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In Yucatec Maya middle schools in the Yucatán, math scores are low and drop out rates are high. Although addressing larger social and economic causes may ameliorate these issues, improving math instruction may be a more immediate, feasible approach. This ethnographic, mixed-methods study explores community approaches to problem-solving relevant to middle school math classrooms. Findings indicate: (1) community members possess specialized, practical mathematical expertise that is overlooked in the research around rural, impoverished students, and (2) formal math instruction misses opportunities to capitalize upon cultural approaches involving autonomy and improvisational expertise. Results are relevant for curriculum reform in the US, México, and beyond. [For the complete proceedings, see ED583608.]

In Yucatec Maya middle schools in the Yucatán, math scores are low and drop out rates are high. Although addressing larger social and economic causes may ameliorate these issues, improving math instruction may be a more immediate, feasible approach. This ethnographic, mixed-methods study explores community approaches to problem-solving relevant to middle school math classrooms. Findings indicate: (1) community members possess specialized, practical mathematical expertise that is overlooked in the research around rural, impoverished students, and (2) formal math instruction misses opportunities to capitalize upon cultural approaches involving autonomy and improvisational expertise. Results are relevant for curriculum reform in the US, México, and beyond. [For the complete proceedings, see ED583608.]

This paper reports on the processes and results of a project on the instruction and practices of mathematics problem solving and strategies. Students in the second, third, and fifth grades were targeted to increase achievement in the area of math problem solving. Problem solving achievement was measured using performance tasks, rubrics, and tests from September 2001 to January 2002. Appendices include a teacher survey, pretests and posttests, scoring rubrics, problem solving steps and strategies, and writing stems. (Contains 26 references.) (KHR)

This study compared the ability of college students with deafness and typical college students to transfer and apply their math computation and problem-solving skills to similar problems presented under different conditions. Thirty-seven students with deafness and 12 hearing students were given 30 math problems to solve that were presented under graphic and word conditions. The problems were matched for similarity and difficulty for the two conditions. The results showed that both groups of students were comparable in transferring and applying their math skills to solve the varied problem sets within the graphic condition and the first problem set of the math word problems. When comparing performance between the graphic and word conditions, the data show that the hearing students performed consistently across both conditions. In contrast, the problem-solving performance of the students with deafness was not consistent across the graphic and word conditions. A comparative analysis with respect to reading abilities showed that students with deafness with reading levels at the 9.3 grade level or higher performed significantly better on more complex and difficult word problems when compared to those in the 7th and 8th grade reading range. (CR)

This study compared the ability of college students with deafness and typical college students to transfer and apply their math computation and problem-solving skills to similar problems presented under different conditions. Thirty-seven students with deafness and 12 hearing students were given 30 math problems to solve that were presented under graphic and word conditions. The problems were matched for similarity and difficulty for the two conditions. The results showed that both groups of students were comparable in transferring and applying their math skills to solve the varied problem sets within the graphic condition and the first problem set of the math word problems. When comparing performance between the graphic and word conditions, the data show that the hearing students performed consistently across both conditions. In contrast, the problem-solving performance of the students with deafness was not consistent across the graphic and word conditions. A comparative analysis with respect to reading abilities showed that students with deafness with reading levels at the 9.3 grade level or higher performed significantly better on more complex and difficult word problems when compared to those in the 7th and 8th grade reading range. (CR)

It is generally considered that diagram use aids efficacy of math word problem solving. While understanding diagrams is considered important in both New Zealand and Japanese secondary schools, there is an additional emphasis in New Zealand schools for students to appreciate their use as tools for problem solving and communication. This study examined whether there are actual differences in the amount and quality of diagrams that students in New Zealand and Japan use when given math word problems to solve. The participants were 614 secondary school students from New Zealand and Japan, aged 13 to 15 years old, who were given one- and two-object math word problems (involving length or non-length components) to solve. The findings were that while the New Zealand students evidenced greater diagram use and provided more correct answers, they produced significantly more high quality diagrams only for the easiest problems given (the one-object problems with length story context). One implication of the findings is that greater emphasis needs to be placed on, and/or more effective strategies need to be used in, developing New Zealand students' skills in the use of diagrams for solving more complex types of problems. (Contains 5 tables.) [This paper has been published as: Manalo, E., & Uesaka, Y. (2006). Quantity and quality of diagrams used in math word problem solving: A comparison between New Zealand and Japanese students. "Refereed papers of the NZARE (New Zealand Association for Research in Education) National Conference 2006" (CD-ROM; ISSN 1176-4902). Wellington, New Zealand: NZARE.]

Understanding a student's problem-solving strategy can have a significant impact on effective math learning using Intelligent Tutoring Systems (ITSs) and Adaptive Instructional Systems (AISs). For instance, the ITS/AIS can better personalize itself to correct specific misconceptions that are indicated by incorrect strategies, specific problems can be designed to improve strategies and frustration can be minimized by adapting to a student's natural way of thinking rather than trying to fit a standard strategy for all. While it may be possible for human experts to identify strategies manually in classroom settings with sufficient student interaction, it is not possible to scale this up to big data. Therefore, we leverage advances in Machine Learning and AI methods to perform scalable strategy prediction that is also fair to students at all skill levels. Specifically, we develop an embedding called MVec where we learn a representation based on the mastery of students. We then cluster these embeddings with a non-parametric clustering method where we progressively learn clusters such that we group together instances that have approximately symmetrical strategies. The strategy prediction model is trained on instances sampled from these clusters. This ensures that we train the model over diverse strategies and also that strategies from a particular group do not bias the DNN model, thus allowing it to optimize its parameters over all groups. Using real world large-scale student interaction datasets from MATHia, we implement our approach using transformers and Node2Vec for learning the mastery embeddings and LSTMs for predicting strategies. We show that our approach can scale up to achieve high accuracy by training on a small sample of a large dataset and also has predictive equality, i.e., it can predict strategies equally well for learners at diverse skill levels. [For the complete proceedings, see ED630829.]

The Influence of Bot Typing Speed and Math Skill on the Use of AI-Tools in Problem Solving

Although current categories of learning disabilities include as specific disabilities calculation and mathematical problem solving [see IDEA reauthorization, 2004, Sec. 300.8(c)(10)], the majority of research focuses on calculation disabilities. Previous studies have shown, however, that deficits in word problem solving difficulties are persistent across the elementary school years even when calculation and reading skills are in the normal achievement range (Swanson et al., 2008). Previous research shows that growth in working memory (WM) is related to growth in word problem solving accuracy for children with math disabilities (MD e.g., Swanson, 2006, Swanson et al., 2008). However, the research is unclear as to the interventions that compensate for WM limitations in children with MD that in turn positively influence performance on problem solving measures. The purpose of this intervention study was to determine whether children with MD improve in problem solving ability as a function of strategy intervention that emphasizes relevant components of word problems. Children with MD and without MD were randomly assigned to one of three treatment groups: verbal strategies, visual strategies or a combination of both verbal and visual strategies. This study addressed three questions:(1) Do cognitive strategies that direct MD children's attention to relevant components of word problems in the context of distracting or irrelevant information enhance mathematical problem solving accuracy when compared to control conditions? (2) Are visual-spatial strategies in isolation or when combined with verbal strategies more effective than verbal strategies in isolation for children with MD? and (3) Do specific cognitive strategies and increases in WM load (number of irrelevant sentences in word problems) play an independent or interactive role in facilitating solution accuracy and transfer? One hundred and twenty (120) children from grades 2 and 3 in a Southern California public school district participated in this study. The primary findings were: both children with and without MD significantly improved word problem solving accuracy at post-test relative to other conditions, (b) pretest/post-test changes in correctly identifying relevant/irrelevant components mediated the effects of treatment on solution accuracy, (c) transfer (tasks independent of training) occurred on CBM of text word problems, verbal WM and calculation, and (d) individual differences in WM capacity interacted with treatment conditions when predicting overall solution accuracy.

Although current categories of learning disabilities include as specific disabilities calculation and mathematical problem solving [see IDEA reauthorization, 2004, Sec. 300.8(c)(10)], the majority of research focuses on calculation disabilities. Previous studies have shown, however, that deficits in word problem solving difficulties are persistent across the elementary school years even when calculation and reading skills are in the normal achievement range (Swanson et al., 2008). Previous research shows that growth in working memory (WM) is related to growth in word problem solving accuracy for children with math disabilities (MD e.g., Swanson, 2006, Swanson et al., 2008). However, the research is unclear as to the interventions that compensate for WM limitations in children with MD that in turn positively influence performance on problem solving measures. The purpose of this intervention study was to determine whether children with MD improve in problem solving ability as a function of strategy intervention that emphasizes relevant components of word problems. Children with MD and without MD were randomly assigned to one of three treatment groups: verbal strategies, visual strategies or a combination of both verbal and visual strategies. This study addressed three questions:(1) Do cognitive strategies that direct MD children's attention to relevant components of word problems in the context of distracting or irrelevant information enhance mathematical problem solving accuracy when compared to control conditions? (2) Are visual-spatial strategies in isolation or when combined with verbal strategies more effective than verbal strategies in isolation for children with MD? and (3) Do specific cognitive strategies and increases in WM load (number of irrelevant sentences in word problems) play an independent or interactive role in facilitating solution accuracy and transfer? One hundred and twenty (120) children from grades 2 and 3 in a Southern California public school district participated in this study. The primary findings were: both children with and without MD significantly improved word problem solving accuracy at post-test relative to other conditions, (b) pretest/post-test changes in correctly identifying relevant/irrelevant components mediated the effects of treatment on solution accuracy, (c) transfer (tasks independent of training) occurred on CBM of text word problems, verbal WM and calculation, and (d) individual differences in WM capacity interacted with treatment conditions when predicting overall solution accuracy.

Although current categories of learning disabilities include as specific disabilities calculation and mathematical problem solving [see IDEA reauthorization, 2004, Sec. 300.8(c)(10)], the majority of research focuses on calculation disabilities. Previous studies have shown, however, that deficits in word problem solving difficulties are persistent across the elementary school years even when calculation and reading skills are in the normal achievement range (Swanson et al., 2008). Previous research shows that growth in working memory (WM) is related to growth in word problem solving accuracy for children with math disabilities (MD e.g., Swanson, 2006, Swanson et al., 2008). However, the research is unclear as to the interventions that compensate for WM limitations in children with MD that in turn positively influence performance on problem solving measures. The purpose of this intervention study was to determine whether children with MD improve in problem solving ability as a function of strategy intervention that emphasizes relevant components of word problems. Children with MD and without MD were randomly assigned to one of three treatment groups: verbal strategies, visual strategies or a combination of both verbal and visual strategies. This study addressed three questions:(1) Do cognitive strategies that direct MD children's attention to relevant components of word problems in the context of distracting or irrelevant information enhance mathematical problem solving accuracy when compared to control conditions? (2) Are visual-spatial strategies in isolation or when combined with verbal strategies more effective than verbal strategies in isolation for children with MD? and (3) Do specific cognitive strategies and increases in WM load (number of irrelevant sentences in word problems) play an independent or interactive role in facilitating solution accuracy and transfer? One hundred and twenty (120) children from grades 2 and 3 in a Southern California public school district participated in this study. The primary findings were: both children with and without MD significantly improved word problem solving accuracy at post-test relative to other conditions, (b) pretest/post-test changes in correctly identifying relevant/irrelevant components mediated the effects of treatment on solution accuracy, (c) transfer (tasks independent of training) occurred on CBM of text word problems, verbal WM and calculation, and (d) individual differences in WM capacity interacted with treatment conditions when predicting overall solution accuracy.

Although current categories of learning disabilities include as specific disabilities calculation and mathematical problem solving [see IDEA reauthorization, 2004, Sec. 300.8(c)(10)], the majority of research focuses on calculation disabilities. Previous studies have shown, however, that deficits in word problem solving difficulties are persistent across the elementary school years even when calculation and reading skills are in the normal achievement range (Swanson et al., 2008). Previous research shows that growth in working memory (WM) is related to growth in word problem solving accuracy for children with math disabilities (MD e.g., Swanson, 2006, Swanson et al., 2008). However, the research is unclear as to the interventions that compensate for WM limitations in children with MD that in turn positively influence performance on problem solving measures. The purpose of this intervention study was to determine whether children with MD improve in problem solving ability as a function of strategy intervention that emphasizes relevant components of word problems. Children with MD and without MD were randomly assigned to one of three treatment groups: verbal strategies, visual strategies or a combination of both verbal and visual strategies. This study addressed three questions:(1) Do cognitive strategies that direct MD children's attention to relevant components of word problems in the context of distracting or irrelevant information enhance mathematical problem solving accuracy when compared to control conditions? (2) Are visual-spatial strategies in isolation or when combined with verbal strategies more effective than verbal strategies in isolation for children with MD? and (3) Do specific cognitive strategies and increases in WM load (number of irrelevant sentences in word problems) play an independent or interactive role in facilitating solution accuracy and transfer? One hundred and twenty (120) children from grades 2 and 3 in a Southern California public school district participated in this study. The primary findings were: both children with and without MD significantly improved word problem solving accuracy at post-test relative to other conditions, (b) pretest/post-test changes in correctly identifying relevant/irrelevant components mediated the effects of treatment on solution accuracy, (c) transfer (tasks independent of training) occurred on CBM of text word problems, verbal WM and calculation, and (d) individual differences in WM capacity interacted with treatment conditions when predicting overall solution accuracy.

A study examined and explored the importance of language in successful math problem-solving. The participants of the study were 24 fourth-grade students in a public elementary school in an urban area. Students were instructed in solving math word problems with a variety of strategies ranging from task specific, procedural methods to teacher-directed explicit strategies. A comparison was made between student performance using task-specific instruction and teacher-directed explicit strategy instruction with special attention paid to the interpretation of language in word problems to determine if there would be a significant difference in levels of performance. The hypothesis that there would not be was rejected. (Contains 44 references; sample math computations, word problem pre- and posttests, and a math story chart form are appended.) (Author/CR)

A study examined and explored the importance of language in successful math problem-solving. The participants of the study were 24 fourth-grade students in a public elementary school in an urban area. Students were instructed in solving math word problems with a variety of strategies ranging from task specific, procedural methods to teacher-directed explicit strategies. A comparison was made between student performance using task-specific instruction and teacher-directed explicit strategy instruction with special attention paid to the interpretation of language in word problems to determine if there would be a significant difference in levels of performance. The hypothesis that there would not be was rejected. (Contains 44 references; sample math computations, word problem pre- and posttests, and a math story chart form are appended.) (Author/CR)

Early middle school students have a strong tendency to adhere to the left-to-right principle when solving math problems, which may lead them to overlook the role of brackets within math expressions (Blando et al., 1989; Gunnarsson et al., 2016). However, perceptual features, such as the spacing between symbols, can direct students’ attention to salient features of math expressions that support efficient problem-solving (Goldstone et al., 2017; Harrison et al., 2020). Similarly, superfluous brackets—brackets that do not change the meaning of notation, but can guide learners’ attention to the correct procedure—can help students achieve higher success rates in solving problems (Hoch & Dreyfus, 2004; Marchini & Papadopoulos, 2011). The current study explores how the presence of superfluous brackets and the position of higher-order operands (i.e., multiplication and division) within an expression affect student performance in an online assignment, as measured by assignment mastery speed and average response time. We will recruit at least 280 students in 6-7th grade through an online learning platform. Students will be randomly assigned to one of six conditions in this 3 (Operand position: left, center, or right) X 2 (Superfluous brackets vs. no brackets) design. Students will solve order of operation expressions that are consistent with their condition: 1) brackets-left, 2) no brackets-left, 3) brackets-center, 4) no brackets-center, 5) brackets-right, and 6) no brackets-right. We hypothesize that students who view problems with superfluous brackets will have quicker mastery speed and shorter average response time than those who don’t. We also hypothesize that students who view problems with the brackets to the furthest left will have quicker mastery speed and shorter average response time compared to students who view problems with brackets in the center or to the furthest right. We also predict that the presence of superfluous brackets may positively impact student performance on expressions with higher-order operands that are in the center or right-most position of the expression (e.g., 2+(4*3)+4 or 2+4+(4*3)) by providing perceptual cues to attend to the order of operations. We will discuss the implications of these findings for perceptual learning theory as well as instructional practice. Overall, the results from this study will further explore the roles of perceptual cues in mathematics notation in math. References: Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors. Journal for Research in Mathematics Education, 20(3), 301-308. Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, E. R., & Landy, D. (2017). Adapting perception, action, and technology for mathematical reasoning. Current Directions in Psychological Science, 26(5), 434-441. Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations?. Educational Studies in Mathematics, 92(1), 91-105. Harrison, A., Smith, H., Hulse, T., & Ottmar, E. R. (2020). Spacing out! Manipulating spatial features in mathematical expressions affects performance. Journal of Numerical Cognition, 6(2), 186-203. Hoch, M., & Dreyfus, T. (2005). Structure sense in high school algebra: The effect of brackets. International Group for the Psychology of Mathematics Education. Marchini, C., & Papadopoulos, I. (2011). Are useless brackets useful tools for teaching?. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education. Aristotle University of Thessaloniki.

Early middle school students have a strong tendency to adhere to the left-to-right principle when solving math problems, which may lead them to overlook the role of brackets within math expressions (Blando et al., 1989; Gunnarsson et al., 2016). However, perceptual features, such as the spacing between symbols, can direct students’ attention to salient features of math expressions that support efficient problem-solving (Goldstone et al., 2017; Harrison et al., 2020). Similarly, superfluous brackets—brackets that do not change the meaning of notation, but can guide learners’ attention to the correct procedure—can help students achieve higher success rates in solving problems (Hoch & Dreyfus, 2004; Marchini & Papadopoulos, 2011). The current study explores how the presence of superfluous brackets and the position of higher-order operands (i.e., multiplication and division) within an expression affect student performance in an online assignment, as measured by assignment mastery speed and average response time. We will recruit at least 280 students in 6-7th grade through an online learning platform. Students will be randomly assigned to one of six conditions in this 3 (Operand position: left, center, or right) X 2 (Superfluous brackets vs. no brackets) design. Students will solve order of operation expressions that are consistent with their condition: 1) brackets-left, 2) no brackets-left, 3) brackets-center, 4) no brackets-center, 5) brackets-right, and 6) no brackets-right. We hypothesize that students who view problems with superfluous brackets will have quicker mastery speed and shorter average response time than those who don’t. We also hypothesize that students who view problems with the brackets to the furthest left will have quicker mastery speed and shorter average response time compared to students who view problems with brackets in the center or to the furthest right. We also predict that the presence of superfluous brackets may positively impact student performance on expressions with higher-order operands that are in the center or right-most position of the expression (e.g., 2+(4*3)+4 or 2+4+(4*3)) by providing perceptual cues to attend to the order of operations. We will discuss the implications of these findings for perceptual learning theory as well as instructional practice. Overall, the results from this study will further explore the roles of perceptual cues in mathematics notation in math. References: Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors. Journal for Research in Mathematics Education, 20(3), 301-308. Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, E. R., & Landy, D. (2017). Adapting perception, action, and technology for mathematical reasoning. Current Directions in Psychological Science, 26(5), 434-441. Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations?. Educational Studies in Mathematics, 92(1), 91-105. Harrison, A., Smith, H., Hulse, T., & Ottmar, E. R. (2020). Spacing out! Manipulating spatial features in mathematical expressions affects performance. Journal of Numerical Cognition, 6(2), 186-203. Hoch, M., & Dreyfus, T. (2005). Structure sense in high school algebra: The effect of brackets. International Group for the Psychology of Mathematics Education. Marchini, C., & Papadopoulos, I. (2011). Are useless brackets useful tools for teaching?. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education. Aristotle University of Thessaloniki.

Early middle school students have a strong tendency to adhere to the left-to-right principle when solving math problems, which may lead them to overlook the role of brackets within math expressions (Blando et al., 1989; Gunnarsson et al., 2016). However, perceptual features, such as the spacing between symbols, can direct students’ attention to salient features of math expressions that support efficient problem-solving (Goldstone et al., 2017; Harrison et al., 2020). Similarly, superfluous brackets—brackets that do not change the meaning of notation, but can guide learners’ attention to the correct procedure—can help students achieve higher success rates in solving problems (Hoch & Dreyfus, 2004; Marchini & Papadopoulos, 2011). The current study explores how the presence of superfluous brackets and the position of higher-order operands (i.e., multiplication and division) within an expression affect student performance in an online assignment, as measured by assignment mastery speed and average response time. We will recruit at least 280 students in 6-7th grade through an online learning platform. Students will be randomly assigned to one of six conditions in this 3 (Operand position: left, center, or right) X 2 (Superfluous brackets vs. no brackets) design. Students will solve order of operation expressions that are consistent with their condition: 1) brackets-left, 2) no brackets-left, 3) brackets-center, 4) no brackets-center, 5) brackets-right, and 6) no brackets-right. We hypothesize that students who view problems with superfluous brackets will have quicker mastery speed and shorter average response time than those who don’t. We also hypothesize that students who view problems with the brackets to the furthest left will have quicker mastery speed and shorter average response time compared to students who view problems with brackets in the center or to the furthest right. We also predict that the presence of superfluous brackets may positively impact student performance on expressions with higher-order operands that are in the center or right-most position of the expression (e.g., 2+(4*3)+4 or 2+4+(4*3)) by providing perceptual cues to attend to the order of operations. We will discuss the implications of these findings for perceptual learning theory as well as instructional practice. Overall, the results from this study will further explore the roles of perceptual cues in mathematics notation in math. References: Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors. Journal for Research in Mathematics Education, 20(3), 301-308. Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, E. R., & Landy, D. (2017). Adapting perception, action, and technology for mathematical reasoning. Current Directions in Psychological Science, 26(5), 434-441. Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations?. Educational Studies in Mathematics, 92(1), 91-105. Harrison, A., Smith, H., Hulse, T., & Ottmar, E. R. (2020). Spacing out! Manipulating spatial features in mathematical expressions affects performance. Journal of Numerical Cognition, 6(2), 186-203. Hoch, M., & Dreyfus, T. (2005). Structure sense in high school algebra: The effect of brackets. International Group for the Psychology of Mathematics Education. Marchini, C., & Papadopoulos, I. (2011). Are useless brackets useful tools for teaching?. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education. Aristotle University of Thessaloniki.

Early middle school students have a strong tendency to adhere to the left-to-right principle when solving math problems, which may lead them to overlook the role of brackets within math expressions (Blando et al., 1989; Gunnarsson et al., 2016). However, perceptual features, such as the spacing between symbols, can direct students’ attention to salient features of math expressions that support efficient problem-solving (Goldstone et al., 2017; Harrison et al., 2020). Similarly, superfluous brackets—brackets that do not change the meaning of notation, but can guide learners’ attention to the correct procedure—can help students achieve higher success rates in solving problems (Hoch & Dreyfus, 2004; Marchini & Papadopoulos, 2011). The current study explores how the presence of superfluous brackets and the position of higher-order operands (i.e., multiplication and division) within an expression affect student performance in an online assignment, as measured by assignment mastery speed and average response time. We will recruit at least 280 students in 6-7th grade through an online learning platform. Students will be randomly assigned to one of six conditions in this 3 (Operand position: left, center, or right) X 2 (Superfluous brackets vs. no brackets) design. Students will solve order of operation expressions that are consistent with their condition: 1) brackets-left, 2) no brackets-left, 3) brackets-center, 4) no brackets-center, 5) brackets-right, and 6) no brackets-right. We hypothesize that students who view problems with superfluous brackets will have quicker mastery speed and shorter average response time than those who don’t. We also hypothesize that students who view problems with the brackets to the furthest left will have quicker mastery speed and shorter average response time compared to students who view problems with brackets in the center or to the furthest right. We also predict that the presence of superfluous brackets may positively impact student performance on expressions with higher-order operands that are in the center or right-most position of the expression (e.g., 2+(4*3)+4 or 2+4+(4*3)) by providing perceptual cues to attend to the order of operations. We will discuss the implications of these findings for perceptual learning theory as well as instructional practice. Overall, the results from this study will further explore the roles of perceptual cues in mathematics notation in math. References: Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors. Journal for Research in Mathematics Education, 20(3), 301-308. Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, E. R., & Landy, D. (2017). Adapting perception, action, and technology for mathematical reasoning. Current Directions in Psychological Science, 26(5), 434-441. Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations?. Educational Studies in Mathematics, 92(1), 91-105. Harrison, A., Smith, H., Hulse, T., & Ottmar, E. R. (2020). Spacing out! Manipulating spatial features in mathematical expressions affects performance. Journal of Numerical Cognition, 6(2), 186-203. Hoch, M., & Dreyfus, T. (2005). Structure sense in high school algebra: The effect of brackets. International Group for the Psychology of Mathematics Education. Marchini, C., & Papadopoulos, I. (2011). Are useless brackets useful tools for teaching?. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education. Aristotle University of Thessaloniki.

Early middle school students have a strong tendency to adhere to the left-to-right principle when solving math problems, which may lead them to overlook the role of brackets within math expressions (Blando et al., 1989; Gunnarsson et al., 2016). However, perceptual features, such as the spacing between symbols, can direct students’ attention to salient features of math expressions that support efficient problem-solving (Goldstone et al., 2017; Harrison et al., 2020). Similarly, superfluous brackets—brackets that do not change the meaning of notation, but can guide learners’ attention to the correct procedure—can help students achieve higher success rates in solving problems (Hoch & Dreyfus, 2004; Marchini & Papadopoulos, 2011). The current study explores how the presence of superfluous brackets and the position of higher-order operands (i.e., multiplication and division) within an expression affect student performance in an online assignment, as measured by assignment mastery speed and average response time. We will recruit at least 280 students in 6-7th grade through an online learning platform. Students will be randomly assigned to one of six conditions in this 3 (Operand position: left, center, or right) X 2 (Superfluous brackets vs. no brackets) design. Students will solve order of operation expressions that are consistent with their condition: 1) brackets-left, 2) no brackets-left, 3) brackets-center, 4) no brackets-center, 5) brackets-right, and 6) no brackets-right. We hypothesize that students who view problems with superfluous brackets will have quicker mastery speed and shorter average response time than those who don’t. We also hypothesize that students who view problems with the brackets to the furthest left will have quicker mastery speed and shorter average response time compared to students who view problems with brackets in the center or to the furthest right. We also predict that the presence of superfluous brackets may positively impact student performance on expressions with higher-order operands that are in the center or right-most position of the expression (e.g., 2+(4*3)+4 or 2+4+(4*3)) by providing perceptual cues to attend to the order of operations. We will discuss the implications of these findings for perceptual learning theory as well as instructional practice. Overall, the results from this study will further explore the roles of perceptual cues in mathematics notation in math. References: Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors. Journal for Research in Mathematics Education, 20(3), 301-308. Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, E. R., & Landy, D. (2017). Adapting perception, action, and technology for mathematical reasoning. Current Directions in Psychological Science, 26(5), 434-441. Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations?. Educational Studies in Mathematics, 92(1), 91-105. Harrison, A., Smith, H., Hulse, T., & Ottmar, E. R. (2020). Spacing out! Manipulating spatial features in mathematical expressions affects performance. Journal of Numerical Cognition, 6(2), 186-203. Hoch, M., & Dreyfus, T. (2005). Structure sense in high school algebra: The effect of brackets. International Group for the Psychology of Mathematics Education. Marchini, C., & Papadopoulos, I. (2011). Are useless brackets useful tools for teaching?. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education. Aristotle University of Thessaloniki.

Early middle school students have a strong tendency to adhere to the left-to-right principle when solving math problems, which may lead them to overlook the role of brackets within math expressions (Blando et al., 1989; Gunnarsson et al., 2016). However, perceptual features, such as the spacing between symbols, can direct students’ attention to salient features of math expressions that support efficient problem-solving (Goldstone et al., 2017; Harrison et al., 2020). Similarly, superfluous brackets—brackets that do not change the meaning of notation, but can guide learners’ attention to the correct procedure—can help students achieve higher success rates in solving problems (Hoch & Dreyfus, 2004; Marchini & Papadopoulos, 2011). The current study explores how the presence of superfluous brackets and the position of higher-order operands (i.e., multiplication and division) within an expression affect student performance in an online assignment, as measured by assignment mastery speed and average response time. We will recruit at least 280 students in 6-7th grade through an online learning platform. Students will be randomly assigned to one of six conditions in this 3 (Operand position: left, center, or right) X 2 (Superfluous brackets vs. no brackets) design. Students will solve order of operation expressions that are consistent with their condition: 1) brackets-left, 2) no brackets-left, 3) brackets-center, 4) no brackets-center, 5) brackets-right, and 6) no brackets-right. We hypothesize that students who view problems with superfluous brackets will have quicker mastery speed and shorter average response time than those who don’t. We also hypothesize that students who view problems with the brackets to the furthest left will have quicker mastery speed and shorter average response time compared to students who view problems with brackets in the center or to the furthest right. We also predict that the presence of superfluous brackets may positively impact student performance on expressions with higher-order operands that are in the center or right-most position of the expression (e.g., 2+(4*3)+4 or 2+4+(4*3)) by providing perceptual cues to attend to the order of operations. We will discuss the implications of these findings for perceptual learning theory as well as instructional practice. Overall, the results from this study will further explore the roles of perceptual cues in mathematics notation in math. References: Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors. Journal for Research in Mathematics Education, 20(3), 301-308. Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, E. R., & Landy, D. (2017). Adapting perception, action, and technology for mathematical reasoning. Current Directions in Psychological Science, 26(5), 434-441. Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations?. Educational Studies in Mathematics, 92(1), 91-105. Harrison, A., Smith, H., Hulse, T., & Ottmar, E. R. (2020). Spacing out! Manipulating spatial features in mathematical expressions affects performance. Journal of Numerical Cognition, 6(2), 186-203. Hoch, M., & Dreyfus, T. (2005). Structure sense in high school algebra: The effect of brackets. International Group for the Psychology of Mathematics Education. Marchini, C., & Papadopoulos, I. (2011). Are useless brackets useful tools for teaching?. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education. Aristotle University of Thessaloniki.

The purpose of this study was to examine child-level pathways in development of prealgebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early calculation, word-problem, and number knowledge at start of Grade 2; calculation accuracy and calculation fluency at end of Grade 2; and prealgebraic knowledge and word-problem solving at end of Grade 4. Important similarities in pathways were identified, but path analysis also indicated that language comprehension is more critical for later word-problem solving than prealgebraic knowledge. We conclude that pathways in development of these forms of 4th-grade mathematics performance are more alike than different, but demonstrate the need to fine-tune instruction for strands of the mathematics curriculum in ways that address individual students' foundational mathematics skills or cognitive processes. [At time of submission to ERIC this article was in press with "Developmental Psychology."]

The purpose of this study is to analyze the complex argumentative structure in undergraduate mathematics classroom conversations during problem solving by taking into consideration students' and teacher' utterances in the classroom using field-independent Toulmin's theory of argumentation. Analyzing students' and teacher' utterances in the class allowed us to reconstruct argumentations evolving in the classroom talk as argumentations in classrooms are generally teacher guided. The analyses contributed to an emerging body of research on classroom conversations. [For the complete proceedings, see ED597799.]

This study investigated the role of strategy instruction and working memory capacity (WMC) on word problem solving accuracy in children with (n = 100) and without (n = 92) math difficulties (MD). Within classrooms, children in Grades 2 and 3 were randomly assigned to one of four treatment conditions: verbal-only strategies (e.g., underlining question sentence), verbal + visual strategies, visual-only strategies (e.g., correctly placing numbers in diagrams), or untreated control. Strategy interventions included 20 sessions in both Year 1 and Year 2. The intent-to-treat as well as the "as-treated" analyses showed that treatment effects were significantly moderated by WMC. In general, treatment outcomes were higher when WMC was set to a high rather than low level. When set to a relatively high WMC level, children with MD performed significantly better under visual-only strategy conditions and children without MD performed better under verbal + visual conditions when compared to control conditions.

This study investigated the role of strategy instruction and working memory capacity (WMC) on word problem solving accuracy in children with (n = 100) and without (n = 92) math difficulties (MD). Within classrooms, children in Grades 2 and 3 were randomly assigned to one of four treatment conditions: verbal-only strategies (e.g., underlining question sentence), verbal + visual strategies, visual-only strategies (e.g., correctly placing numbers in diagrams), or untreated control. Strategy interventions included 20 sessions in both Year 1 and Year 2. The intent-to-treat as well as the "as-treated" analyses showed that treatment effects were significantly moderated by WMC. In general, treatment outcomes were higher when WMC was set to a high rather than low level. When set to a relatively high WMC level, children with MD performed significantly better under visual-only strategy conditions and children without MD performed better under verbal + visual conditions when compared to control conditions.

This document points out the numbers of students who have difficulty in their required college mathematics courses. The four common syndromes identified with pupils who were experiencing a lack of success in mathematics were: (1) a lack of an adequate college or high school background in mathematics; (2) the tendency to feel that because they completed homework problems they really "knew their mathematics,"; (3) the inability to apply appropriate reading-learning strategies to their mathematics studies; and (4) a feeling of hostility over mathematics classes and anxiety over tests. These students are referred to as "underdeveloped learners," who did not know how to study and learn mathematics. Twelve learning strategies that students should learn to apply are listed and described. A remediation strategy for these pupils that combines an understanding of problem solving strategies and problem solving is suggested. A self-instructional packet titled "Problem Solving in Mathematics" that is designed to help students develop a mind set that will enable them to have some degree of success in required math courses is described. This teacher-developed packet is based on the problem solving approach used by Whimby and Lockhead in their text "Problem Solving and Comprehension." (MP)

Non-routine word problem solving is an essential feature of the mathematical development of elementary school students worldwide. Many students experience difficulties in solving these problems due to erroneous problem comprehension. These difficulties could be alleviated by instructing students how to use visual representations that clarify the problem structure and the relations between solution-relevant elements (so-called visual-schematic representations). Research shows that instructional effectiveness depends largely on teachers' mathematical knowledge for teaching. Teachers' knowledge of visual representations is therefore essential to instructing word problem comprehension in this way. As there is little to no literature investigating teachers' practices in this area, the goal of the present study is to examine teachers' use of visual representations to support non-routine word problem solving. Eight mainstream elementary school teachers implemented an innovative approach focused on the use of visual-schematic representations. After a short training, teachers were able to produce these representations during instruction. However, some teachers seemed unclear about what these representations comprise and what function they serve within the word problem solving context. Teachers seemed to base their use of representations on personal preferences rather than on an optimal fit with the word problem characteristics. These aspects need to be addressed in teacher training and professional development programs. This study makes an unique contribution to research in the important and problematic area of word problem solving in regular classrooms. The results of this study are relevant for educational researcher, teachers, and teacher educators who deal with difficulties in instructing mathematical word problems.

This study compared two approaches for teaching sixth-grade middle school students to solve math problems in math, technology education, and special education classrooms. A total of 17 students with disabilities and 76 students without disabilities were taught using either enhanced anchored instruction (EAI) or text-based instruction coupled with applied problems (TBI). Results showed that both EAI and TBI students benefited from instruction in their math class, but EAI students were able to maintain and transfer what they learned in the technology education classroom several weeks later. The performance level of students with disabilities was low in both groups, but additional small-group instruction in special education settings helped several students with disabilities achieve at levels commensurate with their peers without disabilities. (Contains 1 figure and 3 tables.)

This study compared two approaches for teaching sixth-grade middle school students to solve math problems in math, technology education, and special education classrooms. A total of 17 students with disabilities and 76 students without disabilities were taught using either enhanced anchored instruction (EAI) or text-based instruction coupled with applied problems (TBI). Results showed that both EAI and TBI students benefited from instruction in their math class, but EAI students were able to maintain and transfer what they learned in the technology education classroom several weeks later. The performance level of students with disabilities was low in both groups, but additional small-group instruction in special education settings helped several students with disabilities achieve at levels commensurate with their peers without disabilities. (Contains 1 figure and 3 tables.)

NASA CONNECT is an annual series of integrated mathematics, science, and technology instructional distance learning programs for students in grades 6-8. This program is designed for students to learn about the evolution of flight. The program has three components--television broadcast, Web activity, and lesson guide--which are designed as an integrated instructional package. Students learn how the Wright Brothers became the first human beings to successfully design, construct, and fly an airplane and which method the Wright Brothers used in designing their airplane. Students also observe NASA engineers and researchers using problem-solving skills to design wings that will change their shape during flight. By conducting hands-on and Web activities, students make connections between NASA research and the mathematics, science, and technology they learn in the classroom. The educator guide features: (1) program overview; (2) hands-on activity; (3) student worksheets; (4) teacher materials; (5) interactive kite modeler; and (6) resources. (KHR)

For grades 4 to 8

Mathematics is an essential element of physics problem solving, but experts often fail to appreciate exactly how they use it. Math may be the language of science, but math-in-physics is a distinct dialect of that language. Physicists tend to blend conceptual physics with mathematical symbolism in a way that profoundly affects the way equations are used and interpreted. Research with university physics students in classes from algebra-based introductory physics indicates that the gap between what students think they are supposed to be doing and what their instructors expect them to do can cause severe problems. (Contains 7 figures.)

The aim of this study is to examine the potential of cryptographic techniques in enhancing the organization of online group work for solving mathematical problems, while applying differentiated instruction. Engaging students in mathematics often requires additional motivational strategies and compelling incentives for sustained effort. Online group work presents a valuable opportunity for collaboration and intensive communication in solving mathematical problems. However, it also poses challenges, particularly concerning academic integrity and the risk of unauthorized copying. To address these issues, this study proposes the integration of cryptographic protocols with differentiated instruction in online collaborative tasks. Specifically, various levels of problem-solving assistance are made accessible only when the majority of the group members reach a consensus. Assistance is unlocked through the submission of individual cryptographic key segments, assigned by the instructor. A group password-required to access incremental guidance-can be generated only when a sufficient number of key segments have been submitted. This mechanism facilitates progress monitoring and fosters group accountability. The paper illustrates this approach with an example from mathematics instruction, supported by a Python-based software tool designed to aid collaborative learning. The software employs Lagrange interpolation to generate unique key parts for each participant. The method was piloted with six pre-service teachers in Serbia, and the qualitative findings are discussed alongside implications for educational research and practice. Enable Ginger Cannot connect to Ginger Check your internet connection or reload the browserDisable GingerRephraseRephrase with Ginger (Ctrl+Alt+E) Edit in Ginger Ginger is checking your text for mistakes... Disable Ginger in this text fieldDisable Ginger on this website×

The aim of this study is to examine the potential of cryptographic techniques in enhancing the organization of online group work for solving mathematical problems, while applying differentiated instruction. Engaging students in mathematics often requires additional motivational strategies and compelling incentives for sustained effort. Online group work presents a valuable opportunity for collaboration and intensive communication in solving mathematical problems. However, it also poses challenges, particularly concerning academic integrity and the risk of unauthorized copying. To address these issues, this study proposes the integration of cryptographic protocols with differentiated instruction in online collaborative tasks. Specifically, various levels of problem-solving assistance are made accessible only when the majority of the group members reach a consensus. Assistance is unlocked through the submission of individual cryptographic key segments, assigned by the instructor. A group password-required to access incremental guidance-can be generated only when a sufficient number of key segments have been submitted. This mechanism facilitates progress monitoring and fosters group accountability. The paper illustrates this approach with an example from mathematics instruction, supported by a Python-based software tool designed to aid collaborative learning. The software employs Lagrange interpolation to generate unique key parts for each participant. The method was piloted with six pre-service teachers in Serbia, and the qualitative findings are discussed alongside implications for educational research and practice. Enable Ginger Cannot connect to Ginger Check your internet connection or reload the browserDisable GingerRephraseRephrase with Ginger (Ctrl+Alt+E) Edit in Ginger Ginger is checking your text for mistakes... Disable Ginger in this text fieldDisable Ginger on this website×

The aim of this study is to examine the potential of cryptographic techniques in enhancing the organization of online group work for solving mathematical problems, while applying differentiated instruction. Engaging students in mathematics often requires additional motivational strategies and compelling incentives for sustained effort. Online group work presents a valuable opportunity for collaboration and intensive communication in solving mathematical problems. However, it also poses challenges, particularly concerning academic integrity and the risk of unauthorized copying. To address these issues, this study proposes the integration of cryptographic protocols with differentiated instruction in online collaborative tasks. Specifically, various levels of problem-solving assistance are made accessible only when the majority of the group members reach a consensus. Assistance is unlocked through the submission of individual cryptographic key segments, assigned by the instructor. A group password-required to access incremental guidance-can be generated only when a sufficient number of key segments have been submitted. This mechanism facilitates progress monitoring and fosters group accountability. The paper illustrates this approach with an example from mathematics instruction, supported by a Python-based software tool designed to aid collaborative learning. The software employs Lagrange interpolation to generate unique key parts for each participant. The method was piloted with six pre-service teachers in Serbia, and the qualitative findings are discussed alongside implications for educational research and practice. Enable Ginger Cannot connect to Ginger Check your internet connection or reload the browserDisable GingerRephraseRephrase with Ginger (Ctrl+Alt+E) Edit in Ginger Ginger is checking your text for mistakes... Disable Ginger in this text fieldDisable Ginger on this website×

The aim of this study is to examine the potential of cryptographic techniques in enhancing the organization of online group work for solving mathematical problems, while applying differentiated instruction. Engaging students in mathematics often requires additional motivational strategies and compelling incentives for sustained effort. Online group work presents a valuable opportunity for collaboration and intensive communication in solving mathematical problems. However, it also poses challenges, particularly concerning academic integrity and the risk of unauthorized copying. To address these issues, this study proposes the integration of cryptographic protocols with differentiated instruction in online collaborative tasks. Specifically, various levels of problem-solving assistance are made accessible only when the majority of the group members reach a consensus. Assistance is unlocked through the submission of individual cryptographic key segments, assigned by the instructor. A group password-required to access incremental guidance-can be generated only when a sufficient number of key segments have been submitted. This mechanism facilitates progress monitoring and fosters group accountability. The paper illustrates this approach with an example from mathematics instruction, supported by a Python-based software tool designed to aid collaborative learning. The software employs Lagrange interpolation to generate unique key parts for each participant. The method was piloted with six pre-service teachers in Serbia, and the qualitative findings are discussed alongside implications for educational research and practice. Enable Ginger Cannot connect to Ginger Check your internet connection or reload the browserDisable GingerRephraseRephrase with Ginger (Ctrl+Alt+E) Edit in Ginger Ginger is checking your text for mistakes... Disable Ginger in this text fieldDisable Ginger on this website×

The aim of this study is to examine the potential of cryptographic techniques in enhancing the organization of online group work for solving mathematical problems, while applying differentiated instruction. Engaging students in mathematics often requires additional motivational strategies and compelling incentives for sustained effort. Online group work presents a valuable opportunity for collaboration and intensive communication in solving mathematical problems. However, it also poses challenges, particularly concerning academic integrity and the risk of unauthorized copying. To address these issues, this study proposes the integration of cryptographic protocols with differentiated instruction in online collaborative tasks. Specifically, various levels of problem-solving assistance are made accessible only when the majority of the group members reach a consensus. Assistance is unlocked through the submission of individual cryptographic key segments, assigned by the instructor. A group password-required to access incremental guidance-can be generated only when a sufficient number of key segments have been submitted. This mechanism facilitates progress monitoring and fosters group accountability. The paper illustrates this approach with an example from mathematics instruction, supported by a Python-based software tool designed to aid collaborative learning. The software employs Lagrange interpolation to generate unique key parts for each participant. The method was piloted with six pre-service teachers in Serbia, and the qualitative findings are discussed alongside implications for educational research and practice. Enable Ginger Cannot connect to Ginger Check your internet connection or reload the browserDisable GingerRephraseRephrase with Ginger (Ctrl+Alt+E) Edit in Ginger Ginger is checking your text for mistakes... Disable Ginger in this text fieldDisable Ginger on this website×