Learning and Individual Differences 100 (2022) 102230 Contents lists available at ScienceDirect Learning and Individual Differences journal homepage: www.elsevier.com/locate/lindif Spatial processing rather than logical reasoning was found to be critical for mathematical problemsolving Mingxin Yu a, b, c, 1, Jiaxin Cui c, d, 1, Li Wang a, b, 1, Xing Gao c, d, Zhanling Cui c, d, Xinlin Zhou a, b, c, * a State Key Laboratory of Cognitive Neuroscience and Learning, Beijing Normal University, Beijing 100875, China b Advanced Innovation Center for Future Education, Beijing Normal University, China c Research Association for Brain and Mathematical Learning, China d College of Education, Hebei Normal University, Shijiazhuang 050024, China A R T I C L E I N F O A B S T R A C T Keywords: Students' ability to solve mathematical problems is a standard mathematical skill; however, its cognitive cor Logical reasoning relates are unclear. Thus, this study aimed to examine whether spatial processing (mental rotation, paper folding, Spatial processing and the Corsi blocks test) and logical reasoning (abstract and concrete syllogisms) were correlated with math Mathematical problemsolving ematical problemsolving (word problems and geometric proofing) for college students. The regression results showed that after controlling for gender, age, general IQ, language processing, cognitive processing (visual perception, attention, and memory skills), and number sense and arithmetic computation skills, spatial pro cessing skills still predicted mathematical problemsolving and geometry skills in Chinese college students. Contrastingly, logical reasoning measures related to syllogisms did not predict after controlling for these vari ables. Further, notably, it did not correlate significantly with geometry performance when no control variables were included. Our results suggest that spatial processing is a significant component of math skills involving word and geometry problems (even after controlling for multiple key cognitive factors). 1. Introduction 2007). Therefore, to explore the independent prediction of spatial pro cessing and logical reasoning on mathematical problemsolving (espe Over the past few decades, an emphasis on mathematical problem cially word problems and geometry) is necessary to control these other solving has intensified in international mathematical education (Niss related cognitive factors that influence mathematical problemsolving, et al., 2017), including word problems and geometry problems solving. logical reasoning and spatial processing. Additionally, numerous studies have investigated the cognitive mecha nisms underlying mathematical problemsolving (e.g., Boonen et al., 1.1. Spatial processing in mathematical problemsolving 2013; Cummins et al., 1988; Hegarty & Kozhevnikov, 1999). Further more, understanding these mechanisms can help us improve mathe Spatial processing is the ability to represent, transform, generate, matical education. and recall visual information (Linn & Petersen, 1985). This multidi Spatial processing and logical reasoning are two crucial general mensional concept includes a series of psychological manipulations cognitive correlates of mathematical problemsolving (e.g., Chuderski & involving visual information (Uttal et al., 2013). Many tasks have been Jastrzebski, 2018; Duffy et al., 2020; GomezVeiga et al., 2018; Hawes & used to measure spatial processing, including paper folding, three Ansari, 2020; Kleemans et al., 2018; Rothenbusch et al., 2018). How dimensional (3D) mental rotation, and the Corsi block test. Paper ever, mathematical problemsolving, logical reasoning and spatial pro folding is a classic task for measuring mental manipulation and spatial cessing are supported by some other cognitive factors, including visualisation (e.g., Boonen et al., 2013; Boonen et al., 2014; Burte et al., attention, processing speed, memory, language processing, etc. (e.g., 2017; Wei et al., 2012). 3D mental rotation is a typical task for Bull & Sherif, 2001; Fuchs et al., 2006; Fürst & Hitch, 2000; Gutierrez measuring mental rotation ability (e.g., Boonen et al., 2013; Boonen et al., 2019; Knauff et al., 2003; Mayer et al., 1984; Swanson & Kim, et al., 2014; Delgado & Prieto, 2004; Oostermeijer et al., 2014; Tolar * Corresponding author at: State Key Laboratory of Cognitive Neuroscience and Learning, Beijing Normal University, Beijing 100875, China. Email address: [email protected] (X. Zhou). 1 Mingxin Yu, Jiaxin Cui and Li Wang are considered as Cofirst authors. https://doi.org/10.1016/j.lindif.2022.102230 Received 24 March 2021; Received in revised form 22 September 2022; Accepted 9 October 2022 Available online 21 October 2022 10416080/© 2022 Elsevier Inc. All rights reserved. M. Yu et al. Learning and Individual Differences 100 (2022) 102230 et al., 2009). Finally, the Corsi blocks test measures spatial working 1.2. Logical reasoning in mathematical problemsolving memory (e.g., Andersson & Lyxell, 2007; Berg, 2008; Geary et al., 2008). On the one hand, a large number of previous studies have revealed Logical reasoning refers to making deductions and reaching novel the close relationship between spatial processing and mathematical conclusions according to given premises (Markovits & Doyon, 2010), problemsolving. also namely deductive reasoning. Math curricula often treat logical First, there was significant correlation between spatial processing reasoning as a core ability. In the 1980s, educators even began to discuss and mathematical problemsolving, not only for children (e.g., Bates the role of logic in teaching proofs, emphasising that logical reasoning et al., 2021; Gilligan et al., 2018), but also for adults (e.g., Wei et al., should be brought into mathematics teaching and that teachers should 2012; Xie et al., 2020). For example, Gilligan et al. (2018) explored the pay more attention to logical reasoning (DurandGuerrier et al., 2011). developmental relationships between mathematics and spatial skills in China's mathematics curriculum standards in compulsory education children aged 6–10 years and found that overall spatial skills explained (2011 edition) point out that reasoning ability is one of the ten core 5 %–14 % of the variation across three mathematics performance abilities for mathematics. measures (standardized mathematics skills, approximate number sense, Although mathematics and logical reasoning are generally thought and number line estimation skills). In this study, only language and age to be closely related, however, there is essential distinction between were included as control variables. Wei et al. (2012) found a close them, mathematical problemsolving typically rely on mathematical correlation between spatial processing ability and advanced mathe knowledge, but logical reasoning relies on rule derivation. If premise is matical processing of college students, controlling factors such as age, correct or incorrect does not affect the correctness of logic reasoning. gender, general intelligence and language processing. From the perspective of logical thinking, people usually do not Second, the spatial processing can predict future mathematical follow the rules of rational thinking but rather the facts and experiences. achievement (e.g., BlattoVallee et al., 2007; Kyttälä & Björn, 2014). For Many logical reasoning errors have been shown to reflect the lack of example, BlattoVallee et al. (2007) found that spatial relationship application of logic rules in logical thinking (e.g., Brisson et al., 2018; competence (measured with Primary Mental Abilities Spatial Relations Dias & Harris, 1988; Hawkins et al., 1984; Nys et al., 2022; Scribner, Test, Optometric Extension Program, 1995, and Revised Minnesota 1977), including ‘empirical bias’ (Scribner, 1977) and content effect Paper Form Board Test, Likert & Quasha, 1994) could explain over 20 % (Brisson et al., 2018). ‘Empirical bias’ refers to people often making of the variance in problemsolving scores among junior high school, high conclusions based on their personal knowledge and experience (Scrib school, and college students. ner, 1977). For example, Dias and Harris (1988) found that children Third, training studies showed that the training of spatial processing perform worse when the premise of syllogistic reasoning contradicts ability can promote mathematical problemsolving (e.g., Cheng & Mix, empirical facts (e.g., all fish live in trees, Tot is a fish, and Tot lives in 2013; Lowrie et al., 2017; Schmitt et al., 2018). For example, one study trees). In addition to empirical bias, the content effect also shows that showed that training students to improve mental rotation and spatial deductive reasoning performance is influenced by reason content. visualisation in first and sixthgrade students led to significant Brisson et al. (2018) and Nys et al. (2022) showed that logical reasoning enhancement in mathematics scores (Cheng & Mix, 2013). is controlled by the background knowledge of the participants and the Fourth, the close association can also be found between spatial reason content for both adults and children. However, for mathematical processing and different types of mathematical problemsolving, problemsolving, people usually follow the mathematical knowledge including word problem solving (e.g., Männamaa et al., 2012; Mix rather than rule deduction. For example, Studies by mathematics edu et al., 2016) and geometric proofing (e.g., Harris et al., 2021; Karaman & cators have shown that students tend to use empirical arguments rather Toğrol, 2009). For instance, Harris et al. (2021) found that spatial pro than deductive proofs when solving geometric problems (e.g., Balacheff, cessing significantly predicted geometric performance in fifthand 1988; Chazan, 1993; Martin & Harel, 1989). eighthgraders. Furthermore, Mix et al. (2016) researched the re Although the relationship between logical reasoning and mathe lations among various spatial and mathematics skills were assessed in a matics has attracted the extensive attention, only a few empirical studies crosssectional study of 854 children from kindergarten, third, and sixth have shown an association between logical reasoning and mathematical grades (i.e., 5 to 13 years of age). The results showed that the mathe problemsolving. matical tasks that predicted the most significant variance in spatial skill There have been two studies on the association between mathe were place value, word problems, calculation, fraction concepts, and matical problemsolving and relational reasoning, a type of logical algebra. reasoning (Morsanyi et al., 2013, 2017). The two studies showed On the other hand, the close association could be due to the consistent correlation. For example, Morsanyi et al. (2017) showed that involvement of spatial model constructed during problem solving. For relational reasoning is associated with number line estimation and open example, researchers have put forward the concept of “problem space” mathematical word problemsolving for adults after controlling for in problem solving for a long time. The problem space is extracting general intelligence. There have been three studies on the relationship external problem environment as an internal representation, and search between syllogism and mathematical problemsolving, without consis solving path in this mental space (Newell & Simon, 1972). Hawes and tent findings (Duque de Blas et al., 2021; GomezVeiga et al., 2018; Ansari (2020) proposed a spatial modelling account, which indicates Kleemans et al., 2018). Two of the studies found correlation between that spatial visualisation provides a “mental blackboard” of which nu syllogism and mathematical problemsolving (GomezVeiga et al., 2018; merical relations and operations in mathematical problemsolving can Kleemans et al., 2018). For example, Kleemans et al. (2018) showed that be modelled and visualized. syllogistic reasoning in fifthgrade students was related to their perfor The spatial model is a type of structure reflecting the association of mance on geometric problems, controlling for age, sex and general in key problem information (Boonen et al., 2014; Hegarty & Kozhevnikov, telligence (Kleemans et al., 2018). Another study that did not find the 1999). For example, Hegarty and Kozhevnikov (1999) showed that association (Duque de Blas et al., 2021). There have been four studies on schematic spatial representations (representing the quantity relations the association between mathematical problemsolving and conditional between objects and imagining spatial transformations, such as a 15 m reasoning, with inconsistent findings (Duque de Blas et al., 2021; long path with trees spaced 5 m apart) were associated with success in GomezVeiga et al., 2018; Morsanyi et al., 2017; Wong, 2018). Two of mathematical problemsolving. Contrastingly, pictorial representations the studies found correlation (GomezVeiga et al., 2018; Morsanyi et al., (constructing vivid and detailed visual pictures, such as a tree with 2017), but two studies did not (Duque de Blas et al., 2021; Wong, 2018). leaves) were negatively correlated with success. For example, Wong (2018) found that after controlling for intelligence, working memory and language processing, mathematical problem solving ability of fourthgrade students was significantly correlated 2 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 with conditional reasoning, but this significant correlation disappeared presented differently, but the spatial processing was fundamental in after further controlling for calculation and numerical sentence con solving both problems. However, logical reasoning relies on empirical struction ability. knowledge rather than rules, leading to incorrect judgements. Con All the 6 studies controlled general intelligence measured with non trastingly, mathematical problemsolving needs to be based on mathe verbal matrix reasoning. Only two studies (Duque de Blas et al., 2021; matical knowledge. Therefore, logical reasoning is not a necessary Wong, 2018) did not show the association. The only difference between psychological processing component in mathematical problemsolving. the two studies and other four studies is that the two studies controlled In conclusion, the current investigation is to explore the independent additional language processing or additional number sentence con prediction of logical reasoning and spatial processing on mathematical struction and calculation. It seems that how to control covariates could problemsolving, and hypothesised that spatial processing has a unique be critical to check the association. contribution to mathematical problemsolving while logical reasoning Additionally, Duque de Blas et al.'s study (2021) used two types of does not. To verify this hypothesis, we conducted two studies, and their logical reasoning (conditional reasoning and syllogistic reasoning), characteristics are presented below. showing consistent findings for the two types of reasoning. GomezVeiga et al.'s study (2018) also used two types of logical reasoning (conditional 2. Study 1 reasoning and syllogistic reasoning), also showing consistent findings for the two types of reasoning. Therefore, it could be effective to apply Study 1 was designed to examine the independent roles of spatial and one type of reasoning to explore the association between logic reasoning logical reasoning in word problemsolving after controlling other and mathematical problemsolving. Moreover, syllogism could be a factors. common form to logic, because “Scholastic logicians thought that almost all arguments purporting to be logical could be expressed in syllogism” (p427, in a review by Khemlani & JohnsonLaird, 2012). Combining the 2.1. Methods two points, this study focuses on the relationship between logical reasoning ability measured by syllogism and mathematical problem 2.1.1. Participants solving. A total of 360 participants (179 men, 181 women; mean age = 20.84, SD = 1.76; age range = 17.3–27.8 years) were recruited from 10 uni 1.3. Present study versities in China. All the participants were righthanded native Man darin speakers with normal or correctedtonormal visual acuity. Before Numerous studies have shown that spatial processing skills are pre participation, participants provided written informed consent after the dictors of mathematical problemsolving skills, but relatively few studies investigation was fully explained. This study was approved by the have examined logical reasoning, and their results are inconsistent. institutional review board of the State Key Laboratory of Cognitive Thus, the present study explore the independent prediction of logical Neuroscience and Learning. reasoning and spatial processing on mathematical problemsolving, including the important control variables (e.g., attention, general IQ, 2.1.2. Tests memory, language), while addressing a specific question about the Seventeen tests were conducted. The testing data has been uploaded relative importance of spatial vs. logical deductive reasoning for math to the psychological research platform (www.dweipsy.com/lattice). ematical problemsolving. (Wei et al., 2012; Wei et al., 2016; Zhou et al., 2015). An illustration of From a cognitive point of view, some cognitive factors (including the trial for each test is shown in Fig. 1. attention, processing speed, memory, language, etc.), that could be involved when participants performed the tasks to measure mathemat 2.1.2.1. Mathematical word problems. A typical mathematical problem ical problemsolving, logical reasoning and spatial processing (e.g., Bull solving exercise assesses the ability to solve word problems, but not & Sherif, 2001; Fuchs et al., 2006; Fürst & Hitch, 2000; Gutierrez et al., arithmetic problems. All 15 problems involve the application of complex 2019; Knauff et al., 2003; Mayer et al., 1984; Swanson & Kim, 2007). algebra. An applied math problem was presented on the computer First, previous researches show that language comprehension, compu screen in each trial until participants typed their answers to an input box tation, processing speed, working memory for verbal and visualspatial under the problem. The test duration was limited to 6 min. information, and other factors impact mathematical problemsolving. For example, the execution of a math problem is based on the reten 2.1.2.2. Logical reasoning tion of either verbal or visuospatial information (Cornoldi et al., 2012; 2.1.2.2.1. Abstract syllogistic reasoning. Three sentences were pre Re et al., 2016). Meanwhile, processing speed is the best predictor of sented on the screen in each trial, two of which expressed premises, and arithmetical competence in 7yearold students (Bull & Johnston, 1997; the third one expressed the conclusion. All the sentences used mean Fuchs et al., 2006; Swanson & Kim, 2007). Mayer et al. (1984) suggest ingless letters to make the premises abstract. The test consisted of 32 that solving mathematical problems requires good language under trials and was limited to 3 min. standing and relevant arithmetic operations and other processes to 2.1.2.2.2. Concrete syllogistic reasoning. All the sentences used achieve solutions (Mayer et al., 1984). Second, studies have shown that words from their daily lives. Otherwise, the test would be identical to the linguistic processing, spatial processing, memory, attention, among abstract reasoning test. others, may be associated with logical reasoning (e.g., Knauff et al., 2003; Yang et al., 2009). Further, nonverbal matrix reasoning and lin 2.1.2.3. Spatial processing guistic processing may be related to spatial processing (e.g., Colom et al., 2.1.2.3.1. 3D mental rotation. A 3D figure was presented on the top 2004; Gutierrez et al., 2019). Therefore, it is necessary to control some half of the screen in each trial, along with two other 3D figures. One of cognitive factors (including attention, processing speed, memory, lan the lower items was formed by rotating the upper image, and the other guage, etc.), that could be involved when participants performed the was a mirror image of the upper figure. The rotation angle ranged from tasks to measure mathematical problemsolving, logical reasoning and 15◦ to 345◦ . Participants were asked to judge which of the bottom fig spatial processing. Therefore, we can explore the independent predic ures matched the upper figure after the rotation. This test included 180 tion of logical reasoning and spatial processing on mathematical prob trials and lasted 3 min. lemsolving. 2.1.2.3.2. Paper folding. The participants were asked to imagine the Two typical mathematical problems were involved in the current folding and unfolding of pieces of paper. A 2D figure on the top of the investigation: word problems and geometry proofs. The problems were screen in each trial was a square piece of folded paper, and a hole 3 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 Fig. 1. Schematic of tests used in present study. punched in the direction of an arrow. The number of holes created 2.1.2.8.1. Visual memory. There were encoding and recalling ses depended on when the hole was punched (i.e., after onefold, the punch sions. During the encoding session, participants were asked to memorise made two holes). One of the five figures at the bottom of the screen a series of pictures presented on the screen (A total of 40 pictures) (for correctly showed where the holes would be located when the paper was example, a linedrawing tree). During the recalling session, 80 pictures completely unfolded. This test included 18 trials and was limited to 4 (40 are old, and 40 are new) were presented, and participants they made min. a judgment on if they encoded the pictures during encoding session. 2.1.2.3.3. Corsi blocks test. The dots were sequentially presented in Subjects needed to complete all the trials to ending the test. a 3 × 3 grid on a computer screen. Each dot was presented for 1000 ms, 2.1.2.8.2. Digit span. The digits were presented aurally at one digit with a blank interval of 1000 ms between the dots. After all the dots in per second frequency in each trial, and participants were asked to the trial were presented, the participants used a mouse to click the grid remember them. After hearing all digits, they were asked to type them in according to the position and order of the dot presentation. Participants the same order they had been heard, or in the reverse order. The initial finished all 10 trials. The number of dots in each trial ranged from three trial had three digits. As the trials progressed, the number of digits to seven. Each number of dots was presented twice in turn. gradually increased. The test was stopped when the participants pro vided three consecutive incorrect answers. 2.1.2.4. Arithmetic computation. In each trial, an arithmetic problem appeared on the screen. Participants had 15 s to mentally compute the 2.1.2.9. Visual perception answer and type it into the input box. There were 40 trials, including 2.1.2.9.1. Figure matching. Two sets of complex figures were pre addition, subtraction, multiplication, and division problems, with in sented sidebyside on the screen for 400 ms for each trial. The left set tegers or decimals as operators. contained only one figure, and the right set three figures. Participants were asked to judge whether the picture on the left side also appeared on 2.1.2.5. Number sense the right side. The test included 120 trials grouped into three 40trial 2.1.2.5.1. Numerosity comparison. Two dot arrays appeared simul sessions. Subjects needed to complete all the trials to ending the test. taneously on the screen for 200 ms in each trial. There were 120 trials in all. The participants were required to determine which side of the two 2.1.2.10. Response and decision speed dots had the highest number. 2.1.2.10.1. Choice reaction time. A white dot was presented on a black screen to the left or right of the white fixation cross. The partici 2.1.2.6. General IQ pants were asked to judge the left and right positions of the dot and 2.1.2.6.1. Nonverbal matrix reasoning. Each trial contained a figure respond by pressing the corresponding buttons. The test consisted of 30 with a missing segment presented on the screen and 6–8 candidate trials (half with a dot on the left and the other half on the right). segments. Participants were asked to use the mouse to choose which candidate completed the figure according to the figure's inherent regu 2.1.2.11. Language processing larity. In total, 76 trials were conducted. 2.1.2.11.1. Sentence completion. Materials were adapted from recent language examinations used in China for Grades 1 to 12. In each trial, a 2.1.2.7. Attention sentence with one missing word was presented at the centre of the 2.1.2.7.1. Visual searching. Three ‘p’s and two ‘d’s were presented screen. The sentences and choices remained on the screen until partic interspersed in a line for each trial. Each letter had one to four dashes ipants responded. There were 120 trials—this was a 5min test. configured individually or in pairs, above or below each letter. The 2.1.2.11.2. Reading comprehension. The materials used in the test target symbol was a ‘d’ with two dashes, regardless of the location of the were adapted from recent language examinations used in China for dashes (two above, two below, or one above and one below). The test entrance into college. Several paragraphs were presented on the screen included 240 trials and was limited to 4 min. in each trial, and a question about their content was presented below. Participants choose one of the four choices provided. The test consisted 2.1.2.8. Memory of 45 trials and was limited to 8 min. 4 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 2.1.3. Procedure to determine the unique contribution of logical reasoning and spatial The battery contained 17 computerised tests. The participants processing in word problems. Finally, path analysis was conducted. completed the tests in a psychological laboratory. Participants were asked to register their demographic information before the experiment 2.2. Results began. Before each test, the experimenter explained the instructions presented on the computer screen, and the participants completed a The means and standard deviations of the scores and the splithalf practice session before the formal test. All participants completed 17 reliabilities for all 17 tests are presented in Table 1. tests. The entire study lasted 2 h, and the participants were allowed to The Pearson's correlation coefficients among the scores for all 17 rest for half an hour. tests are displayed in Table 2, with a Bonferroni correction. The cor For the numerosity comparison, abstract syllogism reasoning, con rected significance in Table 2 was set at 0.05, corresponding to a pvalue crete syllogism reasoning, 3D mental rotation, figure matching, simple of 0.0003 (0.05/171 correlations). reaction time, and sentence completion, participants input their left/ The partial correlation results with Bonferroni correction (uncor right choices by pressing the ‘Q’ or ‘P’ keys on a computer keyboard, rected p = 0.0003, 0.05/171 correlations) indicated that calculation, respectively. Participants were asked to use a mouse to make their language processing, memory, attention, number sense and general IQ choices for the paper folding, spatial working memory, and reading were also significantly correlated with word problems in addition to comprehension tests, which had more than two choices. For the word spatial processing and logical reasoning (see Table 3). problem, arithmetic computation, and digit span tests, participants Table 4 shows whether logical reasoning or spatial processing had a entered their answers by typing on a numeric keyboard. unique contribution to word problems. A hierarchical multiple regres sion with Bonferroni correction was used (significance set at corrected p 2.1.4. Data analyses < 0.05). Neither type of logical reasoning played a significant role in The indices for all the tests are displayed in Table 1. The timelimited word problems after controlling for other factors other than spatial tests, including abstract syllogism reasoning, concrete syllogism processing. Spatial processing explained 5.2 % of the variance after reasoning, threedimensional mental rotation, visual memory, sentence controlling for logical reasoning and other cognitive factors (corrected p completion, used the adjusted number of correct responses as their < 0.05). score. Unanswered items are not counted as incorrect items. The Finally, Fig. 2 shows the path model of the structural relationships guessing effect in the current investigation was excluded by using the among all main variables. The latent variable ‘spatial processing’ had formula “S = RW/(n − 1)” (S: the adjusted score, R: the number of three manifest measures (threedimensional mental rotation, paper correct responses, W: the number of incorrect responses, n: number of folding, and Corsi blocks test), and logical reasoning had two manifest alternative responses to each item) to adjust the score (Guilford, 1936). measures (abstract and concrete syllogism reasoning). All variables in This was done to control for the guessing effect (Cirino, 2011; Hedden & the model were residuals controlling for age, gender, and other cognitive Yoon, 2006; Salthouse, 1994; Salthouse & Meinz, 1995). factors. The hypothesised model was a good fit for the data (χ 2 (7) = Before the formal analyses, we used the winsorising method to check 5.28, p = 0.626, RMSEA = 0.05, CFI = 1.00, SRMR = 0.02). The sig the extreme values of the data (Hogg, 1979), where the extreme values nificance level of path coefficients in the path model was Bonferroni beyond the three standard deviations were replaced by the values cor corrected and set to 0.05, corresponding to the original alpha of 0.025 responding to plus or minus three standard deviations. (0.05/2 links). Descriptive statistics for each participant were generated for each of the 17 tests (mean, standard deviation, and halfsplit reliability). Sub 2.3. Discussion sequently, Pearson's correlations were used to investigate the relation ships among all 17 tests. Then, partial correlation analyses of all The results show that after controlling for an extensive range of variables with mathematical problemsolving after controlling for age, critical variables, measures of spatial skills significantly predicted word gender, or age, gender, and nonverbal matrix reasoning were conducted. problems in college students. Contrastingly, measures of logical Next, a series of linear hierarchical regression analyses were performed reasoning skills did not. Table 1 Means and standard deviations of scores for all tests in Study 1. Test Index Mean (SD) Splithalf reliability 1. Mathematical word problems Number of correct responses 7.6 (3.0) 0.79 2. Abstract syllogism reasoning Adjusted no. of correct trials 4.0 (5.3) 0.78 3. Concrete syllogism reasoning Adjusted no. of correct trials 6.6 (5.7) 0.80 4. Threedimensional mental rotation Adjusted no. of correct trials 27.0 (9.6) 0.93 5. Paper folding Number of correct responses 8.5 (2.9) 0.86 6. Corsi blocks test Accuracy (%) 83.1 (5.6) 0.96 7. Nonverbal matrix reasoning Number of correct responses 29.0 (5.6) 0.89 8. Visual searching Number of correct responses 47.6 (30.8) 0.96 9. Visual memory Adjusted no. of correct trials 69.7 (8.9) 0.78 10. Digit span (forward) Maximum number of correct trials 9.7 (2.1) / 11. Digit span (backward) Maximum number of correct trials 8.2 (2.0) / 12.1 Figure matching (ACC) Accuracy (%) 77.8 (10.5) 0.88 12.2 Figure matching (RT) Reaction time (ms) 897 (148) 0.98 13. Simple reaction time Reaction time (ms) 366 (66) 0.96 14. Sentence completion Adjusted no. of correct trials 42.0 (6.7) 0.84 15. Reading comprehension Number of correct responses 9.0 (3.4) 0.72 16. Arithmetic computation Number of correct responses 19.7 (5.3) 0.85 17.1 Numerosity comparison (ACC) Accuracy (%) 81.3 (7.5) 0.81 17.2 Numerosity comparison (RT) Reaction time (ms) 534 (91) 0.98 Note: Adjusted no. of correct trials = S = RW/(n − 1) (S: the adjusted score, R: the number of correct responses, W: the number of incorrect responses. n: the number of alternative responses to each item). This adjustment was made to control for the effect of guessing in multiplechoice tests. Accuracy = 100 − (ResponseStandard answer) / (Standard answer + (ResponseStandard answer)) × 100. 5 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 Table 3 17.2 Partial correlations between all control variables, logical reasoning, spatial – ability and mathematical word problems and geometry problems. Predictors Mathematical Plane geometry Solid geometry 0.33* 17.1 word problems – a b a b a b 0.27* 0.13 1. Nonverbal 0.42* 0.36* 0.18 16 – – – – matrix reasoning 2. Abstract 0.29* 0.22* 0.20 0.15 0.21 0.18 − 0.09 syllogism 0.05 0.04 15 reasoning – 3. Concrete 0.34* 0.26* 0.17 0.11 0.15 0.12 syllogism − 0.05 0.33* 0.15 0.11 reasoning 14 – 4. Three 0.33* 0.25* 0.22 0.17 0.07 0.05 dimensional − 0.19* − 0.02 − 0.13 − 0.03 mental rotation 0.37* 5. Paper folding 0.50* 0.36* 0.32* 0.22 0.35* 0.31* 13 – 6. Corsi blocks test 0.15 0.07 − 0.10 − 0.09 0.01 0.02 7. Visual searching 0.27* 0.19* 0.33* 0.25* 0.23 0.20 − 0.05 − 0.18 0.21* 0.53* 0.30* 12.2 0.15 8. Visual memory 0.11 0.04 0.07 0.02 0.10 0.07 9. Digit span 0.14 0.04 0.14 0.03 0.10 0.05 – (forward) − 0.16 10. Digit span 0.27* 0.16 0.28* 0.18 0.21 0.17 0.40* 0.29* 0.22* 0.49* 12.1 0.14 0.03 (backward) – 11.1 0.17 0.05 0.12 0.06 0.15 0.12 Figure matching − 0.14 0.31* 0.35* 0.25* 0.12 0.14 0.01 0.12 (ACC) 11 11.2 0.03 0.04 0.06 0.09 − 0.03 − 0.03 – Figure matching − 0.03 (RT) 0.46* 0.11 0.03 0.12 0.11 0.18 0.05 0.06 12. Simple reaction − 0.15 − 0.09 − 0.03 0.04 − 0.08 − 0.05 10 – time 13. Sentence 0.22* 0.15 0.19 0.11 0.14 0.10 − 0.06 − 0.06 0.21* 0.35* 0.33* 0.36* 0.04 0.15 0.13 0.18 completion 9 – 14. Reading 0.29* 0.21* 0.07 − 0.06 0.06 0.00 comprehension − 0.23* 15. Arithmetic 0.46* 0.34* 0.44* 0.34* 0.31* 0.27* − 0.05 0.22* 0.22* 0.28* 0.24* 0.13 0.11 0.01 0.15 0.05 computation 8 – 16.1 Numerosity 0.22* 0.10 0.11 0.04 0.02 − 0.02 comparison − 0.02 − 0.16 − 0.03 0.26* 0.25* 0.34* 0.30* 0.22* 0.24* 0.44* 0.33* (ACC) 0.17 16.2 Numerosity 0.04 0.05 0.00 0.01 − 0.07 − 0.06 7 – comparison (RT) − 0.12 − 0.04 0.21* 0.20* Note: a/b: Column a represents the control variables are age and gender, and 0.11 0.01 0.12 0.14 0.05 0.06 0.12 0.11 0.16 column b represents the control variables are age, gender and nonverbal matrix 6 – Correlations among all the measure scores based on Pearson's correlations in Study 1. reasoning. − 0.04 − 0.17 − 0.02 0.19* 0.52* 0.21* 0.25* 0.21* 0.42* 0.22* * p < 0.05, using Bonferroni correction. 0.05 0.14 0.17 0.12 5 – However, while paperfolding skill correlated with the ability to − 0.21* solve mathematical word problems, 3D mental rotation and spatial 0.43* 0.25* 0.23* 0.36* 0.33* 0.29* 0.18 0.18 0.11 0.09 0.15 0.03 0.07 0.12 working memory did not correlate after controlling for other factors. 4 – This might be because mathematical word problems require visualising − 0.11 0.20* 0.30* 0.28* 0.20* 0.25* 0.20* 0.23* 0.19* 0.29* the actual problem situation, building a structured spatial model, and 0.17 0.09 0.15 0.01 0.05 0.16 carrying out multiple problem attributes, such as digital information, 3 – complex knowledge diagrams, and multistep operations (Zhang, 2016). − 0.04 0.43* 0.25* 0.28* 0.21* 0.22* 0.27* 0.23* 0.21* 0.40* This process is similar to the numerous problem attributes of operational 0.17 0.11 0.09 0.15 0.05 0.15 0.16 2 – conversion in paper folding (‘punch locations’, ‘number of folds’, and ‘types of folds’) (Burte et al., 2019). Contrastingly, the problem attri − 0.14 0.30* 0.33* 0.35* 0.51* 0.41* 0.26* 0.27* 0.20* 0.28* 0.47* 0.22* 0.15 0.10 0.13 0.17 0.03 0.05 butes of mental rotation may be more about the rotation angle, and the p < 0.05, using Bonferroni correction. 1 – problem attributes of the Corsi blocks test may be more about the memory of the number and location of the points. Therefore, the prob 4. Threedimensional mental rotation 17.1 Numerosity comparison (ACC) lem attributes of paper folding are more than those of mental rotation 17.2 Numerosity comparison (RT) 1. Mathematical word problems 3. Concrete syllogism reasoning 2. Abstract syllogism reasoning and the Corsi blocks test, which are more similar to the structured spatial 7. Nonverbal matrix reasoning 12.1 Figure matching (ACC) 15. Reading comprehension 16. Arithmetic computation model constructed by word problemsolving. Therefore, neither had a 12.2 Figure matching (RT) 11. Digit span (backward) 13. Simple reaction time 14. Sentence completion 10. Digit span (forward) significant effect on the mathematical word problems. Additionally, the results showed that when spatial measures were 8. Visual searching 6. Corsi blocks test 9. Visual memory not included, nonverbal IQ was predicted, while nonverbal IQ was no 5. Paper folding longer significant when spatial measures were included. This result is as follows: Studies have shown a relationship between the nonverbal Table 2 measure of IQ and spatial reasoning. Colom et al. (2004) administered the Advanced Progressive Matrices Test (APM) and the Spatial Rotation * 6 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 Table 4 Hierarchical multiple regression analysis on the relationship between spatial processing and word problems. Predictors Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 β β β β β β Age − 0.003 0.002 0.006 0.014 0.024 0.039 Gender − 0.128 − 0.170* − 0.166* − 0.120* − 0.125* − 0.064 Nonverbal matrix reasoning – 0.288* 0.282* 0.178* 0.167* 0.065 Visual searching – 0.144* 0.142* 0.100 0.087 0.078 Visual memory – 0.000 − 0.008 0.010 0.011 0.029 Digit span (forward) – − 0.045 − 0.042 − 0.039 − 0.036 − 0.035 Digit span (backward) – 0.150* 0.142 0.090 0.071 0.061 Figure matching (ACC) – − 0.050 − 0.065 − 0.078 − 0.088 − 0.091 Figure matching (RT) – 0.090 0.060 0.031 0.035 0.035 Simple reaction time – − 0.061 − 0.083 − 0.067 − 0.065 − 0.025 Sentence completion – 0.053 0.052 0.034 0.016 0.038 Reading comprehension – 0.212* 0.212* 0.223* 0.209* 0.182* Numerosity comparison (ACC) – – 0.063 0.042 0.033 0.031 Numerosity comparison (RT) – – 0.036 0.028 0.029 − 0.005 Arithmetic computation – – – 0.306* 0.276* 0.231* Abstract syllogism reasoning – – – – 0.035 0.007 Concrete syllogism reasoning – – – – 0.126 0.095 Threedimensional mental rotation – – – – – 0.092 Paper folding – – – – – 0.241* Corsi blocks test – – – – – 0.035 R2 = 0.016 R2 = 0.284* R2 = 0.288 R2 = 0.353* R2 = 0.369 R2 = 0.421* (ΔR2 = 0.268*) (ΔR2 = 0.004) (ΔR2 = 0.064*) (ΔR2 = 0.016) (ΔR2 = 0.052*) p 0.052 <0.001 0.379 <0.001 0.013 <0.001 * p < 0.05, Bonferroni correction. Fig. 2. Path model (path coefficients are standardized) in Study 1. *p < 0.05, with Bonferroni correction. Test from the Primary Mental Abilities Battery (PMA) to 239 university Cummins et al., 1988; De Corte et al., 1985). Further, it is necessary to undergraduates. The results showed that men outperformed women in apply mathematical knowledge, such as knowledge of number addition both tests. However, the male advantage on APM turned out to be non and subtraction operations (e.g., Nesher, 2020; Sophian & Vong, 1995). significant when gender differences in spatial rotation were statistically controlled. Therefore, it is suggested that gender differences in PM could 3. Study 2 be a byproduct of its visuospatial format. Moreover, Gutierrez et al. (2019) found that nonverbal matrix reasoning was highly correlated This study was designed to examine the unique role of logical with spatial processing. Furthermore, the study used two tests that reasoning and spatial processing in geometric problemsolving. measure spatial processing and one test that measures nonverbal general reasoning ability: Guay's Visualisation of Views Test, Adapted Version (VVT), Mental Rotations Test (MRT), and Raven's Advanced Progressive 3.1. Methods Matrices Test (APMT). The results showed that the spatial processing scores measured by the VVT and MRT showed a positive correlation 3.1.1. Participants with nonverbal general reasoning ability scores (APMT), supporting the Participants were 209 undergraduate students (106 men, 103 idea that these abilities are linked. women, mean age = 20.96 years, SD = 1.61 years, age range = 17.8 to Reading comprehension and arithmetic are essential components of 27.8 years) from universities in China. Other information is the same as word problems. Mathematical word problems are based on realworld that in Study 1. events and relationships, are stated in natural language, and are based on mathematical operations (Bassok, 2001). Therefore, mathematical 3.1.2. Tests word problems require a certain level of language understanding (e.g., The tests were similar to those in Study 1. However, word problems were replaced with the plane and solid geometric problems (18 total). 7 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 3.1.2.1. Geometric problems 18.2 3.1.2.1.1. Plane and solid geometries. These tests assessed the ability – to solve plane and solid geometric proofs using pen and paper. The proofs were adapted from problems encountered in the entrance ex 0.31* 18.1 aminations of senior high schools and colleges in China. Participants – were required to solve as many plane or solid geometry problems as they 0.20 0.04 could within 15 min. 17 – 3.1.3. Procedure and data analyses − 0.22 0.09 0.03 Same as for Study 1. 16 – − 0.05 0.33* 0.11 0.18 3.2. Results 15 – The means and standard deviations of the scores and the splithalf − 0.12 − 0.09 − 0.14 − 0.06 0.27* reliabilities for all the 18 tests are listed in Table 5. 14 – The Pearson's correlation coefficients among the scores for all 18 − 0.29* tests are displayed in Table 6, with Bonferroni correction. The corrected − 0.13 0.25* 0.57* 13.2 0.09 0.16 alpha for significance in Table 6 was set at 0.05, corresponding to a p – value of 0.0003 (0.05/190 correlations). The results showed that both logical reasoning measures did not correlate with the two measures of − 0.15 − 0.02 0.37* 0.25* 0.40* 0.25* 13.1 0.05 geometry without controlling for the other measures. – The partial correlation results with Bonferroni correction (uncor − 0.15 0.34* rected p = 0.0003, 0.05/190 correlations) indicated that calculation, 0.21 0.05 0.21 0.12 0.18 0.13 12 memory, attention, and general IQ were also significantly correlated – − 0.04 − 0.01 0.39* 0.11 0.01 0.11 0.16 0.23 0.00 Table 5 11 – Means and standard deviations of scores for all tests in Study 2. − 0.05 − 0.06 0.40* 0.36* 0.36* Test Index Mean Splithalf 0.09 0.20 0.12 0.16 0.23 10 (SD) reliability – 1. Plane geometry Score (0− 100) 60.5 / − 0.10 − 0.02 0.28* 0.25* 0.31* (18.8) 0.23 0.23 0.00 0.17 0.10 0.19 2. Solid geometry Score (0–100) 41.7 / 9 – (22.0) − 0.09 − 0.17 − 0.07 3. Abstract syllogism Adjusted no. of correct 3.6 (4.9) 0.80 0.27* 0.34* 0.35* 0.29* 0.35* 0.38* 0.15 0.20 0.21 reasoning trials 8 – 4. Concrete syllogism Adjusted no. of correct 5.3 (5.5) 0.80 reasoning trials − 0.04 − 0.09 − 0.17 − 0.11 0.02 0.02 0.07 0.06 0.16 0.12 0.03 0.13 0.15 5. Threedimensional Adjusted no. of correct 27.5 0.94 7 – mental rotation trials (9.1) 6. Paper folding Number of correct 8.2 (2.7) 0.87 − 0.29* − 0.02 − 0.07 − 0.04 responses 0.34* 0.26* 0.24* 0.35* 0.05 0.20 0.08 0.05 0.15 0.13 7. Corsi blocks test Accuracy (%) 82.1 0.96 6 – (5.9) 8. Nonverbal matrix Number of correct 28.4 0.91 − 0.03 − 0.18 0.35* 0.37* 0.29* 0.02 0.13 0.19 0.19 0.19 0.20 0.00 0.01 0.14 0.19 reasoning responses (5.4) 5 9. Visual searching Number of correct 47.2 0.96 – responses (30.2) − 0.08 0.26* 0.25* 10. Visual memory Adjusted no. of correct 69.8 0.78 0.11 0.22 0.03 0.23 0.16 0.17 0.16 0.18 0.07 0.16 0.17 0.17 0.10 trials (8.7) 4 – 11. Digit span (forward) Maximum number of 9.8 (1.9) / Correlations among all measures for all participants in Study 2. correct trials − 0.05 0.33* 0.24* 0.24* 0.24* 0.17 0.00 0.15 0.19 0.04 0.06 0.17 0.15 0.05 0.07 0.08 0.11 12. Digit span (backward) Maximum number of 8.1 (1.9) / 3 – correct trials 13.1 Figure matching Accuracy (%) 79.2 0.89 − 0.07 − 0.01 0.37* 0.32* (ACC) (9.5) 0.22 0.12 0.13 0.01 0.12 0.22 0.08 0.06 0.20 0.12 0.00 0.07 0.02 0.01 13.2 Figure matching (RT) Reaction time (ms) 921 0.98 2 – (144) − 0.10 − 0.03 14. Simple reaction time Reaction time (ms) 368 (56) 0.97 0.42* 0.24* 0.34* 0.32* 0.33* 0.27* 0.44* 0.20 0.15 0.06 0.12 0.11 0.07 0.15 0.05 0.11 0.01 15. Sentence completion Adjusted no. of correct 40.6 0.84 1 – trials (6.4) p < 0.05, Bonferroni correction. 16. Reading Number of correct 8.3 (3.1) 0.74 18.1 Numerosity comparison (ACC) 18.2 Numerosity comparison (RT) comprehension responses 4. Concrete syllogism reasoning 3. Abstract syllogism reasoning 17. Arithmetic Number of correct 20.1 0.82 8. Nonverbal matrix reasoning 5. Threedimensional mental 13.1 Figure matching (ACC) 16. Reading comprehension 17. Arithmetic computation computation responses (5.5) 13.2 Figure matching (RT) 12. Digit span (backward) 18.1 Numerosity Accuracy (%) 81.8 (7) 0.81 14. Simple reaction time 15. Sentence completion 11. Digit span (forward) comparison (ACC) 9. Visual searching 10. Visual memory 7. Corsi blocks test 18.2 Numerosity Reaction time (ms) 540 (86) 0.98 1. Plane geometry 2. Solid geometry 6. Paper folding comparison (RT) Note: Adjusted no. of correct trials = total correct trials minus total incorrect rotation trials. This adjustment was made to control for the effect of guessing in multiple Table 6 choice tests. Accuracy = 100 − (ResponseStandard answer) / (Standard answer + (ResponseStandard answer)) × 100. * 8 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 with plane geometry, and only calculation was associated considerably processing and geometric problemsolving. Previous studies have also with solid geometry, in addition to spatial processing (see Table 3). found a relationship between spatial processing and geometric problem Next, we conducted hierarchical multiple regression with Bonferroni solving (Delgado & Prieto, 2004; Karaman & Toğrol, 2009). However, correction (significance set at corrected p = 0.05). The regression results they did not control for covariates, such as working memory, simple are presented in Table 7. Logical reasoning had no significant relation reaction time, intelligence, attention, and general cognitive abilities. ship with either plane or solid geometric problems after controlling Therefore, these broad cognitive abilities and logical reasoning were other factors. However, after controlling for all other cognitive factors, considered in the present study. Due to these controls, the unique pre spatial processing was a significant predictor of the plane and solid ge dictive effect of spatial processing on geometric problems can be ometry proofs. observed more clearly. Finally, Fig. 3 shows the path model of the structural relationships One interesting result is significant gender differences in the solid among all main variables. The latent variable ‘geometry problems’ had geometry. Gender showed a substantial relation to the solid geometry the two manifest measures (plane and solid geometry). The hypoth after controlling for all the other cognitive variables, but not to plane esised model was a good fit for the data (χ 2 (11) = 10.20, p = 0.513, geometry. The results are consistent with previous findings that gender RMSEA = 0.07, CFI = 1.00, SRMR = 0.03). Other contents of the model effect was not found as one of the factors associating with the plane are the same as Study 1. geometry performances of the sixthgrade students (Karaman & Toğrol, 2009), but it was found in solid geometry (Pattison & Grieve, 1984). One 3.3. Discussion possibility is that gender differences in solid geometry are due to gender differences in threedimensional mental rotation. Most studies on three In Study 2, mathematical problemsolving changed from word dimensional mental rotation showed male advantage (e.g., Voyer et al., problems to geometric problems. The theoretical viewpoint of pedagogy 1995; Rahe & Jansen, 2022; see a review by Wang et al., 2014). Current generally states that geometric proofs require logical reasoning. investigation showed male advantage only in the mental rotation task. Based on this viewpoint, the present study explores the relationship During solid geometry proofing, participants could also need to mentally between logical reasoning and geometric proofs through experimental rotate the inner line, plane in the solid figure to find numerical relations. research. However, the measures of logical reasoning did not signifi cantly predict geometry performance. Contrastingly, the measures of 4. General discussion spatial skills did, after controlling for a wide range of potentially con founding variables, such as IQ, verbal skills, and basic numeric and This study determined which cognitive mechanism best predicted arithmetic skills. mathematical problemsolving, spatial processing, or logical reasoning. To explain our findings, we believe that geometric proof is still a We hypothesised that spatial processing is more critical than logical process of constructing a structured spatial model, which is similar to reasoning. As expected, spatial processing still correlated with mathe spatial processing rather than logical reasoning. matical problemsolving after controlling for age, gender, other cogni The present study showed a close relationship between spatial tive abilities, and logical reasoning. Contrastingly, logical reasoning did Table 7 Hierarchical multiple regression analysis for the relationship between spatial processing and geometry. Predictors Plane geometry Solid geometry Step 1 Step 2 Step 3 Step 4 Step 5 Step 1 Step 2 Step 3 Step 4 Step 5 β β β β β β β β β β Age 0.058 0.038 0.019 0.025 0.016 − 0.005 − 0.034 − 0.040 − 0.031 − 0.028 Gender − 0.125 − 0.172 − 0.187* − 0.186* − 0.121 − 0.244* − 0.289* − 0.305* − 0.297* − 0.256* Nonverbal matrix – 0.291* 0.292* 0.284* 0.223* – 0.063 0.074 0.060 − 0.017 reasoning Visual searching – 0.236* 0.232* 0.221* 0.204* – 0.145 0.142 0.123 0.103 Visual memory – − 0.094 − 0.089 − 0.086 − 0.073 – 0.012 0.028 0.039 0.076 Digit span (forward) – − 0.053 − 0.061 − 0.058 − 0.030 – − 0.018 − 0.034 − 0.024 − 0.014 Digit span (backward) – 0.144 0.162 0.143 0.121 – 0.124 0.144 0.112 0.087 Figure matching (ACC) – − 0.014 − 0.002 − 0.009 − 0.009 – 0.095 0.125 0.113 0.166 Figure matching (RT) – 0.099 0.149 0.139 0.137 – − 0.084 − 0.029 − 0.051 − 0.047 Simple reaction time – 0.055 0.083 0.085 0.128 – − 0.007 0.020 0.022 0.087 Sentence completion – 0.095 0.103 0.102 0.082 – 0.065 0.082 0.081 0.072 Reading comprehension – − 0.066 − 0.073 − 0.084 − 0.113 − 0.027 − 0.033 − 0.048 − 0.084 Numerosity comparison – – − 0.005 − 0.005 − 0.015 – – − 0.081 − 0.080 − 0.089 (ACC) Numerosity comparison – – − 0.113 − 0.117 − 0.116 – − 0.099 − 0.102 − 0.104 (RT) Abstract syllogism – – – 0.057 0.029 – – – 0.124 0.092 reasoning Concrete syllogism – – – 0.052 0.034 – – – 0.049 0.017 reasoning Threedimensional – – – – 0.088 – – – – − 0.106 mental rotation Paper folding – – – – 0.193* – – – – 0.341* Corsi blocks test – – – – − 0.094 – – – – 0.001 R2 = 0.018 R2 = R2 = 0.243 R2 = 0.249 R2 = R2 = 0.060* R2 = 0.152 R2 = 0.164 R2 = 0.182 R2 = 0.259* 0.236* 0.294* (ΔR2 = (ΔR2 = (ΔR2 = (ΔR2 = (ΔR2 = (ΔR2 = (ΔR2 = (ΔR2 = (ΔR2 = (ΔR2 = 0.018) 0.217*) 0.007) 0.007) 0.045*) 0.060*) 0.092) 0.013) 0.018) 0.076*) p 0.147 <0.001 0.406 0.432 0.008 0.002 0.025 0.236 0.119 <0.001 Note: 1: Threedimensional mental rotation; 2: Paper folding; 3: Corsi blocks test. * p < 0.05, Bonferroni correction. 9 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 Fig. 3. Path model (path coefficients are standardized) in Study 2. *p < 0.05, with Bonferroni correction. not correlate with mathematical problemsolving after controlling for model required by the plane geometry proof may be less structured than covariates. solid geometry proofs and word problems. Compared with paper folding, the processing of mental rotation requires less integrated problem attribute information, so its processing may be similar to plane 4.1. Relation between spatial processing and mathematical problem geometric processing. However, one type of spatial ability, spatial solving working memory, is not an active construction process but a passive information input process, so it may not play a role in problemsolving. This study showed that spatial processing predicts the ability to solve Previous studies have shown that spatial processing is correlated mathematical word and geometry problems, even after controlling for with mathematical problemsolving, and the current investigation age, gender, logical reasoning, and other general cognitive factors. This elaborates on these findings. And the present study systematically result indicates that the spatial model is fundamental for solving controlled for other related cognitive correlates, including arithmetic mathematical problems. This is because of the transformation of ele skills, language, attention, and working memory, which are critical in ments in a situation to a structurally matched symbolic expression in examining the association between spatial processing and mathematics word problemsolving and the transformation of a meaningful spatial problemsolving. Additionally, previous theoretical and empirical structure from geometric problemsolving. Therefore, this study extends studies generally indicate that working memory may play a role in a structured spatial model to explain the role of spatial processing in mathematical problemsolving. Theoretically, working memory plays a mathematical problemsolving. This theoretical model is based on the vital role in mathematics development because many mathematics tasks summary and expansion of previous concepts (Hawes & Ansari, 2020; involve simultaneous information processing and storage (e.g., Newell & Simon, 1972; Rinck, 2005; Rinck et al., 1996, 1997). However, remembering intermediate numbers during word problemsolving) these theories do not consider the characteristics of the structure in (Raghubar et al., 2010; Swanson & Jerman, 2006). Furthermore, mathematical problemsolving, and this study added previous theories empirical studies have shown that the degree of spatial working memory as a new structured spatial model. Furthermore, this study used a new is significantly related to mathematical problemsolving. This inconsis model to explain the relationship between different spatial abilities and tency with our current findings can be explained in two ways. First, the mathematics. participants in the present study were all adults, whereas previous Specifically, our results show a close association between spatial studies typically recruited children and adolescents. Second, the type of processing and mathematical problemsolving. However, the associa math in which spatial working memory was correlated was simple tions between the three spatial tasks and mathematical problemsolving rather than complex. However, the problems encountered in this study differed according to Pearson's correlation. For example, the spatial were complex. Nevertheless, we observed a correlation between spatial working memory measured by the Corsi Block task did not have a pre working memory and visual form perception, the easiest task used in the dictive effect on any mathematical problemsolving measures. Simul current investigation. taneously, paper folding was associated with all mathematical problem solving with the highest correlation coefficient, but threedimensional mental rotation was not correlated with solid geometry. 4.2. Relationship between logical reasoning and mathematical problem The structured spatial model holds that spatial abilities involving solving structural organisation could closely connect with mathematical problemsolving. The higher the structuralisation, the closer the rela Neither experiment showed a close association between logical tionship with mathematical problemsolving. Specifically, the paper reasoning and mathematical problemsolving. Meanwhile, there may be folding task can measure the ability to mentally manipulate objects in two reasons for this conclusion: first, possibly, logical processing is space based on several problem attributes (i.e., the number and location involved in spatial processing, but because spatial processing has of folds and the number of holes). Compared with other tasks, the type of important influence on mathematical problemsolving, after controlling spatial processing required by the paperfolding task could play a spatial processing, the relationship between logical reasoning and particular role in mathematical problemsolving. Both processes involve mathematical problemsolving is insignificant. Indirect evidence for this integrating multiple problem attributes and constructing structured conjecture comes from the mental model theory of logical reasoning and spatial models in mind. Moreover, solid geometry problems are solved brain research. The mental model theory holds that logical reasoning is using a structured spatial model that integrates 3D space, points, lines, involved in constructing a mental model, and this theory is one of the planes, and other problem attributes. Contrastingly, the spatial relations theories accounting for deductive reasoning. Mental model theory between lines and angles in the plane geometry problems can be directly (JohnsonLaird, 1983; JohnsonLaird & Byrne, 1991) posits that logical viewed with fewer problem attributes. Thus, the structured spatial reasoning is performed through the construction and operation of 10 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 mental models. This theory emphasises the cognitive processes involved solid geometry proofs. Therefore, we believe that spatial processing is an in constructing and manipulating spatially organised mental models important cognitive factor that predicts mathematical problemsolving. (JohnsonLaird, 1998). Additionally, the mental model theory is sup ported by evidence from brain research. For example, studies have found activation of brain regions correlated with visualspatial representation Conflict of interest in reasoning, mainly in the parietooccipital cortex (Knauff et al., 2002; Knauff et al., 2003). Knauff et al. (2003) found that the higher the visual We have no known conflict of interest to disclose. spatial manipulation skill, the better the performance of the reasoning task, but the lower the activation of brain regions. This indicates that the Acknowledgments higher the visualspatial operation skill, the fewer resources are needed for visualspatial processing, thus reducing the activation of reasoning This research was supported by the 111 Project (grant number brain regions; this also explains the role of visualspatial manipulation in BP0719032). reasoning (Ruf et al., 2003). Second, as mentioned in the introduction, people often do not base their logical reasoning on logical rules but make Appendix A. Supplementary data logical errors due to the influence of empirical knowledge (e.g., Brisson et al., 2018; Dias & Harris, 1988; Hawkins et al., 1984; Nys et al., 2022; Supplementary data to this article can be found online at https://doi. Scribner, 1977). In mathematical problemsolving, predominantly geo org/10.1016/j.lindif.2022.102230. metric problems, it is necessary to rely on existing mathematical knowledge to solve problems rather than regular reasoning. Therefore, References the mental processing of mathematical problemsolving may not involve logical reasoning. Andersson, U., & Lyxell, B. (2007). Working memory deficit in children with Meanwhile, the results of this study is different from the previous mathematical difficulties: A general or specific deficit? Journal of Experimental Child Psychology, 96, 197–228. theories (DurandGuerrier et al., 2011; Fujita & Jones, 2007; Kleemans Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In et al., 2018) and empirical evidence (Gardner, 1985; Piaget, 1956). D. Pimm (Ed.), Mathematics, teachers, and children (pp. 216–238). London: Hodder In theory, individual mathematical problemsolving skills are and Stoughton. Bassok, M. (2001). Semantic aligments in mathematical word. Problem. In D. Gentner, generally based on logical reasoning abilities (Gardner, 1985; Piaget, K. J. Holyoak, & N. Kokinov (Eds.), The analogical mind: Perspectives from cognitif 1956). A study of 1020 experts found that 99.3 % of respondents science (pp. 401–434). believed that logical reasoning is one of the critical components of the Bates, K. E., GilliganLee, K., & Farran, E. K. (2021). Reimagining mathematics: The role five significant intelligence factors (abstract logical reasoning, problem of mental imagery in explaining mathematical calculation skills in childhood. Mind, Brain, and Education, 15, 189–198. solving, knowledge acquisition, memory, and adaptability) (Carroll, Berg, D. H. (2008). Working memory and arithmetic calculation in children: The 1997). In Gardner's theory of multiple intelligence, logic refers to contributory roles of processing speed, shortterm memory, and reading. Journal of mathematical intelligence. For instance, the ability to use numbers Experimental Child Psychology, 99, 288–308. BlattoVallee, G., Kelly, R. R., Gaustad, M. G., Porter, J., & Fonzi, J. (2007). Visual spatial effectively and rationally (1985). Cattell's threestratum intelligence representation in mathematical problem solving by deaf and hearing students. theory includes general intelligence factors, such as reasoning ability Journal of Deaf Studies and Deaf Education, 12(4), 432–448. (Carroll, 1993). The role of logical reasoning in mathematics is essential; Boonen, A. J. H., van der Schoot, M., van Wesel, F., de Vries, M. H., & Jolles, J. (2013). What underlies successful word problem solving? A path analysis in sixth grade however, its importance is theoretical. students. Contemporary Educational Psychology, 38(3), 271–279. When exploring the relationship between logical reasoning and Boonen, A. J. H., van Wesel, F., Jolles, J., & van der Schoot, M. (2014). The role of visual mathematical problemsolving, previous studies have only controlled representation type, spatial ability, and reading comprehension in word problem solving: An itemlevel analysis in elementary school children. International Journal of for a limited number of relevant influencing factors (DurandGuerrier Educational Research, 68, 15–26. et al., 2011; Fujita & Jones, 2007; Kleemans et al., 2018). Studies Brisson, J., Markovits, H., Robert, S., & Schaeken, W. (2018). Reasoning from an have shown that mathematical problemsolving was supported by incompatibility: False dilemma fallacies and content effects. Memory & Cognition, 46 (5), 657–670. spatial processing, arithmetic, working memory, intelligence, and Bull, R., & Johnston, R. S. (1997). Children’s arithmetical difficulties: Contributions from attention (Fuchs et al., 2015; Kleemans et al., 2018; Tolar et al., processing speed, item identification, and shortterm memory. Journal of 2012). These are also critical for logical reasoning (Kokis et al., 2002). Experimental Child Psychology, 65, 1–24. Therefore, these factors should be controlled when determining the Bull, R., & Sherif, G. (2001). Executive functioning as a predictor of children’s mathematics ability: Inhibition, switching, and working memory. Developmental independent associations between logical reasoning and mathemat Neuropsychology, 19(3), 273–293. ical processing. Burte, H., Gardony, A. L., Hutton, A., & Taylor, H. A. (2017). Think3d!: Improving Additionally, one limitation of this study is that only syllogism was mathematics learning through embodied spatial training. Cognitive Research: Principles and Implications, 2(1), 13. used to assess logical reasoning ability due to limited test time. Other Burte, H., Gardony, A. L., Hutton, A., & Taylor, H. A. (2019). Knowing when to fold 'em: types of logical reasoning, such as relational reasoning, could also be Problem attributes and strategy differences in the paper folding test. Personality and used to check whether they are associated with mathematical problem Individual Differences, 146, 171–181. Carroll, J. B. (1993). Human cognitive abilities: A survey of factoranalytic studies. solving and even other types of mathematical processing in future. In Cambridge: Cambridge University Press. conclusion, much more research investigating spatial vs. logical Carroll, J. B. (1997). Theoretical and technical issues in identifying a factor of general reasoning as predictors of different types of math performance needs to intelligence. New York: Springer. Chazan, D. (1993). High school geometry students' justification for their views of be done in the future, including research using a wider range of logical empirical evidence and mathematical proof. Educational Studies in Mathematics, 24 reasoning measures (e.g., conditional reasoning, relational reasoning). (4), 359–387. Cheng, Y.L., & Mix, K. S. (2013). Spatial training improves children's mathematics ability. Journal of Cognition and Development, 15(1), 2–11. 5. Conclusions Chuderski, A., & Jastrzebski, J. (2018). Much ado about aha!: Insight problem solving is strongly related to working memory capacity and reasoning ability. Journal of This study found that spatial processing, rather than logical Experimental Psychology. General, 147(2), 257–281. Cirino, P. T. (2011). The interrelationships of mathematical precursors in kindergarten. reasoning, was related to mathematical problemsolving abilities Journal of Experimental Child Psychology, 108(4), 713–733. (including word and geometry problems). Specifically, as measured by Cornoldi, C., Drusi, S., Tencati, C., Giofrè, D., & Mirandola, C. (2012). Problem solving abstract and concrete syllogisms, logical reasoning ability was not and working memory updating difficulties in a group of poor comprehenders. correlated with skill in solving word problems or geometry proofs when Journal of Cognitive Education and Psychology, 11, 39–44. Colom, R., Escorial, S., & Rebollo, I. (2004). Sex differences on the progressive matrices controlling for many vital variables. Contrastingly, spatial processing are influenced by sex differences on spatial ability. Personality and Individual ability was significantly related to solving word problems, and plane and Differences, 37(6), 1289–1293. 11 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 Cummins, D. D., Kintsch, W., Reusser, K., & Weimer, R. (1988). The role of Kokis, J. V., Macpherson, R., Toplak, M. E., West, R. F., & Stanovich, K. E. (2002). understanding in solving word problems. Cognitive Psychology, 20(4), 405–438. Heuristic and analytic processing: Age trends and associations with cognitive ability De Corte, E., Verschaffel, L., & De Win, L. (1985). Influence of rewording verbal problems and cognitive styles. Journal of Experimental Child Psychology, 83(1), 26–52. on children's problem representations and solutions. Journal of Educational Kyttälä, M., & Björn, P. M. (2014). The role of literacy skills in adolescents' mathematics Psychology, 77(4), 460. word problem performance: Controlling for visuospatial ability and mathematics Delgado, A. R., & Prieto, G. (2004). Cognitive mediators and sexrelated differences in anxiety. Learning and Individual Differences, 29, 59–66. mathematics. Intelligence, 32(1), 25–32. Likert, R., & Quasha, W. H. (1994). Revised Minnesota paper form board test (2nd ed.). San Dias, M. D. G., & Harris, P. L. (1988). The effect of makebelieve play on deductive Antonio, TX: The Psychological Corporation, Harcourt Brace & Company. reasoning. British Journal of Developmental Psychology, 6(3), 207–221. Linn, M. C., & Petersen, A. C. (1985). Emergence and characterization of sex differences Duffy, G., Sorby, S., & Bowe, B. (2020). An investigation of the role of spatial ability in in spatial ability: A metaanalysis. Child Development, 56(6). representing and solving word problems among engineering students. Journal of Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary Engineering Education, 109, 424–442. students’ mathematics performance. British Journal of Educational Psychology, 87(2), Duque de Blas, G., GómezVeiga, I., & GarcíaMadruga, J. A. (2021). Arithmetic word 170–186. problems revisited: Cognitive processes and academic performance in secondary Männamaa, M., Kikas, E., Peets, K., & Palu, A. (2012). Cognitive correlates of math skills school. Education Sciences, 11(4), 155. in thirdgrade students. Educational Psychology, 32(1), 21–44. DurandGuerrier, V., Boero, P., Douek, N., Epp, S. S., & Tanguay, D. (2011). Examining Markovits, H., & Doyon, C. (2010). Using analogy to improve abstract conditional the role of logic in teaching proof. In Proof and proving in mathematics education (pp. reasoning in adolescents: Not as easy as it looks. European Journal of Psychology of 369–389). Education, 26(3), 355–372. Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P. M., Capizzi, A. M. Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Fletcher, J. M., … (2006). The cognitive correlates of thirdgrade skill in arithmetic, Journal for Research in Mathematics Education, 20(1), 41–51. algorithmic computation, and arithmetic word problems. Journal of Educational Mayer, R. E., Larkin, J. H., & Kadane, J. B. (1984). A cognitive analysis of mathematical Psychology, 98, 29–43. problemsolving ability. In J. R. Stenberg (Ed.), Advances in psychology of human Fuchs, L. S., Fuchs, D., Compton, D. L., Hamlett, C. L., & Wang, A. Y. (2015). Is word intelligence (Vol. 2, pp. 231–273). Hillsdale, NJ: Erlbaum. problem solving a form of text comprehension? Scientific Studies of Reading, 19(3), Mix, K. S., Levine, S. C., Cheng, Y. L., Young, C., Hambrick, D. Z., Ping, R., & 204–223. Konstantopoulos, S. (2016). Separate but correlated: The latent structure of space Fujita, T., & Jones, K. (2007). Learners’ understanding of the definitions and hierarchical and mathematics across development. Journal of Experimental Psychology. General, classification of quadrilaterals: Towards a theoretical framing. Research in 145(9), 1206–1227. Mathematics Education, 9(1), 3–20. Morsanyi, K., Devine, A., Nobes, A., & Szucs, D. (2013). The link between logic, Fürst, A. J., & Hitch, G. J. (2000). Separate roles for executive and phonological mathematics and imagination: Evidence from children with developmental components of working memory in mental arithmetic. Memory and Cognition, 28, dyscalculia and mathematically gifted children. Developmental Science, 16(4), 774–782. 542–553. Gardner, P. (1985). The paradox of moral education: A reassessment. Journal of Morsanyi, K., McCormack, T., & O'Mahony, E. (2017). The link between deductive Philosophy of Education, 19(1), 39–48. reasoning and mathematics. Thinking & Reasoning, 24(2), 234–257. Geary, D. C., Hoard, M. K., Nugent, L., & ByrdCraven, J. (2008). Development of number Nesher, P. (2020). Levels of description in the analysis of addition and subtraction word line representations in children with mathematical learning disability. Developmental problems. In Addition and subtraction (pp. 25–38). Routledge. Neuropsychology, 33, 277–299. Newell, A., & Simon, H. A. (1972). Human problem solving (Vol. 104). Englewood Cliffs, Gilligan, K. A., Hodgkiss, A., Thomas, M. S. C., & Farran, E. K. (2018). The developmental NJ: PrenticeHall. relations between spatial cognition and mathematics in primary school children. Niss, M., Bruder, R., Planas, N., Turner, R., & VillaOchoa, J. A. (2017). Developmental Science, 22(4), 1–19. Conceptualisation of the role of competencies, knowing and knowledge in GomezVeiga, I., Vila Chaves, J. O., Duque, G., & Garcia Madruga, J. A. (2018). A new mathematics education research. In Proceedings of the 13th international congress on look to a classic issue: Reasoning and academic achievement at secondary school. mathematical education (pp. 235–248). Frontiers in Psychology, 9, 400. Nys, B. L., Brisson, J., & Schaeken, W. (2022). Find extra options or reason badly: An Gutierrez, J. C., Holladay, S. D., Arzi, B., Clarkson, C., Larsen, R., & Srivastava, S. (2019). investigation of children’s reasoning with incompatibility statements. Journal of Improvement of spatial and nonverbal general reasoning abilities in female Experimental Child Psychology, 213, Article 105258. veterinary medical students over the first 64 weeks of an integrated curriculum. Oostermeijer, M., Boonen, A. J., & Jolles, J. (2014). The relation between children's Frontiers in Veterinary Science, 6, 141. constructive play activities, spatial ability, and mathematical word problemsolving Guilford, J. P. (1936). The determination of item difficulty when chance success is a performance: A mediation analysis in sixthgrade students. Frontiers in Psychology, 5, factor. Psychometrika, 1(4), 259–264. 782. Harris, D., Logan, T., & Lowrie, T. (2021). Unpacking mathematicalspatial relations: Optometric Extension Program. (1995). Primary mental abilities: Spatial relations and Problemsolving in static and interactive tasks. Mathematics Education Research perceptual speed. In Reprinted by permission of Macmillan/McGraw Hill School Journal., 33(3), 495–511. Publishing Co. Adapted for OEP by S. Groffman & H. Solan. Santa Ana. CA: Optometric Hawes, Z., & Ansari, D. (2020). What explains the relationship between spatial and Extension Program Foundation, Inc. mathematical skills? A review of evidence from brain and behavior. Psychonomic Pattison, P., & Grieve, N. (1984). Do spatial skills contribute to sex differences in Bulletin & Review, 27(3), 465–482. different types of mathematical problems? Journal of Educational Psychology, 76(4), Hawkins, J., Pea, R. D., Glick, J., & Scribner, S. (1984). Merds that laugh don’t like 678–689. mushrooms: Evidence for deductive reasoning by preschoolers. Development Piaget. (1956). General psychological problems of logicomathematical thought. Pychology, 20, 584–594. Re, A., Lovero, A., Cornoldi, C., & Passolunghi, M. C. (2016). Difficulties of children with Hedden, T., & Yoon, C. (2006). Individual differences in executive processing predict ADHD symptoms in solving mathematical problems when information must be susceptibility to interference in verbal working memory. Neuropsychology, 20(5), updated. Research in Developmental Disabilities, 59, 186–193. 511. Raghubar, K. P., Barnes, M. A., & Hecht, S. A. (2010). Working memory and Hegarty, M., & Kozhevnikov, M. (1999). Types of visualspatial representations and mathematics: A review of developmental, individual differences and cognitive mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689. approaches. Learning and Individual Differences, 20, 110–122. Hogg, R. V. (1979). Statistical robustness: One view of its use in application today. The Rahe, M., & Jansen, P. (2022). Sex differences in mental rotation: The role of stereotyped American Statistics, 33(3), 108–115. material, perceived performance and extrinsic spatial ability. Journal of Cognitive JohnsonLaird, P. N. (1983). Mental models: Towards a cognitive science of language, Psychology, 34(3), 400–409. inference and consciousness. Cambridge: Cambridge University Press; Cambridge, MA: Rinck, M., Williams, P., Bower, G. H., & Becker, E. S. (1996). Spatial situation models and Harvard University Press. narrative understanding: Some generalizations and extensions. Discourse Processes, JohnsonLaird, P. N., & Byrne, R. M. J. (1991). Deduction. In (p. 196). Hillsdale, NJ: 21(1), 23–55. Erlbaum. Rinck, M., Hähnel, A., Bower, G. H., & Glowalla, U. (1997). The metrics of spatial JohnsonLaird, P. N. (1998). Imagery, visualization, and thinking. In J. Hochberg (Ed.), situation models. Journal of Experimental Psychology: Learning, Memory, and Perception and cognition at century’s end (pp. 441–467). San Diego, CA: Academic Cognition, 23(3), 622. Press. Rinck, M. (2005). Spatial situation models. In The Cambridge handbook of visuospatial Karaman, & Toğrol. (2009). Relationship between gender, spatial visualization, spatial thinking (pp. 335–382). orientation, flexibility of closure abilities and performance related to plane geometry Rothenbusch, S., Voss, T., Golle, J., & Zettler, I. (2018). Linking teacher and parent subject among sixth grade students. ratings of teachernominated gifted elementary school students to each other and to Khemlani, S., & JohnsonLaird, P. (2012). Theories of the syllogism: A metaanalysis. school grades. Gifted Child Quarterly, 62(2), 230–250. Psychological Bulletin, 138(3), 427–457. Ruf, C., Knauff, M., & Spreer, J. (2003). Reasoning and working memory: Common and Kleemans, T., Segers, E., & Verhoeven, L. (2018). Role of linguistic skills in fifthgrade distinct neuronal processes. Neuropsychologia, 41, 1241–1253. mathematics. Journal of Experimental Child Psychology, 167, 404–413. Salthouse, T. A. (1994). The nature of the influence of speed on adult age differences in Knauff, M., Fangmeier, T., Ruff, C. C., et al. (2003). Reasoning, models, and images: cognition. Developmental Psychology, 30(2), 240–259. Behavioral measures and cortical activity. Journal of Cognitive Neuroscience, 15(4), Salthouse, T. A., & Meinz, E. J. (1995). Aging, inhibition, working memory, and speed. 559–5738. The Journals of Gerontology Series B: Psychological Sciences and Social Sciences, 50(6), Knauff, M., Mulack, T., Kassubek, J., et al. (2002). Spatial imagery in deductive P297–P306. reasoning: A functional MRI study. Cognitive Brain Research, 13, 203–2129. Schmitt, S. A., Korucu, I., Napoli, A. R., Bryant, L. M., & Purpura, D. J. (2018). Using block play to enhance preschool children's mathematics and executive functioning: A randomized controlled trial. Early Childhood Research Quarterly, 44, 181–191. 12 M. Yu et al. Learning and Individual Differences 100 (2022) 102230 Scribner, S. (1977). Modes of thinking and ways of speaking: Culture and logic Voyer, D., Voyer, S., & Bryden, M. P. (1995). Magnitude of sex differences in spatial reconsidered. In P. N. Johnson, & P. C. Watson (Eds.), Thinking (pp. 483–500). New abilities: A metaanalysis and consideration of critical variables. Psychological York: Cambridge University Press. Bulletin, 117(2), 250. Sophian, C., & Vong, K. I. (1995). The parts and wholes of arithmetic story problems: Wang, L., Cohen, A. S., & Carr, M. (2014). Spatial ability at two scales of representation: Developing knowledge in the preschool years. Cognition and Instruction, 13(3), A metaanalysis. Learning and Individual Differences, 36, 140–144. 469–477. Wei, W., Chen, C., & Zhou, X. (2016). Spatial ability explains the male advantage in Swanson, H. L., & Jerman, O. (2006). Math disabilities: A selective metaanalysis of the approximate arithmetic. Frontiers in Psychology, 7, 306. literature. Review of Educational Research, 76, 249–274. Wei, W., Yuan, H., Chen, C., & Zhou, X. (2012). Cognitive correlates of performance in Swanson, H. L., & Kim, K. (2007). Working memory, shortterm memory, and naming advanced mathematics. The British Journal of Educational Psychology, 82(Pt 1), speed as predictors of children’s mathematical performance. Intelligence, 35, 157–181. 151–168. Wong, T. T. Y. (2018). Is conditional reasoning related to mathematical problem Tolar, T. D., Fuchs, L., Cirino, P. T., Fuchs, D., Hamlett, C. L., & Fletcher, J. M. (2012). solving?. Sep Developmental Science, 21(5), Article e12644. Predicting development of mathematical word problem solving across the Xie, F., Zhang, L., Chen, X., & Xin, Z. Q. (2020). Is spatial ability related to mathematical intermediate grades. Journal of Education & Psychology, 104(4), 1083–1093. ability: A metaanalysis. Educational Psychology Review, 32(1), 113–155. Tolar, T. D., Lederberg, A. R., & Fletcher, J. M. (2009). A structural model of algebra Yang, Q., Qiu, J., & Zhang, Q. (2009). Cognitive and brain mechanisms of deductive achievement: Computational fluency and spatial visualisation as mediators of the reasoning: A review. Psychological Science, 3, 646–648. effect of working memory on algebra achievement. Educational Psychology, 29(2), Zhang, L. (2016). Cognitive diagnosis of solving word problems in sixth grade students. 239–266. Southwest University. Master's thesis. Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Zhou, X., Wei, W., Zhang, Y., Cui, J., & Chen, C. (2015). Visual perception can account Newcombe, N. S. (2013). The malleability of spatial skills: A metaanalysis of for the close relation between numerosity processing and computational fluency. training studies. Psychological Bulletin, 139(2), 352–402. Frontiers in Psychology, 6, 1364. 13
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