The main purpose of this study was to test the effects of word-problem (WP) intervention, with versus without embedded language comprehension (LC) instruction, on at-risk 1st graders' WP performance. ...We also isolated the need for a structured approach to WP intervention and tested the efficacy of schema-based instruction at 1st grade. Children (n = 391; Msubscript age = 6.53, SD = 0.32) were randomly assigned to 4 conditions: schema-based WP intervention with embedded language instruction, the same WP intervention but without LC instruction, structured number knowledge (NK) intervention without a structured WP component, and a control group. Each intervention included 45 sessions, each 30 min long. Multilevel models, accounting for classroom and school effects, revealed the efficacy of schema-based WP intervention at 1st grade, with both WP conditions outperforming the NK condition and the control group. Yet, WP performance was significantly stronger for the schema-based condition with embedded LC instruction compared to the schema-based condition without LC instruction. NK intervention conveyed no WP advantage over the control group, even though all 3 intervention conditions outperformed the control group on arithmetic. Results demonstrate the importance of a structured approach to WP intervention, the efficacy of schema-based instruction at 1st grade, and the added value of LC instruction within WP intervention. Results also provide causal evidence on the role of LC in WP solving.
Solving mathematical ( math ) word problems (MWP) automatically is a challenging research problem in natural language processing , machine learning , and education (learning) technology domains, ...which has gained momentum in the recent years. Applications of solving varieties of MWPs can increase the efficacy of teaching-learning systems, such as e-learning systems , intelligent tutoring systems , etc., to help improve learning (or teaching) to solve word problems by providing interactive computer support for peer math tutoring. This article is specifically intended to benefit such teaching-learning systems on arithmetic word problems solving by adding an interactive and intelligent word problem solver to assess an individual's learning outcome. This article presents arithmetic mathematical word problems solver (AMWPS), an educational software application for solving arithmetic word problems involving single equation with single operation . This article is based on a combination of a machine learning based (classification) approach and a rule-based approach. We start with classification of arithmetic word problems into four categories ( Change , Compare , Combine , and Division-Multiplication ) along with their subcategories, followed by the classification of operations (+, -, *, and /) related to different subcategories. Our system processes an input arithmetic word problem, predicts the category and subcategory, predicts the operation, identifies and retrieves the relevant quantities within the problem with respect to answer generation, and formulates and evaluates the mathematical expression to generate the final answer. AMWPS outperformed similar systems on the standard AddSub and SingleOp datasets and produced new state-of-the-art result (94.22% accuracy).
•Developed a novel optimizer inspired by the behavior of Aquila (AO).•Tested AO against classical, CEC2017, CEC2019 test functions and engineering problems.•Compared the AO to other similar ...optimization algorithms.•Demonstrated effectiveness and superiority of the proposed algorithm.
This paper proposes a novel population-based optimization method, called Aquila Optimizer (AO), which is inspired by the Aquila’s behaviors in nature during the process of catching the prey. Hence, the optimization procedures of the proposed AO algorithm are represented in four methods; selecting the search space by high soar with the vertical stoop, exploring within a diverge search space by contour flight with short glide attack, exploiting within a converge search space by low flight with slow descent attack, and swooping by walk and grab prey. To validate the new optimizer’s ability to find the optimal solution for different optimization problems, a set of experimental series is conducted. For example, during the first experiment, AO is applied to find the solution of well-known 23 functions. The second and third experimental series aims to evaluate the AO’s performance to find solutions for more complex problems such as thirty CEC2017 test functions and ten CEC2019 test functions, respectively. Finally, a set of seven real-world engineering problems are used. From the experimental results of AO that compared with well-known meta-heuristic methods, the superiority of the developed AO algorithm is observed. Matlab codes of AO are available at https://www.mathworks.com/matlabcentral/fileexchange/89381-aquila-optimizer-a-meta-heuristic-optimization-algorithm and Java codes are available at https://www.mathworks.com/matlabcentral/fileexchange/89386-aquila-optimizer-a-meta-heuristic-optimization-algorithm.
The purpose of this study was to explore the paths by which word-problem intervention, with versus without embedded prealgebraic reasoning instruction, improved word-problem performance. Students ...with mathematics difficulty (MD; n = 304) were randomly assigned to a business-as-usual condition or 1 of 2 variants of word-problem intervention. The prealgebraic reasoning component targeted relational understanding of the equal sign as well as standard and nonstandard equation solving. Intervention occurred for 16 weeks, 3 times per week, 30 min per session. Sequential mediation models revealed main effects, in which each intervention condition significantly and substantially outperformed the business-as-usual condition, corroborating prior research on the efficacy of schema word-problem intervention. Yet despite comparable effects on word-problem outcomes between the two word-problem conditions, the process by which effects accrued differed: An indirect path via equal-sign understanding and then equation solving was significant only for the word-problem intervention condition with embedded prealgebraic reasoning instruction. Additionally, the effect of this condition on equal-sign reasoning was strong. Given the link between equal-sign reasoning for success with algebra and the importance of algebra for success with advanced mathematics, results suggest an advantage for embedding prealgebraic reasoning instruction within word-problem intervention.
Educational Impact and Implications Statement
This study suggests prealgebraic reasoning is important within math word-problem instruction for third-grade students who experience difficulty with math. Prealgebraic reasoning involves interpreting the equal sign as "the same as" and solving equations (e.g., 3 + __ = 9 or 7 = 13 − __). As students develop strong prealgebraic reasoning, they are better equipped to solve word problems.
Procedural flexibility is an important skill for algebra. Although prior work has focused on measuring students' procedural flexibility using arithmetic problems, word problems may also capture ...students' flexibility because of their open-ended nature. To date, no published study has examined the use of word problems as another measure of procedural flexibility. The present study aims to establish flexibility on fraction word problems as a predictor of algebra learning and to determine whether flexibility demonstrated on fraction arithmetic versus fraction word problems differentially predicts two types of algebra learning (e.g., algebraic feature knowledge and algebra equation-solving). Middle-school students (N = 350) completed fraction arithmetic problems, fraction word problems, and algebra measures at the start and end of the school year. We coded fraction arithmetic and word problems for demonstrated procedural flexibility. Path models showed that overall flexibility significantly predicted both algebraic feature knowledge and equation-solving. When examining flexibility by type, procedural flexibility on arithmetic problems significantly predicted end-of-year algebraic feature knowledge and algebra equation-solving, whereas procedural flexibility on word problems significantly predicted algebraic feature knowledge. Procedural flexibility on arithmetic problems was still predictive of performance when accounting for fraction magnitude skill. These findings build on prior literature that establishes procedural flexibility as a crucial predictor of algebra learning and extends measures of procedural flexibility to include word problems.
Educational Impact and Implications Statement
We found that students' ability to appropriately choose between problem-solving strategies, or their procedural flexibility, was predictive of their knowledge of algebraic features when measured by flexibility on both fraction arithmetic and word problems. Furthermore, students' overall procedural flexibility was predictive of their algebraic equation-solving skills. These results suggest that procedural flexibility with fractions is an important predictor of algebra learning and can be measured both through arithmetic and word problem measures. Practically, results suggest that procedural flexibility is a skill that should be given more consideration in classroom instruction, an area that may be explored in future research.
The current study investigated the relation between spatial ability and children's math performance, as well as the potential mechanisms underlying this relation including numerical magnitude ...knowledge, understanding of arithmetic operations, and word-problem representation. A sample of 221 Asian sixth graders (age 11, 129 boys) completed a series of tasks measuring their spatial abilities, math performance, the potential mechanisms underlying the space-math relation, along with other potential confounding factors such as intelligence and working memory. Results indicated that spatial ability was significantly associated with math performance beyond the effects of controlled variables. Additionally, numerical magnitude knowledge, understanding of arithmetic operations, and word-problem representation were found to partially mediate the relation between spatial ability and math performance. These findings clarify the mechanisms underlying the space-math relation.
Educational Impact and Implications Statement
The current study not only provided additional evidence supporting the relation between spatial ability and math performance but also demonstrated that spatial ability was related to math performance in multiple ways. Spatial ability may facilitate children's representation of numerical magnitude, enables them to better understand arithmetic principles, and allows them to better represent the scenarios during word problem-solving. The findings suggest that focusing on children's spatial ability may be a promising approach to enhancing their math performance in the aforementioned domains.
The purpose of this study was to investigate the effects of 1st-grade number knowledge tutoring with contrasting forms of practice. Tutoring occurred 3 times per week for 16 weeks. In each 30-min ...session, the major emphasis (25 min) was number knowledge; the other 5 min provided practice in 1 of 2 forms. Nonspeeded practice reinforced relations and principles addressed in number knowledge tutoring. Speeded practice promoted quick responding and use of efficient counting procedures to generate many correct responses. At-risk students were randomly assigned to number knowledge tutoring with speeded practice (n = 195), number knowledge tutoring with nonspeeded practice (n = 190), and control (no tutoring, n = 206). Each tutoring condition produced stronger learning than control on all 4 mathematics outcomes. Speeded practice produced stronger learning than nonspeeded practice on arithmetic and 2-digit calculations, but effects were comparable on number knowledge and word problems. Effects of both practice conditions on arithmetic were partially mediated by increased reliance on retrieval, but only speeded practice helped at-risk children compensate for weak reasoning ability.
Adolescents' (n = 342, 169 boys) general algebra and algebra word problems performance were assessed in 9th grade as were intelligence, academic achievement, working memory, and spatial abilities in ...prior grades. The adolescents reported on their academic attitudes and anxiety and their teachers reported on their in-class attentive behavior in 7th to 9th grade. There were no sex differences on the general algebra measure or for mathematics achievement, but boys had an advantage on the algebra word problems measure (d = .51) and for spatial abilities (ds = .29 to .58). Boys had higher mathematics self-efficacy (d = .24 to .33), lower mathematics anxiety (ds = −.31 to −.53) and were less attentive in classrooms (ds = −.28 to −.37). A series of structural equation models revealed the sex difference for algebra word problems was mediated by spatial abilities and mathematics anxiety, controlling myriad confounds.
Public Significance StatementSex differences in mathematics are typically small but larger for word problems. The latter assess the ability to use mathematics in problem-solving situations and are often included on high-stakes tests. Boys' higher spatial abilities appeared to provide them with an advantage and girls' mathematics anxiety a disadvantage in solving algebra word problems.
The purpose of this study was to examine the interplay between basic numerical cognition and domain-general abilities (such as working memory) in explaining school mathematics learning. First graders ...(N = 280; mean age = 5.77 years) were assessed on 2 types of basic numerical cognition, 8 domain-general abilities, procedural calculations, and word problems in fall and then reassessed on procedural calculations and word problems in spring. Development was indexed by latent change scores, and the interplay between numerical and domain-general abilities was analyzed by multiple regression. Results suggest that the development of different types of formal school mathematics depends on different constellations of numerical versus general cognitive abilities. When controlling for 8 domain-general abilities, both aspects of basic numerical cognition were uniquely predictive of procedural calculations and word problems development. Yet, for procedural calculations development, the additional amount of variance explained by the set of domain-general abilities was not significant, and only counting span was uniquely predictive. By contrast, for word problems development, the set of domain-general abilities did provide additional explanatory value, accounting for about the same amount of variance as the basic numerical cognition variables. Language, attentive behavior, nonverbal problem solving, and listening span were uniquely predictive.
The focus of this study was connections among 3 aspects of mathematical cognition at 2nd grade: calculations, word problems, and prealgebraic knowledge. We extended the literature, which is dominated ...by correlational work, by examining whether intervention conducted on calculations or word problems contributes to improved performance in the other domain and whether intervention in either or both domains contributes to prealgebraic knowledge. Participants were 1,102 children in 127 second-grade classrooms in 25 schools. Teachers were randomly assigned to 3 conditions: calculation intervention, word-problem intervention, and business-as-usual control. Intervention, which lasted 17 weeks, was designed to provide research-based linkages between arithmetic calculations or arithmetic word problems (depending on condition) and prealgebraic knowledge. Multilevel modeling suggested calculation intervention improved calculation but not word-problem outcomes; word-problem intervention enhanced word-problem but not calculation outcomes; and word-problem intervention provided a stronger route than calculation intervention to prealgebraic knowledge.
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