies of how learners acquire number systems have focused on how children learn base‐10 systems that are presented in the context of counting routines. Consequently, little is known about the learnability of alternative, historically attested base systems, or how they are learned or used by adult learners. We also noted that understanding the learnability of different systems in adult learners might have consequences for how we understand results from the developmental literature. In particular, if systems with smaller bases are as easy to learn as larger bases, then it is possible that training children on systems with smaller bases would result in earlier learning trajectories, since smaller base systems expose learners to compositional rules more frequently. Also, because smaller bases contain fewer lexical items, they require less rote memorization before rules can be deployed. By contrast, if we were to find that smaller base systems are harder to learn for adults—for example, because of their greater reliance on rules—we might conclude that training children on such systems would be unlikely to accelerate learning. Because our study did not test children directly, it remains to be seen whether these findings generalize to younger learners, owing to differences in their cognitive capacities and learning environments. Our results in adults, while not showing an advantage of smaller bases, also do not suggest that they are harder, though they may involve different learning processes. Given this, future studies should train younger learners with alternative base systems, and ask whether they are able to acquire combinatorial rules earlier in acquisition, if exposed to numbers that require their use.[<reflink idref="bib12" id="ref106">12</reflink>]</p> <p>A second issue that we considered in the Introduction is how learning a counting routine might impact number word learning. Previous studies suggest that there may be long‐term benefits of expertise with counting routines (Koponen et al., [<reflink idref="bib41" id="ref107">41</reflink>]; Martin et al., [<reflink idref="bib50" id="ref108">50</reflink>]), and it has been proposed that learners discover new conceptual content on the basis of the counting routine (Carey, [<reflink idref="bib10" id="ref109">10</reflink>]; [<reflink idref="bib11" id="ref110">11</reflink>]). However, no previous studies have tested whether training on rote routines facilitates learning of combinatorial number rules above and beyond frequent exposure to number words in conversation (though studies have tested how these impact earlier outcomes, like acquisition of small number words, or the cardinal principle; Mix, Sandhofer, Moore, & Russell, [<reflink idref="bib54" id="ref111">54</reflink>]; Gibson et al., [<reflink idref="bib30" id="ref112">30</reflink>]). Our study begins to address this question by testing adult learners, and suggests that embedding training in a count routine may benefit recall of trained words but not rule learning. Further, we found some evidence that embedding number words in a counting routine may even impede the acquisition of number word meanings. In particular, when participants were asked to decide which of two number words represented a larger magnitude, they performed significantly worse if they were trained with a sequential counting routine—with or without combinatorial rules—versus a system with just combinatorial rules alone. These results comport with a growing body of evidence that numerical reasoning is often closely tied to the structure of external representations of number. For example, judgments of magnitude are made differently when comparing symbolic numbers or perceptual magnitudes (Krajcsi et al., [<reflink idref="bib42" id="ref113">42</reflink>]; [<reflink idref="bib44" id="ref114">44</reflink>]; Krajcsi, [<reflink idref="bib43" id="ref115">43</reflink>]). Similarly, research on the history of number highlights how the medium of a number system can impact computation. Such work highlights the effects of visual properties of number symbols on computation, such as size‐ordering, in which longer expressions reliably denote larger cardinalities (Chrisomalis, [<reflink idref="bib16" id="ref116">16</reflink>]), the separate uses of abaci and Roman numerals for calculation and recording of quantities (Schlimm & Neth, [<reflink idref="bib65" id="ref117">65</reflink>]), and the role that instantiating numbers in different media may play in the systematization of number concepts over time constitutive role in the conceptual trajectory of number systems (Overmann, [<reflink idref="bib57" id="ref118">57</reflink>]; [<reflink idref="bib58" id="ref119">58</reflink>]).</p> <hd id="AN0186163275-33">The use of artificial language learning in the study of numerical cognition</hd> <p>Beyond these specific questions that motivated our study, our results also suggest that studies of artificial language learning can provide a fruitful method for further exploration of numerical cognition. A large existing body of work makes empirical claims about the nature of language by teaching artificial languages to linguistically savvy adults (see Kirby, Griffiths, & Smith, [<reflink idref="bib40" id="ref120">40</reflink>], or Culbertson, [<reflink idref="bib19" id="ref121">19</reflink>], for review), but few previous studies have taken a similar approach to number systems. Despite the fact that participants in our study were fully numerate adults with sophisticated number concepts, they exhibited remarkable variability in their learning strategies, suggesting that their expertise did not fully determine their approach to learning new systems. In fact, even when trained in a novel base‐10 system, participants fared no better than when trained with alternative bases. While a small number of participants learned all target rules, others learned only target additive rules, but not multiplicative rules, despite the existence of both additive and multiplicative rules in the English number system. Some participants were appeared to directly adopt strategies inspired by their base‐10 knowledge, though this was less common, and when they did, their particular form of base‐10 strategy differed according to the base system they were learning. Finally, many participants created untrained rule systems. When they did, these rules often resembled those of historically attested systems. The variability of strategies we observed confirms the value of experimental methods for assessing the foundations of compositional rules in number systems. Consequently, our findings set the stage for asking whether similar learning strategies exist in children or innumerate adults (e.g., Gordon, [<reflink idref="bib32" id="ref122">32</reflink>]; Pica, Lemer, Izard, & Dehaene, [<reflink idref="bib60" id="ref123">60</reflink>]; Frank, Everett, Fedorenko, & Gibson, [<reflink idref="bib25" id="ref124">25</reflink>]; Pitt, Gibson, & Piantadosi, [<reflink idref="bib62" id="ref125">62</reflink>]). Drawing on the lessons of this study, in which multiplicative rules and exponential roots may have introduced complexity to learning, future studies might begin with simpler systems in order to target specific tradeoffs, like that between memorizing lexical roots and learning a specific rule. For example, it is possible that exposing learners to a purely additive system, where only lexical learning and rule frequency differ, might result in more obvious effects of base size.</p> <hd id="AN0186163275-34">Open questions and future directions</hd> <p>Our study leaves open a number of interesting questions. First, it leaves open why participants found it easier to learn additive rules than multiplicative rules. One possible factor might be that additive rules were more frequent in participants' training, and also occur more frequently in their familiar base‐10 system. Another possibility is that participants were more likely to intuit the logic of additive rules because they are conceptually more concrete. For example, for an additive composition like 4+2, sets of both 4 and 2 are present, whereas for a multiplicative composition like 2*4, only groups of 4 are present, and the 2 must be interpreted as quantifying the sets of 4 themselves, rather than the objects contained within each set. Interestingly, the emergence of multiplicative rules likely post‐dates that of additive rules in the history of most languages (Hurford, [<reflink idref="bib38" id="ref126">38</reflink>]). On some accounts, multiplicative constructions may only be invented when the expressive limit of a simpler, purely additive system is reached (Damerow, [<reflink idref="bib20" id="ref127">20</reflink>]). As an example, Damerow notes that the largest number in many body tally systems is the last body part within the tally (e.g., Eipo <emph>seselekyaba</emph>, "right little finger," meaning 25). To express larger cardinalities, this number may receive a separate designation, <emph>yupe ton</emph>, "one time counted," from which multiplicative expressions like <emph>betinye ton</emph>, "two times counted" are subsequently coined. The conceptual content of each kind of expression is clearly different: rather than the literal body parts within the tally system, the multiplicative expressions refer to the counting procedure itself.</p> <p>Also left open by our study is how social processes impact the transmission and evolution of counting systems. In our studies, learning occurred via a strict pedagogical process in which there was one teacher (the software) and one learner (the participant). However, in the wild counting, systems have arisen via multigenerational transmission processes, in which each generation learns and transmits their practices resulting in irregularities, systems with mixed bases, and partial generalizations. Future studies should examine how multigeneration communicative transmission chains impact the form that counting systems take, and whether diachronic processes of change favor particular bases, rules, or procedures over others. Such work offers the possibility of uniting work on the learnability of counting systems with existing theories of efficient communication (e.g., Gibson et al., [<reflink idref="bib29" id="ref128">29</reflink>]; Xu, Liu, & Regier, [<reflink idref="bib79" id="ref129">79</reflink>]) while using methods previously deployed for studying the emergence and change of symbolic systems (Kirby et al., [<reflink idref="bib40" id="ref130">40</reflink>]; Hawkins et al., 2022).</p> <p>In addition to assessing the learnability of artificial systems for adult learners, future studies should consider whether naive learners, who have not previously acquired a number system, might show different learning strategies. Naturally, while our studies provide insight into the factors that impact the learnability of different number systems by numerate adults, they do not tell us how these factors impact child learners, where many additional factors likely impact learning. For example, aside from not having prior exposure to a number system, children differ in working memory capacity, their understanding of the social function of number, and also in the practices that they associate with counting. For example, in our study, adults learned number words as syntactic objects that referred to cardinalities (indicated by Arabic numerals), whereas when children learn to count, they associate individual words with objects in a count, which may defocus attention to the relationship between combinatorial number symbols and their cardinal meanings. Studies now in progress are exploring semi‐numerate children's ability to learn different base systems, using methods adapted from the experiments presented here. In doing so, this work will build on recent studies investigating children's understanding of numerical composition (Cheung et al., [<reflink idref="bib15" id="ref131">15</reflink>]; Park et al., 2022), to explore questions of cognitive development that are difficult to assess by observing the acquisition of natural language alone.</p> <p>In this study, we probed the impacts of counting structures and procedures on learning by training numerate adults on a novel number system. We found that participants could learn the words of a new system effectively, regardless of the size of the numeral base. We also found that the rote counting procedure in which these number systems were learned helped learners to memorize each number word without necessarily acquiring the compositional rules underlying them. Instead, participants showed a strong tendency to rely on additive rules, rather than multiplicative rules, to generate novel number words. Further, we found that when number words were learned in random order rather than within a counting procedure, participants performed worse at cued recall of those words, but were better able to access their meanings when comparing the magnitude of number‐word pairs. We conclude that the counting procedure and syntactic structure of a number system not only facilitate different aspects of number‐word learning, and that their interaction may not always be entirely constructive.</p> <hd id="AN0186163275-35">Open Research Badge</hd> <p>This article has earned Open Data, Open Materials, and Preregistered Research Design badges. Data, materials and the preregistered design are available at https://osf.io/rwqk7/. </p> <p>GRAPH:  </p> <ref id="AN0186163275-36"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Importantly, combinations like <emph>okosa okosa okosa</emph> can be represented by a rule, or simply memorized, just as English words up to 100 can be rote learned or generated by rules.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> N.B., these are exponential labels.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref3" type="bt">3</bibl> <bibtext> Note that, given our interest in testing generalization, it was not possible to test larger base systems (e.g., base‐20) since this would require the task to have been roughly four times as long as it was. We, therefore, limited ourselves to bases up to 10 for the current experiments.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref4" type="bt">4</bibl> <bibtext> Note that response time analyses were repeated on reaction time—time from stimulus presentation until the participant started typing—to similar qualitative results.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref43" type="bt">5</bibl> <bibtext> While this suggests that bases with multiplicative rules should be harder, all else equal, this was only one of several features that varied between bases. For example, despite having the same number of multiplicative terms, base‐4 system accuracy was greater than that of base‐3 (<emph>b</emph>= 1.539, <emph>z</emph>= 4.137, <emph>p</emph><.005). One reason might be that the count list in the base‐4 condition was easier to learn because each "decade" contains more exemplars of each rule type, providing more evidence to the learner for compositional rules. Alternatively, base‐4 might be more amenable to rote learning, because the repetition of exemplars in sequence (e.g., 4‐1, 4‐2, 4‐3) continues in longer, uninterrupted chunks.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref84" type="bt">6</bibl> <bibtext> That the "counting up" strategy did not explain as many responses in the base‐10 condition, where it would also be consistent with the expected targets, might be explained by competition with those participants' bias to calque the Arabic numerals (see below).</bibtext> </blist> <blist> <bibl id="bib7" idref="ref87" type="bt">7</bibl> <bibtext> Consistency of rule use varied by participants, bases, and rule types. Using 80% consistency as a threshold, the following number of participants were consistent: Additive rules: base‐2 (16/31), base‐3 (9/30), base‐4 (0/32), base‐5 (0/30), base‐8 (19/30), base‐10 (8/29); Multiplicative rules: base‐2 (0/31), base‐3 (1/30), base‐4 (11/32), base‐5 (18/30), base‐8 (0/30), base‐10 (3/29).</bibtext> </blist> <blist> <bibl id="bib8" idref="ref49" type="bt">8</bibl> <bibtext> In a preregistered analysis, we also asked whether there was a difference in the diversity of forms created, measured using the Simpson's diversity index of forms given for each cardinality. We found there was no difference between the Rules‐Only condition (.803, CI: [0.727, 0.879]) and the Order+Rules condition (.769, CI: [0.676, 0.862]).</bibtext> </blist> <blist> <bibl id="bib9" idref="ref70" type="bt">9</bibl> <bibtext> We had planned to include, as a predictor, the cumulative frequency of both numbers being compared. As observed above in reference to the Training Phase, however, our ability to interpret the frequency predictor was limited by several elements of the task design. Most relevant here, frequency was heavily anticorrelated with the sum of the compared numbers (0.996), making the estimates for each predictor very sensitive to small changes in model parameters. We, therefore, removed frequency from the model.</bibtext> </blist> <blist> <bibtext> This was likely because occasional congruence between word‐length and magnitude in Order‐only trials helped participants respond faster and more accurately than baseline, and not due to a word‐length heuristic. Participants in the Order‐only condition were as inaccurate comparing numbers of the same word length (.633, CI: [0.602, 0.663]) as comparing pairs that misleadingly differed in word length (.631, CI: [0.594, 0.668]), but performed much better when word length was congruent with magnitude (.863, CI: [0.897, 0.83]).</bibtext> </blist> <blist> <bibtext> Note that participants could have used ordinal position to infer relative magnitude (e.g., as in a familiar number system), or may have sometimes translated from the novel system to English or Arabic numbers.</bibtext> </blist> <blist> <bibtext> As we note below, we are currently conducting such studies. We do not include child data here because these studies require substantial modifications to methods and a more detailed review of related developmental studies, and would, therefore, produce an unwieldy report if combined with the current studies.</bibtext> </blist> <blist> <bibtext> There are also roots expressing exponential powers of six, such as <emph>fta</emph> (36) or <emph>taruba</emph> (216).</bibtext> </blist> <blist> <bibtext> English also expresses exponents with new, noncomposed roots, such as <emph>hundred</emph> (100) or <emph>thousand</emph> (1000). Tellingly, while higher exponents are composed, this composition is productive only in erudite speech (e.g., <emph>billion</emph>, <emph>trillion</emph>, <emph>quadrillion</emph>, etc.).</bibtext> </blist> </ref> <ref id="AN0186163275-37"> <title> References </title> <blist> <bibtext> Akhtar, N., & Montague, L. (1999). Early lexical acquisition: The role of cross‐situational learning. 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Header DbId: eric
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An: EJ1475024
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PubType: Academic Journal
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Learning a Novel Number System: The Role of Compositional Rules and Counting Procedures
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Sebastian+Holt%22">Sebastian Holt</searchLink><br /><searchLink fieldCode="AR" term="%22David+Barner%22">David Barner</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Cognitive+Science%22"><i>Cognitive Science</i></searchLink>. 2025 49(6).
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 31
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Adult+Education%22">Adult Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Numbers%22">Numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Adult+Students%22">Adult Students</searchLink><br /><searchLink fieldCode="DE" term="%22Number+Concepts%22">Number Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Multiplication%22">Multiplication</searchLink><br /><searchLink fieldCode="DE" term="%22Memory%22">Memory</searchLink><br /><searchLink fieldCode="DE" term="%22Rote+Learning%22">Rote Learning</searchLink><br /><searchLink fieldCode="DE" term="%22Numeracy%22">Numeracy</searchLink><br /><searchLink fieldCode="DE" term="%22Word+Lists%22">Word Lists</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1111/cogs.70071
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0364-0213<br />1551-6709
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Humans count to indefinitely large numbers by recycling words from a finite list, and combining them using rules--for example, combining sixty with unit labels to generate sixty-one, sixty-two, and so on. Past experimental research has focused on children learning base-10 systems, and has reported that this rule learning process is highly protracted. This raises the possibility that rules are slow to emerge because they are not needed in order to represent smaller numbers (e.g., up to 20). Here, we investigated this possibility in adult learners by training them on a series of artificial number "languages" that manipulated the availability of rules, by varying the numerical base in each language. We found (1) that the size of a base--for example, base-2 versus base-5--had little effect on learning, (2) that learners struggled to acquire multiplicative rules while they learned additive rules more easily, (3) that memory for number words was greater when they were taught as part of a sequential count list, but (4) that learning numbers as part of a rote list may impair the ability to map them to magnitudes.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: Note
  Label: Notes
  Group: Note
  Data: https://osf.io/rwqk7
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1475024
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  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1111/cogs.70071
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 31
    Subjects:
      – SubjectFull: Computation
        Type: general
      – SubjectFull: Numbers
        Type: general
      – SubjectFull: Adult Students
        Type: general
      – SubjectFull: Number Concepts
        Type: general
      – SubjectFull: Multiplication
        Type: general
      – SubjectFull: Memory
        Type: general
      – SubjectFull: Rote Learning
        Type: general
      – SubjectFull: Numeracy
        Type: general
      – SubjectFull: Word Lists
        Type: general
    Titles:
      – TitleFull: Learning a Novel Number System: The Role of Compositional Rules and Counting Procedures
        Type: main
  BibRelationships:
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      – PersonEntity:
          Name:
            NameFull: Sebastian Holt
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          Name:
            NameFull: David Barner
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          Dates:
            – D: 01
              M: 06
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 0364-0213
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              Value: 1551-6709
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              Value: 49
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              Value: 6
          Titles:
            – TitleFull: Cognitive Science
              Type: main
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