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Verbalized Arithmetic Principles Correlate with Mathematics Achievement

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Title: Verbalized Arithmetic Principles Correlate with Mathematics Achievement
Language: English
Authors: Jiaxin Cui, Li Wang, Dawei Li, Xinlin Zhou (ORCID 0000-0002-3530-0922)
Source: British Journal of Educational Psychology. 2024 94(1):41-57.
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: 17
Publication Date: 2024
Document Type: Journal Articles
Reports - Research
Education Level: Higher Education
Postsecondary Education
Descriptors: Verbal Communication, Verbal Development, Mathematics Education, Mathematics Achievement, Undergraduate Students, College Mathematics, Foreign Countries, Language Skills, Cognitive Ability, Number Concepts, Coding, Learning Modalities
Geographic Terms: China (Beijing)
DOI: 10.1111/bjep.12632
ISSN: 0007-0998
2044-8279
Abstract: Background: When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to mathematical vocabulary or educational practice. However, these studies did not exclude the role of symbolic mathematics ability, and almost no research has focused on verbalized mathematical principles. Aims: This study is aimed to investigate whether verbalized mathematics ability independently predicts mathematics achievement. The current study hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability. Sample: A sample of 241 undergraduates (136 males, 105 females, mean age = 21.95, SD = 2.38) in Beijing, China. Methods: A total of 12 tests were used, including a verbalized arithmetic principle test, a mathematics achievement test, and tests on general language (sentence completion test), symbolic mathematical ability (including symbolic arithmetic principles test, simple arithmetic computation and complex arithmetic computation), approximate number sense ability (numerosity comparison test) and several related cognitive covariates (including the non-verbal matrix reasoning, the syllogism reasoning, mental rotation, figure matching and choice reaction time). Results: Results showed that the processing of verbalized arithmetic principles displayed a significant role in mathematics achievement after controlling for general language, related cognitive abilities, approximate number sense ability and symbolic mathematics ability. Conclusions: The results suggest that verbalized mathematics ability was an independent predictor and provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three-component mathematics model.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1410937
Database: ERIC
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  Value: <anid>AN0175303240;6kx01mar.24;2024Feb09.05:47;v2.2.500</anid> <title id="AN0175303240-1">Verbalized arithmetic principles correlate with mathematics achievement </title> <p>Background: When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to mathematical vocabulary or educational practice. However, these studies did not exclude the role of symbolic mathematics ability, and almost no research has focused on verbalized mathematical principles. Aims: This study is aimed to investigate whether verbalized mathematics ability independently predicts mathematics achievement. The current study hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability. Sample: A sample of 241 undergraduates (136 males, 105 females, mean age = 21.95, SD = 2.38) in Beijing, China. Methods: A total of 12 tests were used, including a verbalized arithmetic principle test, a mathematics achievement test, and tests on general language (sentence completion test), symbolic mathematical ability (including symbolic arithmetic principles test, simple arithmetic computation and complex arithmetic computation), approximate number sense ability (numerosity comparison test) and several related cognitive covariates (including the non‐verbal matrix reasoning, the syllogism reasoning, mental rotation, figure matching and choice reaction time). Results: Results showed that the processing of verbalized arithmetic principles displayed a significant role in mathematics achievement after controlling for general language, related cognitive abilities, approximate number sense ability and symbolic mathematics ability. Conclusions: The results suggest that verbalized mathematics ability was an independent predictor and provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three‐component mathematics model.</p> <p>Keywords: mathematical cognition; mathematics achievement; verbalized arithmetic principle; verbalized mathematics</p> <hd id="AN0175303240-2">BACKGROUND</hd> <p>Language is an ideal tool for the expression of knowledge (Kay & Kempton, [<reflink idref="bib26" id="ref1">26</reflink>]). Although mathematical knowledge has developed with the application of specialized symbols and signs (e.g. Arabic digits, +, −, =, >, <), it has a root in language. The mathematical knowledge delivered by mathematical symbols can be equivalent to that delivered by general language (Prather & Alibali, [<reflink idref="bib45" id="ref2">45</reflink>]). For example, the number of '1, 2, 3' was represented by the general language of 'one, two, three' in English or 'yi, er, san' in Chinese. The symbolic mathematical formula 'A + B = B + A' can be verbalized as 'Switching the order in which two numbers are added together does not change their sum'. When mathematical knowledge is expressed in general language, it is called verbalized mathematics (Zhou, [<reflink idref="bib77" id="ref3">77</reflink>]; Zhou & Zeng, [<reflink idref="bib81" id="ref4">81</reflink>]).</p> <p>Previous studies of verbalized mathematics have only focused on mathematical vocabulary or educational practice and have not treated it as an ability independent from symbolic mathematics ability. Hence, whether there exists an independent prediction of verbalized mathematics ability in mathematics achievement is still unknown.</p> <hd id="AN0175303240-3">Verbalized mathematics</hd> <p>Verbalized mathematics is one of the components of the three‐component mathematics model (Zhou, [<reflink idref="bib77" id="ref5">77</reflink>]; Zhou & Zeng, [<reflink idref="bib81" id="ref6">81</reflink>]). This model proposes that mathematics in the brain has three types of representation: symbolic, verbalized and situational. Verbalized mathematics corresponds to the verbalized representation of mathematical knowledge (Cui et al., [<reflink idref="bib8" id="ref7">8</reflink>], [<reflink idref="bib7" id="ref8">7</reflink>]; Li et al., [<reflink idref="bib32" id="ref9">32</reflink>]; Liu et al., [<reflink idref="bib35" id="ref10">35</reflink>], [<reflink idref="bib34" id="ref11">34</reflink>]; Wang et al., [<reflink idref="bib68" id="ref12">68</reflink>]; Wei et al., [<reflink idref="bib70" id="ref13">70</reflink>]; Zhang et al., [<reflink idref="bib73" id="ref14">73</reflink>]; Zhou et al., [<reflink idref="bib79" id="ref15">79</reflink>]) and includes a variety of mathematical knowledge, such as mathematics vocabulary (e.g. more, equation, triangle), mathematics principles (e.g. 'two negatives make a positive', 'the sum of two even numbers must be even'), mathematical rules (e.g. 'the calculation inside the parentheses should be performed first', 'Multiplication and division are performed before addition and subtraction in series of arithmetic computation') and mathematical problem‐solving strategies (e.g. 'Drawing schematic graph can help solve mathematical word problems'). However, word problems were not categorized as verbalized mathematics, but as situational mathematics in the three‐component mathematics theory. Because they use general language to typically express situations and events other than conceptual mathematical knowledge.</p> <p>Mathematical vocabulary that is typically shown with general language is one aspect of verbalized mathematics and has been extensively investigated (Forsyth & Powell, [<reflink idref="bib15" id="ref16">15</reflink>]; Hassinger‐Das et al., [<reflink idref="bib20" id="ref17">20</reflink>]; Lin, [<reflink idref="bib33" id="ref18">33</reflink>]; Peng & Lin, [<reflink idref="bib39" id="ref19">39</reflink>]; Powell et al., [<reflink idref="bib42" id="ref20">42</reflink>], [<reflink idref="bib40" id="ref21">40</reflink>], [<reflink idref="bib44" id="ref22">44</reflink>]; Purpura et al., [<reflink idref="bib48" id="ref23">48</reflink>]; Purpura & Logan, [<reflink idref="bib47" id="ref24">47</reflink>]; Purpura & Reid, [<reflink idref="bib49" id="ref25">49</reflink>]; Toll & Van Luit, [<reflink idref="bib62" id="ref26">62</reflink>], [<reflink idref="bib63" id="ref27">63</reflink>]; Ufer & Bochnik, [<reflink idref="bib65" id="ref28">65</reflink>]). Mathematical vocabulary can be assessed by questions such as 'The difference is the result in (a) addition, (b) subtraction, (c) multiplication, or (d) division' (Powell & Nelson, [<reflink idref="bib43" id="ref29">43</reflink>]). It also can be assessed pictorially, such as using mostly full and mostly empty glasses of water to assess the understanding of 'more than' and 'less than' (Purpura & Logan, [<reflink idref="bib47" id="ref30">47</reflink>]). Different from mathematical vocabulary, mathematical principles reflect the association between different vocabularies. For example, the mathematical principle called the 'commutative laws of multiplication' conceptually contains three typical mathematical vocabularies: multiplier, multiplicand and product.</p> <p>Mathematical reading (e.g. reading math‐related story books, Purpura et al., [<reflink idref="bib48" id="ref31">48</reflink>]), mathematical talks (e.g. 'I am 45 years old, how old are you?' Boonen et al., [<reflink idref="bib3" id="ref32">3</reflink>]) and mathematical self‐explanation (e.g. a student explains the steps in which the triangle sum theorem is applied, Topping et al., [<reflink idref="bib64" id="ref33">64</reflink>]) refer to educational practices that use verbalized mathematics. These practices typically involve mathematical vocabulary, principles, rules and methods/strategies expressed with general language.</p> <hd id="AN0175303240-4">Verbalized mathematics and mathematics achievement</hd> <p>The current study focused on the term 'verbalized mathematics', which covers the mathematics vocabulary and all other language‐based representations of conceptual mathematical knowledge. Previous studies mainly have made efforts to explore the relationship between verbalized mathematics and mathematics achievement from the perspective of mathematical vocabulary and educational practice.</p> <p>The skill of mathematical vocabulary is closely associated with mathematics achievement (Lin, [<reflink idref="bib33" id="ref34">33</reflink>]; Peng & Lin, [<reflink idref="bib39" id="ref35">39</reflink>]; Powell et al., [<reflink idref="bib42" id="ref36">42</reflink>]; Purpura et al., [<reflink idref="bib48" id="ref37">48</reflink>]; Purpura & Logan, [<reflink idref="bib47" id="ref38">47</reflink>]; Purpura & Reid, [<reflink idref="bib49" id="ref39">49</reflink>]; Toll & Van Luit, [<reflink idref="bib62" id="ref40">62</reflink>], [<reflink idref="bib63" id="ref41">63</reflink>]; Ufer & Bochnik, [<reflink idref="bib65" id="ref42">65</reflink>]). For example, Toll and Van Luit ([<reflink idref="bib63" id="ref43">63</reflink>]) found that mathematical vocabulary, but not verbal working memory or symbolic comparison skills, correlated with numeracy development in 4‐ to 5‐year‐old Dutch children. Furthermore, Purpura and Reid ([<reflink idref="bib49" id="ref44">49</reflink>]) suggested that mathematical vocabulary alone could predict numeracy performance in 3‐ to 5‐year‐old preschool children, after controlling for age, gender, parental education, rapid automatized naming and definitional vocabulary. Similarly, Peng and Lin ([<reflink idref="bib39" id="ref45">39</reflink>]) found that mathematical vocabulary in fourth graders made a unique contribution to mathematics performance after controlling for general vocabulary, IQ, working memory and processing speed.</p> <p>Practicing verbalized mathematics has been shown to positively affect mathematics achievement (Aleven & Koedinger, [<reflink idref="bib2" id="ref46">2</reflink>]; Durkin, [<reflink idref="bib12" id="ref47">12</reflink>]; Elliott et al., [<reflink idref="bib13" id="ref48">13</reflink>]; Hilbert et al., [<reflink idref="bib23" id="ref49">23</reflink>]; Murata et al., [<reflink idref="bib37" id="ref50">37</reflink>]; Powell & Driver, [<reflink idref="bib41" id="ref51">41</reflink>]; Purpura et al., [<reflink idref="bib48" id="ref52">48</reflink>]; Rittle‐Johnson et al., [<reflink idref="bib55" id="ref53">55</reflink>]; Sidney et al., [<reflink idref="bib60" id="ref54">60</reflink>]; Topping et al., [<reflink idref="bib64" id="ref55">64</reflink>]). For example, Purpura et al. ([<reflink idref="bib48" id="ref56">48</reflink>]) found that mathematical knowledge in young children could be enhanced by 8‐week dialogic reading interventions that incorporated quantitative and spatial words. Boonen et al. ([<reflink idref="bib3" id="ref57">3</reflink>]) compared the effects of nine types of mathematical speech that teachers can use to facilitate number sense and found that only mathematical speech that uses conventional nominatives (described as 'the use of numbers as labels for age, dates, or time') was positively related to cardinality performance in kindergarteners, after controlling for socio‐economic status and working memory ability.</p> <p>However, although several related studies have been conducted, the impact of verbalized mathematics on mathematics achievement remains unclear. First, the verbalized mathematics tests and mathematics achievement tests used in previous studies did not exclude the role of symbolic mathematics ability. Consider a mathematical vocabulary question (e.g. 'Which of the following represents an algebraic equation? (a) <emph>x</emph> ÷ 7, (b) 13 + 3<emph>x</emph> < 25, (c) <emph>x</emph> = 0, or (d) 3<emph>x</emph> + 1.2') and mathematics achievement that is measured as performance in arithmetic computation (e.g. 'Given than 5<emph>n</emph> + 7 = 42, <emph>n</emph> = __') or word problem solving (e.g. 'The 72 people are equally divided into 4 groups. How many people are in each group?' Peng & Lin, [<reflink idref="bib39" id="ref58">39</reflink>]). All three measures involve the symbolic mathematical knowledge.</p> <p>Second, although mathematical vocabulary has received some attention, almost no research has focused on verbalized mathematical principles as an important type of verbalized mathematics. Verbalized mathematical principles have been reported to be processed differently in the brain from mathematical vocabulary (Liu et al., [<reflink idref="bib35" id="ref59">35</reflink>]; Zhang et al., [<reflink idref="bib73" id="ref60">73</reflink>]). For example, arithmetic vocabulary did not activate regions of the parietal cortex typically activated by mathematics any more than linguistic or tool vocabulary did (Zhang et al., [<reflink idref="bib73" id="ref61">73</reflink>]). However, some parietal regions were activated more by verbalized arithmetic principles than they were by sentence comprehension (Liu et al., [<reflink idref="bib35" id="ref62">35</reflink>]). Moreover, connectivity within the brain between the semantic network (e.g. middle temporal gyrus) and the parietal cortex was related to processing arithmetic principles (Liu et al., [<reflink idref="bib34" id="ref63">34</reflink>]). Thus, verbalized mathematical principles might have a positive effect on mathematics achievement because of the extensive involvement of the semantic network during mathematical processing.</p> <hd id="AN0175303240-5">The current study</hd> <p>Although verbalized mathematics has been shown to correlate with mathematics achievement in studies of mathematical vocabulary (Peng & Lin, [<reflink idref="bib39" id="ref64">39</reflink>]; Powell et al., [<reflink idref="bib42" id="ref65">42</reflink>]; Ufer & Bochnik, [<reflink idref="bib65" id="ref66">65</reflink>]) and educational practice (Boonen et al., [<reflink idref="bib3" id="ref67">3</reflink>]; Purpura et al., [<reflink idref="bib48" id="ref68">48</reflink>]; Topping et al., [<reflink idref="bib64" id="ref69">64</reflink>]), whether verbalized mathematics ability independently predicts mathematics achievement remains unknown. The current study aimed to answer this question using mathematical principles such as verbalized mathematics and controlling for symbolic mathematics ability.</p> <p>We hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability. The rationale for the hypothesis was that verbalized mathematics ability likely facilitates mathematics achievement by promoting conceptual understanding of mathematical knowledge. Brain research has shown that the processing of verbalized mathematical principles is typically supported by the semantic network (Cui et al., [<reflink idref="bib8" id="ref70">8</reflink>], [<reflink idref="bib7" id="ref71">7</reflink>]; Li et al., [<reflink idref="bib32" id="ref72">32</reflink>]; Liu et al., [<reflink idref="bib35" id="ref73">35</reflink>], [<reflink idref="bib34" id="ref74">34</reflink>]; Wang et al., [<reflink idref="bib68" id="ref75">68</reflink>]; Wei et al., [<reflink idref="bib70" id="ref76">70</reflink>]; Zhang et al., [<reflink idref="bib73" id="ref77">73</reflink>]; Zhang & Zhou, [<reflink idref="bib76" id="ref78">76</reflink>]; Zhou et al., [<reflink idref="bib79" id="ref79">79</reflink>]), which suggests that conceptual/semantic processing is the typical component when doing verbalized mathematics. At the same time, behavioural studies have shown that the understanding of conceptual mathematical knowledge is helpful for mathematics achievement (Adeleke, [<reflink idref="bib1" id="ref80">1</reflink>]; Hiebert & Wearne, [<reflink idref="bib22" id="ref81">22</reflink>]; McNeil et al., [<reflink idref="bib36" id="ref82">36</reflink>]; Rittle‐Johnson et al., [<reflink idref="bib54" id="ref83">54</reflink>]; Rittle‐Johnson & Alibali, [<reflink idref="bib53" id="ref84">53</reflink>]; Sidney et al., [<reflink idref="bib60" id="ref85">60</reflink>]). For example, conceptual knowledge transfers better to solving mathematical problems than procedural knowledge (Adeleke, [<reflink idref="bib1" id="ref86">1</reflink>]; Rittle‐Johnson & Alibali, [<reflink idref="bib53" id="ref87">53</reflink>]). Similarly, interventions based on conceptual understanding were able to improve mathematics achievement (Hiebert & Wearne, [<reflink idref="bib22" id="ref88">22</reflink>]; McNeil et al., [<reflink idref="bib36" id="ref89">36</reflink>]; Rittle‐Johnson et al., [<reflink idref="bib54" id="ref90">54</reflink>]; Sidney et al., [<reflink idref="bib60" id="ref91">60</reflink>]).</p> <p>In the current study, verbalized mathematics ability was measured using a verbalized arithmetic principle test, based on knowledge learned in primary school. Mathematics achievement was measured with a self‐adapted test, and individual differences were driven primarily by what had been learned in senior high school (as participants were undergraduates). Therefore, the overlap in mathematical knowledge between these two tests was controlled as little as possible. And the matching of the symbolic arithmetic principle test to the verbalized arithmetic principle test and arithmetic computation tests were used to control for symbolic mathematical ability. The numerosity comparison test was used to control for approximate number sense ability, which is related to mathematics (Cui et al., [<reflink idref="bib9" id="ref92">9</reflink>]).</p> <p>Moreover, general language and several related cognitive abilities were considered. A sentence comprehension task was used to control for general language ability. Other cognitive abilities included general intelligence, which can affect all aspects of cognitive processing (Cowan & Powell, [<reflink idref="bib6" id="ref93">6</reflink>]; Fuchs et al., [<reflink idref="bib16" id="ref94">16</reflink>]; Jordan et al., [<reflink idref="bib25" id="ref95">25</reflink>]), visuospatial processing, which is a significant cognitive factor in mathematics (Boonen et al., [<reflink idref="bib4" id="ref96">4</reflink>]; Kyttala et al., [<reflink idref="bib29" id="ref97">29</reflink>]; Reuhkala, [<reflink idref="bib51" id="ref98">51</reflink>]; Swanson et al., [<reflink idref="bib61" id="ref99">61</reflink>]) and logical reasoning, which is also required thinking in mathematics (Durand‐Guerrier et al., [<reflink idref="bib11" id="ref100">11</reflink>]; Fujita, [<reflink idref="bib18" id="ref101">18</reflink>]; Gardner, [<reflink idref="bib19" id="ref102">19</reflink>]; Rotigel & Fello, [<reflink idref="bib56" id="ref103">56</reflink>]).</p> <hd id="AN0175303240-6">METHODS</hd> <p></p> <hd id="AN0175303240-7">Participants</hd> <p>We recruited 241 undergraduate participants (136 males, mean age: 21.95 ± 2.38 years) from different levels of universities with different majors in China, the score of China's College Entrance Examination ranged from 400 to 700. Only when the sample size is over 200 will correlation results stabilize at a 95% level of confidence (Schonbrodt & Perugini, [<reflink idref="bib58" id="ref104">58</reflink>]). The participants were paid ¥120 CNY (about $18 USD) as compensation for their time (2 h). All participants were right‐handed native Mandarin speakers with normal or corrected‐to‐normal visual acuity. All participants completed formal primary, junior high and senior high school education (12 years). The standard mathematics curriculum covers arithmetic, algebra, statistics and plane and solid geometry. Although they might learn mathematics using different textbooks, all 12‐year education programs follow the national curriculum syllabus.</p> <hd id="AN0175303240-8">Tests</hd> <p>We used 12 computer‐based tests from the 'Online Psychological Experiment System' (OPES, <ulink href="http://www.dweipsy.com/lattice,">http://www.dweipsy.com/lattice,</ulink> Cui et al., [<reflink idref="bib9" id="ref105">9</reflink>], [<reflink idref="bib7" id="ref106">7</reflink>]; Zhang & Zhou, [<reflink idref="bib76" id="ref107">76</reflink>]; Zhou et al., [<reflink idref="bib80" id="ref108">80</reflink>], [<reflink idref="bib78" id="ref109">78</reflink>]). An illustration of a trial for each test is shown in Figure 1. The indices for all 12 tests are displayed in Table 1. The adjusted numbers of the correct trials were used as scores to control for the guessing effect, calculated as the difference between the numbers of correct responses and incorrect responses (Cirino, [<reflink idref="bib5" id="ref110">5</reflink>]; Salthouse, [<reflink idref="bib57" id="ref111">57</reflink>]).</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/6KX/01mar24/bjep12632-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="bjep12632-fig-0001.jpg" title="1 Schematic representation of tests used in the current study." /> </p> <p></p> <p>1 TABLE Means and standard deviations of test scores on index of 12 tests and tests' reliability coefficients.</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left">Test</th><th align="left">Index</th><th align="left">Mean (<italic>SD</italic>)</th><th align="left">Score range</th><th align="left">Split‐half reliability</th></tr></thead><tbody valign="top"><tr><td align="left">1. Mathematics achievement</td><td align="left">Score</td><td align="char" char="(">36.4 (8.9)</td><td align="left">7 to 58</td><td align="char" char=".">.77</td></tr><tr><td align="left">2. Verbalized arithmetic principle</td><td align="left">Adjusted number of correct trials</td><td align="char" char="(">18.1 (6.7)</td><td align="left">0 to 34</td><td align="char" char=".">.91</td></tr><tr><td align="left">3. Symbolic arithmetic principle</td><td align="left">Adjusted number of correct trials</td><td align="char" char="(">18.9 (6.4)</td><td align="left">0 to 32</td><td align="char" char=".">.77</td></tr><tr><td align="left">4. Simple arithmetic computation</td><td align="left">Adjusted number of correct trials</td><td align="char" char="(">51.4 (6.6)</td><td align="left">29 to 66</td><td align="char" char=".">.91</td></tr><tr><td align="left">5. Complex arithmetic computation</td><td align="left">Adjusted number of correct trials</td><td align="char" char="(">28.3 (6.0)</td><td align="left">10 to 43</td><td align="char" char=".">.91</td></tr><tr><td align="left">6. Numerosity comparison (RT)</td><td align="left">Reaction time (millisecond)</td><td align="char" char="(">537 (91)</td><td align="left">243 to 802</td><td align="char" char=".">.99</td></tr><tr><td align="left">Numerosity comparison (ACC)</td><td align="left">Correctness percentage</td><td align="char" char="(">81.8 (7.0)</td><td align="left">52 to 93</td><td align="char" char=".">.78</td></tr><tr><td align="left">7. Choice reaction time</td><td align="left">Reaction time (millisecond)</td><td align="char" char="(">362 (49)</td><td align="left">243 to 531</td><td align="char" char=".">.96</td></tr><tr><td align="left">8. Non‐verbal matrix reasoning</td><td align="left">Number of correct trials</td><td align="char" char="(">30.1 (6.7)</td><td align="left">10 to 50</td><td align="char" char=".">.93</td></tr><tr><td align="left">9. Syllogism reasoning</td><td align="left">Adjusted number of correct trials</td><td align="char" char="(">4.3 (5.3)</td><td align="left">−13 to 19</td><td align="char" char=".">.81</td></tr><tr><td align="left">10. Mental rotation</td><td align="left">Adjusted number of correct trials</td><td align="char" char="(">24.8 (9.1)</td><td align="left">1 to 44</td><td align="char" char=".">.92</td></tr><tr><td align="left">11. Figure matching (RT)</td><td align="left">Reaction time (millisecond)</td><td align="char" char="(">672 (209)</td><td align="left">212 to 1485</td><td align="char" char=".">.95</td></tr><tr><td align="left">Figure matching (ACC)</td><td align="left">Correctness percentage</td><td align="char" char="(">65.8 (8.3)</td><td align="left">36 to 87</td><td align="char" char=".">.72</td></tr><tr><td align="left">12. Sentence completion</td><td align="left">Adjusted number of correct trials</td><td align="char" char="(">38.6 (7.5)</td><td align="left">16 to 59</td><td align="char" char=".">.86</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>: Adjusted number of correct trials: total correct trials minus total incorrect trials. Total score was the maximum score of the subject's response.</p> <p>2 Abbreviations: ACC, accuracy rate; RT, reaction time.</p> <p> <emph>Mathematics achievement test</emph> was a self‐adapted mathematics achievement used in the previous study (Wang et al., [<reflink idref="bib69" id="ref112">69</reflink>]; Zhang et al., [<reflink idref="bib74" id="ref113">74</reflink>]; Zhou et al., [<reflink idref="bib78" id="ref114">78</reflink>]). The problems were put together according to math books and test papers that were typical of those used by Chinese students. For example, a senior high school problem was 'If the set A = {1,2,3,4} and B = {x|x = <emph>n</emph>, <emph>n</emph> ∈ A}, then A ∩ B = (). (a) {1,4}; (b) {2,3}; (c) {9,6}; (d) {1,2}'. Participants were first given a set of three questions at the first‐grade level. If they correctly solved at least two of three problems, the difficulty level of the next set would proceed to the next grade. If they made two mistakes, the level would not change. If they made three mistakes, the difficulty level would drop to the previous grade. The test lasted until the allocated time (18 min) was up or until five sets of questions at the same grade level were completed. The final score was calculated as the sum of the weighted scores for each grade, which was the number of correctly answered questions multiplied by the grade level (1–12). Rules of the score are weighted according to the difficulty of the questions. Because answering questions about knowledge learned in high school should get higher scores than that in primary school. A total of 1701 problems were included in our test database. It significantly correlated with simple arithmetic computation (<emph>r</emph> =.43) and complex arithmetic computation (<emph>r</emph> =.46) in 66 undergraduates, with a good validity (Zhang et al., [<reflink idref="bib74" id="ref115">74</reflink>]).</p> <p>The <emph>verbalized arithmetic principle test</emph> assessed the understanding of five operational principles (the commutative law of addition, associative law of addition and multiplication, commutative law of multiplication, the distributive law and the principle of removing parentheses for an arithmetic expression) taught in primary schools. Participants were asked to decide whether a sentence presented on a computer screen describing an arithmetic principle was true or false. For example, understanding of the distributive law could be assessed by 'True or False? Multiplying a sum of numbers by another number is the same as multiplying each of the numbers being added by that number and then adding the products'. The sentence did not disappear until response. The test contained 40 trials and lasted 3 min. The 40 items to make sure participants could not finish all the trials and avoid the ceiling effect. Within 3 min, the participants were not required to complete all the trials, but as many as they could.</p> <hd id="AN0175303240-10">Covariates: Symbolic mathematics</hd> <p>The <emph>Symbolic Arithmetic Principle Test</emph> was based on the same arithmetic principles as the <emph>Verbalized Arithmetic Principles Test</emph> but expressed using mathematic symbols instead of a verbal description. For example, the distributive law would be assessed by 'True or False? A × (B + C) = A × B + A × C'. The test contained 40 trials and lasted 3 min, with the same procedure as the <emph>Verbalized Arithmetic Principles Test</emph>.</p> <p>The <emph>Simple Arithmetic Computation Test</emph> was used to assess basic computation ability. The minuends were 18 or smaller, and the answers were single‐digit numbers (e.g. 9–4 = () or 16–8 = ()). In each trial, the problem was presented in the upper of the screen and two candidate answers were simultaneously presented underneath. Participants needed to select the correct answer. This test had 92 trials and lasted 2 min.</p> <p>The <emph>Complex Arithmetic Computation Test</emph> included 96 relatively complex subtraction problems. Double‐digit numbers were used for both operands (e.g. 75–28 = ()). Otherwise, this test was the same as the <emph>Simple Arithmetic Computation Test</emph>.</p> <hd id="AN0175303240-11">Covariates: Approximate number sense ability</hd> <p>The <emph>Numerosity Comparison Test</emph> was designed to investigate the understanding of the approximate number system (ANS). In each trial, two dot arrays were presented side by side simultaneously on the screen for 200 ms and participants were required to choose the one that contained more dots while ignoring all other visual properties of the dot arrays. The number of dots in each array ranged from 5 to 32 and the ratio between the numbers of dots in the two arrays ranged from 1.12 to 2.00. The test included 120 trials.</p> <hd id="AN0175303240-12">Covariates: General language</hd> <p>The <emph>Sentence Completion Test</emph> was designed to measure reading‐based semantic processing (Cui et al., [<reflink idref="bib9" id="ref116">9</reflink>]). A sentence with a missing word was presented in the upper part of the screen and participants needed to complete the sentence by selecting one of two candidate words presented underneath. This test had 120 trials and lasted 5 min.</p> <hd id="AN0175303240-13">Covariates: Related cognitive abilities</hd> <p>The <emph>Choice Reaction Time Test</emph> was designed to assess processing speed. For all 30 trials, a fixation cross was first presented in the centre of the screen and a white dot was then presented either to its left or right. Participants needed to press a key that corresponded to the location (right or left) of the white dot as quickly as possible.</p> <p>The <emph>Nonverbal Matrix Reasoning Test</emph> was used to assess general intelligence and was a shortened form of Raven's Progressive Matrices (Raven, [<reflink idref="bib50" id="ref117">50</reflink>]). Participants were asked to select a line drawing from among six to eight choices to complete the missing image of a logical progression of images. The test had 76 trials and lasted 10 min.</p> <p>The <emph>Syllogism Reasoning Test</emph> was designed to assess logical reasoning ability. Three sentences were presented on a screen, the first two expressing premises and the third expressing the conclusion. All sentences used words from daily life. Participants needed to judge if the conclusion was True or False. For example, 'Some airplanes are vehicles, no cars are airplanes, therefore some cars are vehicles'. The test had 32 trials and lasted 3 min.</p> <p>The <emph>Mental Rotation Test</emph> was used to assess spatial processing ability and was adapted from Shepard and Metzler ([<reflink idref="bib59" id="ref118">59</reflink>]). In each trial, a three‐dimensional figure was presented in the upper part of the screen and two others were presented underneath. Participants were asked to choose the bottom figure that matched the top figure as if it had been rotated. The rotation angles ranged from 15° to 345°. This test included 180 trials and lasted 3 min.</p> <p>The <emph>Figure Matching Test</emph> assessed visual perception. In each trial, one picture was presented on the left side of the screen and one to five pictures were presented on the right. Participants needed to judge whether the picture on the left matched any of those on the right. Participants needed to finish all 120 trials presented in random order.</p> <hd id="AN0175303240-14">Data analyses</hd> <p>We first conducted correlation analyses to investigate relationships among the tests. Next, we performed hierarchical linear regression analyses to reveal the relative contribution of a factor or a group of related factors after controlling for others. We included six steps in this analysis based on types of mathematic ability. First, general cognitive processing was entered (general intelligence, visuospatial processing, processing speed and logical reasoning). Next, general language processing was entered. This was independent of general cognitive processing because general language as a tool to express verbalized mathematics was particularly important in the present study (Dowker et al., [<reflink idref="bib10" id="ref119">10</reflink>]; Essien, [<reflink idref="bib14" id="ref120">14</reflink>]; Hecht et al., [<reflink idref="bib21" id="ref121">21</reflink>]; Imbo et al., [<reflink idref="bib24" id="ref122">24</reflink>]; Koponen et al., [<reflink idref="bib28" id="ref123">28</reflink>]; Purpura & Ganley, [<reflink idref="bib46" id="ref124">46</reflink>]; Vukovic & Lesaux, [<reflink idref="bib67" id="ref125">67</reflink>]). The third step was approximate number sense, which separated from symbolic mathematics (Cui et al., [<reflink idref="bib9" id="ref126">9</reflink>]; Zhang et al., [<reflink idref="bib75" id="ref127">75</reflink>]; Zhou et al., [<reflink idref="bib80" id="ref128">80</reflink>]). The arithmetic computation was the fourth step, and the symbolic arithmetic principle was the fifth step. The verbalized arithmetic principle test was then entered in the final step to analyse its respective role in mathematics achievement.</p> <p>We also used a path model to visualize all variables as an integral whole, and provide more details about the indirect relations of tests. Path analysis was performed using maximum likelihood estimation in IBM SPSS Amos 20 software. According to Kline ([<reflink idref="bib27" id="ref129">27</reflink>]), a good model fit is indicated by a non‐significant Chi‐squared value, comparative fit index (CFI) values greater than.90, and standard root mean square residual (SRMR) values less than.08. The hypothesized path model assumes that verbalized mathematics can substantially contribute directly to mathematics achievement.</p> <hd id="AN0175303240-15">RESULTS</hd> <p>The mathematics achievement score for one participant was an outlier based on the assumptions of hierarchical regression. Thus, data from 240 participants were analysed.</p> <p>The means and standard deviations of scores for all tests are shown in Table 1. All tests showed acceptable split‐half reliabilities (.72–.99). The self‐adapted mathematics achievement test had different levels of questions that increased in difficulty. The random division of questions can generate two arrays of questions that do not have matching difficulty levels. This may lead to a lower split‐half reliability (.77) than what is found for other types of tests.</p> <hd id="AN0175303240-16">Correlations among all measures</hd> <p>Pearson's correlation coefficients among all measures are displayed in Table 2. Statistical significance was set at a Bonferroni corrected <emph>p</emph>‐value of.05 (Bonferroni correction: uncorrected <emph>p</emph>‐value × 91, because there were 91 comparisons). Both verbalized and symbolic mathematic principles were closely related to each other (<emph>r</emph> = .561, corrected <emph>p</emph> < .05), as were simple and complex arithmetic computations (<emph>r</emph> = .571, corrected <emph>p</emph> < .05). Scores on the verbalized arithmetic principle test and the symbolic arithmetic principle test were very closely correlated with the mathematics achievement score (<emph>verbalized</emph>: <emph>r</emph> = .548, corrected <emph>p</emph> < .05; <emph>symbolic: r</emph> = .502, corrected <emph>p</emph> < .05).</p> <p>2 TABLE Intercorrelations between all measures.</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left">1</th><th align="left">2</th><th align="left">3</th><th align="left">4</th><th align="left">5</th><th align="left">6.1</th><th align="left">6.2</th><th align="left">7</th><th align="left">8</th><th align="left">9</th><th align="left">10</th><th align="left">11.1</th><th align="left">11.2</th></tr></thead><tbody valign="top"><tr><td align="left">1. Mathematics achievement</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">2. Verbalized arithmetic principle</td><td align="char" char=".">.55*</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">3. Symbolic arithmetic principle</td><td align="char" char=".">.50*</td><td align="char" char=".">.56*</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">4. Simple arithmetic computation</td><td align="char" char=".">.32*</td><td align="char" char=".">.42*</td><td align="char" char=".">.38*</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">5. Complex arithmetic computation</td><td align="char" char=".">.39*</td><td align="char" char=".">.46*</td><td align="char" char=".">.44*</td><td align="char" char=".">.57*</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">6.1 Numerosity comparison (RT)</td><td align="char" char=".">.05</td><td align="char" char=".">−.04</td><td align="char" char=".">−.02</td><td align="char" char=".">−.03</td><td align="char" char=".">−.11</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">6.2 Numerosity comparison (ACC)</td><td align="char" char=".">.12</td><td align="char" char=".">.22*</td><td align="char" char=".">.19</td><td align="char" char=".">.17</td><td align="char" char=".">.10</td><td align="char" char=".">.32*</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">7. Choice reaction time</td><td align="char" char=".">−.08</td><td align="char" char=".">−.11</td><td align="char" char=".">−.11</td><td align="char" char=".">−.20</td><td align="char" char=".">−.26*</td><td align="char" char=".">.33*</td><td align="char" char=".">.04</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">8. Non‐verbal matrix reasoning</td><td align="char" char=".">.36*</td><td align="char" char=".">.41*</td><td align="char" char=".">.37*</td><td align="char" char=".">.36*</td><td align="char" char=".">.32*</td><td align="char" char=".">−.07</td><td align="char" char=".">.26*</td><td align="char" char=".">−.18</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">9. Syllogism reasoning</td><td align="char" char=".">.21</td><td align="char" char=".">.32*</td><td align="char" char=".">.30*</td><td align="char" char=".">.23*</td><td align="char" char=".">.24*</td><td align="char" char=".">−.02</td><td align="char" char=".">.10</td><td align="char" char=".">−.15</td><td align="char" char=".">.22</td><td align="char" char=".">–</td><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">10. Mental rotation</td><td align="char" char=".">.27*</td><td align="char" char=".">.15</td><td align="char" char=".">.26*</td><td align="char" char=".">.17</td><td align="char" char=".">.22*</td><td align="char" char=".">−.03</td><td align="char" char=".">.23*</td><td align="char" char=".">−.24*</td><td align="char" char=".">.37*</td><td align="char" char=".">.19</td><td align="char" char=".">–</td><td align="left" /><td align="left" /></tr><tr><td align="left">11.1 Figure matching (RT)</td><td align="char" char=".">.09</td><td align="char" char=".">.04</td><td align="char" char=".">.08</td><td align="char" char=".">.14</td><td align="char" char=".">−.01*</td><td align="char" char=".">.32*</td><td align="char" char=".">.16</td><td align="char" char=".">.06</td><td align="char" char=".">−.02</td><td align="char" char=".">.01</td><td align="char" char=".">.05</td><td align="char" char=".">–</td><td align="left" /></tr><tr><td align="left">11.2 Figure matching (ACC)</td><td align="char" char=".">.30*</td><td align="char" char=".">.30*</td><td align="char" char=".">.23*</td><td align="char" char=".">.17</td><td align="char" char=".">.23*</td><td align="char" char=".">.12</td><td align="char" char=".">.29*</td><td align="char" char=".">−.04</td><td align="char" char=".">.22</td><td align="char" char=".">.08</td><td align="char" char=".">.14</td><td align="char" char=".">.17</td><td align="char" char=".">–</td></tr><tr><td align="left">12. Sentence completion</td><td align="char" char=".">.34*</td><td align="char" char=".">.50*</td><td align="char" char=".">.39*</td><td align="char" char=".">.35*</td><td align="char" char=".">.34*</td><td align="char" char=".">−.03</td><td align="char" char=".">−.15</td><td align="char" char=".">−.15</td><td align="char" char=".">.03</td><td align="char" char=".">.26</td><td align="char" char=".">.13</td><td align="char" char=".">−.04</td><td align="char" char=".">.17</td></tr></tbody></table> </ephtml> </p> <ulist> <item>3 <emph>Note</emph>: *<emph>p</emph> < .05, corrected with the Bonferroni correction method among all the correlation analyses in the table.</item> <item>4 Abbreviations: ACC, accuracy rate; RT, reaction time.</item> </ulist> <hd id="AN0175303240-17">Hierarchical regression analyses</hd> <p>In order to conduct hierarchical regression analyses, six assumptions need to be met (Osborne & Waters, [<reflink idref="bib38" id="ref130">38</reflink>]; Williams et al., [<reflink idref="bib71" id="ref131">71</reflink>]). The assumptions include linearity in the independent and dependent variables, independence of errors, homoscedasticity of errors, normal distribution of errors, an absence of multicollinearity, and no influential cases biasing the model. One outlier in mathematics achievement scores was removed. All leverage values were less than 2.0, showing the lack of a leverage point and all Cook's distances were less than 1.0, showing the lack of an influential point. Thus, all assumptions were met in the present hierarchical regression analyses.</p> <p>Results of the regression analyses for the role of the verbalized arithmetic principle test in mathematics achievement are shown in Table 3. Statistical significance was set at a Bonferroni corrected <emph>p</emph>‐value of.05 (uncorrected <emph>p</emph>‐value × 6, because there were six steps in each model). Results showed that the scores of the verbalized arithmetic principle test substantially accounted for mathematics achievement after controlling for general language ability, related cognitive abilities, approximate number sense, and symbolic mathematics ability (Δ<emph>R</emph><sups>2</sups> = .055, corrected <emph>p</emph> < .05).</p> <p>3 TABLE Hierarchical regression analyses for the role of verbalized arithmetic principle on mathematics achievement after controlling for general language ability, related cognitive abilities, approximate number sense ability and symbolic mathematics ability.</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left">Mathematics achievement</th></tr><tr><th align="left">Predictors</th><th align="left"><p>Step 1</p><italic>β</italic></th><th align="left"><p>Step 2</p><p><italic>β</italic></p></th><th align="left"><p>Step 3</p><p><italic>β</italic></p></th><th align="left"><p>Step 4</p><p><italic>β</italic></p></th><th align="left"><p>Step 5</p><p><italic>β</italic></p></th><th align="left"><p>Step 6</p><p><italic>β</italic></p></th><th align="left">VIF</th><th align="left"><p><italic>K</italic> = .2</p><p><italic>β</italic></p></th></tr></thead><tbody valign="top"><tr><td align="left">Step 1</td><td align="left">Choice reaction time</td><td align="left">.019</td><td align="left">.036</td><td align="left">.030</td><td align="left">.064</td><td align="left">.041</td><td align="left">.024</td><td align="char" char=".">1.27</td><td align="char" char=".">.025</td></tr><tr><td align="left">Non‐verbal matrix reasoning</td><td align="left">.239*</td><td align="left">.190*</td><td align="left">.209*</td><td align="left">.165</td><td align="left">.126</td><td align="left">.078</td><td align="char" char=".">1.50</td><td align="char" char=".">.065</td></tr><tr><td align="left">Syllogism reasoning</td><td align="left">.118</td><td align="left">.075</td><td align="left">.075</td><td align="left">.049</td><td align="left">.008</td><td align="left">−.023</td><td align="char" char=".">1.18</td><td align="char" char=".">.147*</td></tr><tr><td align="left">Mental rotation</td><td align="left">.131</td><td align="left">.138</td><td align="left">.151</td><td align="left">.140</td><td align="left">.110</td><td align="left">.141</td><td align="char" char=".">1.30</td><td align="char" char=".">.083</td></tr><tr><td align="left">Figure matching (RT)</td><td align="left">.049</td><td align="left">.059</td><td align="left">.054</td><td align="left">.045</td><td align="left">.027</td><td align="left">.025</td><td align="char" char=".">1.20</td><td align="char" char=".">.021</td></tr><tr><td align="left">Figure matching (ACC)</td><td align="left">.209*</td><td align="left">.185*</td><td align="left">.200*</td><td align="left">.169</td><td align="left">.148</td><td align="left">.112</td><td align="char" char=".">1.22</td><td align="char" char=".">.099</td></tr><tr><td align="left">Step 2</td><td align="left">Sentence completion</td><td align="left">–</td><td align="left">.217*</td><td align="left">.222*</td><td align="left">.168</td><td align="left">.108</td><td align="left">.029</td><td align="char" char=".">1.45</td><td align="char" char=".">.036</td></tr><tr><td align="left">Step 3</td><td align="left">Numerosity comparison (RT)</td><td align="left">–</td><td align="left">–</td><td align="left">.060</td><td align="left">.072</td><td align="left">.085</td><td align="left">.104</td><td align="char" char=".">1.39</td><td align="char" char=".">.069</td></tr><tr><td align="left">Numerosity comparison (ACC)</td><td align="left">–</td><td align="left">–</td><td align="left">−.099</td><td align="left">−.098</td><td align="left">−.109</td><td align="left">−.130</td><td align="char" char=".">1.34</td><td align="char" char=".">−.073</td></tr><tr><td align="left">Step 4</td><td align="left">Simple arithmetic computation</td><td align="left">–</td><td align="left">–</td><td align="left">–</td><td align="left">.044</td><td align="left">.021</td><td align="left">.003</td><td align="char" char=".">1.71</td><td align="char" char=".">.026</td></tr><tr><td align="left">Complex arithmetic computation</td><td align="left">–</td><td align="left">–</td><td align="left">–</td><td align="left">.208*</td><td align="left">.139</td><td align="left">.091</td><td align="char" char=".">1.78</td><td align="char" char=".">.089</td></tr><tr><td align="left">Step 5</td><td align="left">Symbolic arithmetic principle</td><td align="left">–</td><td align="left">–</td><td align="left">–</td><td align="left">–</td><td align="left">.304*</td><td align="left">.208*</td><td align="char" char=".">1.68</td><td align="char" char=".">.170*</td></tr><tr><td align="left">Step 6</td><td align="left">Verbalized arithmetic principle</td><td align="left">–</td><td align="left">–</td><td align="left">–</td><td align="left">–</td><td align="left">–</td><td align="left">.330*</td><td align="char" char=".">1.97</td><td align="char" char=".">.251*</td></tr><tr><td align="left" /><td align="left">R<sup>2</sup></td><td align="left">R<sup>2</sup> = .211*</td><td align="left">R<sup>2</sup> = .251*</td><td align="left">R<sup>2</sup> = .259*</td><td align="left">R<sup>2</sup> = .300*</td><td align="left">R<sup>2</sup> = .361*</td><td align="left">R<sup>2</sup> = .416*</td><td align="left" /><td align="left" /></tr><tr><td align="left" /><td align="left">Change of R<sup>2</sup></td><td align="left" /><td align="left">ΔR<sup>2</sup> = .040*</td><td align="left">ΔR<sup>2</sup> = .008</td><td align="left">ΔR<sup>2</sup> = .041*</td><td align="left">ΔR<sup>2</sup> = .061*</td><td align="left">ΔR<sup>2</sup> = .055*</td><td align="left" /><td align="left" /></tr></tbody></table> </ephtml> </p> <p>5 <emph>Note</emph>: *<emph>p</emph> < .05, corrected with Bonferroni correction method (uncorrected <emph>p</emph>‐value × 6).</p> <p>Finally, in order to examine the overlap of two arithmetic computations and principles variables, a ridge regression analysis was used. The analysis showed that when <emph>K</emph> = .20, the coefficient of the ridge path tends to stabilize, and the standardized coefficient is shown in Table 3. The verbalized arithmetic principle contributed significantly to the model formula and has the largest contribution to mathematics achievement, <emph>β</emph> = .251, <emph>p</emph> < .001.</p> <hd id="AN0175303240-18">Path model for mathematics achievement</hd> <p>We constructed a path model to describe the structural relations between all measures (Figure 2). This path model included two layers. Verbalized mathematics ability is different from general language ability, reflecting a deep conceptual understanding of mathematical knowledge. Therefore, general language ability was placed in the first layer and verbalized mathematics was placed in the second layer between general language ability and mathematics achievement. The other related cognitive abilities were the same layer with general language, and the other mathematical abilities were the same layer with verbalized mathematics. The hypothesized model was a good fit for the data, with values of <emph>χ</emph><sups>2</sups>(<reflink idref="bib8" id="ref132">8</reflink>) = 4.188, <emph>p</emph> = .840, CFI = 1.000, and SRMR = .010. The model showed that only verbalized arithmetic principle test substantially contributed to mathematics achievement after Bonferroni correction (uncorrected <emph>p</emph>‐value × 55 because there were 55 comparisons). Sentence completion contributed to performance on the verbalized arithmetic principle test in the path model. This is consistent with the partial correlation analysis in which sentence completion scores significantly correlated with those on the verbalized arithmetic principle test (<emph>r</emph> = .435, <emph>p</emph> < .001), after controlling for related cognitive abilities (choice reaction time, non‐verbal matrix reasoning, syllogism reasoning, mental rotation and figure matching). Notably, although sentence completion correlated significantly with mathematics achievement when controlling for related cognitive abilities (<emph>r</emph> = .225, <emph>p</emph> = .001), this correlation was no longer significant after controlling for the verbalized arithmetic principle test (<emph>r</emph> = .066, <emph>p</emph> = .317).</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/6KX/01mar24/bjep12632-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="bjep12632-fig-0002.jpg" title="2 A path model for the relation of verbalized arithmetic principle and mathematics achievement. Note: *p < .05, corrected with Bonferroni correction method (uncorrected p‐value × 55 comparisons). ACC, accuracy rate; RT, reaction time." /> </p> <p></p> <hd id="AN0175303240-20">DISCUSSION</hd> <p>The current study investigated the unique role of verbalized mathematics (specifically, mathematical principles) in mathematical achievement. Results supported the hypothesis that verbalized mathematics independent of general language, related cognitive abilities and symbolic mathematics ability. This result provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three‐component mathematics model.</p> <p>The present study contributes to the research field in two ways. First, results indicated that verbalized mathematics ability has a unique prediction on mathematics achievement after controlling for general language, related cognitive abilities and symbolic mathematics ability. Similar findings in previous studies of verbalized mathematics could not rule out the possibility that they resulted from symbolic mathematics ability (Lin, [<reflink idref="bib33" id="ref133">33</reflink>]; Peng & Lin, [<reflink idref="bib39" id="ref134">39</reflink>]; Powell et al., [<reflink idref="bib42" id="ref135">42</reflink>]; Purpura et al., [<reflink idref="bib48" id="ref136">48</reflink>]; Purpura & Logan, [<reflink idref="bib47" id="ref137">47</reflink>]; Purpura & Reid, [<reflink idref="bib49" id="ref138">49</reflink>]; Toll & Van Luit, [<reflink idref="bib62" id="ref139">62</reflink>], [<reflink idref="bib63" id="ref140">63</reflink>]; Ufer & Bochnik, [<reflink idref="bib65" id="ref141">65</reflink>]). The current result provides important evidence for the independence of verbalized mathematics in the three‐component mathematics model, instead of being a straightforward derivative of symbolic mathematics. Second, we investigated verbalized mathematics from the perspective of principles rather than vocabulary and indicated that verbalized mathematical principles also correlate with mathematics achievement, consistent with the role of mathematical vocabulary. Thus, this study has extended our knowledge of verbalized mathematics.</p> <hd id="AN0175303240-21">Conceptual understanding of mathematical knowledge</hd> <p>Previous studies showed that the understanding of conceptual mathematical knowledge is helpful for mathematics achievement (Adeleke, [<reflink idref="bib1" id="ref142">1</reflink>]; Hiebert & Wearne, [<reflink idref="bib22" id="ref143">22</reflink>]; McNeil et al., [<reflink idref="bib36" id="ref144">36</reflink>]; Rittle‐Johnson, [<reflink idref="bib52" id="ref145">52</reflink>]; Rittle‐Johnson et al., [<reflink idref="bib54" id="ref146">54</reflink>]; Rittle‐Johnson & Alibali, [<reflink idref="bib53" id="ref147">53</reflink>]; Sidney et al., [<reflink idref="bib60" id="ref148">60</reflink>]). The verbalized arithmetic principle test used in the current study assessed the conceptual processing of mathematical knowledge. For example, one question asked whether the following is True or False: 'Exchanging the positions of multipliers in a successive multiplication expression will not change the result'. To correctly answer this question, participants needed to understand the commutative law of multiplication. Following previous studies on the brain organization of processing verbalized arithmetic principles, it can easily activate the semantic network including the left middle temporal gyrus and angular gyrus (Liu et al., [<reflink idref="bib35" id="ref149">35</reflink>]). The activation could be explained as the conceptual processing of math knowledge. Processing verbalized arithmetic principles can be treated as a type of conceptual understanding of math knowledge. Processing symbolic arithmetic principle also activated the semantic network in the brain (Liu et al., [<reflink idref="bib34" id="ref150">34</reflink>]), which is also a type of conceptual understanding of math knowledge.</p> <p>The conceptual mathematical knowledge tested by the verbalized arithmetic principles test was not directly used in the mathematics achievement test. This study determined the relationship between verbalized mathematics and mathematics achievement after controlling for symbolic mathematics ability. Indeed, we tried to ensure that the problems in the mathematics achievement test and verbalized arithmetic principle test were not similar. First, the verbalized arithmetic principle was not directly tested in the self‐adapted mathematics achievement test by recruited undergraduates. That is the mathematics achievement tests in primary school students must contain a lot of arithmetic principles taught in primary school, which must lead to a significant correlation between the verbalized arithmetic principles test and mathematics achievement. While the mathematics achievement in undergraduates mostly depends on the advanced knowledge learned in high school. The knowledge of arithmetic principles in mathematics achievement could be avoided as much possible as. Second, the knowledge measured by the verbalized arithmetic principle test was controlled in a matched symbolic arithmetic principle test. Each problem in a trial of the symbolic arithmetic principle test corresponded to a problem on the verbalized arithmetic principle test (e.g. 'A + B = B + A' and 'Switching the order of two numbers added together does not change their sum'). Third, arithmetic computation ability, which is fundamental to mathematics achievement (Fuchs et al., [<reflink idref="bib17" id="ref151">17</reflink>]; Zhou et al., [<reflink idref="bib80" id="ref152">80</reflink>]), could also be activated when processing verbalized arithmetic principles and was also controlled for.</p> <hd id="AN0175303240-22">Mathematics and general language</hd> <p>The current study found that the combination of general language and mathematics as verbalized mathematics indeed contributes to mathematics achievement, even after controlling for general cognitive covariates. Although some evidence indicates a dissociation between general language and mathematics (Landerl et al., [<reflink idref="bib30" id="ref153">30</reflink>], [<reflink idref="bib31" id="ref154">31</reflink>]; Vanbinst et al., [<reflink idref="bib66" id="ref155">66</reflink>]; Wilson et al., [<reflink idref="bib72" id="ref156">72</reflink>]), increasing evidence shows that they are related. For example, behavioural evidence has shown that general language ability correlates with mathematics achievement (Dowker et al., [<reflink idref="bib10" id="ref157">10</reflink>]; Essien, [<reflink idref="bib14" id="ref158">14</reflink>]; Hecht et al., [<reflink idref="bib21" id="ref159">21</reflink>]; Imbo et al., [<reflink idref="bib24" id="ref160">24</reflink>]; Koponen et al., [<reflink idref="bib28" id="ref161">28</reflink>]; Purpura & Ganley, [<reflink idref="bib46" id="ref162">46</reflink>]). Recent studies have also indicated a connection between brain areas used in general language processing and those used for mathematics (Cui et al., [<reflink idref="bib7" id="ref163">7</reflink>]; Li et al., [<reflink idref="bib32" id="ref164">32</reflink>]; Liu et al., [<reflink idref="bib35" id="ref165">35</reflink>]; Wang et al., [<reflink idref="bib68" id="ref166">68</reflink>]; Zhou et al., [<reflink idref="bib79" id="ref167">79</reflink>]). One study found that verbalized arithmetic principles produced stronger functional connectivity between the middle temporal gyrus (language region) and the intraparietal sulcus (mathematics region) than did general language or symbolic mathematics alone (Liu et al., [<reflink idref="bib35" id="ref168">35</reflink>]). The current study also suggests that general language is involved in processing verbalized mathematics. This involvement was validated by a path model in which the association between general language and mathematics achievement was fully mediated by scores on the verbalized arithmetic principle test.</p> <hd id="AN0175303240-23">Application in education</hd> <p>Verbalized mathematics has several applications in mathematical education. First, verbalized arithmetic principles still have a unique effect after controlling for symbolic arithmetic principles. More attention should be paid to the function of the verbalized mathematics in mathematical education. Second, education using verbalized mathematics should pay attention to the principles in addition to the vocabulary. An education that emphasizes principles requires teachers and students to attach importance to the connection between mathematical terms and to express mathematical principles with general language. All these together show that verbalized mathematics is very important, and should be used extensively in class. The specific methods of using verbalized mathematics in education can cover a wide range of practices, including mathematical speech, mathematical reading, mathematical self‐explanation, verbalized mathematics in textbooks, verbalized concept maps and so on, which have been shown to promote the conceptual understanding of mathematical knowledge (Hiebert & Wearne, [<reflink idref="bib22" id="ref169">22</reflink>]; McNeil et al., [<reflink idref="bib36" id="ref170">36</reflink>]; Rittle‐Johnson et al., [<reflink idref="bib54" id="ref171">54</reflink>]; Sidney et al., [<reflink idref="bib60" id="ref172">60</reflink>]).</p> <p>Although the correlation is clear, a causal relationship could not be obtained and is an important component of future studies.</p> <hd id="AN0175303240-24">AUTHOR CONTRIBUTIONS</hd> <p> <bold>Jiaxin Cui:</bold> Data curation; funding acquisition; methodology; project administration; writing – review and editing. <bold>Li Wang:</bold> Data curation; formal analysis; methodology; software; visualization. <bold>Dawei Li:</bold> Writing – original draft; writing – review and editing. <bold>Xinlin Zhou:</bold> Conceptualization; funding acquisition; resources; supervision; writing – review and editing.</p> <hd id="AN0175303240-25">ACKNOWLEDGEMENTS</hd> <p>This research was supported by the STI 2030‐Major Projects (2021ZD0200500), a grant from the Natural Science Foundation of China (62277015) and a grant from the Science and Technology Project of Hebei Education Department (ZD2020158).</p> <hd id="AN0175303240-26">CONFLICT OF INTEREST STATEMENT</hd> <p>We have no known conflict of interest to disclose.</p> <hd id="AN0175303240-27">DATA AVAILABILITY STATEMENT</hd> <p>The data that support the findings of this study are available from the corresponding author upon reasonable request.</p> <ref id="AN0175303240-28"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref80" type="bt">1</bibl> <bibtext> Jiaxin Cui and Li Wang contributed equally to this work.</bibtext> </blist> </ref> <ref id="AN0175303240-29"> <title> REFERENCES </title> <blist> <bibtext> Adeleke, M. 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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Verbalized Arithmetic Principles Correlate with Mathematics Achievement
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Jiaxin+Cui%22">Jiaxin Cui</searchLink><br /><searchLink fieldCode="AR" term="%22Li+Wang%22">Li Wang</searchLink><br /><searchLink fieldCode="AR" term="%22Dawei+Li%22">Dawei Li</searchLink><br /><searchLink fieldCode="AR" term="%22Xinlin+Zhou%22">Xinlin Zhou</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-3530-0922">0000-0002-3530-0922</externalLink>)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22British+Journal+of+Educational+Psychology%22"><i>British Journal of Educational Psychology</i></searchLink>. 2024 94(1):41-57.
– 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: 17
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2024
– 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="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Verbal+Communication%22">Verbal Communication</searchLink><br /><searchLink fieldCode="DE" term="%22Verbal+Development%22">Verbal Development</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Undergraduate+Students%22">Undergraduate Students</searchLink><br /><searchLink fieldCode="DE" term="%22College+Mathematics%22">College Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Language+Skills%22">Language Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Cognitive+Ability%22">Cognitive Ability</searchLink><br /><searchLink fieldCode="DE" term="%22Number+Concepts%22">Number Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Coding%22">Coding</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Modalities%22">Learning Modalities</searchLink>
– Name: Subject
  Label: Geographic Terms
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22China+%28Beijing%29%22">China (Beijing)</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1111/bjep.12632
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0007-0998<br />2044-8279
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Background: When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to mathematical vocabulary or educational practice. However, these studies did not exclude the role of symbolic mathematics ability, and almost no research has focused on verbalized mathematical principles. Aims: This study is aimed to investigate whether verbalized mathematics ability independently predicts mathematics achievement. The current study hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability. Sample: A sample of 241 undergraduates (136 males, 105 females, mean age = 21.95, SD = 2.38) in Beijing, China. Methods: A total of 12 tests were used, including a verbalized arithmetic principle test, a mathematics achievement test, and tests on general language (sentence completion test), symbolic mathematical ability (including symbolic arithmetic principles test, simple arithmetic computation and complex arithmetic computation), approximate number sense ability (numerosity comparison test) and several related cognitive covariates (including the non-verbal matrix reasoning, the syllogism reasoning, mental rotation, figure matching and choice reaction time). Results: Results showed that the processing of verbalized arithmetic principles displayed a significant role in mathematics achievement after controlling for general language, related cognitive abilities, approximate number sense ability and symbolic mathematics ability. Conclusions: The results suggest that verbalized mathematics ability was an independent predictor and provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three-component mathematics model.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2024
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1410937
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1410937
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1111/bjep.12632
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 17
        StartPage: 41
    Subjects:
      – SubjectFull: Verbal Communication
        Type: general
      – SubjectFull: Verbal Development
        Type: general
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: Mathematics Achievement
        Type: general
      – SubjectFull: Undergraduate Students
        Type: general
      – SubjectFull: College Mathematics
        Type: general
      – SubjectFull: Foreign Countries
        Type: general
      – SubjectFull: Language Skills
        Type: general
      – SubjectFull: Cognitive Ability
        Type: general
      – SubjectFull: Number Concepts
        Type: general
      – SubjectFull: Coding
        Type: general
      – SubjectFull: Learning Modalities
        Type: general
      – SubjectFull: China (Beijing)
        Type: general
    Titles:
      – TitleFull: Verbalized Arithmetic Principles Correlate with Mathematics Achievement
        Type: main
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            NameFull: Jiaxin Cui
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            NameFull: Li Wang
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            NameFull: Dawei Li
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            NameFull: Xinlin Zhou
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            – D: 01
              M: 01
              Type: published
              Y: 2024
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              Value: 0007-0998
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              Value: 94
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: British Journal of Educational Psychology
              Type: main
ResultId 1