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Copyright of JMD: Journal für Mathematik-Didaktik is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This cross-national study examined students' evaluation of strategies for solving linear equations, as well as the extent to which their evaluation criteria were related to their use of strategies and/or aligned with experts' views about which strategy is the best. A total of 792 middle school and high school students from Sweden, Finland, and Spain participated in the study. Students were asked to solve twelve equations, provide multiple solving strategies for each equation, and select the best strategy among those they produced for each equation. Our results indicate that students' evaluation of strategies was not strongly related to their initial preferences for using strategies. Instead, many students' criteria were aligned with the flexibility goals, in that a strategy that takes advantages of task context was more highly valued than a standard algorithm. However, cross-national differences in strategy evaluation indicated that Swedish and Finnish students were more aligned with flexibility goals in terms of their strategy evaluation criteria, while Spanish students tended to consider standard algorithms better than other strategies. We also found that high school students showed more flexibility concerns than middle school students. Different emphases in educational practice and prior knowledge might explain these cross-national differences as well as the findings of developmental changes in students' evaluation criteria. [ABSTRACT FROM AUTHOR]

Despite the increasing use of technologies in the classroom, there are concerns that technology-enhanced learning environments may hinder students' communication and interaction. In this study, we investigated how tangible technologies can enhance students' multimodal communication and interaction during equation-solving pair work compared to working without such technologies. A tangible app for learning equation solving was developed and tested in fourth- and fifth-grade classrooms with two class teachers and 24 students. Video data of the interventions were analysed using deductive and inductive content analysis. Coded data were also quantified for quantitative analysis. Additionally, teacher interview data were used to compare and contrast the findings. The findings showed that the tangible app better promoted students' multimodal communication and peer interaction than working only with paper and pencil. When working in pairs, tangible-app students interacted with one another much more often and in more ways than their paper-and-pencil peers. The implications of this study are discussed in terms of its contributions to research on tangible technologies for learning, educational technology development, and the use of tangibles in classrooms to support students' multimodal communication and peer interaction. [ABSTRACT FROM AUTHOR]

Copyright of Experimental Technology & Management is the property of Experimental Technology & Management Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this study we aimed to inquire into the impact of the use of early arithmetic strategies by a group of fifth-grade students, on their solving of equations involving two representations of unknowns. Pre- and post-tests consisting of equation-solving items involving two representations of unknowns (number sentences containing empty 'brackets', as in the example 5 + (∙) = 10, or equations containing x, as in 5 + x = 10), were administered to 126 fifth-grade students in a regular class setting. We found a notable difference between students' success rates on these two types of equations and their strategy use. Most students used the inversing strategy (arithmetic operations) after formal instruction on equation solving. Several students even used both the inversing and formal strategies (performing the same operation on both sides) for the same equation. When the unknown x appeared as the subtrahend or the divisor, the success rate dropped dramatically, and students tried to use the formal solving method of performing the same operation on both sides to solve such equations. The findings of this study not only suggest how teachers can be sensitive to students' different interpretations of unknowns, but also highlight the importance of using students' prior sense making to teach equation solving and of helping students gain an in-depth understanding of equation solving and representations of unknowns. [ABSTRACT FROM AUTHOR]

Despite the increasing use of technologies in the classroom, there are concerns that technology-enhanced learning environments may hinder students’ communication and interaction. In this study, we investigated how tangible technologies can enhance students’ multimodal communication and interaction during equation-solving pair work compared to working without such technologies. A tangible app for learning equation solving was developed and tested in fourth- and fifth-grade classrooms with two class teachers and 24 students. Video data of the interventions were analysed using deductive and inductive content analysis. Coded data were also quantified for quantitative analysis. Additionally, teacher interview data were used to compare and contrast the findings. The findings showed that the tangible app better promoted students’ multimodal communication and peer interaction than working only with paper and pencil. When working in pairs, tangible-app students interacted with one another much more often and in more ways than their paper-and-pencil peers. The implications of this study are discussed in terms of its contributions to research on tangible technologies for learning, educational technology development, and the use of tangibles in classrooms to support students’ multimodal communication and peer interaction.

The purpose of this study was to examine the effects of algebraic equation solving intervention for sixth graders with mathematics learning difficulties (MD). A total of 48 students with MD were randomly assigned to either the algebraic equation solving intervention, Mystery Math (n = 24) or control condition (n = 24). The multicomponent intervention was based on the principles of explicit instruction and focused on improving conceptual and procedural knowledge of algebraic equation solving using concrete manipulatives. Students in the intervention group received instruction in pairs, 30 min per session, 3 sessions per week for 5 weeks (i.e., 15 sessions). The results indicated that the main effect of intervention was significant for 2 proximal measures of mathematics vocabulary, and conceptual and procedural understanding of algebraic equation solving with large effect sizes. However, the main effect of intervention was not significant for distal measures of comprehensive pre-algebra skills and whole-number computations. The findings demonstrate that grade-level standards can be successfully taught to students with MD. Implications for practice are discussed. [ABSTRACT FROM AUTHOR]

Online educational technologies offer opportunities for providing individualized feedback and detailed profiles of students' skills. Yet many technologies for mathematics education assess students based only on the correctness of either their final answers or responses to individual steps. In contrast, examining the choices students make for how to solve the equation and the ways in which they might answer incorrectly offers the opportunity to obtain a more nuanced perspective of their algebra skills. To automatically make sense of step‐by‐step solutions, we propose a Bayesian inverse planning model for equation solving that computes an assessment of a learner's skills based on her pattern of errors in individual steps and her choices about what sequence of problem‐solving steps to take. Bayesian inverse planning builds on existing machine learning tools to create a generative model relating (mis)‐understandings to equation solving choices. Two behavioral experiments demonstrate that the model can interpret people's equation solving and that its assessments are consistent with those of experienced teachers. A third experiment uses this model to tailor guidance for learners based on individual differences in misunderstandings, closing the loop between assessing understanding, and using that assessment within an educational technology. Finally, because the bottleneck in applying inverse planning to a new domain is in creating the model of possible student misunderstandings, we show how to combine inverse planning with an existing production rule model to make inferences about student misunderstandings of fraction arithmetic. [ABSTRACT FROM AUTHOR]

Mathematics is the cradle of all creations, without which the world cannot move an inch. Mathematical functions are 'extensively' used in physics, 'structurally' used in graphics, 'practically' used in civil engineering, 'potentially' used in mechanical engineering and in many other fields as well. One fairly common difficulty faced by the engineers and scientists is to find the right function that solves their problem. This involves a lot of time consuming tasks. The idea proposed here is an android application that captures the mathematical expression using a built-in camera which produces a java code that can be utilized for solving the inputs of their needs. Along with the code, it also displays a basic plot of the function, which will be more helpful for selecting the apt solution for any problem. There are many applications in the market which can compute the result of a mathematical equation. But, this application will give an executable java code which can be used for further problem solving. So, the application is unique in its way and this is a new dimension of using image processing for code generation. This application supports polynomial, logarithmic, trigonometric and exponential functions up to four variables. [ABSTRACT FROM AUTHOR]

The zero-product property (ZPP), often stated as ‘if ab = 0, then a = 0 or b = 0,’ is an important concept in secondary algebra (as a tool for solving equations) and abstract algebra (as a property of integral domains). This study analyzes results from a teaching experiment to investigate how an undergraduate mathematics major might intuitively reason with zero-divisors and the ZPP. There are two primary findings. First, a procedurally embodied view of equation solving might preclude students’ attention to the algebraic properties (including the ZPP) that justify the equivalence of two equations. Second, students might not carefully attend to zero-divisors because they are employing the converse of the ZPP instead of the ZPP itself. These findings advance a hypothesis about why students might view abstract algebra as a different subject than school algebra and also affirm the utility of the student-centered theoretical perspective that guided the instructional design and analysis of student activity. [ABSTRACT FROM AUTHOR]

Between two popular teaching methods (i.e., balance method vs. inverse method) for equation solving, the main difference occurs at the operational line (e.g., +2 on both sides vs. −2 becomes +2), whereby it alters the state of the equation and yet maintains its equality. Element interactivity occurs on both sides of the equation in the balance method, but only on one side in the case of the inverse method. Thus, the balance method imposes twice as many interacting elements as the inverse method for each operational line. In two experiments, secondary students were randomly assigned to either the balance method or the inverse method to learn how to solve one-step, two-step, and three-or-more-step linear equations. Test results indicated that the interaction between method and type of equation favored the inverse method for equations involving higher element interactivity. Hence, by managing element interactivity, the efficiency of instruction for equation solving can be improved. [ABSTRACT FROM AUTHOR]

A well known numerical task is the inversion of large symmetric tridiagonal Toeplitz matrices, i.e., matrices whose entries equal a on the diagonal and b on the extra diagonals (a,b∈R). The inverses of such matrices are dense and there exist well known explicit formulas by which they can be calculated in O(n2). In this note we present a simplification of the problem that has proven to be rather useful in everyday practice: If |a|>2|b|, that is, if the matrix is strictly diagonally dominant, its inverse is a band matrix to working precision and the bandwidth is independent of n for sufficiently large n. Employing this observation, we construct a linear time algorithm for an explicit tridiagonal inversion that only uses O(1) floating point operations. On the basis of this simplified inversion algorithm we outline the cornerstones for an efficient parallelizable approximative equation solver. [ABSTRACT FROM AUTHOR]

Researchers interested in mathematical proficiency have recently begun to explore the development of strategic flexibility, where flexibility is defined as knowledge of multiple strategies for solving a problem and the ability to implement an innovative strategy for a given problem solving circumstance. However, anecdotal findings from this literature indicate that students do not consistently use an innovative strategy for solving a given problem, even when these same students demonstrate knowledge of innovative strategies. This distinction, sometimes framed in the psychological literature as competence vs. performance—has not been previously studied for flexibility. In order to explore the competence/performance distinction in flexibility, this study developed and validated measures for potential flexibility (e.g., competence, or knowledge of multiple strategies) and practical flexibility (e.g., performance, use of innovative strategies) for solving equations. The measures were administrated to a sample of 158 Chinese middle school students through a Tri-Phase Flexibility Assessment, in which the students were asked to solve each equation, generate additional strategies, and evaluate own multiple strategies. Confirmatory factor analysis supported a two-factor model of potential and practical flexibility. Satisfactory internal consistency was found for the measures. Additional validity evidence included the significant association with flexibility measured with the previous method. Potential flexibility and practical flexibility were found to be distinct but related. The theoretical and practical implications of the concepts and their measures of potential flexibility and practical flexibility are discussed.

Assimilating multiple interactive elements simultaneously in working memory to allow understanding to occur, while solving an equation, would impose a high cognitive load.Element interactivityarises from the interaction between elements within and across operational and relational lines. Moreover, operating with special features (e.g. negative pronumeral) poses additional challenge to master equation solving skills. In an experiment, 41 8th grade students (girls = 16, boys = 25) sat for a pre-test, attended a session about equation solving, completed an acquisition phase which constituted the main intervention and were tested again in a post-test. The results showed that at post-test, students performed better on one-step equations tapping low rather than high element interactivity knowledge. In addition, students performed better on those one-step equations that contained no special features. Thus, both the degree of element interactivity and the operation with special features affect the challenge posed to 8th grade students on learning how to solve one-step equations. [ABSTRACT FROM AUTHOR]

We examined the use of balance and inverse methods in equation solving. The main difference between the balance and inverse methods lies in the operational line (e.g. +2 on both sides vs −2 becomes +2). Differential element interactivity favours the inverse method because the interaction between elements occurs on both sides of the equation for the balance method but only on one side of the equation for the inverse method. In an experimental study, 63 students (mean age = 13) were randomly allocated to either balance or inverse group to undertake a pre-test, study an instruction sheet, complete acquisition equations, sit for a post-test and a concept test. Procedural knowledge was assessed on performance on practice equations and post-test, whereas conceptual knowledge was assessed on performance on the concept test. The inverse group outperformed the balance group on practice equations but not the post-test. Both the balance and inverse groups scored higher on the inverse concept test than the balance concept test. Positive association between performance on procedural knowledge and performance on conceptual knowledge was found for the inverse group but not the balance group. Overall, the evidence obtained indicates a number of educational implications for implementation. [ABSTRACT FROM PUBLISHER]

Students struggle with planning suitable solution methods in equation solving. Planning suitable solution methods is key to flexibility, a desired skill for equation solving. Comparing solution methods has been shown to foster flexibility. To support the learning benefits of the comparisons, productive classroom talk, which includes the discussion of different solution methods, is recommended. This study examines whether discussions that compare multiple methods include more planning processes than discussions that do not compare multiple solution methods or that are just about one solution method. The content analysis is based on utterances from 172 lessons from 43 classrooms in grades 9 and 10. The hypothesis is tested both across classes using binary logistic regression models and at the class level using paired samples t‑tests. The results show that planning processes are addressed about twice as often when comparing multiple solution methods. Additionally, the study finds that enacting solution methods is the most frequent topic in classroom talk about solving equations., (© The Author(s) 2023.)

Central to equation solving is the maintenance of equivalence on both sides of the equation. However, when the process involves an interaction of multiple elements, solving an equation can impose a high cognitive load. The balance method requires operations on both sides of the equation, whereas the inverse method involves operations on one side only. In an experiment, middle school students (N = 71) were randomly assigned to the balance and inverse methods to complete a pre-test, an acquisition phase and a post-test. Pre-test and post-test comparisons found that the inverse group outperformed the balance group for those equations that involved high element interactivity. Instructional efficiency measures further confirmed that the balance method imposed higher cognitive load. The inverse method was capable of reducing cognitive load due to the interacting elements. [ABSTRACT FROM PUBLISHER]

A new methodology for determining simultaneous chemical and phase equilibrium of mixtures of light gases and aqueous electrolyte solutions with the potential for multiple salt deposition is proposed. The novel aspects of this new approach include, but are not limited to (1) a novel tearing algorithm for determining equilibrium ion solubility limits, (2) rigorous proof that the proposed tearing algorithm generates a Cauchy sequence and is therefore guaranteed to converge to the correct equilibrium ion solubility limits (3) and a unique formulation of the combined chemical and multi-phase equilibrium flash problem that accounts for salt deposition but decouples the chemical and phase equilibrium aspects of the flash. Examples from real EOR and C02 sequestration applications are presented. Results clearly show that the proposed numerical approach is reliable, robust, and efficient and can be used to determine salt deposition in multi-phase flash problems. Several geometric illustrations and numerical details are used to elucidate key points of the proposed approach. [ABSTRACT FROM AUTHOR]

Copyright of Ensayos: Revista de la Facultad de Educacion de Albacete is the property of Ensayos Revista de la Facultad de Educacion de Albacete and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Highlights: [•] Magnitude knowledge of fractions, not whole numbers, predicts algebra performance. [•] Fraction knowledge predicts improvement in equation solving and encoding. [•] Proportional reasoning may be crucial for predicting algebra learning. [ABSTRACT FROM AUTHOR]

Submodule construction is the problem of finding a new submodule which, together with a given submodule, provides a behavior that conforms to a given desired global behavior. A new formulation of this problem and its solution in first-order logic is presented, and it is shown how the known solutions to this problem in the context of various communication paradigms and specification formalisms can be derived. Communication paradigms are: synchronous rendezvous at several interfaces; interleaved rendezvous; input/output automata with complete or partial behavior specifications and with synchronous or interleaved communication. A new algorithm for deriving a progressive solution is also presented. [ABSTRACT FROM AUTHOR]

This paper discusses the approaches to teaching linear equation solving that are embedded in a Standards-based mathematics curriculum (Connected Mathematics Program or CMP) and in a traditional mathematics curriculum (Glencoe Mathematics) in the United States. Overall, the CMP curriculum takes a functional approach to teaching equation solving, while Glencoe Mathematics takes a structural approach. The functional approach emphasizes the important ideas of change and variation in situations and contexts. It also emphasizes the representation of relationships between variables. The structural approach, on the other hand, requires students to work abstractly with symbols and follow procedures in a systematic way. The CMP curriculum may be regarded as a curriculum with a pedagogy that emphasizes predominantly the conceptual aspects of equation solving, while Glencoe Mathematics may be regarded as a curriculum with a pedagogy that emphasizes predominantly the procedural aspects of equation solving. The two curricula may serve as concrete examples of functional and structural approaches, respectively, to the teaching of algebra in general and equation solving in particular. [ABSTRACT FROM AUTHOR]

Abstract: A key learning outcome in problem-solving domains is the development of flexible knowledge, where learners know multiple strategies and adaptively choose efficient strategies. Two interventions hypothesized to improve flexibility in problem solving were experimentally evaluated: prompts to discover multiple strategies and direct instruction on multiple strategies. Participants were 132 sixth-grade students who solved linear equations for three hours. Both interventions improved students'' flexibility in problem solving and did not replace, nor interfere with, one another. Overall, the study provides causal evidence that exposure to multiple strategies leads to improved flexibility in problem solving and that discovery learning and direct instruction are compatible instructional approaches. [Copyright &y& Elsevier]

We investigate the problem of computing the partition function of a probabilistic context-free grammar, and consider a number of applicable methods. Particular attention is devoted to PCFGs that result from the intersection of another PCFG and a finite automaton. We report experiments involving the Wall Street Journal corpus. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the problem of deriving a component X of a system knowing the behavior of the whole system C and the other components A. The component X is derived by solving the parallel automata equation . We present an algorithm for deriving a largest progressive solution to the equation that combined with A does not block any possible action in C and we establish conditions that allow us to characterize all progressive solutions. [Copyright &y& Elsevier]

EqL is a general-purpose language that combines the capabilities of functional and logic programming languages. A program in EqL consists of a collection of conditional, pattern-directed rules, where the conditions are expressed as a conjunction of equations, and the patterns are terms built up of data-constructors and basic values. The computational paradigm in EqL is equation solving. In this paper we describe EqL informally, and present examples illustrating the major features of the language: nondeterminism, deferred evaluation of primitives, and logical variables. This paper also describes the novel aspects of a sequential implementation for EqL: compile-time flattening of equations; and run-time equation-delaying and last-equation optimization. [ABSTRACT FROM AUTHOR]

We present the formal semantics of a novel language, called EqL, for first-order functional and Horn logic programming. An EqL program is a set of conditional pattern-directed rules, where the conditions are expressed as a conjunction of equations. The programming paradigm provided by this language may be called equational programming. The declarative semantics of equations is given in terms of their complete set of solutions, and the operational semantics for solving equations is an extension of reduction, called object refinement. The correctness of the operational semantics is established through soundness and completeness theorems. Examples are given to illustrate the language and its semantics. [ABSTRACT FROM AUTHOR]

This paper discusses the design, implementation and evaluation of linear finite element programs that distribute their computations over a network of workstations. We consider five different algorithms based on direct, iterative and hybrid equation solvers, each of which partitions and maps the model domain across conventional network hardware. A software architecture based on the client-server model distributes the computations and, at the language level, Berkeley sockets enable communication between processes. We evaluate and describe the performance of these algorithms in terms of execution time and speed-up, and we conclude that distributed solvers, particularly those based on substructuring and static condensation, can be effective even on high-latency communication networks. [ABSTRACT FROM AUTHOR]

Researchers interested in mathematical proficiency have recently begun to explore the development of strategic flexibility, where flexibility is defined as knowledge of multiple strategies for solving a problem and the ability to implement an innovative strategy for a given problem solving circumstance. However, anecdotal findings from this literature indicate that students do not consistently use an innovative strategy for solving a given problem, even when these same students demonstrate knowledge of innovative strategies. This distinction, sometimes framed in the psychological literature as competence vs. performance-has not been previously studied for flexibility. In order to explore the competence/performance distinction in flexibility, this study developed and validated measures for potential flexibility (e.g., competence, or knowledge of multiple strategies) and practical flexibility (e.g., performance, use of innovative strategies) for solving equations. The measures were administrated to a sample of 158 Chinese middle school students through a Tri-Phase Flexibility Assessment, in which the students were asked to solve each equation, generate additional strategies, and evaluate own multiple strategies. Confirmatory factor analysis supported a two-factor model of potential and practical flexibility. Satisfactory internal consistency was found for the measures. Additional validity evidence included the significant association with flexibility measured with the previous method. Potential flexibility and practical flexibility were found to be distinct but related. The theoretical and practical implications of the concepts and their measures of potential flexibility and practical flexibility are discussed.