Spatial processing rather than logical reasoning was found to be critical for mathematical problem-solving - Mingxin Yu

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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 problem-solving 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 problem-solving ematical problem-solving (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 problem-solving 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 problem-solving (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 problem-solving, 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 problem-solving (e.g., Boonen et al., 1.1. Spatial processing in mathematical problem-solving 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 problem-solving (e.g., Chuderski & involving visual information (Uttal et al., 2013). Many tasks have been Jastrzebski, 2018; Duffy et al., 2020; Gomez-Veiga 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 problem-solving, 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. E-mail address: [email protected] (X. Zhou). 1 Mingxin Yu, Jiaxin Cui and Li Wang are considered as Co-first 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 1041-6080/© 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 problem-solving 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), problem-solving. 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 problem-solving, 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 (Durand-Guerrier 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 problem-solving 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., Blatto-Vallee et al., 2007; Kyttälä & Björn, 2014). For Many logical reasoning errors have been shown to reflect the lack of example, Blatto-Vallee 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 problem-solving 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 problem-solving (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 sixth-grade 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 problem-solving, problem-solving, 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 fifth-and 1988; Chazan, 1993; Martin & Harel, 1989). eighth-graders. 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 cross-sectional 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 problem-solving. 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 problem-solving 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 problem-solving 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 problem-solving, without consis­ solving path in this mental space (Newell & Simon, 1972). Hawes and tent findings (Duque de Blas et al., 2021; Gomez-Veiga 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 problem-solving (Gomez-Veiga et al., 2018; merical relations and operations in mathematical problem-solving can Kleemans et al., 2018). For example, Kleemans et al. (2018) showed that be modelled and visualized. syllogistic reasoning in fifth-grade 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 problem-solving 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 Gomez-Veiga et al., 2018; Morsanyi et al., 2017; Wong, 2018). Two of mathematical problem-solving. Contrastingly, pictorial representations the studies found correlation (Gomez-Veiga 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 fourth-grade 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 problem-solving 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 problem-solving. 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 problem-solving, and hypothesised that spatial processing has a unique be critical to check the association. contribution to mathematical problem-solving 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. Gomez-Veiga 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 problem-solving after controlling other and mathematical problem-solving. 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 & Johnson-Laird, 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 right-handed native Man­ darin speakers with normal or corrected-to-normal visual acuity. Before Numerous studies have shown that spatial processing skills are pre­ participation, participants provided written informed consent after the dictors of mathematical problem-solving 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 problem-solving, 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 problem-solving. (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 problem-solving, 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 visual-spatial under the problem. The test duration was limited to 6 min. information, and other factors impact mathematical problem-solving. 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 7-year-old 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 problem-solving, 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. lem-solving. 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 one-fold, 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 line-drawing 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 side-by-side 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 40-trial 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 5-min 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 split-half 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 p-value 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 time-limited 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, three-dimensional 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 = R-W/(n − 1)” (S: the adjusted score, R: the number of three manifest measures (three-dimensional 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 half-split 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 problem-solving 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) Split-half 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. Three-dimensional 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. Non-verbal 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 = R-W/(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 multiple-choice tests. Accuracy = 100 − |(Response-Standard answer)| / (Standard answer + |(Response-Standard 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. Non-verbal 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 paper-folding 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 multi-step 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. Three-dimensional 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 problem-solving. 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 Three-dimensional 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 by-product 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 problem-solving. 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 real-world 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 split-half − 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 Split-half 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. Three-dimensional 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. Three-dimensional 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 − |(Response-Standard answer)| / (Standard answer + |(Response-Standard 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 problem-solving. 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 sixth-grade 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 three-dimensional mental rotation. Most studies on three- In Study 2, mathematical problem-solving 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 problem-solving, 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 problem-solving 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 Three-dimensional – – – – 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: Three-dimensional 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 problem-solving 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 problem-solving. 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 problem-solving, 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 problem-solving and the transformation of a meaningful spatial problem-solving. Additionally, previous theoretical and empirical structure from geometric problem-solving. 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 problem-solving. Theoretically, working memory plays a mathematical problem-solving. 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 problem-solving) these theories do not consider the characteristics of the structure in (Raghubar et al., 2010; Swanson & Jerman, 2006). Furthermore, mathematical problem-solving, 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 problem-solving. 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 problem-solving. However, the associa­ math in which spatial working memory was correlated was simple tions between the three spatial tasks and mathematical problem-solving 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 problem-solving measures. Simul­ current investigation. taneously, paper folding was associated with all mathematical problem- solving with the highest correlation coefficient, but three-dimensional 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 problem-solving. The higher the structuralisation, the closer the rela­ Neither experiment showed a close association between logical tionship with mathematical problem-solving. Specifically, the paper- reasoning and mathematical problem-solving. 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 problem-solving, after controlling spatial processing required by the paper-folding task could play a spatial processing, the relationship between logical reasoning and particular role in mathematical problem-solving. Both processes involve mathematical problem-solving 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 (Johnson-Laird, 1983; Johnson-Laird & 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 problem-solving. (Johnson-Laird, 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 visual-spatial representation Conflict of interest in reasoning, mainly in the parieto-occipital 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 visual-spatial operation skill, the fewer resources are needed for visual-spatial 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 visual-spatial 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 problem-solving, 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 problem-solving 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? 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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). 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